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# Sums of Consecutive Prime Squares
Janyarak Tongsomporn, Saeree Wananiyakul, Jörn Steuding
(Date: January 2021)
###### Abstract.
We prove explicit bounds for the number of sums of consecutive prime squares
below a given magnitude
Keywords: prime numbers, sums of squares
MSC Numbers: 11A41, 00A08
## 1\. Motivation and the Main Result
Early last year the authors learned that 2020 can be represented as a sum of
squares of consecutive prime numbers, namely
$2020=17^{2}+19^{2}+23^{2}+29^{2}.$
It is a natural question to ask what the next year with this property will be.
We shall show that such a representation is a rare event.
Indeed, if ${\rm{scp}}(x)$ counts the number of sums of squares of consecutive
primes below $x$, i.e.,
${\rm{scp}}(x)=\sharp\left\\{p_{n}^{2}+p_{n+1}^{2}+\ldots+p_{n-1+m}^{2}\leq
x\,:\,m\in\hbox{{\dubl N}}\right\\},$
where $p_{j}$ denotes the $j$-th prime number in ascending order, then
$\lim_{x\to\infty}{\rm{scp}}(x)/x=0$. The following theorem provides more
precise bounds.
###### Theorem 1.
We have
$2\,{x^{1/2}\over\log
x}<\pi(\sqrt{x})\leq{\rm{scp}}(x)<10.9558\,{x^{2/3}\over(\log x)^{4/3}}\,,$
where the inequality on the far left is valid for $x\geq 289$ and all those to
the right for $x>1$.
Here, as usual, $\pi(N)$ is counting the number of primes $p\leq N$ and
explicit bounds for this prime counting function are the main tool for proving
the inequalities above; we have chosen a recent paper [1] by Pierre Dusart.
The dear reader is invited to improve upon the bounds of our theorem; maybe it
is even possible to prove an asymptotic formula for the number of sums of
consecutive prime squares below a given magnitude. Note that we do not
consider here the question whether or not an integer can have two or even more
such representations or how many of these exist.
Using a computer algebra package one can verify that the next sum of squares
of consecutive primes is given by the expected suspect, namely
$2189=13^{2}+17^{2}+19^{2}+23^{2}+29^{2}.$
A list with all integers below $5000$ that can be written as a sum of
consecutive prime squares can be found in the third and final section.
This year’s prime factorization is $2021=43\cdot 47$ which is a product of two
consecutive primes. Following our approach one can also discuss sums of
products of two consecutive primes; the corresponding bounds should be close
to those found here for sums of consecutive prime squares.
## 2\. Proof of the Theorem
It is convenient to define, for fixed $m\in\hbox{{\dubl N}}$, the counting
function for sums of $m$ consecutive prime squares, i.e.,
${\rm{scp}}_{m}(x)=\sharp\left\\{p_{n}^{2}+p_{n+1}^{2}+\ldots+p_{n-1+m}^{2}\leq
x\right\\}.$
For the lower bound we first observe that the number of squares of prime
numbers $p^{2}$ below or equal to $x$ is given by $\pi(\sqrt{x})$.
In the sequel we shall use the explicit bounds
(1) ${N\over\log N}<\pi(N)<1.2551\,{N\over\log N},$
where the left inequality is valid for $N\geq 17$ and the one on the right for
$N>1$ (see Corollary 5.2 in [1]); of course, the celebrated prime number
theorem provides an asymptotic formula for $\pi(N)$ with main term $N/\log N$,
however, for excluding the related error term for our analysis, we prefer the
version above with the factor $1.2551$. The corresponding range for these
inequalities (resp. the range for $x$ in our theorem) is also useful with
respect to computer experiments.
It follows from (1) that
${\rm{scp}}(x)\geq{\rm{scp}}_{1}(x)=\pi(\sqrt{x})>{\sqrt{x}\over{1\over 2}\log
x},$
which is valid for $x\geq 17^{2}=289$. This proves the lower bound.
The reasoning for the upper bound is a little more advanced. First we note
that for $n={\rm{scp}}(x)$ we have
$mp_{n}^{2}\leq p_{n}^{2}+p_{n+1}^{2}+\ldots+p_{n-1+m}^{2}\leq
x<p_{n}^{2}+p_{n+1}^{2}+\ldots+p_{n-1+m}^{2}+p_{n+m}^{2}.$
Hence, by the inequality in (1),
(2) ${\rm{scp}}_{m}(x)=n\leq\pi(\sqrt{x/m})<1.2551\,{\sqrt{x/m}\over{1\over
2}\log(x/m)}\leq 2.5102\,{(x/m)^{1/2}\over\log x},$
which is valid for $x>m$, which, obviously, is no severe restriction (since
the largest integer $\leq x$ is a trivial upper bound for the length of a sum
of consecutive primes squares $\leq x$).
To continue we shall next bound the length of possible sums of consecutive
prime squares below $x$. For this purpose we shall use an old result due to
Barkley Rosser [2] which has been improved several times, in particular by
Dusart [1], however, we prefer the more simple inequality
$p_{n}>n\log n,$
valid for all $n\in\hbox{{\dubl N}}$; this lower bound is trivial for $n=1$.
We shall use this so-called Rosser theorem for the sum of the squares of the
first primes:
$p_{1}^{2}+p_{2}^{2}+\ldots+p_{M}^{2}>\sum_{2\leq n\leq M}(n\log n)^{2}.$
If we can show that the right hand side is larger than $x$, then the least sum
of $M$ consecutive prime squares already exceeds the given magnitude. Assuming
that this $M$ is the least positive integer with this property, this leads to
a bound for $M$ depending on $x$. This estimate in combination with the
previous one allows us to derive the upper bound of the theorem.
Alternatively, one could also use partial summation here together with the
prime number theorem, however, it is our intention to circumvent error terms.
Obviously, for $M\geq 4$,
$\displaystyle\sum_{2\leq n\leq M}(n\log n)^{2}$ $\displaystyle\geq$
$\displaystyle\sum_{\sqrt{M}\leq n\leq M}n^{2}(\log n)^{2}$
$\displaystyle\geq$ $\displaystyle({\textstyle{1\over 2}}\log
M)^{2}\sum_{\sqrt{M}\leq n\leq M}n^{2}\geq{\textstyle{1\over 12}}M^{3}(\log
M)^{2},$
where we have used in the final step the well-known formula
$1+2^{2}+3^{2}+\ldots+M^{2}={\textstyle{1\over 6}}\,M(M+1)(2M+1)$
and some pen and paper. It thus follows that every sum of consecutive prime
squares below $x$ has less than roughly $x^{1/3}$ summands. For a more precise
bound we observe that substituting
(3) $M=\lfloor 108^{1/3}\,x^{1/3}(\log x)^{-2/3}\rfloor$
into the lower bound above yields a quantity slightly larger than $x$; here
$\lfloor z\rfloor$ denotes the largest integer $\leq z$.
To use this for an upper bound we first observe that (2) implies
${\rm{scp}}(x)=\sum_{1\leq m\leq M}{\rm{scp}}_{m}(x)<2.5102\,{x^{1/2}\over\log
x}\sum_{1\leq m\leq M}m^{-1/2}.$
In general, we have, for $\alpha\in(0,1)$,
$\sum_{1\leq m\leq M}m^{-\alpha}<1+\sum_{2\leq m\leq
M}\int_{m-1}^{m}u^{-\alpha}{\rm{d}}u=1+\int_{1}^{M}u^{-\alpha}{\rm{d}}u={M^{1-\alpha}-\alpha\over
1-\alpha}.$
This in combination with (3) leads to
${\rm{scp}}(x)<5.0204\,{(xM)^{1/2}\over\log
x}\leq{5.0204\cdot(108)^{1/6}}\,{x^{2/3}\over(\log x)^{4/3}}.$
This proves the upper bound of the theorem.
## 3\. Explicit Sums of Consecutive Prime Squares
We conclude with a list of all integers below $5000$ that can be written as a
sum of consecutive prime squares:
4 | 9 | 13 | 25 | 34 | 38 | 49
---|---|---|---|---|---|---
74 | 83 | 87 | 121 | 169 | 170 | 195
204 | 208 | 289 | 290 | 339 | 361 | 364
373 | 377 | 458 | 529 | 579 | 628 | 650
653 | 662 | 666 | 819 | 841 | 890 | 940
961 | 989 | 1014 | 1023 | 1027 | 1179 | 1348
1369 | 1370 | 1469 | 1518 | 1543 | 1552 | 1556
1681 | 1731 | 1802 | 1849 | 2020 | 2189 | 2209
2310 | 2330 | 2331 | 2359 | 2384 | 2393 | 2397
2692 | 2809 | 2981 | 3050 | 3150 | 3171 | 3271
3320 | 3345 | 3354 | 3358 | 3481 | 3530 | 3700
3721 | 4011 | 4058 | 4061 | 4350 | 4489 | 4519
4640 | 4689 | 4714 | 4723 | 4727 | 4852 | 4899
## References
* [1] P. Dusart, Explicit estimates of some functions over primes, Ramanujan J. 45 (2018), 227–251
* [2] B. Rosser, The $n$-th prime is greater than $n\log n$, Proc. London math. Soc. (2) 45 (1938), 21-44
Janyarak Tongsomporn, Saeree Wananiyakul, ${\mathcal{W}}$alailak University,
School of Science, Nakhon Si Thammarat 80 160, Thailand<EMAIL_ADDRESS>
Jörn Steuding, Department of Mathematics, ${\mathcal{W}}$ürzburg University,
Am Hubland, 97 218 Würzburg, Germany<EMAIL_ADDRESS>
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# A doubly relaxed minimal-norm Gauss–Newton method for underdetermined
nonlinear least-squares problems
Federica Pes Department of Mathematics and Computer Science, via Ospedale 72,
09124 Cagliari, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>Giuseppe
Rodriguez11footnotemark: 1
###### Abstract
When a physical system is modeled by a nonlinear function, the unknown
parameters can be estimated by fitting experimental observations by a least-
squares approach. Newton’s method and its variants are often used to solve
problems of this type. In this paper, we are concerned with the computation of
the minimal-norm solution of an underdetermined nonlinear least-squares
problem. We present a Gauss–Newton type method, which relies on two relaxation
parameters to ensure convergence, and which incorporates a procedure to
dynamically estimate the two parameters, as well as the rank of the Jacobian
matrix, along the iterations. Numerical results are presented.
###### keywords:
nonlinear least-squares problem, minimal-norm solution, Gauss–Newton method,
parameter estimation
###### AMS:
65H10, 65F22
## 1 Introduction
Let us assume that
$F(\mathbf{x})=[F_{1}(\mathbf{x}),\ldots,F_{m}(\mathbf{x})]^{T}$ is a
nonlinear twice continuously Frechét-differentiable function with values in
${\mathbb{R}}^{m}$, for any $\mathbf{x}\in{\mathbb{R}}^{n}$. For a given
$\mathbf{b}\in{\mathbb{R}}^{m}$, we consider the nonlinear least-squares data
fitting problem
$\min_{\mathbf{x}\in{\mathbb{R}}^{n}}\|\mathbf{r}(\mathbf{x})\|^{2},\qquad\mathbf{r}(\mathbf{x})=F(\mathbf{x})-\mathbf{b},$
(1)
where $\|\cdot\|$ denotes the Euclidean norm and
$\mathbf{r}(\mathbf{x})=\left[r_{1}(\mathbf{x}),\ldots,r_{m}(\mathbf{x})\right]^{T}$
is the residual vector function between the model expectation $F(\mathbf{x})$
and the vector $\mathbf{b}$ of measured data. The solution to the nonlinear
least-squares problem gives the best model fit to the data in the sense of the
minimum sum of squared errors. A common choice for solving a nonlinear least-
squares problem consists of applying Newton’s method and its variants, such as
the Gauss–Newton method [2, 12, 13].
The Gauss–Newton method is based on the construction of a sequence of linear
approximations to $\mathbf{r}(\mathbf{x})$. Chosen an initial point
$\mathbf{x}^{(0)}$ and denoting by $\mathbf{x}^{(k)}$ the current
approximation, then the new approximation is
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\mathbf{s}^{(k)},\qquad k=0,1,2,\ldots,$
(2)
where the step $\mathbf{s}^{(k)}$ is computed as a solution to the linear
least-squares problem
$\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|J(\mathbf{x}^{(k)})\mathbf{s}+\mathbf{r}(\mathbf{x}^{(k)})\|^{2}.$
(3)
Here $J(\mathbf{x})$ represents the Jacobian matrix of the function
$F(\mathbf{x})$.
The solution to (3) may not be unique: this happens when the matrix
$J(\mathbf{x}^{(k)})$ does not have full column rank, in particular, when
$m<n$. To make the solution unique, the new iterate $\mathbf{x}^{(k+1)}$ is
often obtained by solving the following minimal-norm linear least-squares
problem
$\begin{cases}\displaystyle\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|\mathbf{s}\|^{2}\\\
\displaystyle\mathbf{s}\in\bigl{\\{}\arg\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|J(\mathbf{x}^{(k)})\mathbf{s}+\mathbf{r}(\mathbf{x}^{(k)})\|^{2}\bigr{\\}},\end{cases}$
(4)
where the set in the lower line contains all the solutions to problem (3).
In order to select solutions exhibiting different degrees of regularity, the
term $\|\mathbf{s}\|^{2}$ in (4) is sometimes substituted by the seminorm
$\|L\mathbf{s}\|^{2}$, where $L\in{\mathbb{R}}^{p\times n}$ $(p\leq n)$ is a
matrix which incorporates available a priori information on the solution. The
case $p>n$ can be easily reduced to the previous assumption by performing a
compact $L=QR$ factorization, and substituting $L$ by the triangular matrix
$R$. Typically, $L$ is a diagonal weighting matrix or a discrete approximation
of a derivative operator. For example, the matrices
$D_{1}=\begin{bmatrix}1&-1&&&\\\ &\ddots&\ddots&\\\
&&1&-1\end{bmatrix}\quad\text{ and }\quad D_{2}=\begin{bmatrix}1&-2&1&&&\\\
&\ddots&\ddots&\ddots&\\\ &&1&-2&1\end{bmatrix},$ (5)
of size $(n-1)\times n$ and $(n-2)\times n$, respectively, are approximations
to the first and second derivative operators. When a regularization matrix is
introduced, problem (4) becomes
$\begin{cases}\displaystyle\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|L\mathbf{s}\|^{2}\\\
\displaystyle\mathbf{s}\in\bigl{\\{}\arg\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|J(\mathbf{x}^{(k)})\mathbf{s}+\mathbf{r}(\mathbf{x}^{(k)})\|^{2}\bigr{\\}}.\end{cases}$
(6)
Both (4) and (6) impose some kind of regularity on the update vector
$\mathbf{s}$ for the solution $\mathbf{x}^{(k)}$ and not on the solution
itself. The problem of imposing a regularity constraint directly on the
solution $\mathbf{x}$ of problem (1), i.e.,
$\begin{cases}\displaystyle\min_{\mathbf{x}\in{\mathbb{R}}^{n}}\|\mathbf{x}\|^{2}\\\
\displaystyle\mathbf{x}\in\bigl{\\{}\arg\min_{\mathbf{x}\in{\mathbb{R}}^{n}}\|F(\mathbf{x})-\mathbf{b}\|^{2}\bigr{\\}},\end{cases}$
(7)
is studied in [6, 7, 8, 14]. These papers are based on the application of the
damped Gauss–Newton method to the solution of (7). To ensure the computation
of the minimal-norm solution, at the $k$th iteration, the Gauss–Newton
approximation is orthogonally projected onto the null space of the Jacobian
$J(\mathbf{x}^{(k)})$. In [14], the damping parameter is estimated by the
Armijo–Goldstein principle; we refer to this method as the MNGN algorithm. In
the same paper, this approach is applied to the minimization of a suitable
seminorm, and different regularization techniques are considered under the
assumption that the nonlinear function $F$ is ill-conditioned.
Unfortunately, the algorithms developed in the above papers occasionally lack
to converge. They take the form
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)}-{\mathcal{P}}_{{\mathcal{N}}(J_{k})}\mathbf{x}^{(k)},$
where $\widetilde{\mathbf{s}}^{(k)}$ is the solution of (4), $\alpha_{k}$ is a
step length, and ${\mathcal{P}}_{{\mathcal{N}}(J_{k})}$ is the orthogonal
projector onto the null space of $J_{k}=J(\mathbf{x}^{(k)})$. One reason for
the nonconvergence of such methods is that the projection step may cause the
residual to increase considerably at particular iterations. Moreover, the rank
of $J(\mathbf{x}^{(k)})$ may vary as the iteration progresses, and its
incorrect estimation often leads to the presence of small singular values for
the Jacobian, which amplify computational errors.
This problem of nonconvergence is dealt with in [3], by a method which will be
denoted CKB in the following. The authors consider a convex combination of the
Gauss–Newton approximation and its orthogonal projection, and apply a
relaxation parameter $\gamma_{k}$ to this search direction, chosen according
to a given rule. After some manipulation, the method can be written as
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\widetilde{\mathbf{s}}^{(k)}-\gamma_{k}{\mathcal{P}}_{{\mathcal{N}}(J_{k})}\mathbf{x}^{(k)}.$
(8)
This approach makes the computation of the minimal-norm solution more robust,
but it may not converge in some situation; see Section 4. Moreover, both the
MNGN and the CKB methods suffer from serious convergence problems caused by
the variation of the rank of the Jacobian along the iterations. The rank often
drops to a small value in a neighborhood of the solution, while the two
methods consider a fixed rank, generally assumed to be the smaller dimension
of the Jacobian.
In this paper, we aim at improving the convergence of the methods presented in
[3] and [14]. We do this by first introducing in the MNGN method a technique
to estimate the rank of the matrix $J(\mathbf{x}^{(k)})$ at each iteration.
This procedure has the effect of improving the convergence of the method,
reducing the possibility that the iteration diverges because of error
amplification. Then, we introduce a second relaxation parameter for the
projection term, as well as a strategy to automatically tune it, besides the
usual damping parameter for the Gauss–Newton search direction. This approach
produces, on the average, solutions closer to optimality, i.e., with smaller
norms, than those computed by the CKB method. Furthermore, we consider a model
profile $\overline{\mathbf{x}}$ for the solution, which is useful in
applications where sufficient a priori information on the physical system
under investigation is available.
The paper is structured as follows. In Section 2, we revise the MNGN method
and reformulate Theorem 3.1 from [14] by introducing a model profile for the
solution. Then, we give a theoretical justification for the fact that the
convergence of the method may not be ensured. Section 3 explains how to
estimate the numerical rank of the Jacobian $J(\mathbf{x}^{(k)})$ at each
iteration. In Section 4, we describe an algorithm which introduces a second
parameter to control the size of the correction vector that provides the
minimal-norm solution, and which estimates automatically such parameter. In
Section 5, we extend the discussion to the minimal-$L$-norm solution, where
$L$ is a regularization matrix. Numerical examples can be found in Section 6.
## 2 Nonlinear minimal-norm solution
We begin by recalling the definition of the singular value decomposition (SVD)
of a matrix $J\in{\mathbb{R}}^{m\times n}$ [10], which will be needed later.
The SVD is a matrix decomposition of the form
$J=U\Sigma V^{T},$
where $U=[\mathbf{u}_{1},\dots,\mathbf{u}_{m}]\in{\mathbb{R}}^{m\times m}$ and
$V=[\mathbf{v}_{1},\dots,\mathbf{v}_{n}]\in{\mathbb{R}}^{n\times n}$ are
matrices with orthonormal columns and $\Sigma_{i,j}=0$ for $i\neq j$. The
nonzero diagonal elements of the matrix $\Sigma\in{\mathbb{R}}^{m\times n}$
are the _singular values_
$\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}>0$, with
$r=\mathop{\operator@font rank}\nolimits(J)\leq\min(m,n)$. Let
${\mathcal{N}}(J)$ denote the null space of the matrix $J$. It is well-known
that
${\mathcal{N}}(J):=\left\\{\mathbf{s}\in{\mathbb{R}}^{n}:J\mathbf{s}=0\right\\}=\operatorname{span}\\{\mathbf{v}_{r+1},\ldots,\mathbf{v}_{n}\\}.$
Let us now briefly review the computation of the minimal-norm solution to the
nonlinear problem (1) by the _minimal-norm Gauss–Newton_ (MNGN) method,
presented in [14]. Our aim is showing the reason for the possible lack of
convergence of such method. Here, we extend the discussion from [14] by
introducing a model profile $\overline{\mathbf{x}}\in{\mathbb{R}}^{n}$, which
represents an a priori estimate of the desired solution, and formulate the
problem in the form
$\begin{cases}\displaystyle\min_{\mathbf{x}\in{\mathbb{R}}^{n}}\|\mathbf{x}-\overline{\mathbf{x}}\|^{2}\\\
\displaystyle\mathbf{x}\in\bigl{\\{}\arg\min_{\mathbf{x}\in{\mathbb{R}}^{n}}\|F(\mathbf{x})-\mathbf{b}\|^{2}\bigr{\\}}.\end{cases}$
(9)
We consider an iterative method of the type (2) based on the following first-
order linearization of the problem
$\begin{cases}\displaystyle\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|\mathbf{x}^{(k)}-\overline{\mathbf{x}}+\alpha_{k}\mathbf{s}\|^{2}\\\
\displaystyle\mathbf{s}\in\bigl{\\{}\arg\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|J_{k}\mathbf{s}+\mathbf{r}_{k}\|^{2}\bigr{\\}},\end{cases}$
(10)
where $J_{k}=J(\mathbf{x}^{(k)})$ is the Jacobian of $F$ in $\mathbf{x}^{(k)}$
and $\mathbf{r}_{k}=\mathbf{r}(\mathbf{x}^{(k)})$ is the residual vector.
The damping parameter $\alpha_{k}$ is indispensable to ensure the convergence
of the Gauss–Newton method. We estimate it by the Armijo–Goldstein principle
[1, 9], but it can be chosen by any strategy which guarantees a reduction in
the norm of the residual. In our case, the Armijo condition [1, 5] implies
$f(\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)})\leq
f(\mathbf{x}^{(k)})+\mu\alpha_{k}\nabla
f(\mathbf{x}^{(k)})^{T}\widetilde{\mathbf{s}}^{(k)},$
where $\widetilde{\mathbf{s}}^{(k)}$ is determined by solving (4) and $\mu$ is
a constant in $(0,1)$. Since
$f(\mathbf{x})=\frac{1}{2}\|\mathbf{r}(\mathbf{x})\|^{2}$ and $\nabla
f(\mathbf{x})=J(\mathbf{x})^{T}\mathbf{r}(\mathbf{x})$, it reads
$\|\mathbf{r}(\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)})\|^{2}\leq\|\mathbf{r}_{k}\|^{2}+2\mu\alpha_{k}\mathbf{r}_{k}^{T}J_{k}\widetilde{\mathbf{s}}^{(k)}.$
Note that, as $\widetilde{\mathbf{s}}^{(k)}$ satisfies the normal equations
associated to problem (3), it holds
$J_{k}^{T}\mathbf{r}_{k}=-J_{k}^{T}J_{k}\widetilde{\mathbf{s}}^{(k)}$, so that
$\mathbf{r}_{k}^{T}J_{k}\widetilde{\mathbf{s}}^{(k)}=-\|J_{k}\widetilde{\mathbf{s}}^{(k)}\|^{2}$.
The _Armijo–Goldstein principle_ [2, 9] sets $\mu=\frac{1}{4}$ and determines
the scalar $\alpha_{k}$ as the largest number in the sequence $2^{-i}$,
$i=0,1,\ldots,$ for which it holds
$\|\mathbf{r}_{k}\|^{2}-\|\mathbf{r}(\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)})\|^{2}\geq\frac{1}{2}\alpha_{k}\|J_{k}\widetilde{\mathbf{s}}^{(k)}\|^{2}.$
(11)
The iteration resulting from the solution of (10) is defined by the following
theorem.
###### Theorem 1.
Let $\mathbf{x}^{(k)}\in{\mathbb{R}}^{n}$ and let
$\widetilde{\mathbf{x}}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)}$
be the Gauss–Newton iteration for (1), where the step
$\widetilde{\mathbf{s}}^{(k)}$ is determined by solving (4) and the step
length $\alpha_{k}$ by the Armijo–Goldstein principle. Then, the iteration
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\mathbf{s}^{(k)}$ defined by
(10) is given by
$\mathbf{x}^{(k+1)}=\widetilde{\mathbf{x}}^{(k+1)}-V_{2}V_{2}^{T}\bigl{(}\mathbf{x}^{(k)}-\overline{\mathbf{x}}\bigr{)},$
(12)
where $\mathop{\operator@font rank}\nolimits(J_{k})=r_{k}$ and the columns of
the matrix $V_{2}=[\mathbf{v}_{r_{k}+1},\ldots,\mathbf{v}_{n}]$ are
orthonormal vectors in ${\mathbb{R}}^{n}$ spanning the null space of $J_{k}$.
###### Proof.
The proof follows the pattern of that of Theorem 3.1 in [14]. Let $U\Sigma
V^{T}$ be the singular value decomposition of the matrix $J_{k}$. The upper-
level problem in (10) can be expressed as
$\|\mathbf{x}^{(k)}-\overline{\mathbf{x}}+\alpha_{k}\mathbf{s}\|^{2}=\|V^{T}(\mathbf{x}^{(k)}-\overline{\mathbf{x}}+\alpha_{k}\mathbf{s})\|^{2}=\|\alpha_{k}\mathbf{y}+\mathbf{z}^{(k)}\|^{2},$
with $\mathbf{y}=V^{T}\mathbf{s}$ and
$\mathbf{z}^{(k)}=V^{T}\left(\mathbf{x}^{(k)}-\overline{\mathbf{x}}\right)$.
Replacing $J_{k}$ by its SVD and setting
$\mathbf{g}^{(k)}=U^{T}\mathbf{r}_{k}$, we can rewrite (10) as the following
diagonal linear least-squares problem
$\begin{cases}\displaystyle\min_{\mathbf{y}\in{\mathbb{R}}^{n}}\|\alpha_{k}\mathbf{y}+\mathbf{z}^{(k)}\|^{2}\\\
\displaystyle\mathbf{y}\in\bigl{\\{}\arg\min_{\mathbf{y}\in{\mathbb{R}}^{n}}\|\Sigma\mathbf{y}+\mathbf{g}^{(k)}\|^{2}\bigr{\\}}.\end{cases}$
Solving the lower-level minimization problem uniquely determines the
components $y_{i}=-\sigma_{i}^{-1}g^{(k)}_{i}$, $i=1,\ldots,r_{k}$, while the
entries $y_{i}$, $i=r_{k}+1,\ldots,n$, are left undetermined. Their values can
be found by solving the upper-level problem. From
$\|\alpha_{k}\mathbf{y}+\mathbf{z}^{(k)}\|^{2}=\sum_{i=1}^{r_{k}}\left(-\alpha_{k}\frac{g^{(k)}_{i}}{\sigma_{i}}+z^{(k)}_{i}\right)^{2}+\sum_{i=r_{k}+1}^{n}\left(\alpha_{k}y_{i}+z^{(k)}_{i}\right)^{2},$
we obtain
$y_{i}=-\frac{z^{(k)}_{i}}{\alpha_{k}}=-\frac{1}{\alpha_{k}}\mathbf{v}_{i}^{T}(\mathbf{x}^{(k)}-\overline{\mathbf{x}})$,
$i=r_{k}+1,\ldots,n$. Then, the solution to (10), that is, the next
approximation to the solution of (9), is
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}V\mathbf{y}=\mathbf{x}^{(k)}-\alpha_{k}\sum_{i=1}^{r_{k}}\frac{g^{(k)}_{i}}{\sigma_{i}}\mathbf{v}_{i}-\sum_{i=r_{k}+1}^{n}(\mathbf{v}_{i}^{T}(\mathbf{x}^{(k)}-\overline{\mathbf{x}}))\mathbf{v}_{i},$
where the last summation can be written in matrix form as
$V_{2}V_{2}^{T}\left(\mathbf{x}^{(k)}-\overline{\mathbf{x}}\right)$, and the
columns of $V_{2}=[\mathbf{v}_{r_{k}+1},\ldots,\mathbf{v}_{n}]$ are a basis
for ${\mathcal{N}}(J_{k})$.
It is immediate (see [14, Theorem 3.1]) to prove that
$\widetilde{\mathbf{x}}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)}=\mathbf{x}^{(k)}-\alpha_{k}\sum_{i=1}^{r_{k}}\frac{g^{(k)}_{i}}{\sigma_{i}}\mathbf{v}_{i},$
from which (12) follows. ∎
Summarizing, the MNGN method consists of the iteration
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\mathbf{s}^{(k)},$
where the step is
$\mathbf{s}^{(k)}=\widetilde{\mathbf{s}}^{(k)}-\frac{1}{\alpha_{k}}\mathbf{t}^{(k)},$
with
$\widetilde{\mathbf{s}}^{(k)}=-\sum_{i=1}^{r_{k}}\frac{g^{(k)}_{i}}{\sigma_{i}}\mathbf{v}_{i},\qquad\mathbf{t}^{(k)}=V_{2}V_{2}^{T}\bigl{(}\mathbf{x}^{(k)}-\overline{\mathbf{x}}\bigr{)}.$
(13)
Since ${\mathcal{P}}_{{\mathcal{N}}(J_{k})}=V_{2}V_{2}^{T}$ is the orthogonal
projector onto ${\mathcal{N}}(J_{k})$, the above theorem states that the
$(k+1)$th iterate of the MNGN method is orthogonal to the null space of
$J_{k}$.
Theorem 1 shows that the correction vector $\mathbf{t}^{(k)}$ defined in (13),
which allows to compute the minimal-norm solution at each step, is not damped
by the parameter $\alpha_{k}$. As a result, in some numerical examples, the
method fails to converge because projecting the solution orthogonally to the
null space of $J_{k}$ causes the residual to increase. To understand how this
can happen, a second-order analysis of the objective function is required.
The second-order Taylor approximation to the function
$f(\mathbf{x})=\frac{1}{2}\|\mathbf{r}(\mathbf{x})\|^{2}$ at
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\alpha\mathbf{s}$ is
$f(\mathbf{x}^{(k+1)})\simeq f(\mathbf{x}^{(k)})+\alpha\nabla
f(\mathbf{x}^{(k)})^{T}\mathbf{s}+\frac{1}{2}\alpha^{2}\mathbf{s}^{T}\nabla^{2}f(\mathbf{x}^{(k)})\mathbf{s}.$
(14)
The gradient and the Hessian of $f(\mathbf{x})$, written in matrix form, are
given by
$\nabla
f(\mathbf{x})=J(\mathbf{x})^{T}\mathbf{r}(\mathbf{x}),\qquad\nabla^{2}f(\mathbf{x})=J(\mathbf{x})^{T}J(\mathbf{x})+{\mathcal{Q}}(\mathbf{x}),$
where
${\mathcal{Q}}(\mathbf{x})=\sum_{i=1}^{m}r_{i}(\mathbf{x})\nabla^{2}r_{i}(\mathbf{x}),$
and $\nabla^{2}r_{i}(\mathbf{x})$ is the Hessian matrix of
$r_{i}(\mathbf{x})$. By replacing the expression of $f$ and
$\alpha\mathbf{s}=\alpha\widetilde{\mathbf{s}}-\mathbf{t}$ in (14), where
$\widetilde{\mathbf{s}}$ is the Gauss–Newton step and $\mathbf{t}$ is in the
null space of $J_{k}$, and letting
${\mathcal{Q}}_{k}={\mathcal{Q}}(\mathbf{x}^{(k)})$, the following
approximation is obtained
$\displaystyle\frac{1}{2}\|\mathbf{r}_{k+1}\|^{2}$
$\displaystyle\simeq\frac{1}{2}\|\mathbf{r}_{k}\|^{2}+\alpha\mathbf{r}_{k}^{T}J_{k}\mathbf{s}+\frac{1}{2}\alpha^{2}\mathbf{s}^{T}\left(J_{k}^{T}J_{k}+{\mathcal{Q}}_{k}\right)\mathbf{s}$
$\displaystyle=\frac{1}{2}\|\mathbf{r}_{k}\|^{2}+\alpha\mathbf{r}_{k}^{T}J_{k}\widetilde{\mathbf{s}}+\frac{1}{2}\alpha^{2}\widetilde{\mathbf{s}}^{T}\left(J_{k}^{T}J_{k}+{\mathcal{Q}}_{k}\right)\widetilde{\mathbf{s}}-\alpha\mathbf{t}^{T}{\mathcal{Q}}_{k}\widetilde{\mathbf{s}}+\frac{1}{2}\mathbf{t}^{T}{\mathcal{Q}}_{k}\mathbf{t}.$
The first two terms containing second derivatives (the matrix
${\mathcal{Q}}_{k}$) are damped by the $\alpha$ parameter. If the function $F$
is mildly nonlinear, the third term
$\frac{1}{2}\mathbf{t}^{T}{\mathcal{Q}}_{k}\mathbf{t}$ is negligible. In the
presence of a strong nonlinearity, its contribution to the residual is
significant and may lead to its growth. This shows that a damping parameter is
required to control the step length for both the Gauss–Newton step
$\widetilde{\mathbf{s}}$ and the correction vector $\mathbf{t}$. If a
relaxation parameter is introduced for $\mathbf{t}$, Theorem 1 implies that
the minimal-norm solution of (10) can only be approximated.
###### Remark 2.
We report a simple low dimensional example for which the MNGN method may not
converge. Let us consider the function
$F:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ defined by
$F(\mathbf{x})=\delta^{2}\left[(x_{1}-\gamma)^{2}+(x_{2}-\gamma)^{2}\right]-1,$
depending on the parameters $\delta,\gamma\in{\mathbb{R}}$. Since the Hessian
matrix of the residual is given by
$\nabla^{2}r(\mathbf{x})=\begin{bmatrix}2\delta^{2}&0\\\
0&2\delta^{2}\end{bmatrix},$
the second-order term $\frac{1}{2}\mathbf{t}^{T}{\mathcal{Q}}_{k}\mathbf{t}$
is not negligible, in general, when $\delta$ is relatively large. For example,
setting $\delta=0.7$, $\gamma=2$, and choosing an initial vector
$\mathbf{x}^{(0)}$ with random components in $(-5,5)$, the MNGN method
converges with a large number of the iterations (350 on average). Setting
$\delta=0.75$, the same method does not converge within 500 iterations.
## 3 Estimating the rank of the Jacobian
In order to apply Theorem 1 to computing the minimal-norm solution by (12),
the rank of the Jacobian matrix $J_{k}=J(\mathbf{x}^{(k)})$ should be known in
advance. As the rank may vary along the iterations, we set
$r_{k}=\mathop{\operator@font rank}\nolimits(J_{k})$. The knowledge of $r_{k}$
for each $k=0,1,\ldots$, is not generally available, making it necessary to
estimate its value at each iteration step, to avoid nonconvergence or a
breakdown of the algorithm.
In such situations, it is common to consider the numerical rank
$r_{\epsilon,k}$ of $J_{k}$, sometimes denoted as $\epsilon$-rank, where
$\epsilon$ represents a chosen tolerance. The numerical rank is defined in
terms of the singular values $\sigma_{i}^{(k)}$ of $J_{k}$, as the integer
$r_{\epsilon,k}$ such that
$\sigma_{r_{\epsilon,k}}^{(k)}>\epsilon\geq\sigma_{r_{\epsilon,k}+1}^{(k)}.$
Theorem 1 can be adapted to this setting, by simply replacing at each
iteration the rank $r_{k}$ with the numerical rank $r_{\epsilon,k}$.
Determining the numerical rank is a difficult task for discrete ill-posed
problems, in which the singular values decay monotonically to zero. In such a
case, the numerical rank plays the role of a regularization parameter and is
estimated by suitable methods, which often require information about the noise
level and type; see, e.g., [11, 15].
When the problem is locally rank-deficient, meaning that the rank of
$J(\mathbf{x})$ depends on the evaluation vector $\mathbf{x}$, the numerical
rank $r_{\epsilon,k}$ can be determined, in principle, by choosing a suitable
value of $\epsilon$. Numerical experiments show that a fixed value of
$\epsilon$ does not always lead to a correct estimation of $r_{\epsilon,k}$,
and that it is preferable to determine the $\epsilon$-rank by searching for a
sensible gap between $\sigma_{r_{\epsilon,k}}^{(k)}$ and
$\sigma_{r_{\epsilon,k}+1}^{(k)}$.
To locate such a gap, we adopt a heuristic approach already applied in [4] for
the same purpose, in a different setting. At each step, we compute the ratios
$\rho_{i}^{(k)}=\frac{\sigma_{i}^{(k)}}{\sigma_{i+1}^{(k)}},\qquad
i=1,2,\ldots,q-1,$
where $q=\min(m,n)$. Then, we consider the index set
${\mathcal{I}}_{k}=\left\\{i\in\\{1,2,\ldots,q-1\\}:\rho_{i}^{(k)}>R\text{ and
}\sigma_{i}^{(k)}>\tau\right\\}.$
An index $i$ belongs to ${\mathcal{I}}_{k}$ if there is a significant “jump”
between $\sigma_{i}^{(k)}$ and $\sigma_{i+1}^{(k)}$, and $\sigma_{i}^{(k)}$ is
numerically nonzero. If the set ${\mathcal{I}}_{k}$ is empty, we set
$r_{\epsilon,k}=q$. Otherwise, we consider
$\rho_{j}^{(k)}=\max_{i\in{\mathcal{I}}_{k}}\rho_{i}^{(k)},$ (15)
and we define $r_{\epsilon,k}=j$. This amounts to selecting the largest gap
between “large” and “small” singular values. In our numerical simulations, we
set $R=10^{2}$ and $\tau=10^{-8}$. We observed that the value of these
parameters is not critical for problems characterized by a rank deficient
Jacobian. Estimating the rank becomes increasingly difficult as the gap
between “large” and “small” singular values gets smaller. This condition
usually corresponds to ill-conditioned problems, which require specific
regularization methods.
## 4 Choosing the projection step length
The occasional nonconvergence in the computation of the minimal-norm solution
to a nonlinear least-squares problem was discussed in [3], where the authors
propose an iterative method based on a convex combination of the Gauss–Newton
and the minimal-norm Gauss–Newton iterates, which we denote by CKB. Following
our notation, it can be expressed in the form
$\mathbf{x}^{(k+1)}=\left(1-\gamma_{k}\right)\left[\mathbf{x}^{(k)}+\widetilde{\mathbf{s}}^{(k)}\right]+\gamma_{k}\left[\mathbf{x}^{(k)}+\widetilde{\mathbf{s}}^{(k)}-V_{2}V_{2}^{T}\mathbf{x}^{(k)}\right],$
(16)
where the parameters $\gamma_{k}\in[0,1]$, for $k=0,1,\ldots$, form a sequence
converging to zero. The standard Gauss–Newton method is obtained by setting
$\gamma_{k}=0$, while $\gamma_{k}=1$ leads to the minimal-norm Gauss–Newton
method. In their numerical examples, the authors adopt the sequences
$\gamma_{k}=(0.5)^{k+1}$ and $\gamma_{k}=(0.5)^{2^{k}}$.
It is immediate to rewrite (16) in the form (8), showing that the method
proposed in [3] is equivalent to the application of the undamped Gauss–Newton
method, whose convergence is not theoretically guaranteed [2], with a damped
correction to favor the decrease of the norm of the solution. The numerical
experiments reported in the paper show that the minimization of the residual
is sped up if $\gamma_{k}$ quickly converges to zero, while the norm of the
solution decreases faster if $\gamma_{k}$ has a slower decay. The choice of
the sequence of parameters appears to be critical to tune the performance of
the algorithm, and no adaptive choice for $\gamma_{k}$ is proposed.
In this paper, we propose to introduce a second relaxation parameter,
$\beta_{k}$, to control the step length of the minimal-norm correction
$\mathbf{t}^{(k)}$ defined in (13). The new iterative method is denoted by
MNGN2 and it takes the form
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)}-\beta_{k}\mathbf{t}^{(k)},$
(17)
where $\widetilde{\mathbf{s}}^{(k)}$ is the step vector produced by the
Gauss–Newton method and $\mathbf{t}^{(k)}$ is the projection vector which
makes the norm of $\mathbf{x}^{(k+1)}$ minimal, without changing the value of
the linearized residual.
The second-order analysis reported at the end of Section 2 may be adapted for
the CKB method (8). It shows that neither the CKB nor the MNGN method are
guaranteed to converge, as both the Gauss–Newton search direction and the
projection step should be damped to ensure that the residual decreases. The
MNGN2 method locally converges if $\alpha_{k}$ and $\beta_{k}$ are suitably
chosen, but it will recover the minimal-norm solution only if $\beta_{k}\simeq
1$ for $k$ close to convergence.
Our numerical tests showed that it is important to choose both $\alpha_{k}$
and $\beta_{k}$ adaptively along the iterations. A simple solution is to let
$\beta_{k}=\alpha_{k}$ and estimate $\alpha_{k}$ by the Armijo–Goldstein
principle (11), with
$\mathbf{s}^{(k)}=\widetilde{\mathbf{s}}^{(k)}-\mathbf{t}^{(k)}$ in place of
$\widetilde{\mathbf{s}}^{(k)}$. This approach proves to be effective in the
computation of the minimal-norm solution, but its convergence is often rather
slow. To speed up iteration we propose a procedure to adaptively choose the
value of $\beta_{k}$.
Algorithm 1 Outline of the MNGN2 method.
0: nonlinear function $F$, data vector $\mathbf{b}$,
0: initial solution $\mathbf{x}^{(0)}$, model profile $\overline{\mathbf{x}}$,
tolerance $\eta$ for residual increase
0: approximation $\mathbf{x}^{(k+1)}$ of minimal-norm least-squares solution
1: $k=0$, $\beta=1$
2: repeat
3: $k=k+1$
4: estimate $r_{k}=\mathop{\operator@font rank}\nolimits(J(\mathbf{x}^{(k)}))$
by (15)
5: compute $\widetilde{\mathbf{s}}^{(k)}$ by the Gauss–Newton method (3)
6: compute $\alpha_{k}$ by the Armijo–Goldstein principle (11)
7: compute $\mathbf{t}^{(k)}$ by (13)
8: if $\beta<1$ then
9: $\beta=2\beta$
10: end if
11:
$\widetilde{\mathbf{x}}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)}$
12:
$\widetilde{\rho}_{k+1}=\|F(\widetilde{\mathbf{x}}^{(k+1)})-\mathbf{b}\|+\varepsilon_{M}$
13: $\mathbf{x}^{(k+1)}=\widetilde{\mathbf{x}}^{(k+1)}-\beta\mathbf{t}^{(k)}$
14: $\rho_{k+1}=\|F(\mathbf{x}^{(k+1)})-\mathbf{b}\|$
15: while
$(\rho_{k+1}>\widetilde{\rho}_{k+1}+\delta(\widetilde{\rho}_{k+1},\eta))$ and
($\beta>10^{-8}$) do
16: $\beta=\beta/2$
17: $\mathbf{x}^{(k+1)}=\widetilde{\mathbf{x}}^{(k+1)}-\beta\mathbf{t}^{(k)}$
18: $\rho_{k+1}=\|F(\mathbf{x}^{(k+1)})-\mathbf{b}\|$
19: end while
20: $\beta_{k}=\beta$
21: until convergence
This procedure is outlined in Algorithm 1. Initially, we set $\beta=1$. At
each iteration, we compute the residual at the Gauss–Newton iteration
$\widetilde{\mathbf{x}}^{(k+1)}$ and at the tentative iteration
$\mathbf{x}^{(k+1)}=\widetilde{\mathbf{x}}^{(k+1)}-\beta\mathbf{t}^{(k)}$.
Subtracting the vector $\beta\mathbf{t}^{(k)}$ may cause the residual to
increase. We accept such an increase if
$\|\mathbf{r}(\mathbf{x}^{(k+1)})\|\leq\|\mathbf{r}(\widetilde{\mathbf{x}}^{(k+1)})\|+\delta\bigl{(}\|\mathbf{r}(\widetilde{\mathbf{x}}^{(k+1)})\|,\eta\bigr{)},$
(18)
where $\delta(\rho,\eta)$ is a function determining the maximal increase
allowed in the residual $\rho=\|\mathbf{r}(\widetilde{\mathbf{x}}^{(k+1)})\|$,
and $\eta>0$ is a chosen tolerance. On the contrary, $\beta$ is halved and the
residual is recomputed until (18) is verified or $\beta$ becomes excessively
small. To allow $\beta$ to increase, we tentatively double it at each
iteration (see line 9 in the algorithm) before applying the above procedure.
At line 12 of the algorithm we add the machine epsilon $\varepsilon_{M}$ to
the actual residual $\widetilde{\rho}_{k+1}$ to avoid that
$\delta(\widetilde{\rho}_{k+1},\eta)$ becomes zero.
A possible choice for the value of the residual increase is
$\delta(\rho,\eta)=\eta\rho$, with $\eta$ suitably chosen. Our experiments
showed that it is possible to find, by chance, a value of $\eta$ which
produces good results, but its choice is strongly dependent on the particular
example. We also noticed that, in cases where the residual stagnates,
accepting a large increase in the residual may lead to nonconvergence. In such
situations, a fixed multiple of the residual is not well suited to model its
increase. Indeed, if the residual is large, one is prone to accept only a
small increase, while if the residual is very small, a relatively large growth
may be acceptable.
To overcome these difficulties, we consider $\delta(\rho,\eta)=\rho^{\eta}$,
and choose $\eta$ at each step by the adaptive procedure described in
Algorithm 2. When at least $k_{\text{res}}$ iterations have been performed, we
compute the linear polynomial which fits the logarithm of the last
$k_{\text{res}}$ residuals in the least-squares sense. To detect if the
residual stagnates or increases, we check if the slope $M$ of the regression
line exceeds $-10^{-2}$. If this happens, the value of $\eta$ is doubled. The
effect on the algorithm is to enhance the importance of the decrease of the
residual and reduce that of the norm. To recover a sensible decrease in the
norm, if at a subsequent step the residual reduction accelerates (e.g.,
$M<-\frac{1}{2}$), the value of $\eta$ is halved. In our experiments, we
initialize $\eta$ to $\frac{1}{8}$ and set $k_{\text{res}}=5$.
###### Remark 3.
The adaptive estimation of $\delta(\rho,\eta)$ does not significantly increase
the complexity of Algorithm 1, as line 3 of Algorithm 2 implies the solution
of a $2\times 2$ linear system whose matrix is fixed and can be computed in
advance, while forming the right-hand side requires $4k_{\text{res}}$ floating
point operations.
Algorithm 2 Adaptive determination of the residual increase
$\delta(\rho,\eta)$.
0: actual residual $\rho=\|\mathbf{r}(\widetilde{\mathbf{x}}^{(k+1)})\|$,
starting tolerance $\eta$
0: iteration index $k$, residuals
$\theta_{j}=\|\mathbf{r}(\widetilde{\mathbf{x}}^{(k-k_{\text{res}}+j)})\|$,
$j=1,\ldots,k_{\text{res}}$
0: residual increase $\delta(\rho,\eta)$
1: $M_{\text{min}}=-10^{-2}$, $M_{\text{max}}=-\frac{1}{2}$
2: if $k\geq k_{\text{res}}$ then
3: compute regression line $p_{1}(t)=Mt+N$ of $(j,\log(\theta_{j}))$,
$j=1,\ldots,k_{\text{res}}$
4: if $M>M_{\text{min}}$ then
5: $\eta=2\eta$
6: else if $M<M_{\text{max}}$ then
7: $\eta=\eta/2$
8: end if
9: end if
10: $\delta(\rho,\eta)=\rho^{\eta}$
To detect convergence, we interrupt the iteration as soon as
$\|\mathbf{x}^{(k+1)}-\mathbf{x}^{(k)}\|<\tau\|\mathbf{x}^{(k+1)}\|\qquad\text{or}\qquad\|\alpha_{k}\widetilde{\mathbf{s}}^{(k)}\|<\tau,$
(19)
or when a fixed number of iteration $N_{\text{max}}$ is exceeded. The second
stop condition in (19) detects the slow progress of the relaxed Gauss–Newton
iteration algorithm. This often happens close to the solution. The stop
tolerance is set to $\tau=10^{-8}$.
## 5 Nonlinear minimal-$\boldsymbol{L}$-norm solution
The introduction of a regularization matrix $L\in{\mathbb{R}}^{p\times n}$,
$p\leq n$, in least-squares problems was originally connected to the numerical
treatment of linear discrete ill-posed problems, and in particular to Tikhonov
regularization. The use of a regularization matrix is also justified in
underdetermined least-squares problems to select a solution with particular
features, such as smoothness or sparsity, among the infinitely many possible
solutions.
While in (6) the seminorm $\|L\mathbf{s}\|$ is minimized over all the updating
vectors $\mathbf{s}$ which minimize the linearized residual, here we seek to
compute the minimal-$L$-norm solution to the nonlinear problem (1), that is
the vector $\mathbf{x}$ which solves the constrained problem
$\begin{cases}\displaystyle\min_{\mathbf{x}\in{\mathbb{R}}^{n}}\|L(\mathbf{x}-\overline{\mathbf{x}})\|^{2}\\\
\displaystyle\mathbf{x}\in\bigl{\\{}\arg\min_{\mathbf{x}\in{\mathbb{R}}^{n}}\|F(\mathbf{x})-\mathbf{b}\|^{2}\bigr{\\}}.\end{cases}$
(20)
Similarly to Section 2, we consider an iterative method of the type (2), where
the step $\mathbf{s}^{(k)}$ is the solution of the linearized problem
$\begin{cases}\displaystyle\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|L(\mathbf{x}^{(k)}-\overline{\mathbf{x}}+\alpha\mathbf{s})\|^{2}\\\
\displaystyle\mathbf{s}\in\bigl{\\{}\arg\min_{\mathbf{s}\in{\mathbb{R}}^{n}}\|J_{k}\mathbf{s}+\mathbf{r}_{k}\|^{2}\bigr{\\}}.\end{cases}$
(21)
We will denote the iteration resulting from the solution of (21) as the
_minimal- $L$-norm Gauss–Newton_ (MLNGN) method.
We recall the definition of the generalized singular value decomposition
(GSVD) of a matrix pair $(J,L)$ [10]. Let $J\in{\mathbb{R}}^{m\times n}$ and
$L\in{\mathbb{R}}^{p\times n}$ be matrices with $\mathop{\operator@font
rank}\nolimits(J)=r$ and $\mathop{\operator@font rank}\nolimits(L)=p$. Assume
that $m+p\geq n$ and
$\mathop{\operator@font rank}\nolimits\left(\begin{bmatrix}J\\\
L\end{bmatrix}\right)=n,$
which corresponds to requiring that
$\mathcal{N}(J)\cap\mathcal{N}(L)=\\{0\\}$. The GSVD of the matrix pair
$(J,L)$ is defined as the factorization
$J=U\Sigma_{J}W^{-1},\qquad L=V\Sigma_{L}W^{-1},$
where $U\in{\mathbb{R}}^{m\times m}$ and $V\in{\mathbb{R}}^{p\times p}$ are
matrices with orthonormal columns $\mathbf{u}_{i}$ and $\mathbf{v}_{i}$,
respectively, and $W\in{\mathbb{R}}^{n\times n}$ is nonsingular. If $m\geq
n\geq r$, the matrices $\Sigma_{J}\in{\mathbb{R}}^{m\times n}$ and
$\Sigma_{L}\in{\mathbb{R}}^{p\times n}$ have the form
$\Sigma_{J}=\left[\begin{array}[]{ccc}O_{n-r}&&\\\ &C&\\\ &&I_{d}\\\
\hline\cr\\\ &O_{(m-n)\times
n}&\end{array}\right],\qquad\Sigma_{L}=\left[\begin{array}[]{cc|c}I_{p-r+d}&&\\\
&&O_{p\times d}\\\ &S&\end{array}\right],$
where $d=n-p$,
$\displaystyle C$ $\displaystyle=\mathop{\operator@font
diag}\nolimits(c_{1},\ldots,c_{r-d}),\qquad$ $\displaystyle 0<c_{1}\leq
c_{2}\leq\cdots\leq c_{r-d}<1,$ (22) $\displaystyle S$
$\displaystyle=\mathop{\operator@font
diag}\nolimits(s_{1},\ldots,s_{r-d}),\qquad$ $\displaystyle 1>s_{1}\geq
s_{2}\geq\cdots\geq s_{r-d}>0,$
with $c_{i}^{2}+s_{i}^{2}=1$, for $i=1,\ldots,r-d$. The identity matrix of
size $k$ is denoted by $I_{k}$, while $O_{k}$ and $O_{k\times\ell}$ are zero
matrices of size $k\times k$ and $k\times\ell$, respectively; a matrix block
has to be omitted when one of its dimensions is zero. The scalars
$\gamma_{i}=\frac{c_{i}}{s_{i}}$ are called _generalized singular values_ ,
and they appear in nondecreasing order.
If $r\leq m<n$, the matrices $\Sigma_{J}\in{\mathbb{R}}^{m\times n}$ and
$\Sigma_{L}\in{\mathbb{R}}^{p\times n}$ take the form
$\Sigma_{J}=\left[\begin{array}[]{c|ccc}&O_{m-r}&&\\\ O_{m\times(n-m)}&&C&\\\
&&&I_{d}\end{array}\right],\qquad\Sigma_{L}=\left[\begin{array}[]{cc|c}I_{p-r+d}&&\\\
&&O_{p\times d}\\\ &S&\end{array}\right],$
where the blocks are defined as above.
Let $J_{k}=U\Sigma_{J}W^{-1}$, $L=V\Sigma_{L}W^{-1}$ be the GSVD of the matrix
pair ($J_{k}$,$L$). We indicate by $\mathbf{w}_{i}$ the column vectors of the
matrix $W$, and by $\widehat{\mathbf{w}}^{j}$ the rows of $W^{-1}$, that is
$W=[\mathbf{w}_{1},\ldots,\mathbf{w}_{n}],\qquad
W^{-1}=\begin{bmatrix}\widehat{\mathbf{w}}^{1}\\\ \vdots\\\
\widehat{\mathbf{w}}^{n}\end{bmatrix}.$
We have
${\mathcal{N}}(J_{k})=\operatorname{span}(\mathbf{w}_{1},\ldots,\mathbf{w}_{n-r_{k}})$,
if $r_{k}=\mathop{\operator@font rank}\nolimits(J_{k})$; see [14] for a proof.
###### Theorem 4.
Let $\mathbf{x}^{(k)}\in{\mathbb{R}}^{n}$ and let
$\widetilde{\mathbf{x}}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\widetilde{\mathbf{s}}^{(k)}$
be the Gauss–Newton iteration for (1), where the step
$\widetilde{\mathbf{s}}^{(k)}$ is determined by solving (6) and the step
length $\alpha_{k}$ by the Armijo–Goldstein principle. Then, the iteration
$\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\alpha_{k}\mathbf{s}^{(k)}$ for (21), is
given by
$\mathbf{x}^{(k+1)}=\widetilde{\mathbf{x}}^{(k+1)}-W_{1}\widehat{W}_{1}\bigl{(}\mathbf{x}^{(k)}-\overline{\mathbf{x}}\bigr{)},$
(23)
where $\widehat{W}_{1}\in{\mathbb{R}}^{(n-r_{k})\times n}$ contains the first
$n-r_{k}$ rows of $W^{-1}$, and $W_{1}\in{\mathbb{R}}^{n\times(n-r_{k})}$ is
composed of the first $n-r_{k}$ columns of $W$.
###### Proof.
The proof proceeds analogously to that of Theorem 4.2 in [14]. Replacing
$J_{k}$ and $L$ with their GSVD and setting $\mathbf{y}=W^{-1}\mathbf{s}$,
$\mathbf{z}^{(k)}=W^{-1}\left(\mathbf{x}^{(k)}-\overline{\mathbf{x}}\right)$,
and $\mathbf{g}^{(k)}=U^{T}\mathbf{r}_{k}$, (21) can be rewritten as the
following diagonal least-squares problem
$\begin{cases}\displaystyle\min_{\mathbf{y}\in{\mathbb{R}}^{n}}\|\Sigma_{L}(\alpha_{k}\mathbf{y}+\mathbf{z}^{(k)})\|^{2}\\\
\displaystyle\mathbf{y}\in\bigl{\\{}\arg\min_{\mathbf{y}\in{\mathbb{R}}^{n}}\|\Sigma_{J}\mathbf{y}+\mathbf{g}^{(k)}\|^{2}\bigr{\\}}.\end{cases}$
When $m\geq n$, the diagonal linear system in the constraint is solved by a
vector $\mathbf{y}$ with entries
$y_{i}=\begin{cases}\displaystyle-\frac{g^{(k)}_{i}}{c_{i-n+r_{k}}},\quad&i=n-r_{k}+1,\ldots,p,\\\
-g^{(k)}_{i},&i=p+1,\ldots,n.\end{cases}$
The components $y_{i}$, for $i=1,\ldots,n-r_{k}$, can be determined by
minimizing the norm
$\displaystyle\|\Sigma_{L}(\alpha_{k}\mathbf{y}+\mathbf{z}^{(k)})\|^{2}$
$\displaystyle=\sum_{i=1}^{n-r_{k}}\left(\alpha_{k}y_{i}+z_{i}^{(k)}\right)^{2}$
(24)
$\displaystyle\phantom{=}+\sum_{i=n-r_{k}+1}^{p}\left(-\alpha_{k}\frac{g^{(k)}_{i}}{\gamma_{i-n+r_{k}}}+s_{i-n+r_{k}}z_{i}^{(k)}\right)^{2},$
where $\gamma_{i}=\frac{c_{i}}{s_{i}}$ are the generalized singular values of
the matrix pair $(J_{k},L)$. The minimum of (24) is reached for
$y_{i}=-\frac{1}{\alpha_{k}}z^{(k)}_{i}=-\frac{1}{\alpha_{k}}\widehat{\mathbf{w}}^{i}(\mathbf{x}^{(k)}-\overline{\mathbf{x}})$,
$i=1,\ldots,n-r_{k}$, and the solution to (21), that is, the next
approximation to the solution of (20), is
$\displaystyle\mathbf{x}^{(k+1)}$
$\displaystyle=\mathbf{x}^{(k)}+\alpha_{k}W\mathbf{y}$ (25)
$\displaystyle=\mathbf{x}^{(k)}-\sum_{i=1}^{n-r_{k}}z_{i}^{(k)}\mathbf{w}_{i}-\alpha_{k}\sum_{i=n-r_{k}+1}^{p}\frac{g^{(k)}_{i}}{c_{i-n+r_{k}}}\mathbf{w}_{i}-\alpha_{k}\sum_{i=p+1}^{n}g^{(k)}_{i}\mathbf{w}_{i},$
where the first summation in the right-hand side can be rewritten as
$W_{1}\widehat{W}_{1}(\mathbf{x}^{(k)}-\overline{\mathbf{x}})$. Applying the
same procedure to (6), we obtain
$\widetilde{\mathbf{x}}^{(k+1)}=\mathbf{x}^{(k)}-\alpha_{k}\sum_{i=n-r_{k}+1}^{p}\frac{g^{(k)}_{i}}{c_{i-n+r_{k}}}\mathbf{w}_{i}-\alpha_{k}\sum_{i=p+1}^{n}g^{(k)}_{i}\mathbf{w}_{i},$
from which (23) follows. Since solving (21) for $m<n$ leads to a formula
similar to (25), with $g^{(k)}_{i-n+m}$ in place of $g^{(k)}_{i}$, the
validity of (23) is confirmed. ∎
As in the computation of the minimal-norm solution, the iteration based on
(23) fails to converge without a suitable relaxation parameter $\beta_{k}$ for
the projection vector
$\mathbf{t}^{(k)}=W_{1}\widehat{W}_{1}(\mathbf{x}^{(k)}-\overline{\mathbf{x}})$.
We adopted an iteration similar to (17), choosing $\beta_{k}$ by adapting
Algorithms 1 and 2 to this setting. It is important to note that
$\widetilde{{\mathcal{P}}}_{{\mathcal{N}}(J_{k})}=W_{1}\widehat{W}_{1}$ is an
oblique projector onto ${\mathcal{N}}(J_{k})$.
At the same time, the rank of the Jacobian is estimated at each step by
applying the procedure described in Section 3 to the diagonal elements
$c_{j}^{(k)}$, $j=1,\ldots,q-d$, of the GSVD factor $\Sigma_{J}$ of $J_{k}$;
see (22). In this case, at each step, we compute the ratios
$\rho_{i}^{(k)}=\frac{c_{i+1}^{(k)}}{c_{i}^{(k)}},\qquad i=1,2,\ldots,q-d-1,$
where $q=\min(m,n)$.
Actually, the GSVD routine computes the matrix $W^{-1}$, but the matrix $W$ is
needed for the computation of both the vectors $\widetilde{\mathbf{s}}^{(k)}$
and $\mathbf{t}^{(k)}$. To reduce the computational load, we compute at each
iteration the LU factorization $PW^{-1}=LU$, and we use it to solve the linear
system with two right-hand sides
$W^{-1}\begin{bmatrix}\mathbf{t}^{(k)}&\widetilde{\mathbf{s}}^{(k)}\end{bmatrix}=\begin{bmatrix}\widehat{W}_{1}(\mathbf{x}^{(k)}-\overline{\mathbf{x}})&\mathbf{0}_{n-r}\\\
\mathbf{0}_{r}&\widetilde{\mathbf{y}}\end{bmatrix},$
where $\widetilde{\mathbf{y}}\in{\mathbb{R}}^{r}$ contains the last $r$
components of the vector $\mathbf{y}$ appearing in (25), and $\mathbf{0}_{k}$
denotes the zero vector of size $k$.
## 6 Test problems and numerical results
The MNGN2 method, defined by (17), was implemented in the Matlab programming
language; the software is available from the authors. The developed functions
implement all the variants of the MNGN2 algorithm, as well as the MNGN and CKB
methods developed in [14] and [3], respectively.
In the following, the MNGN2 algorithm (17) will be denoted by different names,
according to the particular implementation. In the method denoted by
MNGN$2_{\alpha}$, we let $\beta_{k}=\alpha_{k}$ in (17), and determine
$\alpha_{k}$ by the Armijo–Goldstein principle. Algorithm 1 is denoted by
MNGN$2_{\alpha\beta}$, when $\delta(\rho,\eta)=\eta\rho$, with a fixed value
of $\eta$. The same algorithm with $\delta(\rho,\eta)=\rho^{\eta}$, and $\eta$
estimated by Algorithm 2, is labeled as MNGN$2_{\alpha\beta\delta}$. The
algorithm (16) developed in [3] is denoted by CKB1 when
$\gamma_{k}=(0.5)^{k+1}$, and by CKB2 when $\gamma_{k}=(0.5)^{2^{k}}$. The
same algorithms are denoted by rCKB1 and rCKB2 when they are applied with the
automatic estimation of the rank of the Jacobian, discussed in Section 3. To
compare the methods and investigate their performance, we performed numerical
experiments on various test problems that highlight particular difficulties in
the computation of the minimal-norm solution. Example 5 illustrates a
situation where the MNGN method either fails or produces unacceptable results,
while the other methods perform well; in Example 6, we investigate the
dependence of the MNGN$2_{\alpha\beta}$ method on the choice of the parameter
$\eta$; Example 7 is the first medium-size test problem we consider, it shows
the importance of the Jacobian rank estimation for the effectiveness of the
algorithms; in Example 8, the methods are compared in the solution of
minimal-$L$-norm problems with different regularization matrices; finally, in
Example 9, we let the dimension of the problem vary and we explore the
dependence of the computed solution on the availability of a priori
information in the form of a model profile.
For each experiment, we repeated the computation 100 times, varying the
starting point $\mathbf{x}^{(0)}$ by letting its components be uniformly
distributed random numbers in $(-5,5)$. The model profile
$\overline{\mathbf{x}}$ was set to the zero vector except in Example 9.
We consider a numerical test a “success” if the algorithm converges according
to condition (19), with stop tolerance $\tau=10^{-8}$ and maximum number of
iterations $N_{\text{max}}=500$. A failure is not a serious problem, in
general, because nonconvergence simply suggests to try a different starting
vector. Anyway, if this happens too often, it increases the computational
load. At the same time, a success of a method does not imply that it recovers
the minimal-norm solution, as the convergence is only local. So, to give an
idea of the performance of the methods, we measure over all the tests the
average of both the number of iterations required and the norm of the
converged solution $\|\widetilde{\mathbf{x}}\|$. We also report the number of
successes.
We note that the computational cost of each iteration is roughly the same for
all the methods considered. Indeed, the additional complexity required by the
MNGN2 algorithms consists of the estimation of the numerical rank
$r_{\epsilon,k}$, of the residual increase $\delta(\rho,\eta)$, and of the
projection parameter $\beta_{k}$. All these computations involve a small
number of floating point operations; see also Remark 3.
###### Example 5.
In this first example we consider a nonlinear model that describes the
behavior of a redundant parallel robot. It is a problem that concerns the
inverse kinematics of position, and is defined by the following function
$F:{\mathbb{R}}^{4}\rightarrow{\mathbb{R}}^{2}$
$F(\mathbf{x})=\begin{bmatrix}(X-A\cos(x_{1}))^{2}+(Y-A\sin(x_{1}))^{2}-x_{2}^{2}\\\
(X-A\cos(x_{3})-H)^{2}+(Y-A\sin(x_{3}))^{2}-x_{4}^{2}\end{bmatrix},$
with the data vector $\mathbf{b}=\mathbf{0}$ in (1). The model describes the
kinematic of a robotic arm moved by 4 motors, whose position is identified by
the unknowns $\\{x_{i}\\}_{i=1}^{4}$, which must reach a point with given
coordinates $(X,Y)$; $A$ and $H$ are parameters describing the system. In our
simulation we assume $(X,Y)=(3,3)$, $A=2$, $H=10$.
The Jacobian matrix of $F$ is
$J(\mathbf{x})=\begin{bmatrix}\dfrac{\partial F_{1}}{\partial
x_{1}}&\dfrac{\partial F_{1}}{\partial x_{2}}&0&0\\\ 0&0&\dfrac{\partial
F_{2}}{\partial x_{3}}&\dfrac{\partial F_{2}}{\partial x_{4}}\end{bmatrix},$
with
$\displaystyle\frac{\partial F_{1}}{\partial x_{1}}$
$\displaystyle=2A(X-A\cos(x_{1}))\sin(x_{1})-2A(Y-A\sin(x_{1}))\cos(x_{1}),$
$\displaystyle\frac{\partial F_{2}}{\partial x_{3}}$
$\displaystyle=2A(X-A\cos(x_{3})-H)\sin(x_{3})-2A(Y-A\sin(x_{3}))\cos(x_{3}),$
$\displaystyle\frac{\partial F_{1}}{\partial x_{2}}$
$\displaystyle=-2x_{2},\quad\frac{\partial F_{2}}{\partial x_{4}}=-2x_{4}.$
The results obtained are reported in Table 1. We see that the MNGN$2_{\alpha}$
and CKB1 methods recover solutions with smaller norms, in the average, but the
first one requires a large number of iterations. The
MNGN$2_{\alpha\beta\delta}$ implementation, with automatic estimation of the
projection step $\beta_{k}$, quickly converges but produces solutions with
slightly larger norms. The CKB2 method leads to solutions with a worse norm,
testifying that the performance of the method in (16) is very sensitive to the
choice of the sequence $\gamma_{k}$. The MNGN method from [14] leads to
solutions far from optimality, and fails in 70% of the tests. This happens in
most of the examples considered in this paper, so we will involve it only in
another experiment.
Table 1: Results for Example 5. method | iterations | $\|\widetilde{\mathbf{x}}\|$ | #success
---|---|---|---
MNGN$2_{\alpha}$ | 239 | 8.7246 | 92
MNGN$2_{\alpha\beta\delta}$ | 38 | 9.0621 | 96
CKB1 | 26 | 8.5515 | 100
CKB2 | 10 | 9.7344 | 100
MNGN | 182 | 17.6329 | 30
###### Example 6.
Here we consider a test problem introduced in [3]. Let
$F:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$ be the nonlinear function defined
by
$F(\mathbf{x})=x_{3}-(x_{1}-1)^{2}-2(x_{2}-2)^{2}-3.$
The equation $F(\mathbf{x})=0$ represents an elliptic paraboloid in
${\mathbb{R}}^{3}$ with vertex $\mathbf{V}=(1,2,3)^{T}$. We remark that the
minimal-norm solution is the point
$\mathbf{x}^{\dagger}\approx(0.859754,1.849178,3.065164)^{T},$
and not the vector $\widehat{\mathbf{x}}$ reported in [3, Sec. 4.2]. Indeed,
$\|\mathbf{x}^{\dagger}\|\approx 3.681558$, whereas
$\|\widehat{\mathbf{x}}\|\approx 3.706359$.
The results obtained are reported in Table 2. The MNGN$2_{\alpha\beta}$ method
is tested with two values of the parameter $\eta$ appearing in the residual
increase $\delta(\rho,\eta)=\eta\rho$; see Algorithm 1. It is clear that it
can lead to accurate solutions only if the parameter is suitably chosen
($\eta=2$). On the contrary ($\eta=8$), it shows a great number of failures.
As in the previous example, the best results are produced by MNGN$2_{\alpha}$,
and MNGN$2_{\alpha\beta\delta}$ reaches very similar solutions but is about 10
times faster. The CKB methods take a smaller number of iterations, but produce
less accurate solutions.
Table 2: Results for Example 6. method | iterations | $\|\widetilde{\mathbf{x}}\|$ | #success
---|---|---|---
MNGN$2_{\alpha\beta}\,(\eta=8)$ | 174 | 3.6903 | 15
MNGN$2_{\alpha\beta}\,(\eta=2)$ | 62 | 3.7120 | 100
MNGN$2_{\alpha}$ | 330 | 3.6816 | 100
MNGN$2_{\alpha\beta\delta}$ | 37 | 3.6832 | 100
CKB1 | 26 | 3.7343 | 100
CKB2 | 10 | 3.7561 | 100
###### Example 7.
Let $F:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}$ be the nonlinear function
$F(\mathbf{x})=\left[F_{1}(\mathbf{x}),F_{2}(\mathbf{x}),\ldots,F_{m}(\mathbf{x})\right]^{T},\qquad
m\leq n,$ (26)
defined by
$F_{i}(\mathbf{x})=\frac{1}{2}S(\mathbf{x})\left(x_{i}^{2}+1\right),\qquad
i=1,\ldots,m,$
where
$S(\mathbf{x})=\sum_{j=1}^{n}\left(\frac{x_{j}-c_{j}}{a_{j}}\right)^{2}-1$
is the $n$-ellipsoid with center $\mathbf{c}=(c_{1},\ldots,c_{n})^{T}$ and
whose semiaxes are the components of the vector
$\mathbf{a}=(a_{1},\ldots,a_{n})^{T}$. The locus of the solutions is the
$n$-ellipsoid.
Setting $y_{i}=x_{i}^{2}+1$, for $i=1,\ldots,m$, and
$z_{j}=\frac{x_{j}-c_{j}}{a_{j}^{2}}$, for $j=1,\ldots,n$, the Jacobian matrix
can be expressed as
$J(\mathbf{x})=S(\mathbf{x})D_{m,n}(\mathbf{x})+\mathbf{y}\mathbf{z}^{T},$
where $D_{m,n}(\mathbf{x})$ is an $m\times n$ diagonal matrix whose main
diagonal consists of the vector $\mathbf{x}$. Indeed,
$\frac{\partial F_{i}}{\partial
x_{k}}=\begin{cases}x_{i}S(\mathbf{x})+\dfrac{x_{i}-c_{i}}{a_{i}^{2}}\left(x_{i}^{2}+1\right),\qquad&k=i,\\\
\dfrac{x_{k}-c_{k}}{a_{k}^{2}}\left(x_{i}^{2}+1\right),\qquad&k\neq
i.\end{cases}$
When $S(\mathbf{x})=0$, $\mathop{\operator@font
rank}\nolimits(J(\mathbf{x}))=1$, so we expect the Jacobian to be rank-
deficient in a neighborhood of the solution.
If $\mathbf{a}=\mathbf{e}=(1,\ldots,1)^{T}$, the locus of the solutions is the
$n$-sphere centered in $\mathbf{c}$ with unitary radius. If
$\mathbf{c}=2\mathbf{e}$, the minimal-norm solution is
$\mathbf{x}^{\dagger}=\left(2-\frac{\sqrt{n}}{n}\right)\mathbf{e},$
while if $\mathbf{c}=(2,0,\ldots,0)^{T}$ it is
$\mathbf{x}^{\dagger}=(1,0,\ldots,0)^{T}$.
Table 3 displays the results for the last case, when $m=8$ and $n=10$. These
results aim at underlining the importance of estimating the rank of the
Jacobian $J_{k}$. The implementations of the MNGN2 algorithm are more or less
equivalent, recovering solutions with almost optimal norm; MNGN$2_{\alpha}$
fails in 17% of the tests. The value of $\eta$ for MNGN$2_{\alpha\beta}$ is
tailored to maximize the performance, which is not possible in practice, while
it is automatically estimated for MNGN$2_{\alpha\beta\delta}$. The MNGN and
CKB methods do not perform well, because of the rank deficiency of the
Jacobian. We also implemented the rank estimation in the algorithms from [3];
the corresponding methods are denoted by rCKB. It happens that rCKB2 produces
results comparable to the MNGN2 methods, confirming that a correct estimation
of the rank is essential for the convergence, while rCKB1 converges only in
32% of the tests and produces solutions with large norms. Again, this shows
that the sequence adopted for the step length in (r)CKB methods is critical
for the effectiveness of the computation.
Table 3: Results for Example 7 with $m=8$, $n=10$, $\mathbf{a}=\mathbf{e}$, and $\mathbf{c}=(2,0,\ldots,0)^{T}$. In MNGN, CKB1, and CKB2, the rank is not estimated. method | iterations | $\|\widetilde{\mathbf{x}}\|$ | #success
---|---|---|---
MNGN$2_{\alpha}$ | 209 | 1.0263 | 83
MNGN$2_{\alpha\beta}\,(\eta=8)$ | 208 | 1.0449 | 99
MNGN$2_{\alpha\beta\delta}$ | 206 | 1.0367 | 97
MNGN | 70 | 2.1083 | 2
CKB1 | 216 | 2.2002 | 32
CKB2 | 20 | 2.1305 | 2
rCKB1 | 160 | 2.1088 | 32
rCKB2 | 197 | 1.0454 | 97
The norms of the solutions, whose average is displayed in Table 3, are
reported in the boxplot in the left pane of Figure 1. In each box, the red
mark is the median, the edges of the blue box are the 25th and 75th
percentiles, and the black whiskers extend to the most extreme data points non
considered to be outliers, which are plotted as red crosses.
Figure 1: Boxplot of the norms of the solutions for Examples 7 (left) and 8
(right). The series, labeled by the methods name, are displayed in the same
order of Table 3 and Table 4, respectively.
###### Example 8.
Let $F$ be a nonlinear function such as (26), with
$F_{i}(\mathbf{x})=S(\mathbf{x})\left(x_{i}-c_{i}\right),\qquad i=1,\ldots,m,$
(27)
and $S(\mathbf{x})$ defined as in the previous example. The first order
derivatives of $F_{i}(\mathbf{x})$ are
$\frac{\partial F_{i}}{\partial
x_{k}}=\begin{cases}\dfrac{2}{a_{i}^{2}}(x_{i}-c_{i})^{2}+S(\mathbf{x}),\qquad&k=i,\\\
\dfrac{2}{a_{k}^{2}}(x_{k}-c_{k})(x_{i}-c_{i}),\qquad&k\neq i.\end{cases}$
Setting $y_{i}=x_{i}-c_{i}$, for $i=1,\ldots,m$, and
$z_{j}=\frac{x_{j}-c_{j}}{a_{j}^{2}}$, for $j=1,\ldots,n$, the Jacobian matrix
can be represented as
$J(\mathbf{x})=S(\mathbf{x})I_{m\times n}+2\mathbf{y}\mathbf{z}^{T},$
where $I_{m\times n}$ includes the first $m$ rows of an identity matrix of
size $n$. The Jacobian turns out to be a diagonal plus rank-1 matrix. This
structure may be useful to reduce complexity when solving large scale
problems.
When $S(\mathbf{x})=0$, the matrix $J(\mathbf{x})$ has rank 1. Indeed, in this
case, the compact SVD of the Jacobian is
$J(\mathbf{x})=\frac{\mathbf{y}}{\|\mathbf{y}\|}(2\|\mathbf{y}\|\|\mathbf{z}\|)\frac{\mathbf{z}^{T}}{\|\mathbf{z}\|},$
so that the only non-zero singular value is $2\|\mathbf{y}\|\|\mathbf{z}\|$.
As in the preceding example, we may assume that the Jacobian is rank-deficient
in the surroundings of a solution.
Figure 2: Solution of problem (27) (Example 8) for $m=2$ and $n=3$, with
$\mathbf{a}=(1,1,1)^{T}$, $\mathbf{c}=(2,0,0)^{T}$, and
$\mathbf{x}^{(0)}=(0,3,3)^{T}$. The locus of the solutions is the sphere and
the line intersection of the two planes. The blue dots are the iterations of
the MNGN$2_{\alpha\beta\delta}$ method, and the red ones correspond to the
rCKB1 method. The black circle encompasses the minimal-norm solution.
The locus of the solutions is the union of the $n$-ellipsoid and the
intersection between the planes $x_{i}=c_{i}$, $i=1,\ldots,m$.
If $\mathbf{a}=\mathbf{e}$ and $\mathbf{c}=2\mathbf{e}$, the minimal-norm
solution $\mathbf{x}^{\dagger}$ depends on the dimensions $m$ and $n$: if
$m<n-\sqrt{n}+\frac{1}{4}$, then it is
$\mathbf{x}^{\dagger}=(\underbrace{2,2,\ldots,2}_{m},\underbrace{0,\ldots,0}_{n-m})^{T},$
otherwise, it is
$\mathbf{x}^{\dagger}=\left(2-\frac{\sqrt{n}}{n}\right)\mathbf{e}.$ (28)
If $\mathbf{c}=(2,0,\ldots,0)^{T}$, it is
$\mathbf{x}^{\dagger}=(1,0,\ldots,0)^{T}$. The case $m=2$, $n=3$, is displayed
in Figure 2, together with the iterations of the algorithms
MNGN$2_{\alpha\beta\delta}$ and rCKB1. In this test, the latter algorithm
converges to a solution of non-minimal norm.
Table 4 illustrates the situation where $\mathbf{a}=\mathbf{e}$,
$\mathbf{c}=(2,0,\ldots,0)^{T}$, $m=8$ and $n=10$. The corresponding boxplot
of the norms of the solutions is displayed in the right pane of Figure 1. The
MNGN$2_{\alpha\beta\delta}$ method is the only one which recovers the correct
solution; MNGN$2_{\alpha}$ gets close to it, but with a very small number of
successes.
Table 4: Results for Example 8 with $m=8$, $n=10$, $\mathbf{a}=\mathbf{e}$, and $\mathbf{c}=(2,0,\ldots,0)^{T}$. method | iterations | $\|\widetilde{\mathbf{x}}\|$ | #success
---|---|---|---
MNGN$2_{\alpha}$ | 215 | 1.5196 | 12
MNGN$2_{\alpha\beta}\,(\eta=8)$ | 11 | 1.9911 | 100
MNGN$2_{\alpha\beta\delta}$ | 47 | 1.0100 | 100
rCKB1 | 27 | 2.0346 | 100
rCKB2 | 11 | 2.0531 | 100
Table 5 reports the results obtained for $\mathbf{a}=\mathbf{e}$ and
$\mathbf{c}=2\mathbf{e}$. In this case, the solution is (28). We applied the
algorithms to both the solution of the minimal-norm problem, and the
computation of the minimal-$L$-norm solution with $L=D_{2}$, i.e., the
discrete approximations of the second derivative (5). Since the solution is
exactly in the null space of $L$, we expect the minimal-$L$-norm solution to
perform well. No algorithm is accurate when $L=I$, as the minimal norm is
$2\sqrt{n}-1=5.3246$. When $L=D_{2}$, the two MNGN2 implementations are
superior to the rCKB methods, as $\|L\mathbf{x}^{\dagger}\|=0$. As in the
previous example, MNGN$2_{\alpha}$ exhibits a large number of failures.
Table 5: Results for Example 8 with $m=8$, $n=10$, $\mathbf{a}=\mathbf{e}$, and $\mathbf{c}=2\mathbf{e}$. | method | iterations | $\|L\widetilde{\mathbf{x}}\|$ | #success
---|---|---|---|---
$L=I$ | MNGN$2_{\alpha}$ | 12 | 5.6569 | 23
| MNGN$2_{\alpha\beta\delta}$ | 45 | 5.4529 | 100
| rCKB1 | 26 | 5.7274 | 100
| rCKB2 | 11 | 5.7520 | 100
$L=D_{2}$ | MNGN$2_{\alpha}$ | 20 | 0.0500 | 26
| MNGN$2_{\alpha\beta\delta}$ | 17 | 0.0765 | 100
| rCKB1 | 27 | 2.1694 | 100
| rCKB2 | 17 | 2.2761 | 100
Since this example is interesting in itself as a test problem, we report some
further comments on it. If $m=n$, the locus of the solutions is the union of
the $n$-ellipsoid and the point $\mathbf{x}=\mathbf{c}$. The spectrum of
$J(\mathbf{x})$ is
$\sigma(J(\mathbf{x}))=\left\\{S(\mathbf{x})+2\mathbf{y}^{T}\mathbf{z},S(\mathbf{x}),\ldots,S(\mathbf{x})\right\\},$
where the eigenvalue $S(\mathbf{x})$ has algebraic multiplicity $n-1$. The
Jacobian matrix is invertible if and only if $S(\mathbf{x})\neq 0$. If this
condition is met, the inverse is obtained by the Sherman–Morrison formula
$J(\mathbf{x})^{-1}=\frac{1}{S(\mathbf{x})}I_{n}-\frac{2}{S(\mathbf{x})(S(\mathbf{x})+2\mathbf{z}^{T}\mathbf{y})}\mathbf{y}\mathbf{z}^{T}.$
###### Example 9.
Let $F$ be the nonlinear function (26) with components
$F_{i}(\mathbf{x})=\begin{cases}S(\mathbf{x}),\qquad&i=1,\\\
x_{i-1}(x_{i}-c_{i}),\qquad&i=2,\ldots,m,\end{cases}$ (29)
and $S(\mathbf{x})$ defined as above. The first order partial derivatives of
$F_{i}(\mathbf{x})$ are
$\frac{\partial F_{i}}{\partial
x_{k}}=\begin{cases}\dfrac{2}{a_{k}^{2}}(x_{k}-c_{k}),\quad&i=1,\
k=1,\ldots,n,\\\ x_{i}-c_{i},\quad&i=2,\ldots,m,\ k=i-1,\\\
x_{i-1},\quad&i=k=2,\ldots,m,\\\ 0,&\text{otherwise}.\end{cases}$
Setting $z_{j}=2\frac{x_{j}-c_{j}}{a_{j}^{2}}$ and $y_{j}=x_{j}-c_{j}$, for
$j=1,\ldots,n$, the Jacobian matrix of $F$ is
$J(\mathbf{x})=\begin{bmatrix}z_{1}&z_{2}&z_{3}&\cdots&z_{m-1}&z_{m}&\cdots&z_{n}\\\
y_{2}&x_{1}&&&&&&\\\ &y_{3}&x_{2}&&&&&\\\ &&\ddots&\ddots&&&&\\\
&&&\ddots&\ddots&&&\\\ &&&&y_{m}&x_{m-1}&&\end{bmatrix}.$ (30)
The locus of the solutions is the intersection between the hypersurface
defined by $S(\mathbf{x})=0$ and by the pairs of planes $x_{i-1}=0$,
$x_{i}-c_{i}=0$, $i=2,\ldots,m$.
Figure 3: Solution of problem (29) (Example 9) for $m=2$ and $n=3$, with
$\mathbf{a}=(1,1,1)^{T}$, $\mathbf{c}=(2,0,0)^{T}$, and
$\mathbf{x}^{(0)}=(\frac{1}{2},3,3)^{T}$. The solutions are in the
intersection between the sphere and the union of the two planes. The blue dots
are the iterations of the MNGN$2_{\alpha\beta\delta}$ method, and the red ones
correspond to the rCKB1 method. The black circle encompasses the minimal-norm
solution.
If $\mathbf{a}=\mathbf{e}=(1,\ldots,1)^{T}$ and $\mathbf{c}=2\mathbf{e}$, the
minimal-norm solution is
$\mathbf{x}^{\dagger}=\left(\xi_{n,m},\underbrace{2,\ldots,2}_{m-1},\underbrace{\xi_{n,m},\ldots,\xi_{n,m}}_{n-m}\right)^{T},$
(31)
with $\xi_{n,m}=2-(n-m+1)^{-1/2}$, while if $\mathbf{c}=(2,0,\ldots,0)^{T}$ it
is $\mathbf{x}^{\dagger}=(1,0,\ldots,0)^{T}$. It is immediate to observe that
in the last situation the Jacobian (30) is rank-deficient at
$\mathbf{x}^{\dagger}$. This case is illustrated in Figure 3, where the
iterations of the MNGN$2_{\alpha\beta\delta}$ and the rCKB1 methods are
reported too. The iterations performed are 20 and 24, respectively; the
computed solutions are substantially coincident.
Table 6 displays the results obtained for the same parameter vectors of Figure
3, when the size of the problem varies, i.e., for $(m,n)=(8k,10k)$, $k=1,2,3$.
The MNGN2 algorithms behave almost optimally, while the rCKB methods lead to
solutions with larger norm. The table shows that the performance is not
significantly affected by the size of the problem. This example suggests that
large scale problems could be faced by the methods discussed, but a suitable
algorithm for the solution of the linearized problem should be adopted, to
reduce the computational complexity of each step. This aspect will be the
object of future research.
Table 6: Results for Example 9 with different size $(m,n)$, $\mathbf{a}=\mathbf{e}$, and $\mathbf{c}=(2,0,\ldots,0)^{T}$. $(m,n)$ | method | iterations | $\|\widetilde{\mathbf{x}}\|$ | #success
---|---|---|---|---
$(8,10)$ | MNGN$2_{\alpha}$ | 167 | 1.0000 | 48
| MNGN$2_{\alpha\beta}\,(\eta=8)$ | 24 | 1.0508 | 100
| MNGN$2_{\alpha\beta\delta}$ | 37 | 1.0659 | 100
| rCKB1 | 44 | 1.4867 | 100
| rCKB2 | 22 | 1.4776 | 100
$(16,20)$ | MNGN$2_{\alpha}$ | 144 | 1.0000 | 36
| MNGN$2_{\alpha\beta}\,(\eta=8)$ | 29 | 1.0170 | 99
| MNGN$2_{\alpha\beta\delta}$ | 34 | 1.0518 | 99
| rCKB1 | 54 | 1.4343 | 100
| rCKB2 | 53 | 1.5269 | 90
$(24,30)$ | MNGN$2_{\alpha}$ | 133 | 1.0000 | 34
| MNGN$2_{\alpha\beta}\,(\eta=8)$ | 34 | 1.0154 | 99
| MNGN$2_{\alpha\beta\delta}$ | 32 | 1.0191 | 96
| rCKB1 | 43 | 1.4446 | 100
| rCKB2 | 52 | 1.4529 | 70
Table 7 investigates the effectiveness of choosing an appropriate model
profile $\overline{\mathbf{x}}$ when applying the MNGN2 algorithms. We
consider the case $\mathbf{a}=\mathbf{e}$, $\mathbf{c}=2\mathbf{e}$, $m=8$,
and $n=10$. The minimal-norm solution $\mathbf{x}^{\dagger}$ is (31), with
$\xi_{8,10}\simeq 1.4226$ and $\|\mathbf{x}^{\dagger}\|\simeq 5.8371$.
When $\overline{\mathbf{x}}=\mathbf{0}$, the solutions produced by the
considered variants of the method are almost optimal, but the number of
iterations is quite large, as well as the number of failures for
MNGN$2_{\alpha\beta}$ (with a suitably chosen $\eta$) and
MNGN$2_{\alpha\beta\delta}$. The model profile
$\overline{\mathbf{x}}=2\mathbf{e}$ reduces the number of iterations and leads
to almost 100% of successes, but the average norm of the solutions is slightly
larger than the optimal one. Choosing $\overline{\mathbf{x}}=1.7\mathbf{e}$, a
value which is roughly halfway between 2 and $\xi_{8,10}$, the extreme values
of $\mathbf{x}^{\dagger}$, restores the optimality of the results. This
confirms that, when a priori information is available, an accurate choice of
the model profile enhances the performance of the algorithms.
Table 7: Results for Example 9 with $m=8$, $n=10$, $\mathbf{a}=\mathbf{e}$, and $\mathbf{c}=2\mathbf{e}$. | method | iterations | $\|\widetilde{\mathbf{x}}\|$ | #success
---|---|---|---|---
$\overline{\mathbf{x}}=\mathbf{0}$ | MNGN$2_{\alpha}$ | 138 | 5.8371 | 100
| MNGN$2_{\alpha\beta}\,(\eta=8)$ | 175 | 5.8374 | 38
| MNGN$2_{\alpha\beta\delta}$ | 94 | 5.8988 | 67
$\overline{\mathbf{x}}=2\mathbf{e}$ | MNGN$2_{\alpha}$ | 37 | 6.1141 | 99
| MNGN$2_{\alpha\beta}\,(\eta=8)$ | 34 | 6.1144 | 98
| MNGN$2_{\alpha\beta\delta}$ | 34 | 6.1144 | 98
$\overline{\mathbf{x}}=1.7\mathbf{e}$ | MNGN$2_{\alpha}$ | 54 | 5.8371 | 100
| MNGN$2_{\alpha\beta}\,(\eta=8)$ | 34 | 5.8394 | 99
| MNGN$2_{\alpha\beta\delta}$ | 40 | 5.8789 | 99
## 7 Conclusions
This paper explores the computation of the minimal-($L$-)norm solution of
nonlinear least-squares problems, and the reasons for the occasional lack of
convergence of Gauss–Newton methods. We propose an automatic procedure to
estimate the rank of the Jacobian along the iteration, and the introduction of
two different relaxation parameters that improve the efficiency of the
iterative method. The first parameter is determined by applying the
Armijo–Goldstein principle, while three techniques are investigated to
estimate the second one. In numerical experiments performed on various test
problems, the new methods prove to be very effective, compared to other
approaches based on a single damping parameter. In particular, the variant
which automatically estimates the projection parameter gives satisfactory
results in all the examples.
## Acknowledgements
The authors are indebted to two anonymous reviewers, whose remarks were
essential for improving both the content and the presentation of this paper.
We thank Maurizio Ruggiu for suggesting the problem reported in Example 5. The
work of the authors was partially supported by the Regione Autonoma della
Sardegna research project “Algorithms and Models for Imaging Science [AMIS]”
(RASSR57257, intervento finanziato con risorse FSC 2014-2020 - Patto per lo
Sviluppo della Regione Sardegna), and the INdAM-GNCS research project
“Tecniche numeriche per l’analisi delle reti complesse e lo studio dei
problemi inversi”. Federica Pes gratefully acknowledges CRS4 (Centro di
Ricerca, Sviluppo e Studi Superiori in Sardegna) for the financial support of
her Ph.D. scholarship.
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|
Leveraging Local Variation of Data in Supervised Deep Learning P. Novello, G.
Poette, D. Lugato, & P.M. Congedo [1,2,3]P. Novello<EMAIL_ADDRESS>
mm/dd/yyyy mm/dd/yyyy
# Leveraging Local Variation in Data: Sampling and Weighting Schemes for
Supervised Deep Learning
G. Poëtte D. Lugato P.M. Congedo CESTA, CEA, Le Barp, 33114, France CMAP,
Ecole Polytechnique, 91120, Palaiseau, France Platon, Inria Paris Saclay,
91120, Palaiseau, France
###### Abstract
In the context of supervised learning of a function by a neural network, we
claim and empirically verify that the neural network yields better results
when the distribution of the data set focuses on regions where the function to
learn is steep. We first traduce this assumption in a mathematically workable
way using Taylor expansion and emphasize a new training distribution based on
the derivatives of the function to learn. Then, theoretical derivations allow
constructing a methodology that we call Variance Based Samples Weighting
(VBSW). VBSW uses labels local variance to weight the training points. This
methodology is general, scalable, cost-effective, and significantly increases
the performances of a large class of neural networks for various
classification and regression tasks on image, text, and multivariate data. We
highlight its benefits with experiments involving neural networks from linear
models to ResNet [19] and Bert [14].
###### keywords:
Supervised learning, importance weighting, learning theory, designs of
experiments
When a Machine Learning (ML) model is used to learn from data, the
distribution of the training data set can have a substantial impact on its
performance. More specifically, in Deep Learning (DL), several works have
hinted at the importance of the training set. In [6, 37], the authors exploit
the observation that a human will benefit more from easy examples than from
harder ones at the beginning of a learning task. They construct a curriculum,
inducing a change in the distribution of the training data set that makes a
neural network achieve better results in an ML problem. With a different
approach, Active Learning [46] modifies the distribution of the training data
dynamically by selecting the data points that will make the training more
efficient. Finally, in Reinforcement Learning, the distribution of experiments
is crucial for the agent to learn efficiently. Moreover, the challenge of
finding a good distribution is not specific to ML. Indeed, in the context of
Monte Carlo estimation of a quantity of interest based on a random variable,
Importance Sampling owes its efficiency to the construction of a second random
variable, which is used instead to improve the estimation of this quantity.
[23] even make a connection between the success of likelihood ratio policy
gradients and importance sampling, which shows that ML and Monte Carlo
estimation, both distribution-based methods, are closely linked.
In this paper, we leverage the importance of the training set distribution to
improve the performances of neural networks in supervised deep learning. We
formalize supervised learning as a task which aims at approximating a function
$f$ with a model $f_{{\bm{\theta}}}$ parametrized by ${\bm{\theta}}$ using
data points drawn from $X\sim d\mathbb{P}_{X}$, $X\in\mathcal{X}$. We build a
new distribution $d\mathbb{P}_{\bar{X}}$ from the training points and their
labels, based on the observation that $f_{{\bm{\theta}}}$ needs more data
points to approximate $f$ on the regions where it is steep. We derive an
illustrative generalization bound involving the derivatives of $f$ that
theoretically corroborates this observation. Therefore, we build
$d\mathbb{P}_{\bar{X}}$ using Taylor expansion of the function $f$, which
links the local behavior of $f$ to its derivatives.
We first focus on the influence of using $d\mathbb{P}_{\bar{X}}$ instead of
$d\mathbb{P}_{X}$ in simple approximation problems. To that end, we build a
methodology for constructing and exploiting $d\mathbb{P}_{\bar{X}}$, that we
call Taylor Based sampling (TBS). We then apply TBS to a more realistic
problem based on the approximation of the solution of Bateman equations.
Solving these equations is an important part of many numerical simulations of
several phenomena (neutronic [8, 15], combustion [9], detonic [34],
computational biology [42], etc.).
Then, we study the benefits of this approach for more general machine learning
problems. In these cases, exploiting $d\mathbb{P}_{\bar{X}}$ is less
straightforward. Indeed, we do not know the derivatives of $f$, and we cannot
obtain labels for new data points sampled from this distribution. To tackle
these problems, we show that variance is an approximation of Taylor expansion
up to a certain order. Then we leverage the link between sampling and
weighting to construct a methodology called Variance Based Sample Weighting
(VBSW). This methodology weights each training data point using the local
variance of their neighbor labels to simulate the new distribution. We
specifically investigate its application in deep learning, where we apply VBSW
within the feature space of a pre-trained neural network. We validate VBSW for
deep learning by obtaining performance improvements on various tasks like
classification and regression of text, from Glue benchmark [51], image, from
MNIST [32] and Cifar10 [29] and multivariate data, from UCI machine learning
repository 111http://archive.ics.uci.edu/ml, for several models ranging from
linear regression to Bert [14] or ResNet20 [19]. We also conduct analyses on
the complementarity of VBSW with other weighting techniques and its robustness
to label noise.
## 1 Related works
This work introduces contributions that rely on different elements. First,
many techniques aim to alter the training distribution to improve the
prediction error of neural networks. Second, finding generalization bounds for
neural networks is the goal of various works in machine learning research.
Finally, the methodology of constructing a sampling distribution for
statistical analysis is used for importance sampling and designs of
experiments.
Modified learning distributions. Some works are dedicated to improving neural
network performances by modifying the training distribution, either by
weighting data points or by inducing sample selection. Active learning [46]
adapts the training strategy to a learning problem by introducing an online
data point selection rule. [16] uses the variational properties of Bayesian
neural network to design a rule that focuses the training on points that will
reduce the prediction uncertainty of the neural network. In [28], the
construction of the selection rule is itself taken as a machine learning
problem. See [46] for a review of more classical active learning methods.
Unlike active learning, and similarly to VBSW, some other methods aim at
introducing diverse a priori evaluations of sample importance. While
curriculum learning [6, 37] starts the training with easier examples, self-
paced learning [30, 22] downscales harder examples. However, some works have
proven that focusing on harder examples at the beginning of the learning could
accelerate it: [48] performs hard example mining to give more importance to
harder examples by selecting them primarily. This work also focuses on
defining hard examples but does so with an original, mathematical way based on
$f$ derivatives and local variance. It also stands out from the aforementioned
techniques for how it modifies the distribution based on this information.
Indeed, it suggests and justifies that a neural network should spend more
learning time on subspaces of $\mathcal{X}$ which contain harder examples.
Generalization bounds. As an argument to motivate our approach, we derive a
generalization bound. The construction of Generalization bounds for the
learning theory of neural networks has motivated many works (see [21] for a
review). In [5, 4], the authors focus on Vapnik Chervonenkis (VC) dimension, a
measure that depends on the number of parameters of neural networks. [2]
introduces a compression approach that aims at reducing the number of model
parameters to investigate its generalization capacities. Probably
Approximately Correct (PAC) Bayes analysis constructs generalization bounds
using a priori and a posteriori distributions over the possible models. It is
investigated, for example, in [40, 3]. [39, 53] links PAC-Bayes theory to the
notion of sharpness of a neural network, i.e. its robustness to small
perturbation. While previous works often mention the sharpness of the model,
our bound includes the derivatives of $f$, which can be seen as an indicator
of the sharpness of the function to learn. Even if it uses elements of
previous works, like the Lipschitz constant of $f_{{\bm{\theta}}}$, our work
does not pretend to tighten and improve the already existing generalization
bounds. It only emphasizes the intuition that the neural network would need
more points to capture sharper functions. In a sense, it investigates the
robustness to perturbations in the input space, not in the parameter space.
Examples weighting. VBSW can be categorized as an examples weighting, or
importance weighting algorithm. The idea of weighting the data set has already
been explored in different ways and for various purposes. Examples weighting
is used in [13] to tackle the class imbalance problem by weighting rarer, so
harder examples. On the contrary, in [33] it is used to solve the noisy label
problem by focusing on cleaner, so easier examples. All these ideas show that
depending on the application, examples weighting can be performed in an
opposed manner. Some works aim at going beyond this opposition by proposing
more general methodologies. In [11], the authors use the variance of the
prediction of each point throughout the training to decide whether it should
be weighted or not. A meta-learning approach is proposed in [44], where the
authors choose the weights after an optimization loop included in the
training. VBSW stands out from the previously mentioned examples weighting
methods because it does not aim at solving dataset-specific problems like
class imbalance or noisy labels. It is built on a more general assumption that
a model would simply need more points to learn more complicated functions. The
resulting weighting scheme verifies recent findings of [52] where authors
conclude that in classification, a good set of weights would put importance on
points close to the decision boundary.
Importance sampling. The challenge of finding a good distribution is not
specific to machine learning. Indeed, in the context of Monte Carlo estimation
of a quantity of interest based on a random variable, importance sampling owes
its efficiency to the construction of a second random variable, which is used
instead to improve the estimation of this quantity. [23] even make a
connection between the success of likelihood ratio policy gradients and
importance sampling, which shows that machine learning and Monte Carlo
estimation, both distribution-based methods, are closely linked. Moreover,
some previously mentioned methods use importance sampling to design the
weights of the data set or to correct the bias induced by the sample selection
[26]. In this work, we construct a new distribution that could be interpreted
as an importance distribution. However, we weigh the data points to simulate
this distribution. It does not aim at correcting a bias induced by this
distribution.
Designs of experiments. Some methodologies are dedicated to the construction
of data sets in the context of statistical analysis. These methodologies are
called designs of experiments. In our case, the construction of a new training
distribution could be seen as a design of experiments for learning. However,
popular designs of experiments used for regression are either space-filling
designs or model-based designs. Space-filling designs, like Latin hypercube
sampling [38] or maximin designs [24], aims at spreading the learning points
to cover the input space as much as possible. Model-based designs use
characteristics of $f_{{\bm{\theta}}}$ to adapt the training distribution.
Such designs can look to maximize the entropy of the prediction [47] or
minimize its uncertainty [25]. These last designs of experiments can be
conducted sequentially, getting close to active learning [45, 35, 12]. Our
methodology does not depend on $f_{{\bm{\theta}}}$, nor aims at filling the
input space. Instead, its goal is to adapt the design of experiments to
characteristics of $f$ in order to reduce the prediction error.
## 2 Link between local variations and learning
Let us first remind some basics on supervised machine learning. We formalize
the supervised machine learning task as approximating a function
$f:\mathbf{S}\subset\mathbb{R}^{n_{i}}\rightarrow\mathbb{R}^{n_{o}}$ with a
machine learning model $f_{{\bm{\theta}}}$ parametrized by ${\bm{\theta}}$,
where $\mathbf{S}$ is a measured sub-space of $\mathbb{R}^{n_{i}}$ depending
on the application. To this end, we are given a training data set of $N$
points, $\\{{\bm{x}}_{1},...,{\bm{x}}_{N}\\}\in\mathbf{S}$, drawn from
${\mathbf{x}}\sim d\mathbb{P}_{{\mathbf{x}}}$ and their point-wise values, or
labels $\\{f({\bm{x}}_{1}),...,f({\bm{x}}_{N})\\}$. Parameters ${\bm{\theta}}$
have to be found in order to minimize an integrated loss function
$J_{{\mathbf{x}}}({\bm{\theta}})=\mathbb{E}[L(f_{{\bm{\theta}}}({\bm{x}}),f({\bm{x}}))]$,
with $L$ the loss function,
$L:\mathbb{R}^{n_{o}}\times\mathbb{R}^{n_{o}}\rightarrow\mathbb{R}$. The data
allow estimating $J_{{\mathbf{x}}}({\bm{\theta}})$ by
$\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})=\sum_{i=1}^{N}\omega_{i}L(f_{{\bm{\theta}}}({\bm{x}}_{i}),f({\bm{x}}_{i}))$,
with $\\{\omega_{1},...,\omega_{N}\\}\in\mathbb{R}$ estimation weights,
generally equal to $\frac{1}{N}$. Then, an optimization algorithm is used to
find a minimum of $\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})$ w.r.t.
${\bm{\theta}}$.
### 2.1 Illustration of the link using derivatives
In the following, we illustrate the intuition with a Generalization Bound (GB)
that include the derivatives of $f$, provided that these derivatives exist.
The goal of the approximation problem is to be able to generalize to points
not seen during the training. The generalization error
$\mathcal{J}_{{\mathbf{x}}}({\bm{\theta}})=J_{{\mathbf{x}}}({\bm{\theta}})-\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})$
thus needs to be as small as possible. Let $S_{i}$, $i\in\\{1,...,N\\}$ be
some sub-spaces of $\mathbf{S}$ such that $\mathbf{S}=\bigcup_{i=1}^{N}S_{i}$,
$\bigcap_{i=1}^{N}S_{i}=$ Ø, and ${\bm{x}}_{i}\in S_{i}$. Suppose that $L$ is
the squared $L_{2}$ error, $n_{i}=n_{o}=1$, $f$ is differentiable,
$f_{{\bm{\theta}}}$ is $K_{{\bm{\theta}}}$-Lipschitz and satisfies the
conditions of Hornik theorem [20]. Provided that $|S_{i}|<1$, we show that
$\mathcal{J}_{{\textnormal{x}}}({\bm{\theta}})\leq\sum_{i=1}^{N}(|f^{\prime}(x_{i})|+K_{{\bm{\theta}}})^{2}\frac{|S_{i}|^{3}}{4}+\mathcal{O}(|S_{i}|^{4}),$
(1)
where $|S_{i}|$ is the volume of $S_{i}$
($|S_{i}|=\int_{S_{i}}d\mathbb{P}_{{\mathbf{x}}}$). The proof can be found in
Appendix A. We see that in the regions where $f^{\prime}({\bm{x}}_{i})$ is
high, quantity $|S_{i}|$ has a stronger impact on the GB. This idea is
illustrated in Figure 1, which visually shows that the generalization bound
increases when $|S_{i}|$ and $f^{\prime}({\bm{x}}_{i})$ are high at the same
time for approximating the function $f:x\rightarrow x^{3}$. Since $|S_{i}|$
can be seen as a metric for how close data points are around ${\bm{x}}_{i}$
(the smaller $|S_{i}|$ is, the closer ${\bm{x}}_{i}$ is to its neighbors), the
GB can be reduced more efficiently by adding more points around ${\bm{x}}_{i}$
in these regions. This bound also involves $K_{{\bm{\theta}}}$, the Lipschitz
constant of the neural network, which has the same impact as
$f^{\prime}({\bm{x}}_{i})$. It also illustrates the link between the Lipschitz
constant and the generalization error, which has been pointed out by several
works like [17], [3] and [43].
Figure 1: Illustration of the GB. The maximum error (the GB), at order
$\mathcal{O}(|S_{i}|^{4})$, is obtained by comparing the maximum variations of
$f_{{\bm{\theta}}}$, and the first order approximation of $f$, whose trends
are given by $K_{{\bm{\theta}}}$ and $f^{\prime}({\bm{x}}_{i})$. We understand
visually that because $f^{\prime}({\bm{x}}_{1})$ and
$f^{\prime}({\bm{x}}_{3})$ are higher than $f^{\prime}({\bm{x}}_{2})$, the GB
is improved more efficiently by reducing $S_{1}$ and $S_{3}$ than $S_{2}$.
### 2.2 A sampling scheme based on Taylor Approximation
Equation (1) formalizes a link between generalization error and derivatives of
$f$. These derivatives are expressed at order $n=1$ for analytical reasons,
but in this work we explore the use of derivatives of order $n>1$. Using
Taylor expansion at order $n$ on $f$ and supposing that $f$ is $n$ times
differentiable:
$f({\bm{x}}+{\bm{\epsilon}})\underset{\mathrm{\|{\bm{\epsilon}}\|\rightarrow
0}}{=}\sum_{0\leq|\bm{k}|\leq
n}\bm{{\epsilon}^{k}}\frac{\partial^{\bm{k}}f({\bm{x}})}{\bm{k}!}+\mathcal{O}(\bm{\epsilon^{n}}).$
The quantity $f({\bm{x}}+{\bm{\epsilon}})-f({\bm{x}})=\sum_{1\leq|\bm{k}|\leq
n}\bm{{\epsilon}^{k}}\frac{\partial^{\bm{k}}f({\bm{x}})}{\bm{k}!}+\mathcal{O}(\bm{\epsilon^{n}})$
gives an indication on how much $f$ changes around ${\bm{x}}$. By neglecting
the orders above $\bm{\epsilon^{n}}$, it is then possible to find the regions
of interest by focusing on $Df^{n}_{{\bm{\epsilon}}}$, defined as:
$Df^{n}_{{\bm{\epsilon}}}({\bm{x}})=\sum_{1\leq|\bm{k}|\leq
n}\bm{{\epsilon}}^{k}\frac{(\partial^{\bm{k}}f({\bm{x}}))^{2}}{\bm{k}!},$ (2)
Where $\bm{k}$ is a multi-index, i.e. $\bm{k}=(k_{1},...,k_{n_{i}})$ is a
vector of $n_{i}$ non negative integers, $|\bm{k}|=\sum_{i=1}^{n_{i}}k_{i}$,
$\bm{k}!=k_{1}!\times...\times k_{n_{i}}!$,
$\bm{{\epsilon}^{k}}=\epsilon_{1}^{k_{1}}\times...\times\epsilon_{n_{i}}^{k_{n_{i}}}$,
$\partial^{\bm{k}}=\frac{\partial^{k_{1}}}{\partial
x_{1}^{k_{1}}}\times...\times\frac{\partial^{k_{n_{i}}}}{\partial
x_{n_{i}}^{k_{n_{i}}}}$. Note that $Df^{n}_{{\bm{\epsilon}}}$ is evaluated
using $(\partial^{\bm{k}}f({\bm{x}}))^{2}$ instead of
$\partial^{\bm{k}}f({\bm{x}})$ for derivatives not to cancel each other. To
avoid these cancellations, the absolute could have been used, but we will see
in Lemma 3.1 that the square value ensures interesting asymptotical
properties. $f$ will be steeper and more irregular in the regions where
${\bm{x}}\rightarrow Df^{n}_{{\bm{\epsilon}}}({\bm{x}})$ is higher. To focus
the training set on these regions, one can use
$\\{Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$
to construct a probability density function (pdf) and sample new data points
from it.
In this part, we empirically verify that using Taylor expansion to construct a
new training distribution has a beneficial impact on the performances of a
neural network. To this end, we construct a methodology, that we call Taylor
Based Sampling (TBS), that generates a new training data set based on the
metric equation (2). To focus the training set on the regions of interest,
i.e. regions of high
$\\{Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$,
we use this metric to construct a probability density function (pdf) - which
is possible since $Df^{n}_{{\bm{\epsilon}}}({\bm{x}})\geq 0$ for all
${\bm{x}}\in\mathbf{S}$. It remains to normalize it but in practice it is
enough considering a distribution $d\mathbb{P}_{\bar{{\mathbf{x}}}}\propto
Df^{n}_{{\bm{\epsilon}}}$. Here, to approximate
$d\mathbb{P}_{\bar{{\mathbf{x}}}}$ we use a Gaussian Mixture Model (GMM) with
pdf $d\mathbb{P}_{\bar{{\mathbf{x}}},GMM}$ that we fit to
$\\{Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$
using the Expectation-Maximization (EM) algorithm. $N^{\prime}$ new data
points $\\{\bar{{\bm{x}}}_{1},...,\bar{{\bm{x}}}_{N^{\prime}}\\}$, can be
sampled, with $\bar{{\mathbf{x}}}\sim d\mathbb{P}_{\bar{{\mathbf{x}}},GMM}$.
Finally, we obtain
$\\{f(\bar{{\bm{x}}}_{1}),...,f(\bar{{\bm{x}}}_{N^{\prime}})\\}$, add it to
$\\{f({\bm{x}}_{1}),...,f({\bm{x}}_{N})\\}$ and train our neural network on
the whole data set.
TBS is described in Algorithm 1. Line 1: The parameter ${\bm{\epsilon}}$, the
number of Gaussian distribution $n_{\text{GMM}}$ and $N^{\prime}$ is chosen in
order to avoid sparsity of
$\\{\bar{{\bm{x}}}_{1},...,\bar{{\bm{x}}}_{N^{\prime}}\\}$ over $\mathbf{S}$.
Line 2: Without a priori information on $f$, we sample the first points
uniformly in a subspace $\mathbf{S}$. Line 3-7: We construct
$\\{Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$,
and then $d\mathbb{P}_{\bar{{\mathbf{x}}},GMM}$ to be able to sample points
accordingly. Line 8: Because the support of a GMM is not bounded, some points
can be sampled outside $\mathbf{S}$. We discard these points and sample until
all points are inside $\mathbf{S}$. This rejection method is equivalent to
sampling points from a truncated GMM. Line 9-10: We construct the labels and
add the new points to the initial data set.
Inputs: ${\bm{\epsilon}}$, $N$, $N^{\prime}$, $n_{\text{GMM}}$, $n$;
Sample $\\{{\bm{x}}_{1},...,{\bm{x}}_{N}\\}$ from
${\mathbf{x}}\sim\mathcal{U}(\mathbf{S})$;
for _$0\leq k\leq n$_ do
Compute
$\\{\partial^{\bm{k}}f({\bm{x}}_{1}),...,\partial^{\bm{k}}f({\bm{x}}_{N})\\}$;
end for
Compute
$\\{Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{n}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$
using equation (2);
Approximate $d\mathbb{P}_{\bar{{\mathbf{x}}}}\propto Df^{n}_{{\bm{\epsilon}}}$
with a GMM using EM algorithm to obtain a density
$d\mathbb{P}_{\bar{{\mathbf{x}}},GMM}$;
Sample $\\{\bar{{\bm{x}}}_{1},...,\bar{{\bm{x}}}_{N^{\prime}}\\}$ using
rejection method to sample inside $\mathbf{S}$;
Compute $\\{f(\bar{{\bm{x}}}_{1}),...,f(\bar{{\bm{x}}}_{N^{\prime}})\\}$;
Add $\\{f(\bar{{\bm{x}}}_{1}),...,f(\bar{{\bm{x}}}_{N^{\prime}})\\}$ to
$\\{f({\bm{x}}_{1}),...,f({\bm{x}}_{N})\\}$;
Algorithm 1 Taylor Based Sampling (TBS)
### 2.3 Taylor based sampling
#### 2.3.1 Application to simple functions
To illustrate the benefits of TBS compared to a uniform, basic sampling (BS),
we apply it to two simple functions: hyperbolic tangent and Runge function. We
chose these functions because they are differentiable and have a clear
distinction between flat and steep regions. These functions are displayed in
Figure 2, as well as the map ${\bm{x}}\rightarrow
Df^{2}_{{\bm{\epsilon}}}({\bm{x}})$.
Figure 2: Left: (left axis) Runge function w.r.t x and (right axis)
$x\rightarrow Df^{2}_{{\bm{\epsilon}}}(x)$. Points sampled using TBS are
plotted on the x-axis and projected on $f$. Right: Same as left, with
hyperbolic tangent function.
All neural networks have been implemented in Python, with Tensorflow [1]. We
use the Python package scikit-learn [41] to construct
$d\mathbb{P}_{\bar{{\mathbf{x}}},GMM}$. The network chosen for this experiment
is a Multi Layer Perceptron (MLP) with one layer of $8$ neurons and relu
activation function, that we trained alternatively with BS and TBS using Adam
optimizer [27] with the defaults tensorflow implementation hyperparameters,
and Mean Squared Error loss function. We first sample
$\\{{\bm{x}}_{1},...,{\bm{x}}_{N}\\}$ according to a regular grid. To compare
the two methods, we add $N^{\prime}$ additional points sampled using BS to
create the BS data set, and then $N^{\prime}$ other points sampled with TBS to
construct the TBS data set. As a result, each data set have the same number of
points $(N+N^{\prime})$. We repeated the method for several values of $n$,
$n_{\text{GMM}}$ and ${\bm{\epsilon}}$, to fine tune these parameters and
finally selected $n=2$, $n_{\text{GMM}}=3$ and ${\bm{\epsilon}}=10^{-3}$.
Sampling | $L_{2}$ error | $L_{\infty}$ error
---|---|---
| $f$: Runge $(\times 10^{-2})$
BS | $1.45\pm 0.62$ | $5.31\pm 0.86$
TBS | $\bm{1.13}\pm 0.73$ | $\bm{3.87}\pm 0.48$
| $f$: tanh $(\times 10^{-1})$
BS | $1.39\pm 0.67$ | $2.75\pm 0.78$
TBS | $\bm{0.95}\pm 0.50$ | $\bm{2.25}\pm 0.61$
Table 1: Comparison between BS and TBS. The metrics used are the $L_{2}$ and
$L_{\infty}$ errors, displayed with a $95\%$ confidence interval.
Table 1 summarizes the $L_{2}$ and the $L_{\infty}$ norm of the error of
$f_{{\bm{\theta}}}$, obtained at the end of the training phase for
$N+N^{\prime}=16$, with $N=8$. Those norms are estimated using the same test
data set of $1000$ points. The values are the means of the $40$ independent
experiments displayed with a $95\%$ confidence interval. These results
illustrate the benefits of TBS over BS. Table 1 shows that TBS does not
significantly improve $L_{2}$ error, but does so for $L_{\infty}$ error, which
may explain the good results of VBSW for classification that we describe in
Section 5. Indeed, the accuracy will not be very sensitive to small output
variations for a classification task since the output is rounded to 0 or 1.
However, a high error increases the risk of misclassification, which can be
limited by the reduction of $L_{\infty}$.
#### 2.3.2 Application to an ODE system
We apply TBS to a more realistic case: the approximation of the resolution of
the Bateman equations, an ODE system. In this system, $u$ is the velocity of
the reacting particles. Depending on the physical field of interest, $u$ may
be distributed according to a Maxwellian distribution (dense gas with chemical
reactions for example) or may be distributed according to a distribution
computed by another part of the code (this is the case in general for
neutronic reactions or collisions in a rarefied plasma).
$\begin{dcases}\partial_{t}u(t)&=v\bm{\sigma_{a}}\cdot\bm{\eta}(t)u(t),\\\
\partial_{t}\bm{\eta}(t)&=v\bm{\Sigma_{r}}\cdot\bm{\eta}(t)u(t),\\\
\end{dcases}\text{, with initial conditions }\begin{dcases}u(0)=u_{0},\\\
\bm{\eta}(0)=\bm{\eta_{0}}.\\\ \end{dcases},$
with
$u\in\mathbb{R}^{+},\bm{\eta}\in(\mathbb{R}^{+})^{M},\bm{\sigma}_{a}^{T}\in\mathbb{R}^{M},\bm{\Sigma}_{r}\in\mathbb{R}^{M\times
M}$. Here, $f:(u_{0},\bm{\eta_{0}},t)\rightarrow(u(t),\bm{\eta}(t))$. For
physical applications, $M$ ranges from tens to thousands, but we consider the
particular case $M=1$ so that $f:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$,
with $f(u_{0},\eta_{0},t)=(u(t),\eta(t))$, and $\sigma_{a}=\sigma_{r}=-0.45$.
The advantage of $M=1$ is that we have access to an analytic, cheap to compute
solution for $f$. Of course, this particular case can also be solved using a
classical ODE solver, which allows us to test it end to end. It can thus be
generalized to higher dimensions ($M>1$).
All neural network training instances have been performed in Python, with
Tensorflow. We used a fully connected neural network with hyperparameters
chosen using a simple grid search. The final values are: 2 hidden layers, relu
activation function, and 32 units for each layer, trained with the Mean
Squared Error (MSE) loss function using Adam optimization algorithm with a
batch size of 50000, for 40000 epochs and on $N+N^{\prime}=50000$ points, with
$N=N^{\prime}$. We trained the model for $(u(t),\eta(t))\in\mathbb{R}$, with
the $N+N^{\prime}$ points sampled uniformly (BS), and compared it to TBS
applied on $N^{\prime}$ after a uniform sampling of $N$ points (TBS). We did
so for several values of $n$, $n_{\text{GMM}}$ and
${\bm{\epsilon}}=\epsilon(1,1,1)$, to fine tune these parameters. We finally
select $\epsilon=5\times 10^{-4}$, $n=2$ and $n_{\text{GMM}}=10$. The data
points used in this case have been sampled with an explicit Euler scheme. Note
that we used this scheme because it is a stable converging accurate scheme if
the time steps for the resolution are fine enough (which we thoroughly
checked). Depending on the application, other schemes could be used (faster
ones, stabler ones etc.). As we here mainly aim at building a database of
solution, we are not constrained by some computational restrictions. So we
decided to use a very simple scheme, easy to handle which can easily produce
accurate solutions, even if costly (as it is only an offline cost). This
experiment has been repeated 50 times to ensure statistical significance of
the results.
Table 2 summarizes the MSE, i.e. the $L_{2}$ norm of the error of
$f_{{\bm{\theta}}}$ and $L_{\infty}$ norm, with
$L_{\infty}({\bm{\theta}})=\underset{{\bm{x}}\in\mathbf{S}}{\max}(|f({\bm{x}})-f_{{\bm{\theta}}}({\bm{x}})|)$
obtained at the end of the training phase. This last metric is important
because the goal in computational physics is not only to be averagely
accurate, which is measured with MSE, but to be accurate over the whole input
space $\mathbf{S}$. Those norms are estimated using a same test data set of
$N_{test}=50000$ points. The values are the means of the $50$ independent
experiments displayed with a $95\%$ confidence interval. These results reflect
an error reduction of 6.6% for $L_{2}$ and of 45.3% for $L_{\infty}$, which
means that TBS mostly improves the $L_{\infty}$ error of $f_{{\bm{\theta}}}$.
Moreover, the $L_{\infty}$ error confidence intervals do not intersect so the
gain is statistically significant for this norm.
Sampling | $L_{2}$ error $(\times 10^{-4})$ | $L_{\infty}$ $(\times 10^{-1})$ | AEG$(\times 10^{-2})$ | AEL$(\times 10^{-2})$
---|---|---|---|---
BS | $1.22\pm 0.13$ | $5.28\pm 0.47$ | - | -
TBS | $\bm{1.14}\pm 0.15$ | $\bm{2.96}\pm 0.37$ | $2.51\pm 0.07$ | $0.42\pm 0.008$
Table 2: Comparison between BS and TBS.
Figure 3(a) shows how the neural network can perform for an average
prediction. Figure 3(b) illustrates the benefits of TBS relative to BS on the
$L_{\infty}$ error (Figure 2b). These 2 figures confirm the previous
observation about the gain in $L_{\infty}$ error. Finally, Figure 3(c)
displays $u_{0},\eta_{0}\rightarrow\underset{0\leq t\leq
10}{\max}D^{n}_{{\bm{\epsilon}}}(u_{0},\eta_{0},t)$ w.r.t. $(u_{0},\eta_{0})$
and shows that $D^{n}_{{\bm{\epsilon}}}$ increases when $U_{0}\rightarrow 0$.
TBS hence focuses on this region. Note that for the readability of these
plots, the values are capped to $0.10$. Otherwise only few points with high
$D^{n}_{{\bm{\epsilon}}}$ are visible. Figure 3(d) displays
$u_{0},\eta_{0}\rightarrow
g_{\theta_{BS}}(u_{0},\eta_{0})-g_{\theta_{TBS}}(u_{0},\eta_{0})$, with
$g_{{\bm{\theta}}}:u_{0},\eta_{0}\rightarrow\underset{0\leq t\leq
10}{\max}\|f(u_{0},\eta_{0},t)-f_{{\bm{\theta}}}(u_{0},\eta_{0},t)\|_{2}^{2}$
where $\theta_{BS}$ and $\theta_{TBS}$ denote the parameters obtained after a
training with BS and TBS, respectively. It can be interpreted as the error
reduction achieved with TBS.
(a)
(b)
(c)
(d)
Figure 3: (a) $t\rightarrow f_{{\bm{\theta}}}(u_{0},\eta_{0},t)$ for randomly
chosen $(u_{0},\eta_{0})$, for $f_{{\bm{\theta}}}$ obtained with the two
samplings. (b) $t\rightarrow f_{{\bm{\theta}}}(u_{0},\eta_{0},t)$ for
$(u_{0},\eta_{0})$ resulting in the highest point-wise error with the two
samplings. (c) $u_{0},\eta_{0}\rightarrow\underset{0\leq t\leq
10}{\max}D^{n}_{{\bm{\epsilon}}}(u_{0},\eta_{0},t)$ w.r.t. $(u_{0},\eta_{0})$.
(d) $u_{0},\eta_{0}\rightarrow
g_{\theta_{BS}}(u_{0},\eta_{0})-g_{\theta_{TBS}}(u_{0},\eta_{0})$,
The highest error reduction occurs in the expected region. Indeed, more points
are sampled where $D^{n}_{{\bm{\epsilon}}}$ is higher. The error is slightly
increased in the rest of $\mathbf{S}$, which could be explained by a sparser
sampling on this region. However, as summarized in Table 2, the average error
loss (AEL) of TBS is around six times lower than the average error gain (AEG),
with $AEG=\mathbb{E}[Z(u_{0},\eta_{0})\mathbf{1}_{Z>0}]$ and
$AEL=\mathbb{E}[Z(u_{0},\eta_{0})\mathbf{1}_{Z<0}]$ where
$Z(u_{0},\eta_{0})=g_{\theta_{BS}}(u_{0},\eta_{0})-g_{\theta_{TBS}}(u_{0},\eta_{0})$.
In practice, AEG and AEL are estimated using uniform grid integration, and
averaged on the $50$ experiments.
## 3 Generalization of Taylor based Sampling
The previous section empirically validated the intuition behind the
construction of a new, more efficient training distribution
$d\mathbb{P}_{\bar{{\mathbf{x}}}}$. However, this new distribution cannot
always be applied as-is for two reasons. Problem 1:
$\\{Df^{2}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{2}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$
cannot be evaluated since it requires to compute the derivatives of $f$, and
it assumes that $f$ is differentiable, which is often not true. Moreover, the
previously described setting, in which we focus on $f$ derivatives, is not
suited to classification tasks where the notion of derivatives is not
straightforward. Problem 2: even if
$\\{Df^{2}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{2}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$
could be computed and new points sampled, we could not obtain their labels to
complete the training data set. In this section, we alleviate this concern to
be able to use insights from $d\mathbb{P}_{\bar{{\mathbf{x}}}}$ in practice.
### 3.1 From Taylor expansion to local variance
To overcome problem 1, we construct a new metric based on statistical
estimation. In this paragraph, $n_{i}>1$ but $n_{o}=1$. The following
derivations can be extended to $n_{o}>1$ by applying it to $f$ element-wise
and then taking the sum across the $n_{o}$ dimensions.
###### Lemma 3.1.
Let ${\mathbf{e}}\sim\mathcal{N}(0,\epsilon{\bm{I}}_{n_{i}})$ with
$\epsilon\in\mathbf{R}^{+}$ and ${\bm{I}}_{n_{i}}$ the identity matrix of
dimension $n_{i}$. Let ${\bm{\epsilon}}=\epsilon(1,...,1)$. Then,
$Var(f({\bm{x}}+{\mathbf{e}}))=Df_{{\bm{\epsilon}}}^{2}({\bm{x}})+\mathcal{O}(\|{\bm{\epsilon}}\|^{3}_{2}).$
The demonstration can be found in Appendix A. Using the unbiased estimator of
variance, we thus define new indices
$\widehat{Df_{{\bm{\epsilon}}}^{2}}({\bm{x}})$ by
$\widehat{Df_{{\bm{\epsilon}}}^{2}}({\bm{x}})=\frac{1}{k-1}\sum_{i=1}^{k}\Big{(}f({\bm{x}}+\bm{\epsilon_{i}})-f({\bm{x}})\Big{)}^{2},$
(3)
with $\\{{\bm{\epsilon}}_{1},...,{\bm{\epsilon}}_{k}\\}$ $k$ samples of
${\bm{\epsilon}}$. The metric
$\widehat{Df^{2}_{{\bm{\epsilon}}}}({\bm{x}})\underset{k\rightarrow\infty}{\rightarrow}Var(f({\bm{x}}+{\bm{\epsilon}}))$
and
$Var(f({\bm{x}}+{\bm{\epsilon}}))=Df^{2}_{{\bm{\epsilon}}}({\bm{x}})+\mathcal{O}(\|{\bm{\epsilon}}\|^{3}_{2})$,
so $\widehat{Df^{2}_{{\bm{\epsilon}}}}({\bm{x}})$ is a biased estimator of
$Df^{2}_{{\bm{\epsilon}}}({\bm{x}})$, with bias
$\mathcal{O}(\|{\bm{\epsilon}}\|^{3}_{2})$. Hence, when
${\bm{\epsilon}}\rightarrow 0$, $\widehat{Df^{2}_{{\bm{\epsilon}}}}({\bm{x}})$
becomes an unbiased estimator of $Df^{2}_{{\bm{\epsilon}}}({\bm{x}})$. It is
possible to compute $\widehat{Df^{2}_{{\bm{\epsilon}}}}({\bm{x}})$ from any
set of points centered around ${\bm{x}}$. Therefore, we evaluate
$\widehat{Df^{2}_{{\bm{\epsilon}}}}({\bm{x}}_{i})$ for each
$i\in\\{1,...,N\\}$ using the set $\mathcal{S}_{k}({\bm{x}}_{i})$ of
$k$-nearest neighbors of ${\bm{x}}_{i}$. We note this metric
$\widehat{Df^{2}}({\bm{x}}_{i})$, where we replace
$f({\bm{x}}_{i}+\bm{\epsilon_{i}})$ by $f({\bm{x}}_{l})$, the values of $f$
for the neighbors of ${\bm{x}}_{i}$
(${\bm{x}}_{l}\in\mathcal{S}_{k}({\bm{x}}_{i})$) and $f({\bm{x}}_{i})$ by
$\frac{1}{k}\sum_{{\bm{x}}_{l}\in\mathcal{S}_{k}({\bm{x}}_{i})}^{k}f({\bm{x}}_{l})$,
the average of $f$ on the neighbors of ${\bm{x}}_{i}$ :
$\widehat{Df^{2}}({\bm{x}}_{i})=\frac{1}{k-1}\sum_{{\bm{x}}_{j}\in\mathcal{S}_{k}({\bm{x}}_{i})}\Big{(}f({\bm{x}}_{j})-\frac{1}{k}\sum_{{\bm{x}}_{l}\in\mathcal{S}_{k}({\bm{x}}_{i})}^{k}f({\bm{x}}_{l})\Big{)}^{2},$
(4)
Equation (4) has several practical advantages. First, $\widehat{Df^{2}}$ can
even be applied to non-differentiable functions and for classification
problems, unlike equation (2). Second, the definition of
$\widehat{Df^{2}}({\bm{x}})$ does not rely on ${\bm{\epsilon}}$, unlike
equation (3). To compute $\widehat{Df^{2}}$, all we need are
$\\{f({\bm{x}}_{1}),...,f({\bm{x}}_{N})\\}$, the points used for the training
of the neural network. In addition, equation (4) can even be applied when the
data points are too sparse for the nearest neighbors of ${\bm{x}}_{i}$ to be
considered as close to ${\bm{x}}_{i}$, which is almost always the case in high
dimension. It can thus be seen as a generalization of
$\widehat{Df_{{\bm{\epsilon}}}^{2}}({\bm{x}})$, which tends towards
$Df_{{\bm{\epsilon}}}^{2}({\bm{x}})$ locally.
### 3.2 From sampling to weighting
To tackle problem 2, recall that the goal of the training is to find
${\bm{\theta}}^{*}=\underset{{\bm{\theta}}}{\operatorname{argmin}}\;\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})$,
with
$\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})=\frac{1}{N}\sum_{i}L(f({\bm{x}}_{i}),f_{{\bm{\theta}}}({\bm{x}}_{i}))$.
With the new distribution based on previous derivations, the procedure is
different. Since the training points are sampled using
$\widehat{Df^{2}_{{\bm{\epsilon}}}}$, we no longer minimize
$\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})$, but
$\widehat{J_{\bar{{\mathbf{x}}}}}({\bm{\theta}})=\frac{1}{N}\sum_{i}L(f(\bar{{\bm{x}}}_{i}),f_{{\bm{\theta}}}(\bar{{\bm{x}}}_{i}))$,
with $\bar{{\mathbf{x}}}\sim d\mathbb{P}_{\bar{{\mathbf{x}}}}$ the new
distribution. However, $\widehat{J_{\bar{{\mathbf{x}}}}}({\bm{\theta}})$
estimates
$J_{\bar{{\mathbf{x}}}}({\bm{\theta}})=\int_{\mathbf{S}}L(f({\bm{x}}),f_{{\bm{\theta}}}({\bm{x}}))d\mathbb{P}_{\bar{{\mathbf{x}}}}.$
Let $p_{{\mathbf{x}}}({\bm{x}})d{\bm{x}}=d\mathbb{P}_{{\mathbf{x}}}$,
$p_{\bar{{\mathbf{x}}}}({\bm{x}})d{\bm{x}}=d\mathbb{P}_{\bar{{\mathbf{x}}}}$
be the pdfs of ${\mathbf{x}}$ and $\bar{{\mathbf{x}}}$ (note that
$Df^{2}_{{\bm{\epsilon}}}\propto p_{\bar{{\mathbf{x}}}}$). Then,
$J_{\bar{{\mathbf{x}}}}({\bm{\theta}})=\int_{\mathbf{S}}L(f({\bm{x}}),f_{{\bm{\theta}}}({\bm{x}}))\frac{p_{\bar{{\mathbf{x}}}}({\bm{x}})}{p_{{\mathbf{x}}}({\bm{x}})}d\mathbb{P}_{{\mathbf{x}}}.$
The straightforward Monte Carlo estimator for this expression of
$J_{\bar{{\mathbf{x}}}}({\bm{\theta}})$ is
$\begin{split}\widehat{J_{\bar{{\mathbf{x}}}}}({\bm{\theta}})&=\frac{1}{N}\sum_{i}L(f({\bm{x}}_{i}),f_{{\bm{\theta}}}({\bm{x}}_{i}))\frac{p_{\bar{{\mathbf{x}}}}({\bm{x}}_{i})}{p_{{\mathbf{x}}}({\bm{x}}_{i})}\propto\frac{1}{N}\sum_{i}L(f({\bm{x}}_{i}),f_{{\bm{\theta}}}({\bm{x}}_{i}))\frac{\widehat{Df^{2}}({\bm{x}}_{i})}{p_{{\mathbf{x}}}({\bm{x}}_{i})}.\end{split}$
(5)
Thus, $J_{\bar{{\mathbf{x}}}}({\bm{\theta}})$ can be estimated with the same
points as $J_{{\mathbf{x}}}({\bm{\theta}})$ by weighting them with
$w_{i}=\frac{\widehat{Df^{2}}({\bm{x}}_{i})}{p_{{\mathbf{x}}}({\bm{x}}_{i})}$.
The expression of $w_{i}$ involves $p_{{\mathbf{x}}}$, the distribution of the
data. Just like for $f$, we do not have access to $p_{{\mathbf{x}}}$. The
estimation of $p_{{\mathbf{x}}}$ is a challenging task by itself, and standard
density estimation techniques such as K-nearest neighbors or Gaussian Mixture
density estimation led to extreme estimated values of
$p_{{\mathbf{x}}}({\bm{x}}_{i})$ in our experiments. Therefore, we decided to
only apply $\omega_{i}=\widehat{Df^{2}}({\bm{x}}_{i})$ as a first-order
approximation. In practice, we re-scale the weights between $1$ and $m$, a
hyperparameter, and then divide them by their sum to avoid affecting the
learning rate.
As a result, we obtain a new methodology based on weighting the training data
set. We call this methodology Variance Based Sample Weighting (VBSW).
## 4 Variance Based Sample Weighting
In this part, we sum up Variance Based Sample Weighting (VBSW) to clarify its
application to machine learning problems. We also study this methodology
through toy experiments.
### 4.1 Methodology
Variance Based Samples Weighting (VBSW) is recapitulated in Algorithm 2. Line
1: $m$ and $k$ are hyperparameters that can be chosen jointly with all other
hyperparameters, e.g. using a random search. Their effects and interactions
are studied and discussed in Sections 4.2 and 5.4. Line 2-3: equation (4) is
applied to compute the weights $w_{i}$ that are used to weight the data set.
Notations $\\{(w_{1},{\bm{x}}_{1}),...,(w_{N},{\bm{x}}_{N})\\}$ denote that
each ${\bm{x}}_{i}$ is weighted by $w_{i}$. To perform a nearest-neighbors
search, we use an approximate nearest neighbor search technique called
hierarchical navigable small world graphs [36] implemented by nmslib [10].
Line 4: Train $f_{{\bm{\theta}}}$ on the weighted data set.
Inputs: $k$, $m$;
Compute
$\\{\widehat{Df^{2}}({\bm{x}}_{1}),...,\widehat{Df^{2}}({\bm{x}}_{N})\\}$
using equation (4);
Construct a new training data set
$\\{(w_{1},{\bm{x}}_{1}),...,(w_{N},{\bm{x}}_{N})\\}$;
Train $f_{{\bm{\theta}}}$ on
$\\{(w_{1},f({\bm{x}}_{1})),...,(w_{N},f({\bm{x}}_{N}))\\}$ ;
Algorithm 2 Variance Based Samples Weighting (VBSW)
### 4.2 Toy experiments & hyperparameter study
VBSW is studied on a Double Moon (DM) classification problem, the Boston
Housing (BH) regression, and Breast Cancer (BC) classification data sets.
(a)
(b)
Figure 4: From left to right: (a) Double Moon (DM) data set. (b) Heat map of
the value of $w_{i}$ for each ${\bm{x}}_{i}$ (red is high and blue is low)
For DM, Figure 4(b) shows that the points with higher $w_{i}$ (in red) are
close to the boundary between the two classes. Indeed, in classification, VBSW
can be interpreted as a local label agreement. This behavior verifies recent
findings of [52] where authors conclude that in classification, a good set of
weights would put importance on points close to the decision boundary.
We train a Multi-Layer Perceptron of $1$ layer of $4$ units, using Stochastic
Gradient Descent (SGD) and binary cross-entropy loss function, on a $300$
points training data set for $50$ random seeds. In this experiment, VBSW, i.e.
weighting the data set with $w_{i}$ is compared to the baseline where no
weights are applied. The results of Table 3 show the improvement obtained with
VBSW.
| VBSW | baseline
---|---|---
DM | 99.4, $\textbf{94.44}\pm 0.78$ | $99$, $92.06\pm 0.66$
BH | 13.31, $\textbf{13.38}\pm 0.01$ | $14.05$, $14.06\pm 0.01$
BC | 99.12, $97.6\pm 0.34$ | $98.25$, $97.5\pm 0.11$
Table 3: best, mean + se for each method. The metric used is accuracy for DM
and BC and Mean Squared Error for BH.
For BH data set, a linear model is trained, and for BC data set, an MLP of $1$
layer and $30$ units, with a train-validation split of $80\%-20\%$. Both
models are trained with Adam [27]. Since these data sets are small and the
models are light, we study the effects of $m$ and $k$ on the error. Moreover,
BH is a regression task and BC a classification task, so it allows studying
the effect of hyperparameters more extensively.
For BH and BC experiments, we conduct a grid search for VBSW on the values of
$m$ and $k$. As a reminder, $m$ is the ratio between the highest and the
lowest weights, and $k$ is the number of neighbor points used to compute the
local variance. We train a linear model for BH and a MLP with $30$ units for
BC with VBSW on a grid of $20$ values of $m$ equally distributed between $2$
and $100$ and $20$ values of $k$ equally distributed between $10$ and $50$. As
a result, we train the model on $400$ pairs of $(m,k)$ values and with $10$
different random seeds for each pair.
Figure 5: Color map of the error, with respect to $m$ and $k$. Left: BH data
set, for the mean of the MSE across $10$ different seeds and right: BC data
set, for the mean of $1-acc$ across these seeds. Blue is lower.
These experiments, illustrated in Figure 5 show that the influence of $m$ and
$k$ on the performances of the model can be different. For BH data set, low
values of $k$ clearly lead to poorer performances. Hyperparameter $m$ seems to
have less impact, although it should be chosen not too far from its lowest
value, $2$. For BC data set, on the contrary, the best performances are
obtained for low values of $k$, while a high value could be chosen for $m$.
These experiments highlight that the impact of $m$ and $k$ can be different
between classification and regression, but it could also be different
depending on the data set. Hence, we recommend considering these
hyperparameters like many others involved in deep learning, selecting their
values using hyperparameters optimization techniques.
It also shows that many different $(m,k)$ pairs lead to error improvement. It
suggests that the weights approximation does not have to be exact for VBSW to
be effective, as stated in Section 5.4.
### 4.3 Cost efficiency of VBSW
VBSW’s computational burden mostly relies on the complexity of the nearest
neighbor search algorithm, which is independent and can be used as a third-
party algorithm. When the data set is not too large, classical techniques like
KDtree [7] can be used. However, when the number of points and the dimension
of the data set increase, approximate nearest neighbors searches may be
necessary to keep satisfying performances. In the previous examples, KDtree is
more than sufficient. However, since we deal with more complex examples in the
following, we directly use nmslib [10], an approximate nearest neighbors
search library for homogeneity of the implementation.
## 5 VBSW for deep learning
The high dimensionality of many deep learning problems makes VBSW difficult to
apply in the form previously described. In this part, we adapt VBSW to such
problems and study its application to various real-world learning tasks. We
also study the robustness of VBSW and its complementarity with other similar
techniques.
### 5.1 Methodology
We mentioned that local variance could be computed using already existing
points. This statement implies finding the nearest neighbors of each point. In
extremely high-dimensional spaces like image spaces, the curse of
dimensionality makes nearest neighbors vacuous. In addition, the data
structure may be highly irregular, and the concept of nearest neighbor may be
misleading. Thus, it would be irrelevant to evaluate $\widehat{Df^{2}}$
directly on this data.
One of the strengths of deep learning is to construct good representations of
the data embedded in lower-dimensional latent spaces. For instance, in
Computer Vision, convolutional neural networks’ deeper layers represent more
abstract features. We could leverage this representational power of neural
networks and simply apply our methodology within this latent feature space.
Variance Based Samples Weighting (VBSW) for deep learning is recapitulated in
Algorithm 3. Here, $\mathcal{M}$ is the initial neural network whose feature
space will be used to project the training data set and apply VBSW. Line 1:
$m$ and $k$ are hyperparameters that can be chosen jointly with all other
hyperparameters, e.g. using a random search. Their effects and interactions
are studied and discussed in Sections 4.2 and 5.4. Line 2: The initial neural
network, $\mathcal{M}$, is trained as usual. Notations
$\\{(\frac{1}{N},{\bm{x}}_{1}),...,(\frac{1}{N},{\bm{x}}_{N})\\}$ is
equivalent to $\\{{\bm{x}}_{1},...,{\bm{x}}_{N}\\}$, because all the weights
are the same ($\frac{1}{N}$). Line 3: The last fully connected layer is
discarded, resulting in a new model $\mathcal{M^{*}}$, and the training data
set is projected in the feature space. Line 4-5: equation (4) is applied to
compute the weights $w_{i}$ that are used to weight the projected data set.
Line 6: The last layer is re-trained (which is often equivalent to fitting a
linear model) using the weighted data set and added to $\mathcal{M^{*}}$ to
obtain the final model $\mathcal{M}_{f}$. As a result, $\mathcal{M}_{f}$ is a
composition of the already trained model $\mathcal{M^{*}}$ and
$f_{{\bm{\theta}}}$ trained using the weighted data set.
Inputs: $k$, $m$, $\mathcal{M}$;
Train $\mathcal{M}$ on the training set
$\\{(\frac{1}{N},{\bm{x}}_{1}),...,(\frac{1}{N},{\bm{x}}_{N})\\}$;
$\\{(\frac{1}{N},f({\bm{x}}_{1})),...,(\frac{1}{N},f({\bm{x}}_{N}))\\}$;
Construct $\mathcal{M^{*}}$ by removing its last layer ;
Compute
$\\{w_{1}=\widehat{Df^{2}}(\mathcal{M^{*}}({\bm{x}}_{1})),...,w_{N}=\widehat{Df^{2}}(\mathcal{M^{*}}({\bm{x}}_{N}))\\}$
using equation (4);
Construct a new training data set
$\\{(w_{1},\mathcal{M^{*}}({\bm{x}}_{1})),...,(w_{N},\mathcal{M^{*}}({\bm{x}}_{N}))\\}$;
Train $f_{{\bm{\theta}}}$ on the training set of inputs
$\\{(w_{1},\mathcal{M^{*}}({\bm{x}}_{1})),...,(w_{N},\mathcal{M^{*}}({\bm{x}}_{N}))\\}$
with outputs $\\{f({\bm{x}}_{1}),...,f({\bm{x}}_{N})\\}$ and add it to
$\mathcal{M^{*}}$. The final model is $\mathcal{M}_{f}$ =
$f_{{\bm{\theta}}}\circ\mathcal{M^{*}}$;
Algorithm 3 Variance Based Samples Weighting (VBSW) for deep learning
### 5.2 Image Classification
In this section, we study the performances of VBSW on MNIST [32] and Cifar10
[29] image classification data sets. For MNIST, we train LeNet [31], with $40$
different random seeds, and then apply VBSW for $10$ different random seeds,
with Adam optimizer and categorical cross-entropy loss. Note that in the
following, Adam is used with the default parameters of its keras
implementation. We record the best value obtained from the $10$ VBSW training.
We follow the same procedure for Cifar10, except that we train a ResNet20 for
$50$ random seeds and with data augmentation and learning rate decay. The
networks have been trained on 4 Nvidia K80 GPUs. The values of the
hyperparameters used can be found in Appendix B. We compare the test accuracy
between LeNet 5 + VBSW, ResNet20 + VBSW, and the initial test accuracies of
LeNet 5 and ResNet20 (baseline) for each of the initial networks.
| VBSW | baseline | gain per model
---|---|---|---
MNIST | 99.09, $\textbf{98.87}\pm 0.01$ | $98.99$, $98.84\pm 0.01$ | 0.15, $\textbf{0.03}\pm 0.01$
Cifar10 | 91.30, $\textbf{90.64}\pm 0.07$ | $91.01$, $90.46\pm 0.10$ | 1.65, $\textbf{0.15}\pm 0.04$
Table 4: best, mean + se for each method. The metric used is accuracy. For a
model $\mathcal{M}$, the gain $g$ for this model is given by
$g=\underset{1\leq i\leq
10}{\operatorname{max}}(acc(\mathcal{M}^{i}_{f})-acc(\mathcal{M}))$ with $acc$
the accuracy and $\mathcal{M}^{i}_{f}$ the VBSW model trained at the $i$-th
random seed.
The results statistics are gathered in Table 4, which also displays statistics
about the gain due to VBSW for each model. The results on MNIST are slightly
but consistently better than for the baseline, by $0.1\%$ for the best with up
to $0.15\%$ of accuracy gain per model. For Cifar10, we get a $0.3\%$ accuracy
improvement for the best model and up to $1.65\%$ accuracy gain, meaning that
among the $50$ ResNet20s, there is one whose accuracy has been improved by
$1.65\%$ using VBSW. Note that applying VBSW took less than 15 minutes on a
laptop with an i7-7700HQ CPU. A visualization of the samples weighted by the
highest $w_{i}$ is given in Figure 6.
Figure 6: Samples from Cifar10 and MNIST with high $w_{i}$. Those pictures are
either unusual or difficult to classify, even for a human (especially for
MNIST).
### 5.3 Text Classification and Regression
In this section, we study the performances of VBSW on RTE and MRPC, two text
classification data sets, and STS-B, a text classification data set, extracted
from the glue benchmark [51]. For this application, we use Bert, a modern
neural network based on transformers [50] that is the state-of-the-art of
text-based machine learning tasks. We do not pre-train Bert, like in the
previous experiments, since it has been originally built for Transfer Learning
purposes. Therefore, its purpose is to be used as-is and then fine-tuned on
any text data set see [14]. However, because of the small size of the data set
and the high number of model parameters, we chose not to fine-tune the Bert
model and only to use the representations of the data sets in its feature
space to apply VBSW. More specifically, we use tiny-bert [49], which is a
lighter version of the initial Bert. We train the linear model with TensorFlow
to be able to add the trained model on top of the Bert model and obtain a
unified model. RTE and MRPC are classification tasks, so we use binary cross-
entropy loss function to train our models. STS-B is a regression task, so the
model is trained with Mean Squared Error. All the models are trained with Adam
optimizer. For each task, we compare the training of the linear model with
VBSW and without VBSW (baseline). The results obtained with VBSW are better
overall, except for Pearson Correlation in STS-B, which is slightly worse than
baseline (Table 5).
| VBSW | baseline
---|---|---
| m1 | m2 | m1 | m2
RTE | 61.73, $\textbf{58.46}\pm 0.15$ | - | $61.01$, $58.09\pm 0.13$ | -
STS-B | 62.31, $\textbf{62.20}\pm 0.01$ | 60.99, $60.88\pm 0.01$ | $61.88$, $61.87\pm 0.01$ | $60.98$, $\textbf{60.92}\pm 0.01$
MRPC | 72.30, $\textbf{71.71}\pm 0.03$ | 82.64, $\textbf{80.72}\pm 0.05$ | $71.56$, $70.92\pm 0.03$ | $81.41$, $80.02\pm 0.07$
Table 5: best, mean + se for each method. For RTE the metric used is accuracy
(m1). For STS-B, metric 1 (m1) is Spearman correlation and metric 2 (m2) is
Pearson correlation. For MRPC, metric 1 (m1) is accuracy and metric 2 (m2) is
F1 score.
### 5.4 Robustness of VBSW
In this section, we assess the robustness of VBSW. First, we focus on the
robustness to label noise. To that end, we train a ResNet20 on Cifar10 with
four different noise levels. We randomly change the label of $p\%$ training
points for four different values of $p$ ($10$, $20$, $30$ and $40$). We then
apply VBSW $30$ times and evaluate the obtained neural networks on a clean
test set. The results are gathered in Table 6.
noise | $10\%$ | $20\%$ | $30\%$ | $40\%$
---|---|---|---|---
original error | $87.43$ | $85.75$ | $84.05$ | $81.79$
VBSW | 87.76, $8\textbf{7.63}\pm 0.01$ | 86.03, $\textbf{85.89}\pm 0.01$ | 84.35, $\textbf{84.18}\pm 0.02$ | 82.48, $\textbf{82.32}\pm 0.02$
Table 6: best, mean + se of the training of a ResNet20 on Cifar10 for
different label noise levels. These results illustrate the robustness of VBSW
to labels noise.
The results show that VBSW is still effective despite label noise. This
specificity must be related to the robustness of VBSW with respect to the
choice of hyperparameters $m$ and $k$, as seen in Section 4.2. Indeed, it
shows that many combinations of $m$ and $k$ improves the performances of the
neural network, and therefore that VBSW is actually robust to error in the
weights evaluation. Its robustness to label noise hence stems from its
robustness to weights evaluation error, since label noise essentially hurts
the accuracy of the weights evaluation.
Although VBSW is robust to label noise, note that the goal of VBSW is not to
address noisy label problem, like discussed in Section 1. It may be more
effective to use a sampling technique tailored specifically for this
situation.
### 5.5 Complementarity of VBSW
Existing techniques based on dataset processing can be used jointly with VBSW,
by applying the first technique during the initial training of the neural
network and then applying VBSW on its feature space. To illustrate this
specificity, we compare VBSW with the recently introduced Active Bias (AB)
[11] and transfer-learning-based curriculum learning (TCL) [18]. AB
dynamically weights the samples based on the variance of the probability of
prediction of each point throughout the training, and TCL creates a curriculum
based on sample difficulty evaluated on previously trained neural networks.
Here, we study the effects of AB and TCL combined with VBSW for the training
of a ResNet20 on Cifar10. Table 7 gathers the results of experiments for
different baselines: vanilla, for regular training with Adam optimizer, AB /
TCL for training with AB / TCL, VBSW for the application of VBSW on top of
regular training, and VBSW + AB / VBSW + CL for initial training with AB / TCL
and the application of VBSW. Unlike in Section 5.2, we do not use data
augmentation nor learning rate decay in order to simplify the experiments.
| accuracy ($\%$) | VBSW gpm
---|---|---
vanilla | $75.88$, $74.55\pm 0.11$ | -
AB | $76.33$, $75.14\pm 0.09$ | -
TCL | $78.54$, $77.46\pm 0.07$ | -
VBSW | $76.57$, $74.94\pm 0.10$ | $0.94$, $0.40\pm 0.03$
AB + VBSW | $76.60$, $75.33\pm 0.09$ | $0.40$, $0.14\pm 0.02$
TCL + VBSW | $\bm{79.86}$, $\bm{78.71}\pm 0.09$ | $\bm{2.19}$, $\bm{1.26}\pm 0.08$
Table 7: Best, mean + se of the training of 60 ResNet20s on Cifar10 for
vanilla, VBSW, AB and AB + VBSW. The gain per model (gpm) $g$ is defined by
$g=\underset{1\leq i\leq
10}{\operatorname{max}}(acc(\mathcal{M}^{i}_{f})-acc(\mathcal{M}))$ with $acc$
the accuracy and $\mathcal{M}^{i}_{f}$ the VBSW model trained at the $i$-th
random seed.
The accuracy obtained with VBSW is quite similar to AB. While TCL yields
better results than VBSW alone, the best accuracy is obtained when they are
used jointly. Overall, the best neural networks are obtained when AB and TCL
are used along with VBSW (AB + VBSW and TCL + VBSW), which demonstrates the
complementarity of VBSW with other dataset processing techniques. Note that
VBSW works much better when applied to a neural network initially trained with
TCL. It means that TCL creates neural network features particularly suited to
VBSW. This lead might be explored in future works.
## 6 Discussion and Perspectives
By studying the training distribution of the neural network, we explored a
practical and classical question that naturally arises when performing
surrogate modeling for approximating computer codes: how to construct the
training set? We found that exploring this question led to findings that are
also relevant for approximation theory, which is an important component of
machine learning.
Hence, the results obtained in this paper are impactful both for machine
learning in numerical simulations and machine learning in general.
### 6.1 Impact for numerical simulations
This work comes from the observation that, on our approximation problems,
neural networks are more efficient when more data are sampled where the
function to learn is steeper. It is an attempt to formalize this observation
and to construct a workable methodology out of it. As a result, the
methodologies for constructing the distribution
$d\mathbb{P}_{\bar{{\mathbf{x}}}}$ can be used as new, principled designs of
experiments.
In the context of numerical simulations, once
$d\mathbb{P}_{\bar{{\mathbf{x}}}}$ is constructed, it is possible to sample
new data from it. It alleviates Problem 2, described in Section 3.2. In
theory, Problem 1 is also solved since we could have access to the derivatives
- either by instrumenting the code with automatic differentiation if we have
access to its implementation or by estimating them with finite differences.
However, the implementation of automatic differentiation can be tedious, and
if the computer code is slow and high dimensional, finite differences may be
unaffordable. In that case, it is possible to use a third methodology based on
the approximation of
$\\{Df^{2}_{{\bm{\epsilon}}}({\bm{x}}_{1}),...,Df^{2}_{{\bm{\epsilon}}}({\bm{x}}_{N})\\}$
using local variance, like VBSW, and the sampling of new points, like TBS.
Finally, the method allows improving the error of neural networks without
increasing the computational cost of their prediction. This achievement is of
interest when they are intended to accelerate numerical simulations.
### 6.2 Impact for machine learning
VBSW is validated on several tasks, complementary with other training
distribution modification frameworks, and robust to noise. It makes it quite
versatile. Moreover, the problem of high dimensionality and irregularity of
$f$, which often arises in deep learning problems, is alleviated by focusing
on the latent space of neural networks. This makes VBSW scalable. As a result,
VBSW can be applied to complex neural networks such as ResNet, or Bert, for
various machine learning tasks.
The experiments support an original view of the learning problem that involves
the local variations of $f$. The studies of Section 2.2, that use the
derivatives of the function to learn to sample a more efficient training data
set, support this approach as well. This view is also bolstered up by
conclusions of [52]. VBSW allows extending this original view to problems
where the derivatives of $f$ are not accessible and sometimes not defined.
Indeed, VBSW comes from Taylor expansion, which is specific to differentiable
functions, but in the end, it can be applied regardless of the properties of
$f$.
Finally, this method is cost-effective. In most cases, it allows to quickly
improve the performances of a neural network using a regular CPU. It is better
than carrying on entirely new training with a wider and deeper neural network.
### 6.3 Further studies
Although VBSW uses theoretically justified approximations concerning TBS, the
actual effect of these approximations should be more thoroughly investigated.
For instance, we could further study the impact of not explicitly using
$p_{{\mathbf{x}}}$, the data distribution, in the weights definitions; the
convergence of the estimator of $Df^{2}_{{\bm{\epsilon}}}$, and in which
context it is adequately approximated; and a more generic derivative-based
generalization bound. In addition, VBSW demonstrated intriguing behaviors,
like its impressive synergy with TCL [18], which would deserve more attention.
## 7 Conclusion
This work is based on the observation that, in supervised learning, a function
$f$ is more difficult to approximate by a neural network in the regions where
it is steep. We mathematically traduced this intuition, derived a
generalization bound to illustrate it, and a methodology, Taylor Based
Sampling, to test it empirically. In order to be able to use these insights
for machine learning problems where $f$ is not available, we constructed a
weighting scheme, Variance Based Samples Weighting (VBSW) that uses the
variance of the training samples’ labels to weight the training data set. VBSW
is simple to use and implement because it only requires computing statistics
on the input space. In Deep Learning, applying VBSW on the data set projected
in an already trained neural network feature space allows reducing its error
by simply re-training its last layer. Although specifically investigated in
deep learning, this method applies to any loss-function-based supervised
learning problem and is scalable, cost-effective, robust, and versatile. It is
validated on several applications, such as glue benchmark with bert for text
classification and regression, and Cifar10 with ResNet for image
classification.
## Appendix A Appendix A: Proofs
### A.1 Illustration of the link using derivatives
(Section 2.1)
We look at approximating $f:{\bm{x}}\rightarrow f({\bm{x}})$,
${\bm{x}}\in\mathbb{R}^{n_{i}}$, $f({\bm{x}})\in\mathbb{R}^{n_{o}}$ with a NN
$f_{{\bm{\theta}}}$. The goal of the approximation problem can be seen as
being able to generalize to points not seen during the training. We thus want
the generalization error $\mathcal{J}_{{\mathbf{x}}}({\bm{\theta}})$ to be as
small as possible. Given an initial data set
$\\{{\bm{x}}_{1},...,{\bm{x}}_{N}\\}$ drawn from ${\mathbf{x}}\sim
d\mathbb{P}_{{\mathbf{x}}}$ and $\\{f({\bm{x}}_{1}),...,f({\bm{x}}_{N})\\}$,
and the loss function $L$ being the squared $L_{2}$ error, recall that the
integrated error $J_{{\mathbf{x}}}({\bm{\theta}})$, its estimation
$\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})$ and the generalization error
$\mathcal{J}_{{\mathbf{x}}}({\bm{\theta}})$ can be written:
$\begin{split}J_{{\mathbf{x}}}({\bm{\theta}})&=\int_{\mathbf{S}}\|f({\bm{x}})-f_{{\bm{\theta}}}({\bm{x}})\|d\mathbb{P}_{{\mathbf{x}}},\\\
\widehat{J_{{\mathbf{x}}}}({\bm{\theta}})&=\frac{1}{N}\sum_{i=1}^{N}\|f_{{\bm{\theta}}}({\bm{x}}_{i})-f({\bm{x}}_{i})\big{\|},\\\
\mathcal{J}_{{\mathbf{x}}}({\bm{\theta}})&=J_{{\mathbf{x}}}({\bm{\theta}})-\widehat{J_{{\mathbf{x}}}}({\bm{\theta}}),\\\
\end{split}$ (6)
where $\|.\|$ denotes the squared $L_{2}$ norm. In the following, we find an
upper bound for $\mathcal{J}_{{\mathbf{x}}}({\bm{\theta}})$. We start by
finding an upper bound for $J_{{\mathbf{x}}}({\bm{\theta}})$ and then for
$\mathcal{J}_{{\mathbf{x}}}({\bm{\theta}})$ using equation (6).
Let $S_{i}$, $i\in\\{1,...,N\\}$ be some sub-spaces of a bounded space
$\mathbf{S}$ such that $\mathbf{S}=\bigcup_{i=1}^{N}S_{i}$,
$\bigcap_{i=1}^{N}S_{i}=$ Ø, and ${\bm{x}}_{i}\in S_{i}$. Then,
$\begin{split}J_{{\mathbf{x}}}({\bm{\theta}})=&\sum_{i=1}^{N}\int_{S_{i}}\|f({\bm{x}})-f_{{\bm{\theta}}}({\bm{x}})\|d\mathbb{P}_{{\mathbf{x}}},\\\
J_{{\mathbf{x}}}({\bm{\theta}})=&\sum_{i=1}^{N}\int_{S_{i}}\|f({\bm{x}}_{i}+{\bm{x}}-{\bm{x}}_{i})-f_{{\bm{\theta}}}({\bm{x}})\|d\mathbb{P}_{{\mathbf{x}}}.\\\
\end{split}$
Suppose that $n_{i}=n_{o}=1$ (${\bm{x}}$ becomes $x$ and ${\mathbf{x}}$
becomes x) and $f$ twice differentiable. Let
$|\mathbf{S}|=\int_{\mathbf{S}}d\mathbb{P}_{{\textnormal{x}}}$. The volume
$|\mathbf{S}|=1$ since $d\mathbb{P}_{{\textnormal{x}}}$ is a probability
measure, and therefore $|S_{i}|<1$ for all $i\in\\{1,...,N\\}$ . Using Taylor
expansion at order 2, and since $|S_{i}|<1$ for all $i\in\\{1,...,N\\}$
$J_{{\textnormal{x}}}({\bm{\theta}})=\sum_{i=1}^{N}\int_{S_{i}}\|f(x_{i})+f^{\prime}(x_{i})(x-x_{i})+\frac{1}{2}f^{\prime\prime}(x_{i})(x-x_{i})^{2}-f_{{\bm{\theta}}}(x)+\mathcal{O}((x-x_{i})^{3})\|d\mathbb{P}_{{\textnormal{x}}}.\\\
$
To find an upper bound for $J({\bm{\theta}})$, we can first find an upper
bound for $|A_{i}(x)|$, with
$A_{i}(x)=f(x_{i})+f^{\prime}(x_{i})(x-x_{i})+\frac{1}{2}f^{\prime\prime}(x_{i})(x-x_{i})^{2}-f_{{\bm{\theta}}}(x)+\mathcal{O}((x-x_{i})^{3})$.
NN $f_{{\bm{\theta}}}$ is $K_{{\bm{\theta}}}-$Lipschitz, so since $\mathbf{S}$
is bounded (so are $S_{i}$), for all $x\in S_{i}$,
$|f_{{\bm{\theta}}}(x)-f_{{\bm{\theta}}}(x_{i})|\leq
K_{{\bm{\theta}}}|x-x_{i}|$. Hence,
$\begin{split}&f_{{\bm{\theta}}}(x_{i})-K_{{\bm{\theta}}}|x-x_{i}|\leq
f_{{\bm{\theta}}}(x)\leq
f_{{\bm{\theta}}}(x_{i})+K_{{\bm{\theta}}}|x-x_{i}|,\\\
&-f_{{\bm{\theta}}}(x_{i})-K_{{\bm{\theta}}}|x-x_{i}|\leq-
f_{{\bm{\theta}}}(x)\leq-
f_{{\bm{\theta}}}(x_{i})+K_{{\bm{\theta}}}|x-x_{i}|,\\\
&f(x_{i})+f^{\prime}(x_{i})(x-x_{i})+\frac{1}{2}f^{\prime\prime}(x_{i}(x-x_{i})^{2})-f_{{\bm{\theta}}}(x_{i})-K_{{\bm{\theta}}}|x-x_{i}|+\mathcal{O}((x-x_{i})^{3})\\\
&\leq A_{i}(x)\leq
f(x_{i})+f^{\prime}(x_{i})(x-x_{i})+\frac{1}{2}f^{\prime\prime}(x_{i})(x-x_{i})^{2}-f_{{\bm{\theta}}}(x_{i})+K_{{\bm{\theta}}}|x-x_{i}|+\mathcal{O}((x-x_{i})^{3}),\\\
&A_{i}(x)\leq
f(x_{i})-f_{{\bm{\theta}}}(x_{i})+f^{\prime}(x_{i})(x-x_{i})+\frac{1}{2}f^{\prime\prime}(x_{i})(x-x_{i})^{2}+K_{{\bm{\theta}}}|x-x_{i}|+\mathcal{O}((x-x_{i})^{3}).\end{split}$
And finally, using triangular inequality,
$\boxed{A_{i}(x)\leq|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|+|f^{\prime}(x_{i})||x-x_{i}|+\frac{1}{2}|f^{\prime\prime}(x_{i})||x-x_{i}|^{2}+K_{{\bm{\theta}}}|x-x_{i}|+\mathcal{O}(|x-x_{i}|^{3}).}$
Now, $\|.\|$ being the squared $L_{2}$ norm:
$\begin{split}J_{{\textnormal{x}}}({\bm{\theta}})=\sum_{i=1}^{N}\int_{S_{i}}\|&f(x_{i})+f^{\prime}(x_{i})(x-x_{i})+\frac{1}{2}f^{\prime\prime}(x_{i})(x-x_{i})^{2}-f_{{\bm{\theta}}}(x)+\mathcal{O}(|x-x_{i}|^{3})\|d\mathbb{P}_{{\textnormal{x}}},\\\
J_{{\textnormal{x}}}({\bm{\theta}})\leq\sum_{i=1}^{N}\int_{S_{i}}&\Bigg{[}\Big{(}|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|\Big{)}+\Big{(}|f^{\prime}(x_{i})||x-x_{i}|+\frac{1}{2}|f^{\prime\prime}(x_{i})||x-x_{i}|^{2}+K_{{\bm{\theta}}}|x-x_{i}|\Big{)}\\\
&+\mathcal{O}(|x-x_{i}|^{3})\Bigg{]}^{2}d\mathbb{P}_{{\textnormal{x}}},\\\
=\sum_{i=1}^{N}\int_{S_{i}}&\Bigg{[}|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|^{2}\\\
&+2|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|\Big{(}|f^{\prime}(x_{i})||x-x_{i}|+\frac{1}{2}|f^{\prime\prime}(x_{i})||x-x_{i}|^{2}+K_{{\bm{\theta}}}|x-x_{i}|\Big{)}\\\
&+\Big{[}\Big{(}|f^{\prime}(x_{i})||x-x_{i}|\Big{)}+\Big{(}\frac{1}{2}|f^{\prime\prime}(x_{i})||x-x_{i}|^{2}+K_{{\bm{\theta}}}|x-x_{i}|\Big{)}\Big{]}^{2}+\mathcal{O}(|x-x_{i}|^{3})\Bigg{]}d\mathbb{P}_{{\textnormal{x}}},\\\
=\sum_{i=1}^{N}\int_{S_{i}}&\Bigg{[}|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|^{2}\\\
&+2|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|\Big{(}|f^{\prime}(x_{i})||x-x_{i}|+\frac{1}{2}|f^{\prime\prime}(x_{i})||x-x_{i}|^{2}+K_{{\bm{\theta}}}|x-x_{i}|\Big{)}\\\
&+\Big{[}|f^{\prime}(x_{i})|^{2}|x-x_{i}|^{2}+2K_{{\bm{\theta}}}|f^{\prime}(x_{i})||x-x_{i}|^{2}+K_{{\bm{\theta}}}^{2}|x-x_{i}|^{2}\Big{]}+\mathcal{O}(|x-x_{i}|^{3})\Bigg{]}d\mathbb{P}_{{\textnormal{x}}},\\\
=\sum_{i=1}^{N}\int_{S_{i}}&\Bigg{[}|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|^{2}\\\
&+2|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|\Big{(}|f^{\prime}(x_{i})||x-x_{i}|+\frac{1}{2}|f^{\prime\prime}(x_{i})||x-x_{i}|^{2}+K_{{\bm{\theta}}}|x-x_{i}|\Big{)}\\\
&+\Big{(}|f^{\prime}(x_{i})|+K_{{\bm{\theta}}}\Big{)}^{2}|x-x_{i}|^{2}+\mathcal{O}(|x-x_{i}|^{3})\Bigg{]}d\mathbb{P}_{{\textnormal{x}}}.\end{split}$
Hornik’s theorem [20] states that given a norm $\|.\|_{p,\mu}=$ such that
$\|f\|^{p}_{p,\mu}=\int_{\mathbf{S}}|f(x)|^{p}d\mu(x)$, with $d\mu$ a
probability measure, for any $\epsilon$, there exists ${\bm{\theta}}$ such
that for a Multi Layer Perceptron, $f_{{\bm{\theta}}}$,
$\|f(x)-f_{{\bm{\theta}}}(x)\|^{p}_{p,\mu}<\epsilon$,
This theorem grants that for any $\epsilon$, with
$d\mu=\sum_{i=1}^{N}\frac{1}{N}\delta(x-x_{i})$, there exists ${\bm{\theta}}$
such that
$\begin{dcases}\|f(x)-f_{{\bm{\theta}}}(x)\|^{1}_{1,\mu}=\sum_{i=1}^{N}\frac{1}{N}|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|\leq\epsilon,\\\
\|f(x)-f_{{\bm{\theta}}}(x)\|^{2}_{2,\mu}=\sum_{i=1}^{N}\frac{1}{N}\big{(}f(x_{i})-f_{{\bm{\theta}}}(x_{i})\big{)}^{2}\leq\epsilon.\\\
\end{dcases}$ (7)
Let’s introduce $i^{*}$ such that $i^{*}=\operatorname{argmin}|S_{i}|$. Note
that for any $i\in\\{1,...,N\\}$, $\mathcal{O}(|S_{i}^{*}|^{4})$ is
$\mathcal{O}(|S_{i}|^{4})$. Now, let’s choose $\epsilon$ such that $\epsilon$
is $\mathcal{O}(|S_{i}^{*}|^{4})$. Then, equation (7) implies that
$\begin{dcases}|f(x_{i})-f_{{\bm{\theta}}}(x_{i})|=\mathcal{O}(|S_{i}|^{4}),\\\
\big{(}f(x_{i})-f_{{\bm{\theta}}}(x_{i})\big{)}^{2}=\mathcal{O}(|S_{i}|^{4}),\\\
\widehat{J_{{\textnormal{x}}}}({\bm{\theta}})=\|f(x)-f_{{\bm{\theta}}}(x)\|^{2}_{2,\mu}=\mathcal{O}(|S_{i}|^{4}).\\\
\end{dcases}$
Thus, we have
$\mathcal{J}_{{\textnormal{x}}}({\bm{\theta}})=J_{{\textnormal{x}}}({\bm{\theta}})-\widehat{J_{{\textnormal{x}}}}({\bm{\theta}})=J_{{\textnormal{x}}}({\bm{\theta}})+\mathcal{O}(|S_{i}|^{4})$
and therefore,
$\mathcal{J}_{{\textnormal{x}}}({\bm{\theta}})\leq\sum_{i=1}^{N}\int_{S_{i}}\Big{[}\Big{(}|f^{\prime}(x_{i})|+K_{{\bm{\theta}}}\Big{)}^{2}|x-x_{i}|^{2}d\mathbb{P}_{{\textnormal{x}}}\Big{]}+\mathcal{O}(|S_{i}|^{4}).$
Finally,
$\boxed{\mathcal{J}_{{\textnormal{x}}}({\bm{\theta}})\leq\sum_{i=1}^{N}(|f^{\prime}(x_{i})|+K_{{\bm{\theta}}})^{2}\frac{|S_{i}|^{3}}{3}+\mathcal{O}(|S_{i}|^{4}).}$
(8)
We see that on the regions where $f^{\prime}(x_{i})+K_{{\bm{\theta}}}$ is
higher, quantity $|S_{i}|$ (the volume of $S_{i}$) has a stronger impact on
the GB. Then, since $|S_{i}|$ can be seen as a metric for the local density of
the data set (the smaller $|S_{i}|$ is, the denser the data set is), the
Generalization Bound (GB) can be reduced more efficiently by adding more
points around $x_{i}$ in these regions. This bound also involves
$K_{{\bm{\theta}}}$, the Lipschitz constant of the NN, which has the same
impact as $f^{\prime}(x_{i})$. It also illustrates the link between the
Lipschitz constant and the generalization error, which has been pointed out by
several works like, for instance, [17], [3] and [43].
### A.2 Problem 1: Unavailability of derivatives
(Section 3.1)
In this paragraph, we consider $n_{i}>1$ but $n_{o}=1$. The following
derivations can be extended to $n_{o}>1$ by applying it to $f$ element-wise.
Let $\mathbf{e}\sim\mathcal{N}(0,\epsilon{\bm{I}}_{n_{i}})$ with
$\epsilon\in\mathbb{R}^{+}$,
${\mathbf{e}}=(\epsilon_{1},...,\epsilon_{n_{i}})$, i.e.
$\epsilon_{i}\sim\mathcal{N}(0,\epsilon)$ and
${\bm{\epsilon}}=\epsilon(1,...,1)$. Using Taylor expansion on $f$ at order
$2$ gives
$f({\bm{x}}+{\mathbf{e}})=f({\bm{x}})+\nabla_{{\bm{x}}}f({\bm{x}})\cdot{\mathbf{e}}+\frac{1}{2}{\mathbf{e}}^{T}\cdot\mathbb{H}_{x}f({\bm{x}})\cdot{\mathbf{e}}+\mathcal{O}(\|{\mathbf{e}}\|^{3}_{2}),$
with $\nabla_{x}f$ and $\mathbb{H}_{x}f({\bm{x}})$ the gradient and the
Hessian of $f$ w.r.t. ${\bm{x}}$. We now compute $Var(f(X+{\mathbf{e}}))$ and
make
$Df_{{\bm{\epsilon}}}^{2}({\bm{x}})=\epsilon\|\nabla_{x}f({\bm{x}})\|^{2}_{F}+\frac{1}{2}\epsilon^{2}\|\mathbb{H}_{{\bm{x}}}f({\bm{x}})\|^{2}_{F}$
appear in its expression to establish a link between these two quantities:
$\begin{split}Var(f({\bm{x}}+{\mathbf{e}}))&=Var\Big{(}f({\bm{x}})+\nabla_{{\bm{x}}}f({\bm{x}})\cdot{\mathbf{e}}+\frac{1}{2}{\mathbf{e}}^{T}\cdot\mathbb{H}_{x}f({\bm{x}})\cdot{\mathbf{e}}+\mathcal{O}(\|{\mathbf{e}}\|^{3}_{2})\Big{)},\\\
&=Var\Big{(}\nabla_{{\bm{x}}}f({\bm{x}})\cdot{\mathbf{e}}+\frac{1}{2}{\mathbf{e}}^{T}\cdot\mathbb{H}_{x}f({\bm{x}})\cdot{\mathbf{e}}\Big{)}+\mathcal{O}(\|{\bm{\epsilon}}\|^{3}_{2}).\end{split}$
Since $\epsilon_{i}\sim\mathcal{N}(0,\epsilon)$,
${\bm{x}}=(x_{1},...,x_{n_{i}})$ and with $\frac{\partial^{2}f}{\partial
x_{i}x_{j}}({\bm{x}})$ the cross derivatives of $f$ w.r.t. $x_{i}$ and
$x_{j}$,
$\begin{split}\nabla_{{\bm{x}}}f({\bm{x}})\cdot{\mathbf{e}}+\frac{1}{2}{\mathbf{e}}^{T}\cdot\mathbb{H}_{x}f({\bm{x}})\cdot{\mathbf{e}}=&\sum_{i=1}^{n_{i}}\epsilon_{i}\frac{\partial
f}{\partial
x_{i}}({\bm{x}})+\frac{1}{2}\sum_{j=1}^{n_{i}}\sum_{k=1}^{n_{i}}\epsilon_{j}\epsilon_{k}\frac{\partial^{2}f}{\partial
x_{j}x_{k}}({\bm{x}}),\\\ \end{split}$
$\begin{split}Var\Big{(}\nabla_{{\bm{x}}}f({\bm{x}})\cdot{\mathbf{e}}+\frac{1}{2}{\mathbf{e}}^{T}\cdot\mathbb{H}_{x}f({\bm{x}})\cdot{\mathbf{e}}\Big{)}=&Var\Big{(}\sum_{i=1}^{n_{i}}\epsilon_{i}\frac{\partial
f}{\partial
x_{i}}({\bm{x}})+\frac{1}{2}\sum_{j=1}^{n_{i}}\sum_{k=1}^{n_{i}}\epsilon_{j}\epsilon_{k}\frac{\partial^{2}f}{\partial
x_{j}x_{k}}({\bm{x}})\Big{)},\\\
=&\sum_{i_{1}=1}^{n_{i}}\sum_{i_{2}=1}^{n_{i}}Cov\Big{(}\epsilon_{i_{1}}\frac{\partial
f}{\partial x_{i_{1}}}({\bm{x}}),\epsilon_{i_{2}}\frac{\partial f}{\partial
x_{i_{2}}}({\bm{x}})\Big{)},\\\
&+\frac{1}{4}\sum_{j_{1}=1}^{n_{i}}\sum_{k_{1}=1}^{n_{i}}\sum_{j_{2}=1}^{n_{i}}\sum_{k_{2}=1}^{n_{i}}Cov\Big{(}\epsilon_{j_{1}}\epsilon_{k_{1}}\frac{\partial^{2}f}{\partial
x_{j_{1}}x_{k_{1}}}({\bm{x}}),\epsilon_{j_{2}}\epsilon_{k_{2}}\frac{\partial^{2}f}{\partial
x_{j_{2}}x_{k_{2}}}({\bm{x}})\Big{)}\\\
&+\sum_{i=1}^{n_{i}}\sum_{j=1}^{n_{i}}\sum_{k=1}^{n_{i}}Cov\Big{(}\epsilon_{i}\frac{\partial
f}{\partial
x_{i}}({\bm{x}}),\epsilon_{j}\epsilon_{k}\frac{\partial^{2}f}{\partial
x_{j}x_{k}}({\bm{x}})\Big{)},\\\
=&\sum_{i_{1}=1}^{n_{i}}\sum_{i_{2}=1}^{n_{i}}\frac{\partial f}{\partial
x_{i_{1}}}({\bm{x}})\frac{\partial f}{\partial
x_{i_{2}}}({\bm{x}})Cov\Big{(}\epsilon_{i_{1}},\epsilon_{i_{2}}\Big{)}\\\
&+\frac{1}{4}\sum_{j_{1}=1}^{n_{i}}\sum_{k_{1}=1}^{n_{i}}\sum_{j_{2}=1}^{n_{i}}\sum_{k_{2}=1}^{n_{i}}\frac{\partial^{2}f}{\partial
x_{j_{1}}x_{k_{1}}}({\bm{x}})\frac{\partial^{2}f}{\partial
x_{j_{2}}x_{k_{2}}}({\bm{x}})Cov\Big{(}\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}}\Big{)}\\\
&+\sum_{i=1}^{n_{i}}\sum_{j=1}^{n_{i}}\sum_{k=1}^{n_{i}}\frac{\partial
f}{\partial x_{i}}({\bm{x}})\frac{\partial^{2}f}{\partial
x_{j}x_{k}}({\bm{x}})Cov\Big{(}\epsilon_{i},\epsilon_{j}\epsilon_{k}\Big{)}.\\\
\end{split}$
In this expression, three quantities have to be assessed :
$Cov(\epsilon_{i_{1}},\epsilon_{i_{2}})$,
$Cov(\epsilon_{i},\epsilon_{j}\epsilon_{k})$ and
$Cov(\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}})$.
First, since $(\epsilon_{1},...,\epsilon_{n_{i}})$ are i.i.d.,
$Cov\Big{(}\epsilon_{i_{1}},\epsilon_{i_{2}}\Big{)}=\begin{dcases}Var(\epsilon_{i})=\epsilon\text{
if }i_{1}=i_{2}=i,\\\ 0\text{ otherwise.}\end{dcases}.$
To assess $Cov(\epsilon_{i},\epsilon_{j}\epsilon_{k})$, three cases have to be
considered.
* •
If $i=j=k$, because $\mathbb{E}[\epsilon_{i}^{3}]=0$,
$\begin{split}Cov(\epsilon_{i},\epsilon_{j}\epsilon_{k})&=Cov(\epsilon_{i},\epsilon_{i}^{2}),\\\
&=\mathbb{E}[\epsilon_{i}^{3}]-\mathbb{E}[\epsilon_{i}]\mathbb{E}[\epsilon_{i}^{2}],\\\
&=0.\end{split}$
* •
If $i=j$ or $i=k$ (we consider $i=k$, and the result holds for $i=j$ by
commutativity),
$\begin{split}Cov(\epsilon_{i},\epsilon_{j}\epsilon_{k})&=Cov(\epsilon_{i},\epsilon_{i}\epsilon_{j}),\\\
&=\mathbb{E}[\epsilon_{i}^{2}\epsilon_{j}]-\mathbb{E}[\epsilon_{i}]\mathbb{E}[\epsilon_{i}\epsilon_{j}],\\\
&=\mathbb{E}[\epsilon_{i}^{2}]\mathbb{E}[\epsilon_{j}],\\\ &=0.\end{split}$
* •
If $i\neq j$ and $i\neq k$, $\epsilon_{i}$ and $\epsilon_{j}\epsilon_{k}$ are
independent and so $Cov(\epsilon_{i},\epsilon_{j}\epsilon_{k})$ = 0.
Finally, to assess
$Cov(\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}})$, four
cases have to be considered:
* •
If $j_{1}=j_{2}=k_{1}=k_{2}=i$,
$\begin{split}Cov(\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}})&=Var(\epsilon_{i}^{2}),\\\
&=2\epsilon^{2}.\end{split}$
* •
If $j_{1}=k_{1}=i$ and $j_{2}=k_{2}=j$,
$Cov(\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}})=Cov(\epsilon_{i}^{2},\epsilon_{j}^{2})=0$
since $\epsilon_{i}^{2}$ and $\epsilon_{j}^{2}$ are independent.
* •
If $j_{1}=j_{2}=j$ and $k_{1}=k_{2}=k$,
$\begin{split}Cov(\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}})&=Var(\epsilon_{j}\epsilon_{k}),\\\
&=Var(\epsilon_{j})Var(\epsilon_{k}),\\\ &=\epsilon^{2}.\end{split}$
* •
If $j_{1}\neq k_{1},j_{2}$ and $k_{2}$,
$\begin{split}Cov(\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}})&=\mathbb{E}[\epsilon_{j_{1}}\epsilon_{k_{1}}\epsilon_{j_{2}}\epsilon_{k_{2}}]-\mathbb{E}[\epsilon_{j_{1}}\epsilon_{k_{1}}]\mathbb{E}[\epsilon_{j_{2}}\epsilon_{k_{2}}],\\\
&=\mathbb{E}[\epsilon_{j_{1}}]\mathbb{E}[\epsilon_{k_{1}}\epsilon_{j_{2}}\epsilon_{k_{2}}]-\mathbb{E}[\epsilon_{j_{1}}]\mathbb{E}[\epsilon_{k_{1}}]\mathbb{E}[\epsilon_{j_{2}}\epsilon_{k_{2}}],\\\
&=0.\end{split}$
All other possible cases can be assessed using the previous results,
commutativity and symmetry of $Cov$ operator. Hence,
$\begin{split}Var\Big{(}\nabla_{{\bm{x}}}f({\bm{x}})\cdot{\mathbf{e}}+\frac{1}{2}{\mathbf{e}}^{T}\cdot\mathbb{H}_{x}f({\bm{x}})\cdot{\mathbf{e}}\Big{)}=&\sum_{i_{1}=1}^{n_{i}}\sum_{i_{2}=1}^{n_{i}}\frac{\partial
f}{\partial x_{i_{1}}}({\bm{x}})\frac{\partial f}{\partial
x_{i_{2}}}({\bm{x}})Cov\Big{(}\epsilon_{i_{1}},\epsilon_{i_{2}}\Big{)}\\\
&+\frac{1}{4}\sum_{j_{1}=1}^{n_{i}}\sum_{k_{1}=1}^{n_{i}}\sum_{j_{2}=1}^{n_{i}}\sum_{k_{2}=1}^{n_{i}}\frac{\partial^{2}f}{\partial
x_{j_{1}}x_{k_{1}}}({\bm{x}})\frac{\partial^{2}f}{\partial
x_{j_{2}}x_{k_{2}}}({\bm{x}})Cov\Big{(}\epsilon_{j_{1}}\epsilon_{k_{1}},\epsilon_{j_{2}}\epsilon_{k_{2}}\Big{)},\\\
=&\sum_{i=1}^{n_{i}}\epsilon\frac{\partial f^{2}}{\partial
x_{i}}({\bm{x}})+\frac{1}{2}\sum_{j=1}^{n_{i}}\sum_{k=1}^{n_{i}}\epsilon^{2}\frac{\partial^{2}f^{2}}{\partial
x_{j}x_{k}}({\bm{x}}),\\\
=&\epsilon\|\nabla_{x}f({\bm{x}})\|^{2}_{F}+\frac{1}{2}\epsilon^{2}\|\mathbb{H}_{{\bm{x}}}f({\bm{x}})\|^{2}_{F},\\\
=&Df_{{\bm{\epsilon}}}^{2}({\bm{x}}).\end{split}$
And finally,
$\boxed{Var(f({\bm{x}}+{\mathbf{e}}))=Df_{{\bm{\epsilon}}}^{2}({\bm{x}})+\mathcal{O}(\|{\bm{\epsilon}}\|^{3}_{2})}$
(9)
If we consider $\widehat{Df^{2}}_{\epsilon}({\bm{x}})$ as defined in equation
(2), on section* 2.2 of the main document,
$\widehat{Df^{2}}_{\epsilon}({\bm{x}})\underset{k\rightarrow\infty}{\rightarrow}Var(f({\bm{x}}+{\bm{\epsilon}}))$
. Since
$Var(f({\bm{x}}+{\bm{\epsilon}}))=Df^{2}_{{\bm{\epsilon}}}({\bm{x}})+\mathcal{O}(\|{\bm{\epsilon}}\|^{3}_{2})$,
$\widehat{Df^{2}}_{\epsilon}({\bm{x}})$ is a biased estimator of
$Df^{2}_{{\bm{\epsilon}}}({\bm{x}})$, with bias
$\mathcal{O}(\|{\bm{\epsilon}}\|^{3}_{2})$. Hence, when $\epsilon\rightarrow
0$, $\widehat{Df^{2}}_{\epsilon}({\bm{x}})$ becomes an unbiased estimator of
$Df^{2}_{{\bm{\epsilon}}}({\bm{x}})$.
## Appendix B Appendix B: Hyperparameters spaces
The values chosen for the hyperparameters experiments are gathered in Table 8.
For Adam optimizer hyperparameters, we kept the default values of Keras
implementation. We chose these hyperparameters after simple grid searches.
Experiment | $m$ | $k$ | learning rate | batch size | epochs | optimizer | random seeds
---|---|---|---|---|---|---|---
double moon | 100 | 20 | $1\times 10^{-3}$ | 100 | 10000 | SGD | 50
Boston housing | 8 | 35 | $5\times 10^{-4}$ | 404 | 50000 | Adam | 10
Breast Cancer | 50 | 35 | $5\times 10^{-2}$ | 455 | 250000 | Adam | 10
MNIST | 40 | 20 | $1\times 10^{-3}$ | 25 | 25 | Adam | 40
Cifar10 | 40 | 20 | $1\times 10^{-3}$ | 25 | 25 | Adam | 50
RTE | 20 | 10 | $3\times 10^{-4}$ | 8 | 10000 | Adam | 50
STS-B | 30 | 30 | $3\times 10^{-4}$ | 8 | 10000 | Adam | 50
MRPC | 75 | 25 | $3\times 10^{-4}$ | 16 | 10000 | Adam | 50
Table 8: Hyperparameters values for experiments.
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|
# Using StyleGAN for Visual Interpretability of Deep Learning Models on
Medical Images
Kathryn Schutte, Olivier Moindrot, Paul Hérent, Jean-Baptiste Schiratti, Simon
Jégou
Owkin, Inc.
New York, NY, USA
Corresponding author<EMAIL_ADDRESS>
###### Abstract
As AI-based medical devices are becoming more common in imaging fields like
radiology and histology, interpretability of the underlying predictive models
is crucial to expand their use in clinical practice. Existing heatmap-based
interpretability methods such as GradCAM only highlight the location of
predictive features but do not explain how they contribute to the prediction.
In this paper, we propose a new interpretability method that can be used to
understand the predictions of any black-box model on images, by showing how
the input image would be modified in order to produce different predictions. A
StyleGAN is trained on medical images to provide a mapping between latent
vectors and images. Our method identifies the optimal direction in the latent
space to create a change in the model prediction. By shifting the latent
representation of an input image along this direction, we can produce a series
of new synthetic images with changed predictions. We validate our approach on
histology and radiology images, and demonstrate its ability to provide
meaningful explanations that are more informative than GradCAM heatmaps. Our
method reveals the patterns learned by the model, which allows clinicians to
build trust in the model’s predictions, discover new biomarkers and eventually
reveal potential biases.
## 1 Introduction
As of September 2020, the FDA had approved 64 AI-based medical devices
(Benjamens et al.,, 2020), and for the first time the Centers for Medicare &
Medicaid Services (CMS) approved the reimbursement of deep-learning powered
stroke detector for brain CT scans (Viz.ai,, 2020). The advances of deep
learning in computer vision (Krizhevsky et al.,, 2012) are especially
promising in medical imaging fields such as radiology (Ardila et al.,, 2019),
histology (Coudray et al.,, 2018), dermatology (Esteva et al.,, 2017) or
ophthalmology (Gulshan et al.,, 2016).
While many deep learning techniques may provide state-of-the-art predictive
performance, _interpretable_ deep learning models are necessary for regulatory
approval, as their ability to explain their predictions can reveal potential
biases and failure modes, as seen in the case of (Oakden-Rayner,, 2017).
Additionally, interpretable models also provide new opportunities for
biomedical investigation, as evidenced in (Courtiol et al.,, 2019). Finally,
such models are able to make inroads with medical experts, as their
explainability helps build confidence in their utility (Holzinger et al.,,
2019). As illustrated by the COVID-19 crisis (Bai et al.,, 2020; Li et al.,,
2020; Wang et al.,, 2020), the go-to method for model interpretation in the
medical imaging field is GradCAM (Selvaraju et al.,, 2017), which produces a
coarse heatmap based on gradient intensity to identify which areas of the
input image are responsible for the prediction. However, these heatmaps only
highlight the location of predictive features but do not explain how they
contribute to the prediction. In an image where information is diffuse, the
heatmap cannot highlight any specific region so GradCAM is not sufficient to
interpret the model predictions.
In this paper, we propose a new interpretability method that generates small
synthetic transformations of the original image that would lead to different
model predictions. We train a generative model called StyleGAN (Karras et
al.,, 2019, 2020) and find the minimal modification in the latent space that
changes the model prediction, which ensures that generated images remain as
close as possible to the original image. Seah et al., (2019) explore a similar
idea by using an older GAN algorithm to create heatmaps highlighting features
of congestive heart failure, but their method cannot be applied to any black-
box model. Fetty et al., (2020) manipulate three attributes of the StyleGAN
latent space in order to enlarge datasets with synthetic images. We validate
our interpretability method on two different imaging modalities and
demonstrate its ability to provide meaningful explanations of the predictions,
and its potential to discover new biomarkers.
## 2 Method
We propose to create StyleGAN-generated visualizations that explain the
predictions of a deep neural network in an interpretable manner. Let $f$ be a
classifier (e.g. a fully convolutional neural network) trained on a dataset
$\mathcal{D}=\left(\mathbf{x}_{i},y_{i}\right)\in\mathcal{X}\times\mathcal{Y}$,
where $\mathcal{X}$ denotes a set of 2D images and $\mathcal{Y}$ a finite set
of labels. Our method consists of three steps. First, the images in
$\mathcal{X}$ are used to train a StyleGAN2 (Karras et al.,, 2019), which is
an improved GAN whose _generator_
$G\,:\,\mathcal{W}\,\rightarrow\,\mathcal{X}$ has a linearly disentangled
intermediate _latent space_ $\mathcal{W}\subset\mathbb{R}^{512}$. The
generator $G$ is used to generate a set of synthetic images
$\left(G\left(\mathbf{w}_{i}\right)\right)$, where the $\mathbf{w}_{i}$ are
sampled in the latent space $\mathcal{W}$. Then, we train (using a Mean
Squared Error loss) a ResNet50 (He et al.,, 2016) _encoder_
$E\,:\,\mathcal{X}\,\rightarrow\,\mathcal{W}$ on the synthetic dataset
$\left(\mathbf{w}_{i},G\left(\mathbf{w}_{i}\right)\right)$ to retrieve the
latent representation $\mathbf{w}_{i}$ from a generated image
$G\left(\mathbf{w}_{i}\right)$. Finally, a logistic regression classifier
$\tilde{f}\left(\mathbf{w_{i}}\right)=\sigma\left(\boldsymbol{\alpha}^{\top}\mathbf{w_{i}}+\boldsymbol{\beta}\right)$
is trained on the latent space $\mathcal{W}$ to predict the estimated labels
$\tilde{y}_{i}=f(G(\mathbf{w_{i}}))$ associated to each latent vector
$\mathbf{w}_{i}\in\mathcal{W}$. Given a new input image
$\mathbf{x}\in\mathcal{X}$, our method translates the latent vector
$\mathbf{w}=E\left(x\right)$ along the direction $\boldsymbol{\alpha}$. We can
then create new images from the latent representation via
$G\left(\mathbf{w}+\lambda\boldsymbol{\alpha}\right)$ associated with a lower
or a higher prediction depending on the value of $\lambda\in\mathbb{R}$.
Figure 1: Our method applied to knee osteoarthritis severity prediction on an
X-ray image. The input image is gradually modified to increase the
osteoarthritis severity. The GradCAM heatmap is computed on the input image to
compare both interpretability methods. X-rays of the patient’s later visits
are displayed to visually assess the clinical relevance of our method.
## 3 Experiments
### 3.1 Knee osteoarthritis severity prediction on X-ray images
We first demonstrate our method by explaining the predictions of an
osteoarthritis severity predictor on X-ray images. The dataset on which the
predictor has been trained consists of 20,123 X-rays of patients suffering
from knee osteoarthritis collected by the Osteoarthritis Initiative (OAI)
(Nevitt et al.,, 2006). Each patient has one to eight 12-month follow-up
X-rays, as well as associated clinical data, including the Kellgren and
Lawrence (KL) grade (Kohn et al.,, 2016). The KL grade describes a degree of
osteoarthritis severity and ranges from 0 to 4: grades 0 and 1 mean no or
doubtful osteoarthritis, while grades 2 to 4 mean mild to severe
osteoarthritis.
The image classifier $f$ is a ResNet50 trained on the multi-class prediction
task. To fit this multi-class setting to our method, we transform it to a
binary classification task by pooling grades 0 and 1 versus grades 2 to 4. The
predictor $f$ obtains 89% test AUC on this binary task, while $\tilde{f}$
obtains 80% test AUC on the latent space. Three radiologists evaluated the
quality of the StyleGAN generator with a Turing test. They reach 58% accuracy
on average, showing that synthetic and real X-rays are almost
indistinguishable.
In Figure 1, our interpretability method is applied to a real X-ray image. The
GradCAM heatmap provides topographical information by showing that the
osteoarthritis features are located in lateral femorotibial space. Our method
provides more than topographical information by showing the gradual emergence
of the different osteoarthritis features as the KL grade increases, such as
the joint space narrowing (red arrow) and osteophytes (blue arrow). By
comparing the synthetic evolution of the image to the real evolution of the
patient at 12, 24 and 72 months after baseline, we observe that the direction
found in the latent space corresponds to a biologically plausible
osteoarthritis progression.
### 3.2 Tumor detection on histology images of metastatic lymph nodes
We apply the same method to histology images, to explain the predictions of a
metastasis detector on Camelyon16 (Bejnordi et al.,, 2017). The dataset
contains 224,166 patch images from breast cancer lymph node whole-slide
images, each with a binary label indicating the presence of tumor cells.
The image classifier $f$ is a ResNet50 trained on this dataset, obtaining 92%
test AUC, while the latent predictor $\tilde{f}$ reaches 95% test AUC. Figure
2 shows our interpretability method on two images: patch B contains tumoral
cells while patch A does not. The GradCAM heatmaps are not relevant here
because the informative features are spread over the entire image. On the
contrary, our approach reveals clinically relevant features. On patch A, it
shows the appearance of tumor cells (blue arrow) and the disappearance of
lymphocytes (red arrow) as the tumor probability increases, and inversely on
patch B.
We can see that the encoder-decoder model is not able to perfectly reconstruct
histology images, as opposed to knee X-rays. A possible explanation is that
the StyleGAN model does not generate images that are under-represented in the
training set. This issue is highlighted in this particular use-case as there
is more variability in the histology images than in the knee X-ray images.
Recently, Yu et al., (2020) propose to overcome this data coverage challenge
by harmonizing adversarial training with reconstructive generation.
Figure 2: Our method applied to tumor probability prediction on two histology
tiles of metastatic lymph nodes. The input image is gradually modified to
increase (on patch A) or decrease (on patch B) the tumor probability. The
GradCAM heatmap is computed on the input images to compare both
interpretability methods.
## 4 Conclusion
In this study we explored the potential of StyleGANs to explain the
predictions of black-box models on medical images. Although heatmap-based
methods dominate the interpretability field, they only highlight the
localization of predictive features in the image. Our method provides an
intuitive way for medical researchers to understand _where_ are located the
predictive features in the image and _how_ they impact the prediction by
showing modified views of the input image that would produce different
predictions. This method shows how the model learned to solve the prediction
task which allows clinicians to build trust in the model’s predictions,
discover new biomarkers and eventually reveal potential biases. In both
experiments, our method proved that the models learned clinically relevant
features.
#### Acknowledgments
We thank Eric W. Tramel for his valuable feedback on the manuscript. We thank
the three radiologists Eric Pessis, François Legoux and Thibaut Emorine for
their participation in the Turing Test.
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|
# Performance analysis of greedy algorithms for minimising a Maximum Mean
Discrepancy
Luc Pronzato
Université Côte d’Azur, CNRS, Laboratoire I3S
Bât. Euclide, Les Algorithmes, 2000 route des lucioles,
06900 Sophia Antipolis, France
<EMAIL_ADDRESS>
###### Abstract
We analyse the performance of several iterative algorithms for the
quantisation of a probability measure $\mu$, based on the minimisation of a
Maximum Mean Discrepancy (MMD). Our analysis includes kernel herding, greedy
MMD minimisation and Sequential Bayesian Quadrature (SBQ). We show that the
finite-sample-size approximation error, measured by the MMD, decreases as
$1/n$ for SBQ and also for kernel herding and greedy MMD minimisation when
using a suitable step-size sequence. The upper bound on the approximation
error is slightly better for SBQ, but the other methods are significantly
faster, with a computational cost that increases only linearly with the number
of points selected. This is illustrated by two numerical examples, with the
target measure $\mu$ being uniform (a space-filling design application) and
with $\mu$ a Gaussian mixture. They suggest that the bounds derived in the
paper are overly pessimistic, in particular for SBQ. The sources of this
pessimism are identified but seem difficult to counter.
keywords Maximum Mean Discrepancy; quantisation; greedy algorithm; sequential
Bayesian quadrature; kernel herding; space-filling design; computer
experiments
## 1 Introduction and motivation
##### Background.
Quantisation of a probability measure $\mu$ is a basic task in many fields,
such as probabilistic integration (Briol et al., , 2019), MCMC computation
(Joseph et al., 2015a, ; Joseph et al., , 2019) or space-filling design in
computer experiments (Joseph et al., 2015b, ; Mak and Joseph, , 2017, 2018;
Pronzato and Zhigljavsky, , 2020), and minimisation of the Maximum Mean
Discrepancy (MMD) defined by a kernel $K$ is a powerful tool for this
task111We do not consider quantisation methods based on Voronoi partitions,
for which one can refer in particular to Graf and Luschgy, (2000).. In
particular, it easily allows iterative constructions that can be stopped when
the discrete approximation obtained is deemed sufficient, a situation where
the number of support points is not fixed in advance.
##### Claims and hint of the contents.
We derive finite-sample-size errors bounds for iterative methods to quantise a
probability measure by minimising the MMD for a given kernel. The methods
considered include gradient-type algorithms (kernel herding), greedy one-step-
ahead minimisation, and Sequential Bayesian Quadrature (SBQ) that sets optimal
weights on the current support at each iteration. Two variants of SBQ are
considered, with and without the constraint that the weights sum to one (the
bound for the unconstrained version is markedly pessimistic but our analysis
reveals a connection with kernel herding and gives some insight for the reason
of this pessimism). We consider the practical situation where the candidate
set is finite; it may correspond in particular to points independently sampled
with $\mu$, with the possibility to use a different set at every iteration
(see Section 6 and Appendix C). The context of a finite candidate set is the
most widely used in practical situations. It allows us to derive simple proofs
that only use (finite-dimensional) linear algebra, but our results can be
extended to the infinite-dimensional (Hilbert space) situation, where the new
support point selected at each iteration is searched within a continuous set;
see, e.g., Chen et al., (2018); Teymur et al., (2021). We show that the
error is $\mathcal{O}(n^{-1})$ for SBQ and for algorithms that use a suitable
step-size sequence and construct nonuniform discrete measures (with a slightly
better constant for SBQ). We show that it is also $\mathcal{O}(n^{-1})$ for
the construction of uniform (empirical) measures provided that the measure
with total mass one minimising the MMD over the candidate set is a probability
measure. We show that the complexity of gradient and greedy one-step-ahead
methods grows linearly with $n$, whereas it grows quadratically for SBQ. Two
variants of kernel herding are considered, with similar performance to SBQ but
slightly lighter calculations.
##### Paper organisation.
Section 2 recalls the background on MMD and Bayesian quadrature. It defines
the notation and introduces the methods that are considered in the rest of the
paper. The performance of kernel herding is analysed in Section 3. The results
presented in Sections 3.1 and 3.2 are not new, but the analysis of this basic
gradient-type algorithm is central to the investigation of the convergence
rate for the other methods, more sophisticated, that we consider in Sections
3.3 (variants of kernel herding), 4 (greedy MMD minimisation) and 5 (SBQ).
Section 6 extends the results of previous sections to the case where the
candidate set corresponds to points independently sampled with $\mu$. Two
illustrative examples are presented in Section 7, one with $\mu$ uniform
(space-filling design), the other with $\mu$ a Gaussian mixture. Section 8
concludes briefly.
## 2 Maximum Mean Discrepancy and Bayesian quadrature
### 2.1 Maximum Mean Discrepancy (MMD)
Let ${\mathscr{X}}$ be a measurable set, equipped with a probability measure
$\mu$. For instance, for application to space-filling design for computer
experiments, ${\mathscr{X}}$ is typically a compact subset of $\mathds{R}^{d}$
and $\mu$ is proportional to the Lebesgue measure on ${\mathscr{X}}$. Let $K$
be a symmetric strictly positive definite (s.p.d.) kernel defined on
${\mathscr{X}}\times{\mathscr{X}}$, uniformly bounded on ${\mathscr{X}}$; that
is,
$\displaystyle K(\mathbf{x},\mathbf{x})\leq\overline{K}<+\infty\,,\ \mbox{ for
all }\mathbf{x}\in{\mathscr{X}}\,,$ (1)
and for any $n\in\mathds{N}$ and any $\mathbf{x}_{1},\ldots,\mathbf{x}_{n}$ in
${\mathscr{X}}$, the $n\times n$ matrix $\mathbf{K}_{n}$ with element $i,j$
equal to $K(\mathbf{x}_{i},\mathbf{x}_{j})$ is p.d. and s.p.d. when the
$\mathbf{x}_{i}$ are all pairwise different. Note that $K$ being s.p.d.
implies that $K^{2}(\mathbf{x},\mathbf{x}^{\prime})\leq
K(\mathbf{x},\mathbf{x})K(\mathbf{x}^{\prime},\mathbf{x}^{\prime})$ for all
$\mathbf{x}$ and $\mathbf{x}^{\prime}$ in ${\mathscr{X}}$ with the inequality
being strict when $\mathbf{x}\neq\mathbf{x}^{\prime}$. Moreover, (1) implies
$\tau_{\gamma}(\mu)=\int_{\mathscr{X}}K^{\gamma}(\mathbf{x},\mathbf{x})\,\mathrm{d}\mu(\mathbf{x})\leq\overline{K}^{\gamma}<+\infty\
\mbox{ for any }\gamma\geq 0$. $K$ defines a Reproducing Kernel Hilbert Space
(RKHS) $\mathcal{H}_{K}$, and we respectively denote by
$\langle\cdot,\cdot\rangle_{K}$ and $\|\cdot\|_{\mathcal{H}_{K}}$ the scalar
product and norm in $\mathcal{H}_{K}$. We do not assume that $\mathcal{H}_{K}$
is finite-dimensional. We say that $K$ is positive ($K\geq 0$) when
$K(\mathbf{x},\mathbf{x}^{\prime})\geq 0$ for all $\mathbf{x}$ and
$\mathbf{x}^{\prime}$ in ${\mathscr{X}}$.
We denote by ${\mathscr{M}}({\mathscr{X}})$ the set of finite signed measures
on ${\mathscr{X}}$, by ${\mathscr{M}}_{[1]}({\mathscr{X}})$ the set of signed
measures with total mass $1$, and by ${\mathscr{M}}^{+}_{[1]}({\mathscr{X}})$
the set of probability measures on ${\mathscr{X}}$ (with thus
$\mu\in{\mathscr{M}}^{+}_{[1]}({\mathscr{X}})$). The reproducing property
implies that, for any $\nu\in{\mathscr{M}}({\mathscr{X}})$, the _energy_ of
$\nu$, defined by
${\mathscr{E}}_{K}(\nu)=\int_{{\mathscr{X}}^{2}}K(\mathbf{x},\mathbf{x}^{\prime})\,\mathrm{d}\nu(\mathbf{x})\,\mathrm{d}\nu(\mathbf{x}^{\prime})$,
satisfies
$\displaystyle{\mathscr{E}}_{K}(\nu)=\int_{{\mathscr{X}}^{2}}\langle
K(\mathbf{x},\cdot),K(\mathbf{x}^{\prime},\cdot)\rangle_{K}\,\mathrm{d}\nu(\mathbf{x})\,\mathrm{d}\nu(\mathbf{x}^{\prime})$
$\displaystyle\leq$
$\displaystyle\int_{{\mathscr{X}}^{2}}\|K(\mathbf{x},\cdot)\|_{\mathcal{H}_{K}}\,\|K(\mathbf{x}^{\prime},\cdot)\|_{\mathcal{H}_{K}}\,\mathrm{d}\nu(\mathbf{x})\,\mathrm{d}\nu(\mathbf{x}^{\prime})$
$\displaystyle=$
$\displaystyle\left[\int_{\mathscr{X}}K^{1/2}(\mathbf{x},\mathbf{x})\,\mathrm{d}\nu(\mathbf{x})\right]^{2}=\tau_{1/2}^{2}(\nu)<+\infty\,.$
For any $\nu\in{\mathscr{M}}({\mathscr{X}})$ and any
$\mathbf{x}\in{\mathscr{X}}$, we denote by
$\displaystyle
P_{K,\nu}(\mathbf{x})=\int_{\mathscr{X}}K(\mathbf{x},\mathbf{x}^{\prime})\,\mathrm{d}\nu(\mathbf{x}^{\prime})$
the _potential_ of $\nu$ at $\mathbf{x}$; $P_{K,\nu}(\cdot)$ is also called
the _kernel imbedding_ of $\nu$ into $\mathcal{H}_{K}$.
For $\mu$ and $\nu$ in ${\mathscr{M}}^{+}_{[1]}({\mathscr{X}})$, any
$f\in\mathcal{H}_{K}$ satisfies the following (Koksma-Hlawka type) inequality:
$\left|\int_{\mathscr{X}}f(\mathbf{x})\,\mathrm{d}\nu(\mathbf{x})-\int_{\mathscr{X}}f(\mathbf{x})\,\mathrm{d}\mu(\mathbf{x})\right|=\left|\langle
f,P_{K,\mu}-P_{K,\nu}\rangle_{K}\right|\leq\|f\|_{\mathcal{H}_{K}}\,\mathsf{MMD}_{K}(\mu,\nu)$,
where
$\displaystyle\mathsf{MMD}_{K}(\mu,\nu)$ $\displaystyle=$
$\displaystyle\sup_{\|f\|_{\mathcal{H}_{K}}=1}\left|\int_{\mathscr{X}}f(\mathbf{x})\,\mathrm{d}\nu(\mathbf{x})-\int_{\mathscr{X}}f(\mathbf{x})\,\mathrm{d}\mu(\mathbf{x})\right|=\|P_{K,\nu}-P_{K,\mu}\|_{\mathcal{H}_{K}}$
(2) $\displaystyle=$
$\displaystyle{\mathscr{E}}_{K}^{1/2}(\nu-\mu)=\left[\int_{{\mathscr{X}}^{2}}K(\mathbf{x},\mathbf{x}^{\prime})\,\mathrm{d}(\nu-\mu)(\mathbf{x})\,\mathrm{d}(\nu-\mu)(\mathbf{x}^{\prime})\right]^{1/2}$
$\displaystyle=$
$\displaystyle\left[{\mathscr{E}}_{K}(\mu)+{\mathscr{E}}_{K}(\nu)-2\,\int_{\mathscr{X}}P_{K,\mu}(\mathbf{x})\,\mathrm{d}\nu(\mathbf{x})\right]^{1/2}$
is called the _Maximum-Mean-Discrepancy_ (MMD) between $\nu$ and $\mu$; see
Sejdinovic et al., (2013, Def. 10). $\mathsf{MMD}_{K}(\mu,\nu)$ defines an
integral pseudometric between probability distributions and a pseudometric
between kernel imbeddings. It defines a metric on
${\mathscr{M}}^{+}_{[1]}({\mathscr{X}})$ when $K$ is characteristic222Since
$K$ is uniformly bounded, this is equivalent to the condition that $K$ be
Conditionally Integrally Strictly Positive Definite (CISPD), that is,
${\mathscr{E}}_{K}(\nu)>0$ for all nonzero signed measure
$\nu\in{\mathscr{M}}_{[0]}({\mathscr{X}})$; see Sriperumbudur et al., (2010,
Def. 6 and Lemma 8); see also Pronzato and Zhigljavsky, (2020) for a
comprehensive survey including the case of singular kernels., which we assume
in the following. This implies in particular that
$\mathsf{MMD}_{K}(\mu,\nu)>0$ for any
$\nu\in{\mathscr{M}}_{[1]}({\mathscr{X}})$, $\nu\neq\mu$.
For a collection $\mathbf{X}_{n}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\\}$
of $n$ points in ${\mathscr{X}}$, called $n$-point design, we denote by
$\xi_{n,e}=(1/n)\sum_{i=1}^{n}\delta_{\mathbf{x}_{i}}$ the associated
empirical measure, with $\delta_{\mathbf{x}}$ the Dirac delta measure at
$\mathbf{x}$. For
$\mathbf{w}_{n}=(\\{\mathbf{w}_{n}\\}_{1},\ldots,\\{\mathbf{w}_{n}\\}_{n})^{\top}\in\mathds{R}^{n}$
a vector of $n$ weights, we denote by $\xi_{n}=\xi(\mathbf{w}_{n})$ the signed
measure
$\displaystyle\xi(\mathbf{w}_{n})=\sum_{i=1}^{n}\\{\mathbf{w}_{n}\\}_{i}\,\delta_{\mathbf{x}_{i}}$
(3)
(so that $\xi_{n,e}=\xi(\mathbf{1}_{n}/n)$, with $\mathbf{1}_{n}$ the
$n$-dimensional vector with all components equal to 1). An important area of
application for MMD minimisation is space-filling design, where the objective
is to build evenly distributed designs on a compact ${\mathscr{X}}$; see, for
example, Pronzato and Müller, (2012); Pronzato, (2017). Minimising
$\mathsf{MMD}_{K}(\mu,\xi_{n,e})$ with $\mu$ uniform over ${\mathscr{X}}$ is
then an effective approach to achieve this goal. One may also minimise
$\mathsf{MMD}_{K}(\mu,\xi(\mathbf{w}_{n}))$ with respect to $\mathbf{X}_{n}$
and $\mathbf{w}_{n}$, and the designs obtained differ depending on the chosen
kernel $K$, the constraints set on $\mathbf{w}_{n}$ and on the optimisation
method that is used. In this paper, we focuss our attention on the
construction of extensive point sequences
$\mathbf{X}_{n}=[\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n}]$, such
that $\mathbf{X}_{k+1}=[\mathbf{X}_{k},\mathbf{x}_{k+1}]$ for all $k$, with
the property that $\mathbf{X}_{n}$ is the support of a measure $\xi_{n}$ which
approximates $\mu$ well in the sense of $\mathsf{MMD}_{K}(\mu,\xi_{n})$.
Analytic expressions for the quantities ${\mathscr{E}}_{K}(\mu)$ and
$P_{K,\mu}(\cdot)$ that appear in (2) are available for particular measures
and particular kernels, see Table 1 of Briol et al., (2019). This includes
the case when $\mu$ is uniform on ${\mathscr{X}}=[0,1]^{d}$ and $K$ is
separable, see for example Table 3.1 of (Pronzato and Zhigljavsky, , 2020),
and separable kernels $K$ based on variants of Brownian motion covariance
yield $L_{2}$ discrepancies (symmetric, centred, wrap-around and so on); see
Hickernell, (1998), Fang et al., (2006, Chap. 3). ${\mathscr{E}}_{K}(\mu)$
and $P_{K,\mu}(\cdot)$ are not available when $\mu$ is a posterior
distribution with unknown normalising constant; in that case, Joseph et al.,
2015a ; Joseph et al., (2019) suggest to construct minimum-energy designs
that minimise ${\mathscr{E}}_{K}(\xi_{n,e})$ for a particular kernel $K$.
Another way is to minimise a kernel Stein discrepancy, that is, to minimise
MMD for the image $K^{\prime}$ of a kernel $K$ under a Stein operator, so that
${\mathscr{E}}_{K^{\prime}}(\mu)=0$ and $P_{K^{\prime},\mu}(\mathbf{x})=0$ for
any $\mathbf{x}$; see Chen et al., (2018); Detommaso et al., (2018); Gorham
and MacKey, (2017); Liu and Wang, (2016); Oates et al., (2017). Throughout
the paper we consider the general framework where $\mathcal{H}_{K}$ is an
infinite-dimensional RKHS and assume that ${\mathscr{E}}_{K}(\mu)$ and
$P_{K,\mu}(\mathbf{x})$ can be easily computed for any
$\mathbf{x}\in{\mathscr{X}}$ (Monte-Carlo methods can always be used as a last
resort).
### 2.2 MMD and optimal weights for discrete measures
For a given design $\mathbf{X}_{n}$, $\mathsf{MMD}_{K}(\mu,\xi_{n})$ is
quadratic in $\mathbf{w}_{n}$, and the optimal weights are easily obtained.
Indeed, (2) gives
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})={\mathscr{E}}_{K}(\xi_{n}-\mu)$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}\\{\mathbf{w}_{n}\\}_{i}\\{\mathbf{w}_{n}\\}_{j}K(\mathbf{x}_{i},\mathbf{x}_{j})-2\,\sum_{i=1}^{n}\\{\mathbf{w}_{n}\\}_{i}P_{K,\mu}(\mathbf{x}_{i})+{\mathscr{E}}_{K}(\mu)\,,$
(4) $\displaystyle=$
$\displaystyle\mathbf{w}_{n}^{\top}\mathbf{K}_{n}\mathbf{w}_{n}-2\,\mathbf{w}_{n}^{\top}\mathbf{p}_{n}(\mu)+{\mathscr{E}}_{K}(\mu)\,,$
where
$\mathbf{p}_{n}(\mu)=[P_{K,\mu}(\mathbf{x}_{1}),\ldots,P_{K,\mu}(\mathbf{x}_{n})]^{\top}$
(alternative expressions for $\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ are given in
Appendix A). Therefore, $\mathbf{w}_{n}^{*}$ that minimises
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ under the constraints
$\\{\mathbf{w}_{n}\\}_{i}\geq 0$ and $\mathbf{1}_{n}^{\top}\mathbf{w}_{n}=1$
is solution of a Quadratic Programming (QP) problem. We assume that the
$\mathbf{x}_{i}$ in $\mathbf{X}_{n}$ are pairwise different, so that
$\mathbf{K}_{n}$ has full rank. Releasing the positivity constraints,
$\widehat{\mathbf{w}}_{n}$ that minimises $\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$
with $\mathbf{1}_{n}^{\top}\mathbf{w}_{n}=1$ is obtained explicitly as
$\displaystyle\widehat{\mathbf{w}}_{n}=\left(\mathbf{K}_{n}^{-1}-\frac{\mathbf{K}_{n}^{-1}\mathbf{1}_{n}\mathbf{1}_{n}^{\top}\mathbf{K}_{n}^{-1}}{\mathbf{1}_{n}^{\top}\mathbf{K}_{n}^{-1}\mathbf{1}_{n}}\right)\mathbf{p}_{n}(\mu)+\frac{\mathbf{K}_{n}^{-1}\mathbf{1}_{n}}{\mathbf{1}_{n}^{\top}\mathbf{K}_{n}^{-1}\mathbf{1}_{n}}$
(5)
(with $\mathbf{w}_{n}^{*}=\widehat{\mathbf{w}}_{n}$ when all components of
$\widehat{\mathbf{w}}_{n}$ are nonnegative). Also, the unconstrained weights
that minimise $\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ are given by
$\displaystyle\widetilde{\mathbf{w}}_{n}=\mathbf{K}_{n}^{-1}\mathbf{p}_{n}(\mu)\,.$
(6)
Throughout the paper, for any measure $\xi_{n}$ supported on $\mathbf{X}_{n}$,
we denote by $\xi_{n}^{*}$, $\widehat{\xi}_{n}$ and $\widetilde{\xi}_{n}$ the
measures with the same support and respective weights $\mathbf{w}_{n}^{*}$,
$\widehat{\mathbf{w}}_{n}$ and $\widetilde{\mathbf{w}}_{n}$, so that
$\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{n})\leq\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{n})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{n}^{*})$
(and
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{n}^{*})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$
if $\xi_{n}\in{\mathscr{M}}_{[1]}^{+}(\mathbf{X}_{n})$).
### 2.3 Incremental MMD minimisation
We consider three families of incremental constructions.
#### 2.3.1 Sequential Bayesian Quadrature (SBQ)
The construction of a design $\mathbf{X}_{n}$ that minimises
$\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{n})$ or
$\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{n})$ is called Bayesian quadrature
(BQ); it can be Sequential (SBQ), see Briol et al., (2015), and we consider
two versions of SBQ. Bounds on their finite-sample-size error are given in
Section 5. Note that
$\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k})=\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k+1})$
(respectively,
$\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k})=\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k+1})$)
when $\widetilde{\xi}_{k}$ and $\widetilde{\xi}_{k+1}$ (respectively,
$\widehat{\xi}_{k}$ and $\widehat{\xi}_{k+1}$) have the same support, so that
SBQ always selects new points whenever possible (i.e., until all eligible
points are exhausted). We may thus assume that the $\mathbf{x}_{i}$ are all
pairwise different, $i=1,\ldots,k$, and that $\mathbf{K}_{k}$ has full rank
for all $k$.
##### (i) Greedy minimisation of
$\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k})$.
The equations (4) and (6) give
$\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k})={\mathscr{E}}_{K}(\mu)-\mathbf{p}_{k}^{\top}(\mu)\mathbf{K}_{k}^{-1}\mathbf{p}_{k}(\mu)$.
We have
$\displaystyle\mathbf{K}_{n+1}=\left[\begin{array}[]{cc}\mathbf{K}_{n}&\mathbf{k}_{n}(\mathbf{x}_{n+1})\\\
\mathbf{k}_{n}^{\top}(\mathbf{x}_{n+1})&K(\mathbf{x}_{n+1},\mathbf{x}_{n+1})\\\
\end{array}\right]\,,$
where
$\mathbf{k}_{n}(\mathbf{x})=[K(\mathbf{x}_{1},\mathbf{x}),\ldots,K(\mathbf{x}_{n},\mathbf{x})]^{\top}$,
$\mathbf{x}\in{\mathscr{X}}$. The calculation of its inverse by
$\displaystyle\mathbf{K}_{n+1}^{-1}=\left(\begin{array}[]{cc}\mathbf{K}_{n}^{-1}+\beta_{n+1}\,\mathbf{u}_{n+1}\mathbf{u}_{n+1}^{\top}&-\beta_{n+1}\,\mathbf{u}_{n+1}\\\
-\beta_{n+1}\,\mathbf{u}_{n+1}^{\top}&\beta_{n+1}\\\ \end{array}\right)\,,$
(10)
with $\mathbf{u}_{n+1}=\mathbf{K}_{n}^{-1}\mathbf{k}_{n}(\mathbf{x}_{n+1})$
and
$\beta_{n+1}=[K(\mathbf{x}_{n+1},\mathbf{x}_{n+1})-\mathbf{k}_{n}^{\top}(\mathbf{x}_{n+1})\mathbf{u}_{n+1}]^{-1}$,
will be used several times for incremental constructions and gives here
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k+1})=\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k})-\frac{\left[\mathbf{p}_{k}^{\top}(\mu)\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x}_{k+1})-P_{K,\mu}(\mathbf{x}_{k+1})\right]^{2}}{K(\mathbf{x}_{k+1},\mathbf{x}_{k+1})-\mathbf{k}_{k}^{\top}(\mathbf{x}_{k+1})\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x}_{k+1})}\,.$
Since $\widetilde{\mathbf{w}}_{k}$ satisfies (6), we get
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k+1})=\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k})-\frac{\left[P_{K,\widetilde{\xi}_{k}}(\mathbf{x}_{k+1})-P_{K,\mu}(\mathbf{x}_{k+1})\right]^{2}}{\min_{\mathbf{w}\in\mathds{R}^{k}}\|K(\mathbf{x}_{k+1},\cdot)-\mathbf{w}^{\top}\mathbf{k}_{k}(\cdot)\|_{\mathcal{H}_{K}}^{2}}\,.$
(11)
This corresponds to the “standard” version of SBQ, which uses general signed
measures $\widetilde{\xi}_{k}$ in ${\mathscr{M}}({\mathscr{X}})$: it selects
$\mathbf{x}_{1}\in\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}}P_{K,\mu}^{2}(\mathbf{x})/K(\mathbf{x},\mathbf{x})$
and then
$\displaystyle\mathbf{x}_{k+1}\in\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}}\frac{\left[P_{K,\widetilde{\xi}_{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})\right]^{2}}{K(\mathbf{x},\mathbf{x})-\mathbf{k}_{k}^{\top}(\mathbf{x})\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x})}\,,\
k\geq 1\,.$ (12)
##### (ii) Greedy minimisation of
$\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k})$.
The equations (4) and (5) give
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k})$ $\displaystyle=$
$\displaystyle{\mathscr{E}}_{K}(\mu)-\mathbf{p}_{k}^{\top}(\mu)\mathbf{K}_{k}^{-1}\mathbf{p}_{k}(\mu)+\frac{(1-\mathbf{p}_{k}^{\top}(\mu)\mathbf{K}_{k}^{-1}\mathbf{1}_{k})^{2}}{\mathbf{1}_{k}^{\top}\mathbf{K}_{k}^{-1}\mathbf{1}_{k}}\,,$
but a simpler expression can be obtained through the introduction of the
reduced kernel $K_{\mu}$ defined by
$\displaystyle
K_{\mu}(\mathbf{x},\mathbf{x}^{\prime})=K(\mathbf{x},\mathbf{x}^{\prime})-P_{K,\mu}(\mathbf{x})-P_{K,\mu}(\mathbf{x}^{\prime})+{\mathscr{E}}_{K}(\mu)\,,\
\mathbf{x},\mathbf{x}^{\prime}\in{\mathscr{X}}\,.$ (13)
Let
$\\{{\mathbf{K}_{\mu}}_{k}\\}_{i,j}=K_{\mu}(\mathbf{x}_{i},\mathbf{x}_{j})$,
$i,j=1,\ldots,k$, so that
${\mathbf{K}_{\mu}}_{k}=\mathbf{K}_{k}-\mathbf{p}_{k}(\mu)\mathbf{1}_{k}^{\top}-\mathbf{1}_{k}\mathbf{p}_{k}^{\top}(\mu)+{\mathscr{E}}_{K}(\mu)\,\mathbf{1}_{k}\mathbf{1}_{k}^{\top}$
and, for any measure $\xi_{k}$ with total mass one,
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})=\mathbf{w}_{k}^{\top}{\mathbf{K}_{\mu}}_{k}\mathbf{w}_{k}\,.$
(14)
As $K$ is characteristic and $\mathbf{K}_{k}$ has full rank,
${\mathbf{K}_{\mu}}_{k}$ is invertible when $\mu$ is not fully supported on
$\mathbf{X}_{k}$. Indeed, let $\mathbf{u}_{k}$ be an eigenvector of
${\mathbf{K}_{\mu}}_{k}$. If $a=\mathbf{u}_{k}^{\top}\mathbf{1}_{k}\neq 0$,
then the measure $\xi_{k}$ with weights $\mathbf{w}_{k}=\mathbf{u}_{k}/a$ has
total mass one and satisfies $\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})=\gamma_{k}>0$
since $\xi_{k}\neq\mu$, so that
$\mathbf{u}_{k}^{\top}{\mathbf{K}_{\mu}}_{k}\mathbf{u}_{k}=a^{2}\gamma_{k}>0$.
If $a=\mathbf{u}_{k}^{\top}\mathbf{1}_{k}=0$, then
$\mathbf{u}_{k}^{\top}{\mathbf{K}_{\mu}}_{k}\mathbf{u}_{k}=\mathbf{u}_{k}^{\top}\mathbf{K}_{k}\mathbf{u}_{k}$,
which is strictly positive since $\mathbf{K}_{k}$ has full rank. Direct
calculation then gives
$\displaystyle\widehat{\mathbf{w}}_{k}={\mathbf{K}_{\mu}}_{k}^{-1}\mathbf{1}_{k}/(\mathbf{1}_{k}^{\top}{\mathbf{K}_{\mu}}_{k}^{-1}\mathbf{1}_{k})\
\mbox{ and }\
\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k})=1/(\mathbf{1}_{k}^{\top}{\mathbf{K}_{\mu}}_{k}^{-1}\mathbf{1}_{k})\,,$
and, using block matrix inversion for ${\mathbf{K}_{\mu}}_{k+1}$,
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k+1})=\left\\{\mathbf{1}_{k}^{\top}{\mathbf{K}_{\mu}}_{k}^{-1}\mathbf{1}_{k}+\frac{\left[\mathbf{1}_{k}^{\top}{\mathbf{K}_{\mu}}_{k}^{-1}{\mathbf{k}_{\mu}}_{k}(\mathbf{x}_{k+1})-1\right]^{2}}{K_{\mu}(\mathbf{x}_{k+1},\mathbf{x}_{k+1})-{\mathbf{k}_{\mu}}_{k}^{\top}(\mathbf{x}_{k+1}){\mathbf{K}_{\mu}}_{k}^{-1}{\mathbf{k}_{\mu}}_{k}(\mathbf{x}_{k+1})}\right\\}^{-1}\,,$
where
${\mathbf{k}_{\mu}}_{k}(\mathbf{x})=[K_{\mu}(\mathbf{x}_{1},\mathbf{x}),\ldots,K_{\mu}(\mathbf{x}_{k},\mathbf{x})]^{\top}$,
$\mathbf{x}\in{\mathscr{X}}$. Straightforward manipulations using (5) give
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k+1})=\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k})-\frac{\left[P_{K,\widehat{\xi}_{k}}(\mathbf{x}_{k+1})-P_{K,\mu}(\mathbf{x}_{k+1})+\widehat{\mathbf{w}}_{k}^{\top}\mathbf{p}_{k}(\mu)-{\mathscr{E}}_{K}(\widehat{\xi}_{k})\right]^{2}}{\min_{\scriptsize\begin{array}[]{l}\mathbf{w}\in\mathds{R}^{k}\\\
\mathbf{1}_{k}^{\top}\mathbf{w}=1\end{array}}\|K(\mathbf{x}_{k+1},\cdot)-\mathbf{w}^{\top}\mathbf{k}_{k}(\cdot)\|_{\mathcal{H}_{K}}^{2}}\,.$
(17)
This version of SBQ selects
$\mathbf{x}_{1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}}K_{\mu}(\mathbf{x},\mathbf{x})$
and then
$\displaystyle\mathbf{x}_{k+1}\in\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}}\frac{\left[P_{K,\widehat{\xi}_{k}}(\mathbf{x}_{)}-P_{K,\mu}(\mathbf{x})+\widehat{\mathbf{w}}_{k}^{\top}\mathbf{p}_{k}(\mu)-\widehat{\mathbf{w}}_{k}^{\top}\mathbf{K}_{k}\widehat{\mathbf{w}}_{k})\right]^{2}}{K(\mathbf{x},\mathbf{x})-\mathbf{k}_{k}^{\top}(\mathbf{x})\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x})+\frac{[1-\mathbf{1}_{k}^{\top}\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x})]^{2}}{\mathbf{1}_{k}^{\top}\mathbf{K}_{k}^{-1}\mathbf{1}_{k}}}\,,\
k\geq 1\,.$ (18)
The expressions (11) and (17) are pivotal to the derivation of finite-sample-
size error bounds for SBQ through the consideration of simplified versions
where $\mathbf{x}_{k+1}$ is chosen by kernel herding, see Section 3.3.1. When
$\mathbf{x}$ is selected at each iteration within a candidate set of size $C$,
see Section 2.4, the complexity of SBQ grows like $\mathcal{O}(n^{2}\,C)$ for
$n$ iterations (the main contribution comes from the denominators in (12) and
(18) which must be calculated for the $C$ candidates, but matrix-vector
multiplications $\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x})$ can be avoided
by using recursive calculation; see Remark 4).
#### 2.3.2 Greedy MMD Minimisation (GM)
To lighten the computations required by SBQ, we can consider the optimal
choice of successive $\mathbf{x}_{k}$ for a predefined sequence of weights
$\mathbf{w}_{k}$. The standard version of Greedy MMD Minimisation (GM) uses
$\mathbf{w}_{k}=\mathbf{1}_{k}/k$ for all $k$, so that $\xi_{k}$ is the
empirical measure $\xi_{k,e}$ supported on $\mathbf{X}_{k}$. It selects
$\mathbf{x}_{1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}}K(\mathbf{x},\mathbf{x})-2\,P_{K,\mu}(\mathbf{x})=\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}}K_{\mu}(\mathbf{x},\mathbf{x})$
and then minimises $\mathsf{MMD}_{K}(\mu,\xi_{k+1,e})$ incrementally: (4)
gives
$\displaystyle\mathbf{x}_{k+1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}}\sum_{i=1}^{k}K(\mathbf{x}_{i},\mathbf{x})+\frac{1}{2}\,K(\mathbf{x},\mathbf{x})-(k+1)\,P_{K,\mu}(\mathbf{x})\,,\
k\geq 1\,.$ (19)
GM will be considered in Section 4. The complexity of MMD grows linearly and
is $\mathcal{O}(n\,C)$ for $n$ iterations when the selection is among $C$
possible candidates. In Section 4.2 we also consider versions with nonuniform
weights: one must then define the weight $w_{k+1}$ to be allocated to the next
point $\mathbf{x}_{k+1}$, not selected yet. A convenient way to proceed, usual
in the area of optimal design of experiments, is to take
$\xi_{k+1}=(1-\alpha_{k+1})\,\xi_{k}+\alpha_{k+1}\delta_{\mathbf{x}_{k+1}}$,
for some step size $\alpha_{k+1}\in[0,1]$. This construction guarantees that
$\xi_{k+1}\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})$ when
$\xi_{k}\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})$; the choice of
$\alpha_{k+1}$ defines the sequence of weights $\mathbf{w}_{k}$ and the point
$\mathbf{x}_{k+1}$ is chosen to minimise $\mathsf{MMD}_{K}(\mu,\xi_{k+1})$.
#### 2.3.3 The Frank-Wolfe algorithm and Kernel Herding (KH)
Another way of proceeding consists in exploiting the convexity of the
functional
$\phi_{K,\mu}(\cdot):\xi\to\phi_{K,\mu}(\xi)=\mathsf{MMD}_{K}^{2}(\mu,\xi)$
using a gradient descent algorithm. This gives a family of methods for which
performance bounds can be easily established by convexity arguments, arguments
that are also applicable to the derivation of performance bounds for the GM
and SBQ algorithms.
For any $\xi,\nu\in{\mathscr{M}}({\mathscr{X}})$, the directional derivative
of $\phi_{K,\mu}(\cdot)$ at $\xi$ in the direction $\nu$ equals
$\displaystyle
F_{\mathsf{MMD}_{K}^{2}}(\xi,\nu)=2\,\left[\int_{{\mathscr{X}}^{2}}K_{\mu}(\mathbf{x},\mathbf{x}^{\prime})\,\mathrm{d}\nu(\mathbf{x})\,\mathrm{d}\xi(\mathbf{x}^{\prime})-{\mathscr{E}}_{K_{\mu}}(\xi)+\int_{\mathscr{X}}P_{K,\mu}(\mathbf{x})\,\mathrm{d}(\xi-\nu)(\mathbf{x})\right]\,,$
so that
$F_{\mathsf{MMD}_{K}^{2}}(\xi,\delta_{\mathbf{x}})=2\,[P_{K,\xi}(\mathbf{x})-P_{K,\mu}(\mathbf{x})+\int_{\mathscr{X}}P_{K,\mu}(\mathbf{x}^{\prime})\,\mathrm{d}\xi(\mathbf{x}^{\prime})-{\mathscr{E}}_{K}(\xi)]$.
Iterations of the Frank-Wolfe algorithm (Frank and Wolfe, , 1956) correspond
to $\xi_{k+1}=(1-\alpha_{k+1})\,\xi_{k}+\alpha_{k+1}\nu_{k+1}$, where
$\nu_{k+1}\in\mathrm{Arg}\min_{\nu\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})}F_{\mathsf{MMD}_{K}^{2}}(\xi_{k},\nu)$
and $\alpha_{k+1}\in[0,1]$. This gives $\nu_{k+1}=\delta_{\mathbf{x}_{k+1}}$,
with
$\displaystyle\mathbf{x}_{k+1}$ $\displaystyle\in$
$\displaystyle\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}}F_{\mathsf{MMD}_{K}^{2}}(\xi_{k},\delta_{\mathbf{x}})=\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}}\left[P_{K,\xi_{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})\right]\,.$
(20)
When $\alpha_{k}=1/k$ for all $k$, $\xi_{k}$ remains uniform on its support
(unless the same $\mathbf{x}$ is chosen several times); see Wynn, (1970) for
an early contribution in optimal design of experiments. The method is also
called conditional-gradient and corresponds to kernel herding (KH) used in
machine learning (Bach et al., , 2012). The algorithm selects
$\mathbf{x}_{1}\in\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}}P_{K,\mu}(\mathbf{x})$
and then
$\displaystyle\mathbf{x}_{k+1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}}\sum_{i=1}^{k}K(\mathbf{x}_{i},\mathbf{x})-k\,P_{K,\mu}(\mathbf{x})\,,\
k\geq 1\,.$ (21)
Notice the similarity (but not full agreement) with (19). In particular, (21)
can be used with singular kernels, which have an intrinsic repelling property
(Pronzato and Zhigljavsky, , 2021) but for which $K(\mathbf{x},\mathbf{x})$ is
not defined, whereas (19) cannot. The complexity of KH is $\mathcal{O}(n\,C)$
for $n$ iterations when the selection is among $C$ eligible candidates. In
Section 3.2 we consider nonuniform weights, including the case where
$\alpha_{k+1}$ is chosen optimally in
$\xi_{k+1}=(1-\alpha_{k+1})\,\xi_{k}+\alpha_{k+1}\delta_{\mathbf{x}_{k+1}}$.
In the area of optimal design of experiments, this corresponds to Fedorov’s
algorithm (1972).
We shall also consider two variants of KH (Section 3.3). First, in a Bayesian
integration application, at iteration $k$ of the algorithm we can exploit the
support of $\xi_{k}$ only, and use one of the optimal measures $\xi_{k}^{*}$,
$\widehat{\xi}_{k}$, or $\widetilde{\xi}_{k}$, with respective weights
$\mathbf{w}_{k}^{*}$, $\widehat{\mathbf{w}}_{k}$ and
$\widetilde{\mathbf{w}}_{k}$, for integration; we shall call this variant
_Off-Line Weight Optimisation_ (OLWO)333It is called Frank-Wolfe Bayesian
quadrature in (Briol et al., , 2015).. Second, we can replace $\xi_{k}$ by
$\xi_{k}^{*}$, $\widehat{\xi}_{k}$, or $\widetilde{\xi}_{k}$, _in the
algorithm itself_ before next iteration; we shall call this variant, closely
related to SBQ, _Integrated Weight Optimisation_ (IWO)444Depending how the
‘optimal’ measure is constructed, this includes the minimum-norm point (Bach
et al., , 2012) and fully-corrective Frank-Wolfe (Lacoste-Julien and Jaggi, ,
2015) algorithms; see Remark 3..
### 2.4 Notation
At each iteration, instead of searching $\mathbf{x}$ in the whole set
${\mathscr{X}}$, we shall use a finite subset
${\mathscr{X}}_{C}=\\{\mathbf{x}^{(1)},\ldots,\mathbf{x}^{(C)}\\}$ of
candidate points in ${\mathscr{X}}$ (typically, $C$ points independently
sampled from $\mu$; see Section 6). We denote
$\mathds{I}_{C}=\\{1,\ldots,C\\}$,
$\displaystyle\overline{K}_{C}$ $\displaystyle=$
$\displaystyle\max_{\mathbf{x}\in{\mathscr{X}}_{C}}K(\mathbf{x},\mathbf{x})\leq\overline{K}\,,$
$\displaystyle\overline{K}_{\mu,C}$ $\displaystyle=$
$\displaystyle\max_{\mathbf{x}\in{\mathscr{X}}_{C}}K_{\mu}(\mathbf{x},\mathbf{x})=\max_{\mathbf{x}\in{\mathscr{X}}_{C}}K(\mathbf{x},\mathbf{x})-2\,P_{K,\mu}(\mathbf{x})+{\mathscr{E}}_{K}(\mu)$
(22)
$\displaystyle\leq\,\overline{K}_{C}+2\,\overline{K}_{C}^{1/2}\tau_{1/2}(\mu)+\tau_{1/2}^{2}(\mu)=\left[\overline{K}_{C}^{1/2}+\tau_{1/2}(\mu)\right]^{2}$
(and $\overline{K}_{\mu,C}\leq\overline{K}_{C}+\tau_{1/2}^{2}(\mu)$ when
$K\geq 0$). We also denote by $\mathbf{K}_{C}$ and ${\mathbf{K}_{\mu}}_{C}$
the $C\times C$ matrices with $i,j$ elements respectively equal to
$K(\mathbf{x}^{(i)},\mathbf{x}^{(j)})$ and
$K_{\mu}(\mathbf{x}^{(i)},\mathbf{x}^{(j)})$, for $\mathbf{x}^{(i)}$,
$\mathbf{x}^{(j)}$ in ${\mathscr{X}}_{C}$; $\mathbf{e}_{j}$ is the $j$-th
canonical basis vector of $\mathds{R}^{C}$. We assume that $\mu$ is not fully
supported on ${\mathscr{X}}_{C}$, so that ${\mathbf{K}_{\mu}}_{C}$ has full
rank; see Section 2.3.1 ($K_{\mu}$ is thus s.p.d. on ${\mathscr{X}}_{C}$).
For any $\xi\in{\mathscr{M}}({\mathscr{X}})$, any $\mathbf{x}\in{\mathscr{X}}$
and any $\alpha\in[0,1]$, we define
$\displaystyle\xi^{+}(\mathbf{x},\alpha)=(1-\alpha)\,\xi+\alpha\,\delta_{\mathbf{x}}\,,$
(23)
so that
$\xi^{+}(\mathbf{x},\alpha)\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C})$
(respectively, ${\mathscr{M}}_{[1]}({\mathscr{X}}_{C})$) when
$\mathbf{x}\in{\mathscr{X}}_{C}$ and
$\xi\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C})$ (respectively,
$\xi\in{\mathscr{M}}_{[1]}({\mathscr{X}}_{C})$).
Any probability measure $\xi$ in ${\mathscr{M}}^{+}_{[1]}({\mathscr{X}}_{C})$,
i.e., supported on ${\mathscr{X}}_{C}$, can be represented as a vector of
weights $\mathbf{\omega}$ in the probability simplex
$\displaystyle{\mathscr{P}}_{C}=\\{\mathbf{\omega}\in\mathds{R}^{C}:\sum_{j=1}^{C}{\mathbf{\omega}}_{j}=1\,,\
{\mathbf{\omega}}_{j}\geq 0\mbox{ for all }j\\}\,.$
Any measure $\xi_{n}$ with $n$ support points in ${\mathscr{X}}_{C}$ can thus
be represented as in (3), with $\mathbf{w}_{n}$ a $n$-dimensional vector of
weights attached to its support ($\mathbf{w}_{n}\in{\mathscr{P}}_{n}$ when
$\xi_{n}\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C})$), and also as a vector
$\mathbf{\omega}_{n}\in\mathds{R}^{C}$
($\mathbf{\omega}_{n}\in{\mathscr{P}}_{C}$ when
$\xi_{n}\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C})$). For any
$\mathbf{x}^{(j)}\in{\mathscr{X}}_{C}$ and
$\xi\in{\mathscr{M}}({\mathscr{X}}_{C})$ with weights $\mathbf{\omega}$, the
vector of weights associated with $\xi^{+}(\mathbf{x}^{(j)},\alpha)$ equals
$\mathbf{\omega}^{+}(\mathbf{x}^{(j)},\alpha)=(1-\alpha)\,\mathbf{\omega}+\alpha\mathbf{e}_{j}$.
We shall denote
$\centerdot$
$\xi^{C}_{*}$ the minimum-MMD measure in
${\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C})$, with weights
$\mathbf{\omega}^{C}_{*}=\mathrm{arg}\min_{\mathbf{\omega}\in{\mathscr{P}}_{C}}\mathbf{\omega}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega}$,
so that (14) gives
$\displaystyle
M_{C}^{2}=\mathsf{MMD}_{K}^{2}(\mu,\xi^{C}_{*})={\mathbf{\omega}^{C}_{*}}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega}^{C}_{*}\,;$
(24)
$\centerdot$
$\widehat{\xi}^{C}$ the minimum-MMD measure in
${\mathscr{M}}_{[1]}({\mathscr{X}}_{C})$, with weights
$\widehat{\mathbf{\omega}}^{C}$ optimal under the total mass constraint
$\mathbf{1}_{C}^{\top}\widehat{\mathbf{\omega}}^{C}$ only:
$\widehat{\mathbf{\omega}}^{C}$ is given by (5) where $\mathbf{1}_{n}$,
$\mathbf{K}_{n}$ and $\mathbf{p}_{n}(\mu)$ are respectively replaced by
$\mathbf{1}_{C}$, $\mathbf{K}_{C}$ and
$\mathbf{p}_{C}(\mu)=[P_{K,\mu}(\mathbf{x}^{(1)}),\ldots,P_{K,\mu}(\mathbf{x}^{(C)})]^{\top}$;
$\centerdot$
$\widetilde{\xi}^{C}$ the minimum-MMD unconstrained measure in
${\mathscr{M}}({\mathscr{X}}_{C})$, with weights
$\widetilde{\mathbf{\omega}}^{C}=\mathbf{K}_{C}^{-1}\mathbf{p}_{C}(\mu)$, see
(6).
In the rest of the paper we derive finite-sample-size error bounds, i.e.,
bounds on $\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})$, for each of the constructions
of Section 2.3 and give a numerical illustration in Section 7. Note that we
are interested in situations where $k\ll C$. We start with the simplest
method, the Frank-Wolfe algorithm, which has already been much studied in the
literature (Section 3). The derivation of the upper bound closely follows
Clarkson, (2010), and those arguments will be central for the derivation of
the bounds for GM and SBQ in Sections 4 and 5.
Table 1 summarises our results (error bounds and complexity) for the main
algorithms considered. The computational complexity of KH grows like
$\mathcal{O}(nC)$ for $n$ iterations; the error bound decreases like
$\mathcal{O}((\log n)/n)$ for the standard version with step size
$\alpha_{k}=1/k$ for all $k$ (which yields uniform weights, Theorem 1) and
decreases like $\mathcal{O}(1/n)$ when $\alpha_{k}=2/(k+1)$ (Theorem 2) or
when $\alpha_{k}$ is optimised (Theorem 3). The error bounds for the variants
OLWO and IWO also decrease like $\mathcal{O}(1/n)$ (Theorem 4) and their
computational complexity grows like $\mathcal{O}(n^{2}C)$. The same results
apply to SBQ (Theorem 8). The numerical experiments in Section 7 indicate that
the error bound $\mathcal{O}(1/n)$ for SBQ is pessimistic, in particular for
the version where $\mathbf{x}_{k+1}$ is given by (12), some explanations for
this pessimism are given in Section 5. GM has the same complexity as KH, with
a slightly better error bound for its standard version (Theorem 5) and similar
bounds for other versions (Theorems 6 and 7). The case where
${\mathscr{X}}_{C}$ is a random set of candidate points, possibly resampled at
every iteration, is considered in Section 6 and Appendix C where we show that
our results on the decrease of the error bound at finite horizon continue to
apply. In some cases, a better bound is obtained when
$\widehat{\xi}^{C}=\xi^{C}_{*}$, i.e., when all weights
$\widehat{\omega}^{C}_{i}$ are nonnegative (which occurs when
$\omega_{i}^{*}>0$ for all $i$). Deriving precise sufficient conditions for
this property is a difficult task (as is the question of positivity of
quadrature weights in general; see Karvonen et al., (2019)), but it is
usually satisfied when the $\mathbf{x}^{(i)}$ in ${\mathscr{X}}_{C}$ are
independently sampled from $\mu$.
Table 1: Error bound and complexity for $n$ iterations of the KH, GM and SBQ algorithms; $A_{C}=[\overline{K}_{C}^{1/2}+\tau_{1/2}(\mu)]^{2}$, $B_{C}=4\,\overline{K}_{C}$ ($A_{C}=\overline{K}_{C}+\tau_{1/2}^{2}(\mu)$ and $B_{C}=2\,\overline{K}_{C}$ when $K$ is positive), $M_{C}^{2}$ is given by (24). Method | Algorithm | | Error bound | Theorem | Complexity
---|---|---|---|---|---
KH | 1 | $\alpha_{k}=1/k$ | $M_{C}^{2}+B_{C}\,\frac{2+\log n}{n+1}$ | 1 | $\mathcal{O}(nC)$
| 1 | $\alpha_{k}=2/(k+1)$ | $M_{C}^{2}+\frac{4\,B_{C}}{n+3}$ | 2 | $\mathcal{O}(nC)$
| 2 | $\alpha_{k}^{*}$ | $M_{C}^{2}+\frac{4\,B_{C}}{n+3}$ | 3 | $\mathcal{O}(nC)$
GM | 4 | $\alpha_{k}=1/k$ | $M_{C}^{2}+A_{C}\,\frac{1+\log n}{n}$ | 5 | $\mathcal{O}(nC)$
| 4 | $\alpha_{k}=2/(k+1)$ | $M_{C}^{2}+\frac{4\,B_{C}}{n+3}$ | 6 | $\mathcal{O}(nC)$
| 5 | $\alpha_{k}^{*}$ | $M_{C}^{2}+\frac{4\,B_{C}}{n+3}$ | 7 | $\mathcal{O}(nC)$
SBQ | Eq. (12) | | $M_{C}^{2}+\frac{4\,\overline{K}}{n+13/3}$ | 8 | $\mathcal{O}(n^{2}C)$
| Eq. (18) | | $M_{C}^{2}+\frac{4\,B_{C}}{n+3}$ | 8 | $\mathcal{O}(n^{2}C)$
## 3 Performance analysis of kernel herding and its variants
### 3.1 Empirical measures
Consider first the case of standard KH, corresponding to Algorithm 1 with
$\alpha_{k}=1/k$. It selects $\xi_{1}=\delta_{\mathbf{x}_{1}}$, with
$\mathbf{x}_{1}\in\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}_{C}}P_{K,\mu}(\mathbf{x})$,
and then $\xi_{k+1}=\xi_{k}^{+}[\mathbf{x}_{k+1},1/(k+1)]$ with
$\mathbf{x}_{k+1}$ given by (21) where ${\mathscr{X}}_{C}$ is substituted for
${\mathscr{X}}$. This choice of $\alpha_{k}$ implies that
$\mathbf{w}_{k}=\mathbf{1}_{k}/k$ for all $k$; that is, $\xi_{k}=\xi_{k,e}$,
the empirical measure on $\mathbf{X}_{k}$. The complexity only grows linearly
and is $\mathcal{O}(n\,C)$ for $n$ iterations: the
$P_{K,\mu}(\mathbf{x}^{(i)})$ are only computed once for all at the beginning,
with complexity $\mathcal{O}(C)$;
$S_{k}(\mathbf{x}^{(i)})=P_{K,\xi_{k}}(\mathbf{x}^{(i)})$ is updated at each
iteration for each $\mathbf{x}^{(i)}$ in ${\mathscr{X}}_{C}$, again with
complexity $\mathcal{O}(C)$. The finite-sample-size error can be bounded as
indicated in Theorem 1. The proof is given in Appendix B.
Algorithm 1 Kernel herding, predefined step sizes $\alpha_{k}$:
$\xi_{k+1}=\mathsf{KH}(\xi_{k},\alpha_{k+1})$
1:$\mu\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})\cap{\mathscr{M}}_{K}^{1/2}({\mathscr{X}})$,
${\mathscr{X}}_{C}\subset{\mathscr{X}}$, $n\in\mathds{N}$;
2:set $S_{0}(\cdot)\equiv 0$ and $\xi_{0}=0$; compute $P_{K,\mu}(\mathbf{x})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
3:a sequence $(\alpha_{k})_{k}$ in $[0,1]$ with $\alpha_{1}=1$;
4:$k\leftarrow 1$
5:while $k\leq n$ do
6: find
$\mathbf{x}_{k}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}S_{k-1}(\mathbf{x})-\,P_{K,\mu}(\mathbf{x})$;
7:
$S_{k}(\mathbf{x})\leftarrow(1-\alpha_{k})\,S_{k-1}(\mathbf{x})+\alpha_{k}\,K(\mathbf{x}_{k},\mathbf{x})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
8:
$\xi_{k}\leftarrow(1-\alpha_{k})\,\xi_{k-1}(\mathbf{x})+\alpha_{k}\,\delta_{\mathbf{x}_{k}}$;
9: $k\leftarrow k+1$
10:end while
11:return $\mathbf{X}_{n}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]$, $\xi_{n}$.
###### Theorem 1.
The empirical measure $\xi_{n}$ generated by Algorithm 1 with $\alpha_{k}=1/k$
for all $k$ satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})\leq
M_{C}^{2}+B_{C}\,\frac{2+\log n}{n+1}\,,\ n\geq 1\,,$ (25)
where $B_{C}=4\,\overline{K}_{C}$ ($B_{C}=2\,\overline{K}_{C}$ when $K$ is
positive) and $M_{C}^{2}$ is given by (24). When
$\widehat{\xi}^{C}=\xi^{C}_{*}$, $\xi_{n}$ satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})\leq
M_{C}^{2}+\frac{B_{C}}{n}\,,\ n\geq 1\,.$ (26)
It may happen that the same $\mathbf{x}^{(j)}$ is selected several (possibly
consecutive) times at line 5 of Algorithm 1. One may refer to Chen et al.,
(2018) for the extension to the case where $\mathbf{x}_{k+1}$ is searched
within the whole set ${\mathscr{X}}$ and the selection is suboptimal with some
bounded error. Chen et al., (2010) show that the error can decrease as
$\mathcal{O}(n^{-2})$ when $\mathcal{H}_{K}$ is finite-dimensional, but Bach
et al., (2012) indicate that one can only expect the rate
$\mathcal{O}(n^{-1})$ in the infinite-dimensional situation; see also Pronzato
and Zhigljavsky, (2020, Appendix A). In the next section, we show that a
better convergence rate than (25), without the log term, can be obtained in
general (without the assumption that $\widehat{\xi}^{C}=\xi^{C}_{*}$) when we
allow $\xi_{k}$ to be nonuniform on $\mathbf{X}_{k}$. The arguments are
similar to those used for the proof of Theorem 1: exploiting the convexity of
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ with respect to the vector of weights
$\mathbf{\omega}_{n}$, we obtain a recurrence equation which imposes a
particular decrease, see Lemma 2 in Appendix B. The same arguments are used
for the other algorithms in the following sections.
### 3.2 Nonuniform weights
Next theorem shows that for a suitable predefined step-size sequence
$(\alpha_{k})_{k}$ in Algorithm 1, the squared MMD decreases as
$\mathcal{O}(n^{-1})$. The proof is in Appendix B.
###### Theorem 2.
The measure $\xi_{n}$ generated with Algorithm 1 with $\alpha_{k}=2/(k+1)$ for
all $k$ satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})\leq
M_{C}^{2}+\frac{4\,B_{C}}{n+3}\,,\ n\geq 1\,.$ (27)
Again, it may happen that the same $\mathbf{x}^{(j)}$ is selected several
times at line 5 of Algorithm 1; that is, there may exist repetitions in
$\mathbf{X}_{k}$. The weights $\\{\mathbf{w}_{n}\\}_{i}$ that $\xi_{n}$
allocates to the $\mathbf{x}_{i}$, $i=1,\ldots,n$, can be computed explicitly.
When $\alpha_{k}=2/(k+1)$ for all $k$, we have
$\xi_{n}=\sum_{i=1}^{n}2i/[n(n+1)]\,\delta_{\mathbf{x}_{i}}$. The distribution
is thus far from being uniform, contrary to the case with $\alpha_{k}=1/k$;
see the right panel of Figure 1. When the condition
$\widehat{\xi}^{C}=\xi^{C}_{*}$ is satisfied in Theorem 1, the bound (26) is
better than (27) and there is no point in using $\alpha_{k}=2/(k+1)$ rather
than $\alpha_{k}=1/k$. Example 1 will illustrate that the decrease of
$\mathsf{MMD}_{K}(\mu,\xi_{k})$ may be worse for nonuniform weights; see
Figure 1-left.
As a further attempt to improve performance, we can select
$\mathbf{x}_{k+1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}[P_{K,\xi_{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})]$
as previously, and then optimise
$\mathsf{MMD}_{K}[\mu,\xi_{k}^{+}(\mathbf{x}_{k+1},\alpha)]$ with respect to
$\alpha$ in $[0,1]$. This function being quadratic in $\alpha$, the optimal
value $\alpha_{k+1}^{*}=\alpha_{k+1}^{*}(\mathbf{x}_{k+1})$ can be obtained
explicitly; the construction is summarised in Algorithm 2 (see the proof of
Theorem 3 in Appendix B for details). Again, the complexity grows linearly
with $n$. The use of the optimal $\alpha$ implies that the same
$\mathbf{x}^{(j)}$ cannot be selected two consecutive times at line 4 of
Algorithm 2.
Algorithm 2 Kernel herding, optimal step sizes:
$\xi_{k+1}=\mathsf{KH}(\xi_{k},\alpha_{k+1}^{*})$
1:$\mu\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})\cap{\mathscr{M}}_{K}^{1/2}({\mathscr{X}})$,
${\mathscr{X}}_{C}\subset{\mathscr{X}}$, $n\in\mathds{N}$;
2:set $S_{0}(\cdot)\equiv 0$ and $\xi_{0}=0$; compute $P_{K,\mu}(\mathbf{x})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
3:$k\leftarrow 1$
4:while $k\leq n$ do
5: find
$\mathbf{x}_{k}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}S_{k-1}(\mathbf{x})-P_{K,\mu}(\mathbf{x})$;
6: if $k=1$ then set $\alpha_{k}^{*}=1$,
$Q_{1}=K(\mathbf{x}_{1},\mathbf{x}_{1})$, $R_{1}=P_{K,\mu}(\mathbf{x}_{1})$;
7: else compute
$A_{k}=Q_{k-1}-R_{k-1}+P_{K,\mu}(\mathbf{x}_{k})-S_{k-1}(\mathbf{x}_{k})$,
8:
$B_{k}=Q_{k-1}-2\,S_{k-1}(\mathbf{x}_{k})+K(\mathbf{x}_{k},\mathbf{x}_{k})$,
9: and $\alpha_{k}^{*}=\min\\{A_{k}/B_{k},1\\}$
10: end if
11: if $\alpha_{k}^{*}=0$ then return
$\mathbf{X}_{k-1}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{k-1}]$, $\xi_{k-1}$ and
stop;
12: end if
13:
$R_{k}\leftarrow(1-\alpha_{k}^{*})\,R_{k-1}+\alpha_{k}^{*}\,P_{K,\mu}(\mathbf{x}_{k})$;
14:
$Q_{k}\leftarrow(1-\alpha_{k}^{*})^{2}\,Q_{k-1}+2\,\alpha_{k}^{*}(1-\alpha_{k}^{*})S_{k-1}(\mathbf{x}_{k})+(\alpha_{k}^{*})^{2}K(\mathbf{x}_{k},\mathbf{x}_{k})$;
15:
$S_{k}(\mathbf{x})\leftarrow(1-\alpha_{k}^{*})\,S_{k-1}(\mathbf{x})+\alpha_{k}^{*}\,K(\mathbf{x}_{k},\mathbf{x})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
16:
$\xi_{k}\leftarrow(1-\alpha_{k}^{*})\,\xi_{k-1}+\alpha_{k}^{*}\,\delta_{\mathbf{x}_{k}}$;
17: $k\leftarrow k+1$
18:end while
19:return $\mathbf{X}_{n}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]$, $\xi_{n}$.
###### Theorem 3.
The measure $\xi_{n}$ generated with Algorithm 2 satisfies (27); when
$\widehat{\xi}^{C}=\xi^{C}_{*}$ it satisfies (26). The optimal $\alpha$ at
iteration $k$ is $\alpha_{k+1}^{*}=\min\\{\widehat{\alpha}_{k+1},1\\}$ with
$\displaystyle\widehat{\alpha}_{k+1}=\widehat{\alpha}_{k+1}(\mathbf{x}_{k+1})=\frac{\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}\left[P_{K,\xi_{k}}(\mathbf{x}_{i})-P_{K,\mu}(\mathbf{x}_{i})\right]+P_{K,\mu}(\mathbf{x}_{k+1})-P_{K,\xi_{k}}(\mathbf{x}_{k+1})}{\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}P_{K,\xi_{k}}(\mathbf{x}_{i})-2\,P_{K,\xi_{k}}(\mathbf{x}_{k+1})+K(\mathbf{x}_{k+1},\mathbf{x}_{k+1})}$
(28)
where
$\mathbf{x}_{k+1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}\left[P_{K,\xi_{n}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})\right]$.
If the algorithm stops at iteration $k$ with $\widehat{\alpha}_{k+1}=0$, then
$\mathsf{MMD}_{K}(\mu,\xi_{k})=\mathsf{MMD}_{K}(\mu,\xi^{C}_{*})$.
It may seem surprising that the bound obtained with optimal step sizes is not
better than when $\alpha_{k}=2/(k+1)$ for all $k$ in Algorithm 1, since the
decrease of MMD is larger in the former case when starting from the same
$\xi_{k}$. However, the global decrease over many iterations with the optimal
$\alpha$ is not necessarily better than with a predefined step-size sequence;
one can refer to Dunn, (1980) for a discussion. A numerical comparison in
provided in Section 7, showing that Algorithm 2 may perform worse than
Algorithm 1; see the left panel of Figure 1.
###### Remark 1.
Dunn and Harshbarger, (1978) and Dunn, (1980) propose other choices of step-
size sequences which we do not consider here. We also do not consider Frank-
Wolfe algorithm with away steps (Wolfe, , 1970; Atwood, , 1973)555See also
Todd and Yildirim, (2007); Ahipaşaoğlu et al., (2008) for a recent use in
the minimum-volume ellipsoid problem., for which
$\xi_{k+1}=\xi_{k}+\alpha_{k+1}(\xi_{k}-\delta_{\mathbf{x}_{j_{k}}})$ moves
away from one of its support points $\mathbf{x}_{j_{k}}$. Here
$\mathbf{x}_{j_{k}}$ is taken in
$\mathrm{Arg}\max_{\mathbf{x}\in\mathrm{supp}(\xi_{k})}F_{\mathsf{MMD}_{K}^{2}}(\xi_{k},\delta_{\mathbf{x}})=\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}}\left[P_{K,\xi_{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})\right]$
and
$\alpha_{k+1}\in[0,\xi_{k}(\mathbf{x}_{j_{k}})/[1-\xi_{k}(\mathbf{x}_{j_{k}})]$
to ensure that $\xi_{k+1}(\mathbf{x}_{j_{k}})\geq 0$; the decision to use an
away step instead of $\xi_{k+1}=\xi_{k}^{+}(\mathbf{x}_{k+1},\alpha)$ with
$\mathbf{x}_{k+1}$ given by (20) can rely on the comparison between the
criterion values obtained, or on the comparison between the absolute values of
the directional derivatives
$|F_{\mathsf{MMD}_{K}^{2}}(\xi_{k},\delta_{\mathbf{x}_{j_{k}}})|$ and
$|F_{\mathsf{MMD}_{K}^{2}}(\xi_{k},\delta_{\mathbf{x}_{k+1}})|$. Neither do we
consider vertex-exchange methods, for which
$\xi_{k+1}=\xi_{k}+\alpha_{k+1}(\delta_{\mathbf{x}_{k+1}}-\delta_{\mathbf{x}_{j_{k}}})$
for $\alpha_{k+1}\in[0,\xi_{k}(\mathbf{x}_{j_{k}})]$; see for instance
Pronzato and Zhigljavsky, (2020, Appendix A.3), Pronzato and Pázman, (2013,
Chap. 9) and the references therein. These methods prove especially efficient
for design problems for which the optimal solution is attained on the boundary
of ${\mathscr{P}}_{C}$, with many components equal to zero, in particular due
to their ability to reduce the support of the current measure (when
$\alpha_{k+1}=1$). The situation is different for the type of problems we have
in mind here, and we can only expect a rate of decrease of the finite-sample-
size error similar to Algorithm 2. Lacoste-Julien and Jaggi, (2015) give a
precise analysis of the convergence of these variants of Frank-Wolfe algorithm
and prove that they have a global linear convergence rate (contrary to the
original Frank-Wolfe algorithm666Linear convergence is obtained for the Frank-
Wolfe algorithm under the condition that $\widehat{\mathbf{\omega}}^{C}$ is in
the interior of ${\mathscr{P}}_{C}$; but even in this favourable case the
result has no practical interest for large $C$; see Pronzato and Zhigljavsky,
(2020, Lemma A4).). However, the pyramidal width defined in the same paper
(eq. (9)) decreases as $C^{-1/2}$ and the constant $\rho$ in the linear
convergence factor $\exp(-\rho k)$ decreases as $1/C$. $\lhd$
### 3.3 KH with off-line and integrated weight optimisation
#### 3.3.1 Off-Line Weight Optimisation (OLWO)
The first KH variant mentioned in Section 2.3.2 (Frank-Wolfe Bayesian
quadrature, Briol et al., (2015)) uses the support
$\mathbf{X}_{k}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{k}]$ obtained at iteration
$k$ with Algorithm 1 or 2, and constructs an optimal measure $\xi_{k}^{*}$,
$\widehat{\xi}_{k}$, or $\widetilde{\xi}_{k}$, respectively in
${\mathscr{M}}_{[1]}^{+}(\mathbf{X}_{k})$,
${\mathscr{M}}_{[1]}(\mathbf{X}_{k})$ or ${\mathscr{M}}(\mathbf{X}_{k})$, with
respective weights $\mathbf{w}_{k}^{*}$, obtained as solution of a QP problem,
$\widehat{\mathbf{w}}_{k}$ given by (5), and $\widetilde{\mathbf{w}}_{k}$
given by (6). Let $\xi_{k}$ be the measure generated by Algorithm 1 or 2, with
support $\mathbf{X}_{k}$; since $\xi_{k}$ is a probability measure,
$\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{k})\leq\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{k})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{k}^{*})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})$
for all $k$, and the bounds of Theorems 1-3 remain valid.
#### 3.3.2 Integrated Weight Optimisation (IWO)
The situation is more complicated for the second variant of Section 2.3.2,
where we substitute
$\nu_{k}\in\\{\xi_{k}^{*},\widehat{\xi}_{k},\widetilde{\xi}_{k}\\}$ for
$\xi_{k}$ _at every iteration_ (the case $\nu_{k}=\xi_{k}^{*}$ corresponds to
the fully-corrective Frank-Wolfe algorithm, we do not detail the minimum-norm
point algorithm, see Remark 3 below). We only consider the situation where the
same choice is applied for all iterations and denote respectively by (i), (ii)
and (iii) the three cases $\nu_{k}=\xi_{k}^{*}$, $\nu_{k}=\widehat{\xi}_{k}$
and $\nu_{k}=\widetilde{\xi}_{k}$ for all $k$. The choice of
$\mathbf{x}_{k+1}$ is the same as for KH, but now there is no $\alpha_{k+1}$
to choose. The construction is summarised in Algorithm 3.
Algorithm 3 Kernel herding + IWO (i), (ii) and (iii)
1:$\mu\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})\cap{\mathscr{M}}_{K}^{1/2}({\mathscr{X}})$,
${\mathscr{X}}_{C}\subset{\mathscr{X}}$, $n\in\mathds{N}$;
2:set $S_{0}(\cdot)\equiv 0$ and $\xi_{0}=0$; compute $P_{K,\mu}(\mathbf{x})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
3:$k\leftarrow 1$
4:while $k\leq n$ do
5: find
$\mathbf{x}_{k}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}S_{k-1}(\mathbf{x})-\,P_{K,\mu}(\mathbf{x})$;
6: compute (i) $\mathbf{w}_{k}=\mathbf{w}_{k}^{*}$ (a QP problem), or (ii)
$\mathbf{w}_{k}=\widehat{\mathbf{w}}_{k}$ (5), or (iii)
$\mathbf{w}_{k}=\widetilde{\mathbf{w}}_{k}$ (6),
7: compute
$S_{k}(\mathbf{x})=\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}\,K(\mathbf{x},\mathbf{x}_{i})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
8: $\xi_{k}=\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}\,\delta_{\mathbf{x}_{i}}$;
9: $k\leftarrow k+1$
10:end while
11:return $\mathbf{X}_{n}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]$, $\xi_{n}$.
Note that $S_{k}(\cdot)=P_{K,\nu_{k}}(\cdot)$ can no longer be computed
recursively, so that the complexity grows faster than linearly: at iteration
$k$, the complexity of the determination of $\mathbf{w}_{k}^{*}$,
$\widehat{\mathbf{w}}_{k}$ or $\widetilde{\mathbf{w}}_{k}$ is
$\mathcal{O}(m(k))$, independently of $C$ (with, in the last two cases,
$m(k)=k^{3}$ in general and $m(k)=k^{2}$ if rank-one updating is used to
compute $\mathbf{K}_{k}^{-1}$ in (5) and (6); see Remark 4); the complexity of
the computation of all $S_{k}(\mathbf{x}^{(i)})$ is $\mathcal{O}(k\,C)$ and
the complexity for $n$ iterations is thus $\mathcal{O}(n^{2}\,C)$ for $n\ll
C$. Kernel herding with IWO satisfies the error bounds in Theorem 4; the proof
is in Appendix B.
###### Theorem 4.
The measure $\xi_{n}$ generated by Algorithm 3-(i) satisfies (27); when
$\widehat{\xi}^{C}=\xi^{C}_{*}$, it satisfies (26). When using Algorithm
3-(ii), $\xi_{n}$ satisfies (27); when $\widehat{\xi}^{C}=\xi^{C}_{*}$, it
satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})\leq
M_{C}^{2}+\frac{B_{C}}{n+2}\,,\ n\geq 2\,,$ (29)
where $B_{C}=4\,\overline{K}_{C}$ ($B_{C}=2\,\overline{K}_{C}$ when $K$ is
positive) and $M_{C}^{2}$ is given by (24). When using Algorithm 3-(iii),
$\xi_{n}$ satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})\leq
M_{C}^{2}+\frac{4\,\overline{K}}{n+\frac{4\,\overline{K}}{\mathsf{MMD}_{K}^{2}(\mu,\xi_{1})}-1}\leq
M_{C}^{2}+\frac{4\,\overline{K}}{n+13/3}\,,\ n\geq 1\,.$ (30)
###### Remark 2.
The measures used in Algorithms 3-(ii) and (iii) are not constrained to belong
to ${\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C})$, so that the algorithm can
still progress when
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi^{C}_{*})=M_{C}^{2}$
(an obvious indication of the pessimism of the bounds in Theorem 4). We show
in Appendix B that the following stopping condition can be added after Step 4
of IWO (ii) and (iii), respectively:
4’-(ii):
if $S_{k-1}(\mathbf{x}_{k})-P_{K,\mu}(\mathbf{x}_{k})\geq
S_{k-1}(\mathbf{x}_{k-1})-P_{K,\mu}(\mathbf{x}_{k-1})$ then return
$\mathbf{X}_{k-1}$, $\xi_{k-1}$ and stop;
4’-(iii):
if $S_{k-1}(\mathbf{x}_{k})-P_{K,\mu}(\mathbf{x}_{k})\geq 0$ then return
$\mathbf{X}_{k-1}$, $\xi_{k-1}$ and stop; $\lhd$
###### Remark 3.
Algorithm 3-(i) corresponds to the fully-corrective Frank-Wolfe algorithm
analysed in (Lacoste-Julien and Jaggi, , 2015). The Minimum-norm point
variant, based on (Wolfe, , 1976), uses a sequence of affine projections based
on the calculation of a sequence $\widehat{\mathbf{\omega}}_{k_{i}}$
restricted to give nonzero weights to subsets $\SS_{k_{i}}$ of
$\SS_{k_{0}}=\mathrm{supp}(\xi_{k})$ (at most $k$ weights
$\widehat{\mathbf{\omega}}_{k_{i}}$ need to be calculated); see Algorithm 5 in
(Lacoste-Julien and Jaggi, , 2015). Since the
$\widehat{\mathbf{\omega}}_{k_{i}}$ can be obtained explicitly through (5),
this construction is simpler than the fully-corrective Frank-Wolfe algorithm.
The bounds in Theorem 4 indicated for Algorithm 3-(i) continue to apply, since
we still have
$\Delta_{C}(\xi_{k+1})\leq(1-\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}$,
$k\geq 1$, for $\alpha_{k+1}=2/(k+2)$, and
$\Delta_{C}(\xi_{k+1})\leq(1-2\,\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}$,
$k\geq 1$, for $\alpha_{k+1}=1/(k+1)$ when $\widehat{\xi}^{C}=\xi^{C}_{*}$;
see the proofs of Theorems 1 and 4. $\lhd$
###### Remark 4.
OLWO and IWO require the repeated computation of optimal weights
$\widehat{\mathbf{w}}_{n}$ or $\widetilde{\mathbf{w}}_{n}$, respectively given
by (5) and (6), for which it is advantageous to use the block matrix inversion
(10). Rank-one Cholesky updates can be used too; the details are omitted. Eq.
(10) also gives
$\displaystyle\mathbf{k}_{n+1}^{\top}(\mathbf{x})\mathbf{K}_{n+1}^{-1}\mathbf{k}_{n+1}(\mathbf{x})$
$\displaystyle=$
$\displaystyle\mathbf{k}_{n}^{\top}(\mathbf{x})\mathbf{K}_{n}^{-1}\mathbf{k}_{n}(\mathbf{x})+\beta_{n+1}\,[(\mathbf{u}_{n+1}^{\top},\
-1)\mathbf{k}_{n+1}(\mathbf{x})]^{2}\,,$
$\displaystyle\mathbf{1}_{n+1}^{\top}\mathbf{K}_{n+1}^{-1}\mathbf{k}_{n+1}(\mathbf{x})$
$\displaystyle=$
$\displaystyle\mathbf{1}_{n}^{\top}\mathbf{K}_{n}^{-1}\mathbf{k}_{n}(\mathbf{x})+\beta_{n+1}\,[(\mathbf{u}_{n+1}^{\top},\
-1)\mathbf{1}_{n+1}][(\mathbf{u}_{n+1}^{\top},\
-1)\mathbf{k}_{n+1}(\mathbf{x})]$
for all $\mathbf{x}$, so that matrix-vector multiplications can be avoided in
SBQ when computing the denominators on the right-hand side of (12) and (18) by
using recursive calculation. $\lhd$
###### Remark 5.
The numerical experiments of Section 7 show that the bound (30) for Algorithm
3-(iii) becomes very loose when $k$ increases. The arguments used in the proof
of Theorem 4 suggest two sources of pessimism. First, we substitute the
inequality (57) for the convexity bound (56) (this approximation is used for
all the methods considered). Second, we ignore the decrease of
$K(\mathbf{x}_{k+1},\mathbf{x}_{k+1})-\mathbf{k}_{k}^{\top}(\mathbf{x}_{k+1})\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x}_{k+1})$
as $k$ increases in the denominator on the right-hand side of (11), and simply
bound it by $\overline{K}$. In the algorithm defined by (32), the denominator
is constant, so that the pessimism of the error bound is mainly due to the
substitution of (57) for (56); this effect is illustrated on Figure 2-left.
Although this indicates that there still exists room for improvement, the
derivation of better bounds seems difficult. Similar statements can be made
for Algorithm 3-(ii), for which numerical experiments show that it performs
similarly to Algorithm 1 for moderate $k$, but tends to converge faster for
large $k$ (see Figure 2-left and Figure 6-left). $\lhd$
## 4 Performance analysis of Greedy MMD Minimisation (GM)
### 4.1 Empirical measures
GM with empirical measures corresponds to Algorithm 4 with $\alpha_{k}=1/k$
for all $k$; it selects
$\mathbf{x}_{1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}K(\mathbf{x},\mathbf{x})-2\,P_{K,\mu}(\mathbf{x})$
and then chooses $\mathbf{x}_{n+1}$ according to (19) with ${\mathscr{X}}_{C}$
substituted for ${\mathscr{X}}$. It corresponds to Algorithm 1 in (Teymur et
al., , 2021), where the authors derive a finite-sample-size error bound using
the RKHS framework. Taking advantage of the finiteness of the candidate set
${\mathscr{X}}_{C}$, we provide a simpler proof using only linear (finite-
dimensional) algebra; see Appendix B. Notice that the bound is smaller than
for KH in Theorem 1. The complexity of Algorithm 4 is $\mathcal{O}(n\,C)$ for
$n$ iterations: the $K(\mathbf{x}^{(i)},\mathbf{x}^{(i)})$ and
$P_{K,\mu}(\mathbf{x}^{(i)})$ are only computed once for all at the beginning,
with complexity $\mathcal{O}(C)$, the $S_{k}(\mathbf{x}^{(i)})$ are updated at
each iteration for all $\mathbf{x}^{(i)}\in{\mathscr{X}}_{C}$, again with
complexity $\mathcal{O}(C)$.
Algorithm 4 Greedy MMD minimisation, predefined step sizes $\alpha_{k}$:
$\xi_{k+1}=\mathsf{GM}(\xi_{k},\alpha_{k+1})$
1:$\mu\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})\cap{\mathscr{M}}_{K}^{1/2}({\mathscr{X}})$,
${\mathscr{X}}_{C}\subset{\mathscr{X}}$, $n\in\mathds{N}$;
2:set $S_{0}(\cdot)\equiv 0$ and $\xi_{0}=0$; compute
$K(\mathbf{x},\mathbf{x})$ and $P_{K,\mu}(\mathbf{x})$ for all
$\mathbf{x}\in{\mathscr{X}}_{C}$;
3:a sequence $(\alpha_{k})_{k}$ in $[0,1]$ with $\alpha_{1}=1$;
4:$k\leftarrow 1$
5:while $k\leq n$ do
6: find
$\mathbf{x}_{k}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}2(1-\alpha_{k})\,S_{k-1}(\mathbf{x})+\alpha_{k}\,K(\mathbf{x},\mathbf{x})-2\,P_{K,\mu}(\mathbf{x})$;
7:
$S_{k}(\mathbf{x})\leftarrow(1-\alpha_{k})\,S_{k-1}(\mathbf{x})+\alpha_{k}\,K(\mathbf{x}_{k},\mathbf{x})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
8:
$\xi_{k}\leftarrow(1-\alpha_{k})\,\xi_{k-1}(\mathbf{x})+\alpha_{k}\,\delta_{\mathbf{x}_{k}}$;
9: $k\leftarrow k+1$
10:end while
11:return $\mathbf{X}_{n}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]$, $\xi_{n}$.
###### Theorem 5.
The measure $\xi_{n}$ generated by Algorithm 4 with $\alpha_{k}=1/k$ for all
$k$ satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})\leq
M_{C}^{2}+A_{C}\,\frac{1+\log n}{n}\,,\ n\geq 1\,,$ (31)
where $M_{C}^{2}$ is given by (24) and
$A_{C}=[\overline{K}_{C}^{1/2}+\tau_{1/2}(\mu)]^{2}$
($A_{C}=\overline{K}_{C}+\tau_{1/2}^{2}(\mu)$ when $K$ is positive).
### 4.2 Nonuniform weights
Consider now the case of general discrete measures $\xi_{n}$ in
${\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C})$, see (3). We show that allowing
nonuniform weights in GM yields a faster decrease of
$\mathsf{MMD}_{K}(\mu,\xi_{n})$. As for KH, we consider iterations of the form
$\xi_{k+1}=\xi_{k}^{+}(\mathbf{x}_{k+1},\alpha_{k+1})$, $k\geq 1$, for some
$\alpha_{k+1}\in[0,1]$ and $\mathbf{x}_{k+1}\in{\mathscr{X}}_{C}$, where
$\xi_{k}^{+}(\mathbf{x},\alpha)$ is defined by (23). We first consider the
same choice $\alpha_{k}=2/(k+1)$ as in Section 3.2. The proof of Theorem 6 is
in Appendix B.
###### Theorem 6.
The measure $\xi_{n}$ generated by Algorithm 4 with $\alpha_{k}=2/(k+1)$ for
all $k$ satisfies (27). When $\widehat{\xi}^{C}=\xi^{C}_{*}$, Algorithm 4 with
$\alpha_{k}=1/k$ for all $k$ yields (26).
###### Remark 6.
As for Algorithm 1, when $\alpha_{k}=2/(k+1)$ in Algorithm 4 the measure
$\xi_{n}$ is not uniform on its support $\mathbf{X}_{n}$. It is uniform when
$\alpha_{k}=1/k$ for all $k$, but the arguments used in the proof of Theorem 6
only give (25), which is worse than (31) obtained by Teymur et al., (2021).
$\lhd$
Consider now GM with optimal step size, which selects $\alpha_{k+1}$ and
$\mathbf{x}_{k+1}$ optimally at each iteration:
$[\mathbf{x}_{k+1},\alpha_{k+1}]\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C},\,\alpha\in[0,1]}\mathsf{MMD}_{K}[\mu,\xi_{k}^{+}(\mathbf{x},\alpha)]$,
with $\xi_{1}=\delta_{\mathbf{x}_{1}}$ and
$\mathbf{x}_{1}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}K(\mathbf{x},\mathbf{x})-2\,P_{K,\mu}(\mathbf{x})$.
As in Algorithm 2, the optimal value of
$\alpha_{k+1}=\alpha(\mathbf{x}_{k+1})$ is obtained explicitly, see the proof
of Theorem 7 in Appendix B for details. The complexity is again
$\mathcal{O}(n\,C)$ for $n$ iterations (it is larger than for Algorithm 2 as
$\alpha(\mathbf{x})$ must be calculated for all
$\mathbf{x}\in{\mathscr{X}}_{C}$).
Algorithm 5 Greedy MMD minimisation, optimal step sizes:
$\xi_{k+1}\mathsf{GM}(\xi_{k},\alpha_{k+1}^{*})$
1:$\mu\in{\mathscr{M}}_{[1]}^{+}({\mathscr{X}})\cap{\mathscr{M}}_{K}^{1/2}({\mathscr{X}})$,
${\mathscr{X}}_{C}\subset{\mathscr{X}}$, $n\in\mathds{N}$;
2:compute $K(\mathbf{x},\mathbf{x})$ and $P_{K,\mu}(\mathbf{x})$ for all
$\mathbf{x}\in{\mathscr{X}}_{C}$;
3:set $S_{0}(\cdot)\equiv 0$, $Q_{0}=R_{0}=0$, $\alpha_{1}(\cdot)\equiv 1$ and
$\xi_{0}=0$;
4:set $A_{0}(\mathbf{x})=P_{K,\mu}(\mathbf{x})$,
$B_{0}(\mathbf{x})=K(\mathbf{x},\mathbf{x})$ for all
$\mathbf{x}\in{\mathscr{X}}_{C}$;
5:$k\leftarrow 1$
6:while $k\leq n$ do
7: find
$\mathbf{x}_{k}\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C}}\alpha^{2}(\mathbf{x})B_{k-1}(\mathbf{x})-2\,\alpha(\mathbf{x})A_{k-1}(\mathbf{x})$;
8: $\alpha_{k}\leftarrow\alpha(\mathbf{x}_{k})$
9:
$R_{k}\leftarrow(1-\alpha_{k})\,R_{k-1}+\alpha_{k}\,P_{K,\mu}(\mathbf{x}_{k})$;
10:
$Q_{k}\leftarrow(1-\alpha_{k})^{2}\,Q_{k-1}+2\,\alpha_{k}(1-\alpha_{k})S_{k-1}(\mathbf{x}_{k})+\alpha_{k}^{2}K(\mathbf{x}_{k},\mathbf{x}_{k})$;
11:
$S_{k}(\mathbf{x})\leftarrow(1-\alpha_{k})\,S_{k-1}(\mathbf{x})+\alpha_{k}\,K(\mathbf{x}_{k},\mathbf{x})$
for all $\mathbf{x}\in{\mathscr{X}}_{C}$;
12:
$\xi_{k}\leftarrow(1-\alpha_{k})\,\xi_{k}+\alpha_{k}\,\delta_{\mathbf{x}_{k}}$;
13: compute
$A_{k}(\mathbf{x})=Q_{k}-R_{k}+P_{K,\mu}(\mathbf{x})-S_{k}(\mathbf{x})$,
$B_{k}(\mathbf{x})=Q_{k}-2\,S_{k}(\mathbf{x})+K(\mathbf{x},\mathbf{x})$
14: and
$\alpha(\mathbf{x})=\max\\{0,\min\\{A(\mathbf{x})/B(\mathbf{x}),1\\}\\}$ for
all $\mathbf{x}\in{\mathscr{X}}_{C}$;
15: if all $\alpha(\mathbf{x})$ equal = 0 then return
$\mathbf{X}_{k}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{k}]$, $\xi_{k}$ and stop;
16: end if
17: $k\leftarrow k+1$
18:end while
19:return $\mathbf{X}_{n}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]$, $\xi_{n}$.
###### Theorem 7.
The measure $\xi_{n}$ generated with Algorithm 5 satisfies (27). When
$\widehat{\xi}^{C}=\xi^{C}_{*}$, it satisfies (26).
Similarly to Theorem 3, one might think that bounds obtained with predefined
step sizes should be overly loose concerning an algorithm for which
$\alpha_{k}$ is optimised at every iteration. However, the observed behaviour
is often similar to that of Algorithm 4, if not worse, see the examples in
Section 7, indicating that optimal but myopic steps are not necessarily
preferable to myopic, non-optimised but suitably chosen steps; see, e.g.,
Zhigljavsky et al., (2012) for an illustration with the steepest descent
algorithm.
###### Remark 7.
As for KH, one may also consider OLWO and IWO variants of GM; see Section 3.3.
The OLWO variant does not raise any particular difficulty as it runs in
parallel and does not affect the algorithm: like in Section 3.3.1 for KH, OLWO
can only improve performance. The situation is different for IWO: the fact
that the next point $\mathbf{x}_{k+1}$ and the step size $\alpha_{k+1}$ must
be selected simultaneously render its use less adapted than with KH, which
maximises the right-hand side of (12) and (18), see the proof of Theorem 4.
$\lhd$
## 5 Performance analysis of SBQ
We suppose again that the successive support points are searched within a
finite candidate set ${\mathscr{X}}_{C}$, and consider the two versions of SBQ
presented in Section 2.3.1 with ${\mathscr{X}}_{C}$ substituted for
${\mathscr{X}}$. We do not detail the algorithm which simply implements (12)
or (18)—with $\mathbf{K}_{k}^{-1}$ calculated recursively as indicated in
Remark 4. The numerical experiments of Section 7 show that the two versions
behave similarly to Algorithms 3-(iii) and (ii), respectively, but have a
slightly higher computational cost.
We also consider a version of SBQ where all previous weights
$\\{\widetilde{\mathbf{w}}_{k}\\}_{i}$ are kept fixed, $i=1,\ldots,k$, and
only the next one is optimised (without constraint) when choosing
$\mathbf{x}_{k+1}$. Since
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{k}+w\,\delta_{\mathbf{x}})=\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})+w^{2}\,K(\mathbf{x},\mathbf{x})+2w\,[P_{K,\xi_{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})]$,
the optimal $w$ is
$w^{*}(\mathbf{x})=[P_{K,\mu}(\mathbf{x})-P_{K,\widetilde{\xi}_{k}}(\mathbf{x})]/K(\mathbf{x},\mathbf{x})$.
This algorithm selects
$\mathbf{x}_{1}\in\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}_{C}}P_{K,\mu}^{2}(\mathbf{x})/K(\mathbf{x},\mathbf{x})$
and then uses
$\displaystyle\mathbf{x}_{k+1}\in\mathrm{Arg}\max_{\mathbf{x}\in{\mathscr{X}}_{C}}\frac{\left[P_{K,\widetilde{\xi}_{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})\right]^{2}}{K(\mathbf{x},\mathbf{x})}\,,\
\xi_{k+1}=\xi_{k}+w^{*}(\mathbf{x}_{k+1})\delta_{\mathbf{x}_{k+1}}\,,\ k\geq
1\,.$ (32)
When $K(\mathbf{x},\mathbf{x})$ is a constant, the choice of
$\mathbf{x}_{k+1}$ is similar to that of KH, see (20). It corresponds to a
Coordinate-Descent (CD) algorithm (see, e.g., Wright, (2015)) operating on
the weights
$\mathbf{\omega}=(\mathbf{\omega}_{1},\ldots,\mathbf{\omega}_{C})\in\mathds{R}^{C}$;
see Section 2.4. Performance bounds for these three versions of SBQ are given
in Theorem 8.
###### Theorem 8.
Suppose that ${\mathscr{X}}_{C}$ is substituted for ${\mathscr{X}}$. Then,
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ satisfies the same bounds as those
indicated in Theorem 4 for Algorithm 3-(ii) when using (18), and satisfies
(30) when using (12), or when using (32) if $K(\mathbf{x},\mathbf{x})$ is a
constant.
Although our numerical experiments indicate that (32) is not competitive
compared to (12), the analysis of its finite sample error helps understanding
the pessimism of the error bound derived for version (12) of SBQ; see Remark 5
and the proof of Theorem 8.
## 6 Random candidate sets
The extension of the results in previous sections to the case where
${\mathscr{X}}_{C}$ corresponds to $C$ points independently sampled from $\mu$
is fairly simple; see Teymur et al., (2021). For instance, (27) becomes
$\displaystyle\mathsf{E}_{\mu}\\{\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})\\}\leq\mathsf{E}_{\mu}\\{M_{C}^{2}\\}+\frac{4\,\mathsf{E}_{\mu}\\{B_{C}\\}}{n+3}\,,\
n\geq 1\,.$
We thus only have to bound the expected values of the constants $A_{C}$,
$B_{C}$ and $M_{C}^{2}$ that intervene in the various bounds that have been
presented. Their values are given in the following lemma, the proof is in
Appendix B.
###### Lemma 1.
Suppose $\mu\in{\mathscr{M}}_{K}^{1}({\mathscr{X}})$ and that the $C$ points
in ${\mathscr{X}}_{C}$ are independently sampled from $\mu$, then
$\displaystyle\mathsf{E}_{\mu}\\{A_{C}\\}$ $\displaystyle\leq$ $\displaystyle
A(\mu)=[\overline{K}^{1/2}+\tau_{1/2}(\mu)]^{2}\
(A(\mu)=\overline{K}+\tau_{1/2}^{2}(\mu)\mbox{ when }K\mbox{ is positive}),$
$\displaystyle\mathsf{E}_{\mu}\\{B_{C}\\}$ $\displaystyle\leq$ $\displaystyle
B=4\,\overline{K}\ (B=2\,\overline{K}\mbox{ when }K\mbox{ is positive}),$
$\displaystyle\mathsf{E}_{\mu}\\{M_{C}^{2}\\}$ $\displaystyle\leq$
$\displaystyle M^{2}(\mu)/C=[\tau_{1}(\mu)-{\mathscr{E}}_{K}(\mu)]/C\,,$
where $M_{C}^{2}$ is given by (24),
$A_{C}=[\overline{K}_{C}^{1/2}+\tau_{1/2}(\mu)]^{2}$ and
$B_{C}=4\,\overline{K}_{C}$ ($A_{C}=\overline{K}_{C}+\tau_{1/2}^{2}(\mu)$ and
$B_{C}=2\,\overline{K}_{C}$ when $K$ is positive).
Teymur et al., (2021) derive a bound similar to (31) (with $A_{C}$ and
$M_{C}^{2}$ replaced by $\mathsf{E}_{\mu}\\{A_{C}\\}$ and
$\mathsf{E}_{\mu}\\{M_{C}^{2}\\}$) for Algorithm 4 with $\alpha_{k}=1/k$ for
all $k$ in the situation where a different sample ${\mathscr{X}}_{C}[k]$ of
$C$ random points is used at each iteration; see also Chen et al., (2019).
The extension to this situation of the approach used in previous sections does
not seem straightforward as the probability simplex ${\mathscr{P}}_{C}$ and
matrices $\mathbf{K}_{C}$ and ${\mathbf{K}_{\mu}}_{C}$ refer to a fixed set
${\mathscr{X}}_{C}$. In Appendix C we provide arguments explaining how our
results extend to the case where ${\mathscr{X}}_{C}={\mathscr{X}}_{C}[k]$
depends on $k$: basically, similar bounds continue to hold provided we
consider the expectation of $\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ and bound the
expected values of the constants involved as in Lemma 1. Note that changing
the candidate set at every iteration implies that we need to calculate
$K(\mathbf{x}_{i},\mathbf{x})$ for all
$\mathbf{x}_{i}\in\mathrm{supp}(\xi_{k})$ and all
$\mathbf{x}\in{\mathscr{X}}_{C}[k]$, and to recalculate
$P_{K,\mu}(\mathbf{x})$ (Algorithms 1 and 2), or $P_{K,\mu}(\mathbf{x})$ and
$K(\mathbf{x},\mathbf{x})$ (Algorithms 4 and 5), for all
$\mathbf{x}\in{\mathscr{X}}_{C}[k]$ at every iteration, with a computational
cost thus growing as $\mathcal{O}(k^{2}\,C)$.
We conclude this section by recalling a result on the MMD of the empirical
measure $\xi_{n,e}$ of a random $n$-point sample from $\mu$; see Mak and
Joseph, (2018, Lemma 2). The proof is given in Appendix B.
###### Theorem 9.
When $\mathbf{x}_{1},\ldots,\mathbf{x}_{n}$ are independently sampled from
$\mu$, then $n\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{n,e})\stackrel{{\scriptstyle\rm
d}}{{\rightarrow}}Z=\sum_{i=1}^{\infty}\lambda_{i}\chi_{1i}^{2}$, where the
$\lambda_{i}$ are the eigenvalues of the operator $T_{K_{\mu}}$ on
$L_{2}({\mathscr{X}},\mu)$ defined by
$T_{K_{\mu}}f(\mathbf{x})=\int_{\mathscr{X}}K_{\mu}(\mathbf{x},\mathbf{x}^{\prime})f(\mathbf{x}^{\prime})\,\mathrm{d}\mu(\mathbf{x}^{\prime})$,
$f\in L_{2}({\mathscr{X}},\mu)$, $\mathbf{x}\in{\mathscr{X}}$, and the
$\chi_{1i}^{2}$ are independent $\chi_{1}^{2}$ random variables.
From Lemma 1 and Theorem 9 we have in particular
$\mathsf{E}_{\mu}\\{\mathsf{MMD}_{K}^{2}(\mu,\xi_{n,e})\\}=M^{2}(\mu)/n=[\tau_{1}(\mu)-{\mathscr{E}}_{K}(\mu)]/n$
and $n^{2}\,\mathsf{var}_{\mu}\\{\mathsf{MMD}_{K}^{2}(\mu,\xi_{n,e})\\}\to
2\,\sum_{i=1}^{\infty}\lambda_{i}^{2}$ as $n\rightarrow\infty$. Although the
bounds obtained in previous sections suggest that the measures obtained with
the algorithms that have been considered do not perform necessarily better
(asymptotically) than i.i.d. samples from $\mu$, the examples in the next
section demonstrate the interest of using KH, GM or SBQ.
## 7 Numerical study
### 7.1 Example 1: space-filling design
For illustration purpose we only consider the case $d=2$ and take $\mu$
uniform on ${\mathscr{X}}=[0,1]^{2}$; ${\mathscr{X}}_{C}$ corresponds to the
first $2^{17}=131\,072$ points of a scrambled Sobol’ sequence in
${\mathscr{X}}$. $K$ is a separable kernel given by the product of uni-
dimensional Matérn 3/2 covariance functions, that is,
$K(\mathbf{x},\mathbf{x}^{\prime})=\prod_{i=1}^{d}K_{3/2,\theta}(x_{i},x_{i}^{\prime})$
with
$\displaystyle
K_{3/2,\theta}(x,x^{\prime})=(1+\sqrt{3}\theta\,|x-x^{\prime}|)\,\exp(-\sqrt{3}\theta\,|x-x^{\prime}|)\,.$
We have
${\mathscr{E}}_{K}(\mu)=\prod_{i=1}^{d}{\mathscr{E}}_{K_{3/2,\theta}}(\mu_{1})$
and $P_{K,\mu}(\mathbf{x})=\prod_{i=1}^{d}P_{K_{3/2,\theta},\mu_{1}}(x_{i})$
with $\mu_{1}$ the uniform measure on $[0,1]$, and
${\mathscr{E}}_{K_{3/2,\theta}}(\mu_{1})$ and $P_{K_{3/2,\theta},\mu_{1}}(x)$
can be computed explicitly; see, e.g., Pronzato and Zhigljavsky, (2020, Table
3.1). Examples of space-filling design based on MMD-minimisation with $d=10$
and recommendations for the choice of $\theta$ are given in the same paper. We
use $\theta=10$ throughout the example.
The left panel of Figure 1 shows the evolution of
$\mathsf{MMD}_{K}(\mu,\xi_{n})$ as a function of $n$ (log-log plot) when
$\xi_{n}$ is generated with Algorithm 1 with $\alpha_{k}=1/k$ and
$\alpha_{k}=2/(k+1)$, or with Algorithm 2. The bound (27) is also shown
($M_{C}^{2}$ is negligible). Although Algorithm 2 uses optimal step-sizes, its
performance is the worst for large $n$ and is never better than that of
Algorithm 1 with $\alpha_{k}=1/k$ (note that the rate of decrease of
$\mathsf{MMD}_{K}(\mu,\xi_{n})$ for Algorithm 2 closely follows
$\mathcal{O}(1/n)$ when $n\gtrsim 100$). Although the bound (27) of Theorem 2
is better than (25) of Theorem 1, $\alpha_{k}=1/k$ yields better performance
than $\alpha_{k}=2/(k+1)$ all along the sequence. This suggests that there is
little interest in using more sophisticated versions of KH than Algorithm 1
with $\alpha_{k}=1/k$. We also computed the MMD for empirical measures
associated with random designs. The average and 2$\sigma$ intervals obtained
for 100 repetitions are presented, showing a decrease that closely follows
$\mathcal{O}(1/n)$, as predicted by Theorem 9. The evolution of
$\mathsf{MMD}_{K}(\mu,\xi_{n})$ obtained for $\xi_{n}$ generated with
Algorithm 4 with $\alpha_{k}=1/k$ and $\alpha_{k}=2/(k+1)$ (respectively, with
Algorithm 5) is visually hardly distinguishable from that obtained with
Algorithm 1 with $\alpha_{k}=1/k$ and $\alpha_{k}=2/(k+1)$ (respectively, with
Algorithm 2) and is not presented.
The right panel of Figure 1 shows the strong non-uniformity of the weights
$\\{\mathbf{w}_{1\,000}\\}_{i}$, $i=1,\ldots,1\,000$, associated with the
measures $\xi_{1\,000}$ generated by Algorithm 1 with $\alpha_{k}=2/(k+1)$ and
by Algorithm 2 (note that most recent points are overweighed for the former
and downweighed for the latter).
Figure 1: Left: upper bound (27) and $\mathsf{MMD}_{K}(\mu,\xi_{n})$ for
$\xi_{n}$ generated with Algorithm 1 with $\alpha_{k}=1/k$ and
$\alpha_{k}=2/(k+1)$, with Algorithm 2 and for the empirical measure
$\xi_{n}=\xi_{n,e}$ of random $n$-point designs (mean value $\pm$ $2\sigma$
over 100 repetitions), $n=1,\ldots,1\,000$. Right: weights
$\\{\mathbf{w}_{1\,000}\\}_{i}$ of $\xi_{1\,000}$.
We consider now the variants (ii) and (iii) of IWO in Algorithm 3.
Unsurprisingly, $\mathsf{MMD}_{K}(\mu,\nu_{n})$ is smaller for
$\nu_{n}=\widetilde{\xi}_{n}$ of variant (iii) than for
$\nu_{n}=\widehat{\xi}_{n}$ of variant (ii) since the weights are
unconstrained in the former case, see the left panel of Figure 2. The two
variants perform similarly for large $n$, however. The bound (30) for
Algorithm 3-(iii) is accurate for small $n$ but very pessimistic for large
$n$; see Remark 5. Algorithm 3-(ii) performs as Algorithm 1 with
$\alpha_{k}=1/k$ for small $n$ ($n\lesssim 30$) but performs significantly
better for larger $n$. The performances are quasi identical when using OLWO of
Section 3.3.1 (Frank-Wolfe Bayesian quadrature, not shown). When we stop
Algorithm 1 (with $\alpha_{k}=1/k$) at $n=200$, all weights
$\\{\widehat{\mathbf{w}}_{n}\\}_{i}$ and
$\\{\widetilde{\mathbf{w}}_{n}\\}_{i}$, $i=1,\ldots,n$, are positive. The
weights are positive too for Algorithm 3-(ii) and (iii), so that variant (ii)
coincides with Algorithm 3-(i), the fully-corrective Frank-Wolfe algorithm
(and also with the minimum-norm point algorithm, see Remark 3). The evolution
of $\mathsf{MMD}_{K}(\mu,\xi_{n})$ for $\xi_{n}$ obtained with the version
(18) of SBQ (with weights whose sum equals one) is indistinguishable from that
obtained with Algorithm 3-(ii). The behaviour of the version (12) of SBQ (with
unconstrained weights) is similar to that of Algorithm 3-(iii) and is rather
typical, see for instance Briol et al., (2015); Huszár and Duvenaud, (2012);
see also Figure 6-left. The Coordinate-Descent variant (32) of SBQ, denoted
SBQ-CD, is clearly not competitive compared to the other algorithms
considered.
Computational times777All calculations are made with Matlab, on a PC with a
clock speed of 1.5 GHz and 16 GB RAM. are shown on the right panel of Figure 2
for Algorithm 1 with $\alpha_{k}=1/k$ and $\alpha_{k}=2/(k+1)$, for Algorithm
2, and for Algorithm 3-(ii) and (iii)888All computational times start with a
positive value at $n=0$ since we account for the calculation of
$P_{K,\mu}(\mathbf{x})$ for all $\mathbf{x}\in{\mathscr{X}}_{C}$. The faster
running time than the theoretical complexity estimates given in the paper can
be explained by the internal vectorisation of operations in Matlab.. The
choice of the sequence $(\alpha_{k})_{k}$ has no influence on the
computational time of Algorithm 1; Algorithm 2 is slightly more demanding, but
its computational time still grows linearly with $n$; the better performance
of IWO shown on the left panel comes with a significant increase of
computational cost (which is similar for OLWO).
Figure 2: Left: upper bounds (27) and (30), and
$\mathsf{MMD}_{K}(\mu,\xi_{n})$ for $\xi_{n}$ generated with Algorithm 1 with
$\alpha_{k}=1/k$, with Algorithm 3-(ii) and (iii), and with version (32) of
SBQ, $n=1,\ldots,200$. Right: computational time $T(n)$ (in s) of $\xi_{n}$
for Algorithm 1 with $\alpha_{k}=1/k$ and $\alpha_{k}=2/(k+1)$, for Algorithm
2 and for Algorithm 3-(ii) and (iii), $n=1,\ldots,200$.
Computational times for Algorithm 4 with $\alpha_{k}=1/k$, Algorithm 5 and the
two versions (12) and (18) of SBQ are shown on the left panel of Figure 3:
Algorithm 4 is as fast as Algorithm 1; Algorithm 5 is slightly slower than
Algorithm 3. The two versions of SBQ are slightly slower than Algorithm 3-(ii)
and (iii) for similar performance.
MMD minimisation with $\mu$ uniform on ${\mathscr{X}}$ is an efficient method
to construct nested space-filling designs; this is one of the main motivations
in (Pronzato and Zhigljavsky, , 2020). The right panel of Figure 3 shows the
25-point design corresponding to the support of the measure $\xi_{n}$
generated with Algorithm 4 with $\alpha_{k}=1/k$, with a covering radius999The
covering radius of a design $\mathbf{X}_{n}$ is defined by
$\operatorname{\mathsf{CR}}(\mathbf{X}_{n})=\max_{\mathbf{x}\in{\mathscr{X}}}\min_{\mathbf{x}_{i}\in\mathbf{X}_{n}}\|\mathbf{x}-\mathbf{x}_{i}\|$;
a small value of $\operatorname{\mathsf{CR}}(\mathbf{X}_{n})$ indicates that
for each point in ${\mathscr{X}}$ there is a design point at proximity, hence
the frequent use of $\operatorname{\mathsf{CR}}(\mathbf{X}_{n})$ as a space-
filling characteristic to be minimised.
$\operatorname{\mathsf{CR}}(\mathbf{X}_{25})\simeq 0.1625$. Algorithm 1 with
$\alpha_{k}=1/k$ yields a very similar design, but with a different ordering
of points and a slightly larger covering radius
$\operatorname{\mathsf{CR}}(\mathbf{X}_{25})\simeq 0.1685$. When using
Algorithm 3-(ii) (respectively, Algorithm 3-(iii)), the support of $\xi_{25}$
has a covering radius $\operatorname{\mathsf{CR}}(\mathbf{X}_{25})\simeq
0.1677$ (respectively, $\operatorname{\mathsf{CR}}(\mathbf{X}_{25})\simeq
0.2024$). This illustrates the fact that a smaller MMD is not necessarily
synonym to better space-filling properties: the optimal weighting of a given
design improves its MMD, but space-filling performance, measured for instance
by the covering radius, is unweighed. In fact, when allocating uniform weights
to the support of $\xi_{n}$ generated with Algorithm 3-(iii), the MMD obtained
is similar to that shown on the left panel of Figure 2 for Algorithm 1 with
$\alpha_{k}=1/k$ (red $\bigstar$), thus much worse than for the original
$\xi_{n}$ (black $\circ$).
Figure 3: Left: computational time $T(n)$ (in s) of $\xi_{n}$ for Algorithm 4
with $\alpha_{k}=1/k$, Algorithm 5 and for the two versions (12) and (18) of
SBQ, $n=1,\ldots,200$. Right: support $\mathbf{X}_{n}$ (ordered) of $\xi_{25}$
generated with Algorithm 4 with $\alpha_{k}=1/k$; the radius of the circles
equals the covering radius of $\mathbf{X}_{n}$ (the smallest value that
permits to cover ${\mathscr{X}}$).
### 7.2 Example 2: Gaussian mixture
Here
$\mu=\sum_{j=1}^{m}\beta_{j}\,\mu_{\mathscr{N}}(\mathbf{a}_{j},\sigma_{j})$
with $\beta_{j}>0$, $\sum_{j=1}^{m}\beta_{j}=1$, where
$\mu_{\mathscr{N}}(\mathbf{a}_{j},\sigma_{j})$ corresponds to the normal
distribution with mean $\mathbf{a}_{j}$ and variance
$\sigma_{j}^{2}\,\mathbf{I}_{d}$, with $\mathbf{I}_{d}$ the identity matrix.
Again, for illustration purpose, we take $d=2$. The difficulty increases with
the number $m$ of components, the problem is also more difficult when the
weights and/or variances of the components differ. We take $m=3$,
$\mathbf{a}_{1}=(-1,1)^{\top}$, $\mathbf{a}_{2}=(1,-1)^{\top}$,
$\mathbf{a}_{3}=(1,1)^{\top}$, $\sigma_{j}=1/2$ for all $j$, and
$\beta_{1}=\beta_{2}=2/7$, $\beta_{3}=3/7$ (this is a slight variation of the
example in Figure 1 of (Teymur et al., , 2021) where the three components have
equal weights). Figure 4 presents a 3-d plot of the probability density
function $\varphi_{\mu}$ (left) and its contour lines (right) together with
the candidate set ${\mathscr{X}}_{C}$ formed by $2^{14}=16\,384$ independent
samples, among which we shall select a subset of $n$ representative points.
Figure 4: Left: 3d-plot of the p.d.f. $\varphi_{\mu}$; right: contour lines of
$\varphi_{\mu}$ and candidate set ${\mathscr{X}}_{C}$ (dots).
We use the Gaussian (or Radial Basis Function) kernel
$\displaystyle
K_{\theta}(\mathbf{x},\mathbf{x}^{\prime})=\exp-(\theta\,\|\mathbf{x}-\mathbf{x}^{\prime}\|^{2})\,,$
(33)
for which direct calculation gives101010The analytic expression of
$P_{K,\mu}(\mathbf{x})$ is also available for $K$ the product of uni-
dimensional Matérn 3/2 kernels as in Section 7.1, though the expression is
more complicated than (34) and involves the error function
$\mathrm{erf}(t)=(2/\sqrt{\pi})\int_{0}^{t}\exp(-x^{2})\mathrm{d}x$; the
experimental results obtained are similar to those presented here for the
Gaussian kernel.
$\displaystyle{\mathscr{E}}_{K_{\theta}}(\mu)$ $\displaystyle=$
$\displaystyle\sum_{j,\ell=1}^{d}\frac{\beta_{j}\beta_{\ell}}{(1+2\theta\sigma_{j}^{2}+2\theta\sigma_{\ell}^{2})^{d/2}}\,\exp\left(-\frac{\theta\,\|\mathbf{a}_{j}-\mathbf{a}_{\ell}\|^{2}}{1+2\theta\sigma_{j}^{2}+2\theta\sigma_{\ell}^{2}}\right)$
$\displaystyle P_{K,\mu}(\mathbf{x})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{d}\frac{\beta_{j}}{(1+2\theta\sigma_{j}^{2})^{d/2}}\,\exp\left(-\frac{\theta\,\|\mathbf{x}-\mathbf{a}_{j}\|^{2}}{1+2\theta\sigma_{j}^{2}}\right),\
\ \mathbf{x}\in\mathds{R}^{d}\,.$ (34)
It is important to choose a suitable order of magnitude for $\theta$, even if
a precise tuning is not essential. This issue is frequently mentioned in the
literature, see for example Huszár and Duvenaud, (2012), but is often
overlooked. As for the construction of space-filling designs where the target
measure $\mu$ is uniform on ${\mathscr{X}}$, see Pronzato and Zhigljavsky,
(2020), we recommend to let $\theta$ depend on the number of points to be
generated. If the target size is $n_{\max}$ points, each point
$\mathbf{x}_{i}$ will “represent” a fraction $1/n_{\max}$ of the $C$ candidate
points, and having a correlation $K_{\theta}(\mathbf{x}_{i},\mathbf{x})>1/2$
with $C/n_{\max}$ points seems reasonable. We thus choose $\theta$ such that
$K_{\theta}(\mathbf{x}_{i},\mathbf{x}_{j})<1/2$ for $(100/n_{\max})\%$ of the
pairs $(\mathbf{x}^{(j)},\mathbf{x}^{(k)})$ in a random sample of 1 000 points
of ${\mathscr{X}}_{C}$ (that is, with obvious notation,
$\theta=-\log(0.5)/Q_{1/n_{\max}}(\|\mathbf{x}^{(j)}-\mathbf{x}^{(k)}\|^{2})$
for the example considered).
Figure 5 shows the first $n_{\max}$ points selected by Algorithms 1 and 4,
both with $\alpha_{k}=1/k$ for all $k$: $n_{\max}=25$ ($\theta\simeq 5.7$) on
the first row, $n_{\max}=200$ ($\theta\simeq 46.4$) on the second. The points
location looks roughly the same for both algorithms when $n_{\max}=25$, the
ordering being however different starting at $n=12$; the designs look also
similar when $n_{\max}=200$ and it is difficult to separate them.
Figure 5: Designs $\mathbf{X}_{n}$ obtained with Algorithms 1 and 4 (both with
$\alpha_{k}=1/k$); $n_{\max}=25$ ($\theta\simeq 5.7$) on the first row,
$n_{\max}=200$ ($\theta\simeq 46.39$) on the second row.
Figure 6 shows the evolution of $\mathsf{MMD}_{K}(\mu,\xi_{n})$ and its upper
bound (27) for $\xi_{n}$ generated with Algorithms 1, 2, 3-(ii), 3-(iii), 4
and 5, and for empirical measures of random designs (empirical mean $\pm$ 2
standard deviations for 100 repetitions), when $n_{\max}=200$ (the designs
constructed algorithmically are not shown, but they all look very similar to
those on the second row of Figure 5). Algorithms 1 and 4 (both with
$\alpha_{k}=1/k$ for all $k$) and 2 and 5 perform similarly; Algorithm 3-(ii)
is only marginally superior; the MMD is significantly smaller for Algorithm
3-(iii) which does not set constraints on $\xi_{n}$. The weights that
$\xi_{n}$ allocates to its support points are positive for Algorithm 3-(ii)
and (iii), with $\sum_{i=1}^{200}\\{\mathbf{w}_{200}\\}_{i}\simeq 0.834$ for
Algorithm 3-(iii). The two versions (12) and (18) of SBQ perform similarly to
Algorithm 3-(iii) and (ii), respectively, and their weights
$\\{\mathbf{w}_{200}\\}_{i}$ are positive too.
Figure 6: Upper bound (27) and $\mathsf{MMD}_{K}(\mu,\xi_{n})$ for $\xi_{n}$
generated with (left) Algorithms 1, 2, 3-(ii), 3-(iii); (right) Algorithms 4
and 5, and for the empirical measure of random designs (mean value $\pm$
$2\sigma$ over 100 repetitions).
Finally, we also evaluate the approximation error by the MMD for the distance
kernel
$K_{D}(\mathbf{x},\mathbf{x}^{\prime})=-\|\mathbf{x}-\mathbf{x}^{\prime}\|$ of
Székely and Rizzo, (2013). Since $P_{K_{D},\mu}(\cdot)$ and the energy
distance ${\mathscr{E}}_{K_{D}}(\mu)$ are not known explicitly, we compute
$\mathsf{MMD}_{K_{D}}(\mu_{C},\xi_{n})$, with $\mu_{C}$ the empirical measure
for the candidate set ${\mathscr{X}}_{C}$. Figure 7 shows
$\mathsf{MMD}_{K_{D}}(\mu_{C},\xi_{n})$ for $\xi_{n}$ generated with
Algorithms 1, 3-(ii) and 4, using the Gaussian kernel (33)—$\xi_{n}$ generated
with Algorithm 3-(iii) cannot be tested since its weights do not sum to
one111111$K_{D}$ is Conditionally Integrally Strictly Positive Definite and
defines a metric between probability measures, but
${\mathscr{E}}_{K_{D}}(\mu_{C}-\xi)$ can be negative for
$\xi\not\in{\mathscr{M}}_{[1]}({\mathscr{X}})$.. The three algorithms appear
to perform similarly and tend to provide better approximations of $\mu$ than
random sampling in terms of $\mathsf{MMD}_{K_{D}}(\mu_{C},\cdot)$ (the
empirical mean $\pm$ two standard deviations for 100 random designs is
presented). Due to their much smaller computational costs, Algorithms 1 and 4
are preferable to Algorithm 3-(ii) (and version (18) of SBQ) in this example.
We also tried to use a Stein kernel, based on the inverse multiquadric kernel
$K_{s,\theta}(\mathbf{x},\mathbf{x}^{\prime})=1/(1+\theta\,\|\mathbf{x}-\mathbf{x}^{\prime}\|^{2})^{s}$,
$\theta>0$, $s\in(0,1)$, instead of (33) to generate $\xi_{n}$, but the values
obtained for $\mathsf{MMD}_{K_{D}}(\mu_{C},\xi_{n})$ were significantly larger
than with (33)121212We used $s=1/2$ and $\theta$ given by the median heuristic
of Garreau et al., (2017); see also Teymur et al., (2021)..
Figure 7: $\mathsf{MMD}_{K_{D}}(\mu_{C},\xi_{n})$ for the empirical measure of
random designs (mean value $\pm$ $2\sigma$ over 100 repetitions) and for
$\xi_{n}$ generated with Algorithms 1, 3-(ii), and 4, all using the Gaussian
kernel (33), with $K_{D}$ the distance kernel of Székely and Rizzo, (2013).
## 8 Conclusions
Bounds on the finite-sample-size approximation error of iterative methods for
the minimisation of an MMD discrepancy have been derived and illustrated by
numerical experiments. These experiments indicate that the bounds give a fair
picture of the decrease rate of the true MMD for some of the methods
considered (Algorithms 2 and 5), but are pessimistic for most of them. This is
particularly true for SBQ with unconstrained weights (12), for which the link
with kernel herding used for the derivation of the error bound gives a
plausible explanation for its marked pessimism; see Remark 5. These numerical
results also indicate that the performances of kernel herding and greedy MMD
minimisation do not improve by considering other step-size sequences than
$1/k$ (which generate empirical measures), and that a variant of kernel
herding with optimised weights, Algorithm 3-(iii), yields performance similar
to standard SBQ for a slightly lower computational cost. Therefore, on the
whole, Algorithm 3-(iii) appears to be the best option when the budget $n$ is
very limited (its complexity is quadratic in $n$), and standard KH or MMD,
with uniform weights, seem generally preferable to more sophisticated methods
for large $n$ (their complexity is linear in $n$).
We have restricted our attention to finite candidate sets. This situation is
at the same time easier and computationally more efficient in terms of
practical implementation, and simpler in terms of analysis since only finite-
dimensional linear algebra is used, but the extension to the Hilbert-space
situation remains possible.
Finally, we have only considered the case where one adds one-point-at-a-time
to the construction. Less myopic methods that select several (say $m>1$)
points at each iterations could also be considered. The extension of our
results to this context, and the development of computationally efficient
methods that avoid the combinatorial explosion due to considering all possible
$C\choose m$ subset selections, deserve further studies. One can refer to
Teymur et al., (2021) for an exciting contribution in this direction.
## Appendix A: alternative expressions for $\mathsf{MMD}_{K}^{2}(\mu,\xi)$
The quadratic form (4) of $\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ and the fact
that $\widetilde{\xi}_{n}$ has unconstrained optimal weights implies that, for
any $\xi_{n}\in{\mathscr{M}}(\mathbf{X}_{n})$,
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})=(\mathbf{w}_{n}-\widetilde{\mathbf{w}}_{n})^{\top}\mathbf{K}_{n}(\mathbf{w}_{n}-\widetilde{\mathbf{w}}_{n})+\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{n})\,.$
Therefore, any measure $\xi$ in ${\mathscr{M}}({\mathscr{X}}_{C})$ with
associated weights $\mathbf{\omega}$ satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi)=\widetilde{g}_{C}(\mathbf{\omega})+\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}^{C})\,,$
(35)
where, for any $\mathbf{\omega}\in\mathds{R}^{C}$, we denote
$\displaystyle\widetilde{g}_{C}(\mathbf{\omega})=\|\mathbf{\omega}-\widetilde{\mathbf{\omega}}^{C}\|_{\mathbf{K}_{C}}^{2}=(\mathbf{\omega}-\widetilde{\mathbf{\omega}}^{C})^{\top}\mathbf{K}_{C}(\mathbf{\omega}-\widetilde{\mathbf{\omega}}^{C})\quad(=\mathsf{MMD}_{K}^{2}(\xi,\widetilde{\xi}^{C}))\,.$
(36)
When $\xi_{n}\in{\mathscr{M}}_{[1]}(\mathbf{X}_{n})$ (i.e.,
$\mathbf{w}_{n}^{\top}\mathbf{1}_{n}=1$), as
$(\mathbf{w}_{n}-\widehat{\mathbf{w}}_{n})^{\top}\mathbf{1}_{n}=0$ and
$\mathbf{K}_{n}(\widehat{\mathbf{w}}_{n}-\widetilde{\mathbf{w}}_{n})\propto\mathbf{1}_{n}$,
see (5) and (6),
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n})$ $\displaystyle=$
$\displaystyle(\mathbf{w}_{n}-\widehat{\mathbf{w}}_{n}+\widehat{\mathbf{w}}_{n}-\widetilde{\mathbf{w}}_{n})^{\top}\mathbf{K}_{n}(\mathbf{w}_{n}-\widehat{\mathbf{w}}_{n}+\widehat{\mathbf{w}}_{n}-\widetilde{\mathbf{w}}_{n})+\mathsf{MMD}_{K}^{2}(\mu,\widetilde{\xi}_{n})\,,$
$\displaystyle=$
$\displaystyle(\mathbf{w}_{n}-\widehat{\mathbf{w}}_{n})^{\top}\mathbf{K}_{n}(\mathbf{w}_{n}-\widehat{\mathbf{w}}_{n})+\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}_{n})\,.$
Therefore, any measure $\xi$ in ${\mathscr{M}}_{[1]}({\mathscr{X}}_{C})$ with
associated weights $\mathbf{\omega}$ satisfies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi)=\widehat{g}_{C}(\mathbf{\omega})+\mathsf{MMD}_{K}^{2}(\mu,\widehat{\xi}^{C})\,,$
(37)
where
$\displaystyle\widehat{g}_{C}(\mathbf{\omega})=\|\mathbf{\omega}-\widehat{\mathbf{\omega}}^{C}\|_{\mathbf{K}_{C}}^{2}=(\mathbf{\omega}-\widehat{\mathbf{\omega}}^{C})^{\top}\mathbf{K}_{C}(\mathbf{\omega}-\widehat{\mathbf{\omega}}^{C})\quad(=\mathsf{MMD}_{K}^{2}(\xi,\widehat{\xi}^{C}))\,.$
(38)
For any measure $\xi\in{\mathscr{M}}({\mathscr{X}}_{C})$ with associated
weights $\mathbf{\omega}\in{\mathscr{P}}_{C}$, we define
$\displaystyle\Delta_{C}(\xi)=\mathsf{MMD}_{K}^{2}(\mu,\xi)-M_{C}^{2}\,,$ (39)
so that (35) implies
$\Delta_{C}(\xi)=\widetilde{g}_{C}(\mathbf{\omega})-\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})$
and, when $\xi\in{\mathscr{M}}_{[1]}({\mathscr{X}}_{C})$, (37) implies
$\Delta_{C}(\xi)=\widehat{g}_{C}(\mathbf{\omega})-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})$.
## Appendix B: proofs
Our derivations of bounds on $\Delta_{C}(\xi_{k})$ given by (39) (i.e., on
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})$) rely on the convexity of
$\widehat{g}_{C}(\cdot)$ and $\widetilde{g}_{C}(\cdot)$ and on the following
lemma.
###### Lemma 2.
Let $(t_{k})_{k}$ and $(\alpha_{k})_{k}$ be two real positive sequences and
$A$ be a strictly positive real.
(i) If $t_{k}$ satisfies
$\displaystyle t_{1}\leq A\ \mbox{ and }\
t_{k+1}\leq(1-\alpha_{k+1})\,t_{k}+A\,\alpha_{k+1}^{2}\,,\ k\geq 1\,,$ (40)
with $\alpha_{k}=1/k$ for all $k$, then $t_{k}\leq A\,(2+\log k)/(k+1)$ for
all $k\geq 1$.
(ii) If $t_{k}$ satisfies (40) with $\alpha_{k}=2/(k+1)$ for all $k$, then
$t_{k}\leq 4\,A/(k+3)$ for all $k\geq 1$.
(iii) If $t_{k}$ satisfies
$\displaystyle t_{1}\leq A\ \mbox{ and }\
t_{k+1}\leq(1-2\,\alpha_{k+1})\,t_{k}+A\,\alpha_{k+1}^{2}\,,\ k\geq 1\,,$ (41)
with $\alpha_{k}=1/k$ for all $k$, then $t_{k}\leq A/k$ for all $k\geq 1$.
(iv) If $t_{k}$ satisfies
$\displaystyle t_{1}\leq A\ \mbox{ and }\ t_{k+1}\leq
t_{k}-\frac{t_{k}^{2}}{A}\,,\ k\geq 1\,,$ (42)
then, $t_{k}\leq A/(k+p_{2})$ for all $k\geq 2$, with $p_{2}=A/t_{2}-2\geq 2$;
moreover, when $t_{1}\leq A/2$, $t_{k}\leq A/(k+p_{1})$ for all $k\geq 1$,
with $p_{1}=A/t_{1}-1\geq 1$.
Proof.
(i) Suppose that $t_{k}$ satisfies (40) with $\alpha_{k}=1/k$ for all $k$. We
show that $t_{k}\leq A\,(2+\log k)/(k+1)$ by induction on $k$. The inequality
is satisfied for $k=1$, assume that it is satisfied for $k\geq 1$. We get
$A\,[2+\log(k+1)]/(k+2)-t_{k+1}\geq A\,a(k)/[(k+2)(k+1)^{2}]$, with
$a(k)=(k+1)^{2}\,\log(1+1/k)+\log k-k\geq 0$, implying that the inequality is
satisfied for all $k$.
(ii) Suppose now that $t_{k}$ satisfies (40) with $\alpha_{k+1}=b/(k+1+q)$ for
all $k$, for some $0<b<q+2$. We prove that $t_{k}\leq A\,a/(k+p)$ for some
$a,p>0$ by induction on $k$. Not all values of $a,b,p,q$ are legitimate, and a
natural objective is to have $a$ and $p$ respectively as small and large as
possible. We show that the best choice is that indicated in Lemma 2. For
$k=1$, since $t_{1}\leq A$, to ensure that $t_{1}\leq A\,a/(p+1)$ we need to
have $p\leq a-1$. Assume that $t_{k}\leq A\,a/(k+p)$, and denote
$\delta_{k}=A\,a/(k+1+p)-t_{k+1}$. It satisfies
$\displaystyle\delta_{k}$ $\displaystyle\geq$ $\displaystyle
A\,\frac{a}{k+1+p}-\left[(1-\alpha_{k+1})\,t_{k}+B_{C}\,\alpha_{k+1}^{2}\right]\geq
A\,\frac{a(k)}{(k+p)(k+1+p)(k+1+q)^{2}}\,,$
where $a(k)$ is a second-degree polynomial in $k$, with leading term $(ab-
a-b^{2})\,k^{2}$. We thus need to choose a pair $(a,b)$ of positive numbers
such that $ab-a-b^{2}\geq 0$; the pair with the smallest value of $a$ is
$(4,2)$. For this choice of $a,b$, we get $a(k)=4\,[k+1+p-(p-q)^{2}]$, which
increases with $k$. We only need to guarantee that $a(1)\geq 0$, which
corresponds to $q+1/2-(1/2)\,\sqrt{4q+9}\leq p\leq q+1/2+(1/2)\,\sqrt{4q+9}$.
Since $a=4$, the largest $p$ allowed is $p=3$, which is admissible for $q=1$.
In that case, $a(k)=4k$ and $\delta_{k}\geq 0$, showing that $t_{k}\leq
4\,A/(k+3)$ for all $k$ when (40) is satisfied with $\alpha_{k}=2/(k+1)$ for
all $k$.
(iii) Suppose that $t_{k}$ satisfies (41) with $\alpha_{k}=1/k$ for all $k$.
We have $t_{1}\leq A$ by hypothesis; the induction hypothesis $t_{k}\leq A/k$
and (41) give $A/(k+1)-t_{k+1}\geq A/[k(k+1)^{2}]>0$, and thus imply that
$t_{k+1}\leq A/(k+1)$.
(iv) The function $t\to f(t)=t-t^{2}/A$ is increasing on $[0,A/2)$ with a
maximum on $[0,A]$ equal to $A/4$ attained for $t=A/2$. We have $t_{2}\leq
f(A/2)=A/4$, and thus $t_{k}\leq A/4$ for all $k\geq 2$. Take $p=A/t_{2}-2$,
so that $p\geq 2$ and $t_{2}=A/(p+2)\leq A/4$. Suppose that $t_{k}\leq
A/(p+k)$; we have $t_{k+1}\leq
A\,[1/(k+p)-1/(k+p)^{2}]=A\,(k+p-1)/(k+p)^{2}=A/(k+p+1)-A/[(k+p)^{2}(k+p+1)]<A/(k+p+1)$,
showing that $t_{k}\leq A/(p+k)$ for all $k\geq 2$.
When $t_{1}\leq A/2$, we take $p=A/t_{1}-1$, which gives $p\geq 1$,
$t_{1}=A/(p+1)\leq A/2$ and $t_{k}<A/2$ for all $k>1$. Assuming that
$t_{k}\leq A/(p+k)$, we get $t_{k+1}<A/(k+p+1)$, showing that $t_{k}\leq
A/(p+k)$ for all $k\geq 1$.
Proof of Theorem 1. The proof is based on (Clarkson, , 2010). For any
$\mathbf{x}^{(j)}\in{\mathscr{X}}_{C}$, the definition (39) of
$\Delta_{C}(\xi)$ gives
$\displaystyle\Delta_{C}[\xi_{k}^{+}(\mathbf{x}^{(j)},\alpha_{k+1})]$
$\displaystyle=$
$\displaystyle\widehat{g}_{C}[\mathbf{\omega}_{k}+\alpha_{k+1}(\mathbf{e}_{j}-\mathbf{\omega}_{k})]-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})$
(43) $\displaystyle=$
$\displaystyle\widehat{g}_{C}(\mathbf{\omega}_{k})-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})+2\,\alpha_{k+1}\,(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})+\alpha_{k+1}^{2}\|\mathbf{e}_{j}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}\,,$
where $\alpha_{k+1}=1/(k+1)$. The definition of
$\widehat{\mathbf{\omega}}^{C}$ implies that
$\mathbf{u}^{\top}\mathbf{K}_{c}\widehat{\mathbf{\omega}}^{C}=\mathbf{u}^{\top}\mathbf{p}_{C}(\mu)$
for any $\mathbf{u}\in\mathds{R}^{C}$ orthogonal to $\mathbf{1}_{C}$, see (5).
Therefore,
$(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}\widehat{\mathbf{\omega}}^{C}=P_{K,\mu}(\mathbf{x}^{(j)})-\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}P_{K,\mu}(\mathbf{x}_{i})$,
and $\mathbf{x}_{k+1}=\mathbf{x}^{(j_{k+1})}$ with
$\displaystyle
j_{k+1}\in\mathrm{Arg}\min_{j\in\mathds{I}_{C}}P_{K,\xi_{k}}(\mathbf{x}^{(j)})-P_{K,\mu}(\mathbf{x}^{(j)})=\mathrm{Arg}\min_{j\in\mathds{I}_{C}}(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})\,,$
in agreement with (21). The convexity of $\widehat{g}_{C}(\cdot)$ implies that
$\displaystyle\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})\geq\widehat{g}_{C}(\mathbf{\omega}_{k})+2\,(\mathbf{\omega}^{C}_{*}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})\geq\widehat{g}_{C}(\mathbf{\omega}_{k})+2\,\min_{j\in\mathds{I}_{C}}(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})\,,$
(44)
where the second inequality follows from
$\mathbf{\omega}^{C}_{*}\in{\mathscr{P}}_{C}$, implying that
$\displaystyle\Delta_{C}(\xi_{k+1})=\Delta_{C}[\xi_{k}^{+}(\mathbf{x}_{k+1},\alpha_{k+1})]$
$\displaystyle\leq$
$\displaystyle(1-\alpha_{k+1})\,[\widehat{g}_{C}(\mathbf{\omega}_{k})-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})]+\alpha_{k+1}^{2}\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}\,.$
The last term can be bounded as follows: as the weights $\mathbf{\omega}_{k}$
belong to ${\mathscr{P}}_{C}$ for all $k$, for all $j\in\mathds{I}_{C}$, we
have
$\displaystyle\|\mathbf{e}_{j}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}\leq\max_{\mathbf{\omega},\mathbf{\omega}^{\prime}\in{\mathscr{P}}_{C}}(\mathbf{\omega}-\mathbf{\omega}^{\prime})^{\top}\mathbf{K}_{C}(\mathbf{\omega}-\mathbf{\omega}^{\prime})$
$\displaystyle=$
$\displaystyle\max_{i,j\in\mathds{I}_{C}}(\mathbf{e}_{i}-\mathbf{e}_{j})^{\top}\mathbf{K}_{C}(\mathbf{e}_{i}-\mathbf{e}_{j})$
(45)
$\displaystyle=\max_{i,j\in\mathds{I}_{C}}K(\mathbf{x}^{(i)},\mathbf{x}^{(i)})+K(\mathbf{x}^{(j)},\mathbf{x}^{(j)})-2\,K(\mathbf{x}^{(i)},\mathbf{x}^{(j)})\leq
B_{C}\,,$
with $B_{C}=4\,\overline{K}_{C}$ ($B_{C}=2\,\overline{K}_{C}$ when $K\geq 0$).
It implies that
$\displaystyle\Delta_{C}(\xi_{k+1})\leq(1-\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}\,.$
(46)
Since
$\displaystyle\Delta_{C}(\xi_{1})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{1})=K(\mathbf{x}_{1},\mathbf{x}_{1})-2\,P_{K,\mu}(\mathbf{x}_{1})+{\mathscr{E}}_{K}(\mu)\leq
B_{C}\,,$ (47)
Lemma 2-(i) gives (25) when $\alpha_{k}=1/k$ for all $k$.
When $\widehat{\xi}^{C}=\xi^{C}_{*}$,
$\widehat{\mathbf{\omega}}^{C}\in{\mathscr{P}}_{C}$, and
$\Delta_{C}(\xi_{k})=\widehat{g}_{C}(\mathbf{\omega}_{k})$. Therefore,
$\displaystyle\min_{j\in\mathds{I}_{C}}(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})\leq(\widehat{\mathbf{\omega}}^{C}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})=-\widehat{g}_{C}(\mathbf{\omega}_{k})\,,$
(48)
and we get
$\Delta_{C}(\xi_{k+1})\leq(1-2\,\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}$
instead of (46). Lemma 2-(iii) gives (26).
Proof of Theorem 2. $\Delta_{C}(\xi_{k})$ satisfies (46) with
$\alpha_{k+1}=2/(k+2)$. Lemma 2-(ii) gives (27).
Proof of Theorem 3. Consider again the proof of Theorem 1. We have
$\displaystyle\Delta_{C}(\xi_{k+1})=\min_{\alpha\in[0,1]}\Delta_{C}[\xi_{k}^{+}(\mathbf{x}^{(j_{k+1})},\alpha)]\leq\Delta_{C}[\xi_{k}^{+}(\mathbf{x}^{(j_{k+1})},\alpha_{k+1})]\leq(1-\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}\,,$
(49)
for any arbitrary choice of $\alpha_{k+1}\in[0,1]$, and $\alpha_{k+1}=2/(k+2)$
for all $k$ has been shown to imply (27) in Theorem 2. When
$\widehat{\xi}^{C}=\xi^{C}_{*}$, we get
$\Delta_{C}(\xi_{k+1})\leq(1-2\,\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}$,
see the proof of Theorem 1. When we use $\alpha_{k}=1/k$ for all $k$, Lemma
2-(iii) implies (26).
The value of $\alpha$ minimising
$\Delta_{C}[\xi_{k}^{+}(\mathbf{x}^{(j_{k+1})},\alpha)]$ is
$\displaystyle\widehat{\alpha}_{k+1}=\frac{(\mathbf{\omega}_{k}-\mathbf{e}_{j_{k+1}})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})}{\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}}\,,$
(50)
so that (44) implies $\widehat{\alpha}_{k+1}>0$ when
$\mathsf{MMD}_{K}(\mu,\xi_{k})>\mathsf{MMD}_{K}(\mu,\xi^{C}_{*})$. The
algorithm can thus be stopped if $\widehat{\alpha}_{k+1}=0$. Since
$(\mathbf{\omega}_{k}-\mathbf{e}_{j_{k+1}})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})=\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}+(\mathbf{e}_{j_{k+1}}-\widehat{\mathbf{\omega}}^{C})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})-\|\mathbf{e}_{j_{k+1}}-\widehat{\mathbf{\omega}}^{C}\|_{\mathbf{K}_{C}}^{2}$,
we have $\widehat{\alpha}_{k+1}\leq
1+(\mathbf{e}_{j_{k+1}}-\widehat{\mathbf{\omega}}^{C})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})/\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}$.
When $\widehat{\mathbf{\omega}}^{C}\in{\mathscr{P}}_{C}$ (that is, when
$\widehat{\mathbf{\omega}}^{C}=\mathbf{\omega}^{C}_{*}$), the definition of
$j_{k+1}$ implies that $\widehat{\alpha}_{k+1}\leq 1$ (since
$\sum_{j=1}^{C}\\{\widehat{\omega}^{C}\\}_{j}(\mathbf{e}_{j}-\widehat{\mathbf{\omega}}^{C})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})=0$
with all $\\{\widehat{\omega}^{C}\\}_{j}\geq 0$). However, nothing guarantees
that $\widehat{\alpha}_{k+1}\leq 1$ in general. Direct calculation gives
$\displaystyle(\mathbf{\omega}_{k}-\mathbf{e}_{j})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})=\sum_{i,\ell=1}^{k}\\{\mathbf{w}_{k}\\}_{i}\\{\mathbf{w}_{k}\\}_{\ell}K(\mathbf{x}_{i},\mathbf{x}_{\ell})-\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}K(\mathbf{x}_{i},\mathbf{x}^{(j)})$
$\displaystyle\hskip
142.26378pt+P_{K,\mu}(\mathbf{x}^{(j)})-\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}P_{K,\mu}(\mathbf{x}_{i})\,,$
(51)
$\displaystyle\|\mathbf{e}_{j}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}=\sum_{i,\ell=1}^{k}\\{\mathbf{w}_{k}\\}_{i}\\{\mathbf{w}_{k}\\}_{\ell}K(\mathbf{x}_{i},\mathbf{x}_{\ell})-2\,\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}K(\mathbf{x}_{i},\mathbf{x}^{(j)})+K(\mathbf{x}^{(j)},\mathbf{x}^{(j)})\,,$
(52)
which together with (50) gives (28). The recursive updating of
$Q_{k}=\sum_{i,\ell=1}^{k}\\{\mathbf{w}_{k}\\}_{i}\\{\mathbf{w}_{k}\\}_{\ell}K(\mathbf{x}_{i},\mathbf{x}_{\ell})$,
$R_{k}=\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}P_{K,\mu}(\mathbf{x}_{i})$ and
$S_{k}(\mathbf{x})=P_{K,\xi_{k}}(\mathbf{x})$ gives Algorithm 2.
Proof of Theorem 4.
(i) When $\nu_{k}=\xi_{k}^{*}$ is substituted for $\xi_{k}$, we have
$\mathbf{x}_{k+1}=\mathbf{x}^{(j_{k+1})}$ with
$j_{k+1}\in\mathrm{Arg}\min_{j\in\mathds{I}_{C}}(\mathbf{e}_{j}-\mathbf{\omega}_{k}^{*})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}^{*}-\widehat{\mathbf{\omega}}^{C})$
and, by construction,
$\Delta_{C}(\xi_{k+1})\leq\Delta_{C}[\nu_{k}^{+}(\mathbf{x}_{k+1},\alpha)]$
for any $\alpha\in[0,1]$, where $\Delta_{C}(\xi)$ is given by (39) and
$\nu_{k}^{+}(\mathbf{x},\alpha)=(1-\alpha)\,\xi_{k}^{*}+\alpha\,\delta_{\mathbf{x}}$.
Consider (43) in the proof of Theorem 1: we have
$\displaystyle\Delta_{C}[\nu_{k}^{+}(\mathbf{x}_{k+1},\alpha)]=\widehat{g}_{C}(\mathbf{\omega}_{k}^{*})-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})+2\,\alpha\,(\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{*})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}^{*}-\widehat{\mathbf{\omega}}^{C})+\alpha^{2}\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{*}\|_{\mathbf{K}_{C}}^{2}\,,$
and the convexity of $\widehat{g}_{C}(\cdot)$ implies
$\displaystyle\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})\geq\widehat{g}_{C}(\mathbf{\omega}_{k}^{*})+2\,(\mathbf{\omega}^{C}_{*}-\mathbf{\omega}_{k}^{*})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}^{*}-\widehat{\mathbf{\omega}}^{C})\geq\widehat{g}_{C}(\mathbf{\omega}_{k}^{*})+2\,(\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{*})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}^{*}-\widehat{\mathbf{\omega}}^{C})\,,$
where the second inequality follows from
$\mathbf{\omega}^{C}_{*}\in{\mathscr{P}}_{C}$ and the choice of $j_{k+1}$.
Using
$\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{*}\|_{\mathbf{K}_{C}}^{2}\leq
B_{C}$, see the proof of Theorem 1, we thus obtain
$\displaystyle\Delta_{C}(\xi_{k+1})\leq\Delta_{C}[\nu_{k}^{+}(\mathbf{x}_{k+1},\alpha_{k+1})]\leq(1-\alpha_{k+1})\,\Delta_{C}(\nu_{k})+B_{C}\,\alpha_{k+1}^{2}\,,\quad
k\geq 1\,,$ (53)
for any predefined $\alpha_{k+1}$. When $\alpha_{k+1}=2/(k+2)$, the induction
used in the proof of Theorem 2 gives (27). When
$\widehat{\xi}^{C}=\xi^{C}_{*}$, the right-hand side of (53) becomes
$(1-2\,\alpha_{k+1})\,\Delta_{C}(\nu_{k})+B_{C}\,\alpha_{k+1}^{2}$, see the
proof of Theorem 1, and if we take $\alpha_{k}=1/k$ for all $k$, Lemma 2-(iii)
implies (26).
(ii) Suppose now that $\nu_{k}=\widehat{\xi}_{k}$ is substituted for $\xi_{k}$
at iteration $k$. Equation (51) gives
$\displaystyle
P_{K,\widehat{\xi}_{k}}(\mathbf{x}_{k+1})-P_{K,\mu}(\mathbf{x}_{k+1})+\widehat{\mathbf{w}}_{k}^{\top}\mathbf{p}_{k}(\mu)-{\mathscr{E}}_{K}(\widehat{\xi}_{k})=(\mathbf{e}_{j_{k+1}}-\widehat{\mathbf{\omega}}_{k})^{\top}\mathbf{K}_{C}(\widehat{\mathbf{\omega}}_{k}-\widehat{\mathbf{\omega}}^{C})\,,$
so that (17) gives
$\displaystyle\Delta_{C}(\xi_{k+1})=\Delta_{C}(\xi_{k})-\frac{\left[(\mathbf{e}_{j_{k+1}}-\widehat{\mathbf{\omega}}_{k})^{\top}\mathbf{K}_{C}(\widehat{\mathbf{\omega}}_{k}-\widehat{\mathbf{\omega}}^{C})\right]^{2}}{\min_{\scriptsize\begin{array}[]{l}\mathbf{w}\in\mathds{R}^{k}\\\
\mathbf{1}_{k}^{\top}\mathbf{w}=1\end{array}}\|K(\mathbf{x}_{k+1},\cdot)-\mathbf{w}^{\top}\mathbf{k}_{k}(\cdot)\|_{\mathcal{H}_{K}}^{2}}\,.$
As long as
$\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})\leq\widehat{g}_{C}(\widehat{\mathbf{\omega}}_{k})$,
that is, $\Delta_{C}(\widehat{\xi}_{k})\geq 0$, (44) with
$\widehat{\mathbf{\omega}}_{k}$ substituted for $\mathbf{\omega}_{k}$ gives
$\displaystyle\Delta_{C}(\xi_{k+1})$ $\displaystyle\leq$
$\displaystyle\Delta_{C}(\xi_{k})-\frac{\left[\widehat{g}_{C}(\widehat{\mathbf{\omega}}_{k})-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})\right]^{2}}{4\,\|K(\mathbf{x}_{k+1},\cdot)-(\mathbf{1}_{k}^{\top}/k)\mathbf{k}_{k}(\cdot)\|_{\mathcal{H}_{K}}^{2}}$
(55)
$\displaystyle=\Delta_{C}(\xi_{k})-\frac{\Delta_{C}^{2}(\xi_{k})}{4\,\left[K(\mathbf{x}_{k+1},\mathbf{x}_{k+1})+\mathbf{1}_{k}^{\top}\mathbf{K}_{k}\mathbf{1}_{k}/k^{2}-2\,\mathbf{1}_{k}^{\top}\mathbf{k}_{k}(\mathbf{x}_{k+1})/k\right]}$
$\displaystyle\leq\Delta_{C}(\xi_{k})-\frac{\Delta_{C}^{2}(\xi_{k})}{4\,B_{C}}\,,$
with $\Delta_{C}(\xi_{1})\leq B_{C}$, see (47). Lemma 2-(iv) with $A=4\,B_{C}$
and $p_{1}=3$ gives (27). When $\widehat{\xi}^{C}=\xi^{C}_{*}$, (48) gives
$\Delta_{C}(\xi_{k+1})\leq\Delta_{C}(\xi_{k})-\Delta_{C}^{2}(\xi_{k})/B_{C}$,
and Lemma 2-(iv) with $A=B_{C}$ and $p_{2}=2$ gives (29).
(iii) Suppose finally that $\nu_{k}=\widetilde{\xi}_{k}$ is substituted for
$\xi_{k}$ at iteration $k$. Since $\widetilde{\xi}_{k}$ is not necessarily in
${\mathscr{M}}_{[1]}({\mathscr{X}}_{C}$), we shall use (36) instead of (38).
The definition of $\widetilde{\mathbf{\omega}}^{C}$ implies
$\mathbf{K}_{C}\widetilde{\mathbf{\omega}}^{C}=\mathbf{p}_{C}(\mu)$, and thus
$\mathbf{x}_{k+1}=\mathbf{x}^{(j_{k+1})}$ with
$\displaystyle
j_{k+1}\in\mathrm{Arg}\min_{j\in\mathds{I}_{C}}P_{K,\nu_{k}}(\mathbf{x}^{(j)})-P_{K,\mu}(\mathbf{x}^{(j)})=\mathrm{Arg}\min_{j\in\mathds{I}_{C}}\mathbf{e}_{j}^{\top}\mathbf{K}_{C}(\widetilde{\mathbf{\omega}}_{k}-\widetilde{\mathbf{\omega}}^{C})\,.$
Let
$\widetilde{\mathbf{g}}_{k}=2\,\mathbf{K}_{C}(\widetilde{\mathbf{\omega}}_{k}-\widetilde{\mathbf{\omega}}^{C})$
denote the gradient of $\widetilde{g}(\cdot)$ at
$\widetilde{\mathbf{\omega}}_{k}$. By construction,
$\mathbf{e}_{j}^{\top}\widetilde{\mathbf{g}}_{k}=0$ for all $j$ with
$\mathbf{x}^{(j)}\in\mathrm{supp}(\widehat{\xi}^{k})$, so that
$\widetilde{\mathbf{\omega}}_{k}^{\top}\mathbf{K}_{C}(\widetilde{\mathbf{\omega}}_{k}-\widetilde{\mathbf{\omega}}^{C})=0$,
and the convexity of $\widetilde{g}(\cdot)$ implies
$\displaystyle\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})\geq\widetilde{g}_{C}(\widetilde{\mathbf{\omega}}_{k})+2\,(\mathbf{\omega}^{C}_{*}-\widetilde{\mathbf{\omega}}_{k})^{\top}\mathbf{K}_{C}(\widetilde{\mathbf{\omega}}_{k}-\widetilde{\mathbf{\omega}}^{C})=\widetilde{g}_{C}(\widetilde{\mathbf{\omega}}_{k})+2\,{\mathbf{\omega}^{C}_{*}}^{\top}\mathbf{K}_{C}(\widetilde{\mathbf{\omega}}_{k}-\widetilde{\mathbf{\omega}}^{C})\,.$
(56)
Since $\mathbf{\omega}^{C}_{*}\in{\mathscr{P}}_{C}$, the definition of
$j_{k+1}$ implies
$\displaystyle\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})\geq\widetilde{g}_{C}(\widetilde{\mathbf{\omega}}_{k})+2\,\mathbf{e}_{j_{k+1}}^{\top}\mathbf{K}_{C}(\widetilde{\mathbf{\omega}}_{k}-\widetilde{\mathbf{\omega}}^{C})\,.$
(57)
Therefore, as long as
$\Delta_{C}(\widetilde{\xi}_{k})=\widetilde{g}_{C}(\widetilde{\mathbf{\omega}}_{k})-\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})\geq
0$, (11) gives
$\displaystyle\Delta_{C}(\xi_{k+1})$ $\displaystyle=$
$\displaystyle\Delta_{C}(\xi_{k})-\frac{\left[\mathbf{e}_{j_{k+1}}^{\top}\mathbf{K}_{C}(\widetilde{\mathbf{\omega}}_{k}-\widetilde{\mathbf{\omega}}^{C})\right]^{2}}{\left[K(\mathbf{x}_{k+1},\mathbf{x}_{k+1})-\mathbf{k}_{k}^{\top}(\mathbf{x}_{k+1})\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x}_{k+1})\right]}$
(58) $\displaystyle\leq$
$\displaystyle\Delta_{C}(\xi_{k})-\frac{\Delta_{C}^{2}(\xi_{k})}{4\,\left[K(\mathbf{x}_{k+1},\mathbf{x}_{k+1})-\mathbf{k}_{k}^{\top}(\mathbf{x}_{k+1})\mathbf{K}_{k}^{-1}\mathbf{k}_{k}(\mathbf{x}_{k+1})\right]}$
$\displaystyle\leq\Delta_{C}(\xi_{k})-\frac{\Delta_{C}^{2}(\xi_{k})}{4\,\overline{K}_{C}}\leq\Delta_{C}(\xi_{k})-\frac{\Delta_{C}^{2}(\xi_{k})}{4\,\overline{K}}\,.$
Since any signed measure supported on ${\mathscr{X}}_{C}$ can be used, we may
start with $\xi_{0}=0$, with weights $\mathbf{\omega}_{0}=\mathbf{0}_{C}$, so
that (11) and the inequality above apply from $k=0$. We have
$\Delta_{C}(\xi_{0})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{0})={\mathscr{E}}_{K}(\mu)\leq\overline{K}$;
therefore, $\Delta_{C}(\xi_{k})\leq 4\,\overline{K}/(k+p_{1})$, $k\geq 1$, for
$p_{1}=4\,\overline{K}/\mathsf{MMD}_{K}^{2}(\mu,\xi_{1})-1\leq
4\,\overline{K}/\Delta_{C}(\xi_{1})-1$; see the proof of Lemma 2-(iv).
$\Delta_{C}(\xi_{0})\leq\overline{K}$ implies $\Delta_{C}(\xi_{1})\leq
3\,\overline{K}/4$, and we can also take $p_{1}=13/3$ in Lemma 2-(iv), which
gives $\Delta_{C}(\xi_{k})\leq 4\,\overline{K}/(k+13/3)$, $k\geq 1$. This
completes the proof of (30).
Stopping conditions for Algorithms 3-(ii) and (iii). Let
$\widehat{\mathbf{g}}_{k}=2\,\mathbf{K}_{C}(\widehat{\mathbf{\omega}}_{k}-\widehat{\mathbf{\omega}}^{C})$
denote the gradient of $\widehat{g}(\cdot)$ at
$\widehat{\mathbf{\omega}}_{k}$. By construction,
$(\mathbf{e}_{j}-\widehat{\mathbf{\omega}}_{k})^{\top}\widehat{\mathbf{g}}_{k}=0$
for all $j$ such that $\mathbf{x}^{(j)}\in\mathrm{supp}(\widehat{\xi}^{k})$,
with
$\displaystyle(\mathbf{e}_{j}-\widehat{\mathbf{\omega}}_{k})^{\top}\widehat{\mathbf{g}}_{k}=2\left\\{P_{K,\widehat{\xi}^{k}}(\mathbf{x}^{(j)})-P_{K,\mu}(\mathbf{x}^{(j)})+\sum_{i=1}^{k}\\{\widehat{\mathbf{w}}_{k}\\}_{i}\left[P_{K,\mu}(\mathbf{x}_{i})-P_{K,\widehat{\xi}^{k}}(\mathbf{x}_{i})\right]\right\\}\,.$
Therefore,
$P_{K,\widehat{\xi}^{k}}(\mathbf{x}^{(j)})-P_{K,\mu}(\mathbf{x}^{(j)})$ equals
a constant $c_{k}$ for all
$\mathbf{x}^{(j)}\in\mathrm{supp}(\widehat{\xi}^{k})$, and the existence of an
$\mathbf{x}\in{\mathscr{X}}_{C}$ such that
$P_{K,\widehat{\xi}^{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})<c_{k}$ implies that
Algorithm 3-(ii) can still progress. Conversely, if
$P_{K,\widehat{\xi}^{k}}(\mathbf{x})-P_{K,\mu}(\mathbf{x})\geq c_{k}$ for all
$\mathbf{x}\in{\mathscr{X}}_{C}$, the algorithm can be stopped. We can thus
add the following line to Algorithm 3-(ii):
4’-(ii):
if $S_{k-1}(\mathbf{x}_{k})-P_{K,\mu}(\mathbf{x}_{k})\geq
S_{k-1}(\mathbf{x}_{k-1})-P_{K,\mu}(\mathbf{x}_{k-1})$ then return
$\mathbf{X}_{k-1}$, $\xi_{k-1}$ and stop;
Similarly,
$\mathbf{e}_{j}^{\top}\widetilde{\mathbf{g}}_{k}=2\left[P_{K,\widehat{\xi}^{k}}(\mathbf{x}^{(j)})-P_{K,\mu}(\mathbf{x}^{(j)})\right]=0$
for all $j$ with $\mathbf{x}^{(j)}\in\mathrm{supp}(\widehat{\xi}^{k})$, with
$\widetilde{\mathbf{g}}_{k}$ the gradient of $\widetilde{g}(\cdot)$ at
$\widetilde{\mathbf{\omega}}_{k}$; see the proof of Theorem 4-(iii).
Therefore,
$P_{K,\widehat{\xi}^{k}}(\mathbf{x}^{(j)})=P_{K,\mu}(\mathbf{x}^{(j)})$ for
all $\mathbf{x}^{(j)}\in\mathrm{supp}(\widetilde{\xi}^{k})$; Algorithm 3-(iii)
can be stopped when $P_{K,\widehat{\xi}^{k}}(\mathbf{x})\geq
P_{K,\mu}(\mathbf{x})$ for all $\mathbf{x}\in{\mathscr{X}}_{C}$ and we can add
the following line to Algorithm 3-(iii):
4’-(iii):
if $S_{k-1}(\mathbf{x}_{k})-P_{K,\mu}(\mathbf{x}_{k})\geq 0$ then return
$\mathbf{X}_{k-1}$, $\xi_{k-1}$ and stop;
Proof of Theorem 5. We have
${\mathscr{E}}_{K}(\nu-\mu)={\mathscr{E}}_{K_{\mu}}(\nu)$ for any
$\nu\in{\mathscr{M}}_{[1]}({\mathscr{X}}_{C})$ with $K_{\mu}$ the reduced
kernel defined by (13); see Damelin et al., (2010), Pronzato and Zhigljavsky,
(2020, Th. 3.5). Let $\mathbf{\omega}_{k}$ be the vector of weights associated
with $\xi_{k}$ at step $k$. Since
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})=\mathbf{\omega}_{k}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega}_{k}$,
see (14), we have
$\displaystyle(k+1)^{2}\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{k+1})=k^{2}\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})+2k\,\mathbf{\omega}_{k}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{e}_{j_{k+1}}+K_{\mu}(\mathbf{x}^{(j_{k+1})},\mathbf{x}^{(j_{k+1})})\,,$
where
$j_{k+1}\in\mathrm{Arg}\min_{j\in\mathds{I}_{C}}2k\,\mathbf{\omega}_{k}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{e}_{j}+K_{\mu}(\mathbf{x}^{(j)},\mathbf{x}^{(j)})$,
and $\mathbf{x}^{(j_{k+1})}$ coincides with (19) when ${\mathscr{X}}_{C}$ is
substituted for ${\mathscr{X}}$. Therefore, for any
$\mathbf{\omega}\in{\mathscr{P}}_{C}$, we have
$\displaystyle(k+1)^{2}\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{k+1})$
$\displaystyle\leq$ $\displaystyle
k^{2}\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})+2k\,\mathbf{\omega}_{k}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega}+\overline{K}_{\mu,C}$
(59) $\displaystyle\leq
k^{2}\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})+2k\,(\mathbf{\omega}_{k}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega}_{k})^{1/2}(\mathbf{\omega}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega})^{1/2}+\overline{K}_{\mu,C}\,,$
where
$\overline{K}_{\mu,C}=\max_{\mathbf{x}\in{\mathscr{X}}_{C}}K_{\mu}(\mathbf{x},\mathbf{x})\leq
A_{C}$, see (22). The inequality (59) is satisfied in particular for
$\mathbf{\omega}^{C}_{*}\in\mathrm{Arg}\min_{\mathbf{\omega}\in{\mathscr{P}}_{C}}\mathbf{\omega}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega}$,
with
${\mathbf{\omega}^{C}_{*}}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{\omega}^{C}_{*}=M_{C}^{2}$,
therefore
$\displaystyle(k+1)^{2}\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{k+1})$
$\displaystyle\leq$ $\displaystyle
k^{2}\,\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})+2k\,M_{C}\,\mathsf{MMD}_{K}(\mu,\xi_{k})+A_{C}\,.$
(60)
We prove (31) by induction on $n$. For $n=1$,
$\mathsf{MMD}_{K}^{2}(\mu,\delta_{\mathbf{x}_{1}})=K_{\mu}(\mathbf{x}_{1},\mathbf{x}_{1})\leq\overline{K}_{\mu,C}\leq
A_{C}$. Suppose that (31) is true for $n$. Then, (60) implies
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{n+1})$ $\displaystyle\leq$
$\displaystyle\frac{1}{(n+1)^{2}}\,\left\\{n^{2}\left(M_{C}^{2}+A_{C}\,\frac{1+\log
n}{n}\right)+2n\,M_{C}\,\left(M_{C}^{2}+A_{C}\,\frac{1+\log
n}{n}\right)^{1/2}+A_{C}\right\\}$
$\displaystyle\leq\frac{1}{(n+1)^{2}}\,\left\\{n^{2}\left(M_{C}^{2}+A_{C}\,\frac{1+\log
n}{n}\right)+n\,\left(2\,M_{C}^{2}+A_{C}\,\frac{1+\log
n}{n}\right)+A_{C}\right\\}$
$\displaystyle=M_{C}^{2}+A_{C}\,\frac{1+\log(n+1)}{n+1}-\frac{M_{C}^{2}+A_{C}\left[(n+1)\log(1+1/n)-1\right]}{(n+1)^{2}}$
$\displaystyle=M_{C}^{2}+A_{C}\,\frac{1+\log(n+1)}{n+1}-\frac{M_{C}^{2}/(n+1)+A_{C}\left[-\log(1-1/(n+1))-1/(n+1)\right]}{n+1}$
$\displaystyle\leq M_{C}^{2}+A_{C}\,\frac{1+\log(n+1)}{n+1}$
since $\log(1+x)\leq x$ for $x>-1$, which concludes the proof of (31).
Proof of Theorem 6. Using (43) and (45), we get
$\displaystyle\min_{\mathbf{x}\in{\mathscr{X}}_{C}}\Delta_{C}[\xi_{k}^{+}(\mathbf{x},\alpha_{k+1})]\leq\widehat{g}_{C}(\mathbf{\omega}_{k})-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})+\min_{j\in\mathds{I}_{C}}2\,\alpha_{k+1}\,(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})+B_{C}\,\alpha_{k+1}^{2}\,,$
and, from the same arguments as in the proof of Theorem 1,
$\displaystyle\Delta_{C}(\xi_{k+1})=\min_{\mathbf{x}\in{\mathscr{X}}_{C}}\Delta_{C}[\xi_{k}^{+}(\mathbf{x},\alpha_{k+1})]\leq(1-\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}\,.$
(61)
When $\alpha_{k}=2/(k+1)$ for all $k$, Lemma 2-(ii) gives (27).
Similarly, when $\widehat{\xi}^{C}=\xi^{C}_{*}$, we have
$\displaystyle\Delta_{C}(\xi_{k+1})=\min_{\mathbf{x}\in{\mathscr{X}}_{C}}\Delta_{C}[\xi_{k}^{+}(\mathbf{x},\alpha_{k+1})]\leq(1-2\,\alpha_{k+1})\,\Delta_{C}(\xi_{k})+B_{C}\,\alpha_{k+1}^{2}\,,$
see the proof of Theorem 1, and Lemma 2-(iii) gives (26).
Proof of Theorem 7. For $\alpha\in[0,1]$, any
$\mathbf{x}^{(j)}\in{\mathscr{X}}_{C}$ satisfies
$\displaystyle\Delta_{C}[\xi_{k}^{+}(\mathbf{x}^{(j)},\alpha)]=\widehat{g}_{C}(\mathbf{\omega}_{k})-\widehat{g}(\mathbf{\omega}^{C}_{*})+2\,\alpha\,(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})+\alpha^{2}\,\|\mathbf{e}_{j}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}\,,$
(62)
see (43), the right-hand side of which is minimum when
$\displaystyle\alpha=\widehat{\alpha}_{k+1,j}=\frac{(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\widehat{\mathbf{\omega}}^{C}-\mathbf{\omega}_{k})}{\|\mathbf{e}_{j}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}}\,.$
By restricting $\alpha$ to $[0,1]$, we obtain
$[\mathbf{x}_{k+1},\alpha_{k+1}]=[\mathbf{x}^{(j_{k+1}^{*})},\alpha_{k+1,j}^{*}]$
with
$\displaystyle\alpha_{k+1,j}^{*}$ $\displaystyle=$
$\displaystyle\max\\{0,\min\\{\widehat{\alpha}_{k+1,j},1\\}\\}$ $\displaystyle
j_{k+1}^{*}$ $\displaystyle=$
$\displaystyle\mathrm{arg}\min_{j\in\mathds{I}_{C}}2\,\alpha_{k+1,j}^{*}\,(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})+(\alpha_{k+1,j}^{*})^{2}\,\|\mathbf{e}_{j}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}\,.$
Using (51) and (52), we obtain that $\widehat{\alpha}_{k+1,j}$ is given by
(28) with $\mathbf{x}^{(j)}$ substituted for $\mathbf{x}_{k+1}$. The recursive
updating of
$Q_{k}=\sum_{i,\ell=1}^{k}\\{\mathbf{w}_{k}\\}_{i}\\{\mathbf{w}_{k}\\}_{\ell}K(\mathbf{x}_{i},\mathbf{x}_{\ell})$,
$R_{k}=\sum_{i=1}^{k}\\{\mathbf{w}_{k}\\}_{i}P_{K,\mu}(\mathbf{x}_{i})$ and
$S_{k}(\mathbf{x})=P_{K,\xi_{k}}(\mathbf{x})$ gives Algorithm 5. As in the
proof of Theorem 3,
$\mathsf{MMD}_{K}(\mu,\xi_{k})>\mathsf{MMD}_{K}(\mu,\xi^{C}_{*})$ implies that
there exists $j\in\mathds{I}_{C}$ such that
$(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})<0$,
and therefore $\widehat{\alpha}_{k+1,j}=\alpha(\mathbf{x}^{(j)})>0$.
Conversely, $\alpha(\mathbf{x})=0$ for all $\mathbf{x}\in{\mathscr{X}}_{C}$
implies that $\mathsf{MMD}_{K}(\mu,\xi_{k})=\mathsf{MMD}_{K}(\mu,\xi^{C}_{*})$
and Algorithm 5 can be stopped.
As $\|\mathbf{e}_{j}-\mathbf{\omega}_{k}\|_{\mathbf{K}_{C}}^{2}\leq B_{C}$ in
(62), see (45), we have
$\displaystyle\min_{\mathbf{x}\in{\mathscr{X}}_{C},\alpha\in[0,1]}\Delta_{C}[\xi_{k}^{+}(\mathbf{x},\alpha)]\leq\widehat{g}_{C}(\mathbf{\omega}_{k})-\widehat{g}_{C}(\mathbf{\omega}^{C}_{*})+\min_{j\in\mathds{I}_{C}}2\,\alpha_{k+1}\,(\mathbf{e}_{j}-\mathbf{\omega}_{k})^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widehat{\mathbf{\omega}}^{C})+B_{C}\,\alpha_{k+1}^{2}$
for any predefined choice of $\alpha_{k+1}$ in $[0,1]$. The rest of the proof
is similar to that of Theorem 3.
Proof of Theorem 8. For any $\xi_{k}$, the value of $\Delta_{C}(\xi_{k+1})$
obtained by SBQ cannot exceed that obtained by IWO applied to KH, which yields
the bounds given in Theorem 8 for (12) and (18).
Denote $\xi^{++}(\mathbf{x},\alpha)=\xi+w\delta_{\mathbf{x}}$ for any
$\xi\in{\mathscr{M}}({\mathscr{X}}_{C})$, $\mathbf{x}\in{\mathscr{X}}_{C}$ and
$w\in\mathds{R}$, so that (32) corresponds to
$\xi_{k+1}=\xi_{k}^{++}(\mathbf{x}_{k+1},w_{k+1})$ with
$[\mathbf{x}_{k+1},w_{k+1}]\in\mathrm{Arg}\min_{\mathbf{x}\in{\mathscr{X}}_{C},\,w}\mathsf{MMD}_{K}^{2}[\xi_{k}^{++}(\mathbf{x},w)]$.
We get
$\displaystyle\Delta_{C}(\xi_{k+1})=\Delta_{C}(\xi_{k})-\frac{[\mathbf{e}_{j_{k+1}}^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widetilde{\mathbf{\omega}}^{C})]^{2}}{K(\mathbf{x}_{k+1},\mathbf{x}_{k+1})}\,,$
where
$j_{k+1}\in\mathrm{Arg}\max_{j\in\mathds{I}_{C}}[\mathbf{e}_{j}^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widetilde{\mathbf{\omega}}^{C})]^{2}/K(\mathbf{x}^{(j)},\mathbf{x}^{(j)})$,
see the proof of Theorem 4-(iii). The convexity of $\widetilde{g}(\cdot)$
implies
$\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})\geq\widetilde{g}_{C}(\mathbf{\omega}_{k})+2\,{\mathbf{\omega}^{C}_{*}}^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widetilde{\mathbf{\omega}}^{C})$,
see (56). Therefore, as long as
$\Delta_{C}(\xi_{k})=\widetilde{g}_{C}(\mathbf{\omega}_{k})-\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})\geq
0$,
$[{\mathbf{\omega}^{C}_{*}}^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widetilde{\mathbf{\omega}}^{C})]^{2}\geq[\widetilde{g}_{C}(\mathbf{\omega}_{k})-\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})]^{2}/4$.
The maximum of
$[\mathbf{\omega}^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widetilde{\mathbf{\omega}}^{C})]^{2}$
with respect to $\mathbf{\omega}\in{\mathscr{P}}_{C}$ is attained on a vertex
of ${\mathscr{P}}_{C}$, which implies that
$[\mathbf{e}_{j_{k+1}}^{\top}\mathbf{K}_{C}(\mathbf{\omega}_{k}-\widetilde{\mathbf{\omega}}^{C})]^{2}\geq[\widetilde{g}_{C}(\mathbf{\omega}_{k})-\widetilde{g}_{C}(\mathbf{\omega}^{C}_{*})]^{2}/4$
when $K(\mathbf{x},\mathbf{x})=\overline{K}_{C}$ for all $\mathbf{x}$, and
$\Delta_{C}(\xi_{k})$ satisfies (58). The conclusion is the same as for
Theorem 4-(iii).
Proof of Lemma 1. $A_{C}$ depends on ${\mathscr{X}}_{C}$, but $A_{C}\leq
A(\mu)$ since $\overline{K}_{C}\leq\overline{K}$; similarly, $B_{C}\leq B$.
Since
$M_{C}^{2}\leq\mathbf{1}_{C}^{\top}{\mathbf{K}_{\mu}}_{C}\mathbf{1}_{C}/C^{2}$,
we get
$\displaystyle\mathsf{E}\\{M_{C}^{2}\\}$ $\displaystyle\leq$
$\displaystyle\frac{1}{C^{2}}\,\mathsf{E}\left\\{\sum_{i,j=1}^{C}K_{\mu}(\mathbf{x}^{(i)},\mathbf{x}^{(j)})\right\\}$
$\displaystyle=\frac{1}{C^{2}}\,\mathsf{E}\left\\{\sum_{i,j=1}^{C}K(\mathbf{x}^{(i)},\mathbf{x}^{(j)})\right\\}-\frac{2}{C}\,\mathsf{E}\left\\{\sum_{i=1}^{C}P_{K,\mu}(\mathbf{x}^{(i)})\right\\}+{\mathscr{E}}_{K}(\mu)$
$\displaystyle=\frac{1}{C^{2}}\,\mathsf{E}\left\\{\sum_{i=1}^{C}K(\mathbf{x}^{(i)},\mathbf{x}^{(i)})\right\\}+\frac{1}{C^{2}}\,\mathsf{E}\left\\{\sum_{\scriptsize\begin{array}[]{l}i,j=1\\\
i\neq
j\end{array}}^{C}K(\mathbf{x}^{(i)},\mathbf{x}^{(j)})\right\\}-{\mathscr{E}}_{K}(\mu)$
$\displaystyle=\frac{\tau_{1}(\mu)}{C}+\frac{C(C-1)}{C^{2}}\,{\mathscr{E}}_{K}(\mu)-{\mathscr{E}}_{K}(\mu)=\frac{\tau_{1}(\mu)-{\mathscr{E}}_{K}(\mu)}{C}\,.\hskip
113.81102pt\mbox{}~{}\hfill\rule{5.69054pt}{5.69054pt}$
Proof of Theorem 9. We have
$\mathsf{MMD}_{K}^{2}(\mu,\xi_{n,e})=\mathbf{1}_{n}^{\top}{\mathbf{K}_{\mu}}_{n}\mathbf{1}_{n}/n^{2}$,
with $\mathsf{E}_{\mu}\\{K_{\mu}(\cdot,X)\\}\equiv 0$ on ${\mathscr{X}}$ and
$\int_{{\mathscr{X}}^{2}}K_{\mu}(\mathbf{x},\mathbf{x}^{\prime})\,\mathrm{d}\mu(\mathbf{x})\mathrm{d}\mu(\mathbf{x}^{\prime})={\mathscr{E}}_{K_{\mu}}(\mu)=0$.
From Serfling, (1980, p. 194), the U-statistic
$U_{n}=2/[n(n-1)]\,\sum_{i<j}K_{\mu}(\mathbf{x}_{i},\mathbf{x}_{j})$ satisfies
$n\,U_{n}\stackrel{{\scriptstyle\rm
d}}{{\rightarrow}}Y=\sum_{i=1}^{\infty}\lambda_{i}(\chi_{1i}^{2}-1)$. The
V-statistic
$V_{n}=(1/n^{2})\,\sum_{i,j}K_{\mu}(\mathbf{x}_{i},\mathbf{x}_{j})=\mathsf{MMD}_{K}^{2}(\mu,\xi_{n,e})$
satisfies
$V_{n}=(1-1/n)\,U_{n}+(1/n^{2})\,\sum_{i}K_{\mu}(\mathbf{x}_{i},\mathbf{x}_{i})$,
with $U_{n}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}0$ and
$(1/n)\,\sum_{i}K_{\mu}(\mathbf{x}_{i},\mathbf{x}_{i})\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}\sum_{i=1}^{\infty}\lambda_{i}$. Therefore,
$n\,V_{n}\stackrel{{\scriptstyle\rm
d}}{{\rightarrow}}Z=\sum_{i=1}^{\infty}\lambda_{i}\chi_{1i}^{2}$.
## Appendix C: multiple random candidate sets
Suppose that $\mathbf{x}_{k+1}$ is selected within ${\mathscr{X}}_{C}[k+1]$ at
iteration $k$. Denote
${\mathscr{X}}_{C_{k+1}}=\cup_{i=1}^{k+1}{\mathscr{X}}_{C}[i]$ and let
${\mathscr{P}}_{C_{k+1}}$ be the corresponding probability simplex in
$\mathds{R}^{C_{k+1}}$ (with $C_{k+1}=(k+1)\,C$ when the
${\mathscr{X}}_{C}[i]$ do not intersect). To a measure $\xi_{k}$ in
${\mathscr{M}}({\mathscr{X}}_{C_{k}})$ (respectively, in
${\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C_{k}})$) corresponds a vector of
weights $\mathbf{\omega}_{k}$ in $\mathds{R}^{C_{k}}$ (respectively, in
${\mathscr{P}}_{C_{k}}$), and we denote by $\mathbf{\omega}_{k}^{\prime}$ the
same vector plunged into $\mathds{R}^{C_{k+1}}$ (respectively, into
${\mathscr{P}}_{C_{k+1}}$); $\xi_{*}^{C[k+1]}$ is the measure in
${\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C}[k+1])$ that minimises
$\mathsf{MMD}(\mu,\xi)$ and $\mathbf{\omega}_{*}^{C[k+1]^{\prime}}$ is the
vector of associated weights in ${\mathscr{P}}_{C_{k+1}}$. Similarly,
$\widehat{\mathbf{\omega}}^{C_{k+1}}$ denotes the vector of weights for the
optimal measure in ${\mathscr{M}}_{[1]}^{+}({\mathscr{X}}_{C_{k+1}})$.
Consider one-step-ahead algorithms (Algorithms 1, 2, 4 and 5) and
$\xi_{k}^{+}[\mathbf{x}_{k+1},\alpha_{k+1}]=(1-\alpha_{k+1})\,\xi_{k}+\alpha_{k+1}\,\delta_{\mathbf{x}_{k+1}}$
constructed at iteration $k$. We have
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{k}^{+}[\mathbf{x}_{k+1},\alpha_{k+1}])-\mathsf{MMD}_{K}^{2}(\mu,\xi_{*}^{C[k+1]})$
$\displaystyle=$
$\displaystyle\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime})-\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{*}^{C[k+1]^{\prime}})$
$\displaystyle+2\,\alpha_{k+1}\,(\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{\prime})^{\top}\mathbf{K}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime}-\widehat{\mathbf{\omega}}^{C_{k+1}})+\alpha_{k+1}^{2}\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{\prime}\|^{2}_{\mathbf{K}_{C_{k+1}}}\,,$
with $\mathbf{e}_{j_{k+1}}$ the basis vector in $\mathds{R}^{C_{k+1}}$
corresponding to $\mathbf{x}_{k+1}\in{\mathscr{X}}_{C}[k+1]$, and
$\|\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{\prime}\|^{2}_{\mathbf{K}_{C_{k+1}}}\leq
B$, see (43), (45) and Lemma 1. The convexity of
$\widehat{g}_{C_{k+1}}(\cdot)$ implies
$\displaystyle\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{*}^{C[k+1]^{\prime}})\geq\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime})+2\,(\mathbf{\omega}_{*}^{C[k+1]^{\prime}}-\mathbf{\omega}_{k}^{\prime})^{\top}\mathbf{K}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime}-\widehat{\mathbf{\omega}}^{C_{k+1}})$
so that
$2\,(\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{\prime})^{\top}\mathbf{K}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime}-\widehat{\mathbf{\omega}}^{C_{k+1}})\leq\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{*}^{C[k+1]^{\prime}})-\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime})$
when $\mathbf{x}_{k+1}$ is chosen by kernel herding. This gives
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{k}^{+}[\mathbf{x}_{k+1},\alpha_{k+1}])-\mathsf{MMD}_{K}^{2}(\mu,\xi_{*}^{C[k+1]})$
$\displaystyle\leq$
$\displaystyle(1-\alpha_{k+1})\,\left[\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})-\mathsf{MMD}_{K}^{2}(\mu,\xi_{*}^{C[k+1]})\right]+B\,\alpha_{k+1}^{2}\,.$
When each ${\mathscr{X}}_{C}[i]$ is made of independent samples from $\mu$, we
have
$\mathsf{E}_{\mu}\\{\mathsf{MMD}_{K}^{2}(\mu,\xi_{*}^{C[i]})\\}=\mathsf{E}_{\mu}\\{M_{C}^{2}\\}$
for all $i$, and therefore, when $\alpha_{k+1}$ is predefined (deterministic),
$\displaystyle\mathsf{E}_{\mu}\\{\Delta(\xi_{k+1})\\}\leq(1-\alpha_{k+1})\,\mathsf{E}_{\mu}\\{\Delta(\xi_{k})\\}+B\,\alpha_{k+1}^{2}\,,$
where we have denoted
$\Delta(\xi)=\Delta_{C[1]}(\xi)=\mathsf{MMD}_{K}^{2}(\mu,\xi)-\mathsf{MMD}_{K}^{2}(\mu,\xi_{*}^{C[1]})$.
From the same arguments as those used in the proof of Theorem 1, we thus
obtain that the measure generated by Algorithm 1 with $\alpha_{k}=1/k$ and
using a set ${\mathscr{X}}_{C}[k]$ composed of $C$ independent samples from
$\mu$ for all $k$ satisfies
$\mathsf{E}_{\mu}\\{\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})\\}\leq\mathsf{E}_{\mu}\\{M_{C}^{2}\\}+B\,(2+\log
n)/(n+1)$, $n\geq 1$, compare with (25). When $\alpha_{k}=2/(k+1)$ in
Algorithm 1, then
$\mathsf{E}_{\mu}\\{\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})\\}\leq\mathsf{E}_{\mu}\\{M_{C}^{2}\\}+4\,B/(n+3)$,
$n\geq 1$, compare with (27). Following the arguments used in the proof of
Theorem 3, see (49), we get the same bound for Algorithm 2. Also, following
the arguments in the proofs of Theorems 6 and 7, similar bounds are obtained
for Algorithms 4 and 5, which extends the results in those theorems to this
situation of multiple random candidate sets. A similar extension applies to
Algorithm 3-(i); see the proof of Theorem 4-(i).
Consider now Algorithm 3-(ii) and (iii) and SBQ. For Algorithm 3-(ii), we have
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{k+1})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})-\frac{\left[(\mathbf{e}_{j_{k+1}}-\mathbf{\omega}_{k}^{\prime})^{\top}\mathbf{K}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime}-\widehat{\mathbf{\omega}}^{C_{k+1}})\right]^{2}}{B}$
and the convexity of $\widehat{g}_{C_{k+1}}(\cdot)$ gives
$\displaystyle\mathsf{MMD}_{K}^{2}(\mu,\xi_{k+1})\leq\mathsf{MMD}_{K}^{2}(\mu,\xi_{k})-\frac{\left[\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime})-\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{*}^{C[k+1]^{\prime}})\right]^{2}}{4\,B}\,.$
Since
$\mathsf{E}_{\mu}\\{\mathsf{MMD}_{K}^{2}(\mu,\xi_{*}^{C[i]})\\}=\mathsf{E}_{\mu}\\{M_{C}^{2}\\}$
for all $i$, Jensen’s inequality gives
$\displaystyle\mathsf{E}_{\mu}\left\\{\left[\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime})-\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{*}^{C[k+1]^{\prime}})\right]^{2}\right\\}$
$\displaystyle\geq$
$\displaystyle\left[\mathsf{E}_{\mu}\left\\{\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{k}^{\prime})-\widehat{g}_{C_{k+1}}(\mathbf{\omega}_{*}^{C[k+1]^{\prime}})\right\\}\right]^{2}=\mathsf{E}_{\mu}^{2}\\{\Delta(\xi_{k})\\}\,.$
We thus obtain
$\displaystyle\mathsf{E}_{\mu}\\{\Delta(\xi_{k+1})\\}\leq\mathsf{E}_{\mu}\\{\Delta(\xi_{k})\\}-\frac{\mathsf{E}_{\mu}^{2}\\{\Delta(\xi_{k})\\}}{4\,B}\,,$
an inequality similar to (55) in the proof of Theorem 4-(ii), and $\xi_{n}$
satisfies an inequality of the form (27) where each term is replaced by its
expected value.
Similar developments yield an inequality similar to (58), with expected values
everywhere, and thus an extension of (30) for Algorithm 3-(iii). Theorem 8,
that indicates that the performance of SBQ cannot be worse that that of
Algorithm 3, continues to apply; the details are omitted.
## Acknowledgments
This work was partly supported by project INDEX (INcremental Design of
EXperiments) ANR-18-CE91-0007 of the French National Research Agency (ANR).
The author would like to thank Chris Oates for sending him a preprint of the
paper (Teymur et al., , 2021) which was deeply inspiring.
The author is grateful to the two anonymous referees for their careful reading
and their comments and suggestions which helped to improve the paper.
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[1]
<EMAIL_ADDRESS>
# $\alpha$-cluster formation in heavy $\alpha$-emitters through nucleonic
self-assembly
J. M. Dong Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou
730000, China School of Physics, University of Chinese Academy of Sciences,
Beijing 100049, China Q. Zhao School of Nuclear Science and Technology,
Lanzhou University, Lanzhou 730000, China L.-J. Wang School of Physical
Science and Technology, Southwest University, Chongqing 400715, China W. Zuo
J. Z. Gu China Institute of Atomic Energy, P. O. Box 275(10), Beijing 102413,
China
###### Abstract
$\alpha$-decay always has enormous impetuses to the development of physics and
chemistry, in particular due to its indispensable role in the research of new
elements. Although it has been observed in laboratories for more than a
century, it remains a difficult problem to calculate accurately the formation
probability $S_{\alpha}$ microscopically. We establish a self-assembly model
that an $\alpha$-particle is generated through a nucleonic self-assembly, and
the corresponding formation probability $S_{\alpha}$ values of some typical
$\alpha$-emitters are calculated without adjustable parameters. The
experimental half-lives, in particular their irregular behavior around a shell
closure, are remarkably well reproduced by half-life laws combined with these
$S_{\alpha}$. In our strategy, the cluster formation is a gradual process in
heavy nuclei, different from the situation that cluster pre-exists in light
nuclei. The present study may pave the way to a fully understanding of
$\alpha$-decay from the perspective of nuclear structure.
###### keywords:
alpha-decay Formation probability Self-assembly Superheavy nuclei
$\alpha$-decay is a typical radioactive phenomenon in which an atomic nucleus
emits a helium nucleus spontaneously. As one of the most important decay modes
for heavy and superheavy nuclei, it was regarded as a quantum-tunneling effect
firstly in the pioneering works of Gamov, Condon and Gurney in 1928 [1, 2],
which provided an extremely significant evidence supporting the probability
interpretation of quantum mechanics in the early stage of nuclear physics.
However, a full understanding of $\alpha$-decay mechanism and hence an
accurate description of the half-life, have not been settled yet. The critical
problem lies in how to understand the mechanism of $\alpha$-cluster formation
and compute the formation probability, known as a long-standing problem for
nuclear physics for more than eighty years that has attracted considerable
interest continuously [3]. The $\alpha$-decay is really understood only if the
formation probability $S_{\alpha}$ can be well determined microscopically.
The investigation of the $\alpha$-formation probability also promotes the
exploration of cluster structures in nuclei. Actually, numerous experimental
observations have already revealed clustering phenomena in some light nuclei,
such as the famous Hoyle state in stellar nucleosynthesis that exhibits a
structure composed of three $\alpha$-particles [4]. The theoretical
exploration of the mechanism of cluster formation has been a hot topic in
nuclear physics [5, 6, 7]. For heavy nuclei, a novel manifestation of
$\alpha$-clustering structure, namely, “$\alpha+^{208}$Pb" states in 212Po was
revealed experimentally by their enhanced $E1$ decays [8]. Yet, it is still an
open question that whether or not the light and heavy nuclei share the same
mechanism of cluster formation.
Importantly, $\alpha$-decay has far-reaching implications in the research of
superheavy nuclei (SHN) [9, 10]. Since an “island of stability" of SHN was
predicted in the 1960s, experimental efforts worldwide have continuously
embarked on such hugely expensive programs since it is always at the exciting
forefront in both chemistry and physics [11, 10]. However, the exact locations
of the corresponding nuclear magic numbers remain unknown, and theoretical
approaches to date do not yield consistent predictions. The direct measurement
of nuclear binding energies and detailed spectroscopic studies of SHN with
$Z>110$ have been beyond experimental capabilities [12, 13, 14], therefore, to
uncover their underlying structural information, one has to resort to the mere
knowledge about measured $\alpha$-decay energies and half-lives [14]. The
formation probability, if available microscopically, is of enormous importance
to change this embarrassing situation in combination with accurately measured
$\alpha$-decay properties.
Because of its fundamental importance, theoretically, the exploration of the
$\alpha$-particle formation can be traced back to 1960 [15], which triggered
extensive investigations with shell models [3, 16, 17, 18], Bardeen-Cooper-
Schriffer (BCS) models [3, 19, 20] and Skyrme energy density functionals [21]
later. The formation amplitude is regarded as the overlap between the
configuration of a parent nucleus and the one described by an
$\alpha$-particle coupled to the daughter nucleus. In particular, 212Po as a
typical $\alpha$-emitter with two protons and two neutrons outside the doubly
magic core 208Pb, was discussed extensively. Nevertheless, these calculations
disagree on the decay width, and underestimate it substantially [3, 20, 22].
To improve the calculations, the shell model combined with a cluster
configuration, was proposed with a treatment of all correlations between
nucleons on the same footing [23]. Yet, these calculations tend to be
difficult to generalize for nuclei more complex than 212Po. Over the past two
decades, several new approaches have been put forward to calculate the
formation probability $S_{\alpha}$ in different frameworks, including the
pairing approach [24], $N_{n}N_{p}$ scheme [25], quantum-mechanical
fragmentation theory [26], cluster-formation models [27, 28, 29], a
quartetting wave function approach [30, 31, 32], internal barrier
penetrability approach [33], statistical method [34], some empirical relations
[35, 36, 37, 38], and extraction combined with experimental data [39, 40, 41,
42, 43, 44]. Although great efforts have been made and considerable progress
has been achieved, no fully satisfactory approach has yet been found until
now.
To explore the $\alpha$-particle formation probability $S_{\alpha}$, we
propose a completely new scenario, termed the nucleonic self-assembly model,
and calculate the $S_{\alpha}$ explicitly without introducing any adjustable
parameter with the help of a self-consistent density functional theory. Self-
assembly is a spontaneous process in which a disordered system of pre-existing
components forms an organized structure as a consequence of interactions,
without extrinsic intervention [45], and this concept is used increasingly in
many disciplines.
Figure 1: (Color online) Schematic illustration of the physical picture for an
$\alpha$-cluster formation in the nucleonic self-assembly model through one of
many pathways. The intermediate configuration (IC) is the parent nucleus with
mass number $A$ but with a blocked neutron level $i_{n}$ and a blocked proton
level $i_{p}$ ($v_{i}^{2}=1$). These four nucleons filling the two single-
particle levels will jump to the unoccupied $i_{n}$\- and $i_{p}$-levels of
the daughter nucleus, and then self-assemble into an $\alpha$-particle
finally.
Before we explore the $\alpha$-cluster formation probability, we first discuss
briefly the proton spectroscopic factor of proton radioactivity where the
component of the last odd-proton can be emitted since the proton is a
quasiparticle inside the nucleons [46]. The initial state is proton
quasiparticle excitations of parent BCS vacuum
$\alpha_{i}^{\dagger}|$BCS$\rangle_{\text{P}}$ and the final state is
$c_{i}^{\dagger}|$BCS$\rangle_{\text{D}}$ with
$c_{i}^{\dagger}|$BCS$\rangle_{\text{D}}=\left[u_{i}^{(\text{D})}\alpha_{i}^{\dagger}+v_{i}^{(\text{D})}\alpha_{\overline{i}}\right]|$BCS$\rangle_{\text{D}}$.
The spectroscopic factor (for spherical nuclei) is then given by
$S_{p}=|_{\text{D}}\langle$BCS$|c_{i}\alpha_{i}^{\dagger}|$BCS$\rangle_{\text{P}}|^{2}\approx(u_{i}^{(\text{D})})^{2}$
[46, 47], where $(u_{i}^{(D)})^{2}$ is the probability that the spherical
orbit of the emitted proton is empty in the daughter nucleus. Accordingly, the
$S_{p}$ can be iconically interpreted as a probability that the odd-proton in
the $i_{p}$-orbit of the parent nucleus jumps to the unoccupied $i_{p}$-orbit
of the daughter nucleus. Intriguingly, one does not need the explicit
wavefunction of the emitted proton. With the inclusion of the calculated
$S_{p}$ from nuclear many-body approaches, the partial half-lives for
spherical proton emitters can be quite well reproduced [48, 49], indicating
the success of the strategy for proton spectroscopic factor.
Inspired by the proton radioactivity, we propose a new strategy for
$\alpha$-decays, i.e., the nucleonic self-assembly model, and a sketch is
exhibited in Fig. 1 to show schematically the physical picture of our model.
However, different from the proton radioactivity where the emitted proton
comes from the blocked proton orbit in the parent nucleus, the neutrons
(protons) inside the emitted $\alpha$-particle could come from any single-
neutron (proton) level in principle. Therefore, we introduce the intermediate
configuration (IC) with mass number $A$ to characterize which levels donate
the complete four nucleons for the $\alpha$-formation, as a key idea of our
strategy. The IC is a state that a single-neutron level ($i_{n}$) and a
single-proton level ($i_{p}$) in the parent nucleus are fully-occupied,
namely, their occupation probabilities $v_{i}^{2}=1$ for $i=i_{n},i_{p}$
(which makes sure there are exactly four nucleons from these two levels to
generate an $\alpha$-particle), being analogous to the blocked odd-proton in
proton radioactivity. In fact, it is a component of the parent state according
to the interpretation of quantum mechanics. And there are many IC states and
hence many pathways to form the $\alpha$-particle, where Fig. 1 just
illustrates one of the pathways, and hence our strategy is obviously
distinguished from other models. These four quasiparticles in the $i_{n}$\-
and $i_{p}$-levels are going to form an $\alpha$-particle and the remaining
$A-4$ nucleons accordingly form a daughter nucleus, and the pathway via this
IC is marked as ($i_{n},i_{p}$) for the sake of the following discussion.
Accordingly, the probability to find a final configuration
$\Psi_{\text{2n-2p}}\Psi_{\text{D}}$ in the wavefunction of the parent nucleus
$\Psi_{\text{P}}$ through a given intermediate configuration
$\Psi_{\text{IC}}$ is
$S_{\text{D}}^{(i_{n},i_{p})}=|\langle\Psi_{\text{P}}|\Psi_{\text{IC}}\rangle|^{2}\cdot|\langle\Psi_{\text{IC}}|\Psi_{\text{2n-2p}}\Psi_{\text{D}}\rangle|^{2},$
(1)
which is the formation probability of the daughter nucleus through this
pathway.
For a deformed superfluid nucleus with nucleons paired by up and down spins,
within the BCS formulation, the overlap integral of
$\langle\Psi_{\text{P}}|\Psi_{\text{IC}}\rangle$ is written as
$\langle\Psi_{\text{P}}|\Psi_{\text{IC}}\rangle=\underset{k}{\Pi}(u_{k}^{(\text{P})}u_{k}^{(\text{IC})}+v_{k}^{(\text{P})}v_{k}^{(\text{IC})}),$
(2)
in which $v_{k}^{2}$ ($u_{k}^{2}$) represents the probability that the two-
fold degenerate $k$-th single-particle level is occupied (unoccupied). Note
that the final state can be written as
$\Psi_{\text{2n-2p}}\Psi_{\text{D}}=S_{i_{n}}^{\dagger}S_{i_{p}}^{\dagger}|$BCS$\rangle_{D}$,
where $S_{i_{n}}^{\dagger}$ ($S_{i_{p}}^{\dagger}$) creates two neutrons
(protons). Therefore,
$\langle\Psi_{\text{IC}}|\Psi_{\text{2n-2p}}\Psi_{\text{D}}\rangle$ expressed
in terms of single-particle properties is given by
$\displaystyle\langle\Psi_{\text{IC}}|\Psi_{\text{2n-2p}}\Psi_{\text{D}}\rangle$
$\displaystyle=$
$\displaystyle\left(\underset{i=i_{n},i_{p}}{\Pi}u_{i}^{(\text{D})}\right)^{3}\underset{k\neq
i}{\Pi}\left[u_{k}^{(\text{D})}u_{k}^{(\text{IC})}+v_{k}^{(\text{D})}v_{k}^{(\text{IC})}\langle\phi_{k}^{(\text{D})}|\phi_{k}^{(\text{IC})}\rangle^{2}\right],$
where $\phi_{k}$ denotes the normalized single-particle wavefunction.
We make the following two assumptions: 1) The formation probability of the
$\alpha$-cluster is identical to that of the daughter nucleus, i.e.,
$S_{\alpha}^{(i_{n},i_{p})}=S_{\text{D}}^{(i_{n},i_{p})}$, that is, the
formation of the $\alpha$-particle is achieved accordingly once the daughter
nucleus is generated. This means the four nucleons escaping from the IC
jumping into the unoccupied $i_{n}$\- and $i_{p}$-levels of the daughter
nucleus with probability $(u_{i_{n}}^{(\text{D})}u_{i_{p}}^{(\text{D})})^{4}$
are expected to self-assemble into an $\alpha$-particle at nuclear surface
spontaneously. Namely, the transition probability from $\Psi_{\text{2n-2p}}$
to an actual $\alpha$-cluster state $\Psi_{\alpha}$ is
$P(\Psi_{\text{2n-2p}}\rightarrow\Psi_{\alpha})=1$. Therefore, this strategy
is referred to as the self-assembly model iconically, which analogies to the
self-assembly of nanostructures where atoms, molecules or nanoscale building
blocks spontaneously organize into ordered structures or patterns without
external intervention [50]. 2) Each pathway for the formation process is
expected to be independent of the others. We sum over all pathways (i.e.,
through different ICs) to eventually achieve the $\alpha$-particle formation
probability via
$\displaystyle S_{\alpha}$ $\displaystyle=$
$\displaystyle\underset{(i_{n},i_{p})}{\sum}S_{\text{D}}^{(i_{n},i_{p})}$ (4)
$\displaystyle=$
$\displaystyle\underset{(i_{n},i_{p})}{\sum}\Bigg{\\{}\underset{k^{\prime}}{\Pi}\left(u_{k^{\prime}}^{(\text{P})}u_{k^{\prime}}^{(\text{IC})}+v_{k^{\prime}}^{(\text{P})}v_{k^{\prime}}^{(\text{IC})}\right)^{2}\cdot\underset{i=i_{n},i_{p}}{\Pi}\left(u_{i}^{(\text{D})}\right)^{6}$
$\displaystyle\cdot\underset{k\neq
i}{\Pi}\left[u_{k}^{(\text{D})}u_{k}^{(\text{IC})}+v_{k}^{(\text{D})}v_{k}^{(\text{IC})}\langle\phi_{k}^{(\text{D})}|\phi_{k}^{(\text{IC})}\rangle^{2}\right]^{2}\Bigg{\\}},$
with Eqs. (1-$\alpha$-cluster formation in heavy $\alpha$-emitters through
nucleonic self-assembly). The dimensionless formation probability here is the
expectation value of the $\alpha$-cluster component that can be emitted. The
stationary-state description of a time-dependent cluster formation process is
a quite good approximation and simplifies the problem enormously [3], which is
widely used at present for $\alpha$-decay. It is valid because half-lives of
$\alpha$-emitters are very long ($10^{-6}-10^{17}$ s) compared with the
“periods" of nuclear motion ($10^{-21}$ s) and hence in the time evolution of
a decaying state the nucleons has a large number of opportunities to get
clustered and to get the clusters dissolved before it can actually escaped
from the nucleus [3].
Our approach involves the structure of both parent and daughter nuclei, but
does not involve an intrinsic state or a localized density distribution of the
$\alpha$-cluster, being significantly different from the standard shell or BCS
models where a Gauss-shaped intrinsic $\alpha$-cluster wavefunction is
introduced. The quartetting wave function approach is a successful method
proving a reasonable behavior for the $S_{\alpha}$ of even-even Po isotopes
[31], which does not involve such an intrinsic $\alpha$-cluster wavefunction
to calculate $S_{\alpha}$ either, and does not employ the overlap of the
wavefunctions between the initial and final states. The wavefunction of the
bound state for the center of mass motion of four correlated nucleons is
obtained by solving the corresponding Schrödinger equation, and then the
$S_{\alpha}$ is calculated by integrating the modular square of this
wavefunction in the region below the Mott density $\rho^{\text{Mott}}\simeq
0.03$ fm-3 since an $\alpha$-like state generates automatically at such low
densities [30, 31, 32]. Different substantially from this quartetting wave
function approach, the $S_{\alpha}$ in our work is still based on the concept
of overlap integrals, and finally can be calculated with the compact
expression of Eq. (4) with the help of existing many-body approaches without
introducing any adjustable parameter.
The single particle properties in Eqs. (2,$\alpha$-cluster formation in heavy
$\alpha$-emitters through nucleonic self-assembly) are determined within the
framework of a covariant density functional (CDF) approach starting from an
interacting Lagrangian density [51, 52, 53, 54]. The nuclear CDF employed in
self-consistent calculations is parameterized by means of about ten coupling
constants that are calibrated to basic properties of nuclear matter and finite
nuclei, which enables one to perform an accurate description of ground state
properties and collective excitations over the whole nuclear chart [52, 53,
54], and has become a standard tool in low energy nuclear structure. The
explicit calculations are carried out based on a standard code DIZ [55] for
deformed nuclei, with the NL3 interaction [56] for the mean-field and the
calibrated D1S Gogny force [58] for the pairing channel. The NL3 parameter set
has been used with enormous success in the description of a variety of ground-
state properties of spherical, deformed and exotic nuclei [56, 57], and the
calibrated D1S Gogny force enables one to well reproduce the odd-even
staggerings on nuclear binding energies [58]. We concentrate on the even-even
Po, Rn and Ra isotopes with spherical or near-spherical shapes, because their
$\alpha$-decays tend to have large branching ratios (100% in most cases) and
their corresponding half-lives were best measured experimentally [59]. On the
other hand, these $\alpha$-decay cases usually do not involve excited states
and angular momentum transfers, and thus serve as an optimal testing ground to
examine our model. Moreover, the values of overlap integrals
$\underset{k}{\Pi}\langle\phi_{k}^{(\text{D})}|\phi_{k}^{(\text{IC})}\rangle$
for these nuclei can be taken as unity. The products in Eqs.
(2,$\alpha$-cluster formation in heavy $\alpha$-emitters through nucleonic
self-assembly) along with the summation in Eq. (4) are truncated at 5 MeV for
the single-nucleon spectra to achieve convergence.
Table 1: The rms deviations $\sqrt{\langle\sigma^{2}\rangle}$ and average deviations $\langle\sigma\rangle$ for the VSF and UDL with $S_{\alpha}=1$ and $S_{\alpha}\neq 1$. Formulas | $\sqrt{\langle\sigma^{2}\rangle}$ | $\langle\sigma\rangle$
---|---|---
VSF ($S_{\alpha}=1$) | 0.360 | 0.296
VSF ($S_{\alpha}\neq 1$) | 0.109 | 0.0861
UDL ($S_{\alpha}=1$) | 0.316 | 0.268
UDL ($S_{\alpha}\neq 1$) | 0.0888 | 0.0812
To assess the validity of our nucleonic self-assembly model, we explore the
role of the formation probability $S_{\alpha}$ in half-life calculations. The
widely accepted formulas, i.e., the semi-empirical Viola-Seaborg formula (VSF)
[60] and the universal decay law (UDL) based on the $R$-matrix expression [61]
are employed, which are respectively given as
$\displaystyle\log_{10}T_{1/2}$ $\displaystyle=$
$\displaystyle\frac{aZ+b}{\sqrt{Q_{\alpha}}}+cZ+d-\log_{10}S_{\alpha},$ (5)
$\displaystyle\log_{10}T_{1/2}$ $\displaystyle=$ $\displaystyle
a\chi^{\prime}+b\rho^{\prime}+c-\log_{10}S_{\alpha},$ (6)
with
$\displaystyle\chi^{\prime}$ $\displaystyle=$ $\displaystyle
2(Z-2)\sqrt{\frac{A_{\alpha d}}{Q_{\alpha}}},A_{\alpha d}=4(A-4)/A,$
$\displaystyle\rho^{\prime}$ $\displaystyle=$ $\displaystyle\sqrt{2A_{\alpha
d}(Z-2)\left[(A-4)^{1/3}+4^{1/3}\right]}.$
$Z$ ($A$) is the proton (mass) number of a given parent nucleus. The decay
energy $Q_{\alpha}$ and half-life $T_{1/2}$ are in units of MeV and second,
respectively. $S_{\alpha}=1$ ($S_{\alpha}\neq 1$) corresponds to the results
without (with) the inclusion of the formation probability.
Figure 2: $\log_{10}T_{1/2}^{\text{Exp.}}-b\rho^{\prime}+\log_{10}S_{\alpha}$
as a function of $\chi^{\prime}$ obtained with the UDL for $S_{\alpha}=1$
(left) and $S_{\alpha}\neq 1$ (right). The coefficient $b$ is fixed at the
fitted value in the two cases, respectively. The straight lines are given by
$a\chi^{\prime}+c$. Here $r$ is the corresponding correlation coefficient.
The fitting procedures are performed in the cases of $S_{\alpha}=1$ and
$S_{\alpha}\neq 1$ respectively to test whether or not the predicted
$S_{\alpha}$ could improve substantially the accuracy of the two formulas. It
is worth pointing out that the UDL has already included the logarithmic
formation amplitude which is assumed to be linearly dependent upon
$\rho^{\prime}$. Therefore, in Eq. (6), the $\rho^{\prime}$-dependent
formation probability is replaced by the presently calculated $S_{\alpha}$.
The root-mean-square (rms) deviations $\sqrt{\langle\sigma^{2}\rangle}$ and
average deviations $\langle\sigma\rangle$ for the two formulas with
$S_{\alpha}=1$ and $S_{\alpha}\neq 1$ are summarized in Table 1. The inclusion
of the $S_{\alpha}$ indeed greatly improves the accuracy of both the VSF and
UDL. The UDL with a solid physical ground but less parameters, works better
than the VSF, and reproduces the available experimental half-lives within a
factor of 2 in the case of $S_{\alpha}=1$. Yet, when the microscopically
calculated $S_{\alpha}$ is included, the deviation of the refitting is reduced
down to around $20\%$. The good agreement between the calculated half-lives
and the experimental data is quite encouraging, indicating the reliability of
the formation probability $S_{\alpha}$ given by the nucleonic self-assembly
model. In Fig. 2, we plot the UDL fittings but replace the half-lives with the
experimental values to more visually reveal the role of $S_{\alpha}$, and that
the inclusion of the $S_{\alpha}$ systematically improves the agreement with
data is exhibited. The highly linear correlation is displayed for
$S_{\alpha}\neq 1$, with a correlation coefficient as high as $r=0.9998$,
suggesting the success of our formation probability and the validity of the
two assumptions.
Figure 3: (Color online) Microscopically calculated formation probability
$S_{\alpha}$ values within the nucleonic self-assembly model for the Po, Rn
and Ra isotopes, compared with those extracted by using the experimental half-
lives in combination with the CM calculated penetration probabilities.
Furthermore, $S_{\alpha}$ is extracted in turn by using the ratio of the
theoretical half-life to the experimentally observed value. The barrier
penetrability of $\alpha$-particle, is achieved theoretically by the WKB
approximation which turns out to work excellently [62], where the potential
barrier is constructed by a simple “Cosh$\textquotedblright$ potential plus
the Coulomb barrier model (CM) [63]. Here the extracted $S_{\alpha}$ should be
considered as a relative value. By selecting an optimal constant assault
frequency, the extracted $S_{\alpha}$ values with varying neutron number $N$
are compared with the results given by the nucleonic self-assembly model in
Fig. 3. In sharp contrast with half-lives, the $S_{\alpha}$ values are located
in a relatively narrow range, leading to the success of the empirical half-
life laws even when $S_{\alpha}$ is not included. The $S_{\alpha}$ values
follow the similar behavior with regard to the Po, Rn and Ra isotopic
chains–that is, gradually drop with increasing neutron number up to the
spherical magic number $N=126$, attributed to the increased stability of
isotopes when approaching the magic number, and then they increase drastically
with neutron number. Such a general trend of the extracted $S_{\alpha}$ is
successfully reproduced within our microscopic method. Typically, the
$S_{\alpha}$ of 212Po, being expected to be large owing to its two protons and
two neutrons outside the shell closure core 208Pb, is about six times larger
than that of its neighbor 210Po. The weight of the cluster component in 212Po
is large, which is exactly what one needs to simultaneously describe the B(E2)
[64] and the absolute $\alpha$-decay width [3, 23] within the shell model plus
a cluster component. The distinct behavior that $S_{\alpha}$ varies abruptly
when the magic number is crossed, confirms that the particularly significant
role of shell effects on $S_{\alpha}$ is reasonably accounted for in Eq. (1)
via the single-particle properties. As one expects, $S_{\alpha}$ reaches its
minimum at the shell closure $N=126$ as the result of the well-known shell
stability that strongly enhances the nuclear binding.
In order to analyze the contribution of each pathway in Eq. (4) to
$S_{\alpha}$, Fig. 4 illustrates
$S_{\alpha}^{(i_{n},i_{p})}=|\langle\Psi_{\text{IC}}|\Psi_{\text{P}}\rangle|^{2}\cdot|\langle\Psi_{\text{2n-2p}}\Psi_{\text{D}}|\Psi_{\text{IC}}\rangle|^{2}$
versus the single neutron and single proton energies
($\varepsilon_{\text{n}}$, $\varepsilon_{\text{p}}$) by taking the typical
nucleus 212Po as an example. The formation probability in Eq. (4) is
predominantly determined by the pathways belonging, in the IC, to the fully-
occupied neutron and proton levels $i_{n}$ and $i_{p}$ slightly above the
Fermi surfaces, i.e., these levels are the major nucleon donors to constitute
the emitted $\alpha$-particle. The contributions from other pathways drop
sharply when the fully-occupied levels ($i_{n},i_{p}$) gradually go away from
these leading ones. As a consequence, the nearly degenerate levels splitting
from the last spherical neutron and proton orbits, are most important for the
formation of the $\alpha$-particle.
Figure 4: Contour plot of $\log_{10}S_{\alpha}^{(i_{n},i_{p})}$ versus the
single-particle energies (in units of MeV) of the $i_{n}$\- and the
$i_{p}$-levels for the illustrative example of 212Po to show the role of each
pathway. The dashed lines denote the corresponding Fermi energies
$\varepsilon_{\text{Fn}}$ and $\varepsilon_{\text{Fp}}$ for neutrons and
protons, respectively.
It is well-known that the concept of $\alpha$-clustering is essential for
understanding the structure of light nuclei. In some cases, light nuclei
behave like molecules composed of clusters of protons and neutrons, such as
the definite $2\alpha$ cluster structure of 8Be [65] reflected in a localized
density distributions. But whether such type of cluster structure exists or
not in heavy nuclei is uncertain. In our strategy, however, a localized
$\alpha$-cluster does not pre-exist inside a parent nucleus, but is generated
during the decay process through a nucleonic self-assembly. Therefore, the
scenario that a continuous formation and breaking of the $\alpha$-cluster
until it escapes randomly from the parent nucleus [3], is supported. As a
result, the mechanism of $\alpha$ formation in heavy $\alpha$-emitters is
different markedly from that in light nuclei.
Since the formation probability $S_{\alpha}$ is highly relevant to the
quantum-mechanical shell effects, the extracted $S_{\alpha}$ with a high-
precision UDL combined with experimentally measured $\alpha$-decay properties,
is of great importance for digging up valuable structural information of SHN
and exotic nuclei. The correlations between the $Q_{\alpha}$ values of SHN
have suggested that the heaviest isotopes reported in Dubna do not lie in a
region of rapidly changing shapes [66]. Therefore, the $S_{\alpha}$ versus
proton number exhibits a behavior analogous to isotopic chains shown in Fig.
3, attributed to the nearby shell closure. By employing the NL3 interaction,
the heaviest nucleus ${}^{294}\text{Og}$ ($Z=118$) and its isotonic neighbor
${}^{292}\text{Lv}$ ($Z=116$), are predicted to be nearly spherical. The
calculated $S_{\alpha}$ is 0.66 for ${}^{292}\text{Lv}$ while it reduces to
0.45 for ${}^{294}\text{Og}$ with the ratio of
$S_{\alpha}(^{292}\text{Lv})/S_{\alpha}(^{294}\text{Og})=1.5$, being indeed
similar to the trend shown in Fig. 3, which is consistent with the fact that
the NL3 interaction itself predicts the adjacent $Z=120$ as a magic number. On
the other hand, with the UDL of Eq. (6) along with experimental data [10], the
extracted ratio of $S_{\alpha}(^{292}\text{Lv})/S_{\alpha}(^{294}\text{Og})$
is as high as $3.2^{+4.3}_{-1.8}$ ($4.5^{+5.8}_{-2.2}$) with half-life data of
${}^{292}\text{Lv}$ from Ref. [67] (Ref. [68]) where the significant
uncertainties are due to the low-statistics data for the measurements, and
agrees marginally with the above theoretical value. Hence the probable magic
nature of $Z=120$ is suggested. The future measurements with a much higher
accuracy for these nuclei together with their isotopes are encouraged, which
would pin down the proton magic number eventually.
Intriguingly, the superallowed $\alpha$-decay to doubly magic
${}^{100}\text{Sn}$ was observed recently, which indicates a much larger
$\alpha$-formation probability than ${}^{212}\text{Po}$ counterpart [69].
Within the framework of the quartetting wave function approach, an enhanced
$\alpha$-cluster formation probability for ${}^{104}\text{Te}$ was found
because the bound state wavefunction of the four nucleons has a large
component at the nuclear surface [32]. Yet, a large $S_{\alpha}$ for
${}^{104}\text{Te}$ is inconsistent with our prediction of $S_{\alpha}=0.24$,
suggesting the onset of an unusual type of nuclear superfluidity for self-
conjugate nuclei, i.e., the proton-neutron pairing which is not well-confirmed
at present. This isoscalar pairing would considerably impact on the single-
particle spectra and hence the $S_{\alpha}$, and its absence in our CDF
calculations leads to the underestimated $S_{\alpha}$. Therefore, as a new way
independent of Ref. [70], combined with precise $\alpha$-decay measurements
for $N\simeq Z$ nuclei, our approach for the $S_{\alpha}$ could enable us to
clarify this abnormal pairing interaction in turn by employing CDF approaches
with the inclusion of a tentative isoscalar superfluidity.
In general, computing the formation probability $S_{\alpha}$ defined in Eqs.
(1-4) by nuclear density functionals with a very high accuracy, is out of
reach at present because of the well-known fact that the single-particle
levels are not well-defined in the concept of the mean-field approximation
especially for well-deformed nuclei. Nevertheless, based on two intuitive
assumptions and without any phenomenological adjustment, our strategy opens a
new perspective to account for the formation mechanism. This is highly
important to help one to uncover the underlying knowledge about superheavy
nuclei and isoscalar pairing, in combination with experimental observations.
For example, the experimentally measured $\alpha$-decay properties of heaviest
nuclei combined with our calculated $S_{\alpha}$ suggest the probable proton-
magic nature of $Z=120$. Moreover, extensive calculations of $S_{\alpha}$
could in turn stimulate the development of nuclear many-body approaches along
with the strong nucleon-nucleon interaction to describe single-particle
structure more reliably.
J. M. Dong thanks B. Zhou, W. Scheid, and C. Qi for helpful comments and
suggestions. This work is supported by the National Natural Science Foundation
of China (Grants Nos. 11775276, 11435014, 11675265, and 11905175), by the
Strategic Priority Research Program of Chinese Academy of Sciences (Grant No.
XDB34000000), by the National Key Program for S&T Research and Development
(Grant No. 2016YFA0400502), by the Youth Innovation Promotion Association of
Chinese Academy of Sciences, and by the Continuous Basic Scientific Research
Project (Grant No. WDJC-2019-13).
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|
# Continuous measurements in probability representation of quantum mechanics
Y. V. Przhiyalkovskiy
Kotelnikov Institute of Radioengineering and Electronics (Fryazino Branch)
of Russian Academy of Sciences,
Vvedenskogo sq., 1, Fryazino, Moscow reg., 141190, Russia
e-mail<EMAIL_ADDRESS>
###### Abstract
The continuous quantum measurement within the probability representation of
quantum mechanics is discussed. The partial classical propagator of the
symplectic tomogram associated to a particular measurement outcome is
introduced, for which the representation of a continuous measurement through
the restricted path integral is applied. The classical propagator for the
system undergoing a non-selective measurement is derived by summing these
partial propagators over the entire outcome set. The elaborated approach is
illustrated by considering non-selective position measurement of a quantum
oscillator and a particle.
## 1 Introduction
The probability representation of quantum mechanics, introduced not so far and
being actively developed in recent years, is attractive due to being one of
the most promising to formulate quantum mechanics in the closest manner to
statistical physics [1, 2, 3, 4]. Essentially, this approach suggests that a
family of probability distributions of a coordinate in linearly and
homogeneously transformed phase space is employed to describe a quantum state
instead of a density matrix. Due to their unambiguous mapping to each other,
it is turns out to be possible to formulate quantum mechanics in terms of such
probability distributions, or so-called quantum tomograms.
The importance of influence the measuring of an observable exerts on a quantum
system could hardly be underestimated from both a theoretical and practical
viewpoint. How a measuring process is reflected within the probability
representation is also of great interest. According to the original quantum
theory, the measurement of an observable performed on the system happens
instantaneously and thus implies the collapse of its state. The same obviously
applies to the quantum tomogram. In real systems, instead, the state the
system had before the measurement transits to the new state in a continuous
way during the measurement. The profound research of this subject in quantum
mechanics has begun in 70-th [5, 6, 7, 8] and is well studied now. In
practice, the issues of decoherence and measurement back-action comprise a
significant part of current research on quantum computing, actively growing in
our days [9, 10, 11]. Moreover, apart from being traditionally considered as a
passive, continuous measurement may even be involved to manipulate a quantum
system [12, 13, 14, 15]. Thus, measuring of a predetermined set of observables
ensures the final state of the particular system to have maximum expected
value of a target operator [12]. Another promising application of active
measuring is the optimal control of quantum evolution, in particular,
employing quantum Zeno and anti-Zeno effects [12, 13, 14] or optimal
acceleration of the Landau-Zener transitions [15]. The most intuitively clear
concept to describe a continuous measurement of a quantum system is based on
the Feynman path integral and was elaborated by Mensky [16, 17]. The main idea
of this approach is that certain paths over which the integral is taken are
more preferred, according to the information the environment gains from the
measured system. Technically, it is attained by inserting a path weight
functional into the path integral to calculate the quantum propagator.
Certainly, the continuous measurement experienced by the system modifies both
the tomogram dynamics and the resultant tomogram. A direct parallel drawn
between traditional quantum mechanics and its probability representation leads
to a differential Fokker-Plank-type equation whose solution determines the
tomogram for every time moment [18]. The other approach to figure out the
evolved tomogram, elaborated so far only for isolated systems, is to use a so-
called classical propagator [19, 20, 21]. Being attributed to a particular
quantum system, the classical propagator is determined by the usual quantum
propagator for that system and hence already incorporates its dynamics. The
central theme we focus our attention in this article on is expansion of the
approach of classical propagators in symplectic tomography to quantum systems
undergoing continuous measurement by application the restricted path integral.
## 2 Symplectic tomography
Symplectic tomography was initially introduced in [18, 22, 23, 24] in the
following way. Consider a classical system with its phase space, and let an
observable $X=\mu q+\nu p$ be the result of a general linear transformation of
the coordinate $q$ and momentum $p$ (hereafter $\hbar=1$). In other words, $X$
is the coordinate in a phase space viewed from a new frame according to the
above transform. Here the real quantities $\mu$ and $\nu$ parametrize this
map, after performing of which the coordinate $X$ is measured. If we now turn
to the quantum counterpart of the system, its symplectic tomogram is then, by
definition, the marginal distribution function of the variable $X$. Namely, it
is the Fourier transformation of the characteristic function $F(k)=\langle
e^{ik\hat{X}}\rangle$ related to the self-adjoint operator
$\hat{X}=\mu\hat{q}+\nu\hat{p}$:
$\mathcal{T}(X,\mu,\nu)=\frac{1}{2\pi}\int\langle e^{ik\hat{X}}\rangle
e^{-ikX}dk$ (1)
where $\langle\hat{A}\rangle=\operatorname{Tr}\\{\hat{\rho}\hat{A}\\}$ is the
quantum mean value. If one performs the subsequent reparametrization
$\mu=\cos{\theta}e^{\lambda}$ and $\nu=\sin{\theta}e^{-\lambda}$, it becomes
clear the physical sense of the above transformation of the phase space as the
rotation and scaling [25]. In particular, for $\lambda=1$ the map reduces to
Radon transformation, and the symplectic tomogram turns out to be an optical
tomogram [26].
The symplectic tomogram defined in this way is positive definite and satisfies
$\int\mathcal{T}(X,\mu,\nu)dX=1.$ (2)
Therefore, the tomogram $\mathcal{T}(X,\mu,\nu)$ indeed turns out to be a
probability distribution for each $\mu$ and $\nu$. The essential feature of
such a tomogram set is that it is equivalent to a quantum state. Consequently,
it is possible to formulate quantum mechanics taking this set of probability
distributions as a system state. It is just this formulation that has been
called the probability representation of quantum mechanics.
### 2.1 Star-product formalism
To begin with, we review the framework of operator symbols [27]. Essentially,
its purpose is to relate the algebras of Hilbert space operators with algebras
of ordinary functions equipped with an associative but non-commutative
product, the so-called star product.
Consider a Hilbert space $\mathbb{H}$ attributed to the quantum system and an
operator $\hat{A}$ acting on vectors of $\mathbb{H}$. Let
$\mathbf{x}_{i}=(x_{i}^{1},x_{i}^{2},\dots,x_{i}^{k})$ be a vector of
parameters. Let also $\hat{\mathcal{U}}(\mathbf{x})$ and
$\hat{\mathcal{D}}(\mathbf{x})$ be operators parameterized by $\mathbf{x}_{i}$
and satisfying the consistency condition
$\operatorname{Tr}\\{\hat{\mathcal{U}}(\mathbf{x}_{1})\hat{\mathcal{D}}(\mathbf{x}_{2})\\}=\delta(\mathbf{x}_{1}-\mathbf{x}_{2})$.
Then, one defines the transformation of the operator $\hat{A}$ into a
C-function $f_{\hat{A}}(\mathbf{x})$ through
$f_{\hat{A}}(\mathbf{x})=\operatorname{Tr}\\{\hat{A}\hat{\mathcal{U}}(\mathbf{x})\\}$
(3)
and the inverse transformation as
$\hat{A}=\int f_{\hat{A}}(\mathbf{x})\hat{\mathcal{D}}(\mathbf{x})d\mathbf{x}$
(4)
where $d\mathbf{x}=dx^{1}dx^{2}\dots dx^{k}$. The operators
$\hat{\mathcal{U}}(\mathbf{x})$ and $\hat{\mathcal{D}}(\mathbf{x})$ are
referred to as the dequantizer and quantizer respectively, and the function
$f_{\hat{A}}(\mathbf{x})$ is then the operator symbol of $\hat{A}$.
The symbol of a multiplication of two operators, $\hat{A}\hat{B}$, can be
easily derived using (3) and (4):
$f_{\hat{A}\hat{B}}(\mathbf{x})=\iint
f_{\hat{A}}(\mathbf{x}_{1})f_{\hat{B}}(\mathbf{x}_{2})M_{\mathbf{x}_{1}\mathbf{x}_{2}}(\mathbf{x})d\mathbf{x}_{1}d\mathbf{x}_{2}$
(5)
where the kernel is
$M_{\mathbf{x}_{1}\mathbf{x}_{2}}(\mathbf{x})=\operatorname{Tr}\left\\{\hat{\mathcal{U}}(\mathbf{x})\hat{\mathcal{D}}(\mathbf{x}_{1})\hat{\mathcal{D}}(\mathbf{x}_{2})\right\\}.$
(6)
Relation (5), which is also convenient to shortly denote by
$f_{\hat{A}\hat{B}}(\mathbf{x})=f_{\hat{A}}(\mathbf{x})\star
f_{\hat{B}}(\mathbf{x}),$ (7)
is called the star product of operator symbols. Using (5), one obtains the
symbol of a commutator
$f_{[\hat{A},\hat{B}]}(\mathbf{x})=\iint
f_{\hat{A}}(\mathbf{x}_{1})f_{\hat{B}}(\mathbf{x}_{2})C_{\mathbf{x}_{1}\mathbf{x}_{2}}(\mathbf{x})d\mathbf{x}_{1}d\mathbf{x}_{2}$
(8)
with the kernel
$C_{\mathbf{x}_{1}\mathbf{x}_{2}}(\mathbf{x})=\operatorname{Tr}\left\\{\hat{\mathcal{U}}(\mathbf{x})[\hat{\mathcal{D}}(\mathbf{x}_{1}),\hat{\mathcal{D}}(\mathbf{x}_{2})]\right\\}=M_{\mathbf{x}_{1}\mathbf{x}_{2}}(\mathbf{x})-M_{\mathbf{x}_{2}\mathbf{x}_{1}}(\mathbf{x}).$
(9)
Similarly, a commutator symbol can be written in a compact manner, just as
$f_{[\hat{A},\hat{B}]}(\mathbf{x})=[f_{\hat{A}}(\mathbf{x}),f_{\hat{B}}(\mathbf{x})]_{\star}$
(10)
where $[~{},~{}]_{\star}$ denotes a star-product commutator:
$[f,g]_{\star}=f\star g-g\star f$.
### 2.2 Symplectic tomography using operator symbols
The symplectic tomography can be conveniently introduced using the framework
of operator symbols [27], briefly reviewed in the preceding section.
Specifically, let the parameters set be $\mathbf{x}=(X,\mu,\nu)$ and then
define the dequantizer and quantizer operators as
$\displaystyle\hat{\mathcal{U}}(\mathbf{x})=\delta(X-\mu\hat{q}-\nu\hat{p}),$
(11a)
$\displaystyle\hat{\mathcal{D}}(\mathbf{x})=\frac{1}{2\pi}\exp{\left(iX-i\mu\hat{q}-i\nu\hat{p}\right)}.$
(11b)
Here $\delta(x)$ denotes the Dirac delta function, in the case of an operator
argument being treated as $\delta{(\hat{A})}=(2\pi)^{-1}\int dke^{ik\hat{A}}$.
A symplectic tomogram is, by definition, a symbol of the density matrix
calculated using above dequantizer operator (11a):
$\mathcal{T}(\mathbf{x})=\operatorname{Tr}\\{\hat{\rho}\hat{\mathcal{U}}(\mathbf{x})\\}.$
(12)
A simple comparison reveals the equivalence of this definition to (1). Having
the tomogram, the density matrix is easily restored by the inverse
transformation using quantizer (11b):
$\hat{\rho}=\int\mathcal{T}(\mathbf{x})\hat{\mathcal{D}}(\mathbf{x})d\mathbf{x}.$
(13)
By a direct calculation, one also derives an explicit expression of the
operator multiplication kernel
$M_{\mathbf{x}_{1}\mathbf{x}_{2}}(\mathbf{x})=\frac{\delta(\mu(\nu_{1}+\nu_{2})-\nu(\mu_{1}+\mu_{2}))}{4\pi^{2}}e^{iX_{1}+iX_{2}}e^{-i\frac{(\nu_{1}+\nu_{2})}{\nu}X}e^{i(\nu_{1}\mu_{2}-\nu_{2}\mu_{1})/2}$
(14)
being needed to carry out the further calculations.
### 2.3 Symplectic tomogram evolution
Consider a quantum system which dynamics is described by a certain Hamiltonian
$\hat{H}$. The evolution of the system state, being mixed in general, is
described by the evolution operator $\hat{U}_{t}=\exp{(-i\widehat{H}t)}$. The
density matrix of the system at time $t>0$ is then expressed through the
initial density matrix at time $t=0$ as
$\hat{\rho}(t)=\hat{U}_{t}\hat{\rho}(0)(\hat{U}_{t})^{\dagger}.$ (15)
Each matrix element $U_{t}(q_{f},q_{i})\equiv\langle
q_{f}|\hat{U}_{t}|q_{i}\rangle$ of the evolution operator, which is the
amplitude of the system transition from the point $q_{i}$ at time $0$ to the
point $q_{f}$ at time $t$, as known, can be expressed through the Feynman path
integral [28]
$U_{t}(q_{f},q_{i})=\int\limits_{q_{i},0}^{q_{f},t}d[q]e^{iS[q]}$ (16)
where $S$ is the action of the system.
It is clear, that at each instant the tomogram of the quantum system can be
expressed through the density matrix at that time,
$\mathcal{T}(\mathbf{x},t)=\operatorname{Tr}\\{\hat{\rho}(t)\hat{\mathcal{U}}\\}.$
(17)
This equation means that the tomogram is the instant average value of the
dequantizer $\hat{\mathcal{U}}$ eigenvalues. Turning to the Heisenberg
picture, the tomogram
$\mathcal{T}(\mathbf{x},t)=\operatorname{Tr}\\{\hat{\rho}(0)\hat{\mathcal{U}}_{t}\\}$
(18)
makes sense of the eigenvalues average of the operator
$\hat{\mathcal{U}_{t}}\equiv(\hat{U}_{t})^{\dagger}\hat{\mathcal{U}}\hat{U}_{t}$,
calculated using the initial density matrix. Hereafter, the latter operator
will be referred to as evolved dequantizer.
Applying the inverse transformation (13) to initial density matrix
$\hat{\rho}(0)=\int\mathcal{T}(\mathbf{x},0)\hat{\mathcal{D}}(\mathbf{x})d\mathbf{x}$,
one can relate the evolved tomogram with the initial one by
$\mathcal{T}(\mathbf{x},t)=\int\mathcal{T}(\mathbf{x}^{\prime},0)\Pi_{t}(\mathbf{x}^{\prime},\mathbf{x})d\mathbf{x}^{\prime}$
(19)
where
$\Pi_{t}(\mathbf{x}^{\prime},\mathbf{x})=\operatorname{Tr}\left\\{\hat{\mathcal{D}}(\mathbf{x}^{\prime})\hat{\mathcal{U}}_{t}(\mathbf{x})\right\\}=\operatorname{Tr}\left\\{\hat{\mathcal{D}}(\mathbf{x}^{\prime})(\hat{U}_{t})^{\dagger}\hat{\mathcal{U}}(\mathbf{x})\hat{U}_{t}\right\\}$
(20)
is referred to as the "classical" propagator (or tomogram propagator), in
contrast to quantum propagator (16) [20]. Substituting then dequantizer and
quantizer operators (11) into (20) yields the explicit expression
$\displaystyle\Pi_{t}(\mathbf{x}^{\prime},\mathbf{x})=$ (21)
$\displaystyle=\frac{1}{4\pi^{2}}\int
k^{2}U_{t}^{*}(q_{f,1}+k\nu,q_{i,2})U_{t}(q_{f,1},q_{i,2}+k\nu^{\prime})e^{ik(X^{\prime}+X)}e^{-ik^{2}\frac{\mu^{\prime}\nu^{\prime}+\mu\nu}{2}}e^{-ik\mu^{\prime}q_{i,2}-ik\mu
q_{f,1}}dq_{f,1}dq_{i,2}dk$
where the propagator homogeneity of degree $-2$ relative to the first
argument,
$\Pi_{t}(k\mathbf{x}^{\prime},\mathbf{x})=k^{-2}\Pi_{t}(\mathbf{x}^{\prime},\mathbf{x})$,
has been used, directly stemming from the absolute homogeneity of the tomogram
$\mathcal{T}(k\mathbf{x})=|k|^{-1}\mathcal{T}(\mathbf{x})$.
## 3 Continuous measurements in symplectic tomography
As long as we consider an isolated quantum system, all paths in the
configuration space connecting the starting and end points must be involved
when Feynman path integral (16) is calculated. A continuous measuring of the
system, however, implies that a certain information about the system is
acquired by environment. Therefore, we have some knowledge about the path
along which the system has passed. This can be quantified by introducing a
path weight functional $w_{a}[q]$, $0\leq w_{a}[q]\leq 1$, where $a(t)$ is the
measurement outcome. In other words, all paths we integrate over are weighted
by $w_{a}[q]$ according to the probability of passing through. Therefore, the
Feynman path integral used to get the transition amplitude is generalized to
[29]
$U_{t,a}(q_{f},q_{i})=\int\limits_{q_{i},0}^{q_{f},t}d[q]w_{a}[q]e^{iS[q]}.$
(22)
It is important to note that the transition amplitude defined in this way is
actually no longer unitary due to the path weighting. Given that the
measurement result is $a(t)$, the system state after the measurement is then
described by density matrix
$\hat{\rho}_{a}(t)=\hat{U}_{t,a}\hat{\rho}(0)(\hat{U}_{t,a})^{\dagger}$ (23)
which is obviously not normalized. Therefore, the tomogram density in the set
of outcomes $\\{a\\}$ is
$\mathcal{T}_{a}(\mathbf{x},t)=\operatorname{Tr}\\{\hat{\rho}_{a}(t)\hat{\mathcal{U}}\\}=\operatorname{Tr}\\{\hat{\rho}(0)\hat{\mathcal{U}}_{t,a}\\}$
(24)
where
$\hat{\mathcal{U}}_{t,a}\equiv(\hat{U}_{t,a})^{\dagger}\hat{\mathcal{U}}\hat{U}_{t,a}$
is the evolved dequantizer operator related to the measurement outcome $a(t)$.
It is important to emphasize that $\mathcal{T}_{a}(\mathbf{x},t)$ is not a
true tomogram, since it was derived using a non-normalized density matrix.
Although, it can become such were it normalized by either the outcome
probability (the discrete spectrum case) or the probability the outcome lies
in a certain range (the continuous spectrum case).
The tomogram density of the system undergone the selective measurement is
related to the initial tomogram $\mathcal{T}(\mathbf{x},0)$ by
$\mathcal{T}_{a}(\mathbf{x},t)=\int\mathcal{T}(\mathbf{x}^{\prime},0)\Pi_{t,a}(\mathbf{x}^{\prime},\mathbf{x})d\mathbf{x}^{\prime}$
(25)
where
$\displaystyle\Pi_{t,a}(\mathbf{x}^{\prime},\mathbf{x})=\operatorname{Tr}\left\\{\hat{\mathcal{D}}(\mathbf{x}^{\prime})(\hat{U}_{t,a})^{\dagger}\hat{\mathcal{U}}(\mathbf{x})\hat{U}_{t,a}\right\\}=$
(26) $\displaystyle=\frac{1}{4\pi^{2}}\int
k^{2}U_{t,a}^{*}(q_{f,1}+k\nu,q_{i,2})U_{t,a}(q_{f,1},q_{i,2}+k\nu^{\prime})e^{ik(X^{\prime}+X)}e^{-ik^{2}\frac{\mu^{\prime}\nu^{\prime}+\mu\nu}{2}}e^{-ik\mu^{\prime}q_{i,2}-ik\mu
q_{f,1}}dq_{f,1}dq_{i,2}dk$
is the partial propagator. Note that in contrast to propagator (21), the
partial propagator maps the initial tomogram to the tomogram density.
If we consider non-selective measurement, the result is presumed to be
unknown. Therefore, to obtain the final density matrix of the system, one must
integrate density matrices (23) over all states relevant to particular
outcomes [17],
$\hat{\rho}(t)=\int\hat{U}_{t,a}\hat{\rho}(0)(\hat{U}_{t,a})^{\dagger}da,$
(27)
where $da$ is the measure in the set of outcomes. Note here, that from (27) it
immediately follows that the generalized unitarity condition
$\int(\hat{U}_{t,a})^{\dagger}\hat{U}_{t,a}da=1$ (28)
must be fulfilled to ensure $\operatorname{Tr}\\{\hat{\rho}(t)\\}=1$. The
tomogram of the system undergone by a non-selective measurement is obtained by
calculating the symbol of matrix (27), yielding
$\tilde{\mathcal{T}}(\mathbf{x},t)=\int\mathcal{T}_{a}(\mathbf{x},t)da=\operatorname{Tr}\\{\hat{\rho}(0)\hat{\tilde{\mathcal{U}}}_{t}\\}$
(29)
where
$\hat{\tilde{\mathcal{U}}}_{t}=\int(\hat{U}_{t,a})^{\dagger}\hat{\mathcal{U}}\hat{U}_{t,a}da$
is the evolved dequantizer for the measured system.
In a similar way as for tomogram density, the application of inverse
transformation (13) to (29) yields the relation between the initial and final
tomograms:
$\tilde{\mathcal{T}}(\mathbf{x},t)=\int\mathcal{T}(\mathbf{x}^{\prime},0)\tilde{\Pi}_{t}(\mathbf{x}^{\prime},\mathbf{x})d\mathbf{x}^{\prime}$
(30)
where the tomogram propagator for the system under non-selective continuous
measurement is just an integral of the partial propagators over the outcome
set
$\tilde{\Pi}_{t}(\mathbf{x}^{\prime},\mathbf{x})=\int\Pi_{t,a}(\mathbf{x}^{\prime},\mathbf{x})da.$
(31)
We especially stress here that since
$\tilde{\Pi}_{t}(\mathbf{x}^{\prime},\mathbf{x})$ maps tomograms, it is a true
propagator, yet it contains the influence of the measuring environment.
In the conclusion consider the case of a non-selective continuous measurement
of a single observable $\hat{A}$. In the traditional formulation of quantum
mechanics, the system density matrix evolves according to the master equation
of the form [29, 30]
$\frac{\partial\hat{\rho}}{\partial
t}=-i[\hat{H},\hat{\rho}]-k[\hat{A},[\hat{A},\hat{\rho}]].$ (32)
Now turn to the probability representation. Replacing the operators by their
symbol functions following (3) and (12), and subsequently using expansion
(30), one thus obtains that the propagator obeys
$\frac{\partial\tilde{\Pi}_{t}(\mathbf{x}^{\prime},\mathbf{x})}{\partial
t}+i[f_{\hat{H}}(\mathbf{x}),\tilde{\Pi}_{t}(\mathbf{x}^{\prime},\mathbf{x})]_{\star}+k[f_{\hat{A}}(\mathbf{x}),[f_{\hat{A}}(\mathbf{x}),\tilde{\Pi}_{t}(\mathbf{x}^{\prime},\mathbf{x})]_{\star}]_{\star}=0$
(33)
with initial condition
$\tilde{\Pi}_{0}(\mathbf{x}^{\prime},\mathbf{x})=\delta\left(\mathbf{x}^{\prime}-\mathbf{x}\right)$.
## 4 Spectral measurement of oscillator position
The harmonic oscillator is the underlying model being one of the most
important in quantum mechanics. The basic investigations of an isolated
oscillator in terms of probability representation of quantum mechanics have
already been performed by now [20]. Nevertheless, in real systems, the
measuring environment will inevitably influence the oscillator, which requires
revising its symplectic tomogram dynamics. In this section, we will
demonstrate this for a driven quantum oscillator undergoing a continuous
spectral measurement of its coordinate.
Assume the Hamiltonian of the oscillator to be
$\hat{H}=\hat{p}^{2}/(2m)+m\omega^{2}\hat{q}^{2}/2-\hat{q}f(t)$ where $f(t)$
is the external force. The action used to calculate the path integral in (22)
is then
$S=\int\limits_{0}^{T}dt\left(\frac{m}{2}\dot{q}^{2}-\frac{m\omega^{2}}{2}q^{2}+fq\right).$
(34)
Let the oscillator position be measured during the time interval $(0,T)$. To
get further, decompose the oscillator trajectory $q(t)$ into the Fourier
series as
$q(t)=\sum_{n=1}^{\infty}q_{n}\sin{(\Omega_{n}t)}$ (35)
where $\Omega_{n}=\pi n/T$. According to the approach of the spectral
measurement, one measures the component amplitudes $q_{n}$ with the errors
$\Delta a_{n}$ [28]. The appropriate choice of a weight functional for such a
measurement is [17, 31]
$w_{a}[q]=\exp\left\\{-\sum\limits_{n=1}^{\infty}\frac{(q_{n}-a_{n})^{2}}{\Delta
a_{n}^{2}}\right\\}$ (36)
where $a_{n}$ is the measurement result for amplitude $q_{n}$.
If the measurement result $\\{a_{n}\\}$ are known, the amplitude
$U_{t,a}(q_{f},q_{i})$ can be obtained by changing the integration over paths
in path integral (22) to integration over their Fourier components.
Substituting weighting functional (36), action (34) (where the path $q(t)$ is
expressed through its Fourier components) into (22) eventually yields
$U_{T,a}(q_{f},q_{i})=\left(\frac{1}{2\pi
i}\prod\limits_{n=1}^{\infty}\left\\{1-\frac{\omega_{e,n}^{2}}{\Omega_{n}^{2}}\right\\}^{-1}\right)^{1/2}\exp\left\\{iS(\eta)-\sum\limits_{n=1}^{\infty}\frac{(\eta_{n}-a_{n})^{2}}{\Delta
a_{n}^{2}}\frac{\Omega_{n}^{2}-\omega^{2}}{\Omega_{n}^{2}-\omega_{e,n}^{2}}\right\\}$
(37)
where
$\omega_{e,n}^{2}=\omega^{2}-\frac{4i}{mT\Delta a_{n}^{2}}$ (38)
and $\eta_{n}=(2/T)\int_{0}^{T}\eta(t)\sin{\Omega_{n}t}dt$ is the Fourier
component amplitudes of the classical trajectory $\eta(t)$ derived from the
equation of motion $d^{2}\eta/dt^{2}+\omega^{2}\eta=f(t)/m$ for a classical
oscillator with initial conditions $\eta(0)=q_{i}$, $\eta(T)=q_{f}$.
In the case of a non-selective coordinate measurement, the tomogram propagator
is obtained by substituting partial quantum propagator (37) into partial
tomogram propagator (26) and subsequently integrating over the outcome set,
following (31). A direct but somewhat tedious calculation then results in
$\displaystyle\tilde{\Pi}_{T}(\mathbf{x}^{\prime},\mathbf{x})=$ (39)
$\displaystyle=\left(\frac{1}{2\pi\sigma^{2}}\right)^{1/2}e^{-\frac{\left(X-X^{\prime}-\bar{X}\right)^{2}}{2\sigma^{2}}}\delta\left(\mu^{\prime}-\mu\cos{\omega
T}+\nu m\omega\sin{\omega T}\right)\delta\left(\nu^{\prime}-\nu\cos{\omega
T}-\mu\frac{\sin{\omega T}}{m\omega}\right).$
We see that a continuous coordinate measurement makes a dependence on
$X^{\prime}$ to be Gaussian with the variance
$\sigma^{2}=2\kappa\left(\nu^{2}(1+\cos^{2}{\omega
T})+\mu\nu\frac{\sin{2\omega T}}{m\omega}+\mu^{2}\frac{\sin^{2}{\omega
T}}{m^{2}\omega^{2}}\right)-4\xi\left(\nu^{2}\cos{\omega
T}+\mu\nu\frac{\sin{\omega T}}{m\omega}\right)$ (40)
and the shifted mean
$\bar{X}=\int\limits_{0}^{T}f(t)\left(\nu\frac{\sin{\omega(T-t)}\cos{\omega
T}+\sin{\omega t}}{\sin{\omega
T}}+\mu\frac{\sin{\omega(T-t)}}{m\omega}\right)dt$ (41)
whenever the oscillator is acted by external force $f(t)$. The coefficients
$\kappa$ and $\xi$ in (40) are determined by the measurement accuracy
$\\{\Delta a_{n}\\}$ and amount to [31]
$\displaystyle\kappa=\frac{2}{T^{2}}\sum\limits_{n=1}^{\infty}\frac{\Omega_{n}^{2}}{\Delta
a_{n}^{2}(\Omega_{n}^{2}-\omega^{2})^{2}},$ (42)
$\displaystyle\xi=\frac{2}{T^{2}}\sum\limits_{n=1}^{\infty}\frac{(-1)^{n}\Omega_{n}^{2}}{\Delta
a_{n}^{2}(\Omega_{n}^{2}-\omega^{2})^{2}}.$
Note that, given that propagator (39) of the measured oscillator has a
Gaussian dependence on $X^{\prime}$, integrating it with the initial tomogram
in (30) blurs its dependence on $X$. Instead, in the limit $\Delta
a_{n}\rightarrow\infty$ when there is actually no measurement, the variance
(40) tends to zero, collapsing the Gaussian in propagator (39) to the
$\delta$-function. The latter means that without the position measuring, the
propagator does not change the dependence of the initial tomogram on $X$ at
all.
## 5 Particle scattering
Another example, quite simple but worth discussing, is a particle scattering
upon measuring its position. To consider the measurement as direct, we further
assume $\Delta a_{n}=\Delta a$ for all $n$. Indeed, it is easy to show that in
this case functional (36) becomes
$w_{a}[q]=\exp\left\\{-\frac{2}{T\Delta
a^{2}}\int\limits_{0}^{T}(q(t)-a(t))^{2}dt\right\\}.$ (43)
A particle can be obviously treated as the particular case of the oscillator
having $\omega=0$. Thus, taking the $\omega\rightarrow 0$ limit in propagator
(39), it reduces to
$\tilde{\Pi}_{T}(\mathbf{x}^{\prime},\mathbf{x})=\left(\frac{1}{2\pi\sigma^{2}}\right)^{1/2}e^{-\frac{\left(X^{\prime}-X-\bar{X}\right)^{2}}{2\sigma^{2}}}\delta\left(\mu^{\prime}-\mu\right)\delta\left(\nu^{\prime}-\nu-\mu\frac{T}{m}\right)$
(44)
becoming the propagator for a particle undergoing a position measurement. In
this case, the variance $\sigma^{2}$ becomes
$\sigma^{2}=\frac{2}{3\Delta
a^{2}}\left(3\nu^{2}+3\nu\mu\frac{T}{m}+\mu^{2}\frac{T^{2}}{m^{2}}\right)$
(45)
where it has been taken into account that coefficients (42) containing the
measurement accuracy are simplified to
$\kappa=-2\xi=\frac{1}{3\Delta a^{2}}.$ (46)
Besides, the mean of $X^{\prime}$, denoted above as $\bar{X}$, is now also
simplified to
$\bar{X}=\int\limits_{0}^{T}f(t)\left(\nu+\mu\frac{T-t}{m}\right)dt.$ (47)
Let, for example, the particle has a position-space wave function
$\Psi_{p}(q)=(\pi l^{2})^{-1/4}\exp\left\\{ipq-\frac{q^{2}}{2l^{2}}\right\\}$
(48)
at initial time $t=0$. This state describes a particle located around the
origin with Gaussian distribution with the deviation $\Delta q=l/\sqrt{2}$ and
having an average impulse $p$ with uncertainty $\Delta p=1/(\sqrt{2}l)$. The
initial tomogram derived from (12) is then
$\mathcal{T}(\mathbf{x},0)=\left(\frac{\pi}{\mu^{2}l^{2}+\nu^{2}l^{-2}}\right)^{1/2}\exp\left\\{-\frac{\left(X-\nu
p\right)^{2}}{\mu^{2}l^{2}+\nu^{2}l^{-2}}\right\\}.$ (49)
Convolution of this tomogram with propagator (44) immediately gives the
evolved particle tomogram at time $t=T$:
$\mathcal{T}(\mathbf{x},T)=\left(\frac{\pi}{2\sigma^{2}+\mu^{2}l^{2}+\left(\nu+\mu\frac{T}{m}\right)^{2}l^{-2}}\right)^{1/2}\exp\left\\{-\frac{\left(X+\bar{X}-\left(\nu+\frac{T}{m}\mu\right)p\right)^{2}}{2\sigma^{2}+\mu^{2}l^{2}+\left(\nu+\mu\frac{T}{m}\right)^{2}l^{-2}}\right\\}$
(50)
where $\sigma^{2}$ is defined in (45). One sees that the continuous
measurement of the particle position results, as expected, in an additive
broadening of the initial distribution of $X$ by $\sigma^{2}$.
It is instructive to examine the change in the entropy of the particle, if its
position has been measured. In general, the symplectic entropy reads [32]
$S(\mu,\nu)=-\int\mathcal{T}(\mathbf{x})\ln\mathcal{T}(\mathbf{x})dX.$ (51)
For a Gaussian-type tomogram, as (50) is, the direct calculation gives
$S_{T}(\mu,\nu)=\frac{1+\ln\pi}{2}+\frac{1}{2}\ln\left[2\sigma^{2}+\mu^{2}l^{2}+\left(\nu+\mu\frac{T}{m}\right)^{2}l^{-2}\right]$
(52)
at time $T$. Hence, the difference between the entropy values for the measured
particle and the unaffected one is
$\Delta S_{T}(\mu,\nu)=\frac{1}{2}\ln\left(1+\frac{4}{3\Delta
a^{2}}\frac{3\nu^{2}+3\nu\mu\frac{T}{m}+\mu^{2}\frac{T^{2}}{m^{2}}}{\mu^{2}l^{2}+\left(\nu+\mu\frac{T}{m}\right)^{2}l^{-2}}\right).$
(53)
In the conclusion, we concern whether propagator (44) of a particle being
under a position measurement is consistent with the equation (33) governing
its time evolution. Indeed, applying the star-product formalism to equation
(33) where $\hat{A}$ is replaced by $\hat{q}$, one thus gets [18]
$\frac{\partial\tilde{\Pi}_{t}}{\partial
t}-\frac{\mu}{m}\frac{\partial\tilde{\Pi}_{t}}{\partial\nu}+f(t)\nu\frac{\partial\tilde{\Pi}_{t}}{\partial
X}-k\nu^{2}\frac{\partial^{2}\tilde{\Pi}_{t}}{\partial X^{2}}=0.$ (54)
By a straightforward substitution, it can be easily verified that propagator
(44) does satisfy this equation for $k=1/\Delta a^{2}$ (note, that in order to
satisfy it, one must take into account that $\Delta a^{2}\sim 1/T$ [33]).
## 6 Conclusion
The subject of the current study is the influence a quantum system experiences
upon a continuous measurement, considered within the framework of the
probability representation of quantum mechanics.
The symplectic tomogram attributed to each isolated quantum system evolves
with it, the resulting tomogram being determined by a classical propagator (or
tomogram propagator). The latter, in turn, incorporates the quantum propagator
of that system. We applied the representation of quantum continuous
measurement through the restricted path integral to modify the classical
propagator. In particular, we have introduced a partial tomogram propagator
for the measurement when the result is known. This partial propagator,
however, determines the tomogram density of the evolved system in the set of
measurement outcomes. If one performs a non-selective measurement, when the
outcome is not known, the tomogram propagator can be calculated just by
integrating the partial propagators over all outcomes.
The spectral position measurement of a driven oscillator is examined as well
as its particular case of a particle. Particularly, the tomogram propagator
for the oscillator under a continuous position measurement is obtained. It is
shown that the coordinate dependence of the tomogram of both the oscillator
and the particle takes the Gaussian form. The latter results in an additional
blurring of the coordinate dependence the tomogram initially had.
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|
# The model-companionship spectrum of set theory, generic absoluteness, and
the Continuum problem
Matteo Viale
###### Abstract.
We show that for $\Pi_{2}$-properties of second or third order arithmetic as
formalized in appropriate natural signatures the apparently weaker notion of
_forcibility_ overlaps with the standard notion of _consistency_ (assuming
large cardinal axioms).
Among such $\Pi_{2}$-properties we mention: the negation of the continuum
hypothesis, Souslin Hypothesis, the negation of Whitehead’s conjecture on free
groups, the non-existence of outer automorphisms for the Calkin algebra, etc…
In particular this gives an a posteriori explanation of the success forcing
(and forcing axioms) met in producing models of such properties.
Our main results relate generic absoluteness theorems for second order
arithmetic, Woodin’s axiom $(*)$ and forcing axioms to Robinson’s notion of
model companionship (as applied to set theory). We also briefly outline in
which ways these results provide an argument to refute CH.
The author acknowledge support from INDAM through GNSAGA and from the project:
_PRIN 2017-2017NWTM8R Mathematical Logic: models, sets, computability. MSC:
_03C10 03E57._ _
## Introduction
### Model completeness, model companionship, and the model companionship
spectrum of a theory
Model companionship and model completeness are model theoretic notions
introduced by Robinson which give a simple first order characterization of the
way algebraically closed fields sits inside the class of rings with no zero-
divisors. We start this paper rushing through the main properties of model
completess and model companionship (we will later on analyze carefully all
these concepts in Section 1). Our aim is to show in a few paragraphs how we
can use these notions to reformulate in a simple model-theoretic terminology
deep generic absoluteness results for second order arithmetic by Woodin and
others, as well as other major results on forcing axioms and Woodin’s Axiom
$(*)$.
The key model-theoretic concept we are interested in is that of existentially
closed model of a first order theory111We adopt the following notational
conventions: $\sqsubseteq$ denotes the substructure relation between
structures; $\mathcal{M}\prec_{n}\mathcal{N}$ indicates that $\mathcal{M}$ is
a $\Sigma_{n}$-elementary substructure of $\mathcal{N}$, we omit the $n$ to
denote full-elementarity; given a first order theory $T$, $T_{\forall}$
denotes the universal sentences which are consequences of $T$, likewise we
interpret $T_{\exists},T_{\forall\exists},\dots$. $T$:
###### Definition 1.
Let $\tau$ be a signature and $T$ be a first order theory. $\mathcal{M}$ is
$T$-existentialy closed ($T$-ec) if for any $\tau$-structure
$\mathcal{N}\sqsupseteq\mathcal{M}$ which is a model of $T$ we have that
$\mathcal{M}\prec_{1}\mathcal{N}.$
A key non-trivial fact is that $\mathcal{M}$ is $T$-ec if and only if it is
$T_{\forall}$-ec.
It doesn’t take long to realize that in signature
$\tau=\left\\{+,\cdot,0,1\right\\}$ the $\tau$-theory $T$ of fields has as its
class of existentially closed models exactly the algebraically closed fields.
Note also that if we let $S$ be the class of rings with no zero-divisors which
are not fields, we still have that the $S$-existentially closed structures are
the algebraically closed fields (even if no field is a model of $S$).
Model completeness and model companionship allow to generalize these features
of the class of rings with no zero divisors to arbitrary first order theories.
###### Definition 2.
Let $\tau$ be a first order signature.
* •
A $\tau$-theory $T$ is _model complete_ if any model of $T$ is $T$-ec.
* •
$T$ is the _model companion_ of a $\tau$-theory $S$ if:
* –
any model of $S$ embeds into a model of $T$ and conversely,
* –
$T$ is model complete.
In particular in signature $\tau=\left\\{+,\cdot,0,1\right\\}$, the theory of
algebraically closed fields is model complete and is the model companion both
of the theory of fields and of the theory of rings with no zero-divisors which
are not fields.
We will also need here the following equivalent characterization of model
completeness: $T$ is model complete whenever
> _For $\mathcal{M},\mathcal{N}$ models of $T$, $\mathcal{M}\prec\mathcal{N}$
> if and only if $\mathcal{M}\prec_{1}\mathcal{N}$ if and only if
> $\mathcal{M}\sqsubseteq\mathcal{N}$_.
Note also that:
* •
any theory $T$ admitting quantifier elimination is model complete;
* •
any model complete theory $T$ is the model companion of itself;
* •
two $\tau$-theories $T$ and $S$ which have no model in common can have the
same model companion, but the model companion of a theory $T$ if it exists is
unique;
* •
if $T^{*}$ is the model companion of $T$ it can be the case that no model of
$T$ is a model of $T^{*}$ and conversely;
* •
there are $\tau$-theories $T$ which do not admit a model companion (for
example this is the case for the theory of groups in signature
$\tau=\left\\{\cdot,1\right\\}$).
Much in the same way as the algebraic closure of a ring $R$ with no zero-
divisors closes off $R$ with respect to solutions to polynomial equations with
coefficients in $R$ and which exist in some superring of $R$ which has no
zero-divisors (and which does not have to be algebraically closed), for a
theory $T$ with model companion $T^{*}$ any model $\mathcal{M}$ of $T$ brings
to a supermodel $\mathcal{N}$ of $T^{*}$ which is obtained by adding (at
least) the solutions to the existential formulae with parameters in
$\mathcal{M}$ which are consistent with the universal fragment of $T$ (in the
case of ring with no zero-divisors the key universal property one has to
maintain is the non-existence of zero-divisors along with the ring axioms).
A key property of model companionship which brought our attention to this
notion is the following (see Section 1 for details):
###### Fact 1.
Let $\tau$ be a first order signature and $T$ be a _complete_ $\tau$-theory
with model companion $T^{*}$. Then $T^{*}$ is axiomatized by
$T^{*}_{\forall\exists}$ and TFAE for a $\Pi_{2}$-sentence $\psi$ for $\tau$:
* •
$T_{\forall}+\psi$ is consistent.
* •
$\psi\in T^{*}$.
In case $T$ is a companionable non-complete theory, further weak hypothesis on
$T$ (which are satisfied by set theory) allow to characterize its model
companion $T^{*}$ as the unique theory axiomatized by the $\Pi_{2}$-sentences
which are consistent with the universal fragment of any completion of $T$ (see
Lemma 1.21).
Unlike other notions of complexity (such as stability, NIP, simplicity) model
companionship and model completeness are very sensitive to the signature in
which one formalizes a first order theory $T$.
###### Notation 1.
For a given signature $\tau$, $\tau^{*}$ is the signature extending $\tau$
with new function symbols222As usual we confuse $0$-ary function symbols with
constants. $f_{\phi}$ and new relation symbols $R_{\phi}$ for any
$\tau$-formula $\phi(x_{0},x_{1},\dots,x_{n})$. $T_{\tau}$ is the
$\tau^{*}$-theory with axioms
$\text{{\sf AX}}^{0}_{\phi}:=\forall\vec{x}[\phi(\vec{x})\leftrightarrow
R_{\phi}(\vec{x})]$ $\text{{\sf AX}}^{1}_{\phi}:=\forall
x_{1},\dots,x_{n}[\exists
y\phi(y,x_{1},\dots,x_{n})\rightarrow\phi(f_{\phi}(x_{1},\dots,x_{n}),x_{1},\dots,x_{n})],$
as $\phi$ ranges over the $\tau$-formulae.
It is clear that any $\tau$-structure admits a unique extension to a
$\tau^{*}$-model of $T_{\tau}$ and any $\tau$-theory $T$ is such $T\cup
T_{\tau}$ admits quantifier elimination, hence is model complete and is its
own model companion relative to signature $\tau^{*}$. This holds regardless of
whether the $\tau$-theory $T$ is model complete or admits a model companion in
signature $\tau$ (cfr. $T$ being the theory of groups in signature
$\left\\{\cdot,1\right\\}$). On the other hand $T$ is stable (simple, NIP) if
and only if so is $T_{\tau}$. This is a serious drawback if one wishes to use
model companionship to gauge the complexity of a mathematical theory $T$,
since model companionship of $T$ is very much dependent on the signature in
which we formalize it: $T$ can trivially be model complete if we formalize it
in a rich enough signature.
We now introduce a simple trick to render model companionship a useful
classification tool for mathematical theories regardless of the signature in
which we give their first order axiomatization. Roughly the idea is to
consider all possible signatures in which a theory can be formalized and pay
attention only to those for which the theory admits a model companion.
###### Definition 3.
Let $\tau$ be a signature and $F_{\tau}$ denote the set of $\tau$-formulae.
Given $A\subseteq F_{\tau}\times 2$, let $\tau_{A}$ be the signature
$\tau\cup\left\\{R_{\phi}:(\phi,0)\in
A\right\\}\cup\left\\{f_{\phi}:(\phi,1)\in A\right\\}$. A $\tau$-theory $T$ is
_$(A,\tau)$ -companionable_ if
$T_{A}=T\cup\left\\{\text{{\sf AX}}^{i}_{\phi},:(\phi,i)\in A\right\\}$
admits a model companion for the signature $\tau_{A}$.
Given a $\tau$-theory $T$ its _$\tau$ -companionship spectrum_ is given by
those $A\subseteq F_{\tau}\times\left\\{0,1\right\\}$ such that $T$ is
$(A,\tau)$-companionable.
Note that $F_{\tau}\times\left\\{i\right\\}$ is always in the companionship
spectrum of $T$, but proving that some $\bar{A}\subsetneq F_{\tau}$ is such
that some $A\subseteq\bar{A}\times 2$ is in the companionship spectrum of $T$
is a (possibly highly) non-trivial and informative result on $T$; model-
companionability for $T$ amounts to say that $T$ is
$(\emptyset,\tau)$-companionable. The $\tau$-companionship spectrum of $T$ is
non-informative if $T$ is model complete in signature $\tau$: in this case the
$\tau$-companionship spectrum of $T$ is $\mathcal{P}\left(F_{\tau}\times
2\right)$.
Note also that even if $T$ is $(\emptyset,\tau)$-companionable there could be
many $A\subseteq F_{\tau}\times 2$ such that $T$ is $A$-companionable and many
$B\subseteq F_{\tau}\times 2$ such that $T$ is not $B$-companionable; in
principle nothing prevents the families of such $A$s and $B$s to be both of
size $2^{|F_{\tau}|}$ and to produce a complex ordering of the
$\tau$-companionship spectrum of $T$ with respect to $\subseteq$.
To better grasp the above considerations, let for a $\tau$-theory $T$ $C_{T}$
be the category whose objects are the $\tau$-models of $T$ and whose arrows
are the $\tau$-morphisms. NIP, stability, simplicity are properties which
consider only the objects in this category, model completeness and model
companionship pay also attention to the arrows of this category. We get a much
deeper insight on the properties of $C_{T}$ if we are able to detect for which
$A\subseteq F_{\tau}\times 2$ $T_{A}$ is model companionable: for any
$A\subseteq F_{\tau}\times 2$ in the passage from $C_{T}$ to $C_{T_{A}}$ we
maintain the same class of objects, but the $\tau_{A}$-morphisms (i.e the
arrows of $C_{T_{A}}$) are just the $\tau$-morphisms between models of $T$
which preseve the formulae in $A$, hence we are possibly destroying many
arrows.
Our definition of $\tau$-companionship spectrum of a mathematical theory is
apparently dependent on the signature $\tau$ in which we formalize it. We may
argue that this is not the case, but to uncover why would bring us far afield
and we defer this task to another paper. We will in this paper confine our
attention to use this notion to analyze first order axiomatizations of set
theory enriched with large cardinal axioms. In this case we can certainly say
that proving that some $A\subsetneq F_{\left\\{\in\right\\}}\times 2$ is in
the $\in$-companionship spectrum of set theory is an informative result:
$\left\\{\in\right\\}$ is a minimal signature in which set theory can be
formalized (in the empty signature we certainly cannot formalize it), hence
any $A\subseteq F_{\left\\{\in\right\\}}\times 2$ for which set theory is
$A$-companionable gives non-trivial information on set theory. Moreover we can
easily verify that any reasonable $\in$-axiomatization of set theory is not
model complete for the $\in$-signature, hence the $\in$-companionship spectrum
of set theory is certainly non-trivial.
### Some of our main results
We can now state in an informative way key parts of our main results.
The first non-trivial result states that for any definable cardinal $\kappa$
there is at least one signature admitting a constant for the cardinal $\kappa$
such that set theory is companionable for this signature.
It is convenient from now on to adopt the following short-hand notation for
structures:
###### Notation 2.
Given a signature $\tau$, $\mathcal{M}=(M,\tau^{\mathcal{M}})$ is a shorthand
for the $\tau$-structure $(M,R^{\mathcal{M}}:R\in\tau)$.
###### Theorem 1.
Let $T\supseteq\mathsf{ZFC}$ be a $\in$-theory, and $\kappa$ be a
$T$-definable cardinal (i.e. such that for some $\in$-formula
$\phi_{\kappa}(x)$ $T$ proves $\exists!x[\phi_{\kappa}(x)\wedge x$_is a
cardinal_ $]$).
Then there is at least one $A_{\kappa}\subsetneq F_{\in}$ such that letting
$\bar{A}_{\kappa}=A_{\kappa}\times 2$:
1. (1)
For all models $(V,\in)$ of $T$
$(H_{\kappa^{+}}^{V},\left\\{\in\right\\}_{\bar{A}_{\kappa}}^{V})\prec_{1}(V,\left\\{\in\right\\}_{\bar{A}_{\kappa}}^{V})$.
2. (2)
$T$ is $\bar{A}_{\kappa}$-companionable.
3. (3)
The model companion $T^{*}_{\kappa}$ of $T_{\bar{A}_{\kappa}}$ for signature
$\left\\{\in\right\\}_{\bar{A}_{\kappa}}$ is the
$\left\\{\in\right\\}_{\bar{A}_{\kappa}}$-theory common to
$H_{\kappa^{+}}^{V}$ as $(V,\in)$ ranges over $\in$-models of $T$ and $\kappa$
is the constant of $\left\\{\in\right\\}^{*}$ given by the formula
$\exists!x[\phi_{\kappa}(x)\wedge x$_is a cardinal_ $]$.
4. (4)
$T^{*}_{\kappa}$ is also axiomatized by the $\Pi_{2}$-sentences for
$\left\\{\in\right\\}_{\bar{A}_{\kappa}}$ which are consistent with
$S_{\forall}$ for any $\left\\{\in\right\\}_{\bar{A}_{\kappa}}$-theory $S$
which is a complete extension of $T_{\bar{A}_{\kappa}}$.
Note that the above theorem allows to put in the companionship spectrum of any
extension of $\mathsf{ZFC}$ at least one $\bar{A}_{\kappa}$ for each definable
cardinal $\kappa$ such as
$\omega,\omega_{1},\dots,\aleph_{\omega},\dots,\aleph_{\omega_{1}},\dots,\kappa,\dots$
for $\kappa$ the least inaccessible, measurable, Woodin, supercompact,
extendible…
In case $\kappa=\omega,\omega_{1}$ we can say much more and prove that for
$\Pi_{2}$-sentences in the appropriate signature forcibility and consistency
overlap (assuming large cardinal axioms).
This gives an a posteriori explanation of the success forcing has met in
proving the consistency of $\Pi_{2}$-properties (according to the right
signature) for second or third order artithmetic: our results show that there
are no other means to prove the consistency of such statements.
###### Theorem 2.
Let $S$ be _any_ extension of
$\mathsf{ZFC}+\text{{suitable} large cardinal axioms}$
in signature $\tau=\left\\{\in\right\\}$. There are $A_{1}\neq A_{2}\subseteq
F_{\left\\{\in\right\\}}$ recursive sets of $\in$-formulae such that (letting
$\bar{A}_{i}=A_{i}\times 2$ for $i=1,2$):
1. (1)
For all models $(V,\in)$ of $S$
$(H_{\omega_{i}}^{V},\left\\{\in\right\\}_{\bar{A}_{i}}^{V})\prec_{1}(V,\left\\{\in\right\\}_{\bar{A}_{i}}^{V})$
for both $i=1,2$.
2. (2)
$S$ is $\bar{A}_{i}$-companionable for both $i=1,2$333With very strong large
cardinal axioms for the case for $T_{\bar{A}_{2}}$, and no large cardinal
axioms in the case for $T_{\bar{A}_{1}}$..
3. (3)
The model companion of $S_{\bar{A}_{1}}$ is the $\tau_{\bar{A}_{1}}$-theory
common to the models $H_{\aleph_{1}}^{V[G]}$ as $(V\in)$ ranges over
$\in$-models of $S$ and $G$ is $V$-generic for some444If one is not at ease
with the (inconsistent) assumption that $V[G]$ exists, this can be
reformulated as: $(V\,in)\models\exists P\,(P\Vdash\psi^{H_{\omega_{1}}})$ and
$(V,\in)\models S$. $P\in V$.
4. (4)
The model companion of $S_{\bar{A}_{2}}$ is the $\tau_{\bar{A}_{2}}$-theory
common to all $H_{\aleph_{2}}^{V[G]}$ for $V[G]$ a forcing extension of $V$
which models $\text{{\sf MM}}^{++}$ and $(V,\in)$ a $\in$-model of $S$ 555With
very strong large cardinal axioms holding in $V$. $\text{{\sf MM}}^{++}$ is
one of the strongest forcing axioms..
5. (5)
$(S_{\bar{A}_{1}})_{\forall}$ and $(S_{\bar{A}_{2}})_{\forall}$ are both
invariant across forcing extensions of $V$ for any $\in$-model $(V,\in)$ of
$S$ (assuming the existence of class many Woodin cardinals in $V$).
###### Corollary 1.
Assume $S$ extends $\mathsf{ZFC}$ with the correct large cardinal axioms.
Let $X$ be any among $\bar{A}_{1},\bar{A}_{2}\subseteq
F_{\left\\{\in\right\\}}\times 2$ as in the previous theorem, and:
* •
$S_{X}$ be the $\tau_{X}$-theory $S\cup\left\\{\text{{\sf
AX}}^{i}_{\phi}:(\phi,i)\in X\right\\}$,
* •
$S^{*}_{X}$ be the model companion theory of $S_{X}$ given by the previous
theorem.
TFAE for any $\Pi_{2}$-sentence $\psi$ for $\tau_{X}$:
1. (A)
$\psi\in S^{*}_{X}$;
2. (B)
$R_{\forall}+\psi$ is consistent for all $\tau_{X}$-theories $R$ which are
complete extensions of $S_{X}$;
3. (C)
(if $X=\bar{A}_{2}$)
$S_{X}\models\exists P\,\left[\Vdash_{P}\psi^{H_{\omega_{2}}}\right];$
(if $X=\bar{A}_{1}$)
$S_{X}\models\exists P\,\left[\Vdash_{P}\psi^{H_{\omega_{1}}}\right].$
In particular the equivalence of B with C shows that _forcibility_ and
_consistency_ overlap for $\Pi_{2}$-sentences in signature
$\left\\{\in\right\\}_{X}$.
We complete this introduction outlining a bit more the significance of the
above results and trying to get a better insight on what are the signatures
$\left\\{\in\right\\}_{\bar{A}_{1}},\left\\{\in\right\\}_{\bar{A}_{2}},\left\\{\in\right\\}_{\bar{A}_{\kappa}}$
mentioned in the theorems.
### What is the right signature for set theory?
The $\in$-signature is certainly sufficient to give by means of $\mathsf{ZFC}$
a first order axiomatization of set theory (with eventually other extra
hypothesis such as large cardinal axioms), but we can see rightaway that it is
not efficient to formalize many basic set theoretic concepts. Consider for
example the notion of ordered pair: on the board we write $x=\langle
y,z\rangle$ to mean that _$x$ is the ordered pair with first component $y$ and
second component $z$_. In set theory this concept is formalized by means of
Kuratowski’s trick stating that
$x=\left\\{\left\\{y\right\\},\left\\{y,z\right\\}\right\\}$. However the
$\in$-formula formalizing the above is:
$\exists t\exists u\;[\forall w\,(w\in x\leftrightarrow w=t\vee
w=u)\wedge\forall v\,(v\in t\leftrightarrow v=y)\wedge\forall v\,(v\in
u\leftrightarrow v=y\vee v=z)].$
It is clear that the meaning of this $\in$-formula is hardly decodable with a
rapid glance (unlike $x=\langle y,z\rangle$), moreover just from the point of
view of its syntactic complexity it is already $\Sigma_{2}$. On the other hand
we do not regard the notion of ordered pair as a complex or doubtful concept
(as is the case for the notion of uncountability, or many of the properties of
the continuum such as its correct place in the hierarchy of uncountable
cardinals, etc…). Other vary basic notions such as: being a function, a binary
relation, the domain or the range of a function, etc.. are formalized already
by rather complicated $\in$-formulae, both from the point of view of
readability for human beings and from the mere computation of their syntactic
complexity according to the Levy hierarchy.
The standard solution adopted by set theorists (e.g. [13, Chapter IV]) is to
regard as elementary all those properties which can be formalized using
$\in$-formulae all of whose quantifiers are bounded to range over the elements
of some set, i.e. the so called $\Delta_{0}$-formulae (see [13, Chapter IV,
Def. 3.5]). We henceforth adopt this point of view and let $B_{0}\subseteq
F_{\left\\{\in\right\\}}$ be the set of such formulae and denote by
$\tau_{\text{{\sf ST}}}$ what according to our previous terminology should
rather be $\left\\{\in\right\\}_{B_{0}\times 2}$. For the sake of convenience
and also to further outline some very nice syntactic features of
$\mathsf{ZFC}$ as formalized in $\tau_{\text{{\sf ST}}}$, let us bring to
front an explicit axiomatization of $T_{B_{0}}$ (which from now on will be
denoted by $T_{\text{{\sf ST}}}$).
###### Notation 3.
__
* •
$\tau_{\mathsf{ST}}$ is the extension of the first order signature
$\left\\{\in\right\\}$ for set theory which is obtained by adjoining predicate
symbols $R_{\phi}$ of arity $n$ for any $\Delta_{0}$-formula
$\phi(x_{1},\dots,x_{n})$, function symbols of arity $k$ for any
$\Delta_{0}$-formula $\theta(y,x_{1},\dots,x_{k})$ and constant symbols for
$\omega$ and $\emptyset$.
* •
$\mathsf{ZFC}^{-}$ is the $\in$-theory given by the axioms of $\mathsf{ZFC}$
minus the power-set axiom.
* •
$T_{\text{{\sf ST}}}$ is the $\tau_{\text{{\sf ST}}}$-theory given by the
axioms
$\forall\vec{x}\,(R_{\forall z\in y\phi}(y,z,\vec{x})\leftrightarrow\forall
z(z\in y\rightarrow R_{\phi}(y,z,\vec{x}))$
$\forall\vec{x}\,[R_{\phi\wedge\psi}(\vec{x})\leftrightarrow(R_{\phi}(\vec{x})\wedge
R_{\psi}(\vec{x}))]$
$\forall\vec{x}\,[R_{\neg\phi}(\vec{x})\leftrightarrow\neg R_{\phi}(\vec{x})]$
$\forall\vec{x}\left[\exists!y\,R_{\phi}(y,\vec{x})\leftrightarrow
R_{\phi}(f_{\phi}(\vec{x}),\vec{x})\right]$
for all $\Delta_{0}$-formulae $\phi(\vec{x})$, together with the
$\Delta_{0}$-sentences
$\forall x\in\emptyset\,\neg(x=x),$ $\omega\text{ is the first infinite
ordinal}$
(the former is an atomic $\tau_{\text{{\sf ST}}}$-sentence, the latter is
expressible as the atomic sentence for $\tau_{\text{{\sf ST}}}$ stating that
$\omega$ is a non-empty limit ordinal all whose elements are successor
ordinals or $0$).
* •
$\mathsf{ZFC}^{-}_{\text{{\sf ST}}}$ is the $\tau_{\text{{\sf ST}}}$-theory
$\mathsf{ZFC}^{-}\cup T_{\text{{\sf ST}}}.$
* •
Accordingly we define $\mathsf{ZFC}_{\text{{\sf ST}}}$.
Note that $T_{\text{{\sf ST}}}$ is axiomatized by $\Pi_{2}$-sentences of
$\tau_{\text{{\sf ST}}}$.
### Levy absoluteness and model companionship results for set theory
Kunen’s [13, Chapter IV] gives a rather convincing summary of the reasons why
it is convenient to formalize set theory using $\tau_{\text{{\sf ST}}}$ rather
than $\in$. We focus here on the role Levy’s absoluteness plays in the search
of $A\subseteq F_{\left\\{\in\right\\}}\times 2$ for which set theory is
$A$-companionable.
###### Lemma 1.
Let $(V,\in)$ be a model of $\mathsf{ZFC}$ and $\kappa$ be an infinite
cardinal for $V$. Then
$(H_{\kappa^{+}}^{V},\tau_{\text{{\sf
ST}}}^{V},A:A\subseteq\mathcal{P}\left(\kappa\right)^{k},\,k\in\mathbb{N})\prec_{1}(V,\tau_{\text{{\sf
ST}}}^{V},A:A\subseteq\mathcal{P}\left(\kappa\right)^{k},\,k\in\mathbb{N})$
Its proof is a trivial variant of the classical result of Levy (which is the
above theorem stated just for the signature $\tau_{\text{{\sf ST}}}$); it is
given in [22, Lemma 5.3].
The upshot is that for any model $V$ of $\mathsf{ZFC}$ and any signature
$\sigma$ such that
$\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}\subseteq\sigma\subseteq\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}\cup\bigcup_{k\in\mathbb{N}}\mathcal{P}\left(\kappa\right)^{k}$
$H_{\kappa^{+}}$ is $\Sigma_{1}$-elementary in $V$ according to $\sigma$. This
is a first indication that for a $\mathsf{ZFC}$-definable cardinal $\kappa$
(e.g. $\kappa=\omega,\omega_{1},\aleph_{\omega},\dots$, more precisely
$\kappa$ being provably in some $T\supseteq\mathsf{ZFC}$ the unique solution
of an $\in$-formula $\phi_{\kappa}(x)$) if $\sigma_{\kappa}=\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$ and $T_{\kappa}$ is the
$\sigma_{\kappa}$-theory given by $\mathsf{ZFC}+\phi_{\kappa}(\kappa)$, we get
that the $\sigma_{\kappa}$-theory common to all of the $H_{\kappa^{+}}^{V}$ as
$V$ ranges over the model of $\mathsf{ZFC}$ is not that far from being
$\mathsf{ZFC}$-ec, since a model of this theory is always a
$\Sigma_{1}$-substructure of some $\sigma_{\kappa}$-model of $\mathsf{ZFC}$.
A second indication that the $\sigma_{\kappa}$-theory of $H_{\kappa^{+}}$ is
close to be the model companion of the $\sigma_{\kappa}$-theory of $V$ is the
fact that the $\Pi_{2}$-sentence for $\sigma_{\kappa}$
$\forall x\exists f:\kappa\to x\text{ surjective function}$
is realized in $H_{\kappa^{+}}^{V}$ for any model $V$ of $\mathsf{ZFC}$ (note
that by Levy’s absoluteness this sentence is consistent with the universal
fragment of the $\sigma_{\kappa}$-theory of $V$, hence by Fact 1 it belongs to
the model companion of set theory for $\sigma_{\kappa}$ — if such a model
companion exists).
In particular if some $T\supseteq\mathsf{ZFC}$ is $\sigma$-companionable for
some $\sigma$ as above, the model companion of $T$ for $\sigma$ should be the
theory of $H_{\kappa^{+}}^{\mathcal{M}}$ for suitably chosen $\mathcal{M}$
which are models of set theory.
A natural question is:
> _Can we cook up $\sigma\supseteq\tau_{\text{{\sf
> ST}}}\cup\left\\{\kappa\right\\}$ so that the $\sigma$-theory of
> $H_{\kappa^{+}}$ is the model companion of the $\sigma$-theory of $V$?_
Theorem 1 answer affirmatively to this question for many natural choices of
$\sigma$ and for all definable cardinals $\kappa$.
### Why the continuum is the second uncountable cardinal
Theorem 2 refines Thm. 1 for the cases $\kappa=\omega,\omega_{1}$. In these
cases our knowledge of the theory of $H_{\kappa^{+}}$ is much more extensive;
moreover most of mathematics can be formalized in $H_{\omega_{1}}$ (all of
second order arithmetic) or in $H_{\omega_{2}}$ (most of third order
arithmetic).
We now want to outline briefly why Thm. 2 provides an interesting
metamathematical argument in favour of strong forcing axioms and against CH.
The considerations of this brief paragraph will be expanded in more details in
a forthcoming paper and have been elaborated jointly with Giorgio Venturi.
Those who are familiar with forcing axioms know that Martin’s maximum and its
bounded forms have been instrumental to prove the consistency of a solution of
many problems of third order arithmetic which are provably undecidable in
$\mathsf{ZFC}$ (or even in $\mathsf{ZFC}$ supplemented by large cardinal
axioms), a sample of these solutions include: the negation of the continuum
hypothesis [8, 6, 25, 21, 16], the negation of Whitehead conjecture on free
abelian groups [18], the non-existence of outer automorphism of the Calkin
algebra [7], the Suslin hypothesis [11], the existence of a five element basis
for uncountable linear order [17]…All statements of the above list (with the
exception of the non-existence of outer automorphism of the Calkin algebra)
and many others can be formalized as $\Pi_{2}$-sentences in signature
$\tau_{\omega_{1}}=\tau_{\text{{\sf ST}}}\cup\left\\{\omega_{1}\right\\}$
(where $\omega_{1}$ is a new constant symbol which is the unique solution of
some formula in one free variable defining the first uncountable cardinal).
For example $\neg\text{{\sf CH}}$ is formalized by
$\forall f\,\left[(f\text{ is a
function}\wedge\operatorname{dom}(f)=\omega_{1})\rightarrow\exists
r\,(r\subseteq\omega\wedge r\not\in\operatorname{ran}(f))\right].$
In particular there has been empiric evidence that forcing axioms produce
models of set theory which maximize the family of $\Pi_{2}$-sentences which
hold true in $H_{\omega_{2}}$ for the signature $\tau_{\omega_{1}}$. Thm. 2
makes this empiric evidence a true mathematical fact: first of all it is
important to note here that (sticking to the notation of Thm. 2)
$\left\\{\in\right\\}_{\bar{A}_{2}}\supseteq\tau_{\omega_{1}}$. Now let $T$ be
a theory as in the assumption of Thm. 2; take (in signature
$\left\\{\in\right\\}_{\bar{A}_{2}}$) any $\Pi_{2}$-sentence $\psi$ which is
consistent with $S_{\forall}$ whenever $S$ is a complete extension of
$T_{\bar{A}_{2}}$; then by A$\Longleftrightarrow$B of Corollary 1 $\psi$ is in
the model companion of $T_{\bar{A}_{2}}$, and Thm. 4(4) (almost) asserts that
$\psi^{H_{\omega_{2}}}$ is derivable from $\text{{\sf MM}}^{++}$. Note that
$\text{{\sf MM}}^{++}$ is one of the strongest forcing axioms.
Another key observation is that (assuming large cardinals) the signature
$\left\\{\in\right\\}_{\bar{A}_{2}}$ is such that the universal fragment of
set theory as formalized in $\left\\{\in\right\\}_{\bar{A}_{2}}$ is _invariant
through forcing extensions of $V$_. What this means is that one _can and must
use forcing_ to establish whether some $\Pi_{2}$-sentence $\psi$ is in the
model companion of set theory according to
$\left\\{\in\right\\}_{\bar{A}_{2}}$.
This is the major improvement of Thm. 2 with respect to Thm. 1: for most of
the signatures $\tau_{\bar{A}_{\kappa}}$ mentioned in Thm. 1 we cannot just
use forcing to establish whether a $\Pi_{2}$-sentence $\psi$ for this
signature is in the model companion of set theory for
$\tau_{\bar{A}_{\kappa}}$.
Let us develop more on this point because it is in our eyes one of the major
advances given by the results of the present paper. Take
$S\supseteq\mathsf{ZFC}$; for a given $X\subseteq
F_{\left\\{\in\right\\}}\times 2$ for which we can prove that $S_{X}$ has a
model companion in signature $\left\\{\in\right\\}_{X}$ we would like to show
that a certain $\Pi_{2}$-sentence $\psi$ for $\left\\{\in\right\\}_{X}$ is in
the model companion of $S_{X}$.
Let us first suppose that $X$ is some $\bar{A}_{\kappa}$ as in Thm. 1. A first
observation is that (with the exception of the $\Delta_{0}$-formulae) all the
formulae in $A_{\kappa}$ define subsets of
$\mathcal{P}\left(\kappa\right)^{n}$ for some $n$, hence
$(H_{\kappa^{+}}^{V},\left\\{\in\right\\}_{\bar{A}_{\kappa}}^{V})\prec_{1}(V,\left\\{\in\right\\}_{\bar{A}_{\kappa}}^{V})$
for any $(V,\in)$ which models $\mathsf{ZFC}$. This gives that if
$(W,\left\\{\in\right\\}_{\bar{A}_{\kappa}}^{W})$ models $(S_{X})_{\forall}$,
then so does
$(H_{\kappa^{+}}^{W},\left\\{\in\right\\}_{\bar{A}_{\kappa}}^{W})$.
A natural strategy to put $\psi$ in the model companion of $S_{X}$ would then
be to start from some complete $T\supseteq S$ and some $(V,\in)$ model of $T$;
then force over $V$ that in some $V[G]$ $\psi^{H_{\kappa^{+}}^{V[G]}}$ holds
true; if $(T_{\bar{A}_{\kappa}})_{\forall}$ holds in $V[G]$, then $\psi$ would
be in the model companion of $S_{\bar{A}_{\kappa}}$ by Thm. 1(4): Levy’s
absoluteness applied to $(V[G],\tau_{\bar{A}_{\kappa}}^{V[G]})$ would yield
that $H_{\kappa^{+}}^{V[G]}\models\psi+(T_{\bar{A}_{\kappa}})_{\forall}$.
Now starting from any model $V$ of $S$ we may be able to design a forcing in
$V$ such that $\psi^{H_{\kappa^{+}}^{V[G]}}$ holds if $G$ is $V$-generic for
this forcing, but it may be the case that $(S_{\bar{A}_{\kappa}})_{\forall}$
fails in $V[G]$; in which case we cannot use $H_{\kappa^{+}}^{V[G]}$ as a
witness that $\psi$ is in the model companion of $S_{\bar{A}_{\kappa}}$.
Remark 1 shows that $(S_{\bar{A}_{\kappa}})_{\forall}$ is not preserved
through forcing extensions whenever $\kappa>\omega_{1}$.
On the other hand for the signatures $\tau_{X}$ for $X$ being the
$\bar{A}_{1}$ or $\bar{A}_{2}$ mentioned in Thm. 2 the above strategy works:
the universal fragment of $(S_{X})_{\forall}$ is preserved through the forcing
extensions of models of $S$; hence $\psi$ will be in the model companion of
$S_{X}$ if for any model $V$ of $S$ we can design a forcing making true $\psi$
in $H_{\kappa^{+}}^{V[G]}$ (for $\kappa=\omega,\omega_{1}$ according to
whether $\psi$ is a formula for $\tau_{\bar{A}_{1}}$ or for
$\tau_{\bar{A}_{2}}$).
Summing up one _may and should only_ use forcing to establish the consistency
with large cardinals of $\psi^{H_{\omega_{2}}}$ for any $\Pi_{2}$-sentence
formalizable in signatures
$\tau_{\omega_{1}}\subseteq\left\\{\in\right\\}_{\bar{A}_{2}}$: the strategy
we outlined above is efficient (as the many applications of forcing axioms
have already shown) and sufficient to compute all $\Pi_{2}$-sentences which
axiomatize the model companion of $S_{\bar{A}_{2}}$, provided $S$ is any set
theory satisfying sufficiently strong large cardinal axioms (by Corollary 1
all other means to produce the consistenty of $\psi$ with the universal
fragment of $S$ are reducible to forcing).
Our take on the above considerations is that if one embraces the standpoint
that the universe of sets should be as large as possible, model companionship
(in particular Fact 1 – actually its more refined version provided by Lemma
1.21 and used in Thm. 2) gives a simple model theoretic property to
instantiate this slogan: all $\Pi_{2}$-sentences talking about $\omega_{1}$
(i.e. expressible in signature $\tau_{\omega_{1}}$) which are not outward
contradictory with the basic properties of $\omega_{1}$ (i.e. with the
universal theory of some model of $\mathsf{ZFC}+$_large cardinals_ in
signature $\tau_{\omega_{1}}$) should hold true in $H_{\omega_{2}}$. This is
what Thm. 2 says to be the case in models of strong forcing axioms such as
$\text{{\sf MM}}^{++}$.
Note that this is exactly parallel to the way one singles out algebraically
closed fields from rings with no zero-divisors: in this set-up one is
interested to solve polynomial equations while preserving the ring axioms and
not adding zero-divisors; the $\Pi_{2}$-sentences for the signature
$\left\\{+,\cdot,0,1\right\\}$ which are consistent with the ring axioms and
the non existence of zero divisors are exactly the axioms of algebraically
closed fields.
Now coming back to CH we already observed that its negation is a
$\Pi_{2}$-sentence for $\tau_{\omega_{1}}$ (hence also for
$\left\\{\in\right\\}_{\bar{A}_{2}}$), but we can actually get more. Caicedo
and Veličković [6] proved that there is a quantifier free
$\tau_{\omega_{1}}$-formula $\phi(x,y,z)$ such that $(\forall x,y\exists
z\phi(x,y,z))^{H_{\omega_{2}}}$ is forcible (by a proper forcing) over any
model of $\mathsf{ZFC}$; moreover if $V\models\mathsf{ZFC}+(\forall x,y\exists
z\phi(x,y,z))^{H_{\omega_{2}}}$, then $V\models 2^{\omega}=\omega_{2}$. In
particular if we accept as true large cardinal axioms and we require that the
correct axiomatization of set theory maximizes the set of $\Pi_{2}$-sentences
for $\tau_{\omega_{1}}$ which may hold for $H_{\aleph_{2}}$, we are bound to
accept that $2^{\omega}=\omega_{2}$ holds true.
### Structure of the paper
It is now a good place to streamline the remainder of this paper and specify
what the reader need to know in order to grasp each of its parts.
* •
Section 1 gives a detailed and self-contained account of model companionship;
the unique result which we are not able to trace elsewhere in the literature
is Lemma 1.21, which isolates a key property of (possibly incomplete) first
order theories granting model companionship results; we apply it in later
parts of this paper to various (possibly recursive or incomplete)
axiomatizations of set theory. Since we expect that many of our readers are
not familiar with model companionship, we decided it was worth including here
the key results (with proofs) on this notion. The reader familiar with these
notions can skim through this section or jump it and refer to its relevant
bits when needed elsewhere.
* •
Section 2 proves Theorem 1.
* •
Section 3 proves the results needed to establish item 5 of Thm. 2. We first
give a self-contained proof of the form of Woodin’s generic absoluteness
results for second order arithmetic we employ in this paper. This identifies
which subsets of $F_{\left\\{\in\right\\}}$ can play the role of $A_{1}$ for
item 5 of Thm. 2. Then we show that the universal theory of $V$ as formalized
in a signature extending $\tau_{\text{{\sf ST}}}$ with predicates for the non-
stationary ideal and for the universally Baire sets cannot be changed using
set sized forcing if there are class many Woodin cardinals. This identifies
which subsets of $F_{\left\\{\in\right\\}}$ can play the role of $A_{2}$ for
item 5 of Thm. 2.
* •
Section 4 deals with Theorem 2 for the signature $\tau_{\bar{A}_{1}}$. We
expand slightly the results of [22]: by taking advantage of Lemma 1.21, we are
able here to generalize also to non complete axiomatizations of set theory the
model companionship results given in [22] for complete set theories.
* •
Section 5 deals with Theorem 2 for the signature $\tau_{\bar{A}_{2}}$.
* •
We conclude the paper with a final section with some comments and open
questions.
Any reader familiar enough with set theory and model theory to follow this
introduction can easily grasp the content of Sections 1, 2. The same applies
for the results of Section 4 provided one accepts as a black-box Woodin’s
generic absoluteness results for second order arithmetic given in Section 3.
The proofs in Section 3 require familiarity with Woodin’s stationary tower
forcing and (in its second part, cfr. Section 3.4) also with Woodin’s
$\mathbb{P}_{\mathrm{max}}$-technology. Section 5 can be fully appreciated
only by readers familiar with forcing axioms, Woodin’s stationary tower
forcing, Woodin’s $\mathbb{P}_{\mathrm{max}}$-technology.
### Acknowledgements
This paper wouldn’t exist without the brilliant idea by Venturi to relate the
study of the generic multiverse of set theory to Robinson’s model
companionship, or without the major breakthrough of Asperò and Schindler
establishing that Woodin’s axioms $(*)$ follows from $\text{{\sf MM}}^{++}$.
I’m grateful to Boban Velickovic for many useful discussions on the scopes and
limits of the results presented here (the necessity of large cardinal
assumptions in the hypothesis of Thm. 3, and Remark 1 are due to him). I’m
also grateful to Philipp Schlicht who realized Thm. 1 could be easily
established by slightly generalizing the proofs of results occurring in
previous drafts of this paper. I’ve had fruitful discussions on the content of
this paper with many people, let me mention and thank David Asperó, Ilijas
Farah, Juliette Kennedy, Menachem Magidor, Ralf Schindler, Jouko Vaananen,
Andres Villaveces, Hugh Woodin. Clearly none of the persons mentioned here has
any responsibility for any error or orror existing in this paper…. It has been
important to have the opportunity to present these results in several set
theory (or logic) seminars among which those in Toronto, Muenster, Jerusalem,
Paris, Bogotà, Chicago, Helsinki, Torino. This research has been performed
mostly while on sabbatical in the Équipe de Logique Mathématique of the
University of Paris 7 in the academic year 2019-2020; as long as possible it
has been a productive and pleasant experience (until the Covid19 pandemic took
place).
## 1\. Existentially closed structures, model completeness, model
companionship
We present this topic expanding on [20, Sections 3.1-3.2]. We decided to
include detailed proofs since the presentation of [20] is (in some occasions)
rather sketchy, and the focus is not exactly ours.
The first objective is to isolate necessary and sufficient conditions granting
that some $\tau$-structure $\mathcal{M}$ embeds into some model of some
$\tau$-theory $T$.
We expand Notation 3 as follows:
###### Notation 1.1.
We feel free to confuse a $\tau$-structure $\mathcal{M}=(M,\tau^{M})$ with its
domain $M$ and an ordered tuple $\vec{a}\in\mathcal{M}^{<\omega}$ with its set
of elements. Moreover we often write $\mathcal{M}\models\phi(\vec{a})$ rather
than $\mathcal{M}\models\phi(\vec{x})[\vec{x}/\vec{a}]$ when $\mathcal{M}$ is
$\tau$-structure $\vec{a}\in\mathcal{M}^{<\omega}$, $\phi$ is a
$\tau$-formula. We let the atomic diagram of a $\tau$-model
$\mathcal{M}=(M,\tau^{M})$ be the family of quantifier free sentences
$\phi(\vec{a})$ in signature $\tau\cup M$ such that
.$\mathcal{M}\models\phi(\vec{a})$.
###### Definition 1.2.
Given $\tau$-theories $T,S$, a $\tau$-sentence $\psi$ separates $T$ from $S$
if $T\vdash\psi$ and $S\vdash\neg\psi$.
$T$ is $\Pi_{n}$-separated from $S$ if some $\Pi_{n}$-sentence for $\tau$
separates $T$ from $S$.
###### Lemma 1.3.
Assume $S,T$ are $\tau$-theories. TFAE:
1. (1)
$T$ is not $\Pi_{1}$-separated from $S$ (i.e. no universal sentence $\psi$ is
such that $T\vdash\psi$ and $S\vdash\neg\psi$).
2. (2)
There is _some_ $\tau$-model $\mathcal{M}$ of $S$ which can be embedded in
some $\tau$-model $\mathcal{N}$ of $T$.
See also [20, Lemma 3.1.1, Lemma 3.1.2, Thm. 3.1.3]
###### Proof.
We assume $T,S$ are closed under logical consequences.
(2) implies (1):
By contraposition we prove $\neg$(1)$\to\neg$(2).
Assume some universal sentence $\psi$ separates $T$ from $S$. Then for any
model of $T$, all its substructures model $\psi$, therefore they cannot be
models of $S$.
(1) implies (2):
By contraposition we prove $\neg$(2)$\to\neg$(1).
Assume that for any model $\mathcal{M}$ of $S$ and $\mathcal{N}$ of $T$
$\mathcal{M}\not\sqsubseteq\mathcal{N}$. We must show that $T$ is
$\Pi_{1}$-separated from $S$.
Given a $\tau$-structure $\mathcal{M}=(M,\tau^{M})$ which models $S$, let
$\Delta_{0}(\mathcal{M})$ be the atomic diagram666We let the atomic diagram of
a $\tau$-model $\mathcal{M}=(M,\tau^{M})$ be the family of quantifier free
formulae in signature $\tau\cup M$ which holds in the natural expansion of
$\mathcal{M}$ to $\tau\cup M$. of $\mathcal{M}$ in the signature $\tau\cup M$.
The theory $T\cup\Delta_{0}(\mathcal{M})$ is inconsistent, otherwise
$\mathcal{M}$ embeds into some model of $T$: let $\bar{\mathcal{Q}}$ be a
$\tau\cup\mathcal{M}$-model of $\Delta_{0}(\mathcal{M})\cup T$ and
$\mathcal{Q}$ be the $\tau$-structure obtained from $\bar{\mathcal{Q}}$
omitting the interpretation of the constants not in $\tau$. Clearly
$\mathcal{Q}$ models $T$. The interpretation of the constants in
$\tau\cup\mathcal{M}$ inside $\bar{\mathcal{Q}}$ defines a $\tau$-substructure
of $\mathcal{Q}$ isomorphic to $\mathcal{M}$.
By compactness (since $\Delta_{0}(\mathcal{M})$ is closed under finite
conjunctions) there is a quantifier free $\tau$-formula
$\psi_{\mathcal{M}}(\vec{x})$ and $\vec{a}\in\mathcal{M}^{<\omega}$ such that
$T+\psi_{\mathcal{M}}(\vec{a})$ is inconsistent. This gives that
$T\vdash\neg\psi_{\mathcal{M}}(\vec{a})$. Since $\vec{a}$ is a family of
constants never occurring in $T$, we get that
$T\vdash\forall\vec{x}\neg\psi_{\mathcal{M}}(\vec{x})$ and
$\mathcal{M}\models\exists\vec{x}\psi_{\mathcal{M}}(\vec{x})$.
The theory
$S\cup\left\\{\neg\exists\vec{x}\psi_{\mathcal{M}}(\vec{x}):\mathcal{M}\models
S\right\\}$
is inconsistent, since $\neg\exists\vec{x}\psi_{\mathcal{M}}(\vec{x})$ fails
in any model $\mathcal{M}$ of $S$.
By compactness there is a finite set of formulae
$\psi_{\mathcal{M}_{1}}\dots\psi_{\mathcal{M}_{k}}$ such that
$S+\bigwedge\left\\{\neg\exists\vec{x}_{i}\psi_{\mathcal{M}_{i}}(\vec{x}_{i}):i=1,\dots,k\right\\}$
is inconsistent. This gives that
$S\vdash\bigvee_{i=1}^{k}\exists\vec{x}_{i}\psi_{\mathcal{M}_{i}}(\vec{x}_{i}).$
The $\tau$-sentence
$\psi:=\bigvee_{i=1}^{k}\exists\vec{x}_{i}\psi_{\mathcal{M}_{i}}(\vec{x}_{i})$
holds in all models of $S$ and its negation
$\bigwedge\left\\{\neg\exists\vec{x}_{i}\psi_{\mathcal{M}_{i}}(\vec{x}_{i}):i=1,\dots,k\right\\}$
is a conjunction of universal sentences (hence —modulo logical equivalence—
universal) derivable from $T$. Hence $\neg\psi$ separates $T$ from $S$.
∎
The following Lemma shows that models of $T_{\forall}$ can always be extended
to superstructures which model $T$.
###### Lemma 1.4.
Let $T$ be a $\tau$-theory and $\mathcal{M}$ be a $\tau$-structure. TFAE:
1. (1)
$\mathcal{M}$ is a $\tau$-model of $T_{\forall}$.
2. (2)
There exists $\mathcal{N}\sqsupseteq\mathcal{M}$ which models $T$.
###### Proof.
(2) implies (1) is trivial.
Conversely:
###### Claim 1.
$T$ is not $\Pi_{1}$-separated from $\Delta_{0}(\mathcal{M})$ (in the
signature $\tau\cup\mathcal{M}$).
###### Proof.
If not there are $\vec{a}\in\mathcal{M}^{<\omega}$, and a quantifier free
$\tau$-formula $\phi(\vec{x},\vec{z})$ such that
$T\vdash\forall\vec{z}\phi(\vec{a},\vec{z}),$
while
$\Delta_{0}(\mathcal{M})\vdash\neg\forall\vec{z}\phi(\vec{a},\vec{z}).$
The latter yields that
$\Delta_{0}(\mathcal{M})\vdash\exists\vec{x}\exists\vec{z}\neg\phi(\vec{x},\vec{z}),$
and therefore also that
$\mathcal{M}\models\exists\vec{x}\exists\vec{z}\neg\phi(\vec{x},\vec{z}).$
On the other hand, since the constants $\vec{a}$ do not appear in any of the
sentences in $T$, we also get that
$T\vdash\forall\vec{x}\forall\vec{z}\phi(\vec{x},\vec{z}).$
This is a contradiction since $\mathcal{M}$ models $T_{\forall}$. ∎
By the Claim and Lemma 1.3 some $\tau\cup\mathcal{M}$-model
$\bar{\mathcal{P}}$ of $\Delta_{0}(\mathcal{M})$ embeds into some
$\tau\cup\mathcal{M}$-model $\bar{\mathcal{Q}}$ of $T$. Let $\mathcal{Q}$ be
the $\tau$-structure obtained from $\bar{\mathcal{Q}}$ omitting the
interpretation of the constants not in $\tau$. Then $\mathcal{Q}$ models $T$
and contains a substructure isomorphic to $\mathcal{M}$. ∎
###### Corollary 1.5 (Resurrection Lemma).
Assume $\mathcal{M}\prec_{1}\mathcal{N}$ are $\tau$-structures. Then there is
$\mathcal{Q}\sqsupseteq\mathcal{N}$ which is an elementary extension of
$\mathcal{M}$.
###### Proof.
Let $T$ be the elementary diagram $\Delta_{\omega}(\mathcal{M})$ of
$\mathcal{M}$ in the signature $\tau\cup\mathcal{M}$. It is easy to check that
any model of $T$ when restricted to the signature $\tau$ is an elementay
extension of $\mathcal{M}$. Since $\mathcal{M}\prec_{1}\mathcal{N}$, the
natural extension of $\mathcal{N}$ to a $\tau\cup\mathcal{M}$-structure
realizes the $\Pi_{1}$-fragment of $T$ in the signature $\tau\cup\mathcal{M}$.
Now apply the previous Lemma. ∎
The Resurrection Lemma motivates the resurrection axioms introduced by Hamkins
and Johnstone in [9], and their iterated versions introduced by the author and
Audrito in [5].
### 1.1. Existentially closed structures
The objective is now to isolate the “generic” models of some universal theory
$T$ (i.e. all axioms of $T$ are universal sentences). These are described by
the $T$-existentially closed models.
###### Definition 1.6.
Given a first order signature $\tau$, let $T$ be any consistent $\tau$-theory.
A $\tau$-structure $\mathcal{M}$ is $T$-existentially closed ($T$-ec) if
1. (1)
$\mathcal{M}$ can be embedded in a model of $T$.
2. (2)
$\mathcal{M}\prec_{\Sigma_{1}}\mathcal{N}$ for all
$\mathcal{N}\sqsupseteq\mathcal{M}$ which are models of $T$.
In general $T$-ec models need not be models777For example let $T$ be the
theory of commutative rings with no zero divisors which are not fields in the
signature $(+,\cdot,0,1)$. Then the $T$-ec structures are exactly all the
algebraically closed fields, and no $T$-ec model is a model of $T$. By Thm.
2.6 $(H_{\omega_{1}},\sigma_{\omega}^{V})$ is $S$-ec for $S$ the
$\sigma_{\omega}$-theory of $V$, but it is not a model of $S$: the
$\Pi_{2}$-sentence asserting that every set has countable transitive closure
is true in $(H_{\omega_{1}},\sigma_{\omega}^{V})$ but denied by $S$. of $T$,
but only of their universal fragment. A standard diagonalization argument
shows that for any theory $T$ there are $T$-ec models, see Lemma 1.9 below or
[20, Lemma 3.2.11].
A trivial observation which will come handy in the sequel is the following:
###### Fact 1.7.
Assume $\mathcal{M}$ is a $T$-ec model and $S\supseteq T$ is such that some
$\mathcal{N}\sqsupseteq\mathcal{M}$ models $S$. Then $\mathcal{M}$ is $S$-ec.
###### Proposition 1.8.
Assume a $\tau$-structure $\mathcal{M}$ is $T$-ec. Then:
1. (1)
$\mathcal{M}\models T_{\forall}$.
2. (2)
$\mathcal{M}$ is also $T_{\forall}$-ec.
3. (3)
If $\mathcal{N}\prec_{\Sigma_{1}}\mathcal{M}$, then $\mathcal{N}$ is also
$T$-ec.
4. (4)
Let $\forall\vec{x}\exists\vec{y}\psi(\vec{x},\vec{y},\vec{a})$ be a
$\Pi_{2}$-sentence with $\psi(\vec{x},\vec{y},\vec{z})$ quantifier free
$\tau$-formula and parameters $\vec{a}$ in $\mathcal{M}^{<\omega}$. Assume it
holds in some $\mathcal{N}\sqsupseteq\mathcal{M}$ which models $T_{\forall}$,
then it holds in $\mathcal{M}$.
5. (5)
Let $S$ be the $\tau$-theory of $\mathcal{M}$. For any $\Pi_{2}$-sentence
$\psi$ in the signature $\tau$ TFAE:
* •
$\psi$ holds in some model of $S_{\forall}$.
* •
$\psi$ holds in $\mathcal{M}$.
###### Proof.
__
(1):
There is at least one super-structure of $\mathcal{M}$ which models $T$, and
any $\psi\in T_{\forall}$ holds in this superstructure, hence in
$\mathcal{M}$.
(2):
Assume $\mathcal{M}\sqsubseteq\mathcal{P}$ for some model $\mathcal{P}$ of
$T_{\forall}$. We must argue that $\mathcal{M}\prec_{1}\mathcal{P}$.
By Lemma 1.4, there is $\mathcal{Q}\sqsupseteq\mathcal{P}$ which models $T$.
Since $\mathcal{M}$ and $\mathcal{Q}$ are both models of $T$ and $\mathcal{M}$
is $T$-ec, we get the following diagram:
$\mathcal{M}$$\mathcal{Q}$$\mathcal{P}$$\scriptstyle{\Sigma_{1}}$$\scriptstyle{\sqsubseteq}$$\scriptstyle{\sqsubseteq}$
Then any $\Sigma_{1}$-formula $\psi(\vec{a})$ with
$\vec{a}\in\mathcal{M}^{<\omega}$ realized in $\mathcal{P}$ holds in
$\mathcal{Q}$, and is therefore reflected to $\mathcal{M}$. We are done by
Tarski-Vaught’s criterion.
(3):
Assume $\mathcal{N}\sqsubseteq\mathcal{P}$ for some model of $T_{\forall}$
$\mathcal{P}$. Let $\Delta_{0}(\mathcal{P})$ be the atomic diagram of
$\mathcal{P}$ in the signature $\tau\cup\mathcal{P}\cup\mathcal{M}$ and
$\Delta_{0}(\mathcal{M})$ be the atomic diagram of $\mathcal{M}$ in the same
signature888We are considering $\mathcal{P}\cup\mathcal{M}$ as the union of
the domains of the structure $\mathcal{P},\mathcal{M}$ amalgamated over
$\mathcal{N}$; in particular we add a new constant for each element of
$\mathcal{P}\setminus\mathcal{N}$, a new constant for each element of
$\mathcal{M}\setminus\mathcal{N}$, a new constant for each element of
$\mathcal{N}$..
###### Claim 2.
$T_{\forall}\cup\Delta_{0}(\mathcal{P})\cup\Delta_{0}(\mathcal{M})$ is a
consistent $\tau\cup\mathcal{M}\cup\mathcal{P}$-theory.
###### Proof.
Assume not. Find $\vec{a}\in(\mathcal{P}\setminus\mathcal{N})^{<\omega}$,
$\vec{b}\in(\mathcal{M}\setminus\mathcal{N})^{<\omega}$,
$\vec{c}\in\mathcal{N}^{<\omega}$ and $\tau$-formulae
$\psi_{0}(\vec{x},\vec{z})$, $\psi_{1}(\vec{y},\vec{z})$ such that:
* •:
$\psi_{0}(\vec{a},\vec{c})\in\Delta_{0}(\mathcal{P})$,
* •:
$\psi_{1}(\vec{b},\vec{c})\in\Delta_{0}(\mathcal{M})$,
* •:
$T\cup\left\\{\psi_{0}(\vec{a},\vec{c}),\psi_{1}(\vec{b},\vec{c})\right\\}$ is
inconsistent.
Then
$T\vdash\neg\psi_{0}(\vec{a},\vec{c})\vee\neg\psi_{1}(\vec{b},\vec{c}).$
Since the constants appearing in $\vec{a},\vec{b},\vec{c}$ are never appearing
in sentences of $T$, we get that
$T\vdash\forall\vec{z}\,(\forall\vec{x}\neg\psi_{0}(\vec{x},\vec{z}))\vee(\forall\vec{y}\neg\psi_{1}(\vec{y},\vec{z})).$
Since $\mathcal{P}$ models $T_{\forall}$, and
$\mathcal{P}\models\psi_{0}(\vec{x},\vec{z})[\vec{x}/\vec{a},\vec{z}/\vec{c}],$
we get that
$\mathcal{P}\models\forall\vec{y}\neg\psi_{1}(\vec{y},\vec{c}).$
Therefore
$\mathcal{N}\models\forall\vec{y}\neg\psi_{1}(\vec{y},\vec{c})$
being a substructure of $\mathcal{P}$, and so does $\mathcal{M}$ since
$\mathcal{N}\prec_{1}\mathcal{M}$. This contradicts
$\psi_{1}(\vec{b},\vec{c})\in\Delta_{0}(\mathcal{M})$. ∎
If $\bar{\mathcal{Q}}$ is a model realizing
$T_{\forall}\cup\Delta_{0}(\mathcal{P})\cup\Delta_{0}(\mathcal{M})$, and
$\mathcal{Q}$ is the $\tau$-structure obtained forgetting the constant symbols
not in $\tau$, we get that:
* •:
$\mathcal{P}$ and $\mathcal{M}$ are both substructures of $\mathcal{Q}$
containing $\mathcal{N}$ as a common substructure;
* •:
$\mathcal{N}\prec_{1}\mathcal{M}\prec_{1}\mathcal{Q}$, since $\mathcal{Q}$
realizes $T_{\forall}$ and $\mathcal{M}$ is $T_{\forall}$-ec.
We can now conclude that if a $\Sigma_{1}$-formula $\psi(\vec{c})$ for
$\tau\cup\mathcal{N}$ with parameters in $\mathcal{N}$ holds in $\mathcal{P}$,
it holds in $\mathcal{Q}$ as well (since $\mathcal{Q}\sqsupseteq\mathcal{P}$),
and therefore also in $\mathcal{N}$ (since $\mathcal{N}\prec_{1}\mathcal{Q}$).
(4):
Observe that for all $\vec{b}\in\mathcal{M}^{<\omega}$,
$\exists\vec{y}\,\psi(\vec{b},\vec{y},\vec{a})$ holds in $\mathcal{N}$, and
therefore in $\mathcal{M}$, since $\mathcal{M}$ is $T$-ec; hence
$\mathcal{M}\models\forall\vec{x}\exists\vec{y}\psi(\vec{x},\vec{y},\vec{a})$.
(5):
First of all note that $\mathcal{M}$ is $S$-ec since $S\supseteq T$ (by Fact
1.7). By Lemma 1.4 (applied to $S_{\forall}+\psi$ and $\mathcal{M}$) any
$\Pi_{2}$-sentence $\psi$ for $\tau$ which holds in some model of
$S_{\forall}$ holds in some model of $S_{\forall}$ which is a superstructure
of $\mathcal{M}$. Now apply 4.
∎
In particular a structure is $T$-ec if and only if it is $T_{\forall}$-ec, and
a $T$-ec structure realizes all $\Pi_{2}$-sentences which are consistent with
its $\Pi_{1}$-theory.
We now show that any structure $\mathcal{M}$ can always be extended to a
$T$-ec structure for any $T$ which is not separated from the $\Pi_{1}$-theory
of $\mathcal{M}$.
###### Lemma 1.9.
[20, Lemma 3.2.11] Given a first order $\tau$-theory $T$, any model of
$T_{\forall}$ can be extended to a $\tau$-superstructure which is $T$-ec.
###### Proof.
Given a model $\mathcal{M}$ of $T$, we construct an ascending chain of
$T_{\forall}$-models as follows. Enumerate all quantifier free $\tau$-formulae
as $\left\\{\phi_{\alpha}(y,\vec{x}_{\alpha}):\alpha<|\tau|\right\\}$. Let
$\mathcal{M}_{0}=\mathcal{M}$ have size $\kappa\geq|\tau|+\aleph_{0}$. Fix
also some enumeration
$\displaystyle\pi:$ $\displaystyle\kappa\to|\tau|\times\kappa^{2}$
$\displaystyle\alpha\mapsto(\pi_{0}(\alpha),\pi_{1}(\alpha),\pi_{2}(\alpha))$
such that $\pi_{2}(\alpha)\leq\alpha$ for all $\alpha<\kappa$ and for each
$\xi<|\tau|$, and $\eta,\beta<\kappa$ there are unboundedly many
$\alpha<\kappa$ such that $\pi(\alpha)=(\xi,\eta,\beta)$.
Let now $\mathcal{M}_{\eta}$ with enumeration
$\left\\{\vec{m}^{\xi}_{\eta}:\xi<\kappa\right\\}$ of
$\mathcal{M}_{\eta}^{<\omega}$ be given for all $\eta\leq\beta$. If
$\mathcal{M}_{\beta}$ is $T$-ec, stop the construction. Else check whether
$T_{\forall}\cup\Delta_{0}(\mathcal{M}_{\beta})\cup\left\\{\exists
y\phi_{\pi_{0}(\alpha)}(y,\vec{m}^{\pi_{1}(\alpha)}_{\pi_{2}(\alpha)})\right\\}$
is a consistent $\tau\cup\mathcal{M}_{\beta}$-theory; if so let
$\mathcal{M}_{\beta+1}$ have size $\kappa$ and realize this theory. At limit
stages $\gamma$, let $\mathcal{M}_{\gamma}$ be the direct limit of the chain
of $\tau$-structures $\left\\{\mathcal{M}_{\beta}:\beta<\gamma\right\\}$. Then
all $\mathcal{M}_{\xi}$ are models of $T_{\forall}$, and at some stage
$\beta\leq\kappa$ $\mathcal{M}_{\beta}$ is $T_{\forall}$-ec (hence also
$T$-ec), since all existential $\tau$-formulae with parameters in some
$\mathcal{M}_{\eta}$ will be considered along the construction, and realized
along the way if this is possible, and all $\mathcal{M}_{\eta}$ are always
models of $T_{\forall}$ (at limit stages the ascending chain of
$T_{\forall}$-models remains a $T_{\forall}$-model). ∎
Compare the above construction with the standard consistency proofs of bounded
forcing axioms as given for example in [3, Section 2]. In the latter case to
preserve $T_{\forall}$ at limit stages we use iteration theorems999Assume $G$
is $V$-generic for a forcing which is a limit of an iteration of length
$\omega$ of forcings $\left\\{P_{n}:n<\omega\right\\}$. In general
$H_{\omega_{2}}^{V[G]}$ is not given by the union of $H_{\omega_{2}}^{V[G\cap
P_{n}]}$, hence a subtler argument is needed to maintain that
$H_{\omega_{2}}^{V[G]}$ preserves $T_{\forall}$..
### 1.2. The Kaiser hull of a first order theory
The Kaiser Hull of a theory $T$ describes the smallest elementary class
containing all the “generic” structures for $T$. For most theories $T$ the
models of the respective Kaiser hulls realize exactly all $\Pi_{2}$-sentences
which are consistent with the universal fragment of any extension of $T$.
###### Definition 1.10.
[20, Lemma 3.2.12, Lemma 3.2.13] Given a theory $T$ in a signature $\tau$, its
Kaiser hull $\mathrm{KH}(T)$ is given by the $\Pi_{2}$-sentences of $\tau$
which holds in all $T$-ec structures.
###### Definition 1.11.
A $\tau$-theory $T$ is $\Pi_{n}$-complete, if it is consistent and for any
$\Pi_{n}$-sentence either $\phi\in T$ or $\neg\phi\in T$.
By Proposition 1.8.5 we get:
###### Fact 1.12.
Given a $\Pi_{1}$-complete first order $\tau$-theory $T$, its Kaiser Hull is a
$\Pi_{2}$-complete $\tau$-theory defined by the request that for any
$\Pi_{2}$-sentence $\psi$
$\psi\in\mathrm{KH}(T)\quad\text{ if and only if
}\quad\left\\{\psi\right\\}\cup T_{\forall}\text{ is consistent}.$
In particular any model of the Kaiser hull of a $\Pi_{1}$-complete $T$
realizes simultaneously all $\Pi_{2}$-sentences which are individually
consistent with $T_{\forall}$.
For theories $T$ of interests to us their Kaiser hull can be described in the
same terms, but the proof is much more delicate. We start with the following
weaker property which holds for arbitrary theories:
###### Fact 1.13.
Given a $\tau$-theory $T$, its Kaiser hull $\mathrm{KH}(T)$ contains the set
of $\Pi_{2}$-sentences $\psi$ for $\tau$ such that for all complete
$S\supseteq T$, $S_{\forall}\cup\left\\{\psi\right\\}$ is consistent.
###### Proof.
Assume $\psi$ is a $\Pi_{2}$-sentence such that for all complete $S\supseteq
T$, $S_{\forall}\cup\left\\{\psi\right\\}$ is consistent. We must show that
$\psi$ holds in all $T$-ec models.
Fix $\mathcal{M}$ an existentially closed model for $T$ (it exists by Lemma
1.9); we must show that $\mathcal{M}\models\psi$. Let
$\mathcal{N}\sqsupseteq\mathcal{M}$ be a model of $T$ and $S$ be the
$\tau$-theory of $\mathcal{N}$. Then $S$ is a complete theory and
$\mathcal{M}\models S_{\forall}$ since $\mathcal{M}\prec_{1}\mathcal{N}$
(being $T$-ec). Since $S\supseteq T$, $\mathcal{M}$ is also $S$-ec (by Fact
1.7). Since $S_{\forall}\cup\left\\{\psi\right\\}$ is consistent, and
$S_{\forall}$ is $\Pi_{1}$-complete, we obtain that $\mathcal{M}$ models
$\psi$, being an $S_{\forall}$-ec model, and using Fact 1.12. ∎
We will show in Lemma 1.21 that the set of $\Pi_{2}$-sentences described in
the Fact provides an equivalent characterization of the Kaiser hull for many
theories admitting a model companion, among which the axiomatizations of set
theory considered in this paper.
### 1.3. Model completeness
It is possible (depending on the choice of the theory $T$) that there are
models of the Kaiser hull of $T$ which are not $T$-ec. Robinson has come up
with two model theoretic properties (model completeness and model
companionship) which describe the case in which the models of the Kaiser hull
of $T$ are exactly the class of $T$-ec models (even in case $T$ is not a
complete theory).
###### Definition 1.14.
A $\tau$-theory $T$ is _model complete_ if for all $\tau$-models $\mathcal{M}$
and $\mathcal{N}$ of $T$ we have that $\mathcal{M}\sqsubseteq\mathcal{N}$
implies $\mathcal{M}\prec\mathcal{N}$.
Remark that theories admitting quantifier elimination are automatically model
complete. On the other hand model complete theories need not be
complete101010For example the theory of algebraically closed fields is model
complete, but algebraically closed fields of different characteristics are
elementarily inequivalent.. However for theories $T$ which are
$\Pi_{1}$-complete, model completeness entails completeness: any two models of
a $\Pi_{1}$-complete, model complete $T$ share the same $\Pi_{1}$-theory,
therefore if $T_{1}\supseteq T$ and $T_{2}\supseteq T$ with $\mathcal{M}_{i}$
a model of $T_{i}$, we can suppose (by Lemma 1.3) that
$\mathcal{M}_{1}\sqsubseteq\mathcal{M}_{2}$. Since they are both models of
$T$, model completeness entails that $\mathcal{M}_{1}\prec\mathcal{M}_{2}$.
###### Lemma 1.15.
[20, Lemma 3.2.7] (Robinson’s test) Let $T$ be a $\tau$-theory. The following
are equivalent:
1. (a)
$T$ is model complete.
2. (b)
Any model of $T$ is $T$-ec.
3. (c)
Each _existential_ $\tau$-formula $\phi(\vec{x})$ in free variables $\vec{x}$
is $T$-equivalent to a universal $\tau$-formula $\psi(\vec{x})$ in the same
free variables.
4. (d)
Each $\tau$-formula $\phi(\vec{x})$ in free variables $\vec{x}$ is
$T$-equivalent to a universal $\tau$-formula $\psi(\vec{x})$ in the same free
variables.
Remark that d (or c) shows that being a model complete $\tau$-theory $T$ is
expressible by a $\Delta_{0}(\tau,T)$-property in any model of $\mathsf{ZFC}$,
hence it is absolute with respect to forcing.
###### Proof.
__
a implies b:
Immediate.
b implies c:
Fix an existential formula $\phi(\vec{x})$ in free variables
$x_{1},\dots,x_{n}$. If $\phi(\vec{x})$ is not consistent with $T$ it is
$T$-equivalent to the trivial formula $\forall y(y\neq y)$ in free variables
$\vec{x}$. Hence we may assume that $T\cup\phi(\vec{x})$ is a consistent
theory. Let $\vec{c}=(c_{1},\dots,c_{n})$ be a finite set of new constant
symbols. Then $T\cup\phi(\vec{c})$ is a consistent
$\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}$-theory.
Let $\Gamma$ be the set of universal $\tau$-formulae $\theta(\vec{x})$ such
that
$T\vdash\forall\vec{x}\,(\phi(\vec{x})\rightarrow\theta(\vec{x})).$
Note that $\Gamma$ is closed under finite conjunctions and disjunctions. Let
$\Gamma(\vec{c})=\left\\{\theta(\vec{c}):\,\theta(\vec{x})\in\Gamma\right\\}$.
Note that $T\cup\Gamma(\vec{c})$ is a consistent
$\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}$-theory, since it holds in any
$\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}$-model of $T\cup\phi(\vec{c})$.
It suffices to prove
(1) $T\cup\Gamma(\vec{c})\models\phi(\vec{c});$
if this is the case, by compactness, a finite subset $\Gamma_{0}(\vec{c})$ of
$\Gamma(\vec{c})$ is such that
$T\cup\Gamma_{0}(\vec{c})\models\phi(\vec{c});$
letting
$\bar{\theta}(\vec{x}):=\bigwedge\left\\{\psi(\vec{x}):\psi(\vec{c})\in\Gamma_{0}(\vec{c})\right\\}$,
the latter gives that
$T\models\forall\vec{x}\,(\bar{\theta}(\vec{x})\rightarrow\phi(\vec{x}))$
(since the constants $\vec{c}$ do not appear in $T$).
$\bar{\theta}(\vec{x})\in\Gamma$ is a universal formula witnessing c for
$\phi(\vec{x})$.
So we prove (1):
###### Proof.
Let $\mathcal{M}$ be a $\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}$-model of
$T\cup\Gamma(\vec{c})$. We must show that $\mathcal{M}$ models
$\phi(\vec{c})$.
The key step is to prove the following:
###### Claim 3.
$T\cup\Delta_{0}(\mathcal{M})\cup\left\\{\phi(\vec{c})\right\\}$ is consistent
(where $\Delta_{0}(\mathcal{M})$ is the
$\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}$-atomic diagram of $\mathcal{M}$
in signature $\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}\cup\mathcal{M}$).
Assume the Claim holds and let $\mathcal{N}$ realize the above theory. Then
$\mathcal{M}\sqsubseteq\mathcal{N}\restriction(\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}).$
Hence
$\mathcal{M}\restriction\tau\sqsubseteq\mathcal{N}\restriction\tau.$
By b
$\mathcal{M}\restriction\tau\prec_{1}\mathcal{N}\restriction\tau.$
Now let $b_{1},\dots,b_{n}\in\mathcal{M}$ be the interpretations of
$c_{1},\dots,c_{n}$ in the
$\tau\cup\left\\{c_{1},\dots,c_{n}\right\\}$-structure $\mathcal{M}$. Then
$\mathcal{N}\restriction\tau\models\phi(x_{1},\dots,x_{n})[b_{1},\dots,b_{n}].$
Since $\phi(\vec{x})$ is $\Sigma_{1}$ for $\tau$ and
$b_{1},\dots,b_{n}\in\mathcal{M}$, we get that
$\mathcal{M}\restriction\tau\models\phi(x_{1},\dots,x_{n})[b_{1},\dots,b_{n}],$
hence
$\mathcal{M}\models\phi(c_{1},\dots,c_{n}),$
and we are done.
So we are left with the proof of the Claim.
###### Proof.
Let $\psi(\vec{x},\vec{y})$ be a quantifier free $\tau$-formula such that
$\psi(\vec{c},\vec{a})\in\Delta_{0}(\mathcal{M})$ for some
$\vec{a}\in\mathcal{M}$.
Clearly $\mathcal{M}$ models $\exists\vec{y}\psi(\vec{c},\vec{y})$.
Then the universal formula
$\neg\exists\vec{y}\psi(\vec{c},\vec{y})\not\in\Gamma(\vec{c})$, since
$\mathcal{M}$ models its negation and $\Gamma(\vec{c})$ at the same time.
This gives that
$T\not\vdash\forall\vec{x}\,(\phi(\vec{x})\rightarrow\neg\exists\vec{y}\psi(\vec{x},\vec{y})),$
i.e.
$T\cup\left\\{\exists\vec{x}\,[\phi(\vec{x})\wedge\exists\vec{y}\psi(\vec{x},\vec{y})]\right\\}$
is consistent.
We conclude that
$T\cup\left\\{\phi(\vec{c})\wedge\psi(\vec{c},\vec{a})\right\\}$
is consistent for any tuple $a_{1},\dots,a_{k}\in\mathcal{M}$ and formula
$\psi$ such that $\mathcal{M}$ models $\psi(\vec{c},\vec{a})$ (since
$\vec{c},\vec{a}$ are constants never appearing in the formulae of $T$).
This shows that
$T\cup\Delta_{0}(\mathcal{M})\cup\left\\{\phi(\vec{c})\right\\}$ is
consistent. ∎
(1) is proved. ∎
c implies d:
We prove by induction on $n$ that $\Pi_{n}$-formulae and $\Sigma_{n}$-formulae
are $T$-equivalent to a $\Pi_{1}$-formula.
c gives the base case $n=1$ of the induction for $\Sigma_{1}$-formulae and
(trivially) for $\Pi_{1}$-formulae.
Assuming we have proved the implication for all $\Sigma_{n}$ formulae for some
fixed $n>0$, we obtain it for $\Pi_{n+1}$-formulae
$\forall\vec{x}\psi(\vec{x},\vec{y})$ (with $\psi(\vec{x},\vec{y})$
$\Sigma_{n}$) applying the inductive assumptions to $\psi(\vec{x},\vec{y})$;
next we observe that a $\Sigma_{n+1}$-formula is equivalent to the negation of
a $\Pi_{n+1}$-formula, which is in turn equivalent to the negation of a
universal formula (by what we already argued), which is equivalent to an
existential formula, and thus equivalent to a universal formula (by c).
d implies a:
By d every formula is $T$-equivalent both to a universal formula and to an
existential formula (since its negation is $T$-equivalent to a universal
formula).
This gives that $\mathcal{M}\prec\mathcal{N}$ whenever
$\mathcal{M}\sqsubseteq\mathcal{N}$ are models of $T$, since truth of
universal formulae is inherited by substructures, while truth of existential
formulae pass to superstructures.
∎
We will also need the following:
###### Fact 1.16.
Let $\tau$ be a signature and $T$ a model complete $\tau$-theory. Let
$\sigma\supseteq\tau$ be a signature and $T^{*}\supseteq T$ a $\sigma$-theory
such that every $\sigma$-formula is $T^{*}$-equivalent to a $\tau$-formula.
Then $T^{*}$ is model complete.
###### Proof.
By the model completeness of $T$ and the assumptions on $T^{*}$ we get that
every $\sigma$-formula is equivalent to a $\Pi_{1}$-formula for
$\tau\subseteq\sigma$. We conclude by Robinson’s test. ∎
Later on we will show that in most cases model complete theories maximize the
family of $\Pi_{2}$-sentences compatible with any $\Pi_{1}$-completion of
their universal fragment. This will be part of a broad family of properties
for first order theories which require a new concept in order to be properly
formulated, that of model companionship.
### 1.4. Model companionship
Model completeness comes in pairs with another fundamental concept which
generalizes to arbitrary first order theories the relation existing between
algebraically closed fields and commutative rings without zero-divisors. As a
matter of fact, the case described below occurs when $T^{*}$ is the theory of
algebraically closed fields and $T$ is the theory of commutative rings with no
zero divisors.
###### Definition 1.17.
Given two theories $T$ and $T^{*}$ in the same language $\tau$, $T^{*}$ is the
_model companion_ of $T$ if the following conditions holds:
1. (1)
Each model of $T$ can be extended to a model of $T^{*}$.
2. (2)
Each model of $T^{*}$ can be extended to a model of $T$.
3. (3)
$T^{*}$ is model complete.
Different theories can have the same model companion, for example the theory
of fields and the theory of commutative rings with no zero-divisors which are
not fields both have the theory of algebraically closed fields as their model
companion.
###### Theorem 1.18.
[20, Thm 3.2.14] Let $T$ be a first order theory. If its model companion
$T^{*}$ exists, then
1. (1)
$T_{\forall}=T^{*}_{\forall}$.
2. (2)
$T^{*}$ is the theory of the existentially closed models of $T_{\forall}$.
###### Proof.
__
1. (1)
By Lemma 1.4.
2. (2)
By Robinson’s test 1.15 $T^{*}$ is the theory realized exactly by the
$T^{*}$-ec models; by Proposition 1.8(2) $\mathcal{M}$ is $T^{*}$-ec if and
only if it is $T^{*}_{\forall}$-ec; by (1) $T^{*}_{\forall}=T_{\forall}$.
∎
An immediate by-product of the above Theorem is that the model companion of a
theory does not necessarily exist, but, if it does, it is unique and is its
Kaiser hull.
###### Theorem 1.19.
[20, Thm. 3.2.9] Assume $T$ has a model companion $T^{*}$. Then $T^{*}$ is
axiomatized by its $\Pi_{2}$-consequences and is the Kaiser hull of
$T_{\forall}$.
Moreover $T^{*}$ is the unique model companion of $T$ and is characterized by
the property of being the unique model complete theory $S$ such that
$S_{\forall}=T_{\forall}$.
###### Proof.
For quantifier free formulae $\psi(\vec{x},\vec{y})$ and
$\phi(\vec{x},\vec{z})$ the assertion
$\forall\vec{x}\,[\exists\vec{y}\psi(\vec{x},\vec{y})\leftrightarrow\forall\vec{z}\phi(\vec{x},\vec{z})]$
is a $\Pi_{2}$-sentence.
Let $T^{**}$ be the theory given by the $\Pi_{2}$-consequences of $T^{*}$.
Since $T^{*}$ is model complete, by Robinson’s test 1.15c, for any
$\Sigma_{1}$-formula $\exists\vec{y}\psi(\vec{x},\vec{y})$ there is a
universal formula $\forall\vec{z}\phi(\vec{x},\vec{z})$ such that
$\forall\vec{x}\,[\exists\vec{y}\psi(\vec{x},\vec{y})\leftrightarrow\forall\vec{z}\phi(\vec{x},\vec{z})]$
is in $T^{**}$.
Again by Robinson’s test 1.15c $T^{**}$ is model complete.
Now assume $S$ is a model complete theory such that $S_{\forall}=T_{\forall}$.
Clearly $T^{*}_{\forall}=T_{\forall}=S_{\forall}$. By Robinson’s test 1.15b
and Proposition 1.8(2), $S_{\forall}$ holds exactly in the $T_{\forall}$-ec
models, but these are exactly the models of $T^{*}$. Hence $T^{*}=S$.
This shows that any model complete theory is axiomatized by its
$\Pi_{2}$-consequences, that the model companion $T^{*}$ of $T$ is unique,
that $T^{*}$ is also the Kaiser hull of $T$ (being axiomatized by the
$\Pi_{2}$-sentences which hold in all $T$-ec-models), and is characterized by
the property of being the unique model complete theory $S$ such that
$T_{\forall}=S_{\forall}$. ∎
Thm. 1.19 provides an equivalent characterization of model companion theories
(which is expressible by a $\Delta_{0}$-property in parameters $T$ and
$T^{*}$, hence absolute for transitive models of $\mathsf{ZFC}$).
Note also that Robinson’s test 1.15d gives an explicit axiomatization of a
model complete theory $T$:
###### Fact 1.20.
Assume $T$ is a model complete $\tau$-theory. Let
$\psi\mapsto\theta_{\psi}^{T}$ be a function assigning to each
$\Sigma_{1}$-formula $\psi(\vec{x})$ for $\tau$ a $\Pi_{1}$-formula
$\theta_{\psi}^{T}(\vec{x})$ which is $T$-equivalent to $\psi(\vec{x})$.
Then $T$ is axiomatized by $T_{\forall}$ and the $\Pi_{2}$-sentences
$\text{{\sf
AX}}_{\psi}^{T}\equiv\forall\vec{x}(\psi(\vec{x})\leftrightarrow\theta_{\psi}^{T}(\vec{x}))$
as $\psi(\vec{x})$ ranges over the $\Sigma_{1}$-formulae for $\tau$.
###### Proof.
First of all
$T^{*}=\left\\{\text{{\sf AX}}_{\psi}^{T}:\psi\text{ a
$\tau$-formula}\right\\}$
is a model complete theory, since $T^{*}$ satisfies Robinson’s test 1.15d. Let
$S=T^{*}+T_{\forall}$. Note that $S$ is also model complete (by Robinson’s
test 1.15d). Moreover $S\subseteq T$ (since $\text{{\sf AX}}_{\psi}^{T}\in T$
for all $\Sigma_{1}$-formulae $\psi$), and $S_{\forall}\supseteq T_{\forall}$
(since $T_{\forall}$ is certainly among the universal consequences of $S$). We
conclude that $S_{\forall}=T_{\forall}$. Therefore $S$ is the model companion
of $T$. $S=T$ by uniqueness of the model companion. ∎
We use the following criteria for model companionship in the proofs of
Theorems 2.6, 4.4, 5.
###### Lemma 1.21.
Let $T,T_{0}$ be $\tau$-theories with $T_{0}$ model complete. Assume that for
every $\Pi_{1}$-sentence $\theta$ for $\tau$ $T+\theta$ is consistent if and
only if so is $T_{0}+\theta$. Then:
1. (1)
$T^{*}=T_{0}+T_{\forall}$ is the model companion of $T$.
2. (2)
$T^{*}$ is axiomatized by the the set of $\Pi_{2}$-sentences $\psi$ for $\tau$
such that $S_{\forall}\cup\left\\{\psi\right\\}$ is consistent for all
$\Pi_{1}$-complete $S\supseteq T$.
3. (3)
$T^{*}$ is axiomatized by the the set of $\Pi_{2}$-sentences $\psi$ for $\tau$
such that for all universal $\tau$-sentences $\theta$
$T_{\forall}+\theta+\psi$ is consistent if and only if so is $T+\theta$.
###### Proof.
By assumption $T_{0}$ is consistent with any finite subset of $T_{\forall}$;
hence, by compactness, $T^{*}=T_{0}+T_{\forall}$ is consistent. By Fact 1.16
$T^{*}$ is model complete.
1. (1)
We need to show that any model of $T^{*}$ embeds into a model of $T$ and
conversely.
Assume $\mathcal{N}$ models $T^{*}$. Then $\mathcal{N}$ models $T_{\forall}$.
By Lemma 1.4 there exists $\mathcal{M}\sqsupseteq\mathcal{N}$ which models
$T$.
Conversely let $\mathcal{M}$ model $T$ and $S$ be the $\tau$-theory of
$\mathcal{M}$. By assumption (and compactness) there is $\mathcal{N}$ which
models $T_{0}+S_{\forall}$ (but this $\mathcal{N}$ may not be a superstructure
of $\mathcal{M}$). Let $S^{*}$ be the $\tau$-theory of $\mathcal{N}$. Then
$S^{*}_{\forall}=S_{\forall}$, since $S_{\forall}$ and $S^{*}_{\forall}$ are
$\Pi_{1}$-complete theories with $S^{*}_{\forall}\supseteq S_{\forall}$.
Moreover $S^{*}\supseteq T^{*}$, since $S_{\forall}\supseteq T_{\forall}$.
###### Claim 4.
The $\tau\cup\mathcal{M}$-theory $S^{*}\cup\Delta_{0}(\mathcal{M})$ is
consistent.
Assume the Claim holds, then $\mathcal{M}$ is a $\tau$-substructure of a model
of $S^{*}\supseteq T^{*}$ and we are done.
###### Proof.
If not there is $\psi(\vec{a})\in\Delta_{0}(\mathcal{M})$ such that
$S^{*}\cup\left\\{\psi(\vec{a})\right\\}$ is inconsistent. This gives that
$S^{*}\vdash\neg\psi(\vec{a}).$
Since none of the constant in $\vec{a}$ occurs in $\tau$, we get that
$S^{*}\vdash\forall\vec{x}\neg\psi(\vec{x}),$
i.e. $\forall\vec{x}\neg\psi(\vec{x})\in S^{*}_{\forall}=S_{\forall}$. But
$\mathcal{M}$ models $S_{\forall}$ and $\forall\vec{x}\neg\psi(\vec{x})$ fails
in $\mathcal{M}$; a contradiction. ∎
2. (2)
Assume $\psi\in T^{*}$ and $S$ is a $\Pi_{1}$-complete extension of $T$, we
must show that $S_{\forall}+\psi$ is consistent: by assumption there is
$\mathcal{N}$ which models
$T_{0}+S_{\forall}=T_{0}+T_{\forall}+S_{\forall}=T^{*}+S_{\forall}$, and we
are done. Conversely assume $R_{\forall}+\psi$ is consistent whenever $R$ is a
$\Pi_{1}$-complete extension of $T$. We must show that $\psi\in T^{*}$: pick
$\mathcal{M}$ model of $T^{*}$ and let $S$ be its theory. The assumptions of
the Lemma (and compactness) grant that $T+S_{\forall}$ is consistent. Since
$S$ is complete $S_{\forall}$ is the $\Pi_{1}$-fragment of $T+S_{\forall}$.
Hence $S_{\forall}+\psi$ is consistent, by our assumption on $\psi$. Therefore
$\mathcal{M}\models\psi$ by Proposition 1.8.
3. (3)
Left to the reader (as the previous item, modulo compactness arguments).
∎
###### Remark 1.22.
We do not know whether the characterization of the model companion of $T$
given in Lemma 1.21(3) can be proved for _all_ theories $T$ admitting a model
companion: following the notation of the Lemma, it is conceivable that some
$\tau$-theory $T$ has a model companion $T^{*}$, but there is some universal
$\tau$-sentence $\theta$ such that for any model $\mathcal{M}$ of $T+\theta$
any superstructure of $\mathcal{M}$ which models $T^{*}$ kills the truth of
$\theta$. In this case some $\Pi_{2}$-sentence in the Kaiser hull of $T$ is
inconsistent with the universal fragment of $T+\theta$.
Note also that if $T^{*}$ is the model companion of $T$ and $\theta$ is a
universal sentence such that $T^{*}+\theta$ is consistent, so is $T+\theta$:
if $\mathcal{M}\models T^{*}+\theta$ there is a superstructure $\mathcal{N}$
of $\mathcal{M}$ which models $T$ (since $T^{*}$ is the model companion of
$T$). Now $\mathcal{M}\prec_{1}\mathcal{N}$, since $\mathcal{M}$ is $T$-ec.
Hence $\mathcal{N}\models\theta$.
### 1.5. Is model companionship a tameness notion?
As we already outlined in the introduction model completeness and model
companionship are “tameness” notion for first order theories which must be
handled with care. We spell out the details in this small section.
###### Proposition 1.23.
Given a signature $\tau$ consider the signature $\tau^{*}$ which adds an
$n$-ary predicate symbol $R_{\phi}$ for any $\tau$-formula
$\phi(x_{1},\dots,x_{n})$ with displayed free variables.
Let $T_{\tau}$ be the following $\tau^{*}$-theory:
* •
$\forall\vec{x}\,(\phi(\vec{x})\leftrightarrow R_{\phi}(\vec{x}))$ for all
quantifier free $\tau$-formulae $\phi(\vec{x})$,
* •
$\forall\vec{x}\,[R_{\phi\wedge\psi}(\vec{x})\leftrightarrow(R_{\psi}(\vec{x})\wedge
R_{\phi}(\vec{x}))]$ for all $\tau$-formulae $\phi(\vec{x}),\psi(\vec{x})$,
* •
$\forall\vec{x}\,[R_{\neg\phi}(\vec{x})\leftrightarrow\neg R_{\phi}(\vec{x})]$
for all $\tau$-formulae $\phi(\vec{x})$,
* •
$\forall\vec{x}\,[\exists yR_{\phi}(y,\vec{x})\leftrightarrow R_{\exists
y\phi}(\vec{x})]$ for all $\tau$-formulae $\phi(y,\vec{x})$.
Then any $\tau$-structure $\mathcal{N}$ admits a unique extension to a
$\tau^{*}$-structure $\mathcal{N}^{*}$ which models $T_{\tau}$. Moreover every
$\tau^{*}$-formula is $T_{\tau}$-equivalent to an atomic $\tau^{*}$-formula.
In particular for any $\tau$-model $\mathcal{N}$, the algebras of its
$\tau$-definable subsets and of the $\tau^{*}$-definable subsets of
$\mathcal{N}^{*}$ are the same.
Therefore for any consistent $\tau$-theory $T$, $T\cup T_{\tau}$ is consistent
and admits quantifier elemination, hence is model complete.
###### Proof.
By an easy induction one can prove that any $\tau$-formula $\phi(\vec{x})$ is
$T_{\tau}$-equivalent to the atomic $\tau^{*}$-formula $R_{\phi}(\vec{x})$.
Another simple inductive argument brings that any $\tau^{*}$-formula
$\phi(\vec{x})$ is $T_{\tau}$-equivalent to the $\tau$-formula obtained by
replacing all symbols $R_{\psi}(\vec{x})$ occurring in $\phi$ by the
$\tau$-formula $\psi(\vec{x})$. Combining these observations together we get
that any $\tau^{*}$-formula is equivalent to an atomic $\tau^{*}$-formula.
$T_{\tau}$ forces the $\mathcal{M}^{*}$-interpretation of any relation symbol
$R_{\phi}(\vec{x})$ in $\tau^{*}\setminus\tau$ to be the
$\mathcal{M}$-interpretation of the $\tau$-formula $\phi(\vec{x})$ to which it
is $T_{\tau}$-equivalent. ∎
Observe that the expansion of the language from $\tau$ to $\tau^{*}$ behaves
well with respect to several model theoretic notions of tameness distinct from
model completeness: for example $T$ is a _stable_ $\tau$-theory if and only if
so is the $\tau^{*}$-theory $T\cup T_{\tau}$, the same holds for NIP-theories,
or for $o$-minimal theories, or for $\kappa$-categorical theories.
The passage from $\tau$-structures to $\tau^{*}$-structures which model
$T_{\tau}$ can have effects on the embeddability relation; for example assume
$\mathcal{M}\sqsubseteq\mathcal{N}$ is a non-elementary embedding of
$\tau$-structures; then $\mathcal{M}^{*}\not\sqsubseteq\mathcal{N}^{*}$: if
the non-atomic $\tau$-formula $\phi(\vec{a})$ in parameter
$\vec{a}\in\mathcal{M}^{<\omega}$ holds in $\mathcal{M}$ and does not hold in
$\mathcal{N}$, the atomic $\tau^{*}$-formula $R_{\phi}(\vec{a})$ holds in
$\mathcal{M}^{*}$ and does not hold in $\mathcal{N}^{*}$.
However if $T$ is a model complete $\tau$-theory, then for
$\mathcal{M}\sqsubseteq\mathcal{N}$ $\tau$-models of $T$, we get that
$\mathcal{M}\prec\mathcal{N}$; this entails that
$\mathcal{M}^{*}\sqsubseteq\mathcal{N}^{*}$, which (by the quantifier
elimination of $T\cup T_{\tau}$) gives that
$\mathcal{M}^{*}\prec\mathcal{N}^{*}$. In particular for a model complete
$\tau$-theory $T$ and $\mathcal{M},\mathcal{N}$ $\tau$-models of $T$,
$\mathcal{M}\sqsubseteq\mathcal{N}$ if and only if
$\mathcal{M}^{*}\sqsubseteq\mathcal{N}^{*}$.
Let us now investigate the case of model companionship. If $T$ is the model
companion of $S$ with $S\neq T$ in the signature $\tau$, $T\cup T_{\tau}$ and
$S\cup T_{\tau}$ are both model complete theories in the signature $\tau^{*}$.
But $T\cup T_{\tau}$ cannot be the model companion of $S\cup T_{\tau}$, by
uniqueness of the model companion, since each of these theories is the model
companion of itself and they are distinct. Moreover if $T$ and $S$ are also
complete, no $\tau^{*}$-model of $S\cup T_{\tau}$ can embed into a
$\tau^{*}$-model of $T\cup T_{\tau}$: since $T$ is the model companion of $S$
and $S\neq T$, $T_{\forall}=S_{\forall}$ and there is some $\Pi_{2}$-sentence
$\psi$ $\forall x\exists y\phi(x,y)$ with $\phi$-quantifer free in $T\setminus
S$. Therefore $\forall x\,R_{\exists y\phi}(x)\in(T\cup
T_{\tau})_{\forall}\setminus(S\cup T_{\tau})_{\forall}$; we conclude by Lemma
1.3, since $T\cup T_{\tau}$ and $S\cup T_{\tau}$ are complete, hence the above
sentence separates $(T\cup T_{\tau})_{\forall}$ from $(S\cup
T_{\tau})_{\forall}$.
### 1.6. Summing up
The results of this section gives that for any $\tau$-theory $T$:
* •
The universal fragment of $T$ describes the family of substructures of models
of $T$, and (in most cases, e.g. if $T$ is $\Pi_{1}$-complete) the $T$-ec
models realize all $\Pi_{2}$-sentences which are “absolutely” consistent with
$T_{\forall}$ (i.e. consistent with the universal fragment of any extension of
$T$).
* •
Model companionship and model completeness describe (almost all) the cases in
which the family of $\Pi_{2}$-sentences which are “absolutely” consistent with
$T$ (as defined in the previous item) describes the elementary class given by
the $T$-ec structures.
* •
One can always extend $\tau$ to a signature $\tau^{*}$ so that $T$ has a
conservative extension to a $\tau^{*}$-theory $T^{*}$ which is model complete,
but this process may be completely uninformative since it may completely
destroy the substructure relation existing between $\tau$-models of $T$
(unless $T$ is already model complete).
* •
On the other hand for certain theories $T$ (as the axiomatizations of set
theory considered in the present paper), one can unfold their “tameness” by
carefully extending $\tau$ to a signature $\tau^{*}$ in which only certain
$\tau$-formulae are made equivalent to atomic $\tau^{*}$-formulae. In the new
signature $T$ can be extended to a conservative extension $T^{*}$ which has a
model companion $\bar{T}$, while this process has mild consequences on the
$\tau^{*}$-substructure relation for models of $T^{*}_{\forall}$ (i.e. for the
pairs of interest of $\tau$-models $\mathcal{M}_{0}\sqsubseteq\mathcal{M}_{1}$
of a suitable fragment of $T$, their unique extensions to $\tau^{*}$-models
$\mathcal{M}^{*}_{i}$ are still models of $T^{*}_{\forall}$ and maintain that
$\mathcal{M}^{*}_{0}\sqsubseteq\mathcal{M}^{*}_{1}$ also for $\tau^{*}$). This
gives useful structural information on the web of relations existing between
$\tau^{*}$-models of $T^{*}_{\forall}$ (as outlined by Theorems 2.6, 4.4, 5).
* •
Our conclusion is that model completeness and model companionship are tameness
properties of elementary classes $\mathcal{E}$ defined by a theory $T$ rather
than of the theory $T$ itself: these model-theoretic notions outline certain
regularity patterns for the substructure relation on models of $\mathcal{E}$,
patterns which may be unfolded only when passing to a signature distinct from
the one in which $\mathcal{E}$ is first axiomatized (much the same way as it
occurs for Birkhoff’s characterization of algebraic varieties in terms of
universal theories).
* •
The results of the present paper shows that if we consider set theory together
with large cardinal axioms as formalized in the signature
$\sigma_{\omega},\sigma_{\omega,\mathbf{NS}_{\omega_{1}}},\sigma_{\omega_{1}}$,
we obtain (until now unexpected) tameness properties for this first order
theory, properties which couple perfectly with well known (or at least
published) generic absoluteness results. The notion of companionship spectrum
gives a model theoretic criterium for selecting these signatures out of the
continuum many signatures which produce definable extensions of
$\mathsf{ZFC}$. Moreover the common practice of set theory (independently of
our results) motivate the choice of signatures for set theory made in the
present paper (signatures which belong to the companionship spectrum of set
theory), and our results validate it.
## 2\. The theory of $H_{\kappa^{+}}$ is the model companion of set theory
In this section we prove Thm. 1 The following piece of notation will be used
all along this section and supplements Notations 1, 3:
###### Notation 2.1.
__
* •
$\sigma_{\text{{\sf ST}}}$ is the signature containing a predicate symbol
$S_{\phi}$ of arity $n$ for any $\in$-formula $\phi$ with $n$-many free
variables.
* •
$\sigma_{\kappa}=\sigma_{\text{{\sf ST}}}\cup\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$ with $\kappa$ a constant symbol.
* •
$T_{\kappa}$ is the $\sigma_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-theory given by the axioms
(2) $\forall x_{1}\dots
x_{n}\,[S_{\psi}(x_{1},\dots,x_{n})\leftrightarrow(\bigwedge_{i=1}^{n}x_{i}\subseteq\kappa^{<\omega}\wedge\psi^{\mathcal{P}\left(\kappa^{<\omega}\right)}(x_{1},\dots,x_{n}))]$
as $\psi$ ranges over the $\in$-formulae.
* •
$\mathsf{ZFC}^{-}_{\kappa}$ is the $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-theory
$\mathsf{ZFC}^{-}_{\text{{\sf ST}}}\cup\left\\{\kappa\text{ is an infinite
cardinal}\right\\};$
* •
$\mathsf{ZFC}^{*-}_{\kappa}$ is the $\sigma_{\kappa}$-theory
$\mathsf{ZFC}^{-}_{\kappa}\cup T_{\kappa};$
* •
Accordingly we define $\mathsf{ZFC}_{\kappa}$, $\mathsf{ZFC}^{*}_{\kappa}$.
###### Notation 2.2.
Given a $\in$-structure $(M,E)$ and $\tau$ a signature extending
$\tau_{\text{{\sf ST}}}$, from now we let $(M,\tau^{M})$ be the unique
extension of $(M,E)$ defined in accordance with Notation 3 which satisfies
$T_{\tau}$. In particular $(M,\tau^{M})$ is a shorthand for
$(M,S^{M}:S\in\tau)$. If $(N,E)$ is a substructure of $(M,E)$ we also write
$(N,\tau^{M})$ as a shorthand for $(N,S^{M}\restriction N:S\in\tau)$.
### 2.1. By-interpretability of the first order theory of $H_{\kappa^{+}}$
with the first order theory of $\mathcal{P}\left(\kappa\right)$
Let’s compare the first order theory of the structure
$(\mathcal{P}\left(\kappa\right),S_{\phi}^{V}:\phi\text{ an atomic
$\tau_{\text{{\sf ST}}}$-formula})$
with that of the $\tau_{\text{{\sf ST}}}$-theory of $H_{\kappa^{+}}$ in models
of $\mathsf{ZFC}_{\text{{\sf ST}}}$. We will show that they are
$\mathsf{ZFC}_{\tau_{\text{{\sf ST}}}}$-provably by-interpretable with a by-
interpetation translating $H_{\kappa^{+}}$ in a $\Pi_{1}$-definable subset of
$\mathcal{P}\left(\kappa^{2}\right)$ and atomic predicates into
$\Sigma_{1}$-relations over this set. This result is the key to the proof of
Thm. 1 and is just outlining the model theoretic consequences of the well-
known fact that sets can be coded by well-founded extensional graphs.
###### Definition 2.3.
Given $a\in H_{\kappa^{+}}$, $R\in\mathcal{P}\left(\kappa^{2}\right)$ codes
$a$, if $R$ codes a well-founded extensional relation on some
$\alpha\leq\kappa$ with top element $0$ so that the transitive collapse
mapping of $(\alpha,R)$ maps $0$ to $a$.
* •
$\mathsf{WFE}_{\kappa}$ is the set of $R\in\mathcal{P}\left(\kappa\right)$
which are a well founded extensional relation with domain $\alpha\leq\kappa$
and top element $0$.
* •
$\text{{\rm Cod}}_{\kappa}:\mathsf{WFE}_{\kappa}\to H_{\kappa^{+}}$ is the map
assigning $a$ to $R$ if and only if $R$ codes $a$.
The following theorem shows that the structure $(H_{\kappa^{+}},\in)$ is
interpreted by means of “imaginaries” in the structure
$(\mathcal{P}\left(\kappa\right),\tau_{\text{{\sf ST}}}^{V})$ by means of:
* •
a universal $\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$-formula (with
quantifiers ranging over subsets of $\kappa^{<\omega}$) defining a set
$\mathsf{WFE}_{\kappa}\subseteq\mathcal{P}\left(\kappa^{2}\right)$.
* •
an equivalence relation $\cong_{\kappa}$ on $\mathsf{WFE}_{\kappa}$ defined by
an existential $\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$-formula
(with quantifiers ranging over subsets of $\kappa^{<\omega}$)
* •
A binary relation $E_{\kappa}$ on $\mathsf{WFE}_{\kappa}$ invariant under
$\cong_{\kappa}$ representing the $\in$-relation as the extension of an
existential $\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$-formula (with
quantifiers ranging over subsets of $\kappa^{<\omega}$)111111See [10, Section
25] for proofs of the case $\kappa=\omega$; in particular the statement and
proof of Lemma 25.25 and the proof of [10, Thm. 13.28] contain all ideas on
which one can elaborate to draw the conclusions of Thm. 2.4..
###### Theorem 2.4.
Assume $\mathsf{ZFC}^{-}_{\kappa}$. The following holds121212Many transitive
supersets of $H_{\kappa^{+}}$ are $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-model of $\mathsf{ZFC}^{-}_{\kappa}$ for
$\kappa$ an infinite cardinal (see [13, Section IV.6]). To simplify notation
we assume to have fixed a transitive $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-model $\mathcal{N}$ of
$\mathsf{ZFC}^{-}_{\kappa}$ with domain $N\supseteq H_{\kappa^{+}}$. The
reader can easily realize that all these statements holds for an arbitrary
model $\mathcal{N}$ of $\mathsf{ZFC}^{-}_{\kappa}$ replacing $H_{\kappa^{+}}$
with its version according to $\mathcal{N}$.:
1. (1)
The map $\mathrm{Cod}_{\kappa}$ and $\mathsf{WFE}_{\kappa}$ are defined by
$\mathsf{ZFC}^{-}_{\kappa}$-provably $\Delta_{1}$-properties in parameter
$\kappa$. Moreover $\text{{\rm Cod}}_{\kappa}:\mathsf{WFE}_{\kappa}\to
H_{\kappa^{+}}$ is surjective (provably in $\mathsf{ZFC}^{-}_{\kappa}$), and
$\mathsf{WFE}_{\kappa}$ is defined by a universal $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-formula with quantifiers ranging over
subsets of $\kappa^{<\omega}$.
2. (2)
There are existential $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-formulae (with quantifiers ranging over
subsets of $\kappa^{<\omega}$), $\phi_{\in},\phi_{=}$ such that for all
$R,S\in\mathsf{WFE}_{\kappa}$, $\phi_{=}(R,S)$ if and only if $\text{{\rm
Cod}}_{\kappa}(R)=\text{{\rm Cod}}_{\kappa}(S)$ and $\phi_{\in}(R,S)$ if and
only if $\text{{\rm Cod}}_{\kappa}(R)\in\text{{\rm Cod}}_{\kappa}(S)$. In
particular letting
$E_{\kappa}=\left\\{(R,S)\in\mathsf{WFE}_{\kappa}:\phi_{\in}(R,S)\right\\},$
$\cong_{\kappa}=\left\\{(R,S)\in\mathsf{WFE}_{\kappa}:\phi_{=}(R,S)\right\\},$
$\cong_{\kappa}$ is a $\mathsf{ZFC}^{-}_{\kappa}$-provably definable
equivalence relation, $E_{\kappa}$ respects it, and
$(\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}},E_{\kappa}/_{\cong_{\kappa}})$
is isomorphic to $(H_{\kappa^{+}},\in)$ via the map $[R]\mapsto\text{{\rm
Cod}}_{\kappa}(R)$.
###### Proof.
A detailed proof requires a careful examination of the syntactic properties of
$\Delta_{0}$-formulae, in line with the one carried in Kunen’s [13, Chapter
IV]. We outline the main ideas, following Kunen’s book terminology for certain
set theoretic operations on sets, functions and relations (such as
$\operatorname{dom}(f),\operatorname{ran}(f)$, $\text{Ext}(R)$, etc). To
simplify the notation, we prove the results for a transitive model $(N,\in)$
which is then extended to a structure $(N,\tau_{\text{{\sf
ST}}}^{N},\kappa^{N})$ which models $\mathsf{ZFC}^{-}_{\kappa}$, and whose
domain contains $H_{\kappa^{+}}$. The reader can verify by itself that the
argument is modular and works for any other model of
$\mathsf{ZFC}^{-}_{\kappa}$ (transitive or ill-founded, containing the “true”
$H_{\kappa^{+}}$ or not).
1. (1)
This is proved in details in [13, Chapter IV]. To define
$\mathsf{WFE}_{\kappa}$ by a universal $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-property over subsets of $\kappa$ and
$\text{{\rm Cod}}_{\kappa}$ by a $\Delta_{1}$-property for $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$ over $H_{\kappa^{+}}$, we proceed as
follows:
* •
_$R$ is an extensional relation with domain contained in $\kappa$ and top
element $0$_ is defined by the $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-atomic formula $\psi_{\mathrm{EXT}}(R)$
$\mathsf{ZFC}^{-}_{\kappa}$-provably equivalent to the
$\Delta_{0}(\kappa)$-formula:
$\displaystyle(R\subseteq\kappa^{2})\wedge$
$\displaystyle\wedge(\text{Ext}(R)\in\kappa\vee\text{Ext}(R)=\kappa)\wedge$
$\displaystyle\wedge\forall\alpha,\beta\in\text{Ext}(R)\,[\forall
u\in\text{Ext}(R)\,(u\mathrel{R}\alpha\leftrightarrow
u\mathrel{R}\beta)\rightarrow(\alpha=\beta)]\wedge$
$\displaystyle\wedge\forall\alpha\in\text{Ext}(R)\,\neg(0\mathrel{R}\alpha).$
* •
$\mathsf{WFE}_{\kappa}$ is defined by the universal $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-formula $\phi_{\mathsf{WFE}_{\kappa}}(R)$
(quantifying only over subsets of $\kappa^{<\omega}$)
$\displaystyle\psi_{\mathrm{EXT}}(R)\wedge$ $\displaystyle\wedge[\forall
f\subseteq\kappa^{2}\,(f\text{ is a function }\rightarrow\exists
n\in\omega\,\neg(\langle f(n+1),f(n)\rangle\in R))].$
Its interpretation is the subset of $\mathcal{P}\left(\kappa^{<\omega}\right)$
of the $\sigma_{\kappa}$-symbol $S_{\phi_{\mathsf{WFE}_{\kappa}}}$.
* •
To define $\text{{\rm Cod}}_{\kappa}$, consider the $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-atomic formula $\psi_{\text{{\rm
Cod}}}(G,R)$ provably equivalent to the $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-formula:
$\displaystyle\psi_{\mathrm{EXT}}(R)\wedge$ $\displaystyle\wedge(G\text{ is a
function})\wedge$
$\displaystyle\wedge(\operatorname{dom}(G)=\text{Ext}(R))\wedge(\operatorname{ran}(G)\text{
is transitive})\wedge$
$\displaystyle\wedge\forall\alpha,\beta\in\text{Ext}(R)\,[\alpha\mathrel{R}\beta\leftrightarrow
G(\alpha)\in G(\beta)].$
Then $\text{{\rm Cod}}_{\kappa}(R)=a$ can be defined either by the existential
$\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$-formula131313Given an $R$
such that $\psi_{\mathrm{EXT}}(R)$ holds, _$R$ is a well founded relation_
holds in a model of $\mathsf{ZFC}^{-}_{\kappa}$ if and only if $\text{{\rm
Cod}}_{\kappa}$ is defined on $R$. In the theory $\mathsf{ZFC}^{-}_{\kappa}$,
$\mathsf{WFE}_{\kappa}$ can be defined using a universal property by a
$\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$-formula quantifying only
over subsets of $\kappa$. On the other hand if we allow arbitrary
quantification over elements of $H_{\kappa^{+}}$, we can express the well-
foundedness of $R$ also using the existential formula $\exists
G\,\psi_{\text{{\rm Cod}}_{\kappa}}(G,R)$. This is why $\mathsf{WFE}_{\kappa}$
is defined by a universal $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-property in the structure
$(\mathcal{P}\left(\kappa\right),\tau_{\text{{\sf ST}}}^{V},\kappa)$, while
the graph of $\text{{\rm Cod}}_{\kappa}$ can be defined by a
$\Delta_{1}$-property for $\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$
in the structure $(H_{\kappa^{+}},\tau_{\text{{\sf ST}}}^{V},\kappa^{V})$.
$\exists G\,(\psi_{\text{{\rm Cod}}}(G,R)\wedge G(0)=a)$
or by the universal $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-formula
$\forall G\,(\psi_{\text{{\rm Cod}}}(G,R)\rightarrow G(0)=a).$
2. (2)
The equality relation in $H_{\kappa^{+}}$ is transferred to the isomorphism
relation between elements of $\mathsf{WFE}_{\kappa}$: if $R,S$ are well-
founded extensional on $\kappa$ with a top-element, the Mostowski collapsing
theorem entails that $\text{{\rm Cod}}_{\kappa}(R)=\text{{\rm
Cod}}_{\kappa}(S)$ if and only if
$(\mathrm{Ext}(R),R)\cong(\mathrm{Ext}(S),S)$. Isomorphism of the two
structures $(\mathrm{Ext}(R),R)\cong(\mathrm{Ext}(S),S)$ is expressed by the
$\Sigma_{1}$-formula for $\tau_{\kappa}$:
$\phi_{=}(R,S)\equiv\exists f\,(f\text{ is a bijection of $\kappa$ onto
$\kappa$ and $\alpha R\beta$ if and only if $f(\alpha)Sf(\beta)$}).$
In particular we get that $S_{\phi_{=}}(R,S)$ holds in $H_{\kappa^{+}}$ for
$R,S\in\mathsf{WFE}_{\kappa}$ if and only if $\text{{\rm
Cod}}_{\kappa}(R)=\text{{\rm Cod}}_{\kappa}(S)$.
Similarly one can express $\text{{\rm Cod}}_{\kappa}(R)\in\text{{\rm
Cod}}_{\kappa}(S)$ by the $\Sigma_{1}$-property $\phi_{\in}$ in
$\tau_{\kappa}$ stating that $(\mathrm{Ext}(R),R)$ is isomorphic to
$(\mathrm{pred}_{S}(\alpha),S)$ for some $\alpha\in\kappa$ with
$\alpha\mathrel{S}0$, where $\mathrm{pred}_{S}(\alpha)$ is given by the
elements of $\mathrm{Ext}(S)$ which are connected by a finite path to
$\alpha$.
Moreover letting $\cong_{\kappa}\subseteq\mathsf{WFE}_{\kappa}^{2}$ denote the
isomorphism relation between elements of $\mathsf{WFE}_{\kappa}$ and
$E_{\kappa}\subseteq\mathsf{WFE}_{\kappa}^{2}$ denote the relation which
translates into the $\in$-relation via $\text{{\rm Cod}}_{\kappa}$, it is
clear that $\cong_{\kappa}$ is a congruence relation over $E_{\kappa}$, i.e.:
if $R_{0}\cong_{\kappa}R_{1}$ and $S_{0}\cong_{\kappa}S_{1}$,
$R_{0}\mathrel{E_{\kappa}}S_{0}$ if and only if
$R_{1}\mathrel{E_{\kappa}}S_{1}$.
This gives that the structure
$(\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}},E_{\kappa}/_{\cong_{\kappa}})$ is
isomorphic to $(H_{\kappa^{+}},\in)$ via the map $[R]\mapsto\text{{\rm
Cod}}_{\kappa}(R)$ (where $\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}}$ is the set
of equivalence classes of $\cong_{\kappa}$ and the quotient relation
$[R]\mathrel{E_{\kappa}/_{\cong_{\kappa}}}[S]$ holds if and only if
$R\mathrel{E_{\kappa}}S$).
This isomorphism is defined via the map $\text{{\rm Cod}}_{\kappa}$, which is
by itself defined by a $\mathsf{ZFC}^{-}_{\kappa}$-provably
$\Delta_{1}$-property for $\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$.
The very definition of $\mathsf{WFE}_{\kappa},\cong_{\kappa},E_{\kappa}$ show
that
$\mathsf{WFE}_{\kappa}=S_{\phi_{\mathsf{WFE}_{\kappa}}}^{N},$
$\cong_{\kappa}=S_{\phi_{\mathsf{WFE}_{\kappa}}(x)\wedge\phi_{\mathsf{WFE}_{\kappa}}(y)\wedge\phi_{=}(x,y)}^{N},$
$E_{\kappa}=S_{\phi_{\mathsf{WFE}_{\kappa}}(x)\wedge\phi_{\mathsf{WFE}_{\kappa}}(y)\wedge\phi_{\in}(x,y)}^{N}.$
∎
### 2.2. Model completeness for the theory of $H_{\kappa^{+}}$
###### Theorem 2.5.
Any $\sigma_{\kappa}$-theory $T$ extending
$\mathsf{ZFC}^{*-}_{\kappa}\cup\left\\{\text{all sets have size
$\kappa$}\right\\}$
is model complete.
###### Proof.
To simplify notation, we conform to the assumption of the previous theorem,
i.e. we assume that the model $(N,\in)$ which is uniquely extended to a model
of $\mathsf{ZFC}^{*-}_{\kappa}+$_every set has size $\kappa$_ on which we work
is a transitive superstructure of $H_{\kappa^{+}}$.
The statement _every set has size $\kappa$_ is satisified by a
$\mathsf{ZFC}^{-}_{\kappa}$-model $(N,\tau_{\text{{\sf ST}}}^{V},\kappa)$ with
$N\supseteq H_{\kappa}^{+}$ if and only if $N=H_{\kappa^{+}}$. From now on we
proceed assuming this equality.
By Robinson’s test 1.15 it suffices to show that for all $\in$-formulae
$\phi(\vec{x})$
$\mathsf{ZFC}^{-}_{\kappa}+\text{ every set has size
$\kappa$}\vdash\forall\vec{x}\,(\phi(\vec{x})\leftrightarrow\psi_{\phi}(\vec{x})),$
for some universal $\sigma_{\kappa}$-formula $\psi_{\phi}$.
We will first define a recursive map $\phi\to\theta_{\phi}$ which maps
$\Sigma_{n}$-formulae $\phi$ for $\left\\{\in,\kappa\right\\}$ quantifying
over all elements of $H_{\kappa^{+}}$ to $\Sigma_{n+1}$-formulae
$\theta_{\phi}$ for $\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$ whose
quantifier range just over subsets of $\kappa^{<\omega}$.
The proof of the previous theorem gave $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-formulae $\theta_{x=y}$, $\theta_{x\in y}$
such that
$S_{\theta_{x=y}}^{H_{\kappa^{+}}}=\cong_{\kappa}=\left\\{(R,S)\in(\mathsf{WFE}_{\kappa})^{2}:\,\text{{\rm
Cod}}_{\kappa}(R)=\text{{\rm Cod}}_{\kappa}(S)\right\\},$ $S_{\theta_{x\in
y}}^{H_{\kappa^{+}}}=E_{\kappa}=\left\\{(R,S)\in(\mathsf{WFE}_{\kappa})^{2}:\,\text{{\rm
Cod}}_{\kappa}(R)\in\text{{\rm Cod}}_{\kappa}(S)\right\\}.$
Specifically (following the notation of that proof)
$\theta_{x=y}=\phi_{\mathsf{WFE}_{\kappa}}(x)\wedge\phi_{\mathsf{WFE}_{\kappa}}(y)\wedge\phi_{=}(x,y),$
$\theta_{x\in
y}=\phi_{\mathsf{WFE}_{\kappa}}(x)\wedge\phi_{\mathsf{WFE}_{\kappa}}(y)\wedge\phi_{\in}(x,y).$
Now for any $\left\\{\in,\kappa\right\\}$-formula $\psi(\vec{x})$, we proceed
to define the $\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$-formula
$\theta_{\psi}(\vec{x})$ letting:
* •
$\theta_{\psi\wedge\psi}(\vec{x})$ be
$\theta_{\psi}(\vec{x})\wedge\theta_{\psi}(\vec{x})$,
* •
$\theta_{\neg\psi}(\vec{x})$ be $\neg\theta_{\psi}(\vec{x})$,
* •
$\theta_{\exists y\psi(y,\vec{x})}(\vec{x})$ be $\exists
y\theta_{\psi}(y,\vec{x})\wedge\phi_{\mathsf{WFE}_{\kappa}}(y)$.
An easy induction on the complexity of the $\tau_{\text{{\sf
ST}}}\cup\left\\{\kappa\right\\}$-formulae $\theta_{\phi}(\vec{x})$ gives that
for any $\left\\{\in,\kappa\right\\}$-definable subset $A$ of
$(H_{\kappa^{+}})^{n}$ which is the extension of some
$\left\\{\in,\kappa\right\\}$-formula $\phi(x_{1},\dots,x_{n})$
$\left\\{(R_{1},\dots,R_{n})\in(\mathsf{WFE}_{\kappa})^{n}:\,(\text{{\rm
Cod}}_{\kappa}(R_{1}),\dots,\text{{\rm Cod}}_{\kappa}(R_{n}))\in
A\right\\}=S_{\theta_{\phi}}^{H_{\kappa^{+}}},$
with the further property that
$S_{\theta_{\phi}}^{H_{\kappa^{+}}}\subseteq(\mathsf{WFE}_{\kappa})^{n}$
respects the $\cong_{\kappa}$-relation141414It is also clear from our argument
that the map $\phi\mapsto\theta_{\phi}$ is recursive (and a careful inspection
reveals that it maps a $\Sigma_{n}$-formula to a $\Sigma_{n+1}$-formula)..
Now every $\sigma_{\kappa}$-formula is $\mathsf{ZFC}^{*-}_{\kappa}$-equivalent
to a $\left\\{\in,\kappa\right\\}$-formula151515The map assigning to any
$\sigma_{\kappa}$-formula a $\mathsf{ZFC}^{*-}_{\kappa}$-equivalent
$\left\\{\in,\kappa\right\\}$-formula can also be chosen to be recursive..
Therefore we can extend $\phi\mapsto\theta_{\phi}$ assigning to any
$\sigma_{\kappa}$-formula $\phi(\vec{x})$ the formula $\theta_{\psi}(\vec{x})$
for some $\left\\{\in,\kappa\right\\}$-formula $\psi(\vec{x})$ which is
$\mathsf{ZFC}^{*-}_{\kappa}$-equivalent to $\phi(\vec{x})$.
Then for any $\left\\{\in,\kappa\right\\}$-formula $\phi(x_{1},\dots,x_{n})$
$H_{\kappa^{+}}\models\phi(a_{1},\dots,a_{n})$ if and only if
$(\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}},E_{\kappa}/_{\cong_{\kappa}})\models\phi([R_{1}],\dots,[R_{n}])$
with $\text{{\rm Cod}}_{\kappa}(R_{i})=a_{i}$ for $i=1,\dots,n$ if and only if
$H_{\kappa^{+}}\models\forall
R_{1},\dots,R_{n}\,[(\bigwedge_{i=1}^{n}\text{{\rm
Cod}}_{\kappa}(R_{i})=a_{i})\rightarrow\theta_{\phi}(R_{1},\dots,R_{n})]$
if and only if
$H_{\kappa^{+}}\models\forall
R_{1},\dots,R_{n}\,[(\bigwedge_{i=1}^{n}\text{{\rm
Cod}}_{\kappa}(R_{i})=a_{i})\rightarrow
S_{\theta_{\phi}}(R_{1},\dots,R_{n})].$
Since this argument can be repeated verbatim for any model of
$\mathsf{ZFC}^{*-}_{\kappa}$+_every set has size $\kappa$_, and any
$\sigma_{\kappa}$-formula is $\mathsf{ZFC}^{*-}_{\kappa}$-equivalent to a
$\left\\{\in,\kappa\right\\}$-formula, we have proved the following:
###### Claim 5.
For any $\sigma_{\kappa}$-formula $\phi(x_{1},\dots,x_{n})$,
$\mathsf{ZFC}^{*-}_{\kappa}+$_every set has size $\kappa$_ proves that
$\forall x_{1},\dots,x_{n}\,[\phi(x_{1},\dots,x_{n})\leftrightarrow\forall
y_{1},\dots,y_{n}\,[(\bigwedge_{i=1}^{n}\text{{\rm
Cod}}_{\kappa}(y_{i})=x_{i})\rightarrow
S_{\theta_{\phi}}(y_{1},\dots,y_{n})]].$
But $\text{{\rm Cod}}_{\kappa}(y)=x$ is expressible by an existential
$\tau_{\text{{\sf ST}}}\cup\left\\{\kappa\right\\}$-formula provably in
$\mathsf{ZFC}^{-}_{\kappa}\subseteq\mathsf{ZFC}^{*-}_{\kappa}$, therefore
$\forall y_{1},\dots,y_{n}\,[(\bigwedge_{i=1}^{n}\text{{\rm
Cod}}_{\kappa}(y_{i})=x_{i})\rightarrow S_{\theta_{\phi}}(y_{1},\dots,y_{n})]$
is a universal $\sigma_{\kappa}$-formula, and we are done. ∎
### 2.3. Proof of Thm. 1
Conforming to the notation of Thm. 1, it is clear that $\sigma_{\kappa}$ is a
signature of the form $\left\\{\in\right\\}_{\bar{A}_{\kappa}}$ whenever
$\kappa$ is a $T$-definable cardinal for some $T$ extending $\mathsf{ZFC}$.
Therefore the following result completes the proof of Thm. 1.
###### Theorem 2.6.
Assume $T\supseteq\mathsf{ZFC}^{*}_{\kappa}$ is a $\sigma_{\kappa}$-theory.
Then $T$ has a model companion $T^{*}$. Moreover for any $\Pi_{2}$-sentence
$\psi$ for $\sigma_{\kappa}$, TFAE:
1. (1)
$\psi\in T^{*}$;
2. (2)
$T\vdash\psi^{H_{\kappa^{+}}}$;
3. (3)
For all universal $\sigma_{\kappa}$-sentences $\theta$, $T+\theta$ is
consistent if and only if so is $T_{\forall}+\theta+\psi$.
###### Proof.
By Thm. 2.5, any $\sigma_{\kappa}$-theory extending
$\mathsf{ZFC}^{*-}_{\kappa}+\emph{every set has size $\kappa$}$
is model complete. Therefore so is
$T^{*}=\left\\{\phi:H_{\kappa^{+}}^{\mathcal{M}}\models\phi,\,\mathcal{M}\models
T\right\\},$
since $H_{\kappa^{+}}^{\mathcal{M}}$ models
$\mathsf{ZFC}^{*-}_{\kappa}$+_every set has size $\kappa$_ for any
$\mathcal{M}$ which models $T$.
We must now show that $T^{*}_{\forall}=T_{\forall}$. Assume
$T^{*}\models\theta$ for some universal sentece $\theta$. Then
$H_{\kappa^{+}}^{\mathcal{M}}\models\theta$ for any model $\mathcal{M}$ of
$T$. Since $H_{\kappa^{+}}^{\mathcal{M}}\prec_{1}\mathcal{M}$ for any such
$\mathcal{M}$, we get that any such $\mathcal{M}$ models $\theta$ as well.
Therefore $T^{*}_{\forall}\subseteq T_{\forall}$. Appealing again to Levy
absoluteness, by a similar argument, we get that $T_{\forall}\subseteq
T^{*}_{\forall}$.
We now show that $T^{*}$ is the set of $\Pi_{2}$-sentences $\phi$ such that:
> For all $\Pi_{1}$-sentences $\phi$ for $\tau$, $T+\theta$ is consistent if
> and only if so is $T_{\forall}+\phi+\theta$.
We prove it establishing that $T$ and $T^{*}$ satisfy the assumption of Lemma
1.21 i.e. for any $\Pi_{1}$-sentence $\theta$ for $\sigma_{\kappa}$ $T+\theta$
is consistent if and only if so is $T^{*}+\theta$.
So assume $T+\theta$ is consistent for some $\Pi_{1}$-sentence $\theta$, we
must show that $T^{*}+\theta$ is also consistent, but this is immediate: by
Levy absoluteness if $\mathcal{M}$ models $\theta$, so does
$H_{\kappa^{+}}^{\mathcal{M}}$.
Conversely assume $T+\theta$ is inconsistent for some $\Pi_{1}$-sentence
$\theta$. Then $T\models\neg\theta$. Again by Levy absoluteness if
$\mathcal{M}$ models $T$, $H_{\kappa^{+}}^{\mathcal{M}}\models\neg\theta$.
Hence $\neg\theta\in T^{*}$ by definition, and $\theta$ is inconsistent with
$T^{*}$. ∎
###### Remark 2.7.
Note that the family of models
$\left\\{H_{\kappa^{+}}^{\mathcal{M}}:\,\mathcal{M}\models T\right\\}$ we used
to define $T^{*}$ may not be an elementary class for $\sigma_{\kappa}$.
Thm. 2.6 can be proved for many other signatures other than $\sigma_{\kappa}$.
It suffices that the signature in question adds new predicates just for
definable subsets of $\mathcal{P}\left(\kappa\right)^{n}$, and also that it
adds family of predicates which are closed under definability (i.e.
projections, complementation, finite unions, permutations) and under the map
$\text{{\rm Cod}}_{\kappa}$. Under these assumptions we can still use Lemma 1
and Fact 1.13 to argue for the evident variations of the proof of Thm. 2.6 to
this set up. However linking these model companionship results to generic
absoluteness as we do in Theorem 2 requires much more care in the definition
of the signature. We will pursue this matter in more details in the next
sections.
### 2.4. A weak version of Theorem 2 for third order arithmetic
We can prove a weak version of Thm. 2 for the theory of $H_{\aleph_{2}}$
appealing to the generic absoluteness results of [23, 4, 5] which establish
the invariance of the theory of $H_{\aleph_{2}}$ in models of strong forcing
axioms with respect to stationary set preserving forcings preserving these
axioms.
Let $\mathsf{ZFC}^{*}_{\omega_{1}}\supseteq\mathsf{ZFC}_{\text{{\sf ST}}}$ be
the $\sigma_{\omega_{1}}=\sigma_{\omega}\cup\left\\{\kappa\right\\}$-theory
obtained adding axioms which force in each of its $\sigma_{\omega_{1}}$-models
$\kappa$ to be interpreted by the first uncountable cardinal, and each
predicate symbol $S_{\phi}$ to be interpreted as the subset of
$\mathcal{P}\left(\omega_{1}^{<\omega}\right)^{n}$ defined by
$\phi^{\mathcal{P}\left(\omega_{1}^{<\omega}\right)}(x_{1},\dots,x_{n})$.
###### Theorem 2.8.
Let $T$ be a $\sigma_{\omega_{1}}$-theory extending
$\mathsf{ZFC}^{*}_{\omega_{1}}+\text{{\sf MM}}^{+++}+\emph{there are class
many superhuge cardinals}.$
TFAE for any $\Pi_{2}$-sentence $\psi$ for $\sigma_{\omega_{1}}$:
1. (1)
$S_{\forall}+\psi$ is consistent for all complete $S$ extending $T$;
2. (2)
$T$ proves that some stationary set preserving forcing notion $P$ forces
$\psi^{\dot{H}_{\omega_{2}}}+\text{{\sf MM}}^{+++}$;
3. (3)
$T\vdash\psi^{H_{\omega_{2}}}$.
See Remarks 2.9(2) for some information on $\text{{\sf MM}}^{+++}$, and 2.9(1)
for informations on superhugeness.
The proof of Theorem 2.8 is a trivial variation of the proof of Theorem 2.6:
###### Proof.
[23, Thm. 5.18] gives that 2.8(3) and 2.8(2) are equivalent. Theorem 2.6
establishes the equivalence of 2.8(3) and 2.8(1). ∎
###### Remark 2.9.
__
1. (1)
$\delta$ is superhuge if it supercompact and this can be witnessed by huge
embeddings. A superhuge cardinal is consistent relative to the existence of a
$2$-huge cardinal.
2. (2)
For a definition of $\text{{\sf MM}}^{+++}$ see [23, Def. 5.19]. We just note
that $\text{{\sf MM}}^{+++}$ is a natural strengthening of Woodin’s axiom
$(*)$ (by the recent breakthrough of Asperò and Schindler [2]) and of Martin’s
maximum (for example any of the standard iterations to produce a model of
Martin’s maximum produce a model of $\text{{\sf MM}}^{+++}$ if the iteration
has length a superhuge cardinal [23, Thm 5.29]).
3. (3)
We can prove exactly the same results of Thm. 2.8 replacing (verbatim in its
statement) $\text{{\sf MM}}^{+++}$ by any of the axioms
$\mathsf{RA}_{\omega}(\Gamma)$ introduced in [5] or the axioms
$\mathsf{CFA}(\Gamma)$ and $\mathsf{BCFA}(\Gamma)$ introduced in [4], provided
in item 2.8(2) _stationary set preserving forcing notion $P$_ is replaced by
$P\in\Gamma$.
4. (4)
We consider Thm. 2.8 weaker than Thm. 2 or Corollary 1, because in Corollary 1
one can choose the theory $T$ to be inconsistent with $\text{{\sf MM}}^{++}$
without hampering its conclusion (for example $T$ could satisfy $\mathsf{CH}$,
a statement denied by $\text{{\sf MM}}^{++}$), and because Corollary 1C holds
for all forcing notions $P$ unlike Thm. 2.8(2). The key point separating these
two results is that the signature $\sigma_{\omega_{1}}$ is too expressive and
renders many statements incompatible with forcing axioms formalizable by
existential (or even atomic) $\sigma_{\omega_{1}}$-sentences (for example such
is the case for $\mathsf{CH}$).
5. (5)
A key distinction between the signature $\sigma_{\omega_{1}}$ and the
signature $\left\\{\in\right\\}_{\bar{A}_{2}}$ considered in Thm. 2 is that
for any $T\supseteq\mathsf{ZFC}+$_appropriate large cardinals_ $\mathsf{CH}$
cannot be $T$-equivalent to a $\Sigma_{1}$-sentence for
$\left\\{\in\right\\}_{\bar{A}_{2}}$ because CH is a statement which can
change its truth value across forcing extensions, while the universal
$\left\\{\in\right\\}_{\bar{A}_{2}}$-sentences maintain the same truth value
across all forcing extensions of a model of $T$, by Thm. 2(5). On the other
hand CH is $\mathsf{ZFC}_{\omega_{1}}$-equivalent to an atomic
$\sigma_{\omega_{1}}$-sentence. $\neg\mathsf{CH}$ is the simplest example of
the type of $\Pi_{2}$-sentences which exemplifies why Thm. 2.8(2) is much
weaker than Thm. 2, and why Thm. 2 for the signature
$\left\\{\in\right\\}_{\bar{A}_{2}}$ needs a different (and as we will see
much more sophisticated) proof strategy than the one we use here to establish
Theorems 2.6 and 2.8.
## 3\. Generic invariance results for signatures of second and third order
arithmetic
We collect here generic absoluteness results results needed to prove Thm. 2.
We prove all these results working in “standard” models of $\mathsf{ZFC}$,
i.e. we assume the models are well-founded. This is a practice we already
adopted in Section 2. We leave to the reader to remove this unnecessary
assumption.
### 3.1. Universally Baire sets and generic absoluteness for second order
number theory
We recall here the properties of universally Baire sets and the generic
absoluteness results for second order number theory we need to prove Thm. 2.
###### Notation 3.1.
$\mathcal{A}\subseteq\bigcup_{n\in\omega}\mathcal{P}\left(\kappa\right)^{n}$
is projectively closed if it is closed under projections, finite unions,
complementation, and permutations (if $\sigma:n\to n$ is a permutation and
$A\subseteq\mathcal{P}\left(\kappa\right)^{n}$,
$\hat{\sigma}[A]=\left\\{(a_{\sigma(0)},\dots,a_{\sigma(n-1}):\,(a_{0},\dots,a_{n-1})\in
A\right\\}$).
Otherwise said, $\mathcal{A}$ is the class of lightface definable subsets of
some signature on $\mathcal{P}\left(\kappa\right)$.
### 3.2. Universally Baire sets
Assuming large cardinals there is a very large sample of projectively closed
families of subsets of $\mathcal{P}\left(\omega\right)$ which are are
“simple”, hence it is natural to consider elements of these families as atomic
predicates.
The exact definition of what is meant by a “simple” subset of $2^{\omega}$ is
captured by the notion of universally Baire set.
Given a topological space $(X,\tau)$, $A\subseteq X$ is nowhere dense if its
closure has a dense complement, meager if it is the countable union of nowhere
dense sets, with the Baire property if it has meager symmetric difference with
an open set. Recall that $(X,\tau)$ is Polish if $\tau$ is a completely
metrizable, separable topology on $X$.
###### Definition 3.2.
(Feng, Magidor, Woodin) Given a Polish space $(X,\tau)$, $A\subseteq X$ is
_universally Baire_ if for every compact Hausdorff space $(Y,\sigma)$ and
every continuous $f:Y\to X$ we have that $f^{-1}[A]$ has the Baire property in
$Y$.
$\mathsf{UB}$ denotes the family of universally Baire subsets of $X$ for some
Polish space $X$.
We adopt the convention that $\mathsf{UB}$ denotes the class of universally
Baire sets and of all elements of $\bigcup_{n\in\omega+1}(2^{\omega})^{n}$
(since the singleton of such elements are universally Baire sets).
The theorem below outlines three simple examples of projectively closed
families of universally Baire sets containing $2^{\omega}$.
###### Theorem 3.3.
Let $T_{0}$ be the $\tau_{\text{{\sf ST}}}$-theory $\mathsf{ZFC_{\text{{\sf
ST}}}}+$_there are infinitely many Woodin cardinals and a measurable above_
and $T_{1}$ be the $\tau_{\text{{\sf ST}}}$-theory $\mathsf{ZFC_{\text{{\sf
ST}}}}+$_there are class many Woodin cardinals_.
1. (1)
[15, Thm. 3.1.12, Thm. 3.1.19] Assume $V$ models $T_{0}$. Then every
projective subset of $2^{\omega}$ is universally Baire.
2. (2)
[15, Thm. 3.3.3, Thm. 3.3.5, Thm. 3.3.6, Thm. 3.3.8, Thm. 3.3.13, Thm. 3.3.14]
Assume $V\models T_{1}$. Then $\mathsf{UB}$ is projectively closed.
To proceed further we now list the standard facts about universally Baire sets
we will need:
1. (1)
[10, Thm. 32.22] $A\subseteq 2^{\omega}$ is universally Baire if and only if
for each forcing notion $P$ there are trees $T_{A},S_{A}$ on
$\omega\times\delta$ for some $\delta>|P|$ such that $A=p[[T_{A}]]$ (where
$p:(2\times\kappa)^{\omega}\to 2^{\omega}$ denotes the projection on the first
component and $[T]$ denotes the body of the tree $T$), and
$P\Vdash T_{A}\text{ and }S_{A}\text{ project to complements},$
by this meaning that for all $G$ $V$-generic for $P$
$V[G]\models(p[[T_{A}]]\cap p[[S_{A}]]=\emptyset)\wedge(p[[T_{A}]]\cup
p[[S_{A}]]=(2^{\omega})^{V[G]})$
2. (2)
Any two Polish spaces $X,Y$ of the same cardinality are Borel isomorphic [12,
Thm. 15.6].
3. (3)
Any Polish space is Borel isomorphic to a Borel subset of $[0;1]^{\omega}$
[12, Thm. 4.14], hence also to a Borel subset of $2^{\omega}$ (by the previous
item).
4. (4)
Given $\phi:\mathbb{N}\to\mathbb{N}$, $\prod_{n\in\omega}2^{\phi(n)}$ is
Polish (it is actually homemomorphic to the union of $2^{\omega}$ with a
countable Hausdorff space) [12, Thm. 6.4, Thm. 7.4].
Hence it is not restrictive to focus just on universally Baire subsets of
$2^{\omega}$ and of its countable products, which is what we will do in the
sequel.
###### Notation 3.4.
Given $G$ a $V$-generic filter for some forcing $P\in V$, $A\in\text{{\sf
UB}}^{V[G]}$ and $H$ $V[G]$-generic filter for some forcing $Q\in V[G]$,
$A^{V[G][H]}=\left\\{r\in(2^{\omega})^{V[G][H]}:V[G][H]\models r\in
p[[T_{A}]]\right\\},$
where $(T_{A},S_{A})\in V[G]$ is any pair of trees as given in item 1 above
such that $p[[T_{A}]]=A$ holds in $V[G]$, and $(T_{A},S_{A})$ project to
complements in $V[G][H]$.
### 3.3. Generic absoluteness for second order number theory
The following generic absoluteness result is the key to establish Thm. 2(5)
for the signature $A_{1}$.
We decide to include a full proof of Woodin’s generic absoluteness results for
second order number theory we use in this paper. The version we need follows
readily from [15, Thm. 3.1.2] and the assumptions that there exists class many
Woodin limits of Woodin; here we reduce these large cardinal assumptions to
the existence of class many Woodin cardinals, while providing an alternative
approach to the proof of some of these result. The theorem below is an
improvement of [24, Thm. 3.1].
###### Theorem 3.5.
Assume in $V$ there are class many Woodin cardinals. Let $\mathcal{A}\in V$ be
a family of universally Baire sets of $V$ and
$\tau_{\mathcal{A}}=\tau_{\text{{\sf ST}}}\cup\mathcal{A}$. Let $G$ be
$V$-generic for some forcing notion $P\in V$.
Then
$(H_{\omega_{1}},\tau_{\mathcal{A}}^{V})\prec(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf
ST}}}^{V[G]},A^{V[G]}:A\in\mathcal{A}).$
###### Proof.
We proceed by induction on $n$ to prove the following stronger assertion:
###### Claim 6.
Whenever $G$ is $V$-generic for some forcing notion $P$ in $V$ and $H$ is
$V[G]$-generic for some forcing notion $Q$ in $V[G]$
$(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf
ST}}}^{V[G]},A^{V[G]}:A\in\mathcal{A})\prec_{n}(H_{\omega_{1}}^{V[G][H]},\tau_{\text{{\sf
ST}}}^{V[G][H]},A^{V[G][H]}:A\in\mathcal{A}).$
###### Proof.
It is not hard to check that for all $A\in\mathcal{A}$,
$A^{V[G]}=A^{V[G][H]}\cap V[G]$ (choose in $V$ a pair of trees $(T,S)$ such
that $A=p[[T]]$ and the pair $(T,S)$ projects to complements in $V[G][H]$, and
therefore also in $V[G]$). Therefore $(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf
ST}}}^{V[G]},A^{V[G]}:A\in\mathcal{A})$ is a $\tau_{\mathcal{A}}$-substructure
of $(H_{\omega_{1}}^{V[G][H]},\tau_{\text{{\sf
ST}}}^{V[G][H]},A^{V[G][H]}:A\in\mathcal{A})$.
This proves the base case of the induction.
We prove the successor step.
Assume that for any $G$ $V$-generic for some forcing $P\in V$ and $H$
$V[G]$-generic for some forcing $Q\in V[G]$
$(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf
ST}}}^{V[G]},A^{V[G]}:A\in\mathcal{A})\prec_{n}(H_{\omega_{1}}^{V[G][H]},\tau_{\text{{\sf
ST}}}^{V[G][H]},A^{V[G][H]}:A\in\mathcal{A}).$
Fix $\bar{G}$ and $\bar{H}$ as in the assumptions of the Claim as witnessed by
forcings $\bar{P}\in V$ and $\bar{Q}\in V[\bar{G}]$.
We want to show that
$(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})\prec_{n+1}(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][\bar{H}]},A^{V[\bar{G}][\bar{H}]}:A\in\mathcal{A}).$
Let $\gamma$ be a Woodin cardinal of $V$ such that
$\bar{P}\ast\dot{\bar{Q}}\in V_{\gamma}$ (where $\dot{\bar{Q}}\in V^{P}$ is
chosen so that $\dot{\bar{Q}}_{G}=\bar{Q}$).
Then $\gamma$ is Woodin also in $V[\bar{G}]$. Let $K$ be $V[\bar{G}]$-generic
for161616$\mathcal{T}^{\omega_{1}}_{\gamma}$ denotes here the countable tower
of height $\gamma$ denoted as $\mathbb{Q}_{<\gamma}$ in [15, Section 2.7].
$(\mathcal{T}^{\omega_{1}}_{\gamma})^{V[\bar{G}]}$ with $\bar{H}\in V[K]$, so
that $V[\bar{G}][K]=V[\bar{G}][\bar{H}][\bar{K}]$ for some $\bar{K}\in
V[\bar{G}][K]$.
Hence we have the following diagram:
$(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})$$(H_{\omega_{1}}^{V[\bar{G}][K]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][K]},A^{V[\bar{G}][K]}:A\in\mathcal{A})$$(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][\bar{H}]},A^{V[\bar{G}][\bar{H}]}:A\in\mathcal{A})$$\scriptstyle{\Sigma_{\omega}}$$\scriptstyle{\Sigma_{n}}$$\scriptstyle{\Sigma_{n}}$
obtained by inductive hypothesis applied both on $V[\bar{G}]$,
$V[\bar{G}][\bar{H}]$ and on $V[\bar{G}][\bar{H}]$,
$V[\bar{G}][\bar{H}][\bar{K}]$, and using the fact that
$(H_{\omega_{1}}^{V[\bar{G}][K]},\tau_{\text{{\sf
UB}}^{V[\bar{G}]}}^{V[\bar{G}][K]})$ is a fully elementary superstructure of
$(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
UB}}^{V[\bar{G}]}}^{V[\bar{G}]})$ [15, Thm. 2.7.7, Thm. 2.7.8].
Let $\phi\equiv\exists x\psi(x)$ be any $\Sigma_{n+1}$ formula for
$\tau_{\mathcal{A}}$ with parameters in $H_{\omega_{1}}^{V[\bar{G}]}$. First
suppose that $\phi$ holds in $(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})$, and fix $\bar{a}\in
V[\bar{G}]$ such that $\psi(\bar{a})$ holds in
$(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})$. Since
$(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})\prec_{n}(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][\bar{H}]},A^{V[\bar{G}][\bar{H}]}:A\in\mathcal{A}),$
we conclude that $\psi(\bar{a})$ holds in
$(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][\bar{H}]},A^{V[\bar{G}][\bar{H}]}:A\in\mathcal{A})$, hence
so does $\phi$.
Now suppose that $\phi$ holds in
$(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][\bar{H}]},A^{V[\bar{G}][\bar{H}]}:A\in\mathcal{A})$ as
witnessed by $\bar{a}\in H_{\omega_{1}}^{V[\bar{G}][\bar{H}]}$.
Since
$(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][\bar{H}]},A^{V[\bar{G}][\bar{H}]}:A\in\mathcal{A})\prec_{n}(H_{\omega_{1}}^{V[\bar{G}][K]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][K]},A^{V[\bar{G}][K]}:A\in\mathcal{A}),$
it follows that $\psi(\bar{a})$ holds in
$(H_{\omega_{1}}^{V[\bar{G}][K]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][K]},A^{V[\bar{G}][K]}:A\in\mathcal{A})$, hence so does
$\phi$. Since
$(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})\prec(H_{\omega_{1}}^{V[\bar{G}][K]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][K]},A^{V[\bar{G}][K]}:A\in\mathcal{A}),$
the formula $\phi$ holds also in
$(H_{\omega_{1}}^{V[\bar{G}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})$.
Since $\phi$ is arbitrary, this shows that
$(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}]},A^{V[\bar{G}]}:A\in\mathcal{A})\prec_{n+1}(H_{\omega_{1}}^{V[\bar{G}][\bar{H}]},\tau_{\text{{\sf
ST}}}^{V[\bar{G}][\bar{H}]},A^{V[\bar{G}][\bar{H}]}:A\in\mathcal{A}),$
concluding the proof of the inductive step for $\bar{G}$ and $\bar{H}$.
Since we have class many Woodin, this argument is modular in $\bar{G},\bar{H}$
as in the assumptions of the inductive step, because we can always find some
Woodin cardinal $\gamma$ of $V$ which remains Woodin in $V[\bar{G}]$ and is of
size larger than the poset in $V[\bar{G}]$ for which $\bar{H}$ is
$V[\bar{G}]$-generic. The proof of the inductive step is completed. ∎
∎
### 3.4. Generic invariance for the universal fragment of the theory of $V$
with predicates for the non-stationary ideal and for universally Baire sets
The results of this section are the key to establish Thm. 2(5) for the
signature $A_{1}$. The proofs require some familiarity with the basics of the
$\mathbb{P}_{\mathrm{max}}$-technology and with Woodin’s stationary tower
forcing.
###### Notation 3.6.
__
* •
$\tau_{\mathbf{NS}_{\omega_{1}}}$ is the signature $\tau_{\text{{\sf
ST}}}\cup\left\\{\omega_{1}\right\\}\cup\left\\{\mathbf{NS}_{\omega_{1}}\right\\}$
with $\omega_{1}$ a constant symbol, $\mathbf{NS}_{\omega_{1}}$ a unary
predicate symbol.
* •
$T_{\mathbf{NS}_{\omega_{1}}}$ is the $\tau_{\mathbf{NS}_{\omega_{1}}}$-theory
given by $T_{\text{{\sf ST}}}$ together with the axioms
$\omega_{1}\text{ is the first uncountable cardinal},$ $\forall
x\;[(x\subseteq\omega_{1}\text{ is non-
stationary})\leftrightarrow\mathbf{NS}_{\omega_{1}}(x)].$
* •
$\mathsf{ZFC}^{-}_{\mathbf{NS}_{\omega_{1}}}$ is the
$\tau_{\mathbf{NS}_{\omega_{1}}}$-theory
$\mathsf{ZFC}^{-}_{\text{{\sf ST}}}+T_{\mathbf{NS}_{\omega_{1}}}.$
* •
Accordingly we define $\mathsf{ZFC}_{\mathbf{NS}_{\omega_{1}}}$.
The following is the key to establish Thm. 2(5) for the signature $A_{2}$.
###### Theorem 3.
Assume $(V,\in)$ models $\mathsf{ZFC}+$_there are class many Woodin
cardinals_. Then the $\Pi_{1}$-theory of $V$ for the language
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ is invariant under
set sized forcings171717Here we consider any $A\subseteq(2^{\omega})^{k}$ in
$\text{{\sf UB}}^{V}$ as a predicate symbol of arity $k$..
Asperó and Veličkovic̀ provided the following basic counterexample to the
conclusion of the theorem if large cardinal assumptions are dropped.
###### Remark 3.7.
Let $\phi(y)$ be the $\Delta_{1}$-property in
$\tau_{\mathbf{NS}_{\omega_{1}}}$
$\exists y(y=\omega_{1}\wedge L_{y+1}\models y=\omega_{1}).$
Then $L$ models this property, while the property fails in any forcing
extension of $L$ which collapses $\omega_{1}^{L}$ to become countable.
In order to prove the Theorem we need to recall some basic terminology and
facts about iterations of countable structures.
#### 3.4.1. Generic iterations of countable structures
###### Definition 3.8.
[14, Def. 1.2] Let $M$ be a transitive countable model of $\mathsf{ZFC}$. Let
$\gamma$ be an ordinal less than or equal to $\omega_{1}$. An iteration
$\mathcal{J}$ of $M$ of length $\gamma$ consists of models $\langle
M_{\alpha}:\,\alpha\leq\gamma\rangle$, sets $\langle
G_{\alpha}:\,\alpha<\gamma\rangle$ and a commuting family of elementary
embeddings
$\langle j_{\alpha\beta}:M_{\alpha}\to
M_{\beta}:\,\alpha\leq\beta\leq\gamma\rangle$
such that:
* •
$M_{0}=M$,
* •
each $G_{\alpha}$ is an $M_{\alpha}$-generic filter for
$(\mathcal{P}\left(\omega_{1}\right)/\mathbf{NS}_{\omega_{1}})^{M_{\alpha}}$,
* •
each $j_{\alpha\alpha}$ is the identity mapping,
* •
each $j_{\alpha\alpha+1}$ is the ultrapower embedding induced by $G_{\alpha}$,
* •
for each limit ordinal $\beta\leq\gamma$, $M_{\beta}$ is the direct limit of
the system
$\left\\{M_{\alpha},j_{\alpha\delta}:\,\alpha\leq\delta<\beta\right\\}$, and
for each $\alpha<\beta$, $j_{\alpha\beta}$ is the induced embedding.
We adopt the convention to denote an iteration $\mathcal{J}$ just by $\langle
j_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma\rangle$, we also stipulate that if
$X$ denotes the domain of $j_{0\alpha}$, $X_{\alpha}$ or $j_{0\alpha}(X)$ will
denote the domain of $j_{\alpha\beta}$ for any $\alpha\leq\beta\leq\gamma$.
###### Definition 3.9.
Let $A$ be a universally Baire sets of reals. $M$ is $A$-iterable if:
1. (1)
$M$ is transitive and such that $H_{\omega_{1}}^{M}$ is countable.
2. (2)
$M\models\mathsf{ZFC}+\mathbf{NS}_{\omega_{1}}$_is precipitous_.
3. (3)
Any iteration
$\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\gamma\right\\}$
of $M$ is well founded and such that $A\cap M_{\beta}=j_{\alpha\beta}(A\cap
M_{0})$ for all $\beta\leq\gamma$.
#### 3.4.2. Proof of Theorem 3
###### Proof.
Let $\phi$ be a $\Pi_{1}$-sentence for
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ which holds in $V$.
Assume that for some forcing notion $P$, $\phi$ fails in $V[h]$ with $h$
$V$-generic for $P$. By forcing over $V[h]$ with the appropriate stationary
set preserving (in $V[h]$) forcing notion (using a Woodin cardinal $\gamma$ of
$V[h]$), we may assume that $V[h]$ is extended to a generic extension $V[g]$
such that $V[g]$ models $\mathbf{NS}_{\omega_{1}}$ is saturated181818A result
of Shelah whose outline can be found in [19, Chapter XVI], or [25], or in an
handout of Schindler available on his webpage.. Since $V[g]$ is an extension
of $V[h]$ by a stationary set preserving forcing and there are in $V[h]$ class
many Woodin cardinals, we get that $V[h]\sqsubseteq V[g]$ with respect to the
signature $\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$. Since
$\Sigma_{1}$-properties are upward absolute and $\neg\phi$ holds in $V[h]$,
$\phi$ fails in $V[g]$ as well.
Let $\delta$ be inaccessible in $V[g]$ and let $\gamma>\delta$ be a Woodin
cardinal.
Let $G$ be $V$-generic for $\mathcal{T}^{\omega_{1}}_{\gamma}$ (the countable
tower $\mathbb{Q}_{<\gamma}$ according to [15, Section 2.7]) and such that
$g\in V[G]$. Let $j_{G}:V\to\operatorname{Ult}(V,G)$ be the induced ultrapower
embedding.
Now remark that $V_{\delta}[g]\in\operatorname{Ult}(V,G)$ is
$B^{V[G]}$-iterable for all $B\in\mathsf{UB}^{V}$ (since
$V_{\eta}[g]\in\operatorname{Ult}(V,G)$ for all $\eta<\gamma$, and this
suffices to check that $V_{\delta}[g]$ is $B^{V[G]}$-iterable for all
$B\in\mathsf{UB}^{V}$, see [14, Thm. 4.10]).
By [14, Lemma 2.8] applied in $\operatorname{Ult}(V,G)$, there exists in
$\operatorname{Ult}(V,G)$ an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\gamma=\omega_{1}^{\operatorname{Ult}(V,G)}\right\\}$
of $V_{\delta}[g]$ such that
$\mathbf{NS}_{\omega_{1}}^{X_{\gamma}}=\mathbf{NS}_{\omega_{1}}^{\operatorname{Ult}(V,G)}\cap
X_{\gamma}$, where $X_{\alpha}=j_{0\alpha}(V_{\delta}[g])$ for all
$\alpha\leq\gamma=\omega_{1}^{\operatorname{Ult}(V,G)}$.
This gives that $X_{\gamma}\sqsubseteq\operatorname{Ult}(V,G)$ for
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$. Since
$V_{\delta}[g]\models\neg\phi$, so does $X_{\gamma}$, by elementarity. But
$\neg\phi$ is a $\Sigma_{1}$-sentence, hence it is upward absolute for
superstructures, therefore $\operatorname{Ult}(V,G)\models\neg\phi$. This is a
contradiction, since $\operatorname{Ult}(V,G)$ is elementarily equivalent to
$V$ for $\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$, and
$V\models\phi$.
A similar argument shows that if $V$ models a $\Sigma_{1}$-sentence $\phi$ for
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ this will remain true
in all of its generic extensions:
Assume $V[h]\models\neg\phi$ for some $h$ $V$-generic for some forcing notion
$P\in V$. Let $\gamma>|P|$ be a Woodin cardinal, and let $g$ be $V$-generic
for191919$\mathcal{T}_{\gamma}$ is the full stationary tower of height
$\gamma$ whose conditions are stationary sets in $V_{\gamma}$, denoted as
$\mathbb{P}_{<\gamma}$ in [15], see in particular [15, Section 2.5].
$\mathcal{T}_{\gamma}$ with $h\in V[g]$ and
$\operatorname{crit}(j_{g})=\omega_{1}^{V}$ (hence there is in $g$ some
stationary set of $V_{\gamma}$ concentrating on countable sets). Then
$V[g]\models\phi$ since:
* •
$V_{\gamma}\models\phi$, since $V_{\gamma}\prec_{1}V$ for
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ by Lemma 1;
* •
$V_{\gamma}^{\operatorname{Ult}(V,g)}=V_{\gamma}^{V[g]}$, since $V[g]$ models
that $\operatorname{Ult}(V,g)^{<\gamma}\subseteq\operatorname{Ult}(V,g)$;
* •
$V_{\gamma}^{\operatorname{Ult}(V,g)}\models\phi$, by elementarity of $j_{g}$,
since $j_{g}(V_{\gamma})=V_{\gamma}^{\operatorname{Ult}(V,g)}$;
* •
$V_{\gamma}^{V[g]}\prec_{\Sigma_{1}}V[g]$ with respect to
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$, again by Lemma 1
applied in $V[g]$.
Now repeat the same argument as before to the $\Pi_{1}$-property $\neg\phi$,
with $V[h]$ in the place of $V$ and $V[g]$ in the place of $V[h]$. ∎
## 4\. Model companionship versus generic absoluteness for the theory of
$H_{\aleph_{1}}$
### 4.1. Model companionship for the theory of $H_{\aleph_{1}}$
###### Notation 4.1.
Let $\tau\supseteq\tau_{\text{{\sf ST}}}$ be a signature.
$\mathsf{ZFC}_{\tau}$ is the theory extending $\mathsf{ZFC}$ with the
replacement schema for all $\tau$-formulae. Accordingly we define
$\mathsf{ZFC}^{-}_{\tau}$.
###### Definition 4.2.
Let $S$ be a $\tau$-theory extending $\mathsf{ZFC}_{\tau}$.
$\tau\supseteq\tau_{\text{{\sf ST}}}$ is a projective signature for $S$ if any
$\tau$-model $\mathcal{M}$ of $S$ interprets:
* •
all predicate symbols of arity $k$ of $\tau\setminus\tau_{\text{{\sf ST}}}$ as
subsets of $(2^{\omega})^{k}$ (as defined in $\mathcal{M}$),
* •
all function symbols of arity $k$ of $\tau\setminus\tau_{\text{{\sf ST}}}$ as
functions from $(2^{\omega})^{k}$ to $2^{\omega}$ (as defined in
$\mathcal{M}$),
* •
all constant symbols of $\tau\setminus\tau_{\text{{\sf ST}}}$ as elements of
$2^{\omega}$ (as defined in $\mathcal{M}$).
Assume $\tau$ is a projective signature for $S\supseteq\mathsf{ZFC}_{\tau}$.
$A\subseteq F_{\tau}$ is $S$-projectively closed if:
1. (A)
$A$ is closed under logical equivalence;
2. (B)
for any $(V,\tau)$ model of $S$, any formula in $A$ defines a subset of
$((2^{\omega})^{V})^{k}$ for some $k\in\omega$;
3. (C)
in any model $(V,\tau)$ of $S$, if $B$ is a definable subset of
$((2^{\omega})^{V})^{k}$ in the structure
$(H_{\omega_{1}}^{V},\tau^{V},R_{\phi}^{V},f_{\phi}^{V}:\phi\in A),$
then $B=R_{\psi}^{V}$ for some $\psi\in A$.
###### Example 4.3.
Given a $\tau_{\text{{\sf ST}}}$-theory $T$ extending
$\mathsf{ZFC}_{\text{{\sf ST}}}$, simple examples of $T$-projectively closed
families for $\tau_{\text{{\sf ST}}}$ (which we will use) are:
1. (1)
The family of lightface definable projective sets of reals.
2. (2)
$\text{{\rm l-UB}}^{T}$, i.e. the $\in$-formulae defining subsets of
$(2^{\omega})^{k}$ (as $k$ varies in the natural numbers) which $T$ proves to
be the extension of some $\in$-formula relativized to $L(\text{{\sf UB}})$
(the smallest transitive model of $\mathsf{ZF}$ containing all the ordinals
and the universally Baire sets).
3. (3)
If $(V,\tau_{\text{{\sf ST}}}^{V})$ models the existence of class many Woodin
cardinals, $X\prec(V_{\theta},\in)$ for a large enough $\theta$, and $T_{X}$
is the $\tau_{\text{{\sf ST}}}\cup(\text{{\sf UB}}^{V}\cap X)$-theory of $V$,
one also get that $\tau_{\text{{\sf ST}}}\cup(\text{{\sf UB}}^{V}\cap X)$ is a
projective signature for $T_{X}$ and $\text{{\sf UB}}^{V}\cap X$ is
$T_{X}$-projectively closed (where a universally Baire subset of
$(2^{\omega})^{k}$ is considered a predicate symbol of arity $k$; note that
$X=V_{\theta}$ — i.e. $\text{{\sf UB}}^{V}\cap X=\text{{\sf UB}}^{V}$ — is
possible).
###### Theorem 4.4.
Let $\tau\supseteq\tau_{\text{{\sf ST}}}$ and $S$ be a $\tau$-theory extending
$\mathsf{ZFC}_{\tau}$ such that $\tau$ is a projective signature for $S$.
Let $A\subseteq F_{\tau}$ be an $S$-projectively closed family for $\tau$ and
$\bar{A}=A\times\left\\{0,1\right\\}.$
Then $S_{\bar{A}}$ has as its model companion in signature $\tau_{\bar{A}}$
$S^{*}_{\bar{A}}=\left\\{\phi:(H_{\omega_{1}}^{V},\tau_{\bar{A}}^{V})\models\phi,\,(V,\in)\models
S\right\\}.$
It is clear that the above theorem combined with the results of Section 3
proves Thm. 2 and Corollary 1 for $A_{1}$. More precisely:
###### Corollary 4.5.
Let $S\supseteq\mathsf{ZFC}+$_there are class many Woodin cardinals_ be a
$\in$-theory. Then for any $A\subseteq F_{\in}$ projectively closed for $S$
and such that $\phi$ defines a universally Baire set of reals for any $\phi$
in $A$ not a $\Delta_{0}$-formula, letting
$\bar{A}=A\times\left\\{0,1\right\\}$, $S+T_{\bar{A}}$ has as model companion
the $\Pi_{2}$-sentences $\psi$ for $\left\\{\in\right\\}_{\bar{A}}$ such that
$S\vdash\psi^{H_{\omega_{1}}}.$
###### Proof.
Let $(V,\tau^{V})$ be a model of $S$.
By Levy’s absoluteness Lemma 1, since $A$ includes just formulae definining
subsets of $(2^{\omega})^{k}$ and the same occurs for the symbols of
$\tau\setminus\tau_{\text{{\sf ST}}}$ in models of $S$,
$(H_{\omega_{1}},\tau^{V},R_{\psi}^{V},f_{\psi}^{V}:\psi\in
A)\prec_{1}(V,\tau^{V},R_{\psi}^{V},f_{\psi}^{V}:\psi\in A);$
hence the structures $(V,\tau^{V},R_{\psi}^{V},f_{\psi}^{V}:\psi\in A)$ and
$(H_{\omega_{1}},\tau^{V},R_{\psi}^{V},f_{\psi}^{V}:\psi\in A)$ share the same
$\Pi_{1}$-theory for the signature $\tau_{\bar{A}}$.
Therefore (by the useful characterization of model companionship given in
Lemma 1.21) it suffices to prove that $S^{*}$ is model complete, where $S^{*}$
is the $\tau_{\bar{A}}$-theory common to
$(H_{\omega_{1}},\tau^{V},R_{\psi}^{V},f_{\psi}^{V}:\psi\in A)$ as
$(V,\tau^{V})$ range over models of $S$.
By Robinson’s test (Lemma 1.15c), it suffices to show that any existential
$\tau_{\bar{A}}$-formula is $S^{*}$-equivalent to a universal
$\tau_{\bar{A}}$-formula.
Let $\psi_{1},\dots,\psi_{k}$ be the formulae in $A$ such that some
$R_{\psi_{i}}$ or some $f_{\psi_{i}}$ appears in $\phi$.
Let $\psi(x_{1},\dots,x_{n})$ be the formula $\phi(\text{{\rm
Cod}}_{\omega}(x_{1}),\dots,\text{{\rm Cod}}_{\omega}(x_{n}))$. Since
$\text{{\rm Cod}}_{\omega}(x)=y$ is a $\Delta_{1}$-definable predicate in the
structure $(H_{\omega_{1}},\tau_{\text{{\sf ST}}})$, we get that
$\psi(x_{1},\dots,x_{n})$ in $A$ since its extension is a subset of
$(2^{\omega})^{k}$ in the structure
$(H_{\omega_{1}},\tau^{V},R_{\psi}^{V},f_{\psi}^{V}:\psi\in A).$
Now for any $a_{1},\dots,a_{n}\in H_{\omega_{1}}$:
$(H_{\omega_{1}},\tau_{\bar{A}}^{V})\models\phi(a_{1},\dots,a_{n})$
if and only if
$(H_{\omega_{1}},\tau_{\bar{A}}^{V})\models\forall r_{1}\dots
r_{n}\bigwedge_{i=1}^{n}\text{{\rm Cod}}_{\omega}(r_{i})=a_{i}\rightarrow
R_{\psi}(r_{1},\dots,r_{n}).$
This yields that
$S^{*}\vdash\forall
x_{1},\dots,x_{n}\,(\phi(x_{1},\dots,x_{n})\leftrightarrow\theta_{\psi}(x_{1},\dots,x_{n})).$
where $\theta_{\phi}(x_{1},\dots,x_{n})$ is the $\Pi_{1}$-formula in the
predicate $R_{\psi}\in\tau_{\bar{A}}$
$\forall y_{1},\dots,y_{n}\,[(\bigwedge_{i=1}^{n}x_{i}=\text{{\rm
Cod}}_{\omega}(y_{i}))\rightarrow R_{\psi}(y_{1},\dots,y_{n})].$
∎
It is also convenient to reformulate these notion is a more semantic way which
is handy when dealing with a fixed complete first order axiomatization of set
theory.
###### Definition 4.6.
Let
$\mathcal{A}\subseteq\bigcup_{n\in\omega}\mathcal{P}\left(\omega\right)^{n}$.
$\mathcal{A}$ is $H_{\omega_{1}}$-closed if any definable subset of
$\mathcal{P}\left(\omega\right)^{n}$ for some $n\in\omega$ in the structure
$(H_{\omega_{1}},\in,U:U\in\mathcal{A})$
is in $\mathcal{A}$.
It is immediate to check that if $T$ is the theory of $(V,\in)$ and
$\mathcal{A}$ is a family of universlly Baire subsets of $V$, $\mathcal{A}$ is
projectively closed for $T$ for the signature $\tau_{\text{{\sf
ST}}}\cup\mathcal{A}$ if and only if it is $H_{\omega_{1}}$-closed.
We get the following:
###### Theorem 4.7.
Assume $(V,\in)$ models $\mathsf{ZFC}+$_there are class many Woodin
cardinals_. Let $\mathcal{A}\subseteq\text{{\sf UB}}^{V}$ be
$H_{\omega_{1}}$-closed and $\tau_{\mathcal{A}}=\tau_{\text{{\sf
ST}}}\cup\mathcal{A}$ be the signature in which each element of $\mathcal{A}$
contained in $\mathcal{P}\left(\omega\right)^{k}$ is a predicate symbol of
arity $k$. Then for any $G$ $V$-generic for some forcing $P\in V$ the
$\tau_{\mathcal{A}}$-theory of $H_{\omega_{1}}^{V}$ is the model companion of
the $\tau_{\mathcal{A}}$-theory of $V[G]$ and
$\left\\{A^{V[G]}:A\in\mathcal{A}\right\\}$ is $H_{\omega_{1}}^{V[G]}$-closed.
###### Proof.
The assumptions grant that
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf
ST}}}^{V},A:A\in\mathcal{A})\prec(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf
ST}}}^{V[G]},A:A^{V[G]}\in\mathcal{A})\prec_{1}(V[G],\tau_{\text{{\sf
ST}}}^{V[G]},A:A^{V[G]}\in\mathcal{A})$
(by Thm. 3.5 and by Lemma 1 applied in $V[G]$). Now the theory of
$H_{\omega_{1}}^{V}$ in signature $\tau_{\mathcal{A}}$ is complete and model
complete, and is also the $\tau_{\mathcal{A}}$-theory of
$H_{\omega_{1}}^{V[G]}$. We conclude that it is the model companion of the
$\tau_{\mathcal{A}}$-theory of $V[G]$. It is also easy to check that
$\left\\{A^{V[G]}:A\in\mathcal{A}\right\\}$ is $H_{\omega_{1}}^{V[G]}$-closed.
∎
## 5\. Model companionship versus generic absoluteness for the theory of
$H_{\aleph_{2}}$
Let UB denote the family of universally Baire sets, and $L(\text{{\sf UB}})$
denote the smallest transitive model of $\mathsf{ZF}$ which contains UB (see
for details Section 3.2).
Our first result shows that in models of large cardinal axioms admitting a
strong form of sharp for UB (what is here called
${\mathbf{MAX}(\mathsf{UB})}$), a strong form of Woodin’s axiom $(*)$ (what is
here called $(*)\text{-}\mathsf{UB}$) can be equivalently formulated as the
assertion that the theory of $H_{\aleph_{2}}$ is the model companion of the
theory of $V$ in a signature admitting a predicate symbol for the non-
stationary ideal on $\omega_{1}$ and predicates for each universally Baire
set.
###### Theorem 4.
Let $\mathcal{V}=(V,\in)$ be a model of
$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{UB})}+\emph{there is a supercompact
cardinal and class many Woodin cardinals},$
and UB denote the family of universally Baire sets in $V$.
TFAE
1. (1)
$(V,\in)$ models $(*)\text{-}\mathsf{UB}$;
2. (2)
$\mathbf{NS}_{\omega_{1}}$ is precipitous202020See [15, Section 1.6, pag. 41]
for a definition of precipitousness and a discussion of its properties. A key
observation is that $\mathbf{NS}_{\omega_{1}}$ being precipitous is
independent of $\mathsf{CH}$ (see for example [15, Thm. 1.6.24]), while
$(*)\text{-}\mathsf{UB}$ entails $2^{\aleph_{0}}=\aleph_{2}$ (for example by
the results of [14, Section 6]). Another key point is that we stick to the
formulation of $\mathbb{P}_{\mathrm{max}}$ as in [14] so to be able in its
proof to quote verbatim from [14] all the relevant results on
$\mathbb{P}_{\mathrm{max}}$-preconditions we will use. It is however possible
to develop $\mathbb{P}_{\mathrm{max}}$ focusing on Woodin’s countable tower
rather than on the precipitousness of $\mathbf{NS}_{\omega_{1}}$ to develop
the notion of $\mathbb{P}_{\mathrm{max}}$-precondition. Following this
approach in all its scopes, one should be able to reformulate Thm. 4(2)
omitting the request that $\mathbf{NS}_{\omega_{1}}$ is precipitous. We do not
explore this venue any further. and the
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}$-theory of $V$ has as
model companion the $\tau_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf
UB}}$-theory of $H_{\omega_{2}}$.
(1) implies (2) does not need the supercompact cardinal.
We give rightaway the definitions of ${\mathbf{MAX}(\mathsf{UB})}$ and
$(*)\text{-}\mathsf{UB}$.
###### Definition 4.
${\mathbf{MAX}(\mathsf{UB})}$: There are class many Woodin cardinals in $V$,
and for all $G$ $V$-generic for some forcing notion $P\in V$:
1. (1)
Any subset of $(2^{\omega})^{V[G]}$ definable in
$(H_{\omega_{1}}^{V[G]}\cup\mathsf{UB}^{V[G]},\in)$ is universally Baire in
$V[G]$.
2. (2)
Let $H$ be $V[G]$-generic for some forcing notion $Q\in V[G]$.
Then212121Elementarity is witnessed via the map defined by $A\mapsto
A^{V[G][H]}$ for $A\in\text{{\sf UB}}^{V[G]}$ and the identity on
$H_{\omega_{1}}^{V[G]}$ (See Notation 3.4 for the definition of
$A^{V[G][H]}$).:
$(H_{\omega_{1}}^{V[G]}\cup\text{{\sf
UB}}^{V[G]},\in)\prec(H_{\omega_{1}}^{V[G][H]}\cup\text{{\sf
UB}}^{V[G][H]},\in).$
We observe that ${\mathbf{MAX}(\mathsf{UB})}$ is a form of sharp for the
family of universally Baire sets which holds if $V$ has class many Woodin
cardinals and is a generic extension obtained by collapsing a supercompact
cardinal to become countable (${\mathbf{MAX}(\mathsf{UB})}$ is a weakening of
the conclusion of [15, Thm 3.4.17]). Moreover if ${\mathbf{MAX}(\mathsf{UB})}$
holds in $V$, it remains true in all further set forcing extensions of $V$. It
is open whether ${\mathbf{MAX}(\mathsf{UB})}$ is a direct consequence of
suitable large cardinal axioms.
We now turn to the definition of $(*)\text{-}\mathsf{UB}$, a natural maximal
strengthening of Woodin’s axiom $(*)$. Key to all results of this section is
an analysis of the properties of generic extensions by
$\mathbb{P}_{\mathrm{max}}$ of $L(\text{{\sf UB}})$. In this analysis
${\mathbf{MAX}(\mathsf{UB})}$ is used to argue (among other things) that all
sets of reals definable in $L(\text{{\sf UB}})$ are universally Baire, so that
most of the results established in [14] on the properties of
$\mathbb{P}_{\mathrm{max}}$ for $L(\mathbb{R})$ can be also asserted for
$L(\text{{\sf UB}})$. We will use various forms of Woodin’s axiom $(*)$ each
stating that $\mathbf{NS}_{\omega_{1}}$ is saturated together with the
existence of $\mathbb{P}_{\mathrm{max}}$-filters meeting certain families of
dense subsets of $\mathbb{P}_{\mathrm{max}}$ definable in $L(\text{{\sf
UB}})$. However in this paper we do not define the
$\mathbb{P}_{\mathrm{max}}$-forcing. The reason is that in the proof of all
our results, we will use equivalent characterizations of the proper forms of
$(*)$ which do not mention at all $\mathbb{P}_{\mathrm{max}}$. We will give at
the proper stage the relevant definitions. Meanwhile we assume the reader is
familiar with $\mathbb{P}_{\mathrm{max}}$ or can accept as a blackbox its
existence as a certain forcing notion; our reference on this topic is [14].
###### Definition 5.
Let $\mathcal{A}$ be a family of dense subsets of $\mathbb{P}_{\mathrm{max}}$.
* •
$(*)\text{-}\mathcal{A}$ holds if $\mathbf{NS}_{\omega_{1}}$ is
saturated222222See [15, Section 1.6, pag. 39] for a discussion of saturated
ideals on $\omega_{1}$. and there exists a filter $G$ on
$\mathbb{P}_{\mathrm{max}}$ meeting all the dense sets in $\mathcal{A}$.
* •
$(*)\text{-}\mathsf{UB}$ holds if $\mathbf{NS}_{\omega_{1}}$ is saturated and
there exists an $L(\text{{\sf UB}})$-generic filter $G$ on
$\mathbb{P}_{\mathrm{max}}$.
Woodin’s definition of $(*)$ [14, Def. 7.5] is equivalent to
$(*)\text{-}\mathcal{A}+$_there are class many Woodin cardinals_ for
$\mathcal{A}$ the family of dense subsets of $\mathbb{P}_{\mathrm{max}}$
existing in $L(\mathbb{R})$.
An objection to Thm. 4 is that it subsumes the Platonist standpoint that there
exists a definite universe of sets. At the prize of introducing another bit of
notation, we can prove a version of Thm. 4 which makes perfect sense also to a
formalist and from which we immediately derive Thm. 2 and Corollary 1 for a
certain recursive set of $\in$-formulae $A_{2}$.
###### Notation 4.
__
* •
$\sigma_{\text{{\sf ST}}}$ is the signature containing a predicate symbol
$S_{\phi}$ of arity $n$ for any $\in$-formula $\phi$ with $n$-many free
variables.
* •
$T_{\text{{\rm l-UB}}}$ is the $\sigma_{\text{{\sf ST}}}$-theory given by the
axioms
$\forall x_{1}\dots
x_{n}\,[S_{\psi}(x_{1},\dots,x_{n})\leftrightarrow(\bigwedge_{i=1}^{n}x_{i}\subseteq\omega^{<\omega}\wedge\psi^{L(\text{{\sf
UB}})}(x_{1},\dots,x_{n}))]$
as $\psi$ ranges over the $\in$-formulae.
* •
$\mathsf{ZFC}^{*-}_{\text{{\rm l-UB}}}$ is the
$\sigma_{\omega}=\sigma_{\text{{\sf ST}}}\cup\tau_{\text{{\sf ST}}}$-theory
$\mathsf{ZFC}^{-}_{\text{{\sf ST}}}\cup T_{\text{{\rm l-UB}}}.$
* •
$\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$ is the signature
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\sigma_{\text{{\sf ST}}}$ (recall Notation
3.6).
* •
$\mathsf{ZFC}^{*-}_{\text{{\rm l-UB}},\mathbf{NS}_{\omega_{1}}}$ is the
$\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$-theory
$\mathsf{ZFC}^{-}_{\mathbf{NS}_{\omega_{1}}}\cup T_{\text{{\rm l-UB}}}.$
* •
Accordingly we define $\mathsf{ZFC}^{*}_{\text{{\rm l-UB}}}$,
$\mathsf{ZFC}^{*}_{\text{{\rm l-UB}},\mathbf{NS}_{\omega_{1}}}$.
A key observation is that $\mathsf{ZFC}^{-}_{\text{{\sf ST}}}$,
$\mathsf{ZFC}^{-}_{\mathbf{NS}_{\omega_{1}}}$, $\mathsf{ZFC}^{*-}_{\text{{\rm
l-UB}}}$, $\mathsf{ZFC}^{*-}_{\text{{\rm l-UB}},\mathbf{NS}_{\omega_{1}}}$ are
all _definable_ extension of $\mathsf{ZFC}^{-}$; more precisely: there are
sets $X\subseteq F_{\left\\{\in\right\\}}\times 2$ such that each of the above
theory is of the form $\mathsf{ZFC}^{-}+T_{X}$ according to Def. 3. The same
applies to $\mathsf{ZFC}_{\text{{\sf ST}}}$,
$\mathsf{ZFC}_{\mathbf{NS}_{\omega_{1}}}$, $\mathsf{ZFC}^{*}_{\text{{\rm
l-UB}}}$, $\mathsf{ZFC}^{*}_{\text{{\rm l-UB}},\mathbf{NS}_{\omega_{1}}}$.
###### Theorem 5.
Let $T$ be any $\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$-theory extending
$\mathsf{ZFC}^{*}_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}+{\mathbf{MAX}(\mathsf{UB})}+\text{ there is a
supercompact cardinal and class many Woodin cardinals}$
Then $T$ has a model companion $T^{*}$.
Moreover TFAE for any for any $\Pi_{2}$-sentence $\psi$ for
$\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$:
1. (A)
$T^{*}\vdash\psi$.
2. (B)
$(V[G],\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}^{V[G]})\models\psi^{H_{\omega_{2}}}$
whenever $(V,\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}^{V})\models T$, $V[G]$
is a forcing extension of $V$, and $V[G]\models(*)\text{-}\mathsf{UB}$.
3. (C)
$T$ proves232323$\dot{H}_{\omega_{2}}$ denotes a canonical $P$-name for
$H_{\omega_{2}}$ as computed in generic extension by $P$.
$\exists P\,(P\text{ is a \emph{stationary set preserving} partial order
}\wedge\Vdash_{P}\psi^{\dot{H}_{\omega_{2}}}).$
4. (D)
$T$ proves
$\exists P\,(P\text{ is a partial order
}\wedge\Vdash_{P}\psi^{\dot{H}_{\omega_{2}}}).$
5. (E)
$T$ proves
$L(\text{{\sf
UB}})\models[\mathbb{P}_{\mathrm{max}}\Vdash\psi^{\dot{H}_{\omega_{2}}}].$
6. (F)
If $(V,\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}^{V})\models T$ and $\psi$ is
$\forall x\exists y\,\phi(x,y)$ with $\phi$ quantifier free
$\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$-formula, then for all242424See Def.
5.16 for the notion of honest consistency. $a\in H_{\omega_{2}}^{V}$
$\exists y\phi(a,y)\text{ is \emph{honestly consistent} according to $V$}.$
7. (G)
For any complete theory
$S\supseteq T,$
$S_{\forall}\cup\left\\{\psi\right\\}$ is consistent.
Note that even if $T\models\text{{\sf CH}}$, $\neg\text{{\sf CH}}$ is in
$T^{*}$ (for example by E above). In particular the model companion $T^{*}$ of
$T$ may have models whose theory of $H_{\aleph_{2}}$ is completely unrelated
to that of models of $T$. Moreover recall again that CH is not expressible as
a $\Pi_{1}$-property in $\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$ for $T$: it
is not preserved by forcing, while $T_{\forall}$ is.
The rest of this section is devoted to proof of Theorems 4 and 5.
Crucial to their proof is the recent breakthrough of Asperó and Schindler [2]
establishing that $(*)$-UB follows from $\text{{\sf MM}}^{++}$.
First of all it is convenient to detail more on ${\mathbf{MAX}(\mathsf{UB})}$
and its use in our proofs.
### 5.1. ${\mathbf{MAX}(\mathsf{UB})}$
From now on we will need in several occasions that
${\mathbf{MAX}(\mathsf{UB})}$ holds in $V$ (recall Def. 4). We will always
explicitly state where this assumption is used, hence if a statement does not
mention it in the hypothesis, the assumption is not needed for its thesis.
We will use both properties of ${\mathbf{MAX}(\mathsf{UB})}$ crucially: (1) is
used in the proof of Lemma 5.8; (2) in the proof of Fact 5.10. Similarly they
are essentially used in Remark 5.13. Specifically we will need
${\mathbf{MAX}(\mathsf{UB})}$ to prove that certain subsets of
$H_{\omega_{1}}$ simply definable using an existential formula quantifying
over UB are coded by a universally Baire set, and that this coding is absolute
between generic extensions, i.e. if
$\left\\{x\in H_{\omega_{1}}^{V}:(H_{\omega_{1}}\cup\text{{\sf
UB}},\tau_{\text{{\sf ST}}}^{V})\models\phi(x)\right\\}$
is coded by $A\in\text{{\sf UB}}^{V}$,
$\left\\{x\in H_{\omega_{1}}^{V[G]}:(H_{\omega_{1}}^{V[G]}\cup\text{{\sf
UB}}^{V[G]},\tau_{\text{{\sf ST}}}^{V[G]}))\models\phi(x)\right\\}$
is coded by $A^{V[G]}\in\text{{\sf UB}}^{V[G]}$ for $\phi$ some
$\tau_{\text{{\sf ST}}}$-formula252525Note that the structures
$(H_{\omega_{1}}\cup\text{{\sf UB}},\in)$, $(H_{\omega_{1}}\cup\text{{\sf
UB}},\tau_{\text{{\sf ST}}}^{V})$ have the same algebra of definable sets,
hence we will use one or the other as we deem most convenient, since any set
definable by some formula in one of these structures is also defined by a
possibly different formula in the other. The formulation of
${\mathbf{MAX}(\mathsf{UB})}$ is unaffacted if we choose any of the two
structures as the one for which we predicate it..
It is useful to outline what is the different expressive power of the
structures $(H_{\omega_{1}},\tau_{\text{{\sf ST}}}^{V},A:A\in\text{{\sf
UB}}^{V})$ and $(H_{\omega_{1}}\cup\text{{\sf UB}}^{V},\tau_{\text{{\sf
ST}}}^{V})$. The latter can be seen as a second order extension of
$H_{\omega_{1}}$, where we also allow formulae to quantify over the family of
universally Baire subsets of $2^{\omega}$; in the former quantifiers only
range over elements of $H_{\omega_{1}}$, but we can use the universally Baire
subsets of $H_{\omega_{1}}$ as parameters. This is in exact analogy between
the comprehension scheme for the Morse-Kelley axiomatization of set theory
(where formulae with quantifiers ranging over classes are allowed) and the
comprehension scheme for Gödel-Bernays axiomatization of set theory (where
just formulae using classes as parameters and quantifiers ranging only over
sets are allowed). To appreciate the difference between the two set-up, note
that that the axiom of determinacy for universally Baire sets is expressible
in
$(H_{\omega_{1}}\cup\text{{\sf UB}},\tau_{\text{{\sf ST}}}^{V})$
by the $\tau_{\text{{\sf ST}}}$-sentence
> _For all $A\subseteq 2^{\omega}$ there is a winning strategy for one of the
> players in the game with payoff $A$_,
while in
$(H_{\omega_{1}},\tau_{\text{{\sf ST}}}^{V},A:A\in\text{{\sf UB}}^{V})$
it is expressed by the axiom schema of $\Sigma_{1}$-sentences for
$\tau_{\text{{\sf ST}}}\cup\left\\{A\right\\}$
> _There is a winning strategy for some player in the game with payoff $A$_
as $A$ ranges over the universally Baire sets.
We will crucially use the stronger expressive power of the structure
$(H_{\omega_{1}}\cup\text{{\sf UB}},\tau_{\text{{\sf ST}}})$ to define certain
universally Baire sets as the extension in $(H_{\omega_{1}}\cup\text{{\sf
UB}},\tau_{\text{{\sf ST}}}^{V})$ of lightface $\Sigma_{1}$-properties
(according to the Levy hierarchy); properties which require an existential
quantifier ranging over all universally Baire sets.
### 5.2. A streamline of the proofs of Theorems 4, 5
Let us give a general outline of these proofs before getting into details.
From now on we assume the reader is familiar with the basic theory of
$\mathbb{P}_{\mathrm{max}}$ as exposed in [14].
###### Notation 5.1.
For a given family of universally Baire sets $\mathcal{A}$,
$\tau_{\mathcal{A}}$ is the signature $\tau_{\text{{\sf ST}}}\cup\mathcal{A}$,
$\tau_{\mathcal{A},\mathbf{NS}_{\omega_{1}}}$ is the signature
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}$.
The key point is to prove (just on the basis that
$(V,\in)\models{\mathbf{MAX}(\mathsf{UB})}+(*)\text{-}\mathsf{UB}$) the model
completeness of the $\tau_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}$-theory
of $H_{\omega_{2}}$ assuming $(*)\text{-}\mathsf{UB}$. To do so we use
Robinson’s test and we show the following:
> Assuming ${\mathbf{MAX}(\mathsf{UB})}$ there is a _special_ universally
> Baire set $\bar{D}_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}$ defined by an
> $\in$-formula _(in no parameters)_ relativized to $L(\text{{\sf UB}})$
> coding a family of $\mathbb{P}_{\mathrm{max}}$-preconditions with the
> following fundamental property:
>
> _For any $\tau_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}$-formula
> $\psi(x_{1},\dots,x_{n})$ mentioning the universally Baire predicates
> $B_{1},\dots,B_{k}$, there is an algorithmic procedure which finds a
> _universal_ $\tau_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}$-formula
> $\theta_{\psi}(x_{1},\dots,x_{n})$ mentioning just the universally Baire
> predicates $B_{1},\dots,B_{k},\bar{D}_{\text{{\sf
> UB}},\mathbf{NS}_{\omega_{1}}}$ such that_
>
> $(H_{\omega_{2}}^{L(\text{{\sf
> UB}})[G]},\sigma_{\left\\{B_{1},\dots,B_{k},\bar{D}_{\text{{\sf
> UB}},\mathbf{NS}_{\omega_{1}}}\right\\},\mathbf{NS}_{\omega_{1}}}^{L(\text{{\sf
> UB}})[G]})\models\forall\vec{x}\,(\psi(x_{1},\dots,x_{n})\leftrightarrow\theta_{\psi}(x_{1},\dots,x_{n}))$
>
> _whenever $G$ is $L(\text{{\sf UB}})$-generic for
> $\mathbb{P}_{\mathrm{max}}$._
Moreover the definition of $\bar{D}_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$ and the computation of
$\theta_{\psi}(x_{1},\dots,x_{n})$ from $\psi(x_{1},\dots,x_{n})$ are just
based on the assumption that $(V,\in)$ is a model of
${\mathbf{MAX}(\mathsf{UB})}$, hence can be replicated mutatis-mutandis in any
model of $\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{UB})}$. We will need that
$(V,\in)$ is a model of ${\mathbf{MAX}(\mathsf{UB})}+(*)\text{-}\mathsf{UB}$
just to argue that in $V$ there is an $L(\text{{\sf UB}})$-generic filter $G$
for $\mathbb{P}_{\mathrm{max}}$ such that262626It is this part of our argument
where the result of Asperò and Schindler establishing the consistency of
$(*)\text{-}\mathsf{UB}$ relative to a supercompact is used in an essential
way. We will address again the role of Asperò and Schindler’s result in all
our proofs in some closing remarks. $H_{\omega_{2}}^{L(\text{{\sf
UB}})[G]}=H_{\omega_{2}}^{V}$. Since in all our arguments we will only use
that $(V,\in)$ is a model of ${\mathbf{MAX}(\mathsf{UB})}$ and (in some of
them also of $(*)\text{-}\mathsf{UB}$), we will be in the position to conclude
easily for the truth of Theorem 4 and 5.
We condense the above information in the following:
###### Theorem 5.2.
There is an $\in$-formula $\phi_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}(x)$
in one free variable such that:
1. (1)
$\mathsf{ZFC}^{*}_{\text{{\rm l-UB}}}+{\mathbf{MAX}(\mathsf{UB})}$ proves that
_$S_{\phi_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}}$ is universally Baire_.
2. (2)
Given predicate symbols $B_{1},\dots,B_{k}$, consider the theory
$T_{B_{1},\dots,B_{k}}$ in signature
$\sigma_{\omega}\cup\left\\{B_{1},\dots,B_{k}\right\\}$ extending
$\mathsf{ZFC}^{*}_{\text{{\rm l-UB}}}+{\mathbf{MAX}(\mathsf{UB})}$ by the
axioms:
$B_{j}\text{ is universally Baire}$
for all predicate symbols $B_{1},\dots,B_{k}$.
There is a recursive procedure assigning to any _existential_ formula
$\phi(x_{1},\dots,x_{k})$ for
$\sigma_{\left\\{B_{1},\dots,B_{k}\right\\},\mathbf{NS}_{\omega_{1}}}$ a
_universal_ formula $\theta_{\phi}(x_{1},\dots,x_{k})$ for
$\sigma_{\left\\{B_{1},\dots,B_{k},S_{\phi_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}}\right\\},\mathbf{NS}_{\omega_{1}}}$ such that
$T_{B_{1},\dots,B_{k}}$ proves that
$\mathbb{P}_{\mathrm{max}}\Vdash[(H_{\omega_{2}}^{L(\text{{\sf
UB}})[\dot{G}]},\tau_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}^{L(\text{{\sf
UB}})[\dot{G}]})\models\forall\vec{x}\;(\phi(x_{1},\dots,x_{k})\leftrightarrow\theta_{\phi}(x_{1},\dots,x_{k}))]$
where $\dot{G}\in L(\text{{\sf UB}})$ is the canonical
$\mathbb{P}_{\mathrm{max}}$-name for the generic filter.
### 5.3. Proofs of Thm. 5, and of (1)$\to$(2) of Thm. 4
Theorem 5, (1)$\to$(2) of Theorem 4 are immediate corollaries of the above
theorem combined with Asperò and Schindler’s proof that $\text{{\sf MM}}^{++}$
implies $(*)\text{-}\mathsf{UB}$, and with Theorem 3.
We start with the proof of (1)$\to$(2) of Thm. 4 assuming Thm. 5.2 and Thm. 3:
###### Proof.
Assume $(V,\in)$ models $(*)\text{-}\mathsf{UB}$. Then there is a
$\mathbb{P}_{\mathrm{max}}$-filter $G\in V$ such that
$H_{\omega_{2}}^{L(\text{{\sf UB}})[G]}=H_{\omega_{2}}^{V}$. By Thm. 5.2 and
Robinson’s test, we get that the first order $\tau_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$-theory of $H_{\omega_{2}}^{L(\text{{\sf
UB}})[G]}$ is model complete. By Levy’s absoluteness (Lemma 1),
$H_{\omega_{2}}^{L(\text{{\sf UB}})[G]}$ is a $\Sigma_{1}$-elementary
substructure of $V$ also according to the signature $\tau_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$. We conclude (by Thm. 1.19), since the two
theories share the same $\Pi_{1}$-fragment. ∎
The proof of the converse implication requires more information on
$\bar{D}_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}$ then what is conveyed in
Thm. 5.2. We defer it to a later stage.
We now prove Thm. 5:
###### Proof.
Let $T^{*}_{\text{{\rm l-UB}},\mathbf{NS}_{\omega_{1}}}$ be the theory given
by the $\Pi_{2}$-sentences $\psi$ for
$\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$ which hold in
$H_{\omega_{2}}^{V[G]}$ whenever $(V,\in)$ models
$\mathsf{ZFC}^{*}_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}+{\mathbf{MAX}(\mathsf{UB})}+\text{ there is a
supercompact cardinal and class many Woodin cardinals}$
and $V[G]$ is a generic extension of $(V,\in)$ by some forcing such that
$V[G]\models(*)\text{-}\mathsf{UB}$.
This theory is consistent: by Schindler and Asperò’s result [2]
$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{UB})}+\text{{\sf MM}}^{++}+\emph{there are
class many Woodin cardinals}$
implies $(*)\text{-}\mathsf{UB}$; $\text{{\sf MM}}^{++}$ is forcible over a
model of $\mathsf{ZFC}+$_there is a supercompact_.
By Thm. 5.2 and Robinson’s test, $T^{*}_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}$ is a model complete theory.
Given a $\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$-theory $T$ extending
$\mathsf{ZFC}^{*}_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}+{\mathbf{MAX}(\mathsf{UB})}+\emph{there is a
supercompact cardinal},$
let
$T^{*}=\left\\{\phi:(V[G],\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}^{V[G]})\models(*)\text{-}\mathsf{UB}+\phi^{H_{\omega_{2}}^{V[G]}},\,(V,\sigma_{\omega,\mathbf{NS}_{\omega_{1}}})\models
T\right\\}.$
We start showing that $T$ and $T^{*}$ satisfy the assumptions of Lemma 1.21.
This immediately gives A$\Longleftrightarrow$G for $T$ and $T^{*}$.
We must show:
* •
$T^{*}$ is model complete.
* •
$T^{*}$ is the model companion of $T$.
* •
For any universal sentence $\theta$, $T+\theta$ is consistent if and only if
so is and $T^{*}+\theta$.
First of all $T^{*}$ is model complete, since it extends $T^{*}_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}$: if $(V,\in)\models T$ and $G$ is such that
$(V[G],\in)\models\text{{\sf MM}}^{++}$, then
$\mathsf{ZFC}^{*}_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}+{\mathbf{MAX}(\mathsf{UB})}+(*)\text{-}\mathsf{UB}+\text{there
are class many Woodin cardinals}.$
holds in $V[G]$ by [2], hence $H_{\omega_{2}}^{V[G]}\models T^{*}_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}$.
We now show that $T^{*}_{\forall}=T_{\forall}$, i.e. that $T^{*}$ is the model
companion of $T$.
Fix a universal $\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$-sentence $\theta$.
Assume $T\vdash\theta$. Fix $V$ a model of $T$. Let $G$ be $V$-generic for
some forcing such that $V[G]\models(*)\text{-}\mathsf{UB}$. By Thm. 3
$V[G]\models\theta$, and by Levy absoluteness
$H_{\omega_{2}}^{V[G]}\models\theta$. Since this argument can be repeated for
all models $V$ of $T$, we get that $\theta\in T^{*}$ (by definition of
$T^{*}$).
The converse implication holds by a similar argument which appeals with the
obvious variations to Levy absoluteness and to Thm. 3 (i.e. we go backward
from $H_{\omega_{2}}^{V[G]}$ to $V$ for any model $V$ of $T$ and any forcing
extension $V[G]$ of $V$ which models $(*)\text{-}\mathsf{UB}$).
Again with the same recipe described above we can prove that for any universal
sentence $\theta$, $T+\theta$ is consistent if and only if so is and
$T^{*}+\theta$. We leave the details to the reader.
We are left with the proof of the remaining equivalence between A, B, C, D, E,
F, G.
A$\Longrightarrow$B:
By definition of $T^{*}$.
B$\Longrightarrow$C:
Given a $\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$-model
$(V,\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}^{V})$ of $T$, by the results of
[8], we can find a stationary set preserving forcing extension $V[G]$ of $V$
which models $\text{{\sf MM}}^{++}$. By the key result of Asperó and Schindler
[2] $V[G]\models(*)\text{-}\mathsf{UB}$. By B
$(V[G],\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}^{V[G]})$ models
$\psi^{H_{\omega_{2}}^{V[G]}}$, and we are done.
C$\Longrightarrow$D:
Trivial.
D$\Longrightarrow$E:
By272727${\mathbf{MAX}(\mathsf{UB})}$ implies that the same assumption used in
the cited theorem for $L(\mathbb{R})$ holds for $L(\text{{\sf UB}})$. [14,
Thm. 7.3], if some $P$ forces $\psi^{\dot{H}_{\omega_{2}}}$, we get that
$L(\text{{\sf
UB}})\models\mathbb{P}_{\mathrm{max}}\Vdash\psi^{\dot{H}_{\omega_{2}}}$.
E$\Longleftrightarrow$F:
By [1, Thm. 2.7, Thm. 2.8].
E$\Longrightarrow$G:
Given some complete $S\supseteq T$, and a model $\mathcal{M}$ of $S$, find
$\mathcal{N}$ forcing extension of $\mathcal{M}$ which models
$\psi^{H_{\omega_{2}}^{\mathcal{N}}}$. By Thm. 3 and Levy’s absoluteness Lemma
1, $H_{\omega_{2}}^{\mathcal{N}}\models\psi+S_{\forall}$, and we are done.
∎
### 5.4. Proof of Thm. 5.2
The rest of this section is devoted to the proof of Thm. 5.2.
What we will do first is to sketch a different proof of Thm. 4.4. This will
give us the key intuition on how to define $\bar{D}_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$.
###### Notation 5.3.
From now on given a family of universally Baire sets $\mathcal{A}$, we let
$\tau_{\mathcal{A}}=\tau_{\text{{\sf ST}}}\cup\mathcal{A}$ in which allsymbols
in $\mathcal{A}$ are interpreted as predicate symbols of the appropriate
arity.
#### 5.4.1. A different proof of Thm. 4.4.
Let $M$ be a countable transitive model of $\mathsf{ZFC}+$_there are class
many Woodin cardinals_. Then it will have its own version of Thm. 4.4. In
particular it will model that the theory of
$(H_{\omega_{1}}^{M},\tau_{\text{{\sf ST}}}^{M},\text{{\sf UB}}^{M})$ is model
complete, and also that $\text{{\sf UB}}^{M}$ is an
$H_{\omega_{1}}$-closed282828Recall Def. 4.6. family of universally Baire sets
in $M$.
Now assume that there is a countable family $\text{{\sf UB}}_{M}$ of
universally Baire sets in $V$ which is $H_{\omega_{1}}$-closed in $V$ and is
such that $\text{{\sf UB}}^{M}=\left\\{B\cap M:B\in\text{{\sf
UB}}_{M}\right\\}$. Then
$(H_{\omega_{1}}^{M},\tau_{\text{{\sf ST}}}^{M},\text{{\sf
UB}}^{M})=(H_{\omega_{1}}^{M},\tau_{\text{{\sf ST}}}^{M},\left\\{B\cap
M:B\in\text{{\sf
UB}}_{M}\right\\})\sqsubseteq(H_{\omega_{1}}^{V},\tau_{\text{{\sf
UB}}_{M}}^{V})$
But $\text{{\sf UB}}_{M}$ being $H_{\omega_{1}}$-closed in $V$ entails that
the first order theory of $(H_{\omega_{1}}^{V},\tau_{\text{{\sf
UB}}_{M}}^{V})$ is model complete. In particular if
$(H_{\omega_{1}}^{M},\tau_{\text{{\sf UB}}_{M}}^{M})$ and
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}_{M}}^{V})$ are elementarily
equivalent, then
$(H_{\omega_{1}}^{M},\tau_{\text{{\sf
UB}}_{M}}^{M})\prec(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}_{M}}^{V}).$
The setup described above is quite easy to realize (for example $M$ could the
transitive collapse of some countable $X\prec V_{\theta}$ for some large
enough $\theta$); in particular for any $a\in H_{\omega_{1}}$ and
$B_{1},\dots,B_{k}\in\text{{\sf UB}}$, we can find $M$ countable transitive
model of a suitable fragment of $\mathsf{ZFC}$ with $a\in H_{\omega_{1}}^{M}$
and $\text{{\sf UB}}_{M}\supseteq\left\\{B_{1},\dots,B_{k}\right\\}$ countable
and $H_{\omega_{1}}$-closed family of UB-sets in $V$, such that:
* •
$\text{{\sf UB}}^{M}=\left\\{B\cap M:B\in\text{{\sf UB}}_{M}\right\\}$;
* •
the first order theory $T_{\text{{\sf UB}}_{M}}$ of
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}_{M}}^{V})$ is model complete;
* •
$(H_{\omega_{1}}^{M},\tau_{\text{{\sf ST}}}^{M},\left\\{B\cap M:B\in\text{{\sf
UB}}_{M}\right\\})$ models $T_{\text{{\sf UB}}_{M}}$.
Letting $B_{M}=\prod\text{{\sf UB}}_{M}$, $(H_{\omega_{1}}\cup\text{{\sf
UB}},\in)$ is able to compute correctly whether $B_{M}$ encodes a set
$\text{{\sf UB}}_{M}$ such that the pair $(\text{{\sf UB}}_{M},M)$ satisfies
the above list of requirements; here we use crucially the fact that being a
model complete theory is a $\Delta_{0}$-property, and also that it is possible
to encode the structure $(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}_{M}}^{V})$
in a single universally Baire set292929See Def. 2.3 for the definition of
$\mathsf{WFE}_{\omega}$ and $\text{{\rm Cod}}_{\omega}$. (for example
$\mathsf{WFE}_{\omega}\times B_{M}$).
In particular $(H_{\omega_{1}}\cup\text{{\sf UB}},\in)$ correctly computes the
set $D_{\text{{\sf UB}}}$ of $M\in H_{\omega_{1}}$ such that there exists a
universally Baire set $B_{M}=\prod\text{{\sf UB}}_{M}$ with the property that
the pair $(M,\text{{\sf UB}}_{M})$ realizes the above set of requirements. By
${\mathbf{MAX}(\mathsf{UB})}$, $\bar{D}_{\text{{\sf UB}}}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\text{{\sf UB}}}]$ is a universally Baire set
$\bar{D}_{\text{{\sf UB}}}$.
Note moreover that $\bar{D}_{\text{{\sf UB}}}$ is defined by a $\in$-formula
$\phi_{\text{{\sf UB}}}(x)$ in no extra parameters; in particular for any
model $\mathcal{W}=(W,E)$ of $\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{UB})}$, we
can define $\bar{D}_{\text{{\sf UB}}}$ in $\mathcal{W}$ and all its properties
outlined above will hold relativized to $\mathcal{W}$.
For fixed universally Baire sets $B_{1},\dots,B_{k}$ the set $D_{\text{{\sf
UB}},B_{1},\dots,B_{k}}$ of $M\in D_{\text{{\sf UB}}}$ such that there is a
witness $\text{{\sf UB}}_{M}$ of $M\in D_{\text{{\sf UB}}}$ with
$B_{1},\dots,B_{k}\in\text{{\sf UB}}_{M}$ is also definable in
$(H_{\omega_{1}}\cup\text{{\sf UB}},\in)$
in parameters $B_{1},\dots,B_{k}$. Hence by ${\mathbf{MAX}(\mathsf{UB})}$
$\text{{\rm Cod}}_{\omega}^{-1}[D_{\text{{\sf
UB}},B_{1},\dots,B_{k}}]=\bar{D}_{\text{{\sf UB}},B_{1},\dots,B_{k}}$ is
universally Baire (note as well that $\bar{D}_{\text{{\sf
UB}},B_{1},\dots,B_{k}}$ belongs to any $L(\text{{\sf UB}})$-closed family
$\mathcal{A}$ containing $B_{1},\dots,B_{k}$).
Now take any $\Sigma_{1}$-formula $\phi(\vec{x})$ for $\tau_{\text{{\sf UB}}}$
mentioning just the universally Baire predicates $B_{1},\dots,B_{k}$. It
doesn’t take long to realize that for all $\vec{a}$ in $H_{\omega_{1}}$
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}}^{V})\models\phi(\vec{a})$
if and only if
$(H_{\omega_{1}}^{M},\tau_{\text{{\sf
UB}}_{M}}^{M})\models\phi(\vec{a})\text{\emph{ for all $M\in D_{\text{{\sf
UB}},B_{1},\dots,B_{k}}$ with $\vec{a}\in H_{\omega_{1}}^{M}$}. }$
But $\bar{D}_{\text{{\sf UB}},B_{1},\dots,B_{k}}$ is universally Baire, so the
above can be formulated also as:
$\forall r\in\bar{D}_{\text{{\sf UB}},B_{1},\dots,B_{k}}[\vec{a}\in
H_{\omega_{1}}^{\text{{\rm Cod}}(r)}\rightarrow(H_{\omega_{1}}^{\text{{\rm
Cod}}(r)},\tau_{\text{{\sf UB}}_{\text{{\rm Cod}}(r)}}^{\text{{\rm
Cod}}(r)})\models\phi(\vec{a})].$
The latter is a $\Pi_{1}$-sentence in the universally Baire parameter
$\bar{D}_{\text{{\sf UB}},B_{1},\dots,B_{k}}$.
This is exactly a proof that Robinson’s test applies to the $\tau_{\text{{\sf
UB}}^{V}}$-first order theory of $H_{\omega_{1}}^{V}$ assuming
${\mathbf{MAX}(\mathsf{UB})}$; i.e. we have briefly sketched a different (and
much more convoluted) proof of the conclusion of Thm. 4.4 (using as hypothesis
Thm. 4.4 itself). What we gained however is an insight on how to prove Theorem
5.2.
We will consider the set $D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}$ of
$M\in D_{\text{{\sf UB}}}$ such that:
* •
$(M,\mathbf{NS}_{\omega_{1}}^{M})$ is a
$\mathbb{P}_{\mathrm{max}}$-precondition which is $B$-iterable for all
$B\in\text{{\sf UB}}_{M}$ (according to [14, Def. 4.1]);
* •
$j_{0\omega_{1}}$ is a $\Sigma_{1}$-elementary embedding of
$H_{\omega_{2}}^{M}$ into $H_{\omega_{2}}^{V}$ for $\tau_{\text{{\sf
UB}}_{M},\mathbf{NS}_{\omega_{1}}}$ whenever
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\omega_{1}\right\\}$
is an iteration of $M$ with
$j_{0\omega_{1}}(\mathbf{NS}_{\omega_{1}}^{M})=\mathbf{NS}_{\omega_{1}}^{V}\cap
j_{0\omega_{1}}(H_{\omega_{2}}^{M})$.
It will take a certain effort to prove that assuming $(*)$-UB:
* •
for any $A\in H_{\omega_{2}}$ and $B\in\text{{\sf UB}}$, we can find $M\in
D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}$ with $B\in\text{{\sf UB}}_{M}$,
$a\in H_{\omega_{2}}^{M}$, and an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\omega_{1}\right\\}$
of $M$ with
$j_{0\omega_{1}}(\mathbf{NS}_{\omega_{1}})=\mathbf{NS}_{\omega_{1}}^{V}\cap
j_{0\omega_{1}}(H_{\omega_{2}}^{M})$ such that $j_{0\omega_{1}}(a)=A$.
* •
$D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}$ is correctly computable in
$(H_{\omega_{1}}\cup\text{{\sf UB}},\in)$.
But this effort will pay off since we will then be able to prove the model
completeness of the theory
$(H_{\omega_{2}},\tau_{\mathbf{NS}_{\omega_{1}}}^{V}\cup\text{{\sf UB}}^{V})$
using Robinson’s test with $\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}]$ in the
place of $\bar{D}_{\text{{\sf UB}}}$ and replicating in the new setting what
was sketched before for $(H_{\omega_{1}},\tau_{\text{{\sf UB}}^{V}}^{V})$.
We now get into the details.
#### 5.4.2. UB-correct models
###### Notation 5.4.
Given a countable family $\mathcal{A}=\left\\{B_{n}:n\in\omega\right\\}$ of
universally Baire sets with each $B_{n}\subseteq(2^{\omega})^{k_{n}}$, we say
that
$B_{\mathcal{A}}=\prod_{n\in\omega}B_{n}\subseteq\prod_{n}(2^{\omega})^{k_{n}}$
is a code for $\left\\{B_{n}:n\in\omega\right\\}$.
Clearly $B_{\mathcal{A}}$ is a universally Baire subset of the Polish space
$\prod_{n}(2^{\omega})^{k_{n}}$.
###### Definition 5.5.
$T_{\mathsf{UB}}$ is the $\in$-theory of
$(H_{\omega_{1}},\tau_{\text{{\sf UB}}}).$
A transitive model of $\mathsf{ZFC}$ $(M,\in)$ is $\mathsf{UB}$-correct if
there is an $H_{\omega_{1}}$-closed (in $V$) family $\mathsf{UB}_{M}$ of
universally Baire sets in $V$ such that:
* •
The map
$\displaystyle\Theta_{M}:$ $\displaystyle\text{{\sf UB}}_{M}\to M$
$\displaystyle A\mapsto A\cap M$
is injective.
* •
$(M,\in)$ models that $\left\\{A\cap M:A\in\mathsf{UB}_{M}\right\\}$ is the
family of universally Baire subsets of $M$.
* •
Letting $T_{\text{{\sf UB}}_{M}}$ be the theory of
$(H_{\omega_{1}},\tau_{\mathsf{ST}}^{V},\mathsf{UB}_{M})$
$(H_{\omega_{1}}^{M},\tau_{\mathsf{ST}}^{M},A\cap
M:A\in\mathsf{UB}_{M})\models T_{\text{{\sf UB}}_{M}}.$
* •
If $M$ is countable, $M$ is $A$-iterable for all $A\in\mathsf{UB}_{M}$.
Remark (by Thm. 4.7) that if $M$ is UB-correct, $T_{\text{{\sf UB}}_{M}}$ is
model complete, since $\text{{\sf UB}}_{M}$ is (in $V$) a
$H_{\omega_{1}}$-closed family of universally Baire sets.
###### Notation 5.6.
$D_{\text{{\sf UB}}}$ denotes the set of countable $\mathsf{UB}$-correct $M$;
$\bar{D}_{\text{{\sf UB}}}=\text{{\rm Cod}}_{\omega}^{-1}[D_{\text{{\sf
UB}}}]$.
For each $M$ $\text{{\sf UB}}_{M}$ is a witness that $M\in D_{\text{{\sf
UB}}}$ and $B_{\text{{\sf UB}}_{M}}=\prod\text{{\sf UB}}_{M}$ is a universally
Baire coding this witness303030The Fact below shows that the map
$M\mapsto(\text{{\sf UB}}_{M},B_{\text{{\sf UB}}_{M}})$, can be chosen in
$L(\text{{\sf UB}})$..
For universally Baire sets $B_{1},\dots,B_{k}$, $E_{\text{{\sf
UB}},B_{1},\dots,B_{k}}$ denotes the set of $M\in D_{\text{{\sf UB}}}$ with
$B_{1},\dots,B_{k}\in\text{{\sf UB}}_{M}$ for some witness $\text{{\sf
UB}}_{M}$ that $M\in D_{\text{{\sf UB}}}$; $\bar{E}_{\text{{\sf
UB}},B_{1},\dots,B_{k}}=\text{{\rm Cod}}_{\omega}^{-1}[E_{\text{{\sf
UB}},B_{1},\dots,B_{k}}]$.
###### Fact 5.7.
$(V,\in)$ models $M$_is countable and UB-correct as witnessed by $\text{{\sf
UB}}_{M}$_ if and only if so does $(H_{\omega_{1}}\cup{\text{{\sf UB}}},\in)$.
Consequently the set $D_{\mathsf{UB}}$ of countable $\mathsf{UB}$-correct $M$
is properly computed in $(H_{\omega_{1}}\cup\text{{\sf UB}},\in)$.
Therefore assuming ${\mathbf{MAX}(\mathsf{UB})}$
$\bar{D}_{\text{{\sf UB}}}=\mathrm{Cod}^{-1}[D_{\mathsf{UB}}]$
is universally Baire.
Moreover there is in $L(\text{{\sf UB}})$ a definable map $M\mapsto\text{{\sf
UB}}_{M}$ assigning to each $M\in D_{\text{{\sf UB}}}$ a countable family
$\text{{\sf UB}}_{M}$ witnessing it.
The same holds for $\bar{E}_{\text{{\sf UB}},B_{1},\dots,B_{k}}$ for given
universally Baire sets $B_{1},\dots,B_{k}$.
###### Proof.
The first part follows almost immediately by the definitions, since the
assertion in parameters $B,M$:
> _ $B=\prod_{n\in\omega}B_{n}$ codes a $H_{\omega_{1}}$-closed family
> $\text{{\sf UB}}_{M}=\left\\{B_{n}:n\in\omega\right\\}$ of sets such that_
>
> * •
>
> $M$_is_ $A$-iterable for all $A\in\text{{\sf UB}}_{M}$,
>
> * •
>
> $M$_models that_ $\left\\{A\cap M:A\in\text{{\sf UB}}_{M}\right\\}$ is its
> family of universally Baire sets and is $H_{\omega_{1}}$-closed,
>
> * •
>
> $(H_{\omega_{1}}^{M},\tau_{\text{{\sf ST}}}^{M},\left\\{A\cap
> M:A\in\mathsf{UB}_{M}\right\\})$_models_ $T_{\text{{\sf UB}}_{M}}$.
>
>
gets the same truth value in $(V,\in)$ and in $(H_{\omega_{1}}\cup\text{{\sf
UB}},\in)$.
We conclude that $D_{\mathsf{UB}}$ has the same extension in $(V,\in)$ and in
$(H_{\omega_{1}}\cup\text{{\sf UB}},\in)$. By ${\mathbf{MAX}(\mathsf{UB})}$
$\bar{D}_{\text{{\sf UB}}}$ is universally Baire.
The existence of class many Woodin cardinals grants that we can always
find313131For example by [12, Thm. 36.9] and [15, Thm. 3.3.14, Thm. 3.3.19]. a
universally Baire uniformization of the universally Baire relation on
$\bar{D}_{\text{{\sf UB}}}\times 2^{\omega}$ given by the pairs $\langle
r,B\rangle$ such that $B=\prod\left\\{B_{n}:n\in\omega\right\\}$ witnesses
$\text{{\rm Cod}}_{\omega}(r)\in D_{\text{{\sf UB}}}$ .
The same argument can be replicated for $\bar{E}_{\text{{\sf
UB}},B_{1},\dots,B_{k}}$. ∎
###### Lemma 5.8.
Assume $\mathbf{NS}_{\omega_{1}}$ is precipitous and there are class many
Woodin cardinals in $V$. Let $\delta$ be an inaccessible cardinal in $V$ and
$G$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$. Then $V_{\delta}$
is $\text{{\sf UB}}^{V[G]}$-correct in $V[G]$ as witnessed by
$\left\\{B^{V[G]}:B\in\text{{\sf UB}}^{V}\right\\}$.
###### Proof.
Let in $V$ $\left\\{(T_{A},S_{A}):A\in\text{{\sf UB}}^{V}\right\\}$ be an
enumeration of pairs of trees $S_{A},U_{A}$ on $\omega\times\gamma$ for a
large enough inaccessible $\gamma>\delta$ such that $T_{A},S_{A}$ projects to
complements in $V[G]$ and $A$ is the projection of $T$. Then $A^{V[G]}$ is
correctly computed as the projection of $T_{A}$ in $V[G]$ for any
$A\in\text{{\sf UB}}^{V}$.
By Thm. 4.7
$(H_{\omega_{1}}^{V},\tau_{\mathsf{ST}}^{V},\mathsf{UB}^{V})\prec(H_{\omega_{1}}^{V[G]},\tau_{\mathsf{ST}}^{V[G]},A^{V[G]}:A\in\mathsf{UB}^{V}),$
$\left\\{A^{V[G]}:A\in\mathsf{UB}^{V}\right\\}$ is a $H_{\omega_{1}}$-closed
family of universally Baire sets in $V[G]$, and $T_{\text{{\sf UB}}^{V}}$ is
also the theory of
$(H_{\omega_{1}}^{V[G]},\tau_{\mathsf{ST}}^{V[G]},A^{V[G]}:A\in\mathsf{UB}^{V})$.
To conclude that $\left\\{A^{V[G]}:A\in\mathsf{UB}^{V}\right\\}$ witnesses in
$V[G]$ that $V_{\delta}$ is $\text{{\sf UB}}^{V[G]}$-correct in $V[G]$ it
remains to argue that $V_{\delta}$ is $B^{V[G]}$-iterable for any
$B\in\text{{\sf UB}}^{V}$.
Let $\mathcal{J}$ be any iteration of $V_{\delta}$ in $V[G]$. Then by standard
results on iterations (see [14, Lemma 1.5, Lemma 1.6]) $\mathcal{J}$ extends
uniquely to an iteration $\bar{\mathcal{J}}$ of $V$ in $V[G]$ such that
* •
$\bar{j}_{\alpha\beta}$ is a proper extension of $j_{\alpha\beta}$ for all
$\alpha\leq\beta\leq\gamma$ (i.e. letting
$\bar{V}_{\alpha}=\bar{j}_{0\alpha}(V)$, we have that
$j_{0\alpha}(V_{\delta})$ is the rank initial segments of elements of
$\bar{V}_{\alpha}$ of rank less than $\bar{j}_{0\alpha}(\delta)$).
* •
$\bar{\mathcal{J}}$ is a well defined iteration of transitive structures.
In particular this shows that $V_{\delta}$ is iterable in $V[G]$.
Now fix $B\in\text{{\sf UB}}^{V}$. We must argue that
$j_{0\alpha}(B)=B^{V[G]}\cap\bar{j}_{0\alpha}(V)$. To simplfy notation we
assume $B\subseteq 2^{\omega}$. Let $(T_{B},S_{B})$ be the pair of trees
selected in $V$ to define $B^{V[G]}$.
Then
$\bar{j}_{0\alpha}(V)\models(\bar{j}_{0\alpha}(T_{B}),\bar{j}_{0\alpha}(S_{B}))$
projects to complements; clearly
$\bar{j}_{0\alpha}[T_{B}]\subseteq\bar{j}_{0\alpha}(T_{B})$,
$\bar{j}_{0\alpha}[S_{B}]\subseteq\bar{j}_{0\alpha}(S_{B})$. Let
$p:(\gamma\times 2)^{\omega}\to 2^{\omega}$ be the projection map.
This gives that
$B^{V[G]}\cap\bar{j}_{0\alpha}(V)=p[[T_{B}]]\cap\bar{j}_{0\alpha}(V)=p[[\bar{j}_{0\alpha}[T_{B}]]]\cap\bar{j}_{0\alpha}(V)\subseteq
p[[\bar{j}_{0\alpha}(T_{B})]]\cap\bar{j}_{0\alpha}(V)=\bar{j}_{0\alpha}(B).$
Similarly
$((2^{\omega})^{V[G]}\setminus
B^{V[G]})\cap\bar{j}_{0\alpha}(V)=p[[S_{B}]]\cap\bar{j}_{0\alpha}(V)\subseteq
p[[\bar{j}_{0\alpha}(S_{B})]]\cap\bar{j}_{0\alpha}(V)=\bar{j}_{0\alpha}((2^{\omega})^{V}\setminus
B).$
By elementarity
$\bar{j}_{0\alpha}((2^{\omega})^{V}\setminus
B)\cup\bar{j}_{0\alpha}(B)=(2^{\omega})\cap\bar{j}_{0\alpha}(V).$
These three conditions can be met only if
$B^{V[G]}\cap\bar{j}_{0\alpha}(V)=\bar{j}_{0\alpha}(B).$
Since $\mathcal{J}$ and $B$ were chosen arbitrarily, we conclude that
$V_{\delta}$ is $B^{V[G]}$-iterable in $V[G]$ for all $B\in\text{{\sf
UB}}^{V}$.
Hence $V_{\delta}$ is $\text{{\sf UB}}^{V[G]}$-correct in $V[G]$ as witnessed
by $\left\\{A^{V[G]}:A\in\text{{\sf UB}}^{V}\right\\}$. ∎
###### Definition 5.9.
Given $M,N$ iterable structures, $M\geq N$ if $M\in(H_{\omega_{1}})^{N}$ and
there is an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma=(\omega_{1})^{N}\right\\}$
of $M$ with $\mathcal{J}\in N$ such that
$\mathbf{NS}_{\gamma}^{M_{\gamma}}=\mathbf{NS}_{\gamma}^{N}\cap M_{\gamma}.$
###### Fact 5.10.
$({\mathbf{MAX}(\mathsf{UB})})$ Assume $\mathbf{NS}_{\omega_{1}}$ is
precipitous and ${\mathbf{MAX}(\mathsf{UB})}$ holds. Then for any iterable $M$
and $B_{1},\dots,B_{k}\in\text{{\sf UB}}$, there is an UB-correct $N\geq M$
with $B_{1},\dots,B_{k}\in\text{{\sf UB}}_{N}$.
###### Proof.
The assumptions grant that whenever $G$ is
$\operatorname{Coll}(\omega,\delta)$-generic for $V$, in $V[G]$ $V_{\delta}$
is $\text{{\sf UB}}^{V[G]}$-correct in $V[G]$ (i.e. Lemma 5.8).
By [14, Lemma 2.8], for any iterable $M\in H_{\omega_{1}}^{V}$ there is in $V$
an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\omega_{1}^{V}\right\\}$
of $M$ such that $\mathbf{NS}_{\omega_{1}}^{V}\cap
M_{\omega_{1}}=\mathbf{NS}_{\omega_{1}}^{M_{\omega_{1}}}$.
By ${\mathbf{MAX}(\mathsf{UB})}$
$(H_{\omega_{1}}^{V}\cup\text{{\sf
UB}}^{V},\in)\prec(H_{\omega_{1}}^{V[G]}\cup\text{{\sf UB}}^{V[G]},\in).$
Therefore we have that in $V[G]$ $\bar{E}_{\text{{\sf
UB}},B_{1},\dots,B_{k}}^{V[G]}$ is exactly $\bar{E}_{\text{{\sf
UB}},B_{1}^{V[G]},\dots,B_{k}^{V[G]}}$.
Hence for each iterable $M\in H_{\omega_{1}}^{V}$ and $B\in\text{{\sf
UB}}^{V}$
$(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf
UB}}^{V}}^{V[G]})\models\exists\,N\geq M\text{ $\text{{\sf
UB}}^{V[G]}$-correct with $B^{V[G]}$ in $\text{{\sf UB}}_{N}$},$
as witnessed by $N=V_{\delta}$, i.e.
$(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf
UB}}^{V}}^{V[G]})\models\exists\,N\geq M\,(\bar{E}_{\text{{\sf
UB}},B_{1},\dots,B_{k}}^{V[G]}(N)).$
Since
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf
UB}}^{V}}^{V})\prec(H_{\omega_{1}}^{V[G]},\tau_{\text{{\sf UB}}^{V}}^{V[G]}),$
we get that for every iterable $M\in H_{\omega_{1}}$ and $B\in\text{{\sf
UB}}^{V}$
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}^{V}}^{V})\models\exists\,N\geq
M\,(\bar{E}_{\text{{\sf UB}},B_{1},\dots,B_{k}}(N)).$
The conclusion follows. ∎
###### Lemma 5.11.
$({\mathbf{MAX}(\mathsf{UB})})$
Let $M\geq N$ be both UB-correct structures, with $\text{{\sf UB}}_{N}$ a
witness of $N$ being UB-correct such that $\bar{D}_{\mathsf{UB}}\in\text{{\sf
UB}}_{N}$. Then
$(H_{\omega_{1}}^{M},\tau_{\mathsf{ST}}^{M},A\cap M:A\in\text{{\sf
UB}}_{M})\prec(H_{\omega_{1}}^{N},\tau_{\mathsf{ST}}^{N},A\cap
N:A\in\text{{\sf UB}}_{M}).$
###### Proof.
Since $N\leq M$, and $N$ is UB-correct with
$\bar{D}_{\mathsf{UB}}\in\text{{\sf UB}}_{N}$ we get that
$(H_{\omega_{1}}^{N},\tau_{\text{{\sf UB}}_{N}}^{N})\models M\in
D_{\mathsf{UB}}\cap N=\text{{\rm Cod}}[\bar{D}_{\mathsf{UB}}\cap N],$
since
$(H_{\omega_{1}}^{N},\tau_{\text{{\sf
UB}}_{N}}^{N})\prec(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}_{N}}^{V})$
and
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}_{N}}^{V})\models M\in
D_{\mathsf{UB}}=\text{{\rm Cod}}[\bar{D}_{\mathsf{UB}}].$
Therefore $N$ models that there is a countable set $\text{{\sf
UB}}_{M}^{N}=\left\\{B_{n}^{N}:n\in\omega\right\\}\in N$ coded by the
universally Baire set in $N$ $B_{\text{{\sf
UB}}_{M}}^{N}=\prod_{n\in\omega}B_{n}^{N}$ such that $\left\\{A\cap
M:A\in\text{{\sf UB}}_{M}^{N}\right\\}\in M$ defines the family of universally
Baire sets according to $M$, and such that $N$ models that $M$ is $B^{N}$
iterable for all $B^{N}\in\text{{\sf UB}}_{M}^{N}$. Now $N$ models that
$\prod_{n\in\omega}B_{n}^{N}$
is a universally Baire set on the appropriate product space. Therefore there
is $B\in\text{{\sf UB}}_{N}$ such that $B\cap N=\prod_{n\in\omega}B_{n}^{N}$.
Clearly $\text{{\sf UB}}_{M}^{N}$ is computable from $B\cap N$. Since
$(H_{\omega_{1}}^{N},\tau_{\text{{\sf
UB}}_{N}}^{N})\prec(H_{\omega_{1}}^{V},\tau_{\text{{\sf UB}}_{N}}^{V}).$
we conclude that in $V$ $B=\prod_{n\in\omega}B_{n}$ codes a set $\text{{\sf
UB}}_{M}=\left\\{B_{n}:n\in\omega\right\\}$ witnessing that $M$ is UB-correct.
This gives that $\text{{\sf UB}}_{M}\subseteq\text{{\sf UB}}_{N}$.
Therefore $(H_{\omega_{1}}^{N},\tau_{\text{{\sf UB}}_{M}}^{N})$ is also a
model of $T_{\text{{\sf UB}}_{M}}$. By model completeness of $T_{\text{{\sf
UB}}_{M}}$ we conclude that
$(H_{\omega_{1}}^{M},\tau_{\text{{\sf
UB}}_{M}}^{M})\prec(H_{\omega_{1}}^{N},\tau_{\text{{\sf UB}}_{M}}^{N}),$
as was to be shown. ∎
### 5.5. Three characterizations of $(*)$-UB
Recall that for a family $\mathcal{A}$ of universally Baire sets
$\tau_{\mathcal{A},\mathbf{NS}_{\omega_{1}}}=\tau_{\omega_{1}}\cup\mathcal{A}$.
###### Definition 5.12.
For a UB-correct $M$ with witness $\text{{\sf UB}}_{M}$,
$T_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}_{M}}$ is the $\tau_{\text{{\sf
UB}}_{M},\mathbf{NS}_{\omega_{1}}}$-theory of $H_{\omega_{2}}^{M}$.
A UB-correct $M$ is _$(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}})$ -ec_ if
$(M,\in)$ models that $\mathbf{NS}_{\omega_{1}}$ is precipitous and there is a
witness $\text{{\sf UB}}_{M}$ that $M$ is UB-correct with the following
property:
> Assume an iterable $N\geq M$ is UB-correct with witness $\text{{\sf
> UB}}_{N}$ such that $B_{\text{{\sf UB}}_{M}}\in\text{{\sf UB}}_{N}$ (so that
> $\text{{\sf UB}}_{M}\subseteq\text{{\sf UB}}_{N}$).
>
> Then for all iterations
>
>
> $\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\gamma=\omega_{1}^{N}\right\\}$
>
> in $N$ witnessing $M\geq N$, we have that $j_{0\gamma}$ defines a
> $\Sigma_{1}$-elementary embedding of
>
> $(H_{\omega_{2}}^{M},\tau_{\mathsf{ST}}^{M},B\cap M:B\in\text{{\sf
> UB}}_{M},\mathbf{NS}_{\omega_{1}}^{M})$
>
> into
>
> $(H_{\omega_{2}}^{N},\tau_{\mathsf{ST}}^{N},B\cap N:B\in\text{{\sf
> UB}}_{M},\mathbf{NS}_{\omega_{1}}^{N}).$
###### Remark 5.13.
A crucial observation is that _“ $x$ is $(\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}})$-ec”_ is a property correctly definable in
$(H_{\omega_{1}}\cup\text{{\sf UB}},\in)$. Therefore (assuming
${\mathbf{MAX}(\mathsf{UB})}$)
$D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}=\left\\{M\in
H_{\omega_{1}}:\,M\text{ is $(\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}})$-ec}\right\\}$
is such that $\bar{D}_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}]$ is a
universally Baire set in $V$. Moreover letting for $V[G]$ a generic extension
of $V$
$D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]}}=\left\\{M\in
H_{\omega_{1}}^{V[G]}:\,M\text{ is $(\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}}^{V[G]})$-ec}\right\\},$
we have that
$\bar{D}_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}^{V[G]}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]}}].$
###### Theorem 5.14.
Assume $V$ models ${\mathbf{MAX}(\mathsf{UB})}$. The following are equivalent:
1. (1)
Woodin’s axiom $(*)$-$\mathsf{UB}$ holds (i.e. $\mathbf{NS}_{\omega_{1}}$ is
saturated, and there is an $L(\mathsf{UB})$-generic filter $G$ for
$\mathbb{P}_{\mathrm{max}}$ such that
$L(\mathsf{UB})[G]\supseteq\mathcal{P}\left(\omega_{1}\right)^{V}$).
2. (2)
Let $\delta$ be inaccessible. Whenever $G$ is $V$-generic for
$\operatorname{Coll}(\omega,\delta)$, $V_{\delta}$ is
$(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]})$-ec in $V[G]$.
3. (3)
$\mathbf{NS}_{\omega_{1}}$ is precipitous and for all $\vec{A}\in
H_{\omega_{2}}$, $B\in\text{{\sf UB}}$, there is an
$(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}})$-ec $M$ with witness $\text{{\sf
UB}}_{M}$, and an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\,\alpha\leq\beta\leq\omega_{1}\right\\}$
of $M$ such that:
* •
$A\in M_{\omega_{1}}$,
* •
$B\in\text{{\sf UB}}_{M}$,
* •
$\mathbf{NS}_{\omega_{1}}^{M_{\omega_{1}}}=\mathbf{NS}_{\omega_{1}}\cap
M_{\omega_{1}}$.
Theorem 5.14 is the key to the proofs of Theorem 5.2 and to the missing
implication in the proof of Theorem 4.
#### 5.5.1. Proof of Theorem 5.2
The theorem is an immediate corollary of the following:
###### Lemma 5.15.
Let $B_{1},\dots,B_{k}$ be new predicate symbols and
$T_{B_{1},\dots,B_{k},\mathbf{NS}_{\omega_{1}}}$ be the
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$-theory
$\mathsf{ZFC}^{*}_{\mathbf{NS}_{\omega_{1}}}+{\mathbf{MAX}(\mathsf{UB})}$
enriched with the sentences asserting that $B_{1},\dots,B_{k}$ are universally
Baire sets.
Let $E_{B_{1},\dots,B_{k}}$ consists of the set of $M\in
D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}$ such that:
* •
$M$ is $B_{j}$-iterable for all $j=1,\dots,k$;
* •
there is $\text{{\sf UB}}_{M}$ witnessing $M\in
D_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}$ with $B_{j}\in\text{{\sf
UB}}_{M}$ for all $j$.
Let also $\bar{E}_{B_{1},\dots,B_{k}}=\text{{\rm
Cod}}_{\omega}^{-1}[E_{B_{1},\dots,B_{k}}]$.
Then $T_{B_{1},\dots,B_{k},\mathbf{NS}_{\omega_{1}}}$ proves that
$\bar{E}_{B_{1},\dots,B_{k}}$ is universally Baire.
Moreover let
$T_{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}},\mathbf{NS}_{\omega_{1}}}$
be the natural extension of $T_{B_{1},\dots,B_{k},\mathbf{NS}_{\omega_{1}}}$
adding a predicate symbol for $\bar{E}_{B_{1},\dots,B_{k}}$ and the axiom
forcing its intepretation to be its definition.
Then
$T_{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}},\mathbf{NS}_{\omega_{1}}}$
models that every $\Sigma_{1}$-formula $\phi(\vec{x})$ for the signature
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$ is
equivalent to a $\Pi_{1}$-formula $\psi(\vec{x})$ in the signature
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\}$.
###### Proof.
$\bar{E}_{B_{1},\dots,B_{k}}$ is universally Baire by
${\mathbf{MAX}(\mathsf{UB})}$, since $E_{B_{1},\dots,B_{k}}$ is definable in
$(H_{\omega_{1}}\cup\mathsf{UB},\in)$ with parameters the universally Baire
sets $B_{1},\dots,B_{k},\bar{D}_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}}$.
Given any $\Sigma_{1}$-formula $\phi(\vec{x})$ for
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
mentioning the universally Baire predicates $B_{1},\dots,B_{k}$, we want to
find a universal formula $\psi(\vec{x})$ such that
$T_{\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\},\mathbf{NS}_{\omega_{1}}}\models\forall\vec{x}(\phi(\vec{x})\leftrightarrow\psi(\vec{x})).$
Let $\psi(\vec{x})$ be the formula asserting:
> _For all $M\in E_{B_{1},\dots,B_{k}}$, for all iterations
> $\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}$
> of $M$ such that:_
>
> * •
>
> $\vec{x}=j_{0\omega_{1}}(\vec{a})$_for some_ $\vec{a}\in M$,
>
> * •
>
> $\mathbf{NS}_{\omega_{1}}^{j_{0\omega_{1}}(M)}=\mathbf{NS}_{\omega_{1}}\cap
> j_{0\omega_{1}}(M)$,
>
>
$(H_{\omega_{2}}^{M},\tau_{\text{{\sf
UB}}_{M},\mathbf{NS}_{\omega_{1}}}^{M})\models\phi(\vec{a}).$
More formally:
$\displaystyle\forall r\,\forall\mathcal{J}$ $\displaystyle\\{$
$\displaystyle[$ $\displaystyle(r\in\bar{E}_{B_{1},\dots,B_{k}})\wedge$
$\displaystyle\wedge\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}\text{
is an iteration of }\text{{\rm Cod}}(r)\wedge$
$\displaystyle\wedge\mathbf{NS}_{\omega_{1}}^{j_{0\omega_{1}}(\text{{\rm
Cod}}(r))}=\mathbf{NS}_{\omega_{1}}\cap j_{0\omega_{1}}(\text{{\rm
Cod}}(r))\wedge$ $\displaystyle\wedge\exists\vec{a}\in\text{{\rm
Cod}}(r)\,(\vec{x}=j_{0\omega_{1}}(\vec{a}))$ $\displaystyle]$
$\displaystyle\rightarrow$ $\displaystyle(H_{\omega_{2}}^{\text{{\rm
Cod}}(r)},\tau_{\text{{\sf UB}}_{\text{{\rm
Cod}}(r)},\mathbf{NS}_{\omega_{1}}}^{\text{{\rm
Cod}}(r)})\models\phi(\vec{a})$ $\displaystyle\\}.$
The above is a $\Pi_{1}$-formula for
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\}$.
(We leave to the reader to check that the property
>
> _$\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}$
> is an iteration of $M$ such that
> $\mathbf{NS}_{\omega_{1}}^{j_{0\omega_{1}}(M)}=\mathbf{NS}_{\omega_{1}}\cap
> j_{0\omega_{1}}(M)$_
is definable by a $\Delta_{1}$-property in parameters $M,\mathcal{J}$ in the
signature $\tau_{\mathbf{NS}_{\omega_{1}}}$).
Now it is not hard to check that:
###### Claim 7.
For all $\vec{A}\in H_{\omega_{2}}$
$(H_{\omega_{2}}^{V},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k})\models\phi(\vec{A})$
if and only if
$(H_{\omega_{2}},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}})\models\psi(\vec{A}).$
###### Proof.
__
$\psi(\vec{A})\rightarrow\phi(\vec{A})$:
Take any $M$ and $\mathcal{J}$ satisfying the premises of the implication in
$\psi(\vec{A})$, Then
$(H_{\omega_{2}}^{M},\tau_{\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}}^{M}}^{M})\models\phi(\vec{a})$ for some $\vec{a}$ such that
$j_{0,\omega_{1}}(\vec{a})=\vec{A}$ and $B_{j}\cap
M_{\omega_{1}}=j_{0\omega_{1}}(B_{j}\cap M)$ for all $j=1,\dots,k$.
Since $\Sigma_{1}$-properties are upward absolute and
$(M_{\omega_{1}},\tau_{\mathbf{NS}_{\omega_{1}}}^{M_{\omega_{1}}},B_{j}\cap
M_{\omega_{1}}:j=1,\dots,k)$ is a
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$-substructure
of $(H_{\omega_{2}},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},B_{j}:j=1,\dots,k)$
which models $\phi(\vec{A})$, we get that $\phi(\vec{A})$ holds for
$(H_{\omega_{2}},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k})$.
$\phi(\vec{A})\rightarrow\psi(\vec{A})$:
Assume
$(H_{\omega_{2}},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k})\models\phi(\vec{A}).$
Take any $(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}})$-ec $M\in V$ and any
iteration
$\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}$
of $M$ witnessing the premises of the implication in $\psi(\vec{A})$, in
particular such that:
* •:
$\vec{A}=j_{0\omega_{1}}(\vec{a})\in M_{\omega_{1}}$ for some $\vec{a}\in M$,
* •:
$\mathbf{NS}_{\omega_{1}}^{M_{\omega_{1}}}=\mathbf{NS}_{\omega_{1}}\cap
M_{\omega_{1}}$,
* •:
$M$ is $B_{j}$-iterable for $j=1,\dots,k$.
Such $M$ and $\mathcal{J}$ exists by Thm. 5.14(3) applied to
$\bar{E}_{B_{1},\dots,B_{k}}$ and $\vec{A}$.
Let $G$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$ with $\delta$
inaccessible. Then in $V[G]$, $V_{\delta}$ is $\text{{\sf
UB}}^{V[G]}$-correct, by Lemma 5.8.
Therefore (since $M$ is $(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]})$-ec
also in $V[G]$ by ${\mathbf{MAX}(\mathsf{UB})}$), $V[G]$ models that
$j_{0\omega_{1}^{V}}$ is a $\Sigma_{1}$-elementary embedding of
$(H_{\omega_{2}}^{M},\tau_{\mathbf{NS}_{\omega_{1}}}^{M},B\cap
M:B\in\mathsf{UB}_{M})$
into
$(H_{\omega_{2}}^{V},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},B:B\in\text{{\sf
UB}}_{M}).$
This grants that
$(H_{\omega_{2}}^{M},\tau_{\mathbf{NS}_{\omega_{1}}}^{M},B\cap
M:B\in\mathsf{UB}_{M})\models\phi(\vec{a}),$
as was to be shown.
∎
The Lemma is proved.
∎
#### 5.5.2. Proof of (2)$\to$(1) of Theorem 4
###### Proof.
Assume $\delta$ is supercompact, $P$ is a standard forcing notion to force
$\text{{\sf MM}}^{++}$ of size $\delta$ (such as the one introduced in [8] to
prove the consistency of Martin’s maximum), and $G$ is $V$-generic for $P$;
then $(*)$-UB holds in $V[G]$ by Asperó and Schindler’s recent breakthrough
[2]. By Thm. 3 $V$ and $V[G]$ agree on the $\Pi_{1}$-fragment of their
$\tau_{\text{{\sf UB}}^{V},\mathbf{NS}_{\omega_{1}}}$-theory, therefore so do
$H_{\omega_{2}}^{V}$ and $H_{\omega_{2}}^{V[G]}$ (by Lemma 1 applied in $V$
and $V[G]$ respectively).
Since $P\in\text{{\sf SSP}}$
$(H_{\omega_{2}}^{V},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},A:A\in\text{{\sf
UB}}^{V})\sqsubseteq(H_{\omega_{2}}^{V[G]},\tau_{\mathbf{NS}_{\omega_{1}}}^{V[G]},A^{V[G]}:A\in\text{{\sf
UB}}^{V}).$
Now the model completeness of $T_{\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}}}$-grants that any of its models (among which $H_{\omega_{2}}^{V}$) is
$(T_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}})_{\forall}$-ec. This gives
that:
$(H_{\omega_{2}}^{V},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},\text{{\sf
UB}}^{V})\prec_{\Sigma_{1}}(H_{\omega_{2}}^{V[G]},\tau_{\mathbf{NS}_{\omega_{1}}}^{V[G]},A^{V[G]}:A\in\text{{\sf
UB}}^{V}).$
Therefore any $\Pi_{2}$-property for $\tau_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$ with parameters in $H_{\omega_{2}}^{V}$ which
holds in
$(H_{\omega_{2}}^{V[G]},\tau_{\mathbf{NS}_{\omega_{1}}}^{V[G]},A^{V[G]}:A\in\text{{\sf
UB}})$
also holds in
$(H_{\omega_{2}}^{V},\tau_{\mathbf{NS}_{\omega_{1}}}^{V},\text{{\sf
UB}}^{V})$.
Hence in $H_{\omega_{2}}^{V}$ it holds characterization (3) of $(*)$-UB given
by Thm. 5.14 and we are done. ∎
#### 5.5.3. Proof of Theorem 5.14
###### Proof.
Schindler and Asperó [1, Def. 2.1] introduced the following:
###### Definition 5.16.
Let $\phi(\vec{x})$ be a $\tau_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$-formula in free variables $\vec{x}$, and
$\vec{A}\in H_{\omega_{2}}^{V}$. $\phi(\vec{A})$ is _honestly consistent_ if
for all universally Baire sets $U\in\mathsf{UB}^{V}$, there is some large
enough cardinal $\kappa\in V$ such that whenever $G$ is $V$-generic for
$\operatorname{Coll}(\omega,\kappa)$, in $V[G]$ there is a $\tau_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$-structure $\mathcal{M}=(M,\dots)$ such that
* •
$M$ is transitive and $U^{V[G]}$-iterable in $V[G]$,
* •
$\mathcal{M}\models\phi(\vec{A})$,
* •
$\mathbf{NS}_{\omega_{1}}^{M}\cap V=\mathbf{NS}_{\omega_{1}}^{V}$.
They also proved the following Theorem [1, Thm. 2.7, Thm. 2.8]:
###### Theorem 5.17.
Assume $V$ models $\mathbf{NS}_{\omega_{1}}$ is precipitous and
${\mathbf{MAX}(\mathsf{UB})}$ holds.
TFAE:
* •
$(*)$-UB holds in $V$.
* •
Whenever $\phi(\vec{x})$ is a $\Sigma_{1}$-formula for $\tau_{\text{{\sf
UB}},\mathbf{NS}_{\omega_{1}}}$ in free variables $\vec{x}$, and $\vec{A}\in
H_{\omega_{2}}^{V}$, $\phi(\vec{A})$ is honestly consistent if and only if it
is true in $H_{\omega_{2}}^{V}$.
We use Schindler and Asperó characterization of $(*)$-UB to prove the
equivalences of the three items of Thm. 5.14 (the proofs of these implications
import key ideas from [2, Lemma 3.2]).
(1) implies (2):
Let $G$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$. By Lemma 5.8,
$V_{\delta}$ is $\text{{\sf UB}}^{V[G]}$-correct in $V[G]$ as witnessed by
$\left\\{B^{V[G]}:B\in\text{{\sf UB}}^{V}\right\\}=\text{{\sf
UB}}_{V}=\left\\{B_{n}^{V[G]}:n\in\omega\right\\}$.
###### Claim 8.
$V_{\delta}$ is $(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]})$-ec as
witnessed by $\text{{\sf UB}}_{V}$.
###### Proof.
Let in $V[G]$ $B_{V}=B_{\text{{\sf UB}}_{V}}=\prod_{n\in\omega}B_{n}^{V[G]}$
be the universally Baire set coding $\text{{\sf UB}}_{V}$.
Let $N\leq V_{\delta}$ in $V[G]$ be $\text{{\sf UB}}^{V[G]}$-correct with
$B_{V}\in\text{{\sf UB}}_{N}$ for some $\text{{\sf UB}}_{N}$ witnessing that
$N$ is $\text{{\sf UB}}^{V[G]}$-correct. Then we already observed that
$\left\\{B^{V[G]}\cap N:B^{V[G]}\in\text{{\sf
UB}}_{V}\right\\}\subseteq\left\\{B\cap N:\,B\in\text{{\sf UB}}_{N}\right\\}$.
Therefore
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf
UB}}_{V}}^{V})=(H_{\omega_{1}}^{V},\tau_{\text{{\sf
UB}}^{V}}^{V})\prec(H_{\omega_{1}}^{N},\tau_{\text{{\sf ST}}}^{N},B^{V[G]}\cap
N:B\in\text{{\sf UB}}^{V}).$
Let
$\mathcal{J}=\left\\{j_{\alpha,\beta}:\alpha\leq\beta\leq\gamma=(\omega_{1})^{N}\right\\}\in
N$
be an iteration witnessing $V_{\delta}\geq N$ in $V[G]$.
We must show that
$j_{0\gamma}:H_{\omega_{2}}^{V}\to H_{\omega_{2}}^{N}$
is $\Sigma_{1}$-elementary for $\tau_{\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}}^{V}}$ between
$(H_{\omega_{2}}^{V},\tau_{\text{{\sf ST}}}^{V},\text{{\sf
UB}}^{V},\mathbf{NS}_{\omega_{1}}^{V})$
and
$(H_{\omega_{2}}^{N},\tau_{\text{{\sf ST}}}^{N},B^{V[G]}\cap N:B\in\text{{\sf
UB}}^{V},\mathbf{NS}_{\omega_{1}}^{N}).$
Let $\phi(a)$ be a $\Sigma_{1}$-formula for
$\tau_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V}}$ in parameter $a\in
H_{\omega_{2}}^{V}$ with $B_{1},\dots,B_{k}\in\text{{\sf UB}}^{V}$ the
universally Baire predicates occurring in $\phi$ such that
$(N,\tau_{\text{{\sf ST}}}^{N},B^{V[G]}\cap N:B\in\text{{\sf
UB}}^{V},\mathbf{NS}_{\omega_{1}}^{N})\models\phi(j_{0\gamma}(a)).$
We must show that
$(H_{\omega_{2}}^{V},\tau_{\text{{\sf ST}}}^{V},\text{{\sf
UB}}^{V},\mathbf{NS}_{\omega_{1}}^{V})\models\phi(a).$
Remark that the iteration $\mathcal{J}$ extends to an iteration
$\bar{\mathcal{J}}=\left\\{\bar{j}_{\alpha,\beta}:\alpha\leq\beta\leq\gamma=(\omega_{1})^{N}\right\\}$
of $V$ exactly as already done in the proof of Lemma 5.8.
Using this observation, let $\bar{M}=\bar{j}_{0\gamma}(V)$; then
$\mathbf{NS}_{\omega_{1}}^{\bar{M}}=\mathbf{NS}_{\omega_{1}}^{N}\cap\bar{M}$.
Now let $H$ be $V$-generic for $\operatorname{Coll}(\omega,\eta)$ with $G\in
V[H]$ for some $\eta>\delta$ inaccessible in $V[G]$.
By ${\mathbf{MAX}(\mathsf{UB})}$ $N$ is $\text{{\sf UB}}^{V[H]}$-correct in
$V[H]$: on the one hand
$D_{\text{{\sf UB}}^{V[H]}}=\text{{\rm Cod}}[\bar{D}_{\text{{\sf
UB}}^{V[G]}}^{V[H]}],$
on the other hand
$N\in\text{{\rm Cod}}[\bar{D}_{\text{{\sf UB}}^{V[G]}}]\subseteq\text{{\rm
Cod}}[\bar{D}_{\text{{\sf UB}}^{V[G]}}^{V[H]}].$
In particular for any $B\in\text{{\sf UB}}_{V}$, $N$ is $B^{V[H]}$-iterable in
$V[H]$.
Therefore in $H_{\omega_{1}}^{V[H]}$ for any $B\in\mathsf{UB}^{V}$, the
statement
> _There exists a
> $\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B,B_{1},\dots,B_{k}\right\\}$-super-
> structure $\bar{N}$ of $j_{0\gamma}(V_{\delta})$ which is
> $\left\\{B^{V[H]},B_{1}^{V[H]},\dots,B_{k}^{V[H]}\right\\}$-iterable and
> which realizes $\phi(j_{0\gamma}(a))$_
holds true as witnessed by $N$.
The following is a key observation:
###### Subclaim 1.
For any $s\in(2^{\omega})^{\bar{M}[H]}$ and $B\in\text{{\sf UB}}^{V}$
$s\in j_{0\gamma}(B)^{\bar{M}[H]}\text{ if and only if }s\in
B^{V[H]}\cap\bar{M}[H].$
###### Proof.
For each $B\in\text{{\sf UB}}^{V}$ find in $V$ trees $(T_{B},S_{B})$ which
project to complement in $V[H]$ and such that $B=p[T_{B}]$. Now since
$\bar{j}_{0,\gamma}[T_{B}]\subseteq\bar{j}_{0,\gamma}(T_{B})$ and
$\bar{j}_{0,\gamma}[S_{B}]\subseteq\bar{j}_{0,\gamma}(S_{B})$, we get that
* •:
$(2^{\omega})^{V[H]}=p[[\bar{j}_{0,\gamma}(T_{B})]]\cup
p[[\bar{j}_{0,\gamma}(S_{B})]]$ (since $(2^{\omega})^{V[H]}$ is already
covered by $p[[\bar{j}_{0,\gamma}[T_{B}]]]\cup
p[[\bar{j}_{0,\gamma}[S_{B}]]]$).
* •:
$\emptyset=p[[\bar{j}_{0,\gamma}(T_{B})]]\cap p[[\bar{j}_{0,\gamma}(S_{B})]]$
by elementarity of $\bar{j}_{0,\gamma}$.
Hence $B^{V[H]}$ is also the projection of $\bar{j}_{0,\gamma}(T_{B})$ and the
pair $(\bar{j}_{0,\gamma}(T_{B}),\bar{j}_{0,\gamma}(S_{B}))$ projects to
complement in $V[H]$.
But this pair belongs to $\bar{M}$, and (by elementarity of
$\bar{j}_{0\gamma}$)
$\bar{M}\models(\bar{j}_{0,\gamma}(T_{B}),\bar{j}_{0,\gamma}(S_{B}))\text{
projects to complements for
$\operatorname{Coll}(\omega,\bar{j}_{0,\gamma}(\eta))$.}$
Since $\eta\leq\bar{j}_{0,\gamma}(\eta)$ we get that
$\bar{M}\models(\bar{j}_{0,\gamma}(T_{B}),\bar{j}_{0,\gamma}(S_{B}))\text{
projects to complements for $\operatorname{Coll}(\omega,\eta)$.}$
Therefore in $V[H]$ $s\in j_{0\gamma}(B)^{\bar{M}[H]}$ if and only if $s\in
p[[\bar{j}_{0,\gamma}(T_{B})]^{V[H]}]\cap M[H]$ if and only if $s\in
p[[T_{B}]^{V[H]}]\cap\bar{M}[H]$ if and only if $s\in B^{V[H]}\cap\bar{M}[H]$.
∎
This shows that
$(\bar{M}[H],\tau_{\text{{\sf
UB}}^{V}}^{\bar{M}[H]})\sqsubseteq(V[H],\tau_{\text{{\sf UB}}^{V}}^{V[H]}).$
Moreover $H_{\omega_{1}}^{\bar{M}[H]}$ and $H_{\omega_{1}}^{V[H]}$ both
realize the theory $T_{\text{{\sf UB}}^{V}}$ of $H_{\omega_{1}}^{V}$ in this
language: on the one hand
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf
UB}}^{V}}^{V})\prec(H_{\omega_{1}}^{\bar{M}},\tau_{\text{{\sf
UB}}^{V}}^{\bar{M}})\prec(H_{\omega_{1}}^{\bar{M}[H]},\tau_{\text{{\sf
UB}}^{V}}^{\bar{M}[H]})$
(the leftmost $\prec$ holds since $j_{0,\gamma}:V\to\bar{M}$ is elementary,
the rightmost $\prec$ holds since $\bar{M}$ models
${\mathbf{MAX}(\mathsf{UB})}$); on the other hand
$(H_{\omega_{1}}^{V},\tau_{\text{{\sf
UB}}^{V}}^{V})\prec(H_{\omega_{1}}^{V[H]},\tau_{\text{{\sf UB}}^{V}}^{V[H]})$
(applying ${\mathbf{MAX}(\mathsf{UB})}$ in $V$).
Since $T_{\text{{\sf UB}}^{V}}$ is model complete, we get that
$H_{\omega_{1}}^{\bar{M}[H]}$ is an elementary $\tau_{\text{{\sf
UB}}^{V}}$-substructure of $H_{\omega_{1}}^{V[H]}$; therefore
$H_{\omega_{1}}^{\bar{M}[H]}$ models
> _There exists a $\tau_{\mathbf{NS}_{\omega_{1}},B,B_{1},\dots,B_{k}}$-super-
> structure $\bar{N}$ of $j_{0\gamma}(V_{\delta})$ which is
>
> $\left\\{\bar{j}_{0\gamma}(B)^{\bar{M}[H]},\bar{j}_{0\gamma}(B_{1})^{\bar{M}[H]},\dots,\bar{j}_{0\gamma}(B_{k})^{\bar{M}[H]}\right\\}$-iterable
> and which realizes $\phi(j_{0\gamma}(a))$._
By homogeneity of $\operatorname{Coll}(\omega,\eta)$, in $\bar{M}$ we get that
any condition in $\operatorname{Coll}(\omega,\eta)$ forces:
> _There exists a $\tau_{\mathbf{NS}_{\omega_{1}},B,B_{1},\dots,B_{k}}$-super-
> structure $\bar{N}$ of $j_{0\gamma}(V_{\delta})$ which is
>
> $\left\\{\bar{j}_{0\gamma}(B)^{\bar{M}[\dot{H}]},\bar{j}_{0\gamma}(B_{1})^{\bar{M}[\dot{H}]},\dots,\bar{j}_{0\gamma}(B_{k})^{\bar{M}[\dot{H}]}\right\\}$-iterable
> and which realizes $\phi(j_{0\gamma}(a))$._
By elementarity of $\bar{j}_{0\gamma}$ we get that in $V$ it holds that:
> There exists an $\eta>\delta$ such that any condition in
> $\operatorname{Coll}(\omega,\eta)$ forces:
>
>> _“There exists a countable super structure $\bar{N}$ of $V_{\delta}$ with
respect to
$\tau_{\mathbf{NS}_{\omega_{1}},\left\\{B,B_{1},\dots,B_{k}\right\\}}$ which
is
$\left\\{B^{V[\dot{H}]},B_{1}^{V[\dot{H}]},\dots,B_{k}^{V[\dot{H}]}\right\\}$-iterable
and which realizes $\phi(a)$”_
This procedure can be repeated for any $B\in\text{{\sf UB}}^{V}$, showing that
$\phi(a)$ is honestly consistent in $V$.
By Schindler and Asperó characterization of $(*)$ we obtain that $\phi(a)$
holds in $H_{\omega_{2}}^{V}$. ∎
(2) implies (3):
Our assumptions grants that the set
$D_{\text{{\sf UB}}}=\left\\{M\in H_{\omega_{1}}^{V}:M\text{ is $\text{{\sf
UB}}^{V}$-correct}\right\\}$
is coded by a universally Baire set $\bar{D}_{\text{{\sf UB}}}$ in $V$.
Moreover we also get that whenever $G$ is $V$-generic for
$\operatorname{Coll}(\omega,\delta)$, the lift $\bar{D}_{\text{{\sf
UB}}}^{V[G]}$ of $\bar{D}_{\text{{\sf UB}}}$ to $V[G]$ codes
$D_{\text{{\sf UB}}^{V[G]}}^{V[G]}=\left\\{M\in H_{\omega_{1}}^{V[G]}:M\text{
is $\text{{\sf UB}}^{V[G]}$-correct}\right\\}.$
By (2) we get that $V_{\delta}\in D_{\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}}^{V[G]}}^{V[G]}$.
By Fact 5.10
$(H_{\omega_{1}}^{V},\tau_{\mathsf{ST}}^{V},\text{{\sf UB}}^{V})\models\text{
for all iterable $M$ there exists an $\text{{\sf UB}}$-correct structure
$\bar{M}\geq M$}.$
Again since
$(H_{\omega_{1}}^{V},\tau_{\mathsf{ST}}^{V},\text{{\sf
UB}}^{V})\prec(H_{\omega_{1}}^{V[G]},\tau_{\mathsf{ST}}^{V[G]},\text{{\sf
UB}}^{V}),$
and the latter is first order expressible in the predicate
$\bar{D}_{\text{{\sf UB}}}\in\text{{\sf UB}}^{V}$, we get that
$(H_{\omega_{1}}^{V[G]},\tau_{\mathsf{ST}}^{V[G]},\text{{\sf
UB}}^{V})\models\text{ for all iterable $M$ there exists an $\text{{\sf
UB}}^{V[G]}$-correct structure $\bar{M}\geq M$}.$
So let $N\leq V_{\delta}$ be in $V[G]$ an $\text{{\sf UB}}^{V[G]}$-correct
structure with $V_{\delta}\in H_{\omega_{1}}^{N}$.
Let
$\mathcal{J}=\left\\{j_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma=\omega_{1}^{N}\right\\}\in
H_{\omega_{2}}^{N}$ be an iteration witnessing $N\leq V_{\delta}$.
Now for any $A\in\mathcal{P}\left(\omega_{1}\right)^{V}$ and $B\in\text{{\sf
UB}}^{V}$
$(H_{\omega_{2}}^{N},\tau_{\mathsf{ST}}^{N},\mathbf{NS}_{\gamma}^{N},B^{V[G]}\cap
N:B\in\text{{\sf UB}}^{V})$
models
> _There exists an $(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]})$-ec
> structure $M$ with $B^{V[G]}\cap N\in\text{{\sf UB}}_{M}$ and an iteration
> $\bar{\mathcal{J}}=\left\\{\bar{j}_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma\right\\}$
> of $M$ such that $\bar{j}_{0\gamma}(A)=j_{0\gamma}(A)$_.
This statement is witnessed exactly by $V_{\delta}$ in the place of $M$ (since
$B=B^{V[G]}\cap V_{\delta}\in\text{{\sf UB}}^{V}$ and $\text{{\sf
UB}}^{V[G]}_{V_{\delta}}=\left\\{B^{V[G]}:\,B\in\text{{\sf
UB}}^{V}\right\\}$), and $\mathcal{J}$ in the place of $\bar{\mathcal{J}}$.
Since $V_{\delta}$ is $(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]})$-ec
in $V[G]$ we get that $j_{0\gamma}\restriction H_{\omega_{2}}^{V}$ is
$\Sigma_{1}$-elementary between $H_{\omega_{2}}^{V}$ and $H_{\omega_{2}}^{N}$
for $\tau_{\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V}}$.
Hence
$(H_{\omega_{2}}^{V},\tau_{\mathsf{ST}}^{V},\mathbf{NS}_{\gamma}^{V},\text{{\sf
UB}}^{V})$
models
> _There exists an $(\mathbf{NS}_{\omega_{1}}^{V},\text{{\sf UB}}^{V})$-ec
> structure $M$ with $B\in\text{{\sf UB}}_{M}$ and an iteration
> $\bar{\mathcal{J}}=\left\\{\bar{j}_{\alpha\beta}:\,\alpha\leq\beta\leq(\omega_{1})^{V}\right\\}$
> of $M$ such that $\bar{j}_{0\omega_{1}}(a)=A$ and
> $\mathbf{NS}_{\omega_{1}}^{\bar{j}_{0\omega_{1}}(M)}=\mathbf{NS}_{\omega_{1}}^{V}\cap\bar{j}_{0\omega_{1}}(M)$_.
(3) implies (1):
We use again Schindler and Asperó characterization of $(*)$.
Assume $\phi(A)$ is honestly consistent for some $\Sigma_{1}$-property
$\phi(x)$ in the language $\tau_{\text{{\sf UB}},\mathbf{NS}_{\omega_{1}}}$
and $A\in\mathcal{P}\left(\omega_{1}\right)^{V}$. Let $B_{1},\dots,B_{k}$ be
the universally Baire predicates in UB mentioned in $\phi(x)$.
By (3) there is in $V$ an $(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}})$-ec $M$
with $B_{1},\dots,B_{k}\in\text{{\sf UB}}_{M}$ and
$a\in\mathcal{P}\left(\omega_{1}\right)^{M}$, and an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\,\alpha\leq\beta\leq\omega_{1}\right\\}$
of $M$ such that $j_{0\omega_{1}}(a)=A$ and
$\mathbf{NS}_{\omega_{1}}^{j_{0\omega_{1}}(M)}=\mathbf{NS}_{\omega_{1}}^{V}\cap
j_{0\omega_{1}}(M)$.
Let $G$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$. Find $N\in
V[G]$ such that $N\models\phi(A)$, $N$ is
$B_{1}^{V[G]},\dots,B_{k}^{V[G]}$-iterable in $V[G]$ and
$\mathbf{NS}_{\omega_{1}}^{N}\cap V=\mathbf{NS}_{\omega_{1}}^{V}$ (this $N$
exists by the honest consistency of $\phi(x)$).
Notice that $\mathcal{J}\in V_{\delta}\subseteq N$ witnesses that $M\geq N$ as
well.
Let $\bar{N}\leq N$ in $V[G]$ be a $\text{{\sf UB}}^{V[G]}$-correct structure
with $B_{\text{{\sf UB}}_{V}}\in\text{{\sf UB}}_{\bar{N}}$ ($\bar{N}$ exists
by Fact 5.10 applied in $V[G]$ to $N$ and $B_{\text{{\sf UB}}_{V}}$). Let
$\mathcal{K}=\left\\{k_{\alpha\beta}:\alpha\leq\beta\leq\bar{\gamma}=\omega_{1}^{\bar{N}}\right\\}\in\bar{N}$
be an iteration witnessing that $\bar{N}\leq N$.
Remark that $H_{\omega_{2}}^{\bar{N}}\models\phi(k_{0\bar{\gamma}}(A))$, since
$\Sigma_{1}$-properties are upward absolute and $k_{0\bar{\gamma}}(N)$ is a
$\tau_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$-substructure
of $H_{\omega_{2}}^{\bar{N}}$.
Also $\left\\{B^{V[G]}:B\in\text{{\sf UB}}_{V}\right\\}\subseteq\text{{\sf
UB}}_{\bar{N}}$ entail that $B_{\text{{\sf UB}}_{M}}^{V[G]}\in\text{{\sf
UB}}_{\bar{N}}$.
Letting
$\bar{\mathcal{J}}=\left\\{\bar{j}_{\alpha\beta}:\alpha\leq\beta\leq\bar{\gamma}\right\\}=k_{0\bar{\gamma}}(\mathcal{J}),$
we get that
$\bar{j}_{0\bar{\gamma}}(a)=k_{0\gamma}(j_{0\bar{\gamma}}(a))=k_{0\gamma}(A)$,
and $\bar{\mathcal{J}}$ is such that $B_{j}^{V[G]}\in\text{{\sf
UB}}_{\bar{N}}$ for all $j=1,\dots,k$ since $B_{\text{{\sf UB}}_{M}}^{V[G]}$
in $\text{{\sf UB}}_{\bar{N}}$.
Since $M$ is $(\mathbf{NS}_{\omega_{1}},\text{{\sf UB}}^{V[G]})$-ec in $V[G]$
by ${\mathbf{MAX}(\mathsf{UB})}$, we get that $\bar{j}_{0\bar{\gamma}}$
defines a $\Sigma_{1}$-elementary embedding of
$(H_{\omega_{2}}^{M},\tau_{\text{{\sf UB}}_{M},\mathbf{NS}_{\omega_{1}}}^{M})$
into
$(H_{\omega_{2}}^{\bar{N}},\tau_{\text{{\sf
UB}}_{M},\mathbf{NS}_{\omega_{1}}}^{\bar{N}}).$
Hence
$(H_{\omega_{2}}^{M},\tau_{\text{{\sf
UB}}_{M},\mathbf{NS}_{\omega_{1}}}^{M})\models\phi(a).$
This gives that
$(H_{\omega_{2}}^{M_{\omega_{1}}},\tau_{\text{{\sf
UB}}_{M},\mathbf{NS}_{\omega_{1}}}^{M_{\omega_{1}}})\models\phi(A)$
(since $j_{0\omega_{1}}(a)=A$), and therefore that
$(H_{\omega_{2}}^{V},\tau_{\text{{\sf
UB}}_{M},\mathbf{NS}_{\omega_{1}}}^{V})\models\phi(A),$
since $M_{\omega_{1}}$ is a substructure of $H_{\omega_{2}}^{V}$ for
$\tau_{\text{{\sf UB}}_{M},\mathbf{NS}_{\omega_{1}}}$.
∎
## 6\. Some questions and comments
### Do we really need ${\mathbf{MAX}(\mathsf{UB})}$ to establish Thm. 2?
It is not at all clear whether the chain of equivalences for $(*)$-UB given in
Thm. 5 could be proved without appealing to ${\mathbf{MAX}(\mathsf{UB})}$.
What we can for sure say is that the equivalence between forcibility and
consistency as given by items D and G of Thm. 5 holds for the signature
$\tau_{\omega_{1}}$ and its $\Pi_{2}$-sentences $\psi$.
More precisely:
###### Theorem 6.
Consider any $\tau_{\omega_{1}}$-theory $S$ extending
$\mathsf{ZFC}_{\text{{\sf ST}}}+\omega_{1}\emph{ is the first uncountable
cardinal $+$ there are class many supercompact cardinals}$
and which _is preserved by any forcing_ (e.g. $S$ itself or $S+T_{\forall}$
for any $T$ extending $S$). Then the Kaiser hull of $S$ is equivalently given
by those $\Pi_{2}$-sentences $\psi$ for $\tau_{\omega_{1}}$ satysfying items D
or G of Thm. 5.
###### Proof.
First assume that $S$ proves that $\psi^{H_{\omega_{2}}}$ is forcible; given a
model $V$ of $S$, by collapsing a supercompact of $V$ to countable one gets
some $V[G]$ which models $S+{\mathbf{MAX}(\mathsf{UB})}$ and satisfies the
same universal sentence for $\tau_{\omega_{1}}$ as $V$ (by Thm. 3). Hence by
forcing over $V[G]$ (which is still a model of $S$), we get to some $V[H]$
which models $\psi^{H_{\omega_{2}}}+{\mathbf{MAX}(\mathsf{UB})}+S$ and
satisfies the same universal sentence for $\tau_{\omega_{1}}$ as $V[G]$. Hence
we get that $\psi$ is consistent with the universal fragment of any
$\tau_{\omega_{1}}$-completion of $S$.
Now assume $\psi$ is consistent with the universal fragment of any completion
of $S$: Any $\tau_{\omega_{1}}$-model $V$ of $S$ can be extended (using
forcing) to a $\tau_{\omega_{1}}$-model $V[G]$ of
$S+{\mathbf{MAX}(\mathsf{UB})}+(*)\text{-}\mathsf{UB}$ which satisfies the
same $\tau_{\omega_{1}}$-universal sentences of $V$ (again by Thm. 3). Since
$\tau_{\omega_{1}}\subseteq\sigma_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}$ and any $\tau_{\omega_{1}}$-model $W$ of $S$
admits a unique extension to $\sigma_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}$-model which interprets correctly the new
predicate symbols, we get that $\psi$ is in the model companion of the
$\sigma_{\text{{\rm l-UB}},\mathbf{NS}_{\omega_{1}}}$-theory of $V[G]$, and
also that this model companion is the $\sigma_{\text{{\rm
l-UB}},\mathbf{NS}_{\omega_{1}}}$-theory of $H_{\omega_{2}}^{V[G]}$. By the
equivalence of B and G of Thm. 5 we get that
$H_{\omega_{2}}^{V[G]}\models\psi$.
Using a similar argument (and appealing to Lemma 1.21 for the unique extension
of $S$ to $\sigma_{\text{{\rm l-UB}},\mathbf{NS}_{\omega_{1}}}$ which
inteprets correctly the new predicate symbols) one can also prove that these
$\Pi_{2}$-sentences $\psi$ for $\tau_{\omega_{1}}$ axiomatize the Kaiser hull
of $S$. We leave the details to the reader. ∎
The above argument is not restricted to $\tau_{\omega_{1}}$ and $S$, but holds
mutatis mutandis for many other signatures contained in
$\sigma_{\omega,\mathbf{NS}_{\omega_{1}}}$ and theories extending
$\mathsf{ZFC}$ with large cardinals; we leave the details to the reader.
Let us also note that for $S$ as above $\mathsf{CH}$ cannot be $S$-equivalent
to a $\Sigma_{1}$-sentence for $\tau_{\omega_{1}}$, because CH is a statement
which can change its truth value across forcing extensions, while the
universal $\tau_{\omega_{1}}$-sentences maintain the same truth value across
all forcing extensions of a model of $T$ by Thm. 3.
### Can we prove model companionship results coupled with generic absoluteness
for the theory of $H_{\aleph_{3}}$?
We can also argue that we cannot hope to find a signature
$\sigma\supseteq\tau_{\text{{\sf
ST}}}\cup\left\\{\omega_{1},\omega_{2}\right\\}$ such that the universal
theory of $V$ in signature $\sigma$ is invariant across forcing extension of
$V$. In particular we cannot hope to get a signature $\sigma$ which makes the
theory of $H_{\aleph_{3}}$ the model companion of the theory of $V$ in this
signature and such that it suffices to use forcing to compute which
$\Pi_{2}$-sentences fall into this model companion theory of $V$ (as we argued
to be the case for the theory of $H_{\aleph_{2}}$ in signature
$\left\\{\in\right\\}_{\bar{A}_{2}}\supseteq\tau_{\text{{\sf
ST}}}\cup\left\\{\omega_{1}\right\\}$). This observation is due to Boban
Veličkovic̀.
###### Remark 1.
$\Box_{\omega_{2}}$ is a $\Sigma_{1}$-statement for
$\tau_{\omega_{2}}=\tau_{\text{{\sf
ST}}}\cup\left\\{\omega_{1},\omega_{2}\right\\}$:
$\displaystyle\exists\left\\{C_{\alpha}:\alpha<\omega_{2}\right\\}$
$\displaystyle[$ $\displaystyle\forall\alpha\in\omega_{2}\,(C_{\alpha}\text{
is a club subset of }\alpha)\wedge$
$\displaystyle\wedge\forall\alpha\in\beta\in\omega_{2}\,(\alpha\in\lim(C_{\beta})\rightarrow
C_{\alpha}=C_{\beta}\cap\alpha)\wedge$
$\displaystyle\wedge\forall\alpha\in\omega_{2}\,(\operatorname{otp}(C_{\alpha})\leq\omega_{1})$
$\displaystyle].$
$\Box_{\omega_{2}}$ is forcible by very nice forcings (countably directed and
$<\omega_{1}$-strategically closed), and its negation is forcible by
$\operatorname{Coll}(\omega_{1},<\delta)$ whenever $\delta$ is Mahlo.
In particular the $\Pi_{1}$-theory for $\tau_{\omega_{2}}$ of any forcing
extension $V[G]$ of $V$ can be destroyed in a further forcing extension
$V[G][H]$ assuming mild large cardinals.
Suppose now we want to find $A_{3}\subseteq F_{\in}$ so to be able to extend
Thm. 2 by:
* •
assuming as base theory $\mathsf{ZFC}+$_suitable large cardinal axioms_
* •
replacing $H_{\aleph_{2}}$ with $H_{\aleph_{3}}$ in all statements of the
theorem pertaining to $A_{3}$,
* •
requiring that $\tau_{\omega_{2}}\subseteq\left\\{\in\right\\}_{\bar{A}_{3}}$.
In this case the best we can hope for is to replace clause 5 of Thm. 2 with a
weaker clause asserting that we consider just forcing notions which do not
change the universal $\left\\{\in\right\\}_{\bar{A}_{3}}$-theory of
$H_{\aleph_{3}}$ (which means restricting our attention to a narrow class of
forcings).
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|
# Quasilinear Schrödinger equations: ground state and infinitely many
normalized solutions ††thanks: This work is supported by
NSFC(11771234,12026227); E-mails<EMAIL_ADDRESS>& zou-
<EMAIL_ADDRESS>
Houwang Li1 & Wenming Zou2
1\. Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China.
2\. Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China.
Abstract
In the present paper, we study the normalized solutions for the following
quasilinear Schrödinger equations:
$-\Delta u-u\Delta u^{2}+\lambda
u=|u|^{p-2}u\quad\text{in}~{}{\mathbb{R}^{N}},$
with prescribed mass
$\int_{{\mathbb{R}^{N}}}u^{2}=a^{2}.$
We first consider the mass-supercritical case $p>4+\frac{4}{N}$, which has not
been studied before. By using a perturbation method, we succeed to prove the
existence of ground state normalized solutions, and by applying the index
theory, we obtain the existence of infinitely many normalized solutions. Then
we turn to study the mass-critical case, i.e., $p=4+\frac{4}{N}$, and obtain
some new existence results. Moreover, we also observe a concentration behavior
of the ground state solutions.
Key words: Quasilinear Schrödinger equation; Normalized solution; Perturbation
method; Index theory.
2010 Mathematics Subject Classification: 35J50, 35J15, 35J60.
## 1 Introduction
We consider the equation
(1.1) $-\Delta u-u\Delta u^{2}+\lambda
u=|u|^{p-2}u,\quad\text{in}~{}{\mathbb{R}^{N}},$
which is usually called Modified Nonlinear Schrödinger equation. Such type of
equations appear as a standing wave version of the following Schrödinger
equations,
(1.2) $\left\\{\begin{aligned}
&i\partial_{t}\phi+\Delta\phi+\phi\Delta(|\phi|^{2})+|\phi|^{p-2}\phi=0,\quad\text{in}~{}{\mathbb{R}}^{+}\times{\mathbb{R}^{N}},\\\
&\phi(0,x)=\phi_{0}(x),\quad\text{in}~{}{\mathbb{R}^{N}}.\end{aligned}\right.$
It is well known that the above Schrödinger equations model many phenomena in
mathematical physics, for instance in the theory of Heisenberg ferromagnets
and magnons [8, 27, 45], in models of superfluid films in fluid mechanics and
plasma physics [28, 34, 44], in dissipative quantum mechanics [21], and in
condensed matter theory [40], which have received considerable attention in
mathematical analysis during the last two decades.
In recent years, the search for the solution with prescribed mass has became a
hot direction, that is to find $u$ such that
(1.3) $\left\\{\begin{aligned} &-\Delta u-u\Delta u^{2}+\lambda
u=|u|^{p-2}u,\quad\text{in}~{}{\mathbb{R}^{N}},\\\
&\int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d}x=a,\end{aligned}\right.$
with $\lambda$ appearing as Lagrange multiplier. From the view of physics,
prescribed mass represents the law of conservation of mass, so it seems to be
great meaningful to study such solutions. Solutions of prescribed mass are
often referred to as normalized solutions, and the present paper is devoted to
such solutions i.e., the solution $u$ of (1.3) with a Lagrange multiplier
$\lambda\in{\mathbb{R}}$.
The existence of normalized solutions to the semilinear Schrödinger equation
(1.4) $-\Delta u+\lambda u=g(u),\quad\text{in}~{}{\mathbb{R}^{N}},$
has been widely studied recently. Mathematically, to obtain the normalized
solutions, one needs to consider the corresponding energy functional on a
$L^{2}$ sphere, which has particular difficulties: the weak limit of the
Palais-Smale sequence may be not contained in the $L^{2}$ sphere (even in the
radial case), and the Palais-Smale sequence does not even need to be bounded.
So the study of normalized solutions of (1.4) is much more complicated than
the study of (1.4) with prescribed $\lambda\in{\mathbb{R}}$. Fortunately, in
[23], using an auxiliary functional and a mini-max theorem from [18], L.
Jeanjean obtained a normalized solution of (1.4). The existence of infinitely
many normalized solutions of (1.4) was later proved by T. Bartsch and S. de
Valeriola in [4] using a new linking geometry for the auxiliary functional
(see also the papers by T. Bartsch and N. Soave [5]). After that, N. Ikoma and
K. Tanaka [22] constructed a deformation theorem suitable for the auxiliary
functional, and then obtained infinitely many normalized solutions of (1.4)
through Krasnoselskii index under a weaker condition on $g(u)$. Soon later, L.
Jeanjean and S. S. Lu [24] obtained infinitely many normalized solutions of
(1.4) under a totally different assumption on $g(u)$ which permits $g(u)$ to
be just continuous. As for the least energy normalized solutions, N. Soave in
[48, 49], by restraining the energy functional on a smaller manifold, obtained
the existence of ground state normalized solutions with
$g(u)=|u|^{p-2}u+\mu|u|^{q-2}u$. For more results on normalized solutions for
scalar equations and systems, we refer to [7, 6, 2, 3, 19, 20, 30, 9].
Now we come back to the Modified Nonlinear Schrödinger equation (1.1). When
considering (1.1) with $\lambda\in{\mathbb{R}}$ fixed, one always study the
functional
(1.5) $E_{\lambda}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+\lambda|u|^{2})+\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p},$
on the space
${\cal H}=\left\\{u\in
W^{1,2}({\mathbb{R}^{N}}):\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}<+\infty\right\\}.$
It is easy to check that $u$ is a weak solution of (1.1) if and only if
$E_{\lambda}^{\prime}(u)\phi=\lim_{t\to
0^{+}}\frac{E_{\lambda}(u+t\phi)-E_{\lambda}(u)}{t}=0,$
for every $\phi\in{\cal C}_{0}^{\infty}({\mathbb{R}^{N}})$. We recall, see
[38] for example, that the value
$22^{*}=\begin{cases}&\frac{4N}{N-2},\quad N\geq 3,\\\ &+\infty,\quad N\leq
2\end{cases}$
corresponds to a critical exponent. Compared to equation (1.4), the search of
solutions of (1.1) presents a major difficulty: the functional associated with
the term $u\Delta u^{2}$
$V(u)=\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}$
is non-differentiable in ${\cal H}$ when $N\geq 2$. To overcome this
difficulty, various arguments have been developed, such as the minimizition
methods [43] where the non-differentiability of $E_{\lambda}$ does not come
into play, the methods of a Nehari manifold approach [35], the methods of
changing variables [38, 15] which transform problem (1.1) into a semilinear
one (1.4), and a perturbation method in a series of papers [36, 39, 37] which
recovers the differentiability by considering a perturbed functional on a
smaller function space.
However, when considering the normalized solution problem (1.3), one would
find that the methods of Nehari manifold approach and changing variables are
no longer applicable, since the parameter $\lambda$ is unknown and the
$L^{2}$-norm $\|u\|_{2}$ must be equal to a given number. So there are very
few results on problem (1.3). Formally, a normalized solution of (1.3) can be
obtained as a critical point of
(1.6) $I(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}$
on the set
(1.7) $\tilde{\cal S}(a):=\left\\{u\in{\cal
H}:\int_{\mathbb{R}^{N}}|u|^{2}=a\right\\},$
that is, a normalized solution of (1.3) is a $u\in\tilde{\cal S}(a)$ such that
there exists a $\lambda\in{\mathbb{R}}$ satisfing
(1.8) $\int_{\mathbb{R}^{N}}\nabla
u\cdot\nabla\phi+2\int_{\mathbb{R}^{N}}(u\phi|\nabla u|^{2}+|u|^{2}\nabla
u\cdot\nabla\phi)+\lambda\int_{\mathbb{R}^{N}}u\phi-\int_{\mathbb{R}^{N}}|u|^{p-2}u\phi=0,$
for any $\phi\in{\cal C}_{0}^{\infty}({\mathbb{R}^{N}})$. To proceed our
paper, we introduce a sharp Gagliardo-Nirenberg inequality [1]:
(1.9)
$\int_{\mathbb{R}^{N}}|u|^{\frac{p}{2}}\leq\frac{C(p,N)}{\|Q_{p}\|_{1}^{\frac{p-2}{N+2}}}\left(\int_{\mathbb{R}^{N}}|u|\right)^{\frac{4N-(N-2)p}{2(N+2)}}\left(\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\right)^{\frac{N(p-2)}{2(N+2)}},\quad\forall~{}u\in{\cal E}^{1},$
where $2<p<22^{*}$,
$C(p,N)=\frac{p(N+2)}{\left[4N-(N-2)p\right]^{\frac{4-N(p-2)}{2(N+2)}}\left[2N(p-2)\right]^{\frac{N(p-2)}{2(N+2)}}},$
and
${\cal E}^{q}:=\left\\{u\in L^{q}({\mathbb{R}^{N}}):\nabla u\in
L^{2}({\mathbb{R}^{N}})\right\\},$
with norm $\|u\|_{{\cal E}^{q}}:=\|\nabla u\|_{2}+\|u\|_{q}$. It is well known
that ${\cal E}^{q}$ is a reflexive Banach space when $1<q<\infty$, and for
Embeeding theorems and more related properties we refer to [29]. Moreover,
$Q_{p}$ optimizes (1.9) and is the unique nonnegative radially symmetric
solution of the following equation [47]:
(1.10) $-\Delta u+1=u^{\frac{p}{2}-1},\quad\text{in}~{}{\mathbb{R}^{N}}.$
Strictly speaking, it has been proved in [47, Theorem 1.3] that $Q_{p}$ has a
compact support in ${\mathbb{R}^{N}}$ and exactly satisfies a Dirichlet-
Neumann free boundary problem. Namely, there exists an $R>0$ such that $Q_{p}$
is the unique positive solution of
(1.11) $\left\\{\begin{aligned} &-\Delta
u+1=u^{\frac{p}{2}-1},\quad\text{in}~{}B_{R},\\\ &u=\frac{\partial u}{\partial
n}=0,\quad\text{on}~{}\partial B_{R}.\end{aligned}\right.$
In what follows, if we say that $u$ is a nonnegative solution of (1.10), then
we mean that $u$ is a solution of (1.11). By replacing $u$ with $u^{2}$ in
(1.9), one immediately obtain the following Gagliardo-Nirenberg-type
inequality,
(1.12)
$\int_{\mathbb{R}^{N}}|u|^{p}\leq\frac{C(p,N)}{\|Q_{p}\|_{1}^{\frac{p-2}{N+2}}}\left(\int_{\mathbb{R}^{N}}|u|^{2}\right)^{\frac{4N-p(N-2)}{2(N+2)}}\left(4\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}\right)^{\frac{N(p-2)}{2(N+2)}}.$
Now we collect some known results about normalized solutions of (1.3). First,
to avoid the nondifferentiability of $V(u)$, M. Colin, L. Jeanjean, M.
Squassina [16] and L. Jeanjean, T. J. Luo [25] considered the minimizition
problem
$\tilde{m}(a)=\inf_{u\in\tilde{\cal S}(a)}I(u)$
with $2<p\leq 4+\frac{4}{N}$. Using ineqality (1.12), one can find that
$\tilde{m}(a)>-\infty$ when $2<p<4+\frac{4}{N}$ and $\tilde{m}(a)=-\infty$
when $p>4+\frac{4}{N}$, since
$\frac{N(p-2)}{2(N+2)}<1\quad\text{ if and only if }\quad p<4+\frac{4}{N}.$
These considerations show that the exponent $4+\frac{4}{N}$ for (1.3) plays
the role of $2+\frac{4}{N}$ in (1.4). After that, X. Y. Zeng and Y. M. Zhang
[52] studied the existence and asymptotic behavior of the minimiziers to
$\inf_{u\in\tilde{\cal S}(a)}I(u)+\int_{\mathbb{R}^{N}}a(x)|u|^{2},$
where $a(x)$ is an infinite pontential well. In addition to these minimizition
approaches, L. Jeanjean, T. J. Luo and Z. Q. Wang [26] obtained another
mountain-pass type normalized solution of (1.3) through the perturbation
method. We remark that all of these results on normalized solution of (1.3)
only considered the mass-subcritical or mass-critical case, i.e., $2<p\leq
4+\frac{4}{N}$.
In this paper, we consider the mass-critical and mass-supercritical case,
i.e., $p\geq 4+\frac{4}{N}$. To the best of our knowledge, the case of mass-
supercritical has not been considered before. Actually, we obtain
###### Theorem 1.1.
Assume that one of the following conditons holds:
* (H1)
$N=1,2$, $p>4+\frac{4}{N}$, $a>0$;
* (H2)
$N=3$, $4+\frac{4}{N}<p<2^{*}$, $a>0$.
Then there exists a radially symmetric positive ground state normalized
solution $u\in W^{1,2}({\mathbb{R}^{N}})\cap L^{\infty}({\mathbb{R}^{N}})$ of
(1.3) in the sense that
$I(u)=\inf\left\\{I(v):v\in\tilde{\cal S}(a),I|_{\tilde{\cal
S}(a)}^{\prime}(v)=0,v\neq 0\right\\}.$
###### Theorem 1.2.
Assume that one of the following conditons holds
* (H1)’
$N=2$, $p>4+\frac{4}{N}$, $a>0$,
* (H2)
$N=3$, $4+\frac{4}{N}<p<2^{*}$, $a>0$.
Then there exists a sequence of normalized solutions $u^{j}\in
W^{1,2}({\mathbb{R}^{N}})\cap L^{\infty}({\mathbb{R}^{N}})$ of (1.3) with
increasing energy $I(u^{j})\to+\infty$.
###### Remark 1.1.
* (1)
We state that the dimension is limited due to a lemma limitition used to
control the Lagrange multipliers, see Lemma 2.2 and Remark 4.1.
* (2)
The difference between Theorem 1.1 and Theorem 1.2 is that we can not prove
the existence of infinitely many solutions when $N=1$, because the failure of
the compact embedding $W^{1,2}({\mathbb{R}})\hookrightarrow\hookrightarrow
L^{q}({\mathbb{R}})$ for $2<q<2^{*}$. However when considering the ground
state, we are able to recover the compactness of bounded sequences using the
symmetric decreasing arrangement, due to the advantage of the associated
minimizition $m_{\mu}(a)$ defined in (3.8).
Now we turn to the mass-critical case, i.e., $p=4+\frac{4}{N}$. Let
$a_{*}=\|Q_{4+\frac{4}{N}}\|_{1}$.
###### Theorem 1.3.
Assume that one of the following conditons holds:
* (H3)
$N\leq 3$, $p=4+\frac{4}{N}$, $a>a_{*}$;
* (H4)
$N\geq 4$, $p=4+\frac{4}{N}$,
$a_{*}<a<\left(\frac{N-2}{N-2-\frac{4}{N}}\right)^{\frac{N}{2}}a_{*}$,
Then there exists a radially symmetric positive ground state normalized
solution $u\in W^{1,2}({\mathbb{R}^{N}})\cap L^{\infty}({\mathbb{R}^{N}})$ of
(1.3) in the sense that
$I(u)=\inf\left\\{I(v):v\in\tilde{\cal S}(a),I|_{\tilde{\cal
S}(a)}^{\prime}(v)=0,v\neq 0\right\\}.$
###### Remark 1.2.
In a very recent paper [51], H. Y. Ye and Y. Y. Yu obtained the existence of
ground state normalized solution of (1.3) under assumption $(H3)$. As one can
see, although Theorem 1.3 contains their existence result, the method we used
in the current paper is totally different from theirs, while as they said in
[51, Remark 1.3], they are unable to handle the case $N\geq 4$. Moreover, they
also consider a asymptotic behavior, but our Theorem 1.4 is more accurate,
since we give a discription of $u_{n}$ when $a\to a_{*}$.
We observe that when $p=4+\frac{4}{N}$, the value $a_{*}$ is a threshold of
the existence of normalized solution of (1.3). Actually, we have
###### Proposition 1.1.
Let $p=4+\frac{4}{N}$ and $N\geq 1$. Then
* (1)
$\tilde{m}(a)=\begin{cases}&0,\quad 0<a\leq a_{*},\\\ &-\infty,\quad
a>a_{*}.\end{cases}$
* (2)
(1.3) has no solutions for any $0<a\leq a_{*}$.
* (3)
(1.3) has at least one radially symmetric positive solution for $a>a_{*}$ and
$a$ is close to $a_{*}$.
###### Remark 1.3.
We state that (1) is a direct conclusion of [16, Theorem 1.9], and (3) is a
direct conclusion of Theorem 1.3 above. Now we prove (2). Since $u$ is a
solution of (1.3), there holds (see Lemma 2.1)
$\int_{\mathbb{R}^{N}}|\nabla u|^{2}+(2+N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\frac{N(2+N)}{4(N+1)}\int_{\mathbb{R}^{N}}|u|^{4+\frac{4}{N}}=0.$
Combining with (1.12), we obtain
$\int_{\mathbb{R}^{N}}|\nabla u|^{2}+(2+N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}\leq(2+N)\left(\frac{a}{a_{*}}\right)^{\frac{2}{N}}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2},$
from which we get $u=0$ for any $0<a\leq a_{*}$, a contradiction since
$\|u\|_{2}=a$.
Inspired by Proposition 1.1, we enlighten a concentration behavior of the
radially symmetric positive solution of (1.3) when $p=4+\frac{4}{N}$ and $a\to
a_{*}$.
###### Theorem 1.4.
Let $p=4+\frac{4}{N}$, $N\geq 1$, and let $u_{n}$ be a radially symmetric
positive solution of (1.3) for $a=a_{n}$ with $a_{n}>a_{*}$ and $a_{n}\to
a_{*}$. Then there exists a sequence $y_{n}\in{\mathbb{R}^{N}}$ such that up
to a subsequence,
(1.13)
$\left[\left(\frac{Na_{*}}{N}\right)^{\frac{1}{2+N}}\varepsilon_{n}\right]^{N}u_{n}^{2}\left(\left(\frac{Na_{*}}{N}\right)^{\frac{1}{2+N}}\varepsilon_{n}x+\varepsilon_{n}y_{n}\right)\to
Q_{4+\frac{4}{N}}\quad\text{in}~{}L^{q}({\mathbb{R}^{N}})$
for $1\leq q<2^{*}$, where
$\varepsilon_{n}=\left(\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla
u_{n}|^{2}\right)^{-(2+N)}\to 0.$
###### Remark 1.4.
Theorem 1.4 gives a description of radially symmetric positive solution of
(1.3) as the mass $a_{n}$ approaches to $a_{*}$ from above. Roughly speaking,
it shows that for $N$ large enough, we have
$u_{n}(x)=\left[\left(\frac{Na_{*}}{N}\right)^{\frac{1}{2+N}}\varepsilon_{n}\right]^{-\frac{N}{2}}Q_{4+\frac{4}{N}}\left(\left(\frac{Na_{*}}{N}\right)^{-\frac{1}{2+N}}\varepsilon_{n}^{-1}(x-\varepsilon_{n}^{-1}y_{n})\right).$
The paper is organized as follows. In Section 2, we give perturbation settings
and an important lemma. In section 3.1, we give some properties of the
associated Pohozaev manifold. In section 3.2 and 3.3, we prove the existence
of ground state and infinitely many critical points for perturbed funtional.
In section 4, we study the convergence of the critical points the perturbated
funtional as $\mu\to 0^{+}$. And the Theorem 1.1 for $N=1$ is proved in
section 3.2; the Theorem 1.1 for $N\geq 2$ and Theorem 1.2 are proved in
section 4. Finally, in section 5, we study the mass-critical case, and prove
Theorems 1.3, 1.4. In the Appendix, we prove some valuable results.
Throughtout the paper, we use standard notations. For simplicity, we write
$\int_{\mathbb{R}^{N}}f$ to mean the Lebesgue integral of $f(x)$ over
${\mathbb{R}^{N}}$. $\|\cdot\|_{p}$ denotes the standard norm of
$L^{p}({\mathbb{R}^{N}})$; We use “$\to$” and “$\rightharpoonup$” to denote
the strong and weak convergence in the related function space respectively;
$C,C_{1},C_{2},\cdots$ will denote positive constants unless specified.
## 2 Preliminary
### 2.1 Perturbation setting
Let $I(u)$ be defined by (1.6). Observe that when $N=1$, $I(u)$ is of calss
${\cal C}^{1}$ in $W^{1,2}({\mathbb{R}})$, so there is no need to perturb
$I(u)$, and in this case the proof will be stated separately in the last of
Section 3.2. Thus we assume $N\geq 2$. To avoid the non-differentiability, we
define for $\mu\in(0,1]$,
(2.1) $I_{\mu}(u):=\frac{\mu}{\theta}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+I(u)$
on the space ${\cal X}:=W^{1,\theta}({\mathbb{R}^{N}})\cap
W^{1,2}({\mathbb{R}^{N}})$ for some fixed $\theta$ satisfying
$\frac{4N}{N+2}<\theta<\min\left\\{\frac{4N+4}{N+2},N\right\\}\quad\text{when}~{}N\geq
3$
and
$2<\theta<3\quad\text{when}~{}N=2.$
Then ${\cal X}$ is a reflexive Banach space. And Lemma A.1 implies
$I_{\mu}\in{\cal C}^{1}({\cal X})$. We will consider $I_{\mu}$ on the
constraint
(2.2) ${\cal S}(a):=\left\\{u\in{\cal
X}:\int_{\mathbb{R}^{N}}|u|^{2}=a\right\\}.$
Recalling the $L^{2}$-norm preserved transform [23]
$u\in{\cal S}(a)\mapsto s\star u(x)=e^{\frac{N}{2}s}u(e^{s}x)\in{\cal S}(a),$
we define
(2.3) $\displaystyle Q_{\mu}(u)$
$\displaystyle:=\frac{\mathrm{d}}{\mathrm{d}s}\big{|}_{s=0}I_{\mu}(s\star u)$
$\displaystyle=(1+\gamma_{\theta})\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+(2+N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\gamma_{p}\int_{\mathbb{R}^{N}}|u|^{p},$
where $\gamma_{p}=\frac{N(p-2)}{2p}$. And again Lemma A.1 implies
$Q_{\mu}\in{\cal C}^{1}({\cal X})$. Then we define the manifold
(2.4) ${\cal Q}_{\mu}(a):=\left\\{u\in{\cal S}(a):Q_{\mu}(u)=0\right\\}.$
We observed that
###### Lemma 2.1.
Any critical point $u$ of $I_{\mu}|_{{\cal S}(a)}$ is contained in ${\cal
Q}_{\mu}(a)$.
###### Proof.
By [11, Lemma 3], there exists a $\lambda\in{\mathbb{R}}$ such that
(2.5) $I_{\mu}^{\prime}(u)+\lambda u=0\quad\text{in}~{}{\cal X}^{*}.$
On the one hand, testing (2.5) with $x\cdot\nabla u$, see [10, Proposition 1]
for details, we obtain
(2.6) $\displaystyle 0$
$\displaystyle=\frac{\theta-N}{\theta}\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\frac{2-N}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+(2-N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}+\frac{N}{p}\int_{\mathbb{R}^{N}}|u|^{p}-\frac{N}{2}\lambda\int_{\mathbb{R}^{N}}|u|^{2}.$
On the other hand, testing (2.5) with $u$, we obtain
(2.7) $0=\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+4\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\int_{\mathbb{R}^{N}}|u|^{p}+\lambda\int_{\mathbb{R}^{N}}|u|^{2}.$
Combining (2.6) and (2.7), we have $Q_{\mu}(u)=0$. Then $u\in{\cal
Q}_{\mu}(a)$. ∎
### 2.2 An important lemma
We need the following result, which are crucially used to control the possible
values of the Lagrange parameters.
###### Lemma 2.2.
Suppose $u\neq 0$ is a critical point of $I_{\mu}|_{{\cal S}(a)}$ with
$0\leq\mu\leq 1$, that is there exists a $\lambda\in{\mathbb{R}}$ such that
$I_{\mu}^{\prime}(u)+\lambda u=0\quad\text{in}~{}{\cal X}^{*}.$
And assume that one of the following conditions holds
* (a)
$1\leq N\leq 3$, $4+\frac{4}{N}\leq p\leq 2^{*}$, $a>0$,
* (b)
$N\geq 4$, $p=4+\frac{4}{N}$,
$0<a<\left(\frac{N-2}{N-2-\frac{4}{N}}\right)^{\frac{N}{2}}a_{*}$,
then $\lambda>0$.
###### Proof.
By combining $Q_{\mu}(u)=0$ and (1.3), we obtain
$\displaystyle\frac{\lambda N(p-2)}{2p}a$
$\displaystyle=\left(1+\frac{N(p-\theta)}{p\theta}\right)\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\frac{2N-(N-2)p}{2p}\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\quad~{}+\frac{4N-(N-2)p}{2p}\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}.$
So if condition (a) holds, we immediately get $\lambda>0$. Now suppose
condition (b) holds. Again from $Q_{\mu}(u)=0$ and (2.7), and using inequality
(1.12), we obtain
$\displaystyle\lambda a$
$\displaystyle=\frac{N(\theta-2)}{2\theta}\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+(N-2)\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}-\frac{N^{2}-2N-4}{4(N+1)}\int_{\mathbb{R}^{N}}|u|^{4+\frac{4}{N}}$
$\displaystyle\geq\left[(N-2)-(N-2-\frac{4}{N})\left(\frac{a}{a_{*}}\right)^{\frac{2}{N}}\right]\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}$ $\displaystyle>0,$
which gives $\lambda>0$. ∎
## 3 The critical points of perturbed functional
In this whole section, we assume $p>4+\frac{4}{N}$.
### 3.1 Properties of ${\cal Q}_{\mu}(a)$
###### Lemma 3.1.
Let $0<\mu\leq 1$, then ${\cal Q}_{\mu}(a)$ is a ${\cal C}^{1}$-submanifold of
codimension 1 in ${\cal S}(a)$, hence a ${\cal C}^{1}$-submanifold of
codimension 2 in ${\cal X}$.
###### Proof.
As a subset of ${\cal X}$, the set ${\cal Q}_{\mu}(a)$ is defined by the two
equations $G(u)=0$, $Q_{\mu}(u)=0$, where
$G(u)=a-\int_{\mathbb{R}^{N}}|u|^{2},$
and clearly $G\in{\cal C}^{1}({\cal X})$. We have to check that
(3.1) $\mathrm{d}(Q_{\mu},G):{\cal X}\to{\mathbb{R}}^{2}\quad\text{is
surjective}.$
If this is not ture, $\mathrm{d}Q_{\mu}(u)$ and $\mathrm{d}G(u)$ are linearly
dependent, i.e., there exists $\nu\in{\mathbb{R}}$ such that
(3.2)
$\displaystyle\quad~{}~{}\theta(1+\gamma_{\theta})\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta-2}\nabla u\cdot\nabla\phi+2\int_{\mathbb{R}^{N}}\nabla
u\cdot\nabla\phi$ $\displaystyle+(2+N)2\int_{\mathbb{R}^{N}}(|u|^{2}\nabla
u\cdot\nabla\phi+u\phi|\nabla
u|^{2})-p\gamma_{p}\int_{\mathbb{R}^{N}}|u|^{p-2}u\phi=2\nu\int_{\mathbb{R}^{N}}u\phi,$
for any $\phi\in{\cal X}$. Similar as Lemma 2.1, taking $\phi=x\cdot\nabla u$
and $\phi=u$, we obtain
(3.3) $\theta(1+\gamma_{\theta})^{2}\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+2\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+(2+N)^{2}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-p\gamma_{p}^{2}\int_{\mathbb{R}^{N}}|u|^{p}=0.$
Since $Q_{\mu}(u)=0$, we get
(3.4)
$\displaystyle(p\gamma_{p}-\theta-\theta\gamma_{\theta})(1+\gamma_{\theta})\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+(p\gamma_{p}-2)\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}+(p\gamma_{p}-2-N)(2+N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}=0,$
which means $u=0$ since $p\gamma_{p}>\theta+\theta\gamma_{\theta}$ and
$p\gamma_{p}>2+N$. That contradics with $u\in{\cal S}(a)$. ∎
Now, we prove the following important lemma.
###### Lemma 3.2.
For any $0<\mu\leq 1$ and any $u\in{\cal X}\setminus\\{0\\}$, the following
statements hold.
* (1)
There exists a unique number $s_{\mu}(u)\in{\mathbb{R}}$ such that
$Q_{\mu}(s_{\mu}(u)\star u)=0$.
* (2)
$I_{\mu}(s\star u)$ is strictly increasing in $s\in(-\infty,s_{\mu}(u))$ and
is strictly decreasing in $s\in(s_{\mu}(u),+\infty)$, and
$\lim_{s\to-\infty}I_{\mu}(s\star
u)=0^{+},\quad\lim_{s\to+\infty}I_{\mu}(s\star u)=-\infty,\quad
I_{\mu}(s_{\mu}(u)\star u)>0.$
* (3)
$s_{\mu}(u)<0$ if and only if $Q_{\mu}(u)<0$.
* (4)
The map $u\in{\cal X}\setminus\\{0\\}\mapsto s_{\mu}(u)\in{\mathbb{R}}$ is of
class ${\cal C}^{1}$.
* (5)
$s_{\mu}(u)$ is an even function with respect to $u\in{\cal
X}\setminus\\{0\\}$.
###### Proof.
* (1)
By direct computation, one can check that
(3.5) $\displaystyle Q_{\mu}(s\star u):$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}I_{\mu}(s\star u)$
$\displaystyle=(1+\gamma_{\theta})\mu
e^{\theta(1+\gamma_{\theta})s}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+e^{2s}\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}+(2+N)e^{(2+N)s}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\gamma_{p}e^{p\gamma_{p}s}\int_{\mathbb{R}^{N}}|u|^{p}$
$\displaystyle=e^{p\gamma_{p}s}\big{[}(1+\gamma_{\theta})\mu
e^{-(p\gamma_{p}-\theta-\theta\gamma_{\theta})s}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+e^{-(p\gamma_{p}-2)s}\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}+(2+N)e^{-(p\gamma_{p}-2-N)s}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\gamma_{p}\int_{\mathbb{R}^{N}}|u|^{p}\big{]}$
Since $p\gamma_{p}>\theta+\theta\gamma_{\theta}$ and $p\gamma_{p}>2+N$ when
$p>4+\frac{4}{N}$, $Q_{\mu}(s\star u)=0$ has only one solution
$s_{\mu}(u)\in{\mathbb{R}}$.
* (2)
From $(1)$, $Q_{\mu}(s\star u)>0$ when $s<s_{\mu}(u)$ and $Q_{\mu}(s\star
u)<0$ when $s>s_{\mu}(u)$. So $I_{\mu}(s\star u)$ is strictly increasing in
$s\in(-\infty,s_{\mu}(u))$ and is strictly decreasing in
$s\in(s_{\mu}(u),+\infty)$. Obviously,
$\lim_{s\to-\infty}I_{\mu}(s\star
u)=0^{+},\quad\lim_{s\to+\infty}I_{\mu}(s\star u)=-\infty,$
which implies that
$I_{\mu}(s_{\mu}(u)\star u)=\max_{s\in{\mathbb{R}}}I_{\mu}(s\star u)>0.$
* (3)
It can be obtained directly from $(2)$.
* (4)
Let $\Phi_{\mu}(s,u)=Q_{\mu}(s\star u)$. Then $\Phi_{\mu}(s_{\mu}(u),u)=0$.
Moreover,
(3.6) $\displaystyle\frac{\partial}{\partial s}\Phi_{\mu}(s,u)$
$\displaystyle=\theta(1+\gamma_{\theta})^{2}\mu
e^{\theta(1+\gamma_{\theta})s}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+2e^{2s}\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}+(2+N)^{2}e^{(2+N)s}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-p\gamma_{p}^{2}e^{p\gamma_{p}s}\int_{\mathbb{R}^{N}}|u|^{p}.$
Combining with $Q_{\mu}(s_{\mu}(u)\star u)=0$, we obtain
(3.7) $\displaystyle\frac{\partial}{\partial s}\Phi_{\mu}(s_{\mu}(u),u)$
$\displaystyle=-(p\gamma_{p}-\theta-\theta\gamma_{\theta})(1+\gamma_{\theta})\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}-(p\gamma_{p}-2)\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}-(p\gamma_{p}-2-N)(2+N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}<0.$
Then the Implict Function Theorem [14] implies that the map $u\mapsto
s_{\mu}(u)$ is of class ${\cal C}^{1}$.
* (5)
Since
$Q_{\mu}(s_{\mu}(u)\star(-u))=Q_{\mu}(-s_{\mu}(u)\star
u)=Q_{\mu}(s_{\mu}(u)\star u)=0,$
by the uniqueness, there is $s_{\mu}(-u)=s_{\mu}(u)$.
∎
### 3.2 Ground state critical point of $I_{\mu}|_{{\cal S}(a)}$
In this subsection, we consider a minimizition problem
(3.8) $m_{\mu}(a):=\inf_{u\in{\cal Q}_{\mu}(a)}I_{\mu}(u).$
From Lemma 2.1, we know that if $m_{\mu}(a)$ is achieved, then the minimizer
is a ground state critical point of $I_{\mu}|_{{\cal S}(a)}$. We have the
following lemma.
###### Lemma 3.3.
The following statements hold.
* (1)
${\cal D}(a):=\inf_{0<\mu\leq 1,u\in{\cal
Q}_{\mu}(a)}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}>0$ is independent of
$\mu$.
* (2)
If $\sup_{n\geq 1}I_{\mu}(u_{n})<+\infty$ for $u_{n}\in{\cal Q}_{\mu}(a)$,
then
$\sup_{n\geq 1}\max\left\\{\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta},\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2},\int_{\mathbb{R}^{N}}|\nabla u|^{2}\right\\}<+\infty.$
###### Proof.
* (1)
For any $u\in{\cal Q}_{\mu}(a)$, by the inequality (1.12), there holds
(3.9) $(2+N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}\leq\gamma_{p}\int_{\mathbb{R}^{N}}|u|^{p}\leq
K(p,N)\gamma_{p}a^{\frac{4N-p(N-2)}{2(N+2)}}\left(\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}\right)^{\frac{N(p-2)}{2(N+2)}}.$
Since $\frac{N(p-2)}{2(N+2)}>1$, we obtain ${\cal D}(a)>0$.
* (2)
For any $u\in{\cal Q}_{\mu}(a)$, there is
(3.10) $\displaystyle I_{\mu}(u)$
$\displaystyle=I_{\mu}(u)-\frac{1}{p\gamma_{p}}Q_{\mu}(u)$
$\displaystyle=\frac{p\gamma_{p}-\theta-\theta\gamma_{\theta}}{\theta
p\gamma_{p}}\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\frac{p\gamma_{p}-2}{2p\gamma_{p}}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\frac{p\gamma_{p}-2-N}{p\gamma_{p}}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}.$
So the conclusion holds.
∎
###### Remark 3.1.
Form (3.10), we see that
$m_{\mu}(a)\geq{\cal D}_{0}(a):=\frac{p\gamma_{p}-2-N}{p\gamma_{p}}{\cal
D}(a)>0,\quad\forall\mu\in(0,1].$
Then we have the following result.
###### Lemma 3.4.
There exists a small $\rho>0$ independent of $\mu$ such that for any
$0<\mu\leq 1$, we have that
$0<\sup_{u\in B_{\mu}(\rho,a)}I_{\mu}(u)<{\cal D}_{0}(a)\quad\text{and}\quad
I_{\mu}(u),Q_{\mu}(u)>0,~{}~{}\forall u\in B_{\mu}(\rho,a),$
where
$B_{\mu}(\rho,a)=\left\\{u\in{\cal S}(a):\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}\leq\rho\right\\}.$
###### Proof.
From the definition of $I_{\mu}$, we have
$\sup_{u\in
B_{\mu}(\rho,a)}I_{\mu}(u)\leq\max\left\\{\frac{1}{\theta},\frac{1}{2},1\right\\}\rho<{\cal
D}_{0}(a),$
where $\rho>0$ is small and is independent of $\mu$. On the other hand, by
inequality (1.12), for any $u\in\partial B_{\mu}(r,a)$ with $0<r<\rho$ for a
smaller $\rho>0$,
(3.11) $\displaystyle\inf_{\partial B_{\mu}(r,a)}I_{\mu}(u)$
$\displaystyle\geq\frac{\mu}{\theta}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}-\frac{K(p,N)}{p}a^{\frac{4N-p(N-2)}{2(N+2)}}\left(\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}\right)^{\frac{N(p-2)}{2(N+2)}}$
$\displaystyle\geq\frac{\mu}{\theta}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+C\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}$ $\displaystyle\geq
C_{1}(a,\theta,p,N)r>0,$ $\displaystyle\inf_{\partial B_{\mu}(r,a)}Q_{\mu}(u)$
$\displaystyle\geq C_{2}(a,\theta,p,N)r>0,$
which finish the proof. ∎
To find a Palais-Smale sequence, we consider an auxilary funtional as the one
in [23],
(3.12) $J_{\mu}(s,u):=I_{\mu}(s\star u):{\mathbb{R}}\times{\cal
X}\to{\mathbb{R}}.$
We study $J_{\mu}$ on the radial space ${\mathbb{R}}\times{\cal S}_{r}(a)$
with
${\cal S}_{r}(a):={\cal S}(a)\cap{\cal X}_{r},\quad{\cal
X}_{r}=W^{1,\theta}_{rad}({\mathbb{R}^{N}})\cap
W^{1,2}_{rad}({\mathbb{R}^{N}}).$
Notice that $J_{\mu}$ is of class ${\cal C}^{1}$. By the Symmetric Critical
Point Principle [42], a Palais-Smale sequence for
$J_{\mu}|_{{\mathbb{R}}\times{\cal S}_{r}(a)}$ is also a Palais-Smale sequence
for $J_{\mu}|_{{\mathbb{R}}\times{\cal S}(a)}$. Denoting the closed sublevel
set by
$I_{\mu}^{c}=\left\\{u\in{\cal S}(a):I_{\mu}(u)\leq c\right\\},$
we introduce the minimax class
(3.13) $\Gamma_{\mu}:=\left\\{\gamma=(\alpha,\beta)\in{\cal
C}([0,1],{\mathbb{R}}\times{\cal S}_{r}(a)):\gamma(0)\in\\{0\\}\times
B_{\mu}(\rho,a),\gamma(1)\in\\{0\\}\times I_{\mu}^{0}\right\\},$
with the associated minimax level
(3.14)
$\sigma_{\mu}(a):=\inf_{\gamma_{\in}\Gamma_{\mu}}\sup_{t\in[0,1]}J_{\mu}(\gamma(t)).$
Then
###### Lemma 3.5.
For any $0<\mu\leq 1$, $m_{\mu}(a)=\sigma_{\mu}(a)$.
###### Proof.
For any $\gamma=(\alpha,\beta)\in\Gamma_{\mu}$, let us consider the function
$f_{\gamma}(t):=Q_{\mu}(\alpha(t)\star\beta(t)).$
We have $f_{\gamma}(0)=Q_{\mu}(\beta(0))>0$, by Lemma 3.4. We claim that
$f_{\gamma}(1)=Q_{\mu}(\beta(1))<0$: indeed, since $I_{\mu}(\beta(1))<0$, we
have that $s_{\mu}(\beta(1))<0$, which means that $Q_{\mu}(\beta(1))<0$ by
Lemma 3.2. Moreover, $f_{\gamma}$ is continuous, and hence we deduce that
there exists $t_{\gamma}\in(0,1)$ such that $f_{\gamma}(t_{\gamma})=0$, namely
$\alpha(t_{\gamma})\star\beta(t_{\gamma})\in{\cal Q}_{\mu}(a)$. So
$\max_{t\in[0,1]}J_{\mu}(\gamma(t))\geq
I_{\mu}(\alpha(t_{\gamma})\star\beta(t_{\gamma}))\geq m_{\mu}(a),$
and consequently $\sigma_{\mu}(a)\geq m_{\mu}(a)$.
On the other hand, if $u\in{\cal Q}_{\mu}(a)\cap{\cal X}_{r}$, then
$\gamma_{u}(t):=(0,((1-t)s_{0}+ts_{1})\star u)\in\Gamma_{\mu},$
where $s_{0}\ll-1$ and $s_{1}\gg 1$. Since
$I_{\mu}(u)\geq\max_{t\in[0,1]}I_{\mu}(((1-t)s_{0}+ts_{1})\star
u)\geq\sigma_{\mu}(a),$
there holds
$m_{\mu}^{r}(a):=\inf_{u\in{\cal Q}_{\mu}(a)\cap{\cal
X}_{r}}I_{\mu}(u)\geq\sigma_{\mu}(a).$
Finally the inequality $m_{\mu}(a)\geq m_{\mu}^{r}(a)$ can be obtained easily
by using the Symmetric decreasing rearrangement, see [32]. ∎
###### Remark 3.2.
For any $0<\mu_{1}<\mu_{2}\leq 1$, since $I_{\mu_{2}}(u)\geq I_{\mu_{1}}(u)$
and $\Gamma_{\mu_{2}}\subset\Gamma_{\mu_{1}}$, there holds
$\sigma_{\mu_{2}}(a)=\inf_{\gamma_{\in}\Gamma_{\mu_{2}}}\sup_{t\in[0,1]}J_{\mu_{2}}(\gamma(t))\geq\inf_{\gamma_{\in}\Gamma_{\mu_{2}}}\sup_{t\in[0,1]}J_{\mu_{1}}(\gamma(t))\geq\inf_{\gamma_{\in}\Gamma_{\mu_{1}}}\sup_{t\in[0,1]}J_{\mu_{1}}(\gamma(t))=\sigma_{\mu_{1}}(a),$
i.e., $\sigma_{\mu}(a)$ is non-decreasing with respect to $\mu\in(0,1]$.
We recall the following definition and theorem from [18].
###### Definition A. [18, Definition 3.1].
Let $B$ be a closed subset of $X$. We say that a class ${\cal F}$ of compact
subsets of $X$ is a homotopy stable family with boundary $B$ provided
* (a)
every set in ${\cal F}$ contains $B$.
* (b)
for any set $A$ in ${\cal F}$ and any $\eta\in{\cal C}([0,1]\times X,X)$
satisfying $\eta(t,x)=x~{}$ for all $(t,x)$ in $(\\{0\\}\times
X)\cup([0,1]\times B)$ we have that $\eta(1,A)\subset{\cal F}$.
We remark that the case $B=\emptyset$ is admissible.
###### Theorem B. [18, Theorem 5.2].
Let $\phi$ be a ${\cal C}^{1}$-functional on a complete connected
$C^{1}$-Finsler manifold $X$ and consider a homotopy stable family ${\cal F}$
with an extended closed boundary $B$. Set $c=c(\phi,{\cal F})$ and let $F$ be
a closed subset of $X$ satisfying
(3.15) $A\cap F\setminus B\neq\emptyset\quad\forall A\in{\cal F}$
and
(3.16) $\sup\phi(B)\leq c\leq\inf\phi(F).$
Then for any sequence of sets $A_{n}\subset{\cal F}$ such that
$\lim_{n\to\infty}\sup_{A_{n}}\phi=c$, there exists a sequence $x_{n}\subset
X\setminus B$ such that
* (1)
$\lim_{n\to\infty}\phi(x_{n})=c$,
* (2)
$\lim_{n\to\infty}\|\mathrm{d}\phi(x_{n})\|=0$,
* (3)
$\lim_{n\to\infty}\text{dist}(x_{n},F)=0$,
* (4)
$\lim_{n\to\infty}\text{dist}(x_{n},A_{n})=0$.
Now we establish a technical result showing the existence of a Palais-Smale
sequence of $\sigma_{\mu}(a)$ with an additional property.
###### Lemma 3.6.
For any fixed $\mu\in(0,1]$, there exists a sequence $u_{n}\in{\cal S}_{r}(a)$
such that
$I_{\mu}(u_{n})\to\sigma_{\mu}(a),\quad I_{\mu}|_{{\cal
S}(a)}^{\prime}(u_{n})\to 0,\quad Q_{\mu}(u_{n})\to 0\quad\text{and}\quad
u_{n}^{-}\to 0\text{ a.e. in }{\mathbb{R}^{N}}.$
###### Proof.
Using Definition Definition A. [18, Definition 3.1], it is easy to check that
${\cal F}=\left\\{A=\gamma([0,1]):\gamma\in\Gamma_{\mu}\right\\}$ is a
homotopy stable family of compact subsets of $X={\mathbb{R}}\times{\cal
S}_{\mu}^{r}$ with boundary $B=(\\{0\\}\times
B_{\mu}(\rho,a))\cup(\\{0\\}\times I_{\mu}^{0})$. Set
$F=\left\\{J_{\mu}\geq\sigma_{\mu}(a)\right\\}$, then the assumptions (3.15)
and (3.16) with $\phi=J_{\mu}$, $c=\sigma_{\mu}(a)$ are satisfied. Therefore,
taking a minimizing sequence
$\left\\{\gamma_{n}=(0,\beta_{n})\right\\}\subset\Gamma_{\mu}$ with
$\beta_{n}\geq 0$ a.e. in ${\mathbb{R}^{N}}$, there exists a Palais-Smale
sequence $\left\\{(s_{n},w_{n})\right\\}\subset{\mathbb{R}}\times{\cal
S}_{r}(a)$ for $J_{\mu}|_{{\mathbb{R}}\times{\cal S}_{r}(a)}$ at level
$\sigma_{\mu}(a)$, that is
(3.17) $\partial_{s}J_{\mu}(s_{n},w_{n})\to
0\quad\text{and}\quad\partial_{u}J_{\mu}(s_{n},w_{n})\to 0\quad\text{as
}n\to\infty,$
with the additional property that
(3.18) $|s_{n}|+\text{dist}_{\cal X}(w_{n},\beta_{n}([0,1]))\to 0\quad\text{as
}n\to\infty.$
Let $u_{n}=s_{n}\star w_{n}$. The first condition in (3.17) reads
$Q_{\mu}(u_{n})\to 0$, while the second condition gives
(3.19) $\displaystyle\|\mathrm{d}I_{\mu}|_{{\cal S}(a)}(u_{n})\|$
$\displaystyle=\sup_{\psi\in T_{u_{n}}{\cal S}(a),\|\psi\|_{\cal X}\leq
1}|\mathrm{d}I_{\mu}(u_{n})[\psi]|$ $\displaystyle=\sup_{\psi\in
T_{u_{n}}{\cal S}(a),\|\psi\|_{\cal X}\leq 1}|\mathrm{d}I_{\mu}(s_{n}\star
w_{n})[s_{n}\star(-s_{n})\star\psi]|$ $\displaystyle=\sup_{\psi\in
T_{u_{n}}{\cal S}(a),\|\psi\|_{\cal X}\leq
1}|\partial_{u}J_{\mu}(s_{n},w_{n})[(-s_{n})\star\psi]|$
$\displaystyle\leq\|\partial_{u}J_{\mu}(s_{n},w_{n})\|\sup_{\psi\in
T_{u_{n}}{\cal S}(a),\|\psi\|_{\cal X}\leq 1}|(-s_{n})\star\psi|$
$\displaystyle\leq C\|\partial_{u}J_{\mu}(s_{n},w_{n})\|\to 0\quad\text{as
}n\to\infty.$
Finally, (3.18) implies that $u_{n}^{-}\to 0$ a.e. in ${\mathbb{R}^{N}}$. ∎
Now we show the compactness of the Palais-Smale sequence obtained in Lemma
3.6.
###### Lemma 3.7.
For any fixed $\mu\in(0,1]$, let $u_{n}$ be a sequence obtained in Lemma 3.6.
Then there exists a $u_{\mu}\in{\cal X}\setminus\\{0\\}$ and a
$\lambda_{\mu}\in{\mathbb{R}}$ such that up to a subsequence,
(3.20) $u_{n}\rightharpoonup u_{\mu}\geq 0\quad\text{in }{\cal X},$ (3.21)
$I_{\mu}(u_{\mu})=\sigma_{\mu}(a)\quad\text{and}\quad
I_{\mu}^{\prime}(u_{\mu})+\lambda_{\mu}u_{\mu}=0.$
Moreover, if $\lambda_{\mu}\neq 0$, we have that
$u_{n}\rightarrow u_{\mu}\quad\text{in }{\cal X}.$
###### Proof.
From Lemma 3.3 and Remark 3.2, we know that $u_{n}$ is bounded in ${\cal
X}_{r}$. Thus by [13, Propositon 1.7.1], we conclude that up to a subsequence,
there exists a $u_{\mu}\in{\cal X}_{r}$ such that
$u_{n}\rightharpoonup u_{\mu}\quad\text{in }{\cal X}\text{ and in
}L^{2}({\mathbb{R}^{N}}),$ $u_{n}\rightarrow u_{\mu}\quad\text{in
}L^{q}({\mathbb{R}^{N}}),~{}\forall q\in(2,2^{*}),$ $u_{n}\rightarrow
u_{\mu}\geq 0\quad\text{a.e. in }{\mathbb{R}}.$
By interpolation and inequality (1.12), we have that
$u_{n}\rightarrow u_{\mu}\quad\text{in }L^{q}({\mathbb{R}^{N}}),~{}\forall
q\in(2,22^{*}).$
We claim that $u_{\mu}\neq 0$. Assume $u_{\mu}=0$, then as $n\to\infty$
$(1+\gamma_{\theta})\mu\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{\theta}+\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}+(2+N)\int_{\mathbb{R}^{N}}|u_{n}|^{2}|\nabla
u_{n}|^{2}=Q_{\mu}(u_{n})+\gamma_{p}\int_{\mathbb{R}^{N}}|u_{n}|^{p}\to 0,$
which implies that $I_{\mu}(u_{n})\to 0$, in contradiction with Remark 3.1. So
$u_{\mu}\neq 0$. By [11, Lemma 3], it follows from $I_{\mu}|_{{\cal
S}(a)}^{\prime}(u_{n})\to 0$ that there exists a sequence
$\lambda_{n}\in{\mathbb{R}}$ such that
(3.22) $I_{\mu}^{\prime}(u_{n})+\lambda_{n}u_{n}\to 0\quad\text{in}~{}{\cal
X}^{*}.$
Hence $\lambda_{n}=\frac{1}{a}I_{\mu}^{\prime}(u_{n})[u_{n}]+o_{n}(1)$ is
bounded in ${\mathbb{R}}$, and we assume, up to a subsequence,
$\lambda_{n}\to\lambda_{\mu}$. Since $u_{n}$ is bounded, we have
$I_{\mu}^{\prime}(u_{n})+\lambda_{\mu}u_{n}\to 0$. From Lemma A.2, we see that
(3.23) $I_{\mu}^{\prime}(u_{\mu})+\lambda_{\mu}u_{\mu}=0.$
Then testing (3.23) with $x\cdot\nabla u$ and $u$, we obtain
$Q_{\mu}(u_{\mu})=0$. It follows that
$Q_{\mu}(u_{n})+\gamma_{p}\int_{\mathbb{R}^{N}}|u_{n}|^{p}\to
Q_{\mu}(u_{\mu})+\gamma_{p}\int_{\mathbb{R}^{N}}|u_{\mu}|^{p}.$
Then using the weak lower semicontinuous property, see [16, Lemma 4.3], there
must be
(3.24) $\mu\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{\theta}\to\mu\int_{\mathbb{R}^{N}}|\nabla u_{\mu}|^{\theta},$ (3.25)
$\int_{\mathbb{R}^{N}}|\nabla u_{n}|^{2}\to\int_{\mathbb{R}^{N}}|\nabla
u_{\mu}|^{2},$ (3.26) $\int_{\mathbb{R}^{N}}|u_{n}|^{2}|\nabla
u_{n}|^{2}\to\int_{\mathbb{R}^{N}}|u_{\mu}|^{2}|\nabla u_{\mu}|^{2}.$
That gives $I_{\mu}(u_{\mu})=\lim_{n\to\infty}I_{\mu}(u_{n})=\sigma_{\mu}(a)$.
Moreover, from (3.24)-(3.26), we obtain
(3.27) $I_{\mu}^{\prime}(u_{n})[u_{n}]\to I_{\mu}^{\prime}(u_{\mu})[u_{\mu}].$
Thus combining (3.27) with (3.22)-(3.23), there holds
$\lambda_{\mu}\|u_{n}\|_{2}^{2}\to\lambda_{\mu}\|u_{\mu}\|_{2}^{2}$. So
$\lambda_{\mu}\neq 0$ implies that $u_{n}\rightarrow u_{\mu}$ in ${\cal X}$. ∎
Based on the above preliminary works, we conclude that
###### Theorem 3.1.
For any fixed $\mu\in(0,1]$, there exists a $u_{\mu}\in{\cal
X}_{r}\setminus\\{0\\}$ and a $\lambda_{\mu}\in{\mathbb{R}}$ such that
$\displaystyle I_{\mu}^{\prime}(u_{\mu})+\lambda_{\mu}u_{\mu}=0,$
$\displaystyle I_{\mu}(u_{\mu})=m_{\mu}(a),\quad Q_{\mu}(u_{\mu})=0,$
$\displaystyle 0<\|u_{\mu}\|_{2}^{2}\leq a,\quad u_{\mu}\geq 0.$
Moreover, if $\lambda_{\mu}\neq 0$, we have that $\|u_{\mu}\|_{2}^{2}=a$,
i.e., $m_{\mu}(a)$ is achieved, and $u_{\mu}$ is a ground state critical point
of $I_{\mu}|_{{\cal S}(a)}$.
###### Proof of Theorem 1.1 for the case $N=1$:.
When $N=1$, there is $W^{1,2}({\mathbb{R}})\hookrightarrow{\cal
C}^{0,\alpha}({\mathbb{R}})$, so $V(u)$ and hence $I(u)$ is of class ${\cal
C}^{1}(W^{1,2}({\mathbb{R}}))$. Then one can follow the process in this
subsection to prove Theorem 1.1 by taking $\mu=0$, but we claim that there
needs some necessary modifications, since the compact embedding
$W^{1,2}_{rad}({\mathbb{R}^{N}})\hookrightarrow\hookrightarrow
L^{q}({\mathbb{R}^{N}})$ for $2<q<2^{*}$ does not hold when $N=1$. However the
compactness still holds for bounded sequences of radially decreasing functions
(see e.g. [13, Propositon 1.7.1]). So we need to confirm that the Palais-Smale
sequence obtained in Lemma 3.6 consists of radially decreasing functions. Then
it is natural to replace the minimizing sequence $\gamma_{n}=(0,\beta_{n})$
choosen in Lemma 3.6 with $\bar{\gamma}_{n}:=(0,\bar{\beta}_{n})$, where
$\bar{\beta}_{n}(t)=|\beta_{n}(t)|^{*}$ is the symmetric decreasing
rearrangement of $\beta_{n}(t)$ at every $t\in[0,1]$. This is a natural
candidate to be minimizing sequence, with $\bar{\beta}_{n}(t)\geq 0$, radially
symmetric and decreasing for every $t\in[0,1]$. In order to check that
$\bar{\gamma}_{n}\in\Gamma_{0}$, we have to check that each $\bar{\beta}_{n}$
is continuous on $[0,1]$, which has been proved in [17] (for more argument we
refer to [48, Remark 5.2]). As a result, Theorem 3.1 with $\mu=0$ holds, and
combining with Lemma 2.2, we obtain Theorem 1.1 immediately. ∎
### 3.3 Infinitely many critical points of $I_{\mu}|_{{\cal S}(a)}$
This subsection concerns the existence of infinitely many radial critical
points of $I_{\mu}|_{{\cal S}(a)}$. Denote $\tau(u)=-u$ and let $Y\subset{\cal
X}$. A set $A\subset Y$ is called $\tau$-invariant if $\tau(A)=A$. A homotopy
$\eta:[0,1]\times Y\to Y$ is $\tau$-equivariant if
$\eta(t,\tau(u))=\tau(\eta(t,u))$ for all $(t,u)\in[0,1]\times Y$. We recall
the following definition.
###### Definition C. [18, Definition 7.1].
Let $B$ be a closed $\tau$-invariant subset of $Y$. A class ${\cal G}$ of
compact subsets of $Y$ is said to be a $\tau$-homotopy stable family with
boundary $B$ provided
* (a)
every set in ${\cal G}$ is $\tau$-invariant,
* (b)
every set in ${\cal G}$ contains $B$,
* (c)
for any set $A\in{\cal G}$ and any $\tau$-equivariant homotopy $\eta\in{\cal
C}([0,1]\times Y,Y)$ satisfying $\eta(t,x)=x$ for all $(t,x)$ in
$(\\{0\\}\times Y)\cup([0,1]\times B)$ we have that $\eta(1,A)\subset{\cal
G}$.
Following the strategy of [24, Section. 5], we consider the functional
$K_{\mu}:{\cal X}\setminus\\{0\\}\to{\mathbb{R}}$ defined by
(3.28) $\displaystyle K_{\mu}(u):$ $\displaystyle=I_{\mu}(s_{\mu}(u)\star u)$
$\displaystyle=\frac{\mu}{\theta}e^{\theta(1+\gamma_{\theta})s_{\mu}(u)}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\frac{1}{2}e^{2s_{\mu}(u)}\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\quad~{}~{}+e^{(2+N)s_{\mu}(u)}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}-\frac{1}{p}e^{p\gamma_{p}s_{\mu}(u)}\int_{\mathbb{R}^{N}}|u|^{p},$
where $s_{\mu}(u)$ is given by Lemma 3.2. Then we see that $K_{\mu}(u)$ is
$\tau$-invariant. Moreover, inspired by [50, Proposition 2.9], there holds
###### Lemma 3.8.
The functional $K_{\mu}$ is of class ${\cal C}^{1}$ and
$\displaystyle K_{\mu}^{\prime}(u)[\phi]$ $\displaystyle=\mu
e^{\theta(1+\gamma_{\theta})s_{\mu}(u)}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta-2}\nabla
u\cdot\nabla\phi+e^{2s_{\mu}(u)}\int_{\mathbb{R}^{N}}\nabla u\cdot\nabla\phi$
$\displaystyle\quad~{}~{}+2e^{(2+N)s_{\mu}(u)}\int_{\mathbb{R}^{N}}(u\phi|\nabla
u|^{2}+|u|^{2}\nabla
u\cdot\nabla\phi)-e^{p\gamma_{p}s_{\mu}(u)}\int_{\mathbb{R}^{N}}|u|^{p-2}u\phi$
$\displaystyle=I_{\mu}^{\prime}(s_{\mu}(u)\star u)[s_{\mu}(u)\star\phi],$
for any $u\in{\cal X}\setminus\\{0\\}$ and $\phi\in{\cal X}$.
###### Proof.
Let $u\in{\cal X}\setminus\\{0\\}$ and $\phi\in{\cal X}$. We estimate the term
$K_{\mu}(u_{t})-K_{\mu}(u)=I_{\mu}(s_{t}\star u_{t})-I_{\mu}(s_{0}\star u),$
where $u_{t}=u+t\phi$ and $s_{t}=s_{\mu}(u_{t})$ with $|t|$ small enough. By
the mean value theorem, we have
$\displaystyle\quad~{}~{}I_{\mu}(s_{t}\star u_{t})-I_{\mu}(s_{0}\star u)\leq
I_{\mu}(s_{t}\star u_{t})-I_{\mu}(s_{t}\star u)$ $\displaystyle=\mu
e^{\theta(1+\gamma_{\theta})s_{t}}\int_{\mathbb{R}^{N}}|\nabla
u_{\eta_{t}}|^{\theta-2}(\nabla
u\cdot\nabla\phi+\eta_{t}|\nabla\phi|^{2})t+e^{2s_{t}}\int_{\mathbb{R}^{N}}(\nabla
u\cdot\nabla\phi+\frac{t}{2}|\nabla\phi|^{2})t$
$\displaystyle\quad~{}+2e^{(2+N)s_{t}}\int_{\mathbb{R}^{N}}\left(u_{\eta_{t}}\phi|\nabla
u_{\eta_{t}}|^{2}+|u_{\eta_{t}}|^{2}(\nabla
u\cdot\nabla\phi+\eta_{t}|\nabla\phi|^{2})\right)t-e^{p\gamma_{p}s_{t}}\int_{\mathbb{R}^{N}}|u_{\eta_{t}}|^{p-2}(u\phi+\frac{\eta_{t}}{2}\phi^{2})t,$
where $|\eta_{t}|\in(0,|t|)$. Similarly,
$\displaystyle\quad~{}~{}I_{\mu}(s_{t}\star u_{t})-I_{\mu}(s_{0}\star u)\geq
I_{\mu}(s_{0}\star u_{t})-I_{\mu}(s_{0}\star u)$ $\displaystyle=\mu
e^{\theta(1+\gamma_{\theta})s_{0}}\int_{\mathbb{R}^{N}}|\nabla
u_{\xi_{t}}|^{\theta-2}(\nabla
u\cdot\nabla\phi+\xi_{t}|\nabla\phi|^{2})t+e^{2s_{0}}\int_{\mathbb{R}^{N}}(\nabla
u\cdot\nabla\phi+\frac{t}{2}|\nabla\phi|^{2})t$
$\displaystyle\quad~{}+2e^{(2+N)s_{0}}\int_{\mathbb{R}^{N}}\left(u_{\xi_{t}}\phi|\nabla
u_{\xi_{t}}|^{2}+|u_{\xi_{t}}|^{2}(\nabla
u\cdot\nabla\phi+\xi_{t}|\nabla\phi|^{2})\right)t-e^{p\gamma_{p}s_{0}}\int_{\mathbb{R}^{N}}|u_{\xi_{t}}|^{p-2}(u\phi+\frac{\xi_{t}}{2}\phi^{2})t,$
where $|\xi_{t}|\in(0,|t|)$. Since $s_{t}\to s_{0}$ as $t\to 0$, it follows
from the two inequalities above that
$\displaystyle\lim_{t\to 0}\frac{K_{\mu}(u_{t})-K_{\mu}(u)}{t}$
$\displaystyle=\mu
e^{\theta(1+\gamma_{\theta})s_{\mu}(u)}\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta-2}\nabla
u\cdot\nabla\phi+e^{2s_{\mu}(u)}\int_{\mathbb{R}^{N}}\nabla u\cdot\nabla\phi$
$\displaystyle\quad~{}~{}+2e^{(2+N)s_{\mu}(u)}\int_{\mathbb{R}^{N}}(u\phi|\nabla
u|^{2}+|u|^{2}\nabla
u\cdot\nabla\phi)-e^{p\gamma_{p}s_{\mu}(u)}\int_{\mathbb{R}^{N}}|u|^{p-2}u\phi.$
Then similarly as Lemma A.1, we see that the Gâteaux derivative of $K_{\mu}$
is bounded linear and continuous. Therefore $K_{\mu}$ is of class ${\cal
C}^{1}$, see [14]. In particular, by changing variables in the integrals, we
have
$K_{\mu}^{\prime}(u)[\phi]=I_{\mu}^{\prime}(s_{\mu}(u)\star
u)[s_{\mu}(u)\star\phi].$
The proof is complete. ∎
To get the particular Palais-Smale sequence of $I_{\mu}|_{{\cal S}(a)}$ as the
one in Lemma 3.6, we need
###### Lemma 3.9.
Let ${\cal G}$ be a $\tau$-homotopy stable family of compact subsets of
$Y={\cal S}_{r}(a)$ with boundary $B=\emptyset$, and set
$d:=\inf_{A\in{\cal G}}\max_{u\in A}K_{\mu}(u).$
If $d>0$, then there exists a sequence $u_{n}\in{\cal S}_{r}(a)$ such that
$I_{\mu}(u_{n})\to d,\quad I_{\mu}|_{{\cal S}(a)}^{\prime}(u_{n})\to 0,\quad
Q_{\mu}(u_{n})=0.$
###### Proof.
Let $A_{n}\in{\cal G}$ be a minimizing sequence of $d$. We define the mapping
$\eta:[0,1]\times{\cal S}(a)\to{\cal S}(a),\quad\eta(t,u)=(ts_{\mu}(u))\star
u,$
which is continuous and satisfies $\eta(t,u)=u$ for all
$(t,u)\in\\{0\\}\times{\cal S}(a)$. Thus, by the definition of ${\cal G}$, one
has
$D_{n}:=\eta(1,A_{n})=\left\\{s_{\mu}(u)\star u:u\in A_{n}\right\\}\in{\cal
G}.$
In particular, $D_{n}\subset{\cal Q}_{\mu}(a)$ for any $n\in{\mathbb{N}}^{+}$.
For any $u\in{\cal S}(a)$ and any $s\in{\mathbb{R}}$, we see that
$Q_{\mu}((s_{\mu}(u)-s)\star(s\star u))=Q_{\mu}(s_{\mu}(u)\star u))=0,$
that is $s_{\mu}(s\star u)=s_{\mu}(u)-s$, which gives $K_{\mu}(s\star
u)=K_{\mu}(u)$. Then it is clear that
$\max_{D_{n}}K_{\mu}=\max_{A_{n}}K_{\mu}\to d$ and thus $D_{n}$ is another
minimizing sequence of $d$. Now, using the minimax principle [18, Theorem
7.2], we obtain a Palais-Smale sequence $v_{n}\in{\cal S}(a)$ for $K_{\mu}$ at
the level $d$ such that
$\text{dist}_{\cal X}(v_{n},D_{n})\to 0.$
Finally, a similar argument as the one in Lemma 3.6 gives that
$u_{n}=s_{n}\star v_{n}$ satisfying that
$I_{\mu}(u_{n})\to d,\quad I_{\mu}|_{{\cal S}(a)}^{\prime}(u_{n})\to 0,\quad
Q_{\mu}(u_{n})=0.$
∎
To construct a sequence of $\tau$-homotopy stable families of compact subsets
of ${\cal S}_{r}(a)$ with boundary $B=\emptyset$, we proceed as in [11,
Section. 8]. Since ${\cal X}$ is separable, there exists a nested sequence of
finite dimensional subspaces of ${\cal X}$, $W_{1}\subset
W_{2}\subset\cdots\subset W_{i}\subset W_{i+1}\subset\cdots\subset{\cal X}$
such that $dim(W_{i})=i$ and the closure of $\cup_{i\in{\mathbb{N}}^{+}}W_{i}$
in ${\cal X}$ is equal to ${\cal X}$. Note that since ${\cal X}$ is dense in
$W^{1,2}({\mathbb{R}^{N}})$, the closure in $W^{1,2}({\mathbb{R}^{N}})$ is
also equal to $W^{1,2}({\mathbb{R}^{N}})$. Since $W^{1,2}({\mathbb{R}^{N}})$
is a Hilbert space, we denote by $P_{i}$ the orthogonal projection from
$W^{1,2}({\mathbb{R}^{N}})$ onto $W_{i}$. We also recall the definition of the
genus of $\tau$-invariant sets due to M. A. Krasnoselskii and refer to [46,
Section. 7].
###### Definition D. (Krasnoselskii genus).
For any nonempty closed $\tau$-invariant set $A\subset{\cal X}$. The genus of
$A$ is defined by
$\mathrm{Ind}(A):=\min\left\\{k\in{\mathbb{N}}^{+}:\exists~{}\phi:A\to{\mathbb{R}}^{k}\setminus\\{0\\},\phi\text{
is odd and continuous }\right\\}.$
We set $\mathrm{Ind}(A)=+\infty$ if such $\phi$ does not exist, and set
$\mathrm{Ind}(A)=0$ if $A=\emptyset$.
Let ${\cal A}(a)$ be the family of compact $\tau$-invariant subsets of ${\cal
S}_{r}(a)$. For each $j\in{\mathbb{N}}^{+}$, set
${\cal A}_{j}(a):=\left\\{A\in{\cal A}(a):\mathrm{Ind}(A)\geq j\right\\}$
and
$c_{\mu}^{j}(a):=\inf_{A\in{\cal A}_{j}(a)}\max_{u\in A}K_{\mu}(u).$
Concerning ${\cal A}_{j}(a)$ and $c_{\mu}^{j}(a)$, we have
###### Lemma 3.10.
* (1)
For any $j\in{\mathbb{N}}^{+}$, ${\cal A}_{j}(a)\neq\emptyset$ and ${\cal
A}_{j}(a)$ is a $\tau$-homotopy stable family of compact subsets of ${\cal
S}_{r}(a)$ with boundary $B=\emptyset$.
* (2)
For any $\mu\in(0,1]$, any $j\in{\mathbb{N}}^{+}$, $c_{\mu}^{j+1}(a)\geq
c_{\mu}^{j}(a)\geq{\cal D}_{0}(a)>0$.
* (3)
For any $j\in{\mathbb{N}}^{+}$, $c_{\mu}^{j}(a)$ is non-decreasing with
respect to $\mu\in(0,1]$.
* (4)
$b_{j}(a):=\inf_{0<\mu\leq 1}c_{\mu}^{j}(a)\to+\infty$ as $j\to+\infty$.
###### Proof.
* (1)
For any $j\in{\mathbb{N}}^{+}$, ${\cal S}_{r}(a)\cap W_{j}\in{\cal A}(a)$. By
the basic properties of the genus, one has
$\mathrm{Ind}({\cal S}_{r}(a)\cap W_{j})=j$
and thus ${\cal A}_{j}(a)\neq\emptyset$. The rest is clear by the properties
of the genus.
* (2)
For any $A\in{\cal A}_{j}(a)$, using the fact that $s_{\mu}(u)\star u\in{\cal
Q}_{\mu}(a)$ for all $u\in A$, we have
$\max_{u\in A}K_{\mu}(u)=\max_{u\in A}I_{\mu}(s_{\mu}(u)\star u)\geq
m_{\mu}(a)\geq{\cal D}_{0}(a)$
and thus $c_{\mu}^{j}(a)\geq{\cal D}_{0}(a)>0$. Since ${\cal
A}_{j+1}(a)\subset{\cal A}_{j}(a)$, it is clear that $c_{\mu}^{j+1}(a)\geq
c_{\mu}^{j}(a)$.
* (3)
For any $0<\mu_{1}<\mu_{2}\leq 1$, any $u\in A\in{\cal A}_{j}(a)$, there holds
$K_{\mu_{2}}(u)=I_{\mu_{2}}(s_{\mu_{2}}(u)\star u)\geq
I_{\mu_{2}}(s_{\mu_{1}}(u)\star u)>I_{\mu_{1}}(s_{\mu_{1}}(u)\star
u)=K_{\mu_{1}}(u),$
which means $c_{\mu_{2}}^{j}(a)\geq c_{\mu_{1}}^{j}(a)$, i.e.,
$c_{\mu}^{j}(a)$ is non-decreasing with respect to $\mu\in(0,1]$.
* (4)
The proof is inspired by that of [11, Theorem 9]. First, we claim that
Claim: for any $M>0$, there exists a small $\delta_{0}=\delta_{0}(a,M)>0$, a
small $r_{0}=r_{0}(a,M)>0$ and a large $k_{0}=k_{0}(a,M)\in{\mathbb{N}}^{+}$
such that for any $0<\mu<\delta_{0}$ and any $k\geq k_{0}$, one has
$I_{\mu}(u)\geq M\quad\text{if}\quad\|P_{k}u\|_{\cal X}\leq
r_{0}~{}\text{and}~{}u\in{\cal Q}_{\mu}^{r}(a).$
Now we check it. By contradiction, we assume that there exists $M_{0}>0$ such
that for any $0<\delta\leq 1$, any $r>0$ and any $k\in{\mathbb{N}}^{+}$ one
can always find $\mu\in(0,\delta]$, $l\geq k$ and $u\in{\cal Q}_{\mu}^{r}(a)$
such that
$\|P_{k}u\|_{\cal X}\leq r\quad\text{but}\quad I_{\mu}(u)<M_{0}.$
As a consequnce, one can obtain a sequence $\mu_{n}\to 0^{+}$, a sequence
$k_{n}\to+\infty$, and a sequence $u_{n}\in{\cal Q}_{\mu_{n}}^{r}(a)$ such
that
$\|P_{k_{n}}u_{n}\|_{\cal X}\leq\frac{1}{n}\quad\text{and}\quad
I_{\mu_{n}}(u_{n})<M_{0}$
for any $n\in{\mathbb{N}}^{+}$. From Lemma 3.3, we know that $u_{n}$ is
bounded in $W^{1,2}({\mathbb{R}^{N}})$. Since $P_{k_{n}}u_{n}$ is also bounded
in ${\cal X}$, we assume that up to a subsequence
$u_{n}\rightharpoonup u~{}\text{
in}~{}W^{1,2}({\mathbb{R}^{N}})\quad\text{and}\quad
P_{k_{n}}u_{n}\rightharpoonup v~{}\text{ in}~{}{\cal X}.$
We show that $u=v$. Indeed, one also has $P_{k_{n}}u_{n}\rightharpoonup v$ in
$W^{1,2}({\mathbb{R}^{N}})$ and
$\displaystyle\|u-v\|_{W^{1,2}({\mathbb{R}^{N}})}^{2}$
$\displaystyle=\lim_{n\to\infty}\left<u_{n}-P_{k_{n}}u_{n},u-v\right>_{W^{1,2}({\mathbb{R}^{N}})}$
$\displaystyle=\lim_{n\to\infty}\left<u_{n},u-v\right>_{W^{1,2}({\mathbb{R}^{N}})}-\lim_{n\to\infty}\left<P_{k_{n}}u_{n},u-v\right>_{W^{1,2}({\mathbb{R}^{N}})}$
$\displaystyle=\left<u,u-v\right>_{W^{1,2}({\mathbb{R}^{N}})}-\lim_{n\to\infty}\left<u_{n},P_{k_{n}}u-P_{k_{n}}v\right>_{W^{1,2}({\mathbb{R}^{N}})}$
$\displaystyle=\left<u,u-v\right>_{W^{1,2}({\mathbb{R}^{N}})}-\left<u,u-v\right>_{W^{1,2}({\mathbb{R}^{N}})}$
$\displaystyle=0,$
where we use the fact that $P_{k_{n}}u\to u$ and $P_{k_{n}}v\to v$ in
$W^{1,2}({\mathbb{R}^{N}})$. Therefore $u=v$ and $u\in{\cal X}$. Since
$\|P_{k_{n}}u_{n}\|_{\cal X}\to 0$, there must be $u=0$. Then combining the
interpolation inequality and the fact that
$\sup_{n\in{\mathbb{N}}^{+}}\int_{\mathbb{R}^{N}}|u_{n}|^{2}|\nabla
u_{n}|^{2}<+\infty$, we obtain $\|u_{n}\|_{p}\to 0$. Futher $u_{n}\in{\cal
Q}_{\mu_{n}}(a)$ gives that
$\mu_{n}\int_{\mathbb{R}^{N}}|\nabla u_{n}|^{\theta}\to
0,\quad\int_{\mathbb{R}^{N}}|\nabla u_{n}|^{2}\to
0,\quad\int_{\mathbb{R}^{N}}|u_{n}|^{2}|\nabla u_{n}|^{2}\to 0,$
which is in contradiction with Lemma 3.3. So we prove the claim.
Then we can prove the conclusion $(4)$. By contradiction, we assume that
$\liminf_{j\to\infty}b_{j}<M\quad\text{for some }M>0.$
Then there exist $\mu\in(0,\delta_{0})$, $k>k_{0}$ such that
$c_{\mu}^{k}(a)<M$. By the definition of $c_{\mu}^{k}(a)$, one can find
$A\in{\cal A}_{k}(a)$ such that
$\max_{u\in A}I_{\mu}(s_{\mu}(u)\star u)=\max_{u\in A}K_{\mu}(u)<M.$
Since Lemma 3.2 implies that the mapping $\varphi:A\to{\cal Q}_{\mu}^{r}(a)$
defined by $\varphi(u)=s_{\mu}(u)\star u$ is odd and continuous, we have
$\bar{A}:=\varphi(A)\subset{\cal Q}_{\mu}^{r}(a)$,
$\max_{u\in\bar{A}}I_{\mu}(u)<M$ and
(3.29) $\mathrm{Ind}(\bar{A})\geq\mathrm{Ind}(A)\geq k>k_{0}.$
On the other hand, it follows from the claim that
$\inf_{u\in\bar{A}}\|P_{k_{0}}u_{n}\|_{\cal X}\geq r_{0}>0$. Setting
$\psi(u)=\frac{P_{k_{0}}u}{\|P_{k_{0}}u_{n}\|_{\cal X}}\quad\text{for any
}u\in\bar{A},$
we obtain an odd continuous mapping $\psi:\bar{A}\to\psi(\bar{A})\subset
W_{k_{0}}\setminus\\{0\\}$ and thus
$\mathrm{Ind}(\bar{A})\leq\mathrm{Ind}(\psi(\bar{A}))\leq k_{0},$
which contradicts (3.29). Therefore we have $b_{j}(a)\to+\infty$ as
$j\to+\infty$.
∎
For any fixed $\mu\in(0,1]$ and any $j\in{\mathbb{N}}^{+}$, by Lemma 3.9 and
3.10, one can find a sequence $u_{n}\in{\cal S}_{r}(a)$ such that
$I_{\mu}(u_{n})\to c_{\mu}^{j}(a),\quad I_{\mu}|_{{\cal
S}(a)}^{\prime}(u_{n})\to 0,\quad Q_{\mu}(u_{n})=0.$
Then similar to Lemma 3.7, we have
###### Lemma 3.11.
There exists a $u_{\mu}^{j}\in{\cal X}\setminus\\{0\\}$ and a
$\lambda_{\mu}^{j}\in{\mathbb{R}}$ such that up to a subsequence,
(3.30) $u_{n}^{j}\rightharpoonup u_{\mu}^{j}\quad\text{in }{\cal X},$ (3.31)
$I_{\mu}(u_{\mu}^{j})=c_{\mu}^{j}(a)\quad\text{and}\quad
I_{\mu}^{\prime}(u_{\mu}^{j})+\lambda_{\mu}^{j}u_{\mu}^{j}=0.$
Moreover, if $\lambda_{\mu}^{j}\neq 0$, we have that
$u_{n}^{j}\rightarrow u_{\mu}^{j}\quad\text{in }{\cal X}.$
Based on the above preliminary works, we conclude that
###### Theorem 3.2.
For any fixed $\mu\in(0,1]$ and any $j\in{\mathbb{N}}^{+}$, there exists a
$u_{\mu}^{j}\in{\cal X}_{r}\setminus\\{0\\}$ and a
$\lambda_{\mu}^{j}\in{\mathbb{R}}$ such that
$\displaystyle I_{\mu}^{\prime}(u_{\mu}^{j})+\lambda_{\mu}^{j}u_{\mu}^{j}=0,$
$\displaystyle I_{\mu}(u_{\mu}^{j})=c_{\mu}^{j}(a),\quad
Q_{\mu}(u_{\mu}^{j})=0,$ $\displaystyle 0<\|u_{\mu}^{j}\|_{2}^{2}\leq a.$
Moreover, if $\lambda_{\mu}^{j}\neq 0$, we have that
$\|u_{\mu}^{j}\|_{2}^{2}=a$, i.e.,
$\left\\{u_{\mu}^{j}:j\in{\mathbb{N}}^{+}\right\\}$ are infinitely many
critical points of $I_{\mu}|_{{\cal S}(a)}$ with increasing energy.
## 4 Convergence issues as $\mu\to 0^{+}$
In this section, letting $\mu\to 0^{+}$, we show that the sequences of
critical points of $I_{\mu}|_{{\cal S}(a)}$ obtained in the Section 3 converge
to critical points of $I|_{\tilde{\cal S}(a)}$.
###### Theorem 4.1.
Let $N\geq 2$. Suppose that $\mu_{n}\to 0^{+}$,
$I_{\mu_{n}}^{\prime}(u_{\mu_{n}})+\lambda_{\mu_{n}}u_{\mu_{n}}=0$ with
$\lambda_{\mu_{n}}\geq 0$ and $I_{\mu_{n}}(u_{\mu_{n}})\to c\in(0,+\infty)$
for $u_{\mu_{n}}\in{\cal S}_{r}(a_{n})$ with $0<a_{n}\leq a$. Then there
exists a subsequence $u_{\mu_{n}}\rightharpoonup u$ in
$W^{1,2}({\mathbb{R}^{N}})$ with $u\neq 0$, $u\in
W^{1,2}_{rad}({\mathbb{R}^{N}})\cap L^{\infty}({\mathbb{R}^{N}})$ and there
exists a $\lambda\in{\mathbb{R}}$ such that
$\quad I^{\prime}(u)+\lambda u=0,\quad I(u)=c\quad\text{and}\quad
0<\|u\|_{2}^{2}\leq a.$
Moreover,
* (1)
if $u_{\mu_{n}}\geq 0$ for each $n\in{\mathbb{N}}^{+}$, then $u\geq 0$,
* (2)
if $\lambda\neq 0$, we have that $\|u\|_{2}^{2}=\lim_{n\to\infty}a_{n}$.
###### Remark 4.1.
We note that the condition $\lambda_{\mu_{n}}\geq 0$ is only used in the
following Step 1 to realize the Morse iteration. If one can prove the
conclusion in Step 1 without this condition, then the conclusion in Theorem
1.1 can be extended to $N=3,4$ with $4+\frac{4}{N}<p<22^{*}$.
###### Proof of Theorem 4.1:.
The proof is inspired by [26, 31]. First, by Lemma 2.1,
$I_{\mu_{n}}^{\prime}(u_{\mu_{n}})+\lambda_{\mu_{n}}u_{\mu_{n}}=0$ implies
that
$Q_{\mu_{n}}(u_{\mu_{n}})=0\quad\text{for each }n\in{\mathbb{N}}^{+}.$
Then from Lemma 3.3, we see that
(4.1) $\sup_{n\geq 1}\max\left\\{{\mu_{n}}\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{\theta},\int_{\mathbb{R}^{N}}|u_{n}|^{2}|\nabla
u_{n}|^{2},\int_{\mathbb{R}^{N}}|\nabla u_{n}|^{2}\right\\}<+\infty,$
and hence $u_{\mu_{n}}$ is bounded in $W^{1,2}({\mathbb{R}^{N}})$. We claim
that $\liminf_{n\to\infty}a_{n}>0$ and hence
$\lambda_{\mu_{n}}=\frac{1}{a_{n}}I_{\mu_{n}}^{\prime}(u_{\mu_{n}})$
$[u_{\mu_{n}}]$ is also bounded in ${\mathbb{R}}$. Indeed, if $a_{n}\to 0$,
then $\|u_{\mu_{n}}\|_{p}\to 0$, and it follows from ${\cal
Q}_{\mu_{n}}(u_{n})=0$ that $I_{\mu_{n}}(u_{\mu_{n}})\to 0$ which contradicts
that $c>0$. Thus, up to a subsequence, $\lambda_{\mu_{n}}\to\lambda$ in
${\mathbb{R}}$, $u_{\mu_{n}}\rightharpoonup u$ in
$W^{1,2}_{rad}({\mathbb{R}^{N}})$, $u_{\mu_{n}}\to u$ in
$L^{q}({\mathbb{R}^{N}})$ for $2<q<22^{*}$, and $u_{\mu_{n}}\to u$ a.e. on
${\mathbb{R}^{N}}$. So if $u_{\mu_{n}}\geq 0$ for each $n\in{\mathbb{N}}^{+}$,
we have that $u\geq 0$. Moreover, a similar argument as Lemma A.2 tells that
$u_{n}\nabla u_{n}\to u\nabla u$ in $(L^{2}_{loc}({\mathbb{R}^{N}}))^{N}$ and
$\nabla u_{\mu_{n}}\to\nabla u$ a.e. on ${\mathbb{R}^{N}}$. Now we prove the
conclusion in several steps.
* Step 1:
We prove that $\|u_{\mu_{n}}\|_{\infty}\leq C$ and $\|u\|_{\infty}\leq C$ for
some positive constant $C$.
We just prove the case $N\geq 3$, the case $N=2$ can be obtained similarly.
Set $T>2$, $r>0$ and
$v_{n}=\begin{cases}T,\quad&u_{n}\geq T,\\\ u_{n},\quad&|u_{n}|\leq T,\\\
-T,\quad&u_{n}\leq-T.\end{cases}$
Let $\phi=u_{\mu_{n}}|v_{n}|^{2r}$, then $\phi\in{\cal X}$. From
$I_{\mu_{n}}^{\prime}(u_{\mu_{n}})+\lambda_{\mu_{n}}u_{\mu_{n}}=0$ and
$\lambda_{\mu_{n}}\geq 0$,we obtain
$\displaystyle\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{p-2}u_{\mu_{n}}\phi$
$\displaystyle=\mu_{\mu_{n}}\int_{\mathbb{R}^{N}}|\nabla
u_{\mu_{n}}|^{\theta-2}\nabla
u_{\mu_{n}}\cdot\nabla\phi+\int_{\mathbb{R}^{N}}\nabla
u_{\mu_{n}}\cdot\nabla\phi$
$\displaystyle\quad~{}~{}+2\int_{\mathbb{R}^{N}}(u_{\mu_{n}}\phi|\nabla
u_{\mu_{n}}|^{2}+|u_{\mu_{n}}|^{2}\nabla
u_{\mu_{n}}\cdot\nabla\phi)+\lambda_{\mu_{n}}\int_{\mathbb{R}^{N}}u_{\mu_{n}}\phi$
$\displaystyle\geq 2\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{2}\nabla
u_{\mu_{n}}\cdot\nabla\phi$
$\displaystyle=2\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{2}|\nabla
u_{\mu_{n}}|^{2}|v_{n}|^{2r}+|u_{\mu_{n}}|^{2}2r|v_{n}|^{2r-2}u_{\mu_{n}}v_{n}\nabla
u_{\mu_{n}}\cdot\nabla v_{n}$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{N}}|v_{n}|^{r}|\nabla
u_{\mu_{n}}^{2}|^{2}+\frac{4}{r}\int_{\mathbb{R}^{N}}|u_{\mu_{n}}^{2}\nabla|v_{n}|^{r}|^{2}$
$\displaystyle\geq\frac{1}{r+4}\int_{\mathbb{R}^{N}}|\nabla(u_{\mu_{n}}^{2}|v_{n}|^{2})|^{2}\geq\frac{C}{(r+2)^{2}}\left(\int_{\mathbb{R}^{N}}|u_{\mu_{n}}^{2}|v_{n}|^{2}|^{2^{*}}\right)^{\frac{2}{2^{*}}}.$
On the other hand, by the interpolation inequality, we have
(4.2) $\displaystyle\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{p-2}u_{\mu_{n}}\phi$
$\displaystyle=\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{p}|v_{n}|^{2r}$
$\displaystyle\leq\left(\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{22^{*}}\right)^{\frac{p-4}{22^{*}}}\left(\int_{\mathbb{R}^{N}}\left(|v_{n}|^{r}|u_{\mu_{n}}|^{2}\right)^{\frac{42^{*}}{22^{*}-p+4}}\right)^{\frac{22^{*}-p+4}{22^{*}}}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}^{N}}\left(|v_{n}|^{r}|u_{\mu_{n}}|^{2}\right)^{\frac{42^{*}}{22^{*}-p+4}}\right)^{\frac{22^{*}-p+4}{22^{*}}}.$
Combining these inequalities, one has
(4.3)
$\left(\int_{\mathbb{R}^{N}}|u_{\mu_{n}}^{2}|v_{n}|^{2}|^{2^{*}}\right)^{\frac{2}{2^{*}}}\leq
C(r+2)^{2}\left(\int_{\mathbb{R}^{N}}\left(|v_{n}|^{r}|u_{\mu_{n}}|^{2}\right)^{\frac{42^{*}}{22^{*}-p+4}}\right)^{\frac{22^{*}-p+4}{22^{*}}}.$
Let $r_{0}:(r_{0}+2)q=22^{*}$ and $d=\frac{2^{*}}{q}>1$ where
$q=\frac{42^{*}}{22^{*}-p+4}$. Taking $r=r_{0}$ in (4.3), and letting
$T\to+\infty$, we obtain
(4.4)
$\|u_{\mu_{n}}\|_{(2+r_{0})qd}\leq\left(C(r_{0}+2)\right)^{\frac{1}{r_{0}+2}}\|u_{\mu_{n}}\|_{(2+r_{0})q}.$
Set $2+r_{i+1}=(2+r_{i})d$ for $i\in{\mathbb{N}}$. Then inductively, we have
(4.5)
$\|u_{\mu_{n}}\|_{(2+r_{0})qd^{i+1}}\leq\prod_{k=0}^{i}\left(C(r_{k}+2)\right)^{\frac{1}{r_{k}+2}}\|u_{\mu_{n}}\|_{(2+r_{0})q}\leq
C_{\infty}\|u_{\mu_{n}}\|_{(2+r_{0})q},$
where $C_{\infty}$ is a positive constant. Taking $i\to\infty$ in (4.5), we
get
$\|u_{\mu_{n}}\|_{\infty}\leq C\quad\text{and}\quad\|u\|_{\infty}\leq C.$
* Step 2:
We prove that $I^{\prime}(u)+\lambda u=0$.
Take $\phi=\psi e^{-u_{\mu_{n}}}$ with $\psi\in{\cal
C}_{0}^{\infty}({\mathbb{R}^{N}})$, $\psi\geq 0$, we have
$\displaystyle 0$
$\displaystyle=\left(I_{\mu_{n}}^{\prime}(u_{\mu_{n}})+\lambda_{\mu_{n}}u_{\mu_{n}}\right)[\phi]$
$\displaystyle=\mu_{n}\int_{\mathbb{R}^{N}}|\nabla
u_{\mu_{n}}|^{\theta-2}\nabla u_{\mu_{n}}\left(\nabla\psi
e^{-u_{\mu_{n}}}-\psi e^{-u_{\mu_{n}}}\nabla
u_{\mu_{n}}\right)+\int_{\mathbb{R}^{N}}\nabla u_{\mu_{n}}\left(\nabla\psi
e^{-u_{\mu_{n}}}-\psi e^{-u_{\mu_{n}}}\nabla u_{\mu_{n}}\right)$
$\displaystyle\quad~{}~{}+2\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{2}\nabla
u_{\mu_{n}}\left(\nabla\psi e^{-u_{\mu_{n}}}-\psi e^{-u_{\mu_{n}}}\nabla
u_{\mu_{n}}\right)+2\int_{\mathbb{R}^{N}}u_{\mu_{n}}\psi
e^{-u_{\mu_{n}}}|\nabla u_{\mu_{n}}|^{2}$
$\displaystyle\quad~{}~{}+\lambda_{\mu_{n}}\int_{\mathbb{R}^{N}}u_{\mu_{n}}\psi
e^{-u_{\mu_{n}}}-\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{p-2}u_{\mu_{n}}\psi
e^{-u_{\mu_{n}}}$ $\displaystyle\leq\mu_{n}\int_{\mathbb{R}^{N}}|\nabla
u_{\mu_{n}}|^{\theta-2}\nabla u_{\mu_{n}}\nabla\psi
e^{-u_{\mu_{n}}}+\int_{\mathbb{R}^{N}}(1+2u_{\mu_{n}}^{2})\nabla
u_{\mu_{n}}\nabla\psi e^{-u_{\mu_{n}}}$
$\displaystyle\quad~{}~{}-\int_{\mathbb{R}^{N}}(1+2u_{\mu_{n}}^{2}-2u_{\mu_{n}})\psi
e^{-u_{\mu_{n}}}|\nabla
u_{\mu_{n}}|^{2}+\lambda_{\mu_{n}}\int_{\mathbb{R}^{N}}u_{\mu_{n}}\psi
e^{-u_{\mu_{n}}}-\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{p-2}u_{\mu_{n}}\psi
e^{-u_{\mu_{n}}}$
Since $\mu_{n}\to 0^{+}$ and $\|u_{\mu_{n}}\|_{\infty}\leq C$, (4.1) implies
${\mu_{n}}\int_{\mathbb{R}^{N}}|\nabla u_{\mu_{n}}|^{\theta-2}\nabla
u_{\mu_{n}}\nabla\psi e^{-u_{\mu_{n}}}\to 0.$
By the weak convergence of $u_{\mu_{n}}$, the Hölder inequality and the
Lebesgue’s dominated convergence theorem we know that
$\int_{\mathbb{R}^{N}}(1+2u_{\mu_{n}}^{2})\nabla u_{\mu_{n}}\nabla\psi
e^{-u_{\mu_{n}}}\to\int_{\mathbb{R}^{N}}(1+2u^{2})\nabla u\nabla\psi e^{-u},$
$\lambda_{\mu_{n}}\int_{\mathbb{R}^{N}}u_{\mu_{n}}\psi
e^{-u_{\mu_{n}}}\to\lambda\int_{\mathbb{R}^{N}}u\psi e^{-u},$
and
$\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{p-2}u_{\mu_{n}}\psi
e^{-u_{\mu_{n}}}\to\int_{\mathbb{R}^{N}}|u|^{p-2}u\psi e^{-u}.$
Moreover, by Fatou’s lemma, there holds
$\liminf_{n\to\infty}\int_{\mathbb{R}^{N}}(1+2u_{\mu_{n}}^{2}-2u_{\mu_{n}})\psi
e^{-u_{\mu_{n}}}|\nabla
u_{\mu_{n}}|^{2}\geq\int_{\mathbb{R}^{N}}(1+2u^{2}-2u)\psi e^{-u}|\nabla
u|^{2}.$
Consequently, one has
(4.6) $\displaystyle 0$ $\displaystyle\leq\int_{\mathbb{R}^{N}}\nabla
u\left(\nabla\psi e^{-u}-\psi e^{-u}\nabla
u\right)+2\int_{\mathbb{R}^{N}}|u|^{2}\nabla u\left(\nabla\psi e^{-u}-\psi
e^{-u}\nabla u\right)$ $\displaystyle\quad~{}~{}+2\int_{\mathbb{R}^{N}}u\psi
e^{-u}|\nabla u|^{2}+\lambda_{\mu_{n}}\int_{\mathbb{R}^{N}}u\psi
e^{-u}-\int_{\mathbb{R}^{N}}|u|^{p-2}u\psi e^{-u}.$
For any $\varphi\in{\cal C}_{0}^{\infty}({\mathbb{R}^{N}})$ with $\varphi\geq
0$. Choose a sequence of non-negative functions $\psi_{n}\in{\cal
C}_{0}^{\infty}({\mathbb{R}^{N}})$ such that $\psi_{n}\to\varphi e^{u}$ in
$W^{1,2}({\mathbb{R}^{N}})$, $\psi_{n}\to\varphi e^{u}$ a.e. in
${\mathbb{R}^{N}}$, and that $\psi_{n}$ is uniformly bounded in
$L^{\infty}({\mathbb{R}^{N}})$. Then we obtain from (4.6) that
(4.7) $0\leq\int_{\mathbb{R}^{N}}\nabla
u\cdot\nabla\varphi+2\int_{\mathbb{R}^{N}}(|u|^{2}\nabla
u\cdot\nabla\varphi+u\varphi|\nabla
u|^{2})+\lambda\int_{\mathbb{R}^{N}}u\varphi-\int_{\mathbb{R}^{N}}|u|^{p-2}u\varphi.$
Similarly by choosing $\phi=\psi e^{u_{\mu_{n}}}$, we get an opposite
inequality. Notice $\varphi=\varphi^{+}-\varphi^{-}$ for any $\varphi\in{\cal
C}_{0}^{\infty}({\mathbb{R}^{N}})$, we get $I^{\prime}(u)+\lambda u=0$.
* Step 3:
We complete the proof.
Similar as Lemma 2.1, we get from $I^{\prime}(u)+\lambda u=0$ that
$Q(u):=Q_{0}(u)=0.$
It follows that
$Q_{\mu_{n}}(u_{\mu_{n}})+\gamma_{p}\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{p}\to
Q(u)+\gamma_{p}\int_{\mathbb{R}^{N}}|u|^{p}.$
Then using the weak lower semicontinuous property, there must be
(4.8) $\mu_{n}\int_{\mathbb{R}^{N}}|\nabla u_{\mu_{n}}|^{\theta}\to
0,\quad\int_{\mathbb{R}^{N}}|\nabla
u_{\mu_{n}}|^{2}\to\int_{\mathbb{R}^{N}}|\nabla
u|^{2},\quad\int_{\mathbb{R}^{N}}|u_{\mu_{n}}|^{2}|\nabla
u_{\mu_{n}}|^{2}\to\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}.$
That gives $I(u)=\lim_{n\to\infty}I_{\mu}(u_{\mu_{n}})=c$. Moreover, from
(4.8), we obtain
(4.9) $I_{\mu_{n}}^{\prime}(u_{\mu_{n}})[u_{\mu_{n}}]\to I^{\prime}(u)[u].$
Thus there holds $\lambda\|u_{\mu_{n}}\|_{2}^{2}\to\lambda\|u\|_{2}^{2}$. So
if $\lambda\neq 0$, we have $\|u\|_{2}^{2}=\lim_{n\to\infty}a_{n}$.
∎
Now we are able to end the proof of Theorem 1.1 and 1.2.
###### Proof of Theorem 1.1 for the case $N\geq 2$:.
From Remark 3.1 and 3.2, we see that
$d^{*}(a):=\lim_{\mu\to 0^{+}}m_{\mu}(a)\in(0,+\infty).$
By Theorem 3.1, we can take $\mu_{n}\to 0^{+}$,
$I_{\mu_{n}}^{\prime}(u_{\mu_{n}})+\lambda_{\mu_{n}}u_{\mu_{n}}=0$ ,
$I_{\mu_{n}}(u_{\mu_{n}})\to d^{*}(a)$ for $u_{\mu_{n}}\in{\cal S}_{r}(a_{n})$
with $0<a_{n}\leq a$ and $u_{\mu_{n}}\geq 0$. Then Lemma 2.2 implies that
$\lambda_{\mu_{n}}>0$. Now Theorem 4.1 gives that there exist $v\neq 0$,
$v\geq 0$, $v\in W^{1,2}_{rad}({\mathbb{R}^{N}})\cap
L^{\infty}({\mathbb{R}^{N}})$ and $\lambda_{0}\in{\mathbb{R}}$ such that
$\quad I^{\prime}(v)+\lambda_{0}v=0,\quad I(v)=d^{*}(a)\quad\text{and}\quad
0<\|v\|_{2}^{2}\leq a.$
Thus by Lemma 2.2, there is $\lambda_{0}>0$. Since
$\lambda_{\mu_{n}}\to\lambda_{0}$, we may say that $\lambda_{\mu_{n}}\neq 0$
for $n$ large. Then $a_{n}=a$ and $\|v\|_{2}^{2}=a$. That is, $v$ is a
nontrivial nonnegative solution of (1.3). To consider the ground state
normalized solution, we define
$d(a):=\inf\left\\{I(v):v\in\tilde{\cal S}(a),I|_{\tilde{\cal
S}(a)}^{\prime}(v)=0,v\neq 0\right\\}.$
Then $d(a)\leq I(v)=d^{*}(a)$. Futher, a similar approach to Lemma 3.3 tells
that $d(a)>0$. We take a sequence $v_{n}\in\tilde{\cal S}(a)$,
$I|_{\tilde{\cal S}(a)}^{\prime}(v_{n})=0$, $v_{n}\neq 0$ and $v_{n}\geq 0$
such that $I(v_{n})\to d(a)$. We can show that (the proof is similar to that
of Theorem 4.1, so we omit it), up to a subsequence, there exist $u\neq 0$,
$u\geq 0$, $u\in W^{1,2}_{rad}({\mathbb{R}^{N}})\cap
L^{\infty}({\mathbb{R}^{N}})$ and $\lambda\in{\mathbb{R}}$ such that
$\quad I^{\prime}(u)+\lambda u=0\quad\text{and}\quad I(u)=d(a).$
Again by Lemma 2.2, there is $\lambda\neq 0$, and hence $\|u\|_{2}^{2}=a$.
That is, $u$ is a minimizer of $d(a)$. Finally, by [37, Lemma 2.6], $u$ is
classical and strictly positive since $u\in L^{\infty}({\mathbb{R}^{N}})$. ∎
###### Proof of Theorem 1.2:.
From Lemma 3.10, we see that
$b_{j}(a)=\lim_{\mu\to 0^{+}}c_{\mu}^{j}(a)\in(0,+\infty)\quad\text{and}\quad
b_{j}(a)\to+\infty.$
By Theorem 3.2, for each $j\in{\mathbb{N}}^{+}$ we can take ${\mu_{n}^{j}}\to
0^{+}$,
$I_{\mu_{n}^{j}}^{\prime}(u_{\mu_{n}^{j}}^{j})+\lambda_{\mu_{n}^{j}}^{j}u_{\mu_{n}^{j}}^{j}=0$,
$I_{\mu_{n}^{j}}(u_{\mu_{n}^{j}}^{j})\to b_{j}(a)$ for
$u_{\mu_{n}^{j}}\in{\cal S}_{r}(a_{n}^{j})$ with $0<a_{n}^{j}\leq a$. And
Lemma 2.2 implies that $\lambda_{\mu_{n}^{j}}^{j}>0$. Now Theorem 4.1 gives
that there exist $u^{j}\neq 0$, $u^{j}\in W^{1,2}_{rad}({\mathbb{R}^{N}})\cap
L^{\infty}({\mathbb{R}^{N}})$ and $\lambda^{j}\in{\mathbb{R}}$ such that
$\quad I^{\prime}(u^{j})+\lambda^{j}u^{j}=0,\quad
I(u^{j})=b_{j}(a)\quad\text{and}\quad 0<\|u^{j}\|_{2}^{2}\leq a.$
Thus by Lemma 2.2, there is $\lambda^{j}>0$. Going back since
$\lambda_{\mu_{n}^{j}}^{j}\to\lambda^{j}$, we may say that
$\lambda_{\mu_{n}^{j}}^{j}\neq 0$ for $n$ large. Then $a_{n}^{j}=a$ and
$\|u^{j}\|_{2}^{2}=a$. That is $\left\\{u^{j}:j\in{\mathbb{N}}^{+}\right\\}$
is a sequence of normalized solutions of (1.3). Moreover,
$I(u^{j})=b_{j}\to+\infty$. ∎
## 5 The mass critical case $p=4+\frac{4}{N}$
In this section we denote $p_{*}=4+\frac{4}{N}$ and assume that $p=p_{*}$. We
still consider $I_{\mu}$, but on an open subset of ${\cal X}$. Let
(5.1) ${\cal O}:=\left\\{u\in{\cal X}:\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}<\frac{N}{4(N+1)}\int_{\mathbb{R}^{N}}|u|^{p_{*}}\right\\},$
and for simplicity, we still denote
${\cal S}(a):=\left\\{u\in{\cal O}:\int_{\mathbb{R}^{N}}u^{2}=a\right\\},$
${\cal Q}_{\mu}(a):=\left\\{u\in{\cal S}(a):Q_{\mu}(u)=0\right\\},$ ${\cal
S}_{r}(a):={\cal S}(a)\cap{\cal X}_{r},\quad{\cal Q}_{\mu}^{r}(a):={\cal
Q}_{\mu}(a)\cap{\cal X}_{r}.$
We have
###### Lemma 5.1.
When $a>a^{*}$, ${\cal S}(a)$ is nonempty.
###### Proof.
Let $u=Q_{p_{*}}^{\frac{1}{2}}$, then from (1.9), we have
(5.2)
$\int_{\mathbb{R}^{N}}|u|^{p_{*}}=\frac{4(N+1)}{N}\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}.$
Let $w_{a}=\left(\frac{a}{a_{*}}\right)^{\frac{1}{2}}u$, then
$\|w_{a}\|_{2}^{2}=a$ and (5.2) implies that
(5.3) $\int_{\mathbb{R}^{N}}w_{a}^{2}|\nabla
w_{a}|^{2}=\frac{N}{4(N+1)}\left(\frac{a}{a_{*}}\right)^{-\frac{2}{N}}\int_{\mathbb{R}^{N}}|w_{a}|^{p_{*}}<\frac{N}{4(N+1)}\int_{\mathbb{R}^{N}}|w_{a}|^{p_{*}},$
that is $w_{a}\in{\cal S}(a)$. ∎
So from now on, we assume $a>a^{*}$. Then noting that when $p=p_{*}$, there is
$p_{*}\gamma_{p_{*}}>\theta+\theta\gamma_{\theta}$ and
$p_{*}\gamma_{p_{*}}=2+N$, we still have
###### Lemma 5.2.
Let $0<\mu\leq 1$, then ${\cal Q}_{\mu}(a)$ is a ${\cal C}^{1}$-submanifold of
codimension 1 in ${\cal S}(a)$, hence a ${\cal C}^{1}$-submanifold of
codimension 2 in ${\cal X}$.
###### Lemma 5.3.
For any $0<\mu\leq 1$ and any $u\in{\cal O}\setminus\\{0\\}$, the following
statements hold.
* (1)
There exists a unique number $s_{\mu}(u)\in{\mathbb{R}}$ such that
$Q_{\mu}(s_{\mu}(u)\star u)=0$.
* (2)
$I_{\mu}(s\star u)$ is strictly increasing in $s\in(-\infty,s_{\mu}(u))$ and
is strictly decreasing in $s\in(s_{\mu}(u),+\infty)$, and
$\lim_{s\to-\infty}I_{\mu}(s\star
u)=0^{+},\quad\lim_{s\to+\infty}I_{\mu}(s\star u)=-\infty,\quad
I_{\mu}(s_{\mu}(u)\star u)>0.$
* (3)
$s_{\mu}(u)<0$ if and only if $Q_{\mu}(u)<0$.
* (4)
The map $u\in{\cal X}\setminus\\{0\\}\mapsto s_{\mu}(u)\in{\mathbb{R}}$ is of
class ${\cal C}^{1}$.
* (5)
$s_{\mu}(u)$ is an even function with respect to $u\in{\cal
X}\setminus\\{0\\}$.
Similarly to Lemma 3.3, there also holds
###### Lemma 5.4.
The following statements hold.
* (1)
${\cal D}(a):=\inf_{0<\mu\leq 1,u\in{\cal
Q}_{\mu}(a)}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla u|^{2}>0$ is independent of
$\mu$.
* (2)
If $\sup_{n\geq 1}I_{\mu}(u_{n})<+\infty$ for $u_{n}\in{\cal Q}_{\mu}(a)$,
then
$\sup_{n\geq 1}\max\left\\{\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta},\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2},\int_{\mathbb{R}^{N}}|\nabla u|^{2}\right\\}<+\infty.$
###### Proof.
The proof is different from the one of Lemma 3.3.
* (1)
For any $u\in{\cal Q}_{\mu}(a)$, using the equality $Q_{\mu}(u)=0$ and (1.12)
we obtain
(5.4) $\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\leq(N+2)\left[\left(\frac{a}{a_{*}}\right)^{\frac{2}{N}}-1\right]\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}.$
On the one hand, when $N\leq 3$ , there holds $p_{*}<2^{*}$. Therefore the
calssical Gagliardo-Nirenberg inequality ([41]) tells that
(5.5) $\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\leq\gamma_{p_{*}}\int_{\mathbb{R}^{N}}|u|^{p_{*}}\leq
C(N)a^{1+\frac{2}{N}-\frac{N}{2}}\left(\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\right)^{\frac{N+2}{2}},$
follows which there is $\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\geq\frac{C(N)}{a^{\frac{4}{N^{2}}+\frac{2}{N}-1}}$. Combining with
(5.4), one obtain
$\inf_{0<\mu\leq 1,u\in{\cal Q}_{\mu}(a)}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}>0.$
On the other hand, when $N\geq 4$, there is $p_{*}>2^{*}$. But using
interpolation inequality and Young inequality we have
(5.6) $\displaystyle(N+2)\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}+\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle\leq\gamma_{p_{*}}\int_{\mathbb{R}^{N}}|u|^{p_{*}}\leq\left(\int_{\mathbb{R}^{N}}|u|^{2^{*}}\right)^{\frac{22^{*}-p_{*}}{2^{*}}}\left(\int_{\mathbb{R}^{N}}|u|^{22^{*}}\right)^{\frac{p_{*}-2^{*}}{2^{*}}}$
$\displaystyle\leq C(N)\left(\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\right)^{\frac{22^{*}-p_{*}}{2}}\left(\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}\right)^{\frac{p_{*}-2^{*}}{2}}$
$\displaystyle\leq(N+2)\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}+C(N)\left(\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\right)^{\frac{22^{*}-p_{*}}{2^{*}+2-p_{*}}},$
which gives that $\int_{\mathbb{R}^{N}}|\nabla u|^{2}\geq C(N)$ and again
$\inf_{0<\mu\leq 1,u\in{\cal Q}_{\mu}(a)}\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}>0.$
* (2)
Since $p_{*}\gamma_{p_{*}}=2+N$, we see from (3.10) that
$\sup_{n\geq 1}\max\left\\{\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta},\int_{\mathbb{R}^{N}}|\nabla u|^{2}\right\\}<+\infty.$
Moreover, $Q_{\mu}(u_{n})=0$ implies that
(5.7) $\displaystyle C$ $\displaystyle\geq\mu\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}+\int_{\mathbb{R}^{N}}|\nabla u|^{2}$
$\displaystyle=\gamma_{p_{*}}\int_{\mathbb{R}^{N}}|u_{n}|^{p_{*}}-(2+N)\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla
u_{n}|^{2}$
$\displaystyle>(2+N)\left(\frac{N}{4(N+1)}-1\right)\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla
u_{n}|^{2},$
which finishes the proof.
∎
First, we will consider a minimizition problem
(5.8) $m_{\mu}(a):=\inf_{u\in{\cal Q}_{\mu}(a)}I_{\mu}(u).$
###### Remark 5.1.
It is easy to see from Lemma 5.4 and (3.10) that
(5.9) $\inf_{0\leq\mu\leq 1}m_{\mu}(a)\geq\frac{N}{2(2+N)}\inf_{0\leq\mu\leq
1,u\in{\cal Q}_{\mu}(a)}\int_{\mathbb{R}^{N}}|\nabla u|^{2}>0.$
On the other hand, to use the convergence Theorem 4.1, we need to give an
uniform upper bound of $m_{\mu}(a)$. Indeed for any fixed $a>a^{*}$, recalling
the function
$w_{a}=\left(\frac{a}{a_{*}}\right)^{\frac{1}{2}}Q_{p_{*}}^{\frac{1}{2}}\in{\cal
S}(a)$ in Lemma 5.1, and let $s_{\mu}:=s_{\mu}(w_{a})$, then from
$Q_{\mu}(s_{\mu}\star w_{a})=0$ we obtain
(5.10) $\displaystyle\quad~{}~{}(1+\gamma_{\theta})\mu
e^{-(2+N-\theta-\theta\gamma_{\theta})s_{\mu}}\left(\frac{a}{a_{*}}\right)^{\frac{\theta}{2}}\int_{\mathbb{R}^{N}}|\nabla
Q_{p_{*}}^{\frac{1}{2}}|^{\theta}+e^{-Ns_{\mu}}\left(\frac{a}{a_{*}}\right)\int_{\mathbb{R}^{N}}|\nabla
Q_{p_{*}}^{\frac{1}{2}}|^{2}$ $\displaystyle=(1+\gamma_{\theta})\mu
e^{-(2+N-\theta-\theta\gamma_{\theta})s_{\mu}}\int_{\mathbb{R}^{N}}|\nabla
w_{a}|^{\theta}+e^{-Ns_{\mu}}\int_{\mathbb{R}^{N}}|\nabla w_{a}|^{2}$
$\displaystyle=\gamma_{p_{*}}\int_{\mathbb{R}^{N}}|w_{a}|^{p_{*}}-(2+N)\int_{\mathbb{R}^{N}}|w_{a}|^{2}|\nabla
w_{a}|^{2}$
$\displaystyle=\frac{N(2+N)}{4(N+1)}\left(1-\left(\frac{a}{a_{*}}\right)^{-\frac{2}{N}}\right)\left(\frac{a}{a_{*}}\right)^{2+\frac{2}{N}}\|Q_{p_{*}}^{\frac{1}{2}}\|_{1}>0,$
it follows that $\sup_{0\leq\mu\leq 1}s_{\mu}<+\infty$. Therefore,
(5.11) $\displaystyle\sup_{0\leq\mu\leq 1}m_{\mu}(a)$
$\displaystyle\leq\sup_{0\leq\mu\leq 1}I_{\mu}(s_{\mu}\star
w_{a})=\sup_{0\leq\mu\leq 1}I_{\mu}(s_{\mu}\star w_{a})-Q_{\mu}(s_{\mu}\star
w_{a})$ $\displaystyle=\sup_{0\leq\mu\leq
1}\frac{2+N-\theta-\theta\gamma_{\theta}}{\theta(2+N)}\mu
e^{\theta(1+\gamma_{\theta})s_{\mu}}\int_{\mathbb{R}^{N}}|\nabla
Q_{p_{*}}^{\frac{1}{2}}|^{\theta}+\frac{N}{2(2+N)}e^{2s_{\mu}}\int_{\mathbb{R}^{N}}|\nabla
Q_{p_{*}}^{\frac{1}{2}}|^{2}$ $\displaystyle<+\infty.$
Now we construct a special Palais-Smale sequence of $I_{\mu}|_{{\cal S}(a)}$
at level $m_{\mu}(a)$. But different from the Section 3.2, in mass-critical
case there is no result as Lemma 3.4, and hence there is no the mountain-pass
type result as Lemma 3.5. So we will not consider $I_{\mu}$ directly. Instead
we study the auxilary functional $K_{\mu}(u)$ defined by (3.28), and we point
out that our approach is inspired by [6] (see also [12]). Similar to [6, Lemma
3.7], we have
###### Lemma 5.5.
Let a sequence $u_{n}\in{\cal S}(a)$ with $u_{n}\to u$ in ${\cal X}$ as
$n\to\infty$. Then if $u\in\partial{\cal O}$, we have
$K_{\mu}(u_{n})\to\infty$ as $n\to\infty$.
###### Proof.
If $u_{n}\to u$ in ${\cal X}$, then there are
$\int_{\mathbb{R}^{N}}|\nabla u_{n}|^{\theta}\to\int_{\mathbb{R}^{N}}|\nabla
u|^{\theta}>0,\quad\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}\to\int_{\mathbb{R}^{N}}|\nabla u|^{2}>0,$
$\int_{\mathbb{R}^{N}}|u_{n}|^{2}|\nabla
u_{n}|^{2}\to\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}>0,\quad\int_{\mathbb{R}^{N}}|u_{n}|^{p_{*}}\to\int_{\mathbb{R}^{N}}|u|^{p_{*}}>0.$
Let $s_{n}=s_{\mu}(u_{n})$. Since $Q_{\mu}(s_{n}\star u_{n})=0$, we obtain
(5.12) $\displaystyle\quad~{}~{}(1+\gamma_{\theta})\mu
e^{-(2+N-\theta-\theta\gamma_{\theta})s_{n}}\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{\theta}+e^{-Ns_{n}}\int_{\mathbb{R}^{N}}|\nabla u_{n}|^{2}$
$\displaystyle=\gamma_{p_{*}}\int_{\mathbb{R}^{N}}|u_{n}|^{p_{*}}-(2+N)\int_{\mathbb{R}^{N}}|u_{n}|^{2}|\nabla
u_{n}|^{2}$
$\displaystyle\to\gamma_{p_{*}}\int_{\mathbb{R}^{N}}|u|^{p_{*}}-(2+N)\int_{\mathbb{R}^{N}}|u|^{2}|\nabla
u|^{2}=0,$
where the last equality comes from $u\in\partial{\cal O}$. It follows that
$s_{n}\to+\infty$. So
(5.13) $\displaystyle K_{\mu}(u_{n})$ $\displaystyle=I_{\mu}(s_{n}\star
u_{n})=I_{\mu}(s_{n}\star u_{n})-Q_{\mu}(s_{n}\star u_{n})$
$\displaystyle=\frac{2+N-\theta-\theta\gamma_{\theta}}{\theta(2+N)}\mu
e^{\theta(1+\gamma_{\theta})s_{n}}\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{\theta}+\frac{N}{2(2+N)}e^{2s_{n}}\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}$ $\displaystyle\to+\infty.$
∎
Recalling the Definition A in Section 3.2, we give directly the following
results without proof, since the proof is very similar to the one of [6,
Proposition 3.9] (see also [12]).
###### Lemma 5.6.
Let ${\cal G}$ be a homotopy stable family of compact subsets of $Y={\cal
S}_{r}(a)$ with boundary $B=\emptyset$, and set
$d:=\inf_{A\in{\cal G}}\max_{u\in A}K_{\mu}(u).$
If $d>0$, then there exists a sequence $u_{n}\in{\cal S}_{r}(a)$ such that as
$n\to\infty$,
$I_{\mu}(u_{n})\to d,\quad I_{\mu}|_{{\cal S}(a)}^{\prime}(u_{n})\to 0,\quad
Q_{\mu}(u_{n})=0.$
Moreover, if one can find a minimizing sequence $A_{n}$ for $d$ with the
property that $u\geq 0$ a.e. for any $u\in A_{n}$, then one can find the
sequence $u_{n}$ satisfying the additional condition
$u_{n}^{-}\to 0,\quad\text{a.e. in}~{}{\mathbb{R}^{N}}.$
###### Remark 5.2.
As pointed out in [6], the set ${\cal O}$ is neither complete, nor connected,
and hence in principle the assumptions of the minimax theorem (such as [18,
Theorem 3.2]) are not satisfied. However, the connectedness assumption can be
avoided considering the restriction of $K_{\mu}$ on the connected component of
${\cal O}$ (if $B\neq\emptyset$, we need to assume that $B$ is contained in a
connected component of ${\cal Q}_{\mu}(a)$). Regarding the completeness, what
is really used in the deformation lemma [18, Lemma 3.7] is that the sublevel
sets $K_{\mu}^{c}:=\left\\{u\in{\cal S}(a):K_{\mu}(u)\leq c\right\\}$ are
complete for every $c\in{\mathbb{R}}$. This follows by Lemma 5.5. Hence the
minmax theorem [18, Theorem 3.2] can be used to obtain the Palais-Smale
sequence. The rest of the process is similar to Lemma 3.9.
###### Lemma 5.7.
For any fixed $\mu\in(0,1]$, there exists a sequence $u_{n}\in{\cal S}_{r}(a)$
such that
$I_{\mu}(u_{n})\to m_{\mu}(a),\quad I_{\mu}|_{{\cal S}(a)}^{\prime}(u_{n})\to
0,\quad Q_{\mu}(u_{n})=0\quad\text{and}\quad u_{n}^{-}\to 0~{}\text{a.e.
in}~{}{\mathbb{R}^{N}}.$
###### Proof.
We use Lemma 5.6 by taking the set ${\cal G}$ of all singletons belonging to
${\cal S}_{r}(a)$. It is clearly a homotopy stable family of compact subsets
of ${\cal S}_{r}(a)$ with boundary $B=\emptyset$. Observe that
$\alpha_{\mu}(a)=\inf_{A\in{\cal G}}\max_{u\in A}K_{\mu}(u)=\inf_{u\in{\cal
S}_{r}(a)}\max_{s\in{\mathbb{R}}}I_{\mu}(s\star u).$
We claim that
$\alpha_{\mu}(a)=m_{\mu}(a).$
Indeed, on the one hand, for any $u\in{\cal S}_{r}(a)$ there exists a
$s_{\mu}(u)$ such that $s_{\mu}(u)\star u\in{\cal Q}_{\mu}(a)$ and
$I_{\mu}(s_{\mu}(u)\star u)=\max_{s\in{\mathbb{R}}}I_{\mu}(s\star u)$. This
implies that
$\alpha_{\mu}(a)=\inf_{u\in{\cal
S}_{r}(a)}\max_{s\in{\mathbb{R}}}I_{\mu}(s\star u)\geq\inf_{u\in{\cal
Q}_{\mu}(a)}I_{\mu}(u)=m_{\mu}(a).$
On the other hand, for any $u\in{\cal Q}_{\mu}^{r}(a)$,
$I_{\mu}(u)=\max_{s\in{\mathbb{R}}}I_{\mu}(s\star u)$, so
$m_{\mu}^{r}(a):=\inf_{u\in{\cal Q}_{\mu}^{r}(a)}I_{\mu}(u)\geq\inf_{u\in{\cal
S}_{r}(a)}\max_{s\in{\mathbb{R}}}I_{\mu}(s\star u)=\alpha_{\mu}(a).$
Finally the inequality $m_{\mu}(a)\geq m_{\mu}^{r}(a)$ can be obtained easily
by the symmetric decreasing rearrangement. Thus, the conclusion follows
directly from Lemma 5.6. ∎
Then as in Section 3.2, we have
###### Theorem 5.1.
Let $p=p_{*}$. For any fixed $\mu\in(0,1]$, there exists a $u_{\mu}\in{\cal
X}_{r}\setminus\\{0\\}$ and a $\lambda_{\mu}\in{\mathbb{R}}$ such that
$\displaystyle I_{\mu}^{\prime}(u_{\mu})+\lambda_{\mu}u_{\mu}=0,$
$\displaystyle I_{\mu}(u_{\mu})=m_{\mu}(a),\quad Q_{\mu}(u_{\mu})=0,$
$\displaystyle 0<\|u_{\mu}\|_{2}^{2}\leq a,\quad u_{\mu}\geq 0.$
Moreover, if $\lambda_{\mu}\neq 0$, we have that $\|u_{\mu}\|_{2}^{2}=a$,
i.e., $m_{\mu}(a)$ is achieved, and $u_{\mu}$ is a ground state critical point
of $I_{\mu}|_{{\cal S}(a)}$.
###### Proof of Theorem 1.3: .
The proof is exactly the same as the one of Theorem 1.1, so we omit the
details. ∎
###### Remark 5.3.
We are not able to obtain multiple solutions as in Section 3.3. Indeed, if we
consider an open subset ${\cal O}$ and follow the strategy in Section 3.3, we
need to prove a result like Lemma 3.10. However for any finite dimensional
subspace $W_{j}$ of ${\cal X}$, using the equivalence of norms in finite
dimensional spaces, we can only obtain that for any $j>0$, there exists a
$a(j)>0$ large enough such that
$\left\\{u\in W_{j}:\|u\|_{2}^{2}=a\right\\}\subset{\cal
O}\quad\text{when}\quad a>a(j),$
which is necessary to prove the nonemptyness of the sets of type ${\cal
A}_{j}$. And another difficulty is that as $\mu\to 0^{+}$, we are unable to
distinguish the energy
$b_{j}(a):=\lim_{\mu\to 0^{+}}c_{\mu}^{j}(a)\quad\text{and}\quad
b_{k}(a):=\lim_{\mu\to 0^{+}}c_{\mu}^{k}(a),$
for $j\neq k$. As a result, we can not distinguish the solutions related to
$b_{j}(a)$ and $b_{k}(a)$.
Recalling Proposition 1.1, we prove the concentration theorem.
###### Proof of the Theorem 1.4: .
Let $u_{n}$ be a radially symmetric positive solution of (1.3) for $a=a_{n}$
with $a_{n}>a_{*}$ and $a_{n}\to a_{*}$. From Lemma 5.4, we see that
(5.14) $\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla
u_{n}|^{2}\geq\frac{C}{\left(\frac{a_{n}}{a_{*}}\right)^{\frac{2}{N}}-1}\to+\infty,$
(5.15) $\frac{\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}}{\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla u_{n}|^{2}}\leq
C\left(\left(\frac{a_{n}}{a_{*}}\right)^{\frac{2}{N}}-1\right)\to 0.$
Since $Q_{\mu}(u_{n})=0$, we know that
(5.16)
$\frac{\int_{\mathbb{R}^{N}}|u_{n}|^{p_{*}}}{\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla
u_{n}|^{2}}\to\frac{4(N+1)}{N}.$
Let $v_{n}(x):=\varepsilon_{n}^{\frac{N}{2}}u_{n}(\varepsilon_{n}x)$ with
$\varepsilon_{n}=\left(\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla
u_{n}|^{2}\right)^{-\frac{1}{2+N}}\to 0^{+}.$
Direct calculations show that $\|v_{n}\|_{2}^{2}=a_{n}\to a_{*}$,
$\int_{\mathbb{R}^{N}}v_{n}^{2}|\nabla v_{n}|^{2}=1$,
$\|v_{n}\|_{p_{*}}^{p_{*}}\to\frac{4(N+1)}{N}$ and
$\varepsilon_{n}^{N}\|\nabla v_{n}\|_{2}^{2}\to 0$. Then $v_{n}^{2}$ is
bounded in ${\cal E}^{p_{*}}$. Moreover, using [33, Lemma I.1], we deduce that
there exist $\delta>0$ and a sequence $y_{n}\in{\mathbb{R}^{N}}$ such that for
some $R>0$,
$\int_{B_{R}(y_{n})}v_{n}^{2}\geq\delta.$
Thus there exists a nonnegative radially symmetric function $v\neq 0$ with
$v^{2}\in{\cal E}^{p_{*}}\cap L^{2}({\mathbb{R}^{N}})$ such that
$v_{n}^{2}(\cdot+y_{n})\rightharpoonup v^{2}\quad\text{in}~{}{\cal
E}^{p_{*}},$ $v_{n}(\cdot+y_{n})\rightharpoonup
v\quad\text{in}~{}L^{2}({\mathbb{R}^{N}}),$ $v_{n}^{2}(\cdot+y_{n})\to
v^{2}\quad\text{in}~{}L^{q}({\mathbb{R}^{N}})~{}\text{for}~{}1<q<2^{*},$
$v_{n}(\cdot+y_{n})\to v\quad\text{a.e. in}~{}{\mathbb{R}^{N}}.$
Since $u_{n}$ solves
$-\Delta u_{n}-u_{n}\Delta u_{n}^{2}+\lambda_{n}u_{n}=u_{n}^{p_{*}-1},$
where the Lagrange multiplier is given by
$\lambda_{n}=\frac{1}{a_{n}}\left(\int_{\mathbb{R}^{N}}|u_{n}|^{p_{*}}-\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}-\int_{\mathbb{R}^{N}}u_{n}^{2}|\nabla u_{n}|^{2}\right),$
and $v_{n}$ satisfies
$-\varepsilon_{n}^{N}\Delta v_{n}-v_{n}\Delta
v_{n}^{2}+\varepsilon_{n}^{2+N}\lambda_{n}v_{n}=v_{n}^{p_{*}-1}.$
Combining (5.15) and (5.16), we deduce that
$\varepsilon_{n}^{2+N}\lambda_{n}\to\frac{4}{Na^{*}}$. Then a similar approach
as Lemma A.2 tells that
(5.17) $-v\Delta v^{2}+\varepsilon_{n}^{2+N}\lambda_{n}v=v^{p_{*}-1}.$
Now setting
(5.18) $\displaystyle w_{n}(x):$
$\displaystyle=\left(\frac{Na^{*}}{4}\right)^{\frac{N}{2+N}}v_{n}^{2}\left(\left(\frac{Na^{*}}{4}\right)^{\frac{1}{2+N}}x+y_{n}\right)$
$\displaystyle=\left[\left(\frac{Na^{*}}{4}\right)^{\frac{1}{2+N}}\varepsilon_{n}\right]^{N}u_{n}^{2}\left(\left(\frac{Na^{*}}{4}\right)^{\frac{1}{2+N}}\varepsilon_{n}x+\varepsilon_{n}y_{n}\right),$
(5.19)
$w(x):=\left(\frac{Na^{*}}{4}\right)^{\frac{N}{2+N}}v^{2}\left(\left(\frac{Na^{*}}{4}\right)^{\frac{1}{2+N}}x\right),$
it is easily seen that $w_{n}\rightharpoonup w$ in ${\cal E}^{p_{*}}$ and
$\|w_{n}\|_{1}=\|v_{n}\|_{2}^{2}=a_{n}$. Moreover, it follows from (5.17) that
$w$ is a solution of (1.10). Thus $w=Q_{p_{*}}$, and hence
$\|w\|_{1}=\|v\|_{2}^{2}=a_{*}$. So we have $v_{n}\to v$ in
$L^{2}({\mathbb{R}^{N}})$, which finishes the proof. ∎
## Appendix A A Appendix
###### Lemma A.1.
In the setting of Section 2.1, $V(u)\in{\cal C}^{1}({\cal X})$.
###### Proof.
The proof is elementry. When $N=2$, since
$W^{1,\theta}({\mathbb{R}}^{2})\hookrightarrow{\cal
C}^{0,\alpha}({\mathbb{R}}^{2})$, it is easily to check that $V(u)\in{\cal
C}^{1}({\cal X})$. Now we set $N\geq 3$. For any $u,\phi\in{\cal X}$,
(A.1)
$\frac{V(u+t\phi)-V(u)}{t}=At+Bt^{2}+Ct^{3}+2\int_{\mathbb{R}^{N}}u\phi|\nabla
u|^{2}+u^{2}\nabla u\cdot\nabla\phi,$
where
$A=\int_{\mathbb{R}^{N}}u^{2}|\nabla\phi|^{2}+\phi^{2}|\nabla
u|^{2}+4u\phi\nabla u\cdot\nabla\phi,$ $B=\int_{\mathbb{R}^{N}}\phi^{2}\nabla
u\cdot\nabla\phi+u\phi|\nabla\phi|^{2}\quad\text{and}\quad
C=\int_{\mathbb{R}^{N}}\phi^{2}|\nabla\phi|^{2}.$
We need to prove $A,B,C$ are finite numbers. Indeed, since
$\frac{4N}{N+2}<\theta<\frac{4N+4}{N+2}<4$, there is
$\theta<\frac{2\theta}{\theta-2}<\frac{\theta N}{N-\theta}$ and hence
(A.2)
$\int_{\mathbb{R}^{N}}u^{2}|\nabla\phi|^{2}\leq\left(\int_{\mathbb{R}^{N}}|u|^{\frac{2\theta}{\theta-2}}\right)^{\frac{\theta-2}{\theta}}\left(\int_{\mathbb{R}^{N}}|\nabla\phi|^{\theta}\right)^{\frac{2}{\theta}}\leq
C\|u\|_{W^{1,\theta}({\mathbb{R}^{N}})}^{\frac{2}{\theta}}\|\phi\|_{W^{1,\theta}({\mathbb{R}^{N}})}^{\frac{2}{\theta}}<\infty,$
and we can handle other terms in a similar way, so $A,B,C$ are finite numbers.
Now by letting $t\to 0$ in (A.1), we immediately get the Frèchet deravetive is
$DV(u)[\phi]=2\int_{\mathbb{R}^{N}}u\phi|\nabla u|^{2}+u^{2}\nabla
u\cdot\nabla\phi.$
Then in a similarly way in (A.2), one can prove that $DV(u)$ is continuous for
$u\in{\cal X}$, so $V(u)\in{\cal C}^{1}({\cal X})$ and $V^{\prime}(u)=DV(u)$.
∎
###### Lemma A.2.
Assume that $I_{\mu}^{\prime}(u_{n})+\lambda u_{n}\to 0$ for some
$\lambda\in{\mathbb{R}}$ with $u_{n}\in{\cal X}$, and that
$u_{n}\rightharpoonup u$ in ${\cal X}$. Then up to a subsequence,
* (1)
$u_{n}\to u$ in ${\cal X}_{loc}:=W^{1,\theta}_{loc}({\mathbb{R}^{N}})\cap
W^{1,2}_{loc}({\mathbb{R}^{N}})$,
* (2)
$u_{n}\nabla u_{n}\to u\nabla u$ in $(L^{2}_{loc}({\mathbb{R}^{N}}))^{N}$,
* (3)
$I_{\mu}^{\prime}(u)+\lambda u=0$.
###### Proof.
The proof is inspired by [29, Lemma 14.3]. Since $u_{\mu_{n}}\rightharpoonup
u$ in ${\cal X}$, we have $\sup_{n}\|u_{n}\|_{\cal X}\leq C_{0}$. For any
$R>1$, we set $\phi\in{\cal C}_{0}^{\infty}({\mathbb{R}^{N}})$ satisfying
$0\leq\phi\leq 1,\quad\phi(x)=\begin{cases}1,\quad&|x|\leq R,\\\
0,\quad&|x|\geq 2R,\end{cases}\quad\text{and}\quad|\nabla\phi|\leq 2.$
Then for any $n,m\in{\mathbb{N}}$,
(A.3) $\displaystyle o(1)_{n}+o(1)_{m}$
$\displaystyle=(I_{\mu}^{\prime}(u_{n})+\lambda
u_{n})[(u_{n}-u_{m})\phi]-(I_{\mu}^{\prime}(u_{m})+\lambda
u_{m})[(u_{n}-u_{m})\phi]$ $\displaystyle=\mu\int_{\mathbb{R}^{N}}(|\nabla
u_{n}|^{\theta-2}\nabla u_{n}-|\nabla u_{m}|^{\theta-2}\nabla
u_{m})\cdot\nabla\left((u_{n}-u_{m})\phi\right)$
$\displaystyle\quad~{}~{}+\int_{\mathbb{R}^{N}}(\nabla u_{n}-\nabla
u_{m})\cdot\nabla\left((u_{n}-u_{m})\phi\right)$
$\displaystyle\quad~{}~{}+2\int_{\mathbb{R}^{N}}(u_{n}|\nabla
u_{n}|^{2}-u_{m}|\nabla u_{m}|^{2})(u_{n}-u_{m})\phi$
$\displaystyle\quad~{}~{}+2\int_{\mathbb{R}^{N}}(u_{n}^{2}\nabla
u_{n}-u_{m}^{2}\nabla u_{m})\cdot\nabla\left((u_{n}-u_{m})\phi\right)$
$\displaystyle\quad~{}~{}-\int_{\mathbb{R}^{N}}(|u_{n}|^{p-2}u_{n}-|u_{m}|^{p-2}u_{m})(u_{n}-u_{m})\phi$
$\displaystyle=:K_{1}+K_{2}+K_{3}+K_{4}+K_{5}.$
Next we estimate $K_{i}$ for $i=1,2,3,4,5$ one by one.
$\displaystyle K_{1}$ $\displaystyle=\mu\int_{B_{R}}(|\nabla
u_{n}|^{\theta-2}\nabla u_{n}-|\nabla u_{m}|^{\theta-2}\nabla
u_{m})\cdot\nabla(u_{n}-u_{m})$
$\displaystyle\quad~{}~{}+\mu\int_{B_{2R}\setminus B_{R}}(|\nabla
u_{n}|^{\theta-2}\nabla u_{n}-|\nabla u_{m}|^{\theta-2}\nabla
u_{m})\cdot\nabla(u_{n}-u_{m})\phi$
$\displaystyle\quad~{}~{}+\mu\int_{B_{2R}\setminus B_{R}}(|\nabla
u_{n}|^{\theta-2}\nabla u_{n}-|\nabla u_{m}|^{\theta-2}\nabla
u_{m})\cdot\nabla\phi(u_{n}-u_{m})$ $\displaystyle\geq C\mu\int_{B_{R}}|\nabla
u_{n}-\nabla u_{m}|^{\theta}+C\mu\int_{B_{2R}\setminus B_{R}}|\nabla
u_{n}-\nabla u_{m}|^{\theta}\phi$
$\displaystyle\quad~{}~{}-C\left(\|u_{n}\|_{\theta}^{\theta-1}+\|u_{m}\|_{\theta}^{\theta-1}\right)\|u_{n}-u_{m}\|_{L^{\theta}(B_{2R})}$
$\displaystyle\geq C\mu\|\nabla u_{n}-\nabla
u_{m}\|_{L^{\theta}(B_{R})}^{\theta}-C\|u_{n}-u_{m}\|_{L^{\theta}(B_{2R})},$
and similarly
$K_{2}\geq C\|\nabla u_{n}-\nabla
u_{m}\|_{L^{2}(B_{R})}^{2}-C\|u_{n}-u_{m}\|_{L^{2}(B_{2R})},$
$K_{3}\geq-C\|u_{n}-u_{m}\|_{L^{\theta}(B_{2R})},$ $K_{4}\geq 2\|u_{n}\nabla
u_{n}-u_{m}\nabla
u_{m}\|_{L^{2}(B_{R})}^{2}-C\|u_{n}-u_{m}\|_{L^{\theta}(B_{2R})},$
$K_{5}\geq-C\|u_{n}-u_{m}\|_{L^{p}(B_{2R})}.$
Substituting these estimates into (A.3), we obtain
(A.4) $\displaystyle\quad~{}\mu\|\nabla u_{n}-\nabla
u_{m}\|_{L^{\theta}(B_{R})}^{\theta}+\|\nabla u_{n}-\nabla
u_{m}\|_{L^{2}(B_{R})}^{2}+\|u_{n}\nabla u_{n}-u_{m}\nabla
u_{m}\|_{L^{2}(B_{R})}^{2}$ $\displaystyle\leq
C\|u_{n}-u_{m}\|_{L^{\theta}(B_{2R})}+C\|u_{n}-u_{m}\|_{L^{2}(B_{2R})}+C\|u_{n}-u_{m}\|_{L^{\theta}(B_{2R})}+o(1)_{n}+o(1)_{m}$
$\displaystyle\to 0,\quad\text{as}~{}n\to\infty,~{}m\to\infty,$
where in the last estimate we use the compact embedding theorem in bounded
domains. Thus for any $R>1$, $u_{n}$ is a Cauchy sequence in
$W^{1,\theta}(B_{R})\cap W^{1,2}(B_{R})$, and $u_{n}\nabla u_{n}$ is also a
Cauchy sequence in $\left(L^{2}(B_{R})\right)^{N}$. So up to a subsequence
$u_{n}\to u$ in ${\cal X}_{loc}$ and $u_{n}\nabla u_{n}\to u\nabla u$ in
$\left(L^{2}_{loc}({\mathbb{R}^{N}})\right)^{N}$. Finally, we need to prove
that for any $\varphi\in{\cal X}$, there holds $(I_{\mu}^{\prime}(u)+\lambda
u)[\varphi]=0$. Since $u_{n}\nabla u_{n}\to u\nabla u$ a.e. in
${\mathbb{R}^{N}}$ and $u_{n}$ is bounded in ${\cal X}$, we obtain that
$|\nabla u_{n}|^{\theta-2}\nabla u_{n}\rightharpoonup|\nabla
u|^{\theta-2}\nabla
u\quad\text{in}~{}L^{\frac{\theta}{\theta-1}}({\mathbb{R}^{N}}),$
$u_{n}|\nabla u_{n}|^{2}\rightharpoonup u|\nabla
u|^{2}\quad\text{in}~{}L^{\frac{4}{3}}({\mathbb{R}^{N}}),$ $u_{n}^{2}\nabla
u_{n}\rightharpoonup u^{2}\nabla
u\quad\text{in}~{}\left(L^{\frac{4}{3}}({\mathbb{R}^{N}})\right)^{N},$
it follows that
$(I_{\mu}^{\prime}(u)+\lambda
u)[\varphi]=\lim_{n\to\infty}(I_{\mu}^{\prime}(u_{n})+\lambda
u_{n})[\varphi]=0.$
∎
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# A note on the $g$ and $h$ control charts
Chanseok Park
Applied Statistics Laboratory
Department of Industrial Engineering
Pusan National University
Busan 46241, Korea Min Wang
Department of Management Science and Statistics
University of Texas at San Antonio
San Antonio, TX 78249, USA
###### Abstract
In this note, we revisit the $g$ and $h$ control charts that are commonly used
for monitoring the number of conforming cases between the two consecutive
appearances of nonconformities. It is known that the process parameter of
these charts is usually unknown and estimated by using the maximum likelihood
estimator and the minimum variance unbiased estimator. However, the minimum
variance unbiased estimator in the control charts has been inappropriately
used in the quality engineering literature. This observation motivates us to
provide the correct minimum variance unbiased estimator and investigate
theoretical and empirical biases of these estimators under consideration.
Given that these charts are developed based on the underlying assumption that
samples from the process should be balanced, which is often not satisfied in
many practical applications, we propose a method for constructing these charts
with unbalanced samples.
Keywords: control charts, geometric distribution, maximum likelihood
estimator, minimum variance unbiased estimator, $g$ and $h$ charts.
## 1 Introduction
In an introductory statistics course, the geometric distribution is defined as
a probability distribution that represents the number of failures (or normal
cases) before observing the first success (or adverse case) in a series of
Bernoulli trials. Based on this distribution, Kaminsky et al. (1992) proposed
Shewhart-type statistical control charts, the so-called $g$ and $h$ charts,
for monitoring the number of conforming cases between the two consecutive
appearances of nonconformities. Since then they have been widely used for
monitoring the control process especially in the healthcare department; see,
for example, Benneyan (1999), Benneyan (2000), to name just a few.
The process parameter in the $g$ and $h$ charts is usually unknown and needs
to be estimated in the control chart procedures. One can employ the maximum
likelihood (ML) estimator and the minimum variance unbiased (MVU) estimator
for the process parameter. Of particular note is that the MVU estimator in the
control charts has been inappropriately used in the quality engineering
literature. This motivates us to obtain the correct MVU estimator and
investigate theoretical biases of the estimators considered in this note.
Furthermore, Monte Carlo simulations are conducted to investigate the
empirical biases of these estimators. Numerical results show that the
theoretical and empirical biases of the existing estimators are severe when
the sample size is small and the value of process parameter is large and that
those of the proposed MVU estimator are always very close to zero for all the
simulated scenarios.
It deserves mentioning that these conventional $g$ and $h$ charts are
developed based on the underlying assumption that samples from the process
should be balanced so that the samples have the same size, whereas such an
assumption can be restrictive and may not be satisfied in many practical
applications. To overcome this issue, we propose the method of how to
construct the $g$ and $h$ charts with unbalanced samples.
The remainder of this note is organized as follows. In Section 2, we briefly
review the geometric and negative binomial distributions and then provide the
correct MVU estimator for the process parameter. In Section 3, we obtain the
parameter estimation with unequal sample sizes and investigate theoretical and
empirical properties of these estimators considered in this note. In Section
4, based on the proposed estimator, we provide a method of constructing the
$g$ and $h$ charts with unbalanced samples. Concluding remarks are provided in
Section 5.
## 2 Basic properties of the geometric and negative binomial distributions
Let $Y_{i}$ be independent and identically distributed (iid) according to the
shifted geometric distribution with location shift $a$ and Bernoulli
probability $p$ for $i=1,2,\ldots$. Then its probability mass function (pmf)
is given by
$f(y)=P(Y_{i}=y)=p(1-p)^{y-a},$ (1)
where $y=a,a+1,\ldots$ and $a$ is the known minimum possible number of events
(usually $a=0,1$). The mean and variance of $Y_{i}$ are respectively given by
$E(Y_{i})=\frac{1-p}{p}+a\quad\textrm{and}\quad\mathrm{Var}(Y_{i})=\frac{1-p}{p^{2}}.$
For notational convenience, we let $T_{n}=\sum_{i=1}^{n}Y_{i}$. Then $T_{n}$
has the (shifted) negative binomial with predefined location shift $na$ and
Bernoulli probability $p$ and its pmf is given by
$g_{n}(t)=P(T_{n}=t)=\binom{t-na+n-1}{n-1}p^{n}(1-p)^{t-na},$ (2)
where $t=na,na+1,\ldots$. The mean and variance of $T_{n}$ are respectively
given by
$E(T_{n})=\frac{n(1-p)}{p}+na\quad\textrm{and}\quad\mathrm{Var}(Y_{i})=\frac{n(1-p)}{p^{2}}.$
It is well known that the method of moments and the method of ML yield the
same estimator of $p$, which is given by
$\hat{p}_{\mathrm{ml}}=\frac{1}{\bar{Y}-a+1},$ (3)
where $\bar{Y}=\sum_{i=1}^{n}Y_{i}/n$. It is worth noting that this estimator
is not unbiased and that we are able to identify the best unbiased estimator
of $p$ summarized in the following theorem.
###### Theorem 1.
The MVU estimator for the parameter of the geometric distribution in (1) is
given by
$\hat{p}_{\mathrm{mvu}}=\frac{n-1}{\sum_{i=1}^{n}Y_{i}-na+n-1}=\frac{(n-1)/n}{\bar{Y}-a+1-1/n}.$
###### Proof.
It is immediate from Lehmann and Casella (1998) that
$T_{n}=\sum_{i=1}^{n}Y_{i}$ is a complete sufficient statistic since the joint
mass functions of iid geometric distributions form an exponential family.
Thus, we can employ the Rao-Blackwell theorem (Rao, 1945; Blackwell, 1947) to
obtain the MVU estimator of $p$ as follows.
Let $\delta={I}(Y_{n}=a)$ where $I(\cdot)$ is the indicator function. Then
$\delta$ is an unbiased estimator of $p$ since
$E(\delta)=P(Y_{n}=a)=p.$
Conditioning the unbiased estimator $\delta$ on the complete sufficient
statistic $T_{n}=t$ and taking the expectation, we can obtain the MVU
estimate, denoted by $\eta(t)$, due to the Rao-Blackwell theorem
$\eta(t)=E\big{[}{I}(Y_{n}=a)\mid
T_{n}=t\big{]}=\frac{P(Y_{n}=a,T_{n}=t)}{P(T_{n}=t)}.$ (4)
Since $Y_{n}=a$ and $T_{n}=Y_{1}+Y_{2}+\cdots+Y_{n}=t$, we have
$T_{n-1}=Y_{1}+Y_{2}+\cdots+Y_{n-1}=t-a$ and $T_{n-1}$ is independent of
$Y_{n}$. Thus, we have
$\eta(t)=\frac{P(Y_{n}=a,T_{n-1}=t-a)}{P(T_{n}=t)}=\frac{P(Y_{n}=a)\cdot
P(T_{n-1}=t-a)}{P(T_{n}=t)}.$
Note that the pmfs of $Y_{n}=a$ and $T_{n-1}=t-a$ are given by $f(a)$ in (1)
and $g_{n-1}(t-a)$ from (2), respectively. Thus, we have
$\eta(t)=\frac{f(a)\cdot
g_{n-1}(t-a)}{g_{n}(t)}=\frac{p\cdot\binom{t-a-(n-1)a+n-2}{n-2}p^{n-1}(1-p)^{t-a-(n-1)a}}{\binom{t-na+n-1}{n-1}p^{n}(1-p)^{t-na}},$
which can be simplified as
$\eta(t)=\frac{n-1}{t-na+n-1}.$ (5)
Using (5), we obtain the MVU estimator of $p$ which is given by
$\hat{p}_{\mathrm{mvu}}=\frac{(n-1)/n}{\bar{Y}-a+1-1/n}.$
This completes the proof. ∎
## 3 Parameter estimation with unequal sample sizes
We assume that there are $m$ samples and each sample has different sample
sizes. We denote the size of the $i$th sample by $n_{i}$ for $i=1,\ldots,m$.
Let $X_{ij}$ be the number of independent Bernoulli trials (cases) until the
first nonconforming case in the $i$th sample for $i=1,\ldots,m$ and
$j=1,\ldots,n_{i}$. We assume that $X_{ij}$’s are iid geometric random
variables with location shift $a$ and Bernoulli probability $p$. Let
$T_{N}=\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}X_{ij}$ and $N=\sum_{i=1}^{m}n_{i}$.
Then it is easily seen from (2) that $T_{N}$ has the negative binomial with
predefined location shift $Na$ and Bernoulli probability $p$ and its pmf is
given by
$g_{N}(t)=P(T_{N}=t)=\binom{t-Na+N-1}{N-1}p^{N}(1-p)^{t-Na},$ (6)
where $t=Na,Na+1,\ldots$. It is immediate from (3) that the ML estimator with
all the samples is given by
$\hat{p}_{\mathrm{ml}}=\frac{1}{\bar{\bar{X}}-a+1},$
where $\bar{\bar{X}}=\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}X_{ij}/N$. By following
Theorem 1, we obtain the MVU estimator of $p$ which is given by
$\hat{p}_{\mathrm{mvu}}=\frac{(N-1)/N}{\bar{\bar{X}}-a+1-1/N}.$
To the best of our knowledge, the MVU estimator $\hat{p}_{\mathrm{mvu}}$ above
has not yet been used in the quality engineering literature. For example,
Benneyan (2001) and Minitab (2020) use the following estimator
$\hat{p}_{\mathrm{b}}$ as the MVU estimator of $p$
$\hat{p}_{\mathrm{b}}=\frac{(N-1)/N}{\bar{\bar{X}}-a+1},$ (7)
which is however not unbiased. As an illustration, consider the case of the
degenerating geometric distribution with $p=1$. Then we have $P(X_{ij}=a)=1$,
so that $\bar{\bar{X}}=a$. Thus, we have $\hat{p}_{\mathrm{ml}}=1$ and
$\hat{p}_{\mathrm{mvu}}=1$, whereas $\hat{p}_{\mathrm{b}}=1-1/N$, indicating
that $\hat{p}_{\mathrm{b}}$ is not unbiased.
It is worth noting that the estimator $\hat{p}_{\mathrm{b}}$ in (7) was
obtained by simply multiplying the ML estimator with the factor $(N-1)/N$,
that is, $\hat{p}_{\mathrm{b}}=\hat{p}_{\mathrm{ml}}\cdot(N-1)/N$. For the
case of the exponential distribution with the density $f(y)=\lambda
e^{-\lambda y}$, which can be regarded as a continuous version of the
geometric distribution, the MVU estimator of $\lambda$ can be obtained by
simply multiplying the unbiasing factor $(N-1)/N$ with the ML estimator; see,
for example, Miyakawa (1984) and Park (2010). However, this technique fails to
the case of the geometric distribution. Also, it is of interest to provide the
inequality relation of the three estimators considered above in the following
theorem.
###### Theorem 2.
For $0<p<1$, we have
$\hat{p}_{\mathrm{b}}<\hat{p}_{\mathrm{mvu}}<\hat{p}_{\mathrm{ml}}.$
###### Proof.
First, we show that $\hat{p}_{\mathrm{b}}<\hat{p}_{\mathrm{mvu}}$. Since the
denominator of $\hat{p}_{\mathrm{b}}$ is always larger than that of
$\hat{p}_{\mathrm{mvu}}$, we have
$\hat{p}_{\mathrm{b}}<\hat{p}_{\mathrm{mvu}}$.
Next, we show that $\hat{p}_{\mathrm{mvu}}<\hat{p}_{\mathrm{ml}}$. To prove
this, we use the fact that the mediant of the two fractions is positioned
between them, that is,
$\frac{a}{c}<\frac{a+b}{c+d}<\frac{b}{d},$
where $a/c<b/d$ and $a,b,c,d>0$. The estimator $\hat{p}_{\mathrm{ml}}$ is the
mediant of $\hat{p}_{\mathrm{mvu}}$ and $(1/N)/(1/N)$, that is,
$\frac{1-1/N}{\bar{\bar{X}}-a+1-1/N}<\frac{1}{\bar{\bar{X}}-a+1}<\frac{1/N}{1/N},$
which completes the proof. ∎
We observe from Theorem 2 that $\hat{p}_{\mathrm{b}}$ tends to underestimate
the true value $p$ and that $\hat{p}_{\mathrm{ml}}$ tends to overshoot the
true value. Since $\hat{p}_{\mathrm{b}}$ and $\hat{p}_{\mathrm{ml}}$ are
biased, a natural question arises: what are the theoretical biases of these
estimators? In what follows, we provide the first moments of these estimators
so that the biases of the estimators are easily obtained by subtracting the
true value of $p$ from their first moments.
###### Theorem 3.
For $0<p<1$, we have
$\displaystyle E(\hat{p}_{\mathrm{ml}})$
$\displaystyle=p^{N}\cdot{{}_{2}}F_{1}(N,N;N+1;1-p)$ and $\displaystyle
E(\hat{p}_{\mathrm{b}})$
$\displaystyle=\left(\frac{N-1}{N}\right)p^{N}\cdot{{}_{2}}F_{1}(N,N;N+1;1-p),$
where ${{}_{2}}F_{1}(\cdot)$ is the Gaussian hypergeometric function.
###### Proof.
If $X_{i}$ has the geometric distribution with location shift $a$ and $p$,
then $X_{i}-a$ also follows the geometric distribution with zero shift.
Without loss of generality, we may thus assume that $a=0$. Since
$\hat{p}_{\mathrm{ml}}=1/(\bar{\bar{X}}+1)=N/(T_{N}+N)$, it is immediate upon
using (6) that we have
$E(\hat{p}_{\mathrm{ml}})=\sum_{t=0}^{\infty}\frac{N}{t+N}\cdot
g_{N}(t)=\sum_{t=0}^{\infty}\frac{N}{t+N}\cdot\binom{t+N-1}{N-1}p^{N}(1-p)^{t},$
that is,
$E(\hat{p}_{\mathrm{ml}})=\frac{Np^{N}}{(1-p)^{N}}\sum_{t=0}^{\infty}\binom{t+N-1}{N-1}\frac{(1-p)^{t+N}}{t+N}.$
Using the identity $(1-p)^{t+N}/(t+N)=\int_{p}^{1}(1-y)^{t+N-1}dy$, we have
$\displaystyle E(\hat{p}_{\mathrm{ml}})$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\sum_{t=0}^{\infty}\binom{t+N-1}{N-1}\int_{p}^{1}(1-y)^{t+N-1}dy$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\int_{p}^{1}\frac{(1-y)^{N-1}}{y^{N}}\left[\sum_{t=0}^{\infty}\binom{t+N-1}{N-1}y^{N}(1-y)^{t}\right]dy.$
(8)
Since $\binom{t+N-1}{N-1}y^{N}(1-y)^{t}$ is the pmf of the negative binomial
distribution, we have
$\sum_{t=0}^{\infty}\binom{t+N-1}{N-1}y^{N}(1-y)^{t}=1.$
Thus, Equation (8) can be further simplified as
$E(\hat{p}_{\mathrm{ml}})=\frac{Np^{N}}{(1-p)^{N}}\int_{p}^{1}{(1-y)^{N-1}}{y^{-N}}dy.$
Using the integration by substitution with $x=1-y$, the above is written as
$\displaystyle E(\hat{p}_{\mathrm{ml}})$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\int_{0}^{1-p}x^{N-1}(1-x)^{-N}dx$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\cdot B_{1-p}(N,1-N),$
where $B_{x}(a,b)$ is the incomplete beta function defined as
$B_{x}(a,b)=\int_{0}^{x}y^{a-1}(1-y)^{b-1}dy.$
It deserves mentioning that the calculation of $B_{1-p}(N,1-N)$ can be complex
because few software packages provide its calculation with negative argument.
To deal with this difficulty, one can use the hypergeometric representation of
the incomplete beta function (Dutka, 1981; Özarslan and Ustaoğlu, 2019) which
is given by
$B_{x}(a,b)=\frac{x^{a}}{a}\cdot{{}_{2}}F_{1}(a,1-b;a+1;x).$ (9)
Here ${{}_{p}}F_{q}(\cdot)$ is the hypergeometric function (Abramowitz and
Stegun, 1964; Seaborn, 1991) and it is defined as
${{}_{p}}F_{q}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};z)=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\cdots(a_{p})_{n}}{(b_{1})_{n}\cdots(b_{q})_{n}}\frac{z^{n}}{n!},$
(10)
where $(a)_{n}$ is the Pochhammer symbol for the rising factorial defined as
$(a)_{0}=1$ and $(a)_{n}=a(a+1)\cdots(a+n-1)$ for $n=1,2,\ldots$. Thus, by
using (9), we have
$E(\hat{p}_{\mathrm{ml}})=p^{N}\cdot{{}_{2}}F_{1}(N,N;N+1;1-p).$ (11)
Note that we can easily obtain $E(\hat{p}_{\mathrm{b}})$ since
$\hat{p}_{\mathrm{b}}=\hat{p}_{\mathrm{ml}}\cdot(N-1)/N$. This completes the
proof. ∎
By using the well-known Euler transformation formula for the hypergeometric
function (Miller and Paris, 2011) which is given by
${{}_{2}}F_{1}(a,b;c;z)=(1-z)^{c-a-b}{{}_{2}}F_{1}(c-a,c-b;c;z),$
we obtain $E(\hat{p}_{\mathrm{ml}})=p\cdot{{}_{2}}F_{1}(1,1;N+1;1-p)$. Then
according the definition of the hypergeometric function in (10), we have
$\displaystyle E(\hat{p}_{\mathrm{ml}})$
$\displaystyle=p\sum_{n=0}^{\infty}\frac{(1)_{n}(1)_{n}}{(N+1)_{k}}\frac{(1-p)^{n}}{n!}$
$\displaystyle=p\sum_{n=0}^{\infty}\frac{N!~{}n!}{(N+n)!}(1-p)^{n}$
$\displaystyle=p+\sum_{n=1}^{\infty}\frac{p(1-p)^{n}}{\binom{N+n}{n}}$
since $(1)_{n}=n!$ and $(N+1)_{n}=(N+n)!/N!$. Then the biases of the
estimators $\hat{p}_{\mathrm{ml}}$ and $\hat{p}_{\mathrm{b}}$ are obtained as
$\displaystyle\mathrm{Bias}(\hat{p}_{\mathrm{ml}})$
$\displaystyle=\sum_{n=1}^{\infty}\frac{p(1-p)^{n}}{\binom{N+n}{n}}$ and
$\displaystyle\mathrm{Bias}(\hat{p}_{\mathrm{b}})$
$\displaystyle=-\frac{p}{N}+\frac{N-1}{N}\sum_{n=1}^{\infty}\frac{p(1-p)^{n}}{\binom{N+n}{n}},$
respectively. It should be noted that the R language provides the hypergeo
package to calculate the hypergeometric function; see Hankin (2016). We can
calculate the theoretical values of the biases and provide these values in
Figure 1 along with the empirical values. It deserves mentioning that the
theoretical bias of $\hat{p}_{\mathrm{mvu}}$ is trivially zero.
In what follows, we provide the second moments of the estimators so that their
variances can be easily obtained using them.
###### Theorem 4.
For $0<p<1$, we have
$\displaystyle E(\hat{p}_{\mathrm{ml}}^{2})$
$\displaystyle=p^{N}\cdot{{}_{3}}F_{2}(N,N,N;N+1,N+1;1-p),$ $\displaystyle
E(\hat{p}_{\mathrm{b}}^{2})$
$\displaystyle=\frac{(N-1)^{2}p^{N}}{N^{2}}\cdot{{}_{3}}F_{2}(N,N,N;N+1,N+1;1-p),$
and $\displaystyle E(\hat{p}_{\mathrm{mvu}}^{2})$
$\displaystyle=p^{N}\cdot{{}_{2}}F_{1}(N-1,N-1;N;1-p).$
###### Proof.
We first note that
$\displaystyle E(\hat{p}_{\mathrm{ml}}^{2})$
$\displaystyle=\sum_{t=0}^{\infty}\left(\frac{N}{t+N}\right)^{2}\cdot
g_{N}(t)$
$\displaystyle=\sum_{t=0}^{\infty}\left(\frac{N}{t+N}\right)^{2}\cdot\binom{t+N-1}{N-1}p^{N}(1-p)^{t}$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\sum_{t=0}^{\infty}\frac{N}{t+N}\cdot\binom{t+N-1}{N-1}\frac{(1-p)^{t+N}}{t+N}.$
By using the identity $(1-p)^{t+N}/(t+N)=\int_{p}^{1}(1-y)^{t+N-1}dy$, we have
$\displaystyle E(\hat{p}_{\mathrm{ml}}^{2})$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\sum_{t=0}^{\infty}\frac{N}{t+N}\cdot\binom{t+N-1}{N-1}\int_{p}^{1}(1-y)^{t+N-1}dy$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\int_{p}^{1}\frac{(1-y)^{N-1}}{y^{N}}\left[\sum_{t=0}^{\infty}\frac{N}{t+N}\binom{t+N-1}{N-1}y^{N}(1-y)^{t}\right]dy.$
The term in the integrand,
$\sum_{t=0}^{\infty}\frac{N}{t+N}\binom{t+N-1}{N-1}y^{N}(1-y)^{t}$, is
essentially the same as the first moment of $\hat{p}_{\mathrm{ml}}$ with
probability $y$. Thus, it follows from (11) that
$\sum_{t=0}^{\infty}\frac{N}{t+N}\binom{t+N-1}{N-1}y^{N}(1-y)^{t}=y^{N}\cdot{{}_{2}}F_{1}(N,N;N+1;1-y),$
which results in
$\displaystyle E(\hat{p}_{\mathrm{ml}}^{2})$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\int_{p}^{1}(1-y)^{N-1}\cdot{{}_{2}}F_{1}(N,N;N+1;1-y)dy$
$\displaystyle=\frac{Np^{N}}{(1-p)^{N}}\int_{0}^{1-p}x^{N-1}\cdot{{}_{2}}F_{1}(N,N;N+1;x)dx.$
(12)
Using the general integral representation for ${{}_{p+k}}F_{q+k}$ in Theorem
38 of Rainville (1960) and Section 2 of Driver and Johnston (2006), we have
$\int_{0}^{1-p}x^{N-1}\cdot{{}_{2}}F_{1}(N,N;N+1;x)dx=\frac{(1-p)^{N}}{N}\cdot{{}_{3}}F_{2}(N,N,N;N+1,N+1;1-p).$
(13)
Substituting (13) into (12), we obtain the first result. The second result is
easily obtained from $\hat{p}_{\mathrm{b}}=\hat{p}_{\mathrm{ml}}\cdot(N-1)/N$.
Next, we have
$\displaystyle E(\hat{p}_{\mathrm{mvu}}^{2})$
$\displaystyle=\sum_{t=0}^{\infty}\left(\frac{N-1}{t+N-1}\right)^{2}\cdot
g_{N}(t)$
$\displaystyle=\sum_{t=0}^{\infty}\left(\frac{N-1}{t+N-1}\right)^{2}\cdot\binom{t+N-1}{N-1}p^{N}(1-p)^{t}$
$\displaystyle=\sum_{t=0}^{\infty}\left(\frac{N-1}{t+N-1}\right)\cdot\binom{t+N-2}{N-2}p^{N}(1-p)^{t}$
$\displaystyle=\frac{(N-1)p^{N}}{(1-p)^{N-1}}\sum_{t=0}^{\infty}\cdot\binom{t+N-2}{N-2}\frac{(1-p)^{t+N-1}}{t+N-1}.$
Using the identity $(1-p)^{t+N-1}/(t+N-1)=\int_{p}^{1}(1-y)^{t+N-2}dy$, we
have
$\displaystyle E(\hat{p}_{\mathrm{mvu}}^{2})$
$\displaystyle=\frac{(N-1)p^{N}}{(1-p)^{N-1}}\int_{p}^{1}\frac{(1-y)^{N-2}}{y^{N-1}}\left[\sum_{t=0}^{\infty}\binom{t+N-2}{N-2}(1-y)^{t}y^{N-1}\right]dy$
$\displaystyle=\frac{(N-1)p^{N}}{(1-p)^{N-1}}\int_{p}^{1}\frac{(1-y)^{N-2}}{y^{N-1}}dy$
$\displaystyle=\frac{(N-1)p^{N}}{(1-p)^{N-1}}\int_{0}^{1-p}x^{N-2}(1-x)^{-(N-1)}dx$
$\displaystyle=\frac{(N-1)p^{N}}{(1-p)^{N-1}}\cdot B_{1-p}(N-1,-N+2).$
Then it is immediate upon using the hypergeometric representation of the
incomplete beta function in (9) that we have the result, which completes the
proof. ∎
It should be noted that based on the Euler transformation formula for the
hypergeometric function, we can rewrite
$\displaystyle E(\hat{p}_{\mathrm{mvu}}^{2})$
$\displaystyle=p^{2}\cdot{{}_{2}}F_{1}(1,1;N;1-p)$
$\displaystyle=p^{2}+\frac{\sum_{n=1}^{N}p^{2}(1-p)^{n}}{\binom{N-1+n}{n}},$
which results in
$\mathrm{Var}(\hat{p}_{\mathrm{mvu}})=\frac{\sum_{n=1}^{N}p^{2}(1-p)^{n}}{\binom{N-1+n}{n}}.$
In addition, we also conduct Monte Carlo simulations to study empirical biases
of these estimators under consideration. For each simulation, we generate
$(n_{1},n_{2})=(1,1)$, $(2,3)$, $(5,5)$, $(10,10)$ samples from the geometric
distribution with Bernoulli probability $p=0.1$, $0.3$, $0.5$, $0.7$, $0.9$
with the location shift $a$ being always zero. To obtain empirical biases and
empirical mean square errors (MSEs), we iterate this experiment $I=10,000$
times. It should be noted that the existing methods are all biased so that it
is more appropriate to compare their empirical MSEs instead of the empirical
variances. The empirical biases and MSEs are provided in Tables 1 and 2. The
values of the theoretical MSEs are easily obtained using Theorems 3 and 4 and
we also plot the these values along with the biases in Figure 1. In the
figure, to compare the empirical and theoretical values, we also superimposed
the empirical values with the legends $\circ$ ($n_{1}=1$, $n_{2}=1$), $\times$
($n_{1}=2$, $n_{2}=3$), and $\bullet$ ($n_{1}=5$, $n_{2}=5$).
Figure 1: Theoretical values of the biases and MSEs of the estimators under consideration. The empirical values are denoted by the legends $\circ$, $\times$, and $\bullet$. Table 1: Empirical biases of $\hat{p}_{\mathrm{b}}$, $\hat{p}_{\mathrm{mvu}}$ and $\hat{p}_{\mathrm{ml}}$. $(n_{1},n_{2})$ | | $(1,1)$ | | $(2,3)$ | | $(5,5)$ | | $(10,10)$
---|---|---|---|---|---|---|---|---
$p=0.1$ | $\hat{p}_{\mathrm{b}}$ | | $-0.01768$ | | $-0.00301$ | | $-0.00122$ | | $-0.00061$
| $\hat{p}_{\mathrm{mvu}}$ | | $-0.00060$ | | $0.00005$ | | $0.00000$ | | $-0.00006$
| $\hat{p}_{\mathrm{ml}}$ | | $0.06464$ | | $0.02124$ | | $0.00975$ | | $0.00462$
$p=0.3$ | $\hat{p}_{\mathrm{b}}$ | | $-0.09153$ | | $-0.02461$ | | $-0.00917$ | | $-0.00492$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.00176$ | | $-0.00044$ | | $0.00133$ | | $-0.00008$
| $\hat{p}_{\mathrm{ml}}$ | | $0.11693$ | | $0.04424$ | | $0.02314$ | | $0.01062$
$p=0.5$ | $\hat{p}_{\mathrm{b}}$ | | $-0.19120$ | | $-0.06032$ | | $-0.02628$ | | $-0.01309$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.00481$ | | $0.00051$ | | $0.00144$ | | $0.00004$
| $\hat{p}_{\mathrm{ml}}$ | | $0.11759$ | | $0.04960$ | | $0.02636$ | | $0.01254$
$p=0.7$ | $\hat{p}_{\mathrm{b}}$ | | $-0.30864$ | | $-0.11016$ | | $-0.05098$ | | $-0.02629$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.00017$ | | $-0.00120$ | | $0.00107$ | | $-0.00114$
| $\hat{p}_{\mathrm{ml}}$ | | $0.08272$ | | $0.03731$ | | $0.02114$ | | $0.00917$
$p=0.9$ | $\hat{p}_{\mathrm{b}}$ | | $-0.43420$ | | $-0.16783$ | | $-0.08233$ | | $-0.04134$
| $\hat{p}_{\mathrm{mvu}}$ | | $-0.00005$ | | $-0.00030$ | | $0.00023$ | | $-0.00049$
| $\hat{p}_{\mathrm{ml}}$ | | $0.03160$ | | $0.01521$ | | $0.00852$ | | $0.00385$
Table 2: Empirical MSEs of $\hat{p}_{\mathrm{b}}$, $\hat{p}_{\mathrm{mvu}}$ and $\hat{p}_{\mathrm{ml}}$. $(n_{1},n_{2})$ | | $(1,1)$ | | $(2,3)$ | | $(5,5)$ | | $(10,10)$
---|---|---|---|---|---|---|---|---
$p=0.1$ | $\hat{p}_{\mathrm{b}}$ | | $0.00579$ | | $0.00236$ | | $0.00105$ | | $0.00050$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.01527$ | | $0.00274$ | | $0.00111$ | | $0.00052$
| $\hat{p}_{\mathrm{ml}}$ | | $0.02609$ | | $0.00412$ | | $0.00140$ | | $0.00058$
$p=0.3$ | $\hat{p}_{\mathrm{b}}$ | | $0.02317$ | | $0.01242$ | | $0.00642$ | | $0.00320$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.06487$ | | $0.01706$ | | $0.00737$ | | $0.00340$
| $\hat{p}_{\mathrm{ml}}$ | | $0.07284$ | | $0.02041$ | | $0.00836$ | | $0.00363$
$p=0.5$ | $\hat{p}_{\mathrm{b}}$ | | $0.05345$ | | $0.02156$ | | $0.01133$ | | $0.00588$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.09775$ | | $0.03027$ | | $0.01341$ | | $0.00636$
| $\hat{p}_{\mathrm{ml}}$ | | $0.08140$ | | $0.03047$ | | $0.01383$ | | $0.00649$
$p=0.7$ | $\hat{p}_{\mathrm{b}}$ | | $0.10844$ | | $0.02922$ | | $0.01413$ | | $0.00724$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.09309$ | | $0.03281$ | | $0.01564$ | | $0.00757$
| $\hat{p}_{\mathrm{ml}}$ | | $0.05957$ | | $0.02808$ | | $0.01468$ | | $0.00734$
$p=0.9$ | $\hat{p}_{\mathrm{b}}$ | | $0.19371$ | | $0.03597$ | | $0.01256$ | | $0.00512$
| $\hat{p}_{\mathrm{mvu}}$ | | $0.04339$ | | $0.01671$ | | $0.00837$ | | $0.00409$
| $\hat{p}_{\mathrm{ml}}$ | | $0.02172$ | | $0.01242$ | | $0.00721$ | | $0.00379$
The values of the empirical biases of $\hat{p}_{\mathrm{b}}$ are always
negative and those of $\hat{p}_{\mathrm{ml}}$ are always positive, which is
expected from Theorem 2, and both biases tend to decrease as the sample sizes
increase. It is worth noting that the bias of $\hat{p}_{\mathrm{b}}$ is really
serious, especially when the sample size is small and the probability $p$ is
large. However, the empirical biases of $\hat{p}_{\mathrm{mvu}}$ are very
close to zero for all the cases as expected from the fact that its theoretical
bias is zero. Numerical results clearly show that the proposed estimator
$\hat{p}_{\mathrm{mvu}}$ outperforms the existing estimators. On the other
hand, the bias of $\hat{p}_{\mathrm{ml}}$ is larger when $p$ is around 0.5.
With $n_{1}=1$ and $n_{2}=1$, the bias of $\hat{p}_{\mathrm{b}}$ can reach
around 0.5 with $p$ close to 1 and that of $\hat{p}_{\mathrm{ml}}$ can reach
around 0.1 with $p$ around 0.5. Considering that the value of $p$ is always in
$(0,1)$, the biases of $\hat{p}_{\mathrm{b}}$ and $\hat{p}_{\mathrm{ml}}$ are
really serious. As $N$ gets larger, the bias gets smaller, whereas the bias of
$\hat{p}_{\mathrm{b}}$ is still severe with a large value of $p$.
## 4 Construction of the $g$ and $h$ control charts
As we did earlier, we let $X_{ij}$ be the number of independent Bernoulli
trials (cases) until the first nonconforming case in the $i$th sample for
$i=1,2,\ldots,m$ and $j=1,2,\ldots,n_{i}$. Then $X_{ij}$’s are iid geometric
random variables with location shift $a$ and $p$. Let $\bar{X}_{k}$ be the
mean of the $k$th sample with sample size $n_{k}$.
Based on the asymptotic theory, we have
$\frac{\bar{X}_{k}-\mu}{\sqrt{\sigma^{2}/n_{k}}}\stackrel{{\scriptstyle\bullet}}{{\sim}}N(0,1),$
where $\mu=E(X_{kj})=(1-p)/p+a$ and
$\sigma^{2}=\mathrm{Var}(X_{kj})=(1-p)/p^{2}$. We can construct the control
chart for average number of events per subgroup (the $h$ chart) with
$\mathrm{CL}\pm g\cdot\mathrm{SE}$ control limits
$\frac{\bar{X}_{k}-\mu_{k}}{\sqrt{\sigma^{2}/n_{k}}}=\pm g,$
which results in the upper control limit (UCL), lower control limit (LCL) and
center line (CL) as follows
$\displaystyle\mathrm{UCL}$
$\displaystyle={\mu}+g\sqrt{\frac{\sigma^{2}}{n_{k}}}=\frac{1-p}{p}+a+g\sqrt{\frac{1-p}{n_{k}p^{2}}},$
$\displaystyle\mathrm{CL}$ $\displaystyle={\mu}=\frac{1-p}{p}+a,$ (14)
$\displaystyle\mathrm{LCL}$
$\displaystyle={\mu}-g\sqrt{\frac{\sigma^{2}}{n_{k}}}=\frac{1-p}{p}+a-g\sqrt{\frac{1-p}{n_{k}p^{2}}}.$
It deserves mentioning that the American Standard uses $g=3$ with an ideal
false alarm rate 0.27% and British Standard uses $g=3.09$ with 0.20%.
By setting up $(n_{k}\bar{X}_{k}-n_{k}\mu_{k})/\sqrt{n_{k}\sigma^{2}}=\pm g$,
we can also construct the control chart for the total number of events per
subgroup (the $g$ chart) and its control limits are given by
$\displaystyle\mathrm{UCL}$
$\displaystyle=n_{k}{\mu}+g\sqrt{n_{k}\sigma^{2}}=n_{k}\left(\frac{1-p}{p}+a\right)+g\sqrt{\frac{n_{k}(1-p)}{p^{2}}},$
$\displaystyle\mathrm{CL}$
$\displaystyle=n_{k}{\mu}=n_{k}\left(\frac{1-p}{p}+a\right),$ (15)
$\displaystyle\mathrm{LCL}$
$\displaystyle=n_{k}{\mu}-g\sqrt{n_{k}\sigma^{2}}=n_{k}\left(\frac{1-p}{p}+a\right)-g\sqrt{\frac{n_{k}(1-p)}{p^{2}}}.$
In practice, the parameters $\mu$ and $\sigma^{2}$ are unknown and can be
estimated by substituting an estimator of $p$ through the relationship
$\mu=1/p-1+a$ and $\sigma^{2}=(1-p)/p^{2}$. However, a care should be taken in
this case. For example, $\hat{p}_{\mathrm{mvu}}$ is unbiased for $p$, but
$1/\hat{p}_{\mathrm{mvu}}$ is not unbiased for $1/p$. We have shown that
$\hat{p}_{\mathrm{ml}}$ is not unbiased for $p$, whereas
$1/\hat{p}_{\mathrm{ml}}$ is actually unbiased for $1/p$. Thus, we estimate
$\mu=1/p-1+a$ using $\hat{\mu}=1/\hat{p}_{\mathrm{ml}}-1+a$, which results in
$\hat{\mu}=\bar{\bar{X}}$. Since
$\bar{\bar{X}}=T_{N}/N=\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}X_{ij}/N$ is a complete
sufficient statistic, $\hat{\mu}=\bar{\bar{X}}$ is the MVU estimator of $\mu$
due to the Lehmann-Scheffé theorem. For more details on this theorem, see
Theorem 7.4.1 of Hogg et al. (2013). It should be noted that
$\hat{\mu}=\bar{\bar{X}}$ is also the ML estimator because of the invariance
property of the ML estimator (for example, see Theorem 7.2.10 of Casella and
Berger, 2002). Thus, it is clear that one should use $\hat{\mu}=\bar{\bar{X}}$
to estimate the CL, which results in $\mathrm{CL}=\bar{\bar{X}}$ ($h$ chart)
and $\mathrm{CL}=n_{k}\bar{\bar{X}}$ ($g$ chart).
To estimate $\sigma^{2}$, we consider the ML estimator of $\sigma^{2}$ by
plugging $\hat{p}_{\mathrm{ml}}$ into $\sigma^{2}=(1-p)/p^{2}$, which results
in
$\hat{\sigma}^{2}_{\mathrm{ml}}=(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1).$ (16)
The MVU estimator of $\sigma^{2}$ is also easily obtained using the Lehmann-
Scheffé theorem with
$E\big{[}(T_{N}/N)\cdot(T_{N}+N)/(N+1)\big{]}=(1-p)/p^{2}$. Then we have
$\hat{\sigma}^{2}_{\mathrm{mvu}}=\frac{N}{N+1}(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1).$
(17)
Using $\hat{\mu}=\bar{\bar{X}}$ and $\hat{\sigma}^{2}_{\mathrm{ml}}$ in (16)
along with (14) and (15), we can construct the ML-based $h$ and $g$ charts as
follows.
* •
$h$ chart:
$\displaystyle\mathrm{UCL}$
$\displaystyle=\bar{\bar{X}}+g\sqrt{\frac{(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)}{n_{k}}},$
$\displaystyle\mathrm{CL}$ $\displaystyle=\bar{\bar{X}},$
$\displaystyle\mathrm{LCL}$
$\displaystyle=\bar{\bar{X}}-g\sqrt{\frac{(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)}{n_{k}}}.$
* •
$g$ chart:
$\displaystyle\mathrm{UCL}$
$\displaystyle=n_{k}\bar{\bar{X}}+g\sqrt{n_{k}(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)},$
$\displaystyle\mathrm{CL}$ $\displaystyle=n_{k}\bar{\bar{X}},$
$\displaystyle\mathrm{LCL}$
$\displaystyle=n_{k}\bar{\bar{X}}-g\sqrt{n_{k}(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)}.$
Also, using $\hat{\mu}=\bar{\bar{X}}$ and $\hat{\sigma}^{2}_{\mathrm{mvu}}$ in
(17) along with (14) and (15), we can construct the MVU-based $h$ and $g$
charts as follows.
* •
$h$ chart:
$\displaystyle\mathrm{UCL}$
$\displaystyle=\bar{\bar{X}}+g\sqrt{\frac{N}{N+1}\frac{(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)}{n_{k}}},$
$\displaystyle\mathrm{CL}$ $\displaystyle=\bar{\bar{X}},$
$\displaystyle\mathrm{LCL}$
$\displaystyle=\bar{\bar{X}}-g\sqrt{\frac{N}{N+1}\frac{(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)}{n_{k}}}.$
* •
$g$ chart:
$\displaystyle\mathrm{UCL}$
$\displaystyle=n_{k}\bar{\bar{X}}+g\sqrt{\frac{n_{k}N}{N+1}(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)},$
$\displaystyle\mathrm{CL}$ $\displaystyle=n_{k}\bar{\bar{X}},$
$\displaystyle\mathrm{LCL}$
$\displaystyle=n_{k}\bar{\bar{X}}-g\sqrt{\frac{n_{k}N}{N+1}(\bar{\bar{X}}-a)(\bar{\bar{X}}-a+1)}.$
It should be noted that Kaminsky et al. (1992) provide the control limits for
the MVU-based $h$ and $g$ charts in their Table 1, but these limits are based
on $\hat{p}_{\mathrm{b}}$ which is not the MVU. Also, one can also construct
the control limits by plugging the MVU estimator $\hat{p}_{\mathrm{mvu}}$ into
(14) and (15). However, like the ML estimator, the MVU estimator has no
invariance property. Thus, in this case, the resulting limits can not be
regarded as the MVU-based limits.
## 5 Concluding remarks
We have revisited the $g$ and $h$ control charts with proper ML and MVU
estimators. We have shown that the MVU estimator has been inappropriately used
in the quality engineering literature and thus provided the correct MVU
estimator along with various statistical properties such as their theoretical
first and second moments which are explicitly expressed as the Gauss
hypergeometric function. Furthermore, based on the new estimators developed in
this note, we provided how to construct the ML-based and MVU-based $h$ and $g$
control charts with unbalanced samples.
Finally, it is worth noting that we have developed the rQCC R package (Park
and Wang, 2020) to construct various control charts. In ongoing work, we plan
to add these control charts in the next update so that practitioners can use
our results more easily.
## Acknowledgment
This research was supported by the National Research Foundation of Korea (NRF)
grant (NRF-2017R1A2B4004169) and the BK21-Plus Program (Major in Industrial
Data Science and Engineering) funded by the Korea government.
## References
* Abramowitz and Stegun (1964) Abramowitz, M. and I. A. Stegun (1964). Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Volume 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, D.C.
* Benneyan (1999) Benneyan, J. C. (1999). Geometric-based $g$-type statistical control charts for infrequent adverse events. In Institute of Industrial Engineers Society for Health Systems Conf. Proc., pp. 175–185.
* Benneyan (2000) Benneyan, J. C. (2000). Number-between $g$-type statistical quality control charts for monitoring adverse events. Health Care Management Science 4, 305–318.
* Benneyan (2001) Benneyan, J. C. (2001). Performance of number-between $g$-type statistical control charts for monitoring adverse events. Health Care Management Science 4, 319–336.
* Blackwell (1947) Blackwell, D. (1947). Conditional expectation and unbiased sequential estimation. Annals of Mathematical Statistics 18, 105–110.
* Casella and Berger (2002) Casella, G. and R. L. Berger (2002). Statistical Inference (Second ed.). Pacific Grove, CA: Duxbury.
* Driver and Johnston (2006) Driver, K. A. and S. J. Johnston (2006). An integral representation of some hypergeometric functions. Electron. Trans. Numer. Anal. 25, 115–120.
* Dutka (1981) Dutka, J. (1981). The incomplete beta function – a historical profile. Archive for History of Exact Sciences 24(1), 11–29.
* Hankin (2016) Hankin, R. K. S. (2016). hypergeo: The Gauss hypergeometric function. https://CRAN.R-project.org/package=hypergeo. R package version 1.2.13 (published on April 7, 2016).
* Hogg et al. (2013) Hogg, R. V., J. W. McKean, and A. T. Craig (2013). Introduction to Mathematical Statistics (7 ed.). Boston, MA: Pearson.
* Kaminsky et al. (1992) Kaminsky, F. C., J. C. Benneyan, and R. D. Davis (1992). Statistical control charts based on a geometric distribution. Journal of Quality Technology 24, 63–69.
* Lehmann and Casella (1998) Lehmann, E. L. and G. Casella (1998). Theory of Point Estimation (second ed.). New York: Springer-Verlag.
* Miller and Paris (2011) Miller, A. R. and R. B. Paris (2011). Euler-type transformations for the generalized hypergeometric function ${}_{r+2}f_{r+1}(x)$. Zeitschrift für angewandte Mathematik und Physik 62, 31–45.
* Minitab (2020) Minitab (2020). Methods and formulas for $g$ chart. Minitab 20 Support. https://support.minitab.com/en-us/minitab/20/ (accessed on December 24, 2020).
* Miyakawa (1984) Miyakawa, M. (1984). Analysis of incomplete data in competing risks model. IEEE Transactions on Reliability 33, 293–296.
* Özarslan and Ustaoğlu (2019) Özarslan, M. and C. Ustaoğlu (2019). Some incomplete hypergeometric functions and incomplete riemann-liouville fractional integral operators. Mathematics 7(5), 483.
* Park (2010) Park, C. (2010). Parameter estimation for reliability of load sharing systems. IIE Transactions 42, 753–765.
* Park and Wang (2020) Park, C. and M. Wang (2020). rQCC: Robust quality control chart. https://CRAN.R-project.org/package=rQCC. R package version 1.20.7 (published on July 5, 2020).
* Rainville (1960) Rainville, E. D. (1960). Special Functions. New York: Macmillan.
* Rao (1945) Rao, C. R. (1945). Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society 37, 81–91.
* Seaborn (1991) Seaborn, J. B. (1991). Hypergeometric Functions and Their Applications. New York: Springer.
|
# Spatial Assembly:
Generative Architecture With
Reinforcement Learning,
Self Play and Tree Search
Panagiotis Tigas
University of Oxford
<EMAIL_ADDRESS>&Tyson Hosmer
Bartlett School of Architecture, UCL
<EMAIL_ADDRESS>
###### Abstract
With this work, we investigate the use of _Reinforcement Learning_ (RL) for
generation of spatial assemblies, by combining ideas from Procedural
Generation algorithms (_Wave Function Collapse_ algorithm (WFC) [8]) and RL
for Game Solving. WFC is a Generative Design algorithm, inspired by Constraint
Solving [3]. In WFC, one defines a set of tiles/blocks and constraints and the
algorithm generates an assembly that satisfies these constraints. Casting the
problem of generation of spatial assemblies as a _Markov Decision Process_
whose states transitions are defined by WFC, we propose an algorithm that uses
_Reinforcement Learning_ and _Self-Play_ to learn a policy that generates
assemblies which maximize objectives set by the designer. Finally, we
demonstrate the use of our Spatial Assembly algorithm in Architecture Design.
## 1 Introduction
We present a novel application of Deep Reinforcement Learning, coupled with a
bespoke Constraint Solving algorithm for learning to generate spatial
assemblies.
Constraint Satisfaction Problems (CSPs) consist of a finite set of rules and
objects, whose composition/combination must satisfy a number of constraints
[2]. CSP solvers have been effective in computation logic problems across many
domains including decision making, game development, logic puzzles [6, 7, 10].
Design innovation through constraint solving has been extensively explored by
Killian et al. [4], whose research has focused on constraints in design
exploration and specifically bidirectional constraint solving methods [4]. Our
approach explores building design as a multi-objective CSP, trained to
evaluate each local decision based on the current state of the assembly to
effectively negotiate evolving socioeconomic and environmental goals.
We begin by modeling architecture design as a CSP, extending the approach of
Texture Synthesis and Model Synthesis [5], and Wave Function Collapse [8, 3],
primarily applied to image-based procedural content creation and modeling in
gaming. The algorithm extracts features and their relations from images and
attempts to recreate similar distributions of those features procedurally
creating images that resemble a prototypical image.
However, using such algorithms to generate assemblies 111assemblies, designs,
and structures will be used interchangeably that optimize certain criteria,
additionally to the constraints solving, is a difficult task because of the
lack of differentiability and their dependency on black-box methods. To
elevate this limitation, we equip the search space of possible assemblies with
an efficient learnable search operator/policy $\pi(a|s)$.
Figure 1: Spatial Assembly process.
## 2 Algorithm
First, we define a set of geometric tiles that form a dictionary $D$ of
building blocks ($D_{i}$ represents the $i$-th tile of the set). Next, we set
a rule of constraints, $C$, according to which these tiles can be combined.
The problem then becomes to sequentially combine the tiles in order to create
structures that are valid (no constraint is invalidated) and maximally cover
the available canvas.
Wave Function Collapse starts with an empty state and selects an initial tile
at random. Meanwhile, for each possible expansion node (expansion node is a
connection point of a tile which is free) it keeps track of the number of
tiles that can be connected which do not invalidate the constraints, termed
entropy. WFC works by selecting the node with the least degrees of freedom
(most constraint node) and expanding the node by randomly selecting a tile
that satisfies the constraints. We can see the problem of generation of an
assembly as solving a Markov Decision Process (MDP), where the state
transitions are defined by WFC algorithm, actions are the tiles from the
dictionary $D$, and rewards are defined according to the designer’s goals.
Algorithm 1 Spatial Assembly - Rollout
1:$S\leftarrow\emptyset$ $\triangleright$ Initialize empty
2:while structure not complete or invalid do
3: node = SelectNode($S$)$\triangleright$ Select the most constraint node
4: tiles = GetValidTiles(node, $C$, $D$) $\triangleright$ Get the set of tiles
that satisfy the constraints
5: Sample $T_{\text{new}}$ from policy $\pi(a|S,\text{tiles})$
6: Update $S$ by connecting the new tile $T_{\text{new}}$ at node
Spatial Assembly algorithm, replaces the random selection of the tiles with
the policy $\pi(a|s)$, which returns a distribution over the available tiles
(action $a$) according to their potential to maximize the future expected
reward. We learn the policy with Proximal Policy Optimization [9], a
Reinforcement Learning algorithm which has enjoyed success in various domains
of Artificial Intelligence. The complete rollout algorithm can be found in
alg. 1.
Training the system occurs as follows. We start generating rollouts with an
initially untrained policy until we reach a terminal state. We evaluate the
terminal state according to the success and reward accordingly. For example,
one reward signal we used was the maximum displacement observed on the final
structure after it got simulated by the physics engine of Unity3D (reward
capturing the structural stability of the assembly). We then use Proximal
Policy Optimization to update the value function and the policy for the next
round. We let the system self-play until convergence. This approach can be
seen as Policy Gradient Search [1].
## 3 Acknowledgements
The authors would like to thank Dave Reeves, Octavian Gheorghiu, and Ziming
He, design masters and tutors at Living Architecture Lab, The Bartlett School
of Architecture, and the students Elahe Arab, Barış Erdinçer, Yifei Jia,
Georgia Kolokoudia (IRSILA project, 2020 cohort), Athina Athiana, Evangelia
Triantafylla, Ming Liu (NOMAS project, 2019 cohort), Jelena Peljevic, Yekta
Tehrani, Shahrzad Fereidouni, Noura Alkhaja (ArchiGO project, 2018 cohort).
## References
* Anthony et al. [2019] T. Anthony, R. Nishihara, P. Moritz, T. Salimans, and J. Schulman. Policy gradient search: Online planning and expert iteration without search trees. _arXiv preprint arXiv:1904.03646_ , 2019.
* Apt [2003] K. Apt. _Principles of constraint programming_. Cambridge university press, 2003.
* Karth and Smith [2017] I. Karth and A. M. Smith. WaveFunctionCollapse is constraint solving in the wild. _Proceedings of the International Conference on the Foundations of Digital Games - FDG ’17_ , pages 1–10, 2017. doi: 10.1145/3102071.3110566. URL http://dl.acm.org/citation.cfm?doid=3102071.3110566.
* Kilian [2005] A. Kilian. Design exploration through bidirectional modeling of constraints. 2005\.
* Merrell [2007] P. Merrell. Example-based model synthesis. In _Proceedings of the 2007 symposium on Interactive 3D graphics and games_ , pages 105–112, 2007.
* Miguel [2012] I. Miguel. _Dynamic flexible constraint satisfaction and its application to AI planning_. Springer Science & Business Media, 2012.
* Modi et al. [2001] P. J. Modi, H. Jung, M. Tambe, W.-M. Shen, and S. Kulkarni. A dynamic distributed constraint satisfaction approach to resource allocation. In _International Conference on Principles and Practice of Constraint Programming_ , pages 685–700. Springer, 2001.
* [8] Mxgmn. mxgmn/wavefunctioncollapse. URL https://github.com/mxgmn/WaveFunctionCollapse.
* Schulman et al. [2017] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization algorithms. _arXiv preprint arXiv:1707.06347_ , 2017.
* Simonis [2005] H. Simonis. Sudoku as a constraint problem. In _CP Workshop on modeling and reformulating Constraint Satisfaction Problems_ , volume 12, pages 13–27. Citeseer, 2005.
## Appendix A Application: Spatial Assembly in Architecture Design
This methodology was applied in three design projects, ArchiGo(2018),
Nomas(2019), and ISIRLA(2020), at Bartlett School of Architecture, Living
Architecture Lab.
The ArchiGo (fig. 3) project was developed by iteratively designing and
testing many spatial part sets with different characteristics and relations
evaluated for their ability to avoid contradictions and meet user-defined
spatial objectives (Figure 2).
In the NOMAS project (fig. 2), we investigate the potential for this method to
re-think housing strategies and invent new spatial languages composed of
simple prefabricated parts. The strategy is demonstrated through the digital
process applied to the physical production of a 3.5-meter-tall spatial
prototype assembled with human labor from coconut fiber composite parts.
IRSILA (fig. 4) applies the methodology to a reconfigurable cultural center
where spatial parts are constructed from smaller prefabricated units assembled
and reconfigured by autonomous distributed robots. Both demonstrate the
potential for buildings with reconfigurable and adaptive life cycles.
Figure 2: NoMAS Project (2018) Figure 3: ArchiGo Project (2019) Figure 4:
IRSILA Project (2020)
|
# First detection of collective oscillations of a stored deuteron beam
with an amplitude close to the quantum limit
J. Slim III. Physikalisches Institut B, RWTH Aachen University, 52056 Aachen,
Germany N.N. Nikolaev L.D. Landau Institute for Theoretical Physics, 142432
Chernogolovka, Russia Moscow Institute for Physics and Technology, 141700
Dolgoprudny, Russia F. Rathmann Institut für Kernphysik, Forschungszentrum
Jülich, 52425 Jülich, Germany A. Wirzba Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany Institute for Advanced
Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany A. Nass Institut
für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany V. Hejny
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany J.
Pretz Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany H. Soltner Zentralinstitut für Engineering, Elektronik und Analytik,
Forschungszentrum Jülich, 52425 Jülich, Germany F. Abusaif Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany A. Aggarwal
Marian Smoluchowski Institute of Physics, Jagiellonian University, 30348
Cracow, Poland A. Aksentev Institute for Nuclear Research, Russian Academy
of Sciences, 117312 Moscow, Russia A. Andres III. Physikalisches Institut B,
RWTH Aachen University, 52056 Aachen, Germany L. Barion University of
Ferrara and Istituto Nazionale di Fisica Nucleare, 44100 Ferrara, Italy G.
Ciullo University of Ferrara and Istituto Nazionale di Fisica Nucleare, 44100
Ferrara, Italy S. Dymov University of Ferrara and Istituto Nazionale di
Fisica Nucleare, 44100 Ferrara, Italy Laboratory of Nuclear Problems, Joint
Institute for Nuclear Research, 141980 Dubna, Russia R. Gebel Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany M. Gaisser III.
Physikalisches Institut B, RWTH Aachen University, 52056 Aachen, Germany K.
Grigoryev Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany D. Grzonka Institut für Kernphysik, Forschungszentrum Jülich, 52425
Jülich, Germany O. Javakhishvili Department of Electrical and Computer
Engineering, Agricultural University of Georgia, 0159 Tbilisi, Georgia A.
Kacharava Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany V. Kamerdzhiev Institut für Kernphysik, Forschungszentrum Jülich,
52425 Jülich, Germany S. Karanth Marian Smoluchowski Institute of Physics,
Jagiellonian University, 30348 Cracow, Poland I. Keshelashvili Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany A. Lehrach
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany P.
Lenisa University of Ferrara and Istituto Nazionale di Fisica Nucleare, 44100
Ferrara, Italy N. Lomidze High Energy Physics Institute, Tbilisi State
University, 0186 Tbilisi, Georgia B. Lorentz GSI Helmholtzzentrum für
Schwerionenforschung, 64291 Darmstadt, Germany A. Magiera Marian
Smoluchowski Institute of Physics, Jagiellonian University, 30348 Cracow,
Poland D. Mchedlishvili High Energy Physics Institute, Tbilisi State
University, 0186 Tbilisi, Georgia F. Müller III. Physikalisches Institut B,
RWTH Aachen University, 52056 Aachen, Germany A. Pesce Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany V. Poncza
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany D.
Prasuhn Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany A. Saleev University of Ferrara and Istituto Nazionale di Fisica
Nucleare, 44100 Ferrara, Italy V. Shmakova Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany Laboratory of Nuclear
Problems, Joint Institute for Nuclear Research, 141980 Dubna, Russia H.
Ströher Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany M. Tabidze High Energy Physics Institute, Tbilisi State University,
0186 Tbilisi, Georgia G. Tagliente Istituto Nazionale di Fisica Nucleare
sez. Bari, 70125 Bari, Italy Y. Valdau Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany T. Wagner Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany C. Weidemann
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany A.
Wrońska Marian Smoluchowski Institute of Physics, Jagiellonian University,
30348 Cracow, Poland M. Żurek Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA
###### Abstract
We investigated coherent betatron oscillations of a deuteron beam in the
storage ring COSY, excited by a detuned radio-frequency Wien filter. The beam
oscillations were detected by conventional beam position monitors. With the
currently available apparatus, we show that oscillation amplitudes down to
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ can be detected. The interpretation
of the response of the stored beam to the detuned radio-frequency Wien filter
is based on simulations of the beam evolution in the lattice of the ring and
realistic time-dependent 3D field maps of the Wien filter. Future measurements
of the electric dipole moment of protons will, however, require control of the
relative position of counter-propagating beams in the sub-picometer range.
Since here the stored beam can be considered as a rarefied gas of uncorrelated
particles, we moreover demonstrate that the amplitudes of the zero-point
(ground state) betatron oscillations of individual particles are only a factor
of about 10 larger than the Heisenberg uncertainty limit. As a consequence of
this, we conclude that quantum mechanics does not preclude the control of the
beam centroids to sub-picometer accuracy. The smallest Lorentz force exerted
on a single particle that we have been able to determine is
$10\text{\,}\mathrm{a}\mathrm{N}$.
††preprint: Draft for PR AB
## I Introduction
The approach to the quantum ground state, the observation of quantum effects
in macroscopic systems, and the possibility to detect displacements of
macroscopic bodies on the nanometer scale, are the subject of intense
theoretical and experimental efforts Schreppler _et al._ (2014); Abbott _et
al._ (2009); Murch _et al._ (2008); Biercuk _et al._ (2010); Rugar _et al._
(2004). A notable example is the detection of gravitational waves using an
interferometric detector with mirrors in the kilogram range Abbott _et al._
(2016). In all-electric proton storage rings, coherent beam displacements down
to the picometer range that are caused by Earth’s gravity pull are in
principle accessible using the spin rotations of the proton as a detector
Abusaif _et al._ (2021). We also mention here the ongoing discussions of the
possibility to detect gravitational waves via perturbations of the beam orbit
in high-energy storage rings ari . Here we report the first detection of
collective oscillations of an intense beam of deuterons in a storage ring with
an amplitude close to the quantum limit. The present study is part of an
international effort to prepare for the search for the permanent electric
dipole moment (EDM) of charged particles. The focus of these studies has been
on systematic effects, e.g., imperfection magnetic fields in storage rings
Saleev _et al._ (2017), and orbit improvements in a machine using beam-based
alignment Wagner _et al._ (2021), thereby advancing the high-precision
frontier in spin dynamics in storage rings. A comprehensive description of
this activity and of the proposed stepwise approach leading to a dedicated
proton EDM storage ring is presented in Ref. Abusaif _et al._ (2021).
Experiments searching for electric dipole moments of charged particles using
storage rings are at the forefront of the incessant quest to find new physics
beyond the Standard Model of particle physics. These investigations bear the
potential to shed light on the origin of the anomalously large matter-
antimatter asymmetry in the Universe Pospelov and Ritz (2005), for which the
combined predictions of the Standard Models of particle physics and of
cosmology fall short of the experimentally observed asymmetry by about seven
to eight orders of magnitude Bernreuther (2002).
The signal for an EDM is the spin precession in electric fields, where it
should be noted that the spins of charged particles can be subjected to large
electric fields only in storage rings. The need to eliminate the
overwhelmingly stronger spin rotations driven by the magnetic moment in
magnetic fields brings to the front an all-electric, so-called frozen spin
proton storage ring Anastassopoulos _et al._ (2016); Abusaif _et al._
(2021). An important advantage of such a machine is the ability to
simultaneously store two counter-propagating proton beams. The concurrent
measurement of the EDM-driven spin rotations of the counter-propagating beams
would allow to cancel major systematic effects. To this end, to reach an
ambitious sensitivity to the proton EDM of
$d_{p}\approx${10}^{-29}\text{\,}\mathrm{e}\,\mathrm{c}\mathrm{m}$$, it is
imperative to control the relative vertical displacement of the centers of
gravity of the two beams to an accuracy of about
$5\text{\,}\mathrm{p}\mathrm{m}$ Abusaif _et al._ (2021). One may wonder
whether such an enormously demanding accuracy is not prohibited by the
Heisenberg uncertainty principle. Towards an ultimate precision search for
EDMs of charged particles, this particular aspect of the systematics of such
measurements had not been investigated so far, and our experiment constitutes
the first step in this direction.
Here we report on the measurement of the amplitude of collectively excited
vertical oscillations of a deuteron beam orbiting in the magnetic storage ring
COSY at a momentum of about
$970\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ Maier (1997).
The data were taken in 2018 in the course of a dedicated experiment in the
framework of systematic beam and spin dynamics studies for the deuteron EDM
experiment (so-called precursor experiment), presently carried out by the JEDI
collaboration at COSY Rathmann _et al._ (2013); Morse _et al._ (2013);
Rathmann _et al._ (2020). One of the central devices in the precursor
experiment is the radio-frequency (RF) Wien filter (WF), shown in Fig. 1,
which was designed to provide a cancellation of the electric and magnetic
forces acting on the particle. In this operation mode, the Wien filter affects
only the particle spins, but does not perturb the beam orbit Slim _et al._
(2016, 2017); Slim (2018). A slightly detuned Wien filter, however, exerts a
non-vanishing Lorentz force on the orbiting beam particles. It is shown
collective beam oscillation excited by the WF with amplitudes down to
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ can be detected with the
currently available equipment. Our approach to measuring ultra-small
displacements complements other measurements of ultra-small forces using
different techniques Schreppler _et al._ (2014); Abbott _et al._ (2009);
Murch _et al._ (2008); Biercuk _et al._ (2010); Rugar _et al._ (2004).
In our experiment, the measurement cycles were much shorter than the intrabeam
interaction time, and the beam attenuation rate was negligibly weak (see
discussion in Weidemann _et al._ (2015)), therefore we treat the beam as a
rarefied gas of uncorrelated particles. Individual particles undergo stable
betatron oscillations around the equilibrium orbit in the horizontal and
vertical planes, driven by focusing magnetic fields. Apart from their
conventional individual betatron motions, all the particles in a bunch
participate in one and the same collective and coherent oscillation that is
driven by the Wien filter. Therefore, the upper bound of the amplitude of the
collective oscillation of the entire beam corresponds to the upper bound of
the oscillation amplitude of a single particle. In our approach, access to
ultrasmall oscillation amplitudes results from the fact that the measured
signal corresponds to a collective response of the electric charge of about
$N=${10}^{9}$$ deuterons in the bunch.
The beam tracking simulations were carried out to predict the response of the
stored beam to the detuned radio-frequency Wien filter. The simulations use
the elements of the ring lattice and realistic time-dependent 3D field maps of
the Wien filter. These field maps describe the spatial variation of complex
electric and magnetic fields, including the fringe field areas. Furthermore,
the tolerances of the elements of the circuit driving the Wien filter are also
taken into consideration.
As a reference value for the Heisenberg uncertainty relation, we take an
estimate of the amplitude of the single-particle zero-point betatron
oscillation amplitude $Q$. Then, our result for the smallest measured
amplitude of the Wien filter-driven single-particle oscillation is only about
a factor of ten larger than the quantum limit of Heisenberg’s uncertainty
relation for vertical single-particle betatron oscillations. The smallest
detected oscillation amplitude is by three orders of magnitude smaller than
the beam size.
We demonstrate for the first time that the accuracy with which periodic beam
oscillation amplitudes can be measured is vastly higher compared to that of
static beam displacements generated by steerers using the same BPM. In a broad
context, any new precision tool is of interest per se, and the latter point
has important implications, for instance, for all-electric frozen-spin EDM
storage rings. Here, one aims to control the interfering radial magnetic
fields by measuring the vertical spacing of the counter-propagating beams. Our
result complements the potential of using beam oscillations to measure this
distance, as discussed in Ref. Hacıömeroğlu _et al._ (2019).
One must distinguish the RF-driven collective oscillations above the quantum
limit $Q$ from the quantum uncertainty of the center of mass of the bunch
circulating in a static ring. Specifically, for a rarefied-gas of $N$
uncorrelated particles, the quantum limit of the centroid of the bunch,
detected by the BPMs, amounts to $Q/\sqrt{N}$.
The paper is organized as follows. In Sec. II, the measurement principle is
introduced, followed by a description of the operation of the radio-frequency
Wien filter in Sec. III. The method to determine the beam oscillations is
discussed in Sec. IV, and the evolution of the beam to the combined effect of
the ring lattice and the Wien filter fields are presented in Sec. V. The time-
dependent field maps of the Wien filter are discussed in Sec.V.1, and the
evaluation of the uncertainties is elaborated in Sec. V.2. Experimental
results are presented in Sec. VI, followed by conclusion and outlook in Sec.
VII.
## II Measurement principle
(a) CAD drawing of the design of the RF Wien filter. 1: RF feed, 2: beam pipe,
3: inner mounting cylinder, 4: inner support structure, 5: lower electrode, 6:
insulator, 7: RF connector, and 8: vacuum vessel. (b) Photograph with a view
along the beam axis showing the gold-plated copper electrodes, which have a
length of $808.8\text{\,}\mathrm{m}\mathrm{m}$.
Figure 1: The waveguide RF Wien filter is mounted inside a cylindrical vessel.
The effective length of the device amounts to
$\ell=$1.16\text{\,}\mathrm{m}$$.The technical details are described in Refs.
Slim _et al._ (2017, 2016).
The Cooler Synchrotron (COSY) Martin _et al._ (1985); Maier (1997) at
Forschungszentrum Jülich is a storage ring with a circumference of
approximately $184\text{\,}\mathrm{m}$. Its principal elements used for the
experiments are indicated in Fig. 2. For the investigations presented here,
the two key devices are the RF Wien filter, based on a parallel-plates
waveguide Slim _et al._ (2016), and a conventional electrostatic beam
position monitor (BPM) that is used to monitor the beam oscillations Forck
_et al._ (2008). The Wien filter generates orthogonal and highly-homogeneous
electric and magnetic fields. In the present experiment, the Wien filter was
operated in the mode with the electric field pointing vertically upward
($y$-direction), whereas the magnetic field points radially outward
($x$-direction), and the beam moves in $z$-direction (see coordinate system in
Fig. 2). The effective length of the Wien filter is
$\ell=$1.16\text{\,}\mathrm{m}$$ (see Refs. Slim _et al._ (2017, 2016) for
further technical details).
Figure 2: Schematic diagram of the cooler synchrotron and storage ring (COSY)
with the main components, especially the focusing/defocusing magnets
(quadrupoles) and the bending magnets (dipoles). Indicated are the position of
the RF Wien filter and the location of the beam position monitor Böhme _et
al._ (2018) (BPM 17), used to observe the beam oscillations. Further
components such as the $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electron
cooler Dietrich _et al._ (2014), the WASA Adam _et al._ (2004) and the JEPO
Müller (2019); Müller _et al._ (2020) polarimeters, and the Siberian snake
Lehrach and Maier (2001) are also shown. The coordinate system used is
indicated.
As a spin rotator for the forthcoming deuteron EDM (precursor) experiment
Rathmann _et al._ (2013); Morse _et al._ (2013); Rathmann _et al._ (2020),
the Wien filter is designed to operate in resonance with the spin precession
of the orbiting deuterons Slim _et al._ (2016, 2017); Slim (2018); Slim _et
al._ (2020), and at a vanishing Lorentz force, given by
$\vec{F}=q\left(\vec{E}+\vec{v}\times\vec{B}\right)\,,$ (1)
where $q$ denotes the elementary charge and $\vec{v}$ represents the velocity
of the beam particles. Unlike in conventional DC Wien filters, the crossed
electric field and magnetic fields ($\vec{E}$ and $\vec{B}$) of the RF Wien
filter are generated simultaneously by exciting the transverse electromagnetic
(TEM) mode. The spin resonance tune mapping technique, developed for the Wien
filter operation in the deuteron EDM experiment at COSY, is described in
Rathmann _et al._ (2020).
When the electric and magnetic fields in the Wien filter are mismatched, i.e.,
when the electric and magnetic forces no longer cancel each other, the RF
fields excite collective beam oscillations at the frequency at which the Wien
filter is operated. In the present experimental set up, a mismatch between
electric and magnetic fields provides a vertically mismatched Lorentz force
(see coordinate system in Fig. 2).
With a vanishing Lorentz force, the beam performs idle vertical (and
horizontal) betatron oscillations
$y(t)=y(0)\sqrt{\frac{\beta_{y}(t)}{\beta_{y}(0)}}\cos\left[\psi_{y}(t)\right]\,,$
(2)
where $\beta_{y}(t)$ is the betatron amplitude function. With the beam
revolution period of $T=2\pi/\omega_{\text{rev}}$, the betatron phase advance
$\psi_{y}(t)$ satisfies $\psi_{y}(t+T)-\psi_{y}(t)=\omega_{y}T=2\pi\nu_{y}$,
where $\nu_{y}$ is the vertical betatron tune given by
$\nu_{y}=\omega_{y}/\omega_{\text{rev}}$.
On the other hand, a mismatched Wien filter exerts stroboscopically, i.e.,
once per turn, a vertical force $F_{y}(n)=F_{y}\cos(n\,\omega_{\text{WF}}T)$
on the stored particle, where $n$ is the turn number and $\omega_{\text{WF}}$
denotes the angular velocity of the RF in the Wien filter (see discussion of
Fig. 12 in Sec. VI). The change of the vertical velocity of the stored
particle, accumulated during the time interval $\Delta t=\ell/v_{z}$ the
particle spends per turn $n$ inside the Wien filter, is given by
$\Delta v_{y}(nT)=\frac{F_{y}(n)\Delta t}{\gamma
m}=-\zeta\omega_{y}\cos(n\,\omega_{\text{WF}}T)\,,$ (3)
where $\gamma$ and $m$ are the Lorentz-factor and the mass of the particle,
respectively. The change $\Delta y$ of the vertical position $y$ in the Wien
filter can be neglected. The coupling of vertical and radial beam oscillations
is negligible (see Sec. IV) and it is sufficient here to treat driven
oscillations in a one-dimensional approximation. Due to the very strong
disparity of synchrotron and fractional WF frequencies, synchro-betatron
coupling can be neglected (see discussion in Syphers _et al._ (1993)).
Furthermore, beam attenuation either by intrabeam scattering or by interaction
with residual gas during the data acquisition cycle is very small (see
Appendix B), justifying the rarefied gas approximation.
According to Eq. (2), the stroboscopic signal of the betatron motion observed
at any point in the ring, follows the harmonic law with angular velocity
$\omega_{y}$, and we invoke the familiar description of the oscillatory motion
in terms of the complex variable $z=y-iv_{y}/\omega_{y}$. With the initial
condition $z(0)=0$, summing $\Delta v_{y}(kT)$ after $n$ turns, the solution
for $z(n)$ behind the Wien filter reads
$\begin{split}z(n)=\frac{i\zeta}{2}&\cdot\left[\frac{\exp(in\omega_{y}T)-\exp(in\,\omega_{\text{WF}}T)}{\exp(i(\omega_{y}-\omega_{\text{WF}})T)-1}\right.\\\
&+\left.\frac{\exp(in\omega_{y}T)-\exp(-in\,\omega_{\text{WF}}T)}{\exp(i(\omega_{y}+\omega_{\text{WF}})T)-1}\right]\,.\end{split}$
(4)
This expression serves as the initial condition for the idle betatron motion
during the subsequent $(n+1)$ turn and so forth. A similar analytic result
holds also for generic AC dipole-driven betatron oscillations, discussed in a
very different context of machine diagnostics in Ref. Miyamoto _et al._
(2008) (see also references therein).
Driven by the mismatched Wien filter, all beam particles participate in one
and the same collective and coherent oscillation, and according to Eq. (4),
the beam as a whole exhibits oscillations at the Wien filter frequency
$\omega_{\text{WF}}$. A lock-in amplifier may be used to selectively measure
the corresponding Fourier component of the beam oscillation
$y=\xi_{y}\cos(n\,\omega_{\text{WF}}T)$ from the output of a beam position
monitor. Its amplitude is given by
$\xi_{y}=\frac{\zeta}{2}\cdot\frac{\sin(2\pi\nu_{y})}{\cos(2\pi\nu_{\text{WF}})-\cos(2\pi\nu_{y})}\,,$
(5)
where the vertical betatron tune $\nu_{y}$, and the Wien filter tune
$\nu_{\text{WF}}$, are given by $\nu_{y}=\omega_{y}/\omega_{\text{rev}}$ and
$\nu_{\text{WF}}=\omega_{\text{WF}}/\omega_{\text{rev}}$, respectively. When
the Wien filter tune is close to the vertical betatron tune, a resonant
enhancement of the beam oscillation amplitude $\xi_{y}$ occurs. Equation (5)
describes Hooke’s law, $F_{y}=k_{\text{H}}\xi_{y}$, and Hooke’s constant is
given by
$k_{\text{H}}=\left|\frac{2\gamma m\omega_{y}}{\Delta
t}\cdot\frac{\cos(2\pi\nu_{\text{WF}})-\cos(2\pi\nu_{y})}{\sin(2\pi\nu_{y})}\right|\,.$
(6)
We invoke an approximate description of the betatron motion by a harmonic
oscillator with constant betatron function and evaluate the Heisenberg
uncertainty limit $Q$ for the betatron oscillation amplitude $\xi_{y}$ in
terms of the zero-point oscillator energy $\tfrac{1}{2}\hbar\omega_{y}$, which
yields
$Q^{2}=\frac{\hbar}{m\gamma\omega_{y}}\,\,.$ (7)
For the present experiment, we obtain
$Q=\frac{82}{\sqrt{\gamma\nu_{y}}}\ {\rm nm}\,.$ (8)
With the actual COSY values for the betatron tune $\nu_{y}$ and the Lorentz-
factor $\gamma$ of the beam (see Table 1, Appendix A), the quantum limit of
the vertical betatron oscillations amounts to
$Q\approx$41\text{\,}\mathrm{n}\mathrm{m}$\,.$ (9)
The interpretation of the measured oscillation amplitudes in terms of the Wien
filter parameters requires numerical simulations of the performance of the
Wien filter as an element of the storage ring Slim _et al._ (2016). The
details relevant to the present study are described below; the corresponding
beam simulations carried out are consistent with the available experimental
results on the properties of COSY Weidemann _et al._ (2015).
## III Wien filter operation
The control of the Lorentz force of the waveguide RF Wien filter is based on
the wave-mismatch principle Slim (2018). An impedance mismatch is introduced
at the load part of the device to deliberately create reflections that
generate a standing wave pattern inside the Wien filter Slim _et al._ (2020).
These standing waves can be represented by the complex-valued field quotient
$Z_{q}$, defined as the ratio of the total electric to the total magnetic
field strength,
$\begin{split}Z_{q}&=\frac{E^{\text{total}}}{H^{\text{total}}}=\frac{E^{+}+E^{-}}{H^{+}-H^{-}}=\frac{E^{+}+\Gamma\cdot
E^{+}}{H^{+}-\Gamma\cdot H^{+}}\\\
&=Z_{\text{w}}\frac{1+\Gamma}{1-\Gamma}=Z_{0}\frac{d}{W}\frac{1+\Gamma}{1-\Gamma}\,,\end{split}$
(10)
where the superscripts ’$+$’ and ’$-$’ refer to the forward and backward
direction of propagation, $Z_{\text{w}}$ is the wave impedance,
$Z_{0}\approx$377\text{\,}\Omega$$ is the vacuum wave impedance,
$d=$100\text{\,}\mathrm{mm}$$ is the distance between the electrodes,
$W=$182\text{\,}\mathrm{mm}$$ is their width Slim _et al._ (2016), and
$\Gamma$ is the reflection coefficient that controls the amplitude and phase
of the reflected wave. During the measurements described here, the Wien filter
was typically operated at a net input RF power of $600\text{\,}\mathrm{W}$.
The field quotient $Z_{q}$ is controlled via a specially designed RF circuit
Slim _et al._ (2020). By altering $\Gamma$ via two variable vacuum
capacitors, called $C_{\text{L}}$ and $C_{\text{T}}$, a wide range of $Z_{q}$
values can be covered, and the matching point corresponding to the minimum
induced vertical beam oscillation amplitude may be determined.
## IV Beam oscillations
In this experiment, the electric field of the Wien filter is oriented
vertically and the magnetic field horizontally. This implies that the
oscillations mainly take place along the $y$-axis [see Eq. (1)]. For the
detection of the vertical beam oscillations, a conventional beam position
monitor has been employed. In order to be most sensitive, BPM 17 located in
the straight section opposite to the Wien filter (see Fig. 2) with a large
vertical $\beta$ function was used, $\beta_{y}^{\rm
BPM}\approx$15.3049\text{\,}\mathrm{m}$$, while at the Wien filter location,
$\beta_{y}^{\rm WF}\approx$2.6784\text{\,}\mathrm{m}$$, as shown in Fig. 3.
The arguments to pick BPM 17 are further discussed below in Sec. V.
Figure 3: Vertical and horizontal beta-functions along the circumference of
COSY Weidemann _et al._ (2015). The vertical dashed lines mark the location
of the Wien filter and of the beam position monitor used during the
measurement of the beam oscillations.
In order to measure small beam oscillations, a technique based on lock-in
amplifiers111HF2LI 50 MHz Lock-in Amplifier, Zurich Instruments AG, 8005
Zurich, Switzerland, https://www.zhinst.com/others/products/hf2li-lock-
amplifier. was developed Meade (1983). These devices operate in the frequency
domain and lock onto a signal whose frequency is set as a reference, which is
particularly useful in an electromagnetically noisy environment. Each
measurement consisted of two subsequent machine cycles of
$3\text{\,}\mathrm{min}$ duration, as depicted in Fig. 14 in Appendix B.
Figure 4: Readout scheme of the COSY BPM 17. The signals of the four
electrodes are fed into lock-in amplifiers. The differential signal of each
electrode is analyzed at the two reference frequencies given by the COSY RF
and the Wien filter frequency. The resulting Fourier amplitudes of the signals
are recorded in the EPICS333Experimental Physics and Industrial Control
System, https://epics.anl.gov/index.php. archiving system of COSY.
A stored beam bunch circulating at a revolution frequency of $f_{\text{rev}}$
that passes through a beam position monitor induces a voltage signal on all
its four electrodes, as indicated in the readout scheme of the BPM, shown in
Fig. 3. For the detection of vertical beam oscillations, only the voltage
signals $U_{\text{t},\,b}$ from the top (t) and bottom (b) electrodes are
considered. These signals are trains of short pulses with the repetition
frequency $f_{\text{rev}}$. In view of Eq. (2) and as far as the Fourier
spectrum of the beam oscillations is concerned, without loss of generality,
the BPM can be considered to be located right behind the Wien filter, and the
induced voltages can be represented by
$\vspace{0.3cm}U_{\text{t,\,b}}=\left[U_{0}\pm\Delta U\left(\Delta
y\right)\right]\cos(\omega_{\text{rev}}t)\,,$ (11)
where the index ’t’ refers to the $+$ sign and the index ’b’ to the $-$ sign,
respectively. The harmonic factor $\cos(\omega_{\text{rev}}t)$ emphasizes the
pulse repetition frequency, although $\cos(\omega_{\text{rev}}t)=1$ for
$t=nT$. The voltage $U_{\text{t,\,b}}$ is non-zero only at the time the beam
passes through the BPM. Here, $U_{0}$ denotes the voltage proportional to the
beam current that is induced when the beam passes exactly through the center
of the BPM, and $\Delta U\left(\Delta y\right)$ represents the voltage
variation induced by a beam that is vertically displaced by $\Delta y$.
For small beam displacements, the beam position monitor operates in its linear
regime, which implies that the induced voltages take the form
$\Delta U\left(\Delta y\right)=\kappa\cdot\Delta y\cdot U_{0}\,,$ (12)
where $\kappa$ is a calibration factor that needs to be determined. At a
momentum of $970\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$,
the revolution frequency of deuterons orbiting in COSY is
$f_{\text{rev}}\approx$750\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$. The Wien
filter is operated at the $k^{\text{th}}$ sideband of the spin-precession
frequency $f_{s}$, given by
$f_{\text{WF}}=\frac{\omega_{\text{WF}}}{2\pi}=(k+\nu_{s})f_{\text{rev}}=k\cdot
f_{\text{rev}}+f_{s}\,.$ (13)
Here $\nu_{s}=G\gamma$ denotes the spin tune, i.e., the number of spin
precessions per revolution, $G\approx-$0.1430$$ is the magnetic anomaly of the
deuteron, and the spin precession frequency $f_{s}=\nu_{s}f_{\text{rev}}$ Slim
_et al._ (2016). It should be noted that in view of $\omega_{\rm rev}T=2\pi$,
trains of beam oscillation pulses do not depend on the actual choice of the
sideband. In the present experiment the Wien filter was operated at $k=-1$,
which corresponds to
$f_{\text{WF}}\approx$871\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ 444For the
considerations presented in this paper, negative and positive frequencies are
considered equivalent..
The induced oscillations of amplitude $\xi_{y}$ contribute to Eq. (11) the
harmonic voltage variation $\Delta U\left(y(t)\right)$. The BPM in conjunction
with the lock-in amplifiers is used to measure at times $t=nT$ the beam
positions at the reference frequencies, i.e., at $f_{\text{WF}}$ and at
$f_{\text{rev}}$. Given that $y(t)$ can be evaluated at the spin precession
frequency, the BPM signals of the upper and lower electrodes can be written as
follows
$\begin{split}U_{\text{t,\,b}}(t)=&\left[U_{0}\pm\Delta U\left(\Delta
y\right)\pm\Delta
U\left(y(t)\right)\right]\cos\left(\omega_{\text{rev}}t\right)=\left[U_{0}\pm\Delta
U\left(\Delta
y\right)\pm\kappa\xi_{y}U_{0}\cos\left(\omega_{\text{s}}t\right)\right]\cos\left(\omega_{\text{rev}}t\right)\\\
=&\left[U_{0}\pm\kappa\Delta
yU_{0}\right]\cos\left(\omega_{\text{rev}}t\right)\pm\frac{1}{2}\kappa\xi_{y}U_{0}\cos\left(\omega_{\Delta}t\right)\pm\frac{1}{2}\kappa\xi_{y}U_{0}\cos\left(\omega_{\Sigma}t\right)\,.\end{split}$
(14)
Here $\omega_{\Delta}$ and $\omega_{\Sigma}$ represent sidebands of the Wien
filter frequency at
$\begin{split}\omega_{\Delta}&=\omega_{\text{rev}}-\omega_{\text{s}}=\omega_{\text{WF}}\rvert_{k=1}\,,\text{and}\\\
\omega_{\Sigma}&=\omega_{\text{rev}}+\omega_{\text{s}}=\omega_{\text{WF}}\rvert_{k=-1}\,.\end{split}$
(15)
(a) Magnitude of the field quotient $|Z_{q}|$, evaluated integrally, where
$|Z_{q}|^{\text{int}}=\int|Z_{q}|\text{d}\ell$. Ideally, with $|Z_{q}|$ close
to $176\text{\,}\mathrm{\SIUnitSymbolOhm}$, the electric and magnetic forces
are equal. (b) Phase of the field quotient $\angle Z_{q}$ evaluated
integrally, where $\angle Z_{q}^{\text{int}}=\int\angle Z_{q}\text{d}\ell$. A
non-vanishing $\angle Z_{q}$ implies a phase shift between the electric and
magnetic fields.
Figure 5: Simulated integral magnitude (a) and phase of the field quotient
$Z_{q}$ (b) at each point of the $C_{\text{L}}$ and $C_{\text{T}}$ grid,
indicated by the blue points, $\ell$ denotes the effective length of the Wien
filter. Besides the matching point [see Eq. (19)], $\left(7\times 6\right)$
grid points were investigated.
In order to measure the beam oscillations, four lock-in amplifiers Meade
(1983) were used, two for the horizontal and two for the vertical direction.
For each direction, one lock-in amplifier detects the Fourier amplitudes at
$f_{\text{rev}}\approx$750\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and a
second one at
$f_{\Sigma}=f_{\text{rev}}+f_{s}\approx$871\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.
The lock-in amplifiers receive reference frequencies from the signal generator
of the Wien filter and from the master oscillator of COSY. The four Fourier
amplitudes of the top and bottom electrodes are determined practically in
real-time, yielding
$\begin{split}A_{\text{t,\,b}}^{\text{rev}}&=U_{0}\pm\kappa\Delta
yU_{0}\,,\text{and}\\\
A_{\text{t,\,b}}^{\Sigma}&=\mp\frac{1}{2}\kappa\xi_{y}U_{0}\,.\end{split}$
(16)
The amplitude of the vertical oscillation $\xi_{y}$ can then be determined
from
$\frac{A_{\text{t}}^{\Sigma}-A_{\text{b}}^{\Sigma}}{A_{\text{t}}^{\text{rev}}+A_{\text{b}}^{\text{rev}}}=\hat{\xi}_{y}=\kappa\frac{U_{0}}{2U_{0}}\xi_{y}=\frac{1}{2}\kappa\xi_{y}\,.$
(17)
The uncalibrated raw asymmetry of the four Fourier amplitudes is denoted by
$\hat{\xi}_{y}$.
The readout scheme, shown in Fig. 3, was used to concurrently record radial
beam oscillations, and the above analysis has been repeated for the
corresponding $\hat{\xi}_{x}$. The main result is that the coupling of
vertical and radial betatron oscillations is negligibly weak,
$|\hat{\xi}_{x}/\hat{\xi}_{y}|<$2\text{\times}{10}^{-2}$$, which justifies
treating the RF-driven beam oscillations as one-dimensional.
The determination of the calibration constant $\kappa$, required to calibrate
the vertical oscillation amplitude, is described in detail in Appendix B. It
amounts to
$\kappa=\left(5.82\pm
0.43\right)\cdot${10}^{-6}\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}\,.$
(18)
During the experiments, the vertical betatron tune of the machine amounted to
about $\nu_{y}\approx 3.6040$.555The numerical values used for the simulation
calculations are listed in Table 1 of Appendix A. The frequency
$f_{\Sigma}\approx$871\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$, at which the
Wien filter is operated, is well separated from the lowest intrinsic spin
resonances666An intrinsic depolarizing resonance is encountered, when the
betatron motion of the particles is in sync with the spin motion, hence, when
the condition
$f_{s}=\nu_{s}f_{\text{rev}}=f_{y}=(nP\pm\nu_{y}^{\prime})f_{\text{rev}}$ is
fulfilled Huang _et al._ (2004), where $n\in\mathbb{N}$, $P$ denotes the
superperiodicity of the lattice, and $\nu_{y}^{\prime}$ the fractional tune.
During the experiments described here, $P=1$ (see also Fig. 3). at
$297\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$,
$453\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$,
$1048\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, and
$1204\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.
The two variable and highly accurate capacitors, $C_{\text{L}}$ and
$C_{\text{T}}$, are driven by stepper motors. They constitute the main
dynamical elements of the driving circuit. Each pair of capacitor values
yields a well-defined field quotient $|Z_{q}|$, as shown in Fig. 5 (a). Away
from the matching point, a phase shift $\angle Z_{q}$ occurs between electric
and magnetic fields, as shown in Fig. 5 (b). The corresponding Lorentz force
leads to the measured beam oscillations, i.e., the function
$\xi_{y}=f\left(C_{\text{L}},C_{\text{T}}\right)$, which can be visualized in
the form of a 2D map, as shown in Fig. 6. The experimental data were taken on
a grid of $(7\times 6)$ points of $C_{\text{L}}$ and $C_{\text{T}}$, with
corresponding grid spacings of $\left(94.5\pm
1.0\right)\,$\mathrm{p}\mathrm{F}$$ for $C_{\text{L}}$ and $\left(95.8\pm
1.0\right)\,$\mathrm{p}\mathrm{F}$$ for $C_{\text{T}}$. Each grid spacing
corresponds to 1000 steps of the corresponding stepper motors. The calibration
of the capacitances $C_{\text{L}}$ and $C_{\text{T}}$ as a function of step
number is discussed in detail in Slim _et al._ (2020). The grid spans over
$C_{\text{L}}\in[318.88,\,885.58]\,$\mathrm{pF}$$ and
$C_{\text{T}}\in[428.99,\,907.79]\,$\mathrm{pF}$$. The uncertainties of the
grid spacings are systematic ones.777The individually measured uncertainties
of the capacitors are actually much smaller than the stated uncertainty of
$1.0\text{\,}\mathrm{p}\mathrm{F}$. However, other factors, such as the
capacitances and inductances of the connectors and cables and their power
dependencies, also contribute to the aforementioned uncertainties.
The map of the measured and calibrated vertical beam oscillations $\xi_{y}$ is
shown in Fig. 6. The parameters of the matching point are given by
$\begin{split}C_{\textnormal{L}}&=\left(697.1\pm
1.0\right)\,$\mathrm{pF}$\,,\text{ and}\\\ C_{\textnormal{T}}&=\left(503.0\pm
1.0\right)\,$\mathrm{pF}$\,,\end{split}$ (19)
and the corresponding minimal detected beam oscillation amplitude at the
location of BPM 17 amounts to
$\xi_{y}^{\text{min}}\big{\rvert}_{\text{BPM}}=(1.08\pm
0.52)\,$\mathrm{\SIUnitSymbolMicro m}$\,.$ (20)
The above accuracy of $\delta\xi_{y}^{\rm
min}\big{\rvert}_{\text{BPM}}=$0.52\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
can be compared to the accuracy of measurements of static distortions of the
beam orbit, which is about $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (see
Table 2). As expected, the accuracy of the beam oscillation amplitudes is
better by a factor of about 40 compared to the static amplitudes measured
using the same BPM.
Upon rescaling the oscillation amplitudes using Eq. (2) with $\beta$ functions
listed in Table 1 to the WF location, we obtain
$\xi_{y}^{\text{min}}\big{\rvert}_{\rm WF}=(0.45\pm
0.22)\,$\mathrm{\SIUnitSymbolMicro m}$\,,$ (21)
which should be compared to the value of
$Q\approx$41\text{\,}\mathrm{n}\mathrm{m}$$, given in Eq. (9). The largest
measured amplitude of driven beam oscillations at a strongly mismatched point
with $600\text{\,}\mathrm{W}$ of input RF power amounts to
$\xi_{y}^{\text{max}}\big{\rvert}_{\text{BPM}}=(66.2\pm
3.1)\,$\mathrm{\SIUnitSymbolMicro m}$\,.$ (22)
Here one must bear in mind that the sensitivity to a periodic signal scales
inversely with the square root of the observation time.888The relevant
discussion is found in Ref. Bagdasarian _et al._ (2014). See also the
observation of white noise suppression by two orders in magnitude when using
$5\text{\,}\mathrm{h}$ signal averaging in a test bench experiment with SQUID
BPMs Hacıömeroğlu _et al._ (2019). The frozen-spin proton EDM experiment aims
at the accumulation of the EDM signal for a duration of about
${10}^{7}\text{\,}\mathrm{s}$ Abusaif _et al._ (2021). The accuracy
$\delta\xi_{y}^{\text{min}}\big{\rvert}_{\text{BPM}}=$0.52\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ [Eq. (20)] corresponds to an averaging time of $96\text{\,}\mathrm{s}$
[see Fig. (14) in Appendix (B)]. With the currently used BPMs and their
readout electronics, together with an extension of the averaging time to
${10}^{7}\text{\,}\mathrm{s}$, there would be a factor of 320 improvement in
sensitivity to coherent beam oscillations, leading to an accuracy of
$1.6\text{\,}\mathrm{nm}$.
In Fig. 7 (a), the data measured at the matching point [Eq. (19)] are shown.
Each sample was recorded by the lock-in amplifiers with an integration time
set to $0.5\text{\,}\mathrm{s}$, corresponding to an average of $5000$
measurements. A Monte Carlo error propagation model was applied to treat the
uncertainties of the still uncalibrated raw position asymmetries
$\hat{\xi}_{y}$ and the calibration coefficient $\kappa$ Aster _et al._
(2018). The results are fitted with a normal distribution, as shown in Fig. 7
(b), from which the mean value $\mu_{{\xi}_{y}}$ and the error of the measured
beam oscillations $\sigma_{{\xi}_{y}}$ are estimated. The latter represents
the systematic error of the measurement. It should be noted that the map shown
in Fig. 6 is actually a function of all the circuit elements. The
uncertainties of $\xi_{y}$ are influenced by the uncertainties of all circuit
elements and also by the ones of the BPM itself, which include its readout
electronics, i.e., the lock-in amplifiers.
Figure 6: Measured amplitudes of beam oscillations $\xi_{y}^{\text{exp}}$ at
BMP 17, plotted on a grid as a function of the variable capacitor values
$C_{\text{L}}$ and $C_{\text{T}}$. To avoid crowding up the map, the error
bars of the data points were omitted, these are shown in Fig. 13 instead. The
parameters of the matching point are given in Eq. (19).
(a) Measured oscillation amplitudes $\xi_{y}$ using data samples of
$0.5\text{\,}\mathrm{s}$ duration, each sample reflects the average of $5000$
measurements of the lock-in amplifiers. (b) Probability density distribution
$f_{\xi_{y}}$ of the measured data, fitted with a Gaussian to determine mean
and standard deviation.
Figure 7: Measured beam oscillations at the matching point [Eq. (19)] of the
map shown in Fig. 6. The samples shown in panel (a) were acquired during a
data taking period of $108\text{\,}\mathrm{min}$, using 36 machine fills
(cycles).
To appreciate the result given in Eq. (20), one can compare the oscillation
amplitude to the $1\sigma$ vertical beam size. The latter has been deduced
from the $1\sigma$ beam emittance $\epsilon_{y}$ and the amplitude of the
$\beta$ function at the position of the BPM, yielding
$\sigma_{y}^{\text{BPM}}=\sqrt{\beta_{y}^{\text{BPM}}\epsilon_{y}}\approx$1.4\text{\,}\mathrm{m}\mathrm{m}$\,.$
(23)
In the present experiment, the beam emittance was not monitored. The above
numerical estimate of $\sigma_{y}^{\text{BPM}}$ is based on rescaling the
experimental result for the $2\sigma$ beam emittance of
$49.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ protons in COSY of
$\epsilon_{y}=(0.92\pm 0.15)\,$\mathrm{\SIUnitSymbolMicro m}$$ Weidemann _et
al._ (2015) to the conditions of the present experiment. It is noteworthy that
with the present equipment it is possible to access coherent beam oscillations
with amplitudes that are more than three orders of magnitude smaller than the
beam size.
## V Beam dynamics simulations
To improve our understanding of the measured results, a computer code was
developed to model the beam dynamics in the COSY storage ring. The modeled
storage ring consists of a sequence of drift regions, quadrupole and dipole
magnets, the Wien filter, and beam position monitors. These elements are
represented by transfer matrices, which are well understood and documented in
the literature Wolski (2014). In the model of the ring, the actual settings of
the beam optics elements of COSY were those used at the time when the
experiment took place. Simulations are based on the Hamiltonian formulation as
presented in Ref. Wolski (2014). The Wien filter is modeled by a time-
dependent matrix that also takes into account the arrival time of the
particles.
### V.1 Time-dependent Wien filter field maps
In order to be able to perform reliable beam simulations, we have placed great
emphasis on good spatial resolution and the accuracy of the 3D field maps
inside the Wien filter999Each field map consists of
$2\text{\times}{10}^{6}$points, 200 points along the $x$ axis
$\left(x\in[$-5\text{\,}\mathrm{m}\mathrm{m}$,$5\text{\,}\mathrm{m}\mathrm{m}$]\right)$,
200 points along the $y$ axis
$\left(y\in[$-5\text{\,}\mathrm{m}\mathrm{m}$,$5\text{\,}\mathrm{m}\mathrm{m}$]\right)$,
and 50 points along the $z$ (Wien filter) axis
$\left(z\in[-\ell/2,+\ell/2]\right)$, where $\ell=$1.16\text{\,}\mathrm{m}$$
is the effective length of the Wien filter., computed using a 3D
electromagnetic simulation tool101010Electromagnetic and circuit simulations
were performed using CST, from Dassault Systèmes, Vélizy-Villacoublay, France,
https://www.3ds.com.. The fringe fields of the Wien filter are included,
because they are of particular importance for the beam oscillations, as will
be discussed later.
(a) 3D electric field distribution of the component $E_{y}$. (b) 3D magnetic
field distribution of the flux density component $B_{x}$.
Figure 8: Examples of the main electric and magnetic field components inside
the waveguide RF Wien filter at the matching point [see Eq. (19)] with an
input RF power of $600\text{\,}\mathrm{W}$. The electric field component in
(a) points vertically upward ($y$-direction), while the component of the
magnetic flux density in (b) points radially outward ($x$-direction).
An example of the computed 3D fields of the Wien filter at the experimentally
determined matching point is shown in Fig. 8. The beam-tracking simulations
use the three vector components of the electric and magnetic fields. The Wien
filter is implemented as an RF kicker, as described by Eq. (3).
Inside COSY there are 32 BPMs available to control the horizontal and vertical
beam position during operation. In order to select one of them with a good
sensitivity to determine the beam oscillations induced by the Wien filter, a
number of particles were tracked, as described above, and the orbit response
induced by a field change at the location of the Wien filter was calculated at
each BPM location111111In the preparatory stage, simulations were carried out
using the Software Toolkit for Charged-Particle and X-Ray Simulations BMAD
Sagan (2006).. As a result, BPM 17, located about $70\text{\,}\mathrm{m}$
downstream of the Wien filter (see Fig. 2), was chosen because it offered good
sensitivity to both radial and vertical beam oscillations.
Figure 8 shows the main field components $E_{y}$ and $B_{x}$. When mismatched,
the Wien filter generates periodic transverse perturbations of the trajectory.
Switching off the Wien filter eliminates such oscillations. The maximum
amplitude of the observed oscillations in the simulation is then taken for
$\xi_{y}$. The two simulated vertical beam oscillation amplitudes of BPM 17
and Wien filter read
$\begin{split}\xi_{y}^{\text{BPM}}&=\left(1.086\pm
0.082\right)\,$\mathrm{\SIUnitSymbolMicro m}$\,,\text{ and}\\\
\xi_{y}^{\text{WF}}&=\left(0.435\pm 0.031\right)\,$\mathrm{\SIUnitSymbolMicro
m}$\,,\end{split}$ (24)
which agree well with the experimentally measured results, given in Eqs. (20)
and (21). A detailed description of the determination of the uncertainties of
the beam simulations is discussed in the next section.
For each and every measured point on the $C_{\text{L}}$ versus $C_{\text{T}}$
grid, a beam dynamics simulation was carried out. For each of these points, a
3D field map of the Wien filter was generated and then used for the beam
tracking simulations. The results of these simulations are shown in Fig. 9,
and are later compared with the results of the measurements.
Figure 9: Simulated amplitudes of beam oscillations $\xi_{y}^{\text{sim}}$ as
a function of the variable capacitor values $C_{\text{L}}$ and $C_{\text{T}}$.
To avoid overcrowding the map, the error bars of the data points were omitted
here and are shown in Fig. 13.
### V.2 Uncertainty evaluation
The accuracy with which the Lorentz force and the resulting amplitudes of the
beam oscillations can be tuned depends on the accuracy with which the field
quotient $Z_{q}$ can be integrally set to the desired value. $Z_{q}$ depends
on the hardware elements in the driving circuit. In order to evaluate the
effects of uncertainties of these elements, extensive coupled circuit
electromagnetic simulations have been conducted, as discussed in Ref. Slim
_et al._ (2020). The uncertainties involved are listed in Table 6 and shown in
Fig. 16 of Ref. Slim _et al._ (2020). As far as the Lorentz force is
concerned, most important are the uncertainties of the fixed inductance
$L_{\text{f}}$ and the fixed resistance $R_{\text{f}}$. Once these
uncertainties are known, one can compute the electric and magnetic fields and
the corresponding Lorentz force, including their corresponding errors. Figure
10 shows a few examples of the main components of the electric and magnetic
fields, computed with the above mentioned circuit uncertainties. As will be
explained below, these 3D fields, together with their uncertainties, are
subsequently used as input to the beam simulations.
(a) Electric field component $E_{y}(z)$ under circuit uncertainties. (b)
Magnetic field component $B_{x}(z)$ under circuit uncertainties.
Figure 10: $200$ examples of the electric and magnetic fields as a function of
$z$ along the beam axis under the circuit uncertainties, specified in the list
of uncertainties in Table 6 of Slim _et al._ (2020).
The algorithm used to compute the uncertainties of the beam simulations is the
polynomial chaos expansion (PCE), as explained in Refs. Slim _et al._ (2017,
2020) and in Appendix C. The PCE has been proven in many applications in
science and engineering to be just as accurate as the computationally much
more expensive Monte-Carlo counterpart Smith (2013); Offermann _et al._
(2015); Adelmann (2018); Slim _et al._ (2017).
To compute the uncertainties $\sigma_{\xi_{y}}$, the PCE algorithm requires a
random set of the simulated $\xi_{y}$, alongside a set of randomized input
parameters according to their uncertainties to generate the output. The set of
$\xi_{y}$ is produced using a number of beam-tracking simulations, where for
each instance, a 3D field map of the Wien filter is generated, according to
the randomized input parameters. An example of the electric and magnetic
fields evaluated at the center of the Wien filter for the matching case [see
Eq. (19)] is shown in Fig. 10. The magnitudes of the fields vary as a function
of the uncertainties of the driving circuit Slim _et al._ (2020). The
numerical tracking of the particles through these fields generates a
collection of different $\xi_{y}$ values that the PCE algorithm can use to
project the output onto orthogonal polynomial functions. These functions serve
as basis functions, from which the expansion coefficients are determined that
are used to generate a large sample of outputs to compute the uncertainties of
the beam simulations.
In Fig. 11 (a), the simulated values of $\xi_{y}$ are shown for the matching
case. The detailed steps to achieve this result are discussed in Appendix C.
As shown in Fig. 11(b), fitting these data to a Gaussian yields a standard
deviation of $\sigma_{\xi_{y}}=$0.082\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
This number is of considerable importance, because, given the uncertainties of
the driving circuit, it sets the lower limit that can be achieved by
minimizing the amplitude of the vertical beam oscillations when more sharply
tuning the driving circuit of the Wien filter. The same procedure is performed
on each point of the map shown in Fig. 9.
(a) Simulated oscillation amplitudes under uncertainties at the matching point
(Eq. (19)). Of the ${10}^{6}$ simulations that were carried out, only
${10}^{4}$ are shown here. (b) Probability density distribution $f_{\xi_{y}}$
of the ${10}^{6}$ simulations from panel (a), fitted by a Gaussian to
determine mean and standard deviation.
Figure 11: Results of the sparce PCE algorithm to compute the uncertainties of
the simulated vertical beam oscillations at BPM 17.
## VI Comparison of simulation and experimental results
The simulations yield the net Lorentz force exerted by the Wien filter on beam
particles and the corresponding oscillation amplitudes for each measured point
of the $C_{\text{L}}$ versus $C_{\text{T}}$ map, shown in Fig. 6. The only
variables in this case are the field maps of the Wien filter itself. After
$1000$ turns, the beam position is computed at the same location in the ring,
where the measurement using BPM 17 took place (see Fig. 2). The net Lorentz
force is a result of local cancellations between the electric and magnetic
field components, as illustrated in Fig. 12 for the matching point given in
Eq. (19) with the minimal measured oscillation amplitude.
In Fig. 12(a), the local Lorentz force is shown along the trajectory for 5
randomly chosen passes though the Wien filter. The trajectory of the same
particle changes from pass to pass, thereby different Wien filter fields and
consequently different values of the Lorentz force $F_{y}$ will be picked up.
As shown in Fig. 12, even at the matching point, the matching is still
imperfect, and the largest local $F_{y}$ contributions are caused by the
fringe fields at the entrance and exit of the Wien filter. Despite the
different location of the particle in the vertical and horizontal phase space
at the entrance of the Wien filter upon subsequent passes, the integration of
these local forces along the particle trajectories exhibits nevertheless a
perfectly harmonic time dependence with the frequency $f_{s}$, as shown in
Fig. 12(b). The points encircled in blue correspond to the randomly selected
passes through the Wien filter, shown in Fig. 12(a).
(a) Local Lorentz force $F_{y}(z)$ exerted on a single deuteron for different
passes though the Wien filter. The turn numbers used here were randomly
selected between 1 to 100. The fields were evaluated at the crosses and the
interconnecting lines are to guide the eye. (b) Integral Lorentz force
$F_{y}(n)$ evaluated along the trajectory. Each point represents an overall
kick exerted per turn $n$. The points marked in blue correspond to the
integrated local Lorentz force of the individual turns shown in panel (a).
Figure 12: Simulation of the local and integrated Lorentz force in the Wien
filter at the matching point of Eq. (19). Depending on the initial coordinates
in the vertical and horizontal phase space, the particle travels along
different trajectories, and therefore picks up different field components
$F_{y}$.
In the left panel of Fig. 13, the amplitude of the simulated Lorentz force
$F_{y}^{\text{sim}}$ is plotted versus the simulated oscillation amplitude
$\xi_{y}^{\text{sim}}$ at the Wien filter position. As expected, it exactly
follows Hooke’s law with a spring constant of $k_{\text{H}}=(151.2\pm
0.2)\,$\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}^{2}$$. In Fig.
13(b), the measured amplitudes are compared with the ones simulated for the
location of BPM 17. The two sets $\xi_{y}^{\text{exp}}$ and
$\xi_{y}^{\text{sim}}$ are in very good agreement with each other. The
horizontal and vertical error bars are derived from the uncertainties of the
measurements and simulations, represented by the width of the distributions,
as shown in Figs. 7(b) and 11(b). It is important to note that the error bars
refer to systematic uncertainties and should not be confused with statistical
ones. This implies that repetitions of either the measurements or the
simulations will neither reduce the systematic error of the readout
electronics of BPM 17, nor will it affect the uncertainties of the elements of
the driving circuit.
(a) Simulated Lorentz force $F_{y}^{\text{sim}}$ at the Wien filter location
as function of the oscillation amplitude $\xi_{y}^{\text{sim}}$, fitted with
the function $F_{y}^{\text{sim}}=a\cdot\xi_{y}^{\text{sim}}+b$. (b) Simulated
beam oscillation amplitude $\xi_{y}^{\text{sim}}$ versus the measured
oscillation amplitude $\xi_{y}^{\text{exp}}$ at the BPM, fitted with the
function $\xi_{y}^{\text{sim}}=c\cdot\xi_{y}^{\text{exp}}+d$.
Figure 13: (a): Simulated amplitude of the Lorentz force at the Wien filter
location as function of the simulated beam oscillation amplitudes
$\xi_{y}^{\text{sim}}$. (b): Simulated versus measured vertical beam
oscillation amplitudes at the location of BPM 17. The horizontal error bars of
the measured amplitudes $\xi_{y}^{\text{exp}}$ originate from the readout
electronics of BPM 17 and the calibration factor $\kappa$ (see Appendix B),
whereas the vertical ones are determined by the circuit uncertainties using
the PCE method, as described in Appendix C.
The fit shown in Fig. 13 yields $\chi^{2}/\text{ndf}=45.5/41$, very close to
unity York _et al._ (2004). The linear fit yields a slope of $0.999\pm
0.018$, which is perfectly consistent with unity. The intercept parameter of
the fit yields $($-0.93$\pm 0.31)\,$\mathrm{\SIUnitSymbolMicro m}$$, and
within three standard deviations, it agrees with zero.
The very good agreement between measurements and simulations reflects our good
understanding of both the electromagnetic fields generated in the Wien filter
and of the underlying beam dynamics in the machine. This point is further
substantiated by comparing the simulated amplitudes at the Wien filter and at
the positions of the BPMs with the estimated amplitudes expected from
rescaling based on the $\beta$ functions121212The uncertainty of the $\beta$
functions amounts to about 10%, as discussed in Ref. Weidemann _et al._
(2015)., taking into account the numerical values, listed in Table 1 of
Appendix A,
$\begin{split}\left.\xi_{y}^{\text{WF}}\right|_{\text{sim}}&=(0.435\pm
0.031)\,$\mathrm{\SIUnitSymbolMicro m}$\,,\\\
\sqrt{\frac{\beta_{y}^{\text{WF}}}{\beta_{y}^{\text{BPM}}}}{\xi_{y}^{\text{BPM}}}|_{\text{sim}}=\left.\xi_{y}^{\text{WF}}\right|_{\text{est}}&=(0.435\pm
0.039)\,$\mathrm{\SIUnitSymbolMicro m}$\,.\end{split}$ (25)
The good agreement between these two numerical values in Eq. (25) indicates
that the observation of the oscillation amplitude at one location in the ring
can be reliably transferred to some other place in the ring by use of Eq. (2).
The above quoted value of
$\left.\xi_{y}^{\text{WF}}\right|_{\text{sim}}=0.435\,$\mathrm{\SIUnitSymbolMicro
m}$$ is about a factor of 10 larger than the quantum limit of the vertical
oscillation amplitude $Q$, given in Eq. (9).
In searches for EDMs in dedicated all-electric storage rings, a continuous
monitoring of the orbits of the two counter-rotating beams is mandatory during
data acquisition within the horizontal spin-coherence time Abusaif _et al._
(2021). When intrabeam scattering can be neglected Weidemann _et al._ (2015),
which is arguably justified within the horizontal spin-coherence time, the
beam can be described as a rarefied gas of particles, i.e., the zero-point
oscillations of individual particles are uncorrelated. We repeat the point
from the introduction that in a static regime, the quantum limit of the center
of mass of a bunch with $N$ particles can be estimated via
$Q_{\text{bunch}}=Q/\sqrt{N}$. For a bunch of $N=${10}^{10}$$ stored
particles, one obtains $Q_{\text{bunch}}\simeq$0.4\text{\,}\mathrm{pm}$$. It
follows that Heisenberg’s uncertainty relation does not present an obstacle to
achieving a sensitivity of $5\text{\,}\mathrm{p}\mathrm{m}$ for the vertical
separation of clockwise and counter-clockwise beams – the real challenge is to
develop compact BPMs with a sensitivity improved by a factor of about 300
compared to those used here Böhme _et al._ (2018).
Finally, a satisfactory agreement has been achieved between Hooke’s constant,
simulated using the electromagnetic fields in the Wien filter and the $\beta$
functions of the COSY lattice, and the theoretical approximation of the no-
lattice model assuming constant $\beta$ functions of Eq. (6), yielding
$\begin{split}k_{\text{H}}^{\text{sim}}&=(151.2\pm
0.2)\,$\mathrm{MeV}\text{\,}{\mathrm{m}}^{-2}$\,,\text{ and}\\\
k_{\text{H}}^{\text{th}}&=$207\text{\,}\mathrm{MeV}\text{\,}{\mathrm{m}}^{-2}$\,.\end{split}$
(26)
The theoretical estimate of $k_{\text{H}}^{\text{th}}$, calculated using the
numerical values listed in Table 1 of Appendix A, is about a factor of 1.4
larger than the simulated one. The given uncertainty of
$k_{\text{H}}^{\text{sim}}$ does not include the systematic scale uncertainty
of the BPM calibration factor $\kappa$ [see Eq. (29) in Appendix A]. At the
matching point [see Eq. (19)], the Lorentz force amounts to
$F_{y}^{\text{WF}}=k_{\text{H}}^{\text{sim}}\cdot\left.\xi_{y}^{\text{WF}}\right|_{\text{sim}}\approx
66\,$\mathrm{eV}\text{\,}{\mathrm{m}}^{-1}$={10.6}\,$\mathrm{aN}$\,,$ (27)
where the intercept parameter has been ignored because of its smallness.
## VII Conclusion and outlook
As part of several studies to investigate the performance of the waveguide RF
Wien filter, exploratory data were taken to provide a benchmark on the
sensitivity to very weak collective vertical beam oscillations of deuterons
stored in the COSY ring. To a good approximation, the beam can be viewed as a
rarefied gas of uncorrelated particles, and the sensitivity limit is
applicable to the classical motion of individual particles, propagating along
the ring circumference in the confining oscillatory potential. Simulations of
the beam dynamics in the COSY ring equipped with an RF Wien filter suggest
that with the present apparatus, the sensitivity to collective beam
oscillations on the sub-micron level is only a factor of about 10 larger than
the amplitude of single-particle zero-point quantum oscillations of the stored
deuterons. From the perspective of future EDM experiments, our finding
confirms that, as far as the Heisenberg uncertainty relation is concerned, a
separation of the centroids of two counter-propagating beams may be determined
to sub-picometer accuracy.
The reported excellent agreement between simulated and experimentally observed
vertical beam oscillations at COSY suggests that a further increase in
sensitivity to collective beam oscillations is possible. Specifically, the
simulation on finer capacitor grids indicates that by further optimization of
the Wien filter settings to $C_{\textnormal{L}}=\left(692.76\pm
1.00\right)$\mathrm{p}\mathrm{F}$$ and $C_{\textnormal{T}}=\left(495.77\pm
1.00\right)$\mathrm{p}\mathrm{F}$$, an oscillation amplitude at the Wien
filter location of $\xi_{y}=\left(0.077\pm
0.032\right)$\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ may be achieved. Thus in
that case, the vertical oscillation amplitude would only be about a factor of
2 away from the quantum limit, with a corresponding Lorentz force of
$F_{y}\sim$3\text{\,}\mathrm{aN}$$.
## Acknowledgments
We would like to thank I. Bekman, B. Breitkreutz, J. Hetzel and R. Stassen for
their support in setting up the COSY accelerator for the experiment. The work
presented here has been performed in the framework of the JEDI collaboration
and is supported by an ERC Advanced Grant of the European Union (proposal
number 694340). In addition, it was supported by the Russian Fund for Basic
Research (Grant No. 18-02-40092 MEGA) and by the Shota Rustaveli National
Science Foundation of the Republic of Georgia (SRNSFG Grant No. DI-18-298:
High precision polarimetry for charged-particle EDM searches in storage
rings).
## Appendix A Quantities used in beam simulations
In order to provide a consistent calculation of all effects in the storage
ring, the beam simulations were carried out using the set of quantities given
in Table 1 as an input. The vertical machine tune $\nu_{y}$ is a result of
simulations with the known COSY lattice, reflecting the actual currents of the
magnetic elements in the machine at the time when the experiment was
conducted. The simulations provide the uncalibrated parameters of the vertical
beam oscillations to about per mill accuracy, and giving the kinematic, ring,
and Wien filter parameters to four digits appears therefore sufficient. It
should be noted that within the simulation calculations carried out in the
context of the present work, all quantities have been computed to double
precision (machine epsilon of $1.11\text{\times}{10}^{-16}$). Of the physical
quantities, the highest sensitivity to the vertical betatron tune is exhibited
by the theoretical estimate for Hooke’s constant,
$\text{d}k_{\text{H}}^{\text{th}}/\text{d}\nu_{y}\approx$2\text{\times}{10}^{3}\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}^{2}$$.
The largest uncertainty contributing to the error of the detected oscillation
amplitudes arises from the calibration factor $\kappa$ of the beam position
monitor, given in Eq. (29). It amounts to about $7.3\%$ and is considered a
systematic scale-factor uncertainty (see Appendix B).
Table 1: Numerical values used for the beam simulations. The genuinely independent input parameters are listed in bold face. The derived quantities are displayed in normal font and are truncated to four decimal places. Quantity | Symbol | Value
---|---|---
deuteron beam momentum | $\bm{p}$ | $970.0000\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$
deuteron mass | $\bm{m}$ | $1875.6128\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}^{2}$
deuteron $G$ factor | $\bm{G}$ | $-0.1430$
Lorentz factor | $\beta$ | $0.4594$
Lorentz factor | $\gamma$ | $1.1258$
COSY circumference | $\bm{L_{\text{COSY}}}$ | $183.4728\text{\,}\mathrm{m}$
revolution frequency | $f_{\text{rev}}$ | $750\,603.7600\text{\,}\mathrm{H}\mathrm{z}$
vertical machine tune | $\nu_{y}$ | $3.6040$
vertical $\beta$ function at BPM 17 | $\beta_{y}^{\text{BPM}}$ | $15.3049\text{\,}\mathrm{m}$
vertical $\beta$ function at WF | $\beta_{y}^{\text{WF}}$ | $2.6784\text{\,}\mathrm{m}$
effective length WF | $\bm{\ell}$ | $1.1600\text{\,}\mathrm{m}$
frequency WF | $\bm{f_{\text{WF}}}$ | $871\,000.0000\text{\,}\mathrm{H}\mathrm{z}$
tune WF | $\nu_{\text{WF}}$ | $1.1604$
## Appendix B Calibration of beam position monitor
The complex amplitudes measured by the lock-in amplifiers describe the
magnitude and phase of each signal, and are here expressed by the
corresponding real and imaginary components, denoted by $X$ and $Y$,
respectively, i.e., $A=X+iY$. Examples of the data recorded at the sum
frequency $f^{\Sigma}$ and at the revolution frequency $f^{\text{rev}}$ are
shown in Fig. 14. The observed weak attenuation of the beam current during a
measurement cycle by less than 7% clearly indicates a weak beam loss by
intrabeam or residual gas interactions, thus justifying our treatment of the
beam as a rarefied gas. The effect of switching on the power amplifiers of the
Wien filter at $t=$60\text{\,}\mathrm{s}$$ is clearly visible. In both panels,
one observes a separation of the quantities recorded by the top and bottom
electrodes in the $\mathrm{\SIUnitSymbolMicro V}$ range for both frequencies
after the Wien filter is switched on. This separation is much more pronounced
at the Wien filter frequency than at the revolution frequency.
(a) Real ($X^{\Sigma}$) and imaginary part ($Y^{\Sigma}$) of the complex
Fourier amplitudes $A^{\Sigma}$ at the Wien filter frequency. (b) Real
($X^{\text{rev}}$) and imaginary part ($Y^{\text{rev}}$) of the complex
Fourier amplitudes $A^{\text{rev}}$ at the revolution frequency.
Figure 14: Fourier amplitudes $A=X+iY$ for the top and bottom electrodes of BPM 17 recorded by the lock-in amplifier as a function of time in the cycle at a strongly mismatched point ($C_{\text{L}}=$907.79\text{\,}\mathrm{p}\mathrm{F}$$ and $C_{\text{T}}=$885.58\text{\,}\mathrm{p}\mathrm{F}$$), at the Wien filter frequency (a), and at the revolution frequency (b). In both panels, the stored beam current is shown in black. The cycle starts right after injection is completed at $t=$0\text{\,}\mathrm{s}$$, beam preparation continues until $t=$55\text{\,}\mathrm{s}$$, and the Wien filter is switched on and data acquisition starts at $t=$60\text{\,}\mathrm{s}$$. At $t=$156\text{\,}\mathrm{s}$$ the Wien filter is switched off and data acquisition stops. Table 2: Current $I$ (in $\%$ of the maximum admissible current) in the vertical steerers to generate bumps and the corresponding position change of the vertical orbit $y$ by $\Delta y$ at the location of BPM 17. $I$ (steerer) [%] | $y$ [$\mathrm{m}\mathrm{m}$] | $\Delta y$ [$\mathrm{m}\mathrm{m}$]
---|---|---
$-5$ | $-$7.756$\pm$0.030$$ | $-$7.466$\pm$0.030$$
$-4$ | $-6.684\pm 0.038$ | $-$6.395$\pm$0.038$$
$-3$ | $-5.629\pm 0.016$ | $-$5.339$\pm$0.016$$
$-2$ | $-4.518\pm 0.020$ | $-$4.229$\pm$0.020$$
$-1$ | $-3.489\pm 0.018$ | $-$3.119$\pm$0.018$$
$\phantom{+}0$ | $-2.439\pm 0.029$ | $-$2.150$\pm$0.029$$
$+1$ | $-1.429\pm 0.020$ | $-$1.140$\pm$0.020$$
$+2$ | $-0.288\pm 0.028$ | $\phantom{+}$0.000$\pm$0.000$$
$+3$ | $+0.798\pm 0.044$ | $+$1.085$\pm$0.044$$
$+4$ | $+1.872\pm 0.014$ | $+$2.160$\pm$0.014$$
$+5$ | $+2.928\pm 0.069$ | $+$3.211$\pm$0.069$$
Figure 15: Calibration curve of BPM 17. The ratio $R$, defined in Eq. (28),
depends on the introduced vertical beam displacement $\Delta y$ at the beam
position monitor.
The quantities $A_{\text{t}}^{\text{rev}}$ and $A_{\text{b}}^{\text{rev}}$,
given in Eqs. (16), are related to a vertical beam displacement $\Delta y$ in
the following way,
$\displaystyle
R=\frac{A_{\text{t}}^{\text{rev}}-A_{\text{b}}^{\text{rev}}}{A_{\text{t}}^{\text{rev}}+A_{\text{b}}^{\text{rev}}}=\kappa\frac{2U_{0}\Delta
y}{2U_{0}}=\kappa\Delta y\,.$ (28)
The calibration constant $\kappa$ is experimentally determined by introducing
local vertical beam bumps in the ring at the location of BPM 17. The orbit
positions $y$ and the orbit displacements $\Delta y$, listed in Table 2,
differ by the position of the unperturbed orbit, and are generated by altering
the current of a set of vertical steerers.
The steerer magnets have well-known conversion factors from current to
magnetic field. The calibration factor $\kappa$ is obtained by fitting the
ratio $R$ from Eq. (28) as a function of the vertical orbit variation $\Delta
y$, exhibiting the nearly linear relationship shown in Fig. 15. The slope
corresponds to
$\kappa=\left(5.82\pm
0.43\right)\cdot${10}^{-6}\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$\,.$
(29)
## Appendix C Simulation uncertainties
The uncertainties of the simulated amplitudes of the beam oscillations are
computed using the Polynomial Chaos Expansion (PCE) algorithm. The
functionality of the algorithm is explained below for one of the simulated
data points of the map shown in Fig. 9.
The PCE algorithm offers an alternative to the well-known Monte-Carlo (MC)
method without compromising the intended accuracy. It uses orthogonal
polynomials to represent randomly changing variables to describe observables
by means of a finite (truncated) series (for more details, see, e.g., Ref.
Slim _et al._ (2017)). When the defined criteria of convergence are met, the
expansion coefficients can be used to generate an arbitrarily large sample of
observables, from which the uncertainties can be computed to the desired
statistical accuracy.
The PCE algorithm has been compared with the MC method in many applications
and has been shown to provide very reliable results Smith (2013). The PCE
requires much fewer simulations to converge compared to the MC method. For
instance, for the present case, 200 beam tracking simulations per point in the
2D the map of beam oscillations, shown in Fig. 9 were sufficient to reach
convergence.
Data: Generates Gaussian-distributed ensemble of uncertain circuit parameters
using the Latin Hypercube Sampling (LHS) scheme $X_{i}$
Result: Compute uncertainty of $\xi_{y}$
Given $X_{i}$, run full wave simulations;
Generate 3D electric and magnetic fields;
Run beam tracking simulations to compute ${\xi_{y}}_{i}$;
Standardize input data $X_{i}\longrightarrow\tilde{X}_{i}$;
Guess hyperbolic truncation norm, $q$-norm;
Start with lowest possible expansion order $p$;
Generate basis functions ${H}_{p}({\tilde{X}_{i}})$ ($p^{\text{th}}$-order
Hermite polynomials);
Generate hyperbolically truncated set of basis functions
${H}_{p}^{q}({\tilde{X}_{i}})$;
Apply Least-Angle Regression (LAR) algorithm;
Estimate optimum sparse set of basis functions
${H}_{p}^{q*}({\tilde{X}_{i}})$;
Compute expansion coefficients $C_{j}$, given
${\xi_{y}}_{i}=\sum_{j}C_{j}{H}_{p}^{q*}({\tilde{X}_{i}})$;
Compute leave-one-out error $\text{LOO}_{\text{err}}$;
Check convergence condition( $\text{LOO}_{\text{err}}<10^{-2}$);
while _not convergent_ do
Enhance model (vary $p$ and $q$);
if _convergent_ then
Generate large sample of $\xi_{y}$;
Estimate statistical parameters;
Terminate algorithm;
else
Enrich input samples $X_{i}$;
Repeat algorithm;
Algorithm 1 Sparse Polynomial Chaos Expansion Slim _et al._ (2017).
In cases where the number of random input variables $m$ is larger than $10$,
the PCE method offers clear advantages over the MC method. The reason is that
the number of basis functions in the PCE method increases enormously as a
consequence of the tensor product of the involved polynomials. Therefore, the
algorithm has been improved further to allow for a reduction of the number of
simulations required. Such an approach is also adopted here, as described in
Algorithm 1. The hyperbolic truncation scheme together with the Least-Angle
Regression (LAR) method form a sparse version of the original algorithm.
An $m$-dimensional set is first created, representing $N$ combinations of
simultaneous random variables. Many methods can be used to generate such sets,
and here the Latin-hypercube sample scheme is adopted Slim _et al._ (2020).
Subsequently, the set is standardized for convergence reasons. Depending on
the distribution of the data, the basis functions, here Hermite polynomials,
are determined. The number of basis functions restricts the lower limit of the
number of simulations (full-wave and tracking) which are usually
computationally expensive. As a rule of thumb, with $N$ basis functions, the
PCE algorithm requires at least $1.5\times N$ (in this case, full-wave)
simulations to converge. The number of basis functions itself can, however, be
reduced by the hyperbolic truncation scheme that eliminates higher-order terms
that do not have a significant impact on the observation objects Sudret and
Der Kiureghian (2000); Sudret (2008). Furthermore, by applying the LAR
algorithm, the number of remaining basis functions can be further reduced
substantially, whereby the problem becomes computationally solvable in a very
efficient fashion.
The matching point, specified in Eq. (19), yields the minimum measurable beam
oscillations, as given by Eq. (20). This experimental result can be estimated
using the beam-tracking calculations. Subsequently, the concrete steps of the
application of the PCE algorithm are discussed.
All the reasonable sources of uncertainties of the circuit are represented by
15 random parameters that are allowed to vary simultaneously. At first, a
sample of ($200\times 15$) entries is generated using the Latin-hypercube
sampling scheme. As an example of this sample, the variation of the three
circuit elements $C_{\textnormal{L}}$, $C_{\textnormal{T}}$, and the load
resistor $R_{\text{f}}$ is shown in Fig. 16(a).
(a) Sample of $C_{\text{L}}$, $C_{\text{T}}$, and $R_{\text{f}}$ used in the
PCE calculations showing a subset of the 15-dimensional input of random
circuit uncertainties. (b) Truncation schemes of the PCE algorithm. (c)
Expansion coefficients on a semi-log scale. 91 coefficients have been selected
after applying the LAR algorithm to the matching point. (d) Comparison between
the tracking results and the PCE with respect to the oscillation amplitude,
determined using the expansion coefficients of (c).
Figure 16: Intermediate results of the PCE algorithm applied at the matching
point [see Eq. (19)]. Quantitative results of the PCE algorithm are summarized
in Table 3.
All the uncertain parameters in the electromagnetic circuit simulations are
used to generate the electric and magnetic fields shown in Fig. 10. These are
subsequently used in the beam-tracking calculations. For the matching point of
the map [Eq. (19)], $N=200$ full-wave simulations were conducted. The import
of these field maps into the beam-tracking calculations resulted in a set of
$N=200$ values of $\xi_{y}$. This set is not directly used to conduct the
statistical analysis. Instead, in conjunction with the input samples, these
data are used as input to the sparse PCE algorithm.
The optimum set of basis function is determined using the LAR algorithm, as
shown in Fig. 16(b). With an expansion order of $p=6$ and a truncation norm
$q=0.35$, executing the PCE algorithm required $91$ basis functions to
converge, reflected by the low value of the leave-one-out error
$\text{LOO}_{\text{err}}=$1.7\text{\times}{10}^{-4}$$. Subsequently, the
expansion coefficients are computed, qualitatively depicted in Fig. 16(c). It
is shown in Fig. 16(d) that the PCE algorithm perfectly reproduces the
tracking results using these expansion coefficients. Finally, these
coefficients are used to reconstruct a larger sample of $\xi_{y}$ to estimate
the error $\sigma_{\xi_{y}}$. Figure 11 shows ${10}^{4}$ of the ${10}^{6}$
reconstructed samples. The PCE parameters used are summarized in Table 3.
The fitting of these results with a Gaussian, as depicted in panel (b) of Fig.
11, yields a standard deviation of
$\sigma_{\xi_{y}}=$0.082\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. The
same technique is repeated for each point in the map.
Table 3: PCE simulation parameters of the matching point in Eq. (19). Parameter | Value
---|---
order of expansion $p$ | $6$
dimension $m$ | $15$
hyperbolic truncation $q$ | $0.35$
leave-one-out error $\text{LOO}_{\text{err}}$ | $1.71\text{\times}{10}^{-4}$
number of used basis functions $P^{\text{LAR}}$ | 91
number of used full-wave simulations $N$ | 200
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|
# Explaining the specific heat of liquids based on instantaneous normal modes
Matteo Baggioli1,2<EMAIL_ADDRESS>Alessio Zaccone3,4
<EMAIL_ADDRESS>1Wilczek Quantum Center, School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 2Shanghai
Research Center for Quantum Sciences, Shanghai 201315. 3Department of Physics
“A. Pontremoli”, University of Milan, via Celoria 16, 20133 Milan, Italy.
4Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE
Cambridge, U.K.
###### Abstract
The successful prediction of the specific heat of solids is a milestone in the
kinetic theory of matter, due to Debye (1912). No such success, however, has
ever been obtained for the specific heat of liquids, which has remained a
mystery for over a century. A theory of specific heat of liquids is derived
here using a recently proposed analytical form of the vibrational density of
states (DOS) of liquids, which takes into account saddle points in the liquid
energy landscape via the so-called instantaneous normal modes (INMs),
corresponding to negative eigenvalues (imaginary frequencies) of the Hessian
matrix. The theory is able to explain the typical monotonic decrease of
specific heat with temperature observed in liquids, in terms of the average
INM excitation lifetime decreasing with $T$ (in accordance with Arrehnius
law), and provides an excellent single-parameter fitting to several sets of
experimental data for atomic and molecular liquids. It also correlates the
height of the liquid energy barrier with the slope of the specific heat in
function of temperature in accordance with the available data. These findings
demonstrate that the specific heat of liquids is controlled by the
instantaneous normal modes, i.e. by localized, unstable (exponentially
decaying) vibrational excitations, and provide the missing connection between
anharmonicity, saddle points in the energy landscape, and the thermodynamics
of liquids.
Historically, one of the overarching goals of the kinetic theory has always
been the rationalization of the specific heat of matter based on its
underlying atomic and molecular structure. Classical thermodynamics, revisited
in light of modern molecular physics, explains the specific heat of atomic and
molecular gases in terms of the equipartition theorem for the various
translational and rotational degrees of freedom of the constituent
atoms/molecules: the result is the well known Dulong-Petit law, $C_{v}=3N/2$
(constant with $T$), for a monoatomic gas.
For condensed matter, things become more interesting and more intertwined with
modern physics. The case of solids has been essentially solved by Debye in
1912 Debye (1912). In his remarkable paper, Debye correctly counted the
contribution of plane waves (acoustic phonons) in the isotropic 3d solid to
the internal energy, from which he derived the law $C(T)\sim T^{3}$ valid for
insulators at low temperature (this does not account for the electronic
contribution in metals which is given by the Sommerfeld theory of electronic
heat capacity and yiels a $C(T)\sim T$ contribution). Furthermore, in the same
paper, Debye presented the famous result for the density of states of phonons
in solids, $g(\omega)\sim\omega^{2}$, obtained from the correct way of summing
plane wave contributions in a spherical 3d space, together with the
ultraviolet cutoff at the Debye wavevector $\omega_{D}$, consistent with
atomic-scale granularity of matter. The correct counting of normal modes in
the spherical shell in $k$-space, that is, the $g(\omega)\sim\omega^{2}$, was
the key step that allowed Debye to arrive at the correct result for the
specific heat of solids and it strongly relied on the linear dispersion
relation $\omega=vk$ for acoustic phonons. Furthermore, the Debye theory also
recovers, correctly once again, the high-temperature limit which is again the
Dulong-Petit law mentioned above.
Therefore, we have satisfactory theories of the specific heat for both gases
and solids, in agreement with experimental observations, which can be found in
any textbooks of statistical physics or solid-state theory Kittel (2004) . In
light of these successes for gases and solids, it is thus all the more
surprising that 100 years after Debye, no satisfactory theory of the specific
heat of liquids is available yet. Experimental data show that the specific
heat of liquids decreases monotonically with temperature upon going from the
glass transition or melting transition temperature to higher temperatures.
This behaviour is puzzling because it is clearly in contrast with what is
observed in solids, where the specific heat is an increasing function of $T$,
and then plateaus at the Dulong-Petit value.
One reason for this state of matters is that the dynamics of atoms and
molecules in liquids is strongly anharmonic, which renders the mathematical
problem a strongly nonlinear one and intractable from first-principles. This
strong anharmonicity also makes concepts such as normal modes, that proved
decisive in the Debye theory of specific heat of solids, of less
straightforward applicability in the case of liquids. In other words, the
basic assumption of Debye theory, i.e. the presence of linearly dispersing
propagating (shear) sound waves at small frequencies, must be abandoned. In
this sense, a correct description of the specific heat of liquids at small
temperatures is inevitably connected to the identification of the low-energy
excitations therein, in analogy with acoustic phonons in solids.
Recent advances in our understanding of the specific heat of liquids include
the interstitialcy argument by Granato, which heuristically explains the
decaying $C(T)$ of liquids in terms of Arrhenius-type relaxation of
“interstitial” defects Granato (2002). Though intuitively appealing and
simple, this model is not supported by the existence of point-defects in
liquids, since there is no underlying regular lattice in liquids that can
provide a topologically meaningful definition of interstitials. A different
explanation of the decaying specific heat of liquids with temperature was
suggested by Wallace on the basis of atomic motions through a vast number of
random valleys in the energy landscape Wallace (1998).
More recently, Trachenko and co-workers proposed a theory of specific heat in
liquids based on standard acoustic phonons Bolmatov _et al._ (2012) and the
k-gap theory Baggioli _et al._ (2020). The theory explains the decaying
$C(T)$ in liquids as due to the gradual depletion of transverse acoustic
phonons (and their shift to higher and higher frequency/momenta) as the
temperature is raised. This approach relies on acoustic phonons, whereas at
lower momenta/energies one has to deal with overdamped modes
($\omega=-i/\tau$), whose importance for liquids has been established and
demonstrated in a broad literature Keyes (1997). The role of these modes is
nevertheless not considered in any of the previous approaches.
Modern theories of the liquid state have attempted to extend the concept of
normal modes from solids to liquids, following pioneering ideas and work by
Zwanzig Zwanzig (1967). This led to the concept of Instantaneous Normal Modes
(INMs), which extends the concept of normal modes to liquids, to include the
above mentioned overdamped modes. In short, the locally anharmonic dynamics of
atoms in liquids leads to many saddle points in the energy landscape. These
saddle points are associated with localized unstable (exponentially decaying)
modes, with purely imaginary frequency. The imaginary frequencies correspond
to negative eigenvalues of the Hessian matrix of the atomistic system. In
simpler terms, the anharmonicity leads to locally unbalanced forces between
atoms (which are constantly pushed away from their bonding minima by the
thermal fluctuations), which then lead to exponentially decaying motions in
time with an Arrhenius-dependent time-scale on $T$, i.e. the INMs:
$e^{i\omega^{*}t}\sim e^{-\Gamma t}$, with $\Gamma\sim e^{-U/k_{B}T}$ and
$\omega^{*}$ is purely imaginary, $\omega^{*}=-i\Gamma$.
As shown by many numerical studies over the past decades, the INMs dominate
the low-frequency and intermediate-frequency sectors of the DOS of liquids
Keyes (1997); Stratt (1995); Rabani _et al._ (1997). At low-frequency, they
coexist with one longitudinal acoustic phonon and one transverse diffusive
mode (momentum-shear diffusion), whereas, at higher frequencies, transverse
acoustic phonons only recently have been shown to exist in liquids and to play
a role in their thermodynamics at larger energies (the so-called $k$-gap)
Khusnutdinoff _et al._ (2020). Interestingly, these modes define the regime
of applicability of hydrodynamics Baggioli (2020), intended as an effective
continuum description of fluids.
In this work, we provide a first-principles theory of the specific heat of
liquids, which, for the first time, effectively takes into account the
intrinsic anharmonicity of liquid dynamics and the fact that the DOS of
liquids (derived analytically in recent work Zaccone and Baggioli (2021)) is
dominated by INMs. The theory provides an excellent fitting to experimental
data of several liquids and correctly recovers the Dulong-Petit law as its
high-temperature limit. The results presented here provide a long-sought
answer to the century-long question about the specific heat of liquids, more
than hundred years after Debye’s theory for solids.
As it is customary for specific heat calculations, one starts from the total
energy of a collection of excitations. For harmonic solids, these excitations
are simple harmonic oscillators with frequencies strictly real; in the case of
liquids, the frequencies can be imaginary (as for the INMs). States with
imaginary frequencies in quantum mechanics are not at all uncommon Zeldovich
(1961), and they arise in nuclear physics – the Gamow states –, and in
particle physics, – the $W$ and $Z^{0}$ bosons Stuart (1995); Sirlin (1991).
These modes are simply called resonances, states with a finite lifetime coming
from a large imaginary part, which contributes and may even dominate the
particle mass and energy Zeldovich (1961); Sirlin (1991). In other contexts,
they take the name of quasinormal modes; they are intimately linked to non-
hermiticity Bender (2007) (i.e. dissipation/relaxation) and experimentally
observed even in astronomic black holes collisions Nollert (1999).
Here we describe a population of INMs as a weakly-interacting Bose gas, with
Hamiltonian given by $\mathcal{H}=\sum_{q\neq
0}\epsilon_{q}b_{q}^{\dagger}b_{q}$ Khomskii (2010) where we do not include
the ground state ($T=0$) terms (which is irrelevant since we will later take a
derivative with respect to $T$). In the above expression, $b_{q}^{\dagger}$
and $b_{q}$ are the bosonic (Bogolyubov) creation and annihilation operators
equipped with standard commutation relations and with associated momentum $q$,
while $\epsilon_{q}$ is the energy Khomskii (2010). We then formally rewrite
$\epsilon_{q}\equiv\hbar\omega_{q}$ for the energy of a single boson, where
$\omega_{q}\equiv|\omega_{q}|$, as appropriate for unstable bosons Zeldovich
(1961); Sirlin (1991); Stuart (1995); Keyes (1997), and we further consider
that $b_{q}^{\dagger}b=n_{q}$ where $n_{q}=(e^{\hbar\omega_{q}/T}-1)^{-1}$ is
the Bose-Einstein (BE) occupation number. Since we have a gauge freedom in
defining the ground-state energy (because it obviously does not contribute to
the specific heat), we define it as $\hbar\omega_{q}/2$ in order to maintain a
formal analogy with the case of solids.
Hence the energy of a collection of weakly-interacting bosons under the above
assumptions can be written as
$E=\sum_{q}\frac{\hbar\omega_{q}}{2}\frac{e^{\hbar\omega_{q}/T}+1}{e^{\hbar\omega_{q}/T}-1}$
(1)
where $q\equiv|\mathbf{q}|$ is the modulus of the momentum, since we are
considering isotropic liquids. In the above, we are working in units such that
$k_{B}=1$.
Using the standard replacement
$\sum_{q}\rightarrow\int\frac{d^{3}q}{(2\pi)^{3}}$, and further introducing
the vibrational density of states, $g(\omega)$, defined via
$\frac{d^{3}q}{(2\pi)^{3}}=g(\omega)d\omega$, we arrive at the following
integral (which can be found in textbooks) for the specific heat Khomskii
(2010):
$C_{V}(T)=3N\int_{0}^{\infty}\left(\frac{\omega}{2T}\right)^{2}\sinh{\left(\frac{\omega}{2T}\right)}^{-2}g(\omega)\,d\omega$
(2)
where we have also set $\hbar=k_{B}=1$.
Upon inserting the normalized Debye DOS,
$g(\omega)=3\omega^{2}/\omega_{D}^{2}$ in the above integral, one readily
recovers the low-$T$ limit of the specific heat as $C_{V}\sim T^{3}$, and the
high-temperature limit as the Dulong-Petit law, $C_{V}\sim 3N$ (in units of
$k_{B}=1$).
Figure 1: The (schematic) theoretical predictions of the model. Left: the
dependence of the liquids specific heat on the amplitude of the INMs
relaxation rate $\Gamma$. Right: The dependence on the characteristic
potential height $U$ for relaxation.
Let us now turn to the case of liquids. The starting point is an overdamped
equation of motion for particle dynamics,
$\frac{d\mathbf{v}}{dt}=-\Gamma\,\mathbf{v},\qquad\text{with}\qquad\Gamma\equiv
1/\tau\,,$ (3)
where $\tau$ is the relaxation time and $\Gamma$ is a damping coefficient (the
relaxation rate), which for strongly anharmonic excitations represents the
(short) lifetime of the excitation. Taking advantage of a generalization of
the Plemelj identity to arbitrary integration pathways in the complex plane,
recently it has been possible to derive an analytical form for the DOS of
liquids that takes INMs into account Zaccone and Baggioli (2021). The final
expression has the following form (modulo a normalization factor to ensure
that $\int g(\omega)d\omega=1$):
$g_{\textit{liq}}(\omega)\sim\frac{\omega}{\omega^{2}+\Gamma^{2}}\,e^{-\omega^{2}/\omega_{D}^{2}}\,,$
(4)
where $\Gamma$ is the characteristic relaxation rate of an INM, which exhibits
a typical Arrhenius dependence on temperature Rabani _et al._ (1997)
$\Gamma(T)\,=\,\Gamma_{0}\,e^{-U/T}.$ (5)
Furthermore, the factor $e^{-\omega^{2}/\omega_{D}^{2}}$ is just a Gaussian
cut-off which implements the “granularity” of matter at the atomic/molecular
scale in terms of the ultraviolet cutoff $\omega_{D}$ and was already
introduced in Ref. Rabani _et al._ (1997). We have checked that the main
results do not depend essentially on the specific form of the cutoff.
The above Eq. (4) has been shown in recent work Zaccone and Baggioli (2021) to
provide an excellent fitting of numerical data of the DOS of Lennard-Jones
systems obtained from molecular dynamics simulations in the literature Zhang
_et al._ (2019); Rabani _et al._ (1997).
These formulae, Eqs. (4)-(5), provide a direct connection between relaxation
and vibration in liquids, and play a decisive role in the following
description of the specific heat.
Upon inserting a normalized form of (4) in (2), it is immediately verified
that the limit $T\rightarrow\infty$ of the integral leads $C_{V}=3N$, i.e. the
Dulong-Petit law.
We now turn to the dimensional form of the specific heat integral (2)
$C_{V}(T)=k_{B}\,\int_{0}^{\infty}\left(\frac{\hbar\,\omega}{2k_{B}T}\right)^{2}\,\sinh\left(\frac{\hbar\omega}{2k_{B}T}\right)^{-2}g(\omega,T)d\omega$
(6)
where $g(\omega,T)$ is given by (4) together with (5).
In (4), acoustic phonons are not explicitly taken into account, because it has
been shown in previous work that they are not crucial to reproduce numerical
data of DOS of Lennard-Jones liquids Zaccone and Baggioli (2021). It is also
important to note that, at $T<\Theta_{D}$ where $\Theta_{D}$ is the Debye
temperature, the BE-related factor
$\sinh\left(\frac{\hbar\,\omega}{2\,k_{B}\,T}\right)^{-2}$ in the integral for
the specific heat effectively gives a very low weight to all high-$\omega$
(phonon-type) excitations, whereas it gives a large weight to low-frequency
excitations such as the INMs. More precisely, high frequencies could
eventually be important only at extremely high temperatures and they cannot
possibly be responsible for the low-temperature (above melting transition)
decay typical of liquids. Indeed, as we will prove, there is no need to take
into account high frequency modes (e.g. emerging shear waves in the k-gap
model Baggioli _et al._ (2020)) to reproduce the experimental trends.
Furthermore, quoting from Born and Huang Born and Huang (1954), at
$T>\Theta_{D}$, the specific heat is not sensitive to the specifics of the
frequency distributions and the Einstein model provides a correct estimate in
terms of high-energy atomic/molecular vibrations with $\omega\sim\omega_{D}$
or larger (intramolecular vibrations). Hence, in this high-temperature regime,
phonons, as collective lattice vibrations, do not exist anymore, while the
high-frequency non-collective (gas-like) vibrations contribute a constant
(independent of $T$) to the specific heat Born and Huang (1954). These
arguments suggest that the influence of the INMs on the specific heat and on
its observed decay with $T$ could possibly be the dominant one.
Illustrative calculations of the specific heat using the above theory are
shown in Fig.1. It is clear from these theoretical calculations that the
temperature dependence of the specific heat is mostly controlled by the
relaxation rate of excitation lifetime $\Gamma$ and its Arrhenius dependence
on $T$. In particular, despite the dimensionful pre-factor $\Gamma_{0}$
produces only a vertical shift in the $C(T)$ function (left panel of Fig.1),
the energy barrier $U$ plays a much more fundamental role. It determines the
curvature of the specific heat; the larger the potential energy $U$, the
slower the temperature decay of the specific heat (right panel of Fig.1).
This Arrhenius dependence was fortuitously captured by Granato’s interstitial
defect argument, although its true physical origin resides in the INMs and in
the many saddle points of the energy landscape. From a physical point of view,
the decay of $C(T)$ with increasing $T$ is caused by the decrease of the
average lifetime of the INM excitations, which is equal to $\Gamma^{-1}$.
Hence, since the heat is stored by the INMs, as the dominant vibrational
excitations in liquids, the fact that their lifetime decreases with increasing
$T$ leads to a lower capability of storing heat in the vibrational
excitations.
This picture is confirmed by the fact that the specific heat is reduced upon
increasing the strength of the INMs relaxation rate $\Gamma_{0}$, i.e. upon
decreasing their lifetime. Moreover, the model directly shows that, by
increasing the characteristic potential height $U$ of the anharmonic liquid
landscape, the specific heat grows. This can be simply explained by the fact
that a higher barrier suppresses the probability of the molecular
rearrangements responsible for the INMs dynamics and therefore makes their
lifetimes longer. This is fully consistent with the emerging picture of heat
being stored in the INMs, in liquids.
Liquids: | Xe | Kr | Ne | Ar | N2
---|---|---|---|---|---
$\omega_{D}^{*}$ [K] | 64 | 72.1 | 74.6 | 93.1 | 86
$U^{*}$ [K] | 226.1 | 162.5 | 33.9 | 116.7 | 102.12
$\Gamma_{0}$ [K] | 240 | 100 | 80 | 60 | 29
Table 1: The numerical values used in the fitting procedure. The symbol ∗
indicates that the values are not obtained from the fit but they are fixed
with the literature data Fenichel and Serin (1966); Moreh _et al._ (1992);
Rutkai _et al._ (2017). The only free parameter is $\Gamma_{0}$.
We now turn to the fitting procedure and the main results of our analysis.
Combining Eq.(4) and Eq.(5), our model displays three physical parameters: the
Debye frequency $\omega_{D}$, the activation energy $U$ and the relaxation
rate prefactor $\Gamma_{0}$. The first two parameters for simple liquids are
well-known and they are fixed to their literature values Fenichel and Serin
(1966); Moreh _et al._ (1992); Rutkai _et al._ (2017) displayed in Table 1.
The activation energy is taken to be equivalent to the height of the Lennard-
Jones energy barrier $\epsilon$. All in all, our fitting procedure involves a
single fitting parameter $\Gamma_{0}$. In Fig. 2, we present a series of
comparisons between the specific heat calculated using (4) inside the specific
heat integral (6) and experimental data of simple liquids of various nature,
but all reasonably well approximated by the Lennard-Jones potential. The
obtained values for the relaxation rate scale $\Gamma_{0}$ are shown in Table
1. In all instances, the fitting is excellent and perfectly captures the
decline of the specific heat with increasing temperature, explained by the
present theory in terms of reduced lifetime of INMs. The results show, as
already anticipated, that, the larger the characteristic energy $U$ (which is
related to $\epsilon$), the larger the specific heat and the slower its
temperature decay. This confirms once more not only the validity of our theory
but also its predictive power able to connect microscopic features, such as
the characteristic potential barrier $U$, to macroscopic thermodynamic
observables, such as the temperature dependence of the specific heat.
In order to emphasize the predictive power of our theory, and the excellent
agreement with the data, we re-present the INMs temperature dependent
relaxation rate $\Gamma(T)$ using the parameters obtained from the fits in
Fig. 3. For all the liquids analyzed, we find a relaxation rate of the order
of $1/$ps. According to transition state theory Hänggi _et al._ (1990), the
molecular hopping (attempt) rate is directly proportional to the INMs
relaxation rate, which corresponds to the (negative) curvature of the
potential landscape. Interestingly, our order of magnitude estimate of the
single fitting parameter $\Gamma_{0}$, coincides with the values reported in
the literature, see for example Rabani _et al._ (1997).
Figure 2: The comparison between the model, Eqs.(4)-(6), and experimental data
for four different liquids. The experimental data are taken from Wallace
(1998); nis . The value of the various parameters is displayed in Table 1.
In summary, the above theory provides a definitive answer to the mystery of
liquid specific heat and ideally completes the agenda of the kinetic theory of
matter, set over 100 years ago by Debye, Einstein, Planck and co-workers.
Figure 3: The temperature dependent INM relaxation rate $\Gamma(T)$ obtained
by using the single-parameter fitting in Table 1. The solid portion of the
curves is the one corresponding to the temperature range of the experimental
data fitted.
As in Debye’s work Debye (1912) for solids, the crucial step for the
successful derivation of the specific heat, also in the case of liquids relies
on finding the correct form of the vibrational density of states (DOS). Debye
derived his famous $T^{3}$ law for the specific heat of solids by correctly
counting 3d plane waves in an isotropic solid, leading to the Debye
vibrational density of states, $\sim\omega^{2}$. Here we did the same for
liquids, where the relevant excitations are not plane waves/phonons but the
instantaneous normal modes (INMs), i.e. overdamped relaxations from saddle
points in the energy landscape. This leads to a DOS for liquids $\sim\omega$
at low frequency Zaccone and Baggioli (2021), whose form is given in Eq.(4).
In turn, this DOS leads to a monotonically decreasing $C(T)$ with increasing
$T$, as a result of Arrhenius-type relaxation of INMs, and recovers the
Dulong-Petit plateau in the high-T limit.
These results fill the gap in our understanding of thermal and vibrational
properties of condensed matter.
Finally, given the success of the theory by Trachenko and co-workers Bolmatov
_et al._ (2012), it is important to draw some comparisons. Given our results,
it is clear that the key point in their treatment is not the presence of
propagating shear waves, which appear at large momenta and frequencies (at
least at momenta larger then $\sqrt{2}k_{g}$), but rather the collection of
overdamped modes below that point. In particular, the k-gap dispersion
relation Baggioli _et al._ (2020) displays purely relaxing modes below
$k=k_{g}$. Not only that, but even between $k_{g}<k<\sqrt{2}k_{g}$, the
acoustic waves are mostly overdamped, and therefore more similar in nature to
INMs than to propagating shear waves. Moreover, our results are in agreement
with those of Ref. Kryuchkov _et al._ (2020) where the heat capacity
decreases by increasing the k-gap momentum. Indeed $k_{g}\sim\Gamma$; a larger
k-gap implies a shorter lifetime for the relaxational modes $\omega=-i\Gamma$
and therefore a lower specific heat as explained by our theory.
Following the ideas of Baggioli _et al._ (2021), it would definitely be
interesting to achieve a more fundamental understanding of this relaxation
time scale based on symmetries rather than microscopic mechanisms, in analogy
to the modern formulation of phonons and Debye theory in terms of the
spontaneous symmetry breaking of spacetime translations.
## Acknowledgments
We thank K.Trachenko for fruitful discussions and useful comments. A.Z.
acknowledges financial support from US Army Research Office, contract nr.
W911NF-19-2-0055. M.B. acknowledges the support of the Shanghai Municipal
Science and Technology Major Project (Grant No.2019SHZDZX01) and of the
Spanish MINECO “Centro de Excelencia Severo Ochoa” Programme under grant
SEV-2012-0249.
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|
# Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs
Keren Censor-Hillel Orr Fischer Tzlil Gonen
François Le Gall Dean Leitersdorf Rotem Oshman Technion, Israel Institute of
Technology, Israel<EMAIL_ADDRESS>University, Israel.
Email<EMAIL_ADDRESS>University, Israel. Email:
<EMAIL_ADDRESS>University, Japan. Email:
<EMAIL_ADDRESS>Israel Institute of Technology, Israel.
Email<EMAIL_ADDRESS>University, Israel. Email:
<EMAIL_ADDRESS>
# Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs
Keren Censor-Hillel Orr Fischer Tzlil Gonen
François Le Gall Dean Leitersdorf Rotem Oshman Technion, Israel Institute of
Technology, Israel<EMAIL_ADDRESS>University, Israel.
Email<EMAIL_ADDRESS>University, Israel. Email:
<EMAIL_ADDRESS>University, Japan. Email:
<EMAIL_ADDRESS>Israel Institute of Technology, Israel.
Email<EMAIL_ADDRESS>University, Israel. Email:
<EMAIL_ADDRESS>
###### Abstract
In this paper we give fast distributed graph algorithms for detecting and
listing small subgraphs, and for computing or approximating the girth. Our
algorithms improve upon the state of the art by polynomial factors, and for
girth, we obtain a constant-time algorithm for additive +1 approximation in
Congested Clique, and the first parametrized algorithm for exact computation
in Congest.
In the Congested Clique model, we first develop a technique for learning small
neighborhoods, and apply it to obtain an $O(1)$-round algorithm that computes
the girth with only an additive $+1$ error. Next, we introduce a new technique
(the partition tree technique) allowing for efficiently listing all copies of
any subgraph, which is deterministic and improves upon the state-of the-art
for non-dense graphs. We give two concrete applications of the partition tree
technique: First we show that for constant $k$, it is possible to solve
$C_{2k}$-detection in $O(1)$ rounds in the Congested Clique, improving on
prior work, which used fast matrix multiplication and thus had polynomial
round complexity. Second, we show that in triangle-free graphs, the girth can
be exactly computed in time polynomially faster than the best known bounds for
general graphs. We remark that no analogous result is currently known for
sequential algorithms.
In the Congest model, we describe a new approach for finding cycles, and
instantiate it in two ways: first, we show a fast parametrized algorithm for
girth with round complexity $\tilde{O}(\min\\{g\cdot n^{1-1/\Theta(g)},n\\})$
for any girth $g$; and second, we show how to find small even-length cycles
$C_{2k}$ for $k=3,4,5$ in $O(n^{1-1/k})$ rounds. This is a polynomial
improvement upon the previous running times; for example, our
$C_{6}$-detection algorithm runs in $O(n^{2/3})$ rounds, compared to
$O(n^{3/4})$ in prior work. Finally, using our improved $C_{6}$-freeness
algorithm, and the barrier on proving lower bounds on triangle-freeness of
Eden et al., we show that improving the current $\tilde{\Omega}(\sqrt{n})$
lower bound for $C_{6}$-freeness of Korhonen et al. by _any_ polynomial factor
would imply strong circuit complexity lower bounds.
## 1 Introduction
A fundamental problem in many computational settings is that of finding cycles
and other small subgraphs within a given graph. This paper focuses on finding
subgraphs in distributed networks that communicate through limited bandwidth.
The motivation for this is two-fold: first, for some subgraphs $H$ there exist
distributed algorithms that perform better on $H$-free graphs, such as
distributed cut and coloring algorithms in triangle-free graphs [16, 27]. The
second reason for which we are interested in these problems is that while
solving them only requires obtaining _local_ knowledge, about small non-
distant neighborhoods, the bandwidth restrictions impose a major hurdle for
collecting this information. This induces a rich landscape of complexities for
subgraph-related problems. We contribute to the effort of characterizing the
complexities of subgraph-related problems by providing new techniques, from
which we derive fast algorithms for such problems in the two key distributed
bandwidth restricted models, namely, Congest and Congested Clique.
In the Congested Clique model, $n$ synchronous nodes can send messages of
$O(\log n)$ bits in an all-to-all fashion. The input graph is an arbitrary $n$
vertex graph, partitioned such that every node receives the edges of a single
vertex as input. Our main contribution in this model is an algorithm for
obtaining a $+1$ approximation for the girth in a _constant_ number of rounds,
where the girth of a graph is the length of its shortest cycle.
###### Theorem 1.
Given a graph $G$ with an unknown girth $g$, there exists a deterministic
$O(1)$ round algorithm in the Congested Clique model which outputs an integer
$a$, such that $g\in\\{a,a+1\\}$.
For comparison, note that the current state-of-the-art algorithm computes the
exact girth in $O(n^{0.158})$ rounds [5]. To obtain our $+1$ approximation
algorithm, we devise two main new methods, which we describe here in a
nutshell. The first is an algorithm in which each node learns its entire
neighborhood up to a radius which is a constant approximation of the girth. To
this end, we prove that we can quickly list all paths of sufficient length, as
well as efficiently distribute them to the nodes that need to learn them. The
second method that we introduce is a way to double the radius of the
neighborhoods that all the nodes know, by having each node acquire the
knowledge held by the farthest nodes in its currently-known neighborhood.
Crucially, both of these procedures can be done in $O(1)$ rounds, and could be
useful for additional applications.
Our second contribution in the Congested Clique model is a _partition tree_
technique which allows for efficiently detecting or listing all copies of any
subgraph with at most $\log n$ nodes, in a deterministic manner. In
particular, our main application of the partition tree technique is to obtain
the following subgraph listing algorithm, which improves upon the state-of-
the-art for non-dense graphs.
###### Theorem 2.
Given a graph $G$ with $n$ nodes and $m$ edges and a graph $H$ with $p\leq\log
n$ nodes and $k$ edges, let $\tilde{m}=\max\\{m,n^{1+1/p}\\}$. There exists a
deterministic Congested Clique algorithm that terminates in
$O(\frac{k\tilde{m}}{n^{1+2/p}}+p)$ rounds and lists all instances of $H$ in
$G$.
We give two concrete applications of this result. The first is fast detection
of even cycles.
###### Corollary 3.
Given a graph $G$ and an integer $k\leq(\log n)/2$, there exists a
deterministic $O(k^{2})$-round algorithm in the Congested Clique model for
detecting cycles of length $2k$.
Note that for constant $k$ the above algorithm completes within $O(1)$ rounds.
Prior work for cycle detection in the Congested Clique model used fast matrix
multiplication (FMM) and thus had polynomial round complexity, apart from
detecting $4$-cycles which was shown to have a constant-round algorithm [5].
The second implication of the partition-tree technique is a fast algorithm for
computing the _exact_ girth in triangle-free graphs. Prior algorithms for
girth in the Congested Clique model are based on fast matrix multiplication
(FMM), a technique that can be no faster than checking for triangle-freeness.
###### Corollary 4.
Given a triangle-free graph $G$ with an unknown girth $g$, there exists a
deterministic $\tilde{O}(n^{1/10})$-round algorithm in the Congested Clique
model which outputs $g$.
This result leverages the fact that graphs without small cycles become
increasingly sparse, and the algorithm of Theorem 2 is efficient on sparse
graphs. We remark that, interestingly, no analogous result going below the
complexity of FMM for girth in triangle-free graphs is known for sequential
algorithms, since the best known sequential algorithms for cycle detection in
sparse graphs (see [3]) are not fast enough. We also note that given further
lower bounds on the girth beyond triangle-freeness, the runtime of our
algorithm improves even further; for instance, if the graph does not contain
any $k$-cycle for $k\in\\{3,4,5\\}$, then our algorithm computes the exact
girth in $\tilde{O}(n^{1/21})$ rounds. We refer to Proposition 1 in Section 4
for a more precise statement.
In the Congest model, $n$ synchronous nodes can send messages of $O(\log n)$
bits to their neighbors only, and the input graph is the communication graph.
In this model, we develop a new approach for finding cycles of a given size. A
key step that is present in all known sublinear-round algorithms for finding
cycles in Congest is the elimination of _high-degree vertices_ : we check
whether there is a cycle that includes a high-degree node, and if we conclude
that there is no such cycle, we can remove the high-degree nodes from the
graph. The remaining graph is much easier to handle, since it has low degree.
In prior work, the high-degree vertices were eliminated by sequentially
enumerating over them and starting a short BFS from each one. Here we
introduce a different method for finding cycles that include a high-degree
node: intuitively, we show that if we start from a _neighbor_ of a small even
cycle, we can quickly find the cycle itself. Since high-degree nodes have many
neighbors, if we sample a uniformly random node in the graph, we are somewhat
likely to hit a neighbor of the high-degree node, and from there we can find
the cycle in constant rounds.
We apply this technique to give a fast algorithm that detects small even
cycles, and a fast parameterized algorithm for computing the _exact_ girth.
Specifically, we obtain the following:
###### Theorem 5.
Given a graph $G$, there exists a randomized algorithm in the Congest model
for detection of $2k$-cycles in $O(n^{1-1/k})$ rounds, for $k=3,4,5$.
This significantly improves upon the running time of
$O(n^{1-1/\Theta(k^{2})})$ of the previous state-of-the-art [11]: for cycles
of length 6,8 or 10, the previous algorithm had running time $O(n^{3/4})$,
$O(n^{5/6})$ or $O(n^{10/11})$, respectively. We believe that going below
round complexity of $O(n^{1-1/k})$ for $C_{2k}$-detection in the Congest model
would require a breakthrough beyond currently known techniques, with potential
ramifications also for the Congested Clique model.
For exact girth, previously, an $O(n)$-round algorithm for exact girth was
known, based on computing all-pairs shortest paths [17]. Our result is as
follows:
###### Theorem 6.
Given a graph $G$ with an unknown girth $g$, there exists a randomized
$O(\min\\{g\cdot n^{1-1/\Theta(g)},n\\})$-round algorithm in the Congest model
which outputs $g$.
“Outputs” here means that the first node that halts outputs the girth. Other
nodes of the graph may halt later, and output larger values. This is
unavoidable, unless we introduce a term in the running time that depends on
the diameter of the graph.
Our final result is an _obstacle_ on proving lower bounds for $C_{6}$-freeness
in Congest. In [21] it was shown that the $C_{2k}$-freeness problem is subject
to a lower bound of $\widetilde{\Omega}(\sqrt{n})$, for any $k$. For
$C_{6}$-freeness, the best known algorithm is our new algorithm here, which
runs in $\tilde{O}(n^{2/3})$ rounds, and there are reasons to believe that
this may be optimal. Unfortunately, we show that proving a lower bound of the
form $\Omega(n^{1/2+\alpha})$, for any constant $\alpha>0$, would imply
breakthrough results in circuit complexity. This result uses ideas from our
improved $C_{6}$-freeness algorithm, and the barrier on proving lower bounds
on triangle-freeness from [11].
###### Related work.
The problem of subgraph-freeness, and in particular cycle detection, has been
extensively studied in the Congested Clique and Congest models. While there
are only a few papers which study girth computation, related problems such as
diameter computation or shortest paths were also extensively studied in these
models.
In the first work to consider girth computation in the sequential setting,
Itai and Rodeh [18] gave algorithms with running time $O(mn)$ and $O(n^{2})$
for computing exact girth and $+1$ approximation of the girth, respectively,
using a BFS approach, and an $O(n^{\omega})$ algorithm for exact girth using
an algebric method, where $\omega$ is the exponent of matrix multiplication.
Later, various trade-offs between running time and additive or multiplicative
approximations for girth were obtained (e.g [24, 30, 28, 29]).
In the Congested Clique model, an $O(n^{0.158})$ round algorithm for exact
girth and a $2^{O(k)}n^{0.158}$ round algorithm for $C_{k}$-detection for any
$k$ was shown by [5] based on matrix multiplication techniques. These
algebraic techniques were later extended by [22, 6, 4].
For general subgraphs, a Congested Clique algorithm for listing all instances
of a subgraph $H$ of size $k$ in $\widetilde{O}(n^{1-2/k})$ rounds was shown
in [9], which was shown to be tight for triangles [25, 19] and later for
cliques of size $k>3$ as well [12]. In this work we give a “sparsity-aware”
version of this result, which has improved performance as the graph becomes
sparser. Previously, distributed “sparsity-aware” algorithm were studied in
the context of distributed sparse matrix multiplication [6, 4] and in the
context of the $k$-machine model [25].
Frischknecht et al. [13] was the first work to consider girth computation in
Congest, and showed that at least $\widetilde{\Omega}(\sqrt{n})$ rounds are
required in order to obtain a $(2-\epsilon)$-approximation of the girth. Peleg
et al. [26] showed an algorithm computing a $(2-1/g)$-approximation of the
girth with round complexity $O(D+\sqrt{gn}\log{n})$, where $g$ is the girth of
the graph. Holzer et al. [17] showed an algorithm for exact girth computation
in $O(n)$ rounds, based on an exact all-pairs shortest path algorithm, and an
algorithm for computing an $(1+\epsilon)$-approximation of the girth in
$O(\min\\{n/g+D\log{(D/g)},n\\})$ rounds.
In the cycle-freeness problem in the Congest setting, Drucker et al. [10]
showed a near tight lower bound of $\widetilde{\Omega}(n)$ for constant sized
odd-length cycles, as well as a lower bound of $\widetilde{\Omega}(n^{1/k})$
for $C_{2k}$-freeness, which was later improved to
$\widetilde{\Omega}(\sqrt{n})$ by Korhonen et al. [21]. A Congest randomized
algorithm for listing all triangles with round complexity
$\widetilde{O}(n^{1/3})$ was shown in [8], which improved the previous
$\widetilde{O}(n^{1/2})$-round algorithm of [7] and the
$\widetilde{O}(n^{3/4})$-round algorithm of [19]. The first sublinear-time
algorithm for $C_{2k}$-freeness for $k\geq 3$ was given in [12] running in
$\widetilde{O}(n^{1-1/(k^{2}-k)})$ rounds, and was later improved in [11] to
round complexity $\widetilde{O}(n^{1-2/(k^{2}-k+2)})$ for odd $k$ and
$\widetilde{O}(n^{1-2/(k^{2}-2k+4)})$ for even $k$.
## 2 Preliminaries
###### Definitions.
Given a graph $H$, the _$H$ -listing_ problem is a problem in which each node
may output a set of $H$-copies, and the goal of the network is that w.h.p. the
union over the sets of outputted $H$-copies by the nodes is exactly the set of
$H$-copies in $G$.
The _Túran number_ of a graph $H$, denoted $\operatorname*{ex}(n,H)$, is the
maximum number of edges $m$ such that there exists a graph $G$ on $n$ vertices
and $m$ edges which contains no copy of $H$.
###### Lemma 1 (Túran number of $C_{2k}$ [14]).
For $k\in\mathbb{N}$, if $G$ is $C_{2k}$-free, then $G$ contains at most
$17kn^{1+1/k}$ edges.
###### Lemma 2 (Túran number for girth [14]).
If $G$ is $C_{i}$-free for all $3\leq i\leq 2k$, then $G$ contains at most
$n^{1+1/k}+n$ edges.
Let $N_{i}(v)$ denote the _graph_ defined by the nodes of hop-distance at most
$i$ from $v$, that is, it includes all such nodes and all the _edges_ incident
to nodes with hop-distance at most $i-1$ from $v$. For sets $A,B\subseteq V$,
denote by $E(A,B)=\\{(a,b)\in E\medspace\mid\medspace a\in A\land b\in B\\}$
the set of edges between $A$ and $B$.
###### Load-Balanced Routing in the Congested Clique Model.
We introduce a useful routing procedure which extends that of [23], and it is
used throughout our results. The routing procedure of [23] routes a set of
messages where each node needs to send and receive at most $O(n)$ messages, in
$O(1)$ rounds. The following shows that it possible to replace the constraint
where each node needs to _send_ at most $O(n)$ messages with one stating that
the messages each node desires to send are based on at most $O(n\log n)$ bits.
This allows us to route messages even when the some nodes are each a source of
$\omega(n)$ messages.
###### Lemma 3 (Load Balanced Routing).
Any routing instance $\mathcal{M}$, in which every node $v$ is the target of
up to $O(n)$ messages, and $v$ locally computes the messages it desires to
send from at most $|R(v)|=O(n\log{n})$ bits, can performed in $O(1)$ rounds.
###### Proof.
For every node $v$, let $s(v)$ be the number of messages $v$ is a source of in
$\mathcal{M}$, and have $v$ broadcast $s(v)$. The network allocates
$h(v)=\lceil s(v)/n\rceil$ _helper nodes_ to $v$ in such a way that each node
$u$ is a helper node of at most $O(1)$ other nodes, and such that all nodes
can locally compute which node helps another node. This is possible as since
each node is the target of at most $O(n)$ messages, then the total number of
messages is at most $(c-1)n^{2}$, for some constant $c$, and therefore
$\sum_{v}h(v)\leq cn$.
Having allocated the helper nodes $h(v)$ to each node $v$, we ensure that
these nodes learn $R(v)$ \- this will later allow them to reproduce the
messages which $v$ desires to send. First, each node $v$ partitions $R(v)$
into $O(n)$ messages of size $\log{n}$ and sends the $i^{th}$ message to node
$i$. Then, node $i$ sends to each node $u\in h(v)$, the message which it got
from $v$. Notice that since each node $u$ is the helper of at most $O(1)$
other nodes, then every node $i$ needs to send to every other node $u$ at most
$O(1)$ messages. Therefore, this takes $O(1)$ rounds of communication since
$O(1)$ messages are sent on every communication link $i-u$.
Every node in $h(v)$ now knows all of $R(v)$, and can thus locally create the
messages $v$ desires to send. Node $v$ splits the messages it desires to send
into $h(v)$ sets $(M_{1}(v),\dots,M_{h(v)}(v))$, each of size at most $O(n)$,
and assigns each of the sets to one of its helper nodes. As the targets do not
change, each node is still the target of up to $O(n)$ messages. Further, every
helper node is now the source of up to $O(n)$ messages. Therefore, it is
possible to apply the routing scheme from [23] in order to have each helper
node route the messages which it is assigned. ∎
Notice that this lemma implies, in a straightforward manner, the following
Corollary 7 which we refer to extensively.
###### Corollary 7.
In the deterministic Congested Clique model, given that each node originally
begins with $O(n\log n)$ bits of input, and at most $O(1)$ rounds have passed
since the initiation of the algorithm, then any routing instance
$\mathcal{M}$, in which every node $v$ is the target of up to $O(n\cdot x)$
messages, can performed in $O(x)$ rounds.
While this is a weaker statement than that of Lemma 3, it is convenient to use
when showing constant-time algorithms in the Congested Clique model, as it
completely circumvents the need for a bound on the number of messages each
node desires to send.
## 3 Deterministic $O(1)$ Round Algorithm for +1 Girth Approximation in the
Congested Clique
In this section we prove Theorem 1: we construct a deterministic $O(1)$ round
algorithm for $+1$ girth approximation in the Congested Clique model. The
algorithm is composed of two phases, each a novel technique on its own, and
through their combination, we achieve the desired result. The first procedure
is based on a _subgraph enumeration_ approach and allows each node to learn
its $\lfloor\frac{g}{20}\rfloor$ hop-neighborhood in $O(1)$ rounds. Formally,
it is shown in the following Theorem 8.
###### Theorem 8.
Given a graph $G$, with $n$ nodes, and an unknown girth $g<\log n$, there
exists an $O(1)$ round algorithm in the Congested Clique model, which either
outputs $g$, or, ensures that every node knows its $\lfloor g/20\rfloor$ hop-
neighborhood.
The latter procedure is based on a _BFS-like_ approach and allows each node to
double the hop-distance of the neighborhood which it knows in $O(1)$ rounds,
_at least_ until the first cycle is encountered. This is stated formally in
Theorem 9
###### Theorem 9.
Let $G$ be a graph with $n$ nodes, an unknown girth $g<\log n$, and with
minimum degree $\delta\geq 2$. Assume that for a given integer parameter
$a>0$, every $v\in V$ knows the edges of $N_{a}(v)$, and that $N_{a}(v)$ is a
tree. There exists an algorithm which completes in $O(1)$ rounds of the
Congested Clique model, and either reports $g$ exactly, or, reaches one of the
following two states: (1) Every $v\in V$ knows $N_{2a}(v)$, _or_ (2) For some
value $b\geq\lceil\frac{g}{2}\rceil-1$ which is agreed upon by all nodes of
the network, every $v\in V$ knows all of $N_{b}(v)$. All nodes know whether
$g$ was reported exactly, and if not, which state was reached.
Thus, by invoking the first algorithm once, and then the latter for a constant
number of times, we achieve an $O(1)$ round algorithm for the approximation
problem. The reason we achieve a $+1$ approximation, and not an exact result,
is due to the fact that the second algorithm stops right before detecting the
shortest cycle in the graph and cannot differentiate whether it is of odd or
even length. In Section 3.2, we formally prove Theorem 1, when $g<\log n$ and
the minimum degree is $\delta\geq 2$, using Theorems 8, 9. The constraints
$g<\log n$ and $\delta\geq 2$ can be quickly overcome by eliminating some
trivial, degenerate cases, and it is shown how to remove these constrains in
Section 3.1. Finally, in Sections 3.3, 3.4, we proceed to our fundamental
technical contributions by showing the proofs of Theorems 8, 9.
### 3.1 Preliminary Preprocessing
Prior to the initiation of the main algorithm which achieves the $+1$
approximation for girth in the Congested Clique model, we perform some
preliminary steps in order to treat trivial or degenerate cases. Specifically,
we take care of the case when $g\geq\log n$, and ensure that the minimum
degree in the graph is $\delta\geq 2$, without changing the girth of the
graph.
We assume that the graph is simple, i.e., does not contain self-loops or
multiple edges. Notice that such cases are trivial.
###### Graphs With High Girth
We note that by [14, Theorem 4.1], if $g\geq\log{n}$ or $G$ has no cycles,
then $m=O(n)$. Thus, in this case, by using the routing algorithm of [23], all
the nodes in the graph can learn the entire graph in $O(1)$ rounds and output
$g$ using local computation. Therefore, for the remainder of the algorithm, we
may assume that $g<\log{n}$.
###### Degenerate Nodes
We remove all nodes that do not participate in any cycles. In particular,
after the removal of these nodes, no node $v$ with $d(v)<2$ remains. This
procedure can be seen a specific case of procedures used in [20, 1, 15].
###### Definition 1 ($1$-degenerate nodes).
A node is called $1$-degenerate if it is marked in the following process. Mark
all nodes with $d(v)<2$; remove all marked nodes; repeat as long as it is
possible to mark nodes.
###### Lemma 4.
A $1$-degenerate node does not participate in any cycle.
###### Proof.
For a cycle $C$, assume by contradiction there is such a node. Let $v$ be the
first node in $C$ removed by the process and let $t$ be the time at which it
is removed. Since no other node was removed prior to time $t$, then both
neighbors of $v$ in $C$ are still part of the graph at time $t$. Therefore,
$d(v)\geq 2$ at time $t$, contradicting the claim that it was removed at that
time. ∎
The network can detect all $1$-degenerate vertices in $G$ in $O(1)$ rounds in
the following manner. Each node $v$ broadcasts $(v.id,d(v),\bigoplus_{(u,v)\in
E}u.id)$, which are overall $O(\log{n})$ bits. Each node locally and
iteratively does the following process until there are no nodes of degree $1$:
_If there is a node $v$ of degree $1$ in the graph, $\bigoplus_{(u,v)\in
E}u.id$ is just the ID of its only neighbor. For that $u$, decrease the degree
of $u$ by $1$ and XOR the third field of $u$ with $v.id$. Remove $v$ from the
graph._
### 3.2 Proving Theorem 1
Here, we prove Theorem 1, in case that $g<\log n$ and the minimum degree is
$\delta\geq 2$.
* Proof of Theorem 1:
We first invoke the algorithm from Theorem 8, in $O(1)$ rounds. Either $g$ was
outputted, or, every node learned its $\lfloor g/20\rfloor$ hop-neighborhood.
Next, we invoke the algorithm from Theorem 9. The nodes now learned new,
larger neighborhoods - regardless of whether the algorithm halted in State 1
(every $v\in V$ knows $N_{2a}(v)$) or State 2 (for some
$b\geq\lceil\frac{g}{2}\rceil-1$, every $v\in V$ knows all of $N_{b}(v)$). If
any node sees a cycle, then it broadcasts the length of the shortest cycle
which it sees and all the nodes terminate and output the minimum of the values
which were broadcast in the network. It is clear that, in this case, the exact
value of $g$ is outputted, since all nodes know the neighborhoods surrounding
them of same radius, and thus if any node saw a cycle, one node must have seen
the shortest cycle in the graph.
Finally, in the case that no cycle was seen so far, we differentiate between
the states at which the algorithm from Theorem 9 can halt at. If it halts at
State 1, then every node learned twice the radius of the neighborhood it
already knew. In such a case, we invoke Theorem 9 again and repeat. Notice
that we can do this at most $O(1)$ times, before either seeing a cycle or
halting at State 2, due to the fact that the nodes originally know their
$\lfloor g/20\rfloor$ hop-neighborhoods. In the case that we eventually halt
at State 2, and no cycles were seen by any node so far, all the nodes output
that the girth is either $2b+1$ or $2b+2$, where $b$ is the radius of the
neighborhoods which they learned. It is clear that if all nodes learned their
$b\geq\lceil\frac{g}{2}\rceil-1$ hop-neighborhoods, and none saw a cycle, then
it must be that $b=\lceil\frac{g}{2}\rceil-1$ and thus either $g=2b+1$ or
$g=2b+2$. ∎
### 3.3 Phase I: Initial Neighborhood Learning
The key procedure of this phase (formally stated above as Theorem 8) consists
of two major steps. In the first step, either each path of length
$\lfloor\frac{g}{10}\rfloor$ in $G$ is detected by at least one node, or $g$
is outputted. This step can be seen as an edge-partition variant of the
listing algorithm in [9]. The second step uses a load-balancing routine in
order to redistribute the information computed in the first step so that each
node $v$ learn its $\lfloor\frac{g}{20}\rfloor$ hop-neighborhood.
###### Step 1: Path Listing.
We next list all paths of length $\lfloor\frac{g}{10}\rfloor$, or output $g$,
in $O(1)$ rounds.
First, each node sends its degree to the rest of the network, and each then
locally calculates the number of edges in the graph $m=\sum_{v}deg(v)/2$. Let
$k\in\mathbb{N}$ be the largest integer such that $m\leq n^{1+1/k}+n$. Then,
each node $v$ is assigned a hard-coded range of $deg(v)$ indices in
$\left\\{1,\dots,m\right\\}$, and locally numbers its edges using these
indices.
If $k\leq 4$, then by Lemma 2 the girth is of size at most $10$, and thus,
trivially, paths of length $\lfloor\frac{g}{10}\rfloor\leq 1$ are known and we
can halt. Thus, from here on, we may assume that $k\geq 5$.
Let $P$ be a partition of the set $\\{1,\dots,m\\}$ into $\lceil
kn^{2/k}/(20e)\rceil$ consecutive segments of size $O\left(\frac{m}{\lceil
kn^{2/k}/(20e)\rceil}+1\right)$, and let $K$ be a family containing all the
possible choices of $\lfloor k/4\rfloor$ segments from $P$ (in this context,
$e$ denotes the mathematical constant). It holds that
$|K|={\lceil kn^{2/k}/(20e)\rceil\choose\lfloor
k/4\rfloor}\leq\left(\frac{e\lceil kn^{2/k}/(20e)\rceil}{\lfloor
k/4\rfloor}\right)^{\lfloor
k/4\rfloor}\leq\left(\frac{n^{2/k}}{2}+1\right)^{\lfloor k/4\rfloor}\leq
n^{\frac{2}{k}\lfloor k/4\rfloor}\leq n,$
where the first inequality holds due to the well-known combinatorial statement
that ${n\choose k}\leq(\frac{ne}{k})^{k}$, for all $n\in\mathbb{N},1\leq k\leq
n$, and in the other inequalities, the fact that $5\leq k<\log n$ is used.
Thus, it is possible to associate each $k_{i}\in K$ with a unique node
$v_{i}$. Each $k_{i}$ is a set of $\lfloor k/4\rfloor$ sets of
$O\left(\frac{m}{\lceil kn^{2/k}/(20e)\rceil}+1\right)$ edges, and so let
$E_{i}$ denote the edges in the sets contained in $k_{i}$. Notice that
$|E_{i}|\leq\lfloor k/4\rfloor\left(\frac{m}{\lceil
kn^{2/k}/(20e)\rceil}+1\right)\leq(k/4)\frac{20en^{1+1/k}}{kn^{2/k}}+k=5en^{1-1/k}+k\leq
n,$
and therefore, by Corollary 7, it is possible for each $v_{i}$ to learn all of
the edges in $E_{i}$ in $O(1)$ rounds of communication.
Finally, every node $v_{i}$ broadcasts the shortest cycle which it witnesses
in $E_{i}$. Notice that every path, $p$, of length at most $\lfloor
k/4\rfloor$, is fully contained inside some $E_{j}$, due to the construction
of $P$, and therefore the corresponding node, $v_{j}$, which now knows all of
$E_{j}$, will witness $p$. Thus, if $g\leq\lfloor k/4\rfloor$, some node will
witness the shortest cycle in the graph and be able to broadcast its length,
$g$. Otherwise, notice that since $m\not\leq n^{1+1/(k+1)}+n$, Lemma 2 implies
that the graph is not $C_{i}$-free for all $i\leq 2(k+1)$, and thus $g\leq
2(k+1)$. Thus all paths of length at most $\lfloor k/4\rfloor\geq\lfloor
g/8-1/4\rfloor\geq\lfloor g/10\rfloor$ have been listed. Notice that
$g/8-1/4\geq g/10$ whenever $g\geq 10$, and this can be assumed, since
otherwise, trivially, paths of length $\lfloor\frac{g}{10}\rfloor<1$ are
known.
If at least one node $v_{i}$ informs about a cycle in $E_{i}$, the minimum
number sent by a node is outputted as $g$, and the algorithm terminates.
Otherwise, it proceeds to the second step.
###### Step 2: Neighborhood Learning.
We desire to redistribute some of the information learned in the previous step
so that each node will know its $\lfloor\frac{g}{20}\rfloor$ hop-neighborhood.
Notice that all paths of length at most $\lfloor g/10\rfloor$ have been
listed. Therefore, also all paths of length at most
$\lfloor\frac{g}{20}\rfloor$ have been listed. We strive to redistribute this
information so that each node $v$ knows all paths of length at most
$\lfloor\frac{g}{20}\rfloor$ which start at $v$, and thus $v$ knows its entire
$\lfloor\frac{g}{20}\rfloor$ hop-neighborhood. Notice that we would like for
each $v$ to know both the nodes in its $\lfloor\frac{g}{20}\rfloor$ hop-
neighborhood, as well as the edges between them.
We begin by ensuring that each $v$ knows every node $u$ in its
$\lfloor\frac{g}{20}\rfloor$ hop-neighborhood. Let $v\in V$ and $u$ be some
node in its $\lfloor\frac{g}{20}\rfloor$ hop-neighborhood. Since
$\lfloor\frac{g}{20}\rfloor<g/2$, then the $\lfloor\frac{g}{20}\rfloor$ hop-
neighborhood of $v$ is a tree. Therefore, there exists exactly one path,
$p_{v,u}$, of length at most $\lfloor\frac{g}{20}\rfloor$ between $v$ and $u$.
In the previous step, we ensured that at least one node $w$ is aware of
$p_{v,u}$. Specifically, notice that it might be the case that many nodes know
about $p_{v,u}$, due to the last step, yet, every node $w$ which knows of this
path also knows all the other nodes $w^{\prime}$ which learned this path
through their $E_{w^{\prime}}$. Thus, it is possible to choose, in a hard-
coded manner, a single node $w$ which will be responsible for informing $v$
that $p_{v,u}$ exists. Having done that, node $w$ desires to convey to $v$ the
message that $u$ is in its $\lfloor\frac{g}{20}\rfloor$ hop-neighborhood, in
addition to the hop-distance between $v$ and $u$ — that is, the length of
$p_{v,u}$. Notice that for each such $u$, node $v$ is destined to receive
exactly one message, and therefore every node in the graph is the target of
$O(n)$ messages. This shows that Corollary 7 may be invoked in order to
deliver all these messages in $O(1)$ rounds.
Now, we desire to inform every $v$ of the edges in its
$\lfloor\frac{g}{20}\rfloor$ hop-neighborhood. Node $v$ now knows all the
nodes $u$ in this neighborhood, as well as the hop-distance to each of them.
Node $v$ sends a message to each such $u$ which is at most
$\lfloor\frac{g}{20}\rfloor-1$ hops away from it, and requests that $u$ send
to $v$ _all_ its incident edges in the graph. Notice that all these edges are
exactly all the edges contained in the $\lfloor\frac{g}{20}\rfloor$ hop-
neighborhood of $v$, and since this neighborhood is a tree, $v$ is the target
of at most $O(n)$ messages. As before, this shows that Corollary 7 may be
invoked in order to deliver all these messages in $O(1)$ rounds.
### 3.4 Phase II: Neighborhood Doubling
The key procedure in this phase (formally stated above as Theorem 9) is an
$O(1)$ round algorithm which doubles the radius of the hop-neighborhood known
to each node, until the nodes know a neighborhood large enough in order to
approximate the girth up to an additive value of 1. The algorithm works along
the following lines. Denote by $F_{a}(v)$, the nodes which are exactly at
distance $a$ from $v$ — we refer to these as the _front-line_ nodes. Each
nodes $v$ initially knows $N_{a}(v)$, and at once attempt to learn all of
$\bigcup_{u\in F_{a}(v)}(N_{a}(u)\setminus N_{a}(v))$, in an efficient manner.
If this step succeeds, then all the nodes reach State 1, and halt. Otherwise,
they coordinate to increase the radii of the neighborhoods which they know by
as much as possible in $O(1)$ rounds, and ultimately arrive at State 2, and
halt.
###### Halting at State 1.
Let $v\in V$ and $u\in F_{a}(v)$. Node $u$ aims to send to node $v$ the edges
in $N_{a}(u)\setminus N_{a}(v)$. Notice that node $u$ can locally compute
these edges as follows. It observes the first node $w$ on the path between
$v,u$. Since $N_{a}(u)$ is a tree, for every node $w^{\prime}\in N_{a}(u)$,
there is exactly one simple path, $p_{u,w^{\prime}}$, which $u$ sees to
$w^{\prime}$. Notice that $w^{\prime}\in N_{a}(v)$ if and only if
$p_{u,w^{\prime}}$ passes through $w$. Thus, node $u$ knows exactly which
edges it desires to send to node $v$. However, before sending them, it first
sends to node $v$ the value $|N_{a}(u)\setminus N_{a}(v)|$.
We now shift back to the perspective of node $v$. It computes and broadcasts
an upper bound on $|\bigcup_{u\in F_{a}(v)}(N_{a}(u)\setminus N_{a}(v))|$, by
calculating $\sum_{u\in F_{a}(v)}|N_{a}(u)\setminus N_{a}(v)|$. If all nodes
broadcast values which are at most $n-1$, then by Corollary 7, it is possible
in $O(1)$ rounds to perform all the routing requests and have each node double
the radius of the neighborhood which it knows. At this point, the nodes
collectively reach State 1 and halt.
Otherwise, at least one node reported a value greater than or equal to $n$.
This implies that for some node $v$, there is a cycle in $N_{2a}(v)$, since at
least two nodes $u,u^{\prime}\in F_{a}(v)$ have simple paths in their $a$ hop-
neighborhoods to the same node $w$. In this case, the nodes proceed to a
second part of the algorithm, which eventually leads to halting at State 2.
###### Halting at State 2.
Our goal, at this stage, is to determine the largest possible value
$i^{\prime}\in\\{1,\dots,a-1\\}$, such that for every node $v$, $\sum_{u\in
F_{a}(v)}|N_{i^{\prime}}(u)\setminus N_{a}(v)|<n$. Once this is achieved, then
the algorithm can complete in a similar manner to that above. To see this,
assume that we have this maximal value $i^{\prime}$. Therefore, all nodes $v$
can learn $N_{a+i^{\prime}}(v)$ in $O(1)$ rounds, similarly to above. If any
cycle is seen, then $g$ is outputted and the algorithm halts. Otherwise, due
to the definition of $i^{\prime}$, there must exist some node $v^{\prime}$
such that $\sum_{u\in F_{a}(v^{\prime})}|N_{i^{\prime}+1}(u)\setminus
N_{a}(v^{\prime})|\geq n$. This implies that there is a cycle in
$N_{a+i^{\prime}+1}(v^{\prime})$, and therefore $2a+2i^{\prime}<g\leq
2a+2i^{\prime}+2$. As such,
$a+i^{\prime}=(2a^{\prime}+2i^{\prime}+2)/2-1\geq\lceil g/2\rceil-1$, and we
may halt at State 2.
We now show how to find $i^{\prime}$. This is trivially possible to accomplish
in $O(a)$ rounds — each node $u$ simply sends to $v$ the values
$\\{|N_{1}(u)\setminus N_{a}(v)|,\dots,|N_{a-1}(u)\setminus N_{a}(v)|\\}$,
node $v$ locally computes the $a-1$ different sums, and broadcasts them.
However, this does not suffice for our goal of an $O(1)$ round algorithm, as
$a$ can be logarithmic in $n$. Instead, let every node $v$ broadcast
$|F_{a}(v)|$, and denote by $v^{\prime}$ the node with maximal
$|F_{a}(v^{\prime})|$, and write $d=\left\lfloor
n/|F_{a}(v^{\prime})|\right\rfloor$. For every node $v$ and $u\in F_{a}(v)$,
node $u$ sends to $v$ the values $\\{|N_{1}(u)\setminus
N_{a}(v)|,\dots,|N_{d}(u)\setminus N_{a}(v)|\\}$, node $v$ computes the $d$
different sums of these values from all $u\in F_{a}(v)$, and broadcast them.
Notice that this takes $O(1)$ rounds, using Corollary 7 as each node wants to
receive at most $O(n)$ messages. Notice that it is now possible in $O(1)$
rounds to compute $\min\\{i^{\prime},d\\}$ — either $i^{\prime}\leq d$, and
thus $\min\\{i^{\prime},d\\}=i^{\prime}$ and we can compute it, or,
$\min\\{i^{\prime},d\\}=d$. If we show that $g\leq 2a+2d$, then if all $v$
learn $N_{a+\min\\{i^{\prime},d\\}}(v)$, this would suffice in order to either
find the exact girth or halt at State 2, as required.
We claim that $g\leq 2a+2d$. To see this, assume that $g>2a+2d$. Since
$g>2a+2d$, there are no cycles in $N_{a+d}(v^{\prime})$. Combining this with
the fact that we assume the minimal degree in $G$ to be at least 2, we can see
that for all $j\neq j^{\prime}\in\left\\{1,\dots,d\right\\}$, it holds that
$|F_{a+j}(v^{\prime})|\geq|F_{a+j-1}(v^{\prime})|$, and
$F_{a+j}(v^{\prime})\cap F_{a+j^{\prime}}(v^{\prime})=\emptyset$. Thus, in
$N_{a+d}(v^{\prime})$ there are at least
$\left(d+1\right)\cdot|F_{a}(v^{\prime})|=\left(\left\lfloor
n/|F_{a}(v^{\prime})|\right\rfloor+1\right)\cdot|F_{a}(v^{\prime})|>n$ nodes,
a clear impossibility. As we have arrived at a contradiction, we get that
$g\leq 2a+2d$, as required.
## 4 Subgraph listing in the Congested Clique model
We show an efficient “sparsity-aware” algorithm to _list_ subgraphs in the
Congested Clique model. Our main result is the following theorem, which is
proven in the following sections.
* Theorem 2.
Given a graph $G$ with $n$ nodes and $m$ edges and a graph $H$ with $p\leq\log
n$ nodes and $k$ edges, let $\tilde{m}=\max\\{m,n^{1+1/p}\\}$. There exists a
deterministic Congested Clique algorithm that terminates in
$O(\frac{k\tilde{m}}{n^{1+2/p}}+p)$ rounds and lists all instances of $H$ in
$G$.
As mentioned in the introduction, we can combine this result with known bounds
on the number of edges in graphs without specific subgraphs, to achieve fast
subgraph _detection_ results. First, by combining Theorem 2 with Lemma 1, we
immediately get Corollary 3: If the graph contains more than $17kn^{1+1/k}$
edges (which can be checked in a single round), then by Lemma 1 we can safely
output that there must exist a cycle of length $2k$. Otherwise, plugging
$p=2k$ and $m=17kn^{1+1/k}$ in Theorem 2 gives that we can detect (and even
list, in this case) the existence of a cycle of length $2k$ within $O(k^{2})$
rounds. Next, by combining Theorem 2 with Lemma 2, we can get the following
result:
###### Proposition 1.
Given a graph $G$ with $n$ nodes, $m$ edges and an unknown girth $g$ such that
$g>\ell$ for some known $\ell$, and defining
$f(x)=n^{1/\lfloor(x-1)/2\rfloor-2/(2\cdot\lfloor(x-1)/2\rfloor+1)}$, there is
a deterministic
$\tilde{O}(\min\\{f(g),f(2\cdot\lfloor(\ell+1)/2\rfloor+1)\\})$ round
algorithm in the Congested Clique model which outputs g.
Proposition 1 first shows that the exact girth can be computed in
$\tilde{O}(f(g))$ rounds — a polynomial improvement over the state-of-the-art
for all graphs with $g\geq 5$. Moreover, if it is known that the graph has
girth greater than $\ell$,111It is possible to phrase a slightly stronger
result which does not require a lower bound on the girth, but rather that for
a specific $k$, which depends on the sparsity of the graph, there will not be
any cycles of length $2k+1$. then the round complexity is additionally
guaranteed to be $\tilde{O}(f(2\cdot\lfloor(\ell+1)/2\rfloor+1))$. For
instance, for any odd value $\ell=2r-1$ we get the upper bound
$\tilde{O}(n^{1/r-2/(2r+1)})$. Taking $r=2$ gives Corollary 4 stated in the
introduction, which improves upon the state-of-the-art for triangle free
graphs.
We note that more claims can be shown using bounds for the Túran numbers of
various other graphs — for example, for detection of $K_{s,t}$ (complete
bipartite graph with $s$ nodes on one side and $t$ on the other) for certain
values of $s,t$.
* Proof of Proposition 1:
Let $k^{\prime}$ be the largest integer such that $m\leq
n^{1+1/k^{\prime}}+n$. If $k^{\prime}\geq(\log n)/2$, then $m=O(n)$ and thus
the entire graph can be learned by one node in $O(1)$ rounds, completing the
proof.
Otherwise, it is known that a cycle of length at most $2k^{\prime}+2$ exists
in the graph, due to Lemma 2 and $m>n^{1+1/(k^{\prime}+1)}+n$ due to the
definition of $k^{\prime}$. Notice that since $m\leq n^{1+1/k^{\prime}}+n$,
then for each $p\leq 2k^{\prime}$, it is possible to list all $C_{p}$ in the
graph in $O(k^{\prime})$ rounds using Theorem 2. Therefore, since
$k^{\prime}<(\log n)/2$, it is possible in $\tilde{O}(1)$ rounds to list all
cycles of length up to $2k^{\prime}$. If a cycle is witnessed at this stage,
then the nodes know the exact girth of the graph and halt.
We arrive at the last case, which is determining whether a cycle of length
$2k^{\prime}+1$ exists. We invoke Theorem 2 to list all $C_{2k^{\prime}+1}$,
which takes $\tilde{O}(n^{1/k^{\prime}-2/(2k^{\prime}+1)})$ rounds, and allows
the nodes to determine the exact girth of the graph. Notice that the girth is
either $2k^{\prime}+1$ or $2k^{\prime}+2$, and thus
$f(g)=f(2k^{\prime}+1)=f(2k^{\prime}+2)=1/k^{\prime}-2/(2k^{\prime}+1)$. The
overall round complexity of the algorithm is thus $\tilde{O}(f(g))$.
We now consider the case where we additionally know that $g>\ell$, and derive
another bound on the complexity that depends only on $\ell$. If $\ell$ is even
then we simply run the above algorithm; the complexity is
$\tilde{O}(f(2\cdot\lfloor(\ell+1)/2\rfloor+1))$ since
$g\geq\ell+1=2\cdot\lfloor(\ell+1)/2\rfloor+1$. Now assume that $\ell$ is odd.
In that case we first check if the graph is $C_{\ell+1}$-free in
$\tilde{O}(1)$ rounds using the algorithm of Corollary 3. If the graph is not
$C_{\ell+1}$-free, then we know that $g=\ell+1$. Otherwise we know that
$g\geq\ell+2$ and we run the above algorithm; the complexity is again
$\tilde{O}(f(2\cdot\lfloor(\ell+1)/2\rfloor+1))$ since
$g\geq\ell+2=2\cdot\lfloor(\ell+1)/2\rfloor+1$. ∎
### 4.1 Partition trees
We introduce the notion of _partition trees_ , as a fundamental tool for
subgraph listing in the _deterministic_ Congested Clique model. Partition
trees are a deterministic load-balancing mechanism that evenly divides the
work of checking whether any copies of a subgraph are present. In prior work,
randomized load-balancing was used for this purpose, but this incurs
logarithmic factors which we cannot tolerate here. Throughout this section,
given a subgraph with $p$ nodes, we frequently refer to the value $x=n^{1/p}$.
We assume that $x$ is an integer, because $p\leq\log n$ implies $x\geq 2$, and
so it is possible to round $x$ to an integer without affecting the round
complexity or correctness.
We start with Definition 2, which defines a $p$-partition tree, which is a
tree structure in which every node represents a partition of the graph $G$.
Then, in Definition 3, we define an $H$-partition tree, in which we require
certain conditions on the number of edges between parts of a $p$-partition,
based on the subgraph $H$ of $p$ nodes which we will want to list.
###### Definition 2 ($p$-partition tree, Figure 1).
Let $G=(V,E)$ be a graph with $n$ nodes and $m$ edges, and let $p\leq\log n$.
A _$p$ -partition tree_ $T=T_{G,p}$ is a tree of $p$ layers (depth $p-1$),
where each non-leaf node has at most $x=n^{1/p}$ children. Each node in the
tree is associated with a partition of $V$ consisting of at most $x$ parts.
We inductively denote all partitions associated with nodes in $T$ as follows.
The partition associated with the root $r$ of $T$ is called the _root
partition_ , and is denoted by $P_{\emptyset}$. Given a node with a partition
denoted by $P_{(\ell_{1},\dots,\ell_{i-1})}$, the partition associated with
its $j$th child, for $0\leq j\leq x-1$, is denoted
$P_{(\ell_{1},\dots,\ell_{i-1},j)}$.
The at most $x$ parts of each partition $P_{(\ell_{1},\dots,\ell_{i})}$ are
denoted by $U_{(\ell_{1},\dots,\ell_{i}),j}$, for $0\leq j\leq x-1$. For each
$0\leq j\leq x-1$, the part $U_{(\ell_{1},\dots,\ell_{i-1}),\ell_{i}}$ is
called the _parent_ of the part $U_{(\ell_{1},\dots,\ell_{i-1},\ell_{i}),j}$,
also denoted as
$U_{(\ell_{1},\dots,\ell_{i-1}),\ell_{i}}=\texttt{parent}(U_{(\ell_{1},\dots,\ell_{i-1},\ell_{i}),j})$.
Figure 1: A partial illustration of a partition tree with $p,x=3$.
###### Definition 3 ($H$-partition tree).
Let $G=(V,E)$ be a graph with $n$ nodes and $m$ edges, and let $H$ be a graph
with $p\leq\log n$ nodes, $\\{z_{0},\dots,z_{p-1}\\}$, and denote
$d_{i}=|\\{\\{z_{i},z_{t}\\}\in E_{H}\mid t<i\\}|$ for each $0\leq i\leq p-1$,
$x=n^{1/p}$ and $\tilde{m}=\max\\{m,nx\\}$. A _$H$ -partition tree_
$T=T_{G,H}$ is a $p$-partition tree with the following additional constraints,
for some constants $c_{1},c_{2}$. .
1. 1.
for every part $U=U_{(\ell_{1},\dots,\ell_{i-1},\ell_{i}),j}$, it holds that
$|E(U,V)|\leq c_{1}m/x+n$, and
2. 2.
for every part $U_{i}=U_{(\ell_{1},\dots,\ell_{i-1},\ell_{i}),j}$, and all of
its ancestor parts $U_{t}=\texttt{parent}(U_{t+1})$ for $t=i-1,\dots 0$, it
holds that $\sum_{t<i,\\{z_{i},z_{t}\\}\in E_{H}}{|E(U,U_{t})|}\leq
c_{2}d_{i}\tilde{m}/x^{2}+n$,
Notice that in Definition 3, we define $\tilde{m}$ as an upper bound on $m$,
the number of edges in the input graph. This is done as if the graph is _too_
sparse, we use a slightly higher bound on the number of edges in order to make
decisions regarding the constraints on the partitions. We note that
$\tilde{m}$ is purely a technicality — we do not _require_ that there be at
least this many edges in the graph. In the following two theorems: we show
that we can construct an $H$-partition tree and use it to efficiently perform
$H$-listing.
###### Theorem 10.
Let $G=(V,E)$ be a graph with $n$ nodes, and let $H$ be a graph with
$p\leq\log n$ nodes. There exists a deterministic Congested Clique algorithm
that completes in $O(1)$ rounds and constructs an $H$-partition tree $T$, such
that $T$ is known to all nodes of $G$ — that is, all nodes know all the
partitions making up $T$.
And second, that given an $H$-partition tree, we can list all instances of $H$
in $G$.
###### Theorem 11.
Let $G=(V,E)$ be a graph with $n$ nodes, let $H$ be a graph with $p\leq\log n$
nodes and $k$ edges, and denote $x=n^{1/p}$ and $\tilde{m}=\max\\{m,nx\\}$.
There exists a deterministic Congested Clique algorithm that completes in
$O(\frac{k\tilde{m}}{n^{1+2/p}}+p)$ rounds and lists all instances of $H$ in
$G$, given an $H$-partition tree $T$ that is known to all nodes.
Thus, Theorems 10 and 11, directly imply Theorem 2.
* Proof of Theorem 10:
In order to show this proof, we construct a set of preliminary partitions in
$O(1)$ rounds, and maintain that it is possible to construct the entire
partition tree using only this set of partitions. By ensuring that these
partitions are globally known, each node can compute the entire tree locally.
###### Constructing a preliminary set of partitions.
We construct a main partition, $R$, with at most $x/2$ parts, and then several
more partitions, _of the entire graph_ , which are refinements of $R$.
Specifically, for every set of $1\leq\ell\leq p-1$ parts, denoted
$\\{Q_{j_{0}},\dots,Q_{j_{\ell-1}}\\}$, from $R$, we create a specific
partition denoted as $M_{\\{j_{0},\dots,j_{\ell-1}\\}}$, which has at most $x$
parts $N_{\\{j_{0},\dots,j_{\ell-1}\\},k}$ for $0\leq k\leq x-1$. Notice that
this is a total of at most $(x/2+1)^{p-1}\leq x^{p-1}=n/x$ different
partitions. We emphasize that each $M_{\\{j_{0},\dots,j_{\ell-1}\\}}$ is a
_partition of the entire graph_ , and not of
$\\{Q_{j_{0}},\dots,Q_{j_{\ell-1}}\\}$.
For each partition, we consider a set of $x$ nodes that are called _the
builder nodes_. We assign some $x$ nodes to build the main partition, denoted
by $B_{\emptyset}$, and then we assign sets of builder nodes to each
additional partition in a mutually disjoint manner. That is, denoting by
$B_{\\{j_{0},\dots,j_{\ell-1}\\}}$ the set of builder nodes for a partition
$M_{\\{j_{0},\dots,j_{\ell-1}\\}}$, gives that
$B_{\\{j_{0},\dots,j_{\ell-1}\\}}\cap
B_{\\{j^{\prime}_{0},\dots,j^{\prime}_{\ell-1}\\}}=\emptyset$ for every two
such additional partitions. Due to the fact that there are at most $n/x$
additional partitions, it is clear that this assignment is possible. The
builder nodes $B_{\emptyset}$ initially construct the main partition in $O(1)$
rounds, and then the additional partitions are constructed concurrently by
their respective builder nodes in $O(1)$ rounds.
For the main partition $R$, the only condition that we maintain is Condition
1, which requires that each of its parts $Q$ satisfies $|E(Q,V)|\leq
c_{1}m/x+n$. To ensure this, each node $v$ sends its degree to all builder
nodes in $B_{\emptyset}$. Then, the builder nodes go over the nodes in an
arbitrary order (known to all nodes) and add them to parts of the (initially
empty) partition, as follows. The first processed node $v$ is added to a part
$Q_{0}$, and a counter is set to $deg(v)$. Then, every following node $v$ is
added to $Q_{0}$ and the counter is increased by $deg(v)$, as long as it does
not exceed $c_{1}m/x+n$. Once adding $v$ to a part would make the counter
exceed the threshold, the next part $Q_{1}$ is started, initialized to contain
$v$ and its counter is $deg(v)$. We continue in this manner until all nodes
are processed.
Notice that this creates at most $x/2$ parts in the partition by choosing
$c_{1}\geq 4$, since each part has at least $c_{1}m/x$ edges out of $2m$
(counting each edge twice for both of its endpoints). Finally, note that the
builder nodes in $B_{\emptyset}$ can inform all other nodes about the
partition $R$ within $O(1)$ rounds, since there are at most $x/2$ parts that
can each be described by their first and last nodes in the globally known
order, and the description of each part can be broadcast to all nodes by a
different builder node in $B_{\emptyset}$.
When constructing the partition $M_{\\{j_{0},\dots,j_{\ell-1}\\}}$, we
maintain three conditions. Primarily, we maintain Condition 1; that is, for
every part $N=N_{\\{j_{0},\dots,j_{\ell-1}\\},k}$ it holds that $|E(N,V)|\leq
c_{1}m/x+n$. Furthermore, similarly to Condition 2, we ensure that for each
part, $N=N_{\\{j_{0},\dots,j_{\ell-1}\\},k}$, $\sum_{0\leq
i<\ell}{|E(N,Q_{j_{i}})|}\leq c_{2}\ell\tilde{m}/x^{2}+n$. Lastly, we ensure
that $M_{\\{j_{0},\dots,j_{\ell-1}\\}}$ is a refinement of $R$.
Each $M=M_{\\{j_{0},\dots,j_{\ell-1}\\}}$ is constructed in a similar manner
to the way in which $R$ was constructed - every node $v$ in the graph sends
some $O(1)$ messages to each builder node in
$B=B_{\\{j_{0},\dots,j_{\ell-1}\\}}$, the builder nodes locally compute the
partition $M$, and then each builder node broadcasts to the entire graph some
part of $M$ in $O(1)$ rounds. To begin construction of $M$, every node $v$
sends the values $deg(v)$, $\sum_{0\leq t<\ell}{deg_{j_{t}}(v)}$ to all nodes
in $B$, where $deg_{j_{t}}(v)$ is the number of neighbors of $v$ in
$Q_{j_{t}}$. Similarly to the construction of the root partition, the builder
nodes in $B$ go over the nodes in a known order and add them one by one to
parts of the (initially empty) partition. In order to promise Condition 1,
that $|E(N,V)|\leq c_{1}m/x+n$ for every part $N$ that is constructed, a
counter is maintained that accumulates the degrees $deg(v)$ of every node $v$
that is added to the current part. If adding a node $v$ would make this
counter exceed the threshold, then a new part is started. To promise
$\sum_{0\leq i<\ell}{|E(N,Q_{j_{i}})|}\leq c_{2}\ell\tilde{m}/x^{2}+n$, for
each part that is being constructed, a second counter is maintained. This
counter accumulates $\sum_{0\leq t<\ell}{deg_{j_{t}}(v)}$ , for every $v$ that
is added to the current part. If adding a node $v$ would make this counter
exceed the threshold $c_{2}\ell\tilde{m}/x^{2}+n$, then a new part is started.
Once a new part is started because adding a node $v$ would make one of the
counters of the previous part exceed its threshold, the new part is
initialized to contain $v$, and its counters are initialized to $deg(v)$ and
$\sum_{0\leq t<\ell}{deg_{j_{t}}(v)}$ , respectively. We continue in this
manner until all nodes are processed. Finally, in order to ensure $M$ is a
refinement of $R$, we split every part in $M$ to parts completely contained in
parts of $R$.
We claim that this creates at most $x$ parts in each $M$ by choosing $c_{1}=8$
and $c_{2}=32$. Starting a new part can only happen due to one of the two
counters exceeding its threshold, or due to a split of a part in order to
ensure $M$ is a refinement of $R$. We first bound the number of parts created
only according to the counters, and then proceed to the parts added due to
splitting the parts of $M$ according to the parts of $R$. For the first
counter, as in the analysis of the root partition, exceeding the threshold
means that the part already has at least $c_{1}m/x$ edges that touch it out of
$2m$ possible edges counted for both endpoints. Therefore the first counter
can exceed the threshold no more than $2x/c_{1}$ times. For the second counter
to exceed its threshold, we have that the part already contains
$c_{2}\ell\tilde{m}/x^{2}$ edges to the relevant parts in $R$. Each of the
corresponding parts in $R$ satisfies Condition 1 — has at most $c_{1}m/x+n$
edges touching it altogether — and so in total there are at most $c_{1}\ell
m/x+\ell n$ edges touching the corresponding parts in $R$. We thus claim the
second counter can exceed its threshold no more than $c_{1}x/c_{2}$ times. To
see why, note that the second counter can exceed its threshold at most a
number of times which is $\frac{c_{1}\ell m/x+\ell
n}{c_{2}\ell\tilde{m}/x^{2}}\leq\frac{2c_{1}\ell\tilde{m}/x}{c_{2}\ell\tilde{m}/x^{2}}=2c_{1}x/c_{2}$.
The final condition, that $M$ is a refinement of $R$, can add to $M$ at most
the number of parts in $R$ \- that is, at most $2x/c_{1}$ additional parts. To
see this, notice that since all the partitions are created by going over all
the nodes in the graph in some predetermined order and creating a new part
once some counter has exceeded its threshold, then each part in $R$ can only
incur a single additional point in time at which the builder nodes have to
start a new part in $M$. Therefore, when setting $c_{1}=8,c_{2}=32$, in total
each $M$ has at most $4x/c_{1}+c_{1}x/c_{2}\leq 3x/4\leq x$ parts.
Finally, as each $M_{\\{j_{0},\dots,j_{\ell-1}\\}}$ has at most $x$ parts, the
builder nodes $B_{\\{j_{0},\dots,j_{\ell-1}\\}}$ can ensure that
$M_{\\{j_{0},\dots,j_{\ell-1}\\}}$ is globally known in $O(1)$ rounds.
###### Locally constructing the entire tree.
We now show that using the preliminary set of partitions, each node can
locally construct the entire partition tree.
We set the root partition as $P_{\emptyset}=R$, and proceed to setting the
remaining layers of the tree. We begin by setting the first layer below the
root partition. Each partition in this layer needs to maintain Condition 2
with respect to at most one part in $P_{\emptyset}$. Assume we need to
construct $P_{(j)}$ for some $0\leq j\leq x-1$. We need to ensure that all of
the parts in $P_{(j)}$ have a bounded number of edges entering part
$U_{\emptyset,j}$. Thus, $M_{\\{j\\}}$ certainly maintains all the required
conditions and we can set $P_{(j)}=M_{\\{j\\}}$. Next, we attempt to build the
$i^{th}$ layer below the root partition. In this layer, every partition
created, $P_{(j_{0},\dots,j_{i-1})}$, has to maintain Condition 2 of Theorem
10 with respect to some subset of the parts
$\\{U_{(j_{0},\dots,j_{k-1}),j_{k}}|0\leq k<i\\}$. However, since every
$M_{\\{j_{0},\dots,j_{\ell-1}\\}}$ is a refinement of $P_{\emptyset}$, then
each part in $\\{U_{(j_{0},\dots,j_{k-1}),j_{k}}|0\leq k<i\\}$ can be replaced
by some part in $P_{\emptyset}$ which contains it, and thus if
$P_{(j_{0},\dots,j_{i-1})}$ maintains the required conditions w.r.t. a
specific set of at most $i$ parts of $P_{\emptyset}$, then it would also
maintain them w.r.t. $\\{U_{(j_{0},\dots,j_{k-1}),j_{k}}|0\leq k<i\\}$. We
have already computed partitions which maintain all the required conditions
with respect to any set of at most $p-1$ parts in $P_{\emptyset}$, and thus
there exists a partition which we already computed in our preliminary set of
partitions which can be used as $P_{(j_{0},\dots,j_{i-1})}$. ∎
* Proof of Theorem 11:
Denote by $\\{z_{0},\dots,z_{p-1}\\}$ the nodes of $H$, and denote
$d_{i}=|\\{\\{z_{i},z_{t}\\}\in E_{H}\mid t<i\\}|$ for each $0\leq i\leq p-1$.
We assign each leaf of the $H$-partition tree $T$ to $x$ different nodes. Note
that there are $x^{p-1}$ leaves, which is at most $n/x$ due to our choice of
$x=n^{1/p}$. We abuse the notation and denote a node in $T$ with the same
notation as we use for the partition that is associated with it. Each leaf
$P_{(\ell_{1},\dots,\ell_{p-1})}$ is thus assigned to $x$ different nodes, and
each part $U_{(\ell_{1},\dots,\ell_{p-1}),j}$ in each leaf partition is
assigned to a different node. For each node $v\in V$, we denote by $U_{v,p-1}$
the part of the leaf partition that it is assigned to. Then, inductively, for
every $i=p-2,\dots 0$, we denote $U_{v,i}=\texttt{parent}(U_{v,i+1})$. Note
that for all $v\in V$ we have that $U_{v,0}$ is a part in the root partition.
We now let every node $v\in V$ learn all the edges in $\bigcup_{t<i\text{ s.t.
}\\{z_{i},z_{t}\\}\in E_{H}}{E(U_{v,i},U_{v,t})}$ and list all the instances
of $H$ that it sees. We need to prove that all instances of $H$ in $G$ are
indeed listed by this approach, and that learning the required edges by all
nodes can be done in $O(\frac{k\tilde{m}}{n^{1+2/p}}+p)$ rounds.
We first show that indeed all instances of $H$ are listed. Let $H^{\prime}$ be
an instance of $H$ in $G$, with nodes
$\\{z^{\prime}_{0},\dots,z^{\prime}_{p-1}\\}$, such that
$\\{z^{\prime}_{i},z^{\prime}_{t}\\}$ is an edge in $H^{\prime}$ if and only
if $\\{z_{i},z_{t}\\}$ is an edge in $H$. Let $U^{0}$ be the part of the root
partition that contains $z^{\prime}_{0}$. Denote by $j_{0}$, where $0\leq
j_{0}\leq x-1$, the index of $U^{0}$ in the root partition, and let
$P^{1}=P_{(j_{0})}$. Let $U^{1}$ be the part of $P^{1}$ that contains
$z^{\prime}_{1}$, and denote by $j_{1}$ the index of $U^{1}$ in $P^{1}$.
Continue inductively, for $i=2,\dots,p-1$: Let
$P^{i}=P_{(j_{0},j_{1}\dots,j_{i-1})}$. Let $U^{i}$ be the part of $P^{i}$
that contains $z^{\prime}_{i}$, and denote by $j_{i}$ the index of $U^{i}$ in
$P^{i}$. We now have a sequence of parts $U^{p-1},U^{p-2},\dots,U^{0}$, and
notice that for every $i$, $0\leq i\leq p-2$, we have that
$U^{i}=\texttt{parent}(U^{i+1})$. This implies that for the node $v\in V$ that
is assigned to part $j_{p-1}$ of the leaf partition $P^{p-1}$, it holds that
$U_{v,i}=U^{i}$ for every $0\leq i\leq p-1$, which means that $H^{\prime}$ is
contain in $\bigcup_{t<i\text{ s.t. }\\{z_{i},z_{t}\\}\in
E_{H}}{E(U_{v,i},U_{v,t})}$, and thus the node $v$ indeed lists the instance
$H^{\prime}$ of $H$ given by $\\{z^{\prime}_{0},\dots,z^{\prime}_{p-1}\\}$, as
needed.
It remains to bound the round complexity of having each node $v\in V$ learn
about all of the edges in $\bigcup_{t<i\text{ s.t. }\\{z_{i},z_{t}\\}\in
E_{H}}{E(U_{v,i},U_{v,t})}$. Since the $H$-partition tree $T$ satisfies
Condition 1 of Theorem 10, we have that the number of edges that each node
needs to learn is bounded by
$\displaystyle\bigcup_{t<i\text{ s.t. }\\{z_{i},z_{t}\\}\in
E_{H}}{E(U_{v,i},U_{v,t})}$ $\displaystyle\leq$
$\displaystyle\sum_{i}{\sum_{t<i,\\{z_{i},z_{t}\\}\in E_{H}}{|E(U,U_{t})|}}$
$\displaystyle\leq$ $\displaystyle\sum_{i}{c_{2}d_{i}\tilde{m}/x^{2}+n}$
$\displaystyle\leq$ $\displaystyle c_{2}(\sum_{i}{d_{i}})\tilde{m}/x^{2}+pn$
$\displaystyle\leq$ $\displaystyle O(\frac{k\tilde{m}}{n^{2/p}}+pn).$
Thus, by Corollary 7, all information can be learned in
$O(\frac{k\tilde{m}}{n^{1+2/p}}+p)$ rounds, and so the algorithm completes in
$O(\frac{k\tilde{m}}{n^{1+2/p}}+p)$ rounds, as claimed. ∎
## 5 Detecting Even Cycles and Computing the Girth in Congest
In this section we present our Congest algorithms for finding small even
cycles and for computing the girth.
### 5.1 Algorithm for Detecting Small Even Cycles
Throughout, we assume the convention that negative indices are taken to be
modulo the cycle size, that is, if we are working with cycles of length
$\ell$, then we denote $u_{-i}=u_{\ell-i}$. Likewise, when nodes choose random
colors, the colors are numbers in $[\ell]=\left\\{0,\ldots,\ell-1\right\\}$,
but for convenience, we sometimes write $-i$ for color $\ell-i$.
Fix $k\in\left\\{2,3,4\right\\}$. We show that we can find a copy of $C_{2k}$,
if there is one, in $O(n^{1-1/k})$ rounds, improving on previous algorithms,
which had running time $n^{1-1/\Theta(k^{2})}$.
We say that a $2k$-cycle $u_{0},\ldots,u_{2k-1}$ is _light_ if each cycle node
$u_{i}$ has degree at most $n^{1/k}$. Otherwise we say that the cycle is
_heavy_.
#### 5.1.1 Finding Light Cycles
Light cycles are easily found as follows: repeat, for $R=\Theta((2k)^{2k})$
iterations, the following steps.
1. 1.
Each node $u\in V$ chooses a random color $c(u)\in[2k]$.
2. 2.
We start a _color-BFS_ to depth $k$, in the subgraph of nodes that have degree
$\leq n^{1/k}$: each node $u$ that has color $c(u)=0$ and $\deg(u)\leq
n^{1/k}$ sends out a BFS token carrying its ID to all its neighbors that have
color 1. Next, nodes with color $b\in\left\\{-1,+1\right\\}$ and degree $\leq
n^{1/k}$ forward all the BFS tokens they receive to their neighbors with color
$2b$; this requires at most $n^{1/k}$ rounds. We proceed similarly: in the
$i$-th step of the BFS, nodes with degree $\leq n^{1/k}$ and color $b\cdot i$,
where $b\in\left\\{-1,+1\right\\}$, forward all the BFS tokens they receive to
their neighbors that have color $b\cdot(i+1)$. This requires at most $n^{i/k}$
rounds. Eventually, nodes with color $b\cdot(k-1)$ for
$b\in\left\\{-1,+1\right\\}$ and degree $\leq n^{1/k}$ forward their BFS
tokens to nodes with color $-k=k$.
3. 3.
If a node with color $k$ receives the same BFS token from a neighbor with
color $k-1$ and a neighbor with color $k+1=-(k-1)$, then it rejects.
##### Correctness.
First, note that the algorithm never rejects unless the graph contains a copy
of $C_{2k}$: for each $i\in\left\\{1,\ldots,k-1\right\\}$, the BFS initiated
by a node $u$ with $c(u)=0$ can only reach a node $v$ with
$c(v)\in\left\\{-i,+i\right\\}$ if there is a path of length $i$ from $u$ to
$v$, whose nodes are colored $0,1,\ldots,i$ (if $c(v)=i$) or $0,-1,\ldots,-i$
(if $c(v)=-i$). Therefore, a node $v$ with color $k$ rejects only if there is
some node $u$ that has two disjoint length-$k$ paths to $v$, or in other
words, node $v$ participates in a $2k$-cycle.
Next, suppose that the graph contains a light $2k$-cycle,
$u_{0},\ldots,u_{2k-1}$. In a given iteration, with probability $1/(2k)^{2k}$,
each cycle node $u_{i}$ chooses $c({u_{i}})=i$. Since the cycle is light, the
number of BFS tokens that reach nodes $u_{i},u_{2k-i}$ where
$i\in\left\\{1,\ldots,k-1\right\\}$ is at most $n^{i/k}$: since each cycle
node has degree at most $n^{1/k}$, there are at most $n^{i/k}$ nodes with
color 0 that have a path of length $i$ to node $u_{i}$ or $u_{2k-i}$, and as
we said above, a given BFS token $u$ can only reach a node $v$ with color $i$
or $-i$ if there is a path of length $i$ from $u$ to $v$ (with ascending or
descending colors). This means that no cycle node is forced to stop
participating in the middle of the BFS because it has too many tokens to
forward. The BFS token of node $u_{0}$ is forwarded in the color-ascending
direction by $u_{1},\ldots,u_{k-1}$ and in the color-descending direction by
$u_{-1},\ldots,u_{-(k-1)}$ until it reaches node $u_{k}$, which rejects.
Since a given iteration succeeds with probability $1/(2k)^{2k}$,222Actually,
the success probability is $1/(2k)^{2k-1}$, because we do not care about
cyclic shifts of the colors on the cycle; but the next step of the algorithm
does depend on getting the correct shift, so for simplicity we stick with the
same number of iterations here as well. after $R=\Theta((2k)^{2k})$
iterations, we succeed with probability $2/3$.
#### 5.1.2 Finding Heavy Cycles
It remains to find cycles where at least one node has degree greater than
$n^{1/k}$. To find such cycles, we exploit the fact that if we choose a random
node in the graph, we have noticeable probability ($1/n^{1-1/k}$) of hitting a
_neighbor_ of the cycle. We show that with the exception of a small number of
“bad” neighbors, if we find a neighbor of the cycle, we can find the cycle
itself.
We first describe a “meta-algorithm” $\mathcal{A}$ that cannot quite be
implemented in Congest, analyze it, and then give an implementation
$\mathcal{A}^{\prime}$ in Congest; the implementation is such that there is a
high-probability event $\mathcal{U}$, conditioned on which $\mathcal{A}$ and
$\mathcal{A}^{\prime}$ are in some sense equivalent.
The meta-algorithm $\mathcal{A}^{\prime}$ proceeds as follows: let
$T_{k}:\left\\{1,\ldots,2k-1\right\\}\rightarrow\mathbb{N}$ be a function. We
repeat the following steps for $R^{\prime}=\Theta(n^{1-1/k})$ iterations:
1. 1.
We choose one uniformly random node $s\in V$.
2. 2.
We carry out $R$ color-coded BFSs starting from $s$, each time using fresh
independently-chosen colors for all the nodes in the graph. The BFS proceeds
to depth $2k$, and it is only allowed to cross an edge $(u,v)$ if
$c(v)=(c(u)+1)\bmod 2k$. If one of the color-BFS instances finds a $2k$-cycle,
we reject.
3. 3.
Each node $u$ chooses a random color $c(u)\in[2k]$.
4. 4.
We start a color-BFS from each neighbor of $s$ that has color $0$, in
parallel. In step $i=1,\ldots,k-1$ of the BFS, nodes colored $i$ or $-i$
(resp.) check if they have received more than $T_{k}(|i|)$ BFS tokens; if they
have at most $T_{k}(|i|)$ tokens, they forward all of them to all neighbors
colored $i+1$ or $-(i+1)$ (resp.), and if they have more than $T_{k}(|i|)$
tokens, they send nothing.
5. 5.
If some node colored $k$ receives the same BFS token from neighbors colored
$k-1$ and $k+1$, then it rejects.
If after $R^{\prime}$ iterations no node has rejected, then all nodes accept.
Note that the running time of the meta-algorithm is $R\cdot
R^{\prime}=\Theta(n^{1-1/k})$ (treating $k$ as a constant).
#### 5.1.3 High Level Overview of the Analysis
When we search for heavy cycles, we sample a uniformly random node $s$, check
if it is part of a $2k$-cycle, and if not, we start a color-coded BFS from
each 0-colored neighbor of $s$. There can be many such neighbors, potentially
leading to congestion; however, we show that if the cycle is colored
correctly, it suffices for each node with color $i\in[2k]$ to forward a
constant number $T_{k}(i)$ of BFS tokens.
Our main concern is that the node $s$ that we sampled is “bad”, in the sense
that it has many short node-disjoint paths to some cycle node $u_{i}$. If we
sample such a “bad neighbor” of $u_{0}$, its 0-colored neighbors could
initiate many BFS instances, which would then reach $u_{i}$ and cause
congestion. See Figure 2(a) for an illustration.
(a) A “bad neighbor” $s\in N(u_{0})$.
(b) “Shared paths”: the edges of the 10-cycle are indicated by double lines.
Figure 2: Illustrations for the proof sketch of the $2k$-cycle algorithm
To bound the probability that we hit a “bad neighbor”, we first rule out any
neighbor of $u_{0}$ that itself participates in a $2k$-cycle. Next, we argue
that if $s$ has many node-disjoint paths to some cycle node $u_{i}$, such that
the path nodes are colored $0,1,\ldots,i$ (so that a BFS can be initiated by
the first path node and flow across the path), then we can charge these paths
against the degree of $u_{i}$, as each path ends at a different neighbor of
$u_{i}$. Since $\deg(u_{0})\geq\deg(u_{i})$, this means only a small constant
fraction of $u_{0}$’s neighbors have many such disjoint paths. When we sample
a random node, we are unlikely to hit a “bad neighbor”. (We are not worried
about non-disjoint paths, as they do not contribute any “new” BFS tokens; see
Lemma 6).
The problem with this argument is that if different neighbors of $u_{0}$
_share_ paths to $u_{i}$, we might be overcounting when we charge each path
against the degree of $u_{i}$. Our solution is to show that there is “not too
much” sharing, otherwise a $2k$-cycle appears — and since we only consider
neighbors of $u_{0}$ that are not on a $2k$-cycle, we know that this is
impossible.
In Figure 2(b), we show an example of one situation that must be ruled out
(among others): consider $k=5$ (i.e., 10-cycles), and suppose that two
distinct neighbors $s,s^{\prime}\in N(u_{0})$ each have at least 10 node-
disjoint paths with the “right colors”, 0-1, to $u_{2}$. Suppose further that
one of these paths is _shared_ , as shown in the figure. In addition, node
$s^{\prime}$ has at least one additional path (the rightmost path in the
figure), which must exist because $s^{\prime}$ has at least 10 node-disjoint
paths to $u_{2}$ (so at least one of these paths avoids all the other nodes
shown in the figure). We see that there is a 10-cycle involving nodes $s$ and
$s^{\prime}$; since we only consider neighbors of $u_{0}$ that do not
themselves participate in a 10-cycle, this situation cannot arise.
#### 5.1.4 Analysis of the Meta-Algorithm
Since nodes reject only if they _find_ a copy of $C_{2k}$ (by having their BFS
token return to them in $2k$ color-coded steps, or by receiving the BFS of
some node at distance $k$ through two node-disjoint paths), if the graph
contains no copy of $C_{2k}$, then all nodes accept. We therefore focus on the
case where the graph does contain a heavy copy of $C_{2k}$, and show that the
meta-algorithm can find it.
###### Lemma 5.
If the graph contains a heavy $2k$-cycle, then with probability $9/10$, some
node rejects.
To prove Lemma 5, we show that for each $k=2,3,4,5$, there is a choice of
$T_{k}:\left\\{1,\ldots,2k-1\right\\}\rightarrow\mathbb{N}$ such that one
iteration of the meta-algorithm detects a heavy copy of $C_{2k}$, if there is
one, with probability $1/O(n^{1-1/k})$ (treating $k$ as a constant).
Therefore, after $R=\Theta(n^{1-1/k})$ iterations, we reject with high
probability.
Let $u_{0},\ldots,u_{2k-1}$ be a heavy cycle, and assume that $u_{0}$ is a
node with the largest degree in the cycle (i.e., $\deg(u_{0})\geq\deg(u_{i})$
for each $i\in\left\\{1,\ldots,2k-1\right\\}$). In particular, since the cycle
is heavy, we have $\deg(u_{0})>n^{1/k}$. We consider two cases:
1. 1.
Node $u_{0}$ has at least $n^{1/k}/100$ neighbors that each belong to some
$2k$-cycle. In this case, when we sample a uniformly random node $s\in V$, we
have probability at least $(n^{1/k}/100)/n=1/(100n^{1-1/k})$ that $s$ is on a
$2k$-cycle; and given that $s$ is indeed on a $2k$-cycle, we will find the
$2k$-cycle with probability $99/100$ after $R$ iterations of color-BFS
(provided we choose a large enough constant in $R$). Therefore, in this case,
we reject with probability $\Omega(1/n^{1-1/k})$.
2. 2.
Node $u_{0}$ has at least $(99/100)n^{1/k}$ neighbors that do not belong to
any $2k$-cycle. We consider the following event $\mathcal{E}_{k}$:
1. (a)
$s\in N(u_{0})$, and
2. (b)
$s$ is not on any $2k$-cycle, and
3. (c)
$c({u_{i}})=i$ for each $i\in[2k]$, and
4. (d)
$s\not\in B_{k}(u_{0})$, where $B_{k}(u_{0})$ is a set of “bad neighbors” of
node $u_{0}$, which is defined in a different way for each $k$.
Next, we consider each $k=2,3,4,5$ separately, define $T_{k}$ and
$B_{k}(u_{0})$, and prove that
1. 1.
The number of “bad neighbors” $B_{k}(u_{0})$ that are not on any $2k$-cycle is
bounded from above by $\alpha_{k}\cdot\deg(u_{0})$, where
$\alpha_{k}\in(0,99/100)$ is some constant fraction that depends only on $k$.
Therefore, $u_{0}$ has $\Omega(n^{1/k})$ neighbors that are not in
$B_{k}(u_{0})$ and are also not on any $2k$-cycle. The probability of hitting
such a neighbor is $\Omega(1/n^{1-1/k})$. Independent of this event, the
probability that the cycle $u_{0},\ldots,u_{2k-1}$ is colored correctly is
constant, and therefore $\mathcal{E}_{k}$ occurs with probability
$1/O(n^{1-1/k})$.
2. 2.
Conditioned on $\mathcal{E}_{k}$, each cycle node $u_{i}$ for
$i\in[2k]\setminus\left\\{0,k\right\\}$ receives no more than $T_{k}(i)$
distinct BFS tokens. This means that conditioned on $\mathcal{E}_{k}$, the
color-BFS completes successfully, causing node $u_{k}$ to reject.
Together we see that we have probability $1/O(n^{1-1/k})$ of detecting
$u_{0},\ldots,u_{2k-1}$ in each of the $R^{\prime}$ iterations, as desired.
This part of the analysis proceeds as follows. We say that a neighbor $s\in
N(u_{0})$ is _free_ if $s$ does not participate in a $2k$-cycle. Let
$N^{\prime}(u_{0})$ denote the free neighbors of $u_{0}$, and let
$\deg^{\prime}(u_{0})=|N^{\prime}(u_{0})|$.
After choosing a neighbor $s\in N(u_{0})$, we check if $s$ participates in a
$2k$-cycle, and if not, we initiate a BFS from every 0-colored neighbor of
$s$. We must show that not too many BFS tokens — at most $T_{k}(i)$ — can
reach a given cycle node $u_{b\cdot i}$ where $b\in\left\\{-1,+1\right\\}$ and
$i\in\left\\{1,\ldots,k-1\right\\}$. Thus, we want to show that the “typical”
free neighbor $s\in N^{\prime}(u_{0})$ does not have many disjoint paths of
length $i+1$ to $u_{i}$, through which BFS tokens can flow to $u_{i}$.
Given $b\in\left\\{-1,+1\right\\},i\in\left\\{1,\ldots,k-1\right\\}$, we say
that a path $\pi=w_{0},\ldots,w_{i-1}$ is an _$(i,b)$ -path of $s$_ if
1. 1.
$w_{0}\in N(s)$ and $w_{i-1}\in N(u_{b\cdot i})$,
2. 2.
The path has “the right colors” so that node $w_{0}$ initiates a BFS that
flows across the path and reaches $u_{b\cdot i}$: that is, $c(w_{j})=b\cdot j$
for each $j=0,\ldots,i-1$.
3. 3.
The path is node-disjoint from the prefix $u_{0},u_{b},\ldots,u_{b\cdot(i-1)}$
of the cycle.
In the sequel, to simplify the presentation, we consider $b=1$; the case
$b=-1$ is symmetric. We simplify our notation by writing “$i$-path” instead of
“$(1,i)$-path”.
Our goal is to show that a large fraction of free neighbors $s\in
N^{\prime}(u_{0})$ have only a small number of node-disjoint $i$-paths, for
each $i=0,\ldots,k-1$, as this ensures that congestion is well-controlled:
###### Lemma 6.
Suppose we have sampled a neighbor $s\in N^{\prime}(u_{0})$ which has no more
than $p$ node-disjoint $i$-paths. Then the number of BFS tokens that arrive at
cycle node $u_{i}$ is bounded by $(p+1)\left[\sum_{j=1}^{i-1}T_{k}(j)\right]$.
###### Proof.
Let
$\pi_{1}=(\pi_{1}^{0},\ldots,\pi_{1}^{i-1}),\ldots,\pi_{p}=(\pi_{p}^{0},\ldots,\pi_{p}^{i-1})$
be a maximal set of node-disjoint $i$ paths from $s$ to $u_{i}$. Suppose for
the sake of contradiction that $u_{i}$ receives
$t>(p+1)\left[\sum_{j=1}^{i-1}T_{k}(j)\right]$ BFS tokens.
Note that for each $j=1,\ldots,i-1$, nodes $\pi_{1}^{j},\ldots,\pi_{p}^{j}$
each forward at most $T_{k}(j)$ tokens. In particular, since the last node of
each path $\pi_{1},\ldots,\pi_{p}$ forwards at most $T_{k}(i-1)$ tokens, and
node $u_{i-1}$ also forwards at most $T_{k}(i-1)$ tokens, we have at least
$t-(p+1)T_{k}(i-1)>(p+1)\left[\sum_{j=1}^{i-2}T_{k}(j)\right]$ BFS tokens that
arrived at $u_{i}$ without passing through $u_{i-1}$ or through any of the
nodes $\pi_{1}^{i-1},\ldots,\pi_{p}^{i-1}$. Next, since each next-to-last node
on $\pi_{1},\ldots,\pi_{p}$, as well as $u_{i-2}$, each forward at most
$T_{k}(i-2)$ tokens, we have at least
$t-(p+1)\left[T_{k}(i-1)+T_{k}(i-2)\right]>(p+1)\left[\sum_{j=1}^{i-3}T_{k}(j)\right]$
BFS tokens that arrived at $u_{i}$ without passing through the last two nodes
on any path $\pi_{1},\ldots,\pi_{p}$, or through $u_{i-2},u_{i-1}$. Continuing
in a similar manner, we eventually see that there must be at least
$t-(p+1)\left[\sum_{j=1}^{i-1}T_{k}(j)\right]>0$ tokens — i.e., at least one
token — that arrived at $u_{i}$ without passing through $u_{0},\ldots,u_{i-1}$
or through any of the nodes on paths $\pi_{1},\ldots,\pi_{p}$; this token must
have been forwarded along some path $\tau=\tau^{0},\ldots,\tau^{i-1}$, where
$\tau^{0}\in N(s)$, $\tau^{i-1}\in N(u_{i})$, and $c(\tau^{j})=j$ for each
$j=0,\ldots,i-1$, and $\tau$ is node-disjoint from all the paths
$\pi_{1},\ldots,\pi_{p}$. Note that $\tau$ is “colored correctly”, otherwise a
BFS token could not flow across it; so $\tau$ is in fact an $i$-path. This
contradicts our assumption that $\pi_{1},\ldots,\pi_{p}$ is a maximal set of
node-disjoint $i$-paths from $s$ to $u_{i}$. ∎
For each $i=1,\ldots,k-1$ and $b\in\left\\{-1,+1\right\\}$, define
$B_{k}^{b,i}(u_{0})=\left\\{s\in N(u_{0})\medspace\mid\medspace\text{$s$ is
free and has at least $d_{k}$ node-disjoint $(b,i)$-paths}\right\\}$
to be the “bad neighbors” of $u_{0}$, where here $d_{k}\in\mathbb{N}$ is some
constant (which depends on $k$). We prove that there are not too many bad
neighbors:
$\sum_{i=1}^{k-1}|B_{k}^{b,i}(u_{0})|<\alpha\deg(u_{i}),$ (1)
where $\alpha<1/4$ is some constant. Since we assume that
$\deg(u_{0})\geq\deg(u_{i})$ and that $\deg^{\prime}(u_{0})\geq\deg(u_{0})/2$
(in this part of the analysis), and accounting for both $b=+1$ and $b=-1$, we
have
$\left|N^{\prime}(u_{0})\setminus\bigcup_{b\in\left\\{-1,+1\right\\}}\bigcup_{i=1}^{k-1}B_{k}^{b,i}(u_{0})\right|>(1-4\alpha)\deg^{\prime}(u_{0})=\Omega(n^{1/k}).$
(2)
By Lemma 6, when we sample a good neighbor, the cycle nodes do not have too
much congestion, and the BFS token of $u_{0}$ is able to reach $u_{k-1}$ and
$u_{k+1}$.
###### Controlling the number of bad neighbors.
Let us again assume $b=+1$ and drop $b$ from our notation.
To prove (1), we observe that any bad neighbor $s\in B_{k}^{i}(u_{0})$
contributes at least $d_{k}$ to the degree of $u_{i}$, as $s$ has at least
$d_{k}$ node-disjoint $i$-paths which connect to $u_{i}$ through $d_{k}$
different neighbors of $u_{i}$. Unfortunately, it could be that two different
bad neighbors $s,s^{\prime}\in B_{k}^{i}(u_{0})$ _share_ some of their
$i$-paths, so we cannot immediately argue that the number of bad paths is
bounded by $\deg(u_{i})/d_{k}$. The bulk of the proof consists of showing that
“not too many” bad neighbors can share “too many” of their $i$-paths, and
therefore we can still show that the number of bad neighbors is
$O(\deg(u_{i}))$. Indeed, we show that “too much sharing” of $i$-paths between
different bad neighbors creates a $2k$-cycle through them, and since we only
consider _free_ neighbors of $u_{0}$, this cannot happen.
We proceed to consider each $k=2,3,4,5$ separately.
##### Analysis for $k=2$ (i.e., 4-cycles).
We set $T_{2}(1)=1$ and $B_{2}(u_{0})=\emptyset$.
Suppose for the sake of contradiction that node $u_{b}$, where
$b\in\left\\{-1,+1\right\\}$, receives more than one BFS token. Then there is
some node $v\neq u_{0}$, whose BFS token $u_{b}$ received; both $u_{0}$ and
$v$ are neighbors of $u_{b}$. In addition, since we only start a BFS from
neighbors of $s$, node $v$ must be a neighbor of $s$. Thus, the graph contains
a $4$-cycle that includes $s$: $u_{0},u_{b},v,s$. This contradicts our
assumption that $s$ is not on a $4$-cycle.
##### Analysis for $k=3$ (i.e., 6-cycles).
We set $T_{3}(1)=T_{3}(2)=3$, and define “bad neighbors” as follows:
$B_{3}(u_{0})=\left\\{v\in N(u_{0})\medspace\mid\medspace|N(v)\cap
N(u_{1})|>3\text{ or }|N(v)\cap N(u_{-1})|>3\right\\}.$
First, observe that $u_{0}$ has at most two bad neighbors that are not on any
$6$-cycle: we show that there is at most one node $v\in N(u_{0})$ which is not
on any $6$-cycle and has $|N(v)\cap N(u_{1})|>3$, and similarly when we
replace $u_{1}$ with $u_{-1}=u_{5}$. Suppose for the sake of contradiction
that there are two nodes $v\neq v^{\prime}$ such that $v,v^{\prime}\in
N(u_{0})$, neither $v$ nor $v^{\prime}$ are on a $6$-cycle, and also
$|N(v)\cap N(u_{1})|>3,|N(v^{\prime})\cap N(u_{1})|>3$. Then there exist nodes
$w\in\left(N(v)\cap
N(u_{1})\right)\setminus\left\\{u_{0},v^{\prime}\right\\}$,
$w^{\prime}\in\left(N(v^{\prime})\cap
N(u_{1})\right)\setminus\left\\{u_{0},v,w\right\\}$ (because after removing at
most 3 nodes from $N(v)\cap N(u_{1})$ or from $N(v^{\prime})\cap N(u_{1})$,
the sets are still not empty). Since we assume the graph contains no self-
loops, we also have $w\neq v,u_{1}$ and $w^{\prime}\neq v^{\prime},u_{1}$, as
$v,v^{\prime}$ are not neighbors of themselves. Therefore the following
6-cycle is in the graph: $v,w,u_{1},w^{\prime},v^{\prime},u_{0}$.
Next we show that conditioned on $\mathcal{E}_{3}$, each cycle node $u_{i}$ or
$u_{-i}$ where $i\in\left\\{1,2\right\\}$ receives at most $T_{3}(|i|)=3$ BFS
tokens. We prove it for $u_{1}$ and $u_{2}$; the proof for $u_{-1}$ and
$u_{-2}$ (resp.) is similar.
* •
$u_{1}$: since $\mathcal{E}_{3}$ requires that $c(u_{1})=1$, node $u_{1}$ only
receives BFS tokens in the first step of the color-BFS; that is, $u_{1}$ only
receives BFS tokens from its own neighbors which are also neighbors of $s$
(and are colored 0). Because $s\not\in B_{3}(u_{0})$ under $\mathcal{E}_{3}$,
there are at most three such BFS tokens.
* •
$u_{2}$: since $\mathcal{E}_{3}$ requires that $c(u_{2})=2$, node $u_{2}$ only
receives BFS tokens in the second step of the color-BFS. We already showed
that $u_{1}$ receives at most three BFS tokens; thus, in order for $u_{2}$ to
receive more than three, the fourth token must come through some node other
than $u_{1}$.
If $u_{1}$ receives no more than three BFS tokens, these include the BFS token
of $u_{0}$, so the fourth token received by $u_{2}$ cannot originate at
$u_{0}$. Therefore there must exist $v\neq u_{0}$ and $w\neq u_{1}$ such that
* –
$v\neq u_{0}$ is the originator of the fourth BFS token received by $u_{2}$:
we have $v\in N(s)$ (and $c(v)=0$, but we do not need this fact).
* –
$w\neq u_{1}$ is the node that forwards $v$’s token to $u_{2}$: we have $w\in
N(v)\cap N(u_{2})$.
However, this means that $s$ has two node-disjoint paths of length two to
$u_{2}$, so it participates in the following $6$-cycle:
$s,u_{0},u_{1},u_{2},w,v$. Under $\mathcal{E}_{3}$ we know that $s$ does not
participate in any $6$-cycle, so this is impossible.
On the other hand, if $u_{1}$ receives more than three BFS tokens, it forwards
no tokens to $u_{2}$. Of the four (or more) tokens received by $u_{2}$, at
least one belongs to some node $v\neq u_{0}$. So again, we have nodes $v\neq
u_{0}$ and $w\neq u_{1}$ such that $v\in N(s)$ and $w\in N(v)\cap N(u_{2})$,
and we get a 6-cycle that includes $s$, as above.
##### Analysis for $k=4$ (i.e., 8-cycles).
We say that a node $s$ is _free_ if it does not participate in any 8-cycle.
Our analysis considers two cases, depending on whether or not a certain
pattern is present in the graph.
With respect to the fixed cycle $u_{0},\ldots,u_{7}$, and given
$b\in\left\\{-1,+1\right\\}$, we define a _$b$ -pattern_ $D$ to be the
following 4-node subgraph, which is node-disjoint from the cycle:
$D=(\left\\{s,s^{\prime},w,w^{\prime}\right\\},\left\\{\left\\{s,w\right\\},\left\\{s,w^{\prime}\right\\},\left\\{s^{\prime},w^{\prime}\right\\}\right\\})$,
such that in addition to the internal edges of $D$, we have
* •
$s,s^{\prime}\in N(u_{0})$,
* •
$w,w^{\prime}\in N(u_{b})$.
Nodes $s,s^{\prime}$ are called the _heads_ of the dangerous pattern, and
$w,w^{\prime}$ are called the _tails_.
###### Observation 1.
If there is a $b$-pattern $D$ with heads $s,s^{\prime}$, at least one of which
is free, and with tails $w,w^{\prime}$, then there cannot exist any free node
$s^{\prime\prime}\in
N(u_{0})\setminus\left\\{s,s^{\prime},w,w^{\prime}\right\\}$ such that
$N(s^{\prime\prime})\cap
N(u_{b})\not\subseteq\left\\{w,w^{\prime},u_{0},s,s^{\prime}\right\\}$.
###### Proof.
Suppose otherwise, and let $w^{\prime\prime}\in N(s^{\prime\prime})\cap
N(u_{b})\setminus\left\\{w,w^{\prime},u_{0},s,s^{\prime}\right\\}$. Then the
following 8-cycle is in the graph:
$s,w^{\prime},s^{\prime},u_{0},s^{\prime\prime},w^{\prime\prime},u_{b},w$,
contradicting our assumption that at least one of the nodes $s,s^{\prime}$ is
free (i.e., does not participate in an 8-cycle).
We verify that this is indeed a simple 8-cycle:
* •
$w^{\prime}\neq s$ because they are distinct nodes of $D$,
* •
$s^{\prime}\not\in\left\\{s,w^{\prime}\right\\}$ for the same reason,
* •
$u_{0}\not\in\left\\{s,w^{\prime},s^{\prime}\right\\}$ because $D$ is disjoint
from the cycle,
* •
$s^{\prime\prime}\not\in\left\\{s,w^{\prime},s^{\prime},u_{0}\right\\}$
because we assumed that $s^{\prime\prime}\in
N(u_{0})\setminus\left\\{s,s^{\prime},w,w^{\prime}\right\\}$ and the graph
contains no self-loops,
* •
$w^{\prime\prime}\not\in\left\\{s,w^{\prime},s^{\prime},u_{0},s^{\prime\prime}\right\\}$
by choice of $w^{\prime\prime}$, together with the fact that
$w^{\prime\prime}\in N(s^{\prime\prime})$ and the graph contains no self-
loops,
* •
$u_{b}\not\in\left\\{s,w^{\prime},s^{\prime},u_{0},s^{\prime\prime},w^{\prime\prime}\right\\}$:
we know that $w,w^{\prime},w^{\prime\prime}\in N(u_{b})$, and the graph
contains no self-loops; we cannot have $u_{b}=u_{0}$ because these are
distinct nodes of our fixed 8-cycle; and we cannot have
$u_{b}\in\left\\{s,s^{\prime}\right\\}$ because $D$ is node-disjoint from the
cycle.
* •
$w\not\in\left\\{s,w^{\prime},s^{\prime},u_{0},s^{\prime\prime},w^{\prime\prime},u_{b}\right\\}$:
we know that $w\not\in\left\\{s,s^{\prime},w^{\prime}\right\\}$ because these
are distinct nodes of $D$; also, $w\not\in\left\\{u_{0},u_{b}\right\\}$
because $D$ is distinct from the 8-cycle; and finally,
$w\not\in\left\\{s^{\prime\prime},w^{\prime\prime}\right\\}$ by choice of
$s^{\prime\prime},w^{\prime\prime}$.
∎
The set of bad neighbors, $B_{4}(u_{0})$, is defined as follows:
1. (I)
Define a _0-1 $b$-path_ from node $v\in N(u_{0})$ to node $u_{2b}$ to be a
path of length 2 between these nodes, $\pi=v,w_{0},w_{1},u_{2b}$, such that
$c(w_{0})=0,c(w_{1})=1,w_{0}\neq u_{0},w_{1}\neq u_{b}$. Two 0-1 paths
$\pi_{1}=v,w_{0}^{1},w_{1}^{1},u_{2b}$ and
$\pi_{2}=v,w_{0}^{2},w_{1}^{2},u_{2b}$ are called _node-disjoint_ if
$w_{0}^{1}\neq w_{0}^{2}$ and $w_{1}^{1}\neq w_{2}^{2}$ (note that because of
differing colors, node-disjoint 0-1 paths cannot share any nodes, except the
two endpoints $v,u_{2b}$).
Any neighbor $v\in N(u_{0})$ that has at least four node-disjoint 0-1 paths to
$u_{2b}$ is added to $B_{4}(u_{0})$.
2. (II)
For each $b\in\left\\{-1,+1\right\\}$, if the graph contains a $b$-pattern
w.r.t. $u_{0},\ldots,u_{7}$, we fix one such pattern arbitrarily, and add its
heads to $B_{4}(u_{0})$.
3. (III)
For each $b\in\left\\{-1,+1\right\\}$, if the graph does not contain a
$b$-pattern w.r.t. $u_{0},\ldots,u_{7}$, then we add to $B_{4}(u_{0})$ any
node $v$ with $|N(v)\cap N(u_{b})|\geq 4$ for $b\in\left\\{-1,+1\right\\}$.
First, we bound the number of free bad neighbors of $u_{0}$ of each type
I-III, and show that $|B_{4}(u_{0})|\leq(3/4)\deg(u_{0})$:
1. 1.
For each $b\in\left\\{-1,+1\right\\}$, there is at most one free node $v\in
N(u_{0})$ that has four node-disjoint 0-1 paths to $u_{2b}$: suppose for the
sake of contradiction that there are two such nodes, $v\neq v^{\prime}$. Then
we have the following paths in the graph:
* •
$v,u_{0},u_{b},u_{2b}$,
* •
A 0-1 path, $v,w_{0},w_{1},u_{2b}$, which is node-disjoint from the previous
path by definition,
* •
A path $v^{\prime},w_{0}^{\prime},w_{1}^{\prime},u_{2b}$ which is node-
disjoint from the previous paths (such a path exists because $v^{\prime}$ has
at least four node-disjoint 0-1 paths to $u_{2b}$, none of which include
$u_{0}$ or $u_{b}$; at least one of these paths avoids $v,w_{0},w_{1}$).
Therefore, the graph includes the following 8-cycle:
$v,u_{0},v^{\prime},w_{0}^{\prime},w_{1}^{\prime},u_{2b},w_{1},w_{0}$,
contradicting our assumption that $v,v^{\prime}$ are free.
2. 2.
For each $b\in\left\\{-1,+1\right\\}$, if the graph contains a $b$-pattern
$D$, then it has exactly two heads, so we add two nodes to $B_{4}(u_{0})$.
3. 3.
For each $b\in\left\\{-1,+1\right\\}$, if the graph does not contain a
$b$-pattern, then for any two neighbors $s,s^{\prime}\in N(u_{0})$, if either
$|N(s)\cap N(u_{b})|\geq 4$ or $|N(s^{\prime})\cap N(u_{b})|\geq 4$, then we
must have $N(s)\cap N(s^{\prime})\cap N(u_{b})=\left\\{u_{0}\right\\}$:
otherwise, if w.l.o.g. we had $|N(s)\cap N(u_{b})|\geq 4$ and also $N(s)\cap
N(s^{\prime})\cap N(u_{b})\supsetneq\left\\{u_{0}\right\\}$, then there would
exist tails, $w\in N(s)\cap N(s^{\prime})\cap
N(u_{b})\setminus\left\\{u_{0}\right\\}$ and $w^{\prime}\in N(s)\cap
N(u_{b})\setminus\left\\{u_{0},w,s\right\\}$, such that
$s,s^{\prime},w,w^{\prime}$ are a $b$-pattern w.r.t. $u_{0},\ldots,u_{7}$.
Let $U$ be the set of nodes $s$ with $|N(s)\cap N(u_{b})|\geq 4$. As we just
said, for any distinct $s,s^{\prime}\in U$, we have $N(s)\cap
N(s^{\prime})\cap N(u_{b})=\left\\{u_{0}\right\\}$, that is, $\left(N(s)\cap
N(u_{b})\right)\cap\left(N(s^{\prime})\cap
N(u_{b})\right)=\left\\{u_{0}\right\\}$. Since we assumed that $u_{0}$ has
maximal degree among $u_{0},\ldots,u_{7}$,
$\left|N(u_{0})\right|\geq\left|N(u_{b})\right|\geq\left|\bigcup_{s\in
U}N(s)\cap N(u_{b})\right|\geq 1+3\cdot|U|.$
We see that $|U|<\deg(u_{0})/3$.
Summing across both $b=-1,+1$, we see that the total number of bad neighbors
is bounded by $6+2\deg(u_{0})/3<(3/4)\deg(u_{0})$, assuming $n$ is large
enough (recall that $\deg(u_{0})\geq n^{1/k}$, so for $n$ large enough we have
$\deg(u_{0})/12>6$).
Next, assume we have sampled a free good neighbor $s\in N(u_{0})\setminus
B_{4}(u_{0})$, and let us bound the number of BFS tokens that each cycle node
can receive. Let $b\in\left\\{-1,+1\right\\}$.
* •
$u_{b}$ can receive at most $5$ BFS tokens: since we assume that $c(u_{b})=b$,
the only BFS tokens received by $u_{b}$ are those sent by 0-colored nodes in
$N(s)\cap N(u_{b})$. We consider two cases:
1. 1.
The graph contains a $b$-pattern with heads $v,v^{\prime}$ and tails
$w,w^{\prime}$: then by definition, since $s$ is not a bad neighbor,
$s\not\in\left\\{v,v^{\prime}\right\\}$. By Observation 1, we have $N(s)\cap
N(u_{b})\subseteq\left\\{v,v^{\prime},w,w^{\prime},u_{0}\right\\}$, so at most
5 BFS tokens can reach $u_{b}$.
2. 2.
The graph does not contain a $b$-pattern: then since $s$ is not a bad
neighbor, we have $|N(s)\cap N(u_{b})|<4$, and hence fewer than 5 BFS tokens
can reach $u_{b}$.
* •
$u_{2b}$ can receive at most 30 BFS tokens: since $s$ is not bad, it has at
most four node-disjoint 0-1 paths to $u_{2b}$. Let
$\pi_{1},\ldots,\pi_{\ell}$, $\ell\leq 4$, be a maximal set of node-disjoint
0-1 paths from $s$ to $u_{2b}$. For each such path
$\pi_{i}=s,w_{0}^{i},w_{1}^{i},u_{2b}$, if node $w_{1}^{i}$ receives more than
5 BFS tokens, it sends none of them; and if it receives at most 5 BFS tokens,
it forwards them to $u_{2b}$. The same goes for the path
$s,u_{0},u_{b},u_{2b}$. Thus, node $u_{2b}$ receives at most 25 tokens from
nodes $\left\\{w_{1}^{i}\right\\}_{i=1,\ldots,\ell}$ and $u_{b}$. We also
“throw in for free” the BFS tokens of nodes
$\left\\{w_{0}^{i}\right\\}_{i=1,\ldots,\ell}$ and $u_{0}$, for a total of at
most 30 tokens received at $u_{2b}$. (These latter tokens may reach $u_{2b}$
through some node other than
$\left\\{w_{1}^{i}\right\\}_{i=1,\ldots,\ell},u_{b}$, and we pessimistically
assume that they do.)
Suppose for the sake of contradiction that $u_{2b}$ receives more than 30
tokens. Then one of these tokens was neither originated by one of the nodes
$\left\\{w_{0}^{i}\right\\}_{i=1,\ldots,\ell},u_{0}$, nor forwarded by one of
the nodes $\left\\{w_{1}^{i}\right\\}_{i=1,\ldots,\ell},u_{b}$. This means
that there is some 0-colored neighbor $x\in N(s)$, such that
$x\not\in\left\\{w_{0}^{i}\right\\}_{i=1,\ldots,\ell}\cup\left\\{u_{0}\right\\}$,
whose token was received by $u_{2b}$, and a 1-colored neighbor $y\in N(s)\cap
N(u_{2b})$, such that
$y\not\in\left\\{w_{1}^{i}\right\\}_{i=1,\ldots,\ell}\cup\left\\{u_{b}\right\\}$,
that forwarded $x$’s token to $u_{2b}$. But then the path $s,x,y,u_{2b}$ is a
0-1 path that is node-disjoint from $\pi_{1},\ldots,\pi_{\ell}$, contradicting
our assumption that this is a maximal set of node-disjoint 0-1 paths from $s$
to $u_{2b}$.
* •
$u_{3b}$ can receive at most 36 BFS tokens: suppose for the sake of
contradiction that $u_{3b}$ receives more than 36 tokens. Since $u_{0}$
originates one token, and nodes $u_{b},u_{2b}$ forward 5 tokens and 30 tokens,
respectively, this means that node $u_{3b}$ receives some token originated by
a neighbor $w_{0}\neq u_{0}$ of $s$, and forwarded first by $w_{1}\neq u_{b}$
and then by $w_{2}\neq u_{2b}$. Therefore the graph contains the 8-cycle
$s,u_{0},u_{b},u_{2b},u_{3b},w_{2},w_{1},w_{0}$, contradicting our assumption
that $s$ is free.
##### Analysis for $k=5$ (i.e., 10-cycles).
Let $B_{5}^{1}(u_{0})$ be the set of free neighbors of $u_{0}$ that have 100
or more different 1-paths to $u_{1}$. (Recall that a 1-path is simply one node
$w_{0}$, colored 0, and connected to both $s$ and $u_{1}$.)
###### Lemma 7.
Suppose nodes $s,s^{\prime}\in B_{5}^{1}(u_{0})$ ($s\neq s^{\prime}$) have a
common 1-path, $w_{0}\in N(s)\cap N(s^{\prime})\cap N(u_{1})$. Then for any
two other nodes $s^{\prime\prime},s^{\prime\prime\prime}\in
B_{5}^{1}(u_{0})\setminus\left\\{s,s^{\prime},w_{0}\right\\}$
($s^{\prime\prime}\neq s^{\prime\prime\prime}$), there is no common 1-path
$w_{0}^{\prime}\in N(s^{\prime\prime})\cap N(s^{\prime\prime\prime})\cap
N(u_{1})\setminus\left\\{s,s^{\prime},w_{0},u_{0}\right\\}$.
###### Proof.
Suppose the lemma is false, and let
$s,s^{\prime},s^{\prime\prime},s^{\prime\prime\prime},w_{0},w_{0}^{\prime}$ be
as in the lemma. Since $s\in B_{5}^{1}(u_{0})$, it has at least 100 1-paths to
$u_{1}$, and at least one of them, call it $x_{0}$, excludes nodes
$s,s^{\prime\prime},s^{\prime\prime\prime},w_{0},w_{0}^{\prime},u_{1}$. Also,
since $s^{\prime\prime\prime}\in B_{5}^{1}(u_{0})$, it has at least one
1-path, call it $y_{0}$, which differs from
$s,s^{\prime},s^{\prime\prime},w_{0},w_{0}^{\prime},u_{1},x_{0}$. Therefore
the following 10-cycle is in the graph:
$s,w_{0},s^{\prime},u_{0},s^{\prime\prime},w_{0}^{\prime},s^{\prime\prime\prime},y_{0},u_{1},x_{0}$.
This contradicts our assumption that $s$ (and also
$s^{\prime},s^{\prime\prime},s^{\prime\prime\prime}$) are free. ∎
###### Corollary 12.
Assuming $\deg^{\prime}(u_{0})>\deg(u_{0})/2>100$, we have
$|B_{5}^{1}(0)|\leq\deg^{\prime}(u_{0})/20$.
###### Proof.
We claim that $|B_{5}^{1}(u_{0})|\leq\deg(u_{2})/50+3$. Since
$\deg(u_{2})\leq\deg(u_{0})\leq 2\deg^{\prime}(u_{0})$, this implies that
$|B_{5}^{1}(u_{0})|\leq\frac{2\deg^{\prime}(u_{0})}{50}+3<\frac{\deg^{\prime}(u_{0})}{20}.$
If no two nodes $s\neq s^{\prime}\in B_{5}^{1}(u_{0})$ have a common 1-path
$w_{0}\in N(s)\cap N(s^{\prime})\cap N(u_{1})$, then each node in
$B_{5}^{1}(u_{0})$ contributes at least 100 unique neighbors of $u_{1}$ which
are not contributed by any other neighbor in $B_{5}^{1}(u_{0})$, and therefore
$\deg(u_{2})\geq 100|B_{5}^{1}(u_{0})|$. Thus, assume there do exist $s\neq
s^{\prime}\in B_{5}^{1}(u_{0})$ with a common 1-path $w_{0}$, and fix such
$s,s^{\prime},w_{0}$. The remaining nodes in
$B_{5}^{1}(u_{0})\setminus\left\\{s,s^{\prime},w_{0}\right\\}$ do not have any
common 1-paths among themselves, except possibly $s,s^{\prime},w_{0},u_{0}$;
but each node in $B_{5}^{1}(u_{0})$ has at least 100 1-paths to $u_{1}$, and
at least 50 of them are not $s,s^{\prime},w_{0},u_{0}$, and as we just said,
are therefore not shared with any other node in
$B_{5}^{1}(u_{0})\setminus\left\\{s,s^{\prime},w_{0}\right\\}$. It follows
that each node in
$B_{5}^{1}(u_{0})\setminus\left\\{s,s^{\prime},w_{0}\right\\}$ contributes at
least 50 unique neighbors of $u_{2}$, and hence
$|B_{5}^{1}(u_{0})|\leq\deg(u_{2})/50+3$.
∎
Let $B_{5}^{2}(u_{0})$ be the set of free neighbors of $u_{0}$ that have 100
or more node-disjoint 2-paths to $u_{2}$.
###### Lemma 8.
Assuming $\deg^{\prime}(u_{0})>\deg(u_{0})/2>1000$, we have
$|B_{5}^{2}(u_{0})|\leq\deg^{\prime}(u_{0})/10$.
###### Proof.
Suppose for the sake of contradiction that
$|B_{5}^{2}(u_{0})|>\deg^{\prime}(u_{0})/10>100$. We claim that no two nodes
in $B_{5}^{2}(u_{0})$ can _share_ a 2-path, that is, there cannot exist
$s,s^{\prime}\in B_{5}^{2}(u_{0})$ and 2 paths $w_{0},w_{1}$ and
$w_{0}^{\prime},w_{1}^{\prime}$ which are _not node-disjoint_ , such that
$w_{0},w_{1}$ is a 2-path from $s$ to $u_{2}$, and
$w_{0}^{\prime},w_{1}^{\prime}$ is a 2-path from $s^{\prime}$ to $u_{2}$.
Suppose there exist such $s,s^{\prime}$ and paths such that
$\left\\{w_{0},w_{1}\right\\}\cap\left\\{w_{0}^{\prime},w_{1}^{\prime}\right\\}\neq\emptyset$.
Since $c(w_{0})=c(w_{0}^{\prime})=0$ and $c(w_{1})=c(w_{1}^{\prime})=1$,
either $w_{0}=w_{0}^{\prime}$ or $w_{1}=w_{1}^{\prime}$.
* •
If $w_{0}=w_{0}^{\prime}$, then there cannot exist any $s^{\prime\prime}\in
B_{5}^{2}(u_{0})\setminus\left\\{s,s^{\prime},w_{0},w_{0}^{\prime},w_{1},w_{1}^{\prime},u_{0},u_{1},u_{2}\right\\}$,
contradicting our assumption about the size of $B_{5}^{2}(u_{0})$: if
$s^{\prime\prime}$ exists, then since it has at least 100 node-disjoint
2-paths to $u_{2}$, at least one of these paths, call it
$w_{0}^{\prime\prime},w_{1}^{\prime\prime}$, excludes nodes
$s,s^{\prime},w_{0},w_{0}^{\prime},w_{1},w_{1}^{\prime},u_{0},u_{1},u_{2}$. In
addition, since $s^{\prime}\in B_{5}^{2}(u_{0})$, it also has at least one
additional 2-path to $u_{2}$, call it $x_{0},x_{1}$, which excludes nodes
$s,s^{\prime\prime},w_{0},w_{0}^{\prime},w_{1},w_{1}^{\prime},u_{0},u_{1},u_{2},w_{0}^{\prime\prime},w_{1}^{\prime\prime}$.
We therefore have the following 10-cycle:
$s^{\prime\prime},w_{0}^{\prime\prime},w_{1}^{\prime\prime},u_{2},x_{1},x_{0},s^{\prime},w_{0}^{\prime}=w_{0},s,u_{0}$.
This contradicts our assumption that $s,s^{\prime},s^{\prime\prime}$ are free
neighbors of $u_{0}$.
* •
If $w_{0}\neq w_{0}^{\prime}$ but $w_{1}=w_{1}^{\prime}$: since $s^{\prime}\in
B_{5}^{2}(u_{0})$, it has at least one additional 2-path to $u_{2}$, call it
$x_{0},x_{1}$, which excludes nodes
$\left\\{s,w_{0},w_{0}^{\prime},w_{1}=w_{1}^{\prime},u_{0},u_{1},u_{2}\right\\}$.
Therefore the following 10-cycle is in the graph:
$s,w_{0},w_{1}=w_{1}^{\prime},w_{0}^{\prime},s^{\prime},x_{0},x_{1},u_{2},u_{1},u_{0}$.
Again, this contradicts our assumption that $s,s^{\prime}$ are free neighbors
of $u_{0}$.
We see that each $s\in B_{5}^{2}(u_{0})$ contributes at least 100 2-paths to
$u_{2}$, which are node-disjoint from the 2-paths contributed by any other
node in $B_{5}^{2}(u_{0})$, and therefore we can charge each node in
$B_{5}^{2}(u_{0})$ with 100 $1$-colored vertices in the neighborhood of
$u_{2}$ (which are not double-charged to any other node in
$B_{5}^{2}(u_{0})$). It follows that $\deg(u_{2})\geq 100B_{5}^{2}(u_{0})$.
Since we assume that $\deg(u_{0})\geq\deg(u_{2})$ and that
$\deg^{\prime}(u_{0})>\deg(u_{0})/2$, we get that
$|B_{5}^{2}(u_{0})|\leq\deg(u_{2})/100\leq\deg(u_{0})/100<\deg^{\prime}(u_{0})/50$,
a contradiction to our assumption that $B^{2}_{5}(u_{0})$ is large. ∎
Let $B_{5}^{3}(u_{0})$ be the set of free neighbors of $u_{0}$ that have at
least 10 node-disjoint 3-paths to $u_{3}$.
###### Lemma 9.
We have $|B_{5}^{3}(u_{0})|\leq 1$.
###### Proof.
Suppose not, and let $s,s^{\prime}\in B_{5}^{3}(u_{0})$ be distinct nodes. Let
$w_{0},w_{1},w_{2}$ be a 3-path of $s$ to $u_{3}$, and let
$w_{0}^{\prime},w_{1}^{\prime},w_{2}^{\prime}$ be a 3-path of $s^{\prime}$ to
$u_{3}$, which avoids nodes $s,w_{0},w_{1},w_{2},u_{0},u_{1},u_{2}$ (such a
path exists, since every node in $B_{5}^{3}(u_{0})$ has at least 10 node-
disjoint 3-paths to $u_{3}$). Then the following 10-cycle is in the graph:
$s,w_{0},w_{1},w_{2},u_{3},w_{2}^{\prime},w_{1}^{\prime},w_{0}^{\prime},s^{\prime},u_{0}$.
Therefore nodes $s,s^{\prime}$ are not free, a contradiction. ∎
For any $k$, the “last node in the proof”, $u_{k-1}$, is the easiest to
handle, using the following observation:
###### Observation 2.
For any $k\geq 2$, if $u_{0},\ldots,u_{2k-1}$ is a $2k$-cycle in the graph,
and $s\in N(u_{0})$ is free, then $s$ does not have a $(k-1)$-path
$w_{0},\ldots,w_{k-2}$ to $u_{k-1}$ which is node-disjoint from
$u_{0},\ldots,u_{k-2}$.
###### Proof.
If such a path existed, then we would have the following $2k$-cycle in the
graph: $s,u_{0},u_{1},\ldots,u_{k-1},w_{k-2},\ldots,w_{0}$. Therefore $s$
would not be a free node, contradicting our assumption. ∎
###### Corollary 13.
If $d_{k}\geq k$, then $B_{k}^{k-1}=\emptyset$.
###### Proof.
Suppose for the sake of contradiction that there is some node $s\in
B_{k}^{k-1}(u_{0})$. Since $s$ has at least $k-1$ node-disjoint 4-paths, at
least one of these paths avoids nodes $u_{0},u_{1},\ldots,u_{k-2}$. By
Observation 2, this cannot be. ∎
### 5.2 Exact Algorithm for Computing the Girth in Congest
We show that we can exactly compute the girth $g$ of a graph in time $g\cdot
n^{1-1/\Theta(g)}$ in Congest. For $g\geq\log n$, we can cap the running time
at $O(n)$, because a graph with girth $\geq\log n$ has $O(n)$ edges; thus, the
running time is $O(\min\left\\{g\cdot n^{1-1/\Theta(g)},n\right\\})$.
We say that a $k$-cycle is _light_ if all of its nodes have degree at most
$n^{\delta_{k}}$, where $\delta_{k}=k/2$ if $k$ is even, and
$\delta_{k}=(k-1)/2$ if $k$ is odd.
The meta-algorithm is as follows: first, we search for triangles, which can be
detected in time $\tilde{O}(n^{1/3})$ using the algorithm of [8]. Any node
that finds a triangle outputs “3” for the girth. We proceed to search for $k$
cycles for $k=4,\ldots$:
1. 1.
Search for light $k$-cycles, by simultaneously starting a depth-$\lceil
k/2\rceil$ BFS on the subgraph of nodes that have degree at most
$n^{\delta_{k}}$: each node $u$ with $\deg(u)\leq n^{\delta_{k}}$ initiates a
BFS, by sending a BFS token to its neighbors; the BFS token carries the ID of
the node that originated it, and the number of hops it has traveled. Nodes
with degree at most $n^{\delta_{k}}$ participate in the BFS by forwarded BFS
tokens that they receive, increasing their hop-count, until a maximum of
$\lceil k/2\rceil$ hops (of course, since we are carrying out a BFS, tokens
are forwarded only once).
If node $u$ receives the BFS token of a node $v$ from two distinct neighbors
of $u$, such that the total number of hops traveled on one side is $\lfloor
k/2\rfloor$ and on the other $\lceil k/2\rceil$, then node $u$ rejects and
outputs $k$.
2. 2.
Search for heavy $k$-cycles, by sampling a uniformly random node $s\in V$,
1. (a)
Carrying out a $k$-round BFS from $s$, to check if $s$ itself is on a
$k$-cycle; if node $s$ receives its own BFS token back from some neighbor, it
halts and outputs k.
2. (b)
Starting a depth-$\lceil k/2\rceil$ BFS from all neighbors of node $s$. Now,
each node is allowed to forward only _one_ BFS token, after which it stops
forwarding tokens. Again, if some node $u$ receives the BFS token of a node
$v$ from two distinct neighbors, with $\lfloor k/2\rfloor$ and $\lceil
k/2\rceil$ hops traveled on the two sides (resp.), it halts and outputs $k$.
We repeat this entire step (sampling $s$, etc.) $R=\Theta(n^{1-\delta_{k}})$
times.
For a given $k$, steps (1)-(2) above are called _phase $k$_ of the algorithm.
Observe that if $k$ is even, then a $k$-cycle $u_{0},\ldots,u_{k-1}$ is
detected when node $u_{k/2}$ receives the BFS token of $u_{0}$ from its
neighbors $u_{k/2-1}$ and $u_{k/2+1}$, with a hop count of $k/2$ on both
sides; if $k$ is odd, then a $k$-cycle $u_{0},\ldots,u_{k-1}$ is detected by
node $u_{(k-1)/2}$, which receives $u_{0}$’s token from $u_{(k-3)/2}$ and in
the next round from $u_{(k+1)/2}$, with hop counts of $(k-1)/2$ and $(k+1)/2$,
respectively; and simultaneously, the cycle is also detected by node
$u_{(k+1)/2}$, which receives $u_{0}$’s token first from $u_{(k+3)/2}$ and
then from $u_{(k-1)/2}$.
###### Lemma 10.
If some node halts in phase $k$, and the graph does not contain any cycle of
length less than $k$, then the graph contains a $k$-cycle.
###### Proof.
Suppose node $u$ outputs $k$, after receiving the token of node $v$ from two
neighbors $w_{1}\neq w_{2}$, with a hop-count of $\lfloor k/2\rfloor$ on
$w_{1}$’s side and $\lceil k/2\rceil$ on $w_{2}$’s side. Then the graph
contains paths $v=x_{0},\ldots,x_{\lfloor k/2\rfloor-1}=w_{1},u$ and
$v=y_{0},\ldots,y_{\lceil k/2\rceil-1}=w_{2},u$, which together form a
$k$-cycle. Moreover, the $k$-cycle is simple, as with the exception of
$v=x_{0}=y_{0}$ and $u$, these paths share no nodes: if there were some
$i,j>0$ such that $x_{i}=y_{j}$, then, taking the minimum such $i$ and, after
fixing $i$, the minimum such $j$, the simple cycle
$x_{0},\ldots,x_{i}=y_{i},y_{i-1},\ldots,y_{0}=x_{0}$ would be in the graph,
and its length would be $i+j$. Either $i<\lfloor k/2\rfloor$ or $j<\lceil
k/2\rceil$ (or both), so $i+j<\lfloor k/2\rfloor+\lceil k/2\rceil=k$, but we
assumed that the graph contains no cycles of length less than $k$.
The remaining case is that in one of the iterations, a sampled node $s$
receives its own token back while carrying out a $k$-round BFS. Then $s$
participates in a $k$-cycle: let $s=v_{0},v_{1},\ldots,v_{\ell}$, $\ell\leq
k-1$, be the path traveled by the token, with node $v_{\ell}$ forwarding the
token back to $s$. Since the graph does not contain any cycles of length less
than $k$, we must have $\ell=k-1$, and all nodes $v_{0},\ldots,v_{k-1}$ must
be distinct. Therefore the $k$-cycle $s=v_{0},\ldots,v_{k-1}$ is in the graph.
∎
###### Lemma 11.
If we reach phase $k$, and the graph contains a $k$-cycle and has no cycles of
length less than $k$, then with probability at least $2/3$, some node rejects
in phase $k$.
###### Proof.
Fix a cycle $u_{0},\ldots,u_{k-1}$ of length $k$. If the cycle is light, it
will be found in step (1) of the algorithm: since each cycle node has degree
at most $n^{\delta_{k}}$, all these nodes participate in the BFS.
If $k$ is even, then nodes $u_{k/2-1}$ and $u_{k/2+1}$ are able to forward the
BFS token of $u_{0}$ to $u_{k/2}$, and it arrives with hop count $k/2$ on both
sides; therefore node $u_{k/2}$ rejects. If $k$ is odd, then node
$u_{(k-3)/2}$ is able to forward the token of $u_{0}$ with hop count
$(k-1)/2$, and node $u_{(k+1)/2}$ is able to forward the token of $u_{0}$ with
hop count $k-(k+1)/2+1=(2k-k-1+2)/2=(k+1)/2$, causing node $u_{(k-1)/2}$ to
reject.
Now suppose that the cycle is heavy, and that node $u_{0}$ has
$\deg(u_{0})\geq n^{\delta_{k}}$. Then when we sample a uniformly random node
$s$, with probability at least $n/\deg(u_{0})\geq n^{1-\delta_{k}}$, we have
$s\in N(u_{0})$. When this occurs, we find the cycle: if node $s$ itself
participates in a $k$-cycle, then its $k$-round BFS will detect the cycle,
because $k$ rounds suffice for the BFS token of $s$ to return to it.
Otherwise, every neighbor of $s$, including $u_{0}$, starts a BFS. If the BFS
of $u_{0}$ is able to traverse both paths $u_{0},u_{1},\ldots,u_{\lfloor
k/2\rfloor}$ and $u_{0},u_{k-1},\ldots,u_{\lfloor k/2\lfloor}$, then node
$u_{\lfloor k/2\rfloor}$ receives it with hop counts $\lfloor k/2\rfloor$ and
$\lceil k/2\rceil$ respectively, and it rejects.
Recall that in order for the BFS token of $u_{0}$ to traverse these paths, it
must be the first token received by each node on the path. We show that the
BFS token of $u_{0}$ cannot be blocked on either side, as that would imply the
presence of a smaller cycle: suppose some node $u_{i}$ or $u_{-i}$,
$i\leq\lfloor k/2\rfloor$, receives the BFS token of a node $w_{0}\neq u_{0}$
before or at the same time as it receives $u_{0}$’s token. Let $i$ be minimal,
and assume w.l.o.g. that $u_{i}$ receives the token ($u_{-i}$ is symmetric,
since we take $i\leq\lfloor k/2\rfloor$). Then there exists a path
$w_{0},w_{1},\ldots,w_{j}=u_{i}$ along which $w_{0}$’s token travels to
$u_{i}$, where
$w_{0},\ldots,w_{j-1}\not\in\left\\{u_{0},\ldots,u_{i}\right\\}$. Also,
$w_{0}\in N(s)$, since only neighbors of $s$ start a BFS. Therefore, the graph
contains the cycle $s,w_{0},\ldots,w_{j}=u_{i},u_{i-1},\ldots,u_{0}$, whose
length is $i+j$. Since $w_{0}$’s token arrives at $u_{i}$ with or before
$u_{0}$’s token, we must have $j\leq i$. And since $i\leq\lfloor k/2\rfloor$,
the length of the other cycle is $i+j\leq 2\lfloor k/2\rfloor\leq k$. We see
that for this to occur, node $s$ must participate in a cycle of length at most
$k$, but we have already ruled out this possibility.
This shows that node $u_{0}$’s token is able to traverse both paths above: it
cannot be blocked until it is forwarded by $u_{\lfloor k/2\rfloor-1}$ and
$u_{\lfloor k/2\rfloor+1}=u_{-\left(\lfloor k/2\rfloor-1\right)}$ to
$u_{\lfloor k/2\rfloor}$, which then rejects. ∎
The correctness of the algorithm are implied by the following:
###### Corollary 14.
If the girth of the graph is $g$, then no node halts in phase $k<g$. Moreover,
with probability at least $2/3$, some node halts in phase $g$ and outputs $g$.
###### Proof.
By the first lemma, we see that when the girth is $g$, no node can halt at any
phase $k<g$. Now consider phase $k=g$: there are no cycles of length less than
$g$, and we already said that we do reach phase $k=g$, so by the second lemma,
with probability at least $2/3$, some node halts. ∎
The running time of the algorithm is characterized as follows: with
probability at least $2/3$, after $g\cdot n^{1-\delta_{g}}=g\cdot
n^{1-1/\lfloor g/2\rfloor}$ rounds, some node halts (and outputs $g$).
However, this is not an upper bound on the _expected_ time until the first
node halts. If we want to bound the expected running time, we can increase the
number of repetitions in each phase $k$ to $\Omega(n^{1-\delta_{g}}\cdot\log
n)$, so that the probability of not halting in phase $g$ is reduced to $1/n$.
The expected running time (until the first node halts) is then given by:
$O(g\cdot n^{1-\delta_{g}}\cdot\log
n)\cdot\left(1-\frac{1}{n}\right)+O(n)\cdot\frac{1}{n}=O(g\cdot
n^{1-\delta_{g}}\cdot\log n).$
(The second term uses the fact that we can cap the running time at $O(n)$
rounds, by switching to learning the entire graph if $g>\log n$.)
### 5.3 Implementation of the Exact Girth and Even Cycle Algorithm in Congest
Our implementation of the meta-algorithm for finding heavy cycles avoids
sampling one node $s$ uniformly at random, because we cannot do so in the
Congest model without incurring an additive overhead of $\Omega(D)$.
Notice that in each iteration of the meta-algorithm, nodes can take on one of
the following roles:
* •
Type $\mathcal{S}$: node $s$, a unique randomly-sampled node selected in step
(1) of the meta-algorithm.
* •
Type $\mathcal{NS}$: neighbors of node $s$. Each neighbor of $s$ that is
colored 0 initiates a BFS in step (4) of the meta-algorithm.
* •
Type $\mathcal{O}$: all other nodes. These nodes forward BFS tokens that reach
them (assuming there are not too many), and reject if they are colored $k$ and
receive the same BFS token along two disjoint paths.
The steps taken by each node in the meta-algorithm depend only on the type it
is assigned.
The implementation $\mathcal{A^{\prime}}$ is similar to the meta-algorithm
$\mathcal{A}$, but it executes each of the $R^{\prime}=\Theta(n^{1-1/k})$
iterations as follows:
1. 1.
Each node $u$ chooses a random priority $p(u)\in[n^{3}]$.
2. 2.
For $2R\cdot R^{\prime}$ rounds, each node $u$ forwards the smallest priority
it has received so far. This priority is stored in the local variable
$\mathit{pmin}(u)$.
3. 3.
If node $u$ has $\mathit{pmin}(u)=p(u)$ (i.e., node $u$ has not heard any
priority smaller than its own), it sets $\mathit{type}(u)=\mathcal{S}$, and
informs all its neighbors. The neighbors $v\in N(u)$ then set
$\mathit{type}(v)=\mathcal{NS}$. Nodes $u$ that do not have
$\mathit{pmin}(u)=p(u)$ and do not have a neighbor $v$ with
$\mathit{pmin}(v)=p(v)$ set $\mathit{type}(u)=\mathcal{O}$.
4. 4.
We now execute steps (2)-(5) of the meta-algorithm, with each node following
the role it was assigned above.
Let $\mathcal{U}$ be the event that there are no collisions in the choice of
priorities, i.e., for each $u\neq v$ we have $p(u)\neq p(v)$. This occurs with
probability at least $1-1/n$. Conditioned on $\mathcal{U}$, let $t$ be the
uniformly-random node that has the smallest priority, $p(t)=\min_{v\in
V}p(v)$, and let $U$ be the $2R\cdot R^{\prime}$-neighborhood of $t$. Then
after step (2), node $t$ is the only node in $U$ that sets
$\mathit{type}(t)=\mathcal{S}$, and its neighbors $v\in N(t)$ are the only
nodes in $U$ that set $\mathit{type}(v)=\mathcal{N}$. Thus, inside $U$, the
execution of $\mathcal{A}^{\prime}$ is equivalent to the execution of the
meta-algorithm $\mathcal{A}$ where we sample $s=t$, a uniformly random node.
Since $\mathcal{A}$’s running time is $R\cdot R^{\prime}$ rounds, this
suffices to ensure that a heavy $2k$-cycle will be detected with high
probability.
## 6 Barrier of $\Omega(n^{1/2+\alpha})$ for Lower Bounds on $C_{6}$-Freeness
In this section, we show that for any $\alpha>0$, an $\Omega(n^{1/2+\alpha})$
lower bound for $C_{6}$-freeness in CONGEST implies strong circuit complexity
lower bounds.
The proof is as follows:
1. 1.
First, we reduce the problem of $C_{6}$-freeness to directed triangle
freeness. We do so by showing that given an algorithm $\mathcal{A}_{1}$ for
solving directed triangle freeness in $O(T_{1}(n))$ rounds, we can solve
$C_{6}$-freeness in $\widetilde{O}(\sqrt{n}\cdot T_{1}(n))$ rounds w.h.p.;
thus, if we can prove a lower bound on $C_{6}$-freeness, we also obtain a
lower bound on directed triangle freeness.
2. 2.
It is already known that proving an $\Omega(n^{\alpha})$ lower bound on
_undirected_ triangle-freeness would imply new and powerful circuit lower
bounds [11]; the same argument also holds for _directed_ triangle-freeness.
For the sake of completeness, we give the full argument in Appendix A below.
The reduction from $C_{6}$ to directed triangles works as follows: the network
runs $R^{\prime}=O(\sqrt{n}\log{n})$ iterations of the heavy cycles finding
procedure of $C_{6}$. By the correctness of the $C_{6}$-finding algorithm, if
there is a $C_{6}$ copy with at least one node $v$ with $\deg(v)\geq\sqrt{n}$,
then the network rejects with high probability. Otherwise, the network removes
all nodes with $\deg(v)\geq\sqrt{n}$ from $G$.
Each node $v\in V$ proceeds to choose a random color $c(v)\in[6]$. Let
$G^{\prime}=(V,E^{\prime})$ be the directed graph where $(u,v)\in E^{\prime}$
if and only if there exists $w\in V$ such that both $(u,w),(w,v)\in E$ and
$c(u)+2\equiv c(w)+1\equiv c(v)\pmod{6}$.
Similarly to Section 5, we say that a $6$-cycle
$\left\\{u_{0},\dots,u_{5}\right\\}$ of $G$ is “colored correctly” if
$c(u_{i})=i$ for each $i=0,\dots,5$. The following claim establishes that such
$6$-cycles exist if and only if $G^{\prime}$ has a directed triangle.
###### Claim 1.
$G^{\prime}$ contains a directed triangle if and only if the colored $G$
contains a $C_{6}$ copy which is colored correctly.
###### Proof.
If $G^{\prime}$ has a directed triangle $(v_{1},v_{2},v_{3})$, then by
definition the colors of its vertices hold $c(v_{1})+4\equiv c(v_{2})+2\equiv
c(v_{3})\pmod{6}$, and there exist $w_{1},w_{2},w_{3}$ of colors
$c(v_{1})+3,c(v_{1})+1,c(v_{1})-1\pmod{6}$ respectively, such that
$(v_{1},w_{1},v_{2},w_{2},v_{3},w_{3})$ is a $6$-cycle. On the other hand, if
$G$ has a well colored $6$-cycle $(v_{1},v_{2},v_{3},v_{4},v_{5},v_{6})$, then
$(v_{1},v_{3},v_{5})$ is a triangle in $G^{\prime}$. ∎
If there is a $C_{6}$-copy in $G$ then it becomes a correctly colored copy of
$C_{6}$ with probability $\geq 1/6^{6}$. Therefore, if $G$ is not
$C_{6}$-free, then after repeating this algorithm $O(1)$ times, with high
probability at least once $G^{\prime}$ has a directed triangle (from Claim 1).
On the other hand, if $G$ is $C_{6}$-free then $G^{\prime}$ never contains a
directed triangle. This shows the correctness of the reduction.
We note that the network $G$ may simulate $G^{\prime}$ in the following sense:
first, every node $v\in V$ can learn its edges in $G^{\prime}$ in
$O(\sqrt{n})$ rounds: recall that since every node with $\deg(v)\geq\sqrt{n}$
was deleted from the graph, so we can afford to have each node broadcasts all
its remaining neighbors in $G$ and their colors; thus, every node learns its
neighbors in $G^{\prime}$. We call a node $w$ a bridge between $u,v$ if it is
a neighbor of both in $G$. We note that every $v$ also learns all its bridges
to all of its neighbors in $G^{\prime}$ in this process. Secondly, if the
network $G$ wishes to simulate an $r$ round protocol on $G^{\prime}$ where at
the end each node $v\in V$ knows its output state, it can do so with
$O(\sqrt{n}\cdot r)$ rounds in the following manner: for $i=1,\dots r$, let
$M_{i}(v)$ be the set of messages $v$ wishes to send in the $i$-th round of
the protocol, and for a message $m$ let $t(m)$ be the target node of that
message. First, every node $v$ sends each of its neighbors $w$ in $G$ the
subset of messages $\left\\{m\in M_{i}(v)\mid\text{$w$ is a bridge between $v$
and $t(m)$}\right\\}$. This can be done in $O(\sqrt{n})$ rounds as
$\deg(w)\leq\sqrt{n}$ and therefore every $w$ is a bridge between $v$ and at
most $O(\sqrt{n})$ other nodes. Following this, for every node $w\in V$ and
message $m$ it received, $w$ sends $m$ to $t(m)$. As for any given node $t$ a
node $w$ received at most $\sqrt{n}$ messages which have target $t$ (at most
one from each of its neighbors), it can send these nodes their messages in
$O(\sqrt{n})$ rounds. Overall, the simulation costs $O(r\cdot\sqrt{n})$
rounds.
Figure 3: Illustration of the reduction. The undotted edges are a $C_{6}$
cycle of $G$, and the dotted edges are a triangle of $G^{\prime}$. We note
that $G^{\prime}$ has a second triangle on vertices $1,3,5$.
In Appendix A, for the sake of completeness, we follow the exact lines of [11]
to show that for any $\alpha>0$, showing a lower bound of $\Omega(n^{\alpha})$
on directed triangle freeness implies strong circuit complexity lower bounds.
## 7 Discussion: Intuition Regarding Round Complexity for $C_{2k}$ Detection
in Congest
We believe that the $O(n^{1-1/k})$ round complexity for $C_{2k}$ detection in
the Congest model is the best that can be achieved, barring some major
improvement and a new approach which could have ramifications also for various
other problems in the Congested Clique model. The reasons are as follows:
###### Listing 6-cycles in graphs with high conductance.
As shown in [7], it takes $\Theta(n^{1-1/k+o(1)})$ rounds in the Congest
model, _in graphs with high conductance_ , to perform subgraph _listing_ for
$C_{2k}$. Therefore, primarily, if a faster algorithm for _detection_ is found
for the general Congest model, it would imply a separation between detection
and listing of $C_{2k}$ in the Congest model with high conductance. Such a
separation could imply new algorithms for other problems related to subgraph
listing also in the Congested Clique model, due to the similarity between the
Congested Clique and Congest with high conductance models.
Further, notice it is highly likely that such an algorithm would function
differently than any existing algorithm for $C_{2k}$ detection in the
Congested Clique model. The currently types of algorithms for $C_{2k}$
detection in the Congested Clique model are split into several categories, as
far as we are aware: (1) based on fast matrix multiplication, as in [5], and
(2) based on sparsity aware listing, as seen in [9, 5, 6, 25] and in this
paper, and tend to leverage the Túran number of $C_{2k}$ in order to list
faster. These types of algorithms fail when moving to the Congest model in
graphs with high conductance. The reason that the algorithms break is because
in the Congest model, if the input graph is sparser, there is less bandwidth
available to the entire network, in contrast with the Congested Clique model.
The first type of algorithms break since it is not known how to efficiently
utilize input sparsity to improve the running time of fast matrix
multiplication. This leads to the case where once the input graph is too
sparse, the message complexity of the algorithm remains the same, and since
the bandwidth available decreases, the round complexity increases. The latter
type of algorithms break since while the Túran number of $C_{2k}$ bounds the
sparsity of the graph, and thus reduces the message complexity for the listing
algorithms, it also implies less bandwidth for the network, effectively
canceling out the effect of the reduction in message complexity.
###### Finding 6-cycles in regular graphs.
Since the Túran number of 6-cycles is $\Theta(n^{4/3})$, when we restrict
attention to _regular_ graphs, the “interesting” degree for $C_{6}$-detection
is $\Theta(n^{1/3})$ (above this degree we know for sure that the graph
contains a 6-cycle, and below this degree the problem becomes easier).
Suppose we assign to each node $u$ a random color $c(u)$ in
$\left\\{0,\ldots,5\right\\}$ (i.e., we perform color-coding). Each cycle is
assigned consecutive colors $0,\ldots,5$ with constant probability, so we may
as well search only for this type of cycle (and then repeat a constant number
of times, to ensure that if there is a cycle, at least in one iteration, it
will be colored correctly). For a given node $u$ with $c(u)=3$, let $L(u)$ be
the 0-colored nodes that can be reached from $u$ by traversing a path of
length 3 with descending colors $3,2,1,0$, and let $R(u)$ be the 0-colored
nodes that can be reached from $u$ by traversing a path of length 3 with
ascending colors $3,4,5,0=6\bmod 6$. The problem of checking if $u$
participates in a well-colored 6-cycle boils down to checking whether
$R(u)\cap L(u)=\emptyset$.
Since we are working with a regular graph of degree $\Theta(n^{1/3})$, in the
“average” case (i.e., a random graph), we will have $|R(u)|,|L(u)|=\Theta(n)$.
Checking whether two sets of size $\Theta(n)$ intersect or not is a famous
problem in two-party communication complexity — the Disjointness problem,
which is well-known to require $\Omega(n)$ bits of communication (or, more
strongly, bits of _information_) between the two players. Intuitively, in
order to check whether $L(u)\cap R(u)=\emptyset$, node $u$ must collect
$\Theta(n)$ bits of information about each set, and since $u$ has degree
$\Theta(n^{1/3})$, this requires $\Theta(n^{2/3})$ rounds.
Unfortunately, despite trying for a long time, we have not been able to make
this intuition into a formal lower bound — and now we see that at least there
is a good reason for that, in the form of the barrier of Section 6.
## 8 Acknowledgments
This project was partially supported by the European Union’s Horizon 2020
Research and Innovation Programme under grant agreement no. 755839, by the
JSPS KAKENHI grants JP16H01705, JP19H04066 JP20H04139 and JP20H00579 and by
the MEXT Q-LEAP grant JPMXS0120319794.
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## Appendix A Barrier of $\Omega(n^{\alpha})$ for Lower Bounds on Directed
Triangle Freeness
Following the exact lines in [11] of the barrier for triangle freeness, we
show that for any $\alpha>0$ showing a lower bound of $\Omega(n^{\alpha})$ on
directed triangle freeness implies strong circuit complexity lower bounds.
This reduction is also very strongly based on the algorithm of [7, 8] for
triangle enumeration.
The first step is to reduce from general graphs to graphs with high
conductance: let $\mathcal{A}_{2}$ be an algorithm that solves directed
triangle freeness in a communication network with conductance $\phi$ and $n$
nodes, where every node is given as input $O(\deg(v))$ edges. Let
$T_{2}(n,\phi)$ be the round complexity of $\mathcal{A}_{2}$. We show that
given such an algorithm we can solve directed triangle freeness in
$O(\mathcal{A}_{2}(n,\operatorname{\mathrm{polylog}}{n})\log{n})$ rounds in
Congest.
###### Theorem 15 (Theorem 1 in [8]).
For $\epsilon\in(0,1)$, and a positive integer $k$, the network can partition
its edges into two sets $E_{r}$,$E_{m}$ satisfying the following conditions:
1. 1.
The conductance of each connected component $G_{i}$ of $E_{m}$ satisfies
$\Phi(G[V_{i}])\geq\phi$, where
$\phi=(\epsilon/\operatorname{\mathrm{polylog}}n)^{20\cdot 3^{k}}$.
2. 2.
$|E_{r}|<\epsilon m$.
This decomposition can be constructed using randomization in $O\left((\epsilon
m)^{1/k}\cdot\left(\frac{\operatorname{\mathrm{polylog}}n}{\epsilon}\right)^{20\cdot
3^{k}}\right)$ rounds w.h.p.
We apply the theorem with $\epsilon=1/6$, taking $k$ to be a large enough
constant so that the round complexity of the decomposition round complexity is
less than $O(n^{\alpha})$. We call the set of vertices of every connected
component of $E_{m}$ a _cluster_. A node is called _good_ if it has more edges
in $E_{m}$ than in $E_{r}$, and otherwise _bad_. We call an edge $e\in E_{m}$
_bad_ if at least one of its endpoints is bad.
###### Lemma 12 ([7]).
The number of bad edges is at most $2\epsilon m$.
Each cluster calculates its size $|U|$, and the number of edges in the
cluster. Since $\phi=\widetilde{O}(1)$, the diameter of each cluster is also
$\widetilde{O}(1)$, and therefore this can be done in $\widetilde{O}(1)$
rounds (for example, by constructing a spanning tree on the cluster, and
collecting the number of edges and nodes up the tree). Then, each cluster runs
$\mathcal{A}_{2}$ in parallel, where the input of each good node is all its
edges (including edges leaving the cluster), and for each bad node, its edges
in the cluster. We note that indeed by the definition of a good node, every
node has $O(\deg_{C}(v))$ edges as input for $\mathcal{A}_{2}$, where
$\deg_{C}(v)$ is $v$’s degree in its cluster. If $\mathcal{A}_{2}$ outputs
that there exists a directed triangle, the cluster rejects and terminates.
Otherwise, each cluster $U$ removes all good edges which are contained in $U$.
The network then recurses on the remain edges until $O(1)$ edges remain. We
note that as $|E_{good}|=m-|E_{bad}|-\epsilon m\geq m-3\epsilon m=m/2$, in
each iteration the network removes half of its edges, and the number of
iterations are at most $O(\log{n})$.
Clearly, if a node rejects then the graph contains a directed triangle. On the
other hand, recall that the input graph of $\mathcal{A}_{2}$ is the edges
adjacent to good nodes in $U$; therefore if $\mathcal{A}_{2}$ returns that
there is no directed triangle, the network may safely remove all edges between
two good nodes, as the triangle is contained in the union of inputs of both
good endpoints.
The following lemma from [11] shows that a dense cluster with good conductance
is able to simulate a _circuit_ , where the size, depth, and input size of the
circuit are related to the size, density and conductance of the cluster:
###### Lemma 13 ([11]).
Let $U$ be a graph $U=(V,E)$ with $|V|=n$ vertices and $m=N$ edges, with
mixing time $\tau_{\textrm{mix}}$. Suppose that for some constant $c$, the
function $f_{N}:\\{0,1\\}^{cN\log{n}}\rightarrow\\{0,1\\}$ is computed by a
circuit $\mathcal{C}$ of depth $P$, consisting of gates with constant fan-in
and fan-out, and at most $s\cdot N\cdot\log{n}$ wires for $s\leq n$. Then
there is an $O(P\cdot s\cdot\tau_{\textrm{mix}}\cdot 2^{O(\sqrt{\log
n}\log\log n)})$-round protocol in the Congest model on $U$ that computes
$f_{N}$ in the network assuming the input is partitioned between the nodes
such that each node has $O(\deg(v)\log{n})$ bits of input.
We consider the following family of functions
$f_{N}:\left\\{0,1\right\\}^{2N\log{N}}\rightarrow\\{0,1\\}$: given an
encoding of a graph333For a graph with $N$ edges
$\\{(u_{i},v_{i})\\}_{i=1}^{N}$, the graph is encoded by the string
$u_{1}.id,v_{1}.id,\dots,u_{N}.id,v_{N}.id$, where each id is padded to
$\lceil\log{N}\rceil$ bits, and the rest of the string is padded in such a
manner that indicates that there are no further edges. with at most $N$ edges,
does the graph contain a directed triangle?
###### Corollary 16.
If directed triangle freeness cannot be solved in less than $c_{1}n^{\alpha}$
rounds for any $c_{1}>0$, then there exist constants $c_{2},c_{3}>0$ such that
there is no family of circuits that solve for all $N$ the function $f_{N}$
with $c_{2}N^{\alpha/4}/2^{c_{3}\sqrt{\log{n}}\log\log{n}}$ depth and at most
$c_{2}N^{1+\alpha/4}/2^{c_{3}\sqrt{\log{n}}\log\log{n}}$ wires.
###### Proof.
Let $\mathcal{F}$ be an infinite family of graphs for which directed triangle
freeness cannot be solved in less than $c_{1}n^{\alpha}$ rounds. Let $c_{4}>0$
be a constant such that the conductance of the clusters obtained by Theorem 15
is less than $\log^{c_{4}}{n}$. Assume by contradiction that for sufficiently
large $c_{2},c_{3}$ there is an infinite family of circuits solving for any
$N$ the function $f_{N}$ with
$c_{2}N^{\alpha/4}/2^{c_{3}\sqrt{\log{n}}\log\log{n}}$ depth and
$c_{2}N^{1+\alpha/4}/2^{c_{3}\sqrt{\log{n}}\log\log{n}}$ wires. Then by Lemma
13 taking $s=n^{\alpha/2}/2^{c_{3}\sqrt{\log{n}}\log\log{n}}$ and
$P=n^{\alpha/2}/2^{c_{3}\sqrt{\log{n}}\log\log{n}}$ (both of which are larger
than $c_{2}(n^{2})^{\alpha/4}/2^{c_{3}\sqrt{\log{n}}\log\log{n}}$) there
exists an algorithm with round complexity $c_{1}n^{\alpha}/\log{n}$ that
solves directed triangle freeness on graphs with conductance at least
$\phi=\log^{c_{4}}{n}$, where the input of each node is $O(\deg(v))$. By the
reduction, we get that there is a $c_{1}>0$ such that directed triangle
freeness can be solved in any network with $c_{1}n^{\alpha}$ rounds, which is
a contradiction. ∎
All together, we see that proving a lower bound of the form
$\Omega(n^{\alpha})$ on directed triangle freeness, for any $\alpha>0$, would
imply _superlinear_ lower bounds on the number of wires in circuits of
_polynomial depth_. Such lower bounds on any explicit function are far beyond
the reach of current circuit complexity techniques; currently, the best lower
bounds even for _logarithmic_ -depth circuits is at most linear in the input
size (see, e.g. [31]).
|
# Synaptic metaplasticity in binarized neural networks
Axel Laborieux, Maxence Ernoult , Tifenn Hirtzlin1, Damien Querlioz1 Centre de
Nanosciences et de Nanotechnologies, Université Paris-SaclayMila, Université
de Montréal
## Summary
Unlike the brain, artificial neural networks, including state-of-the-art deep
neural networks for computer vision, are subject to “catastrophic forgetting”
[1]: they rapidly forget the previous task when trained on a new one.
Neuroscience suggests that biological synapses avoid this issue through the
process of synaptic consolidation and _metaplasticity_ : the plasticity itself
changes upon repeated synaptic events [2, 3]. In this work, we show that this
concept of metaplasticity can be transferred to a particular type of deep
neural networks, binarized neural networks (BNNs) [4], to reduce catastrophic
forgetting. BNNs were initially developed to allow low-energy consumption
implementation of neural networks. In these networks, synaptic weights and
activations are constrained to $\\{-1,+1\\}$ and training is performed using
_hidden_ real-valued weights which are discarded at test time. Our first
contribution is to draw a parallel between the metaplastic states of [2] and
the hidden weights inherent to BNNs. Based on this insight, we propose a
simple synaptic consolidation strategy for the hidden weight. We justify it
using a tractable binary optimization problem, and we show that our strategy
performs almost as well as mainstream machine learning approaches to mitigate
catastrophic forgetting, which minimize task-specific loss functions [5], on
the task of learning pixel-permuted versions of the MNIST digit dataset
sequentially. Moreover, unlike these techniques, our approach does not require
task boundaries, thereby allowing us to explore a new setting where the
network learns from a stream of data. When trained on data streams from
Fashion MNIST or CIFAR-10, our metaplastic BNN outperforms a standard BNN and
closely matches the accuracy of the network trained on the whole dataset.
These results suggest that BNNs are more than a low precision version of full
precision networks and highlight the benefits of the synergy between
neuroscience and deep learning [6].
## Hidden weights as metaplastic states
The problem of forgetting in artificial neural networks results from a
dilemma: synapses need to be updated in order to learn new tasks but also to
be protected against further changes in order to preserve knowledge. In a
foundational neuroscience work, Fusi et al. show than in small Hopfield
networks, catastrophic forgetting can be addressed by introducing a hidden
metaplastic state that controls the plasticity of the synapse [2]. Synapses
can assume only $+1$ or $-1$ weight, with the metaplastic state modulating the
difficulty for the synapse to switch. Therefore, in this scheme, repeated
potentiation of a positive-weight synapse will only affect its metaplastic
state and not its actual weight. Here, we remark that the way that BNNs are
trained is remarkably similar to this situation. In BNNs, synapses can also
only assume $+1$ or $-1$ weight, and they feature a hidden real weight
($W^{\rm h}$), which is updated by backpropagation. The synaptic weight
changes between $+1$ and $-1$ only when $W^{\rm h}$ changes sign, suggesting
that $W^{\rm h}$ can be seen as a metaplastic state modulating the difficulty
for the actual weight to change sign. However, standard BNNs are as prone to
catastrophic forgetting as conventional neural networks. In [2], Fusi et al.
showed that the metaplastic changes should make subsequent affect plasticity
exponentially to mitigate forgetting, whereas $W^{\rm h}$ affects weight
changes only linearly in BNNs. Therefore, in this work, we propose to adapt
the learning process of BNNs so that the larger the magnitude of a hidden
weight $W^{\rm h}$, the more difficult to switch its associated binarized
weight $W^{\rm b}=\mbox{sign}(W^{\rm h})$. Denoting $U_{W}$ the update
provided by the learning algorithm, we implement:
$\displaystyle W^{\rm h}$ $\displaystyle\leftarrow W^{\rm h}-\eta U_{W}\cdot
f_{\rm meta}(m,W^{\rm h})\quad{\rm if}\quad U_{W}W^{\rm h}>0$ $\displaystyle
W^{\rm h}$ $\displaystyle\leftarrow W^{\rm h}-\eta U_{W}\quad{\rm otherwise.}$
As in the metaplasticity model of [2] where synaptic plasticity decreases
exponentially with the metaplastic state, we choose $f_{\rm meta}(m,W^{\rm
h})={\rm tanh^{{}^{\prime}}}(m\cdot W^{\rm h})$ to produce an exponential
decay for large metaplastic states $W^{\rm h}$, where $m$ is an hyperparameter
that controls the consolidation.
## Toy problem study
To validate the interpretation of hidden weights as metaplastic states, we
first focus on a highly simplified binary optimization task that we solve in a
way analogous to the BNN training process. We want the binarized weights
$W^{\rm b}$ to minimize a quadratic loss $\mathcal{L}$, as depicted by the
color map on Fig. 1(a) in two dimensions, with $W^{*}$ as the global optimum.
We assume that $W^{\rm h}$ is updated by loss gradients computed with
binarized weights, similarly to BNNs:
$\displaystyle W^{{\rm h}}_{t+1}=W^{{\rm
h}}_{t}-\eta\frac{\partial\mathcal{L}}{\partial W}(W^{\rm b}_{t}).$
We can show that if the infinite norm of $W^{*}$ is lesser than one, some
hidden weights diverge as $t\to\infty$. This is because $W^{\rm h}$ is updated
by loss gradients computed at the corners of the square, in contrast with
conventional optimization. More importantly, if we define importance of the
binarized weight as the increase of the loss $\Delta\mathcal{L}$ when the
weight is switched to the opposite value, we can prove that the speed of
divergence of the hidden weight is directly linked to the importance of the
binarized weight. For instance, in Fig. 1(a), $W^{\rm b}_{x}$ is more
important than $W^{\rm b}_{y}$ for optimization. Finally, we plot
$\Delta\mathcal{L}$ versus $|W^{\rm h}|$ in Fig. 1(b), (c) for higher
dimensions and for a BNN trained on MNIST and observe that the correspondence
between important weights and hidden weight divergence still holds, justifying
the fact that consolidating synapses with diverging hidden weights as our
proposal does, is a promising route for mitigating catastrophic forgetting.
## Experimental results
### Continual learning benchmark.
We now apply our consolidation strategy to the permuted MNIST benchmark on two
hidden layers perceptrons of varying number of neurons. We show in Fig.
1(d),(e) the average test accuracy as a function of the number of tasks
learned so far. We observe that our technique indeed allows sequential task
learning and performs almost as well as Elastic Weight Consolidation (EWC) [5]
adapted to BNNs (the importance factor is computed with the binarized weights)
over a wide range of hidden layer sizes when learning up to 20 tasks. We
choose $m=1.35$ and $\lambda_{EWC}=5\cdot 10^{-3}$ for EWC.
### Learning from a stream of data.
By construction, our approach does not require to update the importance factor
between two consecutive tasks. Building on this asset, we explore a new
setting, which we call stream learning, and where a task is learned by
learning sub-parts of the full dataset sequentially, with all classes evenly
distributed in each subset. We choose Fashion MNIST (FMNIST) and CIFAR-10 for
our experiments. The architectures used are a perceptron with two hidden
layers of 1,024 units for FMNIST and a VGG-16 convolutional architecture for
CIFAR-10. We plot on Fig. 1(f), (g) the test accuracy reached by those
networks when metaplasticity is used (red) or not (blue). We see that our
approach comes closer to the accuracy reached when the full dataset is learned
at once (straight lines) than the non-metaplastic counterpart. Overall, these
results highlight the benefit of metaplasticity models from neuroscience when
applied to machine learning.
## Acknowledgement
This work was supported by European Research Council Starting Grant NANOINFER
(reference: 715872).
## References
* [1] French, R. M., Trends in cogn. sci. (1999).
* [2] Fusi et al. Neuron (2005).
* [3] Abraham, W. C. Nat. Rev. Neurosci. (2008).
* [4] Courbariaux et al. arXiv:1602.02830 (2016)
* [5] Kirkpatrick et al. PNAS (2017).
* [6] Richards et al. Nat. Neurosci. (2019).
* [7] Zenke et al. PMLR (2017).
Figure 1: (a) Quadratic binarized optimization in two dimensions. (b-c)
Average loss increase when switching $W^{\rm b}$ versus normalized hidden
weight, for the binary quadratic problem (b) and for a BNN on MNIST (c). (d-e)
Permuted MNIST benchmark with our method (d) and EWC (e), where the x axis
labels the number of learned tasks, with one color per network size. Fashion
MNIST (f) and CIFAR-10 (g) test accuracy in the stream learning setting
allowed by our approach (red) compared to a standard BNN (blue). Horizontal
rules denote full dataset training baseline.
|
# Real-Time Limited-View CT Inpainting and Reconstruction with Dual Domain
Based on Spatial Information
# Real-Time Limited-View CT Inpainting and Reconstruction with Dual Domain
Based on Spatial Information
###### Abstract
Low-dose Computed Tomography is a common issue in reality. Current reduction,
sparse sampling and limited-view scanning can all cause it. Between them,
limited-view CT is general in the industry due to inevitable mechanical and
physical limitation. However, limited-view CT can cause serious imaging
problem on account of its massive information loss. Thus, we should
effectively utilize the scant prior information to perform completion. It is
an undeniable fact that CT imaging slices are extremely dense, which leads to
high continuity between successive images. We realized that fully exploit the
spatial correlation between consecutive frames can significantly improve
restoration results in video inpainting. Inspired by this, we propose a deep
learning-based three-stage algorithm that hoist limited-view CT imaging
quality based on spatial information. In stage one, to better utilize prior
information in the Radon domain, we design an adversarial autoencoder to
complement the Radon data. In the second stage, a model is built to perform
inpainting based on spatial continuity in the image domain. At this point, we
have roughly restored the imaging, while its texture still needs to be finely
repaired. Hence, we propose a model to accurately restore the image in stage
three, and finally achieve an ideal inpainting result. In addition, we adopt
FBP instead of SART-TV to make our algorithm more suitable for real-time use.
In the experiment, we restore and reconstruct the Radon data that has been cut
the rear one-third part, they achieve PSNR of 40.209, SSIM of 0.943, while
precisely present the texture.
Index Terms— limited-view CT, deep learning, spatial information, domain
transformation, real-time
## 1 Introduction
Computed Tomography has been successfully applied in medicine, biology,
industry and other fields, providing huge help for industrial production,
medical research and people’s daily life [1]. Nevertheless, the radiation dose
brought by CT scanning may somehow have a negative effect on human body that
cannot be neglect. Thus, it is crucial for CT scanning to lower its radiation
dose [2] in accordance with ALARA (as low as reasonably achievable) [3]. Low-
dose Computed Tomography (LDCT) can be realized through current reduction,
sparse sampling and limited-view scanning. Among these, limited-view CT is
really general because that we often encounter mechanical and physical
restriction in the industry which makes it difficult for the machine to scan
through an object. Despite the general application of limited-view CT, its
imaging leads to some grievous problems like blur [4], artifacts [5, 6, 7, 8]
and low signal-to-noise ratio [9, 1], they undoubtedly have a great influence
on clinical diagnosis. Thus, it is crucial for researchers to fully utilize
the limited prior information to effectively complement the fragmentary data.
Traditional analytical reconstruction algorithms, such as FBP [10], have high
requirements for data integrifty. When the radiation dose is reduced,
artifacts in reconstructed images will increase rapidly [11]. In order to
upgrade the quality of reconstructed images, many researchers have proposed
various algorithms for LDCT imaging reconstruction, and we conclude them into
several paths that are presented in Fig.1 for better comprehension.
Fig. 1: The technology roadmap of prevailing CT inpainting and reconstruction
algorithms, the dash line in this figure refers to the reconstruction step
from the Radon domain to the image domain through FBP or SART-TV.
Iterative Reconstruction Algorithms are represented by the red line in Fig.1,
which can directly reconstruct damaged Radon data into target results in the
image domain. Model-based iterative reconstruction (MBIR) algorithm [12], also
known as statistical image reconstruction (SIR) method, combines the modeling
of some key parameters to perform high-quality reconstruction of LDCT. Using
image priors in MBIR can effectively improve the image reconstruction quality
of LDCT scans [13, 14], while still have the high computational complexity.
In addition to the prior information, various regularization methods have
played a crucial role in iterative algorithms of CT reconstruction. The most
typical regularization method is the total variation (TV) method [15]. In the
light of TV, researchers came up with more reconstruction methods, such as TV-
POCS [16], TGV [17] and SART-TV [18] which was proposed on the basis of SART
[19]. Those algorithms can suppress image artifacts to a certain extent so as
to improve imaging quality. In addition, dictionary learning is often used as
a regularizer in MBIR algorithms [20, 21, 22, 23], and multiple dictionaries
are beneficial to reducing artifacts caused by limited-view CT reconstruction.
With the development of computing power, deep learning-based methods [24, 25,
26, 9, 27, 28, 29] have been applied to the restoration of LDCT reconstructed
images in recent years. The methods can be roughly divided into the below
three categories.
Image Inpainting algorithms are presented by blue lines in Fig.1, they firstly
reconstruct the damaged Radon data into the damaged image with artifacts, then
reduce the artifacts and noises in the image domain. Lots of researchers are
currently using convolutional neural network (CNN) and deep learning
architecture to perform this procedure [30, 1, 31, 32, 5, 33, 34, 6, 35, 7,
36]. Zhang et al [30] proposed a data-driven learning method based on deep
CNN. RED-CNN [1] combines the autoencoder, deconvolutional network and
shortcut connections into the residual encoder-decoder CNN for LDCT imaging.
Kang et al [31] applied deep CNN to the wavelet transform coefficients of LDCT
images, used directional wavelet transform to extract the directional
component of artifacts. Wang et al [33] developed a limited-angle
translational CT (TCT) image reconstruction algorithm based on U-Net [34].
Since Goodfellow et al. proposed Generative Adversarial Nets (GAN) [35] in
2014, GAN has been widely used in various image processing tasks, including
the post-processing of CT images. Xie et al. [7] proposed an end-to-end
conditional GAN with joint loss function, which can effectively remove
artifacts.
Fig. 2: The overall architecture of our proposed three-stage restoration and
reconstruction algorithm for limited-view CT imaging.
Sinogram Inpainting algorithms are presented by green lines in Fig.1, they
firstly restore the missing part in the Radon domain, then reconstruct it into
the image domain to get the final result [37, 38, 39, 40, 41]. Li et al. [37]
proposed an effective GAN-based repairing method named patch-GAN, which trains
the network to learn the data distribution of the sinogram to restore the
missing sinogram data. In another paper [38], Li et al. proposed SI-GAN on the
basis of [32], using a joint loss function combining the Radon domain and the
image domain to repair “ultra-limited-angle” sinogram. In 2019, Dai et al.
[39] proposed a limited-view cone-beam CT reconstruction algorithm. It slices
the cone-beam projection data into the sequence of two-dimensional images,
uses an autoencoder network to estimate the missing part, then stack them in
order and finally use FDK [42] for three-dimensional reconstruction. Anirudh
et al. [40] transformed the missing sinogram into a latent space through a
fully convolutional one-dimensional CNN, then used GAN to complement the
missing part. Dai et al. [41] calculated the geometric image moment based on
the projection-geometric moment transformation of the known Radon data, then
estimated the projection-geometric moment transformation of the unknown Radon
data based on the geometric image moment.
Sinogram Inpainting and Image Refining algorithms are presented by yellow
lines in Fig.1, they firstly restore the missing part in the Radon domain,
then reconstruct the full-view Radon data into the image domain so as to
finely repair the image to obtain higher quality [43, 44, 45, 46, 8]. In 2017,
Hammernik et al. [43] proposed a two-stage deep learning architecture, they
first learn the compensation weights that account for the missing data in the
projection domain, then they formulate the image restoration problem as a
variational network to eliminate coherent streaking artifacts. Zhao et al.
[44] proposed a GAN-based sinogram inpainting network, which achieved
unsupervised training in a sinogram-image-sinogram closed loop. Zhao et al.
[45] also proposed a two-stage method, firstly they use an interpolating
convolutional network to obtain the full-view projection data, then use GAN to
output high-quality CT images. In 2019, Lee et al. [46] proposed a deep
learning model based on fully convolutional network and wavelet transform. In
the latest research, Zhang et al. [8] proposed an end-to-end hybrid domain CNN
(hdNet), which consists of a CNN operating in the sinogram domain, a domain
transformation operation, and a CNN operating in the image domain.
Inspired by the combination of the two stages, we implement Radon data
completion through our proposed adversarial autoencoder (AAE) in stage one. In
the second and third stage, after enriching the information through Radon data
completion, we construct the Radon data into the image domain and realize the
image inpainting in a ”coarse-to-fine” [47] manner.
However, all of the above algorithms merely focus on a single image slice
while neglecting the abundant spatial correlation between consecutive image
slices. Consequently, these algorithms may still have trouble to reach an
ideal level of limited-view CT inpainting and reconstruction that can
precisely presents the image texture. During our investigation of video
inpainting [48, 49], we realize the significance of making full use of spatial
correlation and continuity between consecutive image slices. Therefore, we
propose an origin cascade model in stage two called Spatial-AAE to fully
utilize the spatial continuity, thereby breaking the limitation of two-
dimensional space.
It is also worth mentioning that, unlike other current limited-view CT
inpainting and reconstruction algorithms, we use FBP [10] instead of SART-TV
[18] to speed up the reconstruction process. Besides, our models do not limit
resolution of the input data, therefore can be well generalized to various
datasets. In our experiments, we compare our algorithm with the other four
prevalent algorithms under four sorts of damaged data, exhibiting its
prominent performance and robustness.
The organization of this paper is as follows, Sec II presents the design
details of our proposed algorithm and models, Sec III shows our experimental
results, and we finally conclude our research work in Sec IV.
## 2 Methods
This paper proposes a three-stage restoration and reconstruction algorithm for
limited-view CT imaging, and its overall architecture is shown in Fig.2. In
the first stage, after the limited-view Radon data is preprocessed, we input
it into the Adversarial Autoencoder we designed for data completion to obtain
the full-view Radon data. In the second stage, the output of stage one is
first reconstructed into the image domain, and combined with two consecutive
slices before and after to form a group, then we sent this group into our
proposed Spatial-AAE model to perform image restoration based on spatial
information. It is worth noting that through the above two stages of
restoration and reconstruction, most of the texture in the image ground truth
can be restored, but the result still cannot clearly reflect the precise
details, which may pose obstacles for the practical applications. Therefore,
we built the Refine-AAE high-precision inpainting network in stage three,
utilizing the idea of ”coarse-to-fine” [47] in deep learning to refine the
image in patches. The details of our algorithm are shown below.
### 2.1 Data Preprocessing
In order to provide more prior information, we adopt the data preprocessing
method from paper [5], as shown in Fig.3. For the limited-view Radon data
$\boldsymbol{R}_{lv}$, we first transform it into the image data
$\boldsymbol{I}_{recon}$ through inverse radon transformation, and then
convert the image into the full-view Radon data $\boldsymbol{R}_{fv}$ through
Radon transformation. We crop this full-view Radon data for preliminary
completion of the missing part in the original data, so as to obtain the fused
Radon data $\boldsymbol{R}_{merge}$.
Fig. 3: Procedure of data preprocessing.
### 2.2 Algorithm Pipeline
#### 2.2.1 Stage 1: Limited-view Data Completion in the Radon Domain
For the input limited-view Radon data, we need to apply it as the prior
information to perform angle completion in the first stage. Due to the fact
that U-Net [34] is widely use in medical imaging currently, we propose an
adversarial autoencoder with U-Net as the backbone. Its overall architecture
is shown in Fig.4.
Fig. 4: The overall architecture of our proposed adversarial autoencoder in
stage one.
Fig. 5: (a) is the result of reconstructing the original limited-view Radon
data into the image; (b) is the result of reconstructing the full-view Radon
data generated from stage one into the image; (c) is the ground truth of the
image.
We modified U-Net as the autoencoder in our adversarial autoencoder, which
includes an encoder that downsamples the image to extract the representative
feature and a decoder that upsamples the feature to restore the image. The
precise structure of our autoencoder can be seen from TABLE I, where (Ic, Oc)
represents the in-channel and out-channel of the convolutional layer. In its
convolutional layers, the kernel size is 3$\times$3, the stride and padding
are both 1, and the kernel size is 2$\times$2 in all of its pooling layers. In
all of its deconvolution layers, the kernel size is 2$\times$2, and the stride
is 1. In order to upgrade the model’s ability of restoration, we combine this
autencoder with a discriminator whose structure is the same as the encoder
shown in TABLE I. As can be seen from Sec IV, adding this discriminator can
effectively improve the model’s performance.
Fig. 6: The overall architecture of our proposed Spatial-AAE model in stage
two.
#### 2.2.2 Stage 2: Image Restoration Based on Spatial Information
Fig.5 shows that after we reconstruct the output from stage one into the image
domain, the image texture can be partly restored, while there are still some
artifacts and blurry area that can bring severe obstacles for clinical
diagnosis. Therefore, in the second stage, we propose the Spatial-AAE model
based on the spatial correlation between consecutive image slices to
significantly improve the quality of damaged image.
According to our knowledge, in previous studies of CT imaging restoration and
reconstruction algorithms, scholars seemed to neglect the rich spatial
information between consecutive image slices, and only repaired and
reconstructed images in two-dimensional space. During the process of
investigating and comparing the fields of image inpainting and video
inpainting, we were surprised to find that the third dimension usually
contains rich data coherence and continuity, which is very beneficial for
restoring successive images. Thus, we suppose that the effective use of the
third-dimensional information may remarkably improve the quality of restored
images. Inspired by the utilization of the third-dimensional information in
FastDVDNet [46], we come up with the Spatial-AAE network, whose overall
architecture is shown in Fig.6, it can be divided into Spatial autoencoder and
discriminator.
Table 1: Details of the Autoencoder in Stage One (a) Encoder | | (b) Decoder
---|---|---
Layer | (Ic, Oc) | Layer | (Ic, Oc)
Conv1_1 | (1, 32) | UpConv6 | (512, 256)
Conv1_2 | (32, 32) | Concat | [UpConv6, Conv4]
Pool1 | Maxpool | Conv6_1 | (512, 256)
Conv2_1 | (32, 64) | Conv6_2 | (256, 256)
Conv2_2 | (64, 64) | UpConv7 | (256, 128)
Pool2 | Maxpool | Concat | [UpConv7, Conv3]
Conv3_1 | (64, 128) | Conv7_1 | (256, 128)
Conv3_2 | (128, 128) | Conv7_2 | (128, 128)
Pool3 | Maxpool | UpConv8 | (128, 64)
Conv4_1 | (128, 256) | Concat | [UpConv8, Conv2]
Conv4_2 | (256, 256) | Conv8_1 | (128, 64)
Pool4 | Maxpool | Conv8_2 | (64, 64)
Conv5_1 | (256, 512) | UpConv9 | (64, 32)
Conv5_2 | (512, 512) | Concat | [UpConv9, Conv1]
| | | Conv9_1 | (64, 32)
| | | Conv9_2 | (32, 12)
| | | Conv9_3 | (12, 1)
The input of the spatial autoencoder is five consecutive image slices
$S=\\{\boldsymbol{s}_{i-2},\boldsymbol{s}_{i-1},\boldsymbol{s}_{i},\boldsymbol{s}_{i+1},\boldsymbol{s}_{i+2}\\}$,
we divide them into three sets of data
$S_{1}=\\{\boldsymbol{s}_{i-2},\boldsymbol{s}_{i-1},\boldsymbol{s}_{i}\\},S_{2}=\\{\boldsymbol{s}_{i-1},\boldsymbol{s}_{i},\boldsymbol{s}_{i+1}\\}$
and
$S_{3}=\\{\boldsymbol{s}_{i},\boldsymbol{s}_{i+1},\boldsymbol{s}_{i+2}\\}$.
Then, they are sent into the AE block respectively, and their output is
concatenated as
$S^{\prime}=\\{\boldsymbol{s}^{\prime}_{i-1},\boldsymbol{s}^{\prime}_{i},\boldsymbol{s}^{\prime}_{i+1}\\}$,
this set of data is input into the AE block again to obtain the final restored
result. The spatial autoencoder network can be expressed as (1), where $F$ is
the spatial autoencoder model and $G$ is the AE block. The specific details of
the AE block and discriminator in Fig.6 can be seen from TABLE 1, they are the
same as they are in the AAE model of stage one.
$\boldsymbol{s}^{\prime\prime}_{i}=F(S)=G\left(G(S_{1}),G(S_{2}),G(S_{3})\right)$
(1)
#### 2.2.3 Stage 3: Image Refining on Patches
It can be seen from Fig.7 that after the above two stages of dual-domain
combined inpainting and reconstruction, the original limited-view Radon data
can be restored to a relatively satisfying extent. However, the overall
details are still not precise enough.
Fig. 7: (a) contains the result of reconstructing the full-view Radon data
from stage one output into images; (b) contains the result of reconstructing
the full-view Radon data from stage two output into images; (c) contains the
ground truth of images.
Therefore, in the third stage, we utilize the idea of “coarse to fine” in deep
learning to propose the Refine-AAE model, so as to further refine the texture
of repaired images. The overall structure of the Refine-AAE network can be
seen from Fig.8. Give the input image $\boldsymbol{I}_{input}$, the model
divides it into four patches and concatenate them into a set of sequence
$\\{\boldsymbol{I}_{p1},\boldsymbol{I}_{p2},\boldsymbol{I}_{p3},\boldsymbol{I}_{p4},\\}$,
We send it into the autoencoder for inpainting in patches and obtain the
output as
$\\{\boldsymbol{I}^{{}^{\prime}}_{p1},\boldsymbol{I}^{{}^{\prime}}_{p2},\boldsymbol{I}^{{}^{\prime}}_{p3},\boldsymbol{I}^{{}^{\prime}}_{p4},\\}$.
The model integrates this output into $\boldsymbol{I}_{pred}$ and combines it
with the ground truth $\boldsymbol{I}_{GT}$ into pair for discriminator’s
judgment.
The autoencoder and discriminator in the Refine-AAE model are the same as the
Spatial-AAE model, they can be seen from TABLE I.
Fig. 8: The overall architecture of our proposed Refine-AAE model in stage
three.
### 2.3 Loss Function
In all three stages, we use multi-loss function to optimize the autoencoder
model, it can be expressed as (2).
$l_{AE}=\alpha_{1}l_{MSE}+\alpha_{2}l_{adv}+\alpha_{3}l_{reg}$ (2)
$l_{MSE}$ calculates the mean square error between the restored image and the
ground truth image, it is widely used in various image inpainting tasks
because it can provide an intuitive evaluation for the model’s prediction. The
expression of $l_{MSE}$ can be seen from (3).
$l_{MSE}=\dfrac{1}{W\times
H}\sum_{x=1}^{W}\sum_{y=1}^{H}\left(\boldsymbol{I}_{x,y}^{GT}-G_{AE}(\boldsymbol{I}^{input})_{x,y}\right)^{2}$
(3)
where $G_{AE}$ is the auto-encoder, $\boldsymbol{I}^{GT}$ and
$\boldsymbol{I}^{input}$ are the ground truth image and the input image, $W$
and $H$ are the width and height of the input image respectively. $l_{adv}$
refers to the adversarial loss. The autoencoder can fool the discriminator by
making its prediction as close to the ground truth as possible, so as to
achieve the ideal image restoration outcome. Its expression can be seen from
(4).
$l_{adv}=1-D\left(G_{AE}(\boldsymbol{I}^{input})\right)$ (4)
where $D$ is the discriminator and $G_{AE}$ is the autoencoder. $l_{reg}$ is
the regularization term of our multi-loss function. Since noises may have a
huge impact on the restoration result, we add a regularization term to
maintain the smoothness of the image and also avoid the problem of
overfitting. TV Loss is commonly used in image analysis tasks, it can reduce
the difference between adjacent pixel values in the image to a certain extent.
Its expression can be seen from (5).
$l_{reg}=\dfrac{1}{W\times H}\sum_{x=1}^{W}\sum_{y=1}^{H}\left\|\nabla
G_{AE}(\boldsymbol{I}^{input}_{x,y})\right\|$ (5)
where $G_{AE}$ is the auto-encoder, $\boldsymbol{I}_{input}$ is the input
image, $W$ and $H$ are the width and height of the input image respectively.
$\nabla$ calculates the gradient, $\left\|\right\|$ calculates the norm.
For the optimization of the discriminator, the loss function should enable the
discriminator to better distinguish between real and fake inputs. The loss
function can be seen from (6).
$l_{DIS}=1-D(\boldsymbol{I}^{GT})+D\left(G_{AE}(\boldsymbol{I}^{input})\right)$
(6)
where $D$ is the discriminator, $G^{AE}$ is the auto-encoder,
$\boldsymbol{I}^{GT}$ and $\boldsymbol{I}^{input}$ are the ground truth image
and the input image respectively. The discriminator outputs a scalar between
zero and one, when the output is closer to 1, the discriminator thinks that
the input is more likely to be real. On the opposite, when the output is
closer to 0, it thinks the input is more likely to be fake. Therefore,
$1-D(\boldsymbol{I}^{GT})$ makes the output closer to one when the
discriminator inputs real images, and
$D\left(G_{AE}(\boldsymbol{I}^{input})\right)$ makes the output closer to zero
when the discriminator inputs fake images generated by the autoencoder.
## 3 Experiment
Our experiment data comes from 1000 cases in the LIDC-IDRI [50] dataset. We
divided cases 1 to 200 into the test set, cases 201 to 400 into the validation
set, and cases 401 to 1000 into the training set. The CT imaging (size
512$\times$512) is stored as DCM files in the LIDC-IDRI dataset. After
processing it as an array, we reconstruct it to the Radon domain (size
512$\times$180) through the Radon transformation, and perform post-60-degree
clipping on it as the input data of the overall model. During the training
process, we set the learning rate to 1e-4, using ADAM [51] as our model
optimizer, and Leaky ReLU [52] as the nonlinear activation. For the multi-loss
function, we refer to the method in paper [53], where $\alpha_{1}$,
$\alpha_{2}$ and $\alpha_{3}$ are set to 1, 1e-3, and 2e-8 respectively. It is
worth mentioning that there is no fully connected layer in our model, so it
can flexibly handle input images of different resolutions and be applied to
various datasets. In addition, unlike other deep learning-based algorithms,
the reconstruction part of our algorithm adopts FBP instead of SART-TV which
requires a relatively high level of computational complexity, so our method
can be better applied to practical application scenarios such as clinical
diagnosis. Although FBP takes much shorter time than SART-TV, its
reconstruction results have a certain gap with SART-TV. In order to realize
the practical application value of our algorithm, we manage to compensate the
performance of FBP through the superiority of our model design. Also, we
increase the damage degree of Radon data in 4.2 to test the robustness of our
algorithm. We create four types of damaged Radon data and use this algorithm
to repair and reconstruct them. The experimental results prove that our
algorithm can effectively restore these data, thus owns outstanding
robustness. In 4.1, we conduct ablation experiments on models of each stage to
prove the necessity and effectiveness of our structural design. In 4.2, we
compared our algorithm with other four types of algorithms, and test these
algorithms on four various degrees of damaged data.
### 3.1 Ablation Study
#### 3.1.1 Stage1
We first explore the necessity of fusing data in the Radon domain (refers to
Fig.3). We input the directly cut Radon data and the fused Radon data into the
stage one model shown in Fig.4 for data completion, and compare their outputs
with the Radon ground truth. The experimental results can be seen in TABLE II,
CR stands for the directly cut Radon, MR stands for the merged Radon, RCR
stands for the restored CR from stage1, RMR stands for the restored MR from
stage1.
It can be concluded from TABLE II that the fused Radon data can obtain better
experimental results due to its richer prior information, and provide more
texture for the subsequent image restoration steps. The visualized results can
be seen in Fig.9.
Table 2: Details of Various Data Preprocessing Methods | CR | MR | RCR | RMR
---|---|---|---|---
PSNR | 8.714 | 18.196 | 38.549 | 48.181
SSIM | 0.656 | 0.936 | 0.987 | 0.995
Fig. 9: Visualized results obtained from different data preprocessing methods,
(a) is the directly cut Radon data; (b) is the restored result of (a); (c) is
the fused Radon data; (d) is the restored result of (c); (e) is the Radon
ground truth.
In addition, we also explore the architecture of stage one’s adversarial
autoencoder model, and proved that it is essential to add the discriminator
reasonably. We restore the input data with: (1) The autoencoder shown in TABLE
I (a); (2) Combination of the autoencoder and the discriminator in TABLE I,
their experimental results can be seen from TABLE III.
Table 3: Results of Using Diffrent Model Structure in Stage One | AE | AE + D
---|---|---
PSNR | 40.129 | 48.181
SSIM | 0.983 | 0.995
It can be summarized from the above data that adding a discriminator can
greatly improve the data completion result. It can help stage one model to
improve the sinogram data PSNR by a relatively large margin.
Fig. 10: Visualized results obtained from different model structure in stage
one, (a) is the input fused Radon data; (b) is the restored Radon data from
structure AE; (c) is the restored Radon data from structure AE+D; (d) is the
Radon ground truth.
From the visualized comparison in Fig.10, we can see that if we only use this
single autoencoder, the inpainting result would have a large blurred area, and
adding the discriminator can improve this situation.
#### 3.1.2 STAGE2
For the image restoration task in this stage, we adopt the Spatial-AAE model
described in 3.2 to make full use of spatial information. In order to reflect
the prominence of this structure, we compare this model with the AAE model
from stage one, which does not contain any spatial structure. For the same
input fused Radon data, the experimental results can be seen in TABLE IV. It
can be seen from the results that, due to the fact that the Spatial-AAE model
makes full use of the third-dimensional prior information, it can effectively
improve the overall performance of stage two.
Table 4: Results of Using Different Model Structure in Stage Two | AAE | Spacial-AAE
---|---|---
PSNR | 37.384 | 39.646
SSIM | 0.929 | 0.940
#### 3.1.3 STAGE3
In this stage, the input image is divided and concatenated, and then sent to
the Refine-AAE model for finer inpainting. We believe that the way of
intercepting patches during the training process will have a certain impact on
the experimental results, so we test the following three types of interception
methods (As shown in Fig.11): (1) Randomly crop four patches (size
256$\times$256) from the input image (size 512$\times$512); (2) Crop the four
corners out of the input image; (3) Crop the four corners out of the input
image, and then adjust them into the same pattern through different flipping
method.
All of the methods above get an array of size (4, 256, 256), we input it into
the Refine-AAE model (refers to Fig.7) to finely repair the image, and the
experimental results of these three methods are shown in TABLE V.
Table 5: Results of Using Patch-Cropping Methods in Stage Three | Random Crop | Corner Crop | Corner Crop + Flip
---|---|---|---
PSNR | 40.111 | 40.209 | 40.060
SSIM | 0.942 | 0.943 | 0.942
Fig. 11: Methods of cropping patches in stage three
We can conclude that method (2) achieves the best image restoration result,
this is different from our initial assumption. We originally assumed that
patches generated from method (3) can enable the model to learn the mapping
easier. However, the fact is that method (2) gets the better result. We
suppose this is because different patterns in method (2) play a crucial role
in data enhancement, thus prevent the model from overfitting.
### 3.2 Algorithm Comparison
In order to reflect the superiority of our algorithm, we have compared its
performance with the following four sorts of algorithms: (1) Analytical
reconstruction algorithm FBP; (2) Iterative reconstruction algorithm SART
combined with TV regularization; (3) Image inpainting, after reconstructing
the limited-view Radon data into images through FBP, apply the AAE model to
image restoration; (4) Sinogram inpainting, first use the AAE model to
complement the Radon data, and then adopt FBP to reconstruct it to images. We
also test these algorithms on two types of input data: (1) the directly cut
Radon data; (2) the fused Radon data. For Radon data with its post 60 degrees
being cut off, the performance of the above algorithms is shown in TABLE VI
and Fig.12, MR in this table means input the fused Radon data.
Table 6: Results of Different Algorithms Applied to Different Data Preprocessing Methods Algorithms | PSNR | SSIM
---|---|---
(1) FBP | 11.272 | 0.364
(2) FBP+MR | 12.354 | 0.452
(3) SART-TV | 14.727 | 0.635
(4) SART-TV+MR | 21.518 | 0.807
(5) Image Inpainting (II) | 35.566 | 0.916
(6) Image Inpainting + MR | 36.388 | 0.927
(7) Sinogram Inpainting (SI) | 27.345 | 0.800
(8) Sinogram Inpainting + MR | 28.960 | 0.859
(9) Ours | 40.209 | 0.943
Fig. 12: Histograms of different algorithms applied to different data preprocessing methods. Fig. 13: Visualized results of different algorithms applied to different data preprocessing methods. Fig. 14: Error maps of different algorithms applied to different data preprocessing methods. Fig. 15: Histograms of different algorithms applied to different data preprocessing methods on different input data | CUT-MID-60 | CUT-MID-90 | CUT-MID-120
---|---|---|---
Algorithms | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
(1) FBP | 11.131 | 0.362 | 10.350 | 0.289 | 9.636 | 0.217
(2) FBP + MR | 12.182 | 0.446 | 11.432 | 0.391 | 10.525 | 0.309
(3) SART-TV | 14.758 | 0.610 | 12.945 | 0.515 | 10.492 | 0.372
(4) SART-TV + MR | 21.036 | 0.784 | 17.523 | 0.722 | 13.166 | 0.592
(5) Image Inpainting (II) | 31.717 | 0.895 | 30.157 | 0.873 | 28.507 | 0.846
(6) Image Inpainting + MR | 32.031 | 0.895 | 30.422 | 0.876 | 28.999 | 0.849
(7) Sinogram Inpainting (SI) | 26.834 | 0.793 | 25.673 | 0.763 | 24.606 | 0.705
(8) Sinogram Inpainting + MR | 27.789 | 0.828 | 26.582 | 0.795 | 25.210 | 0.755
(9) Ours | 34.248 | 0.919 | 32.624 | 0.900 | 30.975 | 0.876
Table 7: Results of Different Algorithms Applied to Different Data
Preprocessing Methods on Different Input Data
From the results above, we can see that the idea of merging Radon data brings
additional prior information on every type of algorithm, thus improves their
performance by different margin. Besides, merging Radon data is particularly
helpful for SART-TV. Under the condition of using the same AAE model,
restoration in the image domain is more effective than restoration in the
Radon domain. Our algorithm combines the Radon domain and the image domain,
complements, reconstructs, restores and refines the input limited-view Radon
data, can finally improves the image PSNR to 40.209 and SSIM to 0.943. It
upgrades the quality of CT imaging by a large margin, realizes the accurate
restoration of its texture. Comparison of the visualized results can be seen
in Fig.13, we also present the corresponding error maps in Fig.14 to reflect
the difference in performance between different algorithms. To prove the
robustness of our algorithm, we test various degrees of damage on the input
Radon data, including (1) cut off the middle 60 degrees (1/3 of the original
data); (2) cut off the middle 90 degrees (1/2 of the original data); (3) cut
off the middle 120 degrees (2/3 of the original data). We implement the above
nine algorithms in TABLE VI on these three types of input data, and their
performance is shown in TABLE VII and Fig.15. It can be seen that losing data
in the middle can cause more damage than in the rear. With the increase of the
cropping ratio, the inpainting performance of these algorithms has also been
greatly affected. Our algorithm however, proves its outstanding robustness
under various conditions. Even when cutting the middle 120 degrees off the
original Radon data, our method can still restore the seriously damaged
imaging to PSNR of 30.975. Also, our method can exceed the other methods in
TABLE VII by a large margin under varying degrees of damaged data.
## 4 Conclusion
In order to improve the quality of the seriously damaged limited-view CT
imaging, we propose a three-stage restoration and reconstruction algorithm
based on spatial information, which combines the Radon domain and the image
domain, and utilizes the idea of “coarse to fine” to restore the image with
high definition. In the first stage, we designed an adversarial autoencoder to
complement the limited-view Radon data. In the second stage, we first
reconstruct the Radon data into images through FBP, and then send this image
into the Spatial-AAE model we built to achieve image artifact and noise
reduction based on spatial correlation between consecutive slices. In the
third stage, we propose the Refine-AAE network to finely repair the image in
patches, so as to achieve the accurate restoration of the image texture. For
Radon data with limited angle of 120 degrees (cut off one-third of the full-
view Radon data), our algorithm can increase its PSNR to 40.209, and SSIM to
0.943. At the same time, due to the fact that our model does not restrict
input resolution, can adapt to varying degrees of damage, and also can be
quickly implemented, our algorithm has generalization, robustness and
significant practical application value.
In our future work, we hope to incorporate our three-stage model into an end-
to-end network that can be simultaneously trained and tested. As we all know,
such large amount of parameters may be hard to optimize, we plan to solve this
problem by using tricks such as data augmentation and dropout, while also
lightweight model backbone like MobileNet.
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# Equal-time kinetic equations in a rotational field
Shile Chen Ziyue Wang<EMAIL_ADDRESS>Pengfei Zhuang Physics
Department, Tsinghua University, Beijing 100084, China
###### Abstract
We investigate quantum kinetic theory for a massive fermion system under a
rotational field. From the Dirac equation in curved space we derive the
complete set of kinetic equations for the spin components of the covariant and
equal-time Wigner functions. While the particles are no longer on a mass shell
in general case due to the rotation-spin coupling, there are always only two
independent components, which can be taken as the number and spin densities.
With the help from the off-shell constraint we obtain the closed transport
equations for the two independent components in classical limit and at quantum
level. The classical rotation-orbital coupling controls the dynamical
evolution of the number density, but the quantum rotation-spin coupling
explicitly changes the spin density.
## I Indroduction
From the lattice simulations karsch of quantum chromodynamics (QCD), it is
widely accepted that there exists a phase transition from a hadron gas to a
quark-gluon plasma at high temperature. The experimental efforts of high
energy nuclear collisions at Relativistic Heavy Ion Collider (RHIC) and Large
Hadron Collider (LHC) have provided many sensitive signatures qm2019 of the
new state of matter created in the early stage of the collisions. Considering
the very short lifetime of the collision zone, the highly excited quark-gluon
system spends a considerable fraction of its life in a non-equilibrium state,
and the dynamical tool to treat dissipative processes in nuclear collisions
and the approach to hydrodynamic evolution is in principle quantum transport
theory. In classical transport theory, all the physical currents are connected
with the number distribution function $f$. The quantum mechanical analogue of
$f$ is the Wigner function $W$ which is a $4\times 4$ matrix in spin space
degroot . A relativistic and gauge covariant kinetic theory for quarks and
gluons has been derived, both in a classical framework heinz and as a quantum
kinetic theory elze based on the Wigner functions defined in covariant
degroot and equal-time BialynickiBirula:1991tx phase spaces. Many
applications to the quark-gluon plasma, such as linear color response, color
correlations and collective plasma oscillations heinz2 ; elze2 , have been
discussed in this framework using a semi-classical expansion of the quantum
transport theory.
The study on QCD phase transitions at finite temperature and density is
recently extended to including electromagnetic fields, since the strongest
fields in nature is believed to be generated in nuclear collisions at RHIC and
LHC energies tuchin ; deng . Under such strong electromagnetic fields some
anomalous phenomena for massless quarks, such as chiral magnetic effect
Kharzeev:2007jp ; Fukushima:2008xe , are experimentally discovered in non-
central nuclear collisions star ; alice . Since the created fields drop down
very fast and appear only in the very beginning of the collisions, most of the
theoretical investigations is in the frame of quantum kinetic theory
Hidaka:2016yjf ; Huang:2018wdl ; Gao:2018wmr ; Weickgenannt:2019dks ;
Wang:2019moi ; Wang:2020pej ; kharzeev . Apart from the electromagnetic
fields, the strongest rotational field in nature can also be produced in
nuclear collisions. The maximum magnitude is expected to be about $0.1m_{\pi}$
in noncentral Au+Au collisions at RHIC energy STAR:2017ckg ; Deng:2016gyh . A
direct consequence of such strong rotation is the polarization of final state
hadrons Adam:2019srw through spin-orbital coupling at quark level. Different
from the electromagnetic fields which rapidly decay in time, the angular
momentum conservation during the evolution of the collision will lead to a
more visible rotational effect on the final state. The other advantage of
rotational effect is that it becomes more strong in intermediate nuclear
collisions where there might be new physics related to high baryon density.
There have been a lot of theoretical investigations on the rotational effect
and spin polarization in high energy nuclear collisions Liang:2004ph ;
Becattini:2007sr ; Gao:2012ix ; Csernai:2013bqa ; Jiang:2015cva ;
Becattini:2016gvu ; Florkowski:2017ruc ; Ivanov:2019wzg ; Gao:2018jsi . In
this paper, we aim to set up a quantum kinetic theory in a rotational field.
There are two editions for the quantum kinetic theory in the frame of Wigner
function. One is the covariant version degroot for the Wigner function
$W(x,p)$ defined in $8$-dimensional phase space, and the other is the equal-
time version BialynickiBirula:1991tx for the Wigner function
$W_{0}(x,{\bm{p}})$ in $7$-dimensional phase space. The advantage of the
former is the explicit covariance under a Lorentz transformation, and the
latter is directly related to the physical distributions defined in equal-time
phase space. Of course, $W_{0}$ is not manifestly Lorentz covariant. In both
the covariant and equal-time formalisms, an important aspect of the kinetic
theory is that, the complex kinetic equation can be split up into a constraint
and a transport equation, where the former is a quantum extension of the
classical mass-shell condition, and the latter is a generalization of the
Vlasov-Boltzmann equation. The complementarity of these two ingredients is
essential for a physical understanding of quantum kinetic theory zhuang ;
zhuang2 ; zhuang3 . In this paper, we focus on a complete description of the
equal-time Wigner function in a rotational field, by considering the coupled
constraint and transport equations. We will derive the classical and quantum
transport equations for the two independent spin components, namely the number
density and spin density, by using the semi-classical expansion of the kinetic
equations.
The vortical field ${\bm{\omega}}$ of a system can be either generated self-
consistently by the curl of the medium velocity
${\bm{\omega}}={\bm{\nabla}}\times{\bm{v}}$ or considered as an external
field, depending on the particles we describe in kinetic equations. For light
quarks which are constituents of the medium, the quark vorticity is just the
rotation of the medium, but for heavy flavors which are considered as a probe
of the medium, the vorticity in kinetic equations can be treated as an
external field. We will consider in this paper the latter and neglect the
collision terms among particles, in order to focus on the coupling between
particles and the external rotational field. This version of the mean field
approximation, which treats the particles quantum mechanically, but uses the
classical approximation for the field, is widely used in electrodynamic
kinetic theory BialynickiBirula:1991tx ; Hidaka:2016yjf ; Huang:2018wdl ;
Gao:2018wmr ; Weickgenannt:2019dks ; Wang:2019moi ; Wang:2020pej ; kharzeev .
The paper is organized as follows. In section II we obtain in curved space the
Dirac equation and its non-relativistic limit under a rotational field which
is the basis to derive kinetic equations in the frame of Wigner function. We
then calculate the kinetic equations for the covariant Wigner function
$W(x,p)$ and its spin components in section III. By taking the energy
integration of the covariant kinetic equations we obtain the constraint and
transport equations for the spin components of the equal-time Wigner function
$W_{0}(x,{\bm{p}})$ in section IV. In section V we semi-classically solve the
equal-time equations. We will focus on the coupling between the particle spin
and the rotational field. We summarize the work in section VI.
## II Dirac and Schrödinger Equation in rotational field
The starting point to derive a relativistic or non-relativistic kinetic theory
for quarks in Wigner function formalism is the Dirac equation or Schrödinger
equation. The system under a rotational field can be equivalently regarded as
a system at rest in a rotating frame, as has been discussed in
Ref.Jiang:2016wvv where the rotation of a quark system enhances the chiral
symmetry restoration strongly and Ref.Liu:2018xip where the covariant kinetic
theory for chiral fermions in external electromagnetic field is extended to
curved space systematically. To avoid confusion, we use in the following the
indices $\\{\mu,\nu,\lambda,\sigma\\}$ and $\\{\alpha,\beta,\gamma,\delta\\}$
to separately describe Lorentz vectors and tensors in curved and flat space,
known respectively as coordinate and non-coordinate basis Nakahara:2003nw .
The Lagrangian density for fermions under mean-field approximation in non-
coordinate basis has the following form
$\mathcal{L}=\sqrt{-g}\bar{\psi}\left(i\gamma^{\alpha}\partial_{\alpha}-m\right)\psi,$
(1)
where $\sqrt{-g}$ is related to the coordinate we choose. Considering that, in
coordinate basis the tangent space $T_{p}M$ and cotangent space $T^{*}_{p}M$
are expanded in $\partial_{\mu}$ and $dx^{\mu}$, the coordinate transformation
between the two spaces can be expressed as
$\hat{e}_{\alpha}=e_{\alpha}^{\ \mu}\partial_{\mu},\ \ \ \ \ \ \ \ e_{\alpha\
}^{\ \mu}\in GL(m,\mathbb{R}),$ (2)
where $\\{\hat{e}_{\alpha}\\}$ is required to be orthonormal with respect to
$g\ (=g_{\mu\nu}dx^{\mu}\otimes dx^{\nu})$, which means the relation
$g(\hat{e}_{\alpha},\hat{e}_{\beta})=e_{\alpha\ }^{\ \mu}e_{\beta\ }^{\
\nu}g_{\mu\nu}=\eta_{\alpha\beta}$ or inversely $g_{\mu\nu}=e^{\alpha\ }_{\
\mu}e^{\beta\ }_{\ \nu}\eta_{\alpha\beta}$. With the requirement of local
Lorentz invariance, the Lagrangian density in coordinate basis becomes
$\mathcal{L}=\sqrt{-g}\bar{\psi}\left[i\gamma^{\alpha}e_{\alpha\ }^{\
\mu}\left(\partial_{\mu}+\frac{i}{2}\Gamma^{\alpha\ \beta}_{\ \mu\
}\Sigma_{\alpha\beta}\right)-m\right]\psi$ (3)
with the affine connection $\Gamma^{\alpha\ \beta}_{\ \mu\
}=\eta^{\beta\gamma}e^{\alpha\ }_{\ \nu}(\partial_{\mu}e_{\ \gamma}^{\nu\
}+e^{\ \sigma}_{\gamma\ }\Gamma^{\nu}_{\ \mu\sigma})$.
We now consider a system under rotation with a constant vorticity denoted by
${\bm{\omega}}$. The local velocity of this rotating frame is given by ${\bf
v}={\bm{\omega}}\times{\bm{x}}$, and the space-time metric is written as
$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}1-{\bf v}^{2}&-v_{1}&-v_{2}&-v_{3}\\\
-v_{1}&-1&0&0\\\ -v_{2}&0&-1&0\\\ -v_{3}&0&0&-1\end{matrix}\right),$
$\displaystyle g^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}1&-v_{1}&-v_{2}&-v_{3}\\\
-v_{1}&-1+v_{1}^{2}&v_{1}v_{2}&v_{1}v_{3}\\\
-v_{2}&v_{1}v_{2}&-1+v_{2}^{2}&v_{2}v_{3}\\\
-v_{3}&v_{1}v_{3}&v_{2}v_{3}&-1+v_{3}^{2}\end{matrix}\right),$ (4)
where we have introduced a specific tetrad Jiang:2016wvv ,
$e^{\alpha\ }_{\ \mu}=\delta^{\alpha\ }_{\
\mu}+\delta^{\alpha}_{i}\delta_{\mu}^{0}v_{i},\ \ \ \ e_{\alpha\ }^{\
\mu}=\delta_{\alpha\ }^{\ \mu}-\delta_{\alpha}^{0}\delta_{i}^{\mu}v_{i},$ (5)
and $\bm{v}=\bm{\omega}\times\bm{r}$ is the velocity of the coordinate
transformation. It is worth noticing that, the choice of the tetrad is not
unique, since the degrees of freedom of a $n$-dimensional metric is $(n+1)n/2$
and of the tetrad $n^{2}$. After plunging the chosen tetrad into the
Lagrangian, we obtain
$\mathcal{L}=\bar{\psi}\left[i\gamma^{\mu}\partial_{\mu}+\gamma_{0}{\bm{\omega}}\cdot\left({\bm{x}}\times(-i{\bm{\nabla}})+{\bm{s}}\right)-m\right]\psi$
(6)
with
${\bm{s}}=-\frac{1}{2}\gamma_{0}\gamma_{5}{\bm{\gamma}}=\frac{1}{2}\text{diag}({\bm{\sigma}},{\bm{\sigma}})$.
Under the choice of the space-time metric (II), the higher orders of the
rotational field, namely the terms $\sim{\bm{\omega}}^{2}$ and
${\bm{\omega}}^{3}$, vanish automatically, and only the linear term
$\sim{\bm{\omega}}$ appears in the Lagrangian density. However, since the
velocity of the coordinate transformation $\bm{v}=\bm{\omega}\times\bm{r}$ has
been taken in the non-relativistic form, the Lagrangian (6) is valid only for
small ${\bm{\omega}}$, with $|\omega x|\ll 1$. From the structure of the
Lagrangian, the rotational field ${\bm{\omega}}$ serves as a chemical
potential coupled to the total angular momentum
${\bm{J}}={\bm{x}}\times\hat{\bm{p}}+{\bm{s}}$ which is conserved during the
evolution of the system.
With the known Lagrangian density it is easy to derive the Dirac equation for
quarks in the rotational field,
$\left[i\gamma^{\mu}\partial_{\mu}+\gamma_{0}{\bm{\omega}}\cdot{\bm{J}}-m\right]\psi=0.$
(7)
The corresponding Schrödinger equation can be obtained by considering the non-
relativistic limit of the Dirac equation in a standard way. Considering the
stationary solution of the Dirac equation, $\psi(x)=\psi({\bm{x}})e^{-iEt}$,
the stationary wave function $\psi({\bm{x}})$ satisfies the equation
$\left[\left(\gamma_{0}{\bm{\gamma}}\cdot\hat{\bm{p}}-{\bm{\omega}}\cdot{\bm{J}}\right)+m\gamma_{0}\right]\psi=E\psi.$
(8)
To move to the familiar non-relativistic expression, we separate the quark
energy into the mass and the kinetic energy, $E=m+\epsilon$, write the
stationary wave function in a two-component form,
$\psi({\bm{x}})=(\phi({\bm{x}}),\chi({\bm{x}}))^{T}$, and take the Pauli-Dirac
representation for the $\gamma$-matrix, $\gamma_{0}=\left(\begin{matrix}I&0\\\
0&-I\end{matrix}\right)$ and
${\bm{\gamma}}=\left(\begin{matrix}0&{\bm{\sigma}}\\\
-{\bm{\sigma}}&0\end{matrix}\right)$, the two-component Dirac equation is then
written as
$\displaystyle\hat{\bm{p}}\cdot{\bm{\sigma}}\chi-{\bm{\omega}}\cdot{\bm{J}}\phi=\epsilon\phi,$
$\displaystyle\hat{\bm{p}}\cdot{\bm{\sigma}}\phi-\left({\bm{\omega}}\cdot{\bm{J}}+2m\right)\chi=\epsilon\chi$
(9)
with the total angular momentum
${\bm{J}}={\bm{x}}\times\hat{\bm{p}}+{\bm{\sigma}}/2$ in its two-dimensional
form. From the second equation, the small component $\chi$ can be expressed as
$\chi={\hat{\bm{p}}\cdot{\bm{\sigma}}\over
2m+\epsilon+{\bm{\omega}}\cdot{\bm{J}}}\phi\approx{\hat{\bm{p}}\cdot{\bm{\sigma}}\over
2m}\phi$ (10)
to the first order in $1/m$. Substituting it into the first equation leads to
the Schrödinger equation for the large component $\phi$,
$\left[{\hat{\bm{p}}^{2}\over
2m}-{\bm{\omega}}\cdot{\bm{J}}\right]\phi=\epsilon\phi$ (11)
which is the same as obtained by using non-relativistic Galilean
transformation anandan .
To the second order in $1/m$, the small component $\chi$ becomes
$\chi={1\over 2m}\left(1-{\epsilon+{\bm{\omega}}\cdot{\bm{J}}\over
2m}\right)\hat{\bm{p}}\cdot{\bm{\sigma}}\phi,$ (12)
Taking the commutation relations between $x_{i}$ and $\hat{p}_{j}$ and between
$\sigma_{i}$ and $\sigma_{j}$ and employing the above Schrödinger equation to
the first order in $1/m$, we obtain the Scgrödinger equation to the second
order,
$\left[{\hat{\bm{p}}^{2}\over 2m}-{\hat{\bm{p}}^{4}\over
8m^{3}}-{\bm{\omega}}\cdot{\bm{J}}\right]\phi=\epsilon\phi,$ (13)
the only relativistic correction is to the kinetic energy.
## III Covariant kinetic equations
The core ingredient to describe the transport phenomena of a non-equilibrium
system is the distribution function in phase space. Wigner function is the
quantum analogue to the classical distribution function, and has been widely
adopted in the investigation of quantum transport phenomena, such as spin
polarization Gao:2012ix ; Wang:2020pej in heavy ion collisions and pair
creation in QED systems BialynickiBirula:1991tx ; zhuang ; Sheng:2018jwf . The
covariant Wigner function $W(x,p)$ for fermions is defined as the ensemble
average of the Wigner operator in vacuum state, and the Wigner operator is the
four-dimensional Fourier transform of the covariant density matrix degroot ,
$W(x,p)=\int{d^{4}y\over(2\pi)^{4}}\sqrt{-g(x)}e^{ip^{\mu}y_{\mu}}\langle\psi(x_{+})U(x_{+},x_{-})\bar{\psi}(x_{-})\rangle$
(14)
with $x_{\pm}=x\pm y/2$, where the gauge link is defined as
$U(x_{+},x_{-})=e^{ig\int_{-1/2}^{1/2}dsy^{\mu}A_{\mu}(x+sy)}$ (15)
with the gauge field $A_{\mu}$. Since we focus in this paper on the rotational
effect, we neglect the gauge field and in turn the gauge link.
The covariant kinetic equation in the Wigner function formalism is derived by
calculating the first-order derivatives of the density matrix and using the
Dirac equations for the fields $\psi$ and $\bar{\psi}$. After a
straightforward calculation, we obtain the equation of motion for the Wigner
function in phase space which is equivalent to the equation of motion for the
field in coordinate space,
$\left[\gamma^{\mu}K_{\mu}+{\hbar\over
2}\gamma^{5}\gamma^{\mu}\omega_{\mu}-m\right]W(x,p)=0$ (16)
with the definitions of $K_{\mu}=\Pi_{\mu}+{i\hbar\over 2}D_{\mu}$ and
$\omega_{\mu}=(0,{\bm{\omega}})$, where the extended momentum and derivative
operators in phase space are defined as
$\displaystyle\Pi_{\mu}$ $\displaystyle=$
$\displaystyle(p_{0}+\pi_{0},{\bm{p}}),\ \ \ \
\pi_{0}={\bm{\omega}}\cdot\left({\bm{l}}+{\hbar^{2}\over
4}{\bm{\nabla}}\times{\bm{\nabla}}_{p}\right)+\mu_{B},$ $\displaystyle
D_{\mu}$ $\displaystyle=$ $\displaystyle(d_{t},{\bm{\nabla}}),\ \ \ \ \qquad
d_{t}=\partial_{t}-{\bm{\omega}}\cdot\left({\bm{x}}\times{\bm{\nabla}}+{\bm{p}}\times{\bm{\nabla}}_{p}\right)$
(17)
with the orbital angular momentum ${\bm{l}}={\bm{x}}\times{\bm{p}}$. Note
that, the rotational effect changes only the particle energy from $p_{0}$ to
$p_{0}+\pi_{0}$ and time derivative from $\partial_{t}$ to $d_{t}$, and the
vector momentum ${\bm{p}}$ and space derivative ${\bm{\nabla}}$ are not
modified. In comparison with nuclear collisions at extremely high energy, the
rotational effect will become more important in heavy ion collisions at
intermediate energy where the baryon density becomes high. Aiming at a kinetic
theory in such case, we have included here the baryon chemical potential
$\mu_{B}$ which shifts the particle energy. In order to semi-classically solve
the kinetic equations below, we have displayed the $\hbar-$dependence
explicitly. It is clear that, the highest order quantum correction in the
operators comes from the term $\sim\hbar^{2}$.
Very different from the classical distribution which is a scalar function, the
Wigner function in quantum case is a $4\times 4$ matrix in spin space, it
includes in general case $16$ independent components. Because of their
characteristic properties under Lorentz transformations, it is convenient to
choose the $16$ matrices
$1,i\gamma_{5},\gamma_{\mu},\gamma_{\mu}\gamma_{5},\sigma_{\mu\nu}/2$ as the
basis for an expansion of the Wigner function in spin space,
$W={1\over
4}\left(F+i\gamma^{5}P+\gamma^{\mu}V_{\mu}+\gamma^{\mu}\gamma^{5}A_{\mu}+{1\over
2}\sigma^{\mu\nu}S_{\mu\nu}\right).$ (18)
All the components $\Gamma_{\alpha}=\\{F,P,V_{\mu},A_{\mu},S_{\mu\nu}\\}$ are
real functions, since the basis elements transform under hermitian conjugation
like the Wigner function itself, $W^{+}=\gamma_{0}W\gamma_{0}$. They can thus
be interpreted as phase-space densities. Their physical meaning becomes clear
in the equal-time formalism which will be discussed below.
The expansion (18) decomposes the kinetic equation into $5$ coupled Lorentz
covariant equations for the $5$ spinor components $\Gamma_{\alpha}$. Since
these components are real and the operators $P_{\mu}$ and $D_{\mu}$ are self-
adjoint, one can separate the real and imaginary parts of these $5$ complex
equations,
$\displaystyle 2\Pi^{\mu}V_{\mu}+\hbar\omega^{\mu}A_{\mu}=2mF,$
$\displaystyle\hbar D^{\mu}A_{\mu}=2mP,$ $\displaystyle 4\Pi_{\mu}F-2\hbar
D^{\nu}S_{\nu\mu}-\hbar\epsilon_{\mu\nu\alpha\beta}\omega^{\nu}S^{\alpha\beta}=4mV_{\mu},$
$\displaystyle-\hbar
D_{\mu}P+\epsilon_{\mu\nu\alpha\beta}\Pi^{\nu}S^{\alpha\beta}-\hbar\omega_{\mu}F=2mA_{\mu},$
$\displaystyle\hbar(D_{\mu}V_{\nu}-D_{\nu}V_{\mu})+2\epsilon_{\mu\nu\alpha\beta}\Pi^{\alpha}A^{\beta}+\hbar\epsilon_{\mu\nu\alpha\beta}\omega^{\alpha}V^{\beta}=2mS_{\mu\nu}$
(19)
and
$\displaystyle\hbar D^{\mu}V_{\mu}=0,$ $\displaystyle
2\Pi^{\mu}A_{\mu}+\hbar\omega^{\mu}V_{\mu}=0,$ $\displaystyle\hbar
D_{\mu}F+2\Pi^{\nu}S_{\nu\mu}-\hbar\omega_{\mu}P=0,$ $\displaystyle
4\Pi_{\mu}P+\hbar\epsilon_{\mu\nu\alpha\beta}D^{\nu}S^{\alpha\beta}+2\hbar\omega^{\nu}S_{\mu\nu}=0,$
$\displaystyle
2(\Pi_{\mu}V_{\nu}-\Pi_{\nu}V_{\mu})-\hbar\epsilon_{\mu\nu\alpha\beta}D^{\alpha}A^{\beta}+\hbar(\omega_{\mu}A_{\nu}-\omega_{\nu}A_{\mu})=0.$
(20)
These equations are firstly derived by Vasak, Gyulassy and Eltz vasak for a
QED system and are recently used to describe the chiral magnetic effect of a
quark system in electromagnetic fields Hidaka:2016yjf ; Huang:2018wdl ;
Gao:2018wmr ; Weickgenannt:2019dks ; Wang:2019moi ; Wang:2020pej . These
equations can further be divided into two groups. Those equations with terms
multiplied by $p_{0}$ hidden in $P_{0}$ form the constraint group which links
the Wigner function $W$ and its first order energy moment $p_{0}W$, and the
others form the transport group which describes the evolution of $W$ in phase
space. These will be discussed in more detail in the equal-time formalism.
Similar to the Klein-Gordon equation for the wave function $\psi(x)$ which
describes the plane-wave solution of the Dirac equation satisfying the on-
shell condition $p^{2}-m^{2}=0$, we can obtain the phase-space version of the
Klein-Gordon equation for the Wigner function $W(x,p)$ by acting the kinetic
equation (16) with the operator
$\gamma^{\mu}K_{\mu}+\hbar/2\gamma^{5}\gamma^{\mu}\omega_{\mu}+m$, which leads
to
$\left[K^{\mu}K_{\mu}-{i\over
2}\left[K_{\mu},K_{\nu}\right]\sigma^{\mu\nu}-\hbar\gamma^{5}K^{\mu}\omega_{\mu}+{\hbar^{2}\over
4}\omega^{\mu}\omega_{\mu}-m^{2}\right]W(x,p)=0.$ (21)
We will see in the following that, this equation controls whether the particle
is on the mass shell.
## IV Equal-time kinetic equations
From the definition (14), it is easy to see that the covariant Wigner function
at a fixed time is related to the fields at all times. Therefore, the
covariant kinetic equations in general case cannot be solved as an initial
value problem, and we should go to the equal-time formalism of the kinetic
theory, by doing energy integration of the covariant equations zhuang . The
equal-time Wigner function is defined as
$W_{0}(x,{\bm{p}})=\int{d^{3}{\bm{y}}\over(2\pi)^{3}}e^{-i{\bm{p}}\cdot{\bm{y}}}\langle\psi({\bm{x}}_{+},t)U({\bm{x}}_{+},{\bm{x}}_{-},t)\psi^{+}({\bm{x}}_{-},t)\rangle$
(22)
with ${\bm{x}}_{\pm}={\bm{x}}\pm{\bm{y}}/2$. It is clear that, the equal-time
Wigner function is not Lorentz covariant, and the two Wigner functions are
related to each other through the energy integration,
$W_{0}(x,{\bm{p}})=\int dp_{0}W(x,p)\gamma_{0}.$ (23)
This indicates that, the equal-time Wigner function is the zeroth order energy
moment of the covariant Wigner function. This is the reason why we label the
equal-time Wigner function using the subscript $0$. In general case, particles
moving in a medium are not on the mass shell, and the covariant Wigner
function is equivalent to the collection of all the energy moments zhuang3
$W_{n}(x,{\bm{p}})=\int dp_{0}p_{0}^{n}W(x,p)\gamma_{0}$ (24)
with $n=0,1,2,...$. Only in the quasi-particle approximation where particles
are on the shell and the covariant Wigner function satisfies the on-shell
condition $W(x,p)(p^{2}-m^{2})=0$, the two Wigner functions are equivalent to
each other.
Similar to the covariant scenario, the equal-time Wigner function is
decomposed into $8$ components in spin space,
$W_{0}={1\over
4}\left(f_{0}+\gamma_{5}f_{1}-i\gamma_{0}\gamma_{5}f_{2}+\gamma_{0}f_{3}+\gamma_{5}\gamma_{0}{\bm{\gamma}}\cdot{\bf
g}_{0}+\gamma_{0}{\bm{\gamma}}\cdot{\bf g}_{1}-i{\bm{\gamma}}\cdot{\bf
g}_{2}-\gamma_{5}{\bf\gamma}\cdot{\bf g}_{3}\right),$ (25)
the equal-time components $f_{i}(x,{\bm{p}})$ and ${\bf g}_{i}(x,{\bm{p}})\
(i=0,1,2,3)$ are the zeroth order energy moments of the corresponding
covariant components $\Gamma_{\alpha}(x,p)$.
By taking $p_{0}-$integration of the covariant equations (III) and (III), one
obtains two groups of equal-time kinetic equations,
$\displaystyle\hbar\left(d_{t}f_{0}+{\bm{\nabla}}\cdot{\bf g}_{1}\right)=0,$
$\displaystyle\hbar\left(d_{t}f_{1}+{\bm{\nabla}}\cdot{\bf
g}_{0}\right)=-2mf_{2},$ $\displaystyle\hbar d_{t}f_{2}+2{\bm{p}}\cdot{\bf
g}_{3}=2mf_{1},$ $\displaystyle\hbar d_{t}f_{3}-2{\bm{p}}\cdot{\bf g}_{2}=0,$
$\displaystyle\hbar\left(d_{t}{\bf
g}_{0}+{\bm{\nabla}}f_{1}\right)-2{\bm{p}}\times{\bf
g}_{1}+\hbar{\bm{\omega}}\times{\bf g}_{0}=0,$
$\displaystyle\hbar\left(d_{t}{\bf
g}_{1}+{\bm{\nabla}}f_{0}\right)-2{\bm{p}}\times{\bf
g}_{0}+\hbar{\bm{\omega}}\times{\bf g}_{1}=-2m{\bf g}_{2},$
$\displaystyle\hbar\left(d_{t}{\bf g}_{2}+{\bm{\nabla}}\times{\bf
g}_{3}\right)+2{\bm{p}}f_{3}+\hbar{\bm{\omega}}\times{\bf g}_{2}=2m{\bf
g}_{1},$ $\displaystyle\hbar\left(d_{t}{\bf g}_{3}-{\bm{\nabla}}\times{\bf
g}_{2}\right)-2{\bm{p}}f_{2}+\hbar{\bm{\omega}}\times{\bf g}_{3}=0,$ (26)
and
$\displaystyle 2\int dp_{0}p_{0}F=\hbar{\bm{\nabla}}\cdot{\bf
g}_{2}-2\pi_{0}f_{3}+2mf_{0}-\hbar{\bm{\omega}}\cdot{\bf g}_{3},$
$\displaystyle 2\int dp_{0}p_{0}P=-\hbar{\bm{\nabla}}\cdot{\bf
g}_{3}-2\pi_{0}f_{2}-\hbar{\bm{\omega}}\cdot{\bf g}_{2},$ $\displaystyle 2\int
dp_{0}p_{0}V_{0}=2{\bm{p}}\cdot{\bf
g}_{1}-2\pi_{0}f_{0}+2mf_{3}-\hbar{\bm{\omega}}\cdot{\bf g}_{0},$
$\displaystyle 2\int dp_{0}p_{0}A_{0}=-2{\bm{p}}\cdot{\bf
g}_{0}+2\pi_{0}f_{1}+\hbar{\bm{\omega}}\cdot{\bf g}_{1},$ $\displaystyle 2\int
dp_{0}p_{0}{\bf V}=\hbar{\bm{\nabla}}\times{\bf
g}_{0}-2{\bm{p}}f_{0}+2\pi_{0}{\bf g}_{1}-\hbar{\bm{\omega}}f_{1},$
$\displaystyle 2\int dp_{0}p_{0}{\bf A}=-\hbar{\bm{\nabla}}\times{\bf
g}_{1}-2{\bm{p}}f_{1}+2\pi_{0}{\bf g}_{0}+\hbar{\bm{\omega}}f_{0}-2m{\bf
g}_{3},$ $\displaystyle 2\int dp_{0}p_{0}S^{0i}{\bf
e}_{i}=\hbar{\bm{\nabla}}f_{3}-2{\bm{p}}\times{\bf g}_{3}+2\pi_{0}{\bf
g}_{2}+\hbar{\bm{\omega}}f_{2},$ $\displaystyle\int
dp_{0}p_{0}\epsilon^{ijk}S_{jk}{\bf
e}_{i}={\hbar}{\bm{\nabla}}f_{2}-2\pi_{0}{\bf g}_{3}-2{\bm{p}}\times{\bf
g}_{2}-{\hbar}{\bm{\omega}}f_{3}+2m{\bf g}_{0}.$ (27)
The kinetic equations (IV) and (IV) form, respectively, the transport and
constraint groups. The former is an extension of the Boltzmann equation, it
describes the phase-space evolution of the $8$ equal-time distributions in a
rotational field. The latter is an extension of the on-shell condition
$f(x,p)(p^{2}-m^{2})=0$ associated to the Boltzmann equation. Since particles
are generally not on the mass shell, the off-shell constraints cannot be
neglected arbitrarily, and only the two groups together form a complete
description of the quantum system. This is firstly pointed out by Zhuang and
Heinz for a QED system zhuang ; zhuang2 ; zhuang3 .
The constraints play a tremendous role in calculating some of the physical
distributions. Let’s consider the energy density as an example. From the
energy-moment tensor,
$T_{\mu\nu}(x)=i\left\langle\bar{\psi}(x)\gamma_{\mu}\partial_{\nu}\psi(x)-\bar{\psi}(x)\gamma^{0}\epsilon^{ijk}\delta_{i\mu}\omega_{j}x_{k}\partial_{\nu}\psi(x)\right\rangle,$
(28)
the energy distribution in phase space is the first-order energy moment of the
covariant component $V_{0}$,
$\varepsilon(x,{\bm{p}})=T_{00}(x,{\bm{p}})=\int dp_{0}p_{0}V_{0}(x,p).$ (29)
Without the constraints (IV) which links the zeroth- and first-order energy
moments, there is no way to calculate the energy distribution in the frame of
kinetic theory. With the help from the constraints (IV), $\epsilon$ is a
combination of the equal-time spin components,
$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle{\bm{p}}\cdot{\bf
g}_{1}-\pi_{0}f_{0}+mf_{3}-{\hbar\over 2}{\bm{\omega}}\cdot{\bf g}_{0},$ (30)
where the components $f_{1},f_{3},{\bf g}_{0}$ and ${\bf g}_{1}$ are
controlled by the transport equations (IV).
## V Semi-classical expansion
The equal-time kinetic equations can directly be solved for some non-
perturbative problems like pair production in electromagnetic fields
BialynickiBirula:1991tx ; zhuang ; Sheng:2018jwf . As a systematical way the
semi-classical expansion is widely used in covariant Hidaka:2016yjf ;
Huang:2018wdl ; Gao:2018wmr and equal-time BialynickiBirula:1991tx ; zhuang ;
zhuang2 ; zhuang3 ; guo kinetic theories for massive Wang:2019moi and
massless Hidaka:2016yjf ; Huang:2018wdl ; Gao:2018wmr fermions. We discuss in
this section the semi-classical expansion of the equal-time kinetic equations
(IV) and (IV). Considering the fact that, the rotational field appears only up
to the second order in $\hbar$, the kinetic equations at zeroth, first and
second order of $\hbar$ contain already all the quantum effects from the
rotation on the phase-space distributions.
We take the $\hbar$ expansion for the covariant and equal-time wigner
functions $W(x,p)$ and $W_{0}(x,{\bm{p}})$ and the operator $\Pi_{\mu}$,
$\displaystyle W$ $\displaystyle=$ $\displaystyle W^{(0)}+\hbar
W^{(1)}+\hbar^{2}W^{(2)}+\cdots,$ $\displaystyle W_{0}$ $\displaystyle=$
$\displaystyle W_{0}^{(0)}+\hbar W_{0}^{(1)}+\hbar^{2}W_{0}^{(2)}+\cdots,$
$\displaystyle\Pi_{\mu}$ $\displaystyle=$
$\displaystyle\Pi_{\mu}^{(0)}+\hbar^{2}\Pi_{\mu}^{(2)},\ \ \ \
\Pi_{\mu}^{(0)}=\left(p_{0}+{\bm{\omega}}\cdot{\bm{l}}+\mu_{B},{\bf
p}\right),\ \ \ \
\Pi_{\mu}^{(2)}=\left({\bm{\omega}}\cdot({\bm{\nabla}}\times{\bm{\nabla}}_{p})/4,{\bf
0}\right).$ (31)
Note that the other operator $D_{\mu}$ contains only the classical part.
We first consider the Klein-Gordon equation (21) at the zeroth order in
$\hbar$,
$\left[\Pi^{(0)}_{\mu}\Pi^{(0)\mu}-m^{2}\right]W^{(0)}(x,p)=0.$ (32)
This is just the on-shell condition for classical particles with energy,
$p_{0}=E_{p}^{\pm}=\pm\epsilon_{p}-({\bm{\omega}}\cdot{\bm{l}}+\mu_{B})$ (33)
with $\epsilon_{p}=\sqrt{m^{2}+{\bm{p}}^{2}}$. Different from the kinetic
theory for QED where the electromagnetic fields do not affect the free-
particle shell Hidaka:2016yjf ; Huang:2018wdl ; Gao:2018wmr ;
Weickgenannt:2019dks ; Wang:2019moi ; Wang:2020pej ; zhuang , the rotational
field here changes the shell from $\epsilon_{p}$ to $E_{p}$ due to the
interaction of the orbital angular momentum with the rotational field. The
reason is clear: the electromagnetic fields ${\bf E}$ and ${\bf B}$ are
derivatives of the gauge potential but ${\bm{\omega}}$ appears directly in the
effective gauge potential ${\bm{\omega}}\times{\bm{x}}$ anandan . The
derivative leads to the appearance of ${\bf E}$ and ${\bf B}$ at least at the
first order in $\hbar$, but ${\bm{\omega}}$ starts to contribute at the zeroth
order.
Considering the two elementary solutions of the classical Wigner function,
corresponding to the positive and negative energies,
$W^{(0)}(x,p)=W^{(0)+}(x,p)\delta(p_{0}-E_{p}^{+})+W^{(0)-}(x,p)\delta(p_{0}-E_{p}^{-}),$
(34)
the constraint equations (IV) reduce the number of independent spin components
from $8$ to $2$. The independent components can be chosen to be $f_{0}$ and
${\bf g}_{0}$, and the others can be expressed in terms of them explicitly,
$\displaystyle f_{1}^{(0)\pm}$ $\displaystyle=$
$\displaystyle\pm{1\over\epsilon_{p}}{\bm{p}}\cdot{\bf g}_{0}^{(0)\pm},$
$\displaystyle f_{2}^{(0)\pm}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle f_{3}^{(0)\pm}$ $\displaystyle=$
$\displaystyle\pm{m\over\epsilon_{p}}f_{0}^{(0)\pm},$ $\displaystyle{\bf
g}^{(0)\pm}_{1}$ $\displaystyle=$
$\displaystyle\pm{{\bm{p}}\over\epsilon_{p}}f_{0}^{(0)\pm},$
$\displaystyle{\bf g}^{(0)\pm}_{2}$ $\displaystyle=$ $\displaystyle{1\over
m}{\bm{p}}\times{\bf g}^{(0)\pm}_{0},$ $\displaystyle{\bf g}^{(0)\pm}_{3}$
$\displaystyle=$ $\displaystyle\pm{1\over
m\epsilon_{p}}\left[\epsilon_{p}^{2}{\bf
g}^{(0)\pm}_{0}-{\bm{p}}({\bm{p}}\cdot{\bf g}^{(0)\pm}_{0})\right].$ (35)
It is now the point to understand the physics of the spin components at quasi-
particle level. Expressing the charge current and total angular momentum
tensor in terms of the equal-time Wigner function, it is clear that the
independent components $f_{0}$ and ${\bf g}_{0}$ are, respectively, the
particle number density and spin density, and ${\bf g}_{1}$ is the number
current density BialynickiBirula:1991tx ; zhuang . Taking the classical
relation $f_{1}={\bm{p}}/|{\bm{p}}|\cdot{\bf g}_{0}$ for massless fermions,
$f_{1}$ can be interpreted as the helicity density. The components $f_{3}$ and
$f_{2}$ describe the contribution from spontaneous chiral and $U_{A}(1)$
symmetry breaking of the system to the particle mass guo . From the non-
relativistic limit ${\bf g}_{3}\to{\bf g}_{0}$ and the comparison of the term
$-m/(2m){\bm{\sigma}}\cdot{\bm{\omega}}$ in the Schrödinger equation (11) in
rotational field for particles with effective charge $m$ with the term
$-e/(2m){\bm{\sigma}}\cdot{\bf B}$ in the Schrödinger equation in QED for
particles with charge $e$, ${\bf g}_{3}$ which is known as the magnetic moment
density BialynickiBirula:1991tx ; Bargmann:1959gz in electromagnetic fields
can be understood as the rotational moment density. Considering the classical
relation ${\bf g}_{2}={\bm{p}}\times{\bf g}_{0}/m$, ${\bf g}_{2}$ describes
the spin property in the direction perpendicular to the particle momentum.
Using the above classical relations, the energy density in quasi-particle
approximation is simply expressed in terms of the number distributions with
positive and negative energy,
$\varepsilon(x,{\bm{p}})=E_{p}^{+}f_{0}^{(0)+}(x,{\bm{p}})+E_{p}^{-}f_{0}^{(0)-}(x,{\bm{p}}).$
(36)
Since any derivative is multiplied by a factor of $\hbar$, the classical limit
of the transport equations (IV) cannot describe the phase-space evolution of
the classical components but shows again some of the relations appeared
already in the classical constraints (V). To describe the dynamical evolution
of the equal-time Wigner function, we should go to the first order of the
transport equations (IV),
$\displaystyle d_{t}f^{(0)}_{0}+{\bm{\nabla}}\cdot{\bf g}^{(0)}_{1}=0,$
$\displaystyle d_{t}f^{(0)}_{1}+{\bm{\nabla}}\cdot{\bf
g}^{(0)}_{0}+2mf^{(1)}_{2}=0,$ $\displaystyle
d_{t}f_{2}^{(0)}+2{\bm{p}}\cdot{\bf g}^{(1)}_{3}-2mf^{(1)}_{1}=0,$
$\displaystyle d_{t}f^{(0)}_{3}-2{\bm{p}}\cdot{\bf g}^{(1)}_{2}=0,$
$\displaystyle d_{t}{\bf
g}^{(0)}_{0}+{\bm{\nabla}}f^{(0)}_{1}-2{\bm{p}}\times{\bf
g}^{(1)}_{1}+{\bm{\omega}}\times{\bf g}^{(1)}_{0}=0,$ $\displaystyle d_{t}{\bf
g}^{(0)}_{1}+{\bm{\nabla}}f^{(0)}_{0}-2{\bm{p}}\times{\bf
g}^{(1)}_{0}+{\bm{\omega}}\times{\bf g}^{(0)}_{1}+2m{\bf g}^{(1)}_{2}=0,$
$\displaystyle d_{t}{\bf g}^{(0)}_{2}+{\bm{\nabla}}\times{\bf
g}^{(0)}_{3}+2{\bm{p}}f^{(1)}_{3}+{\bm{\omega}}\times{\bf g}^{(0)}_{2}-2m{\bf
g}^{(1)}_{1}=0,$ $\displaystyle d_{t}{\bf g}^{(0)}_{3}-{\bm{\nabla}}\times{\bf
g}^{(0)}_{2}-2{\bm{p}}f^{(1)}_{2}+{\bm{\omega}}\times{\bf g}^{(0)}_{3}=0.$
(37)
Eliminating the first-order components $f_{i}^{(1)}$ and ${\bf g}_{i}^{(1)}$
by a simple algebra and taking into account the classical relations (V), we
obtain the transport equations for the two independent components
$f_{0}^{(0)}$ and ${\bf g}_{0}^{(0)}$,
$\displaystyle\left[\partial_{t}+\left(\pm{{\bm{p}}\over\epsilon_{p}}+{\bm{x}}\times{\bm{\omega}}\right)\cdot{\bm{\nabla}}-({\bm{\omega}}\times{\bm{p}})\cdot{\bm{\nabla}}_{p}\right]f_{0}^{(0)\pm}=0,$
$\displaystyle\left[\partial_{t}+\left(\pm{{\bm{p}}\over\epsilon_{p}}+{\bm{x}}\times{\bm{\omega}}\right)\cdot{\bm{\nabla}}-({\bm{\omega}}\times{\bm{p}})\cdot{\bm{\nabla}}_{p}\right]{\bf
g}_{0}^{(0)\pm}=-{\bm{\omega}}\times{\bf g}_{0}^{(0)\pm}.$ (38)
The two equations are both in the Vlasov from. The particle velocity appeared
in the free-streaming terms is modified by the rotation induced linear
velocity ${\bm{x}}\times{\bm{\omega}}$, and the classical part of the
rotational potential $-{\bm{\omega}}\cdot{\bm{l}}$ in the Dirac equation leads
to a mean-field force (Coriolis force)
$-{\bm{\nabla}}(-{\bm{\omega}}\cdot{\bm{l}})=-{\bm{\omega}}\times{\bm{p}}$.
For the spin density ${\bf g}_{0}$, there is an extra term
${\bm{\omega}}\times{\bf g}_{0}$ indicating the spin-rotation interaction,
similar to the term ${\bf B}\times{\bf g}_{0}$ in spinor QED. From the
transport equations we obtain the equations of motion of the system,
$\displaystyle\dot{\bm{x}}$ $\displaystyle=$
$\displaystyle\pm\frac{\bm{p}}{\epsilon_{p}}+{\bm{x}}\times{\bf\omega},$
$\displaystyle\dot{\bm{p}}$ $\displaystyle=$
$\displaystyle-{\bm{\omega}}\times{\bm{p}}.$ (39)
Considering positive energy, the total force acting on the particles
${\bf
F}=\epsilon_{p}\ddot{\bm{x}}=-{\bm{\omega}}\times{\bm{p}}-\epsilon_{p}{\bm{\omega}}\times({\bm{x}}\times{\bm{\omega}})$
(40)
contains both the Coriolis force and centrifugal force.
In order to investigate spin induced anomalous phenomena in a rotational
field, one needs to go beyond the classical limit and derive quantum transport
equations. To this end, we consider the Klein-Gordon equation (21) again to
see if quantum particles are still on a mass shell . At the first order in
$\hbar$, the whole operator acting on the Wigner function becomes
$iD^{\mu}\Pi_{\mu}^{(0)}-i/2({\bm{\omega}}\times{\bm{p}})\cdot[{\bm{\gamma}},\gamma_{0}]+\gamma_{5}{\bm{p}}\cdot{\bm{\omega}},$
(41)
which is $\gamma$-matrix dependent. Therefore, there is no longer a common
mass shell for all the spin components. To confirm this conclusion, we try the
quasi-particle solution of the first-order constraint equations with an on-
shell condition $W^{(1)}(x,p)(p_{0}^{2}-{\cal E}_{p}^{2})=0$. Note that, if
the quasi-particle ${\cal E}_{p}$ does exist, it should be different from the
classical energy $E_{p}$ due to the modification by quantum fluctuations.
However, the procedure fails. Massive particles cannot be on the shell when
quantum effect is included. The case here is very different from the chiral
limit where massless particles are always on a shell at any order of $\hbar$
Gao:2018wmr . The second procedure we try is the spin-dependent on-shell
condition $\Gamma_{\alpha}^{(1)}(x,p)(p_{0}^{2}-{\cal E}_{p\alpha}^{2})=0$.
This procedure fails again. Neither a common on-shell nor a component-
dependent on-shell can be the solution of the constraint equations for massive
fermions Wang:2019moi . The quantum effects in a general kinetic theory are
essentially reflected in two aspects, one is the spin, and the other is the
off-shell constraint.
Without the on-shell condition, the constraint equations (IV) at the first
order in $\hbar$ becomes
$\displaystyle\pm 2\epsilon_{p}f^{(1)}_{3}+2\Delta E_{p3}$ $\displaystyle=$
$\displaystyle{\bm{\nabla}}\cdot{\bf
g}^{(0)}_{2}+2mf^{(1)}_{0}-{\bm{\omega}}\cdot{\bf g}^{(0)}_{3},$
$\displaystyle\pm 2\epsilon_{p}f^{(1)}_{2}+2\Delta E_{p2}$ $\displaystyle=$
$\displaystyle-{\bm{\nabla}}\cdot{\bf g}^{(0)}_{3}-{\bm{\omega}}\cdot{\bf
g}^{(0)}_{2},$ $\displaystyle\pm 2\epsilon_{p}f^{(1)}_{0}+2\Delta E_{p0}$
$\displaystyle=$ $\displaystyle 2{\bm{p}}\cdot{\bf
g}^{(1)}_{1}+2mf^{(1)}_{3}-{\bm{\omega}}\cdot{\bf g}^{(0)}_{0},$
$\displaystyle\pm 2\epsilon_{p}f^{(1)}_{1}+2\Delta E_{p1}$ $\displaystyle=$
$\displaystyle 2{\bm{p}}\cdot{\bf g}^{(1)}_{0}-{\bm{\omega}}\cdot{\bf
g}^{(0)}_{1},$ $\displaystyle\pm 2\epsilon_{p}{\bf g}^{(1)}_{1}+2\Delta{\bf
E}_{p1}$ $\displaystyle=$ $\displaystyle{\bm{\nabla}}\times{\bf
g}^{(0)}_{0}+2{\bm{p}}f^{(1)}_{0}-{\bm{\omega}}f^{(0)}_{1},$ $\displaystyle\pm
2\epsilon_{p}{\bf g}^{(1)}_{0}+2\Delta{\bf E}_{p0}$ $\displaystyle=$
$\displaystyle{\bm{\nabla}}\times{\bf
g}^{(0)}_{1}+2{\bm{p}}f^{(1)}_{1}-{\bm{\omega}}f^{(0)}_{0}+2m{\bf
g}^{(1)}_{3},$ $\displaystyle\pm 2\epsilon_{p}{\bf g}^{(1)}_{2}+2\Delta{\bf
E}_{p2}$ $\displaystyle=$
$\displaystyle-{\bm{\nabla}}f^{(0)}_{3}+2{\bm{p}}\times{\bf
g}^{(1)}_{3}-{\bm{\omega}}f_{2}^{(0)},$ $\displaystyle\pm 2\epsilon_{p}{\bf
g}^{(1)}_{3}+2\Delta{\bf E}_{p3}$ $\displaystyle=$
$\displaystyle{\bm{\nabla}}f^{(0)}_{2}-2{\bm{p}}\times{\bf
g}^{(1)}_{2}-{\bm{\omega}}f_{3}^{(0)}+2m{\bf g}^{(1)}_{0}$ (42)
with the energy shifts
$\Delta E_{p\alpha}=2\int dp_{0}(p_{0}-E_{p})\Gamma_{\alpha}^{(1)}.$ (43)
To close the equal-time constraint equations (V) which are related to the
covariant components through the energy shifts (43), we need to consider the
semi-classical expansion of the covariant kinetic equations (III) and (III).
At classical level, the vector component is proportional to $\Pi_{\mu}^{(0)}$,
and both the vector and axial-vector are on the mass shell,
$\displaystyle V_{\mu}^{(0)}$ $\displaystyle=$
$\displaystyle\Pi_{\mu}^{(0)}f^{(0)}\delta(\Pi_{\rho}^{(0)}\Pi^{(0)\rho}-m^{2}),$
$\displaystyle A_{\mu}^{(0)}$ $\displaystyle=$ $\displaystyle
g_{\mu}^{(0)}\delta(\Pi_{\rho}^{(0)}\Pi^{(0)\rho}-m^{2}),$ (44)
where $f(x,p)$ and $g_{\mu}(x,p)$ are arbitrary Lorentz scalar and vector
distributions. After a straightforward but a little bit tedious algebra, the
vector and axial-vector at first order in $\hbar$ can be decomposed into
$\displaystyle V_{\mu}^{(1)}$ $\displaystyle=$
$\displaystyle\Pi_{\mu}^{(0)}f^{(1)}\delta(\Pi_{\rho}^{(0)}\Pi_{(0)\rho}-m^{2})+\Pi_{\mu}^{(0)}\omega^{\nu}A_{\nu}^{(0)}\delta^{\prime}(\Pi_{\rho}^{(0)}\Pi^{(0)\rho}-m^{2}),$
$\displaystyle A_{\mu}^{(1)}$ $\displaystyle=$ $\displaystyle
g_{\mu}^{(1)}\delta(\Pi_{\rho}^{(0)}\Pi^{(0)\rho}-m^{2})+\Pi_{\mu}^{(0)}\omega^{\nu}V_{\nu}^{(0)}\delta^{\prime}(\Pi_{\rho}^{(0)}\Pi^{(0)\rho}-m^{2})-\omega_{\mu}\Pi^{(0)\nu}V_{\nu}^{(0)}\delta^{\prime}(\Pi_{\rho}^{(0)}\Pi^{(0)\rho}-m^{2}),$
(45)
where $\delta^{\prime}$ means the derivative of the $\delta$-function.
Taking together the first order transport and constraint equations (V) and (V)
for the equal-time components and (V) for the covariant components, we
determine uniquely the energy shifts
$\displaystyle\Delta E^{\pm}_{p0}=-\frac{1}{2}{\bm{\omega}}\cdot{\bf
g}_{0}^{(0)\pm},$ $\displaystyle\Delta
E^{\pm}_{p1}=\mp\frac{{\bm{\omega}}\cdot{\bm{p}}}{2\epsilon_{p}}f_{0}^{(0)\pm},$
$\displaystyle\Delta E^{\pm}_{p2}=-{1\over
2m}{\bm{\omega}}\cdot({\bm{p}}\times{\bf g}_{0}^{(0)\pm}),$
$\displaystyle\Delta E^{\pm}_{p3}=\mp{1\over
2m\epsilon_{p}}\left[\epsilon_{p}^{2}{\bm{\omega}}\cdot{\bf
g}_{0}^{(0)\pm}-({\bm{\omega}}\cdot{\bm{p}}({\bm{p}}\cdot{\bf
g}_{0}^{(0)\pm})\right],$ $\displaystyle\Delta{\bf E}^{\pm}_{p0}=-{1\over
2\epsilon_{p}^{2}}\left[m^{2}{\bm{\omega}}+{\bm{p}}({\bm{p}}\cdot{\bf\omega})\right]f_{0}^{(0)\pm},$
$\displaystyle\Delta{\bf E}^{\pm}_{p1}=\mp{{\bm{\omega}}\over
2\epsilon_{p}}{\bm{p}}\cdot{\bf g}_{0}^{(0)},$ $\displaystyle\Delta{\bf
E}^{\pm}_{p2}=\frac{{\bm{p}}^{2}({\bf
p}\times{\bm{\omega}})}{2m\epsilon_{p}^{2}}f_{0}^{(0)\pm}$
$\displaystyle\Delta{\bf E}^{\pm}_{p3}=\mp{1\over
2m\epsilon_{p}}\left[\epsilon_{p}^{2}{\bm{\omega}}-{\bm{p}}({\bm{p}}\cdot{\bm{\omega}})\right]f_{0}^{(0)\pm}.$
(46)
All the energy shifts will disappear when the external field is turned off.
The reason is clear that, without the coupling between the total angular
momentum and the rotational field, particles will keep at the classical shell.
Note that, there are two solutions for any energy shift, corresponding to the
two classical shells $E_{p}^{\pm}$ or $\pm\epsilon_{p}$.
The transport and constraint equations (V) and (V) not only fix the quantum
correction from the off-shell effect to the classical mass shell, but also
reduce the number of independent spin components at quantum level. Again there
are only two independent components. Similar to the classical limit, we can
still choose the number density $f_{0}^{(1)}$ and spin density ${\bf
g}_{0}^{(1)}$ as the independent components, and the others are determined by
them self-consistently,
$\displaystyle f_{1}^{(1)\pm}$ $\displaystyle=$
$\displaystyle\pm{1\over\epsilon_{p}}{\bm{p}}\cdot{\bf g}_{0}^{(1)\pm},$
$\displaystyle f_{2}^{(1)\pm}$ $\displaystyle=$ $\displaystyle-{1\over
2m\epsilon_{p}^{2}}\left[\epsilon_{p}^{2}{\bm{\nabla}}\cdot{\bf
g}_{0}^{(0)\pm}-({\bm{p}}\cdot{\bm{\nabla}})({\bm{p}}\cdot{\bf
g}_{0}^{(0)\pm})\right],$ $\displaystyle f_{3}^{(1)\pm}$ $\displaystyle=$
$\displaystyle\pm{m\over\epsilon_{p}}f_{0}^{(1)\pm}\mp{1\over
2m\epsilon_{p}}{\bm{p}}\cdot({\bm{\nabla}}\times{\bf g}_{0}^{(0)\pm}),$
$\displaystyle{\bf g}^{(1)\pm}_{1}$ $\displaystyle=$
$\displaystyle\pm{{\bm{p}}\over\epsilon_{p}}f_{0}^{(1)\pm}\pm{1\over
2\epsilon_{p}}{\nabla}\times{\bf g}_{0}^{(0)\pm},$ $\displaystyle{\bf
g}_{2}^{(1)\pm}$ $\displaystyle=$ $\displaystyle{1\over m}{\bm{p}}\times{\bf
g}_{0}^{(1)\pm}-{m\over 2\epsilon_{p}^{2}}{\bm{\nabla}}f_{0}^{(0)\pm}+{1\over
2m\epsilon_{p}^{2}}{\bm{p}}\times({\bm{p}}\times{\bm{\nabla}})f_{0}^{(0)\pm},$
$\displaystyle{\bf g}^{(1)\pm}_{3}$ $\displaystyle=$ $\displaystyle\pm{1\over
m\epsilon_{p}}\left[\epsilon^{2}_{p}{\bf
g}_{0}^{(1)}-{\bm{p}}({\bm{p}}\cdot{\bf
g}_{0}^{(1)})\right]\pm{{\bm{p}}\times{\bm{\nabla}}\over
2m\epsilon_{p}}f^{(0)\pm}_{0}+{1\over
2m\epsilon_{p}^{2}}\left[{\bm{p}}^{2}{\bm{\omega}}-{\bm{p}}({\bm{p}}\cdot{\bm{\omega}})\right]f_{0}^{(0)\pm}.$
(47)
There are here three kinds of quantum corrections. The first one is a direct
analogy to the classical relations shown in (V), by simply replacing the
classical components $f_{0}^{(0)}$ and ${\bf g}_{0}^{(0)}$ by the first-order
ones $f_{0}^{(1)}$ and ${\bf g}_{0}^{(1)}$. The second one comes from the
derivative of the classical components, remembering that a derivative in
kinetic equations is always accompanied by a factor of $\hbar$. The third
correction is from the interaction with the external field which appears only
in the rotational moment ${\bf g}_{3}$.
The dynamical evolution of the equal-time Wigner function $W_{0}(x,{\bf p})$
at the first order in $\hbar$ is controlled by the transport equations (IV) at
the second order in $\hbar$,
$\displaystyle d_{t}f^{(1)}_{0}+{\bm{\nabla}}\cdot{\bf g}^{(1)}_{1}=0,$
$\displaystyle d_{t}f^{(1)}_{1}+{\bm{\nabla}}\cdot{\bf
g}^{(1)}_{0}+2mf^{(2)}_{2}=0,$ $\displaystyle
d_{t}f^{(1)}_{2}+2{\bm{p}}\cdot{\bf g}^{(2)}_{3}-2mf^{(2)}_{1}=0,$
$\displaystyle d_{t}f^{(1)}_{3}-2{\bm{p}}\cdot{\bf g}^{(2)}_{2}=0,$
$\displaystyle d_{t}{\bf
g}^{(1)}_{0}+{\bm{\nabla}}f^{(1)}_{1}-2{\bm{p}}\times{\bf
g}^{(2)}_{1}+{\bm{\omega}}\times{\bf g}^{(1)}_{0}=0,$ $\displaystyle d_{t}{\bf
g}^{(1)}_{1}+{\bm{\nabla}}f^{(1)}_{0}-2{\bm{p}}\times{\bf
g}^{(2)}_{0}+{\bm{\omega}}\times{\bf g}^{(1)}_{1}+2m{\bf g}^{(2)}_{2}=0,$
$\displaystyle d_{t}{\bf g}^{(1)}_{2}+{\bm{\nabla}}\times{\bf
g}^{(1)}_{3}+2{\bm{p}}f^{(2)}_{3}+{\bm{\omega}}\times{\bf g}^{(1)}_{2}-2m{\bf
g}^{(2)}_{1}=0,$ $\displaystyle d_{t}{\bf g}^{(1)}_{3}-{\bm{\nabla}}\times{\bf
g}^{(1)}_{2}-2{\bm{p}}f^{(2)}_{2}+{\bm{\omega}}\times{\bf g}^{(1)}_{3}=0.$
(48)
By eliminating the second-order components and taking into account the
classical and first-order kinetic equations (V), (V) and (V), we obtain
finally the transport equations for the two independent quantum distribution
functions, namely the number density $f_{0}^{(1)}$ and spin density ${\bf
g}_{0}^{(1)}$,
$\displaystyle\left[\partial_{t}+\left(\pm{{\bm{p}}\over\epsilon_{p}}+{\bm{x}}\times{\bm{\omega}}\right)\cdot{\bm{\nabla}}-({\bm{\omega}}\times{\bm{p}})\cdot{\bm{\nabla}}_{p}\right]f_{0}^{(1)\pm}=0,$
$\displaystyle\left[\partial_{t}+\left(\pm{{\bm{p}}\over\epsilon_{p}}+{\bm{x}}\times{\bm{\omega}}\right)\cdot{\bm{\nabla}}-({\bm{\omega}}\times{\bm{p}})\cdot{\bm{\nabla}}_{p}\right]{\bf
g}_{0}^{(1)\pm}=-{\bm{\omega}}\times{\bf g}_{0}^{(1)}-{1\over
2\epsilon_{p}^{4}}{\bm{p}}\times({\bm{p}}\times{\bm{\omega}})({\bm{p}}\cdot{\bm{\nabla}})f_{0}^{(0)\pm}.$
(49)
While the number density satisfies the same transport equation as the
classical one, the coupling between the two independent components leads to a
new term on the right-hand side of the quantum transport equation for the spin
density.
Following the way we used to derive transport equations in classical case and
to the first order in $\hbar$, it is not a problem to obtain transport
equations for the second-order components of the Wigner function. As has been
mentioned above, the rotational field appears only up to the second order of
$\hbar$ in the kinetic equations, there should be no more new information when
going beyond the second order.
## VI Summary and outlook
We investigated the quantum kinetic theory for a massive fermion system under
a rotational field in Wigner function formalism. We derived the two groups of
kinetic equations in covariant and equal-time versions, one is the constraint
group which describes the off-shell effect in quantum case, and the other is
the transport group which is the quantum analogy to the classical Boltzmann
equation. For the structure of a quantum kinetic theory, the off-shell
constraint is essentially important. It provides the physical interpretation
for all the equal-time spin components, reduces the number of independent
distribution functions, and closes the transport equations for the number
density and spin density at classical level and quantum level.
The interaction with the external rotational field through total angular
momentum changes significantly the transport properties of the particles. The
classical rotation-orbital coupling controls the dynamical evolution of the
number distribution. It adds a linear velocity ${\bm{x}}\times{\bm{\omega}}$
to the particle velocity, and the induced Coriolis force
${\bm{p}}\times{\bm{\omega}}$ behaves as a mean field force acting on the
particles. Apart from the classical coupling, the quantum rotation-spin
coupling changes the spin distribution but does not affect the number
distribution. While the two distributions are independent in classical limit,
the number density influences the spin density at quantum level.
There are still some questions we need to discuss in the future. One is the
application of the obtained transport equations for the number and spin
distributions. Since an equal-time transport equation can be solved as an
initial value problem, the two transport equations can be used to describe the
evolution of heavy quarks in high energy nuclear collisions to see the
rotational effect on their propagation in hot medium. The collision terms
should be included in a complete kinetic theory. The collisions among
particles control the approaching of a system from non-equilibrium to
equilibrium and will bring in the self-generated vorticity. The finite size
effect is also important for a system under rotation. For a constant rotation,
to guarantee the law of causality, the size of the system is under the
constraint of $R_{max}\omega\leq 1$. This means that, a rotational system
should be finite, and therefore, one should consider the finite size effect on
the Wigner function.
Acknowledgement: The work is supported by Guangdong Major Project of Basic and
Applied Basic Research No. 2020B0301030008 and the NSFC under grant Nos.
11890712 and 12005112.
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|
# Interpretable Models for Granger Causality
Using Self-explaining Neural Networks
Ričards Marcinkevičs
Department of Computer Science
ETH Zürich
Universitätstrasse 6
8092 Zürich, Switzerland
<EMAIL_ADDRESS>&Julia E. Vogt
Department of Computer Science
ETH Zürich
Universitätstrasse 6
8092 Zürich, Switzerland
<EMAIL_ADDRESS>
###### Abstract
Exploratory analysis of time series data can yield a better understanding of
complex dynamical systems. Granger causality is a practical framework for
analysing interactions in sequential data, applied in a wide range of domains.
In this paper, we propose a novel framework for inferring multivariate Granger
causality under nonlinear dynamics based on an extension of self-explaining
neural networks. This framework is more interpretable than other neural-
network-based techniques for inferring Granger causality, since in addition to
relational inference, it also allows detecting signs of Granger-causal effects
and inspecting their variability over time. In comprehensive experiments on
simulated data, we show that our framework performs on par with several
powerful baseline methods at inferring Granger causality and that it achieves
better performance at inferring interaction signs. The results suggest that
our framework is a viable and more interpretable alternative to sparse-input
neural networks for inferring Granger causality.
## 1 Introduction
Granger causality (GC) (Granger, 1969) is a popular practical approach for the
analysis of multivariate time series and has become instrumental in
exploratory analysis (McCracken, 2016) in various disciplines, such as
neuroscience (Roebroeck et al., 2005), economics (Appiah, 2018), and
climatology (Charakopoulos et al., 2018). Recently, the focus of the
methodological research has been on inferring GC under nonlinear dynamics
(Tank et al., 2018; Nauta et al., 2019; Wu et al., 2020; Khanna & Tan, 2020;
Löwe et al., 2020), causal structures varying across replicates (Löwe et al.,
2020), and unobserved confounding (Nauta et al., 2019; Löwe et al., 2020).
To the best of our knowledge, the latest powerful techniques for inferring GC
do not target the effect sign detection (see Section 2.1 for a formal
definition) or exploration of effect variability with time and, thus, have
limited interpretability. This drawback defeats the purpose of GC analysis as
an exploratory statistical tool. In some nonlinear interactions, one variable
may have an exclusively positive or negative effect on another if it
consistently drives the other variable up or down, respectively. Negative and
positive causal relationships are common in many real-world systems, for
example, gene regulatory networks feature inhibitory effects (Inoue et al.,
2011) or in metabolomics, certain compounds may inhibit or promote synthesis
of other metabolites (Rinschen et al., 2019). Differentiating between the two
types of interactions would allow inferring and understanding such inhibition
and promotion relationships in real-world dynamical systems and would
facilitate a more comprehensive and insightful exploratory analysis.
Therefore, we see a need for a framework capable of inferring nonlinear GC
which is more amenable to interpretation than previously proposed methods
(Tank et al., 2018; Nauta et al., 2019; Khanna & Tan, 2020). To this end, we
introduce a novel method for detecting nonlinear multivariate Granger
causality that is interpretable, in the sense that it allows detecting effect
signs and exploring influences among variables throughout time. The main
contributions of the paper are as follows:
1. 1.
We extend self-explaining neural network models (Alvarez-Melis & Jaakkola,
2018) to time series analysis. The resulting autoregressive model, named
generalised vector autoregression (GVAR), is interpretable and allows
exploring GC relations between variables, signs of Granger-causal effects, and
their variability through time.
2. 2.
We propose a framework for inferring nonlinear multivariate GC that relies on
a GVAR model with sparsity-inducing and time-smoothing penalties. Spurious
associations are mitigated by finding relationships that are stable across
original and time-reversed (Winkler et al., 2016) time series data.
3. 3.
We comprehensively compare the proposed framework and the powerful baseline
methods of Tank et al. (2018), Nauta et al. (2019), and Khanna & Tan (2020) on
a range of synthetic time series datasets with known Granger-causal
relationships. We evaluate the ability of the methods to infer the ground
truth GC structure and effect signs.
## 2 Background and Related Work
### 2.1 Granger Causality
Granger-causal relationships are given by a set of directed dependencies
within multivariate time series. The classical definition of Granger causality
is given, for example, by Lütkepohl (2007). Below we define nonlinear
multivariate GC, based on the adaptation by Tank et al. (2018). Consider a
time series with $p$ variables:
$\left\\{{\mathbf{x}}_{t}\right\\}_{t\in\mathbb{Z}^{+}}=\left\\{\left({\textnormal{x}}^{1}_{t}\text{
}{\textnormal{x}}^{2}_{t}\text{ }...\text{
}{\textnormal{x}}^{p}_{t}\right)^{\top}\right\\}_{t\in\mathbb{Z}^{+}}$. Assume
that causal relationships between variables are given by the following
structural equation model:
${\textnormal{x}}^{i}_{t}:=g_{i}\left({\textnormal{x}}^{1}_{1:(t-1)},...,{\textnormal{x}}^{j}_{1:(t-1)},...,{\textnormal{x}}^{p}_{1:(t-1)}\right)+\varepsilon^{i}_{t},\text{
for }1\leq i\leq p,$ (1)
where ${\textnormal{x}}^{j}_{1:(t-1)}$ is a shorthand notation for
${\textnormal{x}}^{j}_{1},{\textnormal{x}}^{j}_{2},...,{\textnormal{x}}^{j}_{t-1}$;
$\varepsilon^{i}_{t}$ are additive innovation terms; and $g_{i}(\cdot)$ are
potentially nonlinear functions, specifying how the future values of variable
${\textnormal{x}}^{i}$ depend on the past values of ${\mathbf{x}}$. We then
say that variable ${\textnormal{x}}^{j}$ does not Granger-cause variable
${\textnormal{x}}^{i}$, denoted as
${\textnormal{x}}^{j}\not\xrightarrow{}{\textnormal{x}}^{i}$, if and only if
$g_{i}(\cdot)$ is constant in ${\textnormal{x}}^{j}_{1:(t-1)}$.
Depending on the form of the functional relationship $g_{i}(\cdot)$, we can
also differentiate between positive and negative Granger-causal effects. In
this paper, we define the effect sign as follows: if $g_{i}(\cdot)$ is
increasing in all ${\textnormal{x}}^{j}_{1:(t-1)}$, then we say that variable
${\textnormal{x}}^{j}$ has a positive effect on ${\textnormal{x}}^{i}$, if
$g_{i}(\cdot)$ is decreasing in ${\textnormal{x}}^{j}_{1:(t-1)}$, then
${\textnormal{x}}^{j}$ has a negative effect on ${\textnormal{x}}^{i}$. Note
that an effect may be neither positive nor negative. For example,
${\textnormal{x}}^{j}$ can ‘contribute’ both positively and negatively to the
future of ${\textnormal{x}}^{i}$ at different delays, or, for instance, the
effect of ${\textnormal{x}}^{j}$ on ${\textnormal{x}}^{i}$ could depend on
another variable.
Granger-causal relationships can be summarised by a directed graph
$\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$, referred to as summary
graph (Peters et al., 2017), where $\mathcal{V}=\left\\{1,...,p\right\\}$ is a
set of vertices corresponding to variables, and
$\mathcal{E}=\left\\{(i,j):{\textnormal{x}}^{i}\xrightarrow{}{\textnormal{x}}^{j}\right\\}$
is a set of edges corresponding to Granger-causal relationships. Let
${\bm{A}}\in\left\\{0,1\right\\}^{p\times p}$ denote the adjacency matrix of
$\mathcal{G}$. The inference problem is then to estimate ${\bm{A}}$ from
observations $\left\\{{\bm{x}}_{t}\right\\}_{t=1}^{T}$, where $T$ is the
length of the time series observed. In practice, we usually fit a time series
model that explicitly or implicitly infers dependencies between variables.
Consequently, a statistical test for GC is performed. A conventional approach
(Lütkepohl, 2007) used to test for linear Granger causality is the linear
vector autoregression (VAR) (see Appendix A).
### 2.2 Related Work
#### 2.2.1 Techniques for Inferring Nonlinear Granger Causality
Relational inference in time series has been studied extensively in statistics
and machine learning. Early techniques for inferring undirected relationships
include time-varying dynamic Bayesian networks (Song et al., 2009) and time-
smoothed, regularised logistic regression with time-varying coefficients
(Kolar et al., 2010). Recent approaches to inferring Granger-causal
relationships leverage the expressive power of neural networks (Montalto et
al., 2015; Wang et al., 2018; Tank et al., 2018; Nauta et al., 2019; Khanna &
Tan, 2020; Wu et al., 2020; Löwe et al., 2020) and are often based on
regularised autoregressive models, reminiscent of the Lasso Granger method
(Arnold et al., 2007).
Tank et al. (2018) propose using sparse-input multilayer perceptron (cMLP) and
long short-term memory (cLSTM) to model nonlinear autoregressive relationships
within time series. Building on this, Khanna & Tan (2020) introduce a more
sample efficient economy statistical recurrent unit (eSRU) architecture with
sparse input layer weights. Nauta et al. (2019) propose a temporal causal
discovery framework (TCDF) that leverages attention-based convolutional neural
networks to test for GC. Appendix B contains further details about these and
other relevant methods.
Approaches discussed above (Tank et al., 2018; Nauta et al., 2019; Khanna &
Tan, 2020) and in Appendix B (Marinazzo et al., 2008; Ren et al., 2020;
Montalto et al., 2015; Wang et al., 2018; Wu et al., 2020; Löwe et al., 2020)
focus almost exclusively on relational inference and do not allow easily
interpreting signs of GC effects and their variability through time. In this
paper, we propose a more interpretable inference framework, building on self
explaining-neural networks (Alvarez-Melis & Jaakkola, 2018), that, as shown by
experiments, performs on par with the techniques described herein.
#### 2.2.2 Stability-based Selection Procedures
The literature on stability-based model selection is abundant (Ben-Hur et al.,
2002; Lange et al., 2003; Meinshausen & Bühlmann, 2010; Sun et al., 2013). For
example, Ben-Hur et al. (2002) propose measuring stability of clustering
solutions under perturbations to assess structure in the data and select an
appropriate number of clusters. Lange et al. (2003) propose a somewhat similar
approach. Meinshausen & Bühlmann (2010) introduce the stability selection
procedure applicable to a wide range of high-dimensional problems: their
method guides the choice of the amount of regularisation based on the error
rate control. Sun et al. (2013) investigate a similar procedure in the context
of tuning penalised regression models.
#### 2.2.3 Self-explaining Neural Networks
Alvarez-Melis & Jaakkola (2018) introduce self-explaining neural networks
(SENN) – a class of intrinsically interpretable models motivated by
explicitness, faithfulness, and stability properties. A SENN with a link
function $g(\cdot)$ and interpretable basis concepts
${\bm{h}}({\bm{x}}):\mathbb{R}^{p}\rightarrow\mathbb{R}^{k}$ follows the form
$f({\bm{x}})=g\left(\theta({\bm{x}})_{1}h({\bm{x}})_{1},...,\theta({\bm{x}})_{k}h({\bm{x}})_{k}\right),$
(2)
where ${\bm{x}}\in\mathbb{R}^{p}$ are predictors; and $\bm{\theta}(\cdot)$ is
a neural network with $k$ outputs. We refer to $\bm{\theta}({\bm{x}})$ as
generalised coefficients for data point ${\bm{x}}$ and use them to ‘explain’
contributions of individual basis concepts to predictions. In the case of
$g(\cdot)$ being sum and concepts being raw inputs, Equation 2 simplifies to
$f({\bm{x}})=\sum_{j=1}^{p}\theta({\bm{x}})_{j}x_{j}.$ (3)
Appendix C lists additional properties SENNs need to satisfy, as defined by
Alvarez-Melis & Jaakkola (2018).
A SENN is trained by minimising the following gradient-regularised loss
function, which balances performance with interpretability:
$\mathcal{L}_{y}(f({\bm{x}}),y)+\lambda\mathcal{L}_{\bm{\theta}}\left(f({\bm{x}})\right),$
(4)
where $\mathcal{L}_{y}(f({\bm{x}}),y)$ is a loss term for the ground
classification or regression task; $\lambda>0$ is a regularisation parameter;
and
$\mathcal{L}_{\bm{\theta}}(f({\bm{x}}))=\left\|\nabla_{{\bm{x}}}f({\bm{x}})-\bm{\theta}({\bm{x}})^{\top}{\bm{J}}^{\bm{h}}_{{\bm{x}}}({\bm{x}})\right\|_{2}$
is the gradient penalty, where ${\bm{J}}^{\bm{h}}_{{\bm{x}}}$ is the Jacobian
of ${\bm{h}}(\cdot)$ w.r.t. ${\bm{x}}$. This penalty encourages $f(\cdot)$ to
be locally linear.
## 3 Method
We propose an extension of SENNs (Alvarez-Melis & Jaakkola, 2018) to
autoregressive time series modelling, which is essentially a vector
autoregression (see Equation 11 in Appendix A) with generalised coefficient
matrices. We refer to this model as generalised vector autoregression (GVAR).
The GVAR model of order $K$ is given by
${\mathbf{x}}_{t}=\sum_{k=1}^{K}\bm{\Psi}_{\bm{\theta}_{k}}\left({\mathbf{x}}_{t-k}\right){\mathbf{x}}_{t-k}+\bm{\varepsilon}_{t},$
(5)
where
$\bm{\Psi}_{\bm{\theta}_{k}}:\mathbb{R}^{p}\rightarrow\mathbb{R}^{p\times p}$
is a neural network parameterised by $\bm{\theta}_{k}$. For brevity, we omit
the intercept term here and in following equations. No specific distributional
assumptions are made on the additive innovation terms $\bm{\varepsilon}_{t}$.
$\bm{\Psi}_{\bm{\theta}_{k}}\left({\mathbf{x}}_{t-k}\right)$ is a matrix whose
components correspond to the generalised coefficients for lag $k$ at time step
$t$. In particular, the component $(i,j)$ of
$\bm{\Psi}_{\bm{\theta}_{k}}\left({\mathbf{x}}_{t-k}\right)$ corresponds to
the influence of ${\textnormal{x}}^{j}_{t-k}$ on ${\textnormal{x}}^{i}_{t}$.
In our implementation, we use $K$ MLPs for
$\bm{\Psi}_{\bm{\theta}_{k}}(\cdot)$ with $p$ input units and $p^{2}$ outputs
each, which are then reshaped into an $\mathbb{R}^{p\times p}$ matrix. Observe
that the model defined in Equation 5 takes on a form of SENN (see Equation 3)
with future time series values as the response, past values as basis concepts,
and sum as a link function.
Relationships between variables
${\textnormal{x}}^{1},...,{\textnormal{x}}^{p}$ and their variability
throughout time can be explored by inspecting generalised coefficient
matrices. To mitigate spurious inference in multivariate time series, we train
GVAR by minimising the following penalised loss function with the mini-batch
gradient descent:
$\frac{1}{T-K}\sum_{t=K+1}^{T}\left\|{\bm{x}}_{t}-\hat{{\bm{x}}}_{t}\right\|^{2}_{2}+\frac{\lambda}{T-K}\sum_{t=K+1}^{T}R\left(\bm{\Psi}_{t}\right)+\frac{\gamma}{T-K-1}\sum_{t=K+1}^{T-1}\left\|\bm{\Psi}_{t+1}-\bm{\Psi}_{t}\right\|_{2}^{2},$
(6)
where $\left\\{{\bm{x}}_{t}\right\\}_{t=1}^{T}$ is a single observed replicate
of a $p$-variate time series of length $T$;
$\hat{{\bm{x}}}_{t}=\sum_{k=1}^{K}\bm{\Psi}_{\hat{\bm{\theta}}_{k}}\left({\bm{x}}_{t-k}\right){\bm{x}}_{t-k}$
is the one-step forecast for the $t$-th time point by the GVAR model;
$\bm{\Psi}_{t}$ is a shorthand notation for the concatenation of generalised
coefficient matrices at the $t$-th time point:
$\left[\bm{\Psi}_{\hat{\bm{\theta}}_{K}}\left({\bm{x}}_{t-K}\right)\text{
}\bm{\Psi}_{\hat{\bm{\theta}}_{K-1}}\left({\bm{x}}_{t-K+1}\right)\text{
}...\text{
}\bm{\Psi}_{\hat{\bm{\theta}}_{1}}\left({\bm{x}}_{t-1}\right)\right]\in\mathbb{R}^{p\times
Kp}$; $R\left(\cdot\right)$ is a sparsity-inducing penalty term; and
$\lambda,\gamma\geq 0$ are regularisation parameters. The loss function (see
Equation 6) consists of three terms: _(i)_ the mean squared error (MSE) loss,
_(ii)_ a sparsity-inducing regulariser, and _(iii)_ the smoothing penalty
term. Note, that in presence of categorically-valued variables the MSE term
can be replaced with e.g. the cross-entropy loss.
The sparsity-inducing term $R(\cdot)$ is an appropriate penalty on the norm of
the generalised coefficient matrices. Examples of possible penalties for the
linear VAR are provided in Table 4 in Appendix A. These penalties can be
easily adapted to the GVAR model. In the current implementation, we employ the
elastic-net-style penalty term (Zou & Hastie, 2005; Nicholson et al., 2017)
$R(\bm{\Psi}_{t})=\alpha\left\|\bm{\Psi}_{t}\right\|_{1}+(1-\alpha)\left\|\bm{\Psi}_{t}\right\|_{2}^{2}$,
with $\alpha=0.5$.
The smoothing penalty term, given by
$\frac{1}{T-K-1}\sum_{t=K+1}^{T-1}\left\|\bm{\Psi}_{t+1}-\bm{\Psi}_{t}\right\|_{2}^{2}$,
is the average norm of the difference between generalised coefficient matrices
for two consecutive time points. This penalty term encourages smoothness in
the evolution of coefficients w.r.t. time and replaces the gradient penalty
$\mathcal{L}_{\bm{\theta}}\left(f\left({\bm{x}}\right)\right)$ from the
original formulation of SENN (see Equation 4). Observe that if the term is
constrained to be $0$, then the GVAR model behaves as a penalised linear VAR
on the training data: coefficient matrices are invariant across time steps.
Figure 1: GVAR generalised coefficients inferred for a time series with linear
dynamics.
Thus, the proposed penalised loss function (see Equation 6) allows controlling
the _(i)_ sparsity and _(ii)_ nonlinearity of inferred autoregressive
dependencies. As opposed to the related approaches of Tank et al. (2018) and
Khanna & Tan (2020), signs of Granger causal effects and their variability in
time can be assessed as well by interpreting matrices
$\bm{\Psi}_{\hat{\bm{\theta}}_{k}}\left({\bm{x}}_{t}\right)$, for $K+1\leq
t\leq T$. Figure 1 shows a plot of generalised coefficients versus time in a
toy linear time series (see Appendix L for details). Observe that for causal
relationships, generalised coefficients are large in magnitude, whereas for
non-causal links, coefficients are shrunk towards 0. Moreover, the signs of
coefficients agree with the true interaction signs ($a_{i}$). We further
support these claims with empirical results in Section 4. In addition, we
provide an ablation study for the loss function in Appendix D.
### 3.1 Inference Framework
Once neural networks $\bm{\Psi}_{\hat{\bm{\theta}}_{k}}$, $k=1,...,K$, have
been trained, we quantify strengths of Granger-causal relationships between
variables by aggregating matrices
$\bm{\Psi}_{\hat{\bm{\theta}}_{k}}\left({\bm{x}}_{t}\right)$ across all time
steps into summary statistics. We aggregate the obtained generalised
coefficients into matrix ${\bm{S}}\in\mathbb{R}^{p\times p}$ as follows:
$S_{i,j}=\max_{1\leq k\leq K}\left\\{\text{median}_{K+1\leq t\leq
T}\left(\left|\left(\bm{\Psi}_{\hat{\bm{\theta}}_{k}}\left({\bm{x}}_{t}\right)\right)_{i,j}\right|\right)\right\\},\text{
for }1\leq i,j\leq p.$ (7)
Intuitively, $S_{i,j}$ are statistics that quantify the strength of the
Granger-causal effect of ${\textnormal{x}}^{i}$ on ${\textnormal{x}}^{j}$
using magnitudes of generalised coefficients. We expect $S_{i,j}$ to be close
to 0 for non-causal relationships and $S_{i,j}\gg 0$ if
${\textnormal{x}}^{i}\rightarrow{\textnormal{x}}^{j}$. Note that in practice
${\bm{S}}$ is not binary-valued, as opposed to the ground truth adjacency
matrix ${\bm{A}}$, which we want to infer, because the outputs of
$\bm{\Psi}_{\hat{\bm{\theta}}_{k}}(\cdot)$ are not shrunk to exact zeros.
Therefore, we need a procedure deciding for which variable pairs $S_{i,j}$ are
significantly different from 0.
To infer a binary matrix of GC relationships, we propose a heuristic
stability-based procedure that relies on time-reversed Granger causality
(TRGC) (Haufe et al., 2012; Winkler et al., 2016). The intuition behind time
reversal is to compare causality scores obtained from original and time-
reversed data: we expect relationships to be flipped on time-reversed data
(Haufe et al., 2012; Winkler et al., 2016). Winkler et al. (2016) prove the
validity of time reversal for linear finite-order autoregressive processes. In
our work, time reversal is leveraged for inferring stable dependency
structures in nonlinear time series.
Algorithm 1 summarises the proposed stability-based thresholding procedure.
During inference, two separate GVAR models are trained: one on the original
time series data, and another on time-reversed data (lines 3-4 in Algorithm
1). Consequently, we estimate strengths of GC relationships with these two
models, as in Equation 7, and choose a threshold for matrix ${\bm{S}}$ which
yields the highest agreement between thresholded GC strengths estimated on
original and time-reversed data (lines 5-9 in Algorithm 1). A sequence of $Q$
thresholds, given by $\bm{\xi}=\left(\xi_{1},...,\xi_{Q}\right)$, is
considered where the $i$-th threshold is an $\xi_{i}$-quantile of values in
${\bm{S}}$. The agreement between inferred thresholded structures is measured
(line 7 in Algorithm 1) using balanced accuracy score (Brodersen et al.,
2010), denoted by $\textrm{BA}\left(\cdot,\cdot\right)$, equal-to the average
of sensitivity and specificity, to reflect both sensitivity and specificity of
the inference results. Other measures can be used for quantifying the
agreement, for example, graph similarity scores (Zager & Verghese, 2008). In
this paper, we utilise BA, because considered time series have sparse GC
summary graphs and BA weighs positives and negatives equally. In practice,
trivial solutions, such as inferring no causal relationships, only self-causal
links or all possible causal links, are very stable. The agreement for such
solutions is set to 0. Thus, the procedure assumes that the true causal
structure is different from these trivial cases. Figure 6 in Appendix E
contains an example of stability-based thresholding applied to simulated data.
Input: One replicate of multivariate time series
$\left\\{{\bm{x}}_{t}\right\\}_{t=1}^{T}$; regularisation parameters $\lambda$
and $\gamma\geq 0$; model order $K\geq 1$; sequence
$\bm{\xi}=\left(\xi_{1},...,\xi_{Q}\right)$,
$0\leq\xi_{1}<\xi_{2}<...<\xi_{Q}\leq 1$.
Output : Estimate $\hat{{\bm{A}}}$ of the adjacency matrix of the GC summary
graph.
1
2Let $\left\\{\tilde{{\bm{x}}}_{t}\right\\}_{t=1}^{T}$ be the time-reversed
version of $\left\\{{\bm{x}}_{t}\right\\}_{t=1}^{T}$, i.e.
$\left\\{\tilde{{\bm{x}}}_{1},...,\tilde{{\bm{x}}}_{T}\right\\}\equiv\left\\{{\bm{x}}_{T},...,{\bm{x}}_{1}\right\\}.$
3Let $\tau\left({\bm{X}},\chi\right)$ be the elementwise thresholding
operator. For each component of ${\bm{X}}$, $\tau\left(X_{i,j},\chi\right)=1$,
if $\left|X_{i,j}\right|\geq\chi$, and $\tau\left(X_{i,j},\chi\right)=0$,
otherwise.
4Train an order $K$ GVAR with parameters $\lambda$ and $\gamma$ by minimising
loss in Equation 6 on $\left\\{{\bm{x}}_{t}\right\\}_{t=1}^{T}$ and compute
${\bm{S}}$ as in Equation 7.
5Train another GVAR on $\left\\{\tilde{{\bm{x}}}_{t}\right\\}_{t=1}^{T}$ and
compute $\tilde{{\bm{S}}}$ as in Equation 7.
6for _$i=1$ to $Q$_ do
7
8 Let $\kappa_{i}=q_{\xi_{i}}({\bm{S}})$ and
$\tilde{\kappa}_{i}=q_{\xi_{i}}(\tilde{{\bm{S}}})$, where $q_{\xi}(X)$ denotes
the $\xi$-quantile of $X$.
9 Evaluate agreement
$\varsigma_{i}=\frac{1}{2}\left[\text{BA}\left(\tau\left({\bm{S}},\kappa_{i}\right),\tau\left(\tilde{{\bm{S}}}^{\top},\tilde{\kappa}_{i}\right)\right)+\text{BA}\left(\tau\left(\tilde{{\bm{S}}}^{\top},\tilde{\kappa}_{i}\right),\tau\left({\bm{S}},\kappa_{i}\right)\right)\right]$.
10 end for
11
12Let $i^{*}=\arg\max_{1\leq i\leq Q}\varsigma_{i}$ and $\xi^{*}=\xi_{i^{*}}$.
13Let $\hat{{\bm{A}}}=\tau\left({\bm{S}},q_{\xi^{*}}({\bm{S}})\right)$.
return $\hat{{\bm{A}}}$.
Algorithm 1 Stability-based thresholding for inferring Granger causality with
GVAR.
To summarise, this procedure attempts to find a dependency structure that is
stable across original and time-reversed data in order to identify significant
Granger-causal relationships. In Section 4, we demonstrate the efficacy of
this inference framework. In particular, we show that it performs on par with
previously proposed approaches mentioned in Section 2.2.
#### 3.1.1 Computational Complexity
Our inference framework differs from the previously proposed cMLP, cLSTM (Tank
et al., 2018), TCDF (Nauta et al., 2019), and eSRU (Khanna & Tan, 2020) w.r.t.
computational complexity. Mentioned methods require training $p$ neural
networks, one for each variable separately, whereas our inference framework
trains $2K$ neural networks. A clear disadvantage of GVAR is its memory
complexity: GVAR has many more parameters, since every MLP it trains has
$p^{2}$ outputs. Appendix F provides a comparison between training times on
simulated datasets with $p\in\\{4,15,20\\}$. In practice, for a moderate order
$K$ and a larger $p$, we observe that training a GVAR model is faster than a
cLSTM and eSRU.
## 4 Experiments
The purpose of our experiments is twofold: (_i_) to compare methods in terms
of their ability to infer the underlying GC structure; and (_ii_) to compare
methods in terms of their ability to detect signs of GC effects. We compare
GVAR to 5 baseline techniques: VAR with $F$-tests for Granger causality111As
implemented in the statsmodels library (Seabold & Perktold, 2010). and the
Benjamini-Hochberg procedure (Benjamini & Hochberg, 1995) for controlling the
false discovery rate (FDR) (at $q=0.05$); cMLP and cLSTM (Tank et al.,
2018)222https://github.com/iancovert/Neural-GC.; TCDF (Nauta et al.,
2019)333https://github.com/M-Nauta/TCDF.; and eSRU (Khanna & Tan,
2020)444https://github.com/sakhanna/SRU_for_GCI.. We particularly focus on the
baselines that, similarly to GVAR, leverage sparsity-inducing penalties,
namely cMLP, cLSTM, and eSRU. In addition, we provide a comparison with
dynamic Bayesian networks (Murphy & Russell, 2002) in Appendix I. The code is
available in the GitHub repository: https://github.com/i6092467/GVAR.
### 4.1 Inferring Granger Causality
Figure 2: GC summary graph adjacency matrix of the Lorenz 96 system with
$p=20$. Dark cells correspond to the absence of a GC relationship; light cells
denote a GC relationship.
We first compare methods w.r.t. their ability to infer GC relationships
correctly on two synthetic datasets. We evaluate inferred dependencies on each
independent replicate/simulation separately against the adjacency matrix of
the ground truth GC graph, an example is shown in Figure 2. Each method is
trained only on one sequence. Unless otherwise mentioned, we use accuracy
(ACC) and balanced accuracy (BA) scores to evaluate thresholded inference
results. For cMLP, cLSTM, and eSRU, the relevant weight norms are compared to
0. For TCDF, thresholding is performed within the framework based on the
permutation test described by Nauta et al. (2019). For GVAR, thresholded
matrices are obtained by applying Algorithm 1. In addition, we look at the
continuously-valued inference results: norms of relevant weights, scores, and
strengths of GC relationships (see Equation 7). We compare these scores
against the true structure using areas under receiver operating characteristic
(AUROC) and precision-recall (AUPRC) curves. For all evaluation metrics, we
only consider off-diagonal elements of adjacency matrices, ignoring self-
causal relationships, which are usually the easiest to infer. Note that our
evaluation approach is different from those of Tank et al. (2018) and Khanna &
Tan (2020); this partially explains some deviations from their results.
Relevant hyperparameters of all models are tuned to maximise the BA score or
AUPRC (if a model fails to shrink any weights to zeros) by performing a grid
search (see Appendix H for details about hyperparameter tuning). In Appendix
M, we compare the prediction error of all models on held-out data.
#### 4.1.1 Lorenz 96 Model
A standard benchmark for the evaluation of GC inference techniques is the
Lorenz 96 model (Lorenz, 1995). This continuous time dynamical system in $p$
variables is given by the following nonlinear differential equations:
$\frac{d{\textnormal{x}}^{i}}{dt}=\left({\textnormal{x}}^{i+1}-{\textnormal{x}}^{i-2}\right){\textnormal{x}}^{i-1}-{\textnormal{x}}^{i}+F,\text{
for }1\leq i\leq p,$ (8)
where ${\textnormal{x}}^{0}:={\textnormal{x}}^{p}$,
${\textnormal{x}}^{-1}:={\textnormal{x}}^{p-1}$, and
${\textnormal{x}}^{p+1}:={\textnormal{x}}^{1}$; and $F$ is a forcing constant
that, in combination with $p$, controls the nonlinearity of the system (Tank
et al., 2018; Karimi & Paul, 2010). As can be seen from Equation 8, the true
causal structure is quite sparse. Figure 2 shows the adjacency matrix of the
summary graph for this dataset (for other datasets, adjacency matrices are
visualised in Appendix G). We numerically simulate $R=5$ replicates with
$p=20$ variables and $T=500$ observations under $F=10$ and $F=40$. The setting
is similar to the experiments of Tank et al. (2018) and Khanna & Tan (2020),
but includes more variables.
Table 1 summarises the performance of the inference techniques on the Lorenz
96 time series under $F=10$ and $F=40$. For $F=10$, all of the methods apart
from TCDF are very successful at inferring GC relationships, even linear VAR.
On average, GVAR outperforms all baselines, although performance differences
are not considerable. For $F=40$, the inference problem appears to be more
difficult (Appendix J investigates performance of VAR and GVAR across a range
of forcing constant values). In this case, TCDF and cLSTM perform surprisingly
poorly, whereas cMLP, eSRU, and GVAR achieve somewhat comparable performance
levels. GVAR attains the best combination of accuracy and BA scores, whereas
cMLP has the highest AUROC and AUPRC. Thus, on Lorenz 96 data, the performance
of GVAR is competitive with the other methods.
Table 1: Performance comparison on the Lorenz 96 model with $F=10$ and $40$. Inference is performed on each replicate separately, standard deviations (SD) are evaluated across 5 replicates. $F$ | Model | ACC($\pm$SD) | BA($\pm$SD) | AUROC($\pm$SD) | AUPRC($\pm$SD)
---|---|---|---|---|---
10 | VAR | 0.918($\pm$0.012) | 0.838($\pm$0.016) | 0.940($\pm$0.016) | 0.825($\pm$0.029)
cMLP | 0.972($\pm$0.005) | 0.956($\pm$0.016) | 0.963($\pm$0.018) | 0.908($\pm$0.049)
cLSTM | 0.970($\pm$0.010) | 0.950($\pm$0.028) | 0.958($\pm$0.029) | 0.925($\pm$0.050)
TCDF | 0.871($\pm$0.012) | 0.709($\pm$0.044) | 0.857($\pm$0.027) | 0.601($\pm$0.053)
eSRU | 0.966($\pm$0.011) | 0.951($\pm$0.021) | 0.963($\pm$0.020) | 0.936($\pm$0.034)
GVAR (ours) | 0.982($\pm$0.003) | 0.982($\pm$0.006) | 0.997($\pm$0.001) | 0.976($\pm$0.016)
40 | VAR | 0.864($\pm$0.008) | 0.585($\pm$0.028) | 0.745($\pm$0.047) | 0.474($\pm$0.036)
cMLP | 0.683($\pm$0.027) | 0.805($\pm$0.017) | 0.979($\pm$0.016) | 0.956($\pm$0.033)
cLSTM | 0.844($\pm$0.012) | 0.656($\pm$0.037) | 0.661($\pm$0.038) | 0.385($\pm$0.063)
TCDF | 0.775($\pm$0.023) | 0.597($\pm$0.029) | 0.679($\pm$0.021) | 0.314($\pm$0.050)
eSRU | 0.867($\pm$0.009) | 0.886($\pm$0.016) | 0.934($\pm$0.021) | 0.834($\pm$0.033)
GVAR (ours) | 0.945($\pm$0.010) | 0.885($\pm$0.046) | 0.970($\pm$0.009) | 0.916($\pm$0.024)
#### 4.1.2 Simulated fMRI Time Series
Another dataset we consider consists of rich and realistic simulations of
blood-oxygen-level-dependent (BOLD) time series (Smith et al., 2011) that were
generated using the dynamic causal modelling functional magnetic resonance
imaging (fMRI) forward model. In these time series, variables represent
‘activity’ in different spatial regions of interest within the brain. Herein,
we consider $R=5$ replicates from the simulation no. 3 of the original
dataset. These time series contain $p=15$ variables and only $T=200$
observations. The ground truth causal structure is very sparse (see Appendix
G). Details about hyperparameter tuning performed for this dataset can be
found in Appendix H.2. This experiment is similar to one presented by Khanna &
Tan (2020).
Table 2 provides a comparison of the inference techniques. Surprisingly, TCDF
outperforms other methods by a considerable margin (cf. Table 1). It is
followed by our method that, on average, outperforms cMLP, cLSTM, and eSRU in
terms of both AUROC and AUPRC. GVAR attains a BA score comparable to cLSTM.
Importantly, eSRU fails to shrink any weights to exact zeros, thus, hindering
the evaluation of accuracy and balanced accuracy scores (marked as ‘NA’ in
Table 2). This experiment demonstrates that the proximal gradient descent
(Parikh & Boyd, 2014), as implemented by eSRU (Khanna & Tan, 2020), may fail
to shrink any weights to 0 or shrinks all of them, even in relatively simple
datasets. cMLP seems to provide little improvement over simple VAR w.r.t.
AUROC or AUPRC. In general, this experiment promisingly shows that GVAR
performs on par with the techniques proposed by Tank et al. (2018) and Khanna
& Tan (2020) in a more realistic and data-scarce scenario than the Lorenz 96
experiment.
Table 2: Performance comparison on simulated fMRI time series. eSRU fails to shrink any weights to exact zeros, therefore, we have omitted accuracy and balanced accuracy score for it. Model | ACC($\pm$SD) | BA($\pm$SD) | AUROC($\pm$SD) | AUPRC($\pm$SD)
---|---|---|---|---
VAR | 0.910($\pm$0.006) | 0.513($\pm$0.015) | 0.615($\pm$0.044) | 0.175($\pm$0.054)
cMLP | 0.846($\pm$0.025) | 0.614($\pm$0.068) | 0.616($\pm$0.068) | 0.191($\pm$0.058)
cLSTM | 0.830($\pm$0.022) | 0.655($\pm$0.053) | 0.663($\pm$0.051) | 0.234($\pm$0.058)
TCDF | 0.899($\pm$0.023) | 0.728($\pm$0.063) | 0.812($\pm$0.041) | 0.368($\pm$0.126)
eSRU | NA | NA | 0.654($\pm$0.057) | 0.190($\pm$0.095)
GVAR (ours) | 0.806($\pm$0.070) | 0.652($\pm$0.045) | 0.687($\pm$0.066) | 0.289($\pm$0.116)
### 4.2 Inferring Effect Sign
So far, we have only considered inferring GC relationships, but not the signs
of Granger-causal effects. Such information can yield a better understanding
of relations among variables. To this end, we consider the Lotka–Volterra
model with multiple species (Bacaër (2011) provides a definition of the
original two-species system), given by the following differential equations:
$\displaystyle\frac{d{\textnormal{x}}^{i}}{dt}$
$\displaystyle=\alpha{\textnormal{x}}^{i}-\beta{\textnormal{x}}^{i}\sum_{j\in
Pa({\textnormal{x}}^{i})}{\textnormal{y}}^{j}-\eta\left({\textnormal{x}}^{i}\right)^{2},\text{
for $1\leq i\leq p$},$ (9) $\displaystyle\frac{d{\textnormal{y}}^{j}}{dt}$
$\displaystyle=\delta{\textnormal{y}}^{j}\sum_{k\in
Pa({\textnormal{y}}^{j})}{\textnormal{x}}^{k}-\rho{\textnormal{y}}^{j},\text{
for $1\leq j\leq p$},$ (10)
where ${\textnormal{x}}^{i}$ correspond to population sizes of prey species;
${\textnormal{y}}^{j}$ denote population sizes of predator species;
$\alpha,\beta,\eta,\delta,\rho>0$ are fixed parameters controlling strengths
of interactions; and $Pa({\textnormal{x}}^{i})$, $Pa({\textnormal{y}}^{j})$
are sets of Granger-causes of ${\textnormal{x}}^{i}$ and
${\textnormal{y}}^{j}$, respectively. According to Equations 9 and 10,
Figure 3: Simulated two-species Lotka–Volterra time series (top) and
generalised coefficients (bottom). Prey have a positive effect on predators,
and vice versa.
the population size of each prey species ${\textnormal{x}}^{i}$ is driven down
by $\left|Pa({\textnormal{x}}^{i})\right|$ predator species (negative
effects), whereas each predator species ${\textnormal{y}}^{j}$ is driven up by
$\left|Pa({\textnormal{y}}^{j})\right|$ prey populations (positive effects).
We simulate the multi-species Lotka–Volterra system numerically. Appendix K
contains details about simulations and the summary graph of the time series.
To infer effect directions, we inspect signs of median generalised
coefficients for trained GVAR models. For cMLP, cLSTM, TCDF, and eSRU, we
inspect signs of averaged weights in relevant layers. For VAR, we examine
coefficient signs. For the sake of fair comparison, we restrict all models to
a maximum lag of $K=1$ (where applicable). In this experiment, we focus on BA
scores for positive (BApos) and negative (BAneg) relationships. Appendix L
provides another example of detecting effect signs with GVAR, on a trivial
benchmark with linear dynamics.
Table 3 shows the results for this experiment. Linear VAR does not perform
well at inferring the GC structure, however, its coefficient signs are
strongly associated with true signs of relationships. cMLP provides a
considerable improvement in GC inference, and surprisingly its input weights
are informative about the signs of GC effects. cLSTM fails to shrink any of
the relevant weights to zero; furthermore, the signs of its weights are not
associated with the true signs. Although eSRU performs better than VAR at
inferring the summary graph, its weights are not associated with effect signs
at all. TCDF performs poorly in this experiment, failing to infer any
relationships apart from self-causation. Our model considerably outperforms
all baselines in detecting effect signs, achieving nearly perfect scores: it
infers more meaningful and interpretable parameter values than all other
models.
These results are not surprising, because the baseline methods, apart from
linear VAR, rely on interpreting weights of relevant layers that, in general,
do not need to be associated with effect signs and are only informative about
the presence or absence of GC interactions. Since the GVAR model follows a
form of SENNs (see Equation 2), its generalised coefficients shed more light
into how the future of the target variable depends on the past of its
predictors. This restricted structure is more intelligible and yet is
sufficiently flexible to perform on par with sparse-input neural networks.
Table 3: Performance comparison on the multi-species Lotka–Volterra system. Next to accuracy and balanced accuracy scores, we evaluate BA scores for detecting positive and negative interactions. Model | ACC($\pm$SD) | BA($\pm$SD) | BApos($\pm$SD) | BAneg($\pm$SD)
---|---|---|---|---
VAR | 0.383($\pm$0.095) | 0.635($\pm$0.060) | 0.845($\pm$0.024) | 0.781($\pm$0.042)
cMLP | 0.825($\pm$0.035) | 0.834($\pm$0.043) | 0.889($\pm$0.031) | 0.846($\pm$0.084)
cLSTM | NA | NA | 0.491($\pm$0.026) | 0.604($\pm$0.042)
TCDF | 0.832($\pm$0.013) | 0.500($\pm$0.012) | 0.538($\pm$0.045) | 0.504($\pm$0.090)
eSRU | 0.703($\pm$0.048) | 0.755($\pm$0.010) | 0.501($\pm$0.025) | 0.650($\pm$0.078)
GVAR (ours) | 0.977($\pm$0.005) | 0.961($\pm$0.014) | 0.932($\pm$0.027) | 0.999($\pm$0.001)
In addition to inferring the summary graph, GVAR allows inspecting variability
of generalised coefficients. Figure 3 provides an example of generalised
coefficients inferred for a two-species Lotka–Volterra system. Although
coefficients vary with time, GVAR consistently infers that the predator
population is driven up by prey and the prey population is driven down by
predators. For the multi-species system used to produce the quantitative
results, inferred coefficients behave similarly (see Figure 12 in Appendix K).
## 5 Conclusion
In this paper, we focused on two problems: (i) inferring Granger-causal
relationships in multivariate time series under nonlinear dynamics and (ii)
inferring signs of Granger-causal relationships. We proposed a novel framework
for GC inference based on autoregressive modelling with self-explaining neural
networks and demonstrated that, on simulated data, its performance is
promisingly competitive with the related methods of Tank et al. (2018) and
Khanna & Tan (2020). Proximal gradient descent employed by cMLP, cLSTM, and
eSRU often does not shrink weights to exact zeros and, thus, prevents treating
the inference technique as a statistical hypothesis test. Our framework
mitigates this problem by performing a stability-based selection of
significant relationships, finding a GC structure that is stable on original
and time-reversed data. Additionally, proposed GVAR model is more amenable to
interpretation, since relationships between variables can be explored by
inspecting generalised coefficients, which, as we showed empirically, are more
informative than input layer weights. To conclude, the proposed model and
inference framework are a viable alternative to previous techniques and are
better suited for exploratory analysis of multivariate time series data.
In future research, we plan a thorough investigation of the stability-based
thresholding procedure (see Algorithm 1) and of time-reversal for inferring
GC. Furthermore, we would like to facilitate a more comprehensive comparison
with the baselines on real-world data sets. It would also be interesting to
consider better-informed link functions and basis concepts (see Equation 2).
Last but not least, we plan to tackle the problem of inferring time-varying GC
structures with the introduced framework.
#### Acknowledgments
We thank Djordje Miladinovic and Mark McMahon for valuable discussions and
inputs. We also acknowledge Jonas Rothfuss and Kieran Chin-Cheong for their
helpful feedback on the manuscript.
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## Appendix A Linear Vector Autoregression
Linear vector autoregression (VAR) (Lütkepohl, 2007) is a time series model
conventionally used to test for Granger causality (see Section 2.1). VAR
assumes that functions $g_{i}(\cdot)$ in Equation 1 are linear:
${\mathbf{x}}_{t}=\bm{\nu}+\sum_{k=1}^{K}\bm{\Psi}_{k}{\mathbf{x}}_{t-k}+\bm{\varepsilon}_{t},$
(11)
where $\bm{\nu}\in\mathbb{R}^{p}$ is the intercept vector;
$\bm{\Psi}_{k}\in\mathbb{R}^{p\times p}$ are coefficient matrices; and
$\bm{\varepsilon}_{t}\sim\mathcal{N}_{p}\left(\bm{0},\bm{\Sigma}_{\bm{\varepsilon}}\right)$
are Gaussian innovation terms. Parameter $K$ is the order of the VAR model and
determines the maximum lag at which Granger-causal interactions occur. In VAR,
Granger causality is defined by zero constraints on the coefficients, in
particular, ${\textnormal{x}}^{i}$ does not Granger-cause
${\textnormal{x}}^{j}$ if and only if, for all lags
$k\in\left\\{1,2,...,K\right\\}$, $\left(\bm{\Psi}_{k}\right)_{j,i}=0$. These
constraints can be tested by performing, for example, $F$-test or Wald test.
Usually a VAR model is fitted using multivariate least squares. In high-
dimensional time series, regularisation can be introduced to avoid inferring
spurious associations. Table 4 shows various sparsity-inducing penalties for a
linear VAR model of order $K$ (see Equation 11), described by Nicholson et al.
(2017). Different penalties induce different sparsity patterns in coefficient
matrices $\bm{\Psi}_{1},\bm{\Psi}_{2},...,\bm{\Psi}_{K}$. These penalties can
be adapted to the GVAR model for the sparsity-inducing term $R(\cdot)$ in
Equation 6.
Table 4: Various sparsity-inducing penalty terms, described by Nicholson et al. (2017), for a linear VAR of order $K$. Herein, $\bm{\Psi}=\begin{bmatrix}\bm{\Psi}_{1}&\bm{\Psi}_{2}&...&\bm{\Psi}_{K}\end{bmatrix}\in\mathbb{R}^{p\times Kp}$ (cf. Equation 11), and $\bm{\Psi}_{k:K}=\begin{bmatrix}\bm{\Psi}_{k}&\bm{\Psi}_{k+1}&...&\bm{\Psi}_{K}\end{bmatrix}$. Different penalties induce different sparsity patterns in coefficient matrices. Model Structure | Penalty
---|---
Basic Lasso | $\left\|\bm{\Psi}\right\|_{1}$
Elastic net | $\alpha\left\|\bm{\Psi}\right\|_{1}+(1-\alpha)\left\|\bm{\Psi}\right\|_{2}^{2},\alpha\in(0,1)$
Lag group | $\sum_{k=1}^{K}\left\|\bm{\Psi}_{k}\right\|_{F}$
Componentwise | $\sum_{i=1}^{p}\sum_{k=1}^{K}\left\|\left(\bm{\Psi}_{k:K}\right)_{i}\right\|_{2}$
Elementwise | $\sum_{i=1}^{p}\sum_{j=1}^{p}\sum_{k=1}^{K}\left\|\left(\bm{\Psi}_{k:K}\right)_{i,j}\right\|_{2}$
Lag-weighted Lasso | $\sum_{k=1}^{K}k^{\alpha}\left\|\bm{\Psi}_{k}\right\|_{1},\alpha\in(0,1)$
## Appendix B Inferring Granger Causality under Nonlinear Dynamics
Below we provide a more detailed overview of the related work on inferring
nonlinear multivariate Granger causality, focusing on the recent machine
learning techniques that tackle this problem.
Kernel-based Methods. Kernel-based GC inference techniques provide a natural
extension of the VAR model, described in Appendix A, to nonlinear dynamics.
Marinazzo et al. (2008) leverage reproducing kernel Hilbert spaces to infer
linear Granger causality in an appropriate transformed feature space. Ren et
al. (2020) introduce a kernel-based GC inference technique that relies on
regularisation – Hilbert–Schmidt independence criterion (HSIC) Lasso GC.
Neural Networks with Non-uniform Embedding. Montalto et al. (2015) propose
neural networks with non-uniform embedding (NUE). Significant Granger causes
are identified using the NUE, a feature selection procedure. An MLP is ‘grown’
iteratively by greedily adding lagged predictor components as inputs. Once
stopping conditions are satisfied, a predictor time series is claimed a
significant cause of the target if at least one of its lagged components was
added as an input. This technique is prohibitively costly, especially, in a
high-dimensional setting, since it requires training and comparing many
candidate models. Wang et al. (2018) extend the NUE by replacing MLPs with
LSTMs.
Neural Granger Causality. Tank et al. (2018) propose inferring nonlinear
Granger causality using structured multilayer perceptron and long short-term
memory with sparse input layer weights, cMLP and cLSTM. To infer GC, $p$
models need to be trained with each variable as a response. cMLP and cLSTM
leverage the group Lasso penalty and proximal gradient descent (Parikh & Boyd,
2014) to infer GC relationships from trained input layer weights.
Attention-based Convolutional Neural Networks. Nauta et al. (2019) introduce
the temporal causal discovery framework (TCDF) that utilises attention-based
convolutional neural networks (CNN). Similarly to cMLP and cLSTM (Tank et al.,
2018), the TCDF requires training $p$ neural network models to forecast each
variable. Key distinctions of the TCDF are _(i)_ the choice of the temporal
convolutional network architecture over MLPs or LSTMs for time series
forecasting and _(ii)_ the use of the attention mechanism to perform
attribution. In addition to the GC inference, the TCDF can detect time delays
at which Granger-causal interactions occur. Furthermore, Nauta et al. (2019)
provide a permutation-based procedure for evaluating variable importance and
identifying significant causal links.
Economy Statistical Recurrent Units. Khanna & Tan (2020) propose an approach
for inferring nonlinear Granger causality similar to cMLP and cLSTM (Tank et
al., 2018). Likewise, they penalise norms of weights in some layers to induce
sparsity. The key difference from the work of Tank et al. (2018) is the use of
statistical recurrent units (SRUs) as a predictive model. Khanna & Tan (2020)
propose a new sample-efficient architecture – economy-SRU (eSRU).
Minimum Predictive Information Regularisation. Wu et al. (2020) adopt an
information-theoretic approach to Granger-causal discovery. They introduce
learnable corruption, e.g. additive Gaussian noise with learnable variances,
for predictor variables and minimise a loss function with minimum predictive
information regularisation that encourages the corruption of predictor time
series. Similarly to the approaches of Tank et al. (2018); Nauta et al.
(2019); Khanna & Tan (2020), this framework requires training $p$ models
separately.
Amortised Causal Discovery & Neural Relational Inference. Kipf et al. (2018)
introduce the neural relational inference (NRI) model based on graph neural
networks and variational autoencoders. The NRI model disentangles the dynamics
and the undirected relational structure represented explicitly as a discrete
latent graph variable. This allows pooling time series data with shared
dynamics, but varying relational structures. Löwe et al. (2020) provide a
natural extension of the NRI model to the Granger-causal discovery. They
introduce a more general framework of the amortised causal discovery wherein
time series replicates have a varying causal structure, but share dynamics. In
contrast to the previous methods (Tank et al., 2018; Nauta et al., 2019;
Khanna & Tan, 2020; Wu et al., 2020), which in this setting, have to be
retrained separately for each replicate, the NRI is trained on the pooled
dataset, leveraging shared dynamics.
## Appendix C Properties of Self-explaining Neural Networks
As defined by Alvarez-Melis & Jaakkola (2018), $g(\cdot)$,
$\bm{\theta}(\cdot)$, and $\bm{h}(\cdot)$ in Equation 2 need to satisfy:
1. 1.
$g(\cdot)$ is monotonic and additively separable in its arguments;
2. 2.
$\frac{\partial g}{\partial z_{i}}>0$ with
$z_{i}=\theta({\bm{x}})_{i}h({\bm{x}})_{i}$, for all $i$;
3. 3.
$\bm{\theta}(\cdot)$ is locally difference-bounded by $\bm{h}(\cdot)$, i.e.
for every ${\bm{x}}_{0}$, there exist $\delta>0$ and $L\in\mathbb{R}$ s.t. if
$\left\|{\bm{x}}-{\bm{x}}_{0}\right\|<\delta$, then
$\left\|\bm{\theta}({\bm{x}})-\bm{\theta}({\bm{x}}_{0})\right\|\leq
L\left\|\bm{h}({\bm{x}})-\bm{h}({\bm{x}}_{0})\right\|$;
4. 4.
$\left\\{h({\bm{x}})_{i}\right\\}_{i=1}^{k}$ are interpretable representations
of ${\bm{x}}$;
5. 5.
$k$ is small.
## Appendix D Ablation Study of the Loss Function
We inspect hyperparameter tuning results for the GVAR model on Lorenz 96 (see
Section 4.1.1) and synthetic fMRI time series (Smith et al., 2011) (see
Section 4.1.2) as an ablation study for the loss function proposed (see
Equation 6). Figures 4 and 5 show heat maps of BA scores (left) and AUPRCs
(right) for different values of parameters $\lambda$ and $\gamma$ for Lorenz
96 and fMRI datasets, respectively. For the Lorenz 96 system, sparsity-
inducing regularisation appears to be particularly important, nevertheless,
there is also an increase in BA and AUPRC from a moderate smoothing penalty.
For fMRI, we observe considerable performance gains from introducing both the
sparsity-inducing and smoothing penalty terms. Given the sparsity of the
ground truth GC structure and the scarce number of observations ($T=200$),
these gains are not unexpected. During preliminary experiments, we ran grid
search across wider ranges of $\lambda$ and $\gamma$ values, however, did not
observe further improvements from stronger regularisation. In summary, these
results empirically motivate the need for two different forms of
regularisation leveraged by the GVAR loss function: the sparsity-inducing and
smoothing penalty terms.
Figure 4: GVAR hyperparameter grid search results for Lorenz 96 time series
(under $F=40$) across 5 values of $\lambda\in[0.0,3.0]$ and
$\gamma\in[0.0,0.02]$. Each cell shows average balanced accuracy (left) and
AUPRC (right) across 5 replicates (darker colours correspond to lower
performance) for one hyperparameter configuration.
Figure 5: GVAR hyperparameter grid search results for simulated fMRI time
series across 5 values of $\lambda\in[0.0,3.0]$ and $\gamma\in[0.0,0.1]$. The
heat map on the left shows average BA scores, and the heat map on the right –
average AUPRCs.
## Appendix E Stability-based Thresholding: Example
Figure 6 shows an example of agreement between dependency structures inferred
on original and time-reversed synthetic sequences across a range of thresholds
(see Algorithm 1). In addition, we plot the BA score for resulting thresholded
matrices evaluated against the true adjacency matrix. As can be seen, the peak
of stability agrees with the highest BA achieved. In both cases, the procedure
described by Algorithm 1 chooses the optimal threshold, which results in the
highest agreement with the true dependency structure (unknown at the time of
inference).
(a) Lorenz 96, $F=10$.
(b) Multi-species Lotka–Volterra.
Figure 6: Agreement ($\blacktriangle$) between GC structures inferred on the
original and time-reversed data across a range of thresholds for one
simulation of the Lorenz 96 (6(a)) and multi-species Lotka–Volterra (6(b))
systems. BA score ($\times$) is evaluated against the ground truth adjacency
matrix.
## Appendix F Comparison of Training & Inference Time
To compare the considered methods in terms of their computational complexity,
we measure training and inference time across three simulated datasets with
$p\in\\{4,15,20\\}$ variables and varying time series lengths. This experiment
was performed on an Intel Core i7-7500U CPU (2.70 GHz × 4) with a GeForce GTX
950M GPU. All models were trained for 1000 epochs with a mini-batch size of
64. In each dataset, the same numbers of hidden layers and hidden units were
used across all models. When applicable, models were restricted to the same
order ($K$). Table 5 contains average training and inference time in seconds
with standard deviations. Observe that for the fMRI and Lorenz 96 datasets,
GVAR is substantially faster than cLSTM and eSRU.
Table 5: Average training and inference time, in seconds, for the methods. Inference was performed on time series generated from the linear model (see Appendix L), simulated fMRI time series (see Section 4.1.2), and the Lorenz 96 system (see Section 4.1.1). Model | Linear ($\bm{p=4,T=500}$, $\bm{K=1}$) | fMRI ($\bm{p=15,T=200}$, $\bm{K=1}$) | Lorenz 96, $\bm{F=10}$ ($\bm{p=20,T=500}$, $\bm{K=5}$)
---|---|---|---
VAR | 0.018($\pm$0.001) | 0.27($\pm$0.01) | 8.5($\pm$0.6)
cMLP | 19.7($\pm$3.5) | 55.1($\pm$4.0) | 94.7($\pm$2.4)
cLSTM | 999.2($\pm$177.2) | 1763.3($\pm$235.3) | 2023.8($\pm$12.0)
TCDF | 11.4($\pm$1.6) | 24.6($\pm$1.7) | 50.1($\pm$2.1)
eSRU | 62.9($\pm$2.6) | 258.3($\pm$31.1) | 671.4($\pm$9.7)
GVAR (ours) | 75.5($\pm$8.3) | 20.4($\pm$0.7) | 197.7($\pm$24.8)
## Appendix G GC Summary Graphs of Simulated Time Series
(a) Lorenz 96.
(b) fMRI.
(c) Multi-species Lotka–Volterra.
Figure 7: Adjacency matrices of Granger-causal summary graphs for Lorenz 96
(see Section 4.1.1), simulated fMRI (see Section 4.1.2), and multi-species
Lotka–Volterra (see Section 4.2) time series. Dark cells correspond to the
absence of a GC relationship, i.e. $A_{i,j}=0$; light cells denote a GC
relationship, i.e. $A_{i,j}=1$.
## Appendix H Hyperparameter Tuning
In our experiments (see Section 4), for all of the inference techniques
compared, we searched across a grid of hyperparameters that control the
sparsity of inferred GC structures. Other hyperparameters were fine-tuned
manually. Final results reported in the paper correspond to the best
hyperparameter configurations. With this testing setup, our goal was to fairly
compare best achievable inferential performance of the techniques.
Tables 6, 7, and 8 provide ranges for hyperparameter values considered in each
experiment. For cMLP and cLSTM (Tank et al., 2018), parameter $\lambda$ is the
weight of the group Lasso penalty; for TCDF (Nauta et al., 2019), significance
parameter $\alpha$ is used to decide which potential GC relationships are
significant; eSRU (Khanna & Tan, 2020) has three different penalties weighted
by $\lambda_{1:3}$. For the stability-based thresholding (see Algorithm 1) in
GVAR, we used $Q=20$ equally spaced values in $[0,1]$ as sequence
$\bm{\xi}$555We did not observe high sensitivity of performance w.r.t.
$\bm{\xi}$, as long as sufficiently many evenly spaced sparsity levels are
considered.. For Lorenz 96 and fMRI experiments, grid search results are
plotted in Figures 4, 8, and 5. Figure 9 contains GVAR grid search results for
the Lotka–Volterra experiment.
### H.1 Lorenz 96
Table 6: Hyperparameter values for Lorenz 96 datasets with $F=10$ and $40$. Herein, $K$ denotes model order (maximum lag). If a hyperparameter is not applicable to a model, the corresponding entry is marked by ‘NA’. Model | $\bm{K}$ | # hidden layers | # hidden units | # training epochs | Learning rate | Mini-batch size | Sparsity hyperparam-s
---|---|---|---|---|---|---|---
VAR | 5 | NA | NA | NA | NA | NA | NA
cMLP | 5 | 2 | 50 | 1,000 | 1.0$\mathrm{e}\text{-}$2 | NA | $F=10$: $\lambda\in[0.5,2.0]$; $F=40$: $\lambda\in[0.0,1.0]$
cLSTM | NA | 2 | 50 | 1,000 | 5.0$\mathrm{e}\text{-}$3 | NA | $F=10$: $\lambda\in[0.1,0.6]$; $F=40$: $\lambda\in[0.2,0.25]$
TCDF | 5 | 2 | 50 | 1,000 | 1.0$\mathrm{e}\text{-}$2 | 64 | $F=10,40$: $\alpha\in[0.0,2.5]$
eSRU | NA | 2 | 10 | 2,000 | 5.0$\mathrm{e}\text{-}$3 | 64 | $F=10,40$: $\lambda_{1:3}\in[0.01,0.1]$
GVAR | 5 | 2 | 50 | 1,000 | 1.0$\mathrm{e}\text{-}$4 | 64 | $F=10,40$: $\lambda\in[0.0,3.0]$, $\gamma\in[0.0,0.025]$
Figure 8: GVAR hyperparameter grid search results for Lorenz 96 time series,
under $F=10$, across 5 values of $\lambda\in[0.0,3.0]$ and
$\gamma\in[0.0,0.02]$. Each cell shows average balanced accuracy (left) and
AUPRC (right) across 5 replicates.
### H.2 fMRI
Table 7: Hyperparameter values for simulated fMRI time series. Model | $\bm{K}$ | # hidden layers | # hidden units | # training epochs | Learning rate | Mini-batch size | Sparsity hyperparam-s
---|---|---|---|---|---|---|---
VAR | 1 | NA | NA | NA | NA | NA | NA
cMLP | 1 | 1 | 50 | 2,000 | 1.0$\mathrm{e}\text{-}$2 | NA | $\lambda\in[0.001,0.75]$
cLSTM | NA | 1 | 50 | 1,000 | 1.0$\mathrm{e}\text{-}$2 | NA | $\lambda\in[0.05,0.3]$
TCDF | 1 | 1 | 50 | 2,000 | 1.0$\mathrm{e}\text{-}$3 | 64 | $\alpha\in[0.0,2.0]$
eSRU | NA | 2 | 10 | 2,000 | 1.0$\mathrm{e}\text{-}$3 | 64 | $\lambda_{1}\in[0.01,0.05]$, $\lambda_{2}\in[0.01,0.05]$, $\lambda_{3}\in[0.01,1.0]$
GVAR | 1 | 1 | 50 | 1,000 | 1.0$\mathrm{e}\text{-}$4 | 64 | $\lambda\in[0.0,3.0]$, $\gamma\in[0.0,0.1]$
### H.3 Lotka-Volterra
Table 8: Hyperparameter values for multi-species Lotka–Volterra time series. Model | $\bm{K}$ | # hidden layers | # hidden units | # training epochs | Learning rate | Mini-batch size | Sparsity hyperparam-s
---|---|---|---|---|---|---|---
VAR | 1 | NA | NA | NA | NA | NA | NA
cMLP | 1 | 2 | 50 | 2,000 | 5.0$\mathrm{e}\text{-}$3 | NA | $\lambda\in[0.2,0.4]$
cLSTM | NA | 2 | 50 | 1,000 | 5.0$\mathrm{e}\text{-}$3 | NA | $\lambda\in[0.0,1.0]$
TCDF | 1 | 2 | 50 | 2,000 | 1.0$\mathrm{e}\text{-}$2 | 256 | $\alpha\in[0.0,2.0]$
eSRU | NA | 2 | 10 | 2,000 | 1.0$\mathrm{e}\text{-}$3 | 256 | $\lambda_{1}\in[0.01,0.05]$, $\lambda_{2}\in[0.01,0.05]$, $\lambda_{3}\in[0.01,1.0]$
GVAR | 1 | 2 | 50 | 500 | 1.0$\mathrm{e}\text{-}$4 | 256 | $\lambda\in[0.0,1.0]$, $\gamma\in[0.0,0.01]$
(a) BA
(b) AUPRC
(c) BApos
(d) BAneg
Figure 9: GVAR hyperparameter grid search results for multi-species
Lotka–Volterra time series across 5 values of $\lambda\in[0.0,1.0]$ and
$\gamma\in[0.0,0.01]$. Heat maps above show balanced accuracies (9(a)), AUPRCs
(9(b)), and balanced accuracies for positive (9(c)) and negative (9(d))
effects.
## Appendix I Comparison with Dynamic Bayesian Networks
We provide a comparison between GVAR and linear Gaussian dynamic Bayesian
networks (DBN). DBNs are a classical approach to temporal structure learning
(Murphy & Russell, 2002). We use R (R Core Team, 2020) package dbnR (Quesada,
2020) to fit DBNs on all datasets considered in Section 4. We use two
structure learning algorithms: the max-min hill-climbing (MMHC) (Tsamardinos
et al., 2006) and the particle swarm optimisation (Xing-Chen et al., 2007).
Table 9 contains average balanced accuracies achieved by DBNs and GVAR for
inferring the GC structure. Not surprisingly, DBNs outperform GVAR on the time
series with linear dynamics, but fail to infer the true structure on Lorenz
96, fMRI, and Lotka–Volterra datasets.
Table 9: Comparison of balanced accuracy scores for GVAR and DBNs. Standard deviations (SD) are taken across 5 independent replicates. Model | Linear | Lorenz 96, $\bm{F=10}$ | Lorenz 96, $\bm{F=40}$ | fMRI | Lotka–Volterra
---|---|---|---|---|---
GVAR (ours) | 0.938($\pm$0.084) | 0.982($\pm$0.006) | 0.885($\pm$0.046) | 0.652($\pm$0.045) | 0.961($\pm$0.014)
DBN (MMHC) | 0.900($\pm$0.105) | 0.821($\pm$0.009) | 0.687($\pm$0.033) | 0.522($\pm$0.023) | 0.586($\pm$0.037)
DBN (PSO) | 0.950($\pm$0.028) | 0.627($\pm$0.011) | 0.514($\pm$0.025) | 0.473($\pm$0.066) | 0.534($\pm$0.027)
## Appendix J The Lorenz 96 System: Further Experiments
Figure 10: Inferential performance of GVAR across a range of forcing constant
values.
In addition to the experiments in Section 4.1.1, we examine the performance of
VAR and GVAR models across a range of forcing constant values $F=0,5,10,25,50$
for the Lorenz 96 system with $p=20$ variables. Figure 10 shows average AUPRCs
with bands corresponding to the 95% CI for the mean. It appears that for both
models, inference is more challenging for lower ($<10$) and higher values of
$F$ ($>20$). This observation is in agreement with the results in Section
4.1.1, where all inference techniques performed worse under $F=40$ than under
$F=10$. Note that herein same GVAR hyperparameters were used across all values
of $F$. It is possible that better inferential performance could be achieved
with GVAR after comprehensive hyperparameter tuning.
## Appendix K The Lotka–Volterra System
The original Lotka–Volterra system (Bacaër, 2011) includes only one predator
and one prey species, population sizes of which are denoted by x and y,
respectively. Population dynamics are given by the following coupled
differential equations:
$\displaystyle\frac{d{\textnormal{x}}}{dt}$
$\displaystyle=\alpha{\textnormal{x}}-\beta{\textnormal{x}}{\textnormal{y}},$
(12) $\displaystyle\frac{d{\textnormal{y}}}{dt}$
$\displaystyle=\delta{\textnormal{y}}{\textnormal{x}}-\rho{\textnormal{y}},$
(13)
where $\alpha,\beta,\delta,\rho>0$ are fixed parameters determining strengths
of interactions.
In this paper, we consider a multiple species version of the system, given by
Equations 9 and 10 in Section 4.2. We simulate the system under
$\alpha=\rho=1.1$, $\beta=\delta=0.2$, $\eta=2.75\times 10^{-5}$,
$\left|Pa({\textnormal{x}}^{i})\right|=\left|Pa({\textnormal{y}}^{j})\right|=2$,
$p=10$, i.e. $2p=20$ variables in total, with $T=2000$ observations. Figure 11
depicts signs of GC effects between variables in a multi-species
Lotka–Volterra with $2p=20$ species and 2 parents per variable. We simulate
this system numerically by using the Runge-Kutta method666Simulations are
based on the implementation available at https://github.com/smkalami/lotka-
volterra-in-python.. We make a few adjustments to the state transition
equations, in particular: we introduce normally-distributed innovation terms
to make simulated data noisy; during state transitions, we clip all population
sizes below 0. Figure 12 shows traces of generalised coefficients inferred by
GVAR: magnitudes and signs of coefficients reflect the true dependency
structure.
Figure 11: Signs of GC relationships between variables in the Lotka–Volterra
system given by Equations 9 and 10, with $p=10$. First ten columns correspond
to prey species, whereas the last ten correspond to predators. Each prey
species is ‘hunted’ by two predator species, and each predator species ‘hunts’
two prey species. Similarly to the other experiments, we ignore self-causal
relationships. Figure 12: Variability of GVAR generalised coefficients
throughout time for a simulation of the multi-species Lotka–Volterra system.
Coefficients for Granger non-causal relationships fluctuate around 0; for
Granger-causal relationships, coefficients are consistently different from 0:
positive for prey $\rightarrow$ predator interactions and negative for
predator $\rightarrow$ prey.
## Appendix L Effect Sign Detection in a Linear VAR
Herein we provide results for the evaluation of GVAR and our inference
framework on a very simple synthetic time series dataset. We simulate time
series with $p=4$ variables and linear interaction dynamics given by the
following equations:
$\displaystyle{\textnormal{x}}_{t}$
$\displaystyle=a_{1}{\textnormal{x}}_{t-1}+\varepsilon_{t}^{{\textnormal{x}}},$
(14) $\displaystyle{\textnormal{w}}_{t}$
$\displaystyle=a_{2}{\textnormal{w}}_{t-1}+a_{3}{\textnormal{x}}_{t-1}+\varepsilon_{t}^{{\textnormal{w}}},$
$\displaystyle{\textnormal{y}}_{t}$
$\displaystyle=a_{4}{\textnormal{y}}_{t-1}+a_{5}{\textnormal{w}}_{t-1}+\varepsilon_{t}^{y},$
$\displaystyle{\textnormal{z}}_{t}$
$\displaystyle=a_{6}{\textnormal{z}}_{t-1}+a_{7}{\textnormal{w}}_{t-1}+a_{8}{\textnormal{y}}_{t-1}+\varepsilon^{{\textnormal{z}}}_{t},$
where coefficients $a_{i}\sim\mathcal{U}\left([-0.8,-0.2]\cup[0.2,0.8]\right)$
are sampled independently in each simulation; and
$\varepsilon^{\cdot}_{t}\sim\mathcal{N}\left(0,0.16\right)$ are additive
innovation terms. This is an adapted version of one of artificial datasets
described by Peters et al. (2013), but without instantaneous effects.
The GC summary graph of the system is visualised in Figure 13. It is
considerably denser than for the Lorenz 96, fMRI, and Lotka–Volterra time
series investigated in Section 4.
Figure 13: The adjacency matrix of the GC summary graph for the model given by
Equation 14.
Similarly to the experiment described in Section 4.2, we infer GC
relationships with the proposed framework and evaluate inference results
against the true dependency structure and effect signs. Table 10 contains
average performance across 10 simulations achieved by GVAR with hyperparameter
values $K=1$, $\lambda=0.2$, and $\gamma=0.5$. In addition, we provide results
for some of the baselines (no systematic hyperparameter tuning was performed
for this experiment).
GVAR attains perfect AUROC and AUPRC in all 10 simulations. In some cases,
stability-based thresholding fails to recover a completely correct GC
structure, nevertheless, average accuracy and balanced accuracy scores are
satisfactory. Signs of inferred generalised coefficients mostly agree with the
ground truth effect signs, as given by coefficients $a_{1:8}$ in Equation 14.
Not surprisingly, linear VAR performs the best on this dataset w.r.t. all
evaluation metrics. Both cMLP and eSRU successfully infer GC relationships,
achieving results comparable to GVAR. However, neither infers effect signs as
well as GVAR. Thus, similarly to the experiment in Section 4.2, we conclude
that generalised coefficients are more interpretable than neural network
weights leveraged by cMLP, TCDF, and eSRU.
To summarise, this simple experiment serves as a sanity check and shows that
our GC inference framework performs reasonably in low-dimensional time series
with linear dynamics and a relatively dense GC summary graph (cf. Figure 7).
Generally, the method successfully infers both the dependency structure and
interaction signs.
Table 10: Performance on synthetic time series with linear dynamics, given by Equation 14. Averages and standard deviations are evaluated across 10 independent simulations. eSRU failed to shrink weights to exact 0s, therefore, we omit accuracy and BA scores for it. | VAR | cMLP | TCDF | eSRU | GVAR (ours)
---|---|---|---|---|---
ACC | 0.975($\pm$0.038) | 0.867($\pm$0.085) | 0.791($\pm$0.056) | NA | 0.950($\pm$0.067)
BA | 0.981($\pm$0.029) | 0.900($\pm$0.064) | 0.688($\pm$0.169) | NA | 0.938($\pm$0.084)
AUROC | 1.000($\pm$0.000) | 1.000($\pm$0.000) | 0.866($\pm$0.114) | 0.972($\pm$0.043) | 1.000($\pm$0.000)
AUPRC | 1.000($\pm$0.000) | 1.000($\pm$0.000) | 0.812($\pm$0.128) | 0.967($\pm$0.046) | 1.000($\pm$0.000)
BApos | 0.995($\pm$0.014) | 0.761($\pm$0.206) | 0.574($\pm$0.235) | 0.613($\pm$0.168) | 0.920($\pm$0.158)
BAneg | 0.990($\pm$0.019) | 0.746($\pm$0.232) | 0.550($\pm$0.169) | 0.622($\pm$0.215) | 0.928($\pm$0.151)
## Appendix M Prediction Error
Herein we evaluate the prediction error of models on held-out data. Last 20%
of time series points were held out to perform prediction on Lorenz 96, fMRI,
and Lotka–Volterra datasets. Root-mean-square error (RMSE) was computed for
predictions across $R=5$ independent replicates:
$\textrm{RMSE}=\frac{1}{p}\sum_{j=1}^{p}\sqrt{\frac{\sum_{t=1}^{T}\left(\hat{x}_{t}^{j}-x_{t}^{j}\right)}{T}},$
(15)
where $\hat{x}_{t}^{j}$ is the one-step forecast made by a model for the
$t$-th point of the $j$-th variable, and $T$ is the length of the held-out
time series segment. Table 11 contains average RMSEs for all models across the
considered datasets. In general, RMSEs are not associated with the inferential
performance of the models (cf. tables 1, 2, and 3). For example, while TCDF
achieves the best inferential performance on fMRI (see Table 2), its
prediction error is higher than for cMLP. This ‘misalignment’ between the
prediction error and the consistency of variable selection is not surprising
and has been discussed before, e.g. by Meinshausen & Bühlmann (2010).
Table 11: RMSEs of models on held-out data. Averages and standard deviations were taken across 5 independent replicates. Model | Lorenz 96, $\bm{F=10}$ | Lorenz 96, $\bm{F=40}$ | fMRI | Lotka–Volterra
---|---|---|---|---
VAR | 0.378($\pm$0.008) | 1.088($\pm$0.021) | 0.970($\pm$0.042) | 0.202($\pm$0.025)
cMLP | 0.336($\pm$0.030) | 0.795($\pm$0.017) | 0.724($\pm$0.037) | 0.098($\pm$0.016)
cLSTM | 0.592($\pm$0.013) | 0.983($\pm$0.014) | 0.874($\pm$0.045) | 0.691($\pm$0.035)
TCDF | 0.536($\pm$0.015) | 1.514($\pm$0.030) | 0.879($\pm$0.022) | 0.111($\pm$0.016)
eSRU | 1.000($\pm$0.010) | 1.006($\pm$0.015) | 1.000($\pm$0.048) | 0.720($\pm$0.035)
GVAR (ours) | 0.572($\pm$0.015) | 1.005($\pm$0.018) | 0.966($\pm$0.047) | 0.119($\pm$0.009)
|
# All-optical linear polarization engineering in single and coupled exciton-
polariton condensates
I. Gnusov<EMAIL_ADDRESS>Skolkovo Institute of Science and
Technology, Moscow, Territory of innovation center “Skolkovo”, Bolshoy
Boulevard 30, bld. 1, 121205, Russia. H. Sigurdsson<EMAIL_ADDRESS>Skolkovo Institute of Science and Technology, Moscow, Territory of innovation
center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia. School of
Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK.
J. D. Töpfer Skolkovo Institute of Science and Technology, Moscow, Territory
of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia.
S. Baryshev Skolkovo Institute of Science and Technology, Moscow, Territory
of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia.
S. Alyatkin Skolkovo Institute of Science and Technology, Moscow, Territory
of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia.
P. G. Lagoudakis Skolkovo Institute of Science and Technology, Moscow,
Territory of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1,
121205, Russia. School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK.
###### Abstract
We demonstrate all-optical linear polarization control of exciton-polariton
condensates in anisotropic elliptical optical traps. The cavity inherent TE-TM
splitting lifts the ground state spin degeneracy with emerging fine structure
modes polarized linear parallel and perpendicular to the trap major axis with
the condensate populating the latter. Our findings show a new type of
polarization control with exciting perspectives in both spinoptronics and
studies on extended systems of interacting nonlinear optical elements with
anisotropic coupling strength and adjustable fine structure.
Introduction. — Exciton-polaritons (polaritons hereafter) arise in the strong
coupling regime between quantum well excitons and cavity photons in
semiconductor microcavities Kavokin _et al._ (2007). Being composite bosons,
they can undergo a power-driven nonequilibrium phase transition into a highly
coherent many-body state referred as polariton condensation Kasprzak _et al._
(2006). An essential characteristic of polaritons is their spin projection
($\pm\hbar$) onto the growth axis of the cavity which corresponds to the right
and left circular polarizations of their photonic part.
The strong nonlinear nature of polaritons through their spin-anisotropic
excitonic Coulomb interactions results in numerous intriguing spinor
condensate properties. This includes spin bistability Pickup _et al._ (2018);
del Valle-Inclan Redondo _et al._ (2019); Sigurdsson (2020) and
multistability Paraïso _et al._ (2010), switches Amo _et al._ (2010); Cerna
_et al._ (2013), optical spin Hall effect Leyder _et al._ (2007), polarized
solitons Hivet _et al._ (2012); Sich _et al._ (2018) and vortices Lagoudakis
_et al._ (2009); Donati _et al._ (2016), bifurcations Ohadi _et al._ (2015),
and topological phases Bleu _et al._ (2016); Sigurdsson _et al._ (2019).
Aforementioned opens great prospects for the utilization of the polariton spin
degree of freedom in future spinoptronic technologies Shelykh _et al._
(2009); Liew _et al._ (2011). Different parts for future polariton based spin
circuitry have already been realized Amo _et al._ (2010); Cerna _et al._
(2013); Gao _et al._ (2015); Dreismann _et al._ (2016); Askitopoulos _et
al._ (2018) with some recent exciting theoretical proposals Sedov _et al._
(2019); Mandal _et al._ (2020), but many challenges are still yet to be
solved. Indeed, optical applications such as data communication or sensing
benefit from precise control over a laser’s polarization and modulation
speeds, ideally using nonresonant excitation schemes like spin-VCSEL
technologies Ostermann and Michalzik (2013); Lindemann _et al._ (2019); Drong
_et al._ (2021). In this spirit, a great deal of effort has been devoted to
generating sources of linearly polarized light such as colloidal nanorods Hu
_et al._ (2001), materials with anisotropic optical properties Wang _et al._
(2015), quantum dots integrated into exotic structures Lundskog _et al._
(2014), and with optical parametric oscillators in the strong coupling regime
Krizhanovskii _et al._ (2006).
Under nonresonant excitation in inorganic semiconductors, spin transfer from
the pumping laser to the condensate is possible by creating a spin-imbalanced
gain media for the circularly polarized polaritons (i.e., optical orientation
of excitons) using an elliptically polarized beam del Valle-Inclan Redondo
_et al._ (2019); Gnusov _et al._ (2020). This allows generating polariton
condensates of high degree of circular polarization aligned with the pump.
However, in such systems the linearly polarized polariton modes experience
isotropic gain, making it not possible to influence the linear polarization of
the condensate under nonresonant excitation Ohadi _et al._ (2012); Baumberg
_et al._ (2008) except in the presence of cavity strain and birefringence
Martín _et al._ (2005); Kłopotowski _et al._ (2006); Kasprzak _et al._
(2007); Balili _et al._ (2007); Read _et al._ (2009); Gnusov _et al._
(2020) or anisotropic confinement Gerhardt _et al._ (2019); Klaas _et al._
(2019) inherent to the engineering of the cavity. The same also applies for
VCSEL cavities, where the linear polarization of the emission is engineered by
etching asymmetric masks Xiang _et al._ (2018); Gayral _et al._ (1998) or
electrodes Choquette and Leibenguth (1994), heating Pusch _et al._ (2017), or
by applying mechanical stress Lindemann _et al._ (2019). Alternatively, in
organic polaritonics, single-molecule Frenkel excitons can be excited by a
linearly polarized pump co-aligned with their dipole moment with condensation
into a mode with the same linear polarization as the pump Plumhof _et al._
(2013). However, control over both circular and linear polarization degrees of
freedom in a polariton condensate through nonresonant all-optical means,
instead of engineering specific cavity systems, remains elusive.
Here, we demonstrate in-situ optical engineering of the linear polarization in
inorganic polariton condensates in a cavity with polarization-dependent
reflectivity, or TE-TM splitting Panzarini _et al._ (1999); Leyder _et al._
(2007). By spatially shaping the nonresonant excitation laser transverse
profile into the form of an ellipse, we are able to fully control the
direction of the condensate linear polarization. Our elliptically shaped
pumping profile induces an anisotropic in-plane trapping potential and gain
media for the condensate. Such an excitation profile along with the cavity TE-
TM splitting leads to condensation (lasing) into a mode of definite linear
polarization parallel to the minor axis of the trap ellipse. The optical
malleability of the trap geometry allows for non-invasive, yet deterministic,
control over the linear polarization of the condensate by just utilizing the
nonresonant excitation laser. Moreover, we investigate the effects of the
anisotropic coupling mechanism between two spatially separated condensates and
identify regions—as a function of coupling strength—of correlated high degree
of random linear polarization between the condensates, and otherwise complete
depolarization.
Figure 1: Spatial in-plane profiles of the (a,b) excitation laser and (c,d)
condensate PL. The excitation laser induces a trapping potential with
horizontal and vertical radii denoted $a,b$ respectively. (e) Momentum
distribution of the condensate PL. Panels (c,d,e) correspond to a condensate
pumped twice above its condensation threshold (i.e., $P=2P_{th}$).
Results. — Our experiments are conducted on an inorganic $2\lambda$
GaAs/AlAs0.98P0.02 microcavity with embedded InGaAs quantum wells Cilibrizzi
_et al._ (2014). The sample is excited nonresonantly by a linearly polarized
continuous wave (CW) laser ($\lambda=783.6$ nm). The optical excitation beam
is chopped using an acousto-optic modulator to form 10 $\mu$s square pulses at
1 kHz repetition rate to diminish heating of the sample held at a temperature
of 4 K. The exciton-cavity mode detuning is $-3$ meV. A reflective, liquid-
crystal spatial light modulator (SLM) transforms the transverse profile of the
pump laser beam to have an elliptically shaped confinement region [see Fig.
1(a) and dashed ellipse in Fig. 1(b)]. We investigate the sample PL in real
[Fig. 1(c,d)] and reciprocal [Fig. 1(e)] space, and record the time- and
space-averaged polarization of the PL by simultaneously detecting all
polarization components Gnusov _et al._ (2020). Our results are independent
on the angle of linear polarization of the pump laser [see Sec. S1 in the
Supplemental Information (SI)].
The polariton condensate can be described by an order parameter written in the
canonical spin-up and spin-down basis $\Psi=(\psi_{+},\psi_{-})^{T}$
corresponding to left- and right-circularly polarized condensate emission,
respectively. It is then convenient to represent the condensate as a
pseudospin on the Poincaré sphere corresponding to the Stokes vector
(polarization) of the emitted light
$\mathbf{S}=(S_{1},S_{2},S_{3})^{T}=\langle\Psi^{\dagger}\boldsymbol{\hat{\sigma}}\Psi\rangle/\langle\Psi^{\dagger}\Psi\rangle$
where $\boldsymbol{\hat{\sigma}}$ is the Pauli matrix vector. The PL is
analyzed in terms of time-averaged Stokes components which are written as,
$S_{1}=\frac{I_{H}-I_{V}}{I_{H}+I_{V}},\
S_{2}=\frac{I_{D}-I_{A}}{I_{D}+I_{A}},\
S_{3}=\frac{I_{\sigma^{+}}-I_{\sigma^{-}}}{I_{\sigma^{+}}+I_{\sigma^{-}}},$
(1)
where $I_{H,V,D,A,\sigma^{+},\sigma^{-}}$ are the time-averaged intensities of
horizontal, vertical, diagonal, antidiagonal, right- and left circular
polarization projections of the emitted light.
Figure 2: (a) Power dependence of the condensate $S_{1,2,3}$ Stokes parameters
(black, red, blue markers) for the annular trap with resultant cyllindrically
symmetric condensate profile. (b) Same but now for a trap/condensate with a
major axis orientated at $90^{\circ}$, (c) $45^{\circ}$, and (d) $0^{\circ}$.
Black lines in the insets depict the orientation of the trap major axis.
We start by exciting with a symmetric ring-shaped pump profile [see inset in
Fig. 2(a)], creating a two-dimensional trap for the polaritons and obtaining
condensation with polaritons dominantly populating the trap ground state by
ramping the pump power above the polariton condensation threshold denoted
$P_{th}$. The optical trap is realized by the strong polariton repulsive
interactions with the background laser-induced cloud of incoherent excitons
which, in the mean field formalism, form a blueshifting potential onto the
polaritons Askitopoulos _et al._ (2013), while at the same time providing
gain to the condensate. Such an optical trapping technique has the advantage
of reducing the overlap between the condensate and uncondensed excitons,
minimizing detrimental dephasing effects. By scanning the excitation position
with the ring-shaped pump profile we locate a spot on our sample with small
degree of polarization $\text{DOP}=\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}$ [see
Fig. 2(a)]. The small $S_{1,2}$ implies that the trap ground state is spin-
degenerate such that from realization to realization random linear
polarization builds up which averages out over many shots. The small $S_{3}$
component confirms that our laser excitation is (to a good degree) linearly
polarized and doesn’t break the spin parity symmetry of the system. We
additionally investigate the condensate pumped with elliptical polarization in
Sec. S2 in the SI.
We then transform the excitation profile to the one shown in Figs. 1(a) and
1(b). Non-uniform distribution of the intensity in the excitation leads to the
formation of an elliptically shaped optical trap denoted by the dashed
ellipse, squeezing the condensate as shown in Fig. 1(d). We now observe a
massive increase of the condensate’s linear polarization components above
1.2$P_{th}$ at the same sample position. The direction of the linear
polarization of the emission is found to follow the trap minor axis. Namely,
for the vertically elongated condensate in Fig. 2(b) we observe an increase of
the $S_{1}$ Stokes component (horizontal polarization). The same effect is
present for the horizontally and diagonally elongated condensates in Figs.
2(c) and 2(d).
Figure 3: (a) Condensate Stokes parameters for different orientations of the
condensate major axis in real space (insets) at $P=1.94P_{th}$. Black lines
depict the major axis of the trap. Colored regions show error of the
measurement. (b) $S_{1}$ and (c) $S_{2}$ for varying pump powers and major
axis orientation.
By rotating the excitation profile with the SLM, we can engineer any desired
linear polarization in the condensate. In Fig. 3(a), we present the measured
polarization components of the condensate as a function of the condensate
major axis angle. We observe a continuous rotation of the condensate
polarization close to the equatorial plane of the Poincaré sphere following
the minor axis of the trap. We also tested a different geometrical
construction of the elliptical excitation profile with the same outcome (see
Sec. S4 in SI). We point out that $\text{DOP}<1$ appears from various
depolarizing effects such as noise due to scattering from the incoherent
reservoir to the condensate Read _et al._ (2009), polariton-polariton
interactions in the condensate causing self-induced Larmor precessions Ryzhov
_et al._ (2020), and mode competition Redlich _et al._ (2016). We also note
that the finite $S_{3}$ component comes from optical elements in the detection
path of our setup.
Figures 3(b) and 3(c) show pump power and trap orientation dependence of the
$S_{1,2}$ Stokes parameters. Interestingly, with increasing pump power we
observe counterclockwise rotation of the pseudospin in the equatorial plane of
the Poincaré sphere. The rotation is approximately $30^{\circ}$ between 1.2
and $2.2P_{th}$. This effect appears due to a small amount of circular
polarization in our pump which creates a spin-imbalanced trapping potential
and gain media which acts as a complex population-dependent out-of-plane
magnetic field $\boldsymbol{\Omega}_{\perp}$ that applies torque on the
condensate pseudospin. This is confirmed through simulations using the
generalised Gross-Pitaevskii equation (see Sec. S10 in SI). Further analysis
on this power dependent trend of the $S_{1,2}$ is beyond the scope of the
current study.
Our observations can be interpreted in terms of photonic TE-TM splitting
acting on the optically confined polaritons which, when the trap
$V(\mathbf{r})$ has broken cylindrical symmetry, leads to fine structure
splitting in the trap transverse modes. This determines a state of definite
polarization which the polaritons condense into. In the noninteracting
(linear) regime the polaritons obey the following Hamiltonian,
$\hat{H}=\frac{\hbar^{2}\boldsymbol{k}^{2}}{2m}-\boldsymbol{\hat{\sigma}}\cdot\boldsymbol{\Omega}+V(\mathbf{r})-\frac{i\hbar\Gamma}{2},$
(2)
where $m$ is the polariton mass, $\mathbf{k}=(k_{x},k_{y})$ is the in-plane
cavity momentum, $\Gamma^{-1}$ is the polariton lifetime, and
$\boldsymbol{\Omega}=\hbar^{2}\Delta\left(k_{x}^{2}-k_{y}^{2},\ 2k_{x}k_{y},\
0\right)^{\text{T}},$ (3)
is the effective magnetic field [see Fig. 4(a)] coming from the TE-TM
splitting of strength $\Delta$ Leyder _et al._ (2007). In the considered case
of an elliptical confinement, which we assume to be harmonic for simplicity
$V(\mathbf{r})=m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2})/2$, the TE-TM
splitting results in an effective magnetic field acting on the polariton
pseudospin which splits the trap spin-levels. For the lowest (fundamental)
harmonic state where most of the polaritons are collected this field can be
written as follows (see Sec. S5 in SI),
$\boldsymbol{\Omega}_{\text{trap}}=\frac{\hbar
m\Delta\delta\omega}{2}\begin{pmatrix}\cos{(2\theta_{\text{min}})}\\\
\sin{(2\theta_{\text{min}})}\\\ 0\end{pmatrix}.$ (4)
Here, $\theta_{\text{min}}$ is the angle of the trap minor axis from the
horizontal, and $\delta\omega=|\omega_{x}-\omega_{y}|\propto|a^{-1}-b^{-1}|$
is the absolute difference between the trap oscillator frequencies along the
major and the minor axis [Fig. 1(d)]. We point out that $\Delta<0$ in our
sample Maragkou _et al._ (2011) (see Sec. S3 in SI).
The direction of the effective magnetic field is controlled by the angle of
our elliptical trap, $\theta_{\text{min}}$ which consequently rotates the
condensate pseudospin in the equatorial plane of the Poincaré sphere such that
it stabilizes antiparallel to the magnetic field
$-\mathbf{S}\parallel\boldsymbol{\Omega}_{\text{trap}}$. This leads to smooth
changes in the $S_{1,2}$ Stokes components of the emitted light as the trap
rotates like shown in Fig. 3.
Figure 4: (a) Distribution of the in-plane effective magnetic field
$\boldsymbol{\Omega}(\mathbf{k})$ (red arrows) in momentum space due to TE-TM
splitting given by Eq. (3). (b) Horizontal (black) and vertical (red)
polarization resolved normalized spectrum of the condensate emission at $k=0$
for a vertically elongated trap ($\theta_{\text{min}}=0$) favouring
condensation into the horizontal mode. Splitting between levels is $\approx
20$ $\mu$eV. (c) Linear polarization components ($S_{1}$ and $S_{2}$) and DOP
for different real-space ellipticities of the condensate. Data was taken at
$P\approx 1.8P_{th}$
The results of our experiment are accurately reproduced through a mean-field
theory using a generalized Gross-Pitaevskii model describing the polariton
condensate spinor order parameter $\Psi$ coupled with a background excitonic
reservoir (see Sec. S6 in SI).
Interestingly, in a recent experiment Gnusov _et al._ (2020) we observed
condensation into the spin ground state of a circular trap, where the fine
structure splitting originated from the cavity birefringence
$\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$. This meant that the condensate
pseudospin stabilized parallel to the magnetic field
$\mathbf{S}\parallel\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$. In the
current experiment however, we instead observe condensation into the excited
spin state, i.e. antiparallel to the magnetic field
$-\mathbf{S}\parallel\boldsymbol{\Omega}_{\text{trap}}$. This can be directly
evidenced in Fig. 4(b) where we show the normalized polarization-resolved
spectrum of a vertically elongated trap which obtains a horizontally polarized
condensate. The horizontal component is higher in energy in Fig. 4(b), in
agreement with Eq. (4).
Performing linear stability analysis on a Gross-Pitaevskii mean field model
(see Sec. S7 in SI) we determine that repulsive polariton-polariton
interactions normally leads to condensation in the fine structure ground state
Read _et al._ (2009). However, the additional presence of an uncondensed
background of excitons (referred as the reservoir) contributes to an effective
attractive mean-field interaction in the condensate Estrecho _et al._ (2018)
which causes the ground state to become unstable, favouring condensation into
the excited state as we observe in the current experiment. Another effect is
the different penetration depths of the linearly polarized polariton modes
(due to their different effective masses) into the excess gain region about
the trap short axis. This leads to higher gain for the fine structure excited
state which facilitates its condensation. Several parameters of the polariton
system such as exciton-photon detuning, the quantum well material, and shape
of the pump profile allow tuning from one stability regime to another which
explains why some experiments show ground-state condensation Gnusov _et al._
(2020) while other, like ours, show exited-state condensation Maragkou _et
al._ (2010). We stress that regardless of whether system parameters favour
condensation into the spin ground- or excited state of the optical trap, the
main result of our study remains valid.
Figure 5: (a,b) Spatially resolved $S_{1}$ for two different realizations of
coupled condensates oriented horizontally and separated by 27.5 $\mu$m, and
(c) orientated vertically at 26.5 $\mu$m. Insets in (a) and (c) depict the
corresponding $k$-space PL. (d,e) Show 100 time-integrated realizations
(shots) of the $S_{1}$ for the left (blue) and right (red) condensate in the
horizontal-horizontal major axis configuration and (f,g) in the vertical-
vertical configuration. The experimental data is taken at $\approx$1.8
$P_{th}$. Distance dependence of the $S_{1}$ from Gross-Pitaevskii simulations
corresponding to (h) horizontal-horizontal and (i) vertical-vertical major
axes configuration. Each datapoint represents one shot (time averaged). Green
and purple backgrounds in (d)-(i) illustrate regions of similar behaviour
between experiment and theory.
In Fig. 4(c) we continuously change the trap ellipticity from a vertically
elongated trap ($a<b$) to a horizontally elongated one ($a>b$). The PL
ellipticity axis denotes the ratio of width of the PL along x- and y-axis [see
Fig. 1(d)]. The pseudospin of the condensate changes from horizontal to
vertical polarization going through a low DOP regime. We stress that the data
in Fig. 4(c) is obtained at a different position of the cavity sample compared
to Figs. 1-3 which leads to finite $S_{1}$ at zero PL ellipticity ($a=b$) even
though $\boldsymbol{\Omega}_{\text{trap}}=0$. This is because of local
birefringence in the cavity mirrors giving rise to an additional static in-
plane magnetic field $\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$.
Therefore, one needs to account for a net field
$\boldsymbol{\Omega}_{\text{net}}=\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})+\boldsymbol{\Omega}_{\text{trap}}$
orientating the condensate pseudospin. The point of low DOP in Fig. 4(c)
corresponds then to near cancellation between the local birefringence and TE-
TM splitting $\boldsymbol{\Omega}_{\text{net}}\approx 0$.
Coupled condensates. — Networks of coupled polariton condensates can be seen
as an attractive platform to study the synchronization pheonomena in laser
arrays, and to investigate the behaviour of complex nonequilibrium many-body
systems and excitations in non-Hermitian lattices Ohadi _et al._ (2017);
Mandal _et al._ (2020); Töpfer _et al._ (2021); Pieczarka _et al._ (2021).
Inspired by these studies, we create two identical, spatially separated,
optical traps utilizing two SLMs resulting in the formation of two coupled
condensates [Fig. 5]. The trap anisotropy [see Fig1(b)] allows polaritons to
escape faster along its major axis[Fig1(e)]. This leads to stronger coupling
when the traps major axes are orientated longitudinally to the coupling
direction, and weaker when orientated transverse (estimated as 3 times weaker
from energy resolved spatial PL). This can be evidenced from the different
visibility in the momentum space interference fringes (implying
synchronization) [see insets in Figs. 5(a) and 5(c)].
Polarization resolving 100 quasi-CW 50 $\mu$s excitation shots, we observe
distinct regimes depending on the condensates separation distance and
orientation. For strongly coupled traps [Figs. 5(a) and 5(b)] at $26.5$ $\mu$m
distance we observe zero DOP in each CW shot [Fig. 5(d)] where the blue and
red curves correspond to the left and right condensate. At a $27.5$ $\mu$m
distance we now observe a strong $S_{1}$ component stochastically flipping
from shot to shot [Fig. 5(e)] with small $S_{2,3}$ (see S9 in SI)).
Interestingly, the $S_{1}$ components of the condensates are almost perfectly
correlated (Pearson correlation coefficient $\rho$ equals 0.99) which implies
that they are strongly coupled. The linear polarization flipping suggests
bistability in our system Sigurdsson (2020), triggered by the spatial coupling
mechanism. This interpretation is supported through Gross-Pitaevskii
simulations on time-delay coupled spinor condensates presented in Fig. 5(h)
(see Sec. S8 in SI). For weakly coupled traps [Fig. 5(c)] we observe
qualitatively different behaviour. Choosing again the same distances, we now
see regimes of strong positive $S_{1}$ component [Fig. 5(f)] and then semi-
depolarized behaviour [Fig. 5(g)]. Due to the weaker spatial coupling the
condensates are no longer strongly correlated in their $S_{1}$ components
($\rho=0.5$ and $0.26$ respectively). We note that the different mean $S_{1}$
values in Fig. 5(g) can be attributed to the position-dependent birefringence
$\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$. We reproduce the experiment
from simulation [Fig. 5(i)] by only decreasing the coupling strength by a
factor of 3. We note that the ballistic (time-delayed) nature of the polariton
condensate coupling Töpfer _et al._ (2020) distinguishes them from
evanescently coupled quantum fluids. Indeed, the distance between the
radiating condensates dictates their interference condition (in analogy to
coupled laser systems) which—in our system—leads to distance-periodic
appearance of the classified polarization regimes as seen in Fig. 5(h) and
5(i). Full dynamical trajectories from simulation, and a wider distance-power
scan, are shown in Secs. S8 and S9 in the SI.
Conclusion. — We have investigated the steady state polarization dynamics of a
polariton condensate in an elliptically shaped trapping potential created
through optical nonresonant linearly polarized injection. We have demonstrated
that the polarization of the condensate is determined by the lifted spin-
degeneracy of the trap levels due to the geometric ellipticity of the trap and
inherent cavity TE-TM splitting. The condensate always forms in a higher
energy spin state of the lowest trap level with a linear polarization that
follows the minor axis of the trap ellipse. By rotating the excitation
profile, we can rotate the condensate linear polarization around the
equatorial plane of the Poincaré sphere. We have extended our system to
coupled condensates, revealing rich physics of synchronization and
desynchronization by tuning the condensate coupling strength through the
optical trap anisotropy and/or spatial separation. Our results pave the way
towards all-optical spin circuitry in spinoptronic applications, and coherent
light sources with on-demand switchable linear polarization.
The data presented in this paper are openly available from the University of
Southampton repository.
Acknowledgements. — The authors acknowledge the support of the UK’s
Engineering and Physical Sciences Research Council (grant EP/M025330/1 on
Hybrid Polaritonics) and by RFBR according to the research project No.
20-02-00919.
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Supplementary Information
## S1 Condensate polarization dependence on the linear polarization of the
excitation laser
The experimental data presented in the main manuscript are acquired using a
horizontally polarized pump laser which excites the optically trapped
polariton condensate. In this supplemental section, we evidence that the
linear polarization direction of the pump laser does not affect our presented
results. In Fig. S1 we show the measured condensate photoluminescence (PL)
Stokes components $S_{1,2,3}$ for varying power and linear polarization
direction of the pump laser, the latter being controlled by a half-waveplate
(HWP) in the excitation path. The four columns in Fig. S1 correspond to
different spatial orientations of the elliptically shaped pump profile (i.e.,
the optical trap). Figures. S1(a-c) are taken for $0^{\circ}$, (d-f)
$-45^{\circ}$, (g-i) $90^{\circ}$, and (j-l) $45^{\circ}$ degrees of the trap
ellipse major axis rotated counterclockwise from the horizontal direction (as
defined in the main manuscript). We observe that the condensate polarization
always dominantly follows the minor axis of the trap ellipse [see Fig.
S1(a),(e),(g), and (k)].
The small amount of $S_{3}$ component emerging for diagonally oriented traps
in Figs. S1(f) and S1(l) is due to optical elements in the detection path of
our setup. For example, different reflectivities of the mirrors for $s$\- and
$p$-polarized light and small birefringence in the cryostat window glass. We
have measured the effective retardance of the detection path in our setup to
be $\approx$0.06$\pi$ at the condensate emission wavelength ($\approx 856$
nm).
Figure S1: Condensate Stokes components for different pump powers and
directions of linear polarization of the excitation laser. The labels H,A,V,D
on the vertical axis denote horizontal, antidiagonal, vertical, and diagonal
polarization respectively. The condensate PL is depicted on the top row with
the black line denoting the trap major axis oriented at (a-c) 0, (d-f) -45,
(g-i) 90, and (j-l) 45 degrees with respect to the cavity plane $x$-axis
(horizontal direction).
## S2 Condensate polarization dependence on the polarization ellipticity of
the excitation laser
In this section we quantitatively investigate the dependence of the condensate
polarization on the pump laser ellipticity. We install a quarter waveplate
(QWP) in the excitation path so that by rotating the QWP, we can control the
ellipticity and handedness of the excitation polarization. In Fig. S2 we show
the measured condensate PL Stokes components $S_{1,2,3}$ depending on the pump
polarization ellipticity and power. Overall, we obtain a similar behavior of
the condensate polarization that was reported for annular optical traps Gnusov
_et al._ (2020).
As expected, circular polarization transfers to the condensate from our
nonresonant excitation through the optical orientation of the background
excitons feeding the condensate. It can also be seen that the linear
polarization of the condensate is sensitive to the pump polarization
ellipticity. This is effect is theoretically modeled and discussed further in
Sec. S10. In agreement with the findings presented in the main manuscript,
when the pump is almost purely linearly polarized ($\text{QWP}\approx 0$) we
observe that the condensate aligns along the short axis of the optical trap
[see e.g. blue coloured region in Fig. S2(a)]. Our additional measurements in
this supplemental section underline the richness of polarization regimes
accessible in polariton condensates where, in this study, we have focused on
anisotropic trapping conditions around $\text{QWP}\approx 0$.
Figure S2: Condensate Stokes components for different pump powers and angles
of the QWP (pump polarization ellipticity). The ellipticty of the pump can be
written in normalized form as $S_{3}^{\text{pump}}=\sin{(2\cdot\text{QWP})}$.
The negative and positive values of the QWP angle correspond to left and right
handedness of the circular polarization. The condensate PL is depicted on the
top row with the black line denoting the trap major axis oriented at (a-c) 0,
(d-f) -45, (g-i) 90, and (j-l) 45 degrees with respect to the cavity plane
$x$-axis (horizontal direction).
## S3 TE-TM splitting
We experimentally measure the TE-TM splitting of the sample by polarization
resolving the lower polariton branch in the linear regime (i.e., below
condensation threshold) along the $k_{y}$ momentum axis. We observe that
vertically polarized polaritons possess higher energy than horizontally
polarized polaritons [see Fig. S3(a)]. The energy difference between these
branches gives the TE-TM splitting which follows the expected parabolic
trajectory [see Fig. S3(b)].
Figure S3: (a) Fitted lower polariton branch in horizontal (blue) and vertical
(red) linear polarization detection. (b) Experimentally obtained TE-TM
splitting for different wavevectors.
## S4 8-point excitation
In this supplemental section, we demonstrate the precise shape of our optical
excitation beam is not important as long as it introduces different
confinement strengths in the two orthogonal spatial directions. Here, we shape
the overall laser profile using 8 Gaussians distributed in the form of an
ellipse [see Fig. S4(b)] leading to the formation of an elliptical condensate
[see Fig. S4(c)]. In agreement with the results presented in the main text,
such an excitation profile also favors the formation of a condensate with
linear polarization aligned along the ellipse minor axis. By rotating the
excitation profile in the cavity plane Fig. S4(a), we observe the same
rotation of the linear polarization of the condensate as in the main text. The
deviations from the sinusoidal fits in Fig. S4(a) occur due to a some
differences in power and shape of the individual Gaussian spots.
Figure S4: (a) Condensate polarization for different spatial orientations of
the 8-Gaussian excitation profile major axis in real space at $P=2P_{th}$. (b)
Excitation laser intensity profile. (c) Condensate PL.
## S5 The single-particle polariton Hamiltonian
In the non-interacting (linear) regime the polaritons obey the following
Hamiltonian (same as Eq. (2) in the main text),
$\hat{H}=\frac{\hbar^{2}k^{2}}{2m}-\boldsymbol{\sigma}\cdot\boldsymbol{\Omega}+V(\mathbf{r})-\frac{i\hbar\Gamma}{2},$
(S1)
where $m$ is the polariton mass, $\mathbf{k}=(k_{x},k_{y})$ is the in-plane
cavity momentum, $\Gamma^{-1}$ is the polariton lifetime,
$\boldsymbol{\sigma}$ is the Pauli matrix vector, and
$\boldsymbol{\Omega}=\hbar^{2}\Delta(k_{x}^{2}-k_{y}^{2},\ 2k_{x}k_{y},\
0)^{T},$ (S2)
is the effective magnetic field [see Fig. S5(e)] coming from the TE-TM
splitting of strength $\Delta$. We will consider that our laser generated
potential in experiment can be approximated by an elliptically shaped harmonic
oscillator (HO),
$V(\mathbf{r})=\frac{1}{2}m\omega_{x}^{2}x^{2}+\frac{1}{2}m\omega_{y}^{2}y^{2}.$
(S3)
Using the shorter momentum operator expression
$p_{x(y)}=-i\hbar\partial_{x(y)}=\hbar k_{x(y)}$ for brevity, our Hamiltonian
becomes:
$\hat{H}=\begin{pmatrix}\dfrac{p_{x}^{2}}{2m}+\dfrac{p_{y}^{2}}{2m}+V(\mathbf{r})-\dfrac{i\hbar\Gamma}{2}&-\Delta(p_{x}-ip_{y})^{2}\\\
-\Delta(p_{x}-ip_{y})^{2\dagger}&\dfrac{p_{x}^{2}}{2m}+\dfrac{p_{y}^{2}}{2m}+V(\mathbf{r})-\dfrac{i\hbar\Gamma}{2}\end{pmatrix}.$
(S4)
We will diagonalize this problem in the basis of the harmonic oscillator modes
written for spin-up and spin-down particles as,
$\displaystyle|\psi_{+}\rangle$
$\displaystyle=\sum_{n_{x},n_{y}}c^{(+)}_{n_{x},n_{y}}|n_{x},n_{y}\rangle$
(S5) $\displaystyle|\psi_{-}\rangle$
$\displaystyle=\sum_{n_{x},n_{y}}c^{(-)}_{n_{x},n_{y}}|n_{x},n_{y}\rangle,$
(S6)
where $|n_{x},n_{y}\rangle=|n_{x}\rangle\otimes|n_{y}\rangle$ are the harmonic
oscillator eigenmodes in the ladder operator $\hat{a}_{x(y)}$ formalism. These
are defined in the standard way through the position and momentum operators,
$\displaystyle x$
$\displaystyle=\sqrt{\frac{\hbar}{2m\omega_{x}}}(\hat{a}^{\dagger}_{x}+\hat{a}_{x}),\qquad
y=\sqrt{\frac{\hbar}{2m\omega_{y}}}(\hat{a}^{\dagger}_{y}+\hat{a}_{y}),$ (S7)
$\displaystyle p_{x}$
$\displaystyle=i\sqrt{\frac{m\hbar\omega_{x}}{2}}(\hat{a}_{x}^{\dagger}-\hat{a}_{x}),\qquad
p_{y}=i\sqrt{\frac{m\hbar\omega_{y}}{2}}(\hat{a}_{y}^{\dagger}-\hat{a}_{y}).$
(S8)
Our Hamiltonian can then be expressed,
$\hat{H}=\begin{pmatrix}\hbar\omega_{x}\left(\hat{a}_{x}^{\dagger}\hat{a}_{x}+\dfrac{1}{2}\right)+\hbar\omega_{y}\left(\hat{a}_{y}^{\dagger}\hat{a}_{y}+\dfrac{1}{2}\right)-\dfrac{i\hbar\Gamma}{2}&-\Delta(p_{x}-ip_{y})^{2}\\\
-\Delta(p_{x}-ip_{y})^{2\dagger}&\hbar\omega_{x}\left(\hat{a}_{x}^{\dagger}\hat{a}_{x}+\dfrac{1}{2}\right)+\hbar\omega_{y}\left(\hat{a}_{y}^{\dagger}\hat{a}_{y}+\dfrac{1}{2}\right)-\dfrac{i\hbar\Gamma}{2}\end{pmatrix},$
(S9)
where the following holds,
$\displaystyle p_{x(y)}^{2}=(p_{x(y)}^{2})^{\dagger}$
$\displaystyle=-\frac{m\hbar\omega_{x(y)}}{2}(\hat{a}_{x(y)}^{\dagger}\hat{a}_{x(y)}^{\dagger}-\hat{a}_{x(y)}^{\dagger}\hat{a}_{x(y)}-\hat{a}_{x(y)}\hat{a}_{x(y)}^{\dagger}+\hat{a}_{x(y)}\hat{a}_{x(y)}),$
(S10) $\displaystyle p_{x}p_{y}=(p_{x}p_{y})^{\dagger}$
$\displaystyle=-\frac{m\hbar\sqrt{\omega_{x}\omega_{y}}}{2}(\hat{a}_{x}^{\dagger}\hat{a}_{y}^{\dagger}-\hat{a}_{x}^{\dagger}\hat{a}_{y}-\hat{a}_{x}\hat{a}_{y}^{\dagger}+\hat{a}_{x}\hat{a}_{y}).$
(S11)
The diagonal harmonic oscillator terms can be written more neatly as,
$E^{(0)}_{n_{x},n_{y}}=\hbar\omega_{x}\left(n_{x}+\frac{1}{2}\right)+\hbar\omega_{y}\left(n_{y}+\frac{1}{2}\right).$
(S12)
The TE-TM terms will operate on our states as follows,
$\displaystyle p_{x}^{2}|n_{x},n_{y}\rangle=$
$\displaystyle-\frac{m\hbar\omega_{x}}{2}(\hat{a}_{x}^{\dagger}\hat{a}_{x}^{\dagger}-\hat{a}_{x}^{\dagger}\hat{a}_{x}-\hat{a}_{x}\hat{a}_{x}^{\dagger}+\hat{a}_{x}\hat{a}_{x})|n_{x},n_{y}\rangle$
$\displaystyle=$
$\displaystyle-\frac{m\hbar\omega_{x}}{2}(\sqrt{n_{x}+1}\sqrt{n_{x}+2}|n_{x}+2,n_{y}\rangle-
n_{x}|n_{x},n_{y}\rangle-(n_{x}+1)|n_{x},n_{y}\rangle+\sqrt{n_{x}}\sqrt{n_{x}-1}|n_{x}-2,n_{y}\rangle),$
(S13) $\displaystyle p_{x}p_{y}|n_{x},n_{y}\rangle=$
$\displaystyle-\frac{m\hbar\sqrt{\omega_{x}\omega_{y}}}{2}(\hat{a}_{x}^{\dagger}\hat{a}_{y}^{\dagger}-\hat{a}_{x}^{\dagger}\hat{a}_{y}-\hat{a}_{x}\hat{a}_{y}^{\dagger}+\hat{a}_{x}\hat{a}_{y})|n_{x},n_{y}\rangle$
$\displaystyle=$
$\displaystyle-\frac{m\hbar\sqrt{\omega_{x}\omega_{y}}}{2}\bigg{(}\sqrt{(n_{x}+1)(n_{y}+1)}|n_{x}+1,n_{y}+1\rangle-\sqrt{n_{x}+1}\sqrt{n_{y}}|n_{x}+1,n_{y}-1\rangle$
$\displaystyle-\sqrt{n_{x}}\sqrt{n_{y}+1}|n_{x}-1,n_{y}+1\rangle+\sqrt{n_{x}n_{y}}|n_{x}-1,n_{y}-1\rangle\bigg{)}.$
(S14)
We will give a special notation to TE-TM terms which do not mix levels,
$\epsilon_{n_{x},n_{y}}=-\frac{m\hbar\Delta}{2}[\omega_{x}(2n_{x}+1)-\omega_{y}(2n_{y}+1)].$
(S15)
We can write a truncated version of our Hamiltonian for just the spins in the
trap ground state $|0,0\rangle$ which reads (i.e., coupling to other HO levels
is neglected),
$\hat{H}\approx\begin{pmatrix}E^{(0)}_{0,0}-\dfrac{i\hbar\Gamma}{2}&\epsilon_{0,0}\\\
\epsilon_{0,0}&E^{(0)}_{0,0}-\dfrac{i\hbar\Gamma}{2}\end{pmatrix}.$ (S16)
The eigenvectors are the horizontally (H) and vertically (V) polarized states
of light with eigenvalues,
$E_{0,0}^{(H,V)}=\frac{\hbar}{2}\left[\omega_{x}+\omega_{y}\mp
m\Delta(\omega_{x}-\omega_{y})\right]-\frac{i\hbar\Gamma}{2}.$ (S17)
We remind that $\Delta<0$ in our cavity sample Maragkou _et al._ (2011) (see
Fig. S3). This expression confirms experimental observations of the cavity
energy resolved emission in Fig. 4(b) in the main text. When the laser induced
trap has a major axis along the vertical direction (i.e.,
$\omega_{x}>\omega_{y}$) then we observe higher frequency in the horizontally
emitted light as opposed to the vertical light, in agreement with
$E_{0,0}^{(H)}>E_{0,0}^{(V)}$. When $\omega_{x}<\omega_{y}$ the vice versa
appears. In Fig. S5 we put $\Gamma=0$ for simplicity and compare the
calculated spectrum obtained from diagonalizing Eq. (S9) (black lines) for
$\text{max}{(n_{x(y)})}=15$ modes against our truncated lowest HO level
Hamiltonian Eq. (S17). The generalization of Eq. (S16) to arbitrary angles of
the potential orientation in the $x$-$y$ plane is straightforward and
presented in Eq. (4) in the main text.
Figure S5: Spectrum of the harmonic oscillator with TE-TM splitting and
$\Gamma=0$ for simplicity. Black lines correspond to diagonalization of Eq.
(S9) for $\text{max}{(n_{x(y)})}=15$ and $\omega_{x}=1.75\omega_{y}=0.55$ ps-1
which were estimated from the full-width-half-maximum of the trapped
condensate, and for a typical polariton mass of $m=0.3$ meV ps2 $\mu$m-2. The
red lines are outcome of Eq. (S17).
## S6 Generalized Gross-Pitaevskii simulations
We will now model the dynamics of the polariton condensate spinor
$\Psi=(\psi_{+},\psi_{-})^{T}$ using the generalized (driven-dissipative)
Gross-Pitaevskii equation, coupled to a semiclassical rate equation describing
a reservoir of low-momentum excitons $\mathbf{X}=(X_{+},X_{-})^{T}$ which
scatter into the condensate Wouters and Carusotto (2007):
$\begin{split}i\hbar{\partial\psi_{\pm}\over\partial
t}&=\left[-{\hbar^{2}\nabla^{2}\over 2m}-\Delta(p_{x}\mp
ip_{y})^{2}+g\left(X_{\pm}+\frac{P}{W}\right)+\alpha|\psi_{\pm}|^{2}+i\hbar{RX_{\pm}-\Gamma\over
2}\right]\psi_{\pm},\\\ {\partial X_{\pm}\over\partial
t}&=P-(R|\psi_{\pm}|^{2}+\Gamma_{R})X_{\pm}+\Gamma_{s}(X_{\mp}-X_{\pm}).\end{split}$
(S18)
Here, $\nabla^{2}$ denotes the two-dimensional Laplacian operator, $\Delta$
the TE-TM splitting, $m$ is the polariton effective mass, $g$ and $\alpha$ are
the interaction constants describing the polariton repulsion off the exciton
density and polariton-polariton repulsion, $R$ governs the stimulated
scattering from the reservoir into the condensate, $\Gamma$ and $\Gamma_{R}$
are the polariton and active exciton decay rates, and $\Gamma_{s}$ is the rate
of spin relaxation. As we are working in continuous wave regime, and the pump
is linearly polarized at all times, we do not need to take into account the
polarization- and time-dependence of a high-momentum (inactive) reservoir
describing excitons that are too energetic to scatter into the condensate
Antón _et al._ (2013). Instead, the contribution of photoexcited high-
momentum excitons to the condensate appears through the blueshift term $P/W$
where $W$ describes the conversion rate of high-momentum excitons into low-
momentum excitons $X_{\pm}$ that sustain the condensate.
We will use the pseudospin formalism (analogous to the Stokes parameters
describing the cavity photons) to describe the polarization of the condensate
(note that in Eq. (1) in the main manuscript we have used the normalized
definition),
$\mathbf{S}=\begin{pmatrix}S_{1}\\\ S_{2}\\\
S_{3}\end{pmatrix}=\Psi^{\dagger}\boldsymbol{\sigma}\Psi.$ (S19)
Here, $\boldsymbol{\sigma}$ is the Pauli matrix vector and the total density
of the condensate is expressed as $S_{0}=|\psi_{+}|^{2}+|\psi_{-}|^{2}$.
We will study the dynamics of Eq. (S18) for three different excitation
profiles shown in Figs. S7(a,d,f). The profiles shown in Fig. S7(d) and S7(f)
represent the experimental configurations shown in Fig. 1(a,b) in the main
manuscript and in Fig. S4(b). We also introduce, for completeness, a third
type of an elliptical excitation profile in the numerical analysis shown in
Fig. S7(a). The three excitation profiles can be written as follows,
$\displaystyle P_{I}(\mathbf{r})$
$\displaystyle=P_{0}\frac{L_{I}^{4}}{\left(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-r_{0}^{2}\right)^{2}+L_{I}^{4}},$
(S20) $\displaystyle P_{II}(\mathbf{r})$
$\displaystyle=P_{0}\frac{L_{II}^{4}}{\left(x^{2}+y^{2}-r_{0}^{2}\right)^{2}+L_{II}^{4}}[1-\eta\cos{(2\varphi+\phi_{\text{maj}})}\sin^{2}{(\pi
r/2r_{0})}],$ (S21) $\displaystyle P_{III}(\mathbf{r})$
$\displaystyle=P_{0}\sum_{n=1}^{8}\frac{L_{III}^{2}}{(x-x_{n})^{2}+(y-y_{n})^{2}+L_{III}^{2}}.$
(S22)
Here, $L_{I,II,III}$ denote the spread (thickness) of the potentials. For
pumps (S20) and (S21) the common radius $r_{0}$ defines the length of the
ellipse minor and major axis. Specifically, the minor and major axis are given
by the parameters $0<a,b<1$ for (S20) and $\eta,\phi_{\text{maj}}$ for (S21).
For the third pump profile (S22) we use coordinates $x_{n},y_{n}$ of the eight
tightly focused pump spots corresponding to the experiment. The laser power
density is given by $P_{0}$.
In Fig. S6 we show the obtained steady state wavefunction $\Psi$ obtained from
random initial conditions while driving the system above the pump threshold
$P_{0}>P_{\text{th}}$ using the first pump profile $P_{I}(\mathbf{r})$. The
threshold $P_{\text{th}}$ is defined as the transition point where the normal
state $|\Psi|=0$ becomes unstable and instead a condensate forms $|\Psi|>0$.
Indeed, choosing parameters corresponding to the experiment we obtain complete
match between experimental observations and simulations. Testing 100 different
random initial conditions we find that the simulated condensate always
converges to a steady state corresponding to the excited spin state of the
trap. The parameters of the simulation are: $m=0.3$ meV ps2 $\mu$m-2;
$\Gamma=\frac{1}{5.5}$ ps-1; $\hbar\alpha=3$ $\mu$eV $\mu$m2, $g=\alpha$;
$\Gamma_{R}=\Gamma/4$; $R=0.67\alpha$; $W=0.05$ ps-1;
$\Gamma_{s}=\Gamma_{R}/2$; $\hbar^{2}\Delta=-0.03$ meV $\mu$m2; $L_{I}=5$
$\mu$m; $L_{II}=6$ $\mu$m; $L_{III}=2$ $\mu$m; $r_{0}=5$ $\mu$m; $\eta=0.2$;
and $P_{0}=4.625$ $\mu$m-2 ps-1 which is around $\approx 20\%$ above threshold
for each configuration.
Figure S6: (Top row) Steady state solution from simulation of Eq. (S18). Here
we used the $P_{I}(\mathbf{r})$ pump profile with a major axis along the
vertical direction, $b/a=1.3$. The results show a condensate forming with
horizontal polarization implying occupation of the excited trap spin state.
(Bottom row) Same simulations but now the pump major axis is orientated
$\pi/4$ from the horizontal resulting in antidiagonally polarized condensate
which again corresponds to the excited spin state of the trap. Figure S7:
(a,d,f) Pump profiles (normalized) corresponding to Eqs. (S20)-(S22). In (a)
we have set $a/b=1.4$ and in (d) $\phi_{\text{maj}}=0$ and $\eta=0.2$. The
green lines are contours which represent the isoenergy lines of the optical
trap. The inset in (e) shows schematically the energy structure of the
linearly polarized modes in the trap s-orbital. In the panels on the right we
show the corresponding condensate $\bar{S}_{0}$ dynamics for the two ansatz
$\Psi^{H,V}$ in Eq. (S23) separately (circles and squares respectively) for
fixed pumping value $P_{0}$ above threshold. The red-blue color denotes the
area-integrated linear polarization $\bar{S}_{1}$. In panels (b,c) and (g,h)
we split the time axis to better show the early and late dynamics. In panel
(b) the early dynamics show that the $\Psi^{H}$ ansatz rises faster and the
condensate saturates into a horizontally polarized state (red circles) with
higher particle number as compared to the vertically polarized state (blue
squares), implying it has larger gain. At later times shown in panel (c), the
horizontal polarized solution always collapses into the vertically polarized
solution, indicating that it is only metastable and that a vertically
polarized condensate survives at long times. In panels (e) and (g) we observe
a qualitatively different early dynamics where now the horizontally polarized
condensate rises slower. Eventually, as we see in (e) and (h), the
horizontally polarized solution destabilizes and converges into a vertically
polarized condensate.
### S6.1 Condensate metastability
We will here scrutinize the early and late condensate dynamics using two
simple initial conditions (ansatz). We will use a trap with a horizontal major
axis (i.e., $\omega_{x}<\omega_{y}$) since all other major axis orientations
are completely generalizable. The two initial conditions for Eq. (S18) are
written,
$\Psi(t=0)=\Psi^{H,V}=Ae^{-c(x^{2}+y^{2})}\begin{pmatrix}1\\\ \pm
1\end{pmatrix}.$ (S23)
The parameter $c$ is chosen to have $\Psi^{H,V}$ localized dominantly within
the trap and $A\ll 1$ is a small number to minimize nonlinear effects in the
initial dynamics. We then solve Eq. (S18) for each initial condition and plot
the spatially integrated particle number $S_{0}$ and $S_{1}$ (normalized)
Stokes parameter as a function of time,
$\bar{S}_{0}(t)=\frac{1}{\mathcal{A}}\int_{\mathcal{A}}S_{0}(\mathbf{r},t)d\mathbf{r}\qquad\bar{S}_{1}(t)=\frac{\int_{\mathcal{A}}S_{1}(\mathbf{r},t)d\mathbf{r}}{\int_{\mathcal{A}}S_{0}(\mathbf{r},t)d\mathbf{r}},$
(S24)
where $\mathcal{A}$ is the area enclosed by the pump profile ridge (i.e.,
$\text{max}\,{P(\mathbf{r})}$).
Simulations using the generalised Gross-Pitaevskii equation (S18) for fixed
power $P_{0}$ (above threshold) and the three pump profiles $P_{I,II,III}$ are
shown in Figs. S7(b,c), and S7(e), and S7(g,h), respectively. We plot the
area-integrated particle number $\bar{S}_{0}$ for the two different initial
conditions $\Psi^{H,V}$ (circles and squares, respectively). The color of the
markers indicates the area-integrated linear polarization $\bar{S}_{1}$ of the
condensate. In Figs. S7(b,c) and S7(g,h) we split the time axis to show better
the early and late dynamics. The inset in Fig. S7(e) shows schematically the
energy splitting between the polarizations. For pump profiles $P_{II}$ and
$P_{III}$ we see that in the early dynamics a vertically polarized condensate
(blue squares) rises faster and saturates at a higher particle number than a
horizontally polarized condensate (red circles). This can be understood from
the fact that the vertical and horizontal polarized modes of the condensate
have different effective masses and, thus, have different penetration depths
into the gain region of the pump. In particular, pumps $P_{II}$ and $P_{III}$
lead to an excess density of reservoir excitons about the short axis of the
ellipse. Since the penetration depth of the confined mode in the potential
well is larger in the direction of the linear polarization axis, this would
increase the overlap of the mode co-polarized with the short-axis of the
potential well with the gain region, i.e. the ’excited state’ in the fine
structure. In the late dynamics [Fig. S7(e) and S7(h)] the horizontal solution
destabilizes and converges into the vertically polarized solution. This is in
agreement with our experimental observations showing robust condensation into
the excited spin state of the optical traps.
Interestingly, for pump $P_{I}$ the early dynamics [Fig. S7(b)] are reversed
with respect to $P_{II,III}$. Now the horizontally polarized condensate rises
faster and saturates at a higher particle number. This pump profile does not
generate a strong excess of excitons about the ellipse short axis and, as a
consequence, the higher energy polaritons, co-polarized with the short axis,
escape (leak) faster from the trap. Nevertheless, in the late dynamics [Fig.
S7(c)] the horizontally polarized solution destabilizes and collapses into the
vertically polarized solution. This observation underlines that the stable
solution of the condensate does not necessarily correspond to the one with
maximum particle number $\bar{S}_{0}$. In the next section we address the
different parameters of our model that determine the stability of the excited
state and the ground state.
## S7 Stability analysis on a two mode problem
Here, we will analyse the stability properties of the condensate by projecting
our order parameter on only the lowest trap state $|0,0\rangle$. Let us here
denote $\psi_{\pm}(\mathbf{r},t)\to\psi_{\pm}(t)$ as the condensate order
parameter describing polaritons only in the HO ground state and neglect
contribution from higher HO modes,
$\displaystyle\begin{split}i&\frac{d\psi_{\pm}}{dt}=\Big{[}\alpha|\psi_{\pm}|^{2}+gX_{\pm}+i\frac{RX_{\pm}-\Gamma}{2}\Big{]}\psi_{\pm}+(\epsilon+i\gamma)\psi_{\mp},\\\
&\frac{dX_{\pm}}{dt}=P-\left(\Gamma_{R}+R|\psi_{\pm}|^{2}\right)X_{\pm}+\Gamma_{s}(X_{\mp}-X_{\pm}).\end{split}$
(S25)
The parameters here have the same meaning as their counterparts in Eq. (S18),
but we stress that we have absorbed $\hbar$ into their definition for brevity
and some will obtain modified values after integrating out the spatial degrees
of freedom depending on the precise shape of the condensate and reservoir. We
have removed the $P/W$ term as it only induces an overall blueshift to both
spins which does not affect the stability properties of the system. The spin-
coupling parameter $\epsilon$ corresponds to $\epsilon_{0,0}$ from Eq. (S15).
We additionally include an imaginary coupling parameter $\gamma$ which
physically represents different linewidths of the horizontal and vertical
polarized modes (i.e., different decay rates).
The two steady state solutions of interest correspond to spin-balanced
reservoirs $X_{+}=X_{-}$ supporting either purely horizontally or vertically
polarized condensate written,
$\Psi_{\text{st}}^{H,V}=\sqrt{\frac{S_{0}^{H,V}}{2}}\begin{pmatrix}1\\\ \pm
1\end{pmatrix}e^{-i\omega^{H,V}t}$ (S26)
where,
$\omega^{H,V}=\frac{\alpha}{2}S_{0}^{H,V}+gX^{H,V}\pm\epsilon_{0,0},\qquad
S_{0}^{H,V}=\frac{P}{\Gamma\mp 2\gamma}-\frac{\Gamma_{R}}{R},\qquad
X^{H,V}=\frac{\Gamma\mp 2\gamma}{R}.$ (S27)
The power to reach the lower threshold solution is
$P_{\text{th}}=\Gamma_{R}(\Gamma-2|\gamma|)/R$. The stability analysis of the
$\Psi_{\text{st}}^{H,V}$ solutions is exactly the same as in Ref. Sigurdsson
(2020) where a $5\times 5$ Jacobian matrix $\mathbf{J}$ corresponding to
linearisation of Eq. (S25) around its steady state solutions was derived. This
allows determining the stability of the solutions in terms of their Jacobian
eigenvalues $\lambda_{n}$ (also known as Lyapunov exponents in nonlinear
dynamics or Bogoliubov elementary excitations in the context of Bose-Einstein
condensates). If a single eigenvalue of $\mathbf{J}$ has a positive real part
then the solution is said to be asymptotically unstable. It was found in Ref.
Sigurdsson (2020) that the relative strength between the mean field energy
coming from the condensates $\alpha|\psi_{\pm}|^{2}$ and the reservoir
blueshift $gX_{\pm}$ played a big role in whether the excited state or the
ground state was stable.
To understand this better, we will first consider the stability of the two
solutions $\Psi_{\text{st}}^{H,V}$ using a more general coupled Gross-
Pitaevskii equations (similar to coupled amplitude oscillators),
$i\frac{d\psi_{\pm}}{dt}=\alpha|\psi_{\pm}|^{2}\psi_{\pm}+\epsilon\psi_{\mp},\qquad\epsilon>0.$
(S28)
Clearly, $\Psi_{\text{st}}^{H,V}$ is a solution of the above equation for any
particle number $S_{0}=S_{0}^{H}=S_{0}^{V}$ with frequency
$\omega^{H,V}=\alpha S_{0}/2\pm\epsilon$. The Lyapunov exponents of these
solutions are written:
$\begin{split}\lambda_{1}^{H,V}&=0,\\\
\lambda_{2}^{H,V}&=\,\,\,\,\sqrt{2\epsilon(-2\epsilon\pm\alpha S_{0})},\\\
\lambda_{3}^{H,V}&=-\sqrt{2\epsilon(-2\epsilon\pm\alpha S_{0})}.\end{split}$
(S29)
We only have three eigenvalues because our two level system (S28) can be
described with the three-dimensional pseudospin state vector [see Eq. (S19)],
analogous to the Stokes vector of light. It is clear that only
$\lambda_{2}^{H,V}$ can have real values greater than zero (the signature of
instability) and there are only two cases when this happens:
1. 1.
If $\alpha>0$ (repulsive particle interactions) then
$\text{Re}{(\lambda_{2}^{\color[rgb]{1,0,0}H})}>0$ when $\alpha
S_{0}>2\epsilon$.
2. 2.
If $\alpha<0$ (attractive particle interactions) then
$\text{Re}{(\lambda_{2}^{\color[rgb]{0,0,1}V})}>0$ when $|\alpha
S_{0}|>2\epsilon$.
This simple result shows that the stability of the excited state and the
ground state changes when the mean field energy $\alpha S_{0}$ exceeds the
fine structure splitting $2\epsilon$. In our case, polariton interactions are
repulsive $\alpha>0$ and the condensate should always form in the ground state
when $S_{0}>2\epsilon/\alpha$ Shelykh _et al._ (2006). Note that
$2\hbar\epsilon\approx 20$ $\mu$eV in our experiment which is small compared
to the typical polariton mean field energies $\alpha S_{0}$. It therefore
appears puzzling that we observe stable excited state condensation when the
above simple consideration dictates that only the ground state should be
stable.
In the following, we will address two different mechanisms that fight against
ground state condensation. First, if the ground state is lossier than the
excited state (i.e., $\gamma/\epsilon<0$) then polaritons will preferentially
condense into the excited state until $S_{0}$ exceeds a critical value
Sigurdsson (2020). However, as we can see from Fig. S7(b,c), even if the
excited state is lossier than the ground state the condensate can still
preferentially populate and stabilize in the excited state. Second, a stable
excited spin state condensation can appear due to an effective attractive
nonlinearity coming from the reservoir [i.e., the term $gX_{\pm}$ in Eq.
(S25)]. Indeed, it is well established that the presence of the condensate
“eats away” the reservoir density analogous to the hole burning effect in
lasers Estrecho _et al._ (2018). In the adiabatic regime where the reservoir
is assumed to adjust to the condensate density dynamics very fast it can be
approximated as follows,
$X_{\pm}=\frac{P}{\Gamma_{R}}\left[1-\frac{R|\psi_{\pm}|^{2}}{\Gamma_{R}}+\mathcal{O}(|\psi_{\pm}|^{4})\right].$
(S30)
The nonlinearity of the condensate can therefore described by an effective
interaction parameter,
$\alpha_{\text{eff}}=\alpha-\frac{gPR}{\Gamma_{R}^{2}}.$ (S31)
To test our hypothesis, we numerically solve the eigenvalues of the Jacobian
for Eq. (S25) and plot their maximum real part for the
$\Psi_{\text{st}}^{H,V}$ solutions in Fig. S8 (red and blue curves,
respectively) as a function of varying $\alpha$ and several values of
$\Gamma_{R}$ [Fig. S8(a)] and $\gamma$ [Fig. S8(b)]. Indeed, we see that there
exist three distinct regimes which we schematically illustrate with the blue-
white-red color gradient:
1. i.
Ground state unstable and excited state stable.
2. ii.
Both states stable.
3. iii.
Ground state stable and excited state unstable.
Figure S8: Maximum real eigenvalues (Lyapunov exponents) from the Jacobian of
Eq. (S25) around the two steady state solution $\Psi_{\text{st}}^{H,V}$ (red
and blue curves, respectively) for $\epsilon=0.01$ ps-1. Regimes where
$\text{Re}{(\lambda_{\text{max}})}>0$ imply instability of the solution. The
color gradient at the top is added for illustration purposes. Parameters:
$\Gamma=1/5$ ps-1, $\Gamma_{s}=\Gamma/4$, $\epsilon=\Gamma/20$,
$R=0.015\Gamma$, $g=5R/6$, and $P=2P_{\text{th}}$. In (a) we fix $\gamma=0$
and in (b) we fix $\Gamma_{R}=0.5\Gamma$.
As expected from Eq. (S31), the stability range of the excited state increases
as $\Gamma_{R}$ decreases [see Fig. S8(a)] due to the nonlinearity
$\alpha_{\text{eff}}$ becoming more negative. Moreover, the stability range of
the excited state also increases when $\gamma/\epsilon$ becomes more negative
corresponding to the ground state becoming lossy [see Fig. S8(b)]. We point
out that it is not possible to determine separately the contribution of
$\alpha S_{0}^{H,V}$ and $gX^{H,V}$ in experiment since we can only measure
the net blueshift in condensate energy. Nevertheless, our experimental results
indicate that the current pump configuration favours the far-left regime in
Fig. S8 where the ground state is unstable. Recently, we reported results
corresponding to the far-right regime where robust ground state condensation
was instead observed Gnusov _et al._ (2020) using the same cavity sample but
somewhat different experimental configuration. How exactly one can tune from
one regime to the other is difficult to tell, but the clearest path would
either involve changing the detuning between the photon and exciton mode. This
is possible because $\alpha\propto|\chi|^{4}$ and $g\propto|\chi|^{2}$ where
$|\chi|^{2}$ is the exciton Hopfield fraction of the polariton quasiparticle.
Another method would be to design an excitation profile $P(\mathbf{r})$ which
changes the mean field rate $R$ of particles scattering into the condensate.
Finally, to see if our hypothesis agrees with the full spatial calculations of
Eq. (S18) we repeat the simulation from Fig. S6 in a new Fig. S9 with the
strength of polariton-polariton interactions doubled, i.e. $\alpha\to 2\alpha$
while keeping all other parameters unchanged. We now find, in agreement with
our predictions, that the steady state solution (tested over 100 random
initial conditions) converges to the spin ground state instead of the excited
state. This can be evidenced from the opposite polarization appearing in the
Stokes components in Fig. S9 as compared to Fig. S6.
Figure S9: Same simulations as in Fig. S6 but with doubled interactions
strength $\alpha\to 2\alpha$ (keeping all other parameters unchanged) such
that the excited spin state now becomes unstable and only the ground state
becomes populated.
## S8 Modeling of coupled optically trapped spinor condensates
In this section we modify Eq. (S25) to describe coupling between two spatially
separated condensates as shown in Fig. 5 in the main manuscript. The inter-
condensate-coupling is sometimes referred to as ballistic coupling because
energetic polaritons escape from each trap and undergo finite-time free-space
propagation before they reach their neighboring condensate. Such coupling is
qualitatively different from evanescent coupling (e.g., tunneling between
Bose-Einstein condensates) since the propagation time of polaritons between
the condensates is comparable to their intrinsic frequencies. This implies
that the coupling between ballistic condensates is time delayed Töpfer _et
al._ (2020) which we can introduce explicitly to Eq. (S25),
$\displaystyle\begin{split}i&\frac{d\psi_{\pm}^{(1,2)}}{dt}=\Big{[}\omega_{0}+\alpha|\psi_{\pm}^{(1,2)}|^{2}+gX_{\pm}^{(1,2)}+i\frac{RX_{\pm}^{(1,2)}-\Gamma}{2}\Big{]}\psi_{\pm}^{(1,2)}+(\epsilon+i\gamma)\psi_{\mp}^{(1,2)}+J\psi_{\pm}^{(2,1)}(t-\tau)+\mathcal{J}\psi_{\mp}^{(2,1)}(t-\tau),\\\
&\frac{dX_{\pm}^{(1,2)}}{dt}=P-\left(\Gamma_{R}+R|\psi_{\pm}^{(1,2)}|^{2}\right)X_{\pm}^{(1,2)}+\Gamma_{s}(X_{\mp}^{(1,2)}-X_{\pm}^{(1,2)}).\end{split}$
(S32)
Here, the indices $(1,2)$ refer to the two different condensates. We have also
introduced the condensate intrinsic energy $\omega_{0}$ since a suitable
rotating reference frame cannot be chosen for time delayed coupling between
oscillators. As was previously demonstrated Töpfer _et al._ (2020), the
strength of the coupling $J$ depends on the separation distance $d$ between
the condensates,
$J(d)=J_{0}|H_{0}^{(1)}(k_{c}d)|,$ (S33)
where $H_{0}^{(1)}$ is the zeroth order Hankel function, $J_{0}\in\mathbb{C}$
quantifies the non-Hermitian coupling strength dictated by the overlap of the
condensates over the optical trap region, and $k_{c}$ is the complex
wavevector of the polaritons propagating outside the optical trap,
$k_{c}=k_{c}^{(0)}+i\frac{\Gamma m}{2\hbar k_{c}^{(0)}}.$ (S34)
From experiment, we have estimated $k_{c}^{(0)}\approx 1.35$ $\mu$m-1 by
spatially filtering the polariton PL outside the pump spots. The imaginary
term in Eq. (S34) describes the additional attenuation of polaritons due to
their finite lifetime. We also account for coupling between the spins of the
two condensates due to the TE-TM splitting which is captured with the
parameter $\mathcal{J}$. The time delay parameter is approximated from the
polariton phase velocity which gives,
$\tau=\frac{2dm}{\hbar k_{c}^{(0)}}.$ (S35)
Figure S10: (a-d) Time averaged Stokes parameters of $\psi_{\pm}^{(1)}$ and
(e-h) $\psi_{\pm}^{(2)}$ from random initial conditions and varying pump power
$P$ and distance $d$. We normalise the pump power in units of the threshold
power for a single condensate $P_{\text{th}}=(\Gamma-2|\gamma|)\Gamma_{R}/R$.
The parameters of the simulation are: $\epsilon=-0.01$ ps-1;
$\gamma=\epsilon/2$; $\Gamma=0.25$ ps-1; $\Gamma_{R}=\Gamma/4$;
$\Gamma_{s}=\Gamma_{R}/2$; $\alpha=0.15\epsilon$; $R=0.05\epsilon$;
$\omega_{0}=5.5\Gamma$; $g=\alpha$; $m=0.3$ meV ps2 $\mu$m-2;
$J_{0}=0.67e^{1.8i}$ ps-1; and $\mathcal{J}=0.2J$. Note that the large value
of $J_{0}$ (as compared to $\epsilon$) is due to the smallness of the Hankel
function in Eq. (S33). At, e.g., $d=27$ $\mu$m we have $|J|=2.64|\epsilon|$.
We show in Fig. S10 results of numerically integrating Eq. (S32) from random
initial conditions. Each pixel in the data is one realization of the
condensate for the given power $P$ and distance $d$. The angled brackets of
the Stokes parameters $\langle S_{n}^{(1,2)}\rangle$ represent time-average.
We applied a constant step size Bogacki-Shampine algorithm Flunkert (2011) (a
3rd order Runge-Kutta). The timestep was chosen $\Delta t=0.05$ ps and the
integration was over $T=5000$ ps for each condensate realization. The results
reveal periodic polarization regimes similar to the phase-flip transitions
recently reported in Töpfer _et al._ (2020). At high powers we observe the
condensates stabilizing into the ground state (horizontal) polarization
[yellow colors in Figs. S10(a,e)] whereas at low power we retrieve stable
excited state (vertical) polarization condensation [blue colors in Figs.
S10(a,e)]. Between these bright yellow and blue regions we observe an
intermediate region (seen as a mixture of blue and yellow datapoints) where
the ground and excited state condensates are both stable and the random
initial condition determines the winner. Such linear polarization bistability
was already reported in Sigurdsson (2020) for a single condensate. We also
observe regions of complete depolarization (sea-green color) which correspond
to condensate destabilization. Comparing Figs. S10(a-d) with S10(e-h) we
observe, in the stable regime, that the condensates are strongly correlated in
polarization (i.e., $\Psi^{(1)}$ and $\Psi^{(2)}$ always co-polarize). Our
theoretical modeling gives good agreement with experimental observations
presented in Fig. 5 in the main manuscript. In Fig. S11 we additionally show
example dynamical trajectories from the unstable and stable regions marked by
the red square and circle in Fig. S10(a), respectively.
Figure S11: Evolution of the Stokes parameters for a single realization of the
condensate at the (a-d) the red square and (e-h) the red circle in Fig. S10,
corresponding to the unstable and stable regimes, respectively.
## S9 Additional experimental data for coupled condensates
Here we present additional experimental data on the coupled elliptical
condensates. In this experiment, we measure the Stokes components for 100
quasi-CW 50 $\mu$s shots. Each experimental point in Fig. S12 represents the
polarization component averaged within one excitation shot. We note that the
$S_{1}$, $S_{2}$ and $S_{3}$ in Figs. S12(c)-(n) are not measured
simultaneously but consequently under the same pumping conditions and at the
same position on the sample.
As expected, when isolated, the individual condensates stay dominantly
polarized linearly parallel to the minor axis of the pump trap, as we describe
in the main text [see Figs. S12(a) and S12(b) for a horizontal trap and
vertical trap, respectively]. However, some fluctuations can be sometimes
observed, for example, in the horizontally elongated trap in Fig. S12(a). This
happens due to some noise in our system as well due to mode competition. It is
worth noting that such fluctuations decrease the values of the Stokes
components presented in Figs. 2-4 in the main text since there we
integrate/average over hundreds of shots.
In Figs. S12(c)-(h) we present the Stokes components for coupled horizontally
elongated ellipses separated by 27.2 $\mu$m in Figs. S12(c)-(e) and 24 $\mu$m
in Figs. S12(f)-(h). Blue and red colors correspond to the ”right” and ”left”
condensate, respectively. For 27.2 $\mu$m, the condensate flips randomly from
horizontal to vertical polarization from shot-to-shot, whereas the $S_{2}$ has
smaller values (less than 0.5) but also flips from shot to shot. Overall the
$S_{3}$ component stays close to zero. For a different separation distance 24
$\mu$m shown in Figs. S12(f)-(h) we observe that all Stokes components are
close to zero in each shot. This means that the condensate pseudospin
fluctuates rapidly in time within one excitation pulse with a zero mean
polarization just like in simulation in Figs. S11(a)-(d). Notice that the
Stokes components still remain correlated indicating the condensate are
coupled together.
We also plot all polarization components for two coupled vertically elongated
condensates [Figs. S12(i)-(n)]. The weaker coupling of such mutual trap
configuration is evidenced through less correlations between the left and
right condensates (i.e., the red and blue curves fluctuate more
independently). For a distance of 26.5$\mu$m both condensates have strong
horizontal polarization — i.e. big $S_{1}$ component and small $S_{2}$ and
$S_{3}$ components. This corresponds to Figs. S11(e)-(h) in simulations. At a
distance of 25 $\mu$m the condensates are in a semi-depolarized regime with
oscillating $S_{1}$ and $S_{2}$ from shot to shot, and small $S_{3}$.
Figure S12: 100 realizations (shots) of the condensates Stokes components
time-integrated in each 50 $\mu$s excitation shot. (a) and (b) shows $S_{1}$
of single horizontally and vertically elongated condensate (trap)
respectively. $S_{1}$, $S_{2}$, and $S_{3}$ for two coupled condensates with
their trap’s major axes orientated longitudinally to the coupling direction
and separated by 27.2 $\mu$m (c)-(e) and 24 $\mu$m (f)-(h), respectively.
$S_{1}$, $S_{2}$, and $S_{3}$ for two coupled condensates with their trap’s
major axes orientated perpendicularly to the coupling direction, separated by
26.5 $\mu$m (i)-(k) and 25 $\mu$m (l)-(n), respectively. Blue and red colors
correspond to right and left condensate respectively. Background green and
purple colors depict different coupling regimes.
## S10 Power dependent pseudospin rotation
In this section we explain the results of Fig. 3(b) and 3(c) in the main
manuscript where we can observe noticeable change in the linear polarization
of the condensate as we increase the power. It manifests as counterclockwise
rotation of the pseudospin in the equatorial plane of the Poincaré sphere.
The explanation for the power dependent torque effect is due to slight
polarization ellipticity in the excitation laser. This creates an imbalance
between the spin-up and spin-down exciton populations in the system and a
consequent out-of-plane effective magnetic field $\boldsymbol{\Omega}_{\perp}$
which rotates the pseudospin. This is confirmed through Gross-Pitaevskii
simulations using Eq. (S25) where we introduce a slight pumping imbalance by
redefining a spin-dependent pumping rate $P\to P_{\pm}$ where $P_{+}\neq
P_{-}$. We present our simulation in Fig. S13 where we show the time-
integrated Stokes (pseudospin) components of the condensate at increasing mean
pump power [$P=(P_{+}+P_{-})/2$] where each datapoint is averaged over 100
random initial conditions. We have set $P_{+}/P_{-}=1.0355$ and other
parameters of the model (specified in the caption) are taken similar to the
ones used in Fig. S8 and S10. The shaded area is one standard deviation in the
pseudospin dynamics (calculated over 5 ns) indicating nonstationary and
stationary behaviour at low and high powers, respectively. We point out that
Eq. (S25) must now include the additional pump induced blueshift $gP_{\pm}/W$
like in Eq. (S18) since it contributes to $\boldsymbol{\Omega}_{\perp}$. The
results show precisely the counterclockwise rotation of the pseudospin in the
$(S_{1},S_{2})$ plane like in Fig. 3(b) and 3(c) in the main manuscript. We
have also confirmed that if $P_{+}<P_{-}$ then the pseudospin rotates
clockwise in the $(S_{1},S_{2})$ plane.
Moreover, this change in the $S_{1}$ and $S_{2}$ can also be evidenced from
the experimental data presented in Fig. S2. There, a small change in the
polarization ellipticity of our excitation beam dramatically affects the
$S_{1}$ and $S_{2}$ distributions.
Figure S13: Generalized Gross-Pitaevskii simulations of the condensate under
spin-imbalanced pumping. (a)-(d) Time-integrated Stokes (pseudospin)
components of the condensate for increasing pump power. Each datapoint is
averaged over 100 random initial conditions. The shaded area is one standard
deviation in the pseudospin dynamics (calculated over 5 ns) indicating
nonstationary and stationary behaviour at low and high powers, respectively.
Parameters are: $P_{+}/P_{-}=1.0355$, $P=(P_{+}+P_{-})/2$,
$P_{\text{th}}=\Gamma\Gamma_{R}/R$, $\Gamma=1/5$ ps-1,
$\Gamma_{R}=0.35\Gamma$, $\Gamma_{s}=\Gamma_{R}$, $\epsilon=\Gamma/20$,
$R=0.015\Gamma$, $W=\Gamma_{R}$, and $g=R$.
## References
* Gnusov _et al._ (2020) I. Gnusov, H. Sigurdsson, S. Baryshev, T. Ermatov, A. Askitopoulos, and P. G. Lagoudakis, Optical orientation, polarization pinning, and depolarization dynamics in optically confined polariton condensates, Phys. Rev. B 102, 125419 (2020).
* Maragkou _et al._ (2011) M. Maragkou, C. E. Richards, T. Ostatnický, A. J. D. Grundy, J. Zajac, M. Hugues, W. Langbein, and P. G. Lagoudakis, Optical analogue of the spin hall effect in a photonic cavity, Opt. Lett. 36, 1095 (2011).
* Wouters and Carusotto (2007) M. Wouters and I. Carusotto, Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons, Phys. Rev. Lett. 99, 140402 (2007).
* Antón _et al._ (2013) C. Antón, T. C. H. Liew, G. Tosi, M. D. Martín, T. Gao, Z. Hatzopoulos, P. S. Eldridge, P. G. Savvidis, and L. Viña, Energy relaxation of exciton-polariton condensates in quasi-one-dimensional microcavities, Phys. Rev. B 88, 035313 (2013).
* Sigurdsson (2020) H. Sigurdsson, Hysteresis in linearly polarized nonresonantly driven exciton-polariton condensates, Phys. Rev. Research 2, 023323 (2020).
* Shelykh _et al._ (2006) I. A. Shelykh, Y. G. Rubo, G. Malpuech, D. D. Solnyshkov, and A. Kavokin, Polarization and propagation of polariton condensates, Physical Review Letters 97, 066402 (2006).
* Estrecho _et al._ (2018) E. Estrecho, T. Gao, N. Bobrovska, M. D. Fraser, M. Steger, L. Pfeiffer, K. West, T. C. H. Liew, M. Matuszewski, D. W. Snoke, A. G. Truscott, and E. A. Ostrovskaya, Single-shot condensation of exciton polaritons and the hole burning effect, Nature Communications 9, 2944 (2018).
* Töpfer _et al._ (2020) J. D. Töpfer, H. Sigurdsson, L. Pickup, and P. G. Lagoudakis, Time-delay polaritonics, Communications Physics 3, 2 (2020).
* Flunkert (2011) V. Flunkert, Delay differential equations, in _Delay-Coupled Complex Systems: and Applications to Lasers_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 153–163.
|
# Diagrammatic Approach to Scattering of Multi-Photon States in Waveguide QED
Kiryl Piasotski ID<EMAIL_ADDRESS>Institut für Theorie der
Statistischen Physik, RWTH Aachen, 52056 Aachen, Germany
and JARA - Fundamentals of Future Information Technology, 52056 Aachen,
Germany Mikhail Pletyukhov ID Institut für Theorie der Statistischen Physik,
RWTH Aachen, 52056 Aachen, Germany
and JARA - Fundamentals of Future Information Technology, 52056 Aachen,
Germany
###### Abstract
We give an exposure to diagrammatic techniques in waveguide QED systems. A
particular emphasis is placed on the systems with delayed coherent quantum
feedback. Specifically, we show that the $N$-photon scattering matrices in
single-qubit waveguide QED systems, within the rotating wave approximation,
admit for a parametrization in terms of $N-1$-photon effective vertex
functions and provide a detailed derivation of a closed hierarchy of
generalized Bethe-Salpeter equations satisfied by these vertex functions. The
advantage of this method is that the above mentioned integral equations hold
independently of the number of radiation channels, their bandwidth, the
dispersion of the modes they are supporting, and the structure of the
radiation-qubit coupling interaction, thus enabling one to study multi-photon
scattering problems beyond the Born-Markov approximation. Further, we
generalize the diagrammatic techniques to the systems containing more than a
single emitter by presenting an exact set of equations governing the generic
two and three-photon scattering operators. The above described theoretical
machinery is then showcased on the example of a three-photon scattering on a
giant acoustic atom, recently studied experimentally [Nat. Phys. 15, 1123
(2019)].
## I Introduction
Waveguide quantum electrodynamics (QED) is a modern field of research focused
on the study of light-matter interactions in one dimension. The confinement of
electromagnetic radiation to a single spatial dimension allows one to achieve
a significant enhancement of the coupling between atoms and fields, as well as
to attain a better matching between the spatial modes of the emitting and
absorbing atoms [1, 2]. Apart from the purely academic interest in the study
of strong light-matter coupling, a great deal of motivation in this field of
research comes from the technological sector, namely from the quantum
computing [3, 8, 4, 5, 6, 7]. One of the most prominent examples is the so-
called quantum network: A system of quantum processors interconnected by
quantum radiation channels propagating quantum information and entanglement
between them [9].
Although photons are the excellent carriers of quantum information, capable of
high fidelity entanglement and information transfer [10, 11, 12], recent
advancement in quantum electronics offers a large variety of alternatives.
Most commonly, nowadays, waveguide QED setups are realized in the experiments
with superconducting quantum circuits, where superconducting transmission
lines act as quantum radiation channels, whereas the Josephson-junction-based
superconducting quantum bits play the role of quantum emitters [13, 14, 15].
Other examples include superconducting qubits coupled to propagating phonons
(surface acoustic waves) employing the piezoelectric effect [16, 17, 18, 19],
and surface plasmons coupled to quantum dots [20].
Theoretically, the atom-field interactions in waveguide QED can often be
accurately treated in the so-called Markovian limit $\gamma\tau\ll 1$, where
$\gamma$ and $\tau$ stand for the characteristic decay rate and time delay in
the system [21]. In the course of the last few decades, a significant number
of theoretical approaches allowing one to tackle the waveguide QED problems
within this limit was proposed. In particular, the Markovian waveguide QED
systems are commonly examined theoretically with help of master equation
based-approaches [1, 22, 23, 24, 25, 26, 27]. Indeed, within the framework of
the associated input-output formalism [1, 28, 29], Lindblad-type equation
approaches allow one to study transmission and reflection characteristics, as
well as the photonic correlations for arbitrary initial photon states,
including coherent, thermal, and Fock ones. Moreover, at a rather modest
expense, a substantial variety of theoretical tools are available for a
derivation of master equations: equation of motion-based methods [1, 22, 23],
path integral techniques [30], SLH formalism [31], to mention just a few.
Recently master equation-based approach was generalized to capture the effects
of temporal modulation of the system parameters [32, 33], such as the light-
matter coupling strength and the transition frequencies of quantum emitters.
Another frequenter method of studying waveguide QED systems within Markov
approximation is the coordinate space Bethe ansatz [34, 35, 36, 41, 37, 38,
39, 40]. Within this approach, one is able to determine the exact stationary
eigenstates of the system’s Hamiltonian in a subspace of a given excitation
number, which, in turn, enables one to calculate stationary observables as
well as photonic correlation functions exactly. Moreover, the Bethe ansatz
technique was shown to be applicable to the studies of real-time dynamics of
few-photon states [36, 42, 43], systems where the photon-photon bound states
occur [44, 41], and systems with delayed coherent feedback [45]. Despite its
scalability to the cases of multiple emitters and emitters with complicated
level structures, this approach is known to be strongly limited by the number
of excitations in the system due to the rapid increase of complexity of the
resulting Bethe wavefunctions [46, 36, 34, 47].
Another group of systematic methods of studying the waveguide QED systems in
the limit of the negligible delay times is comprised of the field theory-based
approaches. For example, in [48, 49, 50], by representing the atomic operators
in terms of the slave fermions, the authors were able to employ the path
integral representation of the field correlation functions, from which the
$S$-matrix may be established by means of the celebrated Lehmann-Symanzik-
Zimmermann reduction formula. Another field theoretical method to be mentioned
is the so-called diagrammatic resummation theory developed in [51, 52, 53,
54]. Within the framework of resummation theory, one sums the perturbative
series of Feynmann diagrams for the matrix elements of the transition operator
to infinite orders, which, in turn, allows one to determine the exact
$S$-matrix, with the help of which all of the stationary observables may be
calculated [52].
Although physics behind waveguide QED systems is easily accessible in the
Markovian limit using a large variety of theoretical tools, there exists a
number of problems forcing one to go beyond this approximation [21, 55, 56,
57, 58]. As it is well known, in the single excitation subspace, all of the
dynamical and stationary information about the system may be easily obtained
(either analytically or numerically) by means of Bethe ansatz for a system of
arbitrary complexity [60, 59, 61, 62]. Primarily, this assertion has to do
with the fact that it is relatively easy to conceive a closed system of
(delay) differential equations governing the evolution of the system. Even
though it is possible to proceed in the same manner in higher excitation
subspaces [63], the calculations become much more cumbersome and lack
systematicity. In the course of the last decade, a number of theoretical
approaches allowing one to overcome these difficulties were put forward. On
the numerical side, for example, significant progress in the dynamical
simulation of the non-Markovian 1D quantum systems was achieved within the
framework of matrix product state ansatz [64, 65, 66, 67]. Despite the
complexity associated with the non-Markovian waveguide QED systems, a few
analytical methods were also recently suggested. In particular, it is a common
practice to generalize the Lindblad-type equation approaches to the non-
Markovian realm [30, 68, 69, 70]. Although generalized master equations can
only provide exact results in the case of linear scatterers, they allow for
systematic approximate treatment of systems with nonlinearities such as
qubits.
Another common approach to waveguide QED problems with delayed coherent
quantum feedback is based on the diagrammatic resummation theory. In recent
years resummation ideas were successfully applied to solve the two-photon
scattering and dynamics in the systems with two distant qubits [71], a single
qubit in front of a mirror [21], a giant acoustic atom [59], and a qubit
coupled to a resonator array [72].
In this paper, we present a systematic generalizattion of the diagrammatic
approach to scattering of multiphoton states in waveguide QED. Our approach is
based on the exact resummation of the perturbation theory for the transition
operator which turns out to be possible due to the conservation of excitation
number guaranteed by the rotating wave approximation. The advantage of our
approach is its insensitivity to the form of light-matter coupling constants,
thus, allowing one to potentially examine any kinds of waveguide QED systems,
including the systems with delayed coherent quantum feedback. We start by
making an exposure of the method by the direct example of $1$-qubit waveguide
QED. This framework lays down a basis for further calculations, in particular
general qubit number two and three-photon scattering theory. We then apply the
developed theory to a weakly coherent pulse scattering on a giant acoustic
atom, an intrinsically non-Markovian system. In particular, we consider the
scattering of a coherent state with small enough coherence parameter
$|\alpha|\ll 1$ chosen such that the terms of order $|\alpha|^{4}$ can be
ignored. With the help of the general methods developed in Section II we
compute an exact output state of radiation and study the particle correlations
in it with the help of the theory of optical coherence. In particular, we
compute the first, second, and third-order coherence functions to the lowest
order in $|\alpha|$ and discuss the impact of the non-Markovianity of the
scatterer on the observable quantities.
## II Scattering Theory
In this section, we first set up the notations used throughout the paper.
Next, we introduce the general formalism in the framework of the single-qubit
waveguide QED. In particular, we extend the findings of the preceding papers
[21, 51, 53, 59] to the realm of non-Markovian models by allowing for
arbitrary momentum dependence of waveguide modes and radiation-qubit
couplings. This development, in turn, allows one to study multi-photon
scattering problems in systems with non-linear dispersion of the modes
supported by the radiation channels, as well as the systems with artificial
feedback loops, such as a qubit placed in front of a mirror or a giant
acoustic atom, for example.
Further on, we generalize the scattering formalism to the systems with a
higher number of qubits, where the quantum feedback loops a naturally present
due to the finiteness of time required for a photon to propagate between a
given pair of distant emitters. Specifically, we discuss two and three-photon
scattering problems on an arbitrary number of emitters, extending the approach
of [71].
Alongside this, we discuss several practical issues, such as the separation of
elastic contributions to the scattering matrices and the generalized cluster
decomposition.
### II.1 Hamiltonian, generalized summation convention, and the $S$ -matrix
Let us consider a collection of $N_{q}$ qubits coupled to a waveguide with
$N_{c}$ radiation channels. The Hamiltonian of such a system assumes the
following form $\mathcal{H}=\mathcal{H}_{0}+\mathcal{V}$, where
$\displaystyle\mathcal{H}_{0}=$
$\displaystyle\sum_{n=1}^{N_{q}}\Omega_{n}\sigma_{+}^{(n)}\sigma_{-}^{(n)}+\sum_{\mu=1}^{N_{c}}\int_{B_{\mu}}dk\omega_{\mu}(k)a^{\dagger}_{\mu}(k)a_{\mu}(k),$
(1)
is the free Hamiltonian, and
$\displaystyle\mathcal{V}=$
$\displaystyle\sum_{\mu=1}^{N_{c}}\sum_{n=1}^{N_{q}}\int_{B_{\mu}}dk[g_{\mu,n}(k)a^{\dagger}_{\mu}(k)\sigma_{-}^{(n)}+\text{h.c.}]$
(2)
is the dipole light-matter interaction in the rotating wave approximation
(RWA). In the above expression, $\Omega_{n}$ is the transition frequency of
the $n^{\text{th}}$ qubit, the dispersion relation $\omega_{\mu}(k)$ and the
bandwidth $B_{\mu}$ characterize the radiation channel $\mu$, while
$a^{\dagger}_{\mu}(k)$ and $a_{\mu}(k)$ stand for the creation and
annihilation field operators of a photon with momentum $k$ and obey the
standard bosonic commutation relations:
$\displaystyle[a_{\mu}(k),a^{\dagger}_{\mu^{\prime}}(k^{\prime})]=$
$\displaystyle\delta_{\mu,\mu^{\prime}}\delta(k-k^{\prime}),$ (3)
$\displaystyle[a_{\mu}^{\dagger}(k),a^{\dagger}_{\mu^{\prime}}(k^{\prime})]=$
$\displaystyle[a_{\mu}(k),a_{\mu^{\prime}}(k^{\prime})]=0.$ (4)
The operators acting on the Hilbert space of the $n^{\text{th}}$ qubit,
$\\{\sigma_{3}^{(n)},\sigma_{+}^{(n)},\sigma_{-}^{(n)}\\}$, are defined
according to
$\sigma^{(n)}_{l}=1_{\mathbb{C}^{2}}^{\otimes(n-1)}\otimes\sigma_{l}\otimes
1_{\mathbb{C}^{2}}^{\otimes(N_{q}-n)}$, with the $\sigma$-matrices being
chosen according to the standard convention
$\displaystyle\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\sigma_{+}=\sigma_{-}^{\dagger}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}.$ (5)
The RWA is justified as long as the characteristic operational frequency
$\omega_{0}$ is such that the condition
$|g_{\mu}^{2}(\omega_{0})/\omega_{0}|\ll 1$ is satisfied [73]. One of the main
benefits of this approximation is the conservation of the total number of
excitations in the system, i.e. an operator
$\mathcal{N}=\sum_{n=1}^{N_{q}}\sigma_{+}^{(n)}\sigma_{-}^{(n)}+\sum_{\mu=1}^{N_{c}}\int_{B_{\mu}}dka_{\mu}^{\dagger}(k)a_{\mu}(k)$
commutes with the full Hamiltonian, $[\mathcal{N},\mathcal{H}]=0$. This
property allows us to simultaneously diagonalize $\mathcal{H}$ and
$\mathcal{N}$. Moreover, the eigenspaces of the Hamiltonian labeled by the
eigenvalues of $\mathcal{N}$ are certainly orthogonal, hence there exists a
direct sum decomposition of the total Hilbert space of the system
$\mathscr{H}=\bigoplus_{N=0}^{\infty}\mathscr{H}_{N},$ (6)
where $\mathscr{H}_{N}$ is the
$D(N,N_{q})=\sum_{l=0}^{\min(N_{q},N)}\frac{N_{q}!}{l!(N_{q}-l)!}$ dimensional
subspace of all possible states with $N$ excitations. Note that here
$D(N,N_{q})$ does not stand for the dimension of $\mathscr{H}_{N}$ in the
strict mathematical sense. Instead, it is a number of ways to distribute $N$
excitations in a system of $N_{q}$ qubits (i.e., each single-photon infinite-
dimensional vector space adds unity to $D(N,N_{q})$). Analogous direct sum
decompositions hold for the Hamiltonian, unitary evolution operator, and the
$S$-matrix (to be introduced later). Moreover, such a decomposition
considerably simplifies the problem since the calculations may be performed in
all of the subspaces separately.
To simplify our further analysis, it is useful to introduce compact notations.
Thus we define a multi-index $s=(\mu,k)$ and the generalized summation
convention: if two multi-indices are repeated, summation over the channel
index $\mu$ and integration with respect to the momentum $k$ (over the
relevant bandwidth $B_{\mu}$) is implied. We also introduce the following
notation for the bare interaction vertex
$v_{s}=\sum_{n=1}^{N_{q}}g_{\mu,n}(k)\sigma^{(n)}_{-}.$ (7)
Using these conventions we rewrite the Hamiltonian as
$\displaystyle\mathcal{H}=$
$\displaystyle\mathcal{H}_{0}+\mathcal{V},\quad\mathcal{H}_{0}=\omega_{s}a^{\dagger}_{s}a_{s}+\sum_{n=1}^{N_{q}}\Omega_{n}\sigma_{+}^{(n)}\sigma_{-}^{(n)},$
(8) $\displaystyle\mathcal{V}=$ $\displaystyle
a^{\dagger}_{s}v_{s}+v_{s}^{\dagger}a_{s}.$ (9)
The scattering matrix, or the $S$-matrix, is the main object of interest in
the present section. It can be generally defined via the so-called transition
operator, or the $T$-matrix, in the following way:
$\displaystyle\mathcal{S}=$ $\displaystyle
1_{\mathscr{H}}-2\pi{i}\delta(\epsilon_{i}-\epsilon_{f})\mathcal{T}(\epsilon_{i}),$
(10) $\displaystyle\mathcal{T}(\epsilon)=$
$\displaystyle\mathcal{V}+\mathcal{V}\mathcal{G}(\epsilon)\mathcal{V},$ (11)
where the $T$-matrix is put on-shell, i.e. $\epsilon=\epsilon_{i}$, and the
energies $\epsilon_{i},\epsilon_{f}$, corresponding to initial and final
states of the system, obey the energy conservation in the scattering
processes, which is mathematically ensured by the delta-function. In (11) we
have denoted by $\mathcal{G}(\epsilon)$ the retarded Green’s operator defined
according to
$\displaystyle\mathcal{G}(\epsilon)=$
$\displaystyle\frac{1}{\epsilon-\mathcal{H}+i\eta}=\sum_{n=0}^{\infty}(\mathcal{G}_{0}(\epsilon)\mathcal{V})^{n}\mathcal{G}_{0}(\epsilon),$
(12) $\displaystyle\mathcal{G}_{0}(\epsilon)=$
$\displaystyle\frac{1}{\epsilon-\mathcal{H}_{0}+i\eta},\quad\eta\rightarrow
0^{+}.$ (13)
### II.2 General properties of the scattering matrix in waveguide QED
Let us consider the scattering problem for the following initial state
$\ket{N_{p}}\otimes\ket{g}$, where $\ket{N_{p}}$ is a $N_{p}$-photon state,
and $\ket{g}=\ket{0}^{\otimes{N_{q}}}$ is the ground state of the scatterer.
Due to the conservation of excitation-number operator, all of the possible
$D(N_{p},N_{q})$ scattering outcomes must contain $N_{p}$ excitations. Due to
the fact that for any system of qubits the ground state $\ket{g}$ is the only
non-decaying (subradiant) subspace, in the long-time limit (a priory assumed
in scattering theory) all of the emitters will definitely decay into the
continuum, leaving us with the only possibility for the system to end up in
the state $\ket{N_{p}^{\prime}}\otimes\ket{g}$ (here $\ket{N_{p}^{\prime}}$ is
again a $N_{p}$-photon state, with potentially redistributed momenta) [74]. If
one wishes to extend the scattering theory to the systems with metastable
ground states, such as e.g. a $\Lambda$ three-level system, one then has to
consider calculating more matrix elements of the transition operator [75, 76].
Since the only matrix elements, we are interested in are diagonal in both the
photon and qubit space, due to the RWA, the only terms contributing to the
perturbation expansion of the $T$-matrix are those containing an even number
of interactions $\mathcal{V}$, thus reducing the number of diagrams by half.
Another important feature to be mentioned is the nilpotency of the photon-
qubit interaction vertex operator
$v_{s_{1}}^{\dagger}...v_{s_{N_{q}+1}}^{\dagger}=v_{s_{1}}...v_{s_{N_{q}+1}}=0$,
which along with the property $v_{s}\ket{g}=\bra{g}v_{s}^{\dagger}=0$ and the
fact that the number of $v$’s and $v^{\dagger}$’s has to be equal in each
graph contributing to the expansion, significantly reduces the number of non-
zero diagrams at each order in perturbation theory. In fact, as we are going
to see in this section, all of the diagrams contributing to the series for any
fixed $N_{q}$ may be constructed out of a finite number of ”clusters”, in turn
allowing, in principle for the exact resummation of the perturbation series.
One can also organize the calculation differently, namely by fixing $N_{p}$
and allowing $N_{q}$ to vary instead. Calculation within this approach is
facilitated in its turn by the fact that
$a_{s_{1}}...a_{s_{N_{p}+1}}\ket{N_{p}}=\bra{N_{p}}a_{s_{1}}^{\dagger}...a_{s_{N_{p}+1}}^{\dagger}=0$
and the fact that the number of $a$’s and $a^{\dagger}$’s in each term of the
perturbation series have to be equal. Although, these two approaches are
clearly dual due to the structure of interaction potential in RWA. This
approach is beneficial when studying few-photon scattering on a large number
of qubits. This assertion has to do with the fact that once the solution of
$N_{p}$ photon scattering problem on $N_{q}=N_{p}$ the scattering of $N_{p}$
particles on $N_{q}>N_{p}$ follows the same lines. Indeed, since $v_{s}$
($v_{s}^{\dagger}$) by itself contains the sum of all of the single-qubit
lowering (raising) operators and the highest number of qubits that the $N_{p}$
photon pulse can excite equals to $N_{p}$, the normal ordered $N_{p}$-photon
$S$-matrix cannot contain projectors on subspaces with higher-excitation
number than $N_{p}$. Using this fact in the following we are going to derive
the generic two and three-photon scattering matrices by considering two and
three particle scattering on two and three atoms respectively.
### II.3 Scattering theory in $N_{q}=1$ waveguide QED
In this subsection, we would like to make a detailed exposition of our general
method by considering the simplest imaginable scenario of a single qubit
coupled to a waveguide. As it was anticipated in Subsection II.2, to solve the
$N_{p}$-photon scattering problem, we shall determine the following matrix
elements of the transition operator
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(1)}(\epsilon)}{N_{p},g}=\Braket{N_{p}^{\prime},g}{\mathcal{V}\mathcal{G}(\epsilon)\mathcal{V}}{N_{p},g}$
(14)
$\displaystyle=\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\mathcal{G}(\epsilon)v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g}$
(15)
$\displaystyle=\sum_{n=0}^{\infty}\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}(\mathcal{G}_{0}(\epsilon)\mathcal{V})^{n}\mathcal{G}_{0}(\epsilon)v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g},$
(16)
where the superscript $(1)$ refers to the $N_{q}=1$ waveguide QED and we have
used the fact that $\Braket{N_{p}^{\prime},g}{\mathcal{V}}{N_{p},g}=0$. Note
that in what follows, whenever an argument of an object is omitted, we
understand that its argument is $\epsilon$, e.g $\mathcal{G}^{(1)}$ stands for
$\mathcal{G}^{(1)}(\epsilon)$, etc, however, when the argument of a given
operator or vertex function is of importance, it would be explicitly stated.
As it was mentioned above, due to RWA, only even terms contribute to the above
geometric series, so that
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(1)}}{N_{p},g}$
$\displaystyle=\sum_{n=0}^{\infty}\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}(\mathcal{G}_{0}\mathcal{V}\mathcal{G}_{0}\mathcal{V})^{n}\mathcal{G}_{0}v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g}.$
(17)
Now, let us consider the $\mathcal{V}\mathcal{G}_{0}\mathcal{V}$ term in the
brackets above
$\displaystyle\mathcal{V}\mathcal{G}_{0}\mathcal{V}=$ $\displaystyle
a^{\dagger}_{s^{\prime}}v_{s^{\prime}}\mathcal{G}_{0}a^{\dagger}_{s}v_{s}+a^{\dagger}_{s^{\prime}}v_{s^{\prime}}\mathcal{G}_{0}v_{s}^{\dagger}a_{s}$
$\displaystyle+$ $\displaystyle
v_{s^{\prime}}^{\dagger}a_{s^{\prime}}\mathcal{G}_{0}v_{s}^{\dagger}a_{s}+v_{s^{\prime}}^{\dagger}a_{s^{\prime}}\mathcal{G}_{0}a^{\dagger}_{s}v_{s}.$
(18)
Clearly, the terms with two $v$s and two $v^{\dagger}$s do not contribute
(they are in fact zero) by the single-qubit nilpotency condition
$v^{2}=(v^{\dagger})^{2}=0$. Among the ”diagonal” terms, the only non-zero
contribution comes from the $v^{\dagger}v$ term since the term $vv^{\dagger}$
annihilates the state $v^{\dagger}\ket{g}$ and gives zero whenever it is
multiplied with $v^{\dagger}v$ term. Hence, we arrive at
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(1)}}{N_{p},g}$
$\displaystyle=\sum_{n=0}^{\infty}\bra{N_{p}^{\prime},g}a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}(\mathcal{G}_{0}v_{s^{\prime}}^{\dagger}a_{s^{\prime}}\mathcal{G}_{0}a^{\dagger}_{s}v_{s})^{n}\mathcal{G}_{0}v^{\dagger}_{s_{1}}a_{s_{1}}\ket{N_{p},g}.$
(19)
Now, by using the commutation relations (3) and (4), one may easily establish
the following intertwining property of bosonic operators [51]
$\displaystyle a_{s}f(\mathcal{H}_{0})$
$\displaystyle=f(\mathcal{H}_{0}-\omega_{s})a_{s},$ (20) $\displaystyle
f(\mathcal{H}_{0})a_{s}^{\dagger}$
$\displaystyle=a_{s}^{\dagger}f(\mathcal{H}_{0}-\omega_{s}),$ (21)
where $f$ is some function admitting for the Maclaurin expansion
$f(z)=\sum_{n=0}^{\infty}f_{n}z^{n}$. So, we see that
$\displaystyle
v_{s^{\prime}}^{\dagger}a_{s^{\prime}}\mathcal{G}_{0}(\epsilon)a^{\dagger}_{s}v_{s}$
$\displaystyle=v_{s}^{\dagger}\mathcal{G}_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+a^{\dagger}_{s^{\prime}}v_{s}^{\dagger}\mathcal{G}_{0}(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s^{\prime}}a_{s},$
(22)
where in the last term of (22) the contraction of the indices $s$ and
$s^{\prime}$ is not assumed. By defining the self-energy operator
$\Sigma^{(1)}$ and the effective potential energy operator $\mathcal{R}^{(1)}$
as
$\displaystyle\Sigma^{(1)}(\epsilon)=$ $\displaystyle
v_{s}^{\dagger}\mathcal{G}_{0}(\epsilon-\omega_{s})v_{s},\quad\mathcal{R}^{(1)}(\epsilon)=a^{\dagger}_{s^{\prime}}\mathcal{R}_{s^{\prime},s}^{(1)}(\epsilon)a_{s},$
(23) $\displaystyle\mathcal{R}_{s^{\prime},s}^{(1)}(\epsilon)=$ $\displaystyle
v_{s}^{\dagger}\mathcal{G}_{0}(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s^{\prime}},$
(24)
and resumming the perturbative series in (19), we conclude that
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(1)}}{N_{p},g}$
$\displaystyle=\Bra{N_{p}^{\prime},g}a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\frac{1}{1-{\mathcal{G}}^{(1)}\mathcal{R}^{(1)}}{\mathcal{G}}^{(1)}v^{\dagger}_{s_{1}}a_{s_{1}}\ket{N_{p},g},$
(25)
where we have introduced the following Green’s operator
$(\mathcal{G}^{(1)})^{-1}(\epsilon)=\mathcal{G}_{0}^{-1}(\epsilon)-\Sigma^{(1)}(\epsilon)$.
By defining the generating operator:
$\displaystyle\mathcal{W}^{(1)}=\mathcal{R}^{(1)}+\mathcal{R}^{(1)}\mathcal{G}^{(1)}\mathcal{W}^{(1)},$
(26)
we arrive at the following result:
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(1)}}{N_{p},g}$
$\displaystyle=\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\mathcal{G}^{(1)}v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g}$
$\displaystyle+\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\mathcal{G}^{(1)}\mathcal{W}^{(1)}\mathcal{G}^{(1)}v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g}.$
(27)
By analyzing the structure of $\mathcal{R}^{(1)}$, we conclude that
$\mathcal{W}^{(1)}$ admits for the following series representation:
$\displaystyle\mathcal{W}^{(1)}=\sum_{n=1}^{\infty}a^{\dagger}_{s_{1}^{\prime}}...a^{\dagger}_{s_{n}^{\prime}}\mathcal{W}^{(1,n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}a_{s_{n}}...a_{s_{1}},$
(28)
where $\mathcal{W}^{(1,n)}$’s are the operator-valued functions, which depend
only on $\epsilon-\mathcal{H}_{0}$ and $2n$ multi-indices
$\\{s_{1}^{\prime},...,s_{n}^{\prime},s_{1},...,s_{n}\\}$.
By inserting (28) in (26) and taking the projections onto the particle
subspaces we arrive at the following hierarchy of integral equations:
$\displaystyle W_{s_{1}^{\prime},s_{1}}^{(1,1)}(\epsilon)=$ $\displaystyle
R_{s_{1}^{\prime},s_{1}}^{(1)}(\epsilon)+R_{s_{1}^{\prime},s}^{(1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W_{s,s_{1}}^{(1,1)}(\epsilon),$
(29) $\displaystyle\vdots$ $\displaystyle
W^{(1,n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}(\epsilon)=$
$\displaystyle
R_{s_{1}^{\prime},s_{1}}^{(1)}\Bigg{(}\epsilon-\sum_{l=2}^{n}\omega_{s_{n}^{\prime}}\Bigg{)}G^{(1)}\Bigg{(}\epsilon-\omega_{s_{1}}-\sum_{l=2}^{n}\omega_{s_{l}^{\prime}}\Bigg{)}W^{(1,n-1)}_{s_{2}^{\prime}...s_{n}^{\prime},s_{2}...s_{n}}(\epsilon-\omega_{s_{1}})$
$\displaystyle+$ $\displaystyle
R_{s_{1}^{\prime},s}^{(1)}\Bigg{(}\epsilon-\sum_{l=2}^{n}\omega_{s_{n}^{\prime}}\Bigg{)}G^{(1)}\Bigg{(}\epsilon-\omega_{s}-\sum_{l=2}^{n}\omega_{s_{l}^{\prime}}\Bigg{)}\Big{[}W^{(1,n)}_{ss_{2}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}(\epsilon)+...+W^{(1,n)}_{s_{2}^{\prime}...s_{n}^{\prime}s,s_{1}...s_{n}}(\epsilon)\Big{]}$
(30) $\displaystyle\vdots$
Here, we have used regular letters $W,G,R$ instead of calligraphic ones to
indicate that these objects are not operators but are rather their projections
on the photonic vacuum (note that $W^{(1,n)}$s are still acting as operators
on the qubit space). The collection of the integral equations above may be
conveniently represented diagrammatically, see Figure 1. Diagrammatic rules
may be formulated as follows. To each dotted line associate the bare
propagator $G_{0}(\epsilon)$. To each wavy line with incoming (outgoing) arrow
associate the bare absorption $v^{\dagger}$ (emission $v$) vertex. Whenever
two wavy lines are contracted together, one has to integrate over all momentum
and sum over all channels. When a given wavy line passes over the propagator,
its argument has to be shifted by the frequency carried by this line (which is
a direct consequence of the intertwining property (20)).
Figure 1: Hierarchy of integral equations governing effective multi-photon
vertex functions in $N_{q}=1$ waveguide QED. Here the $\Sigma^{(1)}$ bubble
corresponds to the self-energy in a subspace of a single excitation used to
the define the dressed Green’s function (represented by a double line) from
the bare Green’s function (depicted by the dashed line). The $W^{(1,n)}$
bubbles (bubbles with $2n$-amputated legs) represent effective $n$-photon
vertex functions describing effective interaction between $n$-photons induced
by non-linearity. Bare absorption $v^{\dagger}$ and emission $v$ vertices are
represented by black dots with incoming and outgoing photon lines
respectively. Diagrammatic rules are as stated in Sec. II.3.
For example, the self-energy diagram in Figure 1 is translated as
$\displaystyle\Sigma^{(1)}(\epsilon)=v_{s}^{\dagger}G_{0}(\epsilon-\omega_{s})v_{s}.$
(31)
Now having identified the equations defining the components of
$\mathcal{W}^{(1)}$, we may express the transition operator as follows
$\mathcal{T}^{(1)}(\epsilon)=\sum_{n=1}^{\infty}T^{(1,n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}(\epsilon)a^{\dagger}_{s_{1}^{\prime}}...a_{s_{n}^{\prime}}^{\dagger}a_{s_{n}}...a_{s_{1}},$
(32)
where we have defined the following functions
$\displaystyle
T^{(1,1)}_{s_{1}^{\prime},s_{1}}(\epsilon)=\Braket{g}{v_{s_{1}^{\prime}}G^{(1)}(\epsilon)v^{\dagger}_{s_{1}}}{g},$
(33) $\displaystyle
T^{(1,n>1)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}(\epsilon)=\Braket{g}{v_{s_{1}^{\prime}}G^{(1)}\Bigg{(}\epsilon-\sum_{l=2}^{n}\omega_{s_{l}^{\prime}}\Bigg{)}W^{(1,n-1)}_{s_{2}^{\prime}...s_{n}^{\prime},s_{2}...s_{n}}(\epsilon)G^{(1)}\Bigg{(}\epsilon-\sum_{l=2}^{n}\omega_{s_{l}}\Bigg{)}v^{\dagger}_{s_{1}}}{g}.$
(34)
Note that we are working with the non-symmetrized forms of operators and
perform the symmetrization only when doing actual calculations. Also note that
the dependence of
$\mathcal{T}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}^{(1,n)}$ on
$\omega_{s}a^{\dagger}_{s}a_{s}$ may be omitted taking into account normal
ordering, when $S$-matrix is going to be finally contracted with the initial
state, all of the $T$-matrix components are going to be eventually projected
onto the photonic vacuum and energies are going to be put on-shell. Using the
above representation of the transition operator we immediately deduce that the
scattering operator takes the following form:
$\mathcal{S}=1_{\mathscr{H}}-2\pi{i}\sum_{n=1}^{\infty}T^{(n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}\Bigg{(}\sum_{l=1}^{n}\omega_{s_{l}}\Bigg{)}\delta\Bigg{(}\sum_{l=1}^{n}\omega_{s_{l}^{\prime}}-\sum_{l=1}^{n}\omega_{s_{l}}\Bigg{)}a^{\dagger}_{s_{1}^{\prime}}...a_{s_{n}^{\prime}}^{\dagger}a_{s_{n}}...a_{s_{1}}.$
(35)
Practically, when one contracts the $S$-matrix with the initial $N_{p}$-photon
state, only first $N_{p}$ terms in the above series contribute. It is,
however, beneficial to represent the $S$-matrix in terms of the direct sum of
$N$-body operators which act solely in the $N$-particle subspaces
$\mathcal{S}=1_{\mathscr{H}}\oplus\bigoplus_{n=1}^{\infty}\mathcal{S}_{n},$
(36)
where $\mathcal{S}_{n}$ may be written as
$\mathcal{S}_{n}=S^{(n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}a^{\dagger}_{s_{1}^{\prime}}...a_{s_{n}^{\prime}}^{\dagger}a_{s_{n}}...a_{s_{1}},$
(37) $\displaystyle
S^{(n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}=\frac{1}{n!}\prod_{l=1}^{n}\delta_{s_{l}^{\prime},s_{l}}-2\pi{i}\sum_{m=1}^{n}\frac{1}{(n-m)!}T^{(n)}_{s_{1}^{\prime}...s_{m}^{\prime},s_{1}...s_{m}}\Bigg{(}\sum_{l=1}^{m}\omega_{s_{l}}\Bigg{)}\delta\Bigg{(}\sum_{l=1}^{n}\omega_{s_{l}^{\prime}}-\sum_{l=1}^{n}\omega_{s_{l}}\Bigg{)}\prod_{r=m+1}^{n}\delta_{s_{r}^{\prime},s_{r}}.$
(38)
Note that the combinatorial prefactors $\frac{1}{(n-m)!},\quad
m\in\\{0,...,n\\}$ and additional $\delta$ functions
$\delta_{s_{r}^{\prime},s_{r}}$ are chosen to take into account the excess
number of $(n-m)!$ Wick contractions.
### II.4 Generalized cluster decomposition
In this section, we would like to illustrate how the generalized cluster
decomposition in waveguide QED, extensively studied in [76-77], naturally
follows from the results of the previous section.
The cluster decomposition is a way of separating the elastic and inelastic
contributions to the $S$-matrix. The elastic contribution physically
corresponds to the scattering channel in which all photons scatter coherently,
i.e. conserve their energy individually in the scattering process. On the
other hand, the inelastic contribution corresponds to the incoherent
scattering. Thereby the photons redistribute their initial energy between one
another via effective photon-photon interaction which is mediated by their
interaction with nonlinear scatterers (e.g., qubits). In contrast, in the case
of a linear scatter (e.g., a cavity mode), the effective photon-photon
interaction is not generated, and the $n$-photon $S$-matrix simply factors out
into the product of $n$ single-particle $S$-matrices.
The first step towards the cluster decomposition of the multi-photon
$S$-matrices is the realization that the multi-photon vertex functions
$W^{(1,n)}$ contain the disconnected components. These, in turn, arise from
the projections of $R^{(1)}_{s^{\prime},s}$ on the ground state of the
scatterer thus resulting in the (quasi)elastic contributions
$\propto\delta(\epsilon-...)$ to $W^{(1,n)}$’s. Not only the separation of
these contributions is crucial for the cluster decomposition but is of evident
importance for both numerical and analytical treatment of the integral
equations (29), (30). It is the easiest to define the connected parts of the
multi-photon vertex functions recursively. First, we observe that
$\displaystyle R_{s^{\prime},s}^{(1)}(\epsilon)=$ $\displaystyle
v_{s}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s^{\prime}}$
(39) $\displaystyle=$
$\displaystyle-i\pi\delta(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s}^{\dagger}v_{s^{\prime}}$
(40) $\displaystyle+$ $\displaystyle
v_{s}^{\dagger}P\frac{1}{\epsilon-\omega_{s^{\prime}}-\omega_{s}}v_{s^{\prime}},$
(41)
where $P$ stands for the Cauchy principle value. Which helps us to define the
connected part of one-particle vertex function as
$\displaystyle W_{s_{1}^{\prime},s_{1}}^{(1,1,C)}(\epsilon)=$ $\displaystyle
W_{s_{1}^{\prime},s_{1}}^{(1,1)}(\epsilon)+i\pi\delta(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{1}})v_{s_{1}}^{\dagger}v_{s_{1}^{\prime}}.$
(42)
By then analyzing the structure of the general equation in the hierarchy (30)
and bearing in mind the equation satisfied by
$W_{s_{1}^{\prime},s_{1}}^{(1,1)}(\epsilon)$ one may easily deduce the
connected part of $n$-photon effective vertex function may be defined in the
following manner
$\displaystyle
W^{(1,n,C)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}(\epsilon)=$
$\displaystyle
W^{(1,n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}(\epsilon)-W_{s_{1}^{\prime},s_{1}}^{(1,1)}\Bigg{(}\epsilon-\sum_{l=2}^{n}\omega_{s_{n}^{\prime}}\Bigg{)}G^{(1)}\Bigg{(}\epsilon-\omega_{s_{1}}-\sum_{l=2}^{n}\omega_{s_{l}^{\prime}}\Bigg{)}W^{(1,n-1)}_{s_{2}^{\prime}...s_{n}^{\prime},s_{2}...s_{n}}(\epsilon-\omega_{s_{1}}).$
(43)
Now, with the help of the definition (42) one may immediately deduce the
following decomposition of the two-body transition operator
$\displaystyle
T^{(1,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega_{s_{1}}+\omega_{s_{2}})$
$\displaystyle=-i\pi\delta(\omega_{s_{1}}-\omega_{s_{2}^{\prime}})T^{(1,1)}_{s_{1}^{\prime},s_{1}}(\omega_{s_{1}})T^{(1,1)}_{s_{2}^{\prime},s_{2}}(\omega_{s_{2}})$
$\displaystyle+T^{(1,2,C)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega_{s_{1}}+\omega_{s_{2}}),$
(44)
where $T^{(1,2,C)}$ is defined in precisely the same way as $T^{(1,2)}$ but
with $W^{(1,1,C)}$ replacing $W^{(1,1)}$. By plugging the representation (44)
into the definition of the two-photon $S$-matrix (38) we immediately arrive at
the following cluster decomposition principle:
$\displaystyle\mathcal{S}_{2}=$
$\displaystyle\Bigg{[}\frac{1}{2}S^{(1)}_{s_{1}^{\prime},s_{1}}S^{(1)}_{s_{2}^{\prime},s_{2}}-2\pi{i}T^{(1,2,C)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega_{s_{1}}+\omega_{s_{2}})$
$\displaystyle\times$
$\displaystyle\delta(\omega_{s_{1}^{\prime}}+\omega_{s_{2}^{\prime}}-\omega_{s_{1}}-\omega_{s_{2}})\Bigg{]}a_{s_{1}^{\prime}}^{\dagger}a_{s_{2}^{\prime}}^{\dagger}a_{s_{2}}a_{s_{1}}.$
(45)
The physical meaning of the above expression is rather clear. The first term,
being a product of one-particle $S$-matrices, describes the coherent
scattering of two particles, i.e. a scattering process in which there is no
effective interactions between the photons. The second term, on the other
hand, is completely connected and describes the incoherent scattering of
photons in which the particles redistribute their initial energy by
interaction. A slightly more involved, but otherwise completely analogous,
calculation may be done in the three-photon sector. As a result, one arrives
at the following cluster decomposition of the three-body $S$-matrix:
$\displaystyle\mathcal{S}_{3}=$
$\displaystyle\Bigg{[}\frac{1}{6}S^{(1)}_{s_{1}^{\prime},s_{1}}S^{(1)}_{s_{2}^{\prime},s_{2}}S^{(1)}_{s_{3}^{\prime},s_{3}}-2\pi{i}T^{(1,2,C)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega_{s_{1}}+\omega_{s_{2}})$
$\displaystyle\times$
$\displaystyle\delta(\omega_{s_{1}^{\prime}}+\omega_{s_{2}^{\prime}}-\omega_{s_{1}}-\omega_{s_{2}})S^{(1)}_{s_{3}^{\prime},s_{3}}$
$\displaystyle-$ $\displaystyle
2\pi{i}T^{(1,3,C)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\omega_{s_{1}}+\omega_{s_{2}}+\omega_{s_{3}})$
$\displaystyle\times$
$\displaystyle\delta(\omega_{s_{1}^{\prime}}+\omega_{s_{2}^{\prime}}+\omega_{s_{3}^{\prime}}-\omega_{s_{1}}-\omega_{s_{2}}-\omega_{s_{3}})\Bigg{]}$
$\displaystyle\times$
$\displaystyle{a}_{s_{1}^{\prime}}^{\dagger}a_{s_{2}^{\prime}}^{\dagger}a_{s_{3}^{\prime}}^{\dagger}a_{s_{3}}a_{s_{2}}a_{s_{1}},$
(46)
where the connected part of the three-photon $T$-matrix may be found in the
Appendix A.
### II.5 Closed form solution in the Markovian limit
Figure 2: Exact diagrammatic representation of the $n$-particle $T$-matrix in
the Markovian limit. The above diagrams also correspond to the non-crossing
approximation discussed in section II.6. As opposed to the exact $n$-photon
$T$-matrix parameterized by the $n-1$-particle effective vertex function
resuming an infinite number of emission and absorption processes in both
direct and exchange channels, this approximation replaces the vertex by a
sequence of $n-2$ and $n-1$ ($n\geq 2$) alternating excitations and
deexcitations of the emitter.
Let us now consider the single-qubit scattering problem in the Markovian
limit. Validity of Markovian approximation demands a number of assumptions:
linear dispersion relation in all of the channels
$\omega_{\mu}(k)=\omega_{0\mu}+v_{\mu}k$, broadband limit
$B_{\mu}=\mathbb{R},\forall\mu\in\\{1,...,N_{c}\\}$ and local couplings (i.e.
independent of frequency) $g_{\mu}(k)=\sqrt{\gamma_{\mu}/\pi}$. Within the
above assumptions, the self-energy diagram reads as
$\Sigma^{(1)}(\epsilon)=-i\sum_{\mu=1}^{N_{c}}\gamma_{\mu}/v_{\mu}\ket{1}\bra{1}$,
which is independent of $\epsilon$ (here we have introduced the projector on
the qubit’s excited state $\ket{1}\bra{1}=\frac{1+\sigma_{3}}{2}$). By
causality, all of the $W$ functions are analytic in momentum (energy)
variables in the lower (upper) half of the complex plane. By exploiting this
analyticity along with the momentum independence of both coupling constants
and self-energy, we can close all of the integration contours in the lower
half of the complex momentum plane to render all of the integrals zero, thus
promoting the integral equations into algebraic ones. As a result, we recover
the solution obtained in [51]. Diagrammatically the above solution is
equivalent to the so-called non-crossing approximation (which becomes exact in
the Markovian limit) and it is represented in Figure 2.
### II.6 Approximation strategies
Figure 3: Diagrammatic representation of $n$-particle transition matrix within
the weak correlation approximation. Here two-particle bubbles correspond to a
single-photon effective vertex functions $W^{(1,1)}$. Opposite to the quasi-
Markovian approximation, this approximation resumms exactly all of the single
particle processes, ignoring all the connected diagrams corresponding to
multiple-particle scattering.
Although it is possible to write an entire hierarchy of exact equations for
the multi-photon vertex functions, which in turn parametrize the exact
$S$-matrix, their analytical solution is in general only possible in the
Markovian limit. To make some progress in solving the scattering problem, one
has to resort to some kind of approximation routine. In this section we
propose a couple of physically motivated resummation approaches, allowing one
to avoid (or partially avoid) the solution of the exact integral equations.
The most basic approximation, which is accurate in the regime
$\gamma\tau\lesssim 1$, where $\tau$ is a typical time delay in the system due
to photons’ propagation between two nearby scatterers, and $\gamma$ is a
typical decay rate of scatterers, is the so called quasi-Markovian
approximation. While in the quantum master equation approach it has a long
tradition (see, e.g., [22, 23]), it has been recently realised [51, 53] that
in the diagrammatic approach it is mathematically equivalent to picking only
non-crossing diagrams shown in Fig. 2 and thereby fully neglecting vertex
corrections. To give a physical explanation of this equivalence, we view
photons propagating in the waveguide as an effective reservoir. It is
intuitively clear that a fast decay of time correlations between these photons
is an essential feature of the Markovian regime (we add here the prefix quasi
in order to indicate that the field values at positions of different
scatterers can still differ from each other by a phase factor). On the other
hand, these time correlations result from correlated virtual processes of
emission and absorption of different photons at different scatterers, which
are mathematically encoded in the vertex corrections. Thus, the neglect of
vertex corrections is equivalent to making the quasi-Markovian approximation.
It is also worth noting that this approximation for a single-photon scattering
matrix coincides with its exact expression, since in the absence of other
photons the correlations in questions are not generated.
Below we give a step-by-step prescription how to implement the quasi-Markovian
approximation in our diagrammatic approach. The exact $n$-photon transition
matrices are represented by a $n-1$-particle effective vertex function in
between two dressed Green’s functions followed by emission/absorption vertices
(see equation (34)). In this approximation, the exact multi-photon vertex
functions are simply replaced with a sequence of $n-1$ bare and $n-2$ dressed
Green’s functions interspersed with $2n-2$ emission and absorption vertices.
In particular, in the $n=2$ case we get
$W^{(1,1)}_{s_{1}^{\prime},s_{1}}(\epsilon)\rightarrow
v_{s_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{1}})v_{s_{1}^{\prime}}$.
In the $n=3$ case we obtain
$W^{(1,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)\rightarrow
v_{s_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}}-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}^{\prime}})v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}}-\omega_{s_{2}^{\prime}})v_{s_{2^{\prime}}}$,
and so on. In this approximation, all of the non-trivial momentum dependence
of the $S$-matrix, comes entirely from the frequency dependence of the self-
energy diagram. Note that the single-photon $T$-matrix is completely
determined by the dressed propagator $G^{(1)}(\epsilon)$ (see equation (33)).
This observation explicitly confirms the statement that the quasi-Markovian
approximation becomes exact for a single propagating photon.
The second approximation routine, which is more accurate than the quasi-
Markovian one for $\gamma\tau\gtrsim 1$, makes a partial account of the vertex
corrections. In particular, this approximation is based on the resummation of
the full single-photon vertex functions $W^{(1,1)}_{s^{\prime},s}(\epsilon)$.
Diagrammatically, this approximation amounts to the replacement of the dotted
lines in Figure 2 by the full single-particle bubbles, the resulting
$n$-photon $T$-matrix is shown in Figure 3. This approximation corresponds to
resummation of all of the direct interaction diagrams (comping entirely from
the single particle sector), completely ignoring the exchange processes
(encompassed by the connected parts of many-particle vertices). In order to
clarify the matters, let us consider an integral equation governing the two-
particle effective vertex shown in Figure 1. Using the diagrammatic rules we
deduce
$\displaystyle W^{(1,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1},s_{2}}(\epsilon)$
$\displaystyle=R^{(1)}_{s_{1}^{\prime},s_{1}}(\epsilon-\omega_{s_{2}^{\prime}})G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}^{\prime}})W^{(1,1)}_{s_{2}^{\prime},s_{2}}(\epsilon-\omega_{1})$
$\displaystyle+R^{(1)}_{s_{1},s}(\epsilon-\omega_{s_{2}^{\prime}})G^{(1)}(\epsilon-\omega_{s}-\omega_{s_{2}^{\prime}})W^{(1,2)}_{ss_{2}^{\prime},s_{1}s_{2}}$
$\displaystyle+R^{(1)}_{s_{1},s}(\epsilon-\omega_{s_{2}^{\prime}})G^{(1)}(\epsilon-\omega_{s}-\omega_{s_{2}^{\prime}})W^{(1,2)}_{s_{2}^{\prime}s,s_{1}s_{2}}.$
(47)
Note that the last line above corresponds to the exchange interaction between
that particles (see also Figure 1). If we ignore the the last term completely
and take into account the equation satisfied by $W^{(1,1)}$ it is easy to show
that
$\displaystyle W^{(1,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1},s_{2}}(\epsilon)$
$\displaystyle=W^{(1,1)}_{s_{1}^{\prime},s_{1}}(\epsilon-\omega_{s_{2}^{\prime}})G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}^{\prime}})W^{(1,1)}_{s_{2}^{\prime},s_{2}}(\epsilon-\omega_{1})$
(48)
is an exact solution of (47).
We expect this approximation to get worse as one increases the number of
photons in the system since this approximation ignores larger and larger
fraction of diagrams with the increase in the number of particles involved in
a scattering process. This approximation is beneficial since an integral
equation for $W^{(1,1)}$ may be frequently solved analytically by means of the
method developed in the supplementary material of [71]. Due to the non-trivial
momentum dependence of $W^{(1,1)}$, we expect this approximation to be better
than a quasi-Markovian one. In what follows we refer to this approximation as
to the weak correlation one. Note that as the quasi-Markovian approximation is
exact in the single-particle sector, the weak correlation approximation is
exact in the two-particle sector, thus in order to test its validity, one has
to consider at least a three-photon scattering problem.
### II.7 Systems with more than one qubit
In this section, we would like to present certain generalizations of the
theory developed in Section II.3. Specifically, we would like to demonstrate
how the above-presented formalism may be extended to the systems with a larger
number of qubits. In particular, we are going to focus our attention on the
systems with two and three emitters. By deriving the equations governing two
and three-photon scattering matrices in two and three-qubit waveguide QED
systems respectively we present the most general two and three-particle
equations, holding independently of the number of qubits (see the discussion
in Section II.2).
#### II.7.1 $N_{q}=2$ waveguide QED
Figure 4: Diagrammatic representation of the replacement required to obtain
$R^{(2)}_{s^{\prime},s}$ from $R^{(1)}_{s^{\prime},s}$. Note that the extra
contribution to the effective potential energy vertex $R^{(2)}$ in two-qubit
systems is not single-particle connected, i.e. it is possible to cut the
intermediate propagator such that the diagram falls into two distinct pieces.
This very fact makes the splitting (56)-(59) into reducible and irreducible
contributions possible.
First, we would like to focus on the case of two emitters coupled to a
waveguide since the generalization of the single-qubit results is the most
apparent in this case. The starting point in the analysis are equations (17)
and (18). As opposed to the single-emitter case, now, clearly, both diagonal
terms
$v_{s^{\prime}}^{\dagger}a_{s^{\prime}}\mathcal{G}_{0}a^{\dagger}_{s}v_{s}$
and
$a^{\dagger}_{s^{\prime}}v_{s^{\prime}}\mathcal{G}_{0}v_{s}^{\dagger}a_{s}$
contribute to the geometric series for the transition operator. Non-diagonal
ones still give zero, since they both annihilate the state
$v^{\dagger}\ket{g}$. By resumming the series and introducing the following
set of objects
$\displaystyle\mathcal{R}^{(2)}=$ $\displaystyle
a_{s_{1}^{\prime}}^{\dagger}\mathcal{R}^{(2)}_{s_{1}^{\prime},s_{1}}a_{s_{1}},$
(49) $\displaystyle\mathcal{R}^{(2)}_{s_{1}^{\prime},s_{1}}=$
$\displaystyle\mathcal{R}^{(1)}_{s_{1}^{\prime},s_{1}}+v_{s_{1}^{\prime}}\mathcal{G}_{0}v_{s_{1}}^{\dagger},$
(50) $\displaystyle\mathcal{W}^{(2)}=$
$\displaystyle\mathcal{R}^{(2)}+\mathcal{R}^{(2)}\mathcal{G}^{(1)}\mathcal{W}^{(2)},$
(51)
we render the matrix elements of the transition operator in the following form
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(2)}}{N_{p},g}$
$\displaystyle=\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\mathcal{G}^{(1)}v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g}$
$\displaystyle+\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\mathcal{G}^{(1)}\mathcal{W}^{(2)}\mathcal{G}^{(1)}v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g}.$
(52)
Since the lowest term in the expansion of $\mathcal{W}^{(2)}$ is again a
single-photon operator
$\displaystyle\mathcal{W}^{(2)}=\sum_{n=1}^{\infty}a^{\dagger}_{s_{1}^{\prime}}...a^{\dagger}_{s_{n}^{\prime}}\mathcal{W}^{(2,n)}_{s_{1}^{\prime}...s_{n}^{\prime},s_{1}...s_{n}}a_{s_{n}}...a_{s_{1}},$
(53)
we see that the hierarchy of equations satisfied by the two-photon vertex
functions is precisely the same as (29)-(30) with $R^{(1)}$ being replaced by
$R^{(2)}$ (see Figure 4).
In order to understand the difference between the single-qubit and the two-
qubit theories, let us have a closer look at the equation defining
$\mathcal{W}^{(2)}$
$\displaystyle\mathcal{W}^{(2)}=$
$\displaystyle\mathcal{R}^{(1)}+a_{s_{1}^{\prime}}^{\dagger}v_{s_{1}^{\prime}}\mathcal{G}_{0}v_{s_{1}}^{\dagger}a_{s_{1}}+\mathcal{R}^{(1)}\mathcal{G}^{(1)}\mathcal{W}^{(2)}$
$\displaystyle+$ $\displaystyle
a_{s_{1}^{\prime}}^{\dagger}v_{s_{1}^{\prime}}\mathcal{G}_{0}v_{s_{1}}^{\dagger}a_{s_{1}}\mathcal{G}^{(1)}\mathcal{W}^{(2)}.$
(54)
Now let us perform the following separation
$\mathcal{W}^{(2)}=\mathcal{W}^{(2,i)}+\mathcal{W}^{(2,r)}$, where the
superscripts $i$ and $r$ stand for the irreducible and reducible contributions
respectively, and $\mathcal{W}^{(2,i)}$ is chosen to satisfy the following
equation (26). This leads one to the following equation satisfied by the
reducible part
$\displaystyle\mathcal{W}^{(2,r)}=$ $\displaystyle
a_{s_{1}^{\prime}}^{\dagger}v_{s_{1}^{\prime}}\mathcal{G}_{0}v_{s_{1}}^{\dagger}a_{s_{1}}+\mathcal{R}^{(1)}\mathcal{G}^{(1)}\mathcal{W}^{(2,r)}$
$\displaystyle+$ $\displaystyle
a_{s_{1}^{\prime}}^{\dagger}v_{s_{1}^{\prime}}\mathcal{G}_{0}v_{s_{1}}^{\dagger}a_{s_{1}}\mathcal{G}^{(1)}\mathcal{W}^{(2,i)}$
$\displaystyle+$ $\displaystyle
a_{s_{1}^{\prime}}^{\dagger}v_{s_{1}^{\prime}}\mathcal{G}_{0}v_{s_{1}}^{\dagger}a_{s_{1}}\mathcal{G}^{(1)}\mathcal{W}^{(2,r)}.$
(55)
The solution of equation (55) may then conveniently parametrized as
$W^{(2,r)}(\epsilon)=\mathcal{V}^{(2)}(\epsilon)\mathcal{G}^{(2)}(\epsilon)\overline{\mathcal{V}}^{(2)}(\epsilon)$,
where
$\displaystyle\mathcal{G}^{(2)}=$
$\displaystyle\mathcal{G}_{0}+\mathcal{G}_{0}\Sigma^{(2)}\mathcal{G}^{(2)},$
(56) $\displaystyle\Sigma^{(2)}=$ $\displaystyle
a_{s}v_{s}^{\dagger}\mathcal{G}^{(1)}v_{s^{\prime}}a^{\dagger}_{s^{\prime}}+a_{s}v_{s}^{\dagger}\mathcal{G}^{(1)}\mathcal{W}^{(2,i)}\mathcal{G}^{(1)}v_{s^{\prime}}a^{\dagger}_{s^{\prime}},$
(57) $\displaystyle\overline{\mathcal{V}}^{(2)}=$ $\displaystyle
v_{s}^{\dagger}a_{s}+v_{s}^{\dagger}a_{s}\mathcal{G}\mathcal{W}^{(2,i)}=:\sum_{n=1}^{\infty}\overline{V}^{(2,n)}_{s_{1}...s_{n}}a_{s_{1}}...a_{s_{n}},$
(58) $\displaystyle\mathcal{V}^{(2)}=$ $\displaystyle
a_{s}^{\dagger}v_{s}+\mathcal{W}^{(2,i)}\mathcal{G}v_{s}a_{s}^{\dagger}=:\sum_{n=1}^{\infty}{V}^{(2,n)}_{s_{1}...s_{n}}a_{s_{1}}^{\dagger}...a_{s_{n}}^{\dagger}.$
(59)
The meaning of the above-defined objects is as follows.
$\mathcal{G}^{(2)}(\epsilon)$ may be thought of as a Green’s operator in the
two-qubit excitation subspace since in practice, it is always projected there.
$\Sigma^{(2)}(\epsilon)$ in its turn may be thought of as a self-energy
operator in the two-qubit excitation subspace. $\mathcal{V}(\epsilon)$ and
$\overline{\mathcal{V}}(\epsilon)$ may be understood as the renormalized
absorption and emission vertex operators.
As it was anticipated above, the equations governing a two-photon scattering
on a pair of qubits hold in the case of a two-photon scattering for a general
$N_{q}$ system. Bearing this in mind, in Figure 5 we present the system of
exact integral equations governing the general two-photon scattering problem
in waveguide QED (i.e. equations defining $W^{(2,1)}$). These equations were
first obtained in [71] in relation with the two-photon scattering problem on
two distant qubits.
#### II.7.2 $N_{q}=3$ waveguide QED
Let us finally dedicate our attention to three-qubit waveguide QED systems.
Due to the enormous increase in the computational complexity, we restrict
ourselves to the first non-trivial subspace within such a setup - a three-
excitation subspace. As before, the starting point of the analysis is the
equation (17). No doubt, both of the ”diagonal” terms $\sim v^{\dagger}v$ and
$\sim vv^{\dagger}$ give a non-zero contribution to the transition operator as
it was the case in the previous section, however, one can clearly see that now
the ”off-diagonal” terms ($vv,\ v^{\dagger}v^{\dagger}$) have to be also taken
into account. By defining the following operators
$\displaystyle\mathcal{D}_{+}=a^{\dagger}_{s^{\prime}}v_{s^{\prime}}\mathcal{G}_{0}a^{\dagger}_{s}v_{s},\quad\mathcal{D}_{-}=v_{s^{\prime}}^{\dagger}a_{s^{\prime}}\mathcal{G}_{0}v_{s}^{\dagger}a_{s},$
(60)
Figure 5: Equations corresponding to the general two-photon scattering
problem. Equation in the first line describes the splitting of the effective
single-particle vertex function $W^{(2,1)}$ in the generic two-photon
scattering problem into its reducible $W^{(2,1,r)}$ and irreducible parts
$W^{(2,1,i)}$. The irreducible part is generated from the fundamental process
of the single-qubit waveguide QED $R^{(1)}$ (as shown in the second line) and
thus satisfies the same integral equation as $W^{(1,1)}$. The reducible part,
steming from non-single-particle connectedness of the additional contribution
to $R^{(2)}$ (see Figure 4), is parametrized by the dressed Green’s function
$G^{(2,0)}$ in two excitation subspace (shown as the curly line) as well as
renormalized emission and absorption vertices (indicated by rectangles). As
one can see the entire system of equations blows down to the solution of the
equation defining the irreducible part of the vertex function, since its
knowledge is sufficient to completely determine both the renormalized vertices
$V^{(2,1)},\ \overline{V}^{(2,1)}$ as well as the self energy $\Sigma^{(2,0)}$
in the two excitation subspace.
bearing in mind the rotating wave approximation, and resuming the series (17)
to incorporate the diagonal terms and again making use of RWA, in the realm of
$N_{q}=3$ waveguide QED, we obtain
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(3)}}{N_{p},g}$
$\displaystyle=\Braket{N_{p}^{\prime},g}{a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\frac{1}{(\mathcal{G}^{(1)})^{-1}-\mathcal{R}^{(3)}}v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g},$
(61) $\displaystyle\mathcal{R}^{(3)}=\mathcal{R}^{(2)}+\mathcal{D},$ (62)
$\displaystyle\mathcal{D}=\mathcal{D}_{+}\frac{1}{(\mathcal{G}^{(1)})^{-1}-\mathcal{R}^{(2)}}\mathcal{D}_{-}$
$\displaystyle=\mathcal{R}^{(2)}+\mathcal{D}_{+}\mathcal{G}^{(1)}\mathcal{D}_{-}+\mathcal{D}_{+}\mathcal{G}^{(1)}\mathcal{W}^{(2)}\mathcal{G}^{(1)}\mathcal{D}_{-}.$
(63)
As usual, we define the $\mathcal{W}^{(3)}$ operator according to
$\displaystyle\mathcal{W}^{(3)}=\mathcal{D}+\mathcal{D}(\mathcal{G}^{(1)}+\mathcal{G}^{(1)}\mathcal{W}^{(2)}\mathcal{G}^{(1)})\mathcal{W}^{(3)}.$
(64)
Which brings the transition operator to the following form
$\displaystyle\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(3)}}{N_{p},g}=\Braket{N_{p}^{\prime},g}{\mathcal{T}^{(2)}+a^{\dagger}_{s_{1}^{\prime}}v_{s_{1}^{\prime}}\mathcal{G}^{(1)}(1+\mathcal{W}^{(2)}\mathcal{G}^{(1)})\mathcal{W}^{(3)}(\mathcal{G}^{(1)}\mathcal{W}^{(2)}+1)\mathcal{G}^{(1)}v^{\dagger}_{s_{1}}a_{s_{1}}}{N_{p},g}.$
(65)
Upon the projection, onto the three-photon subspace, we obtain the following
elegant set of equations governing the generic (see Section II.2) three-
particle $T$-matrix (see Appendix C for the detailed derivation)
$\displaystyle
T^{(3,3)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\epsilon)=$
$\displaystyle\Braket{g}{v_{s_{1}^{\prime}}{G}^{(1)}(\epsilon-\omega_{s_{2}^{\prime}}-\omega_{s_{3}^{\prime}})[W^{(2,2)}_{s_{2}^{\prime}s_{3}^{\prime},s_{2}s_{3}}(\epsilon)+V^{(3,2)}_{s_{2}^{\prime}s_{3}^{\prime}}(\epsilon)G^{(3,0)}(\epsilon)\overline{V}^{(3,2)}_{s_{2}s_{3}}(\epsilon)]{G}^{(1)}(\epsilon-\omega_{s_{2}}-\omega_{s_{3}})v^{\dagger}_{s_{1}}}{g}$
$\displaystyle=:$
$\displaystyle\Braket{g}{v_{s_{1}^{\prime}}{G}^{(1)}(\epsilon-\omega_{s_{2}^{\prime}}-\omega_{s_{3}^{\prime}})[W^{(2,2)}_{s_{2}^{\prime}s_{3}^{\prime},s_{2}s_{3}}(\epsilon)+W^{(3,2)}_{s_{2}^{\prime}s_{3}^{\prime},s_{2}s_{3}}(\epsilon)]{G}^{(1)}(\epsilon-\omega_{s_{2}}-\omega_{s_{3}})v^{\dagger}_{s_{1}}}{g},$
(66) $\displaystyle G^{(3,0)}(\epsilon)=$ $\displaystyle
G^{(1)}(\epsilon)+G^{(1)}(\epsilon)\Sigma^{(3,0)}(\epsilon)G^{(3,0)}(\epsilon),$
(67) $\displaystyle\Sigma^{(3,0)}(\epsilon)=$
$\displaystyle(v_{s}^{\dagger}G_{0}(\epsilon-\omega_{s})v_{s^{\prime}}^{\dagger}+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s}^{\dagger})G^{(1)}(\epsilon-\omega_{s}-\omega_{s^{\prime}})V^{(3,2)}_{ss^{\prime}}(\epsilon)$
(68) $\displaystyle\equiv$
$\displaystyle\overline{V}^{(3,2)}_{ss^{\prime}}(\epsilon)G^{(1)}(\epsilon-\omega_{s}-\omega_{s^{\prime}})(v_{s^{\prime}}G_{0}(\epsilon-\omega_{s})v_{s}+v_{s}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s^{\prime}}),$
(69) $\displaystyle\overline{V}^{(3,2)}_{s_{1}s_{2}}(\epsilon)=$
$\displaystyle
v^{\dagger}_{s_{1}}G^{(2,0)}(\epsilon)\overline{V}^{(2,1)}_{s_{2}}(\epsilon)+\overline{V}^{(3,2)}_{ss_{1}}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W^{(2,1)}_{s,s_{2}}(\epsilon),$
(70) $\displaystyle V^{(3,2)}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)=$
$\displaystyle{V}^{(2,1)}_{s_{1}^{\prime}}(\epsilon)G^{(2,0)}(\epsilon)v^{\dagger}_{s_{2}^{\prime}}+W^{(2,1)}_{s_{1}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s}){V}^{(3,2)}_{s_{2}^{\prime}s}(\epsilon).$
(71)
Here $G^{(2,0)}(\epsilon)$ is the projection of (56) onto the photon vacuum
state. As usual, the above equation may be compactly represented
diagrammatically as shown in Figure 6.
Figure 6: Equations corresponding to the general three-photon scattering
problem. Here the first line defines the effective two-body vertex function
$W^{(3,2)}$ in terms of the dressed Green’s function in three excitation
subspace shown as a double dashed line as well as effective two-photon
absorption/emission vertices depicted by square boxes with a pair of
incoming/outgoing amputated photon legs (note that these vertices correspond
to the absorption/emission of photon pairs by a system as a whole). As it is
shown in the last five lines the effective two-photon emission/absorption
vertices as well as the self-energy bubble defining the Green’s function in
the three-excitation subspace satisfy a closed hierarchy of self-consistent
equations, the validity of which is directly proven in Appendix C. Curly lines
as well as the rectangular boxes with single amputated photon lines are
precisely defined in Figure 5.
## III Application: a giant acoustic atom
In this section, we would like to consider a practical application of the
above-developed theory. In order to showcase our method in full glory, we
would like to focus on the non-Markovian scattering setup - a giant acoustic
atom, extensively studied theoretically and experimentally in [19, 59, 78, 79,
80, 60, 81] (see also [82] for a detailed review).
### III.1 Model and conventions
A giant acoustic atom may be defined as a two-level system coupled to an
acoustical waveguide, in which the radiation is carried by the surface
acoustic waves (SAW), at two distant points $x=\pm R/2$. The Hamiltonian of
such a system may be written as $\mathcal{H}=\mathcal{H_{0}}+\mathcal{V}$
$\displaystyle\mathcal{H}_{0}$
$\displaystyle=\Omega\sigma_{+}\sigma_{-}-iv\sum_{\mu=1,2}\int{dx}c_{\mu}b^{\dagger}_{\mu}(x)\partial_{x}b_{\mu}(x),$
(72) $\displaystyle\mathcal{V}$
$\displaystyle=\sum_{\mu=1,2}(\sqrt{\Gamma_{1}}b^{\dagger}_{\mu}(-R/2)+\sqrt{\Gamma_{2}}b^{\dagger}_{\mu}(R/2))\sigma_{-}+\text{h.c.}$
(73)
Here $b_{\mu}(x),b_{\mu}^{\dagger}(x)$ are the position space field operators
of phonons, the index $\mu$ distinguishes beetween the right $\mu=1$ and left
$\mu=2$ mooving fields, and $c_{\mu}=(-1)^{\mu-1}$. Note that we have made use
of the common assumption of the mode dispersion being linear in the vicinity
of the relevant energy scale $\sim\Omega$ and the the bandwidth be infinite.
Defining the Fourier transformation as
$\displaystyle
b_{\mu}(x)=\frac{1}{\sqrt{2\pi}}\int{dk}e^{ic_{\mu}kx}a_{\mu}(k),$ (74)
we bring the Hamiltonian in to the form (1), (2):
$\displaystyle\mathcal{H}_{0}$
$\displaystyle=\Omega\sigma_{+}\sigma_{-}+v\sum_{\mu=1,2}\int{dk}ka^{\dagger}_{\mu}(k)a_{\mu}(k),$
(75) $\displaystyle\mathcal{V}$
$\displaystyle=\sum_{\mu=1,2}\int{dk}{g}_{\mu}(k)a_{\mu}(k)\sigma_{-}+\text{h.c.},$
(76) $\displaystyle{g}_{\mu}(k)$
$\displaystyle=\sqrt{\frac{\Gamma_{1}}{2\pi}}e^{-ic_{\mu}kR/2}+\sqrt{\frac{\Gamma_{1}}{2\pi}}e^{ic_{\mu}kR/2}.$
(77)
In the following subsections we are going to study the scattering of a
coherent pulse in the form of a wavepacket centered around the frequency
$vk_{0}$ (see Section III.2), for that sake it is convenient to perform the
following time-dependant gauge transformation:
$\displaystyle\mathcal{U}(t)=\exp\Bigg{(}-ivk_{0}\Bigg{[}\int{dk}a^{\dagger}_{\mu}(k)a_{\mu}(k)+\sigma_{+}\sigma_{-}\Bigg{]}t\Bigg{)}.$
(78)
The Hamiltonian transforms as
$\mathcal{H}\rightarrow\tilde{\mathcal{H}}=\mathcal{U}^{\dagger}(t)\mathcal{H}\mathcal{U}(t)-i\mathcal{U}^{\dagger}(t)\frac{d\mathcal{U}(t)}{dt}=:\tilde{\mathcal{H}}_{0}+\tilde{\mathcal{V}}$,
resulting in final form of the Hamiltonian we are going to work with:
$\displaystyle\tilde{\mathcal{H}}_{0}$
$\displaystyle=-\Delta\sigma_{+}\sigma_{-}+v\sum_{\mu=1,2}\int{dk}k\tilde{a}^{\dagger}_{\mu}(k)\tilde{a}_{\mu}(k),$
(79) $\displaystyle\tilde{\mathcal{V}}$
$\displaystyle=\sum_{\mu=1,2}\int{dk}\tilde{g}_{\mu}(k)\tilde{a}_{\mu}(k)\sigma_{-}+\text{h.c.},$
(80) $\displaystyle\tilde{g}_{\mu}(k)$
$\displaystyle=\sqrt{\frac{\Gamma_{1}}{2\pi}}e^{-ic_{\mu}(k+k_{0})R/2}+\sqrt{\frac{\Gamma_{1}}{2\pi}}e^{ic_{\mu}(k+k_{0})R/2},$
(81) $\displaystyle\tilde{a}_{\mu}(k)$
$\displaystyle=a_{\mu}(k+k_{0}),\quad\Delta=\omega_{0}-\Omega,\quad\omega_{0}=vk_{0}.$
(82)
In the following we adopt the system of units such that $v=1$, for simplicity
we will also assume $\Gamma_{1}=\Gamma_{2}=\gamma/2$. Furthermore, we shall
also drop the tilde symbols out of operators for notational convenience.
### III.2 Problem formulation
In order to formulate the scattering problem, we define the following wave-
packet operators
$\displaystyle A_{\mu}^{\dagger}=\int{dk}\varphi_{L}(k)a_{\mu}^{\dagger}(k),$
(83)
where $\varphi_{L}(k)=\sqrt{\frac{2}{\pi L}}\frac{\sin(kL/2)}{k}$ is the
Fourier transform of the rectangular pulse of length $L$, with the property
$\varphi_{L}(k)\sim\sqrt{\frac{2\pi}{L}}\delta(k),\ L\rightarrow\infty$
(obviously, it is possible to choose any other nascent $\delta$-function for
$\varphi_{L}(k)$). Note that since we are working in the frame of reference
rotating with frequency $k_{0}$, peaking of $\varphi_{L}(k)$ at $k=0$
corresponds to a pulse centered at $k=k_{0}$ in the original one. Note that
throughout this section we focus on the zero detuning setup $\Delta=0$, i.e.
the initial $k_{0}$ is chosen to be equal to $\Omega$. The effect of non-zero
detuning of atom from radiation is studied in Appendix D. Definition of these
wave-packet operators is crucial since the eigenstates of the bare Hamiltonian
$\mathcal{H}_{0}$ are not normalizable. Hence, in order to avoid the ascent of
the undefinable quantities such as $\delta(0),(\delta(0))^{2},...$ coming from
the elastic clusters of the $S$-matrix in the calculation of the observables,
one has to work with the normalizable states and take the plane-wave limit
$L\rightarrow\infty$ at the very end. Having defined the Fock creation
operators $A^{\dagger}_{\mu}$, we can define the coherent state as a displaced
vacuum
$\displaystyle\ket{\alpha}_{\mu}=e^{-|\alpha|^{2}/2}e^{\alpha
A^{\dagger}_{\mu}}\ket{\Omega}=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\ket{\Phi^{(n)}_{\mu}},$
(84)
where $\alpha\in\mathbb{C}$ is the so-called coherence parameter, defined such
that $|\alpha|^{2}$ is the average photon number (power), and the normalized
$n$-particle Fock states $\ket{\Phi^{(n)}_{\mu}}$ were defined according to
$\ket{\Phi^{(n)}_{\mu}}=\frac{1}{\sqrt{n!}}\Big{(}A^{\dagger}_{\mu}\Big{)}^{n}\ket{\Omega}.$
(85)
In what follows, we assume that $|\alpha|\ll 1$ in such a way that the terms
of order $|\alpha|^{4}$ are negligible
$\displaystyle\ket{\alpha}_{\mu}\approx$
$\displaystyle{e}^{-|\alpha|^{2}/2}\Bigg{(}\ket{\Omega}+\frac{\alpha}{\sqrt{1!}}\ket{\Phi^{(1)}_{\mu}}+\frac{\alpha^{2}}{\sqrt{2!}}\ket{\Phi^{(2)}_{\mu}}$
$\displaystyle+$
$\displaystyle\frac{\alpha^{3}}{\sqrt{3!}}\ket{\Phi^{(3)}_{\mu}}+\mathcal{O}(|\alpha|^{4})\Bigg{)}\equiv\ket{\alpha}_{\mu}^{(3)}.$
(86)
The factor $e^{-|\alpha|^{2}/2}$ has to be retained until various overlaps of
states are calculated, to ensure the normalization of both the initial and
final states as well as the power conservation (here the normalization is
again assumed in the power perturbative regime, that is up to order
$\mathcal{O}(|\alpha|^{8})$). With this in hands, we formulate the scattering
problem as follows. We assume that the initial state of the system is given by
$\ket{\psi_{i}}=\ket{\alpha}_{1}^{(3)}\otimes\ket{0}$, that is a leftwards-
propagating weakly coherent pulse $\ket{\alpha}_{1}^{(3)}$ is incident on a
two-level system which is initially in its ground state $\ket{0}$. According
to the theory developed above, the final state of the systems has the
following form
$\displaystyle\ket{\psi_{f}}=$
$\displaystyle{e}^{-|\alpha|^{2}/2}\Bigg{(}\ket{\Omega}+\frac{\alpha}{\sqrt{1!}}{\mathcal{S}}_{1}\ket{\Phi^{(1)}_{1}}$
$\displaystyle+$
$\displaystyle\frac{\alpha^{2}}{\sqrt{2!}}{\mathcal{S}}_{2}\ket{\Phi^{(2)}_{1}}+\frac{\alpha^{3}}{\sqrt{3!}}{\mathcal{S}}_{3}\ket{\Phi^{(3)}_{1}}\Bigg{)}\otimes\ket{0}.$
(87)
Inelastic contributions to the $n$-phonon $S$-matrices (discussed in Section
II.4) introduce a non-trivial momentum redistribution of the incident phonons
(note that owing to the linear dispersion relation and our choice of units
energy and momentum may be used interchangeably), leading to the non-trivial
phonon correlations in the final state. As it is well known, the phononic
correlations may be conveniently examined with the help of the so-called
coherence functions introduced by Glauber [83] in 1963. Defining the Fourier
transform of the field operators according to
$\displaystyle a_{\mu}(\tau)=\int{dk}\frac{e^{ik\tau}}{\sqrt{2\pi}}a(k),$ (88)
one constructs the first-order coherence function as
$\displaystyle C_{\mu}^{(1)}(\tau)=$
$\displaystyle\braket{\psi_{f}}{a^{\dagger}_{\mu}(\tau)a_{\mu}(0)}{\psi_{f}}$
$\displaystyle=$
$\displaystyle\Bigg{(}1-|\alpha|^{2}+\frac{|\alpha|^{4}}{2}\Bigg{)}|\alpha|^{2}C^{(1,1)}_{\mu}(\tau)+(1-|\alpha|^{2})\frac{|\alpha|^{4}}{2}C^{(1,2)}_{\mu}(\tau)+\frac{|\alpha|^{6}}{6}C^{(1,3)}_{\mu}(\tau)+\mathcal{O}(|\alpha|^{8}),$
(89)
where we have introduced the $n$-particle Fock state first-order correlation
functions as
$C^{(1,n)}_{\mu}(\tau)=\Braket{\Phi^{(n)}_{1}}{(\mathcal{S}_{n})^{\dagger}a^{\dagger}_{\mu}(\tau)a_{\mu}(0)\mathcal{S}_{n}}{\Phi_{1}^{(n)}}.$
(90)
With the help of the first-order coherence function, one may define the
spectral power density as the Fourier transform of (89):
$S_{\mu}(k)=\int d\tau\frac{e^{-ik\tau}}{2\pi}C^{(1)}_{\mu}(\tau).$ (91)
$S_{\mu}(k)$ is understood as a momentum space distribution of power in the
scattered state of radiation, that is to a given mode $k$ (supported by the
$\mu^{th}$ channel) it associates a certain power $S_{\mu}(k)$. The power
conservation condition
$\sum_{\mu=1,2}\int
dkS_{\mu}(k)\rightarrow\Phi,\quad\Phi=\frac{|\alpha|^{2}}{L},$ (92)
is automatically satisfied due to the unitarity of the $S$-matrix. In a linear
system, where the $n$ body $S$-matrix factorises into the product of single-
particle $S$-matrices, the first order coherence function is a constant, thus
leading to the purely elastic power density $S_{\mu}(k)\propto\delta(k)$.
Since a qubit is an intrinsically non-linear system, we are going to see that
the spectral power density admits for the following decomposition
$S(k)=S^{\text{el}}(k)+S^{\text{inel}}(k)$, where
$S^{\text{el}}(k)\propto\delta(k)$ is the elastic contribution to the spectral
density, and $S^{\text{inel}}(k)$, in turn, is the inelastic part of spectral
power density with a non-trivial momentum dependence.
Further, we define the normalized second and third-order coherence functions:
$\displaystyle C^{(2)}_{\mu,\mu^{\prime}}(\tau)=$
$\displaystyle\frac{\braket{\psi_{f}}{a^{\dagger}_{\mu}(0)a^{\dagger}_{\mu^{\prime}}(\tau)a_{\mu^{\prime}}(\tau)a_{\mu}(0)}{\psi_{f}}}{C^{(1)}_{\mu}(0)C^{(1)}_{\mu^{\prime}}(0)}=\Bigg{(}1-\frac{|\alpha|^{2}}{2}\Bigg{)}\frac{1}{2}C^{(2,2)}_{\mu,\mu^{\prime}}(\tau)+\frac{|\alpha|^{2}}{6}C^{(2,3)}_{\mu,\mu^{\prime}}(\tau)+\mathcal{O}(|\alpha|^{4}),$
(93) $\displaystyle C^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\
\tau^{\prime})=$
$\displaystyle\frac{\braket{\psi_{f}}{a^{\dagger}_{\mu}(0)a^{\dagger}_{\mu^{\prime}}(\tau)a^{\dagger}_{\mu^{\prime\prime}}(\tau^{\prime})a_{\mu^{\prime\prime}}(\tau^{\prime})a_{\mu^{\prime}}(\tau)a_{\mu}(0)}{\psi_{f}}}{C^{(1)}_{\mu}(0)C^{(1)}_{\mu^{\prime}}(0)C^{(1)}_{\mu^{\prime\prime}}(0)}=\frac{1}{6}C^{(3,3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\
\tau^{\prime})+\mathcal{O}(|\alpha|^{2}),$ (94) $\displaystyle
C^{(m,n)}_{\mu_{1},...,\mu_{m}}(\tau_{1},..,\tau_{m-1})=$
$\displaystyle\frac{\braket{\Phi^{(n)}_{1}}{{\mathcal{S}}_{n}^{\dagger}a^{\dagger}_{\mu_{1}}(0)...a^{\dagger}_{\mu_{m}}(\tau_{m-1})a_{\mu_{m}}(\tau_{m-1})...a_{\mu_{1}}(0){\mathcal{S}}_{n}}{\Phi^{(n)}_{1}}}{\prod_{l=1}^{m}C^{(1)}_{\mu_{l}}(0)}.$
(95)
For pairs of phonons, the normalized second-order coherence function is
defined as the arrival probability of the second particle as a function of the
delay $\tau$ following the detection of the first one, normalized by the
individual photon probabilities. Likewise, for particle triples, the
normalized third-order coherence function is defined as the arrival
probability of the third and second particle as a function of delays
$\tau^{\prime},\tau$ following the detection of the first one, normalized by
the individual phonon probabilities. A perfectly coherent source is
characterized by a uniform arrival probability, yielding the correlation
functions of all orders equal to unity. Whenever correlation functions exceed
unity, particle statistics is said to be super-Poissonian and the particles
are said to be bunched together, whereas in the case of correlation functions
falling below unity, the statistics of particles is said to be sub-Poissonian
and particles are correspondingly said to be anti-bunched.
In the following subsections, we are going to use the ideas developed in
Section II.3 in order to compute the above-discussed observables to the lowest
order in $|\alpha|$ exactly.
### III.3 Spectral power density
Let us begin with the analysis of the power spectrum of the SAW scattered by
the giant atom. We start by considering the Fock state first-order coherence
functions defined via (90). In order to establish $C^{(1,1)}_{\mu}(\tau)$ one
first has to find the matrix elements of the single-particle $S$-matrix,
which, in turn, demands the knowledge of the dressed propagator in the single-
excitation subspace $G^{(1)}(\epsilon)$. The self-energy diagram may be
evaluated straightforwardly to yield:
$\Sigma^{(1)}(\epsilon)=-i\gamma(1+e^{i(\epsilon+k_{0})R})\sigma_{+}\sigma_{-},$
(96)
in accordance with reference [59]. With this in hands, the single-phonon
scattering operator may be easily established with the help of equations (33),
(37) and (38):
$\displaystyle\mathcal{S}_{1}=$
$\displaystyle\sum_{\mu^{\prime},\mu}\int{dkdk^{\prime}}S^{(1)}_{\mu^{\prime}k^{\prime},\mu
k}a^{\dagger}_{\mu^{\prime}}(k^{\prime})a_{\mu}(k),$ (97) $\displaystyle
S^{(1)}_{\mu^{\prime}k^{\prime},\mu k}=$
$\displaystyle\delta(k-k^{\prime})S^{(1)}_{\mu^{\prime},\mu}(k)\sigma_{-}\sigma_{+}$
(98) $\displaystyle S^{(1)}_{\mu^{\prime},\mu}(k)=$
$\displaystyle(\delta_{\mu,\mu^{\prime}}-2\pi{i}g_{\mu}^{*}(k)g_{\mu^{\prime}}(k)\tilde{G}^{(1)}(k)).$
(99)
Here the tilde symbol denotes the projection onto the excited state in the
qubit space, and
${G}^{(1)}(\epsilon)={G}_{0}^{-1}(\epsilon)-{\Sigma}^{(1)}(\epsilon)$ in
accordance with the definition in Section II.3. Performing a straightforward
calculation, we obtain
$\displaystyle C^{(1)}_{\mu}(\tau)=$
$\displaystyle|\alpha|^{2}C^{(1,1)}_{\mu}(\tau)+\mathcal{O}(|\alpha|^{4})$
$\displaystyle=$ $\displaystyle\Phi|S^{(1)}_{\mu
0,10}|^{2}+\mathcal{O}(\Phi^{2})$ (100) $\displaystyle\implies$ $\displaystyle
S_{\mu}(k)=\Phi|S^{(1)}_{\mu 0,10}|^{2}\delta(k)+\mathcal{O}(\Phi^{2}).$ (101)
As we can see there exists no inelastic contribution to the spectral density
in single-photon sector. In general the phenomenon of resonance fluorescence
[84], leading to the inelastic power spectrum, is underpinned by the
possibility of the two-level system to emit into the modes other than the
incident one. This naturally allows the photons incident on the atom to
exchange their energy between one another (whilst conserving the total energy)
due to the higher order emission and absorption processes, leading to the
inelastic power spectrum. In the case of the single-photon however, the
particle is ought to conserve its energy individually (as mathematically
prescribed by the $\delta$-function in equation (98)) leading to the purely
elastic spectrum.
In order to obtain the leading order inelastic contribution to the spectral
power density, we thus have to consider the contribution of the multi-particle
states in the expansion (87), so that to allow for the inter-particle
interaction. In this case, the information about the inelastic scattering is
entirely contained in the connected component of the two-phonon $T$-matrix
(see Section II.4), which captures the nonlinear acoustic effects via the
effective single-particle vertex function. Contracting the cluster decomposed
$S$-matrix (45) with the two-particle Fock state we arrive at the following
result
$\displaystyle\sum_{\mu,\mu^{\prime}}\Bigg{(}\frac{1}{\sqrt{2}}\int{dkdk^{\prime}}\varphi(k)\varphi(k^{\prime})S^{(1)}_{\mu,1}(k)S^{(1)}_{\mu^{\prime},1}(k^{\prime}){a}_{\mu}^{\dagger}(k)a_{\mu^{\prime}}^{\dagger}(k^{\prime})-\frac{8\pi^{2}i}{L\sqrt{2}}\int{dk}T^{(2,C)}_{\mu
k,\mu^{\prime}-k,10,10}(0){a}_{\mu}^{\dagger}(k)a_{\mu^{\prime}}^{\dagger}(-k)\Bigg{)}\ket{\Omega}\otimes\ket{0}.$
(102)
Again we would like to emphasize that at this point the plane-wave limit
($L\rightarrow\infty$) may only be taken in the term containing the connected
component of the $S$-matrix since in this limit the first part of the state
(102) is clearly non-normalizable. With the help of the final two-particle
state derived above one may easily establish
$\displaystyle C^{(1,2)}_{\mu}(\tau)=$ $\displaystyle
16\pi^{2}\sum_{\mu^{\prime}}\text{Im}\\{M_{\mu^{\prime},\mu}(0)(S_{\mu^{\prime};1}^{(1)}(0)S_{\mu;1}^{(1)}(0))^{*}\\}$
$\displaystyle+$ $\displaystyle
32\pi^{3}\sum_{\mu^{\prime}}\int{dk}e^{ik\tau}|M_{\mu^{\prime},\mu}(k)|^{2},$
(103)
where $M_{\mu^{\prime},\mu}(k)$ is the symmetric part of $T^{(2,C)}_{\mu
k,\mu^{\prime}-k,10,10}(0)$, i.e.
$M_{\mu^{\prime},\mu}(k)=M_{\mu,\mu^{\prime}}(-k)=(T^{(2,C)}_{\mu^{\prime}k,\mu-k,10,10}(0)+T^{(2,C)}_{\mu-k,\mu^{\prime}k,10,10}(0))/2$.
Performing the Fourier transform of (103) we arrive at the following
expressions for the elastic $S^{\text{el}}(k)$ and inelastic
$S^{\text{inel}}(k)$ spectral power densities valid to order $\Phi^{3}$:
$\displaystyle S_{\mu}^{\text{el}}(k)=$
$\displaystyle\delta(k)\Bigg{(}\Phi|S^{(1)}_{\mu;1}(0)|^{2}+16\pi^{2}\Phi^{2}$
$\displaystyle\times$
$\displaystyle\sum_{\mu^{\prime}}\text{Im}\\{M_{\mu^{\prime},\mu}(0)(S_{\mu^{\prime};1}^{(1)}(0)S_{\mu;1}^{(1)}(0))^{*}\\}\Bigg{)},$
(104) $\displaystyle S_{\mu}^{\text{inel}}(k)=$ $\displaystyle
32\pi^{3}\Phi^{2}\sum_{\mu^{\prime}}|M_{\mu^{\prime},\mu}(k)|^{2}.$ (105)
Figure 7: Inelastic spectral power density (scaled by $\Phi^{2}$) of SAW
scattered by a giant acoustic atom as a function of frequency $\omega=k$ for a
variety of inter-leg separations $\gamma R=1,3,5$. Top panel: exact solution;
Bottom panel: quasi-Markovian approximation. Other model parameters:
$k_{0}R=\pi/4\mod 2\pi$, and $\Delta=0$. Here the limit $L\rightarrow\infty$
was taken.
The inelastic spectral densities are shown in Figure 7 for different inter-leg
separations (R = 1, 3, 5). As one can see, the inelastic power spectrum
develops a couple of sharp peaks near the origin of momentum space followed by
infinitely many smaller side peaks. In order to interpret the nature of these
peaks, we adopt the physical picture discussed in [85, 86, 21] for an ”atom in
front of a mirror” system. One can interpret this system as a leaky cavity
formed by the two connection points of a giant atom. In this picture these
peaks may be thought of as being located at the renormalized excitation
frequencies of the effective cavity broadened by the renormalized decay rates.
The resulting effect is the formation of sharp, bound-state like peaks in the
intensity of the scattered phonons, corresponding to cavity resonances. As the
delay $\gamma R$ increases, the peaks get closer to $\omega=0$, and get
sharper, corresponding to a decrease of the effective cavity linewidth $\simeq
1/(\gamma R)$, that is effective cavity resonances approach the resonance of
the two-level system, thus making atomic connection points better and better
mirrors and hence increasing the quality factor of the effective cavity
(quality factor is a common physical measure of both the rate of excitation
damping and the rate of energy loss of an oscillator or a cavity)
Additionally, spectral densities based on the approximate solution of the
scattering problem in the quasi-Markovian approximation are shown in Figure 7.
As it was mentioned in the section II.6, quasi-Markovian results show a good
agreement with an exact solution in $\gamma{R}\ll 1$ parameter regime. As the
delay time increases $\gamma{R}\geq 1$, the discrepancy between the
approximate and complete solutions becomes dramatic. One can clearly see the
tendency of the quasi-Markovian theory to enhance the scattering into the
incoming frequency states $\omega=0$. Indeed, in contrast to the exact
solution, taking into account infinitely many excursions of phonons to the
qubit, the quasi-Markovian approximation is based on the assumption of a
single scattering event after which each phonon immediately leaves the system,
thus leading to a more elastic result.
#### III.3.1 Second-order coherence
Figure 8: The figure demonstrates the independent components of the normalized
second order coherence function $C^{(2)}_{11},\ C^{(2)}_{12},\ C^{(2)}_{22}$
for a various inter-leg separations $\gamma R=1,3,5$. As before the top panel
corresponds to an exact solution, the bottom one to its quasi-Markovian
approximation, and $k_{0}R=\pi/4\mod 2\pi$, $\Delta=0$. Here the limit
$L\rightarrow\infty$ was taken and by definition $C^{(2)}$ is dimensionless.
Let us now consider the second-order coherence function to the lowest order in
$\Phi$ \- $\mathcal{O}(\Phi^{0})$. Performing the relevant Wick contractions
we arrive at the following simple expression for the normalized second-order
coherence
$\displaystyle
C_{\mu^{\prime},\mu}^{(2)}(\tau)=\Bigg{|}1-\frac{4\pi{i}}{S^{(1)}_{\mu^{\prime},1}(0)S^{(1)}_{\mu,1}(0)}\int{dk}e^{ik\tau}M_{\mu^{\prime},\mu}(k)\Bigg{|}^{2}.$
(106)
The independent components of second-order coherence function’s components are
presented in Figure 8.
As it was mentioned above, the higher-order coherence functions provide the
information about the information about the statistics of radiation scattered
by a giant acoustic atom. Since $C^{(2)}_{1,1}(0)>1$, one can clearly see that
the statistics of back-scattered phonons is super-Posissonian, i.e. the
particles tend to bunch together. Indeed, this result agrees with the physical
expectation that at zero detuning $\Delta=0$ the power extinction
$1-|S_{11}(0)|^{2}$ is enhanced even in the presence of pure dephasing[13,
87]. This assertion also explains the anti-bunching of forwardly scattered
photons $C^{(2)}_{2,2}(0)$. Another clear feature of the second-order
coherence function shown in Figure 8 is presence of long-range quantum
correlations of phonons, i.e. the components of $C^{(2)}$ do not decay to
unity even for delays significantly exceeding the inter-leg separation
$\tau\gg 1$. Note that this correlation effect becomes more and more
pronounced with the increase in $R$, where correlation functions exhibit
slightly damped oscillations around unity.
Another interesting feature of the second-order coherence is the presence of
the non-differentiable peaks at natural multiples of the inter-leg separation
$\tau_{n}=nR,\ n\in\mathbb{N}$. Physically, this property may again be
understood with the help of the simple picture of a leaky cavity formed by the
scatterer. Indeed, once a phonon is trapped between the legs of the atom, it
may bounce off the cavity’s walls back and forth multiple times, leading to
the formation of the non-analytical structures present in the second-order
coherence. The fact that the non-differentiable peaks are more pronounced at
smaller values of $n$ is a direct manifestation of the fact that the quality
factor of the effective cavity is not infinite. This property is indeed of
interest since the second-order coherence function is an experimentally
measurable quantity which makes these sharp peaks a potentially observable
effect.
Alongside the exact solution, the results based on the quasi-Markovian
approximation are presented in Figure 8 (lower panel). The discrepancy between
the two is apparent. Another feature typical of this approximation is the
tendency to overestimate the amplitudes of oscillation as was first pointed
out in the supplemental material of reference [71].
#### III.3.2 Third-order coherence
Figure 9: Independent components
$(\mu,\mu^{\prime},\mu^{\prime\prime})=(1,1,1),\ (1,1,2),\ (1,2,2),\ (2,2,2)$
of the third-order coherence function of phonons scattered by a giant acoustic
atom. Top, central, and bottom panels correspond to the exact solution
$C^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})$, weak
correlation approximation
$\tilde{C}^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})$,
and their difference
$\Delta{C^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})}=C^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})-\tilde{C}^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})$
respectively. Here $\gamma R=5$, $k_{0}R=\pi/4\mod 2\pi$, and $\Delta=0$. Here
the limit $L\rightarrow\infty$ was taken and by definition $C^{(2)}$ is
dimensionless.
Let us finally consider the third-order coherence function. The full
expression for the third-order coherence function
$C^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})$ in terms
of the symmetrized components of two and three-particle transition matrices
may be found in Appendix B. Although in general,
$C^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}$ has $8$ independent components
in the case of two radiation channels, due to our particular choice of the
coupling $\Gamma_{1}=\Gamma_{2}=\gamma/2$, only four independent components
remain:
$\displaystyle C^{(3)}_{1,1,1}(\tau,\tau^{\prime}),\quad
C^{(3)}_{2,2,2}(\tau,\tau^{\prime}),$ (107) $\displaystyle
C^{(3)}_{1,1,2}(\tau,\tau^{\prime})=C^{(3)}_{1,2,1}(\tau,\tau^{\prime})=C^{(3)}_{2,1,1}(\tau,\tau^{\prime}),$
(108) $\displaystyle
C^{(3)}_{1,2,2}(\tau,\tau^{\prime})=C^{(3)}_{2,2,1}(\tau,\tau^{\prime})=C^{(3)}_{2,1,2}(\tau,\tau^{\prime}).$
(109)
Independent components of $C^{(3)}$ for a system with $R=5$, $k_{0}R=\pi/4\mod
2\pi$, and $\Delta=0$ are shown in the top panel of Figure 9. First, we note
that the individual components of third-order coherence at
$\tau,\tau^{\prime}=0$ significantly exceed unity, signifying the bunching of
phonons. This effect can be attributed to the fact that a single two-level
system can only emit and absorb a single quantum of radiation at a time,
which, in turn, significantly increases the probability of simultaneous
detection of a pair of phonons in the waveguide. Moreover, this effect is more
pronounced in those components of the correlation function, describing
correlations with phonons in the channel of incident radiation $\mu=1$. This
artifact again has to do with the fact of the enhancement of power extinction
by an atom at zero detuning.
Another interesting feature of $C^{(3)}$ to be noticed is the presence of
clear peak and downfall structures located on the lines
$\tau^{\prime}-\tau=nR,\ n\in\mathbb{Z}$. Physically
$\delta\tau=\tau^{\prime}-\tau$ corresponds to the average delay time between
the second and third phonon detection events upon detection of the first one
at zero time. The quantization of $\delta\tau$ in the units of deterministic
time delay is the characteristic feature of the system under consideration and
may be potentially observed in future experiments via the observation of
enhancement/diminution of the conditional probability of arrival of the third
particle. In fact, the peak and downfall structures discussed above are non-
differentiable, as it was the case with the second-order coherence function,
and again this phenomenon may be understood with the help of a simple picture
of an effective cavity discussed in Sections III.3.1 and III.3.
Beside the exact solution
${C}^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})$, the
solution based on the weak-correlation (WC) approximation
$\tilde{C}^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}(\tau,\tau^{\prime})$ as
well as the difference between the exact solution and the WC one
$\displaystyle\Delta{C}^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}={C}^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}}-\tilde{C}^{(3)}_{\mu,\mu^{\prime},\mu^{\prime\prime}},$
(110)
are shown in Figure 9 in central and bottom pannels respectively. As one can
anticipate, the WC approximation tends to notably overestimate the amplitude
of the third-order coherence function. This effect is especially apparent in
the vicinity of $\tau,\tau^{\prime}=0$, where the WC approximation
significantly overestimates the phononic bunching. Away from the temporal
origin, though, the approximation becomes adequate and only slightly deviates
from the exact result, as one may infer from the plots in the bottom panel of
Figure 9. The discrepancy between the exact solution and its WC approximation
is not hard to understand. Weak-correlation approximation ignores an infinite
diagrammatic channel, consisting of exchange interaction diagrams between the
phonons, which are, of course, of importance when one studies the statistical
properties of particles.
## IV Conclusions and outlook
In this paper, the diagrammatic theory of scattering and dynamics of multi-
photon states in waveguide QED was developed. In particular, it was shown that
the $N_{p}$-photon scattering matrices in single-qubit waveguide QED may be
conveniently parametrized in terms of effective $N_{p}-1$-photon vertex
functions and the equations satisfied by these vertex functions were
established. Next, certain practical issues related to the direct sum
representation of the $S$-matrix, separation of elastic contributions to
effective vertices, as well as the generalized cluster decomposition were
discussed. Further, a generalization to the waveguide QED systems with more
than a single qubit was given. Specifically, in the case of the two-qubit
systems, it was established that the equations governing multi-photon vertex
functions remain the same as in the case of a single qubit, up to the
inclusion of higher-order vertex corrections. Moreover, we have shown that
once the integral equations governing $N_{q}$-photon scattering matrix in
$N_{q}$ waveguide QED, these equations hold for any system of qubits and
established the generic equations governing $2$ and $3$ photon scattering
operators by considering $2$ and $3$ photon scattering on two and three qubits
respectively.
Next, the diagrammatic theory of scattering was applied to a problem of
scattering of a weakly coherent pulse on the giant acoustic atom. Namely, by
expanding a coherent state perturbatively in a coherence parameter up to third
order in $|\alpha|$ and studying its scattering on the atom, we were able to
establish the first, second, and third-order coherence functions of scattered
radiation. Moreover, a set of approximation routines was suggested along with
the exact method and the two were compared where appropriate. Further, the
statistical properties of scattered surface acoustic waves were studied, and
the effect of the non-Markovian nature of the setup on statistics was
discussed.
In our future work, we are going to present the generalization of the
resummation approach enabling one to study real-time dynamics in waveguide QED
systems. It would be of further interest to extend the present theory to study
scattering in waveguide QED systems containing emitters with more complicated
selection rules such as three and four-level systems. Another generalization
of the theory presented in this paper of potential future interest is the
study of particular 2 and 3-particle scattering problems on systems containing
multiple distant qubits. Furthermore, it would be potentially interesting to
assess the effects of counter-rotating terms, which were ignored throughout
this work, which, however, will require the use of techniques other than the
one discussed in this work.
## Acknowledgments
We thank H. Schoeller for helpful instructions on diagrammatic methods in
field theory applications and, in particular, for pointing to us the
distributional Poincare-Bertrand identity. KP is grateful to A. Samson for
enlightening discussions on numerical methods used throughout the paper. MP
acknowledges the durable exchange of ideas on the subject of study with V.
Gritsev and V. Yudson. This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) via the contract RTG 1995.
## Appendix A Cluster decomposition of three-photon $S$-matrix
The starting point of our analysis in this appendix goes back to the
definition of the three-body component of the $S$-matrix entering its direct
sum representation (throughout this appendix, for simplicity, it is assumed
that $\omega_{\mu}(k)=\omega(k),\quad
B_{\mu}=B,\quad\forall\mu\in\\{1,...,N_{c}\\}$), namely
$\displaystyle\mathcal{S}_{3}=$
$\displaystyle\Bigg{[}\frac{1}{3!}\delta_{s_{1}^{\prime},s_{1}}\delta_{s_{2}^{\prime},s_{2}}\delta_{s_{3}^{\prime},s_{3}}-2\pi{i}T^{(1,1)}_{s_{1}^{\prime},s_{1}}(\omega(k_{1}))\delta(\omega(k_{1}^{\prime})-\omega(k_{1}))\frac{1}{2!}\delta_{s_{2}^{\prime},s_{2}}\delta_{s_{3}^{\prime},s_{3}}$
$\displaystyle-2\pi{i}T^{(1,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega(k_{1})+\omega(k_{2}))\delta(\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime})-\omega(k_{1})-\omega(k_{2}))\frac{1}{1!}\delta_{s_{3}^{\prime},s_{3}}$
$\displaystyle-2\pi{i}T^{(1,3)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))\delta(\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1})-\omega(k_{2})-\omega(k_{3}))\Bigg{]}$
$\displaystyle\times$
$\displaystyle{a_{s_{1}^{\prime}}^{\dagger}}a_{s_{2}^{\prime}}^{\dagger}a_{s_{3}^{\prime}}^{\dagger}a_{s_{3}}a_{s_{2}}a_{s_{1}}.$
(111)
As before we write
$\displaystyle
T^{(1,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega(k_{1})+\omega(k_{2}))=$
$\displaystyle-\pi{i}\delta(\omega(k_{1}^{\prime})-\omega(k_{1}))T^{(1,1)}_{s_{1}^{\prime},s_{1}}(\omega(k_{1}))T^{(1,1)}_{s_{2}^{\prime},s_{2}}(\omega(k_{2}))$
$\displaystyle+T^{(1,2,C)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega(k_{1})+\omega(k_{2})).$
(112)
Here, as before, the equality symbol is understood in the sense of permutation
equivalence and on-shell condition. Now let us start massaging the three-body
transition operator. First of all, one has
$\displaystyle
T^{(1,3)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime};\mu_{1}k_{1},\mu_{2}k_{2},\mu_{3}k_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))=g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3})g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})$
$\displaystyle\qquad\qquad\times\tilde{G}^{(1)}(\omega(k_{1}))\tilde{G}^{(1)}(\omega(k_{1}^{\prime}))F^{(1,2)}(k_{2}^{\prime},k_{3}^{\prime},k_{2},k_{3},\omega(k_{1})+\omega(k_{2})+\omega(k_{3})),$
(113)
where
$F^{(1,2)}(k_{2}^{\prime},k_{3}^{\prime},k_{2},k_{3},\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
is not an entirely connected object defined via
$\displaystyle
F^{(1,2)}(k_{1}^{\prime},k_{2}^{\prime},k_{1},k_{2},\epsilon)=\frac{1}{g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})}\Braket{g}{W^{(1,2)}_{\mu_{1}^{\prime}k_{1}^{\prime}\mu_{2}^{\prime}k_{2}^{\prime},\mu_{1}k_{1}\mu_{2}k_{2}}(\epsilon)}{g}.$
(114)
Separation of elastic contribution (43) translates into the following
decomposition
$\displaystyle
F^{(1,2)}(k_{2}^{\prime},k_{3}^{\prime},k_{2},k_{3},\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
$\displaystyle=$ $\displaystyle
F^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1})+\omega(k_{3})-\omega(k_{3}^{\prime}))F^{(1,1)}(k_{3}^{\prime},k_{3},\omega(k_{1})+\omega(k_{3}))$
$\displaystyle+\overline{F}^{(2,2)}(k_{2}^{\prime},k_{3}^{\prime},k_{2},k_{3},\omega(k_{1})+\omega(k_{2})+\omega(k_{3})).$
(115)
Bearing in mind that
$\overline{F}^{(2,2)}(k_{2}^{\prime},k_{3}^{\prime},k_{2},k_{3},\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
is an analytic function we decompose the three photon $T$-matrix as follows
$\displaystyle
T^{(1,3)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime};\mu_{1}k_{1},\mu_{2}k_{2},\mu_{3}k_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
$\displaystyle=$
$\displaystyle\overline{T}^{(1,3)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime};\mu_{1}k_{1},\mu_{2}k_{2},\mu_{3}k_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
$\displaystyle+$
$\displaystyle\hat{T}^{(1,3)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime};\mu_{1}k_{1},\mu_{2}k_{2},\mu_{3}k_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3})),$
(116)
where we have defined the following objects
$\displaystyle\overline{T}^{(1,3)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime};\mu_{1}k_{1},\mu_{2}k_{2},\mu_{3}k_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
$\displaystyle=$ $\displaystyle
g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3}){g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})}$
$\displaystyle\times\tilde{G}(\omega(k_{1}))\tilde{G}(\omega(k_{1}^{\prime}))$
$\displaystyle\times\overline{F}^{(1,2)}(k_{2}^{\prime},k_{3}^{\prime},k_{2},k_{3},\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
(117)
and
$\displaystyle\hat{T}^{(1,3)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime};\mu_{1}k_{1},\mu_{2}k_{2},\mu_{3}k_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
$\displaystyle=$ $\displaystyle
g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3}){g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})}$
$\displaystyle\times\tilde{G}(\omega(k_{1}))\tilde{G}(\omega(k_{1}^{\prime}))F^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime}))$
$\displaystyle\times\tilde{G}^{(1)}(\omega(k_{1})+\omega(k_{3})-\omega(k_{3}^{\prime}))$
$\displaystyle\times
F^{(1,1)}(k_{3}^{\prime},k_{3},\omega(k_{1})+\omega(k_{3})).$ (118)
In the last equation we have
$\displaystyle
F^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime}))=\hat{G}_{0}(\omega(k_{1}^{\prime})-\omega(k_{2}))$
$\displaystyle\qquad+\overline{F}^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime})),$
(119) $\displaystyle
F^{(1,1)}(k_{3}^{\prime},k_{3},\omega(k_{1})+\omega(k_{3}))=\hat{G}_{0}(\omega(k_{1})-\omega(k_{3}^{\prime}))$
$\displaystyle\qquad+\overline{F}^{(1,1)}(k_{3}^{\prime},k_{3},\omega(k_{1})+\omega(k_{3})).$
(120)
The terms in (118) containing $\hat{G}_{0}$ deserve a special attention since
they can yield delta-functions determining additional conservation of
frequencies.
In particular, the terms with $\hat{G}_{0}\overline{F}^{(1,1)}$ and
$\overline{F}^{(1,1)}\hat{G}_{0}$ together give
$\displaystyle-2\pi{i}\delta(\omega(k_{3}^{\prime})-\omega(k_{3}))T^{(1,1)}_{s_{3}^{\prime},s_{3}}(\omega(k_{3}))$
$\displaystyle\times\Bigg{[}T^{(1,2,C)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega(k_{1})+\omega(k_{2}))$
$\displaystyle-
g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})\tilde{G}^{(1)}(\omega(k_{1}))\tilde{G}^{(1)}(\omega(k_{1}^{\prime}))$
$\displaystyle\qquad\times
P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{2}^{\prime})}\Bigg{)}\Bigg{]}$ (121)
$\displaystyle+g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3}){g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})}$
$\displaystyle\quad\times\tilde{G}(\omega(k_{1}))\tilde{G}(\omega(k_{1}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1})+\omega(k_{3})-\omega(k_{3}^{\prime}))$
$\displaystyle\times\Bigg{[}P\Bigg{(}\frac{1}{\omega(k_{1}^{\prime})-\omega(k_{2})}\Bigg{)}\overline{F}^{(1,1)}(k_{3}^{\prime},k_{3},\omega(k_{1})+\omega(k_{3}))$
$\displaystyle\quad+\overline{F}^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime}))P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{3}^{\prime})}\Bigg{)}\Bigg{]}.$
(122)
The term (121) contributes to the $3=2+1$ cluster of the three-photon
$S$-matrix. In turn, the term (122) is completely connected and non-singular.
This property becomes explicitly visible if we re-express it as
$\displaystyle\frac{g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3})g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})}{\omega(k_{1}^{\prime})-\omega(k_{1})}$
$\displaystyle\times$
$\displaystyle\Bigg{[}\tilde{G}^{(1)}(\omega(k_{1}^{\prime}))\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{3}^{\prime})+\omega(k_{1}^{\prime})-\omega(k_{1}))$
$\displaystyle\quad\times\overline{F}^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1}))$
$\displaystyle-\tilde{G}^{(1)}(\omega(k_{1}))\tilde{G}^{(1)}(\omega(k_{1})+\omega(k_{3})-\omega(k_{1}^{\prime}))\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))$
$\displaystyle\quad\times\overline{F}^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime}))\Bigg{]}.$
(123)
This representation makes it obvious that in the limit $k^{\prime}_{1}\to
k_{1}$ this function is finite.
Now, let us consider the contribution to (118) containing
$\hat{G}_{0}\hat{G}_{0}$. It amounts to
$\displaystyle
g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3})g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})$
$\displaystyle\times\tilde{G}(\omega(k_{1}))\tilde{G}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))$
$\displaystyle\times\hat{G}_{0}(\omega(k_{3}^{\prime})-\omega(k_{3}))\hat{G}_{0}(\omega(k_{1})-\omega(k_{1}^{\prime}))$
$\displaystyle=$
$\displaystyle(-i\pi)^{2}T^{(1,1)}_{s_{1}^{\prime},s_{1}}(\omega(k_{1}))T^{(1,1)}_{s_{2}^{\prime},s_{2}}(\omega(k_{2}))T^{(1,1)}_{s_{3}^{\prime},s_{3}}(\omega(k_{3}))$
$\displaystyle\times\delta(\omega(k_{1}^{\prime})-\omega(k_{1}))\delta(\omega(k_{2}^{\prime})-\omega(k_{2}))$
(124) $\displaystyle-$ $\displaystyle
2\pi{i}g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})\tilde{G}^{(1)}(\omega(k_{1}))$
$\displaystyle\times\tilde{G}^{(1)}(\omega(k_{1}^{\prime}))T^{(1,1)}_{s_{3}^{\prime},s_{3}}(\omega(k_{3}))\delta(\omega(k_{3}^{\prime})-\omega(k_{3}))$
(125) $\displaystyle+$ $\displaystyle
g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3}){g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})}$
$\displaystyle\times\tilde{G}(\omega(k_{3}^{\prime}))\tilde{G}(\omega(k_{1}))\tilde{G}^{(1)}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))$
$\displaystyle\times{P}\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}.$
(126)
The term (124) contributes to the $3=1+1+1$ cluster. The term (125)
contributes to the $3=2+1$ cluster.
The term (126) requires a special consideration. Rewriting
$\displaystyle\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1}))$
$\displaystyle\times
P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}$
$\displaystyle=$
$\displaystyle\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1}))P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}$
$\displaystyle\times\frac{\tilde{G}^{(1)}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))-\tilde{G}^{(1)}(\omega(k_{2}))}{\omega(k_{1})-\omega(k_{1}^{\prime})}$
$\displaystyle+$
$\displaystyle\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{1}))$
$\displaystyle\times
P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}$
$\displaystyle=$
$\displaystyle\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1}))P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}$
$\displaystyle\times\frac{\tilde{G}^{(1)}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))-\tilde{G}^{(1)}(\omega(k_{2}))}{\omega(k_{1})-\omega(k_{1}^{\prime})}$
(127) $\displaystyle+$
$\displaystyle\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{1}))$
$\displaystyle\times\frac{\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))-\tilde{G}^{(1)}(\omega(k_{3}))}{\omega(k_{3}^{\prime})-\omega(k_{3})}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}$
(128) $\displaystyle+$
$\displaystyle\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{1}))$
$\displaystyle\times
P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)},$
(129)
we observe that the terms (127) and (128) terms give together a non-singular
contribution. In fact,
$\displaystyle\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1}))P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}$
$\displaystyle\times\frac{\tilde{G}^{(1)}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))-\tilde{G}^{(1)}(\omega(k_{2}))}{\omega(k_{1})-\omega(k_{1}^{\prime})}$
$\displaystyle+$
$\displaystyle\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{1}))\frac{\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))-\tilde{G}^{(1)}(\omega(k_{3}))}{\omega(k_{3}^{\prime})-\omega(k_{3})}$
$\displaystyle\times
P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}$
$\displaystyle=\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1}))\tilde{G}^{(1)}(\omega(k_{2}))$
$\displaystyle\times\tilde{G}^{(1)}(\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{2}^{\prime}))$
$\displaystyle\times\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{2}^{\prime}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{3}))}{\omega(k_{3}^{\prime})-\omega(k_{3})}$
$\displaystyle\times\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{2}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))}{\omega(k_{1})-\omega(k_{1}^{\prime})}$
$\displaystyle+\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\frac{1}{\omega(k_{1}^{\prime})-\omega(k_{1})}$
$\displaystyle\times\Bigg{(}\tilde{G}^{(1)}(\omega(k_{1}^{\prime}))$
$\displaystyle\times\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{3}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{1}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1}))}{\omega(k_{1}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1})-\omega(k_{3})}$
$\displaystyle-\tilde{G}^{(1)}(\omega(k_{1}))\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{3}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{3}^{\prime}))}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}.$
(130)
In contrast, the term (129) is singular and contributes to the $3=1+1+1$
cluster of the scattering matrix. To show this, we first fully symmetrize
$\displaystyle\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{1}))$
$\displaystyle\times
P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}$
$\displaystyle\to$
$\displaystyle\frac{1}{3}\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{1}))$
$\displaystyle\times\Bigg{[}P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}$
$\displaystyle\quad+P\Bigg{(}\frac{1}{\omega(k_{2}^{\prime})-\omega(k_{2})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{3})-\omega(k_{3}^{\prime})}\Bigg{)}$
$\displaystyle\quad+P\Bigg{(}\frac{1}{\omega(k_{1}^{\prime})-\omega(k_{1})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{2})-\omega(k_{2}^{\prime})}\Bigg{)}\Bigg{]}.$
(131)
Owing to the Poincare-Bertrand distributional identity
$\displaystyle P\Bigg{(}\frac{1}{x}\Bigg{)}P\Bigg{(}\frac{1}{y}\Bigg{)}$
$\displaystyle=P\Bigg{(}\frac{1}{y-x}\Bigg{)}\Bigg{[}P\Bigg{(}\frac{1}{x}\Bigg{)}-P\Bigg{(}\frac{1}{y}\Bigg{)}\Bigg{]}$
$\displaystyle+\pi^{2}\delta(x)\delta(y),$ (132)
we establish the identity
$\displaystyle
P\Bigg{(}\frac{1}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{1})-\omega(k_{1}^{\prime})}\Bigg{)}$
$\displaystyle+$ $\displaystyle
P\Bigg{(}\frac{1}{\omega(k_{2}^{\prime})-\omega(k_{2})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{3})-\omega(k_{3}^{\prime})}\Bigg{)}$
$\displaystyle+$ $\displaystyle
P\Bigg{(}\frac{1}{\omega(k_{1}^{\prime})-\omega(k_{1})}\Bigg{)}P\Bigg{(}\frac{1}{\omega(k_{2})-\omega(k_{2}^{\prime})}\Bigg{)}$
$\displaystyle=$
$\displaystyle\pi^{2}\delta(\omega(k_{3}^{\prime})-\omega(k_{3}))\delta(\omega(k_{1}^{\prime})-\omega(k_{1})),$
(133)
leading us to the result
$\displaystyle\frac{\pi^{2}}{3}\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{1}))$
(134)
$\displaystyle\times\delta(\omega(k_{3}^{\prime})-\omega(k_{3}))\delta(\omega(k_{1}^{\prime})-\omega(k_{1}))$
(135)
in (131).
Combining all of the above results, we arrive at the following decomposition
of the three-photon $T$-matrix
$\displaystyle
T^{(1,3)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))$
$\displaystyle=$
$\displaystyle\frac{(-2\pi{i})^{2}}{6}T^{(1)}_{s_{1}^{\prime},s_{1}}(\omega(k_{1}))T^{(1)}_{s_{2}^{\prime},s_{2}}(\omega(k_{2}))T^{(1)}_{s_{3}^{\prime},s_{3}}(\omega(k_{3}))\delta(\omega(k_{1}^{\prime})-\omega(k_{1}))\delta(\omega(k_{2}^{\prime})-\omega(k_{2}))$
$\displaystyle-$
$\displaystyle\,2\pi{i}T^{(1,2,C)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega(k_{1})+\omega(k_{2}))T^{(1)}_{s_{3}^{\prime},s_{3}}(\omega(k_{3}))\delta(\omega(k_{3}^{\prime})-\omega(k_{3}))+T^{(1,3,C)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3})),$
(136)
with the three-body connected part
$\displaystyle
T^{(1,3,C)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))=g_{\mu_{1}}^{*}(k_{1})g_{\mu_{2}}^{*}(k_{2})g_{\mu_{3}}^{*}(k_{3}){g_{\mu_{1}^{\prime}}(k_{1}^{\prime})g_{\mu_{2}^{\prime}}(k_{2}^{\prime})g_{\mu_{3}^{\prime}}(k_{3}^{\prime})}$
$\displaystyle\times\Bigg{\\{}\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1}))\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{2}^{\prime}))$
$\displaystyle\qquad\times\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{2^{\prime}}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{3}))}{\omega(k_{3}^{\prime})-\omega(k_{3})}\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{2}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{2})+\omega(k_{1})-\omega(k_{1}^{\prime}))}{\omega(k_{1})-\omega(k_{1}^{\prime})}$
$\displaystyle\qquad+\frac{1}{\omega(k_{1}^{\prime})-\omega(k_{1})}\Bigg{[}\tilde{G}^{(1)}(\omega(k_{2}))\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))$
$\displaystyle\qquad\qquad\times\Bigg{(}\tilde{G}^{(1)}(\omega(k_{1}^{\prime}))\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{3}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{1}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1}))}{\omega(k_{1}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1})-\omega(k_{3})}$
$\displaystyle\qquad\qquad\qquad-\tilde{G}^{(1)}(\omega(k_{1}))\frac{(\tilde{G}^{(1)})^{-1}(\omega(k_{3}))-(\tilde{G}^{(1)})^{-1}(\omega(k_{3}^{\prime}))}{\omega(k_{3}^{\prime})-\omega(k_{3})}\Bigg{)}$
$\displaystyle\qquad\qquad+\tilde{G}^{(1)}(\omega(k_{1}^{\prime}))\tilde{G}^{(1)}(\omega(k_{3}))\tilde{G}^{(1)}(\omega(k_{3}^{\prime})+\omega(k_{1}^{\prime})-\omega(k_{1}))\overline{F}^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1}))$
$\displaystyle\qquad\qquad-\tilde{G}^{(1)}(\omega(k_{1}))\tilde{G}^{(1)}(\omega(k_{3}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1})+\omega(k_{3})-\omega(k_{1}^{\prime}))\overline{F}^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime}))\Bigg{]}$
$\displaystyle\qquad+\tilde{G}(\omega(k_{1}))\tilde{G}(\omega(k_{1}^{\prime}))\overline{F}^{(1,1)}(k_{2}^{\prime},k_{2},\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime}))\tilde{G}^{(1)}(\omega(k_{1})+\omega(k_{3})-\omega(k_{3}^{\prime}))\overline{F}^{(1,1)}(k_{3}^{\prime},k_{3},\omega(k_{1})+\omega(k_{3}))\Bigg{\\}}$
$\displaystyle+\overline{T}^{(1,3)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3})).$
(137)
Plugging everything into the definition of the three-photon $S$-matrix we
finally obtain
$\displaystyle\mathcal{S}_{3}$
$\displaystyle=\Bigg{[}\frac{1}{3!}S^{(1)}_{s_{1}^{\prime},s_{1}}S^{(1)}_{s_{2}^{\prime},s_{2}}S^{(1)}_{s_{3}^{\prime},s_{3}}-2\pi{i}T^{(1,2,C)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\omega(k_{1})+\omega(k_{2}))S^{(1)}_{s_{3}^{\prime},s_{3}}\delta(\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime})-\omega(k_{1})-\omega(k_{2})$
$\displaystyle-2\pi{i}T^{(1,3,C)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\omega(k_{1})+\omega(k_{2})+\omega(k_{3}))\delta(\omega(k_{1}^{\prime})+\omega(k_{2}^{\prime})+\omega(k_{3}^{\prime})-\omega(k_{1})-\omega(k_{2})-\omega(k_{3}))\Bigg{]}a_{s_{1}^{\prime}}^{\dagger}a_{s_{2}^{\prime}}^{\dagger}a_{s_{3}^{\prime}}^{\dagger}a_{s_{3}}a_{s_{2}}a_{s_{1}}.$
(138)
## Appendix B Third-order coherence function
In this appendix we present the formula for the third order coherence function
of phonons in the giant atom model. Using the formula (138) with $\omega(k)=k$
along with the notations introduced in the Section III, we obtain the
following result upon contraction with the three-phonon Fock state
$\displaystyle\mathcal{S}_{3}\ket{\Phi^{(3)}_{1}}$
$\displaystyle=\frac{1}{\sqrt{6}}\sum_{\\{\mu^{\prime}_{i}\\}}\Bigg{(}\int_{k_{1}k_{2}k_{3}}\varphi(k_{1})\varphi(k_{2})\varphi(k_{3})S^{(1)}_{\mu_{1}^{\prime},1}(k_{1})S^{(1)}_{\mu_{2}^{\prime},1}(k_{2})S^{(1)}_{\mu_{3}^{\prime},1}(k_{3})a_{\mu_{1}^{\prime}}^{\dagger}(k_{1})a_{\mu_{2}^{\prime}}^{\dagger}(k_{2})a_{\mu_{3}^{\prime}}^{\dagger}(k_{3})\ket{\Omega}$
$\displaystyle-12\pi{i}\int_{k_{1}^{\prime}k_{2}^{\prime}k_{1}k_{2}k_{3}}\varphi(k_{1})\varphi(k_{2})\varphi(k_{3})T^{(2,C)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{1}k_{1},\mu_{2}k_{2}}(k_{1}+k_{2})\delta(k_{1}^{\prime}+k_{2}^{\prime}-k_{1}-k_{2})S^{(1)}_{\mu_{3}^{\prime};1}(k_{3})$
$\displaystyle\times{a}_{\mu_{1}^{\prime}}^{\dagger}(k_{1}^{\prime})a_{\mu_{2}^{\prime}}^{\dagger}(k_{2}^{\prime})a_{\mu_{3}^{\prime}}^{\dagger}(k_{3})\ket{\Omega}$
$\displaystyle-12\pi{i}\Bigg{(}\frac{2\pi}{L}\Bigg{)}^{3/2}\int_{k_{1}^{\prime}k_{2}^{\prime}k_{3}^{\prime}}Q(k_{1}^{\prime},k_{2}^{\prime},k_{3}^{\prime})\delta(k_{1}^{\prime}+k_{2}^{\prime}+k_{3}^{\prime})a_{\mu_{1}^{\prime}}^{\dagger}(k_{1}^{\prime})a_{\mu_{2}^{\prime}}^{\dagger}(k_{2}^{\prime})a_{\mu_{3}^{\prime}}^{\dagger}(k_{3}^{\prime})\ket{\Omega}\Bigg{)},$
(139)
where we have introduced the following symmetrized version of the connected
three-phonon transition operator
$\displaystyle Q(k_{1}^{\prime},k_{2}^{\prime},k_{3}^{\prime})$
$\displaystyle=\frac{1}{6}\Big{[}T^{(3,C)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime},10,10,10}(0)+T^{(3,C)}_{\mu_{1}^{\prime}k_{1}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},10,10,10}(0)+T^{(3,C)}_{\mu_{3}^{\prime}k_{3}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},\mu_{1}^{\prime}k_{1}^{\prime},10,10,10}(0)$
$\displaystyle+T^{(3,C)}_{\mu_{3}^{\prime}k_{3}^{\prime},\mu_{1}^{\prime}k_{1}^{\prime},\mu_{2}^{\prime}k_{2}^{\prime},10,10,10}(0)+T^{(3,C)}_{\mu_{2}^{\prime}k_{2}^{\prime},\mu_{1}^{\prime}k_{1}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime},10,10,10}(0)+T^{(3,C)}_{\mu_{2}^{\prime}k_{2}^{\prime},\mu_{3}^{\prime}k_{3}^{\prime},\mu_{1}^{\prime}k_{1}^{\prime},10,10,10}(0)\Big{]}.$
(140)
Here we have suppressed the dependence of
$Q(k_{1}^{\prime},k_{2}^{\prime},k_{3}^{\prime})$ on $\\{\mu^{\prime}\\}$
since in the giant atom model the coupling constants are independent of the
channel index. Now we consider
$\displaystyle
a_{\mu^{\prime\prime}}(\tau_{3})a_{\mu^{\prime}}(\tau_{2})a_{\mu}(\tau_{1})\mathcal{S}_{3}\ket{\Phi^{(3)}_{1}}$
$\displaystyle=\frac{\sqrt{6}}{L^{3/2}}S^{(1)}_{\mu^{\prime\prime},1}(0)S^{(1)}_{\mu^{\prime},1}(0)S^{(1)}_{\mu,1}(0)\Bigg{(}1-4\pi{i}\Bigg{[}\frac{I^{(1)}_{\mu^{\prime\prime},\mu^{\prime}}(\tau_{3}-\tau_{2})}{S^{(1)}_{\mu^{\prime\prime},1}(0)S^{(1)}_{\mu^{\prime},1}(0)}$
$\displaystyle+\frac{I^{(1)}_{\mu,\mu^{\prime}}(\tau_{2}-\tau_{1})}{S^{(1)}_{\mu,1}(0)S^{(1)}_{\mu^{\prime},1}(0)}+\frac{I^{(1)}_{\mu^{\prime\prime},\mu}(\tau_{3}-\tau_{1})}{S^{(1)}_{\mu^{\prime\prime},1}(0)S^{(1)}_{\mu,1}(0)}\Bigg{]}-12\pi{i}\frac{I^{(2)}(\tau_{3}-\tau_{1},\tau_{2}-\tau_{1})}{S^{(1)}_{\mu^{\prime\prime},1}(0)S^{(1)}_{\mu^{\prime},1}(0)S^{(1)}_{\mu,1}(0)}\Bigg{)},$
(141)
where the following functions were defined
$\displaystyle I^{(1)}_{\mu^{\prime},\mu}(t_{1})=$
$\displaystyle\int_{k}e^{ikt_{1}}M_{\mu^{\prime},\mu}(k),$ (142)
$\displaystyle I^{(2)}(t_{2},t_{1})=$
$\displaystyle\int_{k,q}e^{iqt_{2}}e^{ikt_{1}}Q(-k-q,k,q),$ (143)
where
$M_{\mu,\mu^{\prime}}(k)=(T^{(2,C)}_{\mu^{\prime}k,\mu-k,10,10}(0)+T^{(2,C)}_{\mu-k,\mu^{\prime}k,10,10}(0))/2$,
as before. By introducing the following variables
$\tau^{\prime}=\tau_{3}-\tau_{1}$, $\tau=\tau_{2}-\tau_{1}$, we can
immediately write down the normalized third order coherence function to the
lowest order in $\varphi$ as
$\displaystyle
C^{(3)}_{\mu^{\prime\prime},\mu^{\prime},\mu}(t^{\prime},t)=\Bigg{|}1-4\pi{i}\Bigg{[}\frac{I^{(1)}_{\mu^{\prime\prime},\mu^{\prime}}(t^{\prime}-t)}{S^{(1)}_{\mu^{\prime\prime},1}(0)S^{(1)}_{\mu^{\prime},1}(0)}+\frac{I^{(1)}_{\mu,\mu^{\prime}}(t)}{S^{(1)}_{\mu,1}(0)S^{(1)}_{\mu^{\prime},1}(0)}+\frac{I^{(1)}_{\mu^{\prime\prime},\mu}(t^{\prime})}{S^{(1)}_{\mu^{\prime\prime},1}(0)S^{(1)}_{\mu;1}(0)}\Bigg{]}-12\pi{i}\frac{I^{(2)}(t^{\prime},t)}{S^{(1)}_{\mu^{\prime\prime},1}(0)S^{(1)}_{\mu^{\prime},1}(0)S^{(1)}_{\mu,1}(0)}\Bigg{|}^{2}.$
(144)
## Appendix C Diagrammatic representation of the generic three-body
transition operator
In this Appendix we start our analysis with equation (64). Taking matrix
elements of (64) in the three-particle subspace we arrive at the following
integral equation
$\displaystyle W^{(3,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)=$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle[{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{2}^{\prime}\bar{s}_{1}^{\prime}}(\epsilon)+{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime}}(\epsilon)]{G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}^{\prime}\bar{s}_{2}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}})W^{(2,1)}_{\bar{s}_{2},\bar{s}_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$ $\displaystyle
D^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}\bar{s}_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{1}^{\prime}})W^{(2,1)}_{\bar{s}_{1},\bar{s}_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}\bar{s}_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{2}^{\prime}})W^{(2,1)}_{\bar{s}_{1},\bar{s}_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{2}^{\prime}\bar{s}_{2}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{2}^{\prime}}-\omega_{\bar{s}_{2}})W^{(2,1)}_{\bar{s}_{2},\bar{s}_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}\bar{s}_{2}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{2}})W^{(2,2)}_{\bar{s}_{2}\bar{s}_{1},\bar{s}_{2}^{\prime}\bar{s}_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}\bar{s}_{2}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{2}})W^{(2,2)}_{\bar{s}_{1}\bar{s}_{2},\bar{s}_{2}^{\prime}\bar{s}_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}\bar{s}_{2}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{2}})W^{(2,2)}_{\bar{s}_{1}\bar{s}_{2},\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle{D}^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}\bar{s}_{2}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{2}})W^{(2,2)}_{\bar{s}_{2}^{\prime}\bar{s}_{1},\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})W^{(3,2)}_{\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime},s_{1}s_{2}}(\epsilon),$
(145)
where the projection of $\mathcal{D}$ onto the $2$-particle subspace is given
by
$\displaystyle D^{(2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle=v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}G^{(1)}(\epsilon)v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}}^{\dagger}.$
(146)
Baring this in mind we make the following ansatz
$\displaystyle W^{(3,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle=v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}G^{(3,0)}(\epsilon)v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}}^{\dagger}.$
(147)
This, in turn, leads to the following Dyson equation
$\displaystyle
G^{(3,0)}(\epsilon)=G^{(1)}(\epsilon)+G^{(1)}(\epsilon)\Sigma^{(3,0)}(\epsilon)G^{(3,0)}(\epsilon),$
(148)
where the self-energy in the three-excitation subspace is given by
$\displaystyle\Sigma^{(3,0)}(\epsilon)=$
$\displaystyle[v_{s_{1}^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s_{2}^{\prime}}^{\dagger}+v_{s_{2}^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}^{\dagger}]{G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}^{\prime}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}})W^{(2,1)}_{s_{2},s_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{1}^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s_{1}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{1}^{\prime}})W^{(2,1)}_{s_{1},s_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{2}^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{1}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}^{\prime}})W^{(2,1)}_{s_{1},s_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{2}^{\prime}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{2}^{\prime}}-\omega_{s_{2}})W^{(2,1)}_{s_{2},s_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})W^{(2,2)}_{s_{2}s_{1},s_{2}^{\prime}s_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})W^{(2,2)}_{s_{1}s_{2},s_{2}^{\prime}s_{1}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})W^{(2,2)}_{s_{1}s_{2},s_{1}^{\prime}s_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
v_{s_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}}^{\dagger}{G}^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})W^{(2,2)}_{s_{2}s_{1},s_{1}^{\prime}s_{2}^{\prime}}(\epsilon){G}^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})v_{s_{1}^{\prime}}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{2}^{\prime}}.$
(149)
Let us now consider the expression (65) for the transition operator.
Concentrating on the three-photon subspace we obtain
$\displaystyle
T^{(3,3)}_{s_{1}^{\prime}s_{2}^{\prime}s_{3}^{\prime},s_{1}s_{2}s_{3}}(\epsilon)=$
$\displaystyle\Braket{g}{v_{s_{1}^{\prime}}{G}^{(1)}(\epsilon-\omega_{s_{2}^{\prime}}-\omega_{s_{3}^{\prime}})[W^{(2,2)}_{s_{2}^{\prime}s_{3}^{\prime},s_{2}s_{3}}(\epsilon)+V^{(3,2)}_{s_{2}^{\prime}s_{3}^{\prime}}(\epsilon)G^{(3,0)}(\epsilon)\overline{V}^{(3,2)}_{s_{2}s_{3}}(\epsilon)]{G}^{(1)}(\epsilon-\omega_{s_{2}}-\omega_{s_{3}})v^{\dagger}_{s_{1}}}{g},$
(150) $\displaystyle V^{(3,2)}_{s_{1}^{\prime}s_{2}^{\prime}}=$ $\displaystyle
v_{s_{1}^{\prime}}G_{0}(\epsilon)v_{s_{2}^{\prime}}+W^{(2,1)}_{s_{1}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})v_{s}G_{0}(\epsilon)v_{s_{2}^{\prime}}+W^{(2,1)}_{s_{1}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})v_{s_{2}^{\prime}}G_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+$ $\displaystyle
W^{(2,2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{1}^{\prime}\bar{s}_{2}^{\prime}}(\epsilon)G^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})v_{\bar{s}_{1}^{\prime}}G_{0}(\epsilon-\omega_{\bar{s}_{2}^{\prime}})v_{\bar{s}_{2}^{\prime}}$
$\displaystyle+$ $\displaystyle
W^{(2,2)}_{s_{1}^{\prime}s_{2}^{\prime},\bar{s}_{2}^{\prime}\bar{s}_{1}^{\prime}}(\epsilon)G^{(1)}(\epsilon-\omega_{\bar{s}_{1}^{\prime}}-\omega_{\bar{s}_{2}^{\prime}})v_{\bar{s}_{1}^{\prime}}G_{0}(\epsilon-\omega_{\bar{s}_{2}^{\prime}})v_{\bar{s}_{2}^{\prime}},$
(151) $\displaystyle\overline{V}^{(3,2)}_{s_{1}s_{2}}=$ $\displaystyle
v_{s_{1}}G_{0}(\epsilon)v_{s_{2}}+v_{s_{1}}^{\dagger}G_{0}(\epsilon)v_{s^{\prime}}^{\dagger}G^{(1)}(\epsilon-\omega_{s^{\prime}})W^{(2,1)}_{s^{\prime},s_{2}}(\epsilon)+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s^{\prime}})W^{(2,1)}_{s^{\prime},s_{2}}(\epsilon)$
$\displaystyle+$ $\displaystyle
v^{\dagger}_{\bar{s}_{2}}G_{0}(\epsilon-\omega_{\bar{s}_{2}})v^{\dagger}_{\bar{s}_{1}}G^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{2}})W^{(2,2)}_{\bar{s}_{1}\bar{s}_{2},s_{1}s_{2}}(\epsilon)$
$\displaystyle+$ $\displaystyle
v^{\dagger}_{\bar{s}_{2}}G_{0}(\epsilon-\omega_{\bar{s}_{2}})v^{\dagger}_{\bar{s}_{1}}G^{(1)}(\epsilon-\omega_{\bar{s}_{1}}-\omega_{\bar{s}_{2}})W^{(2,2)}_{\bar{s}_{2}\bar{s}_{1},s_{1}s_{2}}(\epsilon).$
(152)
Now, our goal is to rewrite the renormalized two-particle emission
$V_{s_{1}^{\prime}s_{2}^{\prime}}$ and absorption $\overline{V}_{s_{1}s_{2}}$
vertices, as well as the self-energy in three-excitation subspace
$\Sigma^{(3,3)}$ in terms of full Green’s and vertex functions, as it is
presented in the main text.
First, we note the following identity
$\displaystyle
v_{s_{1}^{\prime}}G_{0}(\epsilon)v_{s_{2}^{\prime}}+W^{(2,1)}_{s_{1}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})v_{s}G_{0}(\epsilon)v_{s_{2}^{\prime}}=v_{s^{\prime}_{1}}G_{0}(\epsilon)v_{s^{\prime}_{2}}+W_{s^{\prime}_{1},s}^{(2,1,i)}(\epsilon)G^{(1)}(\epsilon-\omega_{s})v_{s}G_{0}(\epsilon)v_{s^{\prime}_{2}}$
$\displaystyle+V_{s^{\prime}_{1}}^{(2,1)}(\epsilon)G^{(2,0)}(\epsilon)\overline{V}_{s}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s})v_{s}G_{0}(\epsilon)v_{s^{\prime}_{2}}=v_{s^{\prime}_{1}}G_{0}(\epsilon)v_{s^{\prime}_{2}}+W_{s^{\prime}_{1},s}^{(2,1,i)}G^{(1)}(\epsilon-\omega_{s})v_{s}G_{0}(\epsilon)v_{s^{\prime}_{2}}$
$\displaystyle+V_{s^{\prime}_{1}}^{(2,1)}(\epsilon)G^{(2,0)}(\epsilon)\Sigma^{(2,0)}G_{0}(\epsilon)v_{s^{\prime}_{2}}=v_{s^{\prime}_{1}}G_{0}(\epsilon)v_{s^{\prime}_{2}}+W_{s^{\prime}_{1},s}^{(2,1,i)}(\epsilon)G^{(1)}(\epsilon-\omega_{s})v_{s}G_{0}(\epsilon)v_{s^{\prime}_{2}}$
$\displaystyle+V_{s^{\prime}_{1}}^{(2,1)}(\epsilon)[G^{(2,0)}(\epsilon)-G_{0}(\epsilon)]v_{s^{\prime}_{2}}=V_{s^{\prime}_{1}}^{(2,1)}(\epsilon)G^{(2,0)}(\epsilon)v_{s^{\prime}_{2}}.$
(153)
Analogously
$\displaystyle
v_{s_{1}}^{\dagger}G_{0}(\epsilon)v_{s_{2}}^{\dagger}+v_{s_{1}}^{\dagger}G_{0}(\epsilon)v_{s^{\prime}}^{\dagger}G^{(1)}(\epsilon-\omega_{s^{\prime}})W_{s^{\prime},s_{2}}^{(2,1)}(\epsilon)$
$\displaystyle=v_{s_{1}}^{\dagger}G^{(2,0)}(\epsilon)\overline{V}_{s_{2}}^{(2,1)}(\epsilon).$
(154)
Analysing the structure of equations satisfied by $W^{(2,1)}$ and $W^{(2,2)}$
one easily concludes that
$\displaystyle W^{(2,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle=W^{(2,1)}_{s_{1}^{\prime},s_{1}}(\epsilon)G^{(1)}(\epsilon)W^{(2,1)}_{s_{2}^{\prime},s_{2}}(\epsilon)$
$\displaystyle+W^{(2,1)}_{s_{1}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W^{(2,2)}_{s_{2}^{\prime}s,s_{1}s_{2}}(\epsilon),$
(155) $\displaystyle
W^{(2,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)$
$\displaystyle=W^{(2,1)}_{s_{1}^{\prime},s_{1}}(\epsilon)G^{(1)}(\epsilon)W^{(2,1)}_{s_{2}^{\prime},s_{2}}(\epsilon)$
$\displaystyle+W^{(2,2)}_{s_{1}^{\prime}s_{2}^{\prime},ss_{1}}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W^{(2,1)}_{s,s_{2}}(\epsilon).$
(156)
Multiplying (C) and (C) by
$G^{(1)}(\epsilon)v_{s_{2}}G_{0}(\epsilon)v_{s_{1}}$ from the right and by
$v_{s_{2}^{\prime}}^{\dagger}G_{0}(\epsilon)v_{s_{1}^{\prime}}^{\dagger}G^{(1)}(\epsilon)$
from the left respectively and contracting the relevant indices, we obtain
$\displaystyle
W_{s^{\prime}_{1}s^{\prime}_{2},s_{1}s_{2}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{2}}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{1}}=W_{s^{\prime}_{1},s_{1}}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}})[V_{s^{\prime}_{2}}^{(2,1)}(\epsilon)G^{(2,0)}(\epsilon-\omega_{s_{1}})v_{s_{1}}$
$\displaystyle-
v_{s^{\prime}_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}}]+W_{s^{\prime}_{1},s}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W_{s^{\prime}_{2}s,s_{1}s_{2}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}},$
(157) $\displaystyle
v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1}s^{\prime}_{2},s_{1}s_{2}}^{(2,2)}(\epsilon)=[v_{s^{\prime}_{2}}^{\dagger}G^{(2,0)}(\epsilon-\omega_{s_{2}^{\prime}})\overline{V}_{s_{1}}^{(2,1)}(\epsilon)-v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{1}}^{\dagger}]$
$\displaystyle\times
G^{(1)}(\epsilon-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{2},s_{2}}^{(2,1)}+v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1}s^{\prime}_{2},ss_{1}}^{(2,2)}G^{(1)}(\epsilon-\omega_{s})W_{s,s_{2}}^{(2,1)}(\epsilon).$
(158)
Now, defining the following objects
$\displaystyle K^{(d)}_{s^{\prime}_{1}s^{\prime}_{2}}(\epsilon)$
$\displaystyle=W_{s^{\prime}_{1}s^{\prime}_{2},s_{1}s_{2}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{2}}-\omega_{s_{1}})v_{s_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}}$
$\displaystyle+W_{s^{\prime}_{1},s_{1}}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}})v_{s^{\prime}_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}},$
(159) $\displaystyle\overline{K}^{(d)}_{s_{1}s_{2}}(\epsilon)$
$\displaystyle=v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1}s^{\prime}_{2},s_{1}s_{2}}^{(2,2)}(\epsilon)$
$\displaystyle+v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{2},s_{2}}^{(2,1)}(\epsilon),$
(160)
we establish the following equations
$\displaystyle K^{(d)}_{s^{\prime}_{1}s^{\prime}_{2}}(\epsilon)$
$\displaystyle=W_{s^{\prime}_{1},s_{1}}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}})V_{s^{\prime}_{2}}^{(2,1)}(\epsilon-\omega_{s_{1}})G^{(2,0)}(\epsilon-\omega_{s_{1}})v_{s_{1}}$
$\displaystyle+W_{s^{\prime}_{1},s}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s})[K^{(d)}_{s^{\prime}_{2}s}(\epsilon)-W_{s^{\prime}_{2},s_{1}}^{(2,1)}(\epsilon-\omega_{s})G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s})v_{s}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}}],$
(161) $\displaystyle\overline{K}^{(d)}_{s_{1}s_{2}}(\epsilon)$
$\displaystyle=v_{s^{\prime}_{2}}^{\dagger}G^{(2,0)}(\epsilon-\omega_{s_{2}^{\prime}})\overline{V}_{s_{1}}^{(2,1)}(\epsilon-\omega_{s_{2}^{\prime}})G^{(1)}(\epsilon-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{2},s_{2}}^{(1,0)}(\epsilon)$
$\displaystyle+[\overline{K}^{(d)}_{ss_{1}}(\epsilon)-v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{2}^{\prime}}-\omega_{s})W_{s^{\prime}_{2},s_{1}}^{(2,1)}(\epsilon-\omega_{s})]G^{(1)}(\epsilon-\omega_{s})W_{s,s_{2}}^{(2,1)}(\epsilon).$
(162)
Analogously multiplying (C) and (C) by
$G^{(1)}(\epsilon)v_{s_{1}}G_{0}(\epsilon)v_{s_{2}}$ from the right and by
$v_{s_{1}^{\prime}}^{\dagger}G_{0}(\epsilon)v_{s_{2}^{\prime}}^{\dagger}G^{(1)}(\epsilon)$
from the left respectively, contracting the $s_{1,2}$ indices, and defining
$\displaystyle K^{(e)}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)$
$\displaystyle=W^{(2,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{1}}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{2}},$
(163) $\displaystyle\overline{K}^{(e)}_{s_{1}s_{2}}(\epsilon)$
$\displaystyle=v_{s_{1}^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s_{2}^{\prime}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W^{(2,2)}_{s_{1}^{\prime}s_{2}^{\prime},s_{1}s_{2}}(\epsilon),$
(164)
we deduce
$\displaystyle K^{(e)}_{s^{\prime}_{1}s^{\prime}_{2}}(\epsilon)$
$\displaystyle=W_{s^{\prime}_{1},s_{1}}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}})W_{s^{\prime}_{2},s_{2}}^{(2,1)}(\epsilon-\omega_{s_{1}})G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{1}}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{2}}$
$\displaystyle+W_{s^{\prime}_{1},s}^{(2,1)}(\epsilon)G^{(1)}(\epsilon-\omega_{s})K^{(e)}_{s^{\prime}_{2}s}(\epsilon),$
(165) $\displaystyle\overline{K}^{(e)}_{s_{1}s_{2}}(\epsilon)$
$\displaystyle=v_{s^{\prime}_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s^{\prime}_{2}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1},s_{1}}^{(2,1)}(\epsilon-\omega_{s_{2}^{\prime}})G^{(1)}(\epsilon-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{2},s_{2}}^{(2,1)}(\epsilon)$
$\displaystyle+\overline{K}^{(e)}_{ss_{1}}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W_{s,s_{2}}^{(2,1)}(\epsilon).$
(166)
Further we define
$\displaystyle
K_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)=K_{s_{1}^{\prime}s_{2}^{\prime}}^{(d)}(\epsilon)+K_{s_{1}^{\prime}s_{2}^{\prime}}^{(e)}(\epsilon),$
(167)
$\displaystyle\overline{K}_{s_{1}s_{2}}(\epsilon)=\overline{K}_{s_{1}s_{2}}^{(d)}(\epsilon)+\overline{K}_{s_{1}s_{2}}^{(e)}(\epsilon).$
(168)
With these definitions it is easy to show that the two-particle
emission/absorption vertices are given by
$\displaystyle
V^{(3,2)}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)=V^{(2,1)}_{s_{1}^{\prime}}(\epsilon)G^{(2,0)}(\epsilon)V^{(2,1)}_{s_{2}^{\prime}}(\epsilon)+\tilde{K}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon),$
(169)
$\displaystyle\overline{V}^{(3,2)}_{s_{1}s_{2}}(\epsilon)=\overline{V}^{(2,1)}_{s_{1}}(\epsilon)G^{(2,0)}(\epsilon)\overline{V}^{(2,1)}_{s_{2}}(\epsilon)+\tilde{\overline{K}}_{s_{1}s_{2}}(\epsilon),$
(170)
where
$\displaystyle\tilde{K}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)={K}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)$
$\displaystyle-
V_{s_{1}^{\prime}}^{(2,1)}(\epsilon)G^{(2,0)}(\epsilon)W^{(2,1,i)}_{s_{2}^{\prime},s_{1}}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}})v_{s_{1}},$
(171)
$\displaystyle\tilde{\overline{K}}_{s_{1}s_{2}}(\epsilon)={\overline{K}}_{s_{1}s_{2}}(\epsilon)$
$\displaystyle-
v_{s_{2}^{\prime}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{2}^{\prime}})W^{(2,1,i)}_{s_{2}^{\prime},s_{1}}(\epsilon)G^{(2,0)}(\epsilon)\overline{V}_{s_{2}}^{(2,1)}(\epsilon),$
(172)
obey the following integral equations
$\displaystyle\tilde{K}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)=$
$\displaystyle
W^{(2,1)}_{s_{1}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})V^{(2,1)}_{s_{2}^{\prime}}(\epsilon-\omega_{s})G^{(2,0)}(\epsilon-\omega_{s})V^{(2,1)}_{s}(\epsilon)-V_{s_{1}^{\prime}}^{(2,1)}(\epsilon)G^{(2,0)}(\epsilon)W^{(2,1,i)}_{s_{2}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+$ $\displaystyle
W^{(2,1)}_{s_{1},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})\tilde{K}_{s_{2}^{\prime}s}(\epsilon),$
(173) $\displaystyle\tilde{\overline{K}}_{s_{1}s_{2}}(\epsilon)=$
$\displaystyle\overline{V}^{(2,1)}_{s}(\epsilon)G^{(2,0)}(\epsilon-\omega_{s})V^{(2,1)}_{s_{1}}(\epsilon-\omega_{s})G^{(1)}(\epsilon-\omega_{s})W^{(2,1)}_{s,s_{2}}(\epsilon)-v_{s}^{\dagger}G^{(1)}(\epsilon-\omega_{s})W^{(2,1,i)}_{s,s_{1}}(\epsilon)G^{(2,0)}(\epsilon)\overline{V}^{(2,1)}_{s_{2}}(\epsilon)$
$\displaystyle+$
$\displaystyle\tilde{\overline{K}}_{ss_{1}}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W^{(2,1)}_{s,s_{2}}(\epsilon).$
(174)
Using equations (169), (170) together with (173), (174), we finally arrive at
the following equations
$\displaystyle V^{(3,2)}_{s_{1}^{\prime}s_{2}^{\prime}}(\epsilon)$
$\displaystyle=V_{s_{1}^{\prime}}^{(2,1)}(\epsilon)G^{(2,0)}(\epsilon)v_{s_{2}^{\prime}}$
$\displaystyle+W^{(2,1)}_{s_{1}^{\prime},s}(\epsilon)G^{(1)}(\epsilon-\omega_{s})V^{(3,2)}_{s_{2}^{\prime}s}(\epsilon),$
(175) $\displaystyle\overline{V}^{(3,2)}_{s_{1}s_{2}}(\epsilon)$
$\displaystyle=v_{s_{1}}G^{(2,1)}(\epsilon)\overline{V}_{s_{2}}(\epsilon)$
$\displaystyle+\overline{V}^{(3,2)}_{ss_{1}}(\epsilon)G^{(1)}(\epsilon-\omega_{s})W^{(2,1)}_{s,s_{2}}(\epsilon),$
(176)
which are precisely the equations (70) and (71) stated in the main text.
Now, we turn our attention to the self-energy bubble. Let us show that
equations (68) and (69) hold via direct substitution. One has
$\displaystyle\Sigma^{(3,0)}(\epsilon)$
$\displaystyle=(v_{s^{\prime}_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s^{\prime}_{2}}^{\dagger}+v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{1}}^{\dagger})G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})V_{s^{\prime}_{1}s^{\prime}_{2}}^{(3,2)}(\epsilon)$
$\displaystyle=v_{s^{\prime}_{2}}^{\dagger}[G^{(2,0)}(\epsilon-\omega_{s_{2}^{\prime}})-G_{0}(\epsilon-\omega_{s_{2}^{\prime}})]v_{s^{\prime}_{2}}-v_{s^{\prime}_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s})v_{s^{\prime}_{1}}G_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+v_{s^{\prime}_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s^{\prime}_{2}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})V_{s^{\prime}_{1}}^{(2,1)}(\epsilon-\omega_{s_{2}^{\prime}})G^{(2,0)}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{2}}$
$\displaystyle+v_{s^{\prime}_{1}}^{\dagger}G^{(2,0)}(\epsilon-\omega_{s_{1}^{\prime}})\overline{V}_{s}^{(2,1)}(\epsilon-\omega_{s_{1}^{\prime}})G^{(1)}(\epsilon-\omega_{s}-\omega_{s_{1}^{\prime}})v_{s^{\prime}_{1}}G_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{2},s}^{(2,1)}(\epsilon-\omega_{s_{1}^{\prime}})G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s})v_{s^{\prime}_{1}}G_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+v_{s^{\prime}_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s^{\prime}_{2}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1}s^{\prime}_{2},s^{\prime}s}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s^{\prime}}G_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+v_{s^{\prime}_{1}}^{\dagger}G_{0}(\epsilon-\omega_{s_{1}^{\prime}})v_{s^{\prime}_{2}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1}s^{\prime}_{2},ss^{\prime}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s^{\prime}}G_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1}s^{\prime}_{2},s^{\prime}s}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s^{\prime}}G_{0}(\epsilon-\omega_{s})v_{s}$
$\displaystyle+v_{s^{\prime}_{2}}^{\dagger}G_{0}(\epsilon-\omega_{s_{2}^{\prime}})v_{s^{\prime}_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}^{\prime}}-\omega_{s_{2}^{\prime}})W_{s^{\prime}_{1}s^{\prime}_{2},ss^{\prime}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s^{\prime}}-\omega_{s})v_{s^{\prime}}G_{0}(\epsilon-\omega_{s})v_{s},$
(177) $\displaystyle\Sigma^{(3,0)}(\epsilon)$
$\displaystyle=\overline{V}_{s_{1}s_{2}}^{(3,2)}G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})(v_{s_{1}}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{2}}+v_{s_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}})$
$\displaystyle=v_{s_{2}}^{\dagger}[G^{(2,0)}(\epsilon-\omega_{s_{2}})-G_{0}(\epsilon-\omega_{s_{2}})]v_{s_{2}}-v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s^{\prime}})v_{s^{\prime}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}}$
$\displaystyle+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s^{\prime}})V_{s^{\prime}}^{(2,1)}(\epsilon-\omega_{s_{1}})G^{(2,0)}(\epsilon-\omega_{s_{1}})v_{s_{1}}$
$\displaystyle+v_{s_{2}}^{\dagger}G^{(2,0)}(\epsilon-\omega_{s_{2}})\overline{V}_{s_{1}}^{(2,1)}(\epsilon-\omega_{s_{2}})G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}}$
$\displaystyle+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s_{1}}^{\dagger}G^{(1)}(\epsilon-\omega_{s^{\prime}}-\omega_{s_{1}})W_{s^{\prime},s_{2}}^{(2,1)}(\epsilon-\omega_{s_{1}})G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{1}}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{2}}$
$\displaystyle+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s}^{\dagger}G^{(1)}(\epsilon-\omega_{s}-\omega_{s^{\prime}})W_{ss^{\prime},s_{1}s_{2}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{1}}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{2}}$
$\displaystyle+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s}^{\dagger}G^{(1)}(\epsilon-\omega_{s}-\omega_{s^{\prime}})W_{s^{\prime}s,s_{1}s_{2}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{1}}G_{0}(\epsilon-\omega_{s_{2}})v_{s_{2}}$
$\displaystyle+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s}^{\dagger}G^{(1)}(\epsilon-\omega_{s}-\omega_{s^{\prime}})W_{ss^{\prime},s_{1}s_{2}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}}$
$\displaystyle+v_{s^{\prime}}^{\dagger}G_{0}(\epsilon-\omega_{s^{\prime}})v_{s}^{\dagger}G^{(1)}(\epsilon-\omega_{s}-\omega_{s^{\prime}})W_{s^{\prime}s,s_{1}s_{2}}^{(2,2)}(\epsilon)G^{(1)}(\epsilon-\omega_{s_{1}}-\omega_{s_{2}})v_{s_{2}}G_{0}(\epsilon-\omega_{s_{1}})v_{s_{1}}.$
(178)
This, together with identities (153) and (154) justifies the proposed
representation of the self-energy.
## Appendix D Effect of the non-zero detuning
Figure 10: Spectral power density (scaled by $1/\Phi^{2}$) for of the giant
atom model as a function of $\Delta,\ k$ for various values of $k_{0}R$ and
$\gamma R=5$. Here the dashed black lines indicate the pole position of the
dressed Green’s function in the single excitation subspace. Figure 11: Second
order coherence function for the system with $k_{0}R=3\pi/11,\ \gamma R=5,\
\Delta=0.1,\ 0.5,\ 1.0$ (as before, the the second order coherence function is
dimensionless).
In this appendix we analyse the effect on non-zero detuning of the atom from
radiation on the observable quantities. In particular, we focus on the
spectral power density and the second order coherence function. As it was
discussed in Section III.3, the sharp bound-state-like peaks in the line-shape
of spectral density may be understood with a simple physical picture of an
effective cavity. When one increases (or decreases) the detuning from zero
value, one effectively changes the modes supported by the cavity and thus one
expects the position of the peaks to be shifted. The precise location of the
resonances in the spectral density as function of $\Delta$ and $k$ for various
values of $k_{0}R$ is shown in Figure 10. As we can see, for non-zero
dephasing $k_{0}R\neq 0$ the spectrum is not a symmetric function of $\Delta$.
For negative dephasing we find that emission into the zero modes is enhanced
for a negatively detuned atom $\Delta<0$, whereas the picture is opposite for
positive $k_{0}R$. In general, we can clearly resolve a pair of sharp peaks
which eventually merge together at certain values of parameters (e.g.
$k_{0}R=\Delta=0$). It is interesting to note that the location of this peaks
is almost entirely determined by the poles of the dressed propagator in the
single-excitation subspace
$(G^{(1)}(k))^{-1}=k+\Delta+i\gamma(1+e^{i(k+k_{0})R})=0$. This equation is
solved by
$\displaystyle k=\pm i\frac{R\gamma-iR\Delta-W_{n}(-\gamma
Re^{i(k_{0}-\Delta-i\gamma)R})}{R},$ (179)
where $W_{n}(z)$ is the $n^{\text{th}}$ branch of the Lambert $W$-function,
also known as the product logarithm. The real part of (179) with $n=0,\pm 1$
is plotted as black dashed lines in Figure 10. The vertical lines in Figure 10
represent the discontinuous jumps of the Lambert function across the branch
cut. As one may notice, the formula (179) is indeed in perfect agreement with
numerical results.
Let us now consider the second order coherence function at non-zero detuning.
Second order coherence for the system with $k_{0}R=3\pi/11,\ \gamma R=5,\
\Delta=0.1,\ 0.5,\ 1.0$ is shown in Figure 11. We note that the presence of
detuning does not affect the general trend of strong photon bunching in the
first channel and their corresponding anti-bunching in the second one, as was
discussed in Section III.3. Neither the detuning affects the presence of non-
differentiable peaks occurring at integer multiplies of delay time $R$.
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|
# Deep Learning Models for Calculation of Cardiothoracic Ratio from Chest
Radiographs for Assisted Diagnosis of Cardiomegaly
Tanveer Gupte Mrunmai Niljikar Manish Gawali
Viraj Kulkarni Amit Kharat Aniruddha Pant DeepTek Inc
###### Abstract
We propose an automated method based on deep learning to compute the
cardiothoracic ratio and detect the presence of cardiomegaly from chest
radiographs. We develop two separate models to demarcate the heart and chest
regions in an X-ray image using bounding boxes and use their outputs to
calculate the cardiothoracic ratio. We obtain a sensitivity of 0.96 at a
specificity of 0.81 with a mean absolute error of 0.0209 on a held-out test
dataset and a sensitivity of 0.84 at a specificity of 0.97 with a mean
absolute error of 0.018 on an independent dataset from a different hospital.
We also compare three different segmentation model architectures for the
proposed method and observe that Attention U-Net yields better results than
SE-Resnext U-Net and EfficientNet U-Net. By providing a numeric measurement of
the cardiothoracic ratio, we hope to mitigate human subjectivity arising out
of visual assessment in the detection of cardiomegaly.
## 1 Introduction
Chest X-rays (CXR) are most commonly used for the diagnosis of heart and chest
related pathologies. The research in computer-aided diagnosis of pathology
from chest X-ray has progressed rapidly in the past few years, which has
resulted in the generation of a massive corpus of open-source X-ray data along
with ground truths, and the development of a large number of complex deep
learning algorithms. However, the performance and outputs of complex deep
learning algorithms on these large datasets are often subjective and lack an
intuitive interpretable understanding of pathology.
Cardiomegaly manifests in the form of enlargement of the heart due to
congenital causes, high blood pressure, or as a result of other pathologic
conditions such as congenital heart diseases, valvular diseases, coronary
artery disease, athletic heart, etc. It presents itself with several forms of
primary or acquired cardiomyopathies and may involve enlargement of the right,
left, or both ventricles or the atriaamin_siddiqui_2020 . Diagnosis of
cardiomegaly is primarily made using imaging techniques, which aid in
measuring the heart’s size. Additionally, a quantitative measure called
Cardiothoracic ratio (CTR) is also used to determine the presence of
cardiomegaly. CTR is the ratio of maximal horizontal cardiac diameter (Wh) to
maximal horizontal thoracic (Wt) diameter (inner edge of ribs/edge of pleura)
as shown in figure 1.
$CTR=Wh/Wt$
The range of CTR values between 0.42 and 0.50 indicates a normal condition.
Values lesser than 0.42 or greater than 0.5 imply pathologic conditions. A
higher CTR is suggestive of cardiomegaly.
Figure 1: Width of heart and thorax used for calculation of CTR
We propose an automated method that employs a U-Net-based architecture to
segment the lungs and the heart from a CXR followed by calculation of the
cardiothoracic ratio to determine the presence or absence of cardiomegaly. The
segmentation outputs for the lung and heart region and the automated CTR
calculation makes the overall decision given by the deep learning model
interpretable. Moreover, we conduct three experiments with three different
segmentation-based architectures to decide the best segmentation-based
architecture compatible with the proposed method.
## 2 Related Work
Computer aided-diagnosis has been used to detect cardiomegaly in a chest X-ray
with or without using deep learning techniques. Initially, cardiomegaly was
identified by calculating CTR using the segmentation method2017SPIE10134E..0KD
hasan_lee_kim_lim_2012 ginneken_stegmann_loog_2006 . In one such study,
Candemir et al.candemir2016automatic used pre-segmented images to localize
the heart and lungs on the CXRs to extract the detailed radiographic index
from the heart and lung boundaries. These radiographic indices assisted in the
diagnosis of cardiomegaly. The study used multiple radiographic indexes to
build a classifier that classified patients with cardiomegaly with a
sensitivity of 0.77 at a specificity of 0.76.
Islam et al.islam2017abnormality and Candemir et al.candemir2018deep
proposed the use of simple classification deep learning architectures to
detect cardiomegaly. To predict Cardiomegaly’s presence with more objectivity,
CTR had to be calculated using deep learning techniques. A fully convolutional
neural network was employed to segment CXR images and calculate CTR. Li et
al.8675927 adopted a UNET architectureronneberger2015u that identified and
localized the lungs and heart to get the heart and chest width, which were
then used to calculate CTR. They used a private dataset of 5000
posteroanterior (PA) view CXRs, in which the lungs and heart boundaries were
annotated.
A similar technique was used by Chamveha et al.chamveha2020automated where
they used UNET architecture with VGG-16simonyan2014very as the backbone of
the encoder. Montgomeryjaeger2014two , JSRTshiraishi2000development , and a
subset of the NIH datasetwang2017chestx were used for training the model and
the validation set contained images from CheXpert dataset. Human radiologists
evaluated the obtained CTR measurements, and 76.5% of the AI results were
accepted and included in medical reports without any need for adjustment.
Sogancioglu et al.sogancioglu2020cardiomegaly compared segmentation and
classification approaches and demonstrated that segmentation models perform
better for detecting cardiomegaly on chest radiographs. Moreover, they
indicated that segmentation models needed fewer images to learn and the output
is interpretable as compared to classification models.
## 3 Data and Methods
### 3.1 Data
We aggregated the chest X-ray images from an open-source research dataset
NIHwang2017chestx , two private hospitals (which we name as D1, D2), and a
private dataset which was collected from population screening (which we name
as D3). A team of expert radiologists annotated the CXRs by drawing bounding
boxes around the heart and chest region using the VIA Annotation
Softwaredutta2019via . After drawing the bounding boxes, the CTR is
automatically calculated by the software by considering the relative cardiac
width and thoracic width from the annotated bounding boxes. A total of 2623
CXRs (table 1) were used for the study. For the NIH dataset, out of 30,805
patients with 112,120 radiographs, we randomly sampled 1440 X-rays of 1440
patients. The sampling was done so that cardiomegaly was present in half of
the X-ray images (720) and not present in the other half (720). Both sets had
50% CXRs with an anteroposterior (AP) view (360) and the other half with a PA
view(360). The reason for this sampling was to create enough data variation to
increase the robustness of the model. 1000 X-ray images from D1 and D3 were
used. We used 183 images from D2 to evaluate the model’s performance on an
out-of-source dataset. For training, 1952 CXRs were used. 244 CXRs were used
for validation, and 244 CXRs were used for testing the result.
Dataset | Train | Validation | Test | Total
---|---|---|---|---
NIH | 1152 | 144 | 144 | 1440
D1 + D3 | 800 | 100 | 100 | 1000
D2 | - | - | 183 | 183
Total | 1952 | 244 | 244 + 183 | 2623
Table 1: Distribution of Cardiomegaly CXR’s
### 3.2 Pre-processing and Augmentation
Figure 2: Illustration of the architecture of the segmentation pipeline for
calculation of CTR to determine the presence of Cardiomegaly. The model gives
two segmentation maps for heart and thorax, which are then processed to
calculate CTR.
The CXRs of various dimensions were reduced to 512x512x3 pixels. All the CXRs
were normalized before training such that values of pixels ranging from 0 to
255 were bounded in the range 0 to 1. The model was trained to get a dual-
channel output. The first channel of the output was a heart’s mask and the
second channel was the thorax’s mask.
We experimented with different types of augmentations to upsample the training
datasethussain2017differential . We resorted to geometric augmentations
(except for Gaussian blur) of the CXRs as we wanted the model to detect the
width of the heart and thorax efficiently. We found that combining shearing
(along with the x and y-axis), scaling, a little bit of grid distortion, and
gaussian blurring improved the model’s results. The training and validation
dataset was randomly upsampled by 75%. The augmentation was done so that only
one or two of the augmentations, as mentioned earlier, were performed at a
time. To maintain the output accuracy while augmentation, the output masks for
the heart and thorax were also changed to match the correct outlines of the
heart and thorax on augmented CXR. To avoid randomness while training every
different model, the augmentations of the CXRs were done only once, and the
subsequent data, along with output masks, were stored. The final training
dataset had 1952 original CXRs and 1464 (75% upsampled) augmented CXRs, i.e.,
a total of 3416 CXRs.
### 3.3 Procedure and Post-processing
We used the same training and validation datasets for training on all three
models. Adam optimizerkingma2014adam and binary cross-entropy loss were used
with a learning rate that was reduced on the plateau i.e. the learning rate
was reduced when the metric (validation loss) stopped improving. The model
which had the lowest validation loss was saved and used for analysis. While
inferring the model, for each CXR, the network predicts a region of interest
with bounding coordinates for both heart and chest. These coordinates were
used to determine the width of the chest and heart, which was then used to
calculate the CTR. The ground truth was selected based on annotations, i.e.,
if the CTR for annotated CXRs was more than 0.5, it was labeled as
Cardiomegaly.
The output of the model contained a pixel-wise probability for the input
image. The pixel probability was thresholded to get a clear-cut mask. The
pixel probability of less than the threshold value was converted to 0, and
anything above it was converted to 1. Following the thresholding,
morphological transformationssreedhar2012enhancement ravi2013morphological
were performed on the output masks to reduce the noise and error. The
morphological transformation did not improve the specificity and the
sensitivity of the model; however, it reduced the MAE and RMSE (Root Mean
Squared Error). The transformations that yielded better results were erosion
for two iterations, followed by dilation for one iteration. The change in the
output masks doesn’t seem apparent to human eyes, but we observed MAE
reduction by 12.5% and RMSE by 9%.
The performance of all three models’ was calculated on two independent
datasets. One was a held-out test dataset with the same source as the training
and validation dataset, while the other was an out-of-source dataset D2. We
compared our results for three different UNET ronneberger2015u based
architectures. The first was enhanced UNET with Spatial-attention
gateoktay2018attention khanh2020enhancing (Attention UNET) with Xception
encoderchollet2017xception . The second one was UNET with Squeeze-and-
Excitationhu2018squeeze network blocks incorporated with the ResNext50
xie2017aggregated (SE-ResNext 50) backbone. The third one was simple UNET
with EfficientNet-b4tan2019efficientnet 9150621 as its encoder. All of these
networks were pre-trained with ImageNet5206848 weights.
## 4 Results
Figure 3: Scatter plot of annotated CTR and predicted CTR on hold out test
dataset Figure 4: Scatter plot of annotated CTR and predicted CTR on D2 test
dataset
(a) Radiograph with CTR 0.39; Cardiomegaly absent
(b) Radiograph with CTR 0.57; Cardiomegaly present
Figure 5: Visual presentations of the original, radiologist-annotated(red
boxes) and AI model predicted (yellow boxes) radiographs.
Three different multi-class segmentation architectures were used to train
three different models on the same datasets to determine the best performing
robust model. Although inherently, all were UNET architectures, the encoders
and the intermediate layers were different for each model. Hyperparameter
optimization, image augmentation, and image processing were done to ensure the
best possible results. The comparison of these models on the test dataset and
D2 dataset can be seen in Tables 2 and 3, respectively. Since the results were
close, it was hard to assess which model performs better for cardiomegaly
classification based on CTR. It can be observed that the best performance
classification metrics, i.e., sensitivity, specificity, and F-1 score, vary
for all the models. Also, a trade-off between sensitivity and specificity
cannot be altered as the cut-off for positive prediction of cardiomegaly is
based on the theoretical cut-off value of CTR at 0.5. For classification
performance, UNET SE-Resnext has the lowest performance for the D2 dataset,
lowest F-1 score on the test dataset, making it the weakest contender. On the
test dataset, the F-1 score of the UNET EfficientNet model is only marginally
higher than Attention UNET, whereas the latter model has a significantly
higher f-1 score on the D2 dataset. For regression metrics, however, the
results were explicit. The Mean Absolute Error (MAE) and Root Mean Squared
Error (RMSE) for Attention UNET are the lowest indicating far better masks
resulting in much accurate CTR calculation. Attention UNET was used for
further analysis due to its better performance. The scatter plot of predicted
CTR and annotated CTR can be seen in figure 3 and 4. The misclassification of
cardiomegaly, where CTR is close to 0.5, is inevitable due to inter-reader
variability. The misclassification cases, where the difference between
annotated and predicted CTR is significantly high, are 6 for test datasets and
only 5-6 for D2 datasets.
Metrics | | Attention
---
UNET
| SE-Resnext
---
UNET
| Efficient Net
---
UNET
Sensitivity | 0.96 | 0.87 | 0.94
Specificity | 0.81 | 0.86 | 0.83
F-1 Score | 0.87 | 0.86 | 0.88
MAE | 0.0209 | 0.0206 | 0.0328
RMSE | 0.0312 | 0.0317 | 0.0798
Table 2: Results on the held-out test dataset Metrics | | Attention
---
UNET
| SE-Resnext
---
UNET
| Efficient Net
---
UNET
Sensitivity | 0.87 | 0.8 | 0.93
Specificity | 0.97 | 0.91 | 0.88
F-1 Score | 0.88 | 0.78 | 0.81
MAE | 0.0181 | 0.0206 | 0.0248
RMSE | 0.0282 | 0.0317 | 0.058
Table 3: Results on D2 (out-of-source) dataset
## 5 Conclusion
The diagnosis of cardiomegaly is subjective and varies from radiologist-to-
radiologist. A limitation of the classification approach to detect
cardiomegaly is that the results are not interpretable and lack objectivity.
Furthermore, many radiologists diagnose cardiomegaly only if it’s severe, and
many borderline cases go unnoticedolatunji2019caveats . Using the
classification approach will not improve the correctness of the diagnosis as
it does not CTR into account. In this study, we built a model to detect
changes in the width of the cardiac silhouette. Unlike other pathologies,
cardiomegaly is a quantifiable pathology, and calculating the CTR will help
establish the necessary output. Our adapted Attention UNET architecture
exhibited excellent chest X-ray image segmentation.
To determine cardiomegaly, CTR is usually calculated on the posteroanterior
(PA) view of the CXR rather than anteroposterior due to better visualization
of the cardiac silhouette in PA viewchon2011calculation . We deliberately
chose to use images without taking into account whether they are AP or PA
views. This step was done to correctly identify the cardiac silhouette and
thoracic outline to get the width of both, respectively. The ratio calculated
will guide the radiologists in clinical settings without subjecting them into
considering the established ideology of a better CTR assessment in PA view.
Another challenge in our study was to make the model robust as to make it work
in practical clinical settings. Since different hospitals use different X-ray
machines with various manual and automatic settings before exposing the
radiographic films, the seemingly similar CXRs may show irregularities. Thus a
model trained on one-source of data usually performs poorly on another source
of data. We tackled this problem by incorporating data from multiple sources
while training and using image augmentations. A dataset from an entirely
different hospital setup was kept aside to evaluate out-of-source performance.
The model’s ability to perform well on out-of-source hospital dataset was
demonstrated.
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# Asymptotic behavior of the number of distinct values in a sample from the
geometric stick-breaking process
Pierpaolo De Blasi<EMAIL_ADDRESS>University of Torino and Carlo
Alberto, Torino, Italy Ramsés H. Mena Universidad Nacional Autónoma de
Mexico, Mexico Igor Prünster Bocconi University and BIDSA, Milano, Italy
###### Abstract
Discrete random probability measures are a key ingredient of Bayesian
nonparametric inferential procedures. A sample generates ties with positive
probability and a fundamental object of both theoretical and applied interest
is the corresponding random number of distinct values. The growth rate can be
determined from the rate of decay of the small frequencies implying that, when
the decreasingly ordered frequencies admit a tractable form, the asymptotics
of the number of distinct values can be conveniently assessed. We focus on the
geometric stick-breaking process and we investigate the effect of the choice
of the distribution for the success probability on the asymptotic behavior of
the number of distinct values. We show that a whole range of logarithmic
behaviors are obtained by appropriately tuning the prior. We also derive a
two-term expansion and illustrate its use in a comparison with a larger family
of discrete random probability measures having an additional parameter given
by the scale of the negative binomial distribution.
Keywords: Bayesian Nonparametrics; random probability measure; geometric
stick-breaking process; asymptotic growth rate; occupancy problem.
## 1 Introduction
Discrete random probability measures can be represented by random frequencies
at random locations as
$\tilde{p}(\mathrm{d}x)=\sum_{j\geq 1}w_{j}\delta_{x_{j}}(\mathrm{d}x).$ (1)
The frequencies $(w_{j})_{j\geq 1}$ are $(0,1)$-valued variables such that
$\sum_{j\geq 1}w_{j}=1$ almost surely (a.s.), and the locations
$(x_{j})_{j\geq 1}$ are draws from some distribution on a Polish space
$\mathbb{X}$ endowed with the corresponding Borel $\sigma$-field. Discrete
measures like $\tilde{p}$ are naturally suited to describe the structure a
population made of potentially infinite different species or types, labeled by
$x_{j}$, with certain random proportions modeled through $w_{j}$. Clearly, a
sample drawn from $\tilde{p}$ will exhibit ties with positive probability and
thus the random number of distinct values in a sample of size $n$, here
denoted by $K_{n}$, is of great interest. From a Bayesian nonparametric
perspective the law of $\tilde{p}$ represents the prior distribution.
Inference is carried out by predicting the number of new distinct values in an
additional sample, conditional on an observed sample. See Lijoi et al., 2007b
. According to the applied context at issue the distinct values or species are
interpreted as distinct genes (Lijoi et al., 2007a, ), words (Teh,, 2006),
economic agents (Lijoi et al.,, 2016) etc. Another important statistical use
of discrete random probality measures is in mixture modeling, when a layer is
added to model the data distribution as in
$Y_{i}\sim f(Y_{i}|X_{i}),\quad
X_{1},X_{2},\ldots|\tilde{p}\overset{\text{iid}}{\sim}\tilde{p}$
for some probability kernel $f(y|x)$. Here $\tilde{p}$ acts as mixing
distribution and $K_{n}$ represents the random number of mixture components,
thus providing a flexible way to model unobserved heterogeneity in the
population. The mixture is characterized by the component distribution
$f(y|x_{j})$, usually referred as the $j$th mixture component, and the mixing
weights $w_{j}$. See De Blasi et al., (2015) for a recent review on the
inferential implications of different choices of $\tilde{p}$.
In probability theory, distributional properties of $K_{n}$ are of prime
interest in combinatorial stochastic processes; see e.g. Arratia et al.,
(2003), Pitman, (2006), Gnedin, (2010), Gnedin et al., (2007). The techniques
employed to study the law of $K_{n}$ depend on the construction of the random
frequencies $(w_{j})$. Karlin, (1967) studied the case of fixed frequencies
and derived a key result, which forms the basis to establish general strong
laws for $K_{n}$: it states that the growth of $K_{n}$ is ultimately
determined by how small the small frequencies are, which can be conveniently
expressed by the tail behavior of $(w_{j})$ once decreasingly ordered. In
particular, the faster the decay to zero, the slower $K_{n}$ diverges to
infinity as $n$ increases. There exist essentially two regimes, logarithmic
and polynomial growth. Notable examples are, respectively, the Dirichlet
process (Ferguson,, 1973) and its two parameter extension known as Pitman-Yor
process (Pitman and Yor,, 1997). The associated distributions of the
frequencies in decreasing order, termed Poisson-Dirichlet and the two-
parameter Poisson-Dirichlet, respectively, are not tractable enough for a
direct application of Karlin’s theory. Instead, the distribution of $K_{n}$ is
derived from the Ewens and the Pitman-Ewens sampling formulae, cf. Pitman,
(2006). In the former case $K_{n}$ is asymptotically normal, with both mean
and variance of the order $\log n$. In the latter case the scale of $K_{n}$ is
$n^{\alpha}$, where $\alpha\in(0,1)$ is the discount parameter of the Pitman-
Yor process. The logarithmic growth of the Dirichlet process was first pointed
out in Korwar and Hollander, (1973). Within the logarithmic regime growth
behaviors of $K_{n}$ slower than the logarithm, e.g. $(\log n)^{\alpha}$ with
$\alpha<1$ or even iterated logarithms, can be achieved with so-called
hierarchical processes Camerlenghi et al., (2019); see also Argiento et al.,
(2020); Bassetti et al., (2020). In this paper we are able to identify models
leading to growth rates of the type $(\log n)^{\beta}$ with $\beta>1$,
specifically we establish a growth rate $(\log n)^{m+2}$, for $m$ a
nonnegative integer, for a class of tractable class of discrete random
probability measures. We stress that from a modeling perspective it is crucial
to have tractable models, which cover the whole range of possible growth
rates. See Lijoi et al., 2007c ; De Blasi et al., (2015); Dahl et al., (2017);
Caron and Fox, (2017); Ayed et al., (2019); Di Benedetto et al., (2020) for
motivation and discussion of these issues in diverse application contexts also
beyond exchangeability. Note that a power logarithmic growth of
$\mathrm{E}(K_{n})$ can be obtained by means of a Dirichlet prior with a
somehow artificial sample size-dependent specification of the total mass
parameter; in particular, one needs the total mass parameter to grow with n,
which leads to an increasingly informative prior as more data becomes
available, an unnatural scenario.
The asymptotic evaluation $K_{n}$, together with its limiting distribution,
has been also object of extensive research in the context of regenerative
composition structures; see Gnedin, (2010) for a survey. In this setting the
frequencies $(w_{j})$ are constructed from the range of a multiplicative
subordinator, that is from the exponential transform $1-\exp\\{S(t)\\}$ of a
subordinator $S(t)$. The logarithmic and polynomial regimes can be recovered
from Karlin’s theory according to the variation at zero of the right tail of
the Lévy measure of $S(t)$. The $\log n$ regime corresponds to finite Lévy
measures, that is when $S(t)$ is a compound Poisson process. In this case, the
frequencies $(w_{j})$ can be conveniently defined in terms of a stick-breaking
procedure, or residual allocation scheme, with
$w_{j}=W_{j}\textstyle\prod_{\ell<j}\displaystyle(1-W_{\ell})$ (2)
for $(W_{\ell})_{\ell\geq 1}$ independent and identically distributed (iid)
$(0,1)$-valued random variables with distribution determined by the Lévy
measure. Exploiting the renewal representation of the composition structure,
aymptotics for the moments of $K_{n}$ and a central limit theorem can be
derived; cf. Gnedin, (2004), Gnedin et al., (2009). Gnedin et al., 2006a show
that when the right tail of the Lévy measure is regularly varying at zero with
index $-1<\alpha<0$, the scale of $K_{n}$ is $n^{\alpha}$ and the partition
structure induced by the Pitman-Yor process can be recovered (Gnedin and
Pitman,, 2005). In contrast, when the right tail diverges at zero like a
slowly varying function, e.g. for $S(t)$ a gamma subordinator, a central limit
theorem with mean of the order $(\log n)^{2}$ and variance of the order $(\log
n)^{3}$ is obtained, cf. Gnedin et al., 2006b .
Discrete random probability measures with $w_{j}$ as in (2), not necessarily
with identically distributed $(W_{\ell})_{\ell\geq 1}$, have been proposed in
Ishwaran and James, (2001) as a Bayesian nonparametric model and termed stick-
breaking priors. The Dirichlet and the Pitman-Yor processes belong to this
class, their distinctive property being that the law of $(w_{j})_{j\geq 1}$ is
invariant under size-biased permutation. In this setting, the distribution of
$W_{1}$ is called the structural distribution of $(w_{j})$, and the limiting
behavior of $K_{n}$ in the Dirichlet and the Pitman-Yor process cases can be
also derived using Karlin’s theory from the variation at zero of the
structural distribution; see Gnedin et al., (2007).
In this paper we further broaden the realm of application of the fundamental
result of Karlin in order to derive a two-term expansion of the mean of
$K_{n}$. The expansion relies on de Haan’s regular variation theory and
requires a precise assessment of the tail behavior of $(w_{j})$ together with
a deconditioning argument, cf. Theorem 1. To illustrate the applicability of
this technique, we consider the geometric stick-breaking process, first
proposed in Fuentes-García et al., (2010), which gained quite some popularity
in Bayesian applications (Mena et al.,, 2011; Gutiérrez et al.,, 2014;
Hatjispyros et al.,, 2018). It is a discrete random probability measure (1)
with locations independent of the frequencies and, importantly,
$(w_{j})_{j\geq 1}$ naturally arranged in decreasing order, which facilitates
the evaluation of the tail behavior of the sequence. Specifically the
frequencies are of geometric type,
$w_{j}=p(1-p)^{j-1},\quad j=1,2,\ldots$ (3)
with $p$, the probability of success, random and endowed with a (prior)
distribution $\pi(p)$ on $(0,1)$. In Theorem 2 we derive a two-term expansion
for a choice of $\pi(p)$, the key technical tool being the regular variation
of fractional integrals. As anticipated, the leading term shows that the mean
of $K_{n}$ can covers the whole range of logarithmic behaviors $(\log
n)^{m+2}$, for $m$ a nonnegative integer, upon setting $\pi(p)$ as an
exponential transform of the gamma distribution of shape parameter $m$. From a
practical perspective this result widens the range of achievable asymptotic
behaviors by means of tractable models and also allows a principled prior
elicitation. To illustrate the importance of the second order term in the
expansion, we also consider an extension of the geometric stick-breaking
process, which has an additional parameter $s$ corresponding to the scale of
the negative binomial distribution. Such a construction reduces to the
geometric stick-breaking process when $s=2$ and was exploited by De Blasi et
al., (2020) within a mixture model, to which the present study provides
further theoretical support. The frequencies $(w_{j})_{j\geq 1}$ are still
decreasingly ordered and are available in closed form for any integer $s\geq
2$. The parameter $s$ determines the tail behavior of $(w_{j})_{j\geq 1}$, the
larger $s$ the slower the decay to zero. In order to single out the effect of
$s$ on $K_{n}$, we set $s=3$ and compare the asymptotic behavior of the mean
of $K_{n}$ with that of the geometric stick-breaking case, while keeping
$\pi(p)$ to be uniform. It turns out that $K_{n}$ grows faster for $s=3$, as
predicted by Karlin’s theory, the difference however emerging only in the
second order term of the expansion, cf. Proposition 2. We conjecture that
similar conclusions apply also for $s$ an integer larger than $3$ and other
prior specifications of $\pi(p)$, although we do not pursue it here. It would
be of interest to investigate the asymptotics of higher order moments like the
variance and whether a central limit theorem holds. These are left for future
research.
Layout of the paper. In Section 2 we review Karlin’s theory and establish a
general two-term expansion of the mean of $K_{n}$. In Section 3 we introduce
the geometric stick-breaking process and investigate the impact of the choice
of prior $\pi(p)$ on the asymptotic behavior of $K_{n}$. In Section 4 we deal
with the negative binomial extension and apply the asymptotic expansion of
Section 2 to show that the scale parameter $s$ enters in the second order
term. Some proofs are deferred to the Appendix.
Notation. Let $F(x)$ be a positive nondecreasing function on $\mathbb{R}$
with $F(x)=0$ for $x\leq 0$ and $\alpha\geq 0$. The fractional integral of
order $\alpha$ of $F(x)$ is given by
${}_{\alpha}F(x)=\frac{1}{\Gamma(\alpha+1)}\int_{0}^{x}(x-t)^{\alpha}{f}(t)\mathrm{d}t.$
We use $f\sim g$ for $f/g\to 1$, the limit being clear from the context. When
either $f$ or $g$ is random, the notation $f\sim_{\text{a.s.}}g$ means that
the asymptotic relation holds with probability one. For $x$ a real number,
$\lfloor x\rfloor$ is the integer part of $x$.
## 2 Occupancy problem and regular variation
Let $\tilde{p}$ be a discrete random probability measure (1). Assume
$(w_{j})_{j\geq 1}$ and $(x_{j})_{j\geq 1}$ are independent with
$(x_{j})_{j\geq 1}$ independent and identically distributed form a non atomic
distribution. Then $\tilde{p}$ is a species sampling model (Pitman,, 1995).
The partition induced by a sample from $\tilde{p}$ depends only on the random
frequencies $(w_{j})_{j\geq 1}$ and can be studied in terms of a multinomial
occupancy problem. The theory is well established and dates back to the
seminal paper Karlin, (1967). The main tools are a Poissonization argument and
regular variation theory. We provide a concise overview taking the set-up from
Gnedin et al., (2007).
The multinomial occupancy problem can be described as the experiment of
throwing balls independently at a fixed infinite series of boxes, with
probability $w_{j}$ of hitting the $j$th box. First consider the case of
fixed, or non random, frequencies. As $n$ balls are thrown, their allocation
is captured by the array $X_{n}=(X_{n,j})_{j\geq 1}$ where $X_{n,j}$ is the
number of balls out of the first $n$ that fall in box $j$. $K_{n}$, the number
of occupied boxes, is then given by $K_{n}=\sum\nolimits_{j\geq
1}\mathds{1}(X_{n,j}>0)$ with mean
$\mathrm{E}(K_{n})=\sum\nolimits_{j\geq 1}(1-(1-w_{j})^{n}).$
In general, it is difficult to work with $\mathrm{E}(K_{n})$ since the
indicators in $K_{n}$ are not independent. In the Poissonized version of the
problem the balls are thrown in continuous time at epochs of a unit rate
Poisson process $(P(t),t\geq 0)$, which is independent of
$(X_{n},n=1,2,\ldots)$. The balls then fall in the boxes according to
independent Poisson processes $(X_{j}(t))_{t\geq 0}$, at rate $w_{j}$ for box
$j$. Hence $K(t):=K_{P(t)}=\sum\nolimits_{j\geq 1}\mathds{1}(X_{j}(t)>0)$ and
$\Phi(t):=\mathrm{E}(K(t))=\sum\nolimits_{j\geq 1}(1-\mathrm{e}^{-tw_{j}}).$
Encoding the frequencies into the counting measure
$\nu(\mathrm{d}x)=\sum_{j\geq 1}\delta_{w_{j}}(\mathrm{d}x)$ and integrating
by parts,
$\Phi(t)=\int_{0}^{1}(1-\mathrm{e}^{-tx})\nu(\mathrm{d}x)=t\int_{0}^{1}\mathrm{e}^{-tx}\overrightarrow{\nu}(x)\mathrm{d}x,$
where $\overrightarrow{\nu}(x)=\nu([x,1)),$ the right tail of $\nu$,
represents the number of frequencies $w_{j}$ not smaller than $x$. $\Phi(t)$
provides an approximation of $\mathrm{E}(K_{n})$ for $n$ large according to
$|\mathrm{E}(K_{n})-\Phi(n)|\leq\textstyle\frac{2}{n}\displaystyle\Phi(n)\to
0$ (4)
cf. (Gnedin et al.,, 2007, Lemma 1). The convenience of working with $\Phi(t)$
is that, being $\Phi(t)$ the Laplace-Stieltjes transform of
$\overrightarrow{\nu}(x)$, its behavior as $t\to\infty$ is determined by the
behavior of $\overrightarrow{\nu}(x)$ as $x\to 0$ by an application of the
Tauberian theorem; see Bingham et al., (1987) for a full account on Abel-
Tauberian theorems for Laplace transforms. Hence, ultimately, by regular
variation theory the growth of $\mathrm{E}(K_{n})$, as $n\to\infty$, is
determined by the behavior of $\overrightarrow{\nu}(x)$ at zero. In the case
of random frequencies, the same result holds with the counting measure
$\nu(\mathrm{d}x)$ being replaced by its mean measure, and correspondingly
adapting the meaning of $\overrightarrow{\nu}(x)$. See (Gnedin et al.,, 2007,
Section 7, Page 162) and Section 3 for an illustration.
Here we work under the hypothesis that $\overrightarrow{\nu}(x)$ is slowly
varying at zero, that is $\lim_{x\to 0}\overrightarrow{\nu}(\lambda
x)/\overrightarrow{\nu}(x)=1$ for all $\lambda>0$. According to (Bingham et
al.,, 1987, Theorems 1.7.1’ and 1.7.6) (see also (Gnedin et al.,, 2007,
Proposition 19)), $\Phi(1/x)\sim\overrightarrow{\nu}(x)$ as $x\to 0$, so that
via (4)
$\mathrm{E}(K_{n})\sim\overrightarrow{\nu}\big{(}\textstyle\frac{1}{n}\displaystyle)\quad\mbox{as
}n\to\infty,$
cf. (Karlin,, 1967, Theorem 1’). In Theorem 1 we derive a two term expansion
of $\mathrm{E}(K_{n})$ under the hypothesis that $\overrightarrow{\nu}(x)$ is
a de Haan slowly varying function at zero, that is for a constant $c$ and a
slowly varying function $\ell(x)$ at zero, called the auxiliary function of
$\overrightarrow{\nu}(x)$,
$\frac{\overrightarrow{\nu}(\lambda x)-\overrightarrow{\nu}(x)}{\ell(x)}\to
c\log\lambda,\quad\text{as }x\to 0.$ (5)
###### Theorem 1.
If $\ell(x)$ is slowly varying at zero and $c\geq 0$ satisfy (5) for all
$\lambda>0$, then
$\mathrm{E}(K_{n})=\overrightarrow{\nu}(1/n)-c\gamma\ell(1/n)+o(\ell(1/n)),\quad\text{as
}n\to\infty$
where $\gamma$ is the Euler-Mascheroni constant.
The proof consists in an adaptation to the present setting of (Bingham et
al.,, 1987, Theorem 3.9.1) for the study of the remainder of Tauberian
theorem, $\Phi(1/x)-\overrightarrow{\nu}(x)$, as $x\to 0$, combined with an
application of (4). The proof is reported in the Appendix. In order to apply
this result, one needs to establish the variation of $\overrightarrow{\nu}(x)$
at $0$, so some explicit or at least tractable form of
$\overrightarrow{\nu}(x)$ is in order. In the next two sections we apply the
asymptotic expansion of Theorem 1 to species sampling priors that features
stochastically decreasing frequencies for which $\overrightarrow{\nu}(x)$ is
tractable enough.
## 3 Geometric stick-breaking process
The geometric stick breaking process is a species sampling model with random
frequencies $(w_{j})_{j\geq 1}$ of geometric type,
$w_{j}=p(1-p)^{j-1},\quad j=1,2,\ldots$
with random success probability $p$. The number of frequencies $w_{j}$ not
smaller than $x$, $\max\\{j:\ p(1-p)^{j-1}\geq x\\}$, can be explicitly found
as the solution in $j$ to the equation $p(1-p)^{j-1}=x$. By direct calculation
$\overrightarrow{\nu}(x,p)=\bigg{\lfloor}\frac{\log(x/p)}{\log(1-p)}+1\bigg{\rfloor}\,\mathds{1}_{(p\geq
x)},$
where the notation $\overrightarrow{\nu}(x,p)$ makes the dependence on $p$
explicit. The case of fixed $p$ provides an illustration of Theorem 1.
###### Example 1.
Let $K_{n}$ be the number of distinct values among $n$ iid draws from the
geometric distribution with success probability $p$. Accurate formulae for the
mean and the variance of $K_{n}$ are given in Archibald et al., (2006). Since
$\overrightarrow{\nu}(x,p)\sim\log x/\log(1-p)$ as $x\to 0$,
$\overrightarrow{\nu}(x,p)$ is a de Haan slowly varying function with
auxiliary function $\ell(x)=1$ and $c=1/\log(1-p)$, cf. (5). Hence Theorem 1
yields
$\mathrm{E}(K_{n})=\bigg{\lfloor}\frac{\log(np)}{|\log(1-p)|}+1\bigg{\rfloor}+\frac{\gamma}{|\log(1-p)|}+o(1)\quad\mbox{as
}n\to\infty,$
in accordance with the expansion of (Archibald et al.,, 2006, Theorem 1).
Now return to the random case with $\pi(p)$ on $(0,1)$ denoting the (prior)
distribution of the success probability $p$ in (3). The results about the
expected value of $K_{n}$ now hold with $\nu(\mathrm{d}x)$ being the mean
measure of the counting measure $\sum_{j\geq 1}\delta_{w_{j}}$ and
$\overrightarrow{\nu}(x)$ obtained by averaging the number of frequencies
$w_{j}$ not smaller than $x$ with respect to $\pi(p)$:
$\overrightarrow{\nu}(x)=\int_{0}^{1}\overrightarrow{\nu}(x,p)\pi(p)\mathrm{d}p.$
In the sequel it is convenient to work with
$m(x)=\int_{x}^{1}\frac{\log x-\log p}{\log(1-p)}\pi(p)\mathrm{d}p,$
since $m(x)\leq\overrightarrow{\nu}(x)\leq m(x)+1$. The variation of
$\overrightarrow{\nu}(x)$ in zero can then be studied in terms of $m(x)$. By
the change of variable $t=\log 1/p$,
$m(x)=\int_{0}^{\log
1/x}\bigg{(}\log\frac{1}{x}-t\bigg{)}\pi(\mathrm{e}^{-t}){f}(t)\mathrm{d}t,\quad{f}(t)=\frac{\mathrm{e}^{-t}}{-\log(1-\mathrm{e}^{-t})}$
(6)
Properties of ${f}(t)$ in (6) are collected in Lemma 1, whose proof is
deferred to the Appendix.
###### Lemma 1.
The function ${f}(t)$ defined in (6) is nondecreasing on $\mathbb{R}_{+}$ with
$\lim_{t\to 0}{f}(t)=0$, $\lim_{t\to\infty}{f}(t)=1$ and
$1-{f}(t)\sim\mathrm{e}^{-t}/2$ as $t\to\infty$. Moreover
$\int_{0}^{\infty}(1-{f}(t))\mathrm{d}t=\gamma$, with
$\gamma=-\Gamma^{\prime}(1)-\int_{0}^{\infty}(\log
x)\mathrm{e}^{-x}\mathrm{d}x$ the Euler-Mascheroni constant.
The variation at zero of $m(x)$ is determined by ${f}(t)$ and the success
probability distribution $\pi(p)$. First consider $p$ uniformly distributed on
the unit interval.
###### Proposition 1.
Let $p$ in (3) be uniformly distributed on $(0,1)$. Then
$\displaystyle\overrightarrow{\nu}(x)$ $\displaystyle=\frac{1}{2}\big{(}\log
1/x\big{)}^{2}-\gamma\log 1/x+O(1),\quad x\to 0$
$\displaystyle\mathrm{E}(K_{n})$ $\displaystyle=\frac{1}{2}(\log n)^{2}+o(\log
n),\quad n\to\infty.$
###### Proof.
For $\pi(p)=\mathds{1}_{(0,1)}(p)$, $m(x)$ in (6) is given by
$m(x)=\int_{0}^{\log 1/x}(\log 1/x-t){f}(t)\mathrm{d}t={}_{1}F(\log 1/x),$
where $F(t)=\int_{0}^{t}{f}(s)\mathrm{d}s$ and
${}_{1}F(x)=\int_{0}^{x}(x-t){f}(t)\mathrm{d}t$ is the fractional integral of
order one of $F$. To prove the first statement, it is sufficient to prove it
for $m(x)$ in place of $\overrightarrow{\nu}(x)$. Integrating by parts,
${}_{1}F(x)=\int_{0}^{x}F(t)\mathrm{d}t$. Hence, since $\log 1/x\to\infty$ as
$x\to 0$, we derive an asymptotic expansion of $F(x)$ as $x\to\infty$.
According to Lemma 1, ${f}(x)$ is a distribution function on $\mathbb{R}_{+}$.
Moreover, given $1-{f}(t)\sim\mathrm{e}^{-t}/2$ as $t\to\infty$, the
distribution function ${f}(x)$ has moments of any order and, in particular,
the first moment is equal to the Euler-Mascheroni constant $\gamma$. Then,
$F(x)$ is regularly varying at infinity with exponent $\beta=1$ and, as
$x\to\infty$,
$F(x)=x-\int_{0}^{x}(1-{f}(t))\mathrm{d}t=x-\gamma+\int_{x}^{\infty}(1-{f}(t))\mathrm{d}t=x-\gamma+O(\mathrm{e}^{-x}).$
(7)
Computing $\int_{0}^{x}F(t)\mathrm{d}t$ with the asymptotic expansion
$F(x)\sim x-\gamma+O(\mathrm{e}^{-x})$ leads to
${}_{1}F(x)=\int_{0}^{x}F(t)\mathrm{d}t=\frac{(x-\gamma)^{2}}{2}+O(1),\quad
x\to\infty.$
Substituting $x$ for $\log 1/x$ yields the first statement. In view of the
application of Theorem 1, note that, as $x\to 0$,
$\displaystyle m(\lambda x)-m(x)$
$\displaystyle=\frac{1}{2}\big{(}(\log(\lambda x))^{2}-(\log
x)^{2}\big{)}+\gamma\big{(}\log(\lambda x)-\log x)+O(1)$
$\displaystyle=\frac{1}{2}\big{(}(\log x)^{2}+2\log\lambda\log x-(\log
x)^{2}\big{)}+O(1)=\log\lambda\log x+O(1)$
so that, as $x\to 0$, $(m(\lambda x)-m(x))/\log x\to\log\lambda$. Hence
$\overrightarrow{\nu}(x)$ is a de Haan slowly varying function at zero with
auxiliary function $\ell(x)=\log x$ and $c=1$, cf. (5). An application of
Theorem 1 yields the second statement. ∎
###### Remark 1.
Using only the leading term of the expansion of $m(x)$ in the application of
Theorem 1, we would get the second order term in the asymptotic expansion of
$\mathrm{E}(K_{n})$ wrong, i.e. differing by $\gamma\log n$. Hence, in this
case, an application of Karamata’s Theorem to the evaluation of
$\int_{0}^{x}(x-t){f}(t)\mathrm{d}t$ would be not precise enough, as the
latter would yield $\int_{0}^{x}(x-t){f}(t)\mathrm{d}t\sim\frac{1}{2}xF(x)$
and, in turn, $m(x)\sim\frac{1}{2}(\log\frac{1}{x})^{2}$.
Next we tackle the case of the success probability distribution $\pi(p)$
chosen such that the integrand in (6) behaves like $t^{m}{f}(t)$ for $m$ a
positive integer. This is obtained by setting $p=\mathrm{e}^{-X}$ for
$X\sim\text{ga}(m+1,1)$, a gamma distributed random variable with shape $m+1$
and unit rate. Note $m=0$ yields the uniform distribution considered in
Proposition 1. As detailed in Theorem 2 below, this choice makes
$\mathrm{E}(K_{n})$ grow as a power of $\log n$ with exponent determined by
$m$. Before that, we first provide an illustration of how the arguments used
in Proposition 1 can be adapted to the case $m=1$, paving the way for the
techniques used in the general case.
###### Example 2.
By direct calculation, the density function of $p\stackrel{{\scriptstyle
d}}{{=}}\mathrm{e}^{-X}$, for $X\sim\text{ga}(2,1)$, is $\pi(p)=-\log p$. Then
$m(x)$ in (6) is given by
$m(x)=\int_{0}^{\log 1/x}(\log 1/x-t)t{f}(t)\mathrm{d}t=\int_{0}^{\log
1/x}\int_{0}^{t}s{f}(s)\mathrm{d}s\,\mathrm{d}t.$
where the second equality follows by integration by parts. We first derive an
asymptotic expansion for $\int_{0}^{x}t{f}(t)\mathrm{d}t$ as $x\to\infty$.
Since
$\int_{0}^{x}t{f}(t)\mathrm{d}t=xF(x)-\int_{0}^{x}F(t)\mathrm{d}t=xF(x)-{}_{1}F(x)$
using the asymptotic expansion $F(x)=x-\gamma+O(\mathrm{e}^{-x})$ in (7) we
find
$\int_{0}^{x}t{f}(t)\mathrm{d}t=x(x-\gamma)+O(x\mathrm{e}^{-x})-\frac{(x-\gamma)^{2}}{2}+O(1)=\frac{x^{2}}{2}+O(1)$
so that
$\int_{0}^{x}\int_{0}^{t}s{f}(s)\mathrm{d}s\,\mathrm{d}t=\int_{0}^{x}\bigg{(}\frac{t^{2}}{2}+O(1)\bigg{)}\mathrm{d}x=\frac{x^{3}}{6}+O(x)$
and, in turn,
$m(x)=\frac{1}{6}\Big{(}\log\frac{1}{x}\Big{)}^{3}+O(\log x).$
We look now for the auxiliary function $\ell(x)$ of the slowly varying
function $m(x)$. We have
$\displaystyle 6\big{(}m(\lambda x)-m(x)\big{)}$ $\displaystyle=-(\log
x+\log\lambda)^{3}+(\log x)^{3}+O(\log x)$ $\displaystyle=-3\log\lambda(\log
x)^{2}+O(\log x)$
so that, as $x\to 0$
$\frac{m(\lambda x)-m(x)}{(\log x)^{2}}\to-\frac{1}{2}\log\lambda.$
Hence, the auxiliary function of $\overrightarrow{\nu}(x)$ is found to be
$\ell(x)=(\log x)^{2}$ with $c=-\frac{1}{2}$, cf. (5). Note that, because of
the cancellation of the $(\log 1/x)^{2}$ term in the expansion of $m(x)$ for
$x\to\infty$, the derivation of $\ell(x)$ only requires the leading term of
$m(x)$. The latter can be alternatively obtained by using regular variation
theory. Since $\int_{0}^{x}t{f}(t)\mathrm{d}t$ is regularly varying at
infinity with index $2$, Karamata’s theorem yields
$\int_{0}^{x}(x-t)t{f}(t)\mathrm{d}t\big{/}x\int_{0}^{x}t{f}(t)\mathrm{d}t\to\frac{1}{3}$
as $x\to\infty$. A second application of Karamata’s Theorem yields
$\int_{0}^{x}t{f}(t)\mathrm{d}t\big{/}x^{2}{f}(x)\to\frac{1}{2}$ to conclude
that, as $x\to\infty$,
$\frac{\int_{0}^{x}(x-t)t{f}(t)\mathrm{d}t}{x^{3}{f}(x)}\to\frac{1}{2}\frac{1}{3}=\frac{1}{6}$
Since ${f}(x)\to 1$ as $x\to\infty$, we get $m(x)\sim\frac{1}{6}(\log
1/x)^{3}$. Finally, by applying Theorem 1 we conclude that
$\mathrm{E}(K_{n})=\frac{1}{6}(\log n)^{3}+\frac{1}{2}\gamma(\log
n)^{2}+o(\log^{2}n)$
It is worth stressing that $\pi(p)=-\log p$ yields $\mathrm{E}(p)=1/4$, i.e. a
mass shift to lower values of $p$ compared to $\pi(p)=1$. This, according to
$\overrightarrow{\nu}(x,p)\sim\log x/\log(1-p)$, favors larger values of
$\mathrm{E}(K_{n}|p)$, which explain a faster growth of $\mathrm{E}(K_{n})$.
The asymptotic expansion of $\mathrm{E}(K_{n})$, for $m$ any positive integer,
is derived in the following theorem. The key ingredient consists in expressing
$\int_{0}^{x}t^{m}{f}(t)\mathrm{d}t$ in terms of fractional integrals of
$F(x)$.
###### Theorem 2.
Let $p$ in (3) have distribution $\pi(p)$ defined by $p\stackrel{{\scriptstyle
d}}{{=}}\mathrm{e}^{-X}$ with $X\sim\text{\rm ga}(m+1,1)$ and $m$ a positive
integer. Then
$\displaystyle\overrightarrow{\nu}(x)$ $\displaystyle=\frac{(\log
1/x)^{m+2}}{(m+2)!}+O\big{(}(\log x)^{m}),\quad x\to 0$
$\displaystyle\mathrm{E}(K_{n})$ $\displaystyle=\frac{(\log
n)^{m+2}}{(m+2)!}+\gamma\frac{(\log n)^{m+1}}{(m+1)!}+o((\log n)^{m+1}),\quad
n\to\infty.$
###### Proof.
Since $\pi(p)=(-\log p)^{m}/m!$, $m(x)$ in (6) becomes
$m(x)=\int_{0}^{\log 1/x}(\log
1/x-t)\frac{t^{m}}{m!}{f}(t)\mathrm{d}t=\int_{0}^{\log
1/x}\int_{0}^{t}\frac{s^{m}}{m!}{f}(s)\mathrm{d}s\,\mathrm{d}t.$ (8)
where the second equality follows again by integration by parts. Recall that,
for an integer-valued index, fractional integrals correspond to “ higher order
” primitives of ${f}(x)$: $F(x)={}_{0}F(x)$,
${}_{1}F(x)=\int_{0}^{x}F(t)\mathrm{d}t$ and
${}_{(k+1)}F(x)=\int_{0}^{x}(x-t)\mathrm{d}\,{}_{k}F(t)=\int_{0}^{x}{}_{k}F(t)\mathrm{d}t,\quad
k=1,2,\ldots$
By repeated integration by parts the inner integral in (8) is found to be
$\displaystyle\int_{0}^{x}\frac{t^{m}}{m!}{f}(t)\mathrm{d}t$
$\displaystyle=\sum_{k=0}^{m}\frac{(-1)^{k}}{(m-k)!}x^{m-k}{}_{k}F(x).$
Next, we exploit the asymptotic evaluation (7),
$F(x)=x-\gamma+O(\mathrm{e}^{-x})$ as $x\to\infty$, to find
$\displaystyle{}_{1}F(x)$
$\displaystyle=\frac{(x-\gamma)^{2}}{2}+O(1)=\frac{x^{2}-2\gamma x}{2}+O(1)$
$\displaystyle{}_{2}F(x)$
$\displaystyle=\frac{(x-\gamma)^{3}}{3!}+O(x)=\frac{x^{3}-3\gamma
x^{2}}{3!}+O(x)$ $\displaystyle{}_{k}F(x)$
$\displaystyle=\frac{(x-\gamma)^{k+1}}{(k+1)!}+O(x^{k-1})=\frac{x^{k+1}-(k+1)\gamma
x^{k}}{(k+1)!}+O(x^{k-1}).$
We get
$\displaystyle\int_{0}^{x}\frac{t^{m}}{m!}{f}(t)\mathrm{d}t$
$\displaystyle=\sum_{k=0}^{m}\frac{(-1)^{k}}{(m-k)!}x^{m-k}\bigg{(}\frac{x^{k+1}-(k+1)\gamma
x^{k}}{(k+1)!}+O(x^{k-1})\bigg{)}$
$\displaystyle=x^{m+1}\sum_{k=0}^{m}\frac{(-1)^{k}}{(m-k)!(k+1)!}+\gamma
x^{m}\sum_{k=0}^{m}\frac{(-1)^{k+1}}{(m-k)!k!}+O(x^{m-1}).$
As for the $x^{m+1}$-term we obtain
$\displaystyle\sum_{k=0}^{m}\frac{(-1)^{k}}{(m-k)!(k+1)!}$
$\displaystyle=\sum_{k=1}^{m+1}\frac{(-1)^{k-1}}{(m+1-k)!(k)!}=\frac{1}{(m+1)!}\sum_{k=1}^{m+1}{m+1\choose
k}(-1)^{k-1}$
$\displaystyle=-\frac{1}{(m+1)!}\bigg{(}\sum_{k=0}^{m+1}{m+1\choose
k}(-1)^{k}-1\bigg{)}=\frac{1}{(m+1)!},$
where in the last step we used
$\sum_{k=0}^{m+1}{m+1\choose k}(-1)^{k}=\sum_{k=0}^{m+1}{m+1\choose
k}(-1)^{k}(+1)^{m+1-k}=(-1+1)^{m+1}=0.$
A similar application of the binomial formula shows that the $x^{m}$-term is
zero, namely
$\displaystyle\sum_{k=0}^{m}\frac{(-1)^{k+1}}{(m-k)!k!}$
$\displaystyle=-\sum_{k=0}^{m}\frac{(-1)^{k}}{(m-k)!k!}=-\frac{1}{m!}(-1+1)^{m}=0$
Hence, we have
$\int_{0}^{x}\frac{t^{m}}{m!}{f}(t)\mathrm{d}t=\frac{x^{m+1}}{(m+1)!}+O(x^{m-1}),$
which yields
$\int_{0}^{x}\int_{0}^{t}\frac{s^{m}}{m!}{f}(s)\mathrm{d}s\,\mathrm{d}t=\int_{0}^{x}\bigg{(}\frac{t^{m+1}}{(m+1)!}+O(t^{m-1})\bigg{)}\mathrm{d}t=\frac{x^{m+2}}{(m+2)!}+O(x^{m})$
to conclude that, as $x\to 0$,
$m(x)=\frac{(\log 1/x)^{m+2}}{(m+2)!}+O\big{(}(\log x)^{m}\big{)}.$
In view of $m(x)\leq\overrightarrow{\nu}(x)\leq m(x)+1$, the first statement
is proved. We now proceed to derive the auxiliary function $\ell(x)$ of $m(x)$
and, in turn, of $\overrightarrow{\nu}(x)$. We have
$\displaystyle(m+2)!\big{(}m(\lambda x)-m(x)\big{)}$
$\displaystyle=(-\log(\lambda x))^{m+2}-(-\log x)^{m+2}+O\big{(}(\log
x)^{m}\big{)}$ $\displaystyle=(-1)^{m+2}\big{(}(\log
x+\log\lambda)^{m+2}-(\log x)^{m+2}\big{)}+O\big{(}(\log x)^{m}\big{)}$
$\displaystyle=(-1)^{m+2}(m+2)\log\lambda(\log x)^{m+1}+O\big{(}(\log
x)^{m}\big{)}$
so that, as $x\to 0$,
$\displaystyle\frac{m(\lambda x)-m(x)}{(\log
x)^{m+1}}\to\frac{(-1)^{m+2}}{(m+1)!}\log\lambda.$
So we find $\ell(x)=(\log(x))^{m+1}$ and $c=(-1)^{m+2}/(m+1)!$ in (5). Note
that
$\displaystyle c\ell(1/n)$ $\displaystyle=\frac{(-1)^{m+2}}{(m+1)!}(-\log
n)^{m+1}=\frac{(-1)^{m+2+m+1}}{(m+1)!}(\log n)^{m+1}=-\frac{(\log
n)^{m+1}}{(m+1)!}.$
Finally, an application of Theorem 1 yields the second statement. ∎
###### Remark 2.
The phenomenon observed in Example 2 for $m=1$, namely the cancellation of the
term $(\log 1/x)^{2}$ in the expansion of $m(x)$, applies to any $m\geq 1$
meaning that the term $(\log 1/x)^{m+1}$ cancels out. Since the auxiliary
function $\ell(x)$ is found to be of the order $(\log 1/x)^{m+1}$, we conclude
that for any $m\geq 1$ the leading term in the expansion of $m(x)$ is
sufficient for the derivation of the second order term in the expansion of
$\mathrm{E}(K_{n})$ according to Theorem 1. As observed in Example 2, a double
application of Karamata’s Theorem yields the leading term of $m(x)$ as it
yields, for $x\to\infty$,
$\frac{\int_{0}^{x}(x-t)t^{m}{f}(t)\mathrm{d}t}{x^{m+2}{f}(x)}\to\frac{1}{m+1}\frac{1}{m+2}.$
These calculations can be easily extended to the case of
$p\stackrel{{\scriptstyle d}}{{=}}\mathrm{e}^{-X}$ for
$X\sim\text{ga}(1+\rho,1)$ with $\rho>-1$. Since $\pi(\mathrm{e}^{-t}){f}(t)$
is regularly varying at infinity with index $\rho+1>0$,
$\frac{\int_{0}^{x}(x-t)\pi(\mathrm{e}^{-t}){f}(t)\mathrm{d}t}{x^{\rho+2}{f}(x)}\to\frac{1}{\Gamma(\rho+1)}\frac{1}{\rho+1}\frac{1}{\rho+2}=\frac{1}{\Gamma(\rho+3)}$
so we obtain
$\mathrm{E}(K_{n})\sim\frac{1}{\Gamma(\rho+3)}(\log n)^{\rho+2}.$
However, a more accurate approximation of
$\int_{0}^{x}(x-t)\pi(\mathrm{e}^{-t}){f}(t)\mathrm{d}t$ is necessary in order
to apply Theorem 1 and obtain the second order term in the expansion of
$\mathrm{E}(K_{n})$.
## 4 Negative binomial extension
In the following we use the notation $w_{j}(p)$ for the $j$th frequency as a
function of the parameter $p$. Recall that the asymptotic behavior of
$\mathrm{E}(K_{n})$ depends on the behavior at zero of the tail mean measure
$\overrightarrow{\nu}(x)=\int_{0}^{1}\overrightarrow{\nu}(x,p)\pi(p)\mathrm{d}p,$
where $\pi(p)$ is the success probability distribution, and
$\overrightarrow{\nu}(x,p)=\\#\\{j:w_{j}(p)\geq x\\}$ is the number of
frequencies larger than a threshold $x\in[0,1]$. When the frequencies
$w_{j}(p)$ are decreasing, $\overrightarrow{\nu}(x,p)=\sup\\{j:w_{j}(p)\geq
x\\}$, that is $\overrightarrow{\nu}(x,p)$ is obtained in terms of the inverse
of $w_{j}(p)$ with respect to $j$. In the case of geometric frequencies the
inverse is explicitly found to be $\log(x/p)/\log(1-p)+1$, thus we have
$\overrightarrow{\nu}(x,p)=\lfloor\log(x/p)/\log(1-p)+1\rfloor$ for $p\geq x$
or, equivalently, for $w_{1}(p)\geq x$. In Section 3, the behavior in zero of
$\overrightarrow{\nu}(x)$ was studied in terms of
$m(x)=\int_{0}^{1}\frac{\log(x/p)}{\log(1-p)}\mathds{1}_{(w_{1}(p)\geq
x)}\pi(p)\mathrm{d}p,$
based on the fact that $m(x)\leq\overrightarrow{\nu}(x)\leq m(x)+1$.
In this section we consider a different model for $w_{j}(p)$ that can be seen
as an extension of the geometric case. To this aim, we resort to a derivation
of the geometric weights $w_{j}(p)=p(1-p)^{j-1}$ as a special case of a
general construction of distributions on the positive integers with decreasing
frequencies. Let $\phi(r;p)$ be a probability function for $r=1,2,\ldots$ with
parameter $p\in(0,1)$. Then
$w_{j}(p)=\sum_{r\geq j}\frac{\phi(r;p)}{r},\ \quad j=1,2,\dots,$
form a decreasing sequence, $w_{j}(p)>w_{j+1}(p)$, of positive numbers summing
up to one. As such, $(w_{j}(p))_{j\geq 1}$ defines a new distribution on the
positive integers parametrized by $p$. An interesting instance of $\phi(r;p)$
is given by
$\phi(r;p)=\phi(r;s,p)={r+s-2\choose r-1}p^{s}(1-p)^{r-1},\quad r=1,2,\ldots$
that is the negative binomial distribution shifted by one. The geometric
frequencies are obtained by taking the scale parameter $s=2$. In fact
$\displaystyle w_{j}(p)$ $\displaystyle=\sum_{r\geq
j}p^{2}(1-p)^{r-1}=p^{2}(1-p)^{j-1}\sum_{i\geq 1}(1-p)^{i-1}$
$\displaystyle=p^{2}(1-p)^{j-1}\sum_{i\geq 0}(1-p)^{i}=p(1-p)^{j-1}.$
An explicit expression for $w_{j}(p)$ can be found for any integer $s\geq 2$.
When $s=3$, differentiating with respect to the geometric series one finds
$\displaystyle\frac{2(1-p)}{p^{3}}w_{j}(p)$ $\displaystyle=\sum_{r\geq
j}(r+1)(1-p)^{r}=-{\mathrm{d}\over\mathrm{d}p}\sum_{r\geq
j}(1-p)^{r+1}=-{\mathrm{d}\over\mathrm{d}p}\frac{(1-p)^{j+1}}{p}$
$\displaystyle=\frac{(1-p)^{j}}{p^{2}}\big{(}(j+1)p+(1-p)\big{)}=\frac{(1-p)^{j}}{p^{2}}\big{(}1+jp\big{)}$
to conclude that
$w_{j}(p)=p(1-p)^{j-1}\frac{1+jp}{2},\quad j=1,2,\ldots$ (9)
Similar formulae are derived for $s>3$: one finds that $w_{j}(p)$ is
proportional to the geometric probability $p(1-p)^{j-1}$ multiplied by a
polynomial in $(j,p)$ of order determined by $s$. Details are omitted. For
instance, $s=4$ yields
$w_{j}(p)=p(1-p)^{j-1}\frac{2+2jp+jp^{2}+j^{2}p^{2}}{6},\quad j=1,2,\ldots$
With a closed form expression of $w_{j}(p)$, an asymptotic evaluation of
$\overrightarrow{\nu}(x,p)$ can be derived for $x\to 0$, and in turn, for
$\overrightarrow{\nu}(x)$, so that the asymptotics of $\mathrm{E}(K_{n})$ is
obtained through Theorem 1. Let us restrict attention to $s=3$ with $w_{j}(p)$
as in (9) and $\pi(p)$ the uniform distribution. Our goal is to investigate
the asymptotics of $\mathrm{E}(K_{n})$ in comparison with the geometric case
of Proposition 1. It is reasonable to expect that $\mathrm{E}(K_{n})$ grows at
a faster rate: in fact, the larger $s$, the larger the mean of the negative
binomial distribution $\phi(r;s,p)$, so $w_{j}(p)$ decrease slower in $j$ for
$s=3$ compared to $s=2$, which corresponds to the geometric case. This implies
that $\overrightarrow{\nu}(x,p)$ grows slower as $x\to 0$ for $s=3$ and, in
turn, via a Tauberian theorem $\mathrm{E}(K_{n})$ grows faster as
$n\to\infty$. In Proposition 2 we establish that the asymptotic behavior of
$\mathrm{E}(K_{n})$ differs from the one found in Proposition 1 only in the
second order term of the expansion.
###### Proposition 2.
Let $w_{j}(p)$ be defined as in (9) and $p$ be uniformly distributed on
$(0,1)$. Then
$\displaystyle\overrightarrow{\nu}(x)$
$\displaystyle=\frac{1}{2}\Big{(}\log\frac{1}{x}\Big{)}^{2}+\log\frac{1}{x}\log\log\frac{1}{x}-\gamma\log\frac{1}{x}-(1+\log
2)\log\frac{1}{x}+O\bigg{(}\log\log\frac{1}{x}\bigg{)},\quad x\to 0$
$\displaystyle\mathrm{E}(K_{n})$ $\displaystyle=\frac{1}{2}\big{(}\log
n\big{)}^{2}+\log n\log(\log n)-(1+\log 2)\log n+o(\log n),\quad n\to\infty.$
###### Proof.
Let $m(x,p)\geq 0$ be defined by
$p(1-p)^{m(x,p)}\frac{1}{2}\big{(}1+p+p\,m(x,p)\big{)}=x,$ (10)
which corresponds to the solution in $m$ to the equation $w_{m+1}(p)=x$. Note
that $m(x,p)\geq 0$ when $w_{1}(p)\geq x$, $\overrightarrow{\nu}(x,p)=\lfloor
m(x,p)+1\rfloor\mathds{1}_{(w_{1}(p)\geq x)}$ and, as in the geometric case,
$m(x)\leq\overrightarrow{\nu}(x)\leq m(x)+1$ for
$m(x)=\int_{0}^{1}m(x,p)\mathds{1}_{(w_{1}(p)\geq x)}\mathrm{d}p.$
Equation (11) provides an asymptotic expansion of $m(x,p)$ as $x\to 0$. The
proof is reported in the Appendix and involves the Lambert W function (Corless
et al.,, 1996).
$m(x,p)=\frac{\log x/p}{\log(1-p)}\\\
-\frac{1}{\log(1-p)}\log\bigg{(}\frac{1}{2}\bigg{(}1+p+p\frac{\log
x/p}{\log(1-p)}+\frac{p}{\log(1-p)}\log\frac{-2\log(1-p)}{p}\bigg{)}\bigg{)}\\\
+\frac{1}{\log(1-p)}O\bigg{(}\frac{\log(\log 1/x)}{\log 1/x}\bigg{)}.$ (11)
An heuristic derivation inspired by (Barndorff-Nielsen and Cox,, 1989, Example
3.13) is as follows. In equation (10), we have that for small $x$, $m(x,p)$
will be large and the term $p(1-p)^{m(x,p)}$ is thus dominant. Rewrite the
equation after taking the log and keeping $m(x,p)$ on the left hand side,
$m(x,p)=\frac{\log
x/p}{\log(1-p)}-\frac{1}{\log(1-p)}\log\bigg{(}\frac{1}{2}\big{(}1+p+pm(x,p)\big{)}\bigg{)}.$
It defines a convergent iterative scheme via
$m_{(k)}(x,p)=\frac{\log
x/p}{\log(1-p)}-\frac{1}{\log(1-p)}\log\bigg{(}\frac{1}{2}\big{(}1+p+pm_{(k-1)}(x,p)\big{)}\bigg{)}$
with $m_{(1)}(x,p)$ solution to $p(1-p)^{m(x,p)}=x$, that is
$m_{(1)}(x,p)=\frac{\log x/p}{\log(1-p)}.$
For $k=2$ we get
$m_{(2)}(x,p)=\frac{\log
x/p}{\log(1-p)}-\frac{1}{\log(1-p)}\log\bigg{(}\frac{1}{2}\bigg{(}1+p+p\frac{\log
x/p}{\log(1-p)}\bigg{)}\bigg{)},$
which nearly matches the expansion in (11). Now we use it to evaluate the
behavior of $m(x)$ and, in turn, $\overrightarrow{\nu}(x)$ as $x\to 0$. Note
that
$m(x)=\int_{0}^{1}m(x,p)\mathds{1}_{(w_{1}(p)\geq
x)}\mathrm{d}p=\int_{\tilde{x}}^{1}m(x,p)\mathrm{d}p,$
where $\tilde{x}$ is defined by $w_{1}(\tilde{x})=x$, that is
$\tilde{x}(1+\tilde{x})/2=x$. It is easy to check that $x\leq\tilde{x}\leq 2x$
for any $x$ and $\tilde{x}\sim 2x$ as $x\to 0$. In order to exploit the
derivation of the asymptotic expansion of $m(x)$ as $x\to 0^{+}$ in the
geometric case, cf. proof of Proposition 1, we use (11) as follows:
$m(x)=\int_{x}^{1}\frac{\log
x/p}{\log(1-p)}\mathrm{d}p-\int_{x}^{\tilde{x}}\frac{\log
x/p}{\log(1-p)}\mathrm{d}p+\int_{\tilde{x}}^{1}\bigg{(}m(x,p)-\frac{\log
x/p}{\log(1-p)}\bigg{)}\mathrm{d}p.$
From Proposition 1, as $x\to 0$
$\int_{x}^{1}\frac{\log
x/p}{\log(1-p)}\mathrm{d}p=\frac{1}{2}\Big{(}\log\frac{1}{x}\Big{)}^{2}-\gamma\log\frac{1}{x}+O(1).$
The first statement of the thesis about the behavior of
$\overrightarrow{\nu}(x)$ as $x\to 0$ is then implied by
$m(x)\leq\overrightarrow{\nu}(x)\leq m(x)+1$ and by showing that, as $x\to 0$,
$\displaystyle\int_{x}^{\tilde{x}}\frac{\log x/p}{\log(1-p)}\mathrm{d}p$
$\displaystyle=O(1)$ (12)
$\displaystyle\int_{\tilde{x}}^{1}\bigg{(}m(x,p)-\frac{\log
x/p}{\log(1-p)}\bigg{)}\mathrm{d}p$
$\displaystyle=\log\frac{1}{x}\log\log\frac{1}{x}-(1+\log
2)\log\frac{1}{x}+O\bigg{(}\log\log\frac{1}{x}\bigg{)}.$ (13)
As for (12), the maximum of $(\log x/p)/\log(1-p)$ is attained at $p=p(x)$,
where $p(x)$, the solution to the first order equation in $p$
$-(1-p)\log(1-p)+p\log x/p=0$, goes to zero as $x\to 0$. It can be shown that
that $p(x)\geq 2x$ for $x\leq 1/4$, that is $(\log x/p)/\log(1-p)$ is
increasing for $x\leq p\leq 2x$ and $x$ sufficiently small. Since
$2x\geq\tilde{x}$,
$\int_{x}^{\tilde{x}}\frac{\log
x/p}{\log(1-p)}\mathrm{d}p\leq\int_{x}^{2x}\frac{\log
x/p}{\log(1-p)}\mathrm{d}p\leq x\frac{\log x/(2x)}{\log(1-2x)}=-x\log
2/\log(1-2x)\leq\log 2/2$
so (12) follows. As for (13), note that by the change of variable $t=\log
1/p$,
$\int_{\tilde{x}}^{1}-\frac{1}{\log(1-p)}\mathrm{d}p=\int_{0}^{\log
1/\tilde{x}}f(t)\mathrm{d}t=F(\log 1/\tilde{x}),$
for ${f}(t)$ in (6) and $F(t)$ the primitive of ${f}(t)$. By equation (11),
(7) and $\tilde{x}\sim 2x$ as $x\to 0$, we have that, as $x\to 0$,
$\int_{\tilde{x}}^{1}\bigg{(}m(x,p)-\frac{\log
x/p}{\log(1-p)}\bigg{)}\mathrm{d}p=-\log 2\log 1/x+O(\log(\log 1/x))\\\
+\int_{\tilde{x}}^{1}-\frac{1}{\log(1-p)}\log\bigg{(}1+p+p\frac{\log
x/p}{\log(1-p)}+\frac{p}{\log(1-p)}\log\frac{-2\log(1-p)}{p}\bigg{)}\mathrm{d}p.$
In studying the asymptotic behavior of the integral in the last display, it is
sufficient to focus on
$\int_{\tilde{x}}^{1}-\frac{1}{\log(1-p)}\log\bigg{(}1+p\frac{\log
x/p}{\log(1-p)}\bigg{)}\mathrm{d}p,$
since the extra terms inside the logarithm satisfy
$-2\mathrm{e}^{-1}\leq p+\frac{p}{\log(1-p)}\log\frac{-2\log(1-p)}{p}\leq 1$
for $0\leq p\leq 1$. By the change of variable $t=\log 1/p$
$\int_{\tilde{x}}^{1}-\frac{1}{\log(1-p)}\log\bigg{(}1+p\frac{\log
x/p}{\log(1-p)}\bigg{)}\mathrm{d}p=\int_{0}^{\log
1/\tilde{x}}\log\Big{(}1+\big{(}\log
1/\tilde{x}-t\big{)}{f}(t)\Big{)}\,{f}(t)\mathrm{d}t,$
for ${f}(t)$ defined in (6). Hence, (13) is implied by
$r(x)=\int_{0}^{x}\log\big{(}1+\big{(}x-t\big{)}{f}(t)\big{)}\,{f}(t)\mathrm{d}t=x\log(x)-x+O(\log
x),$ (14)
as $x\to\infty$. We have
$\displaystyle r(x)$ $\displaystyle=\int_{0}^{x}\bigg{(}\log x+\log
f(t)+\log\frac{1+\big{(}x-t\big{)}{f}(t)}{x{f}(t)}\bigg{)}{f}(t)\mathrm{d}t$
$\displaystyle=\log(x)F(x)+\int_{0}^{x}{f}(t)\log{f}(t)\mathrm{d}t+\int_{0}^{x}\log\bigg{(}1+\frac{1-t{f}(t)}{x{f}(t)}\bigg{)}{f}(t)\mathrm{d}t.$
The first term on the right hand side is $\log
x(x-\gamma+O(\mathrm{e}^{-x})),$ as $x\to\infty$, due to the asymptotic
expansion of $F(x)$ in (7). The second term is easily shown to be bounded in
absolute value uniformly in $x$. As for the third term, we are left to show
that
$\int_{0}^{x}\log\bigg{(}1+\frac{1-t{f}(t)}{x{f}(t)}\bigg{)}{f}(t)\mathrm{d}t=-x+\log
x+O(1),$
as $x\to\infty$. To this aim, it is convenient to split the integral as
$\int_{0}^{1}\log\bigg{(}1+\frac{1-t{f}(t)}{x{f}(t)}\bigg{)}{f}(t)\mathrm{d}t+\int_{1}^{x}\log\bigg{(}1+\frac{1-t{f}(t)}{x{f}(t)}\bigg{)}{f}(t)\mathrm{d}t.$
(15)
The first integral in (15) is bounded in $x$ since
$\int_{0}^{1}\log\bigg{(}1+\frac{1-t{f}(t)}{x{f}(t)}\bigg{)}{f}(t)\mathrm{d}t\leq\frac{1}{x}\int_{0}^{1}\big{(}1-t{f}(t)\big{)}\mathrm{d}t,$
whereas the last integral is a positive and finite constant. As for the second
integral in (15), since $1-f(t)\sim\mathrm{e}^{-t}/2$ for $t\to\infty$, cf.
Lemma 1,
$\int_{1}^{x}\log\bigg{(}1+\frac{1-t{f}(t)}{x{f}(t)}\bigg{)}{f}(t)\mathrm{d}t\sim\int_{1}^{x}\log\bigg{(}1+\frac{1-t}{x}\bigg{)}\mathrm{d}t$
as $x\to\infty$ and
$\int_{1}^{x}\log\bigg{(}1+\frac{1-t}{x}\bigg{)}\mathrm{d}t=-x+\log x+1$
by direct calculation. Hence (14), and in turn (13), follow. The proof of the
first statement of the proposition is then complete. The second statement
about the expansion of $\mathrm{E}(K_{n})$, as $n\to\infty$, follows from an
application of Theorem 1. ∎
## Appendix
### Proof of Theorem 1
The proof follows arguments similar to those of (Bingham et al.,, 1987,
Theorem 3.9.1). It consists in evaluating
$\big{(}\Phi(n)-\overrightarrow{\nu}(1/n)\big{)}/\ell(1/n)$ in the
decomposition
$\mathrm{E}(K_{n})=\overrightarrow{\nu}(1/n)+\frac{\Phi(n)-\overrightarrow{\nu}(1/n)}{\ell(1/n)}\ell(1/n)+\mathrm{E}(K_{n})-\Phi(n).$
Indeed, as $\Phi(n)\sim\overrightarrow{\nu}(1/n)$ and $\ell(1/n)$ are slowly
varying,
$|\mathrm{E}(K_{n})-\Phi(n)|\leq\textstyle\frac{2}{n}\displaystyle\Phi(n)=o(\ell(1/n))$,
cf. (4), so the conclusion follows by showing that
$\big{(}\Phi(n)-\overrightarrow{\nu}(1/n)\big{)}/\ell(1/n)\to-c\gamma$. To
this aim,
$\displaystyle\frac{\Phi(1/x)-\overrightarrow{\nu}(x)}{\ell(x)}$
$\displaystyle=\frac{1}{\ell(x)}\bigg{[}\int_{0}^{\infty}\frac{1}{x}\mathrm{e}^{-y/x}\overrightarrow{\nu}(y)\mathrm{d}y-\int_{0}^{\infty}\overrightarrow{\nu}(x)\mathrm{e}^{-\lambda}\mathrm{d}\lambda\bigg{]}$
$\displaystyle=\frac{1}{\ell(x)}\bigg{[}\int_{0}^{\infty}\mathrm{e}^{-\lambda}\overrightarrow{\nu}(\lambda
x)\mathrm{d}\lambda-\int_{0}^{\infty}\overrightarrow{\nu}(x)\mathrm{e}^{-\lambda}\mathrm{d}\lambda\bigg{]}=\int_{0}^{\infty}\frac{\overrightarrow{\nu}(\lambda
x)-\overrightarrow{\nu}(x)}{\ell(x)}\mathrm{e}^{-\lambda}\mathrm{d}\lambda$
$\displaystyle\to\int_{0}^{\infty}c(\log\lambda)\mathrm{e}^{-\lambda}\mathrm{d}\lambda=c\Gamma^{\prime}(1)=-c\gamma,\quad\text{as
}x\to 0,$
where in taking the limit we used the dominated convergence theorem, cf.
global bounds in (Bingham et al.,, 1987, Theorem 3.8.6). ∎
### Proof of Lemma 1
We will use the following integral representation of the Euler-Mascheroni
constant:
$\gamma=\int_{0}^{\infty}\bigg{(}\frac{1}{1-\mathrm{e}^{-x}}-\frac{1}{x}\bigg{)}\mathrm{e}^{-x}\mathrm{d}x.$
By the change of variable $t=-\log(1-\mathrm{e}^{-x})$ so that
$\mathrm{d}t=-\frac{\mathrm{e}^{-x}}{1-\mathrm{e}^{-x}}\mathrm{d}x$ and
$x=-\log(1-\mathrm{e}^{-t})$, we obtain
$\displaystyle\gamma$
$\displaystyle=\int_{0}^{\infty}\bigg{(}\frac{1}{1-\mathrm{e}^{-x}}-\frac{1}{x}\bigg{)}\mathrm{e}^{-x}\mathrm{d}x=\int_{0}^{\infty}\bigg{(}1-\frac{1-\mathrm{e}^{-x}}{x}\bigg{)}\frac{\mathrm{e}^{-x}}{1-\mathrm{e}^{-x}}\mathrm{d}x$
$\displaystyle=\int_{0}^{\infty}\bigg{(}1-\frac{\mathrm{e}^{-t}}{-\log(1-\mathrm{e}^{-t})}\bigg{)}\mathrm{d}t=\int_{0}^{\infty}\big{(}1-{f}(t)\big{)}\mathrm{d}t.$
It is easy to check that $\lim_{t\to 0}{f}(t)=0$ and
$\lim_{t\to\infty}{f}(t)=1$. As for the tail behavior, by the Taylor expansion
of $\log(1+x)=x-x^{2}/2+O(x^{3})$ as $x\to 0$, we find that, as $t\to\infty$,
$\displaystyle 1-{f}(t)$
$\displaystyle=1-\frac{\mathrm{e}^{-t}}{-\log(1-\mathrm{e}^{-t})}\sim
1-\frac{\mathrm{e}^{-t}}{\mathrm{e}^{-t}+\mathrm{e}^{-2t}/2}=\frac{\mathrm{e}^{-2t}/2}{\mathrm{e}^{-t}+\mathrm{e}^{-2t}/2}\sim\frac{\mathrm{e}^{-t}}{2}.$
∎
### Proof of Equation (11)
Let $W(z)$ be the Lambert function defined by $W(z)\mathrm{e}^{W(z)}=z,$ where
$W(z)$ is a multivalued function that has, for $z$ a real number, two
branches, the principal branch $W_{0}(z)$ for $W(z)\geq-1$, and the branch
$W_{-1}(z)$ for $W(z)<-1$. We have that $\lim_{z\to 0^{+}}W_{0}(z)=0$ while
$\lim_{z\to 0^{-}}W_{-1}(z)=-\infty$. In particular, according to (Corless et
al.,, 1996, Section 4), as $z\to 0^{-}$
$W_{-1}(z)=\log(-z)-\log(-\log(-z))+O\bigg{(}\frac{\log(-\log(-z))}{\log(-z)}\bigg{)}.$
(16)
By algebraic manipulation of (10)
$\displaystyle
p(1-p)^{m(x,p)}\frac{1}{2}(1+p+p\,m(x,p))=x;\quad\mathrm{e}^{\log(1-p)m(x,p)}(1+p+p\,m(x,p))=2x/p;$
$\displaystyle(1+p+p\,m(x,p))\log(1-p)\mathrm{e}^{\log(1-p)m(x,p)}=\frac{2x\log(1-p)}{p};$
$\displaystyle(1+p+p\,m(x,p))\log(1-p)\mathrm{e}^{\log(1-p)(1/p+1+m(x,p)}=\frac{2x\log(1-p)}{p}\mathrm{e}^{(1/p+1)\log(1-p)};$
$\displaystyle\frac{\log(1-p)}{p}(1+p+p\,m(x,p))\mathrm{e}^{\frac{\log(1-p)}{p}(1+p+p\,m(x,p))}=\frac{2x\log(1-p)}{p^{2}}\mathrm{e}^{\frac{1+p}{p}\log(1-p)};$
$\displaystyle\frac{\log(1-p)}{p}(1+p+p\,m(x,p))=W(z),$
where, in the last display,
$z=\frac{2x\log(1-p)}{p^{2}}\exp\Big{(}\frac{1+p}{p}\log(1-p)\Big{)}.$ (17)
Solving for $m(x,p)$,
$m(x,p)=\frac{1}{p\log(1-p)}\bigg{(}pW_{-1}(z)-\log(1-p)\bigg{)}-1,$ (18)
where we used the branch $W_{-1}$ of $W(z)$ since $z$ in (17) is $\leq 0$ and
$W(z)\leq-1$. The fact that $W(z)\leq-1$ is easily checked by using
$m(x,p)\geq 0$. In fact
$\displaystyle\frac{1}{p\log(1-p)}\bigg{(}pW(z)-\log(1-p)\bigg{)}-1\geq
0;\quad pW(z)-\log(1-p)\leq p\log(1-p);$ $\displaystyle
pW(z)\leq\log(1-p)(1+p);\quad W(z)\leq\log(1-p)\frac{1+p}{p}$
and $\frac{1+p}{p}\log(1-p)$ decreases from $-1$ to $-\infty$ for $p\in(0,1)$.
From (17) one finds that $z\to 0^{-}$ as $x\to 0^{+}$. In particular, from
$w_{1}(p)>x$, that is $p(1+p)/2>x$, it follows that
$\frac{1+p}{p}\log(1-p)\exp\Big{(}\frac{1+p}{p}\log(1-p)\Big{)}\leq z\leq 0$
and the lower bound is larger than $-\mathrm{e}^{-1}$ for any $p\in(0,1)$.
Hence $\log(-z)<-1$ and $\log(-\log(-z))>0$. By direct calculation
$\log(-z)=\log(1-p)\bigg{(}\frac{\log
x/p}{\log(1-p)}+\frac{1}{\log(1-p)}\log\frac{-2\log(1-p)}{p}+\frac{1+p}{p}\bigg{)}$
and
$\frac{1}{p\log(1-p)}\bigg{(}p\log(-z)-\log(1-p)\bigg{)}-1=\frac{\log
x/p}{\log(1-p)}+\frac{1}{\log(1-p)}\log\frac{-2\log(1-p)}{p}.$
Substitute in (18) $W_{-1}(z)$ for $\log(-z)-\log(-\log(-z))$ according to the
two terms expansion in (16), to find
$\displaystyle\frac{1}{p\log(1-p)}$
$\displaystyle\bigg{(}p\Big{(}\log(-z)-\log(-\log(-z))\Big{)}-\log(1-p)\bigg{)}-1$
$\displaystyle=\frac{1}{p\log(1-p)}\bigg{(}p\log(-z)-\log(1-p)\bigg{)}-1-\frac{1}{\log(1-p)}\log(-\log(-z))$
$\displaystyle=\frac{\log
x/p}{\log(1-p)}+\frac{1}{\log(1-p)}\log\frac{-2\log(1-p)}{p}-\frac{1}{\log(1-p)}\log(-\log(-z))$
$\displaystyle=\frac{\log
x/p}{\log(1-p)}-\frac{1}{\log(1-p)}\log\bigg{(}\frac{p}{2\log(1-p)}\log(-z)\bigg{)}$
$\displaystyle=\frac{\log
x/p}{\log(1-p)}-\frac{1}{\log(1-p)}\log\bigg{(}\frac{p}{2}\bigg{(}\frac{\log
x/p}{\log(1-p)}+\frac{1}{\log(1-p)}\log\frac{-2\log(1-p)}{p}+\frac{1+p}{p}\bigg{)}\bigg{)}$
$\displaystyle=\frac{\log
x/p}{\log(1-p)}-\frac{1}{\log(1-p)}\log\bigg{(}\frac{1}{2}\bigg{(}1+p+p\frac{\log
x/p}{\log(1-p)}+\frac{p}{\log(1-p)}\log\frac{-2\log(1-p)}{p}\bigg{)}\bigg{)}.$
The remainder of the expansion is easily found. ∎
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|
# Game values of arithmetic functions
Douglas E. Iannucci111University of the Virgin Islands. and Urban
Larsson222National University of Singapore<EMAIL_ADDRESS>
###### Abstract
Arithmetic functions in Number Theory meet the Sprague-Grundy function from
Combinatorial Game Theory. We study a variety of 2-player games induced by
standard arithmetic functions, such as Euclidian division, divisors,
remainders and relatively prime numbers, and their negations.
## Introduction
Consider the following situation: two players, Alice and Bob, alternate to
partition a given finite number of positive integers into components of the
form of a non-trivial Euclidian division. Whoever will fail to follow the
rule, because each number is a “1”, loses the game. You are only allowed to
split one number at a time. For example, if Alice starts from the number $7$,
then her options are $1+\cdots+1$, $2+2+2+1$, $3+3+1$, $4+3$, $5+2$ and $6+1$;
here the ‘+’ sign from arithmetic functions becomes the disjunctive sum
operator (a convenient game component separator) in the game setting. By
observing that we may remove any pair of the same numbers (by mimicking
strategy), and we may remove a one unless the option is the terminal position
(since its set of options is empty), the set of options from $7$ simplifies to
$\\{1,2,4+3,5+2,6\\}$. Suppose now that Alice starts playing from the
disjunctive sum $7+2$. By the above analysis its easy to find a winning move
to $2+2+2+1+2$. What if she instead starts from the composite game $7+3$?
We study 2-player normal-play games defined with the nonnegative or positive
integers as the set of positions. The two players alternate turns and if a
player has no move option then he/she loses. At each stage of game, the move
options are the same independent of who is to play. In combinatorial game
theory, this notion is referred to as impartial. Games terminate in a finite
number of moves, and there is a finite number of options from each game
position; i.e., games are short. This allows us to use the famous theory
discovered independently by Sprague [8] and Grundy [4], which generalizes
normal-play nim, analyzed by Bouton [2], into disjunctive sum play of any
finite set of impartial normal-play games.
Arithmetic functions are at the core of number theory, in a similar sense that
nim and Sprague-Grundy are central to the theory of combinatorial games. We
consider arithmetic functions [5] of the form $f:X\rightarrow Y$, where the
set $X$ is either the nonnegative integers, $\mathbb{N}_{0}=\\{0,1,\ldots\\}$,
or the positive integers, i.e., the natural numbers
$\mathbb{N}=\\{1,2,\ldots\\}$, and where typically $Y=2^{X}$ is the set of all
subsets of the nonnegative or positive integers respectively. In some
settings, for example when the arithmetic function is counting the instances
of another arithmetic function, we may take $Y=\mathbb{N}_{0}$; in these
cases, we will refer to $f$ as a counting function, and the games as counting
games.
Arithmetic functions may conveniently be interpreted as rulesets of impartial
games, and here we present old and novel games within a classification scheme.
Each arithmetic function induces a couple of rulesets, and we let
$\mbox{opt}:X\rightarrow Z$, define the set of move options from $n\in X$,
given some arithmetic function $f$, with sometimes an imposed terminal sink,
or a modified codomain, for example, in the powerset games to come,
$Z=2^{2^{X}}$; in the singleton and counting game cases we may take $Z=Y$
(modulo sometimes adjustment for $0$).
More specifically, we interprete the arithmetic functions in terms of heap
games; for each game position, i.e., heap size represented by a number (of
pebbles). There are two main variations.
* •
A player can _move-to_ a number or a disjunctve sum of numbers, induced by the
arithmetic function.
* •
A player can _subtract_ a number or a disjunctive sum of numbers, induced by
the arithmetic function.
We will use the name of the arithmetic function and prepend the letters M or
S, respectively, for the move-to and the subtract versions of a particular
game. The following two examples are excerpts from Section 1.1.
###### Example 1.
If the players may move-to a divisor, we get for example: from $\mathrm{6}$
the options are $1$,$2$ and $3$. From $7$ you may only move to $1$. Here, the
divisors must be proper divisors.
###### Example 2.
If the players may subtract a divisor, we get for example: from $6$, the
options are $5,4,3$ and $0$. From $7$ you can move to $0$ or $6$.
Before this paper, instances where number theory connects with impartial games
might individually have seemed like ‘lucky cases’. However, we feel that the
relatively large number of such examples justifies a more systematic
study.333See classical works such as Winning Ways [1] for results on impartial
games coinciding with number theory.
Let us list some game rules induced by some standard arithmetic functions.
When there is only one single option, we may omit the set brackets.
1. 1.
The aliquot (divisor) games:
1. (a)
maliquot. Move-to a proper divisor of your number; i.e.,
$\mbox{opt}(n)=\\{d:d\mid n,\;0<d<n\\}.$
2. (b)
saliquot. Subtract a divisor of your number; i.e.,
$\mbox{opt}(n)=\\{n-d:d\mid n,\;d>0\\}.$
2. 2.
The aliquant (nondivisor) games:
1. (a)
maliquant. Move-to a nondivisor of your number; i.e.,
$\mbox{opt}(n)=\\{k:1\leq k\leq n,\;k\nmid n\\}.$
2. (b)
saliquant. Subtract a nondivisor of your number; i.e.,
$\mbox{opt}(n)=\\{n-k:1\leq k\leq n,\;k\nmid n\\}.$
3. 3.
The $\tau$-games:444Here $\tau(n)$ counts the natural divisors of $n$:
$\tau(n)=\sum_{d\mid n}1$. The divisor function $\tau$ is multiplicative, with
$\tau(p^{a})=a+1$ for all primes $p$ and natural numbers $a$. In one of our
game settings, we use instead the number of proper divisors, and so let
$\tau^{\prime}=\tau-1$, so that in particular $\tau^{\prime}(1)=0$ and
$\tau^{\prime}(2)=1$ (here we lose multiplicativity).
1. (a)
mtau. Move-to the number of proper divisors of your number; i.e.,
$\mbox{opt}(n)=\tau^{\prime}(n).$
2. (b)
stau. Subtract the number of divisors of your number; i.e.,
$\mbox{opt}(n)=n-\tau(n).$
4. 4.
The totative (relative prime residue) and the nontotative games:555The ‘move-
to’ and ‘subtract’ variations are the same, because $(k,n)=1$ if and only if
$(n-k,n)=1$.
1. (a)
totative. Move-to any relatively prime residue; i.e.,
$\mbox{opt}(n)=\\{k:1\leq k\leq n:(k,n)=1\\}.$
2. (b)
nontotative. Move-to any smaller residue that is not relatively prime to your
number; i.e.,
$\mbox{opt}(n)=\\{k:1\leq k<n:(k,n)>1\\}.$
5. 5.
The totient ($\phi$) games:
1. (a)
totient. Move-to the number of relatively prime residues modulo your number;
i.e.,
$\mbox{opt}(n)=\phi(n).$
2. (b)
nontotient. Instead subtract this number; i.e.,
$\mbox{opt}(n)=n-\phi(n).$
6. 6.
dividing. Divide your number into a maximum number of equal parts, at least
two; i.e.,
$\mbox{opt}(n)=\\{\;\underbrace{k+k+\cdots+k}_{m\;\text{$k$'s}}:km=n,\;m>1\\}.$
7. 7.
dividing-and-remainder. Divide your number into a number of equal parts and a
remainder, which is smaller than the other parts and possibly 0; i.e.,
$\mbox{opt}(n)=\\{\,\underbrace{k+k+\cdots+k+r}_{m\;\text{$k$'s}}\\}:km+r=n,\;m>0,\;0\leq
r<k\,\\}.$
This game has two simpler variations, as defined in Section 3.2.
8. 8.
factoring. Factor your number into at least two components, and at most the
number of prime factors, counting multiplicity; i.e.,
$\mbox{opt}(n)=\\{a_{1}+a_{2}+\cdots+a_{k}:1<a_{1}\leq a_{2}\leq\cdots\leq
a_{k},\,a_{1}a_{2}\cdots a_{k}=n\,\\}.$
Item 7 here is the game in the first paragraph of the paper. The goal of this
paper is to evaluate the nim-values, a.k.a. Sprague-Grundy values, of these
games, and hence winning strategies can be computed, via the nim-sum operator,
in disjunctive sum with any other normal-play game. The nim-values of a
ruleset are defined via the _minimal excludant function_ ,
$\mbox{mex}:2^{X}\rightarrow\mathbb{N}_{0}$, where $X$ is the set of
nonnegative (or positive) integers. Let $A\subset\mathbb{N}_{0}$ be a strict
subset of the nonnegative integers. Then
$\mbox{mex}(A)=\min\\{x:x\in\mathbb{N}_{0}\setminus A\\}$ and
$\mathcal{SG}(n)=\mbox{mex}\\{\mathcal{SG}(x):x\in\mbox{opt}(n)\\}$. Note
that, if there is no move option from $n$, then $\mathcal{SG}(n)=0$. Recall
that the nim-sum is used to compute the nim-value of a disjunctive sum of
games, i.e., $\mathcal{SG}(\sum n_{i})=\bigoplus\mathcal{SG}(n_{i})$, where
‘$\sum$’ is disjunctive sum operator, and ‘$\bigoplus$’ is the sum modulo $2$
without carry of the numbers $n_{i}$ in their binary representations.
If $f$ is a counting function, then there is exactly one option. The game on a
single heap reduces to a trivial she-loves-me-she-loves-me-not game, and in
particular, the $\mathcal{SG}$-function reduces to a binary output, that is,
$\mathcal{SG}(n)\in\\{0,1\\}$. Hence, such rulesets will be referred to as
binary rulesets. The same game, however, played on several heaps, with at
least one non-binary ruleset, can be highly non-trivial, and result in great
complexity. The question of which heap to move on does not have a polynomial
time solution in general, while many arithmetic functions are known to be
intractable. Our inspiration for studying binary counting games came from
Harold Shapiro’s classification for the recurrence of the totient function
[6]. A couple of examples will clarify these type of issues.
###### Example 3.
Let $0$ be the empty heap. Suppose that, from a heap of size $n>0$, the
players can remove the number of divisors of $n$. The option of $n=1$ is $0$.
A heap of size 2 also has $0$ as an option, but $1$ is the option of $3$. The
$\mathcal{SG}$-sequence thus starts: $0,1,1,0,0,1$. The heap of size 5 has $3$
as the option, for which the nim-value is $0$. On one heap, while play is
trivial, the problem of determining the winner is as hard as the complexity of
the sequence.
###### Example 4.
Let $0$ be the empty heap. Suppose that, from a heap of size $n>0$, the
players can move-to the number of proper divisors of $n$. The option of $n=1$
is $0$. The heaps of size two and three have moves to the heap with a single
pebble. The number of proper divisors of $n=4$ is $2$, and hence the option is
$2$. As for all primes, the option of $n=5$ is the heap of size one. Thus, the
$\mathcal{SG}$-sequence starts: $0,1,0,0,1,0$.
###### Example 5.
Consider binary games. Of course, even playing a disjunctive sum of binary
games, gives only binary values. Consider, for example totient, where
$\mathcal{SG}(2+3+4+5)=\mathcal{SG}(2)\oplus\mathcal{SG}(3)\oplus\mathcal{SG}(4)\oplus\mathcal{SG}(5)=1\oplus
0\oplus 0\oplus 1=0$. Hence $2+3+4+5$ is a second player winning position. To
see this in play, suppose that the first player selects the heap of size $4$
and moves to $2+3+\phi(4)+5=2+3+2+5$. Now, $\mathcal{SG}(2+3+2+5)=1\oplus
0\oplus 1\oplus 1=1$, which is a winning position for the player to move, and
indeed, since every move changes the parity, we have automatic, ‘random’
optimal play even if we play a sum of games, _provided_ that they are all
binary. In particular if we play a disjunctive sum of totient games, then the
optimal strategy is to play any move. Hence these games seem less interesting
in that respect, as 2-player games, but suppose that we instead play a
disjunctive sum of the totient game $G$ with the totative game $H$. Now an
efficient algorithm for computing the binary value (see Theorem 8) is
interesting again. What is a sufficient move in the first player winning
position $7_{{\rm totient}}+7_{{\rm totative}}$? (There are exactly three
winning moves.)
Those examples motivate play on arithmetic counting functions. Other examples
of binary games are the fullset games, where each move is defined by playing
to a disjunctive sum of all numbers induced by the arithmetic function.
For the powerset rulesets, the range of the opt function is the set of all
subsets of natural numbers, a generic game on a single heap decomposes to play
on several heaps. Hence, the full $\mathcal{SG}$-function is intrinsically
motivated in the solution of a single game, even if the starting position is a
single heap.
###### Example 6.
Consider the game played from a heap of size $n$, where the options are to
play to any non-empty set of proper divisors of $n$. If $n=6$, then the
options are the single heaps of size $1,2,3$ respectively, the pairs of heaps
$1+2,1+3,2+3$, and the triple $1+2+3$. A heap of size one has no option, and a
heap of size two or three has a heap of size one as option. Hence
$\mathcal{SG}(6)=2$. The nim-value of each prime is one, and so on.
###### Example 7.
Consider the game played from a heap of size $n$, where the options are to
play to any finite set of relatively prime residues smaller than $n$. If
$n=5$, then the options are all nonempty subsets of $\\{1,2,3,4\\}$. In spite
of the relatively large number of options, in this particular case, the
$\mathcal{SG}$-computation becomes easy. A heap of size one has no option and
so $\mathcal{SG}(1)=0$. Therefore, $\mathcal{SG}(2)=1$, and so
$\mathcal{SG}(3)=2$. A heap of size $4$ has $1,3,1+3$ as options, and so
$\mathcal{SG}(4)=1$. By this, obviously $\mathcal{SG}(5)=4$. A heap of size
$6$ has few options, and easily $\mathcal{SG}(6)=1$. A heap of size $7$ has
many options, and likewise easily $\mathcal{SG}(7)=8$, the smallest unused
power of two. This game is revisited in Theorem 17.
In view of the above examples, we use the following classification of games on
arithmetic functions; an _arithmetic game_ satisfies one of these items.
* (i)
Play singletons from the arithmetic property;
* (ii)
Play the number of elements from the arithmetic property;
* (iii)
Play the disjunctive sum of all numbers from the arithmetic property;
* (iv)
Play any non-empty subset of numbers from the arithmetic property, as a
disjunctive sum.
The word “Play …” (read: “Play is defined by…”) is intentionally left open for
interpretation. Here, it will have one out of two meanings; either the players
move-to the numbers, or they subtract the numbers, from the given heap (size).
The items (iii) and (iv) typically split a heap into several heaps to be
played in a disjunctive sum of heaps. Note that (iii) is binary, although it
does not concern counting functions. The rulesets induced by (iii) and (iv)
above are not listed above, but naturally build on items 1, 2 and 4. We define
them in their respective sections.
Some arithmetic functions directly induce a disjunctive sum of games, such as
the division algorithm or the factoring problem. For the ruleset on Euclidian
division from the first paragraph (Section 3.1), we conjecture that the
relative nim-values, $\mathcal{SG}(n)/n$, tend to 0 with increasing heap
sizes.
In Section 1, we study singleton games. In Section 2, we study counting games.
In Section 3, we study dividing games, where division induces a disjunctive
sum of games, and similar for Section 4 with factoring games. In Section 5, we
study disjunctive sum games on the full set induced by the arithmetic
function. In Section 6, we study powerset disjunctive sum games. Section 7 is
devoted to some future direction.
For reference, let us include a table of studied rulesets, in the order of
appearance, including some significant properties. The abbreviations are m-t:
move-to, subtr.: subtraction, div.:divisor, rel.:relative, n.:number, pr.:
problem, disj.: disjunctive. The solution functions are defined in the
respective sections, but let us list them here as well. In particular, we
encounter indexing functions, where numbers with a certain property are
enumerated, starting with $1$ for their smallest member, etc. In the table we
find the following functions $\Omega$: number of prime divisors counted with
multiplicity, $\omega$: the number of prime divisors counted without
multiplicity, $\Omega_{2}$: the number of prime divisors counted with
multiplicity, unless the divisor is 2, which is counted without multiplicity,
$v$: usual 2-valuation, $i_{o}$: index of largest odd divisor, $i_{p}$: index
of smallest prime divisor,
Ruleset | description | arithmetic f. | solution f. | Sec.
---|---|---|---|---
maliquot | m-t div. | aliquot | $\Omega$ | 1.1.1
saliquot | subtr. div. | aliquot | $v$ | 1.1.2
maliquant | m-t non-div. | aliquant | $i_{o}$ | 1.2.1
saliquant | subtr. non-div. | aliquant | partial sol. | 1.2.2
totative | m-t rel. prime | totative | $i_{p}$ | 1.3
nontotative | m-t nonrel. prime | totative | partial sol. | 1.4
totient | m-t n. rel. prime | totient | Shapiro | 2.1.1
nontotient | m-t num. nonrel. prime | totient | | 2.1.2
mtau | m-t n. div. | $\tau$ | observation | 2.2.1
stau | subtr. n. div. | $\tau$ | | 2.2.2
m$\Omega$ | m-t n. prime div. | $\Omega$ | observation | 2.3.1
s$\Omega$ | subtr. n. prime div. | $\Omega$ | | 2.3.2
m$\omega$ | m-t n. dist. prime div. | $\omega$ | observation | 2.3.3
s$\omega$ | subtr. n. dist. prime div. | $\omega$ | | 2.3.4
dividing | m-t disj. sum div. | aliquot | $\Omega_{2}$ | 3.1
div.-and-res. | m-t disj. sum Eucl. div. | Eucl. div. | | 3.2
compl.-grundy | m-t disj. sum Eucl. div. | Eucl. div. | | 3.2
div.-throw-res. | m-t disj. sum Eucl. div. | Eucl. div. | $i_{o}$ | 3.2
res.-throw-div. | m-t residue | Eucl. div. | yes | 3.2
m-factoring | m-t factoring | factoring | $\Omega$ | 4
s-factoring | subtr. factoring | factoring | | 4
fs maliquot | m-t disj. sum all div. | aliquot | square free | 5
ps maliquot | m-t disj. sum div. | aliqout | | 6
ps saliquot | subtr. disj. sum div. | aliqout | $v$ | 6
ps maliquant | m-t disj. sum div. | aliquant | $i_{o}$ | 6
ps saliquant | subtr. disj. sum div. | aliquant | $i_{p}$ | 6
ps totative | m-t disj. sum div. | totative | | 6
ps nontotative | m-t disj. sum div. | totative | | 6
## 1 Singletons
This section concerns items 1,2 and 4 from the introduction, the aliquots, the
aliquants and the totatives.
### 1.1 The aliquots
The first game, maliquot, is ‘nim in disguise’ (think of the prime factors of
a number as the pebbles in a heap) but since the factoring problem is hard,
the game is equally hard. Here, the arithmetic function is
$f(n)=\\{d:d\,|\,n,n\in\mathbb{N}_{0}\\}$. In this section, the set of game
positions is $\mathbb{N}$. Since all nonnegative integers divide 0, we do not
admit 0 to the set of game positions. The second game, saliquot, turns out to
be somewhat more interesting.
Let $n\in\mathbb{N}$. Then $\Omega(n)$ is the number of prime factors of $n$,
counting multiplicities, and $v=v(n)$ is the 2-valuation of $n=2^{v}m$, where
$m$ is odd.
#### 1.1.1 maliquot, move-to a proper divisor
Here, the set of move options from $n$ is $\mbox{opt}(n)=\\{d:d\,|\,n,d\neq
n,n\in\mathbb{N}\\}$.
###### Example 8.
From 6 you have the options $1$, $2$ and $3$. From $7$ you may only move-to
$1$.
The unique terminal position is $1$. It follows that $\mathcal{SG}(1)=0$, and
if $p$ is a prime then $\mathcal{SG}(p)=1$. We have computed the first few
nim-values. $n$ $\mbox{opt}(n)$ $\mathcal{SG}(n)$ 1 $\varnothing$ 0 2 1 1 3 1
1 4 1,2 2 5 1 1 6 1,2,3 2 7 1 1 8 1,2,4 3
###### Theorem 1.
Consider maliquot. Then, for all $n$, $\mathcal{SG}(n)=\Omega(n)$.
###### Proof.
We have that $0=\mathcal{SG}(1)=\Omega(1)$, since there are no options from
$1$, and $1$ does not have any prime factors. Suppose that $n>1$ has $k$ prime
factors, counting multiplicities. Then, for each $x\in\\{1,\ldots,k-1\\}$,
there is a divisor of $n$, corresponding to a move-to a number with $x$ prime
factors. Since you are not allowed to divide by $n$, the number of prime
factors decreases by moving, and so there is no option of nim-value $k$, by
induction. By the mex-rule, the result holds, $\mathcal{SG}(n)=k=\Omega(n)$. ∎
#### 1.1.2 saliquot, subtract a divisor
This game is defined on the nonnegative integers, $\mathbb{N}_{0}$. Here
$\mbox{opt}(n)=\\{n-d:\;d\,|\,n,\,d>0\\}.$
###### Example 9.
The options of $6$ are $5$, $4$, $3$ and $0$. From $7$ you can move to $0$ or
$6$.
Since $0$ is always an option from $n$, it is clear that the nim-value of a
non-zero position is greater than $0$. The initial nim-values are: $n$
$\mbox{opt}(n)$ $\mathcal{SG}(n)$ 0 $\varnothing$ 0 1 0 1 2 0,1 2 3 0,2 1 4
0,2,3 3 5 0,4 1 6 0,3,4,5 2 7 0,6 1 8 0,4,6,7 4 As the table indicates, the
nim-values concern the 2-valuation of $n$.
###### Theorem 2.
Consider saliquot. Then $\mathcal{SG}(0)=0$. Suppose that $n>0$ and let
$n=2^{k}m$, where $2\nmid m$ and $k\geq 0$. Then $\mathcal{SG}(n)=v(n)+1=k+1$.
###### Proof.
The case $0\,|\,0$ is excluded in the definition of opt. Hence there is no
move from 0 and so $\mathcal{SG}(0)=0$. Note that if $n>0$ then $0$ of nim-
value $0$ is an option.
Suppose that $n$ is odd. Then, for all $d\,|\,n$, $n-d$ is even. By induction
the even numbers have nim-value greater than one. Hence, since 0 is an option,
the mex-function gives $0+1=1$ as the nim-value of $n=2^{0}m$.
Suppose that $n$ is even. Then $n-1$, which is odd, is an option. It has nim-
value 1 by induction. (Hence, $\mathcal{SG}(n)\geqslant 2$.) Let
$d=2^{\ell}q\leqslant n$, where we may assume that $\ell>0$, since we are
interested in the even options.
Since $d$ is a divisor of $n$, we have that $0\leq\ell\leqslant k$, and $q\mid
m$, with odd $m>1$. We get $n-d=2^{k}m-2^{\ell}q=2^{\ell}(2^{k-\ell}m-q)$. The
number $x=2^{k-\ell}m-q$ is odd, of nim-value $1$ by induction, unless
$k=\ell$. In this case, if $m=q$, the option is $0$, so suppose $m>q$. Since
both $m$ and $q$ are odd, then the option of $n$ has a greater $2$-valuation
than $n$, i.e $v(n)\geq k+1$. Therefore, no option has $2$-valuation $k$, and
hence by induction no option has nim-value $k+1$.
Since $\ell$ can be chosen freely in the interval $0\leq\ell<k$, by induction
all nim-values $0\leq\mathcal{SG}(n-d)\leq k$ can be reached; since $m>1$, we
may take $q<m$. The result follows. ∎
### 1.2 The aliquants
The aliquant games are somewhat more intricate than the aliquots, but we still
have an explicit solution in the first variation. Here $f(n)=\\{d:d\nmid
n,n\in\mathbb{N}_{0}\\}$.
#### 1.2.1 maliquant: move-to a non-divisor
Since all numbers divide $0$, $0$ does not have any options, and hence
$\mathcal{SG}(0)=0$. On the other hand, $0$ does not divide any nonzero
number, and hence $0$ will be an option from each number. The options are:
$\mbox{opt}(n)=\\{d<n:d\nmid n,n\in\mathbb{N}_{0}\\}.$ $n$ $\mbox{opt}(n)$
$\mathcal{SG}(n)$ 0 $\varnothing$ 0 1 0 1 2 0 1 3 0,2 2 4 0,3 1 5 0,2,3,4 3 6
0,4,5 2 7 0,2,3,4,5,6 4 8 0,3,5,6,7 1 In maliquant, the 2-valuation plays an
opposite role as in saliquot; here only the odd part of $n$ determines the
nim-value. Let $i_{o}:\mathbb{N}\rightarrow\mathbb{N}$ be the index function
for largest odd factor of a given natural number. That is, if $n=2^{k}(2m-1)$,
then $i(n)=m$. Clearly $i_{o}(2m-1)=m$ and $i_{o}(n)\leq(n+1)/2$.
###### Lemma 1.
For all $n\in\mathbb{N}$, the numbers in the set $\\{n,\ldots,2n-1\\}$
contribute all maximal odd factor indices in the set $\\{1,\ldots,n\\}$. That
is, $\\{i_{o}(x):n\leq x\leq 2n-1\\}=\\{1,\ldots,n\\}$.
###### Proof.
We use induction on $n$. Assuming the statement of the lemma holds for all
natural numbers up to $n$, we consider the set $\\{n+1,\dots,2n-1,2n,2n+1\\}$.
As $i_{o}(2n)=i_{o}(n)$ and $i_{o}(2n+1)=n+1$, it follows that
$\\{i_{o}(x):n\leq x\leq 2n+1\\}=\\{1,\ldots,n,n+1\\}$ ∎
###### Theorem 3.
Consider maliquant. Then, $\mathcal{SG}(0)=0$ and, for all $n\in\mathbb{N}$,
$\mathcal{SG}(n)=i_{o}(n)$.
###### Proof.
We use induction on $n\in\mathbb{N}$. If $n$ is odd, then write $n=2m-1$
whence $\\{m-1,\dots,2m-3\\}\subset\mbox{opt}(n)$. By Lemma 1 and induction
hypothesis, we have $\\{\mathcal{SG}(k):m-1\leq k\leq
2m-3\\}=\\{1,\dots,m-1\\}$. By induction hypothesis,
$\mathcal{SG}(k)=i_{o}(k)\leq m/2$ for all $k<m-1$, and hence
$\mathcal{SG}(n)=m=i_{o}(n)$. Even numbers are options, but they have nim-
values smaller than $m/2$, by induction.
If $n$ is even, write $n=2^{k}m$ where $k>0$ and $m$ is odd. Thus it suffices
to prove $\mathcal{SG}(n)=i_{o}(m)$. Note that, for all positive integers $l$,
we have $i_{o}(l)=i_{o}(m)$ if and only if $l=2^{j}m$ for some $j\geq 0$. Thus
if $m=1$ then $\mbox{opt}(n)$ omits all those elements $2^{j}$ with index $1$,
hence $\mathcal{SG}(n)=1$. Otherwise, if $m>1$, we observe that
$\left\\{2^{k-1}m+1,2^{k-1}m+2,\dots,2^{k}m-1\right\\}\subset\mbox{opt}(n).$
If we augment this set with the element $2^{k}m$, then by Lemma 1 the indices
of its elements are $\\{1,2,\dots,2^{k-1}m+1\\}$, which includes
$\\{1,2\dots,i_{o}(m)-1\\}$ as a subset. However, $\mbox{opt}(n)$ contains no
elements of the form $2^{j}m$, and hence $i_{o}(m)$ does not appear among the
elements $i_{o}(x)$ for all $x\in\mbox{opt}(n)$, yet all elements of
$\\{1,\dots,i_{o}(m)-1\\}$ do so appear. Thus by induction hypothesis
$\mathcal{SG}(n)=i_{o}(m)$. ∎
#### 1.2.2 saliquant: subtract a non-divisor
Here we run into some mysterious sequences. We can only prove partial results.
The options are: $\mbox{opt}(n)=\\{n-d:d\nmid n,n\in\mathbb{N}_{0}\\}.$
$n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$ | $n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$
---|---|---|---|---|---
0 | $\varnothing$ | 0 | 10 | 1,2,3,4,6,7 | 2
1 | $\varnothing$ | 0 | 11 | 1,2,3,4,5,6,7,8,9 | 5
2 | $\varnothing$ | 0 | 12 | 1,2,3,4,5,7 | 4
3 | 1 | 1 | 13 | $1,\ldots,11$ | 6
4 | 1 | 1 | 14 | $1,\ldots,6$, 8,9,10,11 | 6
5 | 1,2,3 | 2 | 15 | $1,\ldots,9$,11,13 | 7
6 | 1,2 | 1 | 16 | $1,\ldots,7$, 9,10,11,13 | 7
7 | 1,2,3,4,5 | 3 | 17 | 1,…,15 | 8
8 | 1,2,3,5 | 3 | 18 | 1,…, 8,10,11,13,14 | 4
9 | 1,2,3,4,5,7 | 4 | 19 | 1,…,17 | 9
The odd heap sizes turn out to be simple. We give some more nim-values for
even heap sizes, $n=0,2,\ldots$,
$\mathcal{SG}(n)=0,0,1,1,3,2,4,6,7,4,7,5,10,12,10,13,15,8,13,9,17,17,16,11,22,\ldots$
For even heap sizes $n\geq 2$,
$\mathcal{SG}(n)/n=0,1/4,1/6,3/8,1/5,1/3,3/7,7/16,2/9,7/20,5/22,5/12,6/13,5/14,$
$13/30,15/32,4/17,13/36,9/38,17/40,17/42,4/11,11/46,11/24,\ldots$
Sorting the ratios $\mathcal{SG}(n)/n$ by size, we find that the associated
nim-values $[n,\mathcal{SG}(n)]$ for the smallest ratios,
$[6,1],[10,2],[18,4],[22,5],[34,8],[38,9],[46,11]$ satisfy
$(n-2)/\mathcal{SG}(n)=4$. The half of each heap size in this sequence is odd,
and we get the odd numbers $3,5,9,11,17,19,23,\ldots$. We have not
investigated these patterns further, but we believe that, for all $n$,
$\mathcal{SG}(n)\geq(n-2)/4$. Indeed, by plotting the first 1000 nim-values in
Figure 1, this lower bound appears to continue.
Figure 1: The initial 1000 nim-values of saliquant.
###### Theorem 4.
Consider saliquant. Then $\mathcal{SG}(0)=0$, and if $n$ is odd, then
$\mathcal{SG}(n)=\frac{n-1}{2}$. Moreover, $\mathcal{SG}(n)<n/2$.
###### Proof.
Suppose that the statement holds for all $m<n$. If $n=2x+1$ then each
nonnegative integer smaller than $x$ is represented as a nim-value, and
specifically, for each odd number $2y+1$, with $y<x$, $\mathcal{SG}(2y+1)=y$.
Moreover, each odd number is an option of $x$, since each even integer is a
non-divisor of $n=2x+1$. Therefore, we use that, by induction, each even
number smaller than $n$ has a smaller nim-value and we are done with the first
part of the proof.
Suppose next that $n=2x$. Then, since both 1 and 2 are divisors, we know that
the largest option is smaller than $2x-2$. By induction, the nim-value of any
number smaller than $2x-2$ is smaller than $x-2$, namely
$\mathcal{SG}(2x-3)=\mathcal{SG}(2(x-2)+1)$ is the upper bound for a nim-value
of an option of $2x$. ∎
### 1.3 The totatives: move-to a relatively prime
Here $f(n)=\\{x\mid(x,n)=1\\}$. The totative games are defined by moving to a
relatively prime residue. We list the first few nim-values of totative. $n$
$\mbox{opt}(n)$ $\mathcal{SG}(n)$ 1 $\varnothing$ 0 2 1 1 3 1,2 2 4 1,3 1 5
1,2,3,4 3 6 1,5 1 7 1, …, 6 4 8 1,3,5,7 1 We have the following result. The
solution involves the function $i_{p}$, the index of the smallest prime
divisor of a given number, where the prime 2 has index 1.
###### Theorem 5.
Consider totative. The nim-value of $n>1$ is the index of the smallest prime
divisor of $n$, and $\mathcal{SG}(1)=0$.
###### Proof.
There is no move from $1$, because the only number relatively prime with $1$
is $1$, and options have smaller size than the number. Hence, by the
definition of the mex-function, $\mathcal{SG}(1)=0$. Also,
$\mathcal{SG}(2)=1$, since the only relatively prime number of $2$ is $1$,
which has nim-value $0$, and $i_{p}(2)=1$. Suppose that the result holds for
all numbers smaller than $n$. From $n$ you can only access a smaller number
with no common divisor to $n$. Therefore none of its options has the same
smallest prime divisor. This is one of the properties of the mex-rule.
Thus, the index of the smallest prime divisor of $n$ will be chosen as nim-
value if each prime with a smaller index appears as an option. But, the set of
relatively prime numbers smaller than $n$ contains in particular all the
relatively prime numbers of the smallest prime divisor of $n$, and hence, all
the primes that are smaller than the smallest prime factor of $n$. By
induction, this is the desired set of nim-values, since the move-to 1 (of nim-
value $0$) is always available. ∎
This is sequence A055396 in Sloane [7]: “Smallest prime dividing $n$ is
$a(n)$-th prime $(a(1)=0)$.”
### 1.4 The non-totatives: move-to a non-relatively prime
Here is a table of the first few nim-values of nontotative: $n$
$\mbox{opt}(n)$ $\mathcal{SG}(n)$ $n$ $\mbox{opt}(n)$ $\mathcal{SG}(n)$ 0
$\varnothing$ 0 10 0,2,4,5,6,8 5 1 0 1 11 0 1 2 0 1 12 0,2,3,4,6,8,9,10 6 3 0
1 13 $0$ 1 4 0,2 2 14 0,2,4,6,7,8,10,12, 7 5 0 1 15 0,3,5,6,9,10,12 4 6
0,2,3,4 3 16 0,2,4,6,8,10,12,14 8 7 0 1 17 0 1 8 0,2,4,6 4 18
0,2,3,4,6,8,9,10,12,14,15,16 9 9 0,3,6 2 19 0 1
This sequence does not yet appear in OEIS [7], but curiously enough, a nearby
sequence is A078898, “Number of times the smallest prime factor of $n$ is the
smallest prime factor for numbers $\leq n$; $a(0)=0$, $a(1)=1$.” For $n\geq
2$, $a(n)$ tells in which column of the sieve of Eratosthenes (see A083140,
A083221) $n$ occurs in. Here, $\mathcal{SG}(15)=4\neq 3=a(15)$ is the first
differing entry. In Figure 2 we plot the first $1000$ nim-values.
Figure 2: The initial $1000$ nim-values of nontotative.
Let us sketch a few nim-value subsequences. The primes have nim-value $1$, and
the prime squares have nim-value $2$. The numbers with close prime factors,
‘almost squares’, appear to have almost constant nim-values. On the other
hand, some numbers in arithmetic progressions appear to have nim-values in
almost arithmetic progressions. For all $n$, $\mathcal{SG}(2n)=n$. We give the
exact statements in Theorem 6 below.
For subsequences of the natural numbers, $s$, let the asymptotic relative nim-
value be
$r_{s}=\lim_{n}\frac{\mathcal{SG}(s(n))}{s(n)},$
if it exists. The subsequences of largest relative nim-values, apart from
$s_{0}=2,4,\ldots$ (with $\mathcal{SG}(2n)=n$), are $s_{1}=3,9,\ldots$ and
$s_{2}=5,25,35,55,65,\ldots$, with corresponding first differences
$\Delta_{1}=(6,6,\ldots)$ and $\Delta_{2}=(20,10,20,10,\ldots)$, with nim-
values in $\\{\lfloor(n+1)/4\rfloor\\}$ and $\\{\lfloor n/10\rfloor,\lceil
n/10\rceil\\}$ respectively. Thus $r_{0}=1/2$, $r_{1}=1/4$ and $r_{2}=1/10$,
where $r_{s_{i}}=r_{i}$. The region between ‘prime factorization’ and ‘purely
arithmetic behavior’ is still mysterious. We can identify at least one more
sequence of arithmetic behavior, with $r_{3}\approx 1/17$, but the
descriptions start to get quite technical here. Note that (ix) and (v) imply
(viii); it is handy to state (v) as a separate item as it is used several
places in the proof.
###### Theorem 6.
Consider nontotative. For $n\in\mathbb{N}$,
1. (i)
$\mathcal{SG}(n)=1$ if and only if $n$ is prime;
2. (ii)
$\mathcal{SG}(n)=2$ if and only if $n$ is a prime square;
3. (iii)
$\mathcal{SG}(n)\in\\{3,4\\}$ if and only if $n=p_{i}p_{i+1}$, or $n=8$:
$\mathcal{SG}(n)=3$ if and only if $i$ is odd, with $p_{1}=2$;
4. (iv)
$\mathcal{SG}(n)\in\\{5,6\\}$ if and only if $n=p_{i}p_{i+2}$ or $n=12$:
$\mathcal{SG}(n)=5$ if and only if $i\equiv 1,2\pmod{4}$.
Moreover, for $n\in\mathbb{N}$,
1. (v)
$\mathcal{SG}(2n)=n$;
2. (vi)
If $n\equiv 3\pmod{6}$, then $\mathcal{SG}(n)=\lfloor(n+1)/4\rfloor$;
3. (vii)
if $n\equiv 5,25\pmod{30}$, then $\mathcal{SG}(n)\in\\{\lfloor
n/10\rfloor,\lceil n/10\rceil\\}$.
Lastly, for $n\in\mathbb{N}$,
1. (viii)
$\mathcal{SG}(n)\leq n/2$;
2. (ix)
if $n$ is odd, then $\mathcal{SG}(n)\leq(n+1)/4$.
###### Proof.
Note that $\mathcal{SG}(0)=0$ implies $\mathcal{SG}(n)>0$ if $n>0$. The
induction hypothesis assumes all the items. Not that item (v) takes care of
all cases where the prime $2$ divides $n$, so in all other items, we may
assume that the smallest prime dividing $n$ is greater than $2$.
For (i), the only non-relatively prime number of a prime is 0. Hence
$\mathcal{SG}(p)=1$ if $p$ is prime. If $n=pm$ is not a prime, then there is a
move to the prime divisor $p$, a non-relative prime, and there is a move to 0.
Hence the nim-value is greater than one.
For (ii) we consider prime squares $p^{2}$, and note that each option is of
the form $np$, $0\leq n\leq p-1$. In particular there are moves $p^{2}\mapsto
0$ and $p^{2}\mapsto p$, as noted in the first paragraph. Moreover, by
induction we assume that $\mathcal{SG}(m)=2$ if and only if $1<m<p^{2}$ is a
prime square. Then $m\neq np$, and so, by the minimal exclusive algorithm,
$\mathcal{SG}(p^{2})=2$. For the other direction, we are done with the cases
$0,1$ and primes. Consider the composite $n=pm$, not a prime square, where $p$
is the smallest prime factor. Then there is a move to $p^{2}$, and hence
$\mathcal{SG}(n)\neq 2$.
For (iii) we begin by proving that $\mathcal{SG}(n)\in\\{3,4\\}$ if
$n=p_{i}p_{i+1}$ or $n=8$, and where the nim-value is three if and only if the
index of the smaller prime is odd. The base case is $\mathcal{SG}(2\cdot
3)=3$, and where the exception $n=8$ is by inspection. For the generic case,
each option is of the form $np_{i}$, $0\leq n\leq p_{i+1}-1$ or $np_{i+1}$,
$0\leq n\leq p_{i}-1$. In particular, there is a move to $p_{i}$ (and to
$p_{i+1}$) of nim-value one, and there is a move to $p_{i}^{2}$ of nim-value
$2$. We must show that there is no move to another prime pair of the same
form, i.e., $p_{j}p_{j+1}$, with $j$ of the same parity as $i$. Observe that
there is a move to $p_{i-1}p_{i}$, with $j+1=i$, but there is no move to any
other almost square $p_{j}p_{j+1}$. By induction, this observation suffices to
find a move to nim-value $3$, if $i$ is even. For the other direction, we must
show that $\mathcal{SG}(n)\not\in\\{3,4\\}$ if $n$ is not an almost square. We
are done with the cases $n$ a prime or a prime square. Suppose that $n=px$ is
not of the mentioned form, where $p$ is the smallest prime factor of $n$. The
case $p=2$ is dealt with in item (v), so lets assume $p>2$. Then, there is a
move to $pq$ (non-relative prime with $n$), where $q$ is the smallest prime
larger than $p$, because by assumption, $x>q$. And there is a move to $pq$,
where $q$ is the largest prime smaller than $p$.
For (iv), we study the case $n=p_{i}p_{i+2}$. If $i=1$, then $n=10$, and
$\mathcal{SG}(10)=5$. If $i=2$, then $n=21$, and, by inspection,
$\mathcal{SG}(21)=5$. For the general case, among the options we find
$0,p_{i},p_{i}^{2},p_{i}p_{i+1},p_{i+1}p_{i+2}$. Hence, by the previous
paragraphs, the options attain all nim-values smaller than 5. Next, suppose
that $i\equiv 1,2\pmod{4}$, and we must show that there is no option of the
same form, to create nim-value $5$. Each option is a multiple of one of the
primes $p_{i}$ and $p_{i+2}$. The only possibility would be the option
$p_{i-2}p_{i}$. But $i-2\equiv 0,3\pmod{4}$. Hence no option has nim-value
$5$. The analogous argument suffices to show that no option has nim-value $6$
if $i\equiv 0,3\pmod{4}$, $i\geq 3$. The special case $n=12=2\times 2\times 3$
is not an option, by $i\geq 3$. On the other hand, the argument shows that
there is an option to nim-value $5$. Consider the other direction. Suppose
that $n=p_{i}x$, where $p_{i}>2$ is the smallest prime in the factorization of
$n$. (The case $p=2$ is dealt with below.) If $x>p_{i+2}$ then there is a move
to $p_{i}p_{i+2}$, and if $i\geq 3$, then there is a move to $p_{i-2}p_{i}$.
If $i=2$, then there is a move to $12$ of nim-value $6$. That concludes this
case. If $p_{i}<x<p_{i+2}$, then $x=p_{i+1}$, since $p_{i}$ is the smallest
prime in the decomposition of $n$, and we are done with this case.
For (v), we verify that, for all $n$, $\mathcal{SG}(2n)=n$. The options are of
the forms $2j$, with $0\leq j\leq n$, and so, induction on (v) gives that each
nim-value smaller than $n$ can be reached. Moreover induction on (viii) gives
that nim-value $n$ does not appear among the options.
For (vi) we must prove: if $n\equiv 3\pmod{6}$, then
$\mathcal{SG}(n)=\lfloor(n+1)/4\rfloor$. The claimed nim-value sequence for
the positions $3,9,15,21,27,\ldots$ is
$\lfloor(3+1)/4\rfloor,\lfloor(9+1)/4\rfloor,\lfloor(15+1)/4\rfloor,\ldots$,
which is $1,2,4,5,7,8,\ldots$. Clearly $n$ has each smaller position of the
same form as an option. Precisely, the multiples of 3 are missing in the nim-
value sequence. But, induction on (v), the multiples of $6$ have nim-values
multiples of $3$, and indeed, all multiples of $6$, smaller than $n$ are
options of $n$. By induction on item (ix), since $n$ is odd, the nim-value
$\lfloor(n+1)/4\rfloor$ does not appear among its options. The proof of (vii)
is similar to (vi), but more technical, so we omit it.
Item (viii) follows directly by induction (for example, if $n$ is even then
$n-2$ is the largest option and $\mathcal{SG}(n-2)\leq n/2-1$).
For item (ix), assume that $p>2$ is smallest prime divisor of $n$. The cases
with $p\leq 5$ have already been proved in items (vi) and (vii). Hence $p>5$.
It follows that with $t=\lfloor\frac{n}{2p}\rfloor$, $tp+3<n/2<(t+1)p-3$. It
follows that the nim-value $\lfloor\frac{n+1}{4}\rfloor$ cannot be reached
from $n$, by options of the form in (v). On the other hand it cannot be
reached, by moving to an odd number, since options $n-2p$ or smaller produce
too small nim-values, by induction. ∎
We do not yet know, if all nim-values can be obtained by analogous reasoning.
The initial occurrences (as for $n=12$ in the proof above) of nim-values that
do not follow general patterns may complicate things.
## 2 Counting games
This section concerns rulesets as in item (i) in the introduction. _Binary
games_ have only one option per heap. At each stage of play, the decision
problem reduces to which one of the heaps to make the move. The nim-value of
any sum of binary games is binary, that is, each nim-value $\in\\{0,1\\}$.
Indeed, the nim-value of a given disjunctive sum of binary games is 0 if and
only if the number of heaps of nim-value one is even. Of course, the nim-value
sequence for any given ruleset is valid in the much larger context of all
normal-play combinatorial games.
We begin in Section 2.1.1, by solving totient, and then we sketch a
classification scheme for nontotient. Then we list nim-values of other open
counting games.
### 2.1 Harold Shapiro and the totient games
Recall that Euler’s totient (or $\phi$) function counts the number of
relatively prime residues of a given number. For example $\phi(7)=6$ and
$\phi(6)=2$. Recall that this function is multiplicative. This is where we can
apply a known result by Harold Shapiro “An arithmetic function arising from
the $\phi$ function” [6]. The fundamental theorem for iteration of the Euler
$\phi$ function in his work is as follows. For all $x$, let
$\phi^{i}(n)=\phi^{i-1}(\phi(n))$. Since $\phi^{i}(n)<\phi^{i-1}(n)$ and
$\phi(n)$ is even, if $n>2$, we have that, for all $n$ and some unique $i>0$
(depending on $n$ only)
$\displaystyle\phi^{i}(n)=2.$ (1)
This lets us define the Class of $n$, as $C(n)=i$, when (1) holds, and
otherwise $C(1)=C(2)=0$.
###### Theorem 7 ([6]).
Let $m,n\in\mathbb{N}$. If $n$ is odd, then $C(n)=C(2n)$, and otherwise
$C(n)+1=C(2n)$. In general, if either $m$ or $n$ is odd, then
$C(mn)=C(m)+C(n)$. Otherwise, that is, if both $m$ and $n$ are even, then
$C(mn)=C(m)+C(n)+1$.
For example $\phi(7)^{2}=\phi(6)=2$, so $C(7)=2$. In general, for primes $p$,
$\phi(p)^{2}=\phi(p-1)$, so $C(p)=C(p-1)+1$. Here is an example:
$C(15)=C(3)+C(5)=1+C(4)+1=2+C(2)+C(2)+1=3$, and note that
$\phi(15)=\phi(3)\phi(5)=2\cdot 4=8$. Moreover $\phi(8)=4$ and $\phi(4)=2$, so
indeed $\phi^{3}(15)=2$.
This result lets us compute the nim-values of the first totient ruleset,
totient, simply by recalling the $\phi$-values for the primes.
#### 2.1.1 totient, the move-to variation of the totient game
As mentioned, the nim-value table contains only $0$s and $1$s, and we call
this kind of games ‘binary games’. Indeed, there is only one option from $n$,
namely $\phi(n)$, the number of relatively prime residues of $n$. $n$
$\mbox{opt}(n)$ $\mathcal{SG}(n)$ 1 $\varnothing$ 0 2 1 1 3 2 0 4 2 0 5 4 1 6
2 0 7 6 1 8 4 1
Here are a few more nim-values in the $\mathcal{SG}$-sequence:
$01001,01111,01010,01100,00101,0001$
where the “,” is for readability. Each game component has a forced move, that
if played alone may be regarded as an automaton. Starting from 8, for example,
the iteration of $\phi$ gives the sequence of moves $8\mapsto 4\mapsto
2\mapsto 1$, and the $\mathcal{SG}$-sequence, of course is alternating between
0s and 1s, terminating with the 0 at position 1. If played on a disjunctive
sum of totient, the nim-value sequence is of course also binary, alternating
between 0s and 1s, and it is 0 if and only if there is an even number of heaps
of nim-value 1. Suppose that we play $7_{\rm t}+7_{\rm s}$, where the first
$7$ is totient and the second $7$ is subtraction$\\{1,2\\}$, with nim-value
sequence $0,1,2,0,1,2,0,\ldots$, say with sink number $1$ on both components.
Then there are exactly two winning moves to $6_{\rm t}+7_{\rm s}$ or $7_{\rm
t}+5_{\rm s}$. Intelligent play from a general position $m_{\rm t}+n_{\rm s}$
requires full understanding of totient.
###### Theorem 8.
Consider totient, and let $C$ be as in Theorem 7. Then $\mathcal{SG}(1)=0$,
and for $n>1$, $\mathcal{SG}(n)=C(n)+1\pmod{2}$.
###### Proof.
Use Theorem 7. ∎
In general, thus it suffices to compute the parity of $C(n)$ and, given the
factorization of $n$, apply Theorem 7. For example,
$C(2^{3})=C(2)+C(2^{2})+1=3C(2)+2=2$, and without looking into the table, we
get $\mathcal{SG}(8)=1$. For another example, if $n=2\cdot 3^{7}\cdot 11$,
then $C(n)=7C(3)+C(11)=7+3=10$, since $\phi(3)=2$ and $\phi^{3}(11)=2$.
Therefore $\mathcal{SG}(48114)=(C(48114)+1)\pmod{2}=11\pmod{2}=1$, and we find
a unique winning move $48114_{\rm t}+3_{\rm s}\mapsto 48114_{\rm t}+2_{\rm
s}$, where s is still subtraction$\\{1,2\\}$.
#### 2.1.2 nontotient, the subtraction variation of the totient game.
From a given number $n$, subtract the number of relatively prime numbers
smaller than $n$. We cannot adapt Theorem 8, because it relies on iterations
where you instead move-to this number, and the authors have not yet found a
similarly efficient tool. Let us list the initial options and nim-values. An
alternative way to think of the options is: move-to the number of nonrelative
primes, including the number. The nim-values alternate for heaps that are
powers of primes, starting with $\mathcal{SG}(p^{0})=0$. This happens, because
the number of nonrelatively prime numbers smaller than or equal a prime power
$p^{k}$ is $p^{k-1}$. Hence, $\mathcal{SG}(p^{k})=0$ if and only if $k$ is
even. Since $\phi$ is multiplicative it is easy to compute $f(n)=n-\phi(n)$,
for any $n$, or get a formula for $f$, for any given prime decomposition of
$n$. However, $f$ is not multiplicative, which limits the applicability of
such formulas. In some special cases, we can use the proximity to powers of
primes for fast computation of the nim-value. Take the case of $n=p^{k}q$, for
some distinct primes $p$ and $q$. Then $f(n)=n-\phi(n)=p^{k-1}(q+p-1)$.
Whenever $q+p-1$ is a power of the prime $p$, the nim-value of $n$ is
immediate by the parity of the new exponent. Take for example $p=2$ and $q=7$.
Then $p+q-1=2^{3}$, and so we can find the nim-value of, for example
$n=7168=2^{10}\times 7$ gives the exponent $9+3=12$ and so
$\mathcal{SG}(7168)=0$. Similarly, with $p=3$ and $q=7$, we can easily compute
$\mathcal{SG}(413343)=1$, because $p+q-1=3^{2}$, and $413343=3^{10}\times 7$.
Let ‘$\mathrm{dist}$’ denote the number of iterations of $f$ to an even power
of a prime. We get the following suggestive table of the first few nim-values.
We leave a further classification of $\mathrm{dist}$ as an open problem.
$n$ | $\mbox{opt}(n)$ | dist | $\mathcal{SG}(n)$
---|---|---|---
1 | $\varnothing$ | 0 | 0
2 | 1 | 1 | 1
3 | 1 | 1 | 1
4 | 2 | 0 | 0
5 | 1 | 1 | 1
6 | 4 | 1 | 1
7 | 1 | 1 | 1
8 | 4 | 1 | 1
9 | 3 | 0 | 0
10 | 6 | 2 | 0
11 | 1 | 1 | 1
12 | 8 | 2 | 0
13 | 1 | 1 | 1
14 | 8 | 2 | 0
15 | 7 | 2 | 0
16 | 8 | 0 | 0
### 2.2 The $\tau$-games
The $\mathcal{SG}$-sequences of mtau and stau do not yet appear in OEIS. The
number of divisors is multiplicative in the following sense:
$\tau(n)=(a_{1}+1)\cdots(a_{k}+1)$, where $n=p^{a_{1}}_{1}\cdots
p^{a_{k}}_{k}$.
#### 2.2.1 Move-to the number of proper divisors
Consider mtau, where the single option is the number of proper
divisors.666Note that if we remove the word proper here, then both 1 and 2
become loopy, and thus all games would be drawn. See also Section 7 for some
more reflections on ‘loopy’ or ‘cyclic’ games. A heap of size one has no
option, so the nim-value sequence starts at $\mathcal{SG}(1)=0$. Let us list
the first few nim-values.
$n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$
---|---|---
1 | $\varnothing$ | 0
2 | 1 | 1
3 | 1 | 1
4 | 2 | 0
5 | 1 | 1
6 | 3 | 0
7 | 1 | 1
8 | 3 | 0
9 | 2 | 0
Note that each prime has nim-value $1$ because they have only one proper
divisor. From this small table we may deduce many more nim-values. The first
few 0-positions are
$1,4,6,8,9,10,12,14,15,18,20,21,22,24,25,26,27,28,30,\ldots$
Note that $16$ is the first composite number that is not included, and $36$ is
the second one, and then $48,80,81,100,$ etc. What is special about these
composite numbers?
The sequence of all ones has some resemblance to the sequence of all numbers
with a nonprime number of proper divisors. As mentioned, $16$ and $36$ are the
first composite members of this sequence. These two numbers are the smallest
composite numbers with a composite, i.e., nonprime, number of proper divisors,
such numbers generalize the primes, because primes also have a nonprime number
of proper divisors. We are interested in the smallest number
$n=p^{a_{1}}_{1}\cdots p^{a_{k}}_{k}$, for which
$\tau(n)-1=(a_{1}+1)\cdots(a_{k}+1)-1\in\\{16,36,48,80,\ldots\\}$, that is,
the smallest number $n$, such that
$(a_{1}+1)\cdots(a_{k}+1)\in\\{17,37,49,81,\ldots\\}$. An obvious candidate is
$n=2^{16}$, with $a_{1}=16$, and otherwise $a_{i}=0$. But it turns out that
$n=2^{6}\cdot 3^{6}=46656<2^{16}$ gives $\tau(46656)=(6+1)(6+1)=49$, and this
is indeed the smallest such number. Thus, we have the following observation.
###### Observation 1.
Consider mtau. If $n<46656$, then $\mathcal{SG}(n)=1$ if and only if $n$
consists of a nonprime number of divisors.
We note that neither sequence is listed in OEIS (see Section 2.3.1 for a
similar sequence that is listed).
#### 2.2.2 Subtract the number of divisors
Consider stau. This variation has a $\mathcal{SG}$-sequence beginning with
$\mathcal{SG}(0)=0$. A heap of size one has one divisor, with an option to
zero. A heap of size two has two divisors and hence the option is zero, and so
on: $0,1,1,0,0,1,0,0,1,1,1,0,1,1,0,1,1,0,0,1,1,1$. The ‘1’s occur at
$1,2,5,8,9,10,12,13,15,16,19,20\ldots$
### 2.3 The $\Omega$ and $\omega$-games
The sequence of number of prime factors counted with multiplicity, is called
$\Omega(n)$. Otherwise, when only the distinct primes are counted, it is
called $\omega(n)$.777That is, if $n$ has canonical form
$n=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{k}^{a_{k}}$,
$\Omega(n)=a_{1}+a_{2}+\cdots+a_{k}$ and $\omega(n)=k$.
Somewhat surprisingly, nim-value sequences for the games that count the number
of prime divisors, do not yet appear in OEIS.
#### 2.3.1 Move-to the number of prime divisors
The nim-value sequence of m$\Omega$ starts at a heap of size one, of nim-value
$0$, by definition. Any prime, has a move-to one, so all primes have nim-value
one, a square has a move-to the heap of size 2, and hence has nim-value $0$,
and so on. The nim-value sequence starts:
$0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,1,\ldots$
The indices of the ones is a generalization of the primes:
$2,3,5,7,11,13,16,17,19,23,24,29,31,36,37,40,41\ldots$
The number $64$ is in the sequence, and this distinguishes it from A026478.
Still it is not exactly A167175, since not all numbers with a nonprime number
of prime divisors are included. The sequences coincide until $2^{16}-1$
though, since the first such number to be excluded is $2^{16}$. Via a similar
(but easier) reasoning as in Section 2.2.1, we have the following observation.
###### Observation 2.
Consider m$\Omega$. If $n<2^{16}$, then $\mathcal{SG}(n)=1$ if and only if $n$
consists of a nonprime number of prime divisors, counted with multiplicity.
#### 2.3.2 Subtract the number of prime divisors
Here we consider the ruleset s$\Omega$ ‘subtract the number of prime
divisors’. A heap of size one has nim-value $0$, by definition. A heap of size
two has a move to a heap of size one, and has nim-value one. A heap of size
three has a move to a heap of size two, and has nim-value $0$. The nim-value
sequence starts:
$0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,1,\ldots,$
and the indices of the ones are located at
$2,5,6,9,10,13,14,16,18,20,21,23\ldots$
This sequence does not appear in OEIS.
#### 2.3.3 Move-to the number of distinct prime divisors
The nim-value sequence of m$\omega$ ‘move-to number of distinct prime
divisors’ starts at one, of nim-value zero. The first few nim-values are:
$0,1,1,1,1,0,1,1,1,0,1,0,\ldots$, and the corresponding indices of the ones
are $2,3,4,5,7,8,9,11,13,\ldots$
The first nim-value that distinguishes it from m$\Omega$ is for the heap of
size 4. Since it has only one distinct factor, this game behaves like a prime,
and the nim-value is one. Six is the first number that has more than one
distinct factor. Hence $7!$ is the smallest number with distinct factors, for
which the nim-value is one.
###### Observation 3.
If $n<7!$, then $\mathcal{SG}(n)=1$ if and only if $n$ contains exactly one
distinct factor.
#### 2.3.4 Subtract the number of distinct prime divisors
The nim-value sequence of s$\omega$ ‘subtract the number of distinct prime
divisors’ starts at one, which does not have any prime divisor, and hence of
nim-value zero. Next, two has the option one, three has the option two, and
four has the option three. The first few nim-values are:
$0,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,\ldots,$
with $1$s at indices $2,4,7,9,10,13,14,16,19,20,23,\ldots$ Neither of these
sequences appear in OEIS.
## 3 Dividing games
The ruleset dividing deploys the notion of a disjunctive sum in their
recursive definition. That is, an option is typically, with some exception, a
disjunctive sum of games. A reference that goes into detail of such games is
[3].
### 3.1 The dividing game
For this game, the position is a natural number. The $Y$ in the definition of
opt is not $2^{X}$, but instead consists of disjunctive sums of natural
numbers. A player divides the current number into equal parts and we write “+”
to separate the parts, the new game components. To avoid long chains of
components, we use multiplicative notation, in the sense that $x\times y$
means $y$ copies of $x$ (that is, $x+\ldots+x$). In this notation, addition is
commutative, but multiplication is not. For example $\text{opt}(10)=\\{5\times
2,2\times 5,1\times 10\\}$. The current player moves in precisely one of the
components and leaves the other ones unchanged. For example, a move from $5+5$
is to $5+1\times 5=5$ (because no move is possible from $1\times 5$), and
indeed, by symmetry, this is the only admissible move. The number of options
is $\tau(n)-1$.
$n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$
---|---|---
1 | $\varnothing$ | 0
2 | $1+1$ | 1
3 | $1\times 3$ | 1
4 | $2\times 2,1\times 4$ | 1
5 | $1\times 5$ | 1
6 | $3\times 2,2\times 3,1\times 6$ | 2
7 | $1\times 7$ | 1
8 | $4\times 2,2\times 4,1\times 8$ | 1
Let $\Omega_{2}(n)$ denote the number of prime factors of $n$, where the
powers of 2 are counted without multiplicity, and the powers of odd primes are
counted with multiplicity.
###### Theorem 9.
Consider dividing. For all $n\in\mathbb{N}$, $\mathcal{SG}(n)=\Omega_{2}(n)$.
###### Proof.
$\mathcal{SG}(1)=0$, and $1$ does not have any prime components. Suppose that
$n$ is a power of two. Then $\mathcal{SG}(n)=1$, since each option permits the
mimic strategy. Similarly, if $n$ is a prime, then $\mathcal{SG}(n)=1$.
Suppose that $n=2^{k}p_{1}\cdots p_{j}$, with $k\in\mathbb{N}_{0}$ and each
$p_{i}$ odd. We use induction to prove that $\mathcal{SG}(n)=j$ if $k=0$, and
otherwise $\mathcal{SG}(n)=j+1$. If $k=0$, $n$ can be split into an odd number
of components each having $m$ prime factors for each $m\in[1,j-1]$. Induction
and the nim-sum together with the mex-rule gives the result in this case.
Similarly, if $k>0$, $n$ can be split into an odd number of components of $m$
prime factors for each $m\in[1,j]$, which proves that $\mathcal{SG}(n)=j+1$ in
this case. ∎
###### Example 10.
Suppose the position is $18+7=2\cdot 3^{2}+7$. How do you play to win? The
nim-value is $3\oplus 1=2$, where $\oplus$ denotes the nim-sum. Hence the next
player has a good move. The good move turns the 18-component to nim-value 1,
that is, we divide it into an odd number of even numbers with no odd factor.
This can be done in only one way: you move to $2\times 9+7=2+7$, and clearly
$\mathcal{SG}(2+7)=0$. The next player has exactly two options, but, either
way, you will finish the game in your next move.
### 3.2 The dividing and remainder game
The ruleset divide-and-residue is an extension of dividing, where you are
allowed to divide $n$ to $k$ equal parts $d$ and a remainder $r$ that is
smaller than the parts. Thus, here we have a lot more options (for a generic
game) than dividing, which is obvious by the representation $n=k\times d+r$,
with $0\leq r<d$. By moving we are free to choose any $1\leq d<n$, so we have
$n-1$ options, for all $n>0$. The $\mathcal{SG}$-sequence starts:
$n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$
---|---|---
1 | $\varnothing$ | 0
2 | $1+1$ | 1
3 | $2+1,1\times 3$ | 2
4 | $3+1,2\times 2,1\times 4$ | 1
5 | $4+1,3+2,2\times 2+1,1\times 5$ | 2
6 | $5+1,4+2,3\times 2,2\times 3,1\times 6$ | 3
7 | $6+1,5+2,4+3,3\times 2+1,2\times 3+1,1\times 7$ | 2
8 | $7+1,6+2,5+3,4\times 2,3\times 2+2,2\times 4,1\times 8$ | 3
An even number of heaps of the same sizes reduces to a heap of size one. A
heap of size one in a disjunctive sum, gets removed. We get an equivalent
reduced table:
$n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$
---|---|---
1 | $\varnothing$ | 0
2 | $1$ | 1
3 | $2,1$ | 2
4 | $3,1$ | 1
5 | $4,3+2,1$ | 2
6 | $5,4+2,2,1$ | 3
7 | $6,5+2,4+3,2,1$ | 2
8 | $7,6+2,5+3,2,1$ | 3
9 | $8,7+2,6+3,5+4,3,1$ | 4
10 | $9,8+2,7+3,6+4,3,2,1$ | 3
Note that, when we remove pairs of equal numbers, sometimes we must add the
option ‘1’ to symbolize a move to a terminal position of nim-value $2+2=0$.
From this table we may deduce that the game $7+3$ from the first paragraph in
the paper is indeed a losing position. divide-and-residue has a mysterious
$\mathcal{SG}$-sequence, as depicted in Figure 3.
Figure 3: The initial 20000 nim-values of divide-and-residue. They just about
touch the nim-value $2^{8}=256$.
Here are the 50 first nim-values, of the form [heap size, nim-value]:
$[1,0],[2,1],[3,2],[4,1],[5,2],[6,3],[7,2],[8,3],[9,4],[10,3],[11,4],[12,3],[13,4],[14,3],[15,4],[16,3],\newline
[17,4],[18,5],[19,4],[20,5],[21,3],[22,5],[23,4],[24,2],[25,1],[26,5],[27,6],[28,5],[29,6],[30,2],[31,6],\newline
[32,5],[33,3],[34,8],[35,9],[36,8],[37,9],[38,8],[39,9],[40,8],[41,9],[42,4],[43,9],[44,4],[45,9],[46,8],\newline
[47,9],[48,4],[49,9],[50,4].$
Early nim-values tend to be odd for heaps of even size, and even for those of
odd size. By an elementary argument we get: the heap of size 25 is the largest
heap of nim-value one, and one can prove the analogous statement for a few
more small nim-values. We conjecture that any fixed nim-value occurs finitely
many times.
For the upper bound, the nim-values seem to be bounded by $n^{3/5}$, for
sufficiently large heap sizes $n$. An empirical observation is that the growth
of nim-values appears to be halted at powers of two. For example, the nim-
value $2^{2}$ starts to appear at heap size $9$, but does not increase beyond
$2^{2}+2^{0}$ until the heap of size 27.
###### Conjecture 1.
Consider divide-and-residue. Then each nim-value occurs, and at most a finite
number of times. Moreover $\mathcal{SG}(n)/n\rightarrow 0$, as
$n\rightarrow\infty$.
There is no big surprise that this game is hard, since it is an extension of
grundy’s game [1, 7]. Indeed, the options of divide-and-residue in which the
divisor $d$ is greater than $n/2$ correspond to the rule of splitting a heap
into two unequal parts of grundy’s game. If we define the ruleset complement-
grundy, by requiring that $k\geq 2$ in divide-and-residue, then we can prove
the second statement in Conjecture 1 for this new game. Let us tabular the
first few nim-values, where options are displayed in reduced form:
$n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$
---|---|---
1 | $\varnothing$ | 0
2 | $1$ | 1
3 | $1$ | 1
4 | $1$ | 1
5 | $1$ | 1
6 | $2,1$ | 2
7 | $2,1$ | 2
8 | $2,1$ | 2
9 | $3,2,1$ | 2
10 | $3,2,1$ | 2
11 | $3,3+2,2,1$ | 2
Figure 4 shows that the initial regularity of nim-values is replaced by more
complexity further down the road, although not as severely as for divide-and-
residue. Note that the two games appear to share some geometric properties
such as a local stop of nim-value growth at powers of two, and a bounded
number of occurrences for each nim-value. In this case though, some nim-values
do not appear, such as $12,15,20$ etc. We do not yet know if the omitted nim-
values can be described by some succinct formula, and we do not even know if
the occurrence of each nim-value is finite.
Figure 4: The initial $20000$ nim-values of complement-grundy.
###### Theorem 10.
Consider complement-grundy. Then $\mathcal{SG}(n)/n\rightarrow 0$, as
$n\rightarrow\infty$.
###### Proof.
Consider the nim-value $2^{k}$. If it does not appear, we are done. Suppose it
appears for the first time at heap size $n_{k}$. By the mex rule, if a nim-
value is greater than $2^{k}$, it must have nim-value $2^{k}$ in its set of
options. By the rules of game, this can only happen for a heap of size $m\geq
3\cdot n_{k}$. In particular, this holds for the nim-value $2^{k+1}$, which
occurs for the first time at $m=n_{k+1}$, say. Thus, for nim-values that are
powers of two, we get
$\frac{2}{3}\mathcal{SG}(n_{k})/n_{k}\geq\mathcal{SG}(n_{k+1})/n_{k+1}$. This
upper bound holds for arbitrary nim-values, since the lower bound on where the
power of two $2^{k+1}$ can appear is the same lower bound where any other nim-
value greater than $2^{k}$ may appear. ∎
Two simpler variations of divide-and-residue are:
1) The remainder is not included in the disjunctive sum of an option: divide-
throw-residue.
2) Only the remainders are the options: residue-throw-divisor.
Figure 5: The initial nim-values of divide-throw-residue and residue-throw-
divisor, respectively.
The patterns of the nim-values of these rulesets are displayed in Figure 5.
For variation 1 (to the left), we prove that, for heaps $>1$, the
$\mathcal{SG}$-sequence coincides with OEIS, A003602: If $n=2^{m}(2k-1)$, for
some $m\geq 0$, then $a(n)=k$. The Sprague-Grundy sequence starts at heap of
size one with nim-values as follows
$0,1,2,1,3,2,4,1,5,3,6,2,7,4,8,1,9,5,10,3,11,6,12,\ldots$
It turns out the divide-throw-residue has the same solution as maliquant, the
game where the options are the non-divisor singletons; recall Theorem 3, where
this result is expressed as an index function, $i_{o}$, the index of the
largest odd divisor.
###### Theorem 11.
Consider divide-throw-residue. Then $\mathcal{SG}(n)=i_{o}(n)=k$, if
$n=2^{m}(2k-1)$, for some integer $m\geq 0$.
###### Proof.
Observe that the options in the interval $[\lfloor n/2\rfloor+1,n-1]$ are the
same as for maliquant. Assume first $n$ is even. Then $n/2+n/2$ is an option
in divide-throw-residue, but $n/2$ is not an option in maliquant. However,
$n/2+n/2$ only contributes the nim-value $0$ and may be ignored. Consider next
the disjunctive sum $m+\cdots+m$, with an odd number of components adding up
to $n$. Then there is a power of 2, say $2^{k}$ such that $2^{k}m\in[\lfloor
n/2\rfloor+1,n-1]$, i.e., $2^{k}m\nmid n$. And so, by induction,
$\mathcal{SG}(m)=\mathcal{SG}^{\rm M}(2^{k}m)$, where the M indicates
maliquant. On the other hand, there are options of maliquant of the form
$m\nmid n,m<n/2$. They do not have a match in divide-throw-residue. But, as we
saw in the proof of Theorem 3, they do not contribute to the nim-value
computation in maliquant. The case of odd $n$ is similar. ∎
For variation 2, we observe the following $\mathcal{SG}$-sequence:
$0,1,1,1,2,2,2,2,2,2,\ldots$, i.e., for $n>0$ if $3\cdot 2^{k}$ copies of
$k+1$ have appeared append $3\cdot 2^{k+1}$ copies of $k+2$ as the next nim-
values.
###### Theorem 12.
Consider residue-throw-divisor. For all $n\in\mathbb{N}$, $\mathcal{SG}(n)=k$
if
$n\in\\{3(2^{k-1}-1)+2,\ldots,3(2^{k}-1)+1\\}.$
###### Proof.
We leave this proof to the reader. ∎
## 4 The factoring games
Is there any game that has the number of prime factors of $n\in\mathbb{N}$ as
the sequence of nim-values? The answer is yes, as given by the almost trivial
aliquot game in Section 1.1.1. There is a related game that decomposes into
several components in play, namely to play to any factorization of $n$.
###### Example 11 (m-factoring).
Let $n=12$. Then the set of options is $\\{6+2,3+4,2+2+3\\}$. The unique
winning move is to $2+2+3$, because the nim-values in the set of options are
$1,1,0$, respectively. Hence $\mathcal{SG}(12)=2$.
###### Example 12 (s-factoring).
Let $n=12$. Then the set of options is $\\{6+10,9+8,10+10+9\\}$. The
$\mathcal{SG}$-sequence starts:
$0,0,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1$, where the first heap is
the empty heap and the second 0 is due to that 1 does not have any prime
factors. $\mathcal{SG}(12)=2$.
m-factoring has a simple solution, but s-factoring we do not yet understand.
Recall the omega-functions from Section 2.3.
###### Theorem 13.
Consider m-factoring, and let $n\geqslant 2$, where each option is a non-
trivial disjunctive sum of a factoring of $n$. Then
$\mathcal{SG}(n)=\Omega(n)-1$. If no two distinct components may contain the
same prime number, then $\mathcal{SG}(n)=\omega(n)-1$.
###### Proof.
If $n$ is a prime, then $\mathcal{SG}(n)=0$, because no factoring to smaller
components is possible. If $n$ is composite with $k$ prime factors, then, by
induction, it is possible to play to an option of nim-value $\ell$, for each
$\ell\in\\{1,\ldots,k-2\\}$, by factoring $n$ into one number with $\ell$
prime factors, and $k-\ell$ other prime components. On the other hand, the
nim-value $k-1$ cannot be obtained as a move option, since
$(x_{1}-1)\oplus\cdots\oplus(x_{\ell}-1)\leqslant(x_{1}-1)+\cdots+(x_{\ell}-1)\leqslant
k-2$, if $k=x_{1}+\cdots+x_{\ell}$. The proof of the second part is similar. ∎
## 5 Full set games
In fullset maliquot a player moves to all the proper divisors in a disjunctive
sum.888Obviously we need to exclude the divisor $n\mid n$; the word “proper”
is implicit in the naming. Let us display the first few numbers with their
options and nim-values.
$n$ | $\mbox{opt}(n)$ | $\mathcal{SG}(n)$
---|---|---
1 | $\varnothing$ | 0
2 | $1$ | $1$
3 | $1$ | $1$
4 | $1+2$ | $0$
5 | $1$ | $1$
6 | $1+2+3$ | $1$
7 | $1$ | $1$
8 | $1+2+4$ | $0$
9 | $1+3$ | $0$
The nim-value sequence starts
$0,1,1,0,1,1,1,0,0,1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,\ldots$. The non-unit proper
divisors of $24$ are $2,3,4,6,8$ and $12$. The only square-free ones are $2,3$
and $6$, an odd number. Such observations are relevant for the proof of the
location of the 0s.
###### Theorem 14.
Consider fullset maliquot. Then $\mathcal{SG}(n)\in\\{0,1\\}$, and
$\mathcal{SG}(n)=1$ if and only if $n>1$ is square-free.
Let us indicate the idea of the proof. The nim-value $\mathcal{SG}(4)=0$
because the only non-unit proper divisor, $2$, is square-free, and
$\mathcal{SG}(8)=0$, because there is exactly one square-free proper divisor,
namely 2. In the proof we will use the idea that $\mathcal{SG}(n)=0$ if and
only if $n$ has an even number of square-free proper divisors.
###### Proof.
We induct on the number of divisors.
If $n=p$ is prime, there is an even number, namely 0, of square-free non-unit
proper divisors. The nim-value $\mathcal{SG}(p)=1$ is correct, because the
move to the heap of size one is terminal.
Consider an arbitrary number $n$. Each move will alter the nim-value modulo 2.
We must relate this to the non-unit square-free proper divisors in the
components of the option of $n$. By induction, if this number is even if and
only if $\mathcal{SG}(n)=0$, we are done. Henceforth, we will ignore the
component of a heap of size one, since it has nim-value $0$ and will not
contribute to the disjunctive sum.
Suppose first that $n=p^{2}$ is a perfect square. Then the option is the prime
$p$, and hence $\mathcal{SG}(p^{2})=0$. Indeed, there is an odd number of
square-free non-unit divisors.
If $n=p^{t}$, $t>2$, is any other power of a prime, we must prove that
$\mathcal{SG}(n)=0$. The set of non-unit proper divisors is
$\\{p,\ldots,p^{t-1}\\}$, and hence there is exactly one square-free divisor
in the disjunctive sum $p+\cdots+p^{t-1}$. By induction, we get that each
component, except $p$ has nim-value $0$. This proves this claim.
Next, suppose $n=pq$, where $p$ and $q$ are primes. Then the option is $p+q$
of nim-value $1\oplus 1=0$. Hence $\mathcal{SG}(pq)=1$, and $n$ has an even
number of square-free non-unit proper divisors.
Similarly, if $n=p_{1}\cdots p_{j}$ is a product of distinct primes, then
$\mathcal{SG}(n)=1$. This follows, because the number of proper non-unit
divisors,
$\displaystyle\sum_{1\leq i<j}{j\choose i},$ (2)
is even, where $j$ is the number of prime factors in $n$ (this holds both for
even and odd $n$). And, by induction, each such individual component divisor
has nim-value 1. Note that, by moving in one such divisor, the number of
components in the disjunctive sum changes parity; if moved in a prime, then
the prime is deleted, if moved in $pq$, then this component splits to $p+q$,
and so on.
By combining these observations, we prove the general case of an arbitrary
prime factorization. Assume $n$ contains a square. We must show that
$\mathcal{SG}(n)=0$. By induction, we are concerned only with the square-free
divisor components, and we show that the number of such divisors is odd.
Indeed, if we assume $j$ in (2) is the number of distinct prime factors, then
there is one missing term, namely ${j\choose j}$. Namely, the divisor composed
of all square-free factors must be counted, whenever $n$ contains a square.
Apart from this, no new square-free divisor is introduced. Thus, the number of
such components is odd, and since by induction they have nim-value $1$, the
result $\mathcal{SG}(n)=0$ holds. ∎
We have investigated a few more of the fullset games, including those in the
subclass ‘subtraction’, but not yet found other examples with sufficient
regularity to prove basic correspondence with number theory. Apart from
fullset maliquot, this class, for now, remains a mystery. For example, for
fullset totient, the sequence starts $0,1,0,1,1,0,0,0,0,1,1,1,1,0,1,1,0,0,0$.
The heap of size one has nim-value zero by definition, and the heap of size
two has nim-value one, because one is relatively prime with two.
$\mathcal{SG}(3)=0$, because the option is $1+2$ of nim-value $0\oplus 1=1$.
The sequence of the indices of the ones is $2,4,5,10,11,12,13,15$, and so on.
This sequence does not yet appear in OEIS.
## 6 Powerset games
We study six version of the powerset games on arithmetic functions, and we
begin by listing the first 20 nim-values for the respective ruleset. All start
at a heap of size one, except item 2, which starts at the empty heap (defined
as terminal).
1. 1.
powerset maliquot: move-to an element in the powerset of the proper divisors.
$0,1,1,2,1,2,1,4,2,2,1,4,1,2,2,8,1,4,1.$
2. 2.
powerset saliquot: subtract an element in the powerset of the divisors.
$0,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1.$
3. 3.
powerset maliquant: move-to an element in the powerset of the non-divisors.
$0,0,1,0,2,1,4,8,16,2,32,1,64,4,128,8,256,16,512$
4. 4.
powerset saliquant: subtract an element in the powerset of the non-divisors.
$0,0,1,1,2,1,4,4,8,2,16,8,32,32,64,64,128,8,256,64.$
5. 5.
powerset totative: move-to an element in the powerset of the relatively prime
numbers smaller than the heap.
$0,1,2,1,4,1,8,1,2,1,16,1,32,1,2,1,64,1,128.$
6. 6.
powerset nontotative: move-to an element in the powerset of the non-relatively
prime numbers smaller than the heap.
$0,0,0,1,0,2,0,4,1,8,0,16,0,32,4,64,0,128,0.$
For single heaps, these games tend to have nim-values powers of two. The
intuition of this is that by induction there is plenty opportunity, in a
powerset, to construct any number between the powers of two, by using various
sums of single heaps. We will study the precise behavior in a couple of
instances, namely items 2,3 and 5.
###### Theorem 15.
Consider powerset saliquot. Then $\mathcal{SG}(0)=0$, and
$\mathcal{SG}(n)=2^{p}$, if $2^{p}$ is largest power of two divisor of $n\geq
1$.
###### Proof.
A heap of size zero has nim-value 0 because it is terminal by definition. The
heap of size one has nim-value $1=2^{0}$, because $1\mid 1$. The heap of size
two has nim-value $2=2^{1}$, because $1,2\mid 2$, and
$\mathcal{SG}(2-1)=1,\mathcal{SG}(2-2)=0$. Both these cases satisfy the
largest power of two divisor criterion.
Suppose the statement holds for all numbers smaller than the heap size
$n=2^{p}a$, with $a$ odd, say. We must show that all nim-values less than
$2^{p}$ exist among the options of $n$. For each $q<p$, we will find a number
$0\leq m<n$ with $2^{q}$ largest power of $2$ divisor of $m$, and where $n-m$
is a divisor of $n$. For example with $m=n-2^{q}a\in\mathbb{N}$, then
$n-m=2^{q}a\mid n$, and $m=2^{q}a(2^{p-q}-1)$ has greatest power of two
divisor $2^{q}$. By induction, $\mathcal{SG}(m)=2^{q}$. Let $q$ range between
$0$ and $p-1$. By the rules of powerset, and by using the disjunctive sum
operator, this suffices to establish that all nim-values less than $2^{p}$
exist among the options of $n$.
Next, we must prove that the nim-value $2^{p}$ does not exist among the
options. It suffices to show that no individual heap in an option, which is a
disjunctive sum, is of the same form as $n$. This follows, since, by
induction, all numbers smaller than $n$ have nim-values powers of two, and
apply nim-sum.
A divisor of $n$ is of the form $2^{q}y$, where $y\mid a$ is odd, and where
$q\leq p$.
Suppose first $q=p$. Then $n-2^{p}y=2^{p}y(a/y-1)$. But $a/y$ is odd, and
hence $a/y-1$ is even, so $n-2^{p}y=2^{z}b$, with $z>p$ and $b$ odd, unless
$a=y$ when $n-2^{p}y=0$.
In case $q<p$, we get $n-2^{q}y=2^{q}y(2^{-q}n/y-1)$, and since $2^{-q}n/y-1$
is odd, by induction, the heap is not of the same form (since $q<p$). ∎
Recall the indexing function, $i_{o}$, of largest odd divisor, concerning the
singleton version of maliquant. It applies here as well with some initial
modification; while it looks like one could ‘peel’ off the $2$’s it does not
work due to the irregular set of initial nim-values.
###### Theorem 16.
Consider powerset maliquant. The sequence starts at a heap of size one, and
the first eight nim-values are, $0,0,1,0,2,1,4,8$. Otherwise, if $n=2k+1,k\geq
4$, then $\mathcal{SG}(n)=2^{k}$, and if $n\geq 10$ is even, then
$\mathcal{SG}(n)=\mathcal{SG}(n/2)$.
###### Proof.
The smaller heaps are easy to justify by hand. The heap of size 8 is pivotal.
It achieves nim-value 0, by the option $3+6$, both numbers being nondivisors.
And the nim-values $1,2,4$ may be combined freely by using the nondivisor
heaps $5,6,7$. Hence, the nim-values of the small heaps are verified.
For the base cases, we consider the heaps of sizes $9$ and $10$, of nim-values
$16=2^{4}$, with $2\cdot 4+1=9$ and $2=\mathcal{SG}(5)$ respectively.
For the induction, let us start with a heap of even size, $n=4t+2$, say. It
suffices to show that $\mathcal{SG}(n)=\mathcal{SG}(n/2)$. Observe that each
number between $n/2$ and $n$ is a nondivisor to $n$, and hence may be part of
a disjunctive sum to build desirable nim-values, by induction. Since
$n/2=2t+1$ is odd, each power of two $2^{4},\ldots,2^{t}$ appears among the
nim-values for heap sizes in $[9,n/2]$. By induction, each power of two
$2^{0},\ldots,2^{t-1}$ appears as a nim-value in the heap interval
$I=[n/2-1,\ldots n-1]$. Namely, for $y\in[0,t-1]$, multiply $2y+1$ by $2$
iteratively until $2^{s}(2y+1)\in I$. Thus all numbers smaller than $2^{t}$
appears as options, but note that the nim-value $2^{t}$ appears only as a nim-
value for a divisor of $n$, and hence this is the minimal exclusive. This
proves that $\mathcal{SG}(n)=2^{t}$, if $n$ is even, as desired.
Now, consider odd $n=2t+1$, say. By induction the nim-value of each heap
smaller than $n$ is less than $2^{t}$. In case of $t$ even, the powers of two,
$2^{t/2},\ldots,2^{t-1}$ appear for nim-values of odd heaps in the interval
$[t,\ldots,2t-1]$. And similar to the case for even $n$, the smaller power of
two nim-values can also be found in this interval. Therefore each nim-value
smaller than $2^{t}$ appear as an option of a disjunctive sum of non-divisors
of $n$. Hence, the minimal excusive is $\mathcal{SG}(n)=2^{t}$. ∎
Recall the function $i_{p}$, the index of the smallest prime divisor of $n$,
where the prime 2 has index 1, for the solution of totative, from Section 1.3.
It applies for the powerset game as well.
###### Theorem 17.
Consider powerset totative. Then $\mathcal{SG}(n)=2^{i-1}$, where $i=i_{p}$.
###### Proof.
The nim-value of a heap of size one is 0, since it is terminal. A heap of size
two has a move to the heap of size one, because 1 is relatively prime with all
numbers greater than 1. Hence $\mathcal{SG}(2)=1=2^{0}$. Suppose the statement
holds for all numbers smaller than $n>1$.
If $n$ is even, we must prove that there is a move to nim-value 0, but no move
to nim-value 1. The first part is done in the first paragraph. Hence, let us
show, by induction, that there is no move to nim-value 1. Since all smaller
heaps of odd size have even nim-values, then a disjunctive sum of nim-value 1
must contain a heap of even size. This is impossible, since heaps of even size
are not relatively prime with $n$.
Suppose that $n$ is odd, so that the index of the smallest prime divisor of
$n$ is $i>1$. We must show that $\mathcal{SG}(n)=2^{i-1}$. By induction, each
smaller prime divisor, with index $q<i$ say, has appeared in a heap size
smaller than $n$, with nim-value $2^{q-1}$. Since any disjunctive sum of heap
sizes relatively prime with $n$ is permitted as an option, by induction, each
nim-value smaller than $2^{i-1}$ can be obtained.
Next, we show that there is no option of nim-value $2^{i-1}$. This generalizes
the idea used in the second paragraph. A disjunctive sum of nim-value
$2^{i-1}$ must contain a component of nim-value $2^{i-1}$. But, by induction,
those heap sizes are not relatively prime with $n$. ∎
## 7 Discussion–future work
A natural generalization of counting the number of elements satisfying an
arithmetic function is to instead consider their sum, or partial sums. For
example, consider the sum generalization of the mtau, that is, the option of
$n$ is the sum of the proper divisors of $n$. For example $4$ has the proper
divisors $1$ and $2$ and therefore the option is $3$. Loops and cycles occur
for perfect numbers (those where the sum of proper divisors equals the number)
and (temporarily) increased heap sizes for abundant numbers (those where the
sum of proper divisors is greater than the number). The first loop appears at
$1+2+3=6$ (where ‘+’ is arithmetic sum).
This might at first sight seem to disqualify the Sprague-Grundy
function,999Fraenkel et al have developed a generalized Sprague-Grundy
function for cyclic short games. but in fact, since the game is binary, the
cycles are trivial, in the following sense. If we play a disjunctive sum of
games where one component will not end, then the full game will not end. And
reversely, if no component contains a cycle, but perhaps temporarily
increasing heap sizes, then the full game will end and a winner may be
declared. The nim-value sequence of this ruleset begins at a heap of size one,
as follows: $0,1,1,0,1,\infty,1,0,1,1,1,?$, where the infinity at heap size 6
indicates the loop, $1+2+3=6$.
Let us compute the nim-value for $n=12$, which is indicated by ‘?’ in the
sequence above. The sum of proper divisors is $16$ (temporary increase),
followed by options $15$ and $9$, in the next two moves. The sequence above
indicates that $\mathcal{SG}(9)=1$, and therefore, $\mathcal{SG}(12)=0$. The
recurrence where a number is mapped to the sum of its proper divisors has been
studied in number theory literature, without the games’ twist. It seems well
worthy some more attention.
Even more interesting is the same ruleset but where the player may pick any
partial sum of proper divisors. We have the following table, where for example
the options of a heap of size $4$ are $1,2$ and $1+2$. $n$ $\mbox{opt}(n)$
$\mathcal{SG}(n)$ 1 $\varnothing$ 0 2 $1$ $1$ 3 $1$ $1$ 4 $1,2,3$ $2$ 5 $1$
$1$ 6 $1,2,3,4,5,6$ $\infty_{3}$ 7 $1$ $1$ 8 $1,2,3,4,5,6,7$ $\infty_{3}$ 9
$1,3,4$ $3$
Here $\infty_{3}$, means the nim-value 3, but with an additional option an
infinity, namely $\infty_{3}$. Consider for example the disjunctive sum of
heaps $6+9$. Then every move apart from playing to $\infty_{3}$ is losing. So,
this game is a draw. However, playing instead $6+7$, the first player wins by
moving to $2+7$, $3+7$ or $5+7$. That is, a loopy game component is sensitive
to the disjunctive sum. The $\omega$ game also seems to have an interesting
sum variation, but now we are ready to go and prepare some lunch.
Acknowledgement. This work started when the second author visited the first
author at the University of the Virgin Islands in April 2015. It breaks my
heart to acknowledge that the first author passed away 15 October 2020. Doug
is deeply missed. Many thanks to the referee, whose comments helped to improve
the readability of this paper.
## References
* [1] E. R. Berlekamp, J. H. Conway, R. K. Guy. Winning Ways, Academic Press, London, 1982.
* [2] C. Bouton, “Nim, a game with a complete mathematical theory,” _Annals of Math._ , 2nd ser., 3, no. 1/4, 35–39 (1901–2).
* [3] A. Dailly, E. Duchene, U. Larsson, G. Paris, Partition games, _Discrete Applied Mathematics_ , 285, (2020) 509–525.
* [4] P. Grundy, “Mathematics and games,” _Eureka_ 2, 6–8 (1939).
* [5] G. H. Hardy and E. M. Wright, _An Introduction to the Theory of Numbers_ (5th ed.). Oxford: Clarendon Press (1979) [1938].
* [6] H. Shapiro, An Arithmetic Function Arising from the $\Phi$ Function, _The American Mathematical Monthly_ , 50:1, 18-30 (1943).
* [7] N. Sloane, _The On-Line Encyclopedia of Integer Sequences (OEIS)_ , website at http://oeis.org/.
* [8] R. Sprague, “Über mathematische Kampfspiele,” _Tôhoku J. Math._ , 41, 438–444 (1936).
|
# Comparative Evaluation of 3D and 2D Deep Learning Techniques for Semantic
Segmentation in CT Scans
Abhishek Shivdeo Rohit Lokwani Viraj Kulkarni Amit Kharat Aniruddha Pant
DeepTek Inc
###### Abstract
Image segmentation plays a pivotal role in several medical-imaging
applications by assisting the segmentation of the regions of interest. Deep
learning-based approaches have been widely adopted for semantic segmentation
of medical data. In recent years, in addition to 2D deep learning
architectures, 3D architectures have been employed as the predictive
algorithms for 3D medical image data. In this paper, we propose a 3D stack-
based deep learning technique for segmenting manifestations of consolidation
and ground-glass opacities in 3D Computed Tomography (CT) scans. We also
present a comparison based on the segmentation results, the contextual
information retained, and the inference time between this 3D technique and a
traditional 2D deep learning technique. We also define the area-plot, which
represents the peculiar pattern observed in the slice-wise areas of the
pathology regions predicted by these deep learning models. In our exhaustive
evaluation, 3D technique performs better than the 2D technique for the
segmentation of CT scans. We get dice scores of 79% and 73% for the 3D and the
2D techniques respectively. The 3D technique results in a 5X reduction in the
inference time compared to the 2D technique. Results also show that the area-
plots predicted by the 3D model are more similar to the ground truth than
those predicted by the 2D model. We also show how increasing the amount of
contextual information retained during the training can improve the 3D model’s
performance.
## 1 Introduction
Medical imaging techniques like X-rays, Magnetic Resonance Imaging (MRI),
Computed Tomography (CT), etc., provide precise anatomy of a human body and
thus help detect abnormalities present in the body umar2019review . Effective
and early identification of regions of infection in medical images can play a
crucial role in assisting the doctors for the treatment of various
pathologies. For instance, observing X-rays and finding early signs of
pneumonia, which causes around 50,000 deaths per year in the US
centers2012pneumonia , and treating it in time can save many lives. However,
X-rays compress a 3D volume into a single 2D image, which causes loss of
information. X-rays also lack specificity when pathology regions are concealed
by overlapping tissues, bones, or bad contrast environments when detecting
pathologies like COVID-19 zhang2020viral . Thus, this makes understanding and
identifying a pathology using high-resolution CT scans a sought after medical
diagnosis technique doi2007computer .
CT scans are 3D medical images that comprise several slices or images, similar
to X-rays, stacked upon each other, which combine to give us a volumetric
representation of the interior aspects of our body karatas2014three .
Nevertheless, classifying and marking regions of interest in CT scans needs
significant effort from the radiologists. Hence, automated detection and
segmentation of pathologies in CT scans, to reduce a radiologist’s
involvement, is seen as an essential tool for diagnosing and treating a
disease.
The rapid research and development in machine learning, graphics processing
technologies and the availability of large amounts of data
rodriguez2016general minsky1961steps cockburn2018impact yang2018research
have improved the field of Computer vision lu2020survey [6]. The availability
of high-quality medical image datasets kohli2017medical combined with rapid
advancement in CNN based architectures has led to an increase in the adoption
of deep learning models to assist radiologists in evaluating CT scans. Deep
learning models are used to detect, classify, and segment fractures, tumors,
and other pathologies in CT scans. However, there is a lot of variation in the
contrast of images based on different radiation doses smith2019international
trattner2014standardization given to patients. The quality of CT scanners in
different hospitals, and slice thickness can also differ from scan to scan in
a multi-sourced dataset, making it challenging to train machine learning
models for CT scans. Also, selecting a proper CT window, by manipulating the
Hounsfield unit (HU) values in a CT, can affect the model’s performance
xue2012window . A deep learning model needs to be robust enough to handle
these variations or it may experience a covariance shift when it is tested on
out-of-source data. Studies have tried to mitigate these effects by trying out
image noise reduction methods to reduce the radiation dose of CT imaging
willemink2019evolution yang2018low .
Researchers have adopted both 2D as well as 3D approaches haque2020deep .
Studies conducted by Zhou et al. zhou2018performance compare the segmentation
performance of 2D and 3D deep learning-based approaches using conventional
segmentation metric such as dice score. In a 2D deep learning technique
weston2019automated , the input to the model is a single 2D image, whereas, in
a 3D deep learning technique zhao2020mss , the model takes a 3D volume as its
input. Both these approaches employ a Fully Connected Network (FCN) for
segmentation. As opposed to training slice by slice in 2D FCNs, the 3D FCNs
models analyze volumetric input data and utilize the global features in
between the CT slices. Just like how a word in a sentence gives a clue about
what the next word could be akbik2018contextual hassan2017deep , a CT scan’s
slice can give a clue about the shape of the pathology in its adjacent slices.
This is because, in most of the pathologies, (consolidation and ground-glass-
opacities here) the regions of interest (ROI) or the areas of the pathology in
a scan follow a continuous pattern. As the CT scans are captured in a
particular order, it is observed that continuity of manifestations exists in
the adjacent slices, we can observe the same in Figure 6. Using 3D models for
the segmentation of CT scans is similar to using LSTMs with the attention
module olah2016attention for the formation of a sentence. This contextual
information is lost when we use 2D CT scans because the 2D model predicts the
outcome by considering an individual slice as a single data point, thus their
prediction is not affected by the adjacent slices, which is evident from the
results of our experiments (Figure 7).
In our paper, we propose a 3D technique for the segmentation of consolidation
and ground-glass opacities in CT scans, and also present a comparison between
this 3D technique and a traditional 2D technique. We compare the dice scores
between the predicted masks and the radiologists’ annotations for both these
techniques. We also plot the areas of the predicted masks versus the position
of the slice in the CT scan for both 2D and 3D techniques, and compare it with
the ground truth area-plots. We also compare the inference time for these two
techniques, which is an important factor when a model runs inference in real-
life situations post-deployment.
## 2 Related Work
Recent studies have emphasized the use of deep learning for medical imaging
analysis. The problems solved using deep learning can be broadly classified
into image classification and semantic segmentation. Convolutional Neural
Networks (CNNs) are commonly used for image classification. Badea et al.
badea2016use used LeNet lecun1998gradient and NiN (Network in Network)
lin2013network for classifying burns on the human body from images of size
320 x 240 captured using a camera and achieved an accuracy of 75.91% and
58.01% for classification of Skin vs. Burn and Skin vs. Light Burn vs. Serious
Burn, respectively. Polsinelli et al. polsinelli2020light used SqueezeNet, a
CNN architecture, for classifying CT scan’s slices into COVID-19 or non-
COVID-19 with an accuracy of 85%. The classification process is simpler than
segmentation because in classification all the pixels in a single image need
to be grouped into a single class. While in semantic segmentation, each pixel
needs to be assigned a class.
Image segmentation was initially solved using conventional image processing
approaches. Okada et al.okada2015abdominal proposed an image processing
technique for multi-organ segmentation in which he used statistical shape
modeling and probabilistic atlas and the segmentation of organs was done by
combining the intra-organ information with the inter-organ correlation to get
an average dice coefficients of 92% for the liver, spleen, and kidneys, and a
dice coefficient of 73% and 67% for the pancreas and gallbladder,
respectively. This conventional approach demonstrated highly accurate multiple
organ segmentation techniques for CT scans and presented a detailed evaluation
of the observations.
After the development of encoder-decoder badrinarayanan2017segnet
architectures, they were commonly used for segmentation purposes. U-Net, a 2D
deep learning approach, proposed by Ronneberger et al. ronneberger2015u ,
segmented a single (512, 512) image in under a second on an NVidia Titan GPU.
U-Net was fast, efficient, and accurate and thus was the first widely used
deep learning architecture for image segmentation tasks for medical image data
ronneberger2015u . Christ et al. christ2017automatic cascaded two FCNs to
segment out the liver and its lesions from CT and MRI scans and achieved an
accuracy of around 94% on the validation set in under 100 seconds per volume.
Almotairi et al. almotairi2020liver proposed another deep learning
architecture, SegNet, that employed a trained VGG-16 image classification
network as its encoder, and had a corresponding decoder architecture for
pixel-wise classification at the end, which was able to achieve an accuracy of
99.99% for segmenting a liver tumor. For these 2D approaches, slices of the
MRI and CT scans present in the dataset were treated as individual 2D images,
which means that the 3D volumetric data is transformed into a 2D planar data.
Other such 2D based image segmentation approaches were implemented
zhou2016first long2015fully de2015deep roth2015deep cha2016urinary .
Zhou et al. zhou2016three proposed a segmentation approach in which the 2D
slice-wise results were later combined using 3D majority voting, where a
simple encoder-decoder network was combined to be a part of an all-in-one
network which could segment out complicated multiple organs; it correctly
segmented 89% of the voxels from the CT scans.
However, in recent years, due to improved 3D convolution architectures and
advancements in computational power (GPUs), training of highly complex 3D deep
learning models having a 3D volume as its input, has become much more
accurate, efficient, and faster kayid2018performance . Cicek et al.
cciccek20163d proposed a 3D U-Net architecture that predicted volumetric
segmentation using 2D annotated slices. The average Intersection over Union
(IoU) achieved was 0.863. They cciccek20163d were able to annotate unseen
data as well as densify the sparsely annotated data.
Milletari et al. milletari2016v proposed V-Net: a novel fully convolutional
neural network for volumetric medical image segmentation which gave an average
dice score of 86% to segment out the prostate depicted in 30 MRI scans. These
MRIs were converted into a constant volume of 128 × 128 × 64 using B-spline
interpolation, which alters the global features during the conversion and can
have detrimental effects on the training.
VoxResNet proposed by Chen et al. chen2016voxresnet , which borrows the spirit
of deep residual learning in 2D image recognition tasks, and is extended into
a 3D variant for handling volumetric data, has also been successfully applied
for 3D medical image segmentation tasks. Allan et al. alalwan60efficient
proposed another 3D FCN model architecture called “3D-DenseUNet-569” for liver
and tumor segmentation which used Depthwise Separable Convolution (DS-Conv) as
opposed to traditional convolution.
Although between 2D and 3D, 3D FCN provides us with more accurate results, it
is more complex and requires higher memory along with greater computational
resources li2018h . The higher complexity restrains the model from training a
larger dataset efficiently. Moreover, the high memory footprint leads to
reduced network depth and filter size, which adversely affects the performance
of the model simonyan2014very .
## 3 Data
For our experiments, we obtained 182 CT scans from 2 private Indian hospitals.
These CT scans have non-uniform volumes and are annotated for consolidation
and ground-glass-opacities chung2020ct . Our team of expert radiologists
marked out the regions of infections, in the form of free-hand annotations,
which served as the ground truth for our model. These precise annotations were
done using the ITK-snap tool, an open-source free-hand annotation tool
yushkevich2016itk . Some examples of the CT slices and their superimposed
masks are showcased in the Figure 1 (a) and Figure 1 (b).
Figure 1: (a,b): CT Slice (Left), Free hand annotated CT Slice (Right) Dataset | Number of CT Scans | Number of Slices
---|---|---
Training | 126 | 56387
Validation | 20 | 9992
Test | 36 | 14727
Table 1: Scan-level and Slice-level Dataset splits
The positive class COVID-19 comprised consolidation and ground-glass
opacities. These chest CT scans were divided into train, validation, and test
datasets whose splits are given in Table 1.
The prevalence for all the datasets is 20%, which means that there are 20%
positive slices in the total dataset. We resized all the images to a standard
image size of (512, 512). Windowing, also known as grey-level mapping is the
pre-processing of CT scans in which the grey-scale component of the CT slice
is changed to highlight some particular features. We applied windowing to our
scans lee2018practical , for which we used the information stored in the
metadata from the DICOM files of the CT scans’ slices. The masks were stored
as binary images of size (512, 512). The distribution of the number of slices
in these CT scans for the whole dataset can be visualized from the histogram
in Figure 2
Figure 2: Volume variation in the CT scans for the whole dataset
## 4 Methodology
In this paper, we follow two approaches to address image segmentation of these
CT scans:
### 4.1 2D Approach
Figure 3: 2D Technique
In this approach, we divide the CT scan volume into separate 2D slices (Figure
3). For example, a CT scan of volume (512, 512, 601) containing 601 slices was
divided into 601 individual images of (512, 512). We used the U-net with the
Convolutional Block Attention Module (CBAM) woo2018cbam to focus its
attention on a region of the image and then segment the pathology from that
region of the image. U-Nets are fully convolutional networks having skip
connections between encoder and decoder which provide deconvolution layers
with important features hesamian2019deep . We used Xception
chollet2017xception as the encoder having depthwise separable convolutions
and residual connections. The model was trained using ADAM as its optimizer
with an initial learning rate of 1e-3 and having a learning rate scheduler
which reduced the learning rate to 1/3 the original learning rate, every 5
epochs. We used dice loss as the loss function. Our architecture had 38
million trainable parameters.
### 4.2 3D Approach
In the 3D approach, instead of assigning 2D images as the input for the model,
we provide 3D volumes as to input for the model, and the corresponding stacked
annotated slices as the label. For our 3D approach, we use V-Net
milletari2016v , which is a 3D implementation of U-Net. For our input size of
(512, 512, 32); V-Net had 206 million trainable parameters. We used the dice
loss as the loss function while training. The model was trained using ADAM as
its optimizer with an initial learning rate of 1e-3 and having a learning rate
scheduler which reduced the learning rate to 1/3 the original learning rate
every 5 epochs. The input to V-Net is volumetric data of dimension (x, y, z) =
(512, 512, 32) where x is slice’s height, y is slice’s width, and z is the
number of slices in the depth of the volume. To prepare the input data for
training V-Net, we normalize the volume of the CT scans to the same value,
having a dimension equal to the input dimension of the model. To satisfy this
condition without losing any information, we split CTs into smaller volumes
that match the input dimensions. We can see the CT scan being split into
multiple stacks of slices in Figure 4.
Figure 4: 3D Technique
There are three variables that affect the stacks creation process.
1\. CT Volume: Number of slices in the CT scan
2\. Stack size: Desired number of slices in the sub volume
3.Overlap factor = (Number of overlapping slices/stack size) Where overlapping
slices is the number of slices that are common or overlapping in adjacent
stacks.
For a CT scan having 601 slices, with a stack size of 32, and the number of
overlapping slices as 20 (overlap factor = 0.625), we get a list of 49 stacks
having dimensions (512, 512, 32). The first stack will have indices from 0 to
32, which means that the first input datapoint for V-Net will be a CT sub
volume from the 1st slice to the 32nd slice. The second input data point will
be a sub volume of the CT from index 12 to 44, both inclusive. As we have 20
overlapping slices, the second stack starts from the 12th index and not the
33rd. And so on for the rest of the stacks. For the last stack, however, the
indices are (576, 608), the 7 extra slices are the added paddings to keep the
volume of the stack compatible with the 3D model’s input dimension. During the
inference, we keep the overlap factor as 0, the list of predicted mask volumes
are then stacked together. So during the inference evaluation, we have a total
of 19 stacks grouped. We remove the padding when the predictions are stacked
together to match the volume of the whole CT scan.
## 5 Results
We evaluate the 2D and the 3D model based on these 5 criteria.
### 5.1 Dice Score
We predicted masks for 32 scans in the test set having a prevalence of 20% and
calculated the dice score for the whole dataset. We observed the dice scores
given in Table 2.
Model | Dice Score
---|---
2D Model | 73%
3D Model | 79%
Table 2: Dice Scores for 2D and 3D techniques
We took the average of the dice scores of these 32 scans. For the 2D model, we
arranged the individual slices on top of each other to get the final predicted
volume. For the 3D model, we combined the stacks of an individual CT and then
removed the corresponding padding to match the true label volume.
### 5.2 Slices Predicted masks
Figure 5(a) and 5(b) show the predicted masks at the scan and slice level for
2D and the 3D model. We applied a threshold of 0.2 on the predictions.
Figure 5: a)3D Model, b) 2D Model
Left=Original CT Slice, Middle = True label superimposed, Right = Predicted
Mask superimposed
### 5.3 Area-plots
Here, we find the area of the annotated region with respect to the total area
of the image. We normalize this list of area-ratios by dividing all the values
by the maximum value in the list. Next, we plot these normalized values
according to the slices in the CT scan to get the plot in Figure 6, these
plots are called area-plots. We follow the same procedure for the predicted
masks for 2D and 3D models, by plotting their normalized area-plots according
to the position of the slices in the CT scan in Figure 7, 8 respectively. We
have compared the predicted area-plots for both the 2D and the 3D approach to
the area-plots of the true label.
Figure 6: True Label’s Area-Plot
Figure 7: Area-Plot predicted by 2D Model
Figure 8: Area-Plot predicted by 3D Model
1\. From Figure 6, we observe a continuous pattern in the manifestations of
the pathology in the CT scan. Similar continuous patterns were observed for
the rest of the CT scans, area-plots.
2\. From Figure 7, we observe that there are abrupt variations in the
predicted mask’s area for the slices within a CT scan when we use the 2D
approach.
3\. From Figure 8, we can see the prediction area-plot for a CT scan when
using the 3D approach. We observe a smooth and continuous pattern in the masks
predicted by the 3D model for the slices of that CT scan.
To further prove and understand point number 2, we plotted the predicted masks
for consecutive slices in a CT scan using 2D and the 3D approach.
Figure 9: Predicted masks in four consecutive slices (a, b, c, d).
Left: Original slice, Middle: True Mask, Right: Predicted Mask
2D (top 3 images) and 3D (bottom 3 images)
For the four consecutive positive slices that we chose, the 3D model predicted
all the slices as positive, i.e.(1,1,1,1), whereas, in the case of the 2D
model, the predictions were (0,1,0,0). From Figure 9, we observe that the 3D
approach predicted the masks in all 4 consecutive slices to closely match the
labels. This represents the continuity in the predictions. We also observe
that the shape or the area of the predicted masks do not change abruptly when
we look at the adjacent slices. Although, in the masks predicted by the 2D
approach, we observe that only some of the slices were marked positive. This
represents the discontinuity in the predicted masks’ pattern. It indicates
that the 2D approach does not consider the slices’ information adjacent to the
input slice, thus giving us this abrupt predicted pattern, which is
independent of the slices above and below the current slice.
### 5.4 Inference time
We calculate the inference time for 2D as well as 3D techniques for a CT scan.
Technique | Inference time | Inference time
---|---|---
| With GPU | Without GPU
2D | 70 | 1145
3D | 17 | 229
Table 3: Inference time for 2D and 3D techniques in seconds
Table 3 shows the inference time for a single CT scan having 709 slices with a
stack size of 32 and an image size of (512, 512). We observe that there is a
boost up of 5X in the 3D approach as compared to the 2D approach. For
inference, we used a Tesla T4 GPU with 15GB of memory.
### 5.5 Overlap Variation
In this experiment, we changed the overlap factor to see its effect on the
predictions. We kept the overlap factor as 0, 0.375, and 0.625. The predicted
area-plots are given in Figure 11, 12, 13 respectively and the predicted masks
are given in fig 14(a), 14(b), and 14(c) respectively. Figure 10 shows the
true area-plot for the same CT scan.
Figure 10: True Label’s Area-Plot
Figure 11: Area-Plot predicted by 3D Model, with overlap factor = 0
Figure 12: Area-Plot predicted by 3D Model, with overlap factor = 0.375
Figure 13: Area-Plot predicted by 3D Model, with overlap factor = 0.625
Figure 14: Predicted mask for the same slice having (a) overlap factor = 0,
(b)overlap factor = 0.375, (c) overlap factor = 0.625
Left: Original slice, Middle: True Mask, Right: Predicted Mask
As we increase the overlap factor, the number of overlapping slices increases.
This allows the model to interpret more global features. We observe that the
predicted area-plots in Figure 14 with an overlap factor 0.625 match closely
to the true area-plots in Figure 10. This shift in the pattern from Figure 11
to Figure 13 demonstrates that increasing the amount of contextual information
retained allows the model to perform better thus giving closer predictions to
the ground truth.
## 6 Conclusion
We implement and compare two deep learning approaches for segmenting out
manifestations of consolidation and ground-glass-opacities in CT scans on
three major grounds: predicted masks or the segmentation results, the pattern
observed in the area of the predicted masks in a CT scan, and the inference
time.
We saw that the 3D approach provided us with a better dice score than the 2D
approach, thus proving to be more accurate at segmenting pathology regions. We
also observed from Figure 8 that the area-plots we get using the 3D approach
match closely to the original annotation area-plots against the area-plots of
the 2D approach. We attribute these peculiar predicted patterns in the area-
plots to the contextual information retained in the 3D stack volumes used to
train the 3D model. The independent nature of the input images in the 2D model
could be the reason for a discontinuous pattern in the predicted area-plots of
the 2D approach. When we calculated the inference time, the 3D approach
provided a boost of 5X in inference time compared to that of the 2D approach
This is hugely beneficial for effective and quick diagnosis of patients in
hospital settings especially the Intensive Care Units(ICUs). We conclude that
this 3D stack-based approach is a better choice than the conventional 2D
approach.
Later, when we experimented with the overlap factor for the 3D approach, the
predicted area-plots’ patterns changed (Figure 11, 12, 13) with increasing
overlap factor. However, a higher overlap factor may result in overfitting the
model to the given dataset. The model gets more susceptible to a covariance-
shift in the case of out-of-sample datasets when the overlap factor is high.
Thus, the overlap factor should be treated as a hyperparameter for this 3D
approach and should be set optimally.
Usually, in the case of 3D segmentation for CT scans, the whole CT scan is
compressed using spline interpolation milletari2016v , or other methods, to
the size of input dimension for a 3D model. This removes useful information
when compressed for training and adds noise when decompressed for inference
hahn2016comparison . To counter this alteration in the original data, we split
the CT into smaller stacks to retain the valuable contextual information.
Thus, our batch-based 3D approach is highly adaptable to any CT volume because
of its stack-based nature, where none of the information between the slices in
the CT is altered.
The segmentation results can be improved using more data, and different
augmentation techniques while training. In our case study, the goal was to
compare the 2D and the 3D approaches on equal grounds, and not the
segmentation result itself. In conclusion, the 3D approach that has been
proposed in this paper has outperformed the traditional 2D approach in all the
three criteria that we had set for evaluation.
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|
# A Lightweight Structure Aimed to Utilize Spatial Correlation for Sparse-View
CT Reconstruction
Yitong Liu
<EMAIL_ADDRESS>
&Ken Deng
<EMAIL_ADDRESS>
&Chang Sun
<EMAIL_ADDRESS>
&Hongwen Yang
<EMAIL_ADDRESS>
###### Abstract
Sparse-view computed tomography (CT) is known as a widely used approach to
reduce radiation dose while accelerating imaging through lowered projection
views and correlated calculations. However, its severe imaging noise and
streaking artifacts turn out to be a major issue in the low dose protocol. In
this paper, we propose a dual-domain deep learning-based method that breaks
through the limitations of currently prevailing algorithms that merely process
single image slices. Since the scanned object usually contains a high degree
of spatial continuity, the obtained consecutive imaging slices embody rich
information that is largely unexplored. Therefore, we establish a cascade
model named LS-AAE which aims to tackle the above problem. In addition, in
order to adapt to the social trend of lightweight medical care, our model
adopts the inverted residual with linear bottleneck in the module design to
make it mobile and lightweight (reduce model parameters to one-eighth of its
original) without sacrificing its performance. In our experiments, sparse
sampling is conducted at intervals of 4°, 8° and 16°, which appears to be a
challenging sparsity that few scholars have attempted. Nevertheless, our
method still exhibits its robustness and achieves the state-of-the-art
performance by reaching the PSNR of 40.305 and the SSIM of 0.948, while
ensuring high model mobility. Particularly, it still exceeds other current
methods when the sampling rate is one-fourth of them, thereby demonstrating
its remarkable superiority.
## 1 Introduction
Over the last few decades [1, 2], X-ray Computed Tomography (CT) has
demonstrated its prominent practical value and wide range of applications
including clinical diagnosis, safety inspection and industrial detection [3].
Especially in the past year, due to the global spread of the Corona Virus
Disease 2019 (COVID-19), the term CT has become well-known to the public as an
essential auxiliary technology. However, the radiation dose brought by CT has
a nonnegligible side effect on the human body. Since it has a latent risk of
inducing cancers, radiation dose reduction is becoming more and more crucial
under the principle of ALARA (as low as reasonably achievable) [4, 5, 6, 7].
Generally speaking, there are two approaches to reduce radiation dose. The
approach of tube current (or voltage) reduction [8, 9] lowers the x-ray
exposure in each view but suffers from the increased noise in projections.
Although the approach of projection number reduction [10, 11] (also known as
sparse-view CT) can avoid the former problem and realize the additional
benefit of accelerated scan and calculation, it leads to severe image quality
degradation of increased streaking artifacts brought by its missing
projections. In this paper, we focus on effectively repairing and
reconstructing sparse-view CT so as to acquire high-quality CT images.
Sparse-view CT reconstruction has always been a classic inverse problem which
has attracted wide attention [12]. In the past few decades, iterative
reconstruction methods have become the dominant approach to solve inverse
problems [13, 14, 15, 16]. With the advent of compressed sensing [17] and its
related regularizers, the quality of reconstructed images has been improved to
a certain extent. One of the most typical regularizers is the total variation
(TV) method, algorithms based on which include TV-POCS [18], TGV method [19],
SART [13] and SART-TV [20] etc. In addition, dictionary learning is also
commonly used as a regularizer. For example, [21] constructs a global
dictionary and an iterative adaptive dictionary to solve the problem of low-
dose CT reconstruction.
In recent years, with the improvement of computing power, there comes a rapid
growth in deep learning [22]. Subsequently, neural networks have been widely
applied in image analysis tasks, such as image classification [23], image
segmentation [24, 25, 26], especially inverse problems in image
reconstruction, such as artifacts reduction [27, 28], denoising [29] and
inpainting [30]. Since GAN (Generative Adversarial Networks) was designed
elaborately by Goodfellow in 2014 [31], it has been adopted in many image
processing tasks due to its prominent performance in realistically predicting
image details. Therefore, GANs are also naturally applied to improving the
quality of low-dose CT images [32, 33, 34]. In addition, Ye et al. explored
the relationship between deep learning and classical signal processing methods
in [35], explained the reason why deep learning can be employed in imaging
inverse problems, and provided a theoretical basis for the application of deep
learning in low-dose CT reconstruction.
Some researchers adopt deep learning-based architectures to complement and
restore the limit-view Radon data [36, 37, 32, 38, 39, 40]. Dong et al. [36]
used U-Net [25] to predict the missing Radon data, then reconstruct it to the
image through FBP [41]. Jian Fu et al. [37] built a network that involves the
tight coupling of the deep learning neural network and DPC-CT (Differential
phase-contrast CT) reconstruction algorithm in the domain of DPC projection
sinograms. The estimated result is a complete phase-contrast projection
sinogram. Rushil Anirudh et al. established CTNet [38], a system of 1D and 2D
convolutional neural networks, which operates on the limited-view sinogram to
predict the full-view sinogram, and then fed it to the standard analytical and
iterative reconstruction algorithms to obtain the final result.
Other researchers carried out post-processing on reconstructed images with
deep learning models, so as to remove the artifacts and noises for upgrading
the quality of these images[42, 43, 44, 33, 45, 46, 47]. In 2016, a deep
convolutional neural network [44] was proposed to learn an end-to-end mapping
between the FBP and artifact-free images. In 2018, Yoseob Han and Jong Chul Ye
designed a dual frame and tight frame U-Net [42] which satisfies the frame
condition and performs better for recovery of high frequency edges in sparse-
view CT. In 2019, Xie et al. [33] built an end-to-end cGAN model with joint
function used for removing artifacts from limited-angle CT reconstruction
images. In 2020, Wang et al. [45] developed a limited-angle TCT image
reconstruction algorithm based on U-Net, which could suppress the artifacts
and preserve the structures. Experiments have shown that U-Net-like structures
are efficacious for image artifacts removal and texture restoration [35, 36,
42, 45, 47] .
Since neural networks are capable of predicting unknown data in the Radon and
image domains, a natural idea is to combine these two domains [48, 49, 34, 50,
51, 52] to acquire better restoration results. Specifically, it first
complements the Radon data, and then remove the residual artifacts and noises
on images converted from the full-view Radon data. In 2018, Zhao et al.
proposed SVGAN [34], an artifacts reduction method for low-dose and sparse-
view CT via a single model trained by GAN. In 2019, Liang et al. [49] proposed
a comprehensive network combining projection and image domains. The projection
estimation network is based on Res-CNN structure, and the image domain network
takes the advantage of U-Net. In 2020, Zhu et al. designed ADAPTIVE-NET [50]
to conduct joint denoising on the acquired sinogram and the reconstructed CT
image, while reconstructing CT image in an end-to-end manner. In the past
three years, experiments have proved that this sort of two-stage algorithm is
quite conducive to image quality improvement.
All the current mainstream methods mentioned above make us notice that they
solely process on each single CT image, while neglecting the solid fact the
scanned object is always highly continuous. Consequently, there is abundant
spatial information lies in these obtained consecutive CT images, which is
largely left to be exploited. This enlightens us to propose a novel cascade
model called LS-AAE (Lightweight Spatial Adversarial Autoencoder) that mainly
focus on availably utilizing the spatial information between greatly
correlated images. It has been proved in our experiments that this sort of
structure design manages to efficaciously remove streaking artifacts in
sparse-view CT images, and outruns other prevailing methods with its
remarkable performance.
It is the social trend now to make healthcare mobile and portable. In lots of
deep learning-based methods, however, scholars improve accuracy at the expense
of sacrificing computing resources. Such computational complexity usually
exceeds the capabilities of many mobile and embedded applications. This paper
adopts the inverted residual with linear bottleneck [53] in the module design
to propose a mobile structure that reduce model parameters to one-eighth of
its original without sacrificing its performance.
Although enhancing the sparsity of sparse-view CT can bring benefits of
accelerated scanning and related calculations, it will cause additional
imaging damage. Balancing image quality and X-ray dose level has become a
well-known trade-off problem. Thus, in order to explore the limit of sparsity
in sparse-view CT reconstruction, we conduct sparse sampling at intervals of
4°, 8° and most importantly, 16°. Even under such sampling sparsity, our model
can still exhibit its remarkable robustness and the state-of-the-art
performance.
We introduce our proposed method exhaustively in Section II, the experimental
results and corresponding discussion are described in section III, and
conclusion is stated in section IV.
## 2 Methods
### 2.1 Preliminaries
#### 2.1.1 Utilize Spatial Information for Artifact Removal
As is known to all, consecutive CT images usually contain high spatial
coherency since the scanned object is usually spatially continuous. On account
of that, we can imagine these CT images as adjacent frames in a video which
contains much more information than a still image. This high correlation
within the sequence of images can improve the performance of artifact removal
from two aspects. Firstly, the extension of search regions from two-
dimensional image neighborhoods to three-dimensional spatial neighborhoods
provide extra information which can be used to denoise the reference image.
Secondly, using spatial neighbors helps to reduce streaking artifacts as the
residual error in each image is correlated.
Also, we cannot help but notice that the task of artifact removal between
consecutive images is similar to video denoising. Therefore, after
investigating lots of research work on video denoising [54, 55, 56, 57, 58,
59, 60, 61, 62, 63], we find out that current state-of-the-art methods lay
lots of emphasis on motion estimation due to the strong redundancy along
motion trajectories. To conclude, in order to more effectively remove
streaking artifacts from sparse-view CT images, we need to design a structure
that can not only look into the three-dimensional spatial neighborhood, but
also capture the motion between consecutive images.
#### 2.1.2 Enhance Mobility of Neural Networks
In recent years, lots of research has been invested into tuning deep neural
networks to achieve an optimal balance between efficiency and performance.
Among them, depthwise separable convolutions [64] exhibits its extraordinary
capability and has gradually become an essential building block for numerous
lightweight neural networks [64, 65, 66]. It aims to decompose the standard
convolutional layer into two separate layers, namely the depthwise
convolutional layer and the pointwise convolutional layer. The former layer is
designed to perform lightweight filtering through employing a single
convolutional filter per input channel, the latter one conducts $1\times 1$
convolution to construct new features by computing linear combinations of
input channels.
For the standard convolutional layer with input tensor size $(c_{in},h,w)$,
kernel size $(c_{out},c_{in},k,k)$ and output tensor size $(c_{out},h,w)$, its
computational cost equals to $c_{in}\cdot h\cdot w\cdot(k^{2}\cdot c_{out})$.
However, in depthwise separable convolutions, the depthwise convolutional
layer has a computational cost of $c_{in}\cdot h\cdot w\cdot k^{2}$ since it
merely operates on a single input channel, and the pointwise convolutional
layer has a computational cost of $c_{in}\cdot h\cdot w\cdot c_{out}$.
Therefore, we only need a computational cost of $h\cdot w\cdot
c_{in}\cdot(k^{2}+c_{out})$ for depthwise separable convolutions, which is
almost the one-ninth ($k$ equals to 3 in our case) of the standard
convolution. Most importantly, depthwise separable convolutions manage to
lower the computational complexity to a large extent without sacrificing its
accuracy, which would make it perfect to be inserted into our module design.
### 2.2 Overall Structure
#### 2.2.1 Structure Overview
Figure 1: Structure overview. The sparse-view Radon data $\mathbb{X}$ is first
sent to the neural network $F$ for completion, then the restored full-view
Radon data $\mathbb{X}^{\prime}$ is converted to the image $\mathbb{Y}$, which
is feed into the neural network $G$ for artifacts removal and we can finally
obtain the ideal high-quality image $\mathbb{Y}^{\prime}$.
We can learn from the universal approximation theorem [67] that multilayer
feedforward networks are capable of approximating various continuous
functions. This inspires us to think that neural networks can be used to learn
complex mappings that are difficult to solve through mathematical analysis.
Thus, in this paper, we utilize a deep learning-based structure that combines
the Radon domain and the image domain (Figure 1) to solve the task of sparse-
view CT reconstruction and inpainting.
Firstly, we want to make full use of the prior information in the Radon domain
by converting the sparse-view Radon data $\mathbb{X}$ to the full-view Radon
data $\mathbb{X}^{\prime}$ so as to complement the missing data in some
scanning angles. This process can be represented by the mapping:
$\mathbb{X}\xrightarrow{f}\mathbb{X}^{\prime}$ according to the universal
approximation theorem, where function $f$ can be approximated through our
proposed neural network $F$. After we obtain the full-view Radon data
$\mathbb{X}^{\prime}$, we transform it to the image $\mathbb{Y}$ through FBP.
Although the first stage manages to alleviate the severe imaging damage from
the original sparse-view CT image, there are still lots of streaking artifacts
existing in Y that need to be removed to acquire the high-quality restored
result $\mathbb{Y}^{\prime}$. We represent the restoration process into the
mapping: $\mathbb{Y}\xrightarrow{g}\mathbb{Y}^{\prime}$, where function $g$
can be approximated through our proposed neural network $G$. Through the above
two-stage structure that combines the Radon domain with the image domain, we
can finally get the ideal restored results.
#### 2.2.2 Stage One: Data Completion in the Radon Domain
Figure 2: The diagram of our proposed L-AAE, which is composed of a L-AE and a
discriminator that help restore the image texture.
We first adopt linear interpolation to convert the original sparse-view Radon
data to full-view Radon data so as to satisfy the structural characteristics
of our proposed neural network, which requires the input and output images to
have the same resolution. Then we build a lightweight adversarial autoencoder
(L-AAE) in Figure 2 to restore the Radon data, the structure of its
autoencoder (L-AE) can be seen from Figure 3 and Table 1, which is composed of
the encoder and the decoder that are highly symmetrical.
Figure 3: The detailed structure of the L-AE. Input images are first feed into the encoder for feature extraction and then sent into the decoder for texture restoration, where skip connections are added to merge low-level features. Table 1: Parametric structure of the L-AE Layer | $IC$ | $OC$ | Stride | Input Size | Output Size
---|---|---|---|---|---
Conv1 | 1 | 32 | 2 | 192$\times$512 | 96$\times$256
Block1 | 32 | 16 | 1 | 96$\times$256 | 96$\times$256
Block2_1 | 16 | 32 | 2 | 96$\times$256 | 48$\times$128
Block2_2 | 32 | 32 | 1 | 48$\times$128 | 48$\times$128
Block3_1 | 32 | 64 | 2 | 48$\times$128 | 24$\times$64
Block3_2 | 64 | 64 | 1 | 24$\times$64 | 24$\times$64
Block3_3 | 64 | 64 | 1 | 24$\times$64 | 24$\times$64
Block4_1 | 64 | 128 | 2 | 24$\times$64 | 12$\times$32
Block4_2 | 128 | 128 | 1 | 12$\times$32 | 12$\times$32
Block4_3 | 128 | 128 | 1 | 12$\times$32 | 12$\times$32
Block4_4 | 128 | 128 | 1 | 12$\times$32 | 12$\times$32
Trans Conv1 | 128 | 64 | 2 | 12$\times$32 | 24$\times$64
Block5_1 | 64+64(Concat) | 64 | 2 | 24$\times$64 | 24$\times$64
Block5_2 | 64 | 64 | 1 | 24$\times$64 | 24$\times$64
Block5_3 | 64 | 64 | 1 | 24$\times$64 | 24$\times$64
Trans Conv2 | 64 | 32 | 2 | 24$\times$64 | 48$\times$128
Block6_1 | 32+32(Concat) | 32 | 1 | 48$\times$128 | 48$\times$128
Block6_2 | 32 | 32 | 1 | 48$\times$128 | 48$\times$128
Trans Conv3 | 32 | 16 | 2 | 48$\times$128 | 96$\times$256
Block7 | 16+16(Concat) | 32 | 1 | 96$\times$256 | 96$\times$256
Block8 | 32+32(Concat) | 32 | 1 | 96$\times$256 | 96$\times$256
Trans Conv4 | 32 | 16 | 2 | 96$\times$256 | 192$\times$512
Conv9 | 16 | 1 | 1 | 192$\times$512 | 192$\times$512
We perform four down sampling in the encoder to obtain high-level semantic
features of the input image, which is initially downsampled through conv1
layer with a stride of 2, and the subsequent downsampling is separately
accomplished by the first building block of each unit. Each downsampling will
halve the height and width of the activation map and double the number of
channels.
As for the decoder, we correspondingly conduct four upsampling to restore the
texture of the input image. Deconvolution is adopted here for upsampling with
its kernel size and stride both equal to 2, so that each upsampling will
double the height and width of the activation map and halve the number of
channels. In addition, we add skip connections [68] between the encoder and
decoder feature maps of the same resolution. Since the final feature map of
the encoder has a relatively low resolution due to multiple downsampling, it
will have an undesirable effect on the restoration of the image texture in the
decoder. While the skip connection incorporates low-level features from the
encoder which have a high resolution and contain abundant detailed information
that will help to accurately restore the image texture. This sort of multi-
scale, U-Net-like architectures have been proved to be effective in processing
medical images.
The detailed structure of our building block can be seen from Figure 4, it
adopts the inverted residual with linear bottleneck referring to [53], each
block is composed of three convolutional layers. The first layer expands
(characterized by the expansion factor $exp$) a low-dimensional compressed
representation to high dimension with a kernel size of $1\times 1$. The
intermediate expansion layer adopts lightweight depthwise convolutions
mentioned above so as to significantly decreases the number of operations
(ops) and memory needed while sustaining the same performance. The last layer
projects the feature back to a low-dimensional representation with a linear
convolution like the first layer. All these layers are followed by batch
normalization [69] and ReLU [70] as the non-linear activation except for the
last layer that only followed by a batch normalization layer.
Figure 4: The detailed diagram of the building blocks in L-AAE, which adopt
the inverted residual with linear bottleneck.
In Figure 4, $IC$ and $OC$ stand for the input and output channel of building
blocks respectively. All convolutional layers in all building blocks have a
stride of 1 except for Block2_1, Block3_1 and Block4_1 that have a stride of 2
to conduct downsampling.Expansion factor $exp$ is 1 for Block1, Block7 and
Block8 to avoid large ops and memory cost, we set up $exp$ to be 3 for
Block5_1 and Block6_1, and every block expect these mentioned above have an
$exp$ of 6. Besides, shortcut connections are implemented in blocks that have
the same resolution between its input and output feature maps to enhance
information flow and also improve the ability of a gradient to propagate
across multiplier layers. We adopt $1\times 1$ convolution in shortcuts when
the number of channel in the input and output feature maps is different.
The discriminator in our L-AAE aims to strengthen model’s ability to restore
the detailed texture of images, its structure is almost the same as the
encoder above, except that its Block4_3 and Block4_4 have an $OC$ of 64 and 1
respectively. The output of Block4_4 is flattened and sent to sigmoid function
for probability prediction, which we average to get the final output that
represents the input image’s probability to be a real image. This novel
lightweight AAE enables us to acquire the well restored Radon data that are
complete in every scanning angle, and the computational cost is about 8 times
smaller than that of standard convolutions without sacrificing its accuracy.
#### 2.2.3 Stage Two: LS-AAE – Image Inpainting through spatial information
After stage one, we transform the acquired full-view Radon data to images and
find out that, we successfully enrich the information in the Radon domain and
alleviate streaking artifacts from the original sparse-view CT imaging. Now in
stage two, we will mainly focus on removing artifacts, restoring image to an
ideal level. As mentioned above, we need a neural network that not only look
into the three-dimensional spatial neighborhood, but also capture the motion
between consecutive images, so as to efficaciously utilize the abundant
spatial information between consecutive images to remove artifacts from the
input image.
Generally speaking, motion estimation always brings an additional degree of
complexity that is adverse to model’s implementation in reality. It means that
we need a structure that can manage to deploy motion estimation without much
resource cost, we refer to [71] and its general structure appears to be a
cascaded two-step architecture that inherently embed the motion of objects.
Inspired by this, we propose a model named Lightweight Spatial Adversarial
Autoencoder (LS-AAE) which can be seen from Figure 5. It slightly modifies the
L-AE from Figure 3 as its inpainting block, details are shown in Table 1. The
replacement from 2D convolution to 3D convolution enables our model to look
into the three-dimensional spatial neighborhood for extra information.
Table 2: From 2D convolution to 3D convolution | Layer | $IC$ | $OC$ | Kernel Size | Stride | Padding
---|---|---|---|---|---|---
2D Convolution | Conv1 | 1 | 16 | (3,3) | (2,2) | (1,1)
3D Convolution | Conv1_1 | 1 | 16 | (3,3,3) | (1,2,2) | (1,0,0)
| Conv1_2 | 16 | 32 | (3,3,3) | (2,1,1) | (0,1,1)
As shown in Figure 5, five consecutive images
$\left\\{\mathbb{I}_{i-2},\mathbb{I}_{i-1},\mathbb{I}_{i},\mathbb{I}_{i+1},\mathbb{I}_{i+2}\right\\}$
are sent into the LS-AAE to restore the middle one. We firstly treat these
inputs as triplets of consecutive images
$\left\\{\mathbb{I}_{i-2},\mathbb{I}_{i-1},\mathbb{I}_{i}\right\\}$,
$\left\\{\mathbb{I}_{i-1},\mathbb{I}_{i},\mathbb{I}_{i+1}\right\\}$ and
$\left\\{\mathbb{I}_{i},\mathbb{I}_{i+1},\mathbb{I}_{i+2}\right\\}$, then
enter them into the Inpainting Blocks 1. Subsequently, we obtain the outputs
of these blocks and combine them into triplet
$\left\\{\mathbb{I}^{\prime}_{i-1},\mathbb{I}^{\prime}_{i},\mathbb{I}^{\prime}_{i+1}\right\\}$
which will be sent into Inpainting Block 2 to acquire the ultimate estimation
$\mathbb{I}^{\prime\prime}_{i}$ corresponding to the central image
$\mathbb{I}_{i}$. The LS-AAE digs deep into the three-dimensional space and
implicitly handles motion without any explicit motion compensation stage on
account of the traits of its architecture. Besides, the three Inpainting
Blocks in step one share the same weights so as to avoid memory cost. We also
add a discriminator in stage two to better restore the image texture, the
predicted image $\mathbb{I}^{\prime\prime}_{i}$ and its corresponding ground
truth (the full-view CT imaging) $\mathbb{I}^{GT}_{i}$ are both send into this
discriminator, its structure is exactly the same as it is in stage one..
Figure 5: The diagram of our proposed LS-AAE. It combines an autoencoder that
fully utilizes the spatial correlation between consecutive CT images and a
discriminator that help refine image details.
### 2.3 Network Training
Stage one and stage two are trained separately. For the autoencoders in these
two models, we employ the multi-loss function below, which is consists of
three parts $l_{MSE}$, $l_{Adv}$ and $l_{Reg}$ with their respective
hyperparameters $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$.
${{l}_{AE}}={{\alpha}_{1}}{{l}_{MSE}}+{{\alpha}_{2}}{{l}_{Adv}}+{{\alpha}_{3}}{{l}_{Reg}}$
(1)
$l_{MSE}$ calculates the mean square error between the restored image and its
corresponding ground truth, it is widely used in various image inpainting
tasks since it provides an intuitive evaluation for the model’s prediction.
The expression of $l_{MSE}$ can be seen from Equation (2).
${{l}_{MSE}}=\frac{1}{W\times
H}\sum\limits_{x=1}^{W}{\sum\limits_{y=1}^{H}{{{(\mathbb{I}_{x,y}^{GT}-{{G}_{AE}}{{({{\mathbb{I}}^{Input}})}_{x,y}})}^{2}}}}$
(2)
Where function $G_{AE}$ stands for the autoencoder, $\mathbb{I}^{Input}$ and
$\mathbb{I}^{GT}$ are the input image and its corresponding ground truth, $W$
and $H$ are the width and height of the input image respectively.
$l_{Adv}$ refers to the adversarial loss. The autoencoder manages to fool the
discriminator by making its prediction as close to the ground truth as
possible, so as to achieve the ideal image restoration outcome. Its expression
can be seen from Equation (3).
${{l}_{adv}}=1-D({{G}_{AE}}({\mathbb{I}^{Input}}))$ (3)
Where function $D$ and $G_{AE}$ stands for the discriminator and the
autoencoder respectively, $\mathbb{I}^{Input}$ is the model’s input image.
$l_{Reg}$ is the regularization term of our multi-loss function. Since noises
will have a side effect on our restoration result, we add a regularization
term to maintain the smoothness of the image and also prevent overfitting. TV
Loss is widely used in image analysis tasks, it reduces the variation between
adjacent pixels to a certain extent. Its expression can be seen from Equation
(4).
${{l}_{Reg}}=\frac{1}{W\times
H}\sum\limits_{x=1}^{W}{\sum\limits_{y=1}^{H}{\left\|\left.\nabla{{G}_{AE}}{{({\mathbb{I}^{Input}})}_{x,y}}\right\|\right.}}$
(4)
Where function $G_{AE}$ represents the autoencoder, $I^{Input}$ is the model’s
input image, $W$ and $H$ are the width and height of the input image
respectively. $\nabla$ calculates the gradient, $\left\|\cdot\right\|$ obtains
the norm.
To optimize the discriminator of these two stages, their loss function should
enable them to better distinguish between real and fake inputs. The loss
function $l_{Dis}$ is shown in Equation (5).
${{l}_{D\text{is}}}=1-D\left({\mathbb{I}^{GT}}\right)+D\left({{G}_{AE}}\left({\mathbb{I}^{Input}}\right)\right)$
(5)
Where function $D$ and $G$ stands for the discriminator and the autoencoder
respectively, $\mathbb{I}^{Input}$ and $\mathbb{I}^{GT}$ are the input image
and its corresponding ground truth. The discriminator outputs a scalar between
0 to 1 which represents the probability that the input image is real.
Therefore, minimizing $1-D(\mathbb{I}^{GT})$/maximizing $D(\mathbb{I}^{GT})$
enables the discriminator to recognize real images, while minimizing
$D(G_{AE}(\mathbb{I}^{Input}))$ enables the discriminator to distinguish fake
images that generated from the autoencoder from all input images.
During the training process, we adopt the Adam algorithm [72] for
optimization. the learning rate is set to 1e-4 initially. For the multi-loss
function, $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ are set to 1, 1e-3, and
2e-8 respectively. We implement our whole structure using PyTorch [73] on two
GeForce RTX 2080 Ti.
## 3 Experiments
We adopt the LIDC-IDRI [74] as our dataset, which includes 1018 cases and
approximately 240,000 DCM files of corresponding CT images. Cases 1 to 100 are
divided into test set, cases 101 to 200 are divided into validation set, and
the rest are divided into train set. Such a large amount of data allows us to
train our models from scratch without overfitting. We utilize NumPy to read
from these DCM files and conduct sparse sampling at intervals of 4, 8 and 16
(the corresponding full-view Radon data has 180 projections). Subsequently, we
first analyze our overall structure through a series of ablation studies, and
then compare our experimental results with other current methods to prove its
superiority and robustness.
### 3.1 Ablation Study
With all these innovations we make in our overall structure design, it would
be appropriate for us to conduct corresponding ablation studies to prove their
necessity. In this part, all the experimental results are acquired from
sparse-view CT data with an interval of 4 if there is no specific mention.
#### 3.1.1 The L-AE’s Trade-off between Mobility and Performance
As is known to all, U-Net has extraordinary performance in numerous medical
image processing tasks, [42] implemented it for sparse-view CT imaging
restoration and obtained outstanding restoration results. To testify that our
proposed autoencoder can achieve a good balance between performance and
mobility, we replace it with U-Net in the first stage and compare the
restoration results and model parameters of this stage with ours, as shown in
Table 3. The images mentioned in Table 3 are reconstructed from the Radon data
restored through stage one.
Table 3: U-Net VS. L-AE | Radon PSNR | Radon SSIM | Image PSNR | Image SSIM | Parameters
---|---|---|---|---|---
U-Net | 57.582 | 0.998 | 29.598 | 0.874 | 10.401M
L-AE | 57.66 | 0.998 | 29.609 | 0.874 | 1.675M
As we can see from Table 3, whether in the Radon domain or in the image
domain, L-AE has competitive performance compared with U-Net. Moreover, it
significantly reduces model parameters, making it suitable for situations
where computational resources are extremely limited. This exhibits our model’s
ability in efficiently restoring CT images, thus adapting to the social trend
of deploying portable medical devices.
#### 3.1.2 The Discriminator
We establish discriminators in both two stages, hoping to further improve our
model’s performance in restoring sparse-view CT data through the adversarial
learning between the autoencoders and the discriminators. In order to verify
this point of view, we send the test set into stage one where there is merely
an autoencoder and compare its restoration results with ours, which can be
seen from Table 4. The images mentioned in Table 4 are reconstructed from the
Radon data restored through stage one.
Table 4: The Role of the Discriminator | Radon PSNR | Radon SSIM | Image PSNR | Image SSIM
---|---|---|---|---
L-AE Only | 48.904 | 0.985 | 28.448 | 0.871
L-AAE | 57.660 | 0.998 | 29.609 | 0.874
From the above table, we can realize the significance of our proposed
discriminator, it indeed assists our model to achieve a better level of
restoration under the evaluation of PSNR and SSIM. Its precise structure
(refers to Sec II) also ensures a high degree of mobility, which enables our
overall structure to be portable and accurate at the same time.
#### 3.1.3 The Two-step Architecture – LS-AAE
As we state in Sec II, this sort of cascaded two-step structure inherently
embeds the motion of objects which can largely help to remove image artifacts
due to the strong redundancy between these consecutive images. Consequently,
we design an experiment with reference to [71] to prove this view. In the
second stage, instead of sending five consecutive images into this two-step
LS-AAE, we directly input them into a single Inpainting Block (SIB) that is
slightly modified in the three-dimensional convolution part to handle five
images, that means we adopt a stride of 2 in the Conv1_1 layer (refers to
Table 1). The experimental results can be seen from Table 5 below.
Table 5: Restoration Results of SIB and LS-AAE | Image PSNR | Image SSIM
---|---|---
SIB | 38.972 | 0.941
LS-AAE | 40.305 | 0.948
Now the SIB no longer owns this built-in cascade structure to implicitly
conduct motion estimation, it suffers from a obvious drop in PSNR and SSIM.
Therefore, we can arrive at the conclusion that, LS-AAE manages to effectively
improve model’s capability of restoring CT images with its cascaded two-step
architecture that inherently capture the motion between consecutive images.
#### 3.1.4 The 3D convolution in LS-AAE
We mention in Sec II that, the extension of search regions from two-
dimensional image neighborhoods to three-dimensional spatial neighborhoods
provide extra information for image restoration. Also, extracting spatial
features is conducive to remove streaking artifacts as the residual error in
each image is correlated. In order to realize this extension of search
regions, three-dimensional convolution is employed in every Inpainting Block
of LS-AAE. To verify the cruciality of these three-dimensional convolutions,
we conduct an experiment in which 3D convolution are replaced back to 2D
convolution, where the number of input images is regarded as the number of
input channel (refers to Table 2). The inpainting results of these two models
are shown in Table 6.
Table 6: Restoration Results of 2D and 3D LS-AAE | Image PSNR | Image SSIM
---|---|---
2D LS-AAE | 39.472 | 0.944
3D LS-AAE | 40.305 | 0.948
We can see that the inpainting outcome suffers from a drop about 0.9dB in
PSNR, proving that three-dimensional convolutions assist model in restoring CT
images to a certain extent without significantly consuming computational
resources.
#### 3.1.5 The Image Interval of LS-AAE’s Input
In all the experiments above, we set the image interval between input
consecutive CT images of LS-AAE to the default value of 1. However, we cannot
help but wonder that whether increasing the interval value can help the model
obtain more spatial information, thereby enhancing its ability in removing
image artifacts. In the following experiment, we set this image interval $T$
to 1, 2, 3, 4 and 5 respectively, their corresponding results are shown in
Table 7.
Table 7: The Image Interval’s Effect on Restoration Results | Image PSNR | Image SSIM
---|---|---
$\bm{T=1}$ | 40.305 | 0.948
$T=2$ | 39.961 | 0.948
$T=3$ | 40.032 | 0.948
$T=4$ | 40.147 | 0.948
$T=5$ | 40.195 | 0.950
It can be learnt from Table 7 that this hyperparameter T does not have much
impact on the final restoration result. Spatial correlation seems to be well
utilized when the image interval is set to 1, which would be a decent default
choice.
#### 3.1.6 The Radon Domain VS. the Image Domain
In this paper, we adopt a two-stage structure that combines the Radon domain
and the image domain to obtain high-quality sparse-view CT images. Since each
stage of the overall structure conducts restoration in their separate domains
and both remarkably upgrade the restoration results, this leads us to think,
what role do these two domains play? Subsequently, we feed our test set into
these three structures: L-AAE in stage one that concentrates on the Radon
domain, LS-AAE in stage two that focus on the image domain and of course, our
overall structure that contains these two stages. The quantitative inpainting
results of the above three structures can be referred from Table 8, the
intuitive outcome can also be seen in Figure 6.
Figure 6: The intuitive restoration results obtained by different domains. Table 8: Restoration Results obtained by different domains | Image PSNR | Image SSIM
---|---|---
The Radon Domain | 30.310 | 0.905
The Image Domain | 34.135 | 0.888
Dual Domains | 40.305 | 0.948
It can be seen from above that, restoration in each domain has its pros and
cons. For the Radon domain, it demonstrates its superiority in enhancing the
structural similarity of images so as to perform well under the evaluation of
SSIM. While as for the image domain, it exhibits great ability in alleviating
distortion, thus has a relatively good performance under the evaluation of
PSNR. Naturally, we acquire extraordinary restoration results when combining
these two domains to merge their respective advantages. Besides, we solely
utilize the spatial correlation in the Image domain due to our discovery that,
the spatial information between continuous Radon slices has little impact on
the final inpainting outcome. We suppose this is because the texture in Radon
slices does not have much similarity with CT images, thus cannot be restored
in this way.
### 3.2 Methods Comparison
After verifying the rationality of our overall structural design, we want to
testify its robustness through applying it to sparse-view CT data with a
higher level of sparsity, which means, conducting sparse sampling at intervals
of 4, 8 and even 16 (the corresponding full-view Radon data has 180
projections). In addition, we compare our method with other current ones to
prove its prominent capability of restoring sparse-view CT images and removing
streaking artifacts. The experimental results are shown in Table 9, and the
intuitive outcome can be seen from Figure 7.
Figure 7: The intuitive restoration results of various methods at different sampling intervals. Table 9: Methods Comparison | Interval=4 | Interval=8 | Interval=16
---|---|---|---
PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
FBP | 12.080 | 0.498 | 12.065 | 0.485 | 12.032 | 0.471
SART-TV | 19.179 | 0.665 | 19.061 | 0.632 | 18.777 | 0.602
U-Net | 34.018 | 0.885 | 31.944 | 0.843 | 28.767 | 0.798
Ours | 40.305 | 0.948 | 37.633 | 0.937 | 34.052 | 0.910
As we can see, our method exhibits extraordinary capability of restoring
sparse-view CT imaging, effectively removes streaking artifacts and outruns
other methods by a large margin. Also, it can be applied to extreme sparsity
while still obtaining prominent inpainting outcome. Particularly, our method
still exceeds others when the sampling rate is one-fourth of them, thereby
demonstrating its remarkable robustness and superiority.
## 4 Conclusion
In this paper, we propose a lightweight structure that efficaciously restores
sparse-view CT with its two-stage architecture combining the Radon domain and
the image domain. Most importantly, we groundbreakingly exploit the abundant
spatial information existing between consecutive CT images, so as to achieve a
remarkable restoration outcome even if our method encounters extreme sparsity.
In the first stage, a mobile model named L-AAE is proposed to complement the
original sparse-view CT in the Radon domain, it adopts the inverted residual
with linear bottleneck in order to significantly reduce computational resource
requirements while maintaining outstanding performance. In the second stage,
after reconstructing the restored full-view Radon data into images through
FBP, we establish a lightweight model called LS-AAE. It is designed to
implicitly conduct motion estimation and dig into the three-dimensional
spatial neighborhood with a relatively low memory cost. Therefore, it manages
to concentrates on fully utilizing the strong spatial correlation between
continuous CT images, so as to productively remove streaking artifacts and
finally acquire high-quality restoration results.
Eventually, for the sparse-view CT with a sampling interval of 4, we achieve a
PSNR of 40.305 and a SSIM of 0.948, realizing a remarkable restoration result
that can effectively eliminate image artifacts. In addition, our method also
performs well when it comes to extreme sparsity (the sampling interval is 8 or
even 16), exhibiting its prominent robustness.
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|
Weyl Consistency Conditions from a local Wilsonian Cutoff
Ulrich Ellwanger
University Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
A local UV cutoff $\Lambda(x)$ transforming under Weyl rescalings allows to
construct Weyl invariant kinetic terms for scalar fields including Wilsonian
cutoff functions. First we consider scalar fields in curved space-time with
local bare couplings of any canonical dimension, and anomalous dimensions
which describe their dependence on the UV cutoff. The local component of the
UV cutoff plays the role of an additional coupling, albeit with a trivial
constant $\beta$ function. This approach allows to derive Weyl consistency
conditions for the corresponding anomalous dimensions which assume the form of
an exact gradient flow. For renormalizable theories the Weyl consistency
conditions are initially of the form of an approximate gradient flow for the
$\beta$ functions, and we derive conditions under which it becomes the form of
an exact gradient flow.
## 1 Introduction
Weyl consistency conditions have lead to remarkable insights into quantum
field theories. In the form derived by Osborn and Jack and Osborn in [1, 2, 3]
[JO] the Weyl consistency conditions imply a gradient flow for the
renormalization group flow described by $\beta$ functions in low orders in
perturbation theory, a phenomenon observed earlier in [4, 5]. One motivation
was to derive a $c$ (or $a$) theorem in four dimensions, but of interest are
also the induced relations among coefficients of $\beta$ functions or
anomalous dimensions of composite operators, and the ultraviolet (UV) and
infrared (IR) asymptotic behaviour of quantum field theories.
The derivation of Weyl consistency conditions requires to consider a quantum
field theory in curved space-time described by a background metric
$\gamma_{\mu\nu}$, and to promote couplings $g_{i}$ to local couplings
$g_{i}(x)$. Under local Weyl rescalings
$\delta_{\sigma}\gamma_{\mu\nu}=-2\sigma\gamma_{\mu\nu}$, the local couplings
transform according to their anomalous dimensions or $\beta$ functions. The
Weyl consistency conditions in the form derived in [JO] follow either from the
fact that local Weyl rescalings (being abelian) acting on the vacuum partition
function via Weyl rescalings of $\gamma_{\mu\nu}$ and/or via Weyl rescalings
of the local couplings have to commute, or from the finiteness of $\beta$
functions in dimensional regularization.
Implied relations among coefficients of $\beta$ functions in dimensional
regularization have been studied in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] for
the Standard Model and others. Implications of Weyl consistency conditions for
higher dimensional field theories have been investigated e.g. in [16, 17].
Actually the gradient flow property of the renormalization group flow in $d=4$
is not exact, a priori confined to low orders in perturbation theory. Its
general validity was challenged e.g. in [18], see also the discussion in [19].
By far most of the previous explicit calculations were performed using
dimensional regularization where classical Weyl invariance is broken by the
scale $\mu$, introduced for otherwise dimensionless couplings. Here we
consider a regularization by a Wilsonian cutoff $\Lambda$, by which we
understand a modification of kinetic terms in the action such that modes with
momenta $p^{2}\gg\Lambda^{2}$ are suppressed, i.e. the kinetic terms become
very (e.g. exponentially) large for $p^{2}\gg\Lambda^{2}$.
The implementation of a Wilsonian cutoff in an otherwise Weyl invariant theory
has been considered in [20, 21, 22], with the aim to study exact (functional)
renormalization group equations (RGEs) in 2 and 4 dimensions. The authors
concluded that a cutoff $\Lambda$ has to become local, i.e. $\Lambda(x)$.
However, the authors restricted the dependence of $\Lambda$ on $x$ to be given
by a dilaton (a Weyl compensator) which has dynamics in principle, but can be
gauged away if Weyl invariance remains unbroken. Local couplings have not been
introduced, and Weyl consistency conditions have not been considered. Another
approach to implement a Wilsonian cutoff in a Weyl invariant way has been
proposed in [23], again in the context of exact RGEs. A local renormalization
group equation was considered as early as 1987 in [24] in two-dimensional
curved spacetime sigma models in order to derive consistency conditions on
allowed backgrounds in string theory. In [25] a local cutoff was introduced
for the study of Weyl invariance in the framework of holography, and in [26]
in the context of the quantum renormalization group.
Here we present a way to implement a Wilsonian cutoff respecting Weyl
invariance, based on a local cutoff $\Lambda(x)$ which transforms under Weyl
rescalings. As a consequency all parameters (couplings and masses) transform
under Weyl rescalings according to their canonical plus anomalous dimensions,
except for the renormalization scale $\mu$ itself. Besides $\mu$ all
parameters, couplings and masses, are considered as local. This allows to
deduce a local RGE, a notion used to define the response of the action
including counter terms with respect to Weyl rescalings $\delta_{\sigma}$. It
implies a local RGE satisfied by the vacuum partition function which depends
on the background metric, local couplings and covariant derivatives thereof as
described in [JO]. The local cutoff $\Lambda(x)$ appears in this RGE like a
local coupling, albeit with a trivial constant “$\beta$ function”. For
renormalizable theories, dimensional analysis can be used to relate $\beta$
functions of couplings with respect to the UV cutoff to $\beta$ functions with
respect to a renormalization scale $\mu$.
The purpose of this article is to derive Weyl consistency conditions, and
conditions leading to gradient flows for $\beta$ functions or anomalous
dimensions in the space of couplings enlarged by $\Lambda(x)$. We consider two
distinct cases: Case 1 where a true physical cutoff exists, bare couplings
vary with the cutoff, and one is interested in the variation of the vacuum
partition function with the cutoff. Here the limit of an infinite cutoff may
not exist. Case 2 corresponds to a renormalizable quantum field theory where
the limit of an infinite cutoff exists provided a finite number of counter
terms is added. We show that a gradient flow holds in the space of anomalous
dimensions $\gamma_{n}$ of composite operators ${\cal O}_{n}$ enlarged by
$\Lambda(x)$ in case 1, and under a certain condition in the space of $\beta$
functions in case 2.
In section 2 we introduce Weyl invariant Wilsonian cutoff functions which
allow to derive local RGEs for case 1 and case 2 in section 3. In section 4
these are applied to the vacuum partition function. Weyl consistency
conditions are derived for case 1 in Section 4.1, and for case 2 in Section
4.2.
As a very first application we consider a real scalar field $\varphi$ with
mass $m$ and quartic coupling $g$ in four dimensions in section 5. We compute
the metric relevant for the gradient flow to lowest nontrivial order $g^{1}$,
where only the $\beta$ functions in the subspace $m(x),\Lambda(x)$ contribute.
To this end we employ a new method to compute this metric in momentum space,
in a weak field expansion around flat space time and in fluctuations of local
couplings (including $m(x),\Lambda(x)$) around constant values. The results
for the simple scalar model allow for first consistency checks of the methods.
In Appendix A we describe how to compute the metric in momentum space, in
Appendix B we explain under which condition the component of the metric with
one index corresponding to $\Lambda(x)$ has the form of a gradient in the
space of the remaining couplings. In Appendix C we compute vertices from the
kinetic term including a cutoff function and describe the relevant diagrams. A
summary and conclusions are given in section 6.
## 2 Weyl Invariant Wilsonian Cutoff Functions
In this section we construct a Weyl invariant kinetic term, including a
Wilsonian UV cutoff, for a real scalar field $\varphi$ in $d$ dimensions. The
UV cutoff $\Lambda$ is assumed to be local, i.e. $\Lambda(x)$, and to
transform under Weyl rescalings together with the background metric
$\gamma_{\mu\nu}$ and the scalar field:
$\displaystyle\delta_{\sigma}\gamma_{\mu\nu}=$
$\displaystyle-2\sigma\gamma_{\mu\nu}\;,$ (2.1)
$\displaystyle\delta_{\sigma}\varphi=$
$\displaystyle\sigma\left(\frac{d}{2}-1\right)\varphi\;,$
$\displaystyle\delta_{\sigma}\Lambda=$ $\displaystyle\sigma\Lambda\;.$
The construction of a kinetic term including a Wilsonian cutoff will proceed
stepwise. We start with the known expression for a Weyl covariant
generalization of the covariant Laplacian
$\nabla^{2}-\xi
R\;,\quad\nabla^{2}=\frac{1}{\sqrt{\gamma}}\partial_{\mu}\sqrt{\gamma}\gamma^{\mu\nu}\partial_{\nu}\;,\quad\gamma=\det({\gamma_{\mu\nu}})\;,\quad\xi=\frac{d-2}{4(d-1)}\;,$
(2.2)
which satisfies
$\delta_{\sigma}\left[(\nabla^{2}-\xi R){\cal
O}\right]=\left(\frac{d}{2}+1\right)\sigma\left[(\nabla^{2}-\xi R){\cal
O}\right]$ (2.3)
provided the operator $\cal{O}$ satisfies $\delta_{\sigma}{\cal
O}=\sigma\left(\frac{d}{2}-1\right){\cal O}$ such that, for ${\cal
O}=\varphi$, $\delta_{\sigma}(\sqrt{\gamma}\varphi(\nabla^{2}-\xi
R)\varphi)=0$ (using $\delta_{\sigma}\sqrt{\gamma}=-d\,\sigma\sqrt{\gamma}$).
A Weyl invariant Wilsonian cutoff can be constructed with help of the operator
$D_{\Lambda}=\Lambda^{-2}(\nabla^{2}-\xi R)\;.$ (2.4)
Using $\delta_{\sigma}\Lambda^{-2}=-2\sigma\Lambda^{-2}$, any expression of
the form
$F(D_{\Lambda})$ (2.5)
satisfies
$\delta_{\sigma}\left(F(D_{\Lambda})\varphi\right)=\left(\frac{d}{2}-1\right)\sigma
F(D_{\Lambda})\varphi\;.$ (2.6)
The possibility to construct $F(D_{\Lambda})$ with this property under local
Weyl transformations requires the use of the $x$ dependent cutoff $\Lambda$
transforming as in eq. (2.1). An example is
$F(D_{\Lambda})=e^{-D_{\Lambda}}$ (2.7)
which leads to an exponentially suppressed propagator for large $p^{2}$ in
momentum space. The kinetic part $S_{k}$ of the action reads then
$S_{k}=\frac{1}{2}\int\sqrt{\gamma}d^{d}x\,\varphi(-\nabla^{2}+\xi
R)\,F(D_{\Lambda})\varphi$ (2.8)
and satisfies
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\varphi\frac{\delta}{\delta\varphi}+\Lambda\frac{\delta}{\delta\Lambda}\right\\}S_{k}=0\;.$
(2.9)
## 3 The Action and Local Renormalization Group Equations
As interactions we consider operators ${\cal O}_{n}(\varphi,\nabla)\equiv{\cal
O}_{n}(\varphi,\gamma_{\mu\nu})$ which satisfy
$\left\\{\int\sqrt{\gamma}d^{d}y\,\sigma\left(-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\varphi\frac{\delta}{\delta\varphi}\right)\right\\}\sqrt{\gamma(x)}\,{\cal
O}_{n}(x)=-d_{n}^{0}\sigma(x)\sqrt{\gamma(x)}\,{\cal O}_{n}(x)\;.$ (3.1)
Typically one has ${\cal O}_{n}=\varphi^{n}$ such that $d_{n}^{0}=d-n$. The
interaction part $S_{int}$ involving real scalar fields
$\varphi\equiv\left\\{\varphi_{k}\right\\}$ (indices of fields will be
suppressed for simplicity) reads
$S_{int}=\int\sqrt{\gamma}d^{d}x\sum_{n}g_{n}^{0}(\Lambda)\,{\cal O}_{n}$
(3.2)
where $g_{n}^{0}(\Lambda)$ are bare marginal, relevant or irrelevant couplings
of canonical dimension $d_{n}^{0}$. A Weyl rescaling of a term
$\sim\sqrt{\gamma}\,g_{n}^{0}(\Lambda)\,{\cal O}_{n}$ in $S_{int}$ gives
$\left\\{\int\sqrt{\gamma}d^{d}x\,\sigma\left(-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\varphi\frac{\delta}{\delta\varphi}+\Lambda\frac{\delta}{\delta\Lambda}\right)\right\\}\sqrt{\gamma}\,g_{n}^{0}\,{\cal
O}_{n}=-d_{n}^{0}g_{n}^{0}\sigma\sqrt{\gamma}\,{\cal
O}_{n}+\sqrt{\gamma}\,{\cal
O}_{n}\int\sqrt{\gamma}d^{d}x\sigma\Lambda\frac{\delta
g_{n}^{0}}{\delta\Lambda}\;.$ (3.3)
In the following we consider two different scenarios:
Case 1: One may consider $S_{int}$ as an effective action valid at scales
$<{\Lambda}$ where momenta $>{\Lambda}$ have already been integrated out. Then
$S_{int}$ can include irrelevant operators multiplied by couplings
$g_{n}^{0}(\Lambda)\sim\Lambda^{d_{n}^{0}}$ with $d_{n}^{0}<0$. The aim will
be to derive a local RGE including the dependence of the full action
$S=S_{k}+S_{int}$ on the local cutoff $\Lambda$.
Case 2 corresponds to a renormalizable theory where $g_{n}^{0}$ are bare
marginal or relevant couplings of canonical dimension $d_{n}^{0}$ with $0\leq
d_{n}^{0}<d$. $g_{n}^{0}$ include $\Lambda$ and $\mu$ dependent counter terms
such that Green functions of $\varphi$ are finite for $\Lambda\to\infty$. We
derive a local RGE including the dependence on the local cutoff $\Lambda$
which is valid before the limit $\Lambda\to\infty$ is taken. Subsequently we
consider vacuum diagrams only, thus no sources need to be coupled to
$\varphi$. Hence, apart from a trivial factor originating from the measure,
the path integral allows for redefinitions of $\varphi$. This freedom is used
to remove wave function renormalisation factors $Z$ multiplying the kinetic
term. As a consequence of these field redefinitions the counter terms in
$g_{n}^{0}$ include contributions from the $Z$ factors. (Since $Z$ factors
depend on $\Lambda(x)$ and $g_{i}(x)$ they are local as well. In the case of
several interacting scalars, the treatment of terms $\sim\partial_{\mu}Z(x)$
requires special care.)
In case 1 we assume that the $\Lambda$ dependent bare couplings $g_{n}^{0}$
satisfy scaling relations of the form
$\Lambda\frac{\delta}{\delta\Lambda}g_{n}^{0}=\hat{\gamma}_{n}(\Lambda,g_{i}^{0})\,g_{n}^{0}\;.$
(3.4)
(Throughout this paper $\delta$ denote functional local derivatives.
Appropriate factors of $\int\sqrt{\gamma}$ are understood.) Then the full
action $S=S_{k}+S_{int}$ satisfies a local RGE
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\varphi\frac{\delta}{\delta\varphi}+\Lambda\frac{\delta}{\delta\Lambda}+(d_{n}^{0}-\hat{\gamma}_{n})g_{n}^{0}\frac{\delta}{\delta
g_{n}^{0}}\right\\}S=0\;.$ (3.5)
If we rescale $g_{n}^{0}$ by $\Lambda^{d_{n}^{0}}$ such that $\rho_{n}$ are
dimensionless,
$g_{n}^{0}=\Lambda^{d_{n}^{0}}\rho_{n}\qquad\text{with}\qquad\Lambda\frac{\delta}{\delta\Lambda}\rho_{n}={\gamma}_{n}(\Lambda,\rho_{i}),\qquad\gamma_{n}=(\hat{\gamma}_{n}-d_{n}^{0})\,\rho_{n}\;,$
(3.6)
eq. (3.5) becomes
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\varphi\frac{\delta}{\delta\varphi}+\Lambda\frac{\delta}{\delta\Lambda}-{\gamma}_{n}\frac{\delta}{\delta\rho_{n}}\right\\}S=0\;.$
(3.7)
Turning to case 2, we can relate the terms on the right hand side of eq. (3.3)
to the usual $\beta$ functions of the $\mu$ dependent physical couplings
$g_{i}$. To be specific, we can assume a momentum subtraction scheme where
counter terms are defined by the condition that appropriate one-particle-
irreducible Green functions evaluated at non-exceptional momenta
$-p_{i}^{2}=\mu^{2}$ assume values of corresponding physical couplings
$g_{i}$. We denote the canonical dimensions of $g_{i}$ by $d_{i}$, and apply
naive dimensional analysis to $g_{n}^{0}(\Lambda,g_{i},\mu)$:
$\left\\{\Lambda\frac{\delta}{\delta\Lambda}+\mu\frac{\partial}{\partial\mu}+d_{i}g_{i}\frac{\delta}{\delta
g_{i}}\right\\}g_{n}^{0}=d_{n}^{0}\,g_{n}^{0}\;.$ (3.8)
Next we use the fact that the total derivative of
$g_{n}^{0}(\Lambda,g_{i},\mu)$ with respect to $\mu$ must vanish:
$\mu\frac{\partial
g_{n}^{0}}{\partial\mu}+\mu\frac{dg_{i}}{d\mu}\,\frac{\delta g_{n}^{0}}{\delta
g_{i}}=0$ (3.9)
leading to
$\Lambda\frac{\delta
g_{n}^{0}}{\delta\Lambda}=d_{n}^{0}\,g_{n}^{0}+\left(\mu\frac{dg_{i}}{d\mu}-d_{i}\,g_{i}\right)\frac{\delta
g_{n}^{0}}{\delta g_{i}}\;.$ (3.10)
Thus the right hand side of eq. (3.3) becomes
${\cal O}_{n}\beta_{i}\frac{\delta g_{n}^{0}}{\delta
g_{i}}\qquad\text{with}\qquad\beta_{i}=\mu\frac{dg_{i}}{d\mu}-d_{i}\,g_{i}\;.$
(3.11)
and eq. (3.3) can be written as
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\varphi\frac{\delta}{\delta\varphi}+\Lambda\frac{\delta}{\delta\Lambda}-\beta_{i}\frac{\delta}{\delta
g_{i}}\right\\}S=0\;.$ (3.12)
Here $S_{int}(g_{n}^{0}(\Lambda,g_{i},\mu))$ is considered as
$S_{int}(\Lambda,g_{i},\mu)$, and the cutoff $\Lambda$ is also still present
in the kinetic part $S_{k}$. We recall that eq. (3.12) is valid before the
limit $\Lambda\to\infty$ is taken.
## 4 Gradient Flows from Weyl Consistency Conditions
The aim of this section is to derive consequences of Weyl consistency
conditions which follow from the local RGEs derived in Section 3. The
scenarios case 1 and case 2 are considered separately, although the essential
steps are the same.
### 4.1 A Gradient Flow for Case 1
In this case the action $S(\varphi,\Lambda,\rho_{n})$ satisfies the local RGE
eq. (3.7). We consider the vacuum partition function
${W^{0}}(\gamma_{\mu\nu},\Lambda,\rho_{n})$ as functional of
$\gamma_{\mu\nu},\Lambda,\rho_{n}$ given by
$e^{-{W^{0}}(\gamma_{\mu\nu},\Lambda,\rho_{n})}=\frac{1}{\cal N}\int{\cal
D}\varphi\,e^{-S(\varphi,\gamma_{\mu\nu},\Lambda,\rho_{n})}\;.$ (4.1)
Since $\varphi$ are dummy variables on the right hand side of eq. (4.1), eq.
(3.7) implies the local RGE
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\Lambda\frac{\delta}{\delta\Lambda}-{\gamma}_{n}\,\frac{\delta}{\delta\rho_{n}}\right\\}W^{0}=0$
(4.2)
with ${\gamma}_{n}$ defined in eq. (3.6). $W^{0}$ is invariant under general
coordinate transformations and depends on $\Lambda(x)$, $\rho_{n}(x)$ and
their derivatives.
Since $\Lambda$ is local we express it in the form
$\Lambda(x)=\overline{\Lambda}\,e^{\lambda(x)}$ (4.3)
with $\overline{\Lambda}$=const.; the limit of an infinite cutoff would
correspond to $\overline{\Lambda}\to\infty$, $\lambda(x)$ fixed, but is not
considered here. Under Weyl rescalings we have
$\delta_{\sigma}\lambda=\sigma$, i.e.
$\sigma\Lambda\frac{\delta}{\delta\Lambda}\equiv\sigma\frac{\delta}{\delta\lambda}\;.$
(4.4)
Using $\lambda$ from eq. (4.3) all local variables can be combined into
dimensionless $\chi_{i}$ and corresponding $\beta$ functions
$\tilde{\beta}_{i}$,
$\chi_{i}=\\{\lambda(x),\rho_{n}(x)\\}\qquad\text{and}\qquad\tilde{\beta}_{i}=\\{-1,\gamma_{n}(x)\\}\;.$
(4.5)
Then eq. (4.2) becomes
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}-\tilde{\beta}_{i}\frac{\delta}{\delta\chi_{i}}\right\\}W^{0}=0\;.$
(4.6)
Subsequently we focus on $d=4$ dimensions, and we consider terms in $W^{0}$
involving four derivatives acting on the metric $\gamma_{\mu\nu}$ or
$\chi_{i}$. A complete list of such structures respecting manifest coordinate
invariance is given in [JO]. Five among these structures will play a role
subsequently, given by
$\displaystyle
W^{0}=\int\sqrt{\gamma}d^{4}x\,\left\\{b^{0}\,G+\frac{1}{2}{\cal
G}^{0}_{\chi_{i}\chi_{j}}\,G^{\mu\nu}\,\partial_{\mu}{\chi_{i}}\partial_{\nu}{\chi_{j}}+{\cal
E}^{0}_{\chi_{i}\chi_{j}}\,\partial^{\mu}R\,{\chi_{i}}\partial_{\mu}{\chi_{j}}+\frac{1}{2}\,{\cal
F}^{0}_{\chi_{i}\chi_{j}}\,R\,\partial_{\mu}{\chi_{i}}\partial^{\mu}{\chi_{j}}\right.$
$\displaystyle\left.+\frac{1}{2}{\cal
A}^{0}_{\chi_{i}\chi_{j}}\nabla^{2}{\chi_{i}}\nabla^{2}{\chi_{j}}+\dots\right\\}$
(4.7)
where the coefficients $b^{0},{\cal G}^{0}_{\chi_{i}\chi_{j}},{\cal
E}^{0}_{\chi_{i}\chi_{j}},{\cal F}^{0}_{\chi_{i}\chi_{j}}$ and ${\cal
A}^{0}_{\chi_{i}\chi_{j}}$ are functions of ${\Lambda}$ and ${\rho}_{n}$. The
upper index 0 indicates that these coefficients are generally divergent for
${\overline{\Lambda}}\to\infty$.
$G$ is the Euler density and $G_{\mu\nu}$ the Einstein tensor:
$G=R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}-4R^{\alpha\beta}R_{\alpha\beta}+R^{2},\quad
G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}\gamma_{\mu\nu}R\;.$ (4.8)
Under Weyl rescalings
$\delta_{\sigma}\gamma_{\mu\nu}=-2\sigma\gamma_{\mu\nu}$, $G$ and $G_{\mu\nu}$
transform as
$\delta_{\sigma}G=4\sigma
G-8G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\sigma\;,\quad\delta_{\sigma}G_{\mu\nu}=2(\nabla_{\mu}\nabla_{\nu}\sigma-\gamma_{\mu\nu}\nabla^{2}\sigma)\;.$
(4.9)
Applying the local RGE (4.6) to $W^{0}$ expanded as in (4.7) leads to terms
$\sim\sigma$ and, after partial integrations and using
$\nabla_{\mu}G^{\mu\nu}=0$, to terms $\sim\partial_{\mu}\sigma$ (see [JO]).
One obtains an equation of the form
$\int\sqrt{\gamma}d^{4}x\,\left\\{\sigma X+\partial_{\mu}\sigma
Z^{\mu}\right\\}=0$ (4.10)
where $X$ and $Z^{\mu}$ have to vanish separately. Alternatively, following
[3] one can introduce the operators
$\Delta_{\sigma}^{\gamma}=\int\sqrt{\gamma}d^{4}x\left(-2\sigma\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}\right)\;,\quad\Delta_{\sigma}^{\beta}=\int\sqrt{\gamma}d^{4}x\left(\sigma\tilde{\beta}_{i}\frac{\delta}{\delta\chi_{i}}\right)$
(4.11)
and express eq. (4.6) in the form
$\left(\Delta_{\sigma}^{\gamma}-\Delta_{\sigma}^{\beta}\right)W^{0}=0\;.$
(4.12)
Since Weyl rescalings are abelian one must have
$\left[\Delta_{\sigma}^{\gamma}-\Delta_{\sigma}^{\beta},\Delta_{\sigma^{\prime}}^{\gamma}-\Delta_{\sigma^{\prime}}^{\beta}\right]=0\;.$
(4.13)
Acting with eq. (4.13) on $W^{0}$ leads to the same conditions as we will
obtain from eq. (4.10).
$X$ can be expanded in the same basis
$\\{G,\;G^{\mu\nu}\partial_{\mu}\chi_{i}\partial_{\nu}\chi_{j},\;\dots\\}$ as
$W^{0}$ in (4.7). The required vanishing of each of the corresponding
coefficients in $X$ requires
$\displaystyle\tilde{\beta}_{i}\frac{\delta}{\delta\chi_{i}}b^{0}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\tilde{\beta}_{k}\frac{\delta}{\delta\chi_{k}}{\cal
G}^{0}_{\chi_{i}\chi_{j}}+{\cal
G}^{0}_{\chi_{k}\chi_{j}}\frac{\delta\tilde{\beta}_{k}}{\delta\chi_{i}}+{\cal
G}^{0}_{\chi_{k}\chi_{i}}\frac{\delta\tilde{\beta}_{k}}{\delta\chi_{j}}$
$\displaystyle=$ $\displaystyle 0$ (4.14)
and similar equations for ${\cal E}^{0}_{\chi_{i}\chi_{j}}$, ${\cal
F}^{0}_{\chi_{i}\chi_{j}}$ and ${\cal A}^{0}_{\chi_{i}\chi_{j}}$. It is
straightforward to separate the derivatives with respect to $\lambda$ in these
equations which can then be interpreted as definitions of $\beta$ functions
which describe the dependence of the bare coefficients $b^{0}$ and ${\cal
G}^{0}_{ij}$ on the cutoff ${\Lambda}$:
$\displaystyle\beta_{b}\equiv{\Lambda}\frac{\delta
b^{0}}{\delta{\Lambda}}=\gamma_{n}\frac{\delta b^{0}}{\delta\rho_{n}}$
$\displaystyle\beta_{{\cal G}_{nm}}\equiv{\Lambda}\frac{\delta{\cal
G}^{0}_{\chi_{n}\chi_{m}}}{\delta{\Lambda}}=\gamma_{k}\frac{\delta{\cal
G}^{0}_{\chi_{n}\chi_{m}}}{\delta\rho_{k}}+{\cal
G}^{0}_{\chi_{k}\chi_{m}}\frac{\delta\gamma_{k}}{\delta\rho_{n}}+{\cal
G}^{0}_{\chi_{k}\chi_{n}}\frac{\delta\gamma_{k}}{\delta\rho_{m}}\;.$ (4.15)
More interesting in the following are the consistency conditions which follow
from the vanishing of the coefficients of the terms
$G^{\mu\nu}\partial_{\nu}\chi_{i}$ in $Z^{\mu}$ in eq. (4.10). First, the
action of
$\int\sqrt{\gamma}d^{4}y(-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}})$
on $\sqrt{\gamma}b^{0}G$ gives
$\int\sqrt{\gamma}d^{4}y\left(-2\sigma\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}\right)\sqrt{\gamma}b^{0}G=-8\sqrt{\gamma}b^{0}G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\sigma=8\sqrt{\gamma}G^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}b^{0}=8\sqrt{\gamma}G^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\chi_{i}\frac{\delta
b^{0}}{\delta\chi_{i}}$ (4.16)
where partial integration,
$-2\sigma\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}\sqrt{\gamma}=-4\sigma\sqrt{\gamma}$
and $\nabla_{\mu}G^{\mu\nu}=0$ have been used. Second, the action of
$-\int\sqrt{\gamma}d^{4}y(\sigma\tilde{\beta}_{i}\frac{\delta}{\delta\chi_{i}})$
on the derivatives of $\chi_{i}$ in $\frac{1}{2}{\cal
G}^{0}_{\chi_{i}\chi_{j}}\,G^{\mu\nu}\,\partial_{\mu}{\chi_{i}}\partial_{\nu}{\chi_{j}}$
gives (dropping uninteresting terms)
$-\sqrt{\gamma}G^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\chi_{i}{\cal
G}^{0}_{\chi_{i}\chi_{j}}\tilde{\beta}_{j}+\dots\;.$ (4.17)
These are the only terms proportional to
$\sqrt{\gamma}G^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\chi_{i}$ after
applying the local RGE (4.6) to $W^{0}$, hence the sum of the corresponding
coefficients must vanish:
$8\frac{\delta b^{0}}{\delta\chi_{i}}-{\cal
G}^{0}_{\chi_{i}\chi_{j}}\tilde{\beta}_{j}=0\;.$ (4.18)
This equation is of the form of an exact gradient flow for the $\beta$
functions $\tilde{\beta}_{i}\equiv\\{-1,\gamma_{n}\\}$, with ${\cal
G}^{0}_{\chi_{i}\chi_{j}}$ a metric in the space
$\chi_{i}\equiv\\{\lambda,\rho_{n}\\}$. Due to the introduction of
dimensionless $\chi_{i}$, all components of ${\cal G}^{0}_{\chi_{i}\chi_{j}}$
are dimensionless. However, this form of the gradient flow makes sense only
for a finite cutoff since both the metric ${\cal G}^{0}_{\chi_{i}\chi_{j}}$
and the potential $b^{0}$ will generally be divergent.
Let us consider the components $\chi_{i}=\rho_{n}$ of eq. (4.18),
$8\frac{\delta b^{0}}{\delta\rho_{n}}-{\cal
G}^{0}_{\rho_{n}\rho_{m}}\gamma_{m}+{\cal G}^{0}_{\rho_{n}\lambda}=0$ (4.19)
where we can replace in the arguments of $b^{0}$ and ${\cal G}^{0}$ all
couplings $\rho$ by constants $\bar{\rho}$.
In Appendix B (based on Appendix A) it is shown under which condition ${\cal
G}^{0}_{\rho_{n}\lambda}(\bar{\rho})$ ($={\cal
G}^{0}_{\lambda\rho_{n}}(\bar{\rho}$)) can be written as a gradient of a
function ${\cal B}^{0}(\bar{\rho})$. If this condition is satisfied one finds
an exact gradient flow for $\chi_{i}=\bar{\rho}_{n}$ of the form
$8\frac{\delta\hat{b}^{0}}{\delta\bar{\rho}_{n}}={\cal
G}^{0}_{\rho_{n}\rho_{m}}\gamma_{m}\;,\qquad\hat{b}^{0}={b}^{0}+\frac{1}{8}{\cal
B}^{0}(\bar{\rho})\;.$ (4.20)
We recall again that here we are considering the case of a quantum field
theory with finite cutoff. The dependence of the couplings $\rho_{n}$ on the
cutoff is given by anomalous dimensions, and the dependence of $b^{0}$ and
${\cal G}^{0}_{\rho_{n}\rho_{m}}$ on the cutoff is given by eqs. (4.15). In
the gradient flow for the anomalous dimensions in eq. (4.18) couplings to
terms involving derivatives of $\lambda(x)$ like ${\cal
G}^{0}_{g_{i}\lambda}(\bar{\rho})$ do play a role, but in eq. (4.20) such
couplings appear no longer. Still, calculations of diagrams with vertices
involving $\lambda(x)$ are required in order to compute ${\cal
B}^{0}(\bar{\rho})$ according to the rules given in Appendix B.
### 4.2 Gradient Flows for Case 2
Next we consider a renormalizable quantum field theory in $d=4$ dimensions
where the renormalized action $S(\gamma_{\mu\nu},g_{i},\Lambda,\mu)$ satisfies
the local RGE given in eq. (3.12). $g_{i}$ denote marginal or relevant
couplings, but again it will be useful to work with dimensionless fluctuating
fields $\chi_{i}$ only. Let us assume marginal couplings and masses $m_{i}$
only, and expand $m_{i}(x)$ around constant values $\overline{m}_{i}$:
$m_{i}(x)=\overline{m}_{i}e^{\nu_{i}(x)}\;.$ (4.21)
The $\beta$ functions for $\nu_{i}(x)$ are defined such that
$\beta_{i}\frac{\delta}{\delta g_{i}}$ in eq. (3.11) becomes
$(\gamma_{m_{i}}-1)\frac{\delta}{\delta\nu_{i}}$ with
$\gamma_{m_{i}}=\frac{\mu}{m_{i}}\frac{dm_{i}}{d\mu}\;.$ (4.22)
Again, using $\lambda$ from eq. (4.3) all local variables can be combined into
dimensionless $\chi_{i}$ and corresponding $\beta$ functions
$\tilde{\beta}_{i}$,
$\chi_{i}=\\{\lambda(x),\hat{g}_{i}(x)\\}=\\{\lambda(x),g_{i}(x),\nu_{i}(x)\\}\qquad\text{and}\qquad\tilde{\beta}_{i}=\\{-1,\hat{\beta}_{i}\\}=\\{-1,\beta_{i},\gamma_{m_{i}}-1\\}\;.$
(4.23)
The bare vacuum particion function is given by
$e^{-{W^{0}}(\gamma_{\mu\nu},\overline{\Lambda},\lambda,\hat{g}_{i})}=\frac{1}{\cal
N}\int{\cal
D}\varphi\,e^{-S(\varphi,\gamma_{\mu\nu},\Lambda,g_{i}^{0}(g_{i},\Lambda,\mu),m_{i}^{0}(m_{i},g_{i},\Lambda,\mu))}\;.$
(4.24)
As before we consider terms of fourth order in derivatives acting on
$\gamma_{\mu\nu}$, $\chi_{i}$, accordingly $W^{0}$ can be expanded as in eq.
(4.7). Eq. (3.12) implies the local RGE for $W^{0}$
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\frac{\delta}{\delta\lambda}-\beta_{i}\frac{\delta}{\delta
g_{i}}+(1-\gamma_{m_{i}})\frac{\delta}{\delta\nu_{i}}\right\\}W^{0}=0$ (4.25)
or, in terms of $\chi_{i}$ and $\tilde{\beta}_{i}$ defined in eq. (4.23),
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}-\tilde{\beta}_{i}\frac{\delta}{\delta\chi_{i}}\right\\}W^{0}=0\;.$
(4.26)
The counter terms implicit in $g_{i}^{0}(g_{i},\Lambda,\mu)$ and
$m_{i}^{0}(m_{i},g_{i},\Lambda,\mu)$ cancel subdivergences in the computation
of $W^{0}$; however, superficial divergences for $\overline{\Lambda}\to\infty$
remain. We have to add $W^{ct}$ which contains counter terms for the
coefficients $b^{0}$, ${\cal G}^{0}_{\chi_{i}\chi_{j}}$, ${\cal
E}^{0}_{\chi_{i}\chi_{j}}$, ${\cal F}^{0}_{\chi_{i}\chi_{j}}$ and ${\cal
A}^{0}_{\chi_{i}\chi_{j}}$ in $W^{0}$ all of which are dimensionless. These
counter terms $b^{ct}$, ${\cal G}^{ct}_{\chi_{i}\chi_{j}}$, ${\cal
E}^{ct}_{\chi_{i}\chi_{j}}$, ${\cal F}^{ct}_{\chi_{i}\chi_{j}}$ and ${\cal
A}^{ct}_{\chi_{i}\chi_{j}}$ depend on $\overline{\Lambda}$ such that the limit
$\overline{\Lambda}\to\infty$ exists for $W=W^{0}+W^{ct}$. One obtains
$\displaystyle
e^{-{W}(\gamma_{\mu\nu},\lambda,\hat{g}_{i})}=\left[\frac{1}{\cal N}\int{\cal
D}\varphi\,e^{-S(\varphi,\gamma_{\mu\nu},\Lambda,g_{i}^{0}(g_{i},\Lambda,\mu),m_{i}^{0}(m_{i},g_{i},\Lambda,\mu))+W^{ct}(\gamma_{\mu\nu},\overline{\Lambda},\lambda,\hat{g}_{i})}\right]_{\overline{\Lambda}\to\infty}\;,$
$\displaystyle W=\int\sqrt{\gamma}d^{4}x\,\left\\{b\,G+\frac{1}{2}{\cal
G}_{\chi_{i}\chi_{j}}\,G^{\mu\nu}\,\partial_{\mu}{\chi_{i}}\partial_{\nu}{\chi_{j}}+{\cal
E}_{\chi_{i}\chi_{j}}\,\partial^{\mu}R\,{\chi_{i}}\partial_{\mu}{\chi_{j}}+\frac{1}{2}\,{\cal
F}_{\chi_{i}\chi_{j}}\,R\,\partial_{\mu}{\chi_{i}}\partial^{\mu}{\chi_{j}}\right.$
$\displaystyle\left.+\frac{1}{2}{\cal
A}_{\chi_{i}\chi_{j}}\nabla^{2}{\chi_{i}}\nabla^{2}{\chi_{j}}+\dots\right\\}\;.$
(4.27)
The argument $\lambda$ in $W(\gamma_{\mu\nu},\lambda,\hat{g}_{i})$ indicates
that $W$ still contains derivatives of $\lambda$ in terms like
$\frac{1}{2}{\cal
G}_{\lambda\lambda}\,G^{\mu\nu}\,\partial_{\mu}{\lambda}\,\partial_{\nu}{\lambda}$.
Like all terms in $W^{0}$, ${\cal G}^{0}_{\lambda\lambda}$ has to be computed
before the limit $\overline{\Lambda}\to\infty$ is taken. Being dimensionless,
${\cal G}^{0}_{\lambda\lambda}$ can contain powers of logarithms or remain
finite for $\overline{\Lambda}\to\infty$; after adding ${\cal
G}^{ct}_{\lambda\lambda}$, ${\cal G}_{\lambda\lambda}={\cal
G}^{0}_{\lambda\lambda}+{\cal G}^{ct}_{\lambda\lambda}$ does not have to
vanish for $\overline{\Lambda}\to\infty$. This remains true for all terms
involving derivatives of $\lambda(x)$ in $W$ after renormalization.
$W^{ct}$ does not satisfy the local RGE (4.26). Instead $W^{ct}$ satisfies a
local RGE of the form
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}-\tilde{\beta}_{i}\frac{\delta}{\delta\chi_{i}}\right\\}W^{ct}=-\int\sqrt{\gamma}d^{d}x\,\sigma{\cal
R}_{\beta}$ (4.28)
with
${\cal R}_{\beta}=\beta^{b}G+\beta^{\cal
G}_{\chi_{i}\chi_{j}}G^{\mu\nu}\partial\chi_{i}\partial\chi_{j}+\dots\;.$
(4.29)
(Using recursion relations as in the case of the standard renormalization
theory it should be possible to show that the $\beta$ functions
$\beta^{b},\beta^{\cal G}_{\chi_{i}\chi_{j}}$ etc. are finite for
$\overline{\Lambda}\to\infty$.) Accordingly $W$ satisfies the local RGE
$\int\sqrt{\gamma}d^{d}x\,\sigma\left\\{-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}-\tilde{\beta}_{i}\frac{\delta}{\delta\chi_{i}}\right\\}W=\int\sqrt{\gamma}d^{d}x\,\sigma{\cal
R}_{\beta}\;.$ (4.30)
Hence, instead of the consistency condition eq. (4.10) one obtains
$\int\sqrt{\gamma}d^{4}x\,\left\\{\sigma X+\partial_{\mu}\sigma
Z^{\mu}\right\\}=\int\sqrt{\gamma}d^{d}x\,\sigma{\cal R}_{\beta}$ (4.31)
where $X$ and $Z$ have to be computed as before but in terms of $b$ and ${\cal
G}_{\chi_{i}\chi_{j}}$. For terms $\sim\sigma G$ one obtains now (using
$\frac{\delta b}{\delta\lambda}=0$)
$\beta^{b}=\hat{\beta}_{i}\frac{\delta b}{\delta\hat{g}_{i}}\;,$ (4.32)
for terms $\sim\sigma
G^{\mu\nu}\partial_{\mu}\hat{g}_{i}\partial_{\nu}\hat{g}_{j}$, $\sigma
G^{\mu\nu}\partial_{\mu}\hat{g}_{i}\partial_{\nu}\lambda$ and $\sigma
G^{\mu\nu}\partial_{\mu}\lambda\partial_{\nu}\lambda$ one gets using
$\frac{\delta{\cal G}_{\chi_{i}\chi_{j}}}{\delta\lambda}=0$ and
$\frac{\delta\hat{\beta}_{i}}{\delta\lambda}=0$
$\displaystyle\beta^{\cal G}_{\hat{g}_{i}\hat{g}_{j}}$ $\displaystyle=$
$\displaystyle\hat{\beta}_{k}\frac{\delta{\cal
G}_{\hat{g}_{i}\hat{g}_{j}}}{\delta\hat{g}_{k}}+{\cal
G}_{\hat{g}_{k}\hat{g}_{j}}\frac{\delta\hat{\beta}_{k}}{\delta\hat{g}_{i}}+{\cal
G}_{\hat{g}_{k}\hat{g}_{i}}\frac{\delta\hat{\beta}_{k}}{\delta\hat{g}_{j}}\;,$
$\displaystyle\beta^{\cal G}_{\hat{g}_{i}\lambda}$ $\displaystyle=$
$\displaystyle\hat{\beta}_{k}\frac{\delta{\cal
G}_{\hat{g}_{i}\lambda}}{\delta\hat{g}_{k}}+{\cal
G}_{\hat{g}_{k}\lambda}\frac{\delta\hat{\beta}_{k}}{\delta\hat{g}_{i}}\;,$
$\displaystyle\beta^{\cal G}_{\lambda\lambda}$ $\displaystyle=$
$\displaystyle\hat{\beta}_{k}\frac{\delta{\cal
G}_{\lambda\lambda}}{\delta\hat{g}_{k}}\;.$ (4.33)
For terms proportional to
$\sqrt{\gamma}G^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\chi_{i}$ in eq.
(4.31) one obtains no contribution from the right hand side. The terms with
$\chi_{i}=\lambda,\hat{g}_{i}$ give, respectively,
$8\frac{\delta b}{\delta\lambda}=0={\cal
G}_{\lambda\chi_{j}}\tilde{\beta}_{j}={\cal
G}_{\lambda\hat{g}_{j}}\hat{\beta}_{j}-{\cal G}_{\lambda\lambda}\;,\qquad
8\frac{\delta b}{\delta\hat{g}_{i}}={\cal
G}_{\hat{g}_{i}\chi_{j}}\tilde{\beta}_{j}={\cal
G}_{\hat{g}_{i}\hat{g}_{j}}\hat{\beta}_{j}-{\cal G}_{\hat{g}_{i}\lambda}\;.$
(4.34)
Again, if the condition discussed in Appendix B under which ${\cal
G}_{\hat{g}_{i}\lambda}(\hat{g})$ can be written as a gradient of a function
${\cal B}(\hat{g})$ is satisfied one finds an exact gradient flow of the form
$8\frac{\delta{\hat{b}}}{\delta\hat{g}_{i}}={\cal
G}_{\hat{g}_{i}\hat{g}_{j}}\hat{\beta}_{j}\;,\qquad\hat{b}={b}+\frac{1}{8}{\cal
B}(\hat{g})\;.$ (4.35)
If one applies $\hat{\beta}_{k}\frac{\delta}{\delta\hat{g}_{k}}$ to each term
in eq. (4.34) and uses eqs. (4.32) and (4.33) one obtains after some algebra
$8\frac{\delta\beta^{b}}{\delta\hat{g}_{i}}=\beta^{\cal
G}_{\hat{g}_{i}g_{j}}\hat{\beta}_{j}-\beta^{\cal G}_{\hat{g}_{i}\lambda}\;.$
(4.36)
This version of the Weyl consistency conditions is similar to eq. (3.17a) in
ref. [2]. Under the condition discussed in Appendix B, $\beta^{\cal
G}_{\hat{g}_{i}\lambda}$ can also be written as a gradient of a function
$\beta^{\cal B}(\hat{g})=\frac{\delta}{\delta\hat{g}_{k}}(\hat{\beta}_{k}{\cal
B}(\hat{g}))$ and one can find an exact gradient flow of the form
$8\frac{\delta{\hat{\beta}^{b}}}{\delta\hat{g}_{i}}=\beta^{\cal
G}_{\hat{g}_{i}\hat{g}_{j}}\tilde{\beta}_{j}\;,\qquad\hat{\beta}^{b}=\beta^{b}+\frac{1}{8}\beta^{\cal
B}(\hat{g})\;.$ (4.37)
To sum up this section, Weyl consistency conditions implying gradient flows
have been derived for the following scenarios:
1) For anomalous dimensions $\gamma_{i}$ for bare couplings $\rho_{i}$
associated to operators of any canonical dimensions, in terms of bare
coefficients $b^{0}$ and ${\cal G}^{0}_{\chi_{i}\chi_{j}}$ in the form of eq.
(4.18) in the space $\chi_{i}=\\{\lambda,\rho_{i}\\}$. Under the condition
discussed in Appendix B, one obtains a gradient flow in the form of eq. (4.20)
in the space $\rho_{i}$ without $\lambda$.
2) For $\beta$ functions and anomalous dimensions for physical couplings and
masses in renormalizable theories, approximate gradient flows in terms of
renormalized coefficients $b$ and ${\cal G}_{\chi_{i}\chi_{j}}$
($\chi_{i}=\\{\lambda,\hat{g}_{i}\\}$) and their $\beta$ functions in the form
of eqs. (4.34) and (4.36). Under the condition discussed in Appendix B, exact
gradient flows in the space $\hat{g}_{i}$ are of the form of eqs. (4.35) and
(4.37).
## 5 A Single Massive Scalar
As a first application of the formalism (more will follow in a separate paper)
we consider a single scalar $\varphi$ in $d=4$ dimensions with $g\varphi^{4}$
interaction and mass $m$. Given the Wilsonian cutoff $\Lambda$, quadratic UV
divergences have to be cancelled by counter terms in $m_{0}^{2}$. In order to
get a first idea about the formalism including mass terms we focus first on
Weyl consistency conditions to lowest non-trivial order $\sim g^{1}$ for the
bare couplings $b^{0}$ and ${\cal G}^{0}_{\chi_{i}\chi_{j}}$ (where
$\chi_{i}=\lambda,\nu$) in the form of eq. (4.18).
For the kinetic part of the action we use eq. (2.8) supplemented by a mass
term,
$S_{k}=\frac{1}{2}\int\sqrt{\gamma}d^{4}x\,\left(\varphi(-\nabla^{2}+\frac{1}{6}R)\,F(D_{\Lambda})\varphi+{m_{0}}^{2}\varphi^{2}\right)\,,\qquad
F(D_{\Lambda})=e^{-D_{\Lambda}}\,,$ (5.1)
and the interaction is
$S_{int}=\int\sqrt{\gamma}d^{4}x\,\frac{g^{0}}{4!}{\varphi}^{4}$ (5.2)
where $g^{0}$ is the bare coupling. To ${\cal O}(g^{1})$ eqs. (4.18) read
explicitely
$\displaystyle 8\frac{\delta b^{0}}{\delta\nu}$ $\displaystyle=$
$\displaystyle\left(\gamma_{m}-1\right){\cal G}^{0}_{\nu\nu}-{\cal
G}^{0}_{\nu\lambda}\;,$ $\displaystyle 8\frac{\delta b^{0}}{\delta\lambda}$
$\displaystyle=$ $\displaystyle\left(\gamma_{m}-1\right){\cal
G}^{0}_{\nu\lambda}-{\cal G}^{0}_{\lambda\lambda}\;.$ (5.3)
The calculation of ${\cal G}^{0}_{\chi_{i}\chi_{j}}$ proceeds via weak field
expansions in $\gamma_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ to ${\cal
O}(h_{\mu\nu})$ and in $\chi_{i}$ to ${\cal O}(\chi_{i}(x)^{2})$, see Appendix
A, hence we need the vertices to the appropriate order given in Appendix C.
The required counter terms in $m_{0}^{2}$ to ${\cal O}(g^{1})$ are well known,
but here we have to pay attention to the local nature of $m(x),\Lambda(x)$ in
the form of eqs. (4.3) and (4.21). To this first order in $g$ we have to
${\cal O}(\chi_{i}(x)^{2})$
$\displaystyle m_{0}^{2}$ $\displaystyle=$ $\displaystyle
m^{2}-\frac{g}{32\pi^{2}}\left(\Lambda^{2}-m^{2}\log\left(\Lambda^{2}/\mu^{2}\right)\right)$
(5.4) $\displaystyle\simeq$ $\displaystyle\overline{m}^{2}(1+2\nu+2\nu^{2})$
$\displaystyle-\frac{g}{32\pi^{2}}\left(\overline{\Lambda}^{2}(1+2\lambda+2\lambda^{2})-\overline{m}^{2}(1+2\nu+2\nu^{2})\log\left(\overline{\Lambda}^{2}/\mu^{2}\right)-\overline{m}^{2}(2\lambda+4\nu\lambda)\right)$
and
$\gamma_{m}=\frac{g}{32\pi^{2}}\;.$ (5.5)
Whereas the vertices in $S_{k}$ involving $\lambda,\nu$ from $m_{0}^{2}$ in
eq. (5.4) are momentum independent, relatively complicated momentum dependent
vertices follow from the term
$\sqrt{\gamma}\varphi(-\nabla^{2}+\frac{1}{6}R)\,e^{-D_{\Lambda}}\varphi$
which has to be expanded to ${\cal O}(h_{\mu\nu}(x)\lambda(x)^{2})$. These
vertices are derived in Appendix C.
Diagrams contributing to ${\cal G}^{0}_{\chi_{i}\chi_{j}}$ to ${\cal
O}(g^{1})$ have the topology “$\infty$” with vertices involving the background
fields $h_{\mu\nu},\lambda,\nu$ attached to the loops. The free propagator
$P(q)$ includes the UV cutoff,
$P(q)=1/(q^{2}e^{q^{2}/\overline{\Lambda}^{2}}+m^{2})\;.$ (5.6)
The loop integrals cannot be expressed in terms of standard functions; we will
give the results in an expansion in
$u\equiv\overline{m}^{2}/\overline{\Lambda}^{2}$.
For the free theory we find using the vertices in Appendix C and the procedure
described in Appendix A
${\cal G}_{\nu\nu}^{0}=\frac{1}{16\pi^{2}}\frac{1}{45}+{\cal O}(u)$ (5.7)
whereas ${\cal G}_{\nu\lambda}^{0}$ is suppressed by $u$. The coefficients of
terms suppressed by $u$ depend on the cutoff function $F(D_{\Lambda})$ in
$S_{k}$. With $\beta_{\nu}=-1$ to lowest order it follows from eq. (5.3)
$\frac{\delta b^{0}}{\delta\nu}=-\frac{1}{16\pi^{2}}\frac{1}{360}+{\cal
O}(u)\;.$ (5.8)
$b^{0}$ is divergent, and on dimensional grounds $b^{0}$ has to depend on
$\Lambda/m$. Accordingly
$b^{0}=\frac{1}{16\pi^{2}}\frac{1}{360}(\lambda-\nu)=\frac{1}{16\pi^{2}}\frac{1}{360}\left(\log\left(\Lambda/m\right)+\;\text{const.}\right)\;,$
(5.9)
hence $b^{0}$ requires a counter term $b^{ct}$ such that
$b=b^{0}+b^{ct}=\frac{1}{16\pi^{2}}\frac{1}{360}\left(\log\left(\mu/m\right)+\;\text{const.}\right)\;.$
(5.10)
It follows
$\beta_{b}=\mu\frac{db}{d\mu}=\frac{1}{16\pi^{2}}\frac{1}{360}$ (5.11)
which coincides with the known value for a single scalar field.
To ${\cal O}(g^{1})$ potential quadratic divergences are cancelled by counter
terms. Somewhat astonishingly, finite contributions of ${\cal O}((u)^{0})$
cancel as well. The subleading contributions $\Delta{\cal
G}^{0}_{\chi_{i}\chi_{j}}$ of ${\cal O}(g^{1})$ are
$\displaystyle\Delta{\cal G}_{\nu\nu}^{0}$ $\displaystyle=$
$\displaystyle\frac{g}{(16\pi^{2})^{2}}\frac{u}{45}\log\left(\frac{m^{2}}{\mu^{2}}\right)\;,$
$\displaystyle\Delta{\cal G}_{\nu\lambda}^{0}$ $\displaystyle=$
$\displaystyle\frac{g}{(16\pi^{2})^{2}}\frac{2u}{15}\log\left(\frac{m^{2}}{\mu^{2}}\right)\;,$
$\displaystyle\Delta{\cal G}_{\lambda\lambda}^{0}$ $\displaystyle=$
$\displaystyle-\frac{g}{(16\pi^{2})^{2}}\frac{13u}{45}\log\left(\frac{m^{2}}{\mu^{2}}\right)$
(5.12)
and vanish for $\overline{\Lambda}\to\infty$, hence counter terms $\Delta{\cal
G}^{ct}_{\chi_{i}\chi_{j}}$ are not required. (The logarithms
$\log\left({m^{2}}/{\mu^{2}}\right)$ originate from the counter terms in
$m_{0}^{2}$). The last term has been deduced from eqs. (5.3) using again,
since $b^{0}$ is dimensionless, ${\delta b^{0}}/{\delta\nu}=-{\delta
b^{0}}/{\delta\lambda}$. Given the leading contribution to ${\cal
G}^{0}_{\nu\nu}$ in eq. (5.7) the leading metric in the subspace $\nu,\lambda$
is always positive.
The subleading contributions show no sign for a diagonal metric ${\cal
G}^{0}_{\chi_{i}\chi_{j}}$ in the subspace $\\{\chi_{i}=m,$ $\Lambda\\}$, but
these are scheme dependent, i.e. dependent on the form of the cutoff function
$F(D_{\Lambda})$ in $S_{k}$. Since the Weyl consistency conditions are
covariant under redefinitions in the space $\chi_{i}$ (provided that $\beta$
functions are appropriately redefined), diagonalization of the metric is
straightforward. The cancellation of finite contributions of ${\cal
O}((u)^{0})$ to $\Delta{\cal G}^{0}_{\chi_{i}\chi_{j}}$ may be an artefact of
the low order in the coupling considered here.
## 6 Conclusions and Outlook
In the present article we have derived Weyl consistency conditions in the
framework of a Wilsonian cutoff implemented in the kinetic term for scalar
fields. The crucial tool is a local cutoff $\Lambda(x)$ which transforms under
Weyl rescalings and allows for Weyl invariant kinetic terms. This formalism
provides an alternative to Weyl consistency conditions derived using
dimensional regularization by [JO].
It allows to define the running of bare couplings with the UV cutoff via
anomalous dimensions, a concept which makes no sense in dimensional
regularization. Also it allows to treat couplings of any canonical dimension
on the same footing, see the case 1 above. On the other hand the local field
$\lambda(x)$ describing the $x$-dependence of the cutoff contributes a priori
to the Weyl consistency conditions in the form of an additional local
coupling. In this enlarged space of local fields and corresponding anomalous
dimensions Weyl consistency conditions assume the form of an exact gradient
flow in eq. (4.18) in Subsection 4.1. (For attempts to generalize the AdS/CFT
correspondence towards general QFTs (for which an exact gradient flow is
essential) a local cutoff could be related to a component in the $(d+1)$
dimensional metric corresponding to the extra dimension.)
Within the reduced space corresponding to couplings $\hat{g}$ only, an extra
term involving ${\cal G}_{g_{i}\lambda}$ is still present in the Weyl
consistency conditions which spoils the gradient flow property unless this
term itself can be expressed as a gradient. This extra term depends on
contributions from diagrams $\sim\partial_{\mu}\lambda(x)$ to the vacuum
partition function, and we discuss in Appendix B under which condition ${\cal
G}_{g_{i}\lambda}$ can be expressed as a gradient. It is easy to see that this
condition is satisfied in lowest nontrivial orders in perturbation theory, but
general criteria for its satisfaction remain to be investigated.
In Subsection 4.2 we applied the formalism to renormalizable theories for
which the limit $\overline{\Lambda}\to\infty$ can be taken. Still,
contributions $\sim\partial_{\mu}\lambda(x)$ to the vacuum partition function
remain and, as before, corresponding couplings remain present in the Weyl
consistency conditions. Again, these become a gradient flow only if the
condition discussed in Appendix B is satisfied. Finally we expressed the Weyl
consistency conditions in terms of $\beta$ functions for the coefficients in
the vacuum partition function; in this form they are very similar to the ones
obtained by [JO] except that marginal and relevant couplings appear together.
We have also introduced and verified a method to compute the “metric” ${\cal
G}_{\chi_{i}\chi_{j}}$ in momentum space in a weak field expansion. The
explicit calculations are straightforward but involved due to the cutoff
propagators and the vertices originating from the local cutoff in the kinetic
term. We have computed ${\cal G}_{\chi_{i}\chi_{j}}$ to lowest non-trivial
order in the coupling of a $\varphi^{4}$ theory; further studies of higher
orders, of other models and of other cutoff functions leading to potential
simplifications are desirable in the future.
A way to simplify higher loop calculations would be to add the cutoff function
(2.7) also to the mass term. This would generate additional vertices involving
$\lambda(x)$, but the free cutoff propagator (5.6) simplifies to
$P(q)=e^{-q^{2}/\overline{\Lambda}^{2}}/(q^{2}+m^{2})=\int_{1/\overline{\Lambda}^{2}}^{\infty}d\alpha\,e^{-\alpha
q^{2}}\;.$ (6.1)
Such a Schwinger-like representation allows to execute multi-loop momentum
integrals, and one is left with truncated (overlapping) $\alpha$ integrals.
## Appendix
## Appendix A How to compute ${\cal G}_{\chi_{i}\chi_{j}}$
The aim of this section is to describe how ${\cal G}_{\chi_{i}\chi_{j}}$, the
coefficients of $G^{\mu\nu}\,\partial_{\mu}{\chi_{i}}\partial_{\nu}{\chi_{j}}$
in the vacuum partition function $W$, can be computed in momentum space. (The
following steps hold both for ${\cal G}^{0}_{\chi_{i}\chi_{j}}$ in terms of
$W^{0}$, and ${\cal G}_{\chi_{i}\chi_{j}}$ in terms of $W$.) We assume that
$W$ is computed diagrammatically as function of background fields
$f_{i}(p_{i})$ including the local couplings and the local cutoff, and the
metric $\gamma_{\mu\nu}(q)$.
The first step is an expansion of the metric around flat space,
$\gamma_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, and to consider the terms linear
in $h_{\mu\nu}$:
$W=\int\frac{d^{4}q}{(2\pi)^{4}}h_{\mu\nu}(q)\,T^{\mu\nu}(-q,f_{i})+\dots$
(A.1)
Next, expand $f_{i}(p_{i})$ around constants in space-time,
$f_{i}(p_{i})=\bar{f}_{i}(2\pi)^{4}\delta^{4}(p_{i})+\chi_{i}(p_{i})$, and
keep only terms of second order in $\chi_{i}(p_{i})$ which correspond to the
(Fourier transforms of) the local fields $\chi_{i}$ in section 4. Denoting the
momenta of two fluctuations $\chi_{i}(p_{i})$ by $p$ and $q-p$ using
$\sum_{i}p_{i}=q$, $T^{\mu\nu}(-q)$ is of the form
$T^{\mu\nu}(-q,f_{i})=\int\frac{d^{4}p}{(2\pi)^{4}}\,\chi_{i}(p)\chi_{j}(q-p)\,t^{\mu\nu}_{ij}(p,q)\;.$
(A.2)
We are interested in the terms quartic in derivatives, i.e. quartic in the
momenta $p,q$ in $t^{\mu\nu}_{ij}(p,q)$. These terms can be decomposed into
$t^{\mu\nu}_{ij}(p,q)=K_{1ij}p^{\mu}p^{\nu}+\frac{1}{2}K_{2ij}(p^{\mu}q^{\nu}+q^{\mu}p^{\nu})+K_{3ij}q^{\mu}q^{\nu}+K_{4ij}\eta^{\mu\nu}$
(A.3)
where $K_{1ij},K_{2ij},K_{3ij}$ are polynomials of first order in the Lorentz
invariants $p^{2}$, $pq$, $q^{2}$, and $K_{4ij}$ is of second order in the
same Lorentz invariants. Next we have to expand the terms in $W^{0}$ or $W$ in
eqs. (4.7)/(4.27) to the same order in $h_{\mu\nu}$ and $p,q$. The first term
$\sim G$ drops out since of higher order in $h_{\mu\nu}$, and further terms
not written explicitely in $W^{0}$/$W$ are of higher order in
$\chi_{i}(p_{i})$. The terms quartic in the momenta $p_{i}$ proportional to
$h^{\mu\nu}(q)\chi_{i}(p)\chi_{j}(q-p)$ from the remaining four terms in eq.
(4.27) are of the following form, once expressed in momentum space:
$\displaystyle
G^{\rho\sigma}\,\partial_{\rho}{\chi_{i}}\partial_{\sigma}\,{\chi_{j}}:\frac{1}{2}$
$\displaystyle\left(q^{2}p_{\mu}p_{\nu}-pq(p_{\mu}q_{\nu}+q_{\mu}p_{\nu})+p^{2}q_{\mu}q_{\nu}+\eta_{\mu\nu}((pq)^{2}-p^{2}q^{2})\right)$
$\displaystyle\partial^{\rho}R\,{\chi_{i}}\partial_{\rho}{\chi_{j}}:\frac{1}{2}$
$\displaystyle(-q_{\mu}q_{\nu}q^{2}+\eta_{\mu\nu}q^{2}q^{2})$ $\displaystyle
R\,\partial_{\rho}{\chi_{i}}\partial^{\rho}{\chi_{j}}:-$ $\displaystyle
q_{\mu}q_{\nu}(p^{2}+pq)+\eta_{\mu\nu}q^{2}(p^{2}+pq)$
$\displaystyle\sqrt{\gamma}\nabla^{2}{\chi_{i}}\nabla^{2}{\chi_{j}}:-$
$\displaystyle(p_{\mu}p_{\nu}+p_{\mu}q_{\nu})(2p^{2}+2pq+q^{2})+\eta_{\mu\nu}(\frac{1}{2}p^{2}p^{2}+p^{2}pq+pq^{2}+\frac{1}{2}pqq^{2})$
(A.4)
It follows that ${\cal G}_{\chi_{i}\chi_{j}},\,{\cal
E}_{\chi_{i}\chi_{j}},\,{\cal F}_{\chi_{i}\chi_{j}},\,{\cal
A}_{\chi_{i}\chi_{j}}$ can be obtained from
$t^{\mu\nu}_{ij}(p,q,\bar{\chi}_{k})$, decomposed as in eq. (A.3), as
$\displaystyle{\cal G}_{\chi_{i}\chi_{j}}$ $\displaystyle=$ $\displaystyle
2\frac{d}{dq^{2}}(K_{1ij}-K_{2ij})$ $\displaystyle{\cal A}_{\chi_{i}\chi_{j}}$
$\displaystyle=$ $\displaystyle-\frac{d}{dq^{2}}K_{2ij}$ $\displaystyle{\cal
F}_{\chi_{i}\chi_{j}}$ $\displaystyle=$ $\displaystyle-\frac{d}{d(pq)}K_{3ij}$
$\displaystyle{\cal E}_{\chi_{i}\chi_{j}}$ $\displaystyle=$
$\displaystyle\left(\frac{d}{dq^{2}}\right)^{2}K_{4ij}\;.$ (A.5)
For our purposes the first relation for ${\cal G}_{\chi_{i}\chi_{j}}$ is all
we need.
We note that this approach in momentum space is general, independent from the
UV regularization used to compute the three point functions
$\left<h_{\mu\nu}(q)\chi_{i}(p)\chi_{j}(q-p)\right>$. We have verified it for
a scalar theory with $g(x)\varphi^{4}$ interaction in $d=4-\varepsilon$
dimensions, where the three point functions
$\left<h_{\mu\nu}(q)g(p)g(q-p)\right>$ are calculable to three loop order. We
found ${\cal G}_{gg}$ in agreement with eq. (6.30) in [2].
## Appendix B ${\cal G}_{\lambda g_{i}}$ as a Gradient
The aim of this section is to show under which condition ${\cal G}_{\lambda
g_{i}}={\cal G}_{g_{i}\lambda}$ can be written as a gradient. The reasoning
below applies to ${\cal G}^{0}_{\lambda\rho_{n}}$ in subsection 4.1, to ${\cal
G}^{0}_{\lambda g_{i}}$, to the counterterm ${\cal G}^{ct}_{\lambda g_{i}}$
and hence to the renormalized ${\cal G}_{\lambda g_{i}}$ in subsection 4.2.
For simplicity we use the notation ${\cal G}_{\lambda g_{i}}$ for all cases.
To start with, in momentum space $W$ depends on
$\chi_{i}(p_{i})\equiv\left\\{\Lambda(p_{\lambda}),g_{i}(p_{i})\right\\}$. We
assume $N$ distinct couplings $g_{i}$ ($i=1\dots N$). The momentum dependent
fields have to be expanded around constants in space-time,
$\Lambda(p_{\lambda})=\overline{\Lambda}(2\pi)^{4}\delta^{4}(p_{\lambda})+\widetilde{\lambda}(p_{\lambda})\,,\qquad
g_{i}(p_{i})=\bar{g}_{i}(2\pi)^{4}\delta^{4}(p_{i})+\tilde{g}_{i}(p_{i})\,.$
(B.1)
Following Appendix A, $W$ contains ${\cal G}_{\lambda g_{i}}$ in terms of the
form
$W=\sum_{i=1}^{N}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}p_{\lambda}}{(2\pi)^{4}}\frac{d^{4}p_{i}}{(2\pi)^{4}}\delta^{4}(q+p_{\lambda}+p_{i})\,h_{\mu\nu}(q)\,\widetilde{\lambda}(p_{\lambda})\,\tilde{g}_{i}(p_{i})\,q^{2}\,p^{\mu}_{\lambda}\,p_{i}^{\nu}\,{\cal
G}_{\lambda g_{i}}(\bar{g})+\dots\;.$ (B.2)
Accordingly ${\cal G}_{\lambda g_{i}}$ can be computed in terms of $W$ as
follows: First, expand $W$ to linear order in $h_{\mu\nu}(q)$,
$\widetilde{\lambda}(p_{\lambda})$ and to ${\cal O}(q^{2})$ and ${\cal
O}(p^{\mu}_{\lambda})$ in the corresponding momenta:
$W=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}p_{\lambda}}{(2\pi)^{4}}\,h_{\mu\nu}(q)\,\widetilde{\lambda}(p_{\lambda})\,q^{2}\,p^{\mu}_{\lambda}\,T^{\nu}(-q,g_{i})+\dots\;.$
(B.3)
Here the local couplings $g_{i}$ in $T^{\nu}(-q,g_{i})$ are not yet expanded
around constants in space-time as in eq. (B.1). We assume that
$T^{\nu}(-q,g_{i})$ can be written as a series in $n_{i}$ powers of
$g_{i}(p_{i,j_{i}})$, $j_{i}=1\dots n_{i}$:
$T^{\nu}(-q,g_{i})=\int{\cal D}p\,\sum_{n_{1}\dots
n_{N}}\,\sum_{k}\,p_{k}^{\nu}\,\delta^{4}(q+p+p_{k})\,t_{n_{1}\dots
n_{N},k}\,\prod_{i=1}^{N}g_{i}^{n_{i}}(p_{i,j_{i}})\;.$ (B.4)
In eq. (B.4), $\int{\cal D}p$ denotes the integrals over all
$n_{1}\times$…$\times n_{N}$ momenta, the arguments of the couplings
$g_{i}^{n_{i}}(p_{i,j_{i}})\equiv g_{i}(p_{i,1})\times$…$\times
g_{i}(p_{i,n_{i}})$. The sum over $k$ runs over all possible choices for
$p_{k}$ among these momenta. In general, the coefficients $t_{n_{1}\dots
n_{N},k}$ are different for momenta $p_{k}$ corresponding to different
couplings $g_{i}$. The condition for ${\cal G}_{\lambda g_{i}}$ as a gradient
is the following:
After summation of all diagrams which contribute to $t_{n_{1}\dots n_{N},k}$
to a given order, $t_{n_{1}\dots n_{N},k}$ are symmetric under the exchange of
momenta $p_{k}$ originating from different vertices corresponding to the same
kind of coupling $g_{i}$.
This condition is satisfied if, before the expansion of $T^{\nu}(-q,g_{i})$ to
${\cal O}(p_{k}^{\nu})$ in eq. (B.4), $T^{\nu}(-q,g_{i})$ is symmetric under
the exchange of momenta $p_{k}$ originating from different vertices
corresponding to the same kind of coupling $g_{i}$ which holds at least in low
orders in perturbation theory, but counter examples may exist.
Next one has to expand all couplings around constants $\bar{g}_{i}$ in space-
time. Only the terms linear in the fluctuation $\tilde{g}_{i}(p_{i})$
contribute to eq. (B.2). Inserting the expansion (B.1) for $g_{i}(p_{i})$ into
eq. (B.4), and under the above condition, one obtains $n_{1}$ contributions
$\sim\tilde{g}_{1}(p_{1},j_{1})$ from $g_{1}^{n_{1}}(p_{1,j_{1}})$ with
$j_{1}=1\dots n_{1}$, and $n_{2}$ contributions
$\sim\tilde{g}_{2}(p_{2},j_{2})$ from $g_{2}^{n_{2}}(p_{2,j_{2}})$,
$j_{2}=1\dots n_{2}$ etc..
Consider the contributions involving $\tilde{g}_{1}$. Under the above
condition, all $n_{1}$ contributions
$\sim\tilde{g}_{1}(p_{1,j_{1}}),\;j_{1}=1\dots n_{1},$ have the same
coefficients $t_{n_{1}\dots n_{N},k}$. The momenta $p_{k}^{\nu}$ in eq. (B.4)
correspond to $p_{1,j_{1}}$ since all other momenta vanish, and all momenta
$p_{1,j_{1}}$ can be denoted by $p_{1}$.
The same features hold for the $n_{2}$ contributions
$\sim\tilde{g}_{2}(p_{2},j_{2})$ etc.. Finally $T^{\nu}(-q,g_{i})$ becomes
$\displaystyle T^{\nu}(-q,g_{i})$
$\displaystyle=\sum_{i=1}^{N}\,\int\frac{d^{4}p_{i}}{(2\pi)^{4}}\,p_{i}^{\nu}\,\delta^{4}(q+p+p_{i})\,\tilde{g}_{i}(p_{i})\,\sum_{n_{1}\dots
n_{N}}t_{n_{1}\dots n_{N},i}\,n_{i}\,\bar{g}_{i}^{n_{i}-1}\prod_{j=1,j\neq
i}^{N}\bar{g}_{j}^{n_{j}}$
$\displaystyle=\sum_{i=1}^{N}\,\int\frac{d^{4}p_{i}}{(2\pi)^{4}}\,p_{i}^{\nu}\,\delta^{4}(q+p+p_{i})\,\tilde{g}_{i}(p_{i})\,\sum_{n_{1}\dots
n_{N}}t_{n_{1}\dots
n_{N},i}\,\frac{d}{d\bar{g}_{i}}\,\prod_{j=1}^{N}\bar{g}_{j}^{n_{j}}$
$\displaystyle=\sum_{i=1}^{N}\,\int\frac{d^{4}p_{i}}{(2\pi)^{4}}\,p_{i}^{\nu}\,\delta^{4}(q+p+p_{i})\,\tilde{g}_{i}(p_{i})\,\frac{d}{d\bar{g}_{i}}{\cal
B}(\bar{g})$ (B.5)
where $t_{n_{1}\dots n_{N},i}$ denote the coefficients corresponding to
couplings $g_{i}$, and
${\cal B}(\bar{g})=\sum_{n_{1}\dots n_{N}}t_{n_{1}\dots
n_{N},i}\,\,\prod_{j=1}^{N}\bar{g}_{j}^{n_{j}}\;.$ (B.6)
It remains to replace eq. (B.5) for $T^{\nu}(-q,g_{i})$ into eq. (B.3) which
becomes
$W=\sum_{i=1}^{N}\,\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}p_{\lambda}}{(2\pi)^{4}}\,\frac{d^{4}p_{i}}{(2\pi)^{4}}\,\delta^{4}(q+p+p_{i})\,h_{\mu\nu}(q)\,\widetilde{\lambda}(p_{\lambda})\,\tilde{g}_{i}(p_{i})\,q^{2}\,p^{\mu}_{\lambda}\,p_{i}^{\nu}\frac{d}{d\bar{g}_{i}}\,{\cal
B}(\bar{g})+\dots\;.$ (B.7)
Comparing eq. (B.7) to eq. (B.2) one obtains
${\cal G}_{\lambda g_{i}}(\bar{g})=\frac{d}{d\bar{g}_{i}}{\cal B}(\bar{g})\;,$
(B.8)
i.e. ${\cal G}_{\lambda g_{i}}(\bar{g})$ is the gradient of a function ${\cal
B}(\bar{g})$ defined by
$W=\sum_{i=1}^{N}\,\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}p_{\lambda}}{(2\pi)^{4}}\,\frac{d^{4}p_{i}}{(2\pi)^{4}}\,\delta^{4}(q+p+p_{i})\,h_{\mu\nu}(q)\,\widetilde{\lambda}(p_{\lambda})\,\tilde{g}_{i}(p_{i})\,q^{2}\,p^{\mu}_{\lambda}\,p_{i}^{\nu}\,{\cal
B}(\bar{g})+\dots\;.$ (B.9)
The origin of this property is the expansion of $g_{i}(p_{i})$ around
constants $\bar{g_{i}}$, and the expansion of $W$ to first order in
$\tilde{g}_{i}(p_{i})$ which converts $W$ as function of $g_{i}(p_{i})$ into
derivatives with respect to $\bar{g_{i}}$.
The above reasoning applies also to ${\cal G}^{0}_{\lambda
g_{i}}(\overline{\Lambda})$ and ${\cal G}^{ct}_{\lambda
g_{i}}(\overline{\Lambda})$. After renormalization eq. (B.8) holds for ${\cal
G}_{\lambda g_{i}}(\bar{g})$=$\left[{\cal G}^{0}_{\lambda
g_{i}}(\bar{g},\overline{\Lambda})-{\cal G}^{ct}_{\lambda
g_{i}}(\bar{g},\overline{\Lambda})\right]_{\overline{\Lambda}\to\infty}$ and
${\cal B}(\bar{g})$=$\left[{\cal B}^{0}(\bar{g},\overline{\Lambda})-{\cal
B}^{ct}(\bar{g},\overline{\Lambda})\right]_{\overline{\Lambda}\to\infty}$.
## Appendix C Vertices from $S_{k}$
For the calculation of ${\cal G}_{\chi_{i}\chi_{j}}$,
$\chi_{i}/\chi_{j}=\lambda$, one needs the vertices from $S_{k}$ quadratic in
$\varphi$, up to linear order in $h_{\mu\nu}$ and up to quadratic order in
$\lambda$ in momentum space, for our choice $F(D_{\Lambda})=e^{-D_{\Lambda}}$
(recall $D_{\Lambda}=\Lambda^{-2}(\nabla^{2}-\frac{1}{6}R)$). It is convenient
to split the terms in $S_{k}$ as
$\sqrt{\gamma}\varphi_{l}\left(-\nabla^{2}+\frac{1}{6}R\right)e^{-D_{\Lambda}}\varphi_{r}\equiv\varphi_{l}\,(-\square
e^{-u\square})\,\varphi_{r}+\varphi_{l}\left(V_{l}e^{-D_{\Lambda}}+\overleftarrow{\square}V_{r}\right)\varphi_{r}$
(C.1)
where
$u=1/\overline{\Lambda}^{2}\,,\qquad
V_{l}=\sqrt{\gamma}\left(-\overleftarrow{\nabla}^{2}+\frac{1}{6}R\right)+\overleftarrow{\square}\,,\qquad
V_{r}=e^{-u\square}-e^{-D_{\Lambda}}\;.$ (C.2)
The first term $(-\square e^{-u\square})$ in eq. (C.1) defines the free
propagator including the UV cutoff, $V_{l}$, $V_{r}$ and $e^{-D_{\Lambda}}$
contain vertices, the derivatives in $V_{l}$ act to the left on $\varphi_{l}$
and the derivatives in $V_{r}$ and in $e^{-D_{\Lambda}}$ on $\varphi_{r}$ to
the right.
To linear order in $h_{\mu\nu}$ we have ($h\equiv h_{\mu}^{\ \ \mu}$)
$V_{l}=-\frac{1}{2}\overleftarrow{\square}h-\frac{1}{2}\overleftarrow{\partial}_{\mu}(\partial_{\mu}h)+\frac{1}{2}\overleftarrow{\partial}_{\mu}(\partial_{\nu}h^{\mu\nu})+\overleftarrow{\partial}_{\mu}\overleftarrow{\partial}_{\nu}h^{\mu\nu}+\frac{1}{6}\left((\partial_{\mu}\partial_{\nu}h^{\mu\nu})-(\square
h)\right)\;.$ (C.3)
Subsequently, in momentum space, the derivatives acting to the left have to be
replaced by $i$ times the momentum $l_{1}$ of $\varphi_{l}(l_{1})$ and the
derivatives acting on $h_{\mu\nu}$ by $i$ times the momentum $p_{h}$ of
$h_{\mu\nu}(p_{h})$.
$e^{-D_{\Lambda}}$ can be expanded to linear order in $h^{\mu\nu}$ in the
exponent,
$e^{-D_{\Lambda}}=e^{-\Lambda^{-2}B-\Lambda^{-2}\square}$ (C.4)
with
$B=\frac{1}{2}(\partial_{\mu}h){\partial}_{\mu}-(\partial_{\mu}h^{\mu\nu}){\partial}_{\nu}-h^{\mu\nu}{\partial}_{\mu}{\partial}_{\nu}+\frac{1}{6}\left((\square
h)-(\partial_{\mu}\partial_{\nu}h^{\mu\nu})\right)\;.$ (C.5)
Subsequently we expand $e^{-\Lambda^{-2}B-\Lambda^{-2}\square}$ to linear
order in $h^{\mu\nu}$ using
$e^{C+A}|_{{\cal
O}(C)}=\left(C+\frac{1}{2}[A,C]+\frac{1}{6}[A,[A,C]]+\frac{1}{24}[A,[A,[A,C]]]+\dots\right)e^{A}$
(C.6)
for $A=-\Lambda^{-2}\square$, $C=-\Lambda^{-2}B$. The commutators generate
derivatives acting on $\Lambda^{-2}(x)$ resp. $\lambda(x)$ in $A$. From the
projection of diagrams on ${\cal G}_{\chi_{i}\chi_{j}}$ from Appendix A we
know that we need at most 3 derivatives acting on $\lambda(x)$, therefore we
can truncate the expansion in commutators at this order. Explicitely one
obtains
$\displaystyle e^{-D_{\Lambda}}|_{{\cal O}(B)}=$
$\displaystyle\,\Big{\\{}-\Lambda^{-2}B+\frac{1}{2}\left(\Lambda^{-2}\square\Lambda^{-2}B-\Lambda^{-2}B\Lambda^{-2}\square\right)$
$\displaystyle-\frac{1}{6}\left(\Lambda^{-2}\square(\Lambda^{-2}\square\Lambda^{-2}B-\Lambda^{-2}B\Lambda^{-2}\square)-(\Lambda^{-2}\square\Lambda^{-2}B-\Lambda^{-2}B\Lambda^{-2}\square)\Lambda^{-2}\square\right)$
$\displaystyle+\frac{1}{24}\Big{(}\Lambda^{-2}\square(\Lambda^{-2}\square(\Lambda^{-2}\square\Lambda^{-2}B-\Lambda^{-2}B\Lambda^{-2}\square)-(\Lambda^{-2}\square\Lambda^{-2}B-\Lambda^{-2}B\Lambda^{-2}\square)\Lambda^{-2}\square)$
$\displaystyle-(\Lambda^{-2}\square(\Lambda^{-2}\square\Lambda^{-2}B-\Lambda^{-2}B\Lambda^{-2}\square)-(\Lambda^{-2}\square\Lambda^{-2}B-\Lambda^{-2}B\Lambda^{-2}\square)\Lambda^{-2}\square)\Lambda^{-2}\square\Big{)}\Big{\\}}$
$\displaystyle\times e^{-\Lambda^{-2}\square}$ (C.7)
where, from $\lambda(x)=\log(\Lambda/\overline{\Lambda})$ and to ${\cal
O}(\lambda^{2})$, $\Lambda^{-2}\simeq u(1-2\lambda(x)+2\lambda^{2}(x))$. The
remaining exponential function in eq. (C.7) remains to be expanded to ${\cal
O}(\lambda^{2})$. Again we write it in the form
$e^{C^{\prime}+A^{\prime}}\,,\qquad A^{\prime}=-u\square\;,\qquad
C^{\prime}=-u(-2\lambda(x)+2\lambda^{2}(x))\;.$ (C.8)
Terms of ${\cal O}(C^{\prime})$ can be obtained from eq. (C.6), but we need
also terms of ${\cal O}(C^{\prime 2})$. These can be found in ref. [27]. We
introduce
${\cal B}_{n}=({\cal
L_{A^{\prime}}})^{n-1}C^{\prime}\qquad\text{where}\qquad{\cal
L_{A^{\prime}}}{\cal O}=[A^{\prime},{\cal O}]\;.$ (C.9)
Then the terms of ${\cal O}(C^{\prime 2})$ in the expansion of
$e^{C^{\prime}+A^{\prime}}$ read
$\displaystyle e^{C^{\prime}+A^{\prime}}|_{{\cal O}(C^{\prime 2})}=$
$\displaystyle\;\Bigg{\\{}{\cal B}_{1}\left(\frac{1}{2}{\cal
B}_{1}+\frac{2}{3}{\cal B}_{2}+\frac{3}{4}{\cal B}_{3}\right)+{\cal
B}_{2}\left(\frac{1}{3}{\cal B}_{1}+\frac{1}{2}{\cal B}_{2}+\frac{3}{5}{\cal
B}_{3}\right)$ $\displaystyle+{\cal B}_{3}\left(\frac{1}{4}{\cal
B}_{1}+\frac{2}{5}{\cal B}_{2}+\frac{1}{2}{\cal
B}_{3}\right)\Bigg{\\}}e^{A^{\prime}}$ (C.10)
(note that the order of different ${\cal B}_{n}$ matters). Replacing
$A^{\prime}$ and $C^{\prime}$ from eq. (C.8) one obtains, similarly to eq.
(C.7), a series of Laplacians $\square$ acting on
$(-2\lambda(x)+2\lambda(x)^{2})$ instead of $B$.
It remains to expand all contributions to $e^{-D_{\Lambda}}$ resp. $V_{r}$ to
${\cal O}(\lambda^{2}(x))$, keeping track of the positions of $\square$ and
$B$. In momentum space, all derivatives have to be replaced by $i$ times the
sum of the momenta of $h_{\mu\nu}(p_{h})$ in $B$, and the momenta in
$\varphi_{r}(l_{2})$ and $\lambda(p_{1})$ (or $\lambda(p_{2})$) for terms to
the right of each derivative. Finally the full vertices $V$ are obtained by
adding $\left(V_{l}e^{-D_{\Lambda}}+\square V_{r}\right)=V$ and dropping
unnecessary terms of ${\cal O}(h^{2})$ and ${\cal O}(\lambda^{3})$.
Let us decompose $V$ into
$V=\lambda(p_{1})V^{\lambda}(p_{1})+\lambda(p_{1})\lambda(p_{2})V^{\lambda\lambda}(p_{1},p_{2})+h_{\mu\nu}(p_{h})V^{h}_{\mu\nu}(p_{h})+h_{\mu\nu}(p_{h})\lambda(p_{1})V^{h\lambda}_{\mu\nu}(p_{h},p_{1})$
(C.11)
where it is understood that each component depends in addition on the momenta
$l_{1},l_{2}$ of $\varphi_{l}(l_{1})$, $\varphi_{r}(l_{2})$ and
$V^{\lambda\lambda}(p_{1},p_{2})$ is symmetric in $p_{1},p_{2}$.
A useful test is that $\varphi_{l}(l_{1})V\varphi_{r}(l_{2})$ has to be Weyl
invariant, keeping terms to the appropriate order. From
$\delta_{\sigma}h_{\mu\nu}=-2\sigma\eta_{\mu\nu}$,
$\delta_{\sigma}h=-8\sigma$,
$\delta_{\sigma}\varphi_{l,r}=\sigma\varphi_{l,r}$ and
$\delta_{\sigma}\lambda=\sigma$ one can derive in momentum space
$\delta_{\sigma}(\lambda(q)V^{\lambda}(q)+h(q)V^{h}_{\mu\mu}(p_{h}))\overset{!}{=}-l_{2}l_{2}e^{u\,l_{2}l_{2}}-l_{1}l_{1}e^{u\,l_{1}l_{1}}$
(C.12)
and
$\delta_{\sigma}(\lambda(p_{1})\lambda(q)V^{\lambda\lambda}(p_{1},q)+h(q)\lambda(p_{1})V^{h\lambda}_{\mu\mu}(q,p_{1}))_{{\cal
O}(\lambda)}\overset{!}{=}-\lambda(p_{1})\left(V^{\lambda}(p_{1})_{l_{1}=-l_{2}-p_{1}}+V^{\lambda}(p_{1})_{l_{2}=-l_{1}-p_{1}}\right)$
(C.13)
where $q$ is identified with the momentum of the Weyl mode $\sigma$. (The two
prescriptions in parenthesis on the right hand side are not equivalent since
momentum conservation alone gives $l_{1}+l_{2}+p_{1}+q=0$. These prescriptions
indicate that $q$ disappears from the right hand side if expressed in terms of
the momenta of $\varphi_{l,r}$.)
We have verified that these relations are satisfied for our explicit
expressions for the components of $V$. Below we give the ones for
$V^{\lambda}(p_{1})$ and $V^{h}_{\mu\nu}(p_{h})$; those for
$V^{\lambda\lambda}(p_{1},p_{2})$ and $V^{h\lambda}_{\mu\nu}(p_{h},p_{1})$ are
considerably longer.
$\displaystyle V^{\lambda}(p_{1})$
$\displaystyle=-\frac{1}{12}\big{\\{}(8(l_{2}p_{1})^{3}+12(l_{2}p_{1})^{2}(p_{1}p_{1})+6(l_{2}p_{1})(p_{1}p_{1})^{2}+(p_{1}p_{1})^{3})u^{3}\phantom{xxxxxxxxxxxxxxxxxxx}$
$\displaystyle+(16(l_{2}p_{1})^{2}+16(l_{2}p_{1})(p_{1}p_{1})+4(p_{1}p_{1})^{2})u^{2}+(24(l_{2}p_{1})+12(p_{1}p_{1}))u$
$\displaystyle+24\big{\\}}(l_{1}l_{1})(l_{2}l_{2})u$ (C.14)
$\displaystyle V^{h}_{\mu\nu}(p_{h})$
$\displaystyle=\Big{\\{}\frac{1}{144}(24(l_{1}l_{1})(l_{2}p_{h})^{4}+44(l_{1}l_{1})(l_{2}p_{h})^{3}(p_{h}p_{h})+30(l_{1}l_{1})(l_{2}p_{h})^{2}(p_{h}p_{h})^{2}$
$\displaystyle+9(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})^{3}+(l_{1}l_{1})(p_{h}p_{h})^{4})u^{4}$
$\displaystyle+\frac{1}{36}(12(l_{1}l_{1})(l_{2}p_{h})^{3}+16(l_{1}l_{1})(l_{2}p_{h})^{2}(p_{h}p_{h})+7(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})^{2}+(l_{1}l_{1})(p_{h}p_{h})^{3})u^{3}$
$\displaystyle+\frac{1}{12}(6(l_{1}l_{1})(l_{2}p_{h})^{2}+5(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})+(l_{1}l_{1})(p_{h}p_{h})^{2})u^{2}$
$\displaystyle+\frac{1}{6}(3(l_{1}l_{1})(l_{2}p_{h})+(l_{1}l_{1})(p_{h}p_{h}))u+\frac{1}{2}(l_{1}l_{1})+\frac{1}{2}(l_{1}p_{h})+\frac{1}{6}(p_{h}p_{h})\Big{\\}}-l_{1\mu}l_{1\nu}$
$\displaystyle+\Big{\\{}-\frac{1}{24}(8(l_{1}l_{1})(l_{2}p_{h})^{3}+12(l_{1}l_{1})(l_{2}p_{h})^{2}(p_{h}p_{h})+6(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})^{2}+(l_{1}l_{1})(p_{h}p_{h})^{3})u^{4}$
$\displaystyle-\frac{1}{6}(4(l_{1}l_{1})(l_{2}p_{h})^{2}+4(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})+(l_{1}l_{1})(p_{h}p_{h})^{2})u^{3}$
$\displaystyle-\frac{1}{2}(2(l_{1}l_{1})(l_{2}p_{h})+(l_{1}l_{1})(p_{h}p_{h}))u^{2}-(l_{1}l_{1})u\Big{\\}}l_{2\mu}l_{2\nu}$
$\displaystyle+\Big{\\{}-\frac{1}{144}(8(l_{1}l_{1})(l_{2}p_{h})^{3}+12(l_{1}l_{1})(l_{2}p_{h})^{2}(p_{h}p_{h})+6(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})^{2}+(l_{1}l_{1})(p_{h}p_{h})^{3})u^{4}$
$\displaystyle-\frac{1}{36}(4(l_{1}l_{1})(l_{2}p_{h})^{2}4(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})+(l_{1}l_{1})(p_{h}p_{h})^{2})u^{3}$
$\displaystyle-\frac{1}{12}(2(l_{1}l_{1})(l_{2}p_{h})+(l_{1}l_{1})(p_{h}p_{h}))u^{2}-\frac{1}{6}(l_{1}l_{1})u-\frac{1}{6}\Big{\\}}p_{h\mu}p_{h\nu}-p_{1\mu}p_{h\nu}$
$\displaystyle+\Big{\\{}-\frac{1}{24}(8(l_{1}l_{1})(l_{2}p_{h})^{3}+12(l_{1}l_{1})(l_{2}p_{h})^{2}(p_{h}p_{h})+6(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})^{2}+(l_{1}l_{1})(p_{h}p_{h})^{3})u^{4}$
$\displaystyle-\frac{1}{6}(4(l_{1}l_{1})(l_{2}p_{h})^{2}+4(l_{1}l_{1})(l_{2}p_{h})(p_{h}p_{h})+(l_{1}l_{1})(p_{h}p_{h})^{2})u^{3}$
$\displaystyle-\frac{1}{2}(2(l_{1}l_{1})(l_{2}p_{h})+(l_{1}l_{1})(p_{h}p_{h}))u^{2}-(l_{1}l_{1})u\Big{\\}}p_{2\mu}p_{h\nu}\;.$
(C.15)
The term $\sqrt{\gamma}\,m_{0}^{2}\,\varphi^{2}$ in eq. (5.1) expanded in
$h_{\mu\nu}$ generates vertices proportional to $m_{0}^{2}\,\varphi^{2}$ given
in eq. (5.4), and proportional to $h\,m_{0}^{2}\,\varphi^{2}$ with $m_{0}^{2}$
as in eq. (5.4). However, the latter vertices do not contribute to ${\cal
G}_{\chi_{i}\chi_{j}}$ obtained according to eq. (A.5). The only one-loop
diagram which contributes to ${\cal G}^{0}_{\nu\nu}$ in eq. (5.7) contains two
vertices $V^{\nu}=2\overline{m}^{2}$, and one vertex $V^{h}_{\mu\nu}$ given in
eq. (C.15).
To ${\cal O}(g)$, two-loop diagrams which contribute to $\Delta{\cal
G}^{0}_{\nu\nu}$ consist in vertex-less tadpoles attached to the one-loop
diagram with two vertices $V^{\nu}$ and one vertex $V^{h}_{\mu\nu}$, and a
tadpole containing one vertex $V^{\nu}$ attached to the one-loop diagram with
one vertex $V^{\nu}$ and one vertex $V^{h}_{\mu\nu}$. Quadratic divergences
are cancelled by one-loop diagrams where the tadpoles are replaced by
corresponding counter terms from $m_{0}^{2}$. Altogether finite contributions
for $\overline{\Lambda}^{2}\to\infty$ cancel as well which goes beyond the
required cancellation of subdivergences.
Diagrams which contribute to $\Delta{\cal G}^{0}_{\nu\lambda}$ consist in a
loop with vertices $V^{h}_{\mu\nu}$ and $V^{\lambda}$ with a tadpole
containing one vertex $V^{\nu}$ together with a corresponding counter term,
and a loop with vertices $V^{h}_{\mu\nu}$ and $V^{\nu}$ with a tadpole
containing one vertex $V^{\lambda}$ together with a corresponding counter
term.
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|
# Knowledge Graph for Microdata of Statistics Netherlands
Chang Sun
Institute of Data Science
Maastricht University
<EMAIL_ADDRESS>
## 1 Introduction
Statistics Netherlands (CBS)111https://www.cbs.nl/ hosted a huge amount of
data not only on the statistical level but also on the individual level. They
have collected and maintained data from the whole Dutch population over 100
years. With the development of data science technologies, more and more
researchers request to conduct their research by using high-quality individual
data from CBS (called CBS Microdata) or combining them with other data
sources. CBS Microdata is linkable data at a personal, company and address
level with which researchers can conduct statistical research themselves under
strict conditions [1]. The requester (has to be a researcher) will have a
protected working environment to store the data files, intermediate files, and
output. CBS Microdata is considered as a reliable and informative data source
which covers health, socio-economic, educational, financial and other 14
categories. To a large degree, making great use of these data for research and
scientific purposes can tremendously benefit the whole society.
However, CBS Microdata has been collected and maintained in different ways by
different departments in and out of CBS. The representation, quality, metadata
of datasets are not sufficiently harmonized. Each dataset is briefly described
in one to three sentences on website222https://www.cbs.nl/nl-nl/onze-
diensten/maatwerk-en-microdata. A more detailed description for each dataset
is provided in a PDF file in Dutch on CBS website separately. Due to the lack
of integration of all Microdata sets and a centralized platform to query the
metadata, it is a very time-consuming and costly task for researchers to find
all needed datasets or particular variables. Researchers first need to dive
into all datasets description pages in the specific category. Then,
researchers have to download and read (translate to English if needed) all
lengthy PDF files to know the basic information about the datasets. In this
way, researchers miss the relations between different datasets and are not
able to easily find all needed variables across multiple datasets. Therefore,
a general research question is formulated for this project: Can we convert the
descriptions of all CBS microdata sets into one knowledge graph with high-
quality and comprehensive metadata so that the researchers can easily query
the metadata, explore the relations among multiple datasets, and find the
needed variables? The above general research question can be divided into the
following sub-questions:
1. 1.
Can we extract key information about CBS Microdata from the text (PDF files)?
2. 2.
What are the most suitable ontologies for the CBS Microdata metadata?
3. 3.
Can we use the extracted information to make a knowledge graph on CBS
Microdata metadata?
4. 4.
Can we find relations across different datasets and categories?
## 2 Related Work
Semantic web and linked data technology is not new for the statistics offices.
In 2001, SDMX (Statistical Data and Metadata eXchange) was launched to
standardize and modernize the mechanisms and processes for the exchange of
statistical data and metadata among international organisations and their
member countries [2]. However, there are not many publications or publicly
available softwares describing how to convert statistical (meta)data to a
knowledge graph. EU Open Data Portal333https://data.europa.eu/ provides a
SPARQL tool to query the metadata of their Linked data [3]. They created a
vocabulary for the metadata using the Data Catalogue Vocabulary
(DCAT)444https://www.w3.org/TR/vocab-dcat/ and Dublin Core Terms (DCT)
vocabulary555https://www.dublincore.org/specifications/dublin-core/dcmi-
terms/. Sarker et al. proposed their plan and methods to implement semantic
web technology for Australian Bureau of Statistics in 2017 [4]. It is a proof-
of-concept paper which doesn’t provide any actual implementations. In 2018,
Chaves-Fraga et al. provided a mapping translator from RMLC to R2RML and a
comparative analysis over two different real statistics datasets using Data
Cube Vocabulary [5]. This study focuses on converting CSV to RDF and reducing
the size of the R2RML mapping documents. Existing studies only cover one part
of this project.
## 3 Methodology
Since the descriptions of CBS Microdata are only presented by the text in PDF
files, a heavy data pre-processing job is required before building up the
knowledge graph. The data pre-processing tasks include extracting text from
the diverse layout of PDF files, translating Dutch to English, extracting key
information from the sentences, normalizing extracted information. After pre-
processing, the data description text from PDF files is converted to
structured data (CSV). Furthermore, I converted the CSV data to RDF and
explored the knowledge graph in GraphDB. The whole workflow is presented in
Figure 1. This section will describe all steps in the following subsections.
### 3.1 Automatically download data description files
The data description of each dataset is presented in a PDF file which can be
downloaded separately from CBS Microdata websites. To automatically download
all PDF files, I wrote a Python script to crawl information from the related
CBS websites and catch the downloading links of PDF files. This code is
publicly available on Github repository666https://github.com/sunchang0124/KG-
CBSMicrodata.
### 3.2 Extract text from PDF files
Extracting text accurately from a PDF document is still regarded as a very
challenging task. PDF was designed as an output format that gives a good
viewing layout rather than a data input format. Therefore, most of the content
semantics are lost when a text or word document is converted to PDF. To get a
better converting result, I applied a Python package called
PDFMiner777https://pypi.org/project/pdfminer/ to extract text from PDF
documents. In addition to extracting pure text, it also extracts the
corresponding locations, font names, font sizes, writing direction (horizontal
or vertical) for each text segment. This tool has been developed and well-
maintained since 2008. It is well-recognized in the text mining community
because of it’s good performance.
### 3.3 Translate extracted text from Dutch to English
Many international researchers in the Netherlands are interested in CBS
Microdata. However, the Microdata websites and data descriptions are both
written in Dutch. One additional challenge of this project is to translate
text from Dutch to English. Consider the timeframe of the project, the best
option for this task is Google Translator. I applied for the Google Translator
API in Python.
Figure 1: Workflow and approaches to building the Knowledge Graph on CBS
Microdata
### 3.4 Extract key information from the text (English)
To have a high-quality metadata, a data description which gathers all
information in a text is apparently not enough. Key information such as data
released date, data publisher, subject identifiers, and other metadata
elements need to be extracted from the description text. Text mining
techniques are required to complete this task. I applied two well-known text
mining Python libraries - NLTK888https://www.nltk.org/ and
spaCy999https://spacy.io/ \- to recognize entities from the text. NLTK has a
very good performance on part-of-speech tagging, while spaCy is outstanding on
the name entity recognition. Combining two tools increased the accuracy to
find key terms in the text. In the project, I focused on tagging Noun words
and recognising Organization, Date, and Person. Due to the time limitation and
my Dutch language level, only English text has been processed. Dutch text
could be mined in the future work.
### 3.5 Find suitable vocabularies
Finding a suitable vocabulary is a key to build the knowledge graph for the
metadata of CBS Microdata. As CBS is a national statistics office, I searched
related ontologies and vocabularies in the statistics community. The best
options from what I have observed are Data Catalogue Vocabulary (DCAT) and
Dublin Core Terms (DCT) vocabulary. EU Open Data Portal also applied these two
vocabularies to their metadata for the Linked Data project.
### 3.6 Convert CSV to RDF using R2RML
After matching the vocabularies with extracted key information (metadata), I
applied R2RML [6] to convert CSV (data of metadata) to RDF. I mapped the
dataset, catalog, organization, variables, and keywords (of dataset) as
subjects. Language tag is also used for tagging Dutch and English content.
After RDF is generated successfully, all triples are imported and stored at
GraphDB.
### 3.7 Complete knowledge graph
To complete the knowledge graph, I applied AMIE+ [7] to predict potential
relations between entities. Additionally, I also tried a graph embedding
method using Python Library Gensim. Prediction results will be discussed in
the following section.
## 4 Results
Crawling and downloading all PDF files from CBS Microdata website took less
than 10 minutes including sleeping time. Sleeping time (1-5 seconds) is to
avoid being detected and blocked by the website when the requests are
frequently sent to the website. In total, 505 PDF documents were downloaded on
31st March 2020.
As I discussed in the previous section, text extraction from PDF files still
remains a challenge. At the end, 420 PDF documents (83.2%) were processed
successfully by PDFMiner, while 85 documents failed to be extracted to text.
420 documents were collected from 18 different categories as Table 1 shows.
The two main reasons for the failure of text extraction are the unrecognizable
layout and unable to detect words and paragraphs properly.
Table 1: Checklist for reporting PPDDM studies NO | Category | Num of datasets | Num of variables101010The number of extracted variables names from the data description files. Text extraction from PDF, language translation might cause the number of extracted variable names smaller than the actual number.
---|---|---|---
1 | Labour and social security | 114 | 766
2 | Business | 34 | 198
3 | Population | 49 | 300
4 | Build and live | 24 | 223
5 | Financial and business services | 1 | 12
6 | Health and wellbeing | 62 | 444
7 | Trade and catering | 3 | 36
8 | Income and expenditure | 36 | 298
9 | International trade | 1 | 1
10 | Industry and energy | 8 | 60
11 | Agriculture | 2 | 13
12 | Macroeconomy | 0 | 0
13 | Nature and environment | 2 | 14
14 | Education | 35 | 391
15 | Government and politics | 4 | 3
16 | Prices | 5 | 52
17 | Security and justice | 28 | 237
18 | Traffic and transport | 6 | 44
19 | Leisure and culture | 6 | 23
In total | 420 | 3115
Google Translator API took around 5 minutes to translate all text from Dutch
to English with acceptable results. I evaluated the performance by checking a
random sample of translated text. Most of the sentences and terms were
translated correctly except the ones that were broken from the text
extraction. Unfortunately, I am not able to check the translation accuracy on
a large scale by myself. This could be done in the future work. After
translation, 125 key terms such as Dates, Organizations, Persons, Topics were
extracted from the English text using NLTK and spaCy. Some entities were
mislabelled in the results which required manual correction. For instance, the
full name and abbreviation of some organizations both appear in the data
description which cannot be recognized by spaCy. Another example is the label
names of some variables were mislabelled as Persons or Organizations.
To convert to RDF data, 5 triple maps were created for the dataset, catalog,
organization, variables and keywords using DCAT and DCT vocabularies in the
R2RML mapping file. Due to the page limitation, only the triple maps of the
dataset and publisher are presented in this report (Figure 2). The whole
mapping file can be found in the Github
repository111111https://github.com/sunchang0124/KG-CBSMicrodata. As Figure 2
shows, some elements of metadata such as dct:issued (Date of formal issuance
(e.g., publication) of the item), dct:title, dct:description, dct:identifier,
dct:language, dct:isPartOf, dct:langingPath, dcat:keyword, dct:publisher,
dct:creator can be fulfilled by the extracted key information. At the end,
20242 triples were created by R2RML within 8 seconds and imported to GraphDB
(Figure 3). In GraphDB, some relations can be easily found by querying the
knowledge graph, but they are not discoverable in the 420 lengthy PDF
documents. For example, I found 251 out of 420 datasets share some variables.
43 datasets belong to two categories at the same time, while 4 datasets belong
to three categories.
(a) R2RML triple mapping diagrams for “dataset” and “publisher”
(b) R2RML triple mapping for “publisher” entity
Figure 2: R2RML mapping examples
In the last step, I applied AMIE+ to predict potential relations between
entities. As this knowledge graph is not very complicated, only 9 rules were
found by AMIE+. For instance, ?b <http://purl.org/dc/terms/hasPart> ?a => ?a
<http://purl.org/dc/terms/isPartOf> ?b. I insert 4 out of 9 rules to the
existing graph based on their confidence score. In addition, I also tried a
graph embedding method using Gensim. This method can find two datasets which
are similar to each other but in two different categories because they share
the same keywords or variable names. However, since information might be lost
in the text extraction and translation steps, it’s not very convincing to add
new relations based on the similarity in this case.
Figure 3: Visualise graph based on a part of created triples
## 5 Conclusion
This project converts the descriptions of all CBS microdata sets into one
knowledge graph with comprehensive metadata in Dutch and English. Researchers
can easily query the metadata, explore the relations among multiple datasets,
and find the needed variables. For example, if a researcher searches a dataset
about “Age at Death” in the Health and Well-being category, all information
related to this dataset will appear including keywords and variable names.
“Age at Death” dataset has a keyword - “Death”. This keyword will lead to
other datasets such as “Date of Death”. “Cause of Death”, “Production
statistics Health and welfare” from Population, Business categories, and
Health and well-being categories. This will tremendously save time and costs
for the data requester but also data maintainers. However, there are some
limitations in this short-term project. Firstly, only 83.2% PDF documents were
extracted to text due to several reasons such as different versions of PDF
files, and unrecognizable layout. Second, accuracy of language translation and
entity recognition need to be evaluated on a larger scale and optimized. For
example, several dates can be extracted from the data description of one
dataset. The dates might be the data collecting time, publishing time, or
modifying time. More information needs to be accurately extracted from the
data description documents and map to the metadata vocabulary.
## References
* [1] George Kour and Raid Saabne. Real-time segmentation of on-line handwritten arabic script. In Frontiers in Handwriting Recognition (ICFHR), 2014 14th International Conference on, pages 417–422. IEEE, 2014.
* [2] C. B. voor de Statistiek, “Microdata: Zelf onderzoek doen,” Centraal Bureau voor de Statistiek. https://www.cbs.nl/nl-nl/onze-diensten/maatwerk-en-microdata/microdata-zelf-onderzoek-doen (accessed Apr. 05, 2020).
* [3] D. Mancheva, “Representation of statistical models and standards with linked data technologies,” CROS - European Commission, Oct. 22, 2019. https://ec.europa.eu/eurostat/cros/content/representation-statistical-models-and-standards-linked-data-technologies-0_en (accessed Apr. 05, 2020).
* [4] “Linked Data | Open Data Portal.” https://data.europa.eu/euodp/en/linked-data (accessed Apr. 05, 2020).
* [5] A. Sarkar, M. Mecham, and P. Meadows, “Australian Bureau of Statistics Implementation of Semantic Web Technology,” p. 12.
* [6] C.-F. David, P. Freddy, S.-P. Idafen, and C. Oscar, “Virtual Statistics Knowledge Graph Generation from CSV files,” Stud. Semantic Web, pp. 235–244, 2018, doi: 10.3233/978-1-61499-894-5-235.
* [7] “R2RML: RDB to RDF Mapping Language.” https://www.w3.org/TR/r2rml/ (accessed Apr. 05, 2020).
* [8] “Max-Planck-Institut für Informatik: AMIE.” https://www.mpi-inf.mpg.de/departments/databases-and-information-systems/research/yago-naga/amie/ (accessed Apr. 05, 2020).
|
# Reconstruction of IACT events using deep learning techniques with CTLearn
D. Nieto,1 T. Miener,1 A. Brill2, J. L. Contreras,1 T. B. Humensky,2 and R.
Mukherjee,3 for the CTA Consortium
###### Abstract
Arrays of imaging atmospheric Cherenkov telescopes (IACT) are superb
instruments to probe the very-high-energy gamma-ray sky. This type of
telescope focuses the Cherenkov light emitted from air showers, initiated by
very-high-energy gamma rays and cosmic rays, onto the camera plane. Then, a
fast camera digitizes the longitudinal development of the air shower,
recording its spatial, temporal, and calorimetric information. The properties
of the primary very-high-energy particle initiating the air shower can then be
inferred from those images: the primary particle can be classified as a gamma
ray or a cosmic ray and its energy and incoming direction can be estimated.
This so-called full-event reconstruction, crucial to the sensitivity of the
array to gamma rays, can be assisted by machine learning techniques. We
present a deep-learning driven, full-event reconstruction applied to simulated
IACT events using CTLearn. CTLearn is a Python package that includes modules
for loading and manipulating IACT data and for running deep learning models
with TensorFlow, using pixel-wise camera data as input.
1Instituto de Física de Partículas y del Cosmos and Departamento de EMFTEL,
Universidad Complutense de Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>
2Physics Department, Columbia University, New York, USA;
<EMAIL_ADDRESS>
3Department of Physics and Astronomy, Barnard College, Columbia University,
New York, USA
## 1 Introduction
The ability of deep learning to assist in the analysis of data from imaging
atmospheric Cherenkov telescopes (IACT) was first demonstrated by the
detection of muon rings in real data (Feng & Lin 2016) and by the
classification of gamma-ray and cosmic-ray simulated events (Nieto et al.
2017). Subsequent studies proved its capability to reconstruct the energy and
arrival direction of simulated gamma-ray events (Mangano et al. 2018;
Jacquemont et al. 2020) and to improve IACT sensitivity on real data (Shilon
et al. 2019). CTLearn111https://github.com/ctlearn-project/ctlearn (Nieto et
al. 2019a; Brill et al. 2019) is a high-level, open-source Python package
providing a backend for training deep learning models for IACT event
reconstruction using TensorFlow. CTLearn allows its user to focus on
developing and applying new models while making use of functionality
specifically designed for IACT event reconstruction. The user can customize
the training and built-in models hyperparameters. Hyperparameter optimization
is available through the accompanying CTLearn-optimizer package. Data loading
and pre-processing are performed using an associated external package,
DL1-Data-Handler (Kim et al. 2020). A diagram summarizing CTLearn architecture
is shown in Fig. 1.
Figure 1.: Diagram summarizing CTLearn’s framework design.
## 2 Full-event reconstruction
Model architecture. CTLearn works with any TensorFlow model obeying a generic
signature. In addition, CTLearn includes two built-in models for gamma/hadron
classification of stereoscopic data (Brill et al. 2019) and a third one for
full-event reconstruction of monoscopic data, dubbed _TRN-single-tel_ model
(see Fig. 2, left panel). This last model is based on a deep convolutional
neural network (CNN)-based architecture with residual connections (He et al.
2015) adapted from a thin ResNet (Xie et al. 2019). A squeeze-and-excitation
attention mechanism (Hu et al. 2017) was added into the CNN blocks. The
architecture ends on a selectable fully-connected head that performs either
particle classification or regression (energy or arrival direction
reconstruction).
Figure 2.: _Left)_ _TRN-single-tel_ architecture. Prediction results are
either particle type or energy or arrival direction. _Right)_ Example of ROC
curves for the same given random seed.
Experiments. We trained and tested our _TRN-single-tel_ model with a dataset
of simulated events for the Cherenkov Telescope Array222www.cta-
observatory.org (Acharya et al. 2018, CTA), the next generation ground-based
observatory for gamma-ray astronomy at very-high energies: 2 million images
from events having triggered an array of 4 large-sized telescopes (LSTs) in
monoscopic mode We considered diffuse gamma-ray and proton-initiated events,
simulated within a cone of 10∘ radius - covering the whole field of view (FoV)
of the instrument - in a balanced way. An 80% of the data were used for
training and a 20% for testing. We trained the model on 200k batches of 64
images, validating periodically. The pixel layout in the original images is a
hexagonal lattice, mapped to a Cartesian lattice using bilinear interpolation
(Nieto et al. 2019b). The model was trained independently for each
reconstruction task. Two experiments were conducted: a) training and testing
on all triggered events and b) training and testing on events where the images
of the showers were mostly contained within the field of view of the telescope
(no more than 20% of the total image charge within the two outermost rings of
pixels on the camera). Training was repeated 10 times for each experiment,
setting a different random seed that varied the weight initialization and
training set shuffling.
Results. The _TRN-single-tel_ model successfully learned to perform full-event
reconstruction across the entire FoV of the telescope. The classification
accuracy (0.5 threshold) on the test set of diffuse events was $0.748\pm
0.002~{}(0.756\pm 0.001)$, with an area under the ROC curve of $0.848\pm
0.002~{}(0.856\pm 0.001)$ for the all-events (contained-events) experiment
(average and standard deviation computed from the results of all training
runs). An example of ROC curves can be found in Fig. 2, right panel. The
energy resolution and bias, and angular resolution provided by the model on
the test set of diffuse gamma-ray events can be found in Fig. 3, where the
data points represent the median of all training runs and 16% – 84%
containment bands are shown, illustrating the inference robustness of the
model. We remark that these results come from the reconstruction of diffuse
events and note that the reconstruction for events showing arrival directions
close to the center of the FoV would perform substantially better.
Figure 3.: _Left)_ Energy resolution; _center)_ Energy bias; _right)_ Angular
resolution (all vs. reconstructed energy). Tested on diffuse gamma-ray events.
## 3 Conclusions and outlook
CTLearn’s design and main features have been presented, showing its high level
of configurability and flexibility, and demonstrating its potential for
monoscopic full-event reconstruction using CTA simulated events. Areas where
development is planned or already ongoing are: building models for
stereoscopic full-event reconstruction; exploiting multitask learning;
implementing models that could combine event-level data from a heterogeneous
collection of telescope types, enabling IACT-specific metrics and loss
functions; and ultimately applying full-event reconstruction to real IACT
data.
### Acknowledgments
This work was conducted in the context of the CTA Analysis and Simulations
Working Group. We gratefully acknowledge financial support from the agencies
and organizations listed here:
http://www.cta-observatory.org/consortium_acknowledgments.
DN and JLC acknowledge support from The European Science Cluster of Astronomy
& Particle Physics ESFRI Research Infrastructures funded by the European
Union’s Horizon 2020 research and innovation program under Grant Agreement no.
824064. TM acknowledges support from FPA2017-82729-C6-3-R. AB acknowledges
support from NSF award PHY-1229205. DN acknowledges the support of NVIDIA
Corporation with the donation of a Titan X Pascal GPU used for this research.
The ASP would like to thank the dedicated researchers who are publishing with
the ASP.
## References
* Acharya et al. (2018) Acharya, B., et al. (CTA Consortium) 2018, Science with the Cherenkov Telescope Array (WSP). https://arxiv.org/abs/1709.07997
* Brill et al. (2019) Brill, A., et al. 2019, CTLearn: Deep learning for imaging atmospheric Cherenkov telescopes event reconstruction. URL https://doi.org/10.5281/zenodo.3345947
* Brill et al. (2019) Brill, A., et al. 2019, in 2019 New York Scientific Data Summit (NYSDS), 1. URL https://doi.org/10.1109/NYSDS.2019.8909697
* Feng & Lin (2016) Feng, Q., & Lin, T. T. Y. 2016, Proceedings of the International Astronomical Union, 12, 173–179. URL http://dx.doi.org/10.1017/S1743921316012734
* He et al. (2015) He, K., et al. 2015, arXiv e-prints, arXiv:1512.03385. https://arxiv.org/abs/1512.03385
* Hu et al. (2017) Hu, J., et al. 2017, arXiv e-prints, arXiv:1709.01507. https://arxiv.org/abs/1709.01507
* Jacquemont et al. (2020) Jacquemont, M., et al. 2020, in these proceedings
* Kim et al. (2020) Kim, B., et al. 2020, DL1-Data-Handler: DL1 HDF5 writer, reader, and processor for IACT data. URL https://doi.org/10.5281/zenodo.3979698
* Mangano et al. (2018) Mangano, S., et al. 2018, Lecture Notes in Computer Science, arXiv:1810.00592. URL http://dx.doi.org/10.1007/978-3-319-99978-4
* Nieto et al. (2017) Nieto, D., et al. 2017, in Proceedings of 35th International Cosmic Ray Conference — PoS(ICRC2017), vol. 301, 809. URL https://doi.org/10.22323/1.301.0809
* Nieto et al. (2019a) — 2019a, in Proceedings of 36th International Cosmic Ray Conference — PoS(ICRC2019), vol. 358, 752. https://arxiv.org/abs/1912.09877
* Nieto et al. (2019b) — 2019b, in Proceedings of 36th International Cosmic Ray Conference — PoS(ICRC2019), vol. 358, 753. https://arxiv.org/abs/1912.09898
* Shilon et al. (2019) Shilon, I., et al. 2019, Astroparticle Physics, 105, 44. https://arxiv.org/abs/1803.10698
* Xie et al. (2019) Xie, W., et al. 2019, arXiv e-prints, arXiv:1902.10107. https://arxiv.org/abs/1902.10107
|
# Self-Organizing Intelligent Matter: A blueprint for an AI generating
algorithm
Karol Gregor, Frederic Besse
DeepMind, UK
<EMAIL_ADDRESS>
###### Abstract
We propose an artificial life framework aimed at facilitating the emergence of
intelligent organisms. In this framework there is no explicit notion of an
agent: instead there is an environment made of atomic elements. These elements
contain neural operations and interact through exchanges of information and
through physics-like rules contained in the environment. We discuss how an
evolutionary process can lead to the emergence of different organisms made of
many such atomic elements which can coexist and thrive in the environment. We
discuss how this forms the basis of a general AI generating algorithm. We
provide a simplified implementation of such system and discuss what advances
need to be made to scale it up further.
## 1 Introduction
An AI generating algorithm (Clune, 2019) is a computational system that runs
by itself without outside interventions and after a certain amount of time
generates intelligence (though the general idea is much older than this
reference). Evolution on earth is the only known successful system thus far
that we know of. In this paper we propose a computational framework, and argue
why it might constitute such general algorithm, while being computationally
tractable on current or near future hardware.
Building such system successfully will take many iterations and require a
number of advances. What we hope to provide however, is a general procedure,
where better and better systems arise as a result of improving the elements of
the system and of experimentation rather than a fundamentally new algorithm.
As an example, we had such procedure for supervised learning since the 1980’s
- neural networks trained by back-propagation and stochastic gradient descent.
To reach the current impressive performance, it required a number of clever
improvements, such as rectified non-linearities, convolutions, batch
normalization, attention, residual connections and better optimizers, but the
overall algorithm hasn’t changed.
### 1.1 Evolution
Evolution is the primary process by which our algorithm operates. We describe
it here.
In machine learning, the word evolution is typically used to describe
variations of the following process (Back, 1996). We have a number of
individuals and an objective to optimize. We evaluate the individuals, select
the ones with good values of the objective, and mutate them to produce the
next generation. Over time, individuals that are better at optimizing the
objective appear. The use of the word evolution in this context is perhaps
unfortunate, as this process is quite unlike the evolution observed in nature
(Stanley et al., 2017). The clearest difference we can see is in the outcome.
The former results in a small variation of final individuals that are the best
at the objective. The latter results in a coexistence of huge variation of
individuals with different behaviors - it is open-ended.
Let us therefore discuss the basic operation of natural evolution. We have an
environment built out of elements (atoms) that are organized into bigger units
such as individual bacteria or animals or groups of these. Those classes of
units that propagate (Joyce, 1994) into the future (e.g. replicate - we will
discuss this in a moment) keep existing, while those that don’t propagate
cease to exist. There is no objective based on which units are selected for
propagation. Different collections of units find different means of
propagating.
An important mechanism for the coexistence of a large number of different
solutions is niche construction (NicheConstruction, 2020). Different
collections of individuals modify or form the environment for one another. For
example a bacterium consumes a food and excretes waste products, which
modifies the local environment, being either food or a toxic substance for
other bacteria. Another example is a prey forming a food source for a
predator. These systems are self balancing, for example too many predators
means they can’t find food and die off, and vice versa. This means that the
coexistence of multiple solutions is present (and evolving). Collections of
individuals in such systems have different means of propagating, rather then
being selected by a global objective.
The lack of objective we believe, as argued in (Stanley & Lehman, 2015), is
critical and fundamental to open-ended creation and coexistence of diversity
and runs counter to most developments in machine learning. Conversely, a
presence of an objective in a system likely leads to a collapse of diversity.
To see that attempting to create an objective is problematic, let us therefore
try to suggest one and see what the problems would be. One of the clearest
objectives one might propose is to reward an individual for reproduction.
However, a predator might then simply kill its offspring which would increase
its chance of making another one. We could try to tweak or find other
objectives, but this might lead to unwanted and unforeseen behaviours similar
to the one described above. There are other issues. We don’t actually make
copies of ourselves, but instead mix genes from individuals (sexual
reproduction), which we need to select. What do we actually want to reward? In
addition, most of the time, propagation of species depends on individuals
working together in a group, often sacrificing some members for the good of
the group. What is then a real reproducing unit? This is the reason why the
word ”propagate” (Joyce, 1994) is more appropriate than ”reproduce”. What
really happens is that those classes of groups of elements that are set ”a
certain way” propagate and those that are not, don’t. Note that this does not
contradict the evolution of an intrinsic reward - which is an evolved means of
finding a good policy during the lifetime of a given individual.
Another example of evolutionary process is our society. People don’t have
children in proportion to some objective the society has set, or there isn’t
just one job or hobby that we all converge on and that is the ”best”. People
engage in a large number of jobs and hobbies. At the same time, memes, values,
work practices, company structures and many other emergent concepts propagate.
All these things coexist, both cooperating and competing.
### 1.2 Principles
The field of artificial life aims at producing life and an evolutionary
process inside a computer (Langton, 1997; Ray, 1991; Lenski et al., 2003;
Sims, 1994; Yaeger et al., 2011; Gras et al., 2009; Soros & Stanley, 2014),
see (Aguilar et al., 2014) for a review. In seminal work on Tierra (Ray,
1991), Thomas Ray created an artificial life system in the substrate of
assembly instructions in computer memory. The set of instructions is executed
by a number of heads and one organism corresponds to one head. The system was
initialized by a handcrafted sequence of instructions that when executed, will
copy itself to another part of the memory. The executions undergo mutations.
Some of these result in organisms that are unable to replicate, while some
others get better at it. Some organisms find ways to use other organisms
copying mechanism to copy, forming parasites. Then resistance to parasites
evolves, and hyper-parasites, phenomenons of sociality and cheating are
observed.
This process eventually peters out, and the quest ever since has been to
create a system that is truly open-ended, where complexity keeps increasing
without bounds (Standish, 2003). A number of principles that characterize and
open-ended process has been proposed (Soros & Stanley, 2014; Taylor et al.,
2016) Here we select two that that we find the most important (points 2 and 3)
and introduce two new ones (points 1 and 4).
* •
There should be no built in notion of an individual and no built in operation
for reproduction of an individual. Instead, these should be an emergent
properties of collections of units, composing new collections of units or
themselves.
* •
The evolution of new (here emergent) individuals should create novel
opportunities for the survival of others (Soros & Stanley, 2014).
* •
The potential size and complexity of the individuals’ phenotypes should be (in
principle) unbounded (Soros & Stanley, 2014).
* •
To tractably obtain intelligent agents, fundamental neural operations should
be basic building blocks of the environment.
The first property is absent in majority of works on artificial life, except a
few that we review later below. However, this property is quite close to being
true in Tierra and Avida (Lenski et al., 2003). A given individual (organism),
consisting of a sequence of instructions, actually has to construct a new one
(create the new sequence of instructions). Then however, a new agent is
declared, a new head is allocated for it, and there is a distinction in
operations within and outside of/between the individuals. Properties two and
three should really be consequences of property one, which we will see once we
describe the system in section 2. Property four is introduced for tractability
and again will become clearer below.
There are other approaches that studied diversity in evolution. To prevent
collapse of diversity in objective based evolution, ideas such MAP-Elites
(Mouret & Clune, 2015) or quality diversity (Pugh et al., 2016) have been
proposed, that explicitly try to keep diverse set of solutions. These assume a
small handcrafted space of variables that define what individuals are
different. However, given such space, they show that searching for space of
diverse solutions, leads to a better result on the final objective than
optimizing the objective directly, as the search process is able to step
through solutions that are not obviously directly on the path aimed at the
objective. To remove an objective, (Soros & Stanley, 2014; Brant & Stanley,
2017) propose to give every organism a chance to reproduce but do so if it
satisfies a minimal criterion, such as sufficient amount of energy. In the
latter, they consider a set of mazes and their solvers. A maze is propagated
to the next generation if there are solvers solving it and a solver is
propagated if there are mazes it can solve. This results in a coexistence of
many solutions in the population. The system however behaves somewhat like a
random walk through solvable mazes and it would be good to find a system with
a stronger selection pressure.
Another approach to create an open ended system is to co-evolve one agent that
designs an environment (sets the layout of things and such) and another one
that tries to solve it (Wang et al., 2020; Racaniere et al., 2019). The former
maintains a collection of environment-agent pairs, and a new such pair is
allocated if it is sufficiently far from the pairs in the collection according
to a pre-specified objective that does not depend on the details of the
environment (but on the ability of agents to solve the environment).
In the system that we propose that there is no special agent designing the
environment - there is actually no concept of agent, and instead there is only
an environment where agent-like organisms can emerge and reshape the
environment from within. There neither is any explicit minimal criterion for
an organism since there is no explicit operation of copy of individual, but
such criterion will appear for any emergent individual.
## 2 Proposed system
The real world is built out of elementary particles that interact and compose
bigger entities. Our proposed environment (an aspiring AI generating
algorithm) is as follows. The environment is built out of elements, but at
much coarser scale. Each element contains a neural operation. This can be for
example a matrix multiplication, an outer product, or more likely a sequence
of such operators comprising a mini neural network. The elements interact with
each other through some form of underlying rules, a type of physics, and
through a direct communication of neural states.
There can be various implementations of this system. In section 3 we provide a
grid-world implementation with basic elements lying on a grid, communicating
with propagating signals or via an attention mechanism and with underlying
physics implementing energy and chemical-like exchanges. Another example could
be elements forming rigid pieces in a three dimensional space that can be
attached using joints, that contain the neural operations, interacting through
exchange of signals with nearby attached pieces and that set the torques on
the joints. There could be several types of elements in the system, and not
all need to have neural networks inside.
In section 3 we provide a grid world implementation that highlights a number
of important properties. Along the way we discuss what advances need to be
made to make this system powerful. However, the potential of the proposed
system is unbounded and here are some of the features that this system
supports. Larger units composed of several elements can be formed either by
physical attachment (like a robot) or simply as a set of units that decide to
communicate and form a whole. There is no limit to the potential size of these
units. These units can propagate in a number of ways - they can grow by taking
over other elements in the environment (a colony), they can replicate by
assembling new copies - moving appropriate collected elements to place (e.g. a
robot assembling a copy itself from pieces), or self-assemble, or they can
produce whole different units that either implement specialized functions (a
useful machine) or units that are even better than their predecessors. The
latter likely requires intelligence.
### 2.1 Capacity for intelligence.
In this subsection we discuss why the computational system just proposed has
the capacity to represent general intelligence. We provide two arguments.
First, any neural algorithm in machine learning that we have created, and
likely create in the future, can be written as a sequence of operations, such
as additions, matrix multiplications, outer products and non-linearities,
operating on states which are tensors. An example is the sequence resulting
from the forward, backward, and optimizer operations of a neural network. Auto
ML Zero (Real et al., 2020) realized this, and directly searched for the
sequence of such operators along with connectivity to states on which they
operate and was able to learn basic neural algorithms. Since these operators
are fundamental building elements of our environment, and these elements can
be made to communicate with arbitrary connectivity, all neural algorithms can
be implemented in our system as well.
Second, the human brain is capable of general intelligence, and its
computation closely resembles an online recurrent network (not trained by
back-propagation in time). That is, it can be approximated by a function $F$
that updates neural and synaptic values online
$h_{t},W_{t}=F(h_{t-1},W_{t-1},x_{t},v)$ where $h_{i}$ is a cell state of
neuron $i$ and $W_{ij}$ is the state of the synapse connecting neurons $i$ and
$j$, $x$ is an input and $v$ represent hyper-parameters such as connectivity,
coefficients, and nonlinearities. To represent actual neural computations well
enough (note that we are not interested in faithful representation but
something with similar algorithm and computational power) we likely need to
represent $h_{i}$ and $W_{ij}$ by small vectors, as suggested in (Gregor,
2020; Bertens & Lee, 2019). These computations are local and therefore to
search for them, we need to search for relatively simple functions (with few
hyperparameters). We can search for them automatically, as in Auto ML zero or
other neural architecture search works (Zoph & Le, 2016; Pham et al., 2018;
Liu et al., 2018; Elsken et al., 2018). The important point to remember from
this, is that there likely exists an online neural update, of power at least
as large as that of the human brain, on a system of a similar size in terms of
computational capacity. We could implement the computation of a given element
of the environment by a layer of such neurons. During the run of the
environment, other elements can set the hyper-parameters $v$ as well as $h$,
$W$ of a given element. Those groups that have elements with the right
settings of these parameters, for example $v$’s that implement a good learning
algorithm (the update of $W$), should enable a better propagation of this
group compared to others.
### 2.2 Agent hypothesis.
In our system there is no separation between an agent and an environment
(AgentHypothesis, 2020) \- there is only an environment. The elements
themselves may or may not form units of evolution (Smith, 1986; Szathmáry et
al., 2005) \- entities that multiply, show heredity and where the heredity is
not exact. In the former case, they could move autonomously, collect energy
and replicate, and form bigger aggregates or replicating organisms because
that provides an advantage. However, we propose to aim for the latter, where a
certain minimal number of simpler cooperating units are need to propagate
themselves.
#### 2.2.1 Relation to other works
There is a large amount of works that are related in one way or another to
self-organization, and we can only review certain ones most relevant to our
paper. The study of origins of life aims to understand how life evolved from
non-living things. At some point in time in early earth, there were only
simple combinations of elements: atoms combined into simple molecules, and at
later time, there was the last universal common ancestor (Theobald, 2010) of
all current life, which was already a rather complex organism containing core
structures of bacteria. There are theories of how life emerged in the period
in between, such as an auto-catalytic system enclosed in a membrane (Maturana
& Varela, 1991; Gánti, 2003) or a self-replicating molecule (Joyce, 2002).
Such self replication for was studied in models for example in (Penrose &
Penrose, 1957; Virgo et al., 2012). In the later they consider a physical
system of shapes and succeed at finding length two chains that replicate.
However, they find that the shapes have to have a very specific features,
arguing that such shapes are unlikely to emerge spontaneously from chemistry
and perhaps makes this processes less likely to be the origin of life.
Simple substrates to study artificial life are cellular automata and related
particle systems. (Gardner, 1970; Chan, 2020; Schmickl et al., 2016; Sayama,
2009; Ventrella, 2020; Nichele et al., 2017). In the former, elements are
cells on a grid and in the later they have real valued coordinates and move.
Their behavior is governed by simple rules that update their state based on
their previous state and that of near neighbours. A goal is to find a set of
rules that give rise to life, exhibiting replication, variation and heredity.
This has proven quite difficult and while simple self replicating structures
have been found, they don’t satisfy these criteria of life. It is then
especially difficult to imagine how to obtain an intelligent life from such
simple rules in a tractable fashion. The proposal we put forward in this
paper, is to use general neural networks of section 2.1 as the key parts of
the elements, which can together form large neural networks, but with
comparatively much fewer elements than basic cellular automata/particle
systems. This is the reason for the word ”intelligent” in the title of the
paper. In addition, obtaining complex behavior should be much easier than
tweaking cellular automata rules since we are starting with good learners -
knowing the current capabilities of reinforcement learning agents, which we
can use to jump-start the system.
Cellular automata can be implemented using convolutional neural networks
(Wulff & Hertz, 1993; Gilpin, 2019). In study of morphogenesis (self-assembly
of a given object/pattern), (Mordvintsev et al., 2020) trained a convolutional
neural network that respects the structure of cellular automata to produce a
given pattern starting with single cell or number of other patterns. Similar
approach was pursued using compositional pattern producing networks (Nichele
et al., 2017). These works however are not building artificial life systems.
Finally, there is a body of work on swarm systems and self-assembling robots.
The former studies how can swarms of robots coordinate their behavior to
accomplish tasks such as disaster relief or to study an emergent swarm
behavior. The later, which is the closest in some ways to out proposal,
studies how robots can self-assemble from smaller pieces. In (Weel et al.,
2014), they consider a single type of (simulated) robot piece from which
varying bodies are assembled through a process of evolution. There are some
differences from what we propose: There is an explicit notion of an individual
that is constructed in a central birthing facility according to an evolved
template, an individual has a central brain (rather than composed one), and
individuals replicate if they meet after a certain minimal distance from the
birthing facility. While they do indeed aim for objective free evolution, the
individuals do tend be selected based on how quickly they can get certain
distance from the birthing area. Finally, in (Mathews et al., 2017; Pathak et
al., 2019) they build robotic systems (real and simulated respectively) build
out of pieces that self assemble out of pieces that and at the same time form
a bigger computational system, and are trained to accomplish a task.
## 3 Grid version of SIM and general properties.
This section introduces an instance of the self-organizing intelligent matter
(SIM). It also serves as a discussion of various points relating to SIM and
gives more reasons of why we believe it can form an AI generating algorithm.
As discussed in subsection 2.2, there is no built-in notion of an agent and
there really is just an environment. Seeing that, it feels unnatural to
implement the system on two different platforms as is commonly the case - one
for the physical part - such as a physics simulator, and one for the neural
part - a neural network framework such as TensorFlow, PyTorch or Jax. Instead,
we propose to make such system in a single platform. Because we would like to
produce intelligent behavior, we need to run neural networks efficiently, and
therefore the system needs to be implemented on one of the latter platforms.
Because of its flexibility, we choose Jax.
Jax operates on tensors, which we use to store elements. The elements need to
interact and have the ability to form flexible aggregates of arbitrary size
(property 3, subsection 1.2). We first focus on neural operations and describe
the ideal system: the one we aspire for. We then talk about the steps we took
towards implementing that system.
### 3.1 The ideal system
In our system, we place a neural network at every point of an $m\times m$ grid
($m$ up to 400 in our experiments), but in general, a different and flexible
connectivity can be used. The ideal system would use the networks described in
subsection 2.1 since, as we argued, they are likely able to compose general
learners. At every point in time such network updates activations, weights and
possibly hyperparameters $h_{t},W_{t},v_{t}=F(h_{t-1},W_{t-1},v_{t-1},x_{t})$.
Each network outputs a number of ”actions” that affect other elements - the
cells of the grid. One fundamental action a network can take is to set the
values of $h,W,v$ of neighbouring cells. This action can be executed if
certain conditions are satisfied, such as the cell having enough energy - the
energy concept and its updates in our system are described later (this is a
minimal criterion for an element, not an emergent individual). What does such
action allow? We give a few examples.
* •
The first example is to copy $v$ with a noise, set $W$’s and $h$’s to some
random variables and keep $v$ constant through the lifetime of any cell. This
corresponds to creating a new mini-brain (consisting of one cell) that has a
very similar structure to the source and is untrained - in other words, a
replication operation, but without replicating the learned content - similar
to way children are created.
* •
The second example is to copy all the variables, with some noise. This creates
copy with the same knowledge as the original cell.
* •
The third example is somewhere in between, transferring some knowledge and not
other.
* •
The fourth example is to have a joint set of weights for an aggregate of
cells, and select out those which are used in a given copy, analogous to cell
differentiation in animal bodies or bee specialization in a colony of bees.
* •
The fifth example is setting the parameters (or some of them) to something
quite different - programming them - that causes new aggregates of cells to
perform useful functions for the original such as collecting energy - in
effect creating useful machines for the original aggregate.
* •
In the final example, again, the new cells are programmed, but this time with
the goal of creating new aggregates that are better than the original. The
aggregates can be thought of as both organisms and machines - there is no
distinction between the two in our environment. Here, machines create better
machines by both designing better brains and better bodies. Such process
requires intelligence, which is exactly what we are aiming for. We believe
there will be a point in time in the evolution of the environment when this
will happen, creating a self reinforcing loop of improving intelligence.
Having neural operations as the fundamental elements of the environment is the
reason that makes us believe that such state can be achieved in
computationally tractable fashion in this type of system.
### 3.2 Our proposed steps towards that system
Let us come back to describing the system we built. Since more powerful
algorithms are not yet available, we use standard recurrent neural networks.
We cannot use reinforcement learning to train these networks in a direct way -
RL is an algorithm for optimizing an objective, which we fundamentally don’t
have here. We could try evolving such objective and such approach might be
tractable. Instead we resort to what is usually done in these situations -
simply evolve weights. We use the standard tanh recurrent network and the
operation of setting the weight matrices and biases of new cell is simply copy
with mutation. Alternative representation that is common are compositional
pattern producing networks (Stanley, 2007).
Figure 1: Diagram of the grid version of the system.
The neurons, and all the other variables mentioned below (such as energy and
chemicals), are placed on an $m\times m$ dimensional grid, with $m$ up to 400,
Figure 1 comprising up to 160k networks. The second action that a network can
output is to move to a neighbouring cell, which will swap the content of this
cell with the new one (we experiment with allowing and disallowing this
action).
It is difficult to implement rigid bodies on a grid in Jax. To allow
coordinated movements of larger units, we let the neurons communicate. For
example they can communicate to all the cells belonging to a given group to
move. One way to communicate is via local attention mechanism, where a given
cell can read values of the $h$ of a cell in its neighborhood. This would also
allow an aggregate of cells to implement a deep neural network, having
different units representing different layers of a deep network. A single pass
through such network takes a number of world updates. In our case we opt for a
simpler communication mechanism, because a locally connected computation which
local attention would utilize is not available in Jax. We create four signal
layers that move in four directions (right, up, left, down). Each cell writes
$h$ into all the layers, which then move and each cell can then read the
content of all four signal layers.
Next we introduce a type of physics into the system, other than motion. We
consider fields of energy and fields of $n_{c}$ types of chemicals
$C_{1},\ldots,C_{n_{c}}$ with $n_{c}$ typically $4-9$, at every location of
the grid. A given cell can pull or push all these quantities in each of the
four directions, which costs it energy proportional to the push or pull.
Neighbouring cells can ”fight” for these quantities, and if an energy of a
cell falls below threshold, the cell is declared dead and its weights are set
to zero. The cell also produces enzymes $Z_{1},\ldots,Z_{n_{c}}$ that control
the reactions in the respective order (e.g. $Z_{2}$ controlling
$C_{2}\rightarrow C_{3}$), that cost energy to make and that sum to at most
one. A given reaction releases energy equal to $C_{i}Z_{i}$ except the last
reaction which releases zero energy. As an example, to maximize energy, the
cell should have one enzyme, say $Z_{2}$, pull chemical $C_{2}$ from outside
(assuming some other cell is producing it), convert it to $C_{3}$ and excrete
it to another cell. The cell also has a maximum lifetime and therefore it has
to do something non-trivial (at least produce energy and copy) to propagate
its information. If the population of live elements is below 10% we reawaken
new ones with random weights. We also note that in our implementation, the
elements themselves form autonomous units capable of self-replication,
however, as discussed in section 2.2, in an ideal system they wouldn’t.
There are two other variations that we experimented with. First we wanted to
know how simple can one make the system and still observe and interesting
behavior. We created a pure energy system, where instead of having chemicals,
energy increases at every location up to some threshold. In the second system
we experimented with different implementation. Instead of having a network at
every point of the grid, we consider $n$ (1600) elements on $m\times m$
($200\times 200$) grid that can move around. This time we used attention
mechanism for communication within some distance between elements. We again
only had a background energy, and the elements don’t die or need anything for
them to be ”alive”. They could just ”sit around” (as atoms can). However,
those that are active, can acquire energy by moving (energy gets automatically
absorbed) and copy their weight onto others that have a lower energy. Those
that do that will be the ones that propagate and overtake the system.
## 4 Experiments
We run the system explained in the previous section. More detailed explanation
and the settings of hyper-parameters are in the Appendix. The code will be
released in a near future. What we observe is an exciting diversity among a
series of runs, with snapshots shown in the Figure 2. This is best viewed in
accompanying video 111Videos: Best viewed by downloading (not viewing on the
site): https://bit.ly/3nb6MCI (3.2Gb). Compressed version (more blurry):
https://youtu.be/ifVjzhWL9ro . It also shows the run of the system from the
the start, showing competition among various classes of elements.
Figure 2: Results of runs. Top row. We selected three random weights and
plotted them in RGB channels. This way elements with similar weights will have
similar colour and those with different weights will likely have different
colour. Bottom row. First three chemicals plotting in RGB channels.
Discussion. We see a lot of diversity in the runs. These are best viewed in
accompanying video. We often see coexistence of two species - seen in two
different colours in the same region of space. We also see that the elements
moved the chemicals around creating regions of high and low densities. In the
top left square, we see that the region of low density is occupied by
different species than the region of high density First graph Fraction of
elements alive as a function of time from the start of the run (random new
weights with no evolution in this run only). We see oscillatory behavior
resulting from dynamics of two species. Second graph Energy content of
elements in pure energy system of moving elements that need no energy to
survive. However, those elements that collect energy can copy weights onto
those with less energy, and those that do that propagate.
Looking closely at the Figure 2 top, we see in several regions a stable
coexistence of two species of elements, represented by dots of different color
in the same region of space. These persists for long amount of time, as can be
seen from the videos, meaning they found a way to coexist, creating niches for
one another.
Furthermore, looking at the Figure 2 top left, we see a light yellow region,
containing two species, and an orange region containing different, third
species. We see that the chemicals have been moved to sub-regions of space,
and that the the species of the orange region live in a place with less
chemical density. Thus the elements created niches by modifying density of
chemicals in the environment. The challenge is not simply to be able to
reproduce alone, but in the presence of others.
There are other types of behaviour we observe. In a smaller system, where we
diffused the chemicals everywhere quickly, and turned off evolution for
stability (weights are reset if density is less then 10%) we observed
oscillations in population of two species, Figure 2 middle (and in the video)
is reminiscent of the Lotka-Voltera system, (Lotka, 2002). The behavior in the
pure energy grid system usually results in one species, but there are
instances of two. This is similar for the version of neural elements not
living in a grid. In Figure 2 right we see that in this system, despite not
having any survival notion or an explicit selection for energy, the elements
learn to collect energy, as this allows them to copy their weights onto
others, which causes them to propagate.
## 5 Discussion
In this paper we proposed a framework for achieving intelligence by
evolutionary process in an environment that is built out of interacting
elements implementing computationally efficient and general learning. Only
time will tell whether this framework is capable of such a feat. There are
some critical advances that need to be made, most notably, creating general
neural updates that can be evolved, or otherwise designed. Other directions
for development are a way these elements interact, their embedding in a space
and the underlying physics.
We have created a version of this system, that we believe has all the core
necessary components, or would have if we had more powerful learning updates.
It could form a starting point for the developments outlined in the previous
paragraph. We summarize the core properties of this system that are general to
all instances of SIM. There are elements containing neural networks. These
elements can interact and communicate, forming larger units, in effect
implementing larger networks. There is no objective based on which anything is
selected. Instead, those compositions of units that find ways to propagate,
will. There is no distinction between organisms and machines. The elements can
write into other elements, they can do this by copying, or they can write
other information to build machines that are useful for their creators or they
can create whole new autonomous machines. The machines can create both new
”bodies” - functional compositions utilizing physics, and new brains creating
better algorithms than themselves.
A very intriguing question is what is the necessary physics needed for
explosion of diversity and rise of intelligence. In the real world, the
elements are elementary particles such as quarks and electrons. The former
combine into protons and neutrons (there are few other particles such as
photons), these combine into atoms. Few core atoms, primarily carbon,
hydrogen, oxygen, nitrogen and phosphorus, form a wide diversity of molecules
in the form of proteins and other types of molecules, eventually building
cells as a basic units of life. What are the core rules that could give rise
to complex life in the system we propose, where fundamental units themselves
can already exhibit complex interaction and behavior? Does one need to
introduce a complex chemistry or classical physics? An intriguing possibility,
that we would like to explore in the future is whether any built in complexity
is even necessary? What if we have the brain elements and only a basic rule -
that of energy? Will natural selection, with such general learners,
progressively construct more and more complex structures that outsmart one
another? These are some of the exciting questions we plan to study in the
future.
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## Appendix A Appendix
### A.1 Detailed description of the system.
We describe the system grid implementation in detail here. We consider a grid
of sizes varying from $100\times 100$ to $400\times 400$. At every location of
the grid we have the following variables: Energy (real scalar), chemicals
($n_{c}$ real scalars, typically 4, but up to 9), enzymes ($n_{c}$ real
scalars). We also have signals carrying information about various variables as
we describe below. Finally, we have recurrent neural network variables (we
describe the network next): hidden layer activations of size $n_{h}$
(typically 16), input to hidden weights, hidden to hidden weights, hidden
biases, hidden to actions weights and action biases.
Next, we describe the neural network operations. At every point in time,
except special times described below, we apply a classic recurrent neural
network update to activations while keeping the weights fixed:
$h_{t}=\tanh(W_{x}x+W_{h}h+b_{h});\hat{a}_{t}=W_{a}h_{t}+b_{a}$, where $W$’s
are weights, $h$ are activations and $\hat{a}$ are variables from which
actions are computed (as described below). In the above formula we dropped the
time index from $W$’s as they weren’t updated at this time. The $W$’s were
updated at the following times: If one cell (grid element) took a specific
action (copy + mutate) aimed at neighbour cell, if it had sufficient energy (a
threshold) and if the neighbour cell was empty - meaning having zero energy -
the weights and biases got copied with added noise to the other cell.
As mentioned in the text, we use this update because we haven’t yet discovered
a good forms of general update. In sections 2.1, 3.1 we propose that such
update would take the form $h_{t},W_{t},u_{t}=F(h_{t-1},W_{t-1},u_{t-1})$
where $u_{t}$ are the hyperparameters of the update rules. The simplest
analogue to the previous paragraph would be that at every point in time except
the same special times, $h,W$’s are updated (so the network can learn = update
weights) while $u$’s are fixed. Finally, during the special times, $u$’s are
not merely copied, but the network can generate new $u$’s in the neighbour
cell, as well as $h$ and $W$, allowing the originator cell to program the new
one.
Next let us describe the energy-chemical dynamics within the cell. Energy is a
scalar that is bound between $0$ and some max value. The chemicals transform
in directions $Z_{i}\rightarrow Z_{(i+1)modn_{c}},i=1,\ldots,n_{c}$ in
proportion the amount of enzymes $C_{i}$ respectively and all the reactions
release a fixed amount of energy except for $i=n_{c}$ which does not release
any. The released energy is added to the energy of the cell. If energy falls
below a threshold (by mechanisms described next), the weights and activations
are erased - set to zero. We also use maximum lifetime of a cell after which
the weights and activations are erased. The amount of enzyme as well as flows
of quantities between the cells is controlled by the cell’s actions which we
describe next.
The cell has the following actions. 1. Copy action: from $\hat{a}$ we extract
five components and take softmax to decide where and if to copy (0 - do not
copy, 1-4 copy to one of the four directions). If a copy is successful, a
fixed energy (tending to be large) is removed from the cell. 2. Move action:
This is optional. If enabled, it has the same logic, but this time, the full
content of the two cells (originator and target) is swapped. 3. Energy flow:
The four values determine the proportion of energy the cells want to suck from
each neighbour or push to the neighbour (positive vs negative). The cost of
such action is proportional to the push. The resulting flow between two cells
is obtained by subtracting the values of actions from the two cells, allowing
two cells to fight or cooperate in moving the energy. 4. Chemical flow: The
same logic but for each chemical. There are $4n_{c}$ values, one for each
chemical and direction. 5. Enzyme production: $n_{c}$ actions that create each
enzyme. The total amount of enzyme is normalized to be at most one. There are
two extra features. The energy is normally protected from falling below a
threshold (protecting the cell from accidentally killing itself). The
exception is that if the energy pull from the neighbour is sufficiently large,
it will pull all the energy and kill the other cell (erase its weights and
activations).
Finally there are signals. The cells need to know about what is happening at
the other cells and to communicate to form larger aggregates and networks. We
implemented the following communication. There are four information grids -
one moving in each of the four directions. At every point in time, each grid
moves in its direction. In addition, part of activations $h$ as well as
energy, chemicals and enzymes overwrite to information grids for those cell
for which energy is above threshold. The neural network input is formed by
concatenating the four information grids at a given location.
Finally each run of the system was run on single GPU, and the largest system
size $400\times 400$ at $n_{h}=16$ ($160000$ networks) was determined by the
largest size we could fit into GPU memory.
|
# The split common null point problem for generalized resolvents and
nonexpansive mappings in Banach spaces
Bijan Orouji1, Ebrahim Soori2,∗
###### Abstract.
In this paper, the split common null point problem in two Banach spaces is
considered. Then, using the generalized resolvents of maximal monotone
operators and the generalized projections and an infinite family of
nonexpansive mappings, a strong convergence theorem for finding a solution of
the split common null point problem in two Banach spaces in the presence of a
sequence of errors will be proved.
Keywords: Split common null point problem. Maximal monotone operator.
Generalized projection. Generalized resolvent. Nonexpansive mapping.
∗ Corresponding author
2010 Mathematics Subject Classification. 47H10.
[email protected](B. Orouji); [email protected](E. Soori)
## 1\. Introduction
Let $H_{1}$ and $H_{2}$ be two Hilbert spaces and $C$ and $Q$, two nonempty,
closed and convex subsets of $H_{1}$ and $H_{2}$, respectively. Let
$A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Then the split
feasibility problem (SFP) [8] is: to find $z\in H_{1}$ such that $z\in C\cap
A^{-1}Q$. There exists several generalizations of the SFP: the multiple set
convex sets problem ( MSSFP) [22, 9], the split common fixed point problem
(SCFPP) [10, 23], and the split common null point problem (SCNPP) [7]. (SCNPP)
is as follows: given set-valued mappings $M_{1}:H_{1}\rightarrow 2^{H_{1}}$
and $M_{2}:H_{2}\rightarrow 2^{H_{2}}$, and a bounded linear operator
$A:E\rightarrow F$, find a point $z\in H_{1}$ such that
$z\in M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0),$
where $M_{1}^{-1}0$ and $M_{2}^{-1}0$ are sets of null points of $M_{1}$ and
$M_{2}$, respectively. Many authors have studied the split feasibility problem
and the split common null point problem using nonlinear operators and fixed
points; see, for example, [4, 7, 10, 11, 22, 15, 23, 38]. However, we have not
found many results outside of the framework Hilbert spaces. Note that the
first extension of SFP to Banach spaces is appeared in [28], then this scheme
was later extended to MSSFP in [37]. A very recent generalization for the SFP
is appeared in[29]. The split common null point problem in Banach spaces is
also solved by Takahashi [34, 35, 36].
In this paper, the split common null point problem with generalized resolvents
of maximal monotone operators in two Banach spaces is considered. Then using
the generalized resolvents of maximal monotone operators and the generalized
projections, a strong convergence theorem for finding a solution of the split
null point problem in two banach spaces in the presence of a sequence of
errors is proved.
## 2\. Preliminaries
Let $E$ be a real Banach space with the norm $\|.\|$ and $E^{*}$, the dual
space of $E$. When $\\{x_{n}\\}$ is a sequence in $E$, the strong convergence
of $\\{x_{n}\\}$ to $x\in E$ is denoted by $x_{n}\rightarrow x$ and the weak
convergence to $x\in E$ is denoted by $x_{n}\rightharpoonup x$. A Banach space
$E$ is strictly convex if $\|\frac{x+y}{2}\|<1$, whenever $x,y\in S(E)$,
$x\neq y$ and $S(E)$ is the unite sphere centered at the origin of $E$. $E$ is
said to be uniformly convex if $\delta_{E}(\epsilon)=0$ and
$\delta_{E}(\epsilon)>0$ for all $0<\epsilon\leq 2$ where
$\delta_{E}(\epsilon)$ is the modulus of convexity of $E$ and is defined by
(2.1)
$\displaystyle\delta_{E}(\epsilon)=inf\bigg{\\{}1-\frac{\|x+y\|}{2}:\|x\|,\|y\|\leq
1,\|x-y\|\geq\epsilon\bigg{\\}}$
A uniformly convex Banach space is strictly convex and reflexive. It is also
well known that a uniformly convex Banach space has Kadec Klee property, that
is, $x_{n}\rightharpoonup u$ and $\|x_{n}\|\rightarrow\|u\|$ imply
$x_{n}\rightarrow u$, see [14, 26]. Furthermore, $E$ is called $p$-uniformly
convex if there exists a constant $c>0$ such that $\delta_{E}\geq
c\epsilon^{p}$ for all $\varepsilon\in[0,2]$, where $p$ is a fixed real number
with $p\geq 2$. For example, the $L_{p}$ space is $2$-uniformly convex for
$1<p\leq 2$ and $p$-uniformly convex for $p\geq 2$, see [30]. Let $U=\\{x\in
E:\|x\|=1\\}$. The norm of $E$ is said to be Gateaux differentiable if for
each $x,y\in U$, the limit
(2.2) $\displaystyle\lim_{t_{\rightarrow}0}\frac{\|x+ty\|-\|x\|}{t}$
exists. In this case, $E$ is called smooth. The modulus of smoothness of $E$
is defined by
(2.3) $\displaystyle\rho_{E}(t)=\sup\bigg{\\{}\frac{\|x+y\|+\|x-y\|}{2}-1:x\in
U,\|y\|\leq t\bigg{\\}}.$
If $\displaystyle\lim_{t\rightarrow 0}\frac{\rho_{E}(t)}{t}=0$, then $E$ is
called uniformly smooth. Let $q>1$. If there exists a fixed constant $c>0$
such that $\rho_{E}(t)\leq ct^{q}$, then $E$ is said to be $q$-uniformly
smooth, see [19]. It is well known that a uniformly convex Banach space is
reflexive and strictly convex.
The mapping $J_{E}^{p}$ from $E$ to $2^{E^{*}}$ is defined by
(2.4) $\displaystyle J_{E}^{p}(x)=\\{x^{*}\in E^{*}:\langle
x,x^{*}\rangle=\|x\|\|x^{*}\|,\|x^{*}\|=\|x\|^{p-1}\\},\quad\forall x\in E$
If $p=2$ then $J_{E}^{2}=J_{E}$ is the normalized duality mapping on $E$. Note
that $E$ is smooth if and only if $J_{E}$ is a single-valued mapping of $E$
into $E^{*}$. We also know that $E$ is reflexive if and only if $J_{E}$ is
surjective, and $E$ is strictly convex if and only if $J_{E}$ is one-to-one.
Hence, if $E$ is smooth, strictly convex and reflexive Banach space, then
$J_{E}$ is a single-valued, bijection and in this case, the inverse mapping
$J_{E}^{-1}$ coincides with the duality mapping $J_{{E}^{*}}:E^{*}\rightarrow
2^{E}$, that means $J_{E}^{-1}=J_{{E}^{*}}$. If $E$ is uniformly convex and
uniformly smooth, then is uniformly norm-to-norm continuous on bounded sets of
$E$ and $J_{E}^{-1}=J_{{E}^{*}}$ is also uniformly norm-to-norm continuous on
bounded sets of $E^{*}$. It is known that $E$ is $p$-uniformly convex if and
only if its dual $E^{*}$ is $q$-uniformly smooth where $1<q\leq 2\leq
p<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$. For more details about the mapping
$J_{E}^{p}$ refer to [1, 21, 30, 14, 16, 26, 32, 33].
###### Lemma 2.1.
[12] Let $x,y\in E$ if $E$ is $q$-uniformly smooth, then there is a $c_{q}>0$
so that
$\|x-y\|^{q}\leq\|x\|^{q}-q\langle y,J_{E}^{q}(x)\rangle+c_{q}\|y\|^{q}.$
Suppose that $E$ is a smooth Banach space and $J$ is the duality mapping on
$E$. Define a function $\phi:E\times E\rightarrow\mathbb{R}$ by
(2.5) $\phi_{E}(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y\|^{2}\quad\forall
x,y\in E.$
Observer that, in a Hilbert space H, $\phi(x,y)=\|x-y\|^{2}$ for all $x,y\in
H$. Furthermore, we know that for each $x,y,z,\omega\in E$,
(2.6) $(\|x\|-\|y\|)^{2}\leq\phi(x,y)\leq(\|x\|+\|y\|)^{2};$ (2.7)
$\phi\Big{(}t,J^{-1}\big{(}\lambda
Jx+(1-\lambda)Jy\big{)}\Big{)}\leq\lambda\phi(t,x)+(1-\lambda)\phi(t,y);$
(2.8) $2\langle
x-y,Jz-J\omega\rangle=\phi(x,\omega)+\phi(y,z)-\phi(x,z)-\phi(y,\omega).$
if $E$ is additionally assumed to be a strictly convex Banach space, then
$\phi(x,y)=0\quad\Longleftrightarrow\quad x=y.$
The following lemma is due to Kamimura and Takahashi [20].
###### Lemma 2.2.
[20] Let $E$ be a uniformly convex and smooth Banach space and let
$\\{y_{n}\\},\\{z_{n}\\}$ be two sequences of $E$, if
$\displaystyle\lim_{n\rightarrow\infty}\phi(y_{n},z_{n})=0$ and either
$\\{y_{n}\\}$ or $\\{z_{n}\\}$ is bounded, then
$\displaystyle\lim_{n\rightarrow\infty}(y_{n}-z_{n})=0$.
Suppose that $C$ is a nonempty, closed and convex subset of a smooth, strictly
convex, and reflexive Banach space $E$. Then for any $x\in E$, there exists a
unique element $z\in C$ such that
(2.9) $\phi(z,x)=\min_{y\in C}\phi(y,x).$
The mapping $\Pi_{C}:E\rightarrow C$ defined by $z=\Pi_{C}x$ is called the
generalized projection of $E$ onto $C$. For example, see [2, 3, 20].
###### Lemma 2.3.
[18] Let $E$ be a smooth, strictly convex and reflexive Banach space and $C$
be a nonempty, closed and convex subset of $E$ and let $x\in E$ and $z\in C$.
Then, the following condition hold:
* ($1$)
$\phi(z,\Pi_{C}x)+\phi(\Pi_{C}x,x)\leq\phi(z,x)\quad\forall x\in C,y\in E$;
* ($2$)
$z=\Pi_{C}x\Longleftrightarrow\langle y-z,Jx-Jz\rangle\leq 0\quad\forall y\in
C$.
Suppose that $M$ is a mapping of $E$ into $2^{E^{*}}$ for the Banach space
$E$. The effective domain of $M$ is denote by $dom(M)$, that is,
$dom(M)=\\{x\in E:Mx\neq\emptyset\\}$. A multi-valued mapping $M$ on $E$ is
said to be monotone if $\langle x-y,u^{*}-v^{*}\rangle\geq 0$ for all $x,y\in
dom(M),u^{*}\in Mx$ and $v^{*}\in My$. A monotone operator $M$ on $E$ is said
to be maximal if it’s graph is not property contained in the graph of any
other monotone operator on $E$.
###### Theorem 2.4.
[6, 27] Let $E$ be a uniformly convex and smooth Banach space and let $J$ be
the duality mapping of $E$ into $E^{*}$. Let $M$ be a monotone operator of $E$
into $2^{E^{*}}$. Then $M$ is maximal if and only if for any $r>0$,
$R(J+rM)=E^{*},$
where $R(J+rM)$ is the range of $J+rM$.
Let $E$ be a uniformly convex Banach space with a Gateaux differentiable norm
and let $M$ be a maximal monotone operator of $E$ into $2^{E^{*}}$. For all
$x\in E$ and $r>0$, we consider the following equation
$Jx\in Jx_{r}+rMx_{r}$
This equation has a unique solution $x_{r}$. In fact, it is obvious from
Theorem 3.1 that there exists a solution $x_{r}$ of $Jx\in Jx_{r}+rMx_{r}$.
Assume that $Jx\in Ju+rMu$ and $Jx\in Jv+rMv$. Then there exist $\omega_{1}\in
Mu$ and $\omega_{2}\in Mv$ such that $Jx=Ju+r\omega_{1}$ and
$Jx=Jv+r\omega_{2}$. So, we have that
$\displaystyle 0$ $\displaystyle=\langle u-v,Jx-Jx\rangle$
$\displaystyle=\langle u-v,Ju+r\omega_{1}-(Jv+r\omega_{2})\rangle$
$\displaystyle=\langle u-v,Ju-Jv+r\omega_{1}-r\omega_{2}\rangle$
$\displaystyle=\langle u-v,Ju-Jv\rangle+\langle
u-v,r\omega_{1}-r\omega_{2}\rangle$
$\displaystyle=\phi(u,v)+\phi(v,u)+r\langle u-v,\omega_{1}-\omega_{2}\rangle$
$\displaystyle\geq\phi(u,v)+\phi(v,u)$
and hence $0=\phi(u,v)=\phi(v,u)$. Since $E$ is strictly convex, we have
$u=v$. We defined $J_{r}^{M}$ by $x_{r}=J_{r}^{M}x$, such that $J_{r}^{M},r>0$
are called the generalized resolvents of $M$. The set of null points of $M$ is
defined by $M^{-1}0=\\{z\in E:0\in Mz\\}$. We know that $M^{-1}0$ is closed
and convex; see[26]. Furthermore
(2.10) $\displaystyle\langle J_{r}^{M}x-y,J(x-J_{r}^{M}x)\rangle\geq 0,$
hold for all $x\in E$ and $y\in M^{-1}0$;see [15].
###### Lemma 2.5.
[13] Let $E$ be a real reflexive, strictly convex and smooth Banach space,
$M:E\rightarrow 2^{E^{*}}$ be a maximal monotone with $M^{-1}0\neq\emptyset$,
then for any $x\in E,y\in M^{-1}0$ and $r>0$, we have
$\phi(y,J_{r}^{M}x)+\phi(J_{r}^{M}x,x)\leq\phi(y,x).$
where $J_{r}^{M}:E\rightarrow E$ is defined by $J_{r}^{M}:=(J+rM)^{-1}J$.
###### Definition 2.6.
[24] $M$ is called upper semicontinuous if for any closed subset $C$ of
$E^{*}$, $M^{-1}(C)$ is closed.
###### Theorem 2.7.
[24] Let $M:E\rightarrow 2^{E^{*}}$ be a maximal monotone operator with
$dom(M)=E$. Then, $M$ is upper semicontinuous.
A Banach space $E$ is said to satisfy the Opial condition, if whenever a
sequence $\\{x_{n}\\}$ in $E$ converges weakly to $x_{0}\in E$, then
(2.11)
$\liminf_{n\rightarrow\infty}\|x_{n}-x_{0}\|<\liminf_{n\rightarrow\infty}\|x_{n}-x\|,\quad\forall
x\in E,x\neq x_{0}$
###### Definition 2.8.
Let $C$ be a nonempty convex subset of a Banach space,
$\\{T_{i}\\}_{i\in\mathbb{N}}$ a sequence of nonexpansive mappings of $C$ into
itself and $\\{\lambda_{i}\\}$ a real sequence such that $0\leq\lambda_{i}\leq
1$ for every $i\in\mathbb{N}$. Following [31], for any $n\geq 1$, we define a
mapping $W_{n}$ of $C$ into itself as follows,
$\displaystyle U_{n,n+1}:=I,$ $\displaystyle
U_{n,n}:=\lambda_{n}T_{n}U_{n,n+1}+(1-\lambda_{n})I,$
$\displaystyle\>\;\vdots$ (2.12) $\displaystyle
U_{n,k}:=\lambda_{k}T_{k}U_{n,k+1}+(1-\lambda_{k})I,$
$\displaystyle\>\;\vdots$ $\displaystyle
U_{n,2}:=\lambda_{2}T_{2}U_{n,3}+(1-\lambda_{2})I,$ $\displaystyle W_{n}:=$
$\displaystyle U_{n,1}:=\lambda_{1}T_{1}U_{n,2}+(1-\lambda_{1})I,$
The following results hold for the mappings $W_{n}$.
###### Theorem 2.9.
[31] Let $C$ be a nonempty closed convex subset of a strictly convex Banach
space. Let $\\{T_{i}\\}_{i\in\mathbb{N}}$ be a sequence of nonexpansive
mappings of $C$ into itself such that
$\bigcap_{i=1}^{\infty}\rm{Fix}(T_{i})\neq\emptyset$ and let
$\\{\lambda_{i}\\}$ be a real sequence such that $0\leq\lambda_{i}\leq b<1$
for every $i\in\mathbb{N}$. For any $n\in\mathbb{N}$, let $W_{n}$ be the
$W$-mapping of $C$ into itself generated by $T_{n},T_{n-1},...,T_{1}$ and
$\lambda_{n},\lambda_{n-1},...,\lambda_{1}$. Then
* (1)
$W_{n}$ is asymptotically regular and nonexpansive and
$\rm{Fix}(W_{n})=\bigcap_{i=1}^{n}\rm{Fix}(T_{i})$, for all $n\in\mathbb{N}$.
* (2)
for every $x\in C$ and for each positive integer $j$,
$\displaystyle\lim_{n\rightarrow\infty}U_{n,j}x$ exists.
* (3)
The mapping $W:C\rightarrow C$ defined by
$Wx:=\displaystyle\lim_{n\rightarrow\infty}W_{n}x=\displaystyle\lim_{n\rightarrow\infty}U_{n,1}x$,
for every $x\in C,$ is a nonexpansive mapping satisfying
$\rm{Fix}(W)=\bigcap_{i=1}^{\infty}\rm{Fix}(T_{i})$ and it is called the
$W$-mapping generated by $\\{T_{i}\\}_{i\in\mathbb{N}}$, and
$\\{\lambda_{i}\\}_{i\in\mathbb{N}}$.
###### Theorem 2.10.
[25] Let $\\{T_{i}\\}_{i=1}^{\infty}$ be a sequence of nonexpansive mappings
of $C$ into itself such that
$\bigcap_{i=1}^{\infty}\rm{Fix}(T_{i})\neq\emptyset,\\{\lambda_{i}\\}$ be a
real sequence such that $0<\lambda_{i}\leq b<1,\;(i\geq 1)$. If $D$ is any
bounded subset of $C$, then $\displaystyle\lim_{n\rightarrow\infty}\sup_{x\in
D}\|Wx-W_{n}x\|=0$.
###### Lemma 2.11.
[5] Let $E$ be a strictly convex Banach space and $C\subseteq E$ be a nonempty
and convex subset of $E$. let $T:C\rightarrow E$ be a nonexpansive mapping.
Then $Fix(T)$ is convex.
## 3\. Main results
###### Theorem 3.1.
Let $E$ and $F$ be two $2$-uniformly convex and uniformly smooth real Banach
spaces that satisfy the Opial condition. Let $J_{E}$ and $J_{F}$ be the
duality mappings on $E$ and $F$, respectively. Suppose that $C$ is nonempty,
closed and convex subset of $E$. Let $A:E\longrightarrow F$ be a bounded
linear operator such that $A\neq 0$ with the adjoint operator $A^{*}$. Let
$M_{1}$ be a maximal monotone operator of $E$ into $2^{E^{*}}$ such that
$M_{1}^{-1}0\neq\emptyset$ and $M_{2}$ be a maximal monotone operator of $F$
into $2^{F^{*}}$ such that $M_{2}^{-1}0\neq\emptyset$. Suppose that
$S:C\longrightarrow E$ is a nonexpansive mapping and
$\\{T_{i}\\}_{i=1}^{\infty}:C\longrightarrow C$, a family of nonexpansive
mappings. For every $n\in\mathbb{N}$, let $W_{n}$ be a $W-mapping$ generated
by Definition 2.8. Let $J_{\lambda}^{M_{1}}$ and $Q_{\mu}^{M_{2}}$ be the
generalized resolvents of $M_{1}$ and $M_{2}$ for $\lambda>0$ and $\mu>0$,
respectively. Suppose that $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq C$.
Let $x_{1}\in C$ and let $\\{x_{n}\\}$, $\\{u_{n}\\}$ and $\\{y_{n}\\}$ be the
sequences generated by
(3.1)
$\displaystyle\begin{cases}u_{n}=J_{E}^{-1}\Big{(}(1-\alpha_{n})J_{E}x_{n}+\alpha_{n}J_{E}\Pi_{C}J_{E}^{-1}(\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n})\Big{)};\\\
z_{n}=J_{\lambda_{n}}^{M_{1}}(u_{n}+e_{n});\quad\quad\omega_{n}=Q_{\mu_{n}}^{M_{2}}(Az_{n});\\\
y_{n}=\Pi_{C}J_{E}^{-1}\Big{(}J_{E}z_{n}-\gamma
A^{*}J_{F}(Az_{n}-\omega_{n})\Big{)};\\\ C_{n}=\\{z\in
C,\big{\langle}\omega_{n}-Az,J_{F}(Az_{n}-\omega_{n})\big{\rangle}\geq
0\\};\\\ D_{n}=\\{z\in E,\phi_{E}(z,z_{n})\leq\phi_{E}(z,u_{n}+e_{n})\\};\\\
Q_{n}=\\{z\in E,\langle x_{n}-z,J_{E}x_{1}-J_{E}x_{n}\rangle\geq 0\\};\\\
x_{n+1}=\Pi_{C_{n}\cap Q_{n}\cap D_{n}}x_{1},\quad\forall
n\in\mathbb{N}.\end{cases}$
where $J_{\lambda_{n}}^{M_{1}}=(J_{E}+\lambda_{n}M_{1})^{-1}J_{E}$ and
$Q_{\mu_{n}}^{M_{2}}=(J_{F}+\mu_{n}M_{2})^{-1}J_{F}$ such that
$\\{\lambda_{n}\\},\\{\mu_{n}\\}\subseteq(0,\infty)$ and $a\in\mathbb{R}$
satisfy in $0<a\leq\lambda_{n},\mu_{n},\forall n\in\mathbb{N}$ and
$0<\gamma<\frac{2}{c\parallel A\parallel^{2}}$ with $c>0$. Let
$\\{\alpha_{n}\\},\\{\sigma_{n}\\}$ be real sequences in $(0,1)$ satisfied in
the conditions:
* (i)
$\displaystyle\lim_{n\rightarrow\infty}\alpha_{n}=0$;
* (ii)
$\displaystyle\lim_{n\rightarrow\infty}\dfrac{\|J_{E}x_{n}-J_{E}u_{n}\|}{\alpha_{n}}=0$;
* (iii)
$\displaystyle\lim_{n\rightarrow\infty}\sigma_{n}=1$.
Consider the error sequence $\\{e_{n}\\}\subseteq E$ such that
* (iv)
$\displaystyle\lim_{n\rightarrow\infty}\|e_{n}\|=0$.
Let $\Omega=M_{1}^{-1}0\cap
A^{-1}(M_{2}^{-1}0)\cap\big{(}\bigcap_{i=1}^{\infty}Fix(T_{i})\big{)}$.
Suppose that one of the following two conditions holds:
* (v)
the sequence $\\{x_{n}\\}$ is bounded,
or
* (vi)
$\Omega\neq\emptyset$.
Then
1. (a)
$\Omega\neq\emptyset$ if and only if the sequence $\\{x_{n}\\}$ is bounded,
2. (b)
the sequence $\\{x_{n}\\}$ converges strongly to a point $\omega_{0}\in\Omega$
where $\omega_{0}=\Pi_{\Omega}x_{1}$.
###### Proof.
(a) Let $\Omega\neq\emptyset$. First, it will be checked that $C_{n}\cap
D_{n}\cap Q_{n}$ is closed and convex for all $n\in\mathbb{N}$. For any $z\in
D_{n}$, it is realized that
$\displaystyle\phi_{E}(z,z_{n})\leq\phi_{E}(z,u_{n}+e_{n})$
$\displaystyle\Leftrightarrow$ $\displaystyle\|z\|^{2}+\|z_{n}\|^{2}-2\langle
z,J_{E}z_{n}\rangle\leq\|z\|^{2}+\|u_{n}+e_{n}\|^{2}-2\langle
z,J_{E}(u_{n}+e_{n})\rangle$ (3.2) $\displaystyle\Leftrightarrow$
$\displaystyle\|u_{n}+e_{n}\|^{2}-\|z_{n}\|^{2}+2\langle
z,J_{E}z_{n}\rangle-2\langle z,J_{E}(u_{n}+e_{n})\rangle\geq 0.$
Because $E$ is a real Banach space, the inner product of $E$ is linear in both
components and jointly continuous. Therefore, it is easily observed from (3)
that $D_{n}$ is closed and convex for all $n\in\mathbb{N}$. Further, since $A$
is a bounded linear operator, it is obvious that $C_{n}$ is closed and convex
for all $n\in\mathbb{N}$. Also, it is evident that $Q_{n}$ is closed and
convex for all $n\in\mathbb{N}$. Consequently, $C_{n}\cap D_{n}\cap Q_{n}$ is
closed and convex for all $n\in\mathbb{N}$.
Now, it will be shown that $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq
C_{n}$ for all $n\in\mathbb{N}$. In fact, since $Q_{\mu_{n}}^{M_{2}}$ be the
resolvent of $M_{2}$, we have from (2.10) for all $z\in A^{-1}(M_{2}^{-1}0)$
that
$\langle
Q_{\mu_{n}}^{M_{2}}Az_{n}-Az,J_{F}(Az_{n}-Q_{\mu_{n}}^{M_{2}}Az_{n})\rangle\geq
0,\quad\forall n\in\mathbb{N},$
then $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq C_{n}$ for all
$n\in\mathbb{N}$. Next, it will be demonstrated that $M_{1}^{-1}0\cap
A^{-1}(M_{2}^{-1}0)\subseteq D_{n}$ for all $n\in\mathbb{N}$. Indeed, since
$M_{1}$ is maximal monotone, hence from Lemma 2.5, it is implied for all $z\in
M_{1}^{-1}0$ and $\lambda_{n}>0$ that
$\phi_{E}(z,z_{n})=\phi_{E}\big{(}z,J_{\lambda_{n}}^{M_{1}}(u_{n}+e_{n})\big{)}\leq\phi_{E}(z,u_{n}+e_{n}),\quad\forall
n\in\mathbb{N},$
hence, $M_{1}^{-1}0\subseteq D_{n}$ for all $n\in\mathbb{N}$, and therefore
$M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq D_{n}$ for all $n\in\mathbb{N}$.
Now, it will be shown by induction that $M_{1}^{-1}0\cap
A^{-1}(M_{2}^{-1}0)\subseteq Q_{n}$ for all $n\in\mathbb{N}$. Since $\langle
x_{1}-z,J_{E}x_{1}-J_{E}x_{1}\rangle\geq 0$ for all $z\in E$, it is obvious
that $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq Q_{1}=E$. Suppose that
$M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq Q_{k}$ for some
$k\in\mathbb{N}$. Then $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq C_{k}\cap
D_{k}\cap Q_{k}$. From the fact that $x_{k+1}=\Pi_{C_{k}\cap D_{K}\cap
Q_{k}}x_{1}$, it is implied from Lemma 2.3 that
$\langle x_{k+1}-z,J_{E}x_{1}-J_{E}x_{k+1}\rangle\geq 0,\quad\forall z\in
C_{k}\cap D_{k}\cap Q_{k}.$
Since $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq C_{k}\cap D_{k}\cap
Q_{k}$, it is concluded that
$\langle x_{k+1}-z,J_{E}x_{1}-J_{E}x_{k+1}\rangle\geq 0,\quad\forall z\in
M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0).$
So $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq Q_{k+1}$. Hence by induction,
$M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\\\ \subseteq Q_{n}$ for all
$n\in\mathbb{N}$. Thus
$M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq C_{n}\cap D_{n}\cap
Q_{n},\quad\forall n\in\mathbb{N}.$
Hence $C_{n}\cap D_{n}\cap Q_{n}\neq\emptyset$. This implies that
$\\{x_{n}\\}$ is well defined.
By Theorem 2.7 and Definition 2.6, the set $M_{1}^{-1}0\cap
A^{-1}(M_{2}^{-1}0)$ is closed. Next, we show that $M_{1}^{-1}0$ and
$A^{-1}(M_{2}^{-1}0)$ are convex subsets of $E$. Indeed, it is observed that
for all $x_{1},x_{2}\in M_{1}^{-1}0$ and for all $t\in[0,1]$
$\displaystyle\langle 0-v,tx_{1}+(1-t)x_{2}-u\rangle=$ $\displaystyle t\langle
0-v,x_{1}-u\rangle+(1-t)\langle 0-v,x_{2}-u\rangle$ $\displaystyle\geq
0,\quad\forall u\in Dom(M_{1}),v\in M_{1}u,$
hence, by the maximal monotonicity of $M_{1}$, it is implied that
$tx_{1}+(1-t)x_{2}\in M_{1}^{-1}0$. Also, for all $x_{1},x_{2}\in
A^{-1}(M_{2}^{-1}0)$ and for all $t\in[0,1]$ we have
$\displaystyle\langle 0-v,tAx_{1}+(1-t)Ax_{2}-u\rangle=$ $\displaystyle
t\langle 0-v,Ax_{1}-u\rangle+(1-t)\langle 0-v,Ax_{2}-u\rangle$
$\displaystyle\geq 0,\quad\forall u\in Dom(M_{2}),v\in M_{2}u,$
and since $A$ is linear then by the maximal monotonicity of $M_{2}$, it is
implied that $tx_{1}+(1-t)x_{2}\in A^{-1}(M_{2}^{-1}0)$. Thus,
$M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)$ is a convex subset of $E$ and therefore,
by Lemma 2.11, $\Omega$ is closed and convex.
Since $\Omega$ is a nonempty, closed and convex subset of $E$, there exists
$\omega_{0}\in\Omega$ such that $\omega_{0}=\Pi_{\Omega}x_{1}$. Since
$x_{n+1}=\Pi_{C_{n}\cap D_{n}\cap Q_{n}}x_{1}$, it is concluded that
$\phi_{E}(x_{n+1},x_{1})\leq\phi_{E}(y,x_{1}),\quad\forall y\in C_{n}\cap
D_{n}\cap Q_{n},$
and since $\omega_{0}\in\Omega\subseteq\ M_{1}^{-1}0\cap
A^{-1}(M_{2}^{-1}0)\subseteq C_{n}\cap D_{n}\cap Q_{n}$, it is observed that
(3.3) $\phi_{E}(x_{n+1},x_{1})\leq\phi_{E}(\omega_{0},x_{1}).$
This means that $\\{x_{n}\\}$ is bounded.
Conversely, suppose $\\{x_{n}\\}$ is bounded. First, we show that
$\displaystyle\lim_{n\rightarrow\infty}\phi_{E}(x_{n+1},x_{n})=0$. Since
$x_{n+1}\in C_{n}\cap D_{n}\cap Q_{n}$ hence
$\displaystyle\phi_{E}(x_{n+1},x_{n})$
$\displaystyle=\phi_{E}(x_{n+1},\Pi_{C_{n-1}\cap D_{n-1}\cap Q_{n-1}}x_{1})$
$\displaystyle\leq\phi_{E}(x_{n+1},x_{1})-\phi_{E}(\Pi_{C_{n-1}\cap
D_{n-1}\cap Q_{n-1}}x_{1},x_{1})$ (3.4)
$\displaystyle=\phi_{E}(x_{n+1},x_{1})-\phi_{E}(x_{n},x_{1}).$
and hence
$\phi_{E}(x_{n},x_{1})+\phi_{E}(x_{n+1},x_{n})\leq\phi_{E}(x_{n+1},x_{1}),$ so
(3.5) $\phi_{E}(x_{n},x_{1})\leq\phi_{E}(x_{n+1},x_{1}),$
therefore, it is concluded from (2.6), (3.5) that
$\\{\phi_{E}(x_{n},x_{1})\\}$ is bounded and nondecreasing. Then, there exists
the limit of $\\{\phi_{E}(x_{n},x_{1})\\}.$ Using (3), it is realized that
(3.6) $\lim_{n\longrightarrow\infty}\phi_{E}(x_{n+1},x_{n})=0.$
From Lemma 2.2, it is implied that
(3.7) $\lim_{n\longrightarrow\infty}\|x_{n+1}-x_{n}\|=0.$
Next, we show that
$\displaystyle\lim_{n\rightarrow\infty}\phi_{E}(x_{n+1},u_{n})=0$. From the
inequality (2.7), it is implied that
$\displaystyle\phi_{E}(x_{n+1},u_{n})=$
$\displaystyle\phi_{E}\big{(}x_{n+1},J_{E}^{-1}\big{(}(1-\alpha_{n})J_{E}x_{n}$
$\displaystyle+\alpha_{n}J_{E}\Pi_{C}J_{E}^{-1}(\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n})\big{)}\big{)}$
$\displaystyle\leq$ $\displaystyle(1-\alpha_{n})\phi_{E}(x_{n+1},x_{n})$
$\displaystyle+\alpha_{n}\phi_{E}\big{(}x_{n+1},\Pi_{C}J_{E}^{-1}(\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n})\big{)}$
$\displaystyle\leq$ $\displaystyle(1-\alpha_{n})\phi_{E}(x_{n+1},x_{n})$
$\displaystyle+\alpha_{n}\phi_{E}\big{(}x_{n+1},J_{E}^{-1}(\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n})\big{)}$
$\displaystyle\leq$ $\displaystyle(1-\alpha_{n})\phi_{E}(x_{n+1},x_{n})$
$\displaystyle+\alpha_{n}\sigma_{n}\phi_{E}(x_{n+1},W_{n}x_{n})+\alpha_{n}(1-\sigma_{n})\phi_{E}(x_{n+1},Sx_{n}),$
and since from (2.6), $\\{\phi_{E}(x_{n+1},W_{n}x_{n})\\}$ and
$\\{\phi_{E}(x_{n+1},Sx_{n})\\}$ are bounded. hence, using $(i)$ and (3.6), it
is concluded that
(3.8) $\lim_{n\rightarrow\infty}\phi_{E}(x_{n+1},u_{n})=0.$
Therefore, it is realized from Lemma 2.2 that
(3.9) $\lim_{n\rightarrow\infty}\|x_{n+1}-u_{n}\|=0.$
Hence, it is followed from (3.7) and (3.9) that
(3.10) $\lim_{n\rightarrow\infty}\|x_{n}-u_{n}\|=0.$
Furthermore, since $E$ is uniformly smooth and $J_{E}$ is uniformly
continuous, it is implied from (3.10) that
(3.11) $\lim_{n\rightarrow\infty}\|J_{E}x_{n}-J_{E}u_{n}\|=0.$
Also, since $x_{n+1}\in D_{n}$ then
(3.12) $\phi_{E}(x_{n+1},z_{n})\leq\phi_{E}(x_{n+1},u_{n}+e_{n})$
Notice that
$\displaystyle\phi_{E}(x_{n+1},u_{n}+e_{n})-\phi_{E}(x_{n+1},u_{n})=$
$\displaystyle\|u_{n}+e_{n}\|-\|u_{n}\|$ (3.13) $\displaystyle+2\langle
x_{n+1},J_{E}u_{n}-J_{E}(u_{n}+e_{n})\rangle$
Since $J_{E}$ is uniformly continuous on each bounded subset of $E$ and
$\displaystyle\lim_{n\rightarrow\infty}\|e_{n}\|=0$, we know from (3.8) and
(3.12) that
$\displaystyle\lim_{n\rightarrow\infty}\phi_{E}(x_{n+1},u_{n}+e_{n})=0$, which
implies that
(3.14) $\lim_{n\rightarrow\infty}\phi_{E}(x_{n+1},z_{n})=0.$
Therefore, it is implied from Lemma 2.2 that
(3.15) $\lim_{n\rightarrow\infty}\|x_{n+1}-z_{n}\|=0.$
Hence, it is followed from (3.7) and (3.15) that
(3.16) $\lim_{n\rightarrow\infty}\|x_{n}-z_{n}\|=0.$
From (3.10) and (3.16) it is implied that
(3.17) $\lim_{n\rightarrow\infty}\|u_{n}-z_{n}\|=0.$
Let $z\in C_{n}\cap D_{n}\cap Q_{n}$. Using Lemma 2.1, it is concluded that
$\displaystyle\phi_{E}(z,y_{n})=$
$\displaystyle\phi_{E}\big{(}z,\Pi_{C}J_{E}^{-1}\big{(}J_{E}z_{n}-\gamma
A^{*}J_{F}(Az_{n}-\omega_{n})\big{)}\big{)}$ $\displaystyle\leq$
$\displaystyle\phi_{E}\big{(}z,J_{E}^{-1}\big{(}J_{E}z_{n}-\gamma
A^{*}J_{F}(Az_{n}-\omega_{n})\big{)}\big{)}$ $\displaystyle=$
$\displaystyle\|z\|^{2}+\|J_{E}z_{n}-\gamma
A^{*}J_{F}(Az_{n}-\omega_{n})\|^{2}$ $\displaystyle-2\langle
z,J_{E}z_{n}-\gamma A^{*}J_{F}(Az_{n}-\omega_{n})\rangle$ $\displaystyle=$
$\displaystyle\|z\|^{2}+\|J_{E}z_{n}-\gamma
A^{*}J_{F}(Az_{n}-\omega_{n})\|^{2}-2\langle z,J_{E}z_{n}\rangle$
$\displaystyle+2\gamma\langle z,A^{*}J_{F}(Az_{n}-\omega_{n})\rangle$
$\displaystyle\leq$ $\displaystyle\|z\|^{2}+\|J_{E}z_{n}\|^{2}-2\gamma\langle
Az_{n},J_{F}(Az_{n}-\omega_{n})\rangle$
$\displaystyle+c\gamma^{2}\|A\|^{2}\|Az_{n}-\omega_{n}\|^{2}-2\langle
z,J_{E}z_{n}\rangle+2\gamma\langle Az,J_{F}(Az_{n}-\omega_{n})\rangle$
$\displaystyle=$ $\displaystyle\phi_{E}(z,z_{n})-2\gamma\langle
Az_{n}-Az,J_{F}(Az_{n}-\omega_{n})\rangle$
$\displaystyle+c\gamma^{2}\|A\|^{2}\|Az_{n}-\omega_{n}\|^{2}$ $\displaystyle=$
$\displaystyle\phi_{E}(z,z_{n})-2\gamma\langle
Az_{n}-\omega_{n},J_{F}(Az_{n}-\omega_{n})\rangle$
$\displaystyle-2\gamma\langle\omega_{n}-Az,J_{F}(Az_{n}-\omega_{n})\rangle+c\gamma^{2}\|A\|^{2}\|Az_{n}-\omega_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\phi_{E}(z,z_{n})-2\gamma\|Az_{n}-\omega_{n}\|^{2}-2\gamma\langle\omega_{n}-Az,J_{F}(Az_{n}-\omega_{n})\rangle$
$\displaystyle+c\gamma^{2}\|A\|^{2}\|Az_{n}-\omega_{n}\|^{2}$ (3.18)
$\displaystyle\leq$
$\displaystyle\phi_{E}(z,z_{n})-\gamma(2-c\gamma\|A\|^{2})\|Az_{n}-\omega_{n}\|^{2},$
hence, it is implied from the condition $0<\gamma<\dfrac{2}{c\|A\|^{2}}$ that
(3.19) $\phi_{E}(z,y_{n})\leq\phi_{E}(z,z_{n}).$
Also, using (3.19) and (3.14), it is concluded that
(3.20) $\lim_{n\rightarrow\infty}\phi_{E}(x_{n+1},y_{n})=0.$
Therefore, it is followed from (3.7), (3.20) and Lemma 2.2 that
(3.21) $\lim_{n\rightarrow\infty}\|x_{n}-y_{n}\|=0.$
Next, it is evaluated that
$\displaystyle\lim_{n\rightarrow\infty}\|Az_{n}-\omega_{n}\|=0.$ Indeed,
putting $z=x_{n+1}$, from (3), it is observed that
(3.22)
$\gamma(2-c\gamma\|A\|^{2})\|Az_{n}-\omega_{n}\|^{2}\leq\phi_{E}(x_{n+1},z_{n})-\phi_{E}(x_{n+1},y_{n})\leq\phi_{E}(x_{n+1},z_{n}),$
for all $n\in\mathbb{N}$. Since $0<\gamma<\frac{2}{c\parallel
A\parallel^{2}}$, it is concluded from (3.14) that
(3.23) $\lim_{n\rightarrow\infty}\|Az_{n}-\omega_{n}\|=0.$
Since $\\{x_{n}\\}$ is bounded, from (3.10), (3.16), (3.21) and (3.23), the
sequences $\\{u_{n}\\},$ $\\{z_{n}\\},$ $\\{y_{n}\\}$ and $\\{\omega_{n}\\}$
are bounded.
From the condition (iii) and the fact that
$\\{\|J_{E}Sx_{n}-J_{E}W_{n}x_{n}\|\\}$ is bounded, it is observed that
$\displaystyle\lim_{n\rightarrow\infty}\|$
$\displaystyle\big{(}\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n}\big{)}-J_{E}W_{n}x_{n}\|$
$\displaystyle=\lim_{n\rightarrow\infty}\|(1-\sigma_{n})J_{E}Sx_{n}-(1-\sigma_{n})J_{E}W_{n}x_{n}\|$
$\displaystyle=\lim_{n\rightarrow\infty}(1-\sigma_{n})\|J_{E}Sx_{n}-J_{E}W_{n}x_{n}\|=0,$
hence, from the continuity of $J_{E}\Pi_{C}J_{E}^{-1}$, it is understood that
$\displaystyle\lim_{n\rightarrow\infty}\|J_{E}\Pi_{C}J_{E}^{-1}\big{(}\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n}\big{)}-J_{E}\Pi_{C}J_{E}^{-1}(J_{E}W_{n}x_{n})\|$
(3.24)
$\displaystyle=\lim_{n\rightarrow\infty}\|J_{E}\Pi_{C}J_{E}^{-1}\big{(}\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n}\big{)}-J_{E}(W_{n}x_{n})\|=0.$
Now, it is shown that
$\displaystyle\lim_{n\rightarrow\infty}\|J_{E}u_{n}-J_{E}W_{n}x_{n}\|=0$,
$\displaystyle\|J_{E}u_{n}-J_{E}W_{n}x_{n}\|=$
$\displaystyle\big{\|}\big{(}(1-\alpha_{n})J_{E}x_{n}$
$\displaystyle+\alpha_{n}J_{E}\Pi_{C}J_{E}^{-1}\big{(}\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n})\big{)}$
$\displaystyle-J_{E}W_{n}x_{n}\big{\|}$ $\displaystyle=$
$\displaystyle\big{\|}\big{(}(1-\alpha_{n})J_{E}x_{n}-(1-\alpha_{n})J_{E}W_{n}x_{n}\big{)}$
$\displaystyle+\alpha_{n}\big{(}J_{E}\Pi_{C}J_{E}^{-1}\big{(}\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n}\big{)}$
$\displaystyle-J_{E}W_{n}x_{n}\big{)}\big{\|}$ $\displaystyle\leq$
$\displaystyle(1-\alpha_{n})\|J_{E}x_{n}-J_{E}u_{n}\|$
$\displaystyle+(1-\alpha_{n})\|J_{E}u_{n}-J_{E}W_{n}x_{n}\|$
$\displaystyle+\alpha_{n}\big{\|}\big{(}J_{E}\Pi_{C}J_{E}^{-1}\big{(}\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n}\big{)}$
$\displaystyle-J_{E}W_{n}x_{n}\big{)}\big{\|},$
which implies that
$\displaystyle\alpha_{n}\|J_{E}u_{n}-J_{E}W_{n}x_{n}\|\leq$
$\displaystyle(1-\alpha_{n})\|J_{E}x_{n}-J_{E}u_{n}\|$
$\displaystyle+\alpha_{n}\big{\|}\big{(}J_{E}\Pi_{C}J_{E}^{-1}\big{(}\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n}\big{)}$
$\displaystyle-J_{E}W_{n}x_{n}\big{)}\big{\|},$
therefore,
$\displaystyle\|J_{E}u_{n}-J_{E}W_{n}x_{n}\|\leq$
$\displaystyle(1-\alpha_{n})\frac{\|J_{E}x_{n}-J_{E}u_{n}\|}{\alpha_{n}}+\big{\|}\big{(}J_{E}\Pi_{C}J_{E}^{-1}\big{(}\sigma_{n}J_{E}W_{n}x_{n}$
$\displaystyle+(1-\sigma_{n})J_{E}Sx_{n}\big{)}-J_{E}W_{n}x_{n}\big{)}\big{\|}$
now, using (3) and the condition (ii), it is concluded that
$\lim_{n\rightarrow\infty}\|J_{E}u_{n}-J_{E}W_{n}x_{n}\|=0,$
hence, because $E^{*}$ is uniformly smooth, it is induced that
$\lim_{n\rightarrow\infty}\|u_{n}-W_{n}x_{n}\|=0,$
therefore, it is deduced from (3.10) that
(3.25) $\lim_{n\rightarrow\infty}\|x_{n}-W_{n}x_{n}\|=0.$
Since $\\{x_{n}\\}$ is bounded, there exists a subsequence $\\{x_{n_{k}}\\}$
which converges weakly to a point $x^{*}\in E$. First, we show that
$x^{*}\in\bigcap_{i=1}^{\infty}Fix(T_{i})$. To see that, by Theorems 2.9 and
2.10, the mapping $W:C\rightarrow C$ satisfies
(3.26) $\lim_{n\rightarrow\infty}\|W_{n}x^{*}-Wx^{*}\|=0.$
Moreover, from Theorem 2.9, it is followed that
$Fix(W)=\cap_{i=1}^{\infty}Fix(T_{i})$. Assume that
$x^{*}\notin\cap_{i=1}^{\infty}Fix(T_{i})$ then $x^{*}\neq Wx^{*}$ and using
(3.25), (3.26) and Opial’s property of Banach space, it is concluded that
$\displaystyle\liminf_{k\rightarrow\infty}\|x_{n_{k}}-x^{*}\|<$
$\displaystyle\liminf_{k\rightarrow\infty}\|x_{n_{k}}-Wx^{*}\|$
$\displaystyle\leq$
$\displaystyle\liminf_{k\rightarrow\infty}\big{(}\|x_{n_{k}}-W_{n_{k}}x_{n_{k}}\|+\|W_{n_{k}}x_{n_{k}}-W_{n_{k}}x^{*}\|$
$\displaystyle+\|W_{n_{k}}x^{*}-Wx^{*}\|\big{)}$ $\displaystyle\leq$
$\displaystyle\liminf_{k\rightarrow\infty}\|x_{n_{k}}-x^{*}\|.$
which is a contradiction. Therefore, $x^{*}\in\cap_{i=1}^{\infty}Fix(T_{i})$.
Next, it will be checked that $x^{*}\in M_{1}^{-1}0$. From (3.10) and the fact
that $\\{x_{n_{k}}\\}$ converges weakly to $x^{*}$, there exists a subsequence
$\\{u_{n_{k}}\\}$ of $\\{u_{n}\\}$ converging weakly to $x^{*}$ and therefore
from (3.17), it is induced that
$\\{J_{\lambda_{n_{k}}}^{M_{1}}(u_{n_{k}}+e_{n_{k}})\\}$ converges weakly to
$x^{*}$. Also, from $(iv)$ and (3.17), it is implied that
$\lim_{n\rightarrow\infty}\|(u_{n}+e_{n})-J_{\lambda_{n}}^{M_{1}}(u_{n}+e_{n})\|=0,$
hence, since $E$ is uniformly smooth, it is understood that
(3.27)
$\lim_{n\rightarrow\infty}\|J_{E}(u_{n}+e_{n})-J_{E}J_{\lambda_{n}}^{M_{1}}(u_{n}+e_{n})\|=0.$
Since $J_{\lambda_{n}}^{M_{1}}$ is the generalized resolvent of $M_{1}$, it is
observed that
$\frac{J_{E}(u_{n}+e_{n})-J_{E}J_{\lambda_{n}}^{M_{1}}(u_{n}+e_{n})}{{\lambda_{n}}}\in
M_{1}J_{\lambda_{n}}^{M_{1}}(u_{n}+e_{n}),\quad\forall n\in\mathbb{N}.$
From the monotonicity of $M_{1}$, it is deduced that
(3.28)
$\Big{\langle}r-J_{\lambda_{n_{k}}}^{M_{1}}(u_{n_{k}}+e_{n_{k}}),t^{*}-\frac{J_{E}(u_{n_{k}}+e_{n_{k}})-J_{E}J_{\lambda_{n_{k}}}^{M_{1}}(u_{n_{k}}+e_{n_{k}})}{{\lambda_{n_{k}}}}\Big{\rangle}\geq
0,$
for all $(r,t^{*})\in M_{1}$. From (3.27) and the condition
$0<a\leq\lambda_{n_{k}}$, it is followed that $\langle
r-x^{*},t^{*}-0\rangle\geq 0,$ for all $(r,t^{*})\in M_{1}.$ Since $M_{1}$ is
maximal monotone, we have $x^{*}\in M_{1}^{-1}0$.
Next, we show that $x^{*}\in A^{-1}(M_{2}^{-1}0)$. From (3.16) and the fact
that $\\{x_{n_{k}}\\}$ converges weakly to $x^{*}$, there exists a subsequence
$\\{z_{n_{k}}\\}$ of $\\{z_{n}\\}$ converging weakly to $x^{*}$ and since $A$
is bounded and linear, we also have that $\\{Az_{n_{k}}\\}$ converges weakly
to $Ax^{*}$. Therefore, from (3.23), we have
$\\{Q_{\mu_{n_{k}}}^{M_{2}}Az_{n_{k}}\\}$ converges weakly to $Ax^{*}$. Also,
since $F$ is uniformly smooth, it is induced from (3.23) that
(3.29) $\lim_{n\rightarrow\infty}\|J_{F}Az_{n}-J_{F}\omega_{n}\|=0.$
Since $Q_{\mu_{n}}^{M_{2}}$ is the generalized resolvent of $M_{2}$, it is
understood that
$\frac{J_{F}Az_{n}-J_{F}Q_{\mu_{n}}^{M_{2}}Az_{n}}{{\mu_{n}}}\in
M_{2}Q_{\mu_{n}}^{M_{2}}Az_{n},\quad\forall n\in\mathbb{N}.$
From the monotonicity of $M_{2}$, it follows that
(3.30)
$\Big{\langle}b-Q_{\mu_{n_{k}}}^{M_{2}}Az_{n_{k}},f^{*}-\frac{J_{F}Az_{n_{k}}-J_{F}Q_{\mu_{n_{k}}}^{M_{2}}Az_{n_{k}}}{{\mu_{n_{k}}}}\Big{\rangle}\geq
0,\quad\forall(b,f^{*})\in M_{2}.$
From (3.29) and the condition $0<a\leq\mu_{n_{k}}$, it is concluded that
$\langle b-Ax^{*},f^{*}-0\rangle\geq 0,$ for all $(b,f^{*})\in M_{2}$. Since
$M_{2}$ is maximal monotone, it is implied that $x^{*}\in
A^{-1}(M_{2}^{-1}0)$.
Therefore, $x^{*}\in\Omega=M_{1}^{-1}0\cap
A^{-1}(M_{2}^{-1}0)\cap\big{(}\bigcap_{i=1}^{\infty}Fix(T_{i})\big{)}$ hence,
$\Omega\neq\emptyset$.
(b) Now, let $\bar{x}$ be an arbitrary element of $\omega_{\omega}(x_{n})$
(the set of all weak limit point of the sequence $\\{x_{n}\\}$). Then there
exists another subsequence $\\{x_{n_{i}}\\}$ of $\\{x_{n}\\}$ which converges
weakly to $\bar{x}$. Clearly, repeating the same argument, it is implied that
$\bar{x}\in\Omega$. It is claimed that $\bar{x}=x^{*}$. Indeed, suppose that
$\bar{x}\neq x^{*}$. Obviously, from (3.7), the sequences $\\{x_{n}\\}$ is
cauchy and hence the sequences $\\{\|x_{n}-\bar{x}\|\\}$ and
$\\{\|x_{n}-x^{*}\|\\}$ are convergent. Again, using Opial’s property, it is
concluded that
$\displaystyle\lim_{n\rightarrow\infty}\|x_{n}-\bar{x}\|$
$\displaystyle=\liminf_{i\rightarrow\infty}\|x_{n_{i}}-\bar{x}\|<\liminf_{i\rightarrow\infty}\|x_{n_{i}}-x^{*}\|=\lim_{n\rightarrow\infty}\|x_{n}-x^{*}\|$
$\displaystyle=\liminf_{k\rightarrow\infty}\|x_{n_{k}}-x^{*}\|<\liminf_{k\rightarrow\infty}\|x_{n_{k}}-\bar{x}\|=\lim_{n\rightarrow\infty}\|x_{n}-\bar{x}\|,$
this is a contradiction and thus $\bar{x}=x^{*}$. Therefore,
$\omega_{\omega}(x_{n})$ is singleton. Thus $\\{x_{n}\\}$ converges weakly to
$x^{*}\in\Omega$. Since norm is weakly lower semicontinuous, it is implied
from (3.3) that
$\displaystyle\phi_{E}(\omega_{0},x_{1})$
$\displaystyle=\phi_{E}(\Pi_{\Omega}x_{1},x_{1})\leq\phi_{E}(x^{*},x_{1})\qquad\qquad$
$\displaystyle=\|x^{*}\|^{2}-2\langle x^{*},J_{E}x_{1}\rangle+\|x_{1}\|^{2}$
$\displaystyle\leq\liminf_{k\rightarrow\infty}(\|x_{n_{k}}\|^{2}-2\langle
x_{n_{k}},J_{E}x_{1}\rangle+\|x_{1}\|^{2})$
$\displaystyle=\liminf_{k\rightarrow\infty}\phi_{E}(x_{n_{k}},x_{1})$
$\displaystyle\leq\limsup_{k\rightarrow\infty}\phi_{E}(x_{n_{k}},x_{1})\leq\phi_{E}(\omega_{0},x_{1}),$
hence, from the definition of $\Pi_{\Omega}x_{1}$, it is understood that
$\omega_{0}=x^{*}$ and
$\lim_{k\rightarrow\infty}\phi_{E}(x_{n_{k}},x_{1})=\phi_{E}(x^{*},x_{1})=\phi_{E}(\omega_{0},x_{1}).$
So, it is deduced that
$\displaystyle\lim_{k\rightarrow\infty}\|x_{n_{k}}\|=\|\omega_{0}\|$. From the
Kadec-Klee property of $E$, it is concluded that
$\displaystyle\lim_{k\rightarrow\infty}x_{n_{k}}=\omega_{0}$. Therefore,
$\displaystyle\lim_{k\rightarrow\infty}x_{n}=\omega_{0}$. Thus $\\{x_{n}\\}$
converges strongly to $x^{*}$ where $x^{*}=\Pi_{\Omega}x_{1}$. ∎
## 4\. Applications and numerical example
In this section, using Theorem 3.1, a new strong convergence theorem in Banach
spaces will be demonstrated. Let $E$ be a Banach space and $f$ be a proper,
lower semicontinuous and convex function of $E$ into $(-\infty,\infty]$.
Recall the definition of the subdifferential $\partial f$ of $f$ as follows:
$\partial f(x)=\\{z^{*}\in E^{*}:f(x)+\langle y-x,z^{*}\rangle\leq
f(y),\,\,\forall y\in E\\}$
for all $x\in E$. It is known that $\partial f$ is a maximal monotone operator
by Rocfellar [21]. Let $C$ be a nonempty, closed and convex subset of $E$ and
$i_{C}$ be the indicator function of $C$, i.e.,
$i_{C}(x)=\left\\{\begin{array}[]{r}0\quad x\in C,\\\ \,\infty\quad x\notin
C.\end{array}\right.$
Then $i_{C}$ is a proper, lower semicontinuous and convex function on $E$ and
hence, the subdifferential $\partial i_{C}$ of $i_{C}$ is a maximal monotone
operator. Therefore, the generalized resolvent $j_{\lambda}$ of $\partial
i_{C}$ for $\lambda>0$ is defined as follows:
(4.1) $J_{\lambda}x=(J+\lambda\partial i_{C})^{-1}Jx,\quad\forall x\in E.$
For any $x\in E$ and $u\in C$, the following relations are hold:
$\displaystyle u=$ $\displaystyle J_{\lambda}x\Longleftrightarrow Jx\in
Ju+\lambda\partial i_{C}u$
$\displaystyle\Longleftrightarrow\frac{1}{\lambda}(Jx-Ju)\in\partial i_{C}u$
$\displaystyle\Longleftrightarrow i_{C}y\geq\langle y-u,\frac{1}{\lambda}(Jx-
Ju)\rangle+i_{C}u,\,\,\,\forall y\in E$ $\displaystyle\Longleftrightarrow
0\geq\langle y-u,\frac{1}{\lambda}(Jx-Ju)\rangle,\,\,\,\forall y\in C$
$\displaystyle\Longleftrightarrow\langle y-u,Jx-Ju\rangle\leq 0,\,\,\,\forall
y\in C$ $\displaystyle\Longleftrightarrow u=\Pi_{C}x.$
Next, using Theorem (3.1), a strong convergence theorem for finding minimizers
of convex functions in two Banach spaces is demonstrated.
###### Theorem 4.1.
Let $E$ and $F$ be two $2$-uniformly convex and uniformly smooth real Banach
spaces that satisfies the Opial condition and let $J_{E}$ and $J_{F}$ be the
duality mappings on $E$ and $F$, respectively. Let $C$ and $Q$ be nonempty,
closed and convex subsets of $E$ and $F$ respectively. Let $A:E\longrightarrow
F$ be a bounded linear operator such that $A\neq 0$ and with the adjoint
operator $A^{*}$. Suppose that $S:C\longrightarrow E$ be a nonexpansive
mapping and $\\{T_{i}\\}_{i=1}^{\infty}:C\longrightarrow C$ a family of
nonexpansive mappings. For every $n\in\mathbb{N}$, let $W_{n}$ be a
$W-mapping$ generated by Definition 2.8. Let $x_{1}\in C$ and let
$\\{x_{n}\\}$, $\\{u_{n}\\}$ and $\\{y_{n}\\}$ be the sequences generated by
$\displaystyle\begin{cases}u_{n}=J_{E}^{-1}\Big{(}(1-\alpha_{n})J_{E}x_{n}+\alpha_{n}J_{E}\Pi_{C}J_{E}^{-1}(\sigma_{n}J_{E}W_{n}x_{n}+(1-\sigma_{n})J_{E}Sx_{n})\Big{)};\\\
z_{n}=\Pi_{C}(u_{n}+e_{n});\quad\quad\omega_{n}=\Pi_{Q}(Az_{n});\\\
y_{n}=\Pi_{C}J_{E}^{-1}\Big{(}J_{E}z_{n}-\gamma
A^{*}J_{F}(Az_{n}-\omega_{n})\Big{)};\\\ C_{n}=\\{z\in
C,\big{\langle}\omega_{n}-Az,J_{F}(Az_{n}-\omega_{n})\big{\rangle}\geq
0\\};\\\ D_{n}=\\{z\in E,\phi_{E}(z,z_{n})\leq\phi_{E}(z,u_{n}+e_{n})\\};\\\
Q_{n}=\\{z\in E,\langle x_{n}-z,J_{E}x_{1}-J_{E}x_{n}\rangle\geq 0\\};\\\
x_{n+1}=\Pi_{C_{n}\cap Q_{n}\cap D_{n}}x_{1},\quad\forall n\in\mathbb{N}.\\\
\end{cases}$
where $0<\gamma<\frac{2}{c\parallel A\parallel^{2}}$ with $c>0$. Let
$\\{\alpha_{n}\\},\\{\sigma_{n}\\}$ be real sequences in $(0,1)$ satisfying
the conditions
* ($i$)
$\displaystyle\lim_{n\rightarrow\infty}\alpha_{n}=0$;
* ($ii$)
$\displaystyle\lim_{n\rightarrow\infty}\dfrac{\|J_{E}x_{n}-J_{E}u_{n}\|}{\alpha_{n}}=0$;
* (iii)
$\displaystyle\lim_{n\rightarrow\infty}\sigma_{n}=1$;
and the error sequence $\\{e_{n}\\}\subseteq E$ such that
* ($iv$)
$\displaystyle\lim_{n\rightarrow\infty}\|e_{n}\|=0$.
Let $\Omega=C\cap A^{-1}Q\cap\big{(}\bigcap_{i=1}^{\infty}Fix(T_{i})\big{)}$.
Suppose that one of the following two conditions is hold:
* (v)
the sequence $\\{x_{n}\\}$ is bounded,
or
* (vi)
$\Omega\neq\emptyset$.
Then
1. (a)
$\Omega\neq\emptyset$ if and only if the sequence $\\{x_{n}\\}$ is bounded,
2. (b)
The sequence $\\{x_{n}\\}$ converges strongly to a point $\omega_{0}\in\Omega$
where $\omega_{0}=\Pi_{\Omega}x_{1}$.
Next, Theorem 3.1 will be illustrated by an example:
A numerical example Let $E=F=\mathbb{R}$, the set of real numbers, with the
inner product defined by $\langle x,y\rangle=xy,\forall x,y\in\mathbb{R}$, and
usual norm $|\cdot|$. Suppose that $C=[0,1]$ and the mapping
$A:\mathbb{R}\rightarrow\mathbb{R}$ is defined by $A(x)=-2x,\forall
y\in\mathbb{R}$. Let $T_{i}:C\rightarrow C$ be the identity function for each
$i\in\mathbb{N}$ and hence the mapping $W_{n}:C\rightarrow C$ is the identity
function for each $n\in\mathbb{N}$. Also suppose that
$S:C\rightarrow\mathbb{R}$ is the identity function. Let
$M_{1},M_{2}:\mathbb{R}\rightarrow 2^{\mathbb{R}}$ be defined by
$M_{1}(x)=\\{2x\\},\forall x\in\mathbb{R}$ and $M_{2}(y)=\\{3y\\},\forall
y\in\mathbb{R}$. Then $M_{1}^{-1}0\cap A^{-1}(M_{2}^{-1}0)\subseteq C$ and
$\Omega\neq\emptyset$. Let $\\{\alpha_{n}\\}$ and $\\{\sigma_{n}\\}$ be
arbitrary real sequences in $(0,1)$ such that
$\displaystyle\lim_{n\rightarrow\infty}\alpha_{n}=0$ and
$\displaystyle\lim_{n\rightarrow\infty}\sigma_{n}=1$. Let
$\lambda_{n}=\mu_{n}=0.25,\>e_{n}=n^{-1}$ for each $n\in\mathbb{N}$, and
$\gamma=0.1$. Then the sequences
$\\{x_{n}\\},\\{u_{n}\\},\\{z_{n}\\},\\{\omega_{n}\\}$ and $\\{y_{n}\\}$
generated by (3.1) az follows: given initial value $x_{1}\in C$
$\displaystyle\begin{cases}u_{n}=x_{n}\\\
z_{n}=\dfrac{2}{3}(x_{n}+n^{-1});\quad\quad\omega_{n}=\dfrac{-16}{21}(x_{n}+n^{-1});\\\
y_{n}=\Pi_{C}\big{(}\dfrac{116}{210}(x_{n}+n^{-1})\big{)};\\\
C_{n}=\big{\\{}z\in C,z\leq\dfrac{16(x_{n}+n^{-1})}{42}\big{\\}};\\\
D_{n}=\big{\\{}z\in E,z\leq\dfrac{5x_{n}+2n^{-1}}{6}\big{\\}};\\\
Q_{n}=\\{z\in E,(x_{n}-z)(x_{1}-x_{n})\geq 0\\};\\\ x_{n+1}=\Pi_{C_{n}\cap
Q_{n}\cap D_{n}}x_{1},\quad\forall n\in\mathbb{N}.\\\ \end{cases}$
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# Improved Coefficients for the Karagiannidis–Lioumpas
Approximations and Bounds to the Gaussian $Q$-Function
Islam M. Tanash * — and Taneli Riihonen * — Manuscript received November 19,
2020; revised December 18, 2020 and January 8, 2021; accepted January 9, 2021.
Date of publication January 18, 2021; date of current version . This work was
partially supported by the Academy of Finland under Grant 326448. The
associate editor coordinating the review of this letter and approving it for
publication was A.-A. A. Boulogeorgos. (Corresponding author: Islam M.
Tanash.)The authors are with Tampere University, Tampere 33720, Finland
(e-mail<EMAIL_ADDRESS>[email protected]).Digital Object
Identifier 10.1109/LCOMM.2021.3052257
###### Abstract
We revisit the Karagiannidis–Lioumpas (KL) approximation of the $Q$-function
by optimizing its coefficients in terms of absolute error, relative error and
total error. For minimizing the maximum absolute/relative error, we describe
the targeted uniform error functions by sets of nonlinear equations so that
the optimized coefficients are the solutions thereof. The total error is
minimized with numerical search. We also introduce an extra coefficient in the
KL approximation to achieve significantly tighter absolute and total error at
the expense of unbounded relative error. Furthermore, we extend the KL
expression to lower and upper bounds with optimized coefficients that minimize
the error measures in the same way as for the approximations.
###### Index Terms:
Communication theory, error probability.
## I Introduction
Karagiannidis and Lioumpas presented in [1] a relatively tight, yet
analytically tractable, approximation for the Gaussian $Q$-function[2] as
follows:
$\displaystyle\begin{split}Q(x)&\triangleq\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp\left(-{\textstyle\frac{1}{2}}t^{2}\right)\,\mathrm{d}t\\\
&\approx
a\exp\left(-b\,x^{2}\right)\cdot\frac{1-\exp\left(-c\,x\right)}{x}\triangleq\tilde{Q}(x)\end{split}$
(1)
for which their original study sets
$(a,b,c)=(\frac{1}{B\sqrt{2\pi}},\frac{1}{2},\frac{A}{\sqrt{2}})$ and proposes
for error minimization example coefficient values $A=1.98$ and $B=1.135$
rendering $(a,c)\approx(0.3515,1.4001)$.
Despite drawing some criticism [3] shortly after publication, the
‘Karagiannidis–Lioumpas (KL) approximation’ has gradually established itself
as one of the most usable substitutes for the Gaussian $Q$-function in
communication theory problems and the paper [1] has received a large number of
citations; it is only fitting to begin calling the expression after its
inventors.
A diverse set of applications for the KL approximation can be found in [4, 5,
6, 7, 8] to name but a few prominent articles. In general, the approximation
is often used in the calculation of average bit or symbol error probability as
a tractable replacement for the Gaussian $Q$-function such that analysis can
be carried out and completed in a closed form at the cost of making results
tight approximations instead of exact ones. This usually involves integrating
something like, e.g., $f(\tilde{Q}(x(\gamma)))$, where even simple functions
$f(q)$ and $x(\gamma)$, which are derived from the communication system under
study, may forbid exact analysis using the actual $Q$-function[9, 10, 11]. One
should note especially that the approximation is always used in an
intermediate step of analytical derivations and it is not meant for numerical
probability computations per se — instead, rational Chebyshev functions [12]
are perfect to that end.
This Letter is inspired by the fact that the original study [1] presents
explicit values of $a$ and $c$ for only one approximation (which has low
integrated total error when $b=\frac{1}{2}$, to be exact). However, the KL
approximation family is actually much more versatile, whereby new coefficients
can be acquired in terms of other criteria for better accuracy depending on
the application. The KL expression can be also repurposed to achieve lower and
upper bounds (that are also tight approximations) and, in certain cases,
coefficients admit explicit values. Furthermore, by introducing the extra
coefficient $b$ in (1) that originally was $b=\frac{1}{2}$ and permitting
$b<\frac{1}{2}$, we achieve significantly improved accuracy in terms of
absolute and total error.
The objective of this Letter is to apply the KL expression of the $Q$-function
to derive improved approximations and bounds which are global and tight over
$x\geq 0$ by optimizing the coefficients $(a,b,c)$ in respect to their minimum
global absolute or relative error or minimum integrated total error. Like[11]
for another popular expression[9], we present new formulation that minimizes
the maximum global error of (1) by constructing a set of equations, which
describes the corresponding error function, and solve them numerically to find
the optimized coefficients. The total error is optimized with exhaustive
search for reference. In general, when optimizing one of the three criteria,
better performance will be achieved at the expense of decreased accuracy in
terms of the others.
The new coefficients solved herein are applicable as one-to-one replacements
for the original ones of [1] adopted into the analysis of [4, 5, 6, 7, 8] and
many other studies. Literature is rich in approximations/bounds for the
$Q$-function and, typically, the application’s mathematics define, which ones
are tractable for it. Whenever (1) is preferred, our coefficients offer
variety to tailor accuracy for the application or to use bounds. The tractable
series expansion of the KL expression proposed in [13] can likewise be used
with these substitutes (but without guarantee that bounds remain global).
Consequently, they are useful also in the contexts of, e.g., [14, 15, 16, 17,
18, 19, 20] and many others.
## II Preliminaries
The case $x\geq 0$ is presumed throughout this Letter with little loss of
generality because the relation $Q(x)=1-Q(-x)$ extends all the considered
functions to the negative real axis. In fact, this is the main motive for
optimizing approximations and bounds also subject to an additional constraint
$\tilde{Q}(0)=\frac{1}{2}$ that makes their extensions continuous at the
origin like $Q(x)$.
This study solves optimized approximations and bounds for three criteria and
for combinations thereof, viz. $\min_{(a,b,c)}d_{\mathrm{max}}$ (‘minimax
absolute error’), $\min_{(a,b,c)}r_{\mathrm{max}}$ (‘minimax relative error’)
and $\min_{(a,b,c)}d_{\mathrm{tot}}$ (‘integrated total error’ [1]), where
$\displaystyle d_{\mathrm{max}}\triangleq\max_{x\geq
0}|d(x)|,\,r_{\mathrm{max}}\triangleq\max_{x\geq
0}|r(x)|,\,d_{\mathrm{tot}}\triangleq\int_{0}^{\infty}|d(x)|\,\mathrm{d}x,$
and the error functions are defined as
$\displaystyle d(x)$ $\displaystyle\triangleq\tilde{Q}(x)-Q(x),$ (2)
$\displaystyle r(x)$
$\displaystyle\triangleq\frac{d(x)}{Q(x)}=\frac{\tilde{Q}(x)}{Q(x)}-1.$ (3)
For baseline reference, the coefficients originally given in [1] render
$d_{\mathrm{max}}\approx 0.00789$, $r_{\mathrm{max}}\approx 0.119$, and
$d_{\mathrm{tot}}\approx 0.00385$.
As implied above, the presented approximations and bounds will be global ones,
i.e., tight over the whole non-negative real axis (for all $x\geq 0$). The
error functions converge to explicit values, which may be local extrema, at
both ends of this range:
$\displaystyle\lim_{\mathclap{x\to 0}}d(x)$
$\displaystyle=ac-{\textstyle\frac{1}{2}},$
$\displaystyle\>\>\>\>\lim_{\mathclap{x\to 0}}r(x)$ $\displaystyle=2ac-1,$ (4)
$\displaystyle\lim_{\mathclap{x\to\infty}}d(x)$ $\displaystyle=0,$
$\displaystyle\>\>\>\>\lim_{\mathclap{x\to\infty}}r(x)$
$\displaystyle=\begin{cases}\infty,&\text{if }b<{\textstyle\frac{1}{2}},\\\
a\sqrt{2\pi}-1,&\text{if }b={\textstyle\frac{1}{2}},\\\ -1,&\text{if
}b>{\textstyle\frac{1}{2}}.\end{cases}$
The last limit shows especially that global approximations and bounds in terms
of relative error exist if and only if we set $b=\frac{1}{2}$. However, as a
novel fact, our study demonstrates that absolute error and total error can be
instead significantly reduced by permitting $b<\frac{1}{2}$. Therefore, two
scenarios of approximations for the absolute and total error are considered in
this Letter, i.e., approximations with $b=\frac{1}{2}$ or $b<\frac{1}{2}$.
Local error extrema may occur also at critical points, where the derivatives
of the continuous error functions vanish. Denoting differentiation with an
apostrophe, they are given by
$\displaystyle
d^{\prime}(x)=\tilde{Q}^{\prime}(x)-Q^{\prime}(x),\,\,\,r^{\prime}(x)=\frac{\tilde{Q}^{\prime}(x)Q(x)-\tilde{Q}(x)Q^{\prime}(x)}{[Q(x)]^{2}},$
where
$\displaystyle\tilde{Q}^{\prime}(x)$
$\displaystyle=-\dfrac{a\left(\left(2bx^{2}+1\right)\left(\mathrm{e}^{cx}-1\right)-cx\right)\mathrm{e}^{-bx^{2}-cx}}{x^{2}},$
(5) $\displaystyle Q^{\prime}(x)$
$\displaystyle=-\frac{1}{\sqrt{2\pi}}\exp\left(-{\textstyle\frac{1}{2}}x^{2}\right).$
(6)
Two variations of approximations are considered herein: $d(0)=r(0)=0$ and
$d(0)=-d_{\mathrm{max}}$ (resp. $r(0)=-r_{\mathrm{max}}$). The former case
maintains the continuity of the $Q$-function when extending to $x<0$ and
results in $c=\frac{1}{2\,a}$, when substituted in $\lim_{{x\to 0}}d(x)$
(resp. $\lim_{{x\to 0}}r(x)$) that is given in (4). The latter case provides
slightly better accuracy at the cost of discontinuity occurring at $x=0$ and
results in $c=\sqrt{\frac{\pi}{2}}$ in the cases of relative error, by solving
$\lim_{{x\to 0}}r(x)=-r_{\mathrm{max}}$ with
$\lim_{{x\to\infty}}r(x)=-r_{\mathrm{max}}$ that are defined in (4).
## III Alternative Improved Coefficients for (1)
In this section, we describe the methodologies to solve the new coefficients
$(a,b,c)$ for the KL expression. They are optimized either in the minimax
sense or in terms of the integrated total error to yield an approximation, an
upper bound or a lower bound. All the 17 thus-obtained improved/alternative
coefficient sets and accuracy thereof are listed in Table I.
### III-A Global Uniform Approximations and Bounds
The minimax optimization problems are solved in terms of both absolute and
relative errors defined in (2) and (3), respectively, by constructing a set of
nonlinear equations. This set describes the resulting error function, which
should be uniform with equal values for all the extrema points. Each extremum
point yields two equations, where one expresses its value and the other sets
the derivative of the error function to zero at that point. In addition, one
equation (for $d(x)$) or two equations (for $r(x)$) is/are obtained from
evaluating the limits at the two endpoints of the considered range,
$[0,\infty]$, per (4).
The resulting sets of equations, which have equal number of equations and
unknowns, can be solved straightforwardly by any numerical tool for the
considered variations to find the optimized sets of coefficients that satisfy
$\min_{(a,b,c)}d_{\mathrm{max}}$ for the absolute error and
$\min_{(a,b,c)}r_{\mathrm{max}}$ for the relative error. We used iteratively
random initial guesses for the unknowns in this approach, namely $(a,b,c)$,
$d_{\mathrm{max}}$ or $r_{\mathrm{max}}$, and the location of the extrema
($x_{k}$), until fsolve in Matlab converged to the solution, which is
confirmed by substitution. The formulations for the minimax
approximations/bounds are described below.
#### III-A1 Approximations in Terms of Absolute Error
The coefficients $(a,b,c)$ are optimized for approximations in terms of the
absolute error by formulating a set of equations as
$\displaystyle\begin{cases}d^{\prime}(x_{k})=0,&\text{for }k=1,2\text{ or
}1,2,3,\\\ d(x_{k})=(-1)^{k+1}\,d_{\mathrm{max}},&\text{for }k=1,2\text{ or
}1,2,3,\\\ \begin{cases}a\,c=\frac{1}{2},&\text{when }d(0)=0,\\\
a\,c=\frac{1}{2}-d_{\mathrm{max}},&\text{when
}d(0)=-d_{\mathrm{max}},\end{cases}\end{cases}$ (7)
where $x_{k}$ is an extremum point. The number of the error function’s extrema
depends on the value of $b$; if $b$ is fixed to $\frac{1}{2}$, then we have
two extrema, whereas if $b$ is allowed to be any positive value, then we need
three separate extrema.
#### III-A2 Lower Bounds in Terms of Absolute Error
For the lower bounds, we need to find the optimized coefficients which
minimize the global absolute error for $d(x)\leq 0$ when $x\geq 0$. The value
of $b$ must always equal to $\frac{1}{2}$. The tightest resulting uniform
error function will start from $d(0)=-d_{\mathrm{max}}$, with its maximum
equal to zero and its minimum equal to $-d_{\mathrm{max}}$ so that we can
formulate a set of equations as
$\displaystyle\begin{cases}d^{\prime}(x_{1})=d^{\prime}(x_{2})=0,\\\
d(x_{1})=0,\,d(x_{2})=-d_{\mathrm{max}},\\\
a\,c=\frac{1}{2}-d_{\mathrm{max}}.\end{cases}$ (8)
When $d(0)=0$, we get $a=\sqrt{\frac{\pi}{32}}$ and $c=\sqrt{\frac{8}{\pi}}$
by imposing $d^{\prime}(0)=0$ (only in this case), which produces
$a\,c^{2}=\sqrt{2/\pi}$, and solving with $c=\frac{1}{2\,a}$ that results from
setting $d(0)=0$.
#### III-A3 Upper Bounds in Terms of Absolute Error
The set of equations becomes
$\displaystyle\begin{cases}d^{\prime}(x_{1})=d^{\prime}(x_{2})=d^{\prime}(x_{3})=0,\\\
d(x_{1})=d(x_{3})=d_{\mathrm{max}},\,d(x_{2})=0,\\\
a\,c=\frac{1}{2}.\end{cases}$ (9)
In particular, we shape the uniform error function to have three extrema with
$d(x)\geq 0$ when $x\geq 0$ in which its maxima are equal to
$d_{\mathrm{max}}$ and its minimum is equal to zero. The corresponding error
function must always start from $d(0)=0$.
#### III-A4 Approximations in Terms of Relative Error
The targeted uniform error function in terms of the relative error consists of
only one maximum point and converges to $-r_{\mathrm{max}}$ as $x$ tends to
infinity, which results in $-r_{\mathrm{max}}=a\sqrt{2\,\pi}-1$ according to
(4). Therefore, we can formulate the set of equations as
$\displaystyle\begin{cases}r^{\prime}(x_{1})=0,\,r(x_{1})=r_{\mathrm{max}},\\\
\begin{cases}a\,c=\frac{1}{2},&\text{when }r(0)=0,\\\
a\,c=\frac{1-r_{\mathrm{max}}}{2},&\text{when
}r(0)=-r_{\mathrm{max}},\end{cases}\\\
a=\frac{1-r_{\mathrm{max}}}{\sqrt{2\,\pi}}.\end{cases}$ (10)
#### III-A5 Lower Bounds in Terms of Relative Error
We need to find the optimized coefficients, $a$ and $c$, in the minimax sense
for $r(x)\leq 0$ when $x\geq 0$ which converges to $-r_{\mathrm{max}}$ as $x$
tends to infinity. The resulting error function can either start from
$r(0)=-r_{\mathrm{max}}$ to formulate a set of equations as
$\displaystyle\begin{cases}r^{\prime}(x_{1})=r(x_{1})=0,\\\
a\,c=\frac{1-r_{\mathrm{max}}}{2},\,a=\frac{1-r_{\mathrm{max}}}{\sqrt{2\,\pi}},\end{cases}$
(11)
or from $r(0)=0$ yielding $a=\sqrt{\frac{\pi}{32}}$ and
$c=\sqrt{\frac{8}{\pi}}$ like with the corresponding lower bound in terms of
absolute error.
#### III-A6 Upper Bound in Terms of Relative Error
We must ensure that $r(x)\geq 0$ when $x\geq 0$ for the uniform error
function. The resulting error function has only one maximum point and
converges to zero as $x$ tends to infinity. Therefore,
$a=\frac{1}{\sqrt{2\pi}}$ and $c=\sqrt{\frac{\pi}{2}}$ as proposed earlier in
[21] and $b$ is known to be equal to $\frac{1}{2}$. The optimized upper bound
in terms of relative error is also optimal in terms of absolute error and
integrated total error for the case where $b=\frac{1}{2}$.
### III-B Numerical Optimization in Terms of Total Error
Instead of defining
$d_{\mathrm{tot}}\triangleq\int_{0}^{R}|d(x)|\,\mathrm{d}x$ like in [1] and so
making optimized coefficients specific to the value chosen for $R$ and limited
to the range $[0,R]$, we measure total error with $R\to\infty$ and obtain
globally optimized approximations and bounds. In particular, we optimized the
coefficients for the two variations of the approximations with or without
setting $b=\frac{1}{2}$ by performing an extensive search, where we evaluated
the target metric ($d_{\mathrm{tot}}$) over wide one/two/three-dimensional
grids for the unknowns $a$, $(a,b)$, $(a,c)$, or $(a,b,c)$ with granularity of
$0.000001$ and selected the grid point with the minimum total error for each
variation. This renders four sets of optimized coefficients. Extra constraint
checks guarantee $d(x)<0$ for the lower bound and $d(x)>0$ for the upper
bound.
## IV Numerical Results and Conclusions
TABLE I: New coefficients for (1) and approximation error thereof Figure 1:
The improved approximations and bounds compared to the KL approximation with
the original coefficients [1] and to expressions from [9] and [10].
We summarize the improved coefficients for the minimax approximations and
bounds and for the total absolute error in Table I and illustrate their error
functions in Fig. 1, together with the original KL approximation from[1] and
reference approximations and bounds from [9] and [10].111The labels having the
form $Xy$-$n$ in the results refer to the approximations and bounds as
follows: $X$ is U for upper bounds, A for approximations, and L for lower
bounds; whereas $y$ is d for absolute error, r for relative error, and t for
total error; in addition, $n$ refers to rank of the coefficients according to
the accuracy of the absolute error of each variation in an ascending order.
The numerical results show that the improved coefficients of the proposed KL
approximations and bounds are optimal subject to their optimization targets,
yet expressed precisely in implicit form as solutions to systems of nonlinear
equations as opposed to relying on numerical search to minimize error
measures. In some specific cases, a part or even all of the three coefficients
can be expressed as explicit constants. The best approximation/bound from
Table I for a specific application is chosen by contrasting requirements
against Fig. 1, provided that the KL expression (1) is suitable for it to
begin with.
As an ultimate conclusion, the presented data suggests good alternatives to
the original coefficients given in [1] for the case of $b=\frac{1}{2}$: In
some applications, the accuracy of the KL approximation might be improved by
choosing instead $A=1.95$, $B=1.113$ (a compromise between all A$y$-$n$) for
decreasing both absolute and relative error by round $15$% at the cost of
increasing total error by round $65$%; or $A=2.03$, $B=1.162$ (At-4) for
decreasing absolute error and total error by round $10$% and $25$%,
respectively, at the cost of increasing relative error by round $15$%.
Sometimes it may also be useful to choose $A=B\sqrt{\pi}\approx 1.88$,
$B=1.061$ (Ar-5) for minimizing relative error (with round $50$% reduction)
subject to zero error at the origin. In contrast, when primarily minimizing
absolute error, accuracy can be improved significantly by generalizing the KL
approximation to allow any positive $b$: Namely, the choice $a=0.32$,
$b=0.4703$, $c=1.5625$ (Ad-2) guarantees zero error at the origin while
decreasing absolute error and total error as much as round $90$% and $65$%,
respectively, at the cost of making relative error unbounded for large
arguments.
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# The properties of clusters, and the orientation of magnetic fields relative
to filaments, in magnetohydrodynamic simulations of colliding clouds
C. L. Dobbs1, J. Wurster2
1 School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter,
EX4 4QL, UK
2 Scottish Universities Physics Alliance (SUPA), School of Physics and
Astronomy, University of St. Andrews, North Haugh,
St Andrews, Fife KY16 9SS, UK
E-mail<EMAIL_ADDRESS>
###### Abstract
We have performed Smoothed Particle Magneto-Hydrodynamics (SPMHD) calculations
of colliding clouds to investigate the formation of massive stellar clusters,
adopting a timestep criterion to prevent large divergence errors. We find that
magnetic fields do not impede the formation of young massive clusters (YMCs),
and the development of high star formation rates, although we do see a strong
dependence of our results on the direction of the magnetic field. If the field
is initially perpendicular to the collision, and sufficiently strong, we find
that star formation is delayed, and the morphology of the resulting clusters
is significantly altered. We relate this to the large amplification of the
field with this initial orientation. We also see that filaments formed with
this configuration are less dense. When the field is parallel to the
collision, there is much less amplification of the field, dense filaments
form, and the formation of clusters is similar to the purely hydrodynamical
case. Our simulations reproduce the observed tendency for magnetic fields to
be aligned perpendicularly to dense filaments, and parallel to low density
filaments. Overall our results are in broad agreement with past work in this
area using grid codes.
###### keywords:
general, ISM: clouds, stars: formation, galaxies: star clusters: general
††pagerange: The properties of clusters, and the orientation of magnetic
fields relative to filaments, in magnetohydrodynamic simulations of colliding
clouds–References††pubyear: 2012
## 1 Introduction
Recent works have shown that young massive clouds (YMCs) can form through the
collision of molecular clouds (Dobbs et al., 2020; Liow & Dobbs, 2020). Dobbs
et al. (2020) showed that YMCs are able to form on timescales of 1-2 Myr, in
line with observed age spreads (Longmore et al., 2014). Observationally, there
is evidence of cloud cloud collisions in our Galaxy from red and blue shifted
CO velocities in molecular clouds along the line of sight, in some cases at
the sites of massive young clusters (Furukawa et al., 2009; Fukui et al.,
2014; Fukui et al., 2017; Kuwahara et al., 2020). Dobbs et al. (2015) also
found in galaxy scale simulations that such collisions of massive clouds,
although infrequent, do occur. Liow & Dobbs (2020) carried out a parameter
study showing high density, low turbulence and high velocities promote YMC
formation. They also determined the properties of clusters which formed,
showing that for the cloud masses used ($10^{4}-10^{5}$ M⊙) the properties are
comparable to lower mass YMCs in our Galaxy (Portegies Zwart et al., 2010),
though high velocities led to more elongated clouds and larger cloud radii, at
least in the earliest stages of evolution. These previous simulations however
are all purely hydrodynamical. Whether such clusters still form when magnetic
fields are present, and still have the same properties, is an open question.
A number of studies have examined the effects of magnetic fields in
simulations of colliding flows of interstellar gas (Heitsch et al., 2009;
Inoue & Inutsuka, 2009; Körtgen & Banerjee, 2015; Wu et al., 2017; Wu et al.,
2020; Klassen et al., 2017; Fogerty et al., 2016; Fogerty et al., 2017;
Zamora-Avilés et al., 2018; Seifried et al., 2020). Heitsch et al. (2009)
carry out simulations investigating molecular cloud formation, and show that
there is a strong dependence on the initial field direction, with the
collision only inducive to producing molecular clouds when the field is
parallel to the direction of flow. More recent works have included self
gravity and shown the impact of magnetic fields on core and star formation.
Körtgen & Banerjee (2015) find that magentic fields delay core and star
formation, although Zamora-Avilés et al. (2018) find that star formation
occurs earlier with strong magnetic fields, due to suprresion of the non-
linear thin shell instability. Sakre et al. (2020) model core formation and
suggest that stronger fields provide support to allow the formation of more
massive cores (see also Inoue et al. 2018). Sakre et al. (2020) also
investigate field direction and find that starting with an initial field
parallel to the collision produces a more disordered field compared to when
the initial field is perpendicular. Wu et al. (2020) also find that less
fragmentation occurs in models with stronger fields, and there are fewer
stars. Although some studies now explicitly include sink particles (Inoue et
al., 2018; Fogerty et al., 2016; Zamora-Avilés et al., 2018; Fukui et al.,
2020), few investigate the role of magnetic fields on cluster formation.
Filaments are widespread both in the neutral ISM and star forming regions, and
as such many of the above studies also investigate the relation of magnetic
fields to filaments. Recent observations now reveal the alignment of magnetic
fields with structures in the ISM. In particular, observations appear to show
that the magnetic field is typically aligned parallel to filaments in HI
(McClure-Griffiths et al., 2006; Clark et al., 2014; Planck collaboration
XXXII, 2016), and low density molecuar gas (Heyer & Brunt, 2012; Heyer et al.,
2020). Whereas in higher density molecular gas, the field is more likely to be
perpendicular to the filaments (Alves et al., 2008; Heyer & Brunt, 2012;
Planck collaboration XXXV, 2016). Fissel, et al. (2019) show examples of both
parallel and perpendicular alignment within the Vela cloud, traced by 12CO and
13CO respectively. Simulations of turbulence (Soler et al., 2013; Klassen et
al., 2017; Xu et al., 2019) and shock compressed layers (Chen et al., 2016)
also show a tendency for the magnetic field to be aligned perpendicular to the
field at high density, and parallel otherwise. The dependence of the field
orientation may be simply a density criterion (Soler et al., 2013; Chen et
al., 2016), or additionally related to the mass to flux ratio (Seifried et
al., 2020). Inoue & Inutsuka (2016) find that the alignment of the magnetic
field with the filament depends on the initial angle between the shock wave
and the magnetic field.
In this paper we perform simualtions of colliding clouds with magnetic fields,
though our focus is on the formation of massive clusters rather than
individual stars. In Section 2 we describe our method and initial conditions,
and in particular the timestep constraint we apply to ensure the magnetic
divergence remains low. We descibe the morphologies of the collisions, star
formation rates, the relation of magnetic field to filaments and the
properties of clusters formed in Section 3. In Section 4 we compare to
previous work, and in Section 5 we present our conclusions.
## 2 Method
### 2.1 Details of Simulations
We have performed these calculations using Phantom (Price et al., 2018), which
is a publicly available Smoothed Particle Magneto-Hydrodynamics (SPMHD) code.
Sink particles are included according to the method described in Bate et al.
(1995). Magnetic fields are evolved as the magnetic variable $B/\rho$.
Stability of the magnetic fields is ensured using the source term correction
(Børve et al., 2001) and the divergence is constrained using hyperbolic
divergence cleaning (Tricco & Price, 2012; Tricco et al., 2016). Unlike
previous work, we apply a modified timestep criteria based upon the divergence
cleaning method, which we describe in Section 2.2. This timestep constraint,
which is in addition to the usual Courant and acceleration criteria (Price et
al., 2018), ensures that the timesteps are small enough to prevent large
increases in the divergence. For simplicity, and so that when we vary the
velocity field or magnetic field everything else is unchanged, we employ an
isothermal equation of state, adopting a temperature of 20 K.
We set up the initial conditions for our simulations in a similar way to Dobbs
et al. (2020) and Liow & Dobbs (2020), though with a few differences. We
simulate two ellipsoidal colliding clouds, which are colliding head on along
their minor axes. The clouds have dimensions of $80\times 30\times 30$ pc.
Both clouds have masses of $1.5\times 10^{5}$ M⊙. All the simulations we
present use 6.1 million particles. Although not shown, we initially ran
simulations with one tenth the number of particles, which show very similar
results to those we display here. The initial setup of the clouds differs from
our previous simulations in two main ways. The first is that the two clouds
lie within a low density medium, which is one hundredth of the density of the
clouds. This is the same approach as Wurster et al. (2019), who modelled
isolated clouds within a low density medium. The magnetic field permeates both
the clouds and this surrounding medium, preventing the magnetic field becoming
unstable at the edge of the cloud, and removing the need for more complex
magnetic boundary conditions. For simplicity, the surrounding medium has
periodic boundary conditions (satisfying the need for magnetic boundaries),
where magnetohydrodynamic forces are periodic across the boundary but
gravitational forces are not. Including the low density medium ensures that
any increase in the field does not occur at the edge of the simulation region,
since the field does not evolve significantly in the low density region. The
extent of the low density medium is $\pm 120$ pc in the $x$ dimension, and
$\pm 46$ pc in the $y$ and $z$ dimensions. Secondly, following the setup of
the initial conditions in Wurster et al. (2019), the particles are initially
allocated on a grid in both the clouds and the low density surrounding medium,
rather than randomly.
Run | $\sigma$ | Virial | Magnetic field | Direction of field
---|---|---|---|---
| (km s-1) | parameter ($\alpha$) | strength (G) | relative to collision
BWXLowturb | 2.5 | 0.4 | $2.5\times 10^{-7}$ | parallel
BSXLowturb | 2.5 | 0.4 | $2.5\times 10^{-6}$ | parallel
BWYLowturb | 2.5 | 0.4 | $2.5\times 10^{-7}$ | perpendicular
BSYLowturb | 2.5 | 0.4 | $2.5\times 10^{-6}$ | perpendicular
HydroLowturb | 2.5 | 0.4 | 0 | -
BWXHighturb | 5 | 1.7 | $2.5\times 10^{-7}$ | parallel
BSXHighturb | 5 | 1.7 | $2.5\times 10^{-6}$ | parallel
BWYHighturb | 5 | 1.7 | $2.5\times 10^{-7}$ | perpendicular
BSYHighturb | 5 | 1.7 | $2.5\times 10^{-6}$ | perpendicular
HydroHighturb | 5 | 1.7 | 0 | -
Table 1: Table showing the initial configurations for the simulations
performed in this paper. The virial parameter is calculated as the ratio of
the kinetic energy to the gravitational potential energy.
As for the previous simulations (Liow & Dobbs, 2020), we apply a turbulent
velocity field to each cloud. The velocity field is set up to follow a
Gaussian distribution, which produces a power spectrum consistent with
$P(k)\propto k^{-4}$, Burger’s turbulence. In our previous work, we showed
that quite high velocities are required to form massive clusters over short
timescales. Here we only consider one set of collision velocities, and set up
each cloud with a velocity of 21.75 km s-1, such that the total relative
velocity between the clouds is 43.5 km s-1. This velocity is chosen so that
the collision has a significant effect on star formation, i.e. the collision
enhances the star formation rate above that which would occur for isolated
clouds (Dobbs et al., 2020), but is still consistent with the highest
velocities observed for colliding streams in the Milky Way (Motte et al.,
2014; Fukui et al., 2015, 2018). In Liow & Dobbs (2020) we show results with
different cloud dimensions and collision velocities, but here we focus on
varying the magnetic field strength and orientation.
We apply two different field strengths, of $2.5\times 10^{-6}$ and $2.5\times
10^{-7}$ G, and align these fields either parallel or perpendicular to the
collision. We note that particularly our weaker field strength is
unrealistically low compared to observations, but is intended for comparisons.
Our higher field strengths are at the low end of observed field strengths
(e.g. Heiles & Troland 2005; Crutcher et al. 2010). We discuss in Section 5
how we expect the trends we observe to extend to higher strengths. The Alfvén
velocity is $\sim$0.06 and 0.6 km s-1 for the weak and strong fields
respectively, so significantly lower than both the turbulent velocity field,
and collision velocity. We also carry out purely hydrodynamical simulations as
well for comparison. We vary the velocity dispersion of the turbulence, which
produces clouds which are unbound and bound initially. We show the simulations
presented in Table 1. As indicated in Table 1 the main variables in the
simulations are the level of turbulence, which changes the virial parameter,
the magnetic field strength and the orientation of the magnetic field.
We insert sink particles once the density reaches a critical density of
$10^{-18}$ g cm-3 and the criteria in Bate et al. (1995) (e.g. converging
flows) are fulfilled, using an accretion radius of 0.001 pc. With this
resolution, each sink particle typically represents a small group of stars.
Artificial viscosity is included with a switch for the $\alpha$ parameter
(Cullen & Dehnen, 2010). As recommended for strong shocks (Price & Federrath,
2010), we take $\beta=4$. The artificial resistivity is described in Price et
al. (2018).
### 2.2 Divergence cleaning timestep contraint
The magnetic field is evolved as
$\rho\frac{\text{d}}{\text{d}t}\left(\frac{\bm{B}}{\rho}\right)=\left(\bm{B}\cdot\bm{\nabla}\right)\bm{v}-\nabla\psi,$
(1)
where $\rho$ is the gas density, $\bm{B}$ is the magnetic field, $\bm{v}$ is
the velocity and $\psi$ is a scalar field used for divergence cleaning. We
assume units of the magnetic field such that the Alfvén speed is
$v_{\text{A}}=\left|\bm{B}\right|/\sqrt{\rho}$ (Price & Monaghan, 2004). As
per Tricco et al. (2016), the evolved cleaning parameter is
$\psi/c_{\text{h}}$, where $c_{\text{h}}$ is the characteristic speed,
referred to as the ‘wave cleaning speed.’ The evolution of the parameter is
given by111We have modified this slightly from Tricco et al. (2016) to
explicily include the overcleaning parameter $\sigma$, which has a slightly
different definition here. We explicitly note that this $\sigma$ is not the
velocity dispersion.
$\frac{\text{d}}{\text{d}t}\left(\frac{\psi}{c_{\text{h}}}\right)=-\sigma
c_{\text{h}}\left(\bm{\nabla}\cdot\bm{B}\right)-\frac{\sigma
c_{\text{h}}}{h}\left(\frac{\psi}{c_{\text{h}}}\right)-\frac{1}{2}\left(\frac{\psi}{c_{\text{h}}}\right)\left(\bm{\nabla}\cdot\bm{v}\right),$
(2)
where $c_{\text{h}}=\sqrt{v_{\text{A}}^{2}+c_{\text{s}}^{2}}$ with
$c_{\text{s}}$ being the sound speed and $h$ is a scale length (equal to the
smoothing length in SPH). To ensure that the cleaning is resolved, a new
timestep criteria is introduced. In SPH, the new timestep for particle $i$,
given by
$\text{d}t_{\text{clean},i}=\frac{C_{\text{cour}}h_{i}}{2\sigma_{ij}c_{\text{h},i}},$
(3)
where $C_{\text{cour}}=0.3$ is the tradional coefficient for the Courant
condition.
Tricco et al. (2016) introduced the ‘overcleaning’ parameter
$\sigma_{ij}\equiv\sigma$ to control the cleaning. Optimally, $\sigma=1$,
however, larger values could be chosen to reduce divergence errors, albeit at
the accompying cost of shorter timesteps (recall (3)). The divergence error is
monitored by the dimensionless value
$\epsilon_{\text{divB}}=\frac{h|\bm{\nabla}\cdot\bm{B}|}{|\bm{B}|},$ (4)
and they suggest increasing $\sigma$ if the mean value is $>10^{-2}$.
In most simulations where the magnetic field is reasonably well-behaved (e.g.
Orszag & Tang, 1979; Ryu & Jones, 1995; Wurster et al., 2019),
$\left.\epsilon_{\text{divB}}\right|_{\text{mean}}<10^{-2}$ is satisfied using
the default value of $\sigma=1$. However, in very dynamic regions, such as the
interface between colliding flows as presented here, this criteria is
violated; away from the interface, however, this criteria is satisfied.
Therefore, $\sigma>1$ is required for the stability of the magnetic field at
the interface. This requires a careful choice of $\sigma$ prior to starting
the simulation such that it is large enough to properly clean the magnetic
divergence, but low enough such that computational resources are not wasted.
To circumvent this and to prevent extra computational expense away from the
interface, we dynamially calculate $\sigma_{ij}$ based upon a particle’s local
environment. Specifically,
$\sigma_{ij}=\min\left[{\sigma_{\text{max}},\max\left(\sigma,fh_{i}\left.\frac{\left|\bm{\nabla}\cdot\bm{B}\right|}{\left|\bm{B}\right|}\right|_{i},fh_{j}\left.\frac{\left|\bm{\nabla}\cdot\bm{B}\right|}{\left|\bm{B}\right|}\right|_{j}\right)}\right],$
(5)
where $\sigma_{\text{max}}$ is a parameter defining the maximum permitted
$\sigma_{ij}$, $f$ is a scalar, and this minimising operation occurs over all
of $i$’s neighbours, particles $j$. Note that we keep a constant value of
$\sigma$ in the wave cleaning equation, (2). The advantage to this method is
that we do not need to guess a value of $\sigma$ prior to starting the
simulation, and $\sigma_{ij}$ will only increase as needed and where it is
needed, which will save computational resources given our use of individual
timestepping (Bate et al., 1995). Emperical tests of colliding flows similar
to those presented here suggest $f=10$ and $\sigma_{\text{max}}=512$.
Although $\sigma_{ij}$ is dynamically calculated, careful consideration must
be made of $\sigma$ and $\sigma_{\text{max}}$. In the Ryu & Jones MHD shock
tube tests (Ryu & Jones, 1995),
$\left.\epsilon_{\text{divB}}\right|_{\text{max}}<10^{-2}$ meaning that the
new algorithm has no impact and it is safe to use the default values. However,
in the Orszag–Tang vortex (Orszag & Tang, 1979) with 128 particles in the
$x$-direction, the mean value of $\epsilon_{\text{divB}}$ is $\lesssim 0.005$,
while $\left.\epsilon_{\text{divB}}\right|_{\text{max}}\sim\mathcal{O}(1)$. In
this test, setting $\sigma_{\text{max}}=512$ has a trivial affect on the
results (including the value of
$\left.\epsilon_{\text{divB}}\right|_{\text{max}}$), except the simulation
runs $\approx 10$ times slower when performed with global timestepping due to
the decreased d$t_{\text{clean}}$; in this case, it is optimal to set
$\sigma_{\text{max}}=1$, essentially turning off the dynamic calculation of
$\sigma_{ij}$.
Therefore, this new timestep criterion is required when modelling magnetic
fields at strong, chaotic shocks (such as the colliding flows presented here)
to permit a reliable evolution of the magnetic field. However, it should be
disabled when modelling well-behaved magnetic fields.
## 3 Results
### 3.1 Morphology of shocked region, clusters and magnetic field
In this section we look at the overall evolution for the collisions of clouds
with different magnetic field strengths and orientations. We first discuss the
results from the simulations with the clouds with higher virial ratios. The
evolution of the BWXHighturb model is shown in Figure 1, top row. The clouds
start colliding at around 0.5 Myr. The collision leads to a few main central
filaments which are perpendicular to the direction of the collision, as seen
in the left panel of Figure 1. The shapes, number and structure of these
initial filaments are due to the initial turbulent velocity fields of the
colliding clouds. These filaments are gravitationally unstable and as such
sink particles form along the filaments. As the collision progresses, a more
substantial central structure emerges, the number of sink particles increases,
and the distribution of sink particles becomes more clustered rather than
filamentary. At the final time frame, 2.4 Myr, the sink particles become
particularly concentrated to the uppermost region of the collision interface,
leading to a more evident cluster here.
Figure 1: The evolution is shown for the collision of the clouds with the
higher virial parameter (BXWHighturb, top panels), and lower virial parameter
(BXWLowturb, lower panels). In both cases, the magnetic field is $2.5\times
10^{-7}$ G and parallel to the direction of the collision.
Figure 2: The column density, and distribution of sink particles is shown for
the collisions with higher virial parameter clouds. The magnetic field is
parallel (top row) and perpendicular (lower) to the direction of the
collision, and initial field strength is weaker ($2.5\times 10^{-7}$ G) in the
left panels, and stronger ($2.5\times 10^{-6}$ G) in the right hand panels.
Figure 3: This figure shows the same as Figure 2,but with the magnetic field
vectors overlaid.
Figure 4: The collisions of clouds in the purely hydrodynamical models is
shown for the higher virial parameter clouds (left) and lower virial parameter
clouds (right). Both look similar to the MHD simulations, except when the
field is stronger and perpendicular to the collision.
The evolution of the other simulations with higher virial parameter is very
similar to that shown in Figure 1, with the exception of the run with a strong
field perpendicular to the collision (BSYHighturb). The morphology of the
collision for these MHD runs is shown in Figure 2, at a time of 2.3 Myr. As
shown in Figure 2, the simulations with fields parallel to the collision
(aligned along the filament) show very similar morphologies (top row), as does
the simulation with a weak field perpendicular to the collision (lower left),
although there are some small differences in the morphology of the gas, and
the spatial distributions of the sink particles. However the model with a
strong field perpendicular to the collision (BSYHighturb) shows a very
different morphology. Here the presence of a filament along the collision
interface is less clear, and instead sink particles are strongly grouped into
distinct clusters. Most of the sink particles are congregated in a cluster in
the upper region where the clouds have collided. The evolution of the star
formation in this model is also quite different compared to the others, with
fewer stars forming earlier compared to the other simulations.
We show the magnetic field for these models at the same time frame as Figure 2
in Figure 3. As expected, the field is stronger in the models where the
initial field strength is higher. However we also see that the field has
evolved to higher values in the case where the field is originally
perpendicular to the collision. The field is clearly strongest, and has been
considerably more amplified in the simulation with the strong field
perpendicular to the collision. It seems likely that the high magnetic field
in the shocked region where the clouds collide is the reason for the
difference in morphology, and the resulting difference in star formation. By
contrast the model with a weak magnetic field parallel to the collision
(BWXHighturb) shows little amplification of the field in the denser, shocked
regions. The field also becomes more disordered in the region of the shock.
The field becomes most random in the cases where it is initially parallel to
the shock, and the shock appears to increase the component of the field along
the shock whereas in the models where the field is already aligned with the
shock, the effect is more simply to amplify the field in that direction.
We show the equivalent simulation without magnetic fields, HydroHighturb, in
Figure 4 (left panel). The morphology of the gas and distribution of sink
particles is fairly similar to the models with weak magnetic fields, and the
model with a strong field parallel to the shock, although the sink particles
appear more concentrated to one main cluster in the hydrodynamical model
compared to the MHD cases. The concentration of sink particles into one
cluster is more similar to the model with the strong field perpendicular to
the shock, BSYHighturb, although otherwise the morphology of the gas and stars
is quite different, and there appears to be much more star formation and many
more sink particles in the hydrodynamical model (and indeed the other MHD
models) compared to the BSYHighturb model.
We now present results for the models where the clouds have lower virial
parameters. In Figure 1 (lower row) we show the equivalent evolution for the
simulation with a low magnetic field parallel to the direction of the
collision (BWXLowturb). Compared to the case with the higher virial parameter
clouds, there is a clearer shocked region, and clearer filaments where sink
particles are forming. The morphology of the gas, is not too dissimilar to
previous grid code simualtions (e.g. Körtgen & Banerjee 2015). The
distribution of sink particles shows a clear elongated structure. In the last
panel, the cluster of sink particles appears to have contracted and is less
elongated in the direction perpendicular to the collision, due to gravity
acting on the sinks and gas. The evolution of this model appears fairly
similar to those presented in Liow & Dobbs (2020) with $\alpha<1$. Similar to
Liow & Dobbs (2020), with a higher virial parameter, the sinks are more
dispersed, although in the models here there is still a fairly clear cluster
at the location of the shock interface.
Figure 5: The column density, and distribution of sink particles is shown for
the collisions with lower virial parameter clouds. The magnetic field is
parallel (top row) and perpendicular (lower) to the direction of the
collision, and initial field strength is weaker ($2.5\times 10^{-7}$ G) in the
left panels, and stronger ($2.5\times 10^{-6}$ G) in the right hand panels.
In Figure 5 we show the gas surface density plots for the simulations with
lower virial parameters and different magnetic field strengths and directions.
Again the morphologies of the gas and the sink particles distributions appear
similar for three of the simulations, but different for the case with a strong
field perpendicular to the direction of the collision (BSYLowturb). Similar to
the higher virial parameter simulations, with the exception of BSYLowturb,
there is an elongated distribution of sink particles along the shock. By
contrast for the BSYLowturb model, far fewer sink particles appear to form,
and they tend to be concentrated into a smaller cluster region. The
hydrodynamical model, shown in Figure 4 is very similar to the magnetic field
models with the exception of BSYLowturb, suggesting that the morphology of
these other MHD models is closer to the hydrodynamical case than BSYLowturb.
We do not show the magnetic field vectors for the lower virial parameter
models, however the behaviour of the magnetic field is very similar to that
shown in Figure 3. The field is strongest in the simulation with a strong
field perpendicular to the direction of the collision. There is some
amplification of the field for the models with a strong field parallel to the
collision, and for the weak field perpendicular to the collision, but the
field strength appears relatively unchanged with the weak field parallel to
the collision.
In reality, the magnetic field may not be aligned either directly
perpendicular or parallel to the collision. We ran a further model where
instead the field is aligned at 45 degrees to the direction of the collision.
This model shows behaviour that is in between those presented here, i.e. the
morphology and distribution of sinks appears in between the cases with a
strong field parallel, and perpendicular to the collision, and the magnetic
field is amplified to a level in between these two cases.
### 3.2 Star formation rates
Figure 6: The star formation rates are plotted for the collisions where the
clouds have higher virial parameters (top) and lower virial parameters
(lower).
We show in Figure 6 the star formation rates for the different models. The top
panel shows the star formation rates in the models with the higher virial
parameter. The figure shows that the magnetic field does not appear to impede
star formation in most of the models, with the star formation rates extremely
similar to the purely hydrodynamical case. This is not so surprising since the
morphology of these runs is quite similar. The hydrodynamical model has a
slight increase compared to the other models but it is not clear this is
particularly significant. The model with a strong field perpendicular to the
collision (BSYHighturb) however shows quite a different behaviour in the star
formation rate. The star formation increases at $0.5-1$ Myr later compared to
the other models. The star formation rate still reaches values as high as the
models though, and actually appears to accelerate faster than the other models
after the initial delay. We see from Figure 6 that for the panels in Figures
2-5, shown at a time of 2.3 Myr, star formation has been ongoing for around
1.3 Myr.
In the lower panel of Figure 6 we show the star formation rates for the models
with a lower virial parameter. Again the star formation rates are very similar
(almost identical) for all the models except the model with a strong field
perpendicular to the collision, even the purely hydrodynamic simulation. This
again reflects that the morpohology, and sink distributions of all these
models are very similar. Again the model with a strong field perpendicular to
the collision (BSYHighturb) shows a delay in the star formation rate, but then
the star formation rate increases to values as high, or even higher than the
other models. As previously, for a model with a strong field which lies
neither parallel or perpendicular to the collision, the star formation rate
lies between the BSY models and the other MHD and hydrodynamical models.
### 3.3 Magnetic field density relation
Figure 7: The magnitude of the magnetic field is plotted against density for
the collision of clouds with virial parameters (top) and lower virial
parameters (lower). The dashed line shows a $B\propto\rho^{1/2}$ relation. The
dotted lines show the 95th and 5th percentile lines.
In Figure 7 we show the magnetic density relation for the higher virial
parameter (top) and lower virial parameter models (lower) at the same times as
shown in Figures 2 and 5. The figures show a region between densities of
$10^{-21}$ and $10^{-17}$ g cm-3 where the magnetic field scales with the
density with a relation slightly shallower than $B\propto\rho^{1/2}$. The
initial cloud densities are $\sim 2\times 10^{-22}$ g cm-3, with the
background density around 100 times lower. Below densities of $10^{-21}$ g
cm-3, the magnetic field is roughly constant with density, although the
behaviour is more noisy at low densities. Above densities of $10^{-17}$ g
cm-3, there are relatively few particles to reliably infer a relation. The
behaviour of the magnetic field with density is consistent with both Mocz et
al. (2017) and Wurster et al. (2019), even though they model different
environments of a turbulent molecular cloud, and disc formation around
protostars in a smaller scale region of a molecular cloud. They find a
$B\propto\rho^{1/2}$ correlation at higher densities, whilst the magnetic
field is relatively independent at lower densities. The transition however
occurs at lower densities in our simulations, where the gas exhibits a lower
range of densities, compared to Wurster et al. (2019). Similar to Wurster et
al. (2019) and Mocz et al. (2017), we see that the magnetic field density
correlation is largely independent of our initial conditions, where we are
varying initial magnetic field strength, and the initial level of turbulence.
We do see a slight tendency for the relation to extend to lower densities in
the case with the weakest field parallel to the collision, which perhaps
suggests in environments with relatively weaker fields and lower densities the
relation extends to lower densities, in agreement with seeing an offset in the
flattening of $B$ at low densities, compared with the results of Wurster et
al. (2019) and Mocz et al. (2017).
Unlike Wurster et al. (2019), where the magnetic field strengths converge
above densities of $10^{-18}$ g cm-3, we do see offsets in the magnetic field
strength for the different models, although the models with a strong field
parallel (BSX) and weak field perpendicular (BWY) to the shock are quite
similar. The difference is perhaps not surprising, since the evolution is
dominated by a strong shock, and the simulations are far from reaching any
equilibrium in terms of the gas density distribution, dynamics, and magnetic
field. The field strength is highest when the initial magnetic field is
strong, and where the field is perpendicular to the collision and experiences
amplification. The models which satisfy both, or neither of these properties,
represent the outliers for the range of field strengths we use, and across the
full range of magnetic field orientations.
### 3.4 Magnetic field and filaments
Figure 8: These panels show the column density for the central shocked region
(at earlier times in the collision compared to the other figures) for models
BSXHighturb (left) and BSYHighturb (right), and the orientaion of the field
with respsect to the filaments.
In this section we consider further the evolution of the magnetic field, the
impact of the magnetic field on the evolution of the collision, and the
relation of the magnetic field to the filaments formed at the shock interface.
The filaments formed in our simulations typically have lengths in the range
1-10 pc, and large aspect ratios ($>10$). In Figure 8, we show close ups of
the shock region for the models with a strong field initially parallel (left,
BSY), and perpendicular (right, BSX) to the direction of collision. Both
panels are from the models with the higher virial parameter and higher level
of turbulence. At this time (1.1 Myr), the clouds have collided, but there is
relatively little fragmentation of the filaments formed from the shock at this
point. In the case where the field is perpendicular to the collision, but
parallel to the shock, the shock induces an increase in the strength of the
field in this direction, as expected for a fast shock (Ryu & Jones, 1995;
Fukui et al., 2020). Theoretically, we would expect the magnetic field
component parallel to the shock to increase by the same amount as the density.
We see that both the density and magnetic field strength increase by a factor
of several 10’s. The large increase in magnetic pressure leads to a broader
central filament or shocked region, rather than the narrow dense filaments
seen in the other models. Unlike the other models, no sink particles have
formed at this point, with the magnetic pressure preventing gravitational
collapse (though sink particles do form later).
In the left hand panel, the field strengths for the model with a strong
magnetic field initially parallel to the collision, are much weaker. As
expected from theory, there would be no increase in the field if it is
perpendicular to the shock (parallel to the collision). As seen by comparing
the panels in Figure 8, the field is more perpendicular to the filament in the
left hand panel, but parallel to the filament in the right hand panel. For the
left hand panel, there is likely some component of the field parallel to the
shock, simply because the turbulent velocity field means the filaments formed
from the shock are not completely perpendicular. As such this component will
experience some amplification, and in some places the field acquires some
component along the direction of the filament. For the left hand panel, the
central filament has undergone some fragmentation and a few sink particles
have formed along the central filament.
A similar phenomenon whereby the field is parallel to the weaker filaments,
and perpendicular to the denser filaments, was noted in Wurster et al. 2019.
Here we look at this a little more quantitatively. We do this simply by
selecting the gas in the main central filaments in Figure 8, and comparing the
magnetic field in the $y$ direction with the magnitude of the magnetic field.
For the BSX model (left), the density in the filaments increases to $10^{-18}$
g cm-3, the $y$ component of the field is on average $\sim 10^{-5}$ G, whilst
the magnitude of the field is $\sim 3\times 10^{-5}$ G. For the BSY model
(right), the density in the filaments increases to $10^{-20}$ g cm-3, the $y$
component of the field is on average $\sim 1.2\times 10^{-4}$ G, whilst the
magnitude is $\sim 1.5\times 10^{-4}$ G.
After the time shown in Figure 8, the field becomes more aligned with the
filaments in the BSX model, but it is not that long before there is more
widespread fragmentation, the filaments seen in Figure 8 are less clear, and
the field generally becomes more disordered. Similarly for the BSY model, the
field becomes more disordered as many sink particles start to form. For the
weaker field models, the field direction is similar to the original
orientation of the field.
In Table 2 we have summarised possible outcomes, in terms of magnetic field
orientation relative to filaments, and the relative density of the filament,
and what initial setup or conditions correspond to this outcome. The Table
assumes that the filaments are formed as the result of a high velocity
collision, so if the filament formed by an alternative mechanism, it is
possible that other inferences could be made. However we see that for our
models, a low density filament with magnetic field parallel to the filament
occurs if there is a strong magnetic field. The field cannot be perpendicular
because that would lead to a high density filament. If the filament is high
density, and the field parallel, then the field must be weak, because in this
orientation, the field will be amplified, which will lead to a lower density
filament if the field is initially strong. For a high density filament with
magnetic field perpendicular to the filament, the most likely case is that the
field is initially weak, so has little effect on the formation of the
filament. Alternatively if the field is strong, the field must be strongly
perpendicular to the filament. Interestingly we cannot obtain a relation
between magnetic field orientation and filament density, because there is no
one-to-one mapping between these two parameters, as in the high density case
there are multiple initial conditions which produce a field perpendicular to
the filament.
| Field parallel | Field perpendicular
---|---|---
| to filament | to filament
Low | Strong field | Not an outcome
density | |
High | Weak field | Weak field, or strong field
density | (field unimportant) | with no parallel component
Table 2: Possible outcomes for our simulations are shown according to the
column and row labels, and the implication for the field strength.
### 3.5 Properties of clusters
In this section we study the properties of the clusters formed in the
different models. We use the DBSCAN program (Ester et al., 1996) to find the
3D distribution of sink particles, as described in Liow & Dobbs (2020). DBSCAN
is a clustering technique which groups together points with similar densities.
We use the same maximum separation distance as Liow & Dobbs (2020), 0.5 pc. We
list the properties of the most massive cluster found in each simulation in
Table 3. These properties are listed at a time of 2.3 Myr, which is the same
time as shown in Figures 2 and 5. The properties of the clusters for the
clouds with different initial magnetic field configurations are easiest to
describe for the lower virial parameter cloud cases (lower 5 entries). We see
that the properties of the clusters are very similar for all the models,
except that with a strong field perpendicular to the collision (BSYHighturb),
where a 5 times smaller cluster has formed. This is not surprising since by
eye (Figure 2), the distribution of sink particles appears very similar in all
models except the BSYHighturb model, where the distribution is completely
different. By eye, there appear fewer sink particles in the main cluster in
BSYHighturb, which although we need to take into account their masses, would
suggest a lower mass, smaller radius cluster. In Figure 9 we plot the
distribution of sink particles for the BSXLowturb, BWYLowturb, and BSYLowturb
models, and again it is clear that the BSXLowturb and BWYLowturb models have
very similar distributions of sink particles, and likewise the HydroLowturb
and BWXLowturb look very similar to these two panels although not shown.
For the models with the higher virial parameter clouds (top 5 rows in Table
3), there is more variation in the cluster properties. Again the model with
the strong field perpendicular to the collision, BSYHighturb, forms the
smallest cluster, which is again unsurprising given the differences in
morphology between this and other models, and the lower star formation rates
for most of the duration of the simulations. Again, the distribution of sink
particles in this model (Figure 9, right panel) is very different from the
other models (top left and top centre panels). There is surprising variation
in the cluster properties for the other runs. Figure 9 shows the sink particle
distribution for the BSXHighturb and BWYHighturb models. Here we see that the
main accumulation of sink particles towards the top of the panel is identified
as a single cluster in the BSXHighturb model, but only a subsection of this
region is identified in the BWYHighturb model, hence a less massive cluster is
picked out.
The likely difference between the two sets of models is that for the lower
turbulence clouds, the sink particles tend to be located close together for
all the models, and the DBSCAN algorithm readily finds a similar mass and size
cluster in each case. For the higher turbulence clouds, the higher velocities
leads to sink particles, and groups of sink particles which are slightly more
disparate to each other. As we see for the BWYHighturb model (Figure 9 top
middle panel), the large distribution at the top of the panel is separated
into about 3 different groups, one of which is picked out as the most massive
cluster. In the BSXHighturb (Figure 9, top left panel), and likewise
BWXHighturb models, these groups are selected as a single massive cluster. So
for the higher turbulence cases, there is a similar distribution of sink
particles, but the substructuring within them is slightly different. For the
models with the magnetic field parallel to the collision, the substructuring
is less pronounced, whereas for the BWYHighturb model with the field
perpendicular to the collision, the substructuring is more evident. However it
is difficult to say whether the difference is due to the magnetic field, as
for the purely hydrodynamical model, the DBSCAN algorithm also only picks out
a smaller subcluster (though larger than the BWYHighturb model), or in part
due to the random variations in the distributions of the sink particles in
different models.
Overall we see that the addition of a magnetic field tends to make a large
difference to the cluster properties if the field is stronger and
perpendicular to the collision. Otherwise, if the field is parallel, or
weaker, the distribution of sink particles is relatively unchanged on large
scales, although we do see differences on smaller scales which may manifest in
producing different substructure. For our models with bound clouds (lower
virial parameter), we still readily produce massive clusters ($>10^{4}$ M⊙) in
all the models, which is probably not surprising given that they are strongly
bound clouds. We also see that these clusters form on a timescale of $\sim$
1.3 Myr. For the higher virial parameter clouds (except for the strong field
parallel to the collision, BSYHighturb), we still see large congregations of
sink particles with $\sim 10^{4}$ M⊙, though they are not always detected as a
single cluster by our algorithm. For the strong field parallel to the
collision cases, we see less massive clusters, but we note that the star
formation is delayed in these instances. If we instead compare at the same
time since star formation commences, for the higher turbulence model, the mass
of the cluster is 1.4 $\times 10^{4}$ M⊙, so more comparable to the other
models. However the morphology of the BSY model is still very different, and
the resulting most massive cluster is denser and has a smaller half mass
radius compared to the other models.
Run | Mass (104 M⊙) | Radius (pc)
---|---|---
BWXHighturb | 1.62 | 1.2
BSXHighturb | 1.66 | 1.3
BWYHighturb | 0.87 | 0.39
BSYHighturb | 0.33 | 0.12
HydroHighturb | 1.1 | 0.58
BWXLowturb | 5 | 1.3
BSXLowturb | 5.1 | 1.29
BWYLowturb | 4.96 | 1.35
BSYLowturb | 1.2 | 1.0
HydroLowturb | 5.4 | 1.28
Table 3: The properties are listed for the most massive cluster found in each
simulation. As for Liow & Dobbs (2020) the radius listed is the half mass
radius.
Figure 9: The sink particles are shown for the high virial parameter models
(top) and low virial paramters (lower). The red points show the sink particles
which are identified as the most massive cluster, as picked out by the DBSCAN
algorithm.
## 4 Discussion - comparison with previous work
In this section we compare the results of our work to previous simulations.
Previous work in this area including magnetic fields has tended to use grid
based methods. Our finding of a large dependence on the initial direction of
the magnetic field is in agreement with a number of previous works (Heitsch et
al., 2009; Inoue & Inutsuka, 2009; Fogerty et al., 2016). The earlier works
(Heitsch et al., 2009; Inoue & Inutsuka, 2009) did not include self gravity
and focused on molecular cloud formation, but showed that gas densities only
reached values comparable to molecular clouds when the field was parallel to
the direction of collision. Similar to our work, the morphology of the gas
resembles the hydro case when the field is parallel to the collision, but very
different when the field is perpendicular. Körtgen & Banerjee (2015) and
Fogerty et al. (2016) also see a delay in star formation with an inclined
field compared to a field parallel to the collision, similar to the delay we
see. Unlike our work, previous simulations see clearer differences with
stronger and weaker fields when the field is parallel to the collision
(Körtgen & Banerjee, 2015; Sakre et al., 2020; Heitsch et al., 2009), although
we do not probe such high magnetic field strengths, where such differences may
become more apparent. We also see that in our model with a strong field
perpendicular to the collision, fragmentation is strongly compressed, in
agreement with Wu et al. (2020).
In terms of the mass and time of sinks that form, previous simulations find
different results. Inoue et al. (2018) find that the additional magnetic
pressure leads to massive stars forming, whereas Fogerty et al. (2016) and
Körtgen & Banerjee (2015) find the opposite. Zamora-Avilés et al. (2018) find
stars form earlier but Körtgen & Banerjee (2015) and Fogerty et al. (2016) see
a delay. We clearly see a delay in agreement with the latter works. Our sink
particles represent clusters rather than individual stars. At equivalent
times, we see lower mass clusters in the runs with a strong perpendicular
field. However if we take the time since the first sinks formed, the situation
is less clear, and there is some indication that in the perpendicular case,
denser if not necessarily more massive clusters can form.
Our simulations naturally produce filaments where the shock occurs from the
colliding clouds. We see a tendency for magnetic fields to be more aligned
with filaments when the magnetic field impedes the formation of dense gas and
stars. The field is instead perpendicular to the filaments formed when the
magnetic field has little effect. The densities of the filaments tend to be
lower in the first case, and higher in the second case, in agreement with
observations. This finding is comparable to Soler et al. (2013), who
determined the alignment of the magnetic field in different density filaments
formed from shocks in turbulent gas, and related the alignment of the field to
the divergence of the velocity field (Soler & Hennebelle, 2017). In our
simulations we have a much simpler setup whereby we are modelling individual
colliding streams of gas. However we may expect that the outcome for each
filament formed in turbulence will have a similar dependence on the initial
field strength and angle in our simulations. This idea was approached more
rigorously by Inoue & Inutsuka (2009) and Inoue & Inutsuka (2016) who
analytically relate the resultant density of the shock look to the initial
angle and strength of the magnetic field prior to a shock. We at least see
qualitatively similar behaviour in our models, although we note that there is
not necessarily a one to one relationship between filament density and
magnetic field properties.
## 5 Conclusions
We have performed SPMHD simulations of colliding clouds with magnetic fields
to investigate the formation of massive clusters. We apply a timestep
criterion which prevents large divergence errors. Although this criteria is
not usually necessary in SPMHD calculations, we found it was required in the
more extreme conditions of colliding clouds. Our simulations show that
magnetic fields do not inhibit the formation of massive clusters from cloud
cloud collisions, and YMCs can still form on timescales of 1–2 Myr, and the
conclusions from our previous work (Dobbs et al., 2020; Liow & Dobbs, 2020)
hold. Even in the case where the impact of the magnetic field is strongest,
whilst we see a delay in star formation, we then see star formation rates
which are similar to the other models.
Similar to previous work, we find that the initial orientation of the field
has a strong effect on the outcome of the collision, and in our case the
resultant clusters which form. As shown in Inoue & Inutsuka (2009), this can
be related to the conditions of the magnetic field across a shock. If the
field is initially parallel to the collision, it has little effect on the
evolution even in our stronger field case. Thus dense filaments can form in
the gas, and the magnetic field is aligned perpendicular to the filament, as
seen in observations. If the field is initially perpendicular to the collision
(parallel to the shock), then it is significantly amplified by the shock. As
such the magnetic pressure prevents dense filaments forming and delays star
formation, and leads to comparably lower density filaments. The magnetic field
is aligned along the filament, again in agreement with observations. Thus the
formation of the filaments determines the orientation of the field with
respect to the filament. Despite the differences with orientation and field
strength, we still see a $B\propto\rho^{1/2}$ relation across our models,
though the relations are offset from each other.
The influence of the magnetic field on the filaments leads to a corresponding
impact on the clusters which form. In the cases where the field has little
effect, namely when the field is parallel to the collision, or perpendicular
but low strength, the star formation rates and clusters which form have
similar properties to the purely hydrodynamic case. In our model where the
field has a strong effect (perpendicular, higher strength), the formation of
sink particles is delayed, and the resulting appearance and properties of the
clusters which form are quite different. At the same absolute time, the
clusters are considerably smaller compared to the other models. At the same
duration since star formation commences, the clusters are still less massive
though not by so much, but also quite dense.
Our simulations do not include stellar feedback, which we leave to future
work. We would not expect feedback to have a large effect on the gas over the
short timescales which our clusters form (e.g. Howard et al. 2018), though
ionisation may start operating at relatively early times and may also have an
impact on the magnetic field (Troland et al., 2016). We also have used fairly
modest magnetic field strengths. We would expect the trends we find to
continue to higher magnetic field strengths, although we don’t necessarily
relate our simulations to more extreme environments such as the Galactic
Centre, where the dynamics and interstellar radiation field are also very
different.
Finally, we are not aware of simulations similar to those presented here which
have been carried out with SPH, rather than a grid code. It is encouraging
that our simulations produce results, and even filament morphologies, which
are in agreement with previous grid code results.
## Data availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
## Acknowledgments
We thank the referee for providing helpful comments, particularly with regards
some of the observations. We thank Daniel J. Price and Matthew R. Bate for
useful discussions regarding the divergence cleaning timestep. We also thank
Kong Liow and Alex Pettitt for comments. Calculations for this paper were
performed on the ISCA High Performance Computing Service at the University of
Exeter, and used the DiRAC Complexity system, operated by the University of
Leicester IT Services, which forms part of the STFC DiRAC HPC Facility
(www.dirac.ac.uk ). This equipment is funded by BIS National E-Infrastructure
capital grant ST/K000373/1 and STFC DiRAC Operations grant ST/K0003259/1.
DiRAC is part of the National E-Infrastructure. CLD acknowledges funding from
the European Research Council for the Horizon 2020 ERC consolidator grant
project ICYBOB, grant number 818940.
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|
# Scale-invariant cosmology in de Sitter gauge theory
Tomi S. Koivisto<EMAIL_ADDRESS>Laboratory of Theoretical Physics,
Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia
National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143
Tallinn, Estonia Luxi Zheng<EMAIL_ADDRESS>Laboratory of Theoretical
Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411
Tartu, Estonia
###### Abstract
The Planck mass and the cosmological constant determine the minimum and the
maximum distances in the physical universe. A relativistic theory that takes
into account a fundamental distance limit $\ell$ on par with the fundamental
speed limit $c$, is based on the de Sitter extension of the Lorentz symmetry.
This article proposes a new de Sitter gauge theory of gravity which allows the
consistent cosmological evolution of the $\ell$. The theory is locally
equivalent to Dirac's scale-invariant version of general relativity, and
suggests a novel non-singular extension of cosmology.
## I Introduction
In the standard $\Lambda$CDM model of cosmology Aghanim:2018eyx , the
background universe is dS (de Sitter). The dS geometry can be seen as a
4-dimensional hyperboloid of curvature $R_{\Lambda}$ and the radius
$\ell_{\Lambda}=\sqrt{3/\Lambda}=\sqrt{12/R_{\Lambda}}$ embedded in a
5-dimensional Minkowski space. The dS scale introduces a horizon, the maximum
proper distance up to which any signal can reach.
At the other end of scales, the space has a resolution limit given by the
Planck length $\ell_{P}$, since the wavelengths of photons required to probe
smaller distances would have enclosed the photon's energy within its
Schwarzschild radius. There are more refined thought experiments that lead to
the existence of a minimum length, and it is either assumed or predicted in
most of the approaches to quantum gravity Garay:1994en .
An observer-independent scale $\ell$ is naturally incorporated into the
physical theory of the universe by postulating the spacetime symmetry SO(4,1)
instead of the usual ISO(3,1). Analogously to the Galilean group being the
$c\rightarrow\infty$ contraction limit of the Poincaré group ISO(3,1), the
latter is the contraction limit $\ell\rightarrow\infty$ of the dS group
SO(4,1) Dyson:1972sd . An extension of the relativity principle that describes
the kinematics with a finite limiting distance $\ell$ has been formulated as
the projective special Licata:2017dpm , the doubly special
AmelinoCamelia:2002wr and the dS special Aldrovandi:2006vr relativity. The
gravitational theory is obtained by localisation of the symmetry
Westman:2014yca .
In this paper we propose a dS gauge theory with a dynamical dS scale
$\ell(x)$. Since the Planck length $\ell_{P}$ is defined as
$\ell_{P}=\sqrt{\frac{\hbar G}{c^{3}}}\,,$ (1)
where from now on we will set the speed of light $c=1$ and the Planck constant
$\hbar=1$ to unity, the dynamical Planck length $\ell=\ell(x)$ could
equivalently be considered as the dynamical (squareroot of the) Newton's
constant $G=G(x)$. A well-known realisation of this aspect of the theory is
scalar-tensor gravity, which promotes the gravitational coupling into a
dynamical scalar field Brans:1961sx .
Indeed, we will arrive at an action which is equivalent to the conformally
coupled scalar-tensor gravity Dirac:1973gk , and wherein the $\ell$ plays the
role of a dilaton field. While the dilaton is usually introduced in the
context of Weyl gauge theory Blagojevic:2002du ; Blagojevic:2013xpa ;
Scholz:2017pfo , the kinematical origin of scale invariance in a dS gauge
theory seems not to have been clarified previously.
It was shown long ago that dS gravity can be reduced to Einstein's gravity
with a cosmological constant MacDowell:1977jt ; Pagels:1983pq , and nowadays
it has been well understood that the implied symmetry-breaking is but a
realisation of Cartan's original geometrical construction Wise:2006sm . A fine
introduction to the dynamical symmetry breaking in Cartan geometry, and the
most general polynomial form of such a theory, were presented in
Westman:2014yca . A physical observer requires the further breaking
Gielen:2012fz of SO(4,1)$\rightarrow$SO(3,1)$\rightarrow$SO(3), where the
final step could be the geometrical origin of cold dark matter Zlosnik:2018qvg
, the CDM part of the $\Lambda$CDM Aghanim:2018eyx .
This framework provides a robust approach also to the problems with the
$\Lambda$ Pagels:1983pq , whilst paving the way towards reconciliation of
gravity and quantum mechanics by lifting the kinematics of dS special
relativity Aldrovandi:2006vr to the dynamics of dS general relativity
Aldrovandi:2007bi . The proper formulation of a dS gauge theory as a Cartan
geometry where the homogeneous model spaces are flat and their scale $\ell$ is
a function of the coordinates in the quotient dS spacetime has been already
developed by Jennen and Pereira Jennen:2014mba ; Jennen:2015bxa ;
Jennen:2016pqw .
In this article we propose a completion and generalisation of their theory and
explore its cosmological solutions. We begin in Section II by reviewing the
gauge theory of the dS group in Cartan geometry, now specifically adapted to
cosmology. In Section III we study the cosmological background solutions and
derive a family of exact solutions in the presence of perfect fluid which, as
will be argued, should be non-minimally coupled to the dS distance scale
$\ell$. Despite the apparently non-minimal coupling, the theory is
phenomenologically viable and does not lead to drastic violations of the
equivalence principle. We clarify this in Section IV, where it is shown that
particles move along geodesics of an integrable Weyl geometry. For
completeness we also consider a scalar field coupled to the dS gauge theory,
in order to confirm the consistency of the cosmological set-up beyond the
perfect fluid parameterisation. The conclusions are stated in Section V.
## II The dS gauge theory
The dS space is considered as the 4-dimensional quotient of the dS group by
the Lorentz group. Consequently, in this Section we will be referring to
various distinct sets of coordinates. For clarity, the following table
coordinates | algebra | metric
---|---|---
$\\{x^{i}\\}_{i=1,2,3}$ | $\mathfrak{so}(3)$ | $\delta_{ij}=\text{diag}(1,1,1)$
$\\{x^{a}\\}_{a=0,1,2,3}$ | $\mathfrak{so}(3,1)$ | $\eta_{ab}=\text{diag}(-1,1,1,1)$
$\\{X^{A}\\}_{A=0,\dots,4}$ | $\mathfrak{so}(4,1)$ | $\eta_{AB}=\text{diag}(-1,1,1,1,1)$
$\\{x^{\mu}\\}_{\mu=0,1,2,3}$ | ${\mathfrak{gl}_{\parallel}(3,1)}$ | $g_{\mu\nu}$ locally $\eta_{\mu\nu}$
summarises our conventions.
The generators of the dS algebra $\mathfrak{so}(4,1)$ satisfy the commutation
relations
$[\Omega_{AB},\Omega_{CD}]=2\left(\eta_{D[A}\Omega_{B]C}-\eta_{C[A}\Omega_{B]D}\right)\,,$
(2)
with $\eta_{AB}$ as given above. The 10 distinct generators
$\Omega_{AB}=-\Omega_{BA}$ can be interpreted as spacetime rotations in 5
dimensions, whilst our spacetime has the 4-dimensional tangent space with the
coordinates $\\{x^{a}\\}_{a=0,1,2,3}$ and the metric $\eta_{ab}$. Therefore we
consider the 4-dimensional rotations to the generated by $\Omega_{ab}$ that
coincide with the corresponding $\Omega_{AB}$, but define the rotations around
the 5${}^{\text{th}}$ dimension as
$\Pi_{a}=\ell^{-1}\Omega_{4a}\,.$ (3)
The 4 generators $\Pi_{a}$ will be interpreted as (generalised) translations.
The algebra inherited from (2) by the new generators is
$\displaystyle\left[\Omega_{ab},\Omega_{cd}\right]$ $\displaystyle=$
$\displaystyle 2\left(\eta_{d[a}\Omega_{b]c}-\eta_{c[a}\Omega_{b]d}\right)\,,$
(4a) $\displaystyle\left[\Pi_{a},\Omega_{bc}\right]$ $\displaystyle=$
$\displaystyle 2\eta_{a[b}\Pi_{c]}\,,$ (4b)
$\displaystyle\left[\Pi_{a},\Pi_{b}\right]$ $\displaystyle=$
$\displaystyle-\ell^{-2}\Omega_{ab}\,.$ (4c)
The $\ell$ is a dimensionful parameter that quantifies how much boost along
the 5${}^{\text{th}}$ dimension is needed for a unit translation. In the limit
$1/\ell\rightarrow 0$, (4) reduces to the Poincaré algebra
$\mathfrak{iso}(3,1)$ and the $\Pi_{a}$ become ordinary translations.
In general, the form of $\Pi_{a}$ will depend upon the geometry of the
symmetry breaking. To illustrate the embedding of the hyperboloid, let us
consider here the flat slicing
$\displaystyle X^{0}$ $\displaystyle=$
$\displaystyle\ell\sinh{(t/\ell})+e^{t/\ell}\delta_{ij}x^{i}x^{j}/2\ell\,,$
(5a) $\displaystyle X^{i}$ $\displaystyle=$ $\displaystyle e^{t/\ell}x^{i}\,,$
(5b) $\displaystyle X^{4}$ $\displaystyle=$
$\displaystyle\ell\cosh{(t/\ell})-e^{t/\ell}\delta_{ij}x^{i}x^{j}/2\ell\,,$
(5c)
since then the induced metric $g_{ab}$ has the most commonly used cosmological
(isotropic and homogeneous) form
$\eta_{AB}\textrm{d}X^{A}\textrm{d}X^{B}=g_{ab}\textrm{d}x^{a}\textrm{d}x^{b}=-\textrm{d}t^{2}+e^{2t/\ell}\delta_{ij}\textrm{d}x^{i}\textrm{d}x^{j}\,.$
By inverting (5),
$\displaystyle t\equiv x^{0}$ $\displaystyle=$
$\displaystyle\ell\log{(X^{0}+X^{4})}-\ell\log{(\ell)}\,,$ (6a) $\displaystyle
x^{i}$ $\displaystyle=$ $\displaystyle\ell X^{i}/(X^{0}+X^{4})\,.$ (6b)
We find the conformal relations between the basis vectors,
$\displaystyle\frac{\partial}{\partial X^{a}}$ $\displaystyle=$ $\displaystyle
e^{-t/\ell}\partial_{a}\,,$ (7a) $\displaystyle\frac{\partial}{\partial
X^{4}}$ $\displaystyle=$ $\displaystyle
e^{-t/\ell}\left(\partial_{t}-\ell^{-1}x^{i}\partial_{i}\right)\,.$ (7b)
Using the dictionary (5,6,7) we can easily write down the orbital generators,
$\displaystyle\Omega_{i0}$ $\displaystyle=$ $\displaystyle
2x_{[i}\partial_{0]}+(t+\ell\alpha)\partial_{i}\,,$ (8a)
$\displaystyle\Omega_{ij}$ $\displaystyle=$ $\displaystyle
2x_{[i}\partial_{j]}\,,$ (8b) $\displaystyle\Pi_{0}$ $\displaystyle=$
$\displaystyle\partial_{t}-\alpha\ell^{-1}x^{i}\partial_{i}\,,$ (8c)
$\displaystyle\Pi_{i}$ $\displaystyle=$
$\displaystyle\beta\partial_{i}-\ell^{-1}x_{i}\left(\partial_{t}-\ell^{-1}x^{k}\partial_{k}\right)\,,$
(8d)
where we used the short-hands
$\displaystyle\alpha$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(1+\ell^{-2}\delta_{jk}{x^{j}x^{k}}-e^{-2t/\ell}\right)=e^{-t/\ell}X^{0}/\ell\,,$
$\displaystyle\beta$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(1-\ell^{-2}\delta_{jk}{x^{j}x^{k}}+e^{-2t/\ell}\right)=e^{-t/\ell}X^{4}/\ell\,.$
As a cross-check, we verified that (4) is satisfied by (8).
An interesting conformal structure is manifest in the stereographic embedding
Aldrovandi:2006vr alternative to the flat slicing (5), though it will be
shown elsewhere that the Beltrami geometry Guo:2003qm is convenient for the
representations111In both pictures the $\Omega_{ab}$ have their usual form,
the “transvection” $\Pi_{a}$ becoming a translation contaminated with, in the
stereographic coordinates the special conformal Pereira:2013zxa , and in the
Beltrami coordinates (as well as in the flat slicing) the ordinary conformal
transformation..
To gauge the $\mathfrak{so}(4,1)$, we now introduce the connection 1-form
$\bm{A}=\frac{1}{2}\bm{A}^{AB}\Omega_{AB}=\left(\frac{1}{2}\bm{A}^{ab}{}_{\mu}\Omega_{ab}+\bm{A}^{a}{}_{\mu}\Pi_{a}\right)\textrm{d}x^{\mu}\,.$
(10)
The connection determines the dS-covariant (exterior) derivative
$\textrm{D}=\textrm{d}+\bm{A}$, which further generates the field strength
2-form $\textrm{D}^{2}=\textrm{d}\bm{A}+\bm{A}\wedge\bm{A}\equiv\bm{F}$ and
the 3-form identity $\textrm{D}^{3}=\textrm{D}\bm{F}=0$. It is crucial to note
that because of the definition (3), we have $\bm{A}^{a}=\ell\bm{A}^{4a}$ and
consequently Jennen:2014mba ; Jennen:2015bxa ; Jennen:2016pqw
$\ell
A^{a4}{}_{\mu,\alpha}=A^{a}{}_{\mu,\alpha}-\log{\ell}_{,\alpha}A^{a}{}_{\mu}\,.$
(11)
As a result, the components of the field strength are slightly modified,
$\displaystyle F^{a}{}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\left(A^{a}{}_{[\nu,\mu]}+A^{a}{}_{b[\mu}A^{b}{}_{\nu]}\right)-2\log{\ell}_{,[\mu}A^{a}{}_{\nu]}\,,\,\,\,$
(12a) $\displaystyle F^{ab}{}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\left(A^{ab}{}_{[\nu,\mu]}+A^{a}{}_{c[\mu}A^{cb}{}_{\nu]}-\ell^{-2}A^{[a}{}_{\mu}A^{b]}{}_{\nu}\right)\,.\,\,\,$
(12b)
Thus, a novel term that depends on the dynamics of the scale field $\ell$, now
appears in the translation gauge field strength.
In addition to the connection 1-form (10), a symmetry-breaking scalar field
$\xi^{a}$ is required. Otherwise one cannot introduce the coframe field
$\bm{\mathrm{e}}^{a}$, which is the 1-form defined by
$\bm{\mathrm{e}}^{a}=\bm{A}^{a}+\textrm{D}\xi^{a}\,.$ (13)
From this object, we further obtain the torsion 2-form
$\bm{T}^{a}=\textrm{D}\bm{\mathrm{e}}^{a}$ and the 3-form identity
$\textrm{D}\bm{T}^{a}=\bm{F}^{a}{}_{b}\wedge\bm{\mathrm{e}}^{b}$. One sees
that the translation gauge field strength coincides with the torsion 2-form
once the Lorentz curvature $\bm{F}^{a}{}_{b}=0$ is taken to vanish, since from
(13) we have that $\bm{T}^{a}=\textrm{D}\bm{A}^{a}+\bm{F}^{a}{}_{b}\xi^{b}$.
Assuming that the coframe field has an inverse, all the standard ingredients
of gravitational geometry can now be constructed. In the language of Ref.
Westman:2014yca , our fundamental fields are $V^{A}$ and $\bm{A}^{AB}$, and
$\ell$ corresponds to the norm of the $V^{A}$ and the $\xi^{a}$ to the rest of
its independent components, such that the definition (13) ensures the co-
covariance of the components of
$\bm{\mathrm{e}}^{a}=\mathrm{e}^{a}{}_{\mu}\textrm{d}x^{\mu}$ and its inverse.
Then we can freely project the tangent space indices to spacetime indices and
vice versa.
In particular, we obtain the the spacetime torsion tensor
$T^{\alpha}{}_{\mu\nu}$, and can then construct an action for a translation
gauge theory in terms of the invariant $T$ known as the torsion scalar,
$T=\frac{1}{4}T^{\alpha}{}_{\mu\nu}T_{\alpha}{}^{\mu\nu}+\frac{1}{2}T^{\alpha}{}_{\mu\nu}T^{\nu\mu}{}_{\alpha}-T^{\nu}{}_{\mu\nu}T^{\alpha\mu}{}_{\alpha}\,.$
(14)
If we denote $\bar{\bm{T}}^{a}$ the torsion 2-form in the limit when the
evolution of dS scale is neglected, (12b) tells that
$\bm{T}^{a}=\bar{\bm{T}}^{a}-\textrm{d}\log{\ell}\wedge\bm{A}^{a}$. Plugging
this into (14) we obtain that
$T=\bar{T}+4\log{\ell}_{,\mu}\bar{T}^{\mu}-6\left(\partial\log{\ell}\right)^{2}\,,$
(15)
where we defined $T_{\mu}=T^{\alpha}{}_{\mu\alpha}$. The equivalent result was
reported in Jennen:2015bxa . However, our action integral over this scalar,
$I_{\text{dS}}=-\frac{1}{2}\int\textrm{d}^{4}x\mathrm{e}\left[\ell^{-2}\bar{T}+4\ell^{-3}\ell_{,\mu}\bar{T}^{\mu}-6\ell^{-4}\left(\partial{\ell}\right)^{2}\right]\,,$
(16)
has now different scalings for each of the terms, due to the dilatonic role of
the dS scale. This action turns out to be equivalent to the conformally
coupled scalar-tensor theory Dirac:1973gk . By recalling that the metric Ricci
scalar ${R}$ is related to the torsion scalar via
${R}=-\bar{T}-2\text{}\partial_{\mu}(\mathrm{e}\bar{T}^{\alpha})$, we can
rewrite (16) in the much more conventional (though pedantically speaking, ill-
defined due to higher derivatives) scalar-tensor form
$I_{\text{dS}}=\frac{1}{2}\int\textrm{d}^{4}x\mathrm{e}\left[\ell^{-2}{R}+6\ell^{-4}\left(\partial{\ell}\right)^{2}\right]\,.$
(17)
It is well-known that this theory is invariant under the Weyl rescalings222A
complete classification of scale invariance(s) in the general metric-affine
geometry was given in Ref. Iosifidis:2018zwo . Scale transformations in
torsional geometry have been considered in e.g. Maluf:1985fj ; Maluf:2011kf ;
Bamba:2013jqa ; Wright:2016ayu ; Lucat:2017wtu ; Barnaveli:2018dxo ;
Formiga:2019frd . Dirac:1973gk
$g_{\mu\nu}\rightarrow f^{2}g_{\mu\nu}\,,\quad\ell\rightarrow f\ell\,.$ (18)
This scale invariance allows to reduce the theory explicitly to general
relativity in the gauge $f=\ell_{P}/\ell$, but the symmetry (18) means that is
an equivalence holds regardless of the gauge choice.
### II.1 On alternative formulations
It could also be interesting to reconsider the geometrical foundation
BeltranJimenez:2019tjy of the above formulation. In particular, since the
model spaces are characterised by different scales, it may not be justified to
consider the generators to be independent of the coordinates $x^{\mu}$. In
particular, as we see in the cosmology-motivated example (8), the spacetime
dependence enters into the generators via the $\ell=\ell(x)$. The potential
problem with this could however be avoided by reformulating the theory in a
torsion-free geometry. Then one would to begin, instead of (3), with
generators defined by the opposite scaling
$\displaystyle\hat{\Pi}_{a}$ $\displaystyle=$
$\displaystyle\Omega_{4a}=\hat{\eta}_{ab}\Pi^{b}\,,$ (19a)
$\displaystyle\hat{\Omega}_{ab}$ $\displaystyle=$
$\displaystyle\ell^{2}\Omega_{AB}\delta^{A}_{a}\delta^{B}_{b}=\hat{\eta}_{ac}\Omega^{c}{}_{b}\,.$
(19b)
The same algebra (4) in terms of the newly defined generators has to be then
written in terms of the conformally rescaled metric
$\hat{\eta}_{\mu\nu}=\ell^{2}\eta_{ab}$ as
$\displaystyle\left[\hat{\Omega}_{ab},\hat{\Omega}_{cd}\right]$
$\displaystyle=$ $\displaystyle
2\left(\hat{\eta}_{d[a}\hat{\Omega}_{b]c}-\hat{\eta}_{c[a}\hat{\Omega}_{b]d}\right)\,,$
(20a) $\displaystyle\left[\hat{\Pi}_{a},\hat{\Omega}_{bc}\right]$
$\displaystyle=$ $\displaystyle 2\hat{\eta}_{a[b}\hat{\Pi}_{c]}\,,$ (20b)
$\displaystyle\left[\hat{\Pi}_{a},\hat{\Pi}_{b}\right]$ $\displaystyle=$
$\displaystyle-\ell^{-2}\hat{\Omega}_{ab}\,.$ (20c)
This indeed suggests a relation to the Weyl gauge theory and a rationale for
the emergence of the scale symmetry (18). In this basis, the theory can be
formulated consistently using the stereographic projection where the induced
metric is conformally flat and the rotations are $\ell$-independent. In the
end, the $\textrm{d}\log{\ell}$ term does not appear in the
$\hat{\bm{T}}^{a}$, but a corresponding term is found in the
$\hat{\bm{F}}^{a}{}_{b}$, and one can write down the usual quadratic curvature
action that reduces to the dS general relativity. We will not pursue here the
details of this formulation (presumably resulting in the equivalent (17)). In
fact, the geometry of the canonical version would be both torsion-free and
flat Koivisto:2019jra ; BeltranJimenez:2020sih , but such a formulation would
require the enlarging of the gauge group and be superfluous for the present
purpose.
## III Cosmological solution
Cosmologies inspired by the Jennen-Pereira model Jennen:2015bxa were analysed
as a dynamical system by Otalora Otalora:2014aoa . However, the class of
models studied therein includes neither the particular case of (16) nor the
version of Ref. Jennen:2015bxa (obtained from 15), because, firstly, the sign
of the scalar field kinetic term in Ref. Otalora:2014aoa was flipped, and
secondly, because the trace-coupling was considered as a function of the
scalar field333On more general scalar-torsion modified gravity, see e.g.
Bamba:2013jqa ; Hohmann:2019gmt ; Emtsova:2019qsl ; Flathmann:2019khc ;
Golovnev:2018wbh ; Raatikainen:2019qey ; Bahamonde:2020cfv ; Hohmann:2020dgy .
However, already linear perturbations Hohmann:2020vcv indicate
Golovnev:2018wbh ; Raatikainen:2019qey that generic such models are not
viable. This stems from their Lorentz violation Li:2010cg .. We shall now
explore the cosmology of the dS gauge theory (16), and find that is
qualitatively different from models of scalar-torsion modified gravity.
The line element in the flat Friedmann-Lemaître-Robertson-Walker cosmology is
$\textrm{d}s^{2}=-n^{2}(t)\textrm{d}t^{2}+a^{2}(t)\delta_{ij}\textrm{d}x^{i}\textrm{d}x^{j}\,,$
(21)
where $n$ is the lapse function and $a$ the scale factor. In addition to these
two metric components (of which $n$ can always be trivialised by simply a
redefinition of $t$), we have the dS scale $\ell$ which may now evolve in
time. We'll denote the expansion rates as follows:
scale factor | variable | rate
---|---|---
temporal | $n(t)$ | $N=\dot{n}/n$
spatial | $a(t)$ | $H=\dot{a}/a$
dimensional | $\ell(t)$ | $L=\dot{\ell}/\ell$
Including a matter source $I_{M}$ describing a perfect fluid with the energy
density $\rho_{M}$ and pressure $p_{M}$ coupled to the dS gravity (16), the
cosmological mini-superspace action becomes
$I=I_{\text{dS}}+I_{M}=-\int\textrm{d}t\left[\frac{3a^{3}}{\ell^{2}n}\left(H-L\right)^{2}+na^{3}\rho_{M}\right]\,.$
(22)
The 1${}^{\text{st}}$ and the 2${}^{\text{nd}}$ Friedmann equations (obtained
from the variations of $I$ wrt $n$ and $a$, respectively), can now be written
as
$\displaystyle 3H^{2}$ $\displaystyle=$ $\displaystyle{\left(\ell
n\right)^{2}}\left(\rho_{\ell}+\rho_{M}\right)\,,$ (23a)
$\displaystyle-2\dot{H}-3H^{2}+2NH$ $\displaystyle=$ $\displaystyle\left(\ell
n\right)^{2}\left(p_{\ell}+p_{M}\right)\,,$ (23b)
where
$\displaystyle\left(\ell n\right)^{2}\rho_{\ell}$ $\displaystyle=$
$\displaystyle-3L^{2}+6HL\,,$ $\displaystyle\left(\ell n\right)^{2}p_{\ell}$
$\displaystyle=$ $\displaystyle-2\dot{L}-4HL+L^{2}+2NL\,.$
In vacuum, $\rho_{M}=0$, with the time slicing $n=1$, the solutions are
$\ell/a=\text{constant}$. This already yields the insight into the theory that
only the relative calibration of the two scale factors is fixed in vacuum, and
neither of the scale factors alone.
To properly couple matter sources to dS gravity with an evolving $\ell$, we
should take into account the scaling of the energy density with $\ell$. For
the purposes of background cosmology, the energy density of matter with an
equation of state $w_{M}=p_{M}/\rho_{M}$ is then given by, up to a constant,
$\rho_{M}\sim a^{-3(1+w_{M})}\ell^{-1+3w_{M}}\,.$ (24)
This prescription results in the scaling one would expect from physical
arguments in the cases of radiation or dust in the matter sector or, spatial
curvature or a cosmological constant in the geometric sector. In particular,
the effective action for a point particle, studied in more detail in Section
IV.1, suggests the scaling $\rho_{M}\sim\ell^{-1}a^{-3}$ when $w_{M}=0$, and
the scale invariance of the radiation is compatible with that $\rho_{M}\sim
a^{-4}$, independently of $\ell$, when $w_{M}=1/3$. (Furthermore, the energy
density due to a cosmological term is $\sim\ell^{-4}$ and independent of $a$,
whilst the effective energy of a spatial curvature term $\sim(\ell a)^{-2}$.)
Since, according to (24), the field $\ell$ now couples non-minimally to
matter, its equation of motion acquires a source term and reads
$\displaystyle\dot{H}-\dot{L}$ $\displaystyle+$
$\displaystyle\left(2H-L-N\right)\left(H-L\right)$ (25) $\displaystyle=$
$\displaystyle\left(\frac{1}{6}-\frac{1}{2}w_{M}\right)\left(\ell
n\right)^{2}\rho_{M}\,.$
It should be noted that in general, when $L\neq 0$, the energy densities obey
the modified continuity equations,
$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\left(\ell^{2}{\rho}_{\ell}\right)+3H\ell^{2}\left(\rho_{\ell}+p_{\ell}\right)$
$\displaystyle=$ $\displaystyle-2L\ell^{2}\rho_{M}\,,\,\,$ (26a)
$\displaystyle\dot{\rho}_{M}+3H\left(1+w_{M}\right)\rho_{M}$ $\displaystyle=$
$\displaystyle L\left(1-3w_{M}\right)\rho_{M}\,.\,\,$ (26b)
It is easy to see that the $\ell^{2}=8\pi G$, $L=0$ is a solution to the
Friedmann equations (23). Thus it is clear that the model defined above at
least contains viable solutions that describe the standard cosmological
background evolution. In the case of a possible time-evolution of $\ell$, more
general solutions exist to the system of equations.
To investigate such more general solutions, we begin with the power-law ansatz
$n=1\,,\quad a\sim t^{\alpha}\,,\quad\ell\sim t^{\lambda}\,.$ (27)
By plugging this ansatz into the 1${}^{\text{st}}$ Friedmann equation (23a),
we readily see that the power-laws must have the relation
$\lambda=\frac{1}{1+3w_{M}}\left[3\left(1+w_{M}\right)\alpha-2\right]\,.$ (28)
The solution that gives back the expansion law of general relativity is
$\lambda=0$ which implies $\alpha=2/(3+3w_{M})$, but this is only one amongst
the 1-parameter family of solutions parameterised by $\lambda$. Remarkably,
these solutions satisfy also the 2${}^{\text{nd}}$ Friedmann equation (23b),
and consequently they satisfy identically the equation of motion (25) as well.
Accelerating solutions exist. For a background fluid with $w_{M}>-1/3$, the
universe accelerates as if dominated by a quintessence-like field given that
$\lambda>-1$, and further, the universe super-accelerates if
$\ell<-2/(1+3w_{M})$. In general, the universe expands as if was filled with a
fluid that has the equation of state
$w=\frac{\rho_{\ell}+\rho_{M}}{p_{\ell}+p_{M}}=\frac{w_{M}-\left(\frac{1}{3}+w_{M}\right)\lambda}{1+\left(\frac{1}{3}+w_{M}\right)\lambda}\,.$
(29)
More general cosmological solutions, sourced by perfect fluids with
$\dot{w}_{M}\neq 0$ (which can also effectively describe several distinct
perfect fluid components), could be studied numerically.
It is worthy to point out that the dS coupling prescription (24) is
essentially the unique viable possibility. We will briefly comment upon some
alternative prescriptions, omitting the details of the derivations. The
Jennen-Pereira model Jennen:2015bxa with the standard coupling prescription
(i.e. $\rho_{M}\sim a^{-3(1+w_{M})}$) is not compatible with cosmological
evolution444More precisely, the Friedmann equations would be consistent only
for stiff fluid matter $w_{M}=1$. This was first pointed out to us by Sergio
Bravo Medina.. If this model is supplemented with the $\ell$-dependent
cosmological constant (the sign has to be negative), cosmological evolution
can be recovered such that in the standard Friedmann equation $G\rightarrow
G/(1+3w_{M})$, and therefore in the radiation dominated era the effective
gravitational coupling would be $G/2$, that appears too drastical modification
to allow viable early universe phenomena such as nucleosynthesis and the
formation of the cosmic microwave background. Yet, one could further adjust
the model by retaining the minimal matter coupling but taking into account the
$\ell$-dependence of the gravitational coupling. In such a prescription a
radiation-dominated era is not only phenomenologically excluded, but
incompatible with the Friedmann equations in the first place.
Thus, it turns out that the dS matter coupling prescription (24) that we
justified by physical principles, could actually have been formally deduced by
requiring the existence of viable cosmological background solutions.
### III.1 On the relevance of the solution
It is lluminating to show that the family of solutions is indeed equivalent
under the symmetry (18). The invariant combination of the metric and the scale
field is $\hat{g}_{\mu\nu}=(\ell_{P}/\ell)^{2}g_{\mu\nu}$, and correspondingly
we denote $\hat{a}=(\ell_{P}/\ell)a$, and $\hat{t}$ the time coordinate when
the lapse function is $\hat{n}=(\ell_{P}/\ell)$. It is then straightforward to
compute the invariant Hubble rate and its time derivative,
$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle(\ell/\ell_{P})\left(H-L\right)\,,$ (30a)
$\displaystyle\frac{\textrm{d}}{\textrm{d}\hat{t}}\hat{H}$ $\displaystyle=$
$\displaystyle(\ell/\ell_{P})^{2}\left[\dot{H}-\dot{L}-(H-L)L\right]\,.$ (30b)
Plugging in the relations (28) and (29) we obtain the result for the expansion
rate that corresponds to the effective equation of state $\hat{w}=w_{M}$. In
terms of the ``hatted'' variables, the Friedmann equations (23) of course
assume their standard form.
The gauge freedom allows a radically different re-interpretation of the
expanding universe. In an extreme case, we can understand all the
observational data in a static universe (obtained by setting $\alpha=0$ in the
above family of solutions), where instead the gravitational coupling as well
as the masses of particles are evolving in time (according to
$\lambda=-2/(1+3w_{M})$). For example, the observed cosmological redshift of
photons is then not due to the stretching of the wavelengths together with the
spatial scale factor $a(t)$, but it is due to the shrinking of the dimensional
scale factor $\lambda(t)$. In this description of the universe, we clearly
have no curvature singularity and therefore the cosmological spacetime appears
to be non-singular and extendable to $t\rightarrow-\infty$. Going backward in
time from the present, the dS scale grows indefinitely, and the big bang
would-be-singularity occurs at the point wherein the dS scale becomes infinite
and the hyperboloid flattens out (this is the contraction limit
$\mathfrak{so}(4,1)\rightarrow\mathfrak{iso}(3,1)$), and continuing this naive
extrapolation to still earlier times, the geometry becomes that of anti-dS
with the radius now shrinking indefinitely as we wind backwards towards
$t\rightarrow-\infty$. In this frame, both the metric and the total curvature
invariants are identically zero, though the torsion scalar (14) is $T=6L^{2}$
and thus diverges at $t=0$. The physical matter quantities remain finite. The
radiation555If dust is present at such a primordial stage, its energy density
momentarily disappears at $t=0$. This might be relevant in regards the initial
conditions for the geometric dark matter discovered in Zlosnik:2018qvg . We
note that at least the naive prescription (24) excludes sources with
$w_{M}>1/3$, since their energy density would diverge. energy density and the
pressure are always constant in this static universe frame, as seen from (24).
The possible relevance of the dS kinematics to a new cosmological paradigm has
been foreseen in some discussions Aldrovandi:2004km ; Araujo:2015oqa . Though
no solutions were presented, and the focus was on the opposite contraction
limit $\ell\rightarrow\infty$, the main insight that the conformal property of
the (apparently singular) transition point could be the key in connecting two
aeons in sir Penrose's conformal cyclic cosmology Araujo:2015oqa , is strongly
corroborated by our exact cosmological solution in the consistent dS gauge
gravity (16). By adopting Willem de Sitter's own, projective view of the dS
geometry McInnes:2003xm ; Ong:2016vwr , the solution might naturally be
enclosed into the eternal return of the aeon that is our unique universe.
In the more mainstream context of string theory, the existence of negative
energy vacua seems to be not only a generic prediction in the landscape of
myriad universes, but a requirement for the consistent definition of an
S-matrix, and it has proven quite a challenge to find ways that may lead to
positive vacuum energies compatible with the one observed universe
Kachru:2003sx ; Dasgupta:2019gcd . Previous attempts at realising a non-
singular anti-dS to dS transition have resorted to rather complicated
mechanisms requiring various new indgredients for their realisation
Biswas:2011qe ; Gupt:2013poa . It is remarkable that we seem to consistently
predict the desired non-singular transition, based on an action that is
locally equivalent to general relativity, but underpinned by the principles of
dS gauge theory. -It should be noted that the reinterpretation of the
cosmological expansion as a variation of mass scales is of course well-known
in the context of Fierz-Jordan-Brans-Dicke theory, and in particular, the
possibility that the big bang singularity is a field coordinate singularity
(i.e. removable by a change of variables) was introduced and clarified by
Wetterich Wetterich:2013jsa ; Wetterich:2013aca ; Wetterich:2020oyy (in the
context of his theory of Variable Gravity Wetterich:2013jsa , which is not a
mere reformulation of general relativity). The removal of black hole
singularities was also considered, employing more general than conformal
change of field coordinates Domenech:2019syf .
Finally, let us mention that Hohmann et al Hohmann:2018shl have recently
pointed out that the theory (17) formally contains vacuum solutions with
wormholes, despite their local equivalence with the standard vacuum solutions.
The physicality of such wormholes hinges on global, topological issues. As
Hohmann et al Hohmann:2018shl explained, the key point is that the solutions
may be related by improper Weyl transformations (18), where the factor $f$ may
vanish or become infinite at some points (in other words, the Jacobian of the
field coordinate transformation is not defined at those points). It is
precisely in this sense that the dS gauge theory (16) is inequivalent to
general relativity, and can thus accommodate a more general variety of
physically distinct solutions.
## IV Implications to matter
At the level of background cosmology it was sufficient to exploit the perfect
fluid parameterisation (24) for matter sources, but the question may remain
whether the proposed dS coupling prescription is consistent for more
fundamental field theory description of massive matter fields. To address this
question, we consider the action for a point particle and for a scalar field.
### IV.1 Point particle
Consider the massive point particle action,
$I_{pp}=\int m\textrm{d}s\,,$ (31)
where the line element for a time-like curve $x^{\mu}$ is
$\textrm{d}s=\sqrt{-g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}}$ and the
mass $m$ is related to the fundamental scale $\ell$ as $m(x)=m_{0}/\ell(x)$,
$m_{0}$ being the dimensionless constant of proportionality. We consider small
variations $\delta x^{\mu}$ of the curve $x^{\mu}(\tau)$ parameterised by an
arbitrary parameter $\tau$,
$\delta I_{pp}=\int\left[\delta
m\frac{\textrm{d}s}{\textrm{d}\tau}-\frac{m}{2\textrm{d}s}\delta\left(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right)\right]\textrm{d}\tau\,.$
(32)
The first term we can write as $\int\delta m{\textrm{d}s}=\int
m_{,\mu}\dot{x}^{\mu}\textrm{d}s$ and for the second term we apply integration
by parts. Adding the two terms we get
$\displaystyle\delta I_{pp}$ $\displaystyle=$
$\displaystyle\int\Big{[}mg_{\mu\nu}\frac{\textrm{d}^{2}x^{\nu}}{\textrm{d}s^{2}}+m\frac{\textrm{d}x^{\alpha}}{2\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}\left(2g_{\mu(\nu,\alpha)}-g_{\alpha\nu,\mu}\right)$
$\displaystyle+$
$\displaystyle\frac{\textrm{d}x^{\alpha}}{2\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}\left(2g_{\mu(\nu}m_{,\alpha)}-g_{\alpha\nu}m_{,\mu}\right)\Big{]}\delta
x^{\mu}\textrm{d}s=0\,.$
Since this holds for arbitrary variations of the path $\delta x^{\mu}$, we
get, by raising one index and dividing by $m$,
$\displaystyle\frac{\textrm{d}^{2}x^{\alpha}}{\textrm{d}s^{2}}$
$\displaystyle+$
$\displaystyle\frac{1}{2}g^{\alpha\beta}\left(g_{\beta\mu,\nu}+g_{\beta\nu,\mu}-g_{\mu\beta,\mu}\right)\frac{\textrm{d}x^{\mu}}{\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}$
$\displaystyle=$ $\displaystyle-$
$\displaystyle\frac{1}{2m}\left(\delta^{\alpha}_{\mu}m_{,\nu}+\delta^{\alpha}_{\nu}m_{,\mu}-g_{\mu\nu}{m}^{,\alpha}\right)\frac{\textrm{d}x^{\mu}}{\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}\,.$
Since $\log{m}_{,\alpha}=-\log{\ell}_{,\alpha}$, this can be written as
$\ddot{x}^{\alpha}+\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\mu}=0\,,$
(33)
where the overdot now denotes the derivative wrt the proper time $\tau=s$ and
the connection is
$\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}=\left\\{{}^{\phantom{i}\alpha}_{\mu\nu}\right\\}-\left(\delta^{\alpha}_{(\mu}\log{\ell}_{,\nu)}-\frac{1}{2}g_{\mu\nu}\log{\ell}^{,\alpha}\right)\,.$
(34)
Thus, we predict that the matter moves along the geodesics of a Weyl
connection, for which the Weyl gauge field $\ell_{\mu}=\ell_{,\mu}$ is pure
gauge and thus its curvature vanishes
$F_{\mu\nu}=2\ell_{[\mu,\nu]}=2\ell_{,[\mu\nu]}=0$. Therefore there is no
second clock effect. For more details and the extension of the integrable Weyl
(sometimes called semi-metric) geometry to generic non-metric geometry, see
BeltranJimenez:2020sih .
In terms of the arbitrary parameter $\tau$, (33) generalises to
$\ddot{x}^{\alpha}+\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\mu}=-\dot{s}^{2}\frac{\textrm{d}^{2}\tau}{\textrm{d}s^{2}}\dot{x}^{\alpha}\,.$
(35)
We see that the reparameterisation $\tau=as+b$ where $a,b$ are constants, does
not change the form of the equation (33). We also note that the projective
transformation of the connection by a one-form $p_{\mu}$ can be compensated by
the reparameterisation that satisfies
$\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}\rightarrow\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}+\delta^{\alpha}_{\nu}p_{\mu}\,,\quad
p_{\mu}\dot{x}^{\mu}=\dot{s}^{2}\frac{\textrm{d}^{2}\tau}{\textrm{d}s^{2}}\,.$
(36)
A reparameterisation of the curve thus corresponds to a projective
transformation of the affine geometry. The curve abstracted from its
parameterisation i.e. the projective equivalence class of the geodesic is
called a path.
A more elementary modelling of matter fields would begin with spinor fields,
but in the end the classical approximation relevant to our purposes is given
by the point particle action where the $m\sim\ell^{-1}$ is inherited from the
spinor mass term. Spinor fields and gauge fields can be coupled to dS gravity
elegantly with polynomial Lagrangians Pagels:1983pq ; Westman:2012zk .
### IV.2 Scalar field
Considering a self-interacting scalar field, our coupling prescription
suggests the Lagrangian
$I_{\phi}=-\int\textrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2\ell^{2}}\left(\partial\phi\right)^{2}+\ell^{-4}V(\phi)\right]\,.$
(37)
In the cosmological setting, this gives the total action
$I=I_{\text{dS}}+I_{\phi}$ as
$I=-\int\textrm{d}ta^{3}\left[\frac{3}{\ell^{2}n}\left(H-L\right)^{2}-\frac{1}{2\ell^{2}}\frac{\dot{\phi}^{2}}{n}+\frac{n}{\ell^{4}}{V(\phi)}\right]\,.$
(38)
The scalar field contribution to the Friedmann equations is then given by
$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\dot{\phi}}/{\ell
n}\right)^{2}+{\ell^{-4}}{V(\phi)}\,,$ (39a) $\displaystyle p_{\phi}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left({\dot{\phi}}/{\ell
n}\right)^{2}-{\ell^{-4}}{V(\phi)}\,.$ (39b)
The Klein-Gordon equation is obtained by the variation wrt the scalar field,
$\ddot{\phi}+\left(3H-2L-N\right)\dot{\phi}+n^{2}\ell^{-6}V^{\prime}(\phi)=0\,,$
This equation, by multiplying with $\dot{\phi}$ and rearranging the terms,
reduces to (26a).
To verify the consistency of the system, we consider also the equation of
motion
$\displaystyle\dot{H}-\dot{L}$ $\displaystyle+$
$\displaystyle\left(2H-L-\frac{\dot{N}}{N}\right)\left(H-L\right)$
$\displaystyle=$
$\displaystyle\frac{1}{6}\left(-\dot{\phi}^{2}+4N^{2}\ell^{-2}V(\phi)\right)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{6}-\frac{1}{2}w_{\phi}\right)\left(\ell
N\right)^{2}\rho_{\phi}\,,$
which is in full agreement with (25). Therefore this equation is, as it
should, degenerate with the Friedmann equations.
We note that even though a massless scalar field $V(\phi)\approx 0$ is a
perfect fluid with the stiff fluid equation of state $w_{\phi}=1$, the
heuristic perfect fluid coupling prescription (24), which would naively
suggest the kinetic term to scale as $\ell^{2}$, is not the correct one to
adopt for a proper field theoretical description. The quadratic kinetic term
$\sim(\partial\phi)^{2}$ inherits the scaling dimension
$\sim[\phi^{2}]\sim[\ell^{-2}]$ from the dimension of the scalar field, and
this is the fundamental rationale that determines the coupling.
## V Discussion
Motivated by the fact that our universe has fundamental limiting scales both
in the infrared and in the ultraviolet ends of the spectrum, we developed a dS
gauge theory of gravity which incorporates the new kinematic invariant $\ell$,
besides the invariant $c$ of Einstein's relativity. The Cartan-geometric
construction, illustrated upon the dS hyperboloid embedded into a spacetime
with one extra dimension, was based upon nothing but the standard gauge field,
a connection 1-form and a symmetry-breaking scalar field. Gravity was realised
as a gauge theory of translations in the sense that the action is quadratic in
the translation gauge field strength $\bm{F}^{a}$ whilst the homogeneous model
spaces are flat $\bm{F}^{a}{}_{b}=0$ (though, as mentioned in II.1, it would
be possible to formulate a more canonical version of translation gauge
theory).
A key insight was that the theory (16) exhibits the rescaling invariance (18).
Thus, the calibration of the scale $\ell$ is arbitrary, and it can changed
without affecting the physics given the accompanying rescaling of the metric.
Incidentally, the theory thus realises the foundational motivations of both
the dS and the Weyl gauge theories. On one hand, the description of our
universe requires observer-independent scales. On the other hand, absolute
scales are physically meaningless. Thus, the new dS theory may provide a
conceptually improved framework to the century-old problem of introducing
scales into physics. From a formal point of view, the orthogonal symmetry is
considerably neater than the Weyl extension of the Poincaré symmetry. An
obvious direction to pursue in the future is the incorporation of the two
limiting scales, $\ell_{P}$ and $\ell_{\Lambda}$ independently, via the
completion of the dS to the conformal symmetry666It is natural to speculate
that Dirac’s original motivation for (17), the large number hypothesis
Dirac:1973gk ; Ray:2007cc , could be vindicated by exploiting the additional
freedom provided by another scalar field, thus yielding the satisfactory
explanation of various other scales in physics..
Let us comment on our theory also in view of the so-called ``teleparallel''
models of gravity. Modifications of gravity in that context are by now well-
known to violate Lorentz symmetry, resulting in extra degrees of freedom which
typically have strong coupling and other unwanted problems777Such concerns
have been raised earlier in the literature Kopczynski_1982 ; Li:2010cg , and
the current state of art in the problematics of the extra degrees of freedom
is reviewed in Blixt:2020ekl ; Golovnev:2020zpv .. In contrast, the new theory
developed in this article, though formulated in terms of a flat connection
$\bm{A}^{a}{}_{b}$ i.e. in a ``teleparallel'' geometry, is not based on a
violation but on an extension of the Lorentz symmetry. Thus our approach also
seems to suggest a way of generating viable ``teleparallel'' gravity models.
However, since the theory (16) we arrived at has the metric scalar-tensor
equivalent (17), it still remains an open question whether a consistent
``genuinely teleparallel'' modification of gravity is possible.
The aim of this article was to explore the cosmology of the dS gauge theory
with time-evolving distance scale $\ell=\ell(t)$. We derived the family of
exact solutions which is characterised by the effective equation of state (29)
for the gravity-fluid system. Incidentally, it turned out that the coupling
prescription (24) that follows from the physical interpretation of the dS
scale $\ell$, is actually necessary for the existence of realistic
cosmological solutions (including even those which reduce to the standard
solutions in general relativity). We considered the possible reinterpretation
of observations in the frame where the universe does not expand but the dS
scale is evolving in time. This is, to our knowledge (despite the often-made
claims otherwise in the vast literature on ``teleparallel'' cosmology), the
first description of the cosmic geometry de facto in terms torsion, without
the metric curvature playing its usual role. We argued that such a novel
description could allow the consistent extension of cosmology beyond the big
bang, and believe that the theory and its cosmological implications merit
further investigation.
###### Acknowledgements.
TK would like to thank Manuel Hohmann and Hardi Veermäe for insightful
discussions on scale invariance, and Sergio Bravo Medina and David Mota for an
earlier collaboration on a related topic. This work was supported by the
Estonian Research Council grants PRG356 ``Gauge Gravity'' and MOBTT86, and by
the European Regional Development Fund CoE program TK133 ``The Dark Side of
the Universe''.
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|
# Cooperative NOMA-Based User Pairing for URLLC: A Max-Min Fairness Approach
Fateme Salehi, Naaser Neda, Mohammad-Hassan Majidi, and Hamed Ahmadi F. Salehi
is with the Faculty of Electrical and Computer Engineering, University of
Birjand, Birjand, Iran (e-mail: [email protected]).N. Neda
_(Corresponding author)_ is with the Faculty of Electrical and Computer
Engineering, University of Birjand, Birjand, Iran (e-mail:
[email protected]).M.-H. Majidi is with the Faculty of Electrical and
Computer Engineering, University of Birjand, Birjand, Iran (e-mail:
[email protected]).H. Ahmadi is with the Department of Electronic
Engineering, University of York, United Kingdom (e-mail:
[email protected]).Part of this paper has been presented in EuCNC 2021
[1].Manuscript received January 19, 2021; revised May 3, 2021 and September 1,
2021; accepted September 25, 2021.
###### Abstract
In this paper, cooperative non-orthogonal multiple access (C-NOMA) is
considered in short packet communications with finite blocklength (FBL) codes.
The performance of a decode-and-forward (DF) relaying along with selection
combining (SC) and maximum ratio combining (MRC) strategies at the receiver
side is examined. We explore joint user pairing and resource allocation to
maximize fair throughput in a downlink (DL) scenario. In each pair, the user
with a stronger channel (strong user) acts as a relay for the other one (weak
user), and optimal power and blocklength are allocated to achieve max-min
throughput. To this end, first, only one pair is considered, and optimal
resource allocation is explored. Also, a suboptimal algorithm is suggested,
which converges to a near-optimal solution. Finally, the problem is extended
to a general scenario, and a suboptimal C-NOMA-based user pairing is proposed.
Numerical results show that the proposed C-NOMA scheme in both SC and MRC
strategies significantly improves the users’ fair throughput compared to the
NOMA and OMA. It is also investigated that the proposed pairing scheme based
on C-NOMA outperforms the Hybrid NOMA/OMA scheme from the average throughput
perspective, while the fairness index degrades slightly.
###### Index Terms:
finite blocklength, short packet communication, URLLC, cooperative NOMA, max-
min fairness, user pairing.
## I Introduction
The ever-increasing new demands such as tactile internet, high-resolution
video streaming, virtual/augmented reality, autonomous vehicles, etc., with
various requirements, may be somewhat challenging in terms of reliability and
latency. Unlike most of the existed mobile networks designed for traditional
mobile broadband (MBB) services, Internet-of-Things (IoT) attempts to connect
plentiful devices with the least human intervention. IoT applications are
divided into massive machine-type communications (mMTC) and ultra-reliable
low-latency communications (URLLC). The first one consists of many low-cost
devices with massive connections and high battery lifetime requirements. On
the other hand, URLLC requirements are most related to mission-critical
services in which the importance of uninterrupted and robust data exchange is
far greater than anything else.
Short packets with FBL codes are considered to reduce the transmission delay
and support low-latency communication. In the FBL regime communication, in
contrast to Shannon’s capacity for infinite blocklength, decoding error
probability at the receiver is not negligible owing to short blocklength [2].
Polyanskiy et al. succeeded in deriving an exact approximation of the FBL
regime’s information rate at the AWGN channel [3]. Following that, research in
this context developed to MIMO channel with quasi-static fading [4] and a
quasi-static fading channel with retransmissions [5]. Furthermore, the effect
of short packets on the spectrum sharing, and scheduling of delay-sensitive
packets was considered in [6] and [7], respectively. In [8], massive MIMO
adoption to maximize the achievable uplink data rate for industrial
applications was advocated for both MRC and zero-forcing (ZF) receivers. In
[9], the resource allocation for a secure mission-critical IoT communication
system was studied under finite blocklength, and two optimization problems
with the aim of weighted throughput maximization and total transmit power
minimization were addressed. The authors in [10] proposed a cross-layer
framework for optimizing user association, packet offloading rates, and
bandwidth allocation for mission-critical IoT scenarios.
The NOMA performance in the FBL regime was studied in [11, 12, 13, 14]. In
[11], optimal power and blocklength allocation was considered in a high
signal-to-noise ratio (SNR) scenario, and the amount of NOMA transmission
delay reduction was determined compared to OMA in a closed-form. In [12],
transmission rate and power allocation of the NOMA scheme were optimized to
maximize the effective throughput of the strong user, while the throughput of
the other user was guaranteed at a certain level. The transmitter’s energy
with a hybrid transmission scheme that combines the time division multiple
access (TDMA) and NOMA was minimized in [13] subject to heterogeneous latency
constraints at receivers. In [14], an optimal power allocation algorithm was
proposed to achieve max-min throughput under energy, reliability, and delay
constraints in a DL-NOMA transmission and compared with its optimal OMA
counterpart.
Relaying is a well-known technique to increase capacity and reliability. In
[15], relaying performance in the FBL regime was studied, and its advantages
over the direct transmission were investigated. The throughput and effective
capacity of a relaying system in the FBL regime were obtained in [16] at the
presence of a quasi-static fading channel and some assumptions on average
channel state information (CSI) at the transmitter. In [17], under the
assumption of outdated CSI at the source, the authors maximized the FBL
throughput of a two-hop relaying system while guaranteeing a reliability
constraint.
Ding et al. in [18] proposed the cooperative NOMA transmission scheme, a
cooperative relaying technique in the NOMA system which fully exploits the
prior knowledge available by applying the successive interference cancellation
(SIC) strategy. Followed by that, they introduced a two-stage relay selection
strategy in the C-NOMA network [19]. In [20], a buffer-aided C-NOMA scheme,
where the intended users are equipped with buffers for cooperation, was
proposed to adaptively select a direct or cooperative transmission mode, based
on the instantaneous CSI and the buffer state. In [21], the authors proposed
threshold-based selective C-NOMA, where the strong user forwards the symbols
of weak user only if the signal-to-interference-plus-noise ratio (SINR) is
greater than the pre-determined threshold value, to increase the data
reliability of conventional C-NOMA networks. In [22], the authors investigated
C-NOMA scheme in short-packet communications with flat Rayleigh fading
channels and derived the average block error rate (BLER) of the central user
and the cell-edge user theoretically for both SC and MRC strategies.
Optimization problems of average throughput and max-min throughput were
studied in [23] with power and blocklength allocation between users under
delay and consumed energy constraints by full search method with high
complexity, but users’ reliability was not guaranteed. In [24], Ren et al.
considered optimal power and blocklength allocation in OMA, NOMA, relaying,
and C-NOMA transmissions schemes to minimize the weak user’s decoding error
probability; meanwhile, the reliability of the strong user’s performance was
guaranteed at a certain level. Both [23] and [24] have considered a two-user
scenario.
In [25], a joint user pairing and power allocation problem was explored in a
DL-NOMA network to optimize the achievable sum rate with minimum rate
constraint for each user. In [26], a two-step user-pairing scheme maximizing
the achievable diversity gain for an OFDM-based relaying NOMA system with
fixed-rate transmission was proposed by selecting one near user and one far
user for each subcarrier where far users cannot communicate with the base
station (BS) directly. Zhang et al. in [27] investigated a distance-based user
pairing in the C-NOMA network, where the locations of the source and typical
user are fixed, and the candidate users for pairing follow the distribution of
homogeneous Poisson Point Process, and two close-to-user pairing and close-to-
source pairing schemes were proposed. The authors in [28] considered user
pairing policy and power control scheme jointly in a DL C-NOMA system, where
the objective is maximizing the achievable sum-rate of the whole system while
guaranteeing a certain quality of service (QoS) for all users. In [29], a
joint user pairing and subchannel assignment algorithm was proposed in a DL
C-NOMA network that pairs a strong user with a weak user and assigns them a
subchannel simultaneously, while a Stackelberg game is employed to allocate
power among the users by the BS. All of the works in [25, 26, 27, 28, 29]
investigate the user pairing problem in conventional communication with
infinite blocklength. Moreover, the last two works do not take into
consideration the geometric distance between the paired nodes in the C-NOMA
scheme.
In this work, we consider a DL C-NOMA network in the short packet
communications scenario. It is assumed that the paired users, their channel
gain difference is high. The strong user, which performs SIC and detects the
weak user’s data, acts as a relay. The weak user, which receives its data via
BS and relay separately can implement SC or MRC to detect its data. To the
best of the authors’ knowledge, this is the first work to address the problem
of joint user pairing, blocklength and power allocation in a critical IoT
scenario.
Our main contributions in this work are summarized as follows:
1. 1.
We obtain each user’s decoding error probability in the C-NOMA transmission
scheme for both SC and MRC protocols with the CSI at the transmitter (CSIT)
assumption. The MRC protocol is considered for the first time in the FBL
regime with different blocklengths.
2. 2.
To guarantee the quality of service (QoS) of the weak user and to improve
fairness, joint power and blocklength optimization is done in both NOMA and
relay phases to maximize the minimum throughput of two users in different
combining scenarios, under latency, reliability, and energy constraints.
3. 3.
A suboptimal solution with near-optimal performance is proposed to decrease
the complexity of the optimal resource allocation, and their computational
complexity is determined.
4. 4.
The problem is extended to a multi-user scenario, and a novel joint suboptimal
C-NOMA-based user pairing and resource allocation scheme is proposed.
Meanwhile, the simulation results show its comparable performance to the
exhaustive-search optimal algorithm.
The remainder of this paper is organized as follows. In Section II, the system
model and direct transmission analysis in the FBL regime are presented.
Performance analysis of the C-NOMA transmission consist of SC and MRC
strategies is provided in Section III. Problem formulation with a focus on one
pair is considered in Section IV. The optimal and one suboptimal solution are
proposed for the problem in Section V. The problem is extended to a multi-user
scenario and, one user pairing scheme is proposed in Section VI. Numerical
results are presented in Section VII. Finally, Section VIII concludes the
paper.
## II Preliminaries Issues
### II-A System Model
As shown in Fig. 1(a), the URLLC users with different QoS requirements are
paired into disjoint clusters. For simplicity, we first just focus on one
pair. Section VI will provide more details of user pairing. Here we consider a
cooperative relaying scenario in a DL system with one BS and two NOMA users in
each C-NOMA pair. In phase I, i.e., NOMA phase, BS transmits a NOMA frame of
length $m^{\textrm{I}}$ symbols, which consists of two users’ data ($N_{1}$
bits, user 1’s data and $N_{2}$ bits, user 2’s data). User 1, the strong user,
performs the SIC technique and decodes user 2’s data and sends that to user 2
in a frame of length $m^{\textrm{II}}$ symbols in phase II, i.e., relaying
phase. The instantaneous channel coefficients of BS-user 1, BS-user 2, and
user 1-user 2 links representing small scale fading and large scale fading are
denoted as $h_{1}$, $h_{2}$, and $h_{1,2}$, respectively. It is assumed that
the channels are quasi-static Rayleigh fading. Hence, they are constant during
one frame and vary independently from one frame to the next one.
According to the power domain NOMA principle, in a two-user scenario, BS
transmits $\sum_{i=1}^{2}\sqrt{p_{i}^{\textrm{I}}}x_{i}$, where $x_{i}$ is the
message of user $i$, $i\in\\{1,2\\}$, and $p_{i}^{\textrm{I}}$ refers to the
allocated power of user $i$ in phase I. So, the received signal at user $i$ is
given by
$y_{i}^{\textrm{I}}=(\sqrt{p_{1}^{\textrm{I}}}x_{1}+\sqrt{p_{2}^{\textrm{I}}}x_{2})h_{i}+n_{i}$,
where $n_{i}$ is the complex additive white Gaussian noise with variance
$\sigma^{2}$. Without loss of generality, it is assumed that
$|h_{1}|^{2}>|h_{2}|^{2}$, and more power should be allocated to user 2.
Therefore, user 1 can perform the SIC technique to remove the interference,
while user 2 suffers from the interference and cannot cancel it. If $x_{2}$ is
decoded correctly by user 1, it is re-encoded and transmitted (denoted by
$\sqrt{p_{2}^{\textrm{II}}}x_{2}^{\prime}$). 111One should notice that $x_{2}$
is user 2’s data with rate ${N_{2}}/{m^{\textrm{I}}}$, while $x_{2}^{\prime}$
is the same data with rate ${N_{2}}/{m^{\textrm{II}}}$. Consequently, the
received signal at user 2 in the relaying phase is
$y_{2}^{\textrm{II}}=\sqrt{p_{2}^{\textrm{II}}}x_{2}^{\prime}~{}h_{1,2}+n_{1,2}$.
Let $p_{2}^{\textrm{II}}$ show the allocated power to user 2 by the relay
(user 1) in phase II, and $n_{1,2}$ is the complex additive white Gaussian
noise with variance $\sigma^{2}$ . To implement this scheme, user 1 must know
whether SIC is successful or not. To this end, we suppose that BS sends the
channel coding information of both user 1 and user 2 to user 1 via an error-
free dedicated channel. The channel coding can help to diagnose whether the
decoded data is correct or not. Thus, user 1 knows whether the SIC is
successful or not [24].
### II-B Direct Transmission Analysis in the FBL Regime
According to [3], the achievable data rate $R$ for a finite blocklength of $m$
symbols $(m\geq 100)$, and an acceptable BLER $\varepsilon$ , has an exact
approximation as
$R\approx C-\sqrt{\frac{V}{m}}\frac{Q^{-1}(\varepsilon)}{\ln 2}$ (1)
where $C=\log_{2}(1+\gamma)$ is the Shannon capacity, $\gamma$ is the SNR/SINR
ratio, $Q^{-1}(\cdot)$ refers to the inverse Gaussian Q-function
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-\frac{t^{2}}{2}}\,dt$, and
$V=1-(1+\gamma)^{-2}$ is the channel dispersion. In the FBL regime, even with
perfect CSI, the transmission is not error-free and the decoding error
probability is given by
$\varepsilon\approx Q(f(\gamma,R,m)).$ (2)
where $f(\gamma,R,m)\overset{\Delta}{=}\frac{(C-R)\ln 2}{\sqrt{V/m}}$.
Figure 1: (a) system model, (b) frame structure.
(a)
(b)
## III Performance Analysis of C-NOMA Transmission
It is assumed that the receivers have access to perfect CSI, and BS and each
of the users have one antenna. Also, user 2 can employ various combining
strategies, including SC and MRC. In phase I, user 2 directly detects $x_{2}$
by considering $x_{1}$ as interference. The decoding error probability of
$x_{2}$ at user 2 in phase I is denoted by $\varepsilon_{2,2}^{\textrm{I}}$ ,
which is approximated based on (2) by
$\varepsilon_{2,2}^{\textrm{I}}\approx
Q(f(\gamma_{2,2}^{\textrm{I}},R_{2,2}^{\textrm{I}},m^{\textrm{I}}))$ (3)
where
$\gamma_{2,2}^{\textrm{I}}=p_{2}^{\textrm{I}}|h_{2}|^{2}/(p_{1}^{\textrm{I}}|h_{2}|^{2}+\phi\sigma^{2})$
and $R_{2,2}^{\textrm{I}}=N_{2}/m^{\textrm{I}}$ are the received SINR and the
achievable rate of user 2 related to detecting $x_{2}$ in phase I,
respectively. $\phi>1$ reflects the SNR/SINR loss due to the imperfect CSI.
222Invoking [30], the effect of channel estimation error on data rate can be
equivalent to noise enhancement, which depends on the velocity of the devices.
For devices with slow or medium velocity, $\phi$ is close to 1. Since $x_{2}$
is detected directly, $\varepsilon_{2,2}^{\textrm{I}}$ is the overall error
probability of user 2 in phase I, i.e.,
$\varepsilon_{2}^{\textrm{I}}=\varepsilon_{2,2}^{\textrm{I}}$ . On the
opposite, user 1 performs SIC, meaning it first decodes $x_{2}$ while treats
$x_{1}$ as interference. Similarly, the decoding error probability of $x_{2}$
at user 1 in phase I, which is denoted by $\varepsilon_{1,2}^{\textrm{I}}$, is
approximated as
$\varepsilon_{1,2}^{\textrm{I}}\approx
Q(f(\gamma_{1,2}^{\textrm{I}},R_{1,2}^{\textrm{I}},m^{\textrm{I}}))$ (4)
where
$\gamma_{1,2}^{\textrm{I}}=p_{2}^{\textrm{I}}|h_{1}|^{2}/(p_{1}^{\textrm{I}}|h_{1}|^{2}+\phi\sigma^{2})$
and $R_{1,2}^{\textrm{I}}=N_{2}/m^{\textrm{I}}$ are the received SINR and the
achievable rate of user 1 related to detecting $x_{2}$ in phase I,
respectively. If user 1 decodes and removes $x_{2}$ successfully, then $x_{1}$
can be detected without interference. Accordingly, the decoding error
probability of $x_{1}$ at user 1 in phase I, i.e.,
$\varepsilon_{1,1}^{\textrm{I}}$ , is denoted by
$\varepsilon_{1,1}^{\textrm{I}}\approx
Q(f(\gamma_{1,1}^{\textrm{I}},R_{1,1}^{\textrm{I}},m^{\textrm{I}}))$ (5)
where $\gamma_{1,1}^{\textrm{I}}=p_{1}^{\textrm{I}}|h_{1}|^{2}/\phi\sigma^{2}$
and $R_{1,1}^{\textrm{I}}=N_{1}/m^{\textrm{I}}$ are the received SINR and the
achievable rate of user 1 related to detecting $x_{1}$ in phase I,
respectively. By assuming that $x_{1}$ is detected when SIC is successful and
the fact that in URLLC services, $\varepsilon$ is usually in order of
$10^{-5}\sim 10^{-9}$ [31], the overall decoding error probability at user 1
in phase I can be approximated as
$\varepsilon_{1}^{\textrm{I}}=\varepsilon_{1,2}^{\textrm{I}}+(1-\varepsilon_{1,2}^{\textrm{I}})\varepsilon_{1,1}^{\textrm{I}}\approx\varepsilon_{1,2}^{\textrm{I}}+\varepsilon_{1,1}^{\textrm{I}}.$
(6)
Since it is assumed that channels are half-duplex, the relayed signal is not
received at user 1. Hence, the overall decoding error probability at user 1 is
denoted as $\varepsilon_{1}=\varepsilon_{1}^{\textrm{I}}$ . In contrast, the
overall decoding error probability of user 2 depends on user 1 performance and
thus the signal of phase II and combining strategy, where the following
subsections derive the equations individually for SC and MRC strategies.
### III-A Selection Combining (SC)
In this protocol, user 2 does not combine the NOMA phase and relaying phase
signals, but decodes transmitted messages from BS and relay (user 1)
separately and selects the correctly decoded packet. First, the received
message from user 1 in the relaying phase is decoded. If decoding is failed or
no signal is received from user 1, then the transmitted message from BS in the
NOMA phase is decoded. To differentiate the packets, the packet ID is inserted
in the packet head for each device. Therefore, an error occurs when both
transmissions are unsuccessful. Decoding error probability of $x_{2}^{\prime}$
by user 2 in phase II, i.e., $\varepsilon_{2,2}^{\textrm{II}}$, is given by
$\varepsilon_{2,2}^{\textrm{II}}\approx
Q(f(\gamma_{2,2}^{\textrm{II}},R_{2,2}^{\textrm{II}},m^{\textrm{II}}))$ (7)
where $\gamma_{2,2}^{\textrm{II}}=p_{2}^{\textrm{II}}|h_{1,2}|^{2}/\sigma^{2}$
and $R_{2,2}^{\textrm{II}}=N_{2}/m^{\textrm{II}}$ are the received SNR and the
achievable rate of user 2 related to detecting $x_{2}^{\prime}$ in phase II,
respectively. One should note that the phase II signal will be transmitted if
the message of user 2 is decoded correctly in phase I, so the overall decoding
error probability of user 2 in phase II is approximated as
$\varepsilon_{2}^{\textrm{II}}=\varepsilon_{1,2}^{\textrm{I}}+(1-\varepsilon_{1,2}^{\textrm{I}})\varepsilon_{2,2}^{\textrm{II}}\approx\varepsilon_{1,2}^{\textrm{I}}+\varepsilon_{2,2}^{\textrm{II}}.$
(8)
Finally, the overall decoding error probability of user 2 in SC strategy is
formulated as
$\varepsilon_{2}=\varepsilon_{2}^{\textrm{I}}\varepsilon_{2}^{\textrm{II}}\approx\varepsilon_{2,2}^{\textrm{I}}(\varepsilon_{1,2}^{\textrm{I}}+\varepsilon_{2,2}^{\textrm{II}}).$
(9)
### III-B Maximum Ratio Combining (MRC)
By applying MRC protocol at user 2, since the coding rate of BS-user 2 and
user 1-user 2 links are not equal, the determinative link is the bottleneck
link, i.e., the link with the lowest coding rate. Therefore, the combined
signal with the MRC protocol has a frame of length
$m^{\text{C}}=\max\\{m^{\text{I}},m^{\text{II}}\\}$ symbols and the following
SINR
$\gamma_{2,2}^{\text{C}}=\frac{m^{\text{I}}}{m^{\text{C}}}\gamma_{2,2}^{\text{I}}+\frac{m^{\text{II}}}{m^{\text{C}}}\gamma_{2,2}^{\text{II}}.$
(10)
The probability that user 2 fails in MRC signal decoding is given by
$\varepsilon_{2,2}^{\textrm{C}}\approx
Q(f(\gamma_{2,2}^{\textrm{C}},R_{2,2}^{\textrm{C}},m^{\textrm{C}}))$ (11)
where $R_{2,2}^{\textrm{C}}=N_{2}/m^{\textrm{C}}$ is the achievable rate of
user 2 in the combined packet with MRC protocol.
User 2 fails when either its message is decoded correctly by none of them in
phase I, or user 1 decodes $x_{2}$ correctly, but the combined signal is not
decoded correctly. Hence, the overall decoding error probability of user 2 in
the MRC strategy is given by
$\varepsilon_{2}=\varepsilon_{1,2}^{\textrm{I}}\varepsilon_{2,2}^{\textrm{I}}+(1-\varepsilon_{1,2}^{\textrm{I}})\varepsilon_{2,2}^{\textrm{C}}.$
(12)
## IV Problem Formulation
In the considered URLLC system, the two users are served with the aim of
fairness during two phases with a total $D_{\max}$ symbols period. If channel
feedback is available at the transmitter side, users’ data rates can be set
according to their instantaneous channel conditions. That being the case, a
suitable criterion is max-min fairness [32]. The throughput of user $i$,
$T_{i}$, is defined as the average bits per each channel use (or complex
symbol), which is decoded correctly at the receiver;
$T_{i}\overset{\Delta}{=}\frac{m^{\text{I}}}{D_{\max}}R_{i,i}^{\text{I}}(1-\varepsilon_{i})$
(13)
where $1-\varepsilon_{i}$ is the reliability of user $i$ and a predefined
value for each URLLC use case.
In the C-NOMA scheme, the superposition coding is performed in the NOMA phase,
such that the BS enables to transmit users’ signals simultaneously with
different powers within a frame of length $m^{\text{I}}$. User 1 after
decoding user 2’s data, sends it in the relaying phase within a frame of
length $m^{\text{II}}$. In Fig. 1(b) the frame structure of C-NOMA is
observed. Therefore, the desired optimization problem is formulated as
$\displaystyle\max_{{\left\\{{{p_{i}^{\rm{I}}},{p_{2}^{\rm{II}}},{m^{j}}}\right\\}}_{\scriptsize{i=1,2,}\hfill\atop\scriptsize{j={\rm{I}},{\rm{II}}}\hfill}}\min\left\\{{{T_{1}},{T_{2}}}\right\\}$
(14a) $\displaystyle{\rm{s.t.}}\quad$
$\displaystyle{m^{\rm{I}}}\left({p_{1}^{\rm{I}}+p_{2}^{\rm{I}}}\right)+{m^{\rm{II}}}p_{2}^{\rm{II}}\leq{D_{\max}}{P_{\rm{ave}}},$
(14b) $\displaystyle
0<p_{1}^{\rm{I}}+p_{2}^{\rm{I}}\leq{\kappa_{\rm{p}}}{P_{\rm{ave}}},~{}p_{i}^{\rm{I}}>0,~{}i\in\left\\{{1,2}\right\\},$
(14c) $\displaystyle 0\leq
p_{2}^{\rm{II}}\leq{\kappa_{\rm{p}}}{P_{\rm{ave}}},$ (14d)
$\displaystyle{\varepsilon_{i}}\leq{\varepsilon_{i}}^{\rm{th}},~{}i\in\left\\{{1,2}\right\\},$
(14e) $\displaystyle{m^{\rm{I}}}+{m^{\rm{II}}}={D_{\max}}.$ (14f)
Optimization parameters consist of blocklength and power allocated to two
users in phases I and II. Constraint (14b) indicates the system’s total energy
consumption budget. Constraints (14c) and (14d) are the general power
constraints, where $P_{\text{ave}}$ is the average power, and
$\kappa_{\text{p}}$ is the peak to average power ratio (PAPR) factor.
Constraint (14e) guarantees that the decoding error probability of user $i$
does not violate $\varepsilon_{i}^{\text{th}}$. Moreover, the latency
constraint is stated by (14f).
## V Problem Solving
This section will solve the optimization problem in (14) for the SC and MRC
strategies. To facilitate this issue, we first have to analyze the constraints
and specify their optimal status. Let us first consider the constraint (14e)
on the acceptable BLER of the two users. Since each URLLC use case needs
specific reliability, allocating more resources to achieve a BLER lower than
the required $\varepsilon_{i}^{\text{th}}$, wastes the rare resources.
Moreover, according to (1), the lower desired error probability, the lower
data rate. Therefore, $\varepsilon_{i}=\varepsilon_{i}^{\rm{th}}$ is an
optimal choice. About constraint (14b), invoking [14, Proposition 1], the
acceptable data rate (i.e., $R>0$ ) in (1), is a monotonically increasing
function of the corresponding SNR/SINR. Using the contradiction method, one
can prove that to maximize the throughput, the energy constraint holds with
equality [24], i.e.,
$m^{\rm{I}}\left(p_{1}^{\rm{I}}+p_{2}^{\rm{I}}\right)+m^{\rm{II}}p_{2}^{\rm{II}}=D_{\max}P_{\rm{ave}}$.
In addition, the following proposition indicates the ratio of optimal consumed
energy in two transmission phases.
###### Proposition 1
At the optimal solution, the total consumed energy of the two users in phase I
is always greater than the consumed energy in phase II, i.e.,
${m^{\rm{I}}}P_{\rm{sum}}>{m^{\rm{II}}}p_{2}^{\rm{II}}$, where
$P_{\rm{sum}}\overset{\Delta}{=}p_{1}^{\rm{I}}+p_{2}^{\rm{I}}$. (Refer to
Appendix A for proof.)
Furthermore, invoking [14, Proposition 2], at the optimum point of Problem
(14), throughputs of the two users are equal, i.e., $T_{1}=T_{2}$ . Following
the above discussion, we provide a solution for the optimization problem in
(14) with both SC and MRC strategies.
### V-A Optimal Design of Max-Min Fairness in C-NOMA
Since at the optimal solution $T_{1}=T_{2}$, equation
$R_{2,2}^{\rm{I}}=\frac{1-\varepsilon_{1}^{{\rm{th}}}}{1-\varepsilon_{2}^{{\rm{th}}}}R_{1,1}^{\rm{I}}$
can be derived from (13). Moreover, the message of user 2 contains the same
number of bits in both phases, so it can be concluded that
$R_{2,2}^{{\rm{II}}}=\frac{m^{\rm{I}}}{m^{\rm{II}}}R_{2,2}^{\rm{I}}$.
Consequently, the optimization problem in (14) is rewritten as follows
$\displaystyle\mathop{\max}\limits_{\left\\{{{m^{\rm{I}}},p_{1}^{\rm{I}},p_{2}^{\rm{II}}}\right\\}}$
$\displaystyle{T_{1}}=\frac{m^{\rm{I}}}{D_{\max}}\left({1-\varepsilon_{1}^{\rm{th}}}\right)R_{1,1}^{\rm{I}}$
(15a) $\displaystyle{\rm{s.t.}}\quad$
$\displaystyle{m^{\rm{I}}}{P_{\rm{sum}}}+{m^{\rm{II}}}p_{2}^{\rm{II}}={D_{\max}}{P_{\rm{ave}}}$
(15b) $\displaystyle
0<{P_{\rm{sum}}}\leq{\kappa_{\rm{p}}}{P_{\rm{ave}}},~{}0<p_{1}^{\rm{I}}<\tfrac{P_{\rm{sum}}}{2}$
(15c) $\displaystyle 0\leq
p_{2}^{{\rm{II}}}\leq{\kappa_{\rm{p}}}{P_{\rm{ave}}},~{}{m^{\rm{I}}}{P_{\rm{sum}}}>{m^{\rm{II}}}p_{2}^{\rm{II}}$
(15d)
$\displaystyle{\varepsilon_{i}}={\varepsilon_{i}}^{\rm{th}},~{}i\in\left\\{{1,2}\right\\}$
(15e) $\displaystyle{m^{\rm{I}}}+{m^{\rm{II}}}={D_{\max}}.$ (15f)
The restriction on $p_{1}^{\rm{I}}$ in (15c) is applied based on the
assumption that $|h_{1}|^{2}>|h_{2}|^{2}$. So, to perform SIC correctly in the
NOMA phase, it is necessary that $p_{2}^{\text{I}}>p_{1}^{\text{I}}$. This
problem can be solved using exhaustive linear search; however, we shorten more
the search range of $p_{1}^{\rm{I}}$ to reduce the computational complexity.
The main idea can be summarized as follows:
* •
First, by considering user 1’s decoding error probability, i.e.,
$\varepsilon_{1}\approx\varepsilon_{1,2}^{\textrm{I}}+\varepsilon_{1,1}^{\textrm{I}}$
, the $p_{1}^{\rm{I}}$ bound that guarantees
${\varepsilon_{1}}\leq{\varepsilon_{1}}^{\rm{th}}$ is determined. According to
our previous work in [14], $\varepsilon_{1}$ is convex in $p_{1}^{\rm{I}}$ and
at most two values hold the
${\varepsilon_{1}}(p_{1}^{\rm{I}})={\varepsilon_{1}}^{{\rm{th}}}$. With
$R_{1,1}^{\rm{I}}=0$ and constant values of $m^{\rm{I}}$ and $P_{\rm{sum}}$,
we obtain the possible solutions that keep this equality in the range of
$0<p_{1}^{\rm{I}}<\tfrac{P_{\rm{sum}}}{2}$. Clearly,
$\varepsilon_{1,1}^{\textrm{I}}$ is a monotonically decreasing function of
$p_{1}^{\rm{I}}$, so it is derived that
$p_{1}^{{\rm{I,min}}}=\arg\\{{\varepsilon_{1}}(p_{1}^{\rm{I}})\approx\varepsilon_{1,1}^{\rm{I}}(p_{1}^{\rm{I}})=\varepsilon_{1}^{{\rm{th}}}\\}$.
On the other hand, $\varepsilon_{1,2}^{\textrm{I}}$ a monotonically increasing
function of $p_{1}^{\rm{I}}$ yields to
$p_{1}^{{\rm{I,max}}}=\arg\\{{\varepsilon_{1}}(p_{1}^{\rm{I}})\approx\varepsilon_{1,2}^{\rm{I}}(p_{1}^{\rm{I}})=\varepsilon_{1}^{{\rm{th}}}\\}$.
Hence, the search region of $p_{1}^{\rm{I}}$ is given by
$p_{1}^{\rm{I,min}}\leq p_{1}^{\rm{I}}\leq p_{1}^{\rm{I,max}}$.
* •
Since the decoding error probability is a monotonically increasing function of
the transmission rate, for each value of $p_{1}^{\rm{I}}$ in the feasible
range, $R_{1,1}^{\rm{I}}$ is increased until user 1’s decoding error
probability equals to $\varepsilon_{1}^{\rm{th}}$. One should note that
$R_{1,1}^{\rm{I}}\leq C(\gamma_{1,1}^{\rm{I}})$.
* •
Only those $p_{1}^{{\rm{I,min}}}\leq p_{1}^{\rm{I}}\leq p_{1}^{{\rm{I,max}}}$
that satisfy ${\varepsilon_{2}}(p_{1}^{\rm{I}})={\varepsilon_{2}}^{{\rm{th}}}$
could be acceptable. Since the decoding error probability of user 2 in both SC
and MRC strategies, respectively in (9) and (12), are increasing function of
$p_{1}^{\rm{I}}$, the transmit power can be obtained using the bisection
search method.
* •
After the full search on the values of $m^{\rm{I}}$ and $P_{\rm{sum}}$, among
the feasible solutions, the answer that maximizes $T_{1}$ is optimal.
Based on the above analysis, the algorithm for solving Problem (15) is
proposed in Algorithm 1. It first determines the local maximum of $T_{1}$,
i.e., ${T_{0}}^{\dagger}$, by taking constant $m^{\rm{I}}$ and checking all
possible values of $P_{\rm{sum}}$ and $p_{1}^{\rm{I}}$. In each iteration, the
bisection search is adopted to find the desired $p_{1}^{\rm{I}}$. By repeating
this process on all possible $m^{\rm{I}}$ with a positive integer value, the
global maximum of $T_{1}$, i.e., ${T_{0}}^{*}$, is found. Thus, using a three-
dimensional (3-D) exhaustive linear search, the globally optimal solution is
achieved.
1Input: total blocklength $D_{\max}$, overall BLER of user $i$
${\varepsilon_{i}}^{\rm{th}}$, BS average power $P_{\rm{ave}}$, required
accuracy $\epsilon$.
2 Output: optimum power $p{{}_{1}^{\rm{I}*}}$, $p{{}_{2}^{\rm{I}*}}$,
$p{{}_{2}^{\rm{II}*}}$, and blocklength ${m^{\rm{I}*}}$, ${m^{\rm{II}*}}$, and
fair throughput ${T_{1}}={T_{2}}={T_{0}}^{*}$.
3 for _${m^{\rm{I}}}=1:{D_{\max}}$_ do
4 for _${P_{\rm{sum}}}=0:\Delta p:{\kappa_{\rm{p}}}{P_{\rm{ave}}}$_ do
5 Set ${m^{\rm{II}}}:={D_{\max}}-{m^{\rm{I}}}$ and
$p_{2}^{\rm{II}}:={{\left({{D_{\max}}{P_{\rm{ave}}}-{m^{\rm{I}}}{P_{\rm{sum}}}}\right)}/{m^{\rm{II}}}}$.
6 if _$0\leq p_{2}^{\rm{II}}\leq{\kappa_{\rm{p}}}{P_{\rm{ave}}}$ &
${m^{\rm{I}}}{P_{\rm{sum}}}\geq{m^{\rm{II}}}p_{2}^{\rm{II}}$_ then
7 Calculate $p_{1}^{{\rm{I}},\min}$ and $p_{1}^{{\rm{I}},\max}$.
8 Set $p_{1}^{\rm{I}}:=p_{1}^{\rm{I,min}}$.
9 while _${\varepsilon_{2}} <\varepsilon_{2}^{\rm{th}}$_ do
10 Set $p_{1}^{\rm{I}}:=\min\left({p_{1}^{\rm{I}}+\Delta
p,p_{1}^{\rm{I,max}}}\right)$.
11 Find
$R{{}_{1,1}^{{\rm{I}}^{\dagger}}}=\arg\left\\{{{\varepsilon_{1}}=\varepsilon_{1}^{\rm{th}}}\right\\}$
via bisection method with accuracy $\epsilon$.
12 Calculate $\varepsilon_{2}$ by (9)/(12) for SC/MRC.
13
14 end while
15 Set $p_{1}^{{\rm{I,lb}}}:=p_{1}^{\rm{I}}-\Delta p$ and
$p_{1}^{\rm{I,ub}}:=p_{1}^{\rm{I}}$.
16 Find
$p{{}_{1}^{{\rm{I}}^{\dagger}}}\in\left[{p_{1}^{\rm{I,lb}},p_{1}^{\rm{I,ub}}}\right]$
that satisfies ${\varepsilon_{2}}=\varepsilon_{2}^{\rm{th}}$ via bisection
method with accuracy $\epsilon$.
17
18 end if
19
20 end for
21 Set
$R{{}_{1,1}^{{\rm{I}}^{{\ddagger}}}}:=\max\left\\{{R{{}_{1,1}^{{\rm{I}}^{\dagger}}}\left|{{\varepsilon_{2}}=\varepsilon_{2}^{\rm{th}}}\right.}\right\\}$
and
${T_{0}}^{\dagger}:={{\left({1-\varepsilon_{1}^{\rm{th}}}\right){m^{\rm{I}}}R{{}_{1,1}^{\rm{I}^{{\ddagger}}}}}/{D_{\max}}}$.
22
23 end for
24Set ${T_{0}}^{*}:={\max\\{{{T_{0}}^{\dagger}}\\}}$.
Return
$\left\\{{{m^{\rm{I}*}},p{{}_{1}^{\rm{I}*}},p{{}_{2}^{\rm{II}*}}}\right\\}=\arg\max\\{{{T_{0}}^{\dagger}}\\}$,
${m^{\rm{II}*}}={D_{\max}}-{m^{\rm{I}*}}$,
$p{{}_{2}^{\rm{I}*}}=\frac{\left({{D_{\max}}{P_{\rm{ave}}}-{m^{\rm{II}}}^{*}p{{}_{2}^{\rm{II}*}}}\right)}{{m^{\rm{I}}}^{*}}-p{{}_{1}^{\rm{I}*}}$.
Algorithm 1 Optimum Power and Blocklength Allocation Algorithm in the C-NOMA
Scheme with SC/MRC Strategy
### V-B Suboptimal Design of Max-Min Fairness in C-NOMA
Although the search bounds of the optimum solution of Problem (15) stated in
Algorithm 1 have been limited, the computational complexity is still high. Now
we propose a suboptimal solution to this problem. If phase II transmission is
not successful, part of the resources will go to waste, which in turn, will
cause the system throughput reduction below the NOMA scheme’s one. Therefore,
to avoid this condition and decrease the decoding error probability in phase
II, $x_{2}^{\prime}$ is transmitted with the maximum power, i.e.,
$p_{2}^{\rm{II}}=\kappa_{\rm{p}}{P_{\rm{ave}}}$. Hence, the summation of two
users’ transmit power in phase I is calculated as
$P_{\rm{sum}}=\left[\left(D_{\max}P_{\rm{ave}}-m^{\rm{II}}p_{2}^{\rm{II}}\right)/m^{\rm{I}}\right]^{+}$,
where ${\left[x\right]^{+}}\overset{\Delta}{=}\max\left\\{{x,0}\right\\}$.
Then, as before, the local maximum of $T_{1}$, i.e., ${T_{0}}^{\dagger}$, is
obtained by searching on the possible values of $p_{1}^{\rm{I}}$ within the
range of $\left[{p_{1}^{{\rm{I,}}\min},p_{1}^{{\rm{I,}}\max}}\right]$. By
repeating this process on all possible integer values of $m^{\rm{I}}$ that
satisfy ${m^{\rm{I}}}{P_{{\rm{sum}}}}\geq{m^{\rm{II}}}p_{2}^{\rm{II}}$, the
global maximum of $T_{1}$, i.e., ${T_{0}}^{*}$, is found. If
$m^{\rm{I}}=D_{\max}$ , or equivalently $m^{\rm{II}}=0$, then
$P_{\rm{sum}}=P_{\rm{ave}}$. In this case, signal transmission in phase II
does not occur, and the C-NOMA scheme is transformed into the NOMA. This
suboptimal algorithm which is a special case of Algorithm 1, needs a two-
dimensional (2-D) linear search on
$\left\\{p_{1}^{\rm{I}},m^{\rm{I}}\right\\}$. The numerical results in section
VII demonstrate that the performance of the suboptimal solution is slightly
worse than the optimal one, while has much lower computational complexity.
### V-C Computational Complexity
The computational complexity of Algorithm 1 is calculated as follows. In the
first step, to obtain the bounds of $p_{1}^{\rm{I}}$, a linear search with
complexity $\Omega_{1}$ is applied. In the next step, $R_{1,1}^{\rm{I}}$ is
derived via the bisection method with complexity around
${\log_{2}}({{\varepsilon_{1}^{{\rm{th}}}}/\epsilon})$ where $\epsilon$ is the
desired accuracy. Besides, the complexity of computing $\varepsilon_{2}$ is
denoted as $\Omega_{2}$. This step is performed at most
${K_{1}}={{(p_{1}^{{\rm{I,max}}}-p_{1}^{{\rm{I,min}}})}/{\Delta p}}$ times
where ${\Delta p}$ is the search step, so its complexity is denoted as
${K_{1}}\left({\log_{2}}({{\varepsilon_{1}^{{\rm{th}}}}/\epsilon})+\Omega_{2}\right)$.
In the last step, finding $p_{1}^{\rm{I}}$ via the bisection search method has
complexity around ${\log_{2}}({{\varepsilon_{2}^{{\rm{th}}}}/\epsilon})$.
These three steps are repeated on the possible values of $P_{\rm{sum}}$ and
$m^{\rm{I}}$, respectively ${K_{2}}={{{\kappa_{\rm{p}}}{P_{\rm{ave}}}}/{\Delta
p}}$ and $D_{\max}$ times. Therefore, the worst-case complexity of Algorithm 1
is ${\cal
O}\left({{K_{2}}{D_{\max}}\left({\Omega_{1}}+{K_{1}}({\log_{2}}({{\varepsilon_{1}^{\rm{th}}}/\epsilon})+{\Omega_{2}})+{\log_{2}}({{\varepsilon_{2}^{{\rm{th}}}}/\epsilon})\right)}\right)$.
Likewise, the computational complexity of the suboptimal algorithm is
determined based on the above analysis. However, since $p_{2}^{\rm{II}}$ is a
constant value, $P_{\rm{sum}}$ is removed from the search process. Hence, the
worst-case complexity of this algorithm is ${\cal
O}\left({D_{\max}}\left({\Omega_{1}}+{K_{1}}({\log_{2}}({{\varepsilon_{1}^{{\rm{th}}}}/\epsilon})+{\Omega_{2}})+{\log_{2}}({{\varepsilon_{2}^{{\rm{th}}}}/\epsilon})\right)\right)$.
Although the number of iterations of the proposed algorithms for both SC and
MRC techniques is equal, the number of basic operations related to computing
the user 2’s decoding error, i.e., $\varepsilon_{2}$ , is different. According
to (9), calculation of $\varepsilon_{2}$ in the SC technique just includes one
summation and one multiplication; while, calculation of $\varepsilon_{2}$ in
the MRC technique, regarding (12), requires three summations (one is due to
$\gamma_{2,2}^{\textrm{C}}$) and two multiplications.
## VI Extension to Multi-User Scenario
This section considers a more general situation shown in Fig. 1(a) when there
are more than two users in a cell.
### VI-A Problem Formulation
Let us denote the total number of users as $2K$, and the set of users as
${\cal K}=\left\\{{1,2,\ldots,2K}\right\\}$. We assume that the users’ channel
gains are arranged in descending order, i.e.,
$|{h_{1}}|^{2}>|{h_{2}}|^{2}>\cdots>|{h_{2K}}|^{2}$. To implement the NOMA
scheme, users are grouped into some clusters. While NOMA distinguishes the
users in one cluster, the various clusters become distinct by the OMA
technique. Usually, in practice, to decrease the receiver’s complexity, the
number of users in each cluster is not considered more than four. Here we form
clusters with two users and apply the C-NOMA scheme in each pair. Since for
relaying, the two users need to be in the coverage area of each other; pairing
is done concerning their relative locations. The number of 2-user clusters is
$K$ in the considered network, but the number of possible pairing states is
completely random respecting the network topology and is denoted by $Q$. The
throughput function of pairing in State $q$, where $q=1,\ldots,Q$, is defined
as follows
${f_{q}}\left({{{\bf{A}}^{q}},p_{i}^{\rm{I}},p_{j}^{\rm{II}},m_{i,j}^{\rm{I}}}\right)=a_{i,j}^{q}T_{0}^{i,j};\quad
i,j\in{\cal K}.$ (16)
Let ${{\bf{A}}^{q}}={\left[{a_{i,j}^{q}}\right]_{2K\times 2K}}$ be the pairing
matrix in State $q$. Here $a_{i,j}^{q}$ denotes the link between users $i$ and
$j$ in State $q$ where
$a_{i,j}^{q}=\left\\{\begin{array}[]{l}1,~{}{\textrm{ if users }}i{\textrm{
and }}j{\textrm{ are paired,}}\\\ 0,~{}{\textrm{
otherwise.}}\end{array}\right.$ (17)
The goal is to find the optimum pairing that maximizes the minimum throughput
of the cell users. Thus, the optimization problem can be formulated as
$\displaystyle\mathop{\max}\limits_{q=1,\ldots,Q}$
$\displaystyle\mathop{\min}\limits_{{{\bf{A}}^{q}}=\left[{a_{i,j}^{q}}\right]}{f_{q}}\left({{{\bf{A}}^{q}},p_{i}^{\rm{I}},p_{j}^{\rm{II}},m_{i,j}^{\rm{I}}}\right)$
(18a) $\displaystyle{\rm{s.t.}}\quad$ $\displaystyle
a_{i,j}^{q}=a_{j,i}^{q};~{}i,j\in{\cal K}$ (18b)
$\displaystyle\sum\nolimits_{j\in{\cal K}\backslash i}{a_{i,j}^{q}}\leq
1,~{}i\in{\cal K}$ (18c) $\displaystyle\sum\nolimits_{i\in{\cal K}\backslash
j}{a_{i,j}^{q}}\leq 1,~{}j\in{\cal K}.$ (18d)
Constraint (18b) shows that the pairing matrix ${\bf{A}}^{q}$ is symmetric.
Moreover, constraints (18c) and (18d) indicate that users $i$ and $j$ cannot
belong to more than one pair. The inter-programming problem of Problem (18)
that applies the C-NOMA scheme in each pair is expressed as follows
$\displaystyle T_{0}^{i,j}=$
$\displaystyle\mathop{\max}\limits_{\left\\{{m_{i,j}^{\rm{I}},p_{i}^{\rm{I}},p_{j}^{{\rm{II}}}}\right\\}}\min\left\\{{{T_{i}},{T_{j}}}\right\\},~{}\forall
a_{i,j}^{q}=1$ (19a) $\displaystyle{\rm{s.t.}}\quad$ $\displaystyle
m_{i,j}^{\rm{I}}\left({p_{i}^{\rm{I}}+p_{j}^{\rm{I}}}\right)+m_{i,j}^{\rm{II}}p_{j}^{\rm{II}}=\tfrac{{D_{\max}}{P_{\rm{ave}}}}{K}$
(19b) $\displaystyle
0<p_{i}^{\rm{I}}+p_{j}^{\rm{I}}\leq{\kappa_{\rm{p}}}{P_{\rm{ave}}}$ (19c)
$\displaystyle 0\leq p_{j}^{\rm{II}}\leq{\kappa_{\rm{p}}}{P_{\rm{ave}}}$ (19d)
$\displaystyle{\varepsilon_{i}}={\varepsilon_{i}}^{\rm{th}},~{}{\varepsilon_{j}}={\varepsilon_{j}}^{\rm{th}}$
(19e) $\displaystyle m_{i,j}^{\rm{I}}+m_{i,j}^{\rm{II}}={D_{\max}}.$ (19f)
Here it is assumed that $|{h_{i}}|^{2}>|{h_{j}}|^{2}$ so
$p_{i}^{\rm{I}}<p_{j}^{\rm{I}}$. Constraint (19b) indicates that the total
system’s energy consumption is distributed equally among the pairs. For
solving Problem (18), it is needed that problem (19) is solved for all the
potential pair-users in State $q$, i.e., $\forall
a_{i,j}^{q}=1;~{}i,j=1,\ldots,2K$. Hence, to find the optimum pairing, the
inter-programming problem has to be solved $QK$ times. By an exhaustive search
over all the neighboring users, every two users are paired that the minimum
achieved throughput in the cell is maximized. The complexity of the exhaustive
search (i.e., the number of iterations needed to find the optimal pairing) is
almost high, resulting in excessive scheduling delay with a large number of
users. To alleviate the computational complexity, a suboptimal pairing
algorithm is proposed in the following subsection.
### VI-B The proposed C-NOMA pairing
Here, a suboptimal solution for Problem (18) is proposed. The objective is to
maximize the throughput of the weakest user among all $2K$ users by allocating
them into different pairs according to the geographic locations. To implement
the proposed user pairing, the graph matrix of the network topology has to be
obtained first. For this purpose, each user ought to find all the users in its
coverage area with radius $r_{0}$. Since the aim is leveraging C-NOMA to
increase reliability and system capacity, the priority is with C-NOMA pairs
starting from the weakest user. Users that are far from others and do not have
a chance to exploit the C-NOMA technique use NOMA or OMA instead, depending on
their channel condition. Finally, the users that have not been scheduled in
C-NOMA pairs are rearranged to form hybrid NOMA/OMA pairs, as will be
described in the following subsection. Algorithm 2 expresses the proposed
C-NOMA-based user pairing in detail. The fact that how frequently the pairing
process is executed mainly depends on the URLLC use case. For example, in
factory automation with fixed or slow speed devices, the algorithm does not
need to perform in each frame. Moreover, it should be noted that these
computations are performed at the BS with the assumption of CSIT, and the
results are sent to the users.
1Input: sorted DL channel gains in descending order
${|{h_{1}}|^{2}}>{|{h_{2}}|^{2}}>\cdots>{|{h_{2K}}|^{2}}$ and the
corresponding D2D channel gains, device coverage radius $r_{0}$, inputs of
Algorithm 1.
2 Output: the user pairing ${\bf{A}}={\left[{a_{i,j}}\right]_{2K\times 2K}}$.
3 Determine the graph matrix of the network topology.
4 Set $i:=2K$.
5 while _$i\geq 1$_ do
// allocating C-NOMA pairs
6 if _user $i$ has not been paired _ then
7 Find the set of unpaired adjacent users of user $i$, ${{\bm{\psi}}_{i}}$.
8 if _${\rm{length}}\left({{\bm{\psi}}_{i}}\right)\neq 0$_ then
9 for _$l=1:{\rm{length}}\left({{\bm{\psi}}_{i}}\right)$_ do
10 Calculate $T_{0}^{i,{{\bm{\psi}}_{i}}(l)}$ by Algorithm 1.
11
12 end for
13 Set
$\left[{T_{0}^{*},index}\right]:=\max\left\\{{T_{0}^{i,{{\bm{\psi}}_{i}}(l)}}\right\\}$
and $j:={{\bm{\psi}}_{i}}(index)$.
14 Pair users $i$ and $j$, i.e. ${a_{i,j}}:=1$.
15
16 end if
17
18 end if
19 Set $i:=i-1$.
20
21 end while
22Set $i:=2K$ and $j:=1$.
23 while _$i >j$_ do
// allocating hybrid pairs
24 if _user $i$ has not been paired _ then
25 while _user $j$ has been paired _ do
26 Set $j:=j+1$.
27
28 end while
29 Pair users $i$ and $j$, i.e. ${a_{i,j}}:=1$.
30 Set $j:=j+1$.
31
32 end if
33 Set $i:=i-1$.
34
35 end while
Return: ${\bf{A}}={\left[{a_{i,j}}\right]_{2K\times 2K}}$.
Algorithm 2 Joint suboptimal C-NOMA-based user pairing and resource allocation
### VI-C Hybrid pairing
To describe the hybrid pairing, let us first consider the NOMA user pairing
scheme proposed in [33]. Pursuant to this, the first strong user is paired
with the first weak user; the second strong user is paired with the second
weak user, and so on. Accordingly, all the users are paired. The fact is that
principle of NOMA is to select users with a high difference in their channel
gains. In particular, NOMA’s performance diminishes when the difference in
channel gains among the users is small. For example, in Fig. 2, user-pairs 6
and 7, which have almost the same channel conditions, may decrease the
spectral efficiency and system capacity due to the unsuccessful SIC. Hence, it
is sensible that such non-suitable pairs are omitted from NOMA scheduling, and
their clustering continues with OMA. In this method, the BS adaptively
switches between the NOMA and OMA transmission modes according to the
instantaneous strength of wireless channels and hence the performance of the
NOMA/OMA user pairing, and each of them that meets the max-min fairness
criterion is selected as the access scheme.
We discussed the hybrid pairing for scheduling the users that are isolated or
left unpaired in the proposed C-NOMA-based user pairing. However, these two
basic schemes, namely NOMA and hybrid user pairing, can independently be
implemented and are considered as benchmark schemes in our simulations.
Figure 2: The 2-user NOMA pairing scheme [33].
## VII Numerical Results
In this section, the proposed C-NOMA scheme’s performance along with SC and
MRC strategies are evaluated through the numerical results based on our
analytical solutions. A heterogeneous network consists of URLLC users with
different reliability requirements is considered. PAPR factor and required
accuracy in Algorithm 1 are considered as $\kappa_{\rm{p}}=1.2$ and
$\epsilon=10^{-15}$, respectively. Also, it is assumed that
$P_{\rm{ave}}=10{\rm{~{}W}}$ and $D_{\max}=200$ channel uses, unless otherwise
stated. The numerical results are provided based on fixed channel gains with
two users and random channel gains with more than two users, which are
presented in the following two subsections.
### VII-A Two-user Network with Fixed Channel Gains
Throughout this subsection, to provide insight into the relationships between
the proposed and the benchmark schemes, the channel gains of the two users are
set to be fixed. For instance, it is assumed that
$|{h_{1}}|^{2}/{\sigma^{2}}=0.8$ and $|{h_{2}}|^{2}/{\sigma^{2}}=0.1$. We
investigate the performance of the proposed schemes in two various relaying
link status. Meaning, when the two users are near to each other and the
relaying link is strong, it is assumed that
$|{h_{1,2}}|^{2}/{\sigma^{2}}=0.5$, and when the two users are far from each
other, and the relaying link is poor, it is assumed that
$|{h_{1,2}}|^{2}/{\sigma^{2}}=0.01$. Meanwhile, users BLER are considered as
$\varepsilon_{1}^{\rm{th}}=10^{-7}$ and $\varepsilon_{2}^{\rm{th}}=10^{-5}$.
In Fig. 3, the effect of total blocklength, $D_{\max}$, on the fair throughput
in the proposed C-NOMA with SC and MRC strategies is assessed in two relaying
link modes. Also, the optimal NOMA and OMA results in our previous work [14]
are shown for comparison. It is observed that in the strong relaying link
mode, both combining strategies applied to the C-NOMA effectively improve the
fair throughput compared to the NOMA/OMA. It is also observed that the MRC
receiver outperforms the SC receiver, regardless of the blocklength. Because
in the combined signal with MRC protocol, SINR increases, so the decoding
error probability of user 2 decreases. Hence, it is possible that by less
blocklength allocation to phase II, the reliability performance of user 2 can
still be guaranteed at the desired level. As a result, more blocklength is
allocated to phase I. Hence, users’ data rates and system fair throughput
increase.
On the other hand, in a poor relaying link, the C-NOMA scheme (in both
combining strategies) has exactly the same performance as the NOMA. In fact,
in this case, the optimal decision is in favor of the direct link, and the
C-NOMA is transformed into the NOMA. However, in a realistic wireless channel,
mixed conditions occur together, and C-NOMA outperforms the NOMA on average.
Moreover, it is observed that suboptimal solutions in both SC and MRC
receivers converge to the near-optimal solutions.
In Fig. 4, the effect of average total power, $P_{\textrm{ave}}$, on the fair
throughput is investigated. In the strong relaying link mode, the C-NOMA’s
superiority with MRC receiver is notable against the SC receiver and the
NOMA/OMA scheme. In addition, the C-NOMA with SC strategy outperforms the NOMA
in low power/SNR ranges, while it coincides with the NOMA on average powers
greater than 20 W. This could be justified by the fact that in SC strategy,
the signals do not combine, and transmission in phase II assures the success
of user 2’s packet decoding. Hence, in low SNRs where the weak user’s
probability of successful decoding in phase I is not too high, the reliability
is increased by retransmission in phase II. However, in high SNRs, where the
allocated power of user 2 in the NOMA phase guarantees the reliability, phase
II transmission is pointless. Therefore, in this case, transmission via a
single phase is optimal in comparison with two-phase, and the proposed scheme
performs like the NOMA. Moreover, in the poor relaying link mode, the C-NOMA
scheme always complies with the NOMA. As a result, from the complexity
perspective, the C-NOMA usage with SC strategy seems sensible just in low SNR
regimes.
Figure 3: Maximum fair throughput achieved by the C-NOMA and NOMA schemes
versus $D_{\max}$, when $P_{\rm{ave}}=10{\rm{~{}W}}$. Figure 4: Maximum fair
throughput achieved by the C-NOMA and NOMA schemes versus $P_{\rm{ave}}$, when
$D_{\max}=200$.
### VII-B Multi-user Network with Random Channel Gains
Here, we assume that the BS is located at the center of a cell with radius of
$300$ m. The system bandwidth is set as $B=1$ MHz, which is equivalent to a DL
transmission duration $0.2$ ms for a blocklength of $200$ channel uses, and
satisfies the low-latency criterion of URLLC standards. The noise power
spectral density is $-173$ dBm/Hz, and small-scale channel coefficients are
Rayleigh fading with ${\cal C}{\cal N}\left({0,1}\right)$ distribution. Large-
scale path loss is modeled as $L=35.3+37.6{\log_{10}}d({\rm{m}})$ dB [24]. The
total number of independent channel generations is set as $1000$.
Fig. 5 illustrates the average achievable fair throughput versus the number of
users, $2K$ , for the proposed C-NOMA pairing with SC and MRC strategies. We
compare it with exhaustive search method and the method proposed in [26],
which is based on pairing one near user and one far user. In that method,
first we sort the $K^{2}$ D2D channel gains of any near-far pair and delete
the $K-1$ weakest channels. Then, by considering the number of deleted channel
gains of each far user, first a near user is paired to the far user with the
largest-number deleted channel and last the far user with the smallest-number.
The selection criteria is to maximize the throughput of the far user. To be
comparable with our proposed method, unlike [26], we assume that both near and
far users can communicate with the BS directly, and SC/MRC combining schemes
are performed at the far user. Moreover, the NOMA and hybrid pairing schemes
are illustrated as benchmark. It demonstrates that the proposed C-NOMA pairing
scheme (in both MRC and SC techniques) converges to a near-optimal solution.
While, the near-far pairing method achieves the lower performance in both
combining strategies. On the other hand, the NOMA pairing scheme stated in
[33] yields the lowest throughput, especially in the presence of a large
number of users, and as expected, the hybrid pairing scheme outperforms the
NOMA pairing.
To evaluate the fairness of the proposed C-NOMA-based user pairing, Fig. 6
indicates Jain’s fairness index for the proposed scheme and the benchmarks.
Jain’s fairness index is defined as [34]
$J=\frac{{\left({\sum\nolimits_{k=1}^{K}{T_{k}^{*}}}\right)}^{2}}{K\sum\nolimits_{k=1}^{K}{T{{{}_{k}^{*}}^{2}}}},$
(20)
where $T_{k}^{*}$ indicates the optimal fair throughput of pair $k$. Jain’s
fairness index is bounded in $[0,1]$ which equal users’ throughput obtains the
maximum value. As Fig. 6 illustrates, the hybrid pairing scheme is fairer
comparing to the C-NOMA-based and the NOMA pairing schemes. The reason is that
in the C-NOMA-based pairing schemes, i.e., the proposed, near-far, and
exhaustive search methods, creating C-NOMA pairs for all the users is not
probable. Hence, unavoidably, some users are scheduled in hybrid NOMA/OMA
pairs. Since the C-NOMA users will achieve more throughput than the users with
hybrid pairing, the fairness will degrade in these schemes.
Moreover, regarding the logic behind the hybrid pairing, it will always be
fairer than the NOMA pairing. Interestingly, the C-NOMA-based pairing schemes
(with both combining strategies) result in more fairness relative to the NOMA
pairing in the presence of a large number of users. This is due to the fact
that the denser the network is, the more users will experience the same
channel. This will cause more failures in NOMA scheduling, so the C-NOMA
pairing will obtain more fairness in that case.
Figure 5: Average fair throughput achieved by the different pairing schemes
versus the number of users. Figure 6: Fairness comparison between the
different pairing schemes versus the number of users.
## VIII Conclusion and Future Works
In this paper, the combination of NOMA with the cooperative relaying technique
(i.e., C-NOMA) was considered in short packet communications to guarantee high
reliability and low latency. The performance of two relaying strategies, i.e.,
SC and MRC, was presented in terms of decoding error probability in a quasi-
static channel. Besides, the necessity to provide QoS of all users with
critical services motived us to consider max-min fairness as a design
criterion in URLLC systems. To this end, first, an optimization problem was
formulated for a two-user DL C-NOMA system, and optimal power, blocklength,
and transmission rate were determined under the total energy consumption,
reliability, and delay constraints. To decrease the computational complexity,
a suboptimal algorithm was proposed with near-optimal performance. Numerical
results showed that the proposed C-NOMA scheme improves the users’ fair
throughput significantly, compared to the NOMA scheme. Moreover, it was
demonstrated that the C-NOMA scheme with MRC strategy outperforms SC strategy.
Finally, the problem was extended to a multi-user scenario, and a pairing
scheme based on C-NOMA was proposed. Monte Carlo simulations showed that the
proposed C-NOMA pairing scheme performs close to the optimal solution, with
less computational complexity. Further, the simulation results verify the
supremacy of the proposed user pairing (with both SC and MRC techniques) over
the near-far pairing method proposed in [26], as well as the NOMA and hybrid
OMA/NOMA pairing schemes in boost the average fair throughput despite
degrading the fairness index slightly.
The presented work in this paper can be extended from different directions.
Two of the main potential extensions that remain for future works are
developing the proposed pairing algorithm for the case of statistical CSI, and
considering a distributed method for network coordination. The statistical CSI
knowledge can remove the shortage of the out-dated CSI and the feedback
overhead due to CSIT. Despite distributed method for network coordination
might look to be more suitable for URLLC, it requires deep investigation as
they normally introduce different types of overhead which may cause additional
delay.
## Appendix A Proof of Proposition 1
We prove Proposition 1 by the contradiction method. We consider the optimal
solution of problem (15) as
$\left\\{{p{{{}_{1}^{\rm{I}{\dagger}}}},p{{{}_{2}^{\rm{I}{\dagger}}}},p{{{}_{2}^{\rm{II}{\dagger}}}},{m^{\rm{I}{\dagger}}},{m^{\rm{II}{\dagger}}}}\right\\}$,
where
${m^{\rm{I}{\dagger}}}({p_{1}^{{\rm{I}}{\dagger}}+p_{2}^{{\rm{I}}{\dagger}}})<{m^{\rm{II}{\dagger}}}p_{2}^{{\rm{II}}{\dagger}}$.
It can achieve the maximum value of $\min\left\\{{{T_{1}},{T_{2}}}\right\\}$,
which is denoted by $T_{0}^{\dagger}$. We increase $p_{1}^{{\rm{I}}{\dagger}}$
and $p_{2}^{{\rm{I}}{\dagger}}$ by multiplying in a scalar value $\alpha>1$ to
attain ${p_{1}^{\rm{I}}}^{*}=\alpha{p_{1}^{\rm{I}}}^{\dagger}$ and
${p_{2}^{\rm{I}}}^{*}=\alpha{p_{2}^{\rm{I}}}^{\dagger}$. It can be verified
that the following equation holds,
$\begin{array}[]{c}{m^{\rm{I}{\dagger}}}\left({p{{}_{1}^{\rm{I}{\dagger}}}+p{{}_{2}^{\rm{I}{\dagger}}}}\right)+{m^{\rm{II}{\dagger}}}p{{}_{2}^{\rm{II}{\dagger}}}=\\\
{m^{\rm{I}{\dagger}}}\left({{p_{1}^{\rm{I}}}^{*}+{p_{2}^{\rm{I}}}^{*}}\right)+{m^{\rm{II}{\dagger}}}{p_{2}^{\rm{II}}}^{*}={D_{\max}}{P_{\rm{ave}}}\end{array}$
We note that since $\alpha>1$, so
${p_{1}^{\rm{I}}}^{*}>{p_{1}^{\rm{I}}}^{\dagger}$ and
${p_{2}^{\rm{I}}}^{*}>{p_{2}^{\rm{I}}}^{\dagger}$. Hence, we have
${\gamma_{1,1}^{\rm{I}}}^{*}>{\gamma_{1,1}^{\rm{I}}}^{\dagger}$ and
$\displaystyle{\gamma_{2,2}^{\rm{I}}}^{*}$
$\displaystyle=\frac{{{p_{2}^{\rm{I}}}^{*}{{\left|{h_{2}}\right|}^{2}}}}{{p_{1}^{\rm{I}}}^{*}{{\left|{h_{2}}\right|}^{2}}+{\sigma^{2}}}=\frac{p_{2}^{{\rm{I}}{\dagger}}{{\left|{h_{2}}\right|}^{2}}}{p_{1}^{{\rm{I}}{\dagger}}{{\left|{h_{2}}\right|}^{2}}+\frac{\sigma^{2}}{\alpha}}$
$\displaystyle>\frac{p_{2}^{{\rm{I}}{\dagger}}{{\left|{h_{2}}\right|}^{2}}}{p_{1}^{{\rm{I}}{\dagger}}{{\left|{h_{2}}\right|}^{2}}+{\sigma^{2}}}={\gamma_{2,2}^{\rm{I}}}^{\dagger}$
This means as ${p_{i}}^{{\rm{I}}{\dagger}}$, $i\in\left\\{{1,2}\right\\}$,
increases to ${p_{i}^{{\rm{I}}}}^{*}$, the corresponding SNR/SINR increases,
which results in an increase in $R_{i,i}^{\rm{I}}$ and finally $T_{i}$
increases (Invoking [14, Appendix A], the allowed $R_{i,i}^{\rm{I}}$ is a
monotonically increasing function of $\gamma_{i,i}^{\rm{I}}$.). On the other
hand, $R_{i,i}^{\rm{I}}$ and so $T_{i}$ are clearly increasing functions of
${m^{\rm{I}}}$. Then, we can construct a new solution
$\left\\{{p_{1}^{\rm{I}}}^{*},{p_{2}^{\rm{I}}}^{*},{p_{2}^{\rm{II}}}^{*},{m^{\rm{I}}}^{*},{m^{\rm{II}}}^{*}\right\\}$,
where corresponds to $T_{0}^{*}$. Also,
${m^{\rm{I}}}^{*}={m^{{\rm{I}}{\dagger}}}+\Delta m$ and
${m^{\rm{II}}}^{*}={m^{{\rm{II}}{\dagger}}}-\Delta m$ with $\Delta m>0$. As
before, this solution satisfies
${m^{\rm{I}}}^{*}\left({{p_{1}^{\rm{I}}}^{*}+{p_{2}^{\rm{I}}}^{*}}\right)+{m^{\rm{II}}}^{*}{p_{2}^{\rm{II}}}^{*}={D_{\max}}{P_{\rm{ave}}}$.
Since ${p_{i}^{\rm{I}}}^{*}>p_{i}^{{\rm{I}}{\dagger}}$ and
${m^{\rm{I}}}^{*}>{m^{{\rm{I}}{\dagger}}}$, we have
$T_{0}^{*}>T_{0}^{\dagger}$. This contradicts the assumption that
$T_{0}^{\dagger}$ is an optimal solution. So, we can always find a proper
$\alpha$ and $\Delta m$ such that
${m^{\rm{I}}}^{*}\left({{p_{1}^{\rm{I}}}^{*}+{p_{2}^{\rm{I}}}^{*}}\right)>{m^{\rm{II}}}^{*}{p_{2}^{\rm{II}}}^{*}$.
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Fateme Salehi received the B.Sc. and M.Sc. degrees in communication
engineering from University of Birjand (Birjand, Iran) in 2010 and 2012
respectively. Currently she is a Ph.D. student in communication engineering in
Department of Electrical and Computer Eng. at the University of Birjand, Iran.
Since March 2021, she has joined KTH Royal Institute of Technology as a
visiting researcher. Her research interests include signal processing, channel
estimation, MIMO-OFDM and NOMA systems, resource management in URLLC, and the
Internet-of-Things.
---
Naaser Neda received the B.S. degree in electrical Eng. from the University of
Tehran and the M.S. degree in communication Eng. from Sharif University of
Technology (SUT), both in Tehran, Iran, in 1990 and 1994 respectively. He
received the PhD degree in communication Eng. from the University of Surrey
(CCSR), Guildford, UK, in 2003. He is currently an Associate Professor of
communication engineering with the Department of Electrical and Computer Eng.
at the University of Birjand, Iran. His research interests include signal
processing for communication systems, physical layer of CDMA/MCCDMA/OFDM
networks, sensor networks, and NOMA.
---
Mohammad-Hassan Majidi received the B.S. degree in electrical Eng. from the
Shahid Bahonar University of Kerman and the M.S. degree in communication Eng.
from Imam Hossein Comprehensive University of Tehran, both in Iran, in 2003
and 2006 respectively. He received the PhD degree in telecommunication
engineering from the Department of Telecommunications, Ecole Superieure
d’Electricite (Supelec), Gif-sur-Yvette, France, in 2013. He is currently an
Assistant Professor of communication engineering with the Department of
Electrical and Computer Eng. at the University of Birjand, Iran. His research
interests include signal processing for communication systems, joint channel
and data detection, cryptography and secure communication.
---
Hamed Ahmadi received the Ph.D. degree from the National University of
Singapore, in 2012. He was an Agency for Science Technology and Research
(A-STAR) funded Ph.D. student at the Institute for Infocomm Research (I2R),
National University of Singapore. Since then, he has been working with
different academic and industrial positions in Ireland and U.K. He is
currently an Assistant Professor with the Department of Electronic
Engineering, University of York, U.K. He is also an Adjunct Assistant
Professor with the school of Electrical and Electronic Engineering, University
College Dublin, Ireland. He has published more than 50 peer-reviewed book
chapters, journal, and conference papers. His current research interests
include design, analysis, and optimization of wireless communications
networks, airborne networks, wireless network virtualization, blockchain, the
Internet-of-Things, cognitive radio networks, and the application of machine
learning in small cell and self-organizing networks. He is a member of
Editorial Board of IEEE ACCESS, Frontiers in Blockchain, and Wireless Networks
(Springer). He is a Fellow of the U.K., Higher Education Academy and Networks
working Group Co-Chair and a management committee member of COST Action 15104
(IRACON).
---
|
# Unified description of galactic dynamics and the cosmological constant
Mariano Cadoni<EMAIL_ADDRESS>Andrea P. Sanna<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria,
09042, Monserrato, Italy Istituto Nazionale di Fisica Nucleare (INFN),
Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
(August 27, 2024)
###### Abstract
We explore the phenomenology of a two-fluid cosmological model, where the
field equations of general relativity (GR) are sourced by baryonic and cold
dark matter. We find that the model allows for a unified description of small
and large scale, late-time cosmological dynamics. Specifically, in the static
regime we recover the flattening of galactic rotation curves by requiring the
matter density profile to scale as $1/r^{2}$. The same behavior describes
matter inhomogeneities distribution at small cosmological scales. This traces
galactic dynamics back to structure formation. At large cosmological scales,
we focus on back reaction effects of the spacetime geometry to the presence of
matter inhomogeneities. We find that a cosmological constant with the observed
order of magnitude, emerges by averaging the back reaction term on spatial
scales of order $100\ \text{Mpc}$ and it is related in a natural way to matter
distribution. This provides a resolution to both the cosmological constant and
the coincidence problems and shows the existence of an intriguing link between
the small and large scale behavior in cosmology.
The $\Lambda$-Cold Dark Matter (CDM) model represents our current best
understanding of the observed properties of the universe, by assuming that
only $\sim 5\%$ of its energy content is constituted by baryonic matter, while
the remaining $\sim 95\%$ is exotic. Specifically, $\sim 30\%$ is associated
to non-baryonic CDM, $\sim 65\%$ to dark energy in the form of a cosmological
constant (CC) $\Lambda$ Aghanim:2018eyx . The former allows for a simple
explanation of a wide variety of observations, ranging from the flattening of
rotation curves in disk galaxies and the internal dynamics of galaxy clusters
to cosmological structure formation and evolution, the abundances of light
elements and the power spectrum of the cosmic microwave background radiation
Aghanim:2018eyx ; Bertone:2004pz ; Cuoco:2003cu . The CC accounts instead for
the observed accelerated expansion of the universe Riess:1998cb ;
Perlmutter:1998np .
Despite these successes, several questions remain open. The most pressing ones
are perhaps the understanding of the nature of the dark components, the
explanation of the origin of galactic and cluster dynamics in terms of their
formation and explaining why the CC has the observed value and why its energy
density is so closed to that of matter in the present epoch (the cosmological
constant and the coincidence problems)Peebles:2002gy .
In this letter, we build on the results of Ref. Cadoni:2020jxe , which are
sufficiently general to be applied to various scenarios where cosmology is
sourced by a two-fluid system. Working in the standard $\Lambda$CDM framework
and using baryonic matter and CDM (whose existence is here assumed) as sources
of the gravitational field, we tackle some of the aforementioned problems of
late-time cosmology. Specifically, our model reproduces, in the static regime,
the flattening of galactic rotation curves, whereas at small cosmological
scales explains local inhomogeneities and structure formation. This last
result, in particular, traces the origin of galactic dynamics back to
structure formation. At large cosmological scales, back reaction effects of
the geometry to the presence of matter inhomogeneities are investigated. It is
shown that, when averaged on spatial scales of order $100\ \text{Mpc}$, they
reproduce an effective cosmological constant, whose order of magnitude agrees
with observations (this solves the CC problem). The origin of $\Lambda$ is
thus linked to matter distribution, which solves the coincidence problem.
The model.—Our model of late-time cosmology is GR sourced by a two-fluid
system, consisting of baryonic and cold dark matter. We adopt the standard
description and we model them as two pressureless perfect fluids, interacting
only gravitationally one with each other, with densities $\rho_{B}$ and
$\rho_{DM}$ and 4-velocities $U_{\mu}$ and $W_{\mu}$ respectively. The stress-
energy tensor is then :
$T_{\mu\nu}=\rho_{B}U_{\mu}U_{\nu}+\rho_{DM}W_{\mu}W_{\nu}.$ (1)
It is known that Eq. (1) can be recast as the stress-energy tensor of an
anisotropic fluid by an appropriate rotation of $U_{\mu}$ and $W_{\mu}$
Bayin:1985cd . In the current case, this transformation yields
$T_{\mu\nu}=\rho u_{\mu}u_{\nu}+p_{\parallel}w_{\mu}w_{\nu},$ (2)
describing an anisotropic fluid with zero tangential pressure $p_{\perp}$.
$\rho\simeq\rho_{B}+\rho_{DM}$ (for $W^{\mu}U_{\mu}\simeq 1$),
$u_{\mu}u^{\mu}=-w_{\nu}w^{\nu}=-1$ and ${p_{\parallel}}$ can be interpreted
as an effective radial pressure stabilizing the dark matter halo. This is
conceptually equivalent to the description of a fluid of collisionless
particles, where an effective pressure term can be associated to the stress
tensor modeling the anisotropy of the velocity distributions Jeans22 ;
Binney:1982jf ; Herrera:1997plx . This is particularly suited for dark matter,
which is believed to be made of collisionless particles. The two-fluid
approach has the advantage to provide such a description in a natural and
straightforward way.
To describe both the galactic and cosmological regime, we use the following
general, spherically symmetric, spacetime metric (we use units with the speed
of light $c=1$)
$ds^{2}=a^{2}(t)\left[-e^{\alpha(t,r)}dt^{2}+e^{\beta(t,r)}dr^{2}+r^{2}d\Omega^{2}\right],$
(3)
where $\alpha(t,r)$ and $\beta(t,r)$ are metric functions, $a$ is the
cosmological scale factor and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}$.
The resulting independent Einstein field and conservation equations are (we
use ${}^{\prime}=\partial_{r},\,\dot{\,}=\partial_{t}$):
$3\frac{\dot{a}^{2}}{a^{2}}e^{-\alpha}+\frac{e^{-\beta}}{r^{2}}\left(-1+e^{\beta}+r\beta^{\prime}\right)+\frac{\dot{a}}{a}\dot{\beta}e^{-\alpha}=8\pi
Ga^{2}\rho;$ (4)
$\frac{\dot{a}}{a}\alpha^{\prime}+\frac{\dot{\beta}}{r}=0;$ (5)
$\frac{\left(1+r\alpha^{\prime}\right)-e^{\beta}}{e^{\beta}a^{2}r^{2}}-e^{-\alpha}\left(2\frac{\ddot{a}}{a^{3}}-\frac{\dot{a}}{a^{3}}\dot{\alpha}-\frac{\dot{a}^{2}}{a^{4}}\right)=8\pi
G{p_{\parallel}}$ (6)
$\dot{\rho}+\frac{\dot{a}}{a}\left(3\rho+{p_{\parallel}}\right)+\frac{\dot{\beta}}{2}\left(\rho+{p_{\parallel}}\right)=0;$
(7)
${p_{\parallel}}^{\prime}+\frac{\alpha^{\prime}}{2}\left(\rho+{p_{\parallel}}\right)+\frac{2}{r}{p_{\parallel}}=0.$
(8)
In particular, in the static case, Eq. (8) is the generalization of the
Newtonian hydrostatic equilibrium equation
$p^{\prime}=-\frac{\partial_{r}e^{\alpha}}{2}\rho=-\Phi^{\prime}\rho=-\frac{Gm(r)\rho}{r^{2}},$
(9)
where $\Phi(r)$ is the gravitational potential. In fact, using the weak field
result $e^{\alpha}\simeq 1+2\Phi(r)$, Eq. (8) gives the Tolmann-Oppenheimer-
Volkoff-like equation:
$\partial_{r}\left(r^{2}{p_{\parallel}}\right)=-r^{2}\partial_{r}\Phi\left(\rho+{p_{\parallel}}\right).$
(10)
Galactic regime.— The galactic regime is obtained by taking the static limit
of Eqs. (4)-(8), which implies the scale factor being constant (we set $a=1$),
and the metric functions depending on $r$ only. Integration of Eq. (4) in this
case gives
$e^{-\beta}=1-\frac{2GM(r)}{r},$ (11)
where $M(r)\equiv 4\pi\int\rho(r)r^{2}dr$ is the Misner-Sharp (MS) mass of the
system. The remaining equations are solved together with a given profile for
the matter density $\rho=\rho(r)$. We are looking for solutions reproducing
the flattening of rotation curves at galactic scales. This can be achieved by
choosing the following matter density profile
$\rho=\frac{\sigma}{r^{2}},$ (12)
with $\sigma$ a constant. This gives the MS mass $M(r)=4\pi\sigma r$. In fact,
virializing the galactic motion, we get the velocities $v^{2}=8\pi G\sigma$.
We skip a constant term in $M(r)$, which represents the contribution of the
mass contained in the central regions of the galaxy. Here, we are considering
only galactic scales $\gg\text{kpc}$, where, according to observations (see
e.g. Refs. Rubin:1980zd ; Bosma:1981zz ), rotation curves starts flattening.
With the density profile (12), the field equations can be integrated to give
$\displaystyle\alpha=2\ln\left[\mathcal{C}\ln\left(\frac{r}{L}\right)\right];$
(13a) $\displaystyle{p_{\parallel}}=-\frac{\sigma}{r^{2}}+\frac{1-8\pi
G\sigma}{4\pi G}\frac{1}{r^{2}\ln\left(r/L\right)},$ (13b)
with $\mathcal{C}$ and $L$ integration constants. These are galactic
parameters and could be determined, together with $\sigma$, by combining
rotation curves and gravitational lensing (see, e.g. Ref. Faber:2005xc ).
We note that
${p_{\parallel}}=-\rho=-\frac{\sigma}{r^{2}},\quad\alpha=0$ (14)
also solves the field equations. This solution, giving an equation of state
(EOS) ${p_{\parallel}}=-\rho$ and a negative pressure, dominates for
$r\to\infty$, i.e in the transition to the cosmological regime. Conversely,
the second, positive, term in Eq. (13b) dominates at smaller (galactic)
scales. Physically, this means that, in this regime, the hydrostatic
equilibrium of dark matter halos is obtained by contrasting the gravitational
pull with a positive radial pressure. On the other hand, at large distances,
in the transition to the cosmological regime, the hydrostatic equilibrium is
not reached in the usual intuitive way. It is a local equilibrium in which
both sides of Eq. (10) separately vanish. This is possible only if the EOS is
${p_{\parallel}}=-\rho$ and the pressure is negative. As it is already evident
from the form of the EOS, this is strongly related to the generation of the
cosmological constant in the cosmological regime (see below).
The existence of solution (14) is a peculiar feature of fluids with
anisotropies. Static isotropic fluids do not allow for solutions with
$\alpha^{\prime}=0$ and $p,\,\rho$ satisfying $p=-\rho$.
Our static solution, describing hydrostatic equilibrium of the DM medium,
models a spacetime with a conical singularity (the relevance of this kind of
solution for DM has been already noted in Muckprivate ). In the simplest case,
given by Eq. (14), the metric is:
$ds=-dt^{2}+\frac{dr^{2}}{(1-8\pi G\sigma)}+r^{2}d\Omega^{2}.$ (15)
The solution is physically acceptable in the weak field limit when the the
deficit angle is very small:
$\sigma\ll\frac{1}{8\pi G}.$ (16)
In this limit, the spacetime can be well approximated by flat Minkowski space.
The flattening of galactic rotation curves is observed when the acceleration
drops below $a_{0}\sim\ell^{-1}$ (with $\ell$ the size of the cosmological
horizon). This implies that condition (16) is satisfied for $r\ll l$, which
covers not only galactic, but also larger scales where our universe appears
inhomogeneous, with the density of inhomogeneities scaling as $1/r^{2}$
Cadoni:2020jxe . The behavior $\rho\sim 1/r^{2}$ is thus responsible not only
for the flattening of galactic rotation curves, but also well describes
structure distribution at small cosmological scales Cadoni:2020izk . This
means, physically, that the dynamical properties of galaxies are inherited
from structure formation.
The condition $r\ll l$ holds true also at scales $\mathcal{R}\sim 100\
\text{Mpc}$, where our universe begins to appear homogeneous and isotropic. It
breaks down for $r\sim\ell$, i.e. in the cosmological regime where the static
approximation is no longer valid and we have to consider the full form of the
metric (3).
Cosmological regime.—When $a\neq 1$ and the assumption of staticity of the
metric functions is dropped, the system of equations (4)-(8) describes the
cosmological regime of our model. The matter density determines the metric
function $\beta(t,r)$, whereas the back reaction of the metric to the presence
of matter is codified in the function $\alpha(t,r)\neq\text{constant}$
Cadoni:2020jxe . In the decoupling limit, when the back reaction can be
neglected, the cosmological dynamics is described by the standard Friedmann-
Lemaitre-Robertson-Walker (FLRW) cosmology and decouples completely from
inhomogeneities Cadoni:2020izk .
In the cosmological regime, exact solutions of the field equations (4)-(8) can
be found using a method similar to that used in Ref. Cadoni:2020jxe . One
first integrates Eq. (5), defines the rescaled quantities $\hat{\rho}\equiv
3e^{\alpha}(3-r\alpha^{\prime})^{-1}\rho,\,\hat{p}_{\parallel}\equiv
e^{\alpha}{p_{\parallel}}$ and then uses an ansatz to separate the standard
FLRW dynamics from that of inhomogenities and the back reaction:
$\displaystyle a^{2}\hat{\rho}(t,r)\equiv
a^{2}{\rho}^{(1)}(t)+\frac{3e^{\alpha}}{3-r\alpha^{\prime}}\left(\rho^{(2)}(r)+\rho^{(3)}(t,r)\right),$
$\displaystyle a^{2}\hat{p}_{\parallel}(t,r)\equiv
a^{2}{p}_{\parallel}^{(1)}(t)+e^{\alpha}\left({p}_{\parallel}^{(2)}(r)+{p}_{\parallel}^{(3)}(t,r)\right).$
(17)
The system (4)-(8) splits in the usual FLRW equations for $a$, sourced by
${\rho}^{(1)}$ and ${p}_{\parallel}^{(1)}$, together with the solutions for
$\beta$ and $\alpha$ (see Ref. Cadoni:2020jxe for the calculation details) .
The solutions for $\beta$ is also here given by Eq. (11), with the MS mass
being $M(t,r)\equiv M^{(1)}(t)+M^{(2)}(r)+M^{(3)}(t,r)$, where $M^{(1)}$ is an
integration function, while $M^{(2,3)}$ are the MS masses associated to
$\rho^{(2,3)}$ respectively. Finally, the solution for $\alpha$ turns out to
be
$\displaystyle\alpha(t,r)=\mathcal{A}(t)+2G\
\frac{a}{\dot{a}}\int\frac{\dot{M}}{r^{2}}\left(1-\frac{2GM}{r}\right)dr,$
(18)
with $\mathcal{A}(t)$ integration function. We note that this solution, and in
particular the $(r,t)$-dependent terms, in general, removes the conical
singularity of the static solution.
The next step is to describe cosmology near the transition scale $\mathcal{R}$
to homogeneity and isotropy. In this situation, we cannot simply use the
general solution written above, since it describes also inhomogeneities and
their interaction with the cosmological dynamics encoded in the scale factor
$a$. Also the decoupling limit does not seem appropriate because it completely
neglects the back reaction. The simplest way to circumvent this problem is
first to split $\alpha$ into functions depending only on $r$ and $t$, i.e.
$\alpha\equiv\alpha_{r}(r)+\alpha_{t}(t)$. Then, we expand the solutions near
the decoupling limit, i.e. $r\alpha_{r}^{\prime}=0$, and near the present
epoch of our universe, i.e. $a^{2}=1$. Finally, we perform the spatial average
of the resulting $r$-dependent quantities. Keeping only the leading terms in
the expansions, Eqs. (17) give ($\alpha_{t}$ can be absorbed by a rescaling of
$t$)
$\displaystyle a^{2}\rho(t)=\frac{3}{8\pi
G}\left(\frac{\dot{a}}{a}\right)^{2}+a^{2}\langle\rho^{(3)}\rangle_{r};$ (19)
$\displaystyle a^{2}p(t)=\frac{1}{8\pi
G}\left[\left(\frac{\dot{a}}{a}\right)^{2}-2\frac{\ddot{a}}{a}\right]-a^{2}\Bigg{\langle}\frac{M^{(3)}}{4\pi
r^{3}}\Bigg{\rangle}_{r},$ (20)
where $\rho^{(3)}$, for consistency reasons, is function of $r$ only and the
spatial averaging is performed on spatial scales of order $100\ \text{Mpc}$
(see Cadoni:2020jxe for further details). When $\rho^{(3)}\sim 1/r^{2}$,
these equations describe standard FLRW cosmology with the averaged back
reaction term $\langle\rho^{(3)}\rangle_{r}$ playing the role of a
cosmological constant,
$\displaystyle\Lambda\sim-8\pi G\langle\rho^{(3)}\rangle_{r},$ (21)
corresponding to a perfect fluid with equation of state $p=-\rho$. It is
important to stress that spatial averaging at length scales of order $100\
\text{Mpc}$ solves also the conical singularity issue. In fact, the weak field
condition (16) is satisfied and the $t=\text{constant}$ sections of our metric
are regular, flat Minkowski spacetime. Notice that the emergence of a
cosmological constant at large scales could have been guessed directly from
the existence of the static solution (14), which dominates at large $r$.
Let us now evaluate the order of magnitude of the cosmological constant. In
Ref. Cadoni:2020jxe it is argued that $\rho^{(3)}\sim\rho^{(2)}$, since
$\rho^{(3)}$ is the energy density of the back reaction of the geometry to the
presence of matter inhomogeneities, described by $\rho^{(2)}$. As we have
previously seen, modelling both dark matter at galactic scales and the
structure formation at small cosmological scales Cadoni:2020izk ;
Cadoni:2020jxe is consistent with $\rho^{(2)}$ scaling as $1/r^{2}$. We have
therefore $\rho^{(3)}\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}-\frac{1}{r^{2}}$, where the minus sign is due
to the fact that $\rho^{(3)}$ should give rise to an attractive force and must
behave as an inverse power of $r$, hence $\Lambda\sim 8\pi
G\langle\rho^{(2)}\rangle_{r}$. The spatial average can be easily computed in
terms of the total mass $M$ of dark matter inside a sphere of radius $R$ of
order $100\ \text{Mpc}$: $\langle\rho^{(2)}\rangle_{r}=3M/4\pi R$. We have
$M\sim 10^{18}\ M_{\odot}$ and $\Omega_{\Lambda 0}\sim 1$, giving the correct
order of magnitude of the observed cosmological constant Aghanim:2018eyx . The
fact that the energy densities associated to matter and $\Lambda$ are of the
same order of magnitude at the present epoch does not appear here as a
coincidence. This solves the coincidence problem of the standard $\Lambda$CDM
cosmological scenario.
Conclusions.—In the present work, we have presented and explored the
phenomenology of a cosmological model sourced by baryonic and cold dark matter
in late-time cosmology. Our model allows for a unified description of galactic
dynamics and cosmological dynamics as well as structure formation. The
flattening of the rotation curves of galaxies, the formation and distribution
of structures at small cosmological scales and the cosmological constant all
have the same origin in the distribution of matter inhomogeneities and in back
reaction of the geometry to the presence of the latter. The observed galactic
dynamics and structure formation and distribution are correctly reproduced by
assuming the presence of an anisotropic component of the pressure and of an
$1/r^{2}$ scaling of matter density. The observed order of magnitude of the
cosmological constant is explained as the average of the back reaction of the
geometry at scales where our universe starts appearing homogeneous and
isotropic. The results of our work show the existence of an intriguing link
between the small and large scale behavior in cosmology. Hints on this
direction come also from the presence of the fundamental acceleration scale
$a_{0}$ in the baryonic Tully-Fisher relation, which is of the order of
magnitude of the present Hubble acceleration McGaugh:2004aw .
## References
* (1) N. Aghanim et al. [Planck], “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020), A6, arXiv:1807.06209 [astro-ph.CO].
* (2) G. Bertone, D. Hooper and J. Silk, “Particle dark matter: Evidence, candidates and constraints,” Phys. Rept. 405 (2005), 279-390, arXiv:hep-ph/0404175 [hep-ph].
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* (4) A. G. Riess et al. [Supernova Search Team], “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” Astron. J. 116 (1998) 1009–1038, arXiv:astro-ph/9805201 [astro-ph].
* (5) S. Perlmutter et al. [Supernova Cosmology Project Collaboration], “Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae,” Astrophys. J. 517 (1999) 565, arXiv:astro-ph/9812133v1.
* (6) P. J. E. Peebles and B. Ratra, “The Cosmological Constant and Dark Energy,” Rev. Mod. Phys. 75 (2003), 559-606, arXiv:astro-ph/0207347 [astro-ph].
* (7) M. Cadoni and A. P. Sanna, “Emergence of a Cosmological Constant in Anisotropic Fluid Cosmology,” arXiv:2012.08335 [gr-qc].
* (8) S. S. Bayin, “Anisotropic fluids and cosmology,” Astrophys. J. 303 (1986), 101-110.
* (9) J. H. Jeans, “The Motions of Stars in a Kapteyn Universe,” Mon. Not. R. Astron. Soc. 82 (1922), 122-132.
* (10) J. Binney, “Dynamics of elliptical galaxies and other spheroidal components,” Ann. Rev. Astron. Astrophys. 20 (1982), 399-429.
* (11) L. Herrera and N. O. Santos, “Local anisotropy in self-gravitating systems,” Phys. Rept. 286 (1997), 53-130.
* (12) V. C. Rubin, N. Thonnard and W. K. Ford, Jr., “Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R = 4kpc) to UGC 2885 (R = 122 kpc),” Astrophys. J. 238 (1980), 471.
* (13) A. Bosma, “21-cm line studies of spiral galaxies. 2. The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types,” Astron. J. 86 (1981), 1825.
* (14) T. Faber and M. Visser, “Combining rotation curves and gravitational lensing: How to measure the equation of state of dark matter in the galactic halo,” Mon. Not. Roy. Astron. Soc. 372 (2006), 136-142, arXiv:astro-ph/0512213 [astro-ph].
* (15) W. Mück, Private communication.
* (16) M. Cadoni, A. P. Sanna and M. Tuveri, “Anisotropic fluid cosmology: An alternative to dark matter?,” Phys. Rev. D 102 (2020) no.2, 023514, arXiv:2002.06988 [gr-qc].
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|
# On BiHom-associative dialgebras
Ahmed Zahari
Université de Haute Alsace,
IRIMAS-Département de Mathématiques,
6, rue des Frères Lumière F-68093 Mulhouse, France.
Ibrahima BAKAYOKO
Département de Mathématiques, Université de N’Zérékoré,
BP 50 N’Zérékoré, Guinée. e-mail address<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
Abstract. The aim of this paper is to introduce and study BiHom-associative
dialgebras. We give various constructions and study its connections with
BiHom-Poisson dialgebras and BiHom-Leibniz algebras. Next we discuss the
central extensions of BiHom-diassociative and we describe the classification
of $n$-dimensional BiHom-diassociative algebras for $n\leq 4$. Finally, we
discuss their derivations.
AMS Subject Classification: .
Keywords: BiHom-associative dialgebra, BiHom-Poisson dialgebra, centroid,
averaging operator, Nijenhuis operator, Rota-Baxter operator, Extension,
Derivation, Classification.
## 1 Introduction
The associative dialgebras (also known as diassociative algebras) has been
introduced by Loday in 1990 (see [6] and references therein) as a
generalization of associative algebras. They are a generalization of
associative algebras in the sens that they possess two associative
multiplications and obey to three other conditions; when the two associative
low are equal we recover associative algebra. One of his motivation were to
find an algebra whose commutator give rises to Leibniz algebra as it is the
case in the relation between Lie and associative algebra. Another motivation
come from the research of an obstruction to the periodicity in algebraic
K-theory. Now, these algebras found their applications in classical geometry,
non-commutative geometry and physics.
The centroid plays an important role in understanding forms of an algebra. It
is an element in the classification of associative and diassociative algebras.
They occurs naturally is in the study of derivations of an algebra. The
centroid and averaging operators are used in the deformation of algebra in
order to generate another algebraic structure. The Nijenhuis operator on an
associative algebra was introduced in [16] to study quantum bi-Hamiltonian
systems while the notion Nijenhuis operator on a Lie algebra originated from
the concept of Nijenhuis tensor that was introduced by Nijenhuis in the study
of pseudo-complex manifolds and was related to the well known concepts of
Schouten-Nijenhuis bracket , the Frolicher-Nijenhuis bracket [1], and the
Nijenhuis-Richardson bracket. The associative analog of the Nijenhuis relation
may be regaded as the homogeneous version of Rota-Baxter relation[23].
BiHom-algebraic structures were introduced in 2015 by G. Graziani, A.
Makhlouf, C. Menini and F. Panaite in [7] from a categorical approach as an
extension of the class of Hom-algebras. Since then, other interesting BiHom-
type algebraic structures of many Hom-algebraic structures has been
intensively studied as BiHom-Lie colour algebras structures [17],
Representations of BiHom-Lie algebras [30], BiHom-Lie superalgebra structures
[28], $\\{\sigma,\tau\\}$-Rota-Baxter operators, infinitesimal Hom-bialgebras
and the associative (Bi)Hom-Yang-Baxter equation [22], The construction and
deformation of BiHom-Novikov algebras [27], On n-ary Generalization of BiHom-
Lie algebras and BiHom-Associative Algebras [3], Rota-Baxter operators on
BiHom-associative algebras and related structures [19].
The goal of this paper is to introduce, classify and study structures, central
extensions and derivations of BiHom-associative algebras. The paper is
organized as follows. In section 2, we define BiHom-associative dialgebras,
give some constructions using twisting, direct sum, elements of centroid,
averaging operator, Nijenhuis operator and Rota-Baxter relation. We give a
connection between BiHom-associative dialgebras and BiHom-Leibniz algebras. We
introduce action of a BiHom-Leibniz algebra onto another and give a Leibniz
structure on the semidirect structure. Then, we show that the semidirect sum
of BiHom-Leibniz algebras associated to BiHom-associative dialgebras is the
same that the BiHom-Leibniz algebra associated to the semidirect of BiHom-
associative dialgebras. Finally, we introduce BiHom-associative dialgebras and
show that any BiHom-associative dialgebra carries a structure of BiHom-Poisson
dialgebra. In section 3, we introduce the notion of central extension of
BiHom-associative dialgebras and define $2$-cocycles and $2$-coboundaries of
BiHom-associative dialgebras with coefficients in a trivial BiHom-module. Then
we establish relationship between $2$-cocycles and central extensions. Section
4, is devoted to the classification of $n$-dimensional BiHom-associative
dialgebras for $n\leq 4$. We dedicated Section 5 to the derivations of BiHom-
associative dialgebras.
## 2 Structure of BiHom-associative dialgebras
###### Definition 2.1.
A BiHom-associative dialgebras is a $5$-truple
$(A,\dashv,\vdash,\alpha,\beta)$ consisting of a linear space $A$ linear maps
$\dashv,\vdash,:A\times A\longrightarrow A$ and $\alpha,\beta:A\longrightarrow
A$ satisfying, for all $x,y,z\in A$ the following conditions :
$\displaystyle\alpha\circ\beta$ $\displaystyle=$
$\displaystyle\beta\circ\alpha,$ (2.1) $\displaystyle(x\dashv
y)\dashv\beta(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\dashv(y\dashv z),$
(2.2) $\displaystyle(x\dashv y)\dashv\beta(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\dashv(y\vdash z),$ (2.3) $\displaystyle(x\vdash
y)\dashv\beta(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\vdash(y\dashv z),$
(2.4) $\displaystyle(x\dashv y)\vdash\beta(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\vdash(y\vdash z),$ (2.5) $\displaystyle(x\vdash
y)\vdash\beta(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\vdash(y\vdash z).$
(2.6)
We called $\alpha$ and $\beta$ ( in this order ) the structure maps of A.
###### Example 2.2.
Any Hom-associative dialgebra [10] or any associative dialgebra is a BiHom-
associative dialgebra by setting $\beta=\alpha$ or $\alpha=\beta=id$.
###### Example 2.3.
Let $(A,\dashv,\vdash,\alpha,\beta)$ a BiHom-associative dialgebra. Consider
the module of $n\times n$-matrices
$\mathcal{M}_{n}(D)=\mathcal{M}_{n}(\mathbb{K})\otimes D$ with the linear maps
${\bf\alpha}(A)=(\alpha(a_{ij}))$, ${\bf\beta}(A)=(\beta(a_{ij})$ for all
$A\in\mathcal{M}_{n}(D)$ and the products $(a\triangleleft
b)_{ij}=\sum_{k}a_{ik}\dashv b_{kj}$ and $(a\triangleright
b)_{ij}=\sum_{k}a_{ik}\vdash b_{kj}$. Then,
$(\mathcal{M}_{n}(D),\triangleleft,\triangleright,{\bf\alpha},{\bf\beta})$ is
a BiHom-associative dialgebra.
###### Definition 2.4.
A morphism $f:({D},\dashv,\vdash,\alpha,\beta)$ and
$({D}^{\prime},\dashv^{\prime},\vdash^{\prime},\alpha^{\prime},\beta^{\prime})$
be a BiHom-associative dialgebras is a linear map
$f:{D}\rightarrow{D}^{\prime}$ such that $\alpha^{\prime}\circ
f=f\circ\alpha,\,\beta^{\prime}\circ f=f\circ\beta$ and $f(x\dashv
y)=f(x)\dashv^{\prime}f(y),\quad f(x\vdash y)=f(x)\vdash^{\prime}f(y)$, for
all $x,y\in{D}.$
###### Definition 2.5.
A BiHom-associative dialgebra $(A,\dashv,\vdash,\alpha,\beta)$ in which
$\alpha$ and $\beta$ are morphism is said to be a multiplicative BiHom-
associative dialgebra.
If moreover, $\alpha$ and $\beta$ are bijective (i.e. automorphisms), then
$(A,\dashv,\vdash,\alpha,\beta)$ is said to be a regular BiHom-associative
dialgebra.
We prove in the following proposition that any BiHom-associative dialgebra
turn to another one via morphisms.
###### Theorem 2.6.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$\alpha^{\prime},\beta^{\prime}:D\rightarrow D$ two morphisms of BiHom-
associative dialgebras such that the maps
$\alpha,\alpha^{\prime},\beta,\beta^{\prime}$ commute pairewise. Then
$D_{(\alpha^{\prime},\beta^{\prime})}=(D,\triangleleft:=\dashv(\alpha^{\prime}\otimes\beta^{\prime}),\triangleright:=\vdash(\alpha^{\prime}\otimes\beta^{\prime}),\alpha\alpha^{\prime},\beta\beta^{\prime})$
is a BiHom-associative dialgebra.
###### Proof.
We prove only one axiom and leave the rest to the reader. For any $x,y,z\in
D$,
$\displaystyle(x\triangleleft
y)\triangleleft\beta\beta^{\prime}(z)-\alpha\alpha^{\prime}(x)\triangleleft(y\triangleright
z)$ $\displaystyle=$
$\displaystyle\alpha^{\prime}(\alpha^{\prime}(x)\dashv\beta^{\prime}(y))\dashv\beta^{\prime}\beta\beta^{\prime}(z)-\alpha^{\prime}\alpha\alpha^{\prime}(x)\dashv\beta^{\prime}(\alpha^{\prime}(y)\vdash\beta^{\prime}(z))$
$\displaystyle=$
$\displaystyle(\alpha^{\prime}\alpha^{\prime}(x)\dashv\alpha^{\prime}\beta^{\prime}(y))\dashv\beta\beta^{\prime}\beta^{\prime}(z)-\alpha\alpha^{\prime}\alpha^{\prime}(x)\dashv(\alpha^{\prime}\beta^{\prime}(y)\vdash\beta^{\prime}\beta^{\prime}(z)).$
The left hand side vanishes by (2.3). And, this ends the proof. ∎
###### Corollary 2.7.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a multiplicative BiHom-associative
dialgebra. Then
$(D,\dashv\circ(\alpha^{n}\otimes\beta^{n}),\vdash\circ(\alpha^{n}\otimes\beta^{n}),\alpha^{n+1},\beta^{n+1})$
is also a multiplicative BiHom-associative dialgebra.
###### Proof.
It suffises to take $\alpha^{\prime}=\alpha^{n}$ and
$\beta^{\prime}=\beta^{n}$ in Theorem 2.6. ∎
###### Corollary 2.8.
Let $(D,\dashv,\vdash,\alpha)$ be a multiplicative Hom-associative dialgebra
and $\beta:D\rightarrow D$ an endomorphism of $D$. Then
$(D,\dashv\circ(\alpha\otimes\beta),\vdash\circ(\alpha\otimes\beta),\alpha^{2},\beta)$
is also a Hom-associative dialgebra.
###### Proof.
It suffises to take $\alpha^{\prime}=\alpha$ and replace $\beta$ by $Id_{D}$,
and $\beta^{\prime}$ by $\beta$ in Theorem 2.6. ∎
Any regular Hom-associative dialgebra give rises to associative dialgebra as
stated in the next corollary.
###### Corollary 2.9.
If $(D,\dashv,\vdash,\alpha,\beta)$ is a regular BiHom-associative dialgebra,
then
$(D,\dashv\circ(\alpha^{-1}\otimes\beta^{-1}),\vdash\circ(\alpha^{-1}\otimes\beta^{-1}))$
is an associative dialgebra.
###### Proof.
We have to take $\alpha^{\prime}=\alpha^{-1}$ and $\beta^{\prime}=\beta^{-1}$
in Theorem 2.6. ∎
###### Corollary 2.10.
Let $(D,\dashv,\vdash)$ be an associative dialgebra and $\alpha:D\rightarrow
D$ and $\beta:D\rightarrow D$ a pair of commuting endomorphisms of $D$. Then
$(D,\dashv\circ(\alpha\otimes\beta),\vdash\circ(\alpha\otimes\beta),\alpha,\beta)$
is a BiHom-associative dialgebra.
###### Proof.
We have to take $\alpha=\beta=Id_{D}$ and replace $\alpha^{\prime}$ by
$\alpha$, and $\beta^{\prime}$ by $\beta$ in Theorem 2.6. ∎
###### Definition 2.11.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra. For any
integers $k,l$, an even linear map $\theta:D\rightarrow D$ is called an
element of $(\alpha^{k},\beta^{l})$-centroid on $D$ if
$\displaystyle\alpha\circ\theta$ $\displaystyle=$
$\displaystyle\theta\circ\alpha,\quad\beta\circ\theta=\theta\circ\beta,$ (2.7)
$\displaystyle\theta(x)\dashv\alpha^{k}\beta^{l}(y)$ $\displaystyle=$
$\displaystyle\theta(x)\dashv\theta(y)=\alpha^{k}\beta^{l}(x)\dashv\theta(y),$
(2.8) $\displaystyle\theta(x)\vdash\alpha^{k}\beta^{l}(y)$ $\displaystyle=$
$\displaystyle\theta(x)\vdash\theta(y)=\alpha^{k}\beta^{l}(x)\vdash\theta(y),$
(2.9)
for all $x,y\in D$.
The set of elements of centroid is called centroid.
###### Proposition 2.12.
Let $(A,\dashv,\vdash,\alpha,\beta)$ a BiHom-associative dialgebra and
$\phi:A\rightarrow A$ and $\psi:A\rightarrow A$ be a paire of commuting
elements of cenroid. Let us defined
$x\triangleleft y:=\phi(x)\dashv y\quad\mbox{and}\quad x\triangleright
y:=\psi(x)\vdash y.$
Then, $(A,\triangleleft,\triangleright,\alpha,\beta)$ is a BiHom-associative
dialgebra if and only if
$Im(\phi-\psi)\in Z_{\dashv}(A):=\\{x\in A/x\dashv y=0,\forall y\in
A\\}\;\mbox{and}\;Im(\phi-\psi)\in Z_{\vdash}(A):=\\{x\in A/y\vdash
x=0,\forall y\in A\\}$.
###### Proof.
We only prove axioms (2.3) and (2.5), the three other comes from BiHom-
associativity. So for any $x,y,z\in A$,
$\displaystyle(x\triangleleft
y)\triangleleft\beta(z)-\alpha(x)\triangleleft(y\triangleright z)$
$\displaystyle=$ $\displaystyle(\phi(x)\dashv
y)\dashv\phi\beta(z)-\phi\alpha(x)\dashv(y\vdash\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv
y)\dashv\beta\phi(z)-\alpha\phi(x)\dashv(y\vdash\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv y)\vdash\beta\phi(z)-(\phi(x)\dashv
y)\vdash\beta\psi(z))$ $\displaystyle=$ $\displaystyle(\phi(x)\dashv
y)\vdash\beta(\phi-\psi)(z)$ $\displaystyle=$
$\displaystyle\alpha\phi(x)\dashv(y\vdash(\phi-\psi)(z)).$
and
$\displaystyle(x\triangleleft
y)\triangleright\beta(z)-\alpha(x)\triangleright(y\triangleright z)$
$\displaystyle=$ $\displaystyle(\phi(x)\dashv
y)\vdash\beta\psi(z)-\psi\alpha(x)\vdash(y\vdash\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv
y)\vdash\beta\psi(z)-\alpha\psi(x)\vdash(y\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv y)\vdash\beta\psi(z)-(\psi(x)\dashv
y)\vdash\beta\psi(z)$ $\displaystyle=$ $\displaystyle[(\phi(x)-\psi(x))\dashv
y]\vdash\beta\psi(z).$
A study of cancellation of the two equalities allows to conclude. ∎
###### Proposition 2.13.
Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra, and
$\phi:A\rightarrow A$ and $\psi:A\rightarrow A$ be a paire of commuting
elements of cenroid. Let us defined
$x\dashv y:=\phi(x)\cdot y\quad\mbox{and}\quad x\vdash y:=\psi(x)\cdot y.$
Then, $(A,\dashv,\vdash,\alpha,\beta)$ is a BiHom-associative dialgebra if and
only if $Im(\phi-\psi)$ is contained in the set of isotropic vectors.
###### Proof.
We only prove axioms (2.3) and (2.5), the three other comes from BiHom-
associativity. So for any $x,y,z\in A$,
$\displaystyle(x\dashv y)\dashv\beta(z)-\alpha(x)\dashv(y\vdash z)$
$\displaystyle=$ $\displaystyle(\phi(x)y)\phi\beta(z)-\phi\alpha(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\phi(z)-\alpha\phi(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\phi(z)-(\phi(x)y)\beta\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta(\phi-\psi)(z)$ $\displaystyle=$
$\displaystyle\alpha\phi(x)(y(\phi-\psi)(z)).$
and
$\displaystyle(x\dashv y)\vdash\beta(z)-\alpha(x)\vdash(y\vdash z)$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\psi(z)-\psi\alpha(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\psi(z)-\alpha\psi(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\psi(z)-(\psi(x)y)\beta\psi(z)$
$\displaystyle=$ $\displaystyle[(\phi(x)-\psi(x))y]\beta\psi(z).$
A study of cancellation of the two equalities allow to conclude. ∎
###### Remark 2.14.
Proposition 2.13 may be seen as a consequence of Proposition 2.12.
###### Proposition 2.15.
Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and
$(M,\ast_{L},\ast_{R},\alpha_{M},\beta_{M})$ an $A$-BiHom-bimodule i.e. $M$ is
a vector space, $\alpha_{M}:M\rightarrow M$ and $\beta_{M}:M\rightarrow M$ are
two linear maps, and $\ast_{L}:A\rightarrow M$ and $\ast_{R}:M\rightarrow A$
two bilinear maps such that
$\displaystyle\alpha(x)\ast_{L}(y\ast_{L}m)$ $\displaystyle=$
$\displaystyle(x\cdot y)\ast_{L}\beta_{M}(m)$ (2.10)
$\displaystyle\alpha(x)\ast_{L}(m\ast_{R}y)$ $\displaystyle=$
$\displaystyle(x\ast_{L}m)\ast_{R}\beta(y)$ (2.11)
$\displaystyle\alpha_{M}(m)\ast_{R}(x\cdot y)$ $\displaystyle=$
$\displaystyle(m\ast_{R}x)\ast_{R}\beta(y).$ (2.12)
Suppose that $f:M\rightarrow A$ is a morphism of $A$-BiHom-bimodule i.e. $f$
is linear such that $\alpha\circ f=f\circ\alpha_{M}$, $\beta\circ
f=f\circ\beta_{M}$ and
$\displaystyle f(x\ast_{L}m)$ $\displaystyle=$ $\displaystyle x\cdot f(m)$
(2.13) $\displaystyle f(m\ast_{R}x)$ $\displaystyle=$ $\displaystyle f(m)\cdot
x.$ (2.14)
Then, $(M,\triangleleft,\triangleright,\alpha_{M},\beta_{M})$ is a BiHom-
associative dialgebra with
$m\triangleleft n=f(m)\ast_{R}n\quad\mbox{and}\quad m\triangleright
n=m\ast_{R}f(n),$
for all $m,n\in M$.
###### Proof.
We only prove axiom (2.6), the other being proved similarly. For any $x,y,z\in
A$,
$\displaystyle(m\triangleleft n)\triangleright\beta_{M}(p)$ $\displaystyle=$
$\displaystyle(f(m)\ast_{L}n)\ast_{R}f\beta_{M}(p)$ $\displaystyle=$
$\displaystyle(f(m)\ast_{L}n)\ast_{R}\beta f(p).$
By (2.11),
$\displaystyle(m\triangleleft n)\triangleright\beta_{M}(p)$ $\displaystyle=$
$\displaystyle\alpha f(m)\ast_{L}(n\ast_{R}f(p))$ $\displaystyle=$
$\displaystyle f\alpha_{M}(m)\ast_{L}(n\triangleright p)$ $\displaystyle=$
$\displaystyle\alpha_{M}(m)\triangleleft(n\triangleright p).$
This completes the proof. ∎
###### Remark 2.16.
Any $(\alpha^{0},\beta^{0})$-element of centroid of a BiHom-associative
algebra is a morphism of BiHom-bimodule.
Thanks to the above remark, we have what follows :
###### Corollary 2.17.
Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and let $\theta$
be an element of cenroid on $A$. Then,
$(A,\triangleleft,\triangleright,\alpha,\beta)$ is a BiHom-associative
dialgebra with
$x\triangleleft y=\theta(x)\cdot y\quad\mbox{and}\quad x\triangleright
y=x\cdot\theta(y),$
for any $x,y\in A.$
###### Proposition 2.18.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$R:D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$ i.e. $R$ is
linear and $\alpha\circ R=R\circ\alpha$ , $\beta\circ R=R\circ\beta$, and
$\displaystyle R(x)\dashv R(y)$ $\displaystyle=$ $\displaystyle R(R(x)\dashv
y+x\dashv R(y))$ (2.16) $\displaystyle R(x)\vdash R(y)$ $\displaystyle=$
$\displaystyle R(R(x)\vdash y+x\vdash R(y))$ (2.17)
Then, $(D,\triangleleft,\triangleright,\alpha,\beta)$ is also a BiHom-
associative algebra with
$\displaystyle x\triangleleft y=R(x)\dashv y+x\dashv R(y),$ (2.18)
$\displaystyle x\triangleright y=R(x)\vdash y+x\vdash R(y),$ (2.19)
for all $x,y\in D$.
###### Proof.
We only prove axiom (2.6), the other being proved in a similar way. Thus, For
any $x,y,z\in A$,
$\displaystyle\qquad(x\triangleleft
y)\triangleleft\beta(z)-\alpha(x)\triangleleft(y\triangleright z)=$
$\displaystyle=(x\dashv R(y)+R(x)\dashv y)\dashv R\beta(z)+R(R(x)\dashv
y+x\dashv R(y))\dashv\beta(z)$ $\displaystyle\quad-\alpha(x)\dashv
R(R(y)\dashv z+y\dashv R(z))-R\alpha(x)\dashv(R(y)\vdash z+y\vdash R(z))$
$\displaystyle=(x\dashv R(y))\dashv\beta R(z)+(R(x)\dashv y)\dashv\beta
R(z)+(R(x)\dashv R(y))\dashv\beta(z)$
$\displaystyle\quad-\alpha(x)\dashv(R(y)\vdash R(z))-\alpha R(x)\dashv(y\vdash
R(z))\alpha R(x)\dashv(R(y)\vdash z).$
The left hand side vanishes by axiom (2.6). This ends the proof. ∎
###### Corollary 2.19.
Let $(D,\dashv,\vdash,\alpha,\beta)$ BiHom-associative dialgebra and
$R:D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$. Then,
$(D,\ast,\alpha,\beta)$ is a BiHom-associative algebra with $x\ast
y=x\triangleleft y+x\triangleright y$.
###### Corollary 2.20.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$R:D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$. Then,
$(D,[-,-],\alpha,\beta)$ is a BiHom-Lie algebra with
$[x,y]=x\ast y-\alpha^{-1}\beta(y)\ast\alpha\beta^{-1}(x),$
with $x\ast y=x\triangleleft y+x\triangleright y$.
As in the previous proposition, it is well known that a Nijenhuis operator on
an associative algebra allows to define another associative algebra. In the
next result, we try to establish an analoq of this result for BiHom-
associative dialgebras.
###### Proposition 2.21.
Let $(D,\dashv,\vdash,\alpha,\beta)$ BiHom-associative dialgebra and
$N:D\rightarrow D$ a Nijenhuis operator on $D$ i.e. $N$ is linear and
$\alpha\circ N=N\circ\alpha$ , $\beta\circ N=N\circ\beta$, and
$\displaystyle N(x)\dashv N(y)$ $\displaystyle=$ $\displaystyle N(N(x)\dashv
y+x\dashv N(y)-N(x\cdot y))$ (2.20) $\displaystyle N(x)\vdash N(y)$
$\displaystyle=$ $\displaystyle N(N(x)\vdash y+x\vdash N(y)-N(x\cdot y))$
(2.21)
Then, $(D,\triangleleft,\triangleright,\alpha,\beta)$ is also a BiHom-
associative algebra with
$\displaystyle x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),$
(2.22) $\displaystyle x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash
y),$ (2.23)
for all $x,y\in D$.
###### Proof.
We only prove axiom (2.4) for the products $\triangleleft$ and
$\triangleright$. The others are leave to the reader.
$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=N\Big{(}N(x)\vdash y+x\dashv y-N(x\dashv
y)\Big{)}\dashv\beta(z)+\Big{(}N(x)\dashv y+x\vdash N(y)-N(x\vdash
y)\Big{)}\dashv N\beta(z)$ $\displaystyle\quad-N\Big{(}(N(x)\vdash y+x\vdash
N(y)-N(x\vdash y))\dashv\beta(z)\Big{)}-N\alpha(x)\vdash\Big{(}N(y)\dashv
z+y\dashv N(z)-N(y\dashv z)\Big{)}$ $\displaystyle-\alpha(x)\vdash
N\Big{(}N(y)\dashv z+y\dashv N(z)-N(y\dashv
z)\Big{)}+N\Big{(}\alpha(x)\vdash(N(y)\dashv z+y\dashv N(z)-N(y\dashv
z))\Big{)}.$
By (2.20) and (2.21), we have
$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=(N(x)\vdash N(y))\dashv\beta(z)+(N(x)\dashv y)\dashv\beta
N(z)+(x\vdash N(y))\dashv\beta N(z)-N(x\vdash y))\dashv N\beta(z)$
$\displaystyle\quad-N\Big{(}(N(x)\vdash
y)\dashv\beta(z)\Big{)}-N\Big{(}(x\vdash
N(y))\dashv\beta(z)\Big{)}-N\Big{(}N(x\vdash y)\dashv\beta(z)\Big{)}$
$\displaystyle-\alpha N(x)\vdash(N(y)\dashv z)-\alpha N(x)\vdash(y\dashv
N(z))+N\alpha(x)\vdash N(y\dashv z))$
$\displaystyle-\alpha(x)\vdash(N(y)\dashv
N(z))+N\Big{(}\alpha(x)\vdash(N(y)\dashv
z\Big{)}+N\Big{(}\alpha(x)\vdash(y\dashv N(z))\Big{)}-N\Big{(}\alpha(x)\vdash
N(y\dashv z))\Big{)}.$
By (2.4), we have
$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=-N(x\vdash y))\dashv N\beta(z)-N\Big{(}(N(x)\vdash
y)\dashv\beta(z)\Big{)}-N\Big{(}(x\vdash
N(y))\dashv\beta(z)\Big{)}-N\Big{(}N(x\vdash y)\dashv\beta(z)\Big{)}$
$\displaystyle\quad+N\alpha(x)\vdash N(y\dashv
z))+N\Big{(}\alpha(x)\vdash(N(y)\dashv
z\Big{)}+N\Big{(}\alpha(x)\vdash(y\dashv N(z))\Big{)}-N\Big{(}\alpha(x)\vdash
N(y\dashv z))\Big{)}.$
Using again (2.20) and (2.21), it comes
$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=-N\Big{(}N(x\vdash y)\dashv\beta(z)+(x\vdash y)\dashv\beta
N(z)-N((x\vdash y)\dashv\beta(z))\Big{)}$ $\displaystyle-N\Big{(}(N(x)\vdash
y)\dashv\beta(z)\Big{)}-N\Big{(}(x\vdash
N(y))\dashv\beta(z)\Big{)}-N\Big{(}N(x\vdash y)\dashv\beta(z)\Big{)}$
$\displaystyle\quad+N\Big{(}\alpha(x)\vdash(N(y)\dashv z)+\alpha(x)\vdash
N(y\dashv z)-N(\alpha(x)\vdash(y\dashv z))\Big{)}$
$\displaystyle+N\Big{(}\alpha(x)\vdash(N(y)\dashv
z\Big{)}+N\Big{(}\alpha(x)\vdash(y\dashv N(z))\Big{)}-N\Big{(}\alpha(x)\vdash
N(y\dashv z))\Big{)}.$
The left hand side vanishes by (2.4). ∎
###### Corollary 2.22.
If $(D,\dashv,\vdash,\alpha)$ is a Hom-associative dialgebra and
$N:D\rightarrow D$ a Nijenhuis operator on $D$, then
$(D,\triangleleft,\triangleright,\alpha)$ is also a Hom-associative algebra
with
$\displaystyle x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),$
$\displaystyle x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash y),$
for all $x,y\in D$.
###### Corollary 2.23.
If $(D,\dashv,\vdash,\alpha,\beta)$ is an associative dialgebra and
$N:D\rightarrow D$ a Nijenhuis operator on $D$, then
$(D,\triangleleft,\triangleright,\alpha,\beta)$ is also an associative algebra
with
$\displaystyle x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),$
$\displaystyle x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash y),$
for all $x,y\in D$.
The next proposition asserts that the twist of the products of any BiHom-
associative dialgebra by an averaging operator gives rise to another BiHom-
associative dialgebra.
###### Proposition 2.24.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$\theta:D\rightarrow D$ an injective averaging operator on $D$ i.e. $\theta$
is an injective linear map such that $\alpha\circ\theta=\theta\circ\alpha$ ,
$\beta\circ\theta=\theta\circ\beta$, and
$\displaystyle\theta(x)\dashv\theta(y)$ $\displaystyle=$
$\displaystyle\theta(\alpha^{k}\beta^{l}(x)\dashv\theta(y))=\theta(\theta(x)\dashv\alpha^{k}\beta^{l}(y)),$
(2.24) $\displaystyle\theta(x)\vdash\theta(y)$ $\displaystyle=$
$\displaystyle\theta(\alpha^{k}\beta^{l}(x)\vdash\theta(y))=\theta(\theta(x)\vdash\alpha^{k}\beta^{l}(y)),$
(2.25)
for any $x,y\in D$. Then, $(D,\triangleleft,\triangleright,\alpha,\beta)$ is
also a BiHom-associative algebra with
$\displaystyle x\triangleleft y=\theta(x)\dashv\alpha^{k}\beta^{l}(y))$ (2.26)
$\displaystyle x\triangleright y=\alpha^{k}\beta^{l}(x)\vdash\theta(y),$
(2.27)
for all $x,y\in D$.
###### Proof.
We only prove one identity, the others have a similar proof. For any $x,y,z\in
D$, one has :
$\displaystyle\qquad\theta[(x\triangleleft
y)\triangleright\beta(z)-\alpha(x)\triangleright(y\triangleleft z)]=$
$\displaystyle=\theta[\theta(\theta(x)\dashv\alpha^{k}\beta^{l}(y)))\vdash\alpha^{k}\beta^{l+1}(z))-\theta\alpha(x)\vdash(\theta(y)\dashv\alpha^{k}\beta^{l}(z)))]$
$\displaystyle=\theta[(\theta(x)\dashv\theta(y)\vdash\alpha^{k}\beta^{l+1}(z)]-\theta\alpha(x)\vdash\theta(\theta(y)\dashv\alpha^{k}\beta^{l}(z)))]$
$\displaystyle=(\theta(x)\vdash\theta(y))\dashv\beta\theta(z)-\alpha\theta(x)\vdash(\theta(y)\dashv\theta(z)).$
Which vanishes by axiom (2.4), and the conclusion holds by injectivity. ∎
At this moment, we introduce ideals for BiHom-associative dialgebra in order
to give another construction of BiHom-associative dialgebras.
###### Definition 2.25.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$D_{o}$ a subset of $D$. We say that $D_{o}$ is a BiHom-subalgebra of $D$ if
$D_{o}$ is stable under $\alpha$ and $\beta$, and $x\dashv y,x\vdash y\in
D_{o}$, for any $x,y\in D_{o}$.
###### Example 2.26.
If $\varphi:D_{1}\rightarrow D_{2}$ is a homomorphism of BiHom-associative
dialgebras, the image $Im\varphi$ is a BiHom-subalgebra of $D_{2}$.
###### Definition 2.27.
A two side BiHom-ideal of a BiHom-associative dialgebra
$(D,\dashv,\vdash,\alpha,\beta)$ is subspace $I$ such that $\alpha(I)\subset
I,x\ast y,y\ast x\in I$ for all $x\in D,y\in I$ with $\ast=\dashv$ and
$\vdash$. Note that $I$ is called the left and right BiHom-ideal if $x\dashv
y,x\vdash y$ and $y\dashv x,y\vdash x$ are in $I$, respectively, for all $x\in
D.y\in I$.
###### Example 2.28.
i) Obviously $I=\\{0\\}$ and $I=D$ are two-sided ideals.
ii) If $\varphi:D_{1}\rightarrow D_{2}$ is a homomorphism of BiHom-associative
dialgebras, the kernel $Ker\varphi$ is a two sided ideal in $D_{1}$.
iii) If $I_{1}$ and $I_{2}$ are two sided ideals of $D$, then so is
$I_{1}+I_{2}$.
In the below proposition, we prove that BiHom-associative dialgebras are
closed under direct summation, and give a condition for which a linear map
becomes a morphism.
###### Proposition 2.29.
Let $({A},\dashv_{A},\vdash_{A},\alpha_{A},\beta_{A})$ and
$({B},\dashv_{B},\vdash_{B},\alpha_{B},\beta_{B})$ be two BiHom-associative
dialgebras. Then there exists a BiHom-associative dialgebra structure on
${A}\oplus{B}$ with the bilinear maps
$\triangleleft,\triangleright:({A}\oplus{B})^{\otimes
2}\rightarrow{A}\oplus{B}$ given by
$(a_{1}+b_{2})\dashv(a_{2}+b_{2})=a_{1}\dashv_{A}a_{2}+b_{2}\dashv_{B}b_{2},$
$(a_{1}+b_{1})\vdash(a_{2}+b_{2})=a_{1}\vdash_{A}a_{2}+b_{A}\vdash_{B}b_{2}$
and the linear maps
$\alpha=\alpha_{A}+\alpha_{B},\,\beta=\beta_{A}+\beta_{B}:{A}\oplus{B}\rightarrow{A}\oplus{B}$
given by
$(\alpha_{A}+\alpha_{B})(a+b)=\alpha_{A}(a)+\alpha_{B}(b),\,(\beta_{A}+\beta_{B})(a+b)=\beta_{A}(a)+\beta_{B}(b),\,\forall(a,b)\in({A}\times{B}).$
Moreover, if $\xi:{A}\rightarrow{B}$ is a linear map. Then
$\xi:({A},\dashv_{A},\vdash_{A},\alpha_{A},\beta_{A})$ to
$({B},\dashv_{B},\vdash_{B},\alpha_{B},\beta_{B})$ is a morphism if and only
if its graph $\Gamma_{\xi}=\\{(x,\xi(x)),x\in A\\}$ is a BiHom-subalgebra of
$({A}\oplus{B},\triangleleft,\triangleright,\alpha,\beta)$.
###### Proof.
The proof of the first part of the proposition comes from a simple
computation.
Let us suppose that
$\xi:({A},\dashv_{A},\vdash_{A},\alpha_{A},\beta_{A})\rightarrow({B},\dashv_{B},\vdash_{B},\alpha_{B},\beta_{B})$
is a morphism of BiHom-associative dialgebras. Then
$(u+\xi(u))\dashv(v+\xi(v))=(u\dashv_{A}v+\xi(u)\dashv_{B}\xi(v))=(u\dashv_{A}v+\xi(u\dashv_{A}v)$
$(u+\xi(u))\vdash(v+\xi(v))=(u\vdash_{A}v+\xi(u)\vdash_{B}\xi(v))=(u\vdash_{A}v+\xi(u\vdash_{A}v).$
Thus the graph $\Gamma_{\xi}$ is closed under the operations $\dashv$ and
$\vdash$.
Furthermore since $\xi\circ\alpha_{A}=\alpha_{B}\circ\xi,$ and
$\xi\circ\beta_{A}=\beta_{B}\circ\xi,$ we have
$(\alpha_{A}\oplus\alpha_{B})(u,\xi(u))=(\alpha_{A}(u),\alpha_{B}\circ\xi(u))=(\alpha_{A}(u),\xi\circ\alpha_{A}(u)).$
and
$(\beta_{A}\oplus\beta_{B})(u,\xi(u))=(\beta_{A}(u),\beta_{B}\circ\xi(u))=(\beta_{A}(u),\xi\circ\beta_{A}(u)),$
implies that $\Gamma_{\xi}$ is closed $\alpha_{A}\oplus\alpha_{B}$ and
$\beta_{A}\oplus\beta_{B}.$ Thus, $\Gamma_{\xi}$ is a BiHom-subalgebra of
$({A}\otimes{B},\dashv,\vdash,\alpha,\beta).$
Conversely, if the graph $\Gamma_{\xi}\subset{A}\oplus{B}$ is a BiHom-
subalgebra of $({A}\oplus{B},\dashv,\vdash,\alpha,\beta)$ then we
$(u+\xi(u))\dashv(v+\xi(v))=(u\dashv_{A}v+\xi(u)\dashv_{B}\xi(v))\in\Gamma_{\xi}$
$(u+\xi(u))\vdash(v+\xi(v))=(u\vdash_{A}v+\xi(u)\vdash_{B}\xi(v))\in\Gamma_{\xi}.$
Furthermore,
$(\alpha_{A}\oplus\alpha_{B})(\Gamma_{\xi})\subset\Gamma_{\xi},\,(\beta_{A}\oplus\beta_{B})(\Gamma_{\xi})\subset\Gamma_{\xi},$
implies
$(\alpha_{A}\oplus\alpha_{B})(u,\xi(u))=(\alpha_{A}(u),\alpha_{B}\circ\xi(u))\in\Gamma_{\xi},\,(\beta_{A}\oplus\beta_{B})(u,\xi(u))=(\beta_{A}(u),\beta_{B}\circ\xi(u))\in\Gamma_{\xi},$
which is equivalent to the condition
$\alpha_{B}\circ\xi(u)=\xi\circ\alpha_{A}(u),$ i.e
$\alpha_{B}\circ\xi=\xi\circ\alpha_{A}.$ Similary,
$\beta_{B}\circ\xi=\xi\circ\beta_{A}$. Therefore, $\xi$ is a morphism BiHom-
associative dialgebras. ∎
###### Proposition 2.30.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and $I$
be a two sided BiHom-ideal of $(D,\dashv,\vdash,\alpha,\beta)$. Then,
$(D/I,\left[\cdot,\cdot\right],\overline{\dashv},\overline{\vdash},\overline{\alpha},\overline{\beta})$
is a BiHom-associative dialgebra where
$\overline{x}\;\overline{\dashv}\;\overline{y}:=\overline{x\dashv
y},\;\;\overline{x}\;\overline{\vdash}\;\overline{y}:=\overline{x\vdash
y},\;\;\overline{\alpha}(\overline{x}):=\overline{\alpha(x)},\;\;\overline{\beta}(\overline{x}):=\overline{\beta(x)},$
for all $\overline{x},\overline{y}\in A/I.$
###### Proof.
We only prove left associativity, the other being proved similarly. For all
$\overline{x},\overline{y},\overline{z}\in D/I$, we have
$\displaystyle(\overline{x}\overline{\vdash}\overline{y})\overline{\vdash}\overline{\beta}(\overline{z})-\overline{\alpha}(\overline{x})\overline{\vdash}(\overline{y}\overline{\vdash}\overline{z})=\overline{(x\vdash
y)\vdash\beta(z)-\alpha(x)\vdash(y\vdash z)}=0.$
Then,
$(D/I,\overline{\dashv},\overline{\vdash},\overline{\alpha},\overline{\beta})$
is BiHom-associative dialgebra. ∎
Now, let us recall the definition of BiHom-Lie algebra.
###### Definition 2.31.
[7] $A$ BiHom-Lie algebra $(L,\left[\cdot,\cdot\right],\alpha,\beta)$ is a
$4$-tuple in where L is linear space, $\alpha,\beta:A\rightarrow A$,are linear
maps and $\left[\cdot,\cdot\right]:L\otimes L\rightarrow L$ is a bilinear
maps, such that, for all $x,y,z\in L$ :
$\alpha\circ\beta=\beta\circ\alpha,$ (2.28)
$\alpha(\left[x,y\right])=\left[\alpha(x),\alpha(y)\right],\,\text{and},\,\beta(\left[x,y\right])=\left[\beta(x),\beta(y)\right],$
(2.29)
$\left[\beta(x),\alpha(y)\right])=-\left[\beta(y),\alpha(x)\right],\,(\text{BiHom-
skew-symetry}),$ (2.30)
$\left[\beta^{2}(x),\left[\beta(y),\alpha(z)\right]\right]+\left[\beta^{2}(y),\left[\beta(z),\alpha(x)\right]\right]+\left[\beta^{2}(z),\left[\beta(x),\alpha(y)\right]\right]=0,$
(2.31)
(BiHom-Jacobi identity).
The maps $\alpha$ and $\beta$ (in this order) are called the structure maps of
L.
###### Definition 2.32.
A morphism between two BiHom-Lie algebras
$f:(L,[-,-],\alpha,\beta)\rightarrow(L^{\prime},[-,-]^{\prime},\alpha^{\prime},\beta^{\prime})$
is a linear map $f:L\rightarrow L^{\prime}$ such that $\alpha^{\prime}\circ
f=f\circ\alpha,\,\beta^{\prime}\circ f=f\circ\beta$ and
$f(\left[x,y\right])=\left[f(x),f(y)\right]^{\prime}$, for all $x,y\in L.$
The following lemma asserts that the commutator of any BiHom-associative
algebra gives rise to BiHom-Lie.
###### Lemma 2.33.
[7] Let $(A,\cdot,\alpha,\beta)$ be a regular BiHom-associative algebra. Then
$L(A)=(A,[-,-],\alpha,\beta)$
is a regular BiHom-Lie algebra, where
$[x,y]=x\cdot y-\alpha^{-1}\beta(y)\cdot\alpha\beta^{-1}(x),$
for any $x,y\in A.$
###### Proposition 2.34.
Let $(L,[-,-],\alpha,\beta)$ be a BiHom-Lie algebra and $N:L\rightarrow L$ be
a Nijenhuis operator on $L$ i.e. $\alpha\circ N=N\circ\alpha$, $\beta\circ
N=N\circ\beta$ and
$\displaystyle[N(x),N(y)]=N([N(x),y]+[x,N(y)]-N([x,y]))$
for any $x,y\in L$. Then, $(L,[-,-]_{N},\alpha,\beta)$ is a BiHom-Lie algebra
with
$\displaystyle[x,y]_{N}=[N(x),y]+[x,N(y)]-N([x,y])$
for all $x,y\in L$.
###### Proof.
It follows from direct computation. ∎
###### Corollary 2.35.
Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and
$N:A\rightarrow A$ be a Nijenhuis operator on $A$ i.e. $\alpha\circ
N=N\circ\alpha$, $\beta\circ N=N\circ\beta$ and
$\displaystyle N(x)\cdot N(y)=N(N(x)\cdot y+x\cdot N(y)-N(x\cdot y))$
for any $x,y\in A$. Let us denote by $L(A)$ the BiHom-Lie algebra associated
with $A$ as in Proposition 2.33. Then, $(A,[-,-]_{N},\alpha,\beta)$ is a
BiHom-Lie algebra.
###### Corollary 2.36.
Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and
$N:A\rightarrow A$ be a Nijenhuis operator on $A$ i.e. $\alpha\circ
N=N\circ\alpha$, $\beta\circ N=N\circ\beta$ and
$\displaystyle N(x)\cdot N(y)=N(N(x)\cdot y+x\cdot N(y)-N(x\cdot y))$
for any $x,y\in A$. Then, $(A,\\{-,-\\},\alpha,\beta)$ is a BiHom-Lie algebra
with
$\displaystyle\\{x,y\\}=x\ast_{N}y-\alpha^{-1}\beta(y)\ast_{N}\alpha\beta^{-1}(x)$
and
$\displaystyle x\ast_{N}y=N(x)\cdot y+x\cdot N(y)-N(x\cdot y)$
for all $x,y\in A$.
###### Proof.
It is similar to the one of Proposition 2.21. And the Lemma 2.33 will end the
proof. ∎
###### Remark 2.37.
The BiHom-Lie algebra generated by Corollary 2.35 and Corollary 2.36 are
equal.
###### Proposition 2.38.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra.
Then,for all $x,y\in D$, the bracket
$[x,y]=[x,y]_{L}+[x,y]_{R},$
where
$\displaystyle[x,y]_{L}$ $\displaystyle=$ $\displaystyle x\dashv
y-\alpha^{-1}\beta(y)\dashv\alpha\beta^{-1}(x),$ $\displaystyle{[x,y]_{R}}$
$\displaystyle=$ $\displaystyle x\vdash
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x),$
is a BiHom-Lie bracket if and only if
$\displaystyle\alpha(x)\dashv(y\vdash z)=(x\dashv y)\vdash\beta(z),$ (2.32)
$\displaystyle\alpha(x)\dashv(y\dashv z)=(x\vdash y)\vdash\beta(z).$ (2.33)
###### Proof.
It is essentialy based on Lemma 2.33. That is, an expansion of BiHom-Jacobi
identity leads to $48$ terms including $8$ terms which cancel pairewise by
axiom (2.2), $4$ terms cancel pairewise by axiom (2.3), $12$ terms cancel
pairewise by axiom (2.4), $6$ terms cancel pairewise by axiom (2.5) and $6$
terms cancel pairewise by axiom (2.6).
For the of the $12$ terms, $8$ terms cancel pairewise by axiom (2.32) and $4$
terms cancel pairewise by axiom (2.33). ∎
###### Definition 2.39.
A (right ) BiHom-Leibniz algebra is a $4$-tuple
$(L,\left[\cdot,\cdot\right],\alpha,\beta)$, where L is a linear space,
$\left[\cdot,\cdot\right]:L\times L\rightarrow L$ is a bilinear map and
$\alpha,\beta:L\rightarrow L$ are linear maps satisfying
$\left[\left[x,y\right],\alpha\beta(z),\right]=\left[\left[x,\beta(z)\right],\alpha(y)\right]+\left[\alpha(x),\left[y,\alpha(z)\right]\right],$
(2.34)
for all $x,y,z\in L$.
###### Example 2.40.
Let $L$ be a two-dimensional vector space and $\left\\{e_{1},e_{2}\right\\}$
be a basis of $L$. Then, $(L,[-,-],\alpha,\beta)$ is a BiHom-Leibniz algebra
with
$\left[e_{1},e_{2}\right]=ae_{1},\left[e_{2},e_{2}\right]=be_{1},\;\alpha(e_{2})=\beta(e_{2})=e_{1},a,b\in\mathbb{R}.$
Now, we introduce BiHom-Poisson dialgebras and study its connection with
BiHom-associative dialgebras.
###### Definition 2.41.
A BiHom-Poisson dialgebra is a BiHom-associative dialgebra
$({P},\dashv,\vdash,\alpha,\beta)$ and a BiHom-Leibniz algebra
$({P},[-,-],\alpha,\beta)$ such that
$\displaystyle{[x\dashv y,\alpha\beta(z)]}$ $\displaystyle=$
$\displaystyle\alpha(x)\dashv[y,\alpha(z)]+[x,\beta(z)]\dashv\alpha(y),$
$\displaystyle{[x\vdash y,\alpha\beta(z)]}$ $\displaystyle=$
$\displaystyle\alpha(x)\vdash[y,\alpha(z)]+[x,\beta(z)]\vdash\alpha(y),$
$\displaystyle\\{\alpha\beta(x),y\dashv z\\}$ $\displaystyle=$
$\displaystyle\beta(y)\vdash[\alpha(x),z]+[\beta(x),y]\dashv\beta(z)=[\alpha\beta(x),y\vdash
z],$
are satisfied for $x,y,z\in{P}$.
###### Theorem 2.42.
Let $({D},\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra. Then,
$P(D)=(D,[-,-],\dashv,\vdash,\alpha,\beta)$
is a BiHom-Poisson dialgebra, where $[x,y]=x\dashv y-y\vdash x$, for any
$x,y\in{D}$.
###### Proof.
By Theorem 2.46 $P(D)$ is a BiHom-Leibniz algebra. Moreover, for any $x,y,z\in
D$,
$\displaystyle[x\dashv
y,\alpha\beta(z)]-\alpha(x)\dashv[y,\alpha(z)]-[x,\beta(z)]\dashv\alpha(y)=$
$\displaystyle=(x\dashv
y)\dashv\alpha\beta(z)-\alpha^{-1}\beta\alpha\beta(z)\vdash\alpha\beta^{-1}(x\dashv
y)-\alpha(x)\dashv(y\dashv\alpha(z)-\alpha^{-1}\beta\alpha(z)\vdash\alpha\beta^{-1}(y))$
$\displaystyle-(x\dashv\beta(z)-\alpha^{-1}\beta\beta(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y)$
$\displaystyle=(x\dashv
y)\dashv\alpha\beta(z)-\beta^{2}(z)\vdash(\alpha\beta^{-1}(x)\dashv\alpha\beta^{-1}(y))-\alpha(x)\dashv(y\dashv\alpha(z)$
$\displaystyle\quad+\alpha(x)\dashv(\beta(z)\vdash\alpha\beta^{-1}(y))-(x\dashv\beta(z))\dashv\alpha(y)+(\alpha^{-1}\beta^{2}(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y).$
The last three axioms are proved analagously. This completes the proof. ∎
###### Theorem 2.43.
Let $(P,\dashv,\vdash,[-,-],\alpha,\beta)$ be a BiHom-Poisson dialgebra and
$\alpha^{\prime},\beta^{\prime}:D\rightarrow D$ two morphisms of BiHom-Poisson
dialgebras such that the maps $\alpha,\alpha^{\prime},\beta,\beta^{\prime}$
commute pairewise. Then
$P_{(\alpha^{\prime},\beta^{\prime})}=(D,\,\triangleleft:=\dashv(\alpha^{\prime}\otimes\beta^{\prime}),\;\triangleright:=\vdash(\alpha^{\prime}\otimes\beta^{\prime}),\;\\{-,-\\}:=[-,-](\alpha^{\prime}\otimes\beta^{\prime}),\;\alpha\alpha^{\prime},\;\beta\beta^{\prime}),$
is a BiHom-Poisson dialgebra.
###### Proof.
It is essentialy based on the one of Theorem 2.6. ∎
Now, we introduce action of BiHom-Leibniz algebra on another one.
###### Definition 2.44.
Let $D$ and $L$ be two BiHom-Leibniz algebras. An action of $D$ on $L$
consists of a pair of bilinear maps, $D\times L\rightarrow
L,(x,a)\mapsto[x,a]$ and $L\times D\rightarrow[x,a]$, such that
$\displaystyle\left[\alpha(x),\left[a,\alpha(b)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[x,a\right],\alpha\beta(b),\right]-\left[\left[x,\beta(b)\right],\alpha(a)\right]$
(2.35) $\displaystyle\left[\alpha(a),\left[x,\alpha(b)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[a,x\right],\alpha\beta(b),\right]-\left[\left[a,\beta(b)\right],\alpha(x)\right]$
(2.36) $\displaystyle\left[\alpha(a),\left[b,\alpha(x)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[a,b\right],\alpha\beta(x),\right]-\left[\left[a,\beta(x)\right],\alpha(b)\right]$
(2.37) $\displaystyle\left[\alpha(a),\left[x,\alpha(y)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[a,x\right],\alpha\beta(y),\right]-\left[\left[a,\beta(y)\right],\alpha(x)\right]$
(2.38) $\displaystyle\left[\alpha(x),\left[a,\alpha(y)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[x,a\right],\alpha\beta(y),\right]-\left[\left[x,\beta(y)\right],\alpha(a)\right]$
(2.39) $\displaystyle\left[\alpha(x),\left[y,\alpha(a)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[x,y\right],\alpha\beta(a),\right]-\left[\left[x,\beta(a)\right],\alpha(y)\right]$
(2.40)
for all $x,y\in D,a,b\in L$.
###### Lemma 2.45.
Given a BiHom-Leibniz action of $D$ on $L$, we can consider the semidirect
product Leibniz algebra $L\rJoin D$, which consists of vector space $D\oplus
L$ together with the Leibniz bracket given by
$\displaystyle[(x,a),(y,b)]=([x,y]+[x,b]+[a,y],[a,b])$ (2.41)
for all $(x,a),(x,b)\in D\times L$.
###### Proof.
$\displaystyle[\alpha(x,a),[(y,b),\alpha(z,c)]]$ $\displaystyle=$
$\displaystyle[(\alpha(x),\alpha(a)),([y,\alpha(z)]+[y,\alpha(c)]+[b,\alpha(z)],[b,\alpha(c)])]$
$\displaystyle=$
$\displaystyle\Big{(}[\alpha(x),[y,\alpha(z)]]+[\alpha(x),[y,\alpha(c)]]+[\alpha(x),[b,\alpha(z)]]+[\alpha(x),[b,\alpha(c)]]$
$\displaystyle+[\alpha(a),[y,\alpha(z)]]+[\alpha(a),[y,\alpha(z)]]+[\alpha(a),[y,\alpha(c)]]+[\alpha(a),[b,\alpha(z)],$
$\displaystyle[\alpha(a),[b,\alpha(c)]]\Big{)}.$
$\displaystyle{[[(x,a),(y,b)],\alpha\beta(z,c)]}$ $\displaystyle=$
$\displaystyle[([x,y]+[x,b]+[a,y]),[a,b]),(\alpha\beta(z),\alpha\beta(c))]$
$\displaystyle=$
$\displaystyle([[x,y],\alpha\beta(z)]+[[x,b],\alpha\beta(z)]+[[a,y],\alpha\beta(z)]+[[x,y],\alpha\beta(c)]$
$\displaystyle+[[x,b],\alpha\beta(c)]+[[a,y],\alpha\beta(c)]+[[a,b],\alpha\beta(c)],[[a,b],\alpha\beta(c)].$
$\displaystyle{[[(x,a),\beta(z,c)],\alpha(y,b)]}$ $\displaystyle=$
$\displaystyle[([x,\beta(z)]+[x,\beta(c)]+[a,\beta(z)],[a,\beta(c)]),(\alpha(y),\alpha(b))]$
$\displaystyle=$
$\displaystyle([[x,\beta(z)],\alpha(y)]+[[x,\beta(c)],\alpha(y)]+[[a,\beta(z)],\alpha(y)]+[[x,\beta(z)],\alpha(b)]$
$\displaystyle+[[x,\beta(c)],\alpha(b)]+[[a,\beta(z)],\alpha(b)]+[[a,\beta(c)],\alpha(y)],[[a,\beta(c)],\alpha(b)]).$
Using axioms in Definition 2.35, it follows that
${[[(x,a),(y,b)],\alpha\beta(z,c)]}={[[(x,a),\beta(z,c)],\alpha(y,b)]}+[\alpha(x,a),[(y,b),\alpha(z,c)]].$
Which proves the proposition. ∎
###### Theorem 2.46.
Let $({D},\dashv,\vdash,\alpha,\beta)$ be a regular BiHom-associative
dialgebra. Then the bracket defined by $\left[x,y\right]=x\dashv
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x)$, defines a structure of BiHom-
Leibniz algebra on ${D}$, and denoted ${\bf Lb}(D)$.
###### Proof.
For any $x,y,z\in{D}$, we have
$\displaystyle[[x,y],\alpha\beta(z)]$ $\displaystyle=$ $\displaystyle(x\dashv
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(z)$
$\displaystyle-\alpha^{-1}\beta\alpha\beta(z)\vdash(x\dashv
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1})$ $\displaystyle=$
$\displaystyle(x\dashv
y)\dashv\alpha\beta(z)-(\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(z)$
$\displaystyle-\beta^{2}(z)\vdash(\alpha\beta^{-1}(x)\dashv\alpha\beta^{-1}(y)+\beta^{2}(z)\vdash(y\vdash\alpha^{2}\beta^{-2}(x)).$
$\displaystyle{[[x,\beta(z)],\alpha(y)]}$ $\displaystyle=$
$\displaystyle(x\dashv\beta(z)-\alpha^{-1}\beta^{2}(z)\vdash\alpha\beta^{-1}(x)\dashv\alpha(y)$
$\displaystyle-\alpha^{-1}\beta\alpha(z)\vdash(\alpha\beta^{-1}(x\dashv
y+\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1})$ $\displaystyle=$
$\displaystyle(x\dashv\beta(z))\dashv\alpha(y)-(\alpha^{-1}\beta^{2}(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y)$
$\displaystyle-\beta(y)\vdash(\alpha\beta^{-1}(x)\dashv\alpha(z))+\beta(y)\vdash(\beta(z)\vdash\alpha^{2}\beta^{-2}(x)).$
$\displaystyle{[\alpha(x),[y,\alpha(z)]]}$ $\displaystyle=$
$\displaystyle\alpha(x)\dashv(y\dashv\alpha(z)-\alpha^{-1}\beta\alpha(z)\vdash\alpha\beta^{-1}(y))-\alpha^{-1}\beta(y\dashv\alpha(z)$
$\displaystyle-\beta(z)\vdash\alpha\beta^{-1}(y))\vdash\alpha\beta^{-1}\alpha(x)$
$\displaystyle=$
$\displaystyle\alpha(x)\dashv(y\dashv\alpha(z))-\alpha(x)\dashv(\beta(z)\vdash\alpha\beta^{-1}(y))$
$\displaystyle-(\alpha^{-1}\beta(y)\dashv\beta(z))\vdash\alpha^{2}\beta^{-1}(x)-(\alpha^{-1}\beta^{2}(z)\vdash
y)\vdash\alpha^{2}\beta^{-1}(x).$
By axioms in Definition 2.1, the conclusion holds. ∎
In the relations contained in the below definition, we omitted the subsript
for simplifying the typography.
###### Definition 2.47.
Let $D$ and $L$ be dialgebras. An action of $D$ on $L$ consists of four linear
maps, two of them denoted by the symbol $\dashv$ and other two by $\vdash$,
$\displaystyle\dashv:D\otimes L\rightarrow L,$ $\displaystyle\dashv:L\otimes
D\rightarrow L,$ $\displaystyle\dashv:D\otimes L\rightarrow L,$
$\displaystyle\dashv:L\otimes D\rightarrow L$
such that the following $30$ equalities hold :
${(01)}\quad(x\dashv a)\dashv\beta(b)=\alpha(x)\dashv(a\dashv b)$, | $\qquad\qquad{(16)}\quad(a\dashv x)\dashv\beta(y)=\alpha(a)\dashv(x\dashv y)$,
---|---
${(02)}\quad(x\dashv a)\dashv\beta(b)=\alpha(x)\dashv(a\vdash b)$, | $\qquad\qquad{(17)}\quad(a\dashv x)\dashv\beta(y)=\alpha(a)\dashv(x\vdash y)$,
${(03)}\quad(x\vdash a)\dashv\beta(b)=\alpha(x)\vdash(a\dashv b)$, | $\qquad\qquad{(18)}\quad(a\vdash x)\dashv\beta(y)=\alpha(a)\vdash(x\dashv y)$,
${(04)}\quad(x\dashv a)\vdash\beta(b)=\alpha(x)\vdash(a\vdash b)$, | $\qquad\qquad{(19)}\quad(a\dashv x)\vdash\beta(y)=\alpha(a)\vdash(x\vdash y)$,
${(05)}\quad(x\vdash a)\vdash\beta(b)=\alpha(x)\vdash(a\vdash b)$, | $\qquad\qquad{(20)}\quad(a\vdash x)\vdash\beta(y)=\alpha(a)\vdash(x\vdash y)$,
${(06)}\quad(a\dashv x)\dashv\beta(b)=\alpha(a)\dashv(x\dashv b)$, | $\qquad\qquad{(21)}\quad(x\dashv a)\dashv\beta(y)=\alpha(x)\dashv(a\dashv y)$,
${(07)}\quad(a\dashv x)\dashv\beta(b)=\alpha(a)\dashv(x\vdash b)$, | $\qquad\qquad{(22)}\quad(x\dashv a)\dashv\beta(y)=\alpha(x)\dashv(a\vdash y)$,
${(08)}\quad(a\vdash x)\dashv\beta(b)=\alpha(a)\vdash(x\dashv b)$, | $\qquad\qquad{(23)}\quad(x\vdash a)\dashv\beta(y)=\alpha(x)\vdash(a\dashv y)$,
${(09)}\quad(a\dashv x)\vdash\beta(b)=\alpha(a)\vdash(x\vdash b)$, | $\qquad\qquad{(24)}\quad(x\dashv a)\vdash\beta(y)=\alpha(x)\vdash(a\vdash y)$,
${(10)}\quad(a\vdash x)\vdash\beta(b)=\alpha(a)\vdash(x\vdash b)$, | $\qquad\qquad{(25)}\quad(x\vdash a)\vdash\beta(y)=\alpha(x)\vdash(a\vdash y)$,
${(11)}\quad(a\dashv b)\dashv\beta(x)=\alpha(a)\dashv(b\dashv x)$, | $\qquad\qquad{(26)}\quad(x\dashv y)\dashv\beta(a)=\alpha(x)\dashv(y\dashv a)$,
---|---
${(12)}\quad(a\dashv b)\dashv\beta(x)=\alpha(a)\dashv(b\vdash x)$, | $\qquad\qquad{(27)}\quad(x\dashv y)\dashv\beta(a)=\alpha(x)\dashv(y\vdash a)$,
${(13)}\quad(a\vdash b)\dashv\beta(x)=\alpha(a)\vdash(b\dashv x)$, | $\qquad\qquad{(28)}\quad(x\vdash y)\dashv\beta(a)=\alpha(x)\vdash(y\dashv a)$,
${(14)}\quad(a\dashv b)\vdash\beta(x)=\alpha(a)\vdash(b\vdash x)$, | $\qquad\qquad{(29)}\quad(x\dashv y)\vdash\beta(a)=\alpha(x)\vdash(y\vdash a)$,
${(15)}\quad(a\vdash b)\vdash\beta(x)=\alpha(a)\vdash(b\vdash x)$, | $\qquad\qquad{(30)}\quad(x\vdash y)\vdash\beta(a)=\alpha(x)\vdash(y\vdash a)$,
for all $x,y\in D,a,b\in L$. The action is called trivial if these four maps
are trivial.
###### Example 2.48.
i) Any BiHom-associative dialgebra may be seen as acting on itself ii)Given a
homomorphism $\varphi:D\rightarrow L$ of BiHom-associative dialgebras, then
there is an action of $D$ on $L$ via the maps $x\triangleleft
a:=\varphi(x)\dashv a,\;x\triangleright
a:=\varphi(x)\triangleright,a\;a\triangleleft
x:=a\vdash\varphi(x)\;\;\mbox{and}\;\;a\triangleright x:=a\vdash\varphi(x)$.
iii)If $\psi:L\rightarrow D$ is an isomorphism of BiHom-associative
dialgebras, then there is an action of $D$ on $L$ via the maps $x\triangleleft
a:=\psi^{-1}(x)\dashv a,\;x\triangleright a:=\psi^{-1}(x)\triangleright
a,\;a\triangleleft x:=a\vdash\psi^{-1}(x)\;\;\mbox{and}\;\;a\triangleright
x:=a\vdash\psi^{-1}(x)$.
iv) If $I$ is an ideal of $D$, then the left and the right product yield an
action of $D$ on I.
###### Lemma 2.49.
Given two regular BiHom-associative dialgebras $D$ and $L$ together with an
action of $D$ on $L$, there is an action an action ${\bf Lb}(D)$ on ${\bf
Lb}(L)$ given by
$\displaystyle[x,a]$ $\displaystyle=$ $\displaystyle x\dashv
a-\alpha^{-1}\beta(a)\dashv\alpha\beta^{-1}(x),$ $\displaystyle{[a,x]}$
$\displaystyle=$ $\displaystyle a\vdash
x-\alpha^{-1}\beta(x)\vdash\alpha\beta^{-1}(a),$
for all $x\in{\bf Lb}(D)$, $a\in{\bf Lb}(L)$.
###### Proof.
For all $x\in{\bf Lb}(D)$, $a\in{\bf Lb}(L)$,
$\displaystyle[[x,a],\alpha\beta(b)]$ $\displaystyle=$ $\displaystyle(x\dashv
a-\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(b)$
$\displaystyle-\alpha^{-1}\beta\alpha\beta(b)\vdash\alpha\beta^{-1}(x\dashv
a-\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))$ $\displaystyle=$
$\displaystyle(x\dashv
a)\dashv\beta\alpha(b)-(\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))\dashv\beta\alpha(b)$
$\displaystyle-\beta^{2}(b)\vdash(\alpha\beta^{-1}(x)\dashv\beta^{-1}\alpha(a))+\beta^{2}(b)\vdash(a\vdash\alpha^{2}\beta^{-2}(x)).$
On the other hand,
$\displaystyle\qquad[[x,\beta(b)],\alpha(a)]+[\alpha(x),[a,\alpha(b)]]=$
$\displaystyle=\Big{(}x\dashv\beta(b)-\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x)\Big{)}\dashv\alpha(a)-\alpha^{-1}\beta\alpha(a)\vdash\alpha\beta^{-1}\Big{(}x\dashv\beta(b)-\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x)\Big{)}$
$\displaystyle\quad+\alpha(x)\dashv\Big{(}a\dashv\alpha(b)-\alpha^{-1}\beta\alpha(b)\vdash\alpha\beta^{-1}(a)\Big{)}-\alpha^{-1}\beta\Big{(}a\dashv\alpha(b)-\alpha^{-1}\beta\alpha(b)\vdash\alpha\beta^{-1}(a)\Big{)}\vdash\alpha\beta^{-1}\alpha(x)$
$\displaystyle=(x\dashv\beta(b))\dashv\alpha(a)-(\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x))\dashv\alpha(a)-\beta(a)\vdash(\alpha\beta^{-1}(x)\dashv\alpha(b))$
$\displaystyle\quad+\beta(a)\vdash(\beta(b)\vdash\alpha^{2}\beta^{-2}(x))+\alpha(x)\dashv(a\dashv\alpha(b))-\alpha(x)\dashv(\beta(b)\vdash\alpha\beta^{-1}(a))$
$\displaystyle\quad-(\alpha^{-1}\beta(a)\dashv\beta(b))\vdash\beta^{-1}\alpha^{2}(x)+(\alpha^{-1}\beta^{2}(b)\vdash
a)\vdash\beta^{-1}\alpha^{2}(x).$
Using axioms (2.3), (2.5), it comes
$\displaystyle\qquad[[x,\beta(b)],\alpha(a)]+[\alpha(x),[a,\alpha(b)]]=$
$\displaystyle=-(\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x))\dashv\alpha(a)-\beta(a)\vdash(\alpha\beta^{-1}(x)\dashv\alpha(b))$
$\displaystyle\quad+\alpha(x)\dashv(a\dashv\alpha(b))+(\alpha^{-1}\beta^{2}(b)\vdash
a)\vdash\alpha^{2}\beta^{-1}(x).$
By comparing, we get the attended result. The five other axioms are proved in
the same way. ∎
###### Lemma 2.50.
Let $D$ and $L$ be two regular BiHom-associative dialgebras together with an
action of $D$ on $L$. There is a BiHom-associative dialgebra structure on
$L\rJoin D$ which consists with vector space $L\oplus D$ and
$\displaystyle(a,x)\triangleleft(b,y)$ $\displaystyle=$ $\displaystyle(a\dashv
b+a\dashv y+x\dashv b,x\dashv y),$ $\displaystyle(a,x)\triangleright(b,y)$
$\displaystyle=$ $\displaystyle(a\vdash b+a\vdash y+x\vdash b,x\vdash y),$
for any $(a,x),(b,y)\in L\times D$
###### Proof.
For any $a,b,c\in L,x,y,z\in D$, one has
$\displaystyle\qquad\Big{(}(a,x)\triangleleft(b,y)\Big{)}\triangleleft\beta(c,z)-\alpha(a,x)\triangleleft\Big{(}(b,y)\triangleright(c,z)\Big{)}=$
$\displaystyle=(a\dashv b+a\dashv y+x\dashv b,x\dashv
y)\triangleleft(\beta(c),\beta(z))-(\alpha(a),\alpha(x))\triangleleft(b\vdash
c+b\vdash z+y\vdash c,y\vdash z)$ $\displaystyle=\Big{(}(a\dashv b+a\dashv
y+x\dashv b)\dashv\beta(c)+(a\dashv b+a\dashv y+x\dashv
b)\dashv\beta(z)+(x\dashv y)\dashv\beta(c),\;\;(x\dashv
y)\dashv\beta(z)\Big{)}$ $\displaystyle\quad-\Big{(}\alpha(a)\dashv(b\vdash
c+b\vdash z+y\vdash c)+\alpha(a)\dashv(y\vdash z)+\alpha(x)\dashv(b\vdash
c+b\vdash z+y\vdash c),\;\;\alpha(x)\dashv(y\vdash z)\Big{)}$
$\displaystyle=\Big{(}(a\dashv b)\dashv\beta(c)+(a\dashv
y)\dashv\beta(c)+(x\dashv b)\dashv\beta(c)+(a\dashv b)\dashv\beta(z)+(a\dashv
y)\dashv\beta(z)+(x\dashv b)\dashv\beta(z)$ $\displaystyle\quad+(x\dashv
y)\dashv\beta(c)-\alpha(a)\dashv(b\vdash c)-\alpha(a)\dashv(b\vdash
z)-\alpha(a)\dashv(y\vdash c)-\alpha(a)\dashv(y\vdash
z)-\alpha(x)\dashv(b\vdash c)$ $\displaystyle\quad-\alpha(x)\dashv(b\vdash
z)-\alpha(x)\dashv(y\vdash c),\;\;(x\dashv
y)\dashv\beta(z)-\alpha(x)\dashv(y\vdash z)\Big{)}.$
The left hand side vanishes by axiom (2.3) and axioms
$(02),(07),(12),(17),(22),(27)$ in Definition 2.47. The other axioms are
proved in the same way. ∎
###### Theorem 2.51.
Let $D$ and $L$ be two regular BiHom-associative dialgebras together with an
action of $D$ on $L$. Then, ${\bf Lb}(L\rJoin D)={\bf Lb}(L)\rJoin{\bf
Lb}(D)$.
###### Proof.
By lemma 2.49, ${\bf Lb}(D)$ acts on ${\bf Lb}(L)$, so it makes sense to
consider the semidirect product Leibniz algebra ${\bf Lb}(L)\rJoin{\bf
Lb}(D)$. It is clear that ${\bf Lb}(L\rJoin D)$ and ${\bf Lb}(L)\rJoin{\bf
Lb}(D)$ are egal as vector space, so we only need to verify that they share
the same bracket. Let $(a,x),(b,y)\in L\times D$. If we use the bracket in
${\bf Lb}(L)\rJoin{\bf Lb}(D)$, we get :
$\displaystyle[(a,x),(b,y)]$ $\displaystyle=$
$\displaystyle([a,b]+[x,b]+[a,y],[x,y])$ $\displaystyle=$
$\displaystyle(a\dashv y-b\vdash+x\dashv b-b\vdash x+a\dashv y-y\vdash
a,x\dashv y-y\vdash x).$
On the other hand, if we use the Leibniz bracket in ${\bf Lb}(L\rJoin D)$
(Lemma 2.50), we get
$\displaystyle\\{(a,x),(b,y)\\}$ $\displaystyle=$
$\displaystyle(a,x)\triangleleft(b,y)-(b,y)\triangleright(a,x)$
$\displaystyle=$ $\displaystyle(a\dashv b+x\dashv b+a\dashv y,x\dashv
y)-(b\vdash a+y\vdash a+b\vdash x,y\vdash x),$
So the brackets are equal. ∎
## 3 Central extensions
This section concerns the central extension of BiHom-associative dialgebras in
relation with cocycles.
###### Definition 3.1.
Let $(D_{i},\dashv_{i},\vdash_{i},\alpha_{i},\beta_{i}),i=1,2,3$ be three
BiHom-associative dialgebras. The BiHom-associative dialgebra $D_{2}$ is
called the extension of $D_{3}$ by $D_{1}$ if there are homomorphisms
$\phi:D_{1}\rightarrow D_{2}$ and $\psi:D_{2}\rightarrow D_{3}$ such that the
following sequence
$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}D_{3}\rightarrow
0$
is exact.
###### Definition 3.2.
An extension is called trivial if there exists a BiHom-ideal $I$ of $D_{2}$
complementary to $Ker\psi$ i.e.
$D_{2}=Ker\psi\oplus I$
It may happen that there exist several extensions of $D_{3}$ by $D_{1}$. To
classify extensions the notion of equivalent extensions is defined.
###### Definition 3.3.
Two sequences
$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}D_{3}\rightarrow
0$
and
$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi^{\prime}}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi^{\prime}}}{{\rightarrow}}D_{3}\rightarrow
0$
are equivalent extensions if there exists a associative dialgebra isomorphism
$f:D_{2}\rightarrow D^{\prime}_{2}$ such that
$f\circ\phi=\phi^{\prime}\quad\mbox{and}\quad\psi^{\prime}\circ f=\psi.$
###### Definition 3.4.
An extension
$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}D_{3}\rightarrow
0$
is called central if the kernel of $\psi$ is contained in the center
$Z(D_{2})$ of $D_{2}$, i.e. $Ker\psi\subset Z(D)$.
Now, we introduce $2$-cocycle on BiHom-associative dialgebra with values in a
BiHom-module.
###### Definition 3.5.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$(M,\alpha_{M},\beta_{M})$ a BiHom-module over the same field that $D$. A pair
$\Theta=(\theta_{1},\theta_{2})$ of bilinear maps $\theta_{1}:D\times
D\rightarrow V$ and $\theta_{2}:D\times D\rightarrow V$ is called a
$2$-cocycle on $D$ with values in $V$ if $\theta_{1}$ and $\theta_{2}$ satisfy
$\displaystyle\theta_{1}(x\dashv y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{1}(\alpha(x),y\dashv z),$ (3.1)
$\displaystyle\theta_{1}(x\dashv y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{1}(\alpha(x),y\vdash z),$ (3.2)
$\displaystyle\theta_{2}(x\vdash y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{2}(\alpha(x),y\vdash z),$ (3.3)
$\displaystyle\theta_{2}(x\dashv y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{2}(\alpha(x),y\vdash z),$ (3.4)
$\displaystyle\theta_{1}(x\vdash y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{2}(\alpha(x),y\dashv z),$ (3.5)
for all $x,y,z\in D$.
The set of all $2$-cocycles on $D$ with values in $M$ is denoted $Z^{2}(D,M)$,
which a vector space.
In the below lemma, we give a special type of $2$-cocycles which are called
$2$-coboundaries.
###### Lemma 3.6.
Let $\nu:D\rightarrow V$ be a linear map, and define
$\varphi_{1}(x,y)=\nu(x\dashv y)$ and $\varphi_{2}(x,y)=\nu(x\vdash y)$. Then,
$\Phi=(\varphi_{1},\varphi_{2})$ is a $2$-cocycle on $D$.
###### Proof.
We will prove one equality, the others being proved in the same way. For any
$x,y,z\in D$, one has
$\displaystyle\varphi_{1}(\alpha(x),y\dashv z)$ $\displaystyle=$
$\displaystyle\nu(\alpha(x)\dashv(y\dashv z))=\nu((x\dashv y)\dashv\beta(z))$
$\displaystyle=$ $\displaystyle\nu(\alpha(x)\dashv(y\vdash
z))=\varphi_{1}(\alpha(x),y\vdash z).$
This finishes the proof. ∎
The set of all $2$-coboundaries is denoted by $B^{2}(D,M)$ and it is a
subgroup of $Z^{2}(D,M)$. The group $H^{2}(D,M)=Z^{2}(D,M)/B^{2}(D,M)$ is said
to be a second cohomology group of $D$ with values in $M$. Two cocycles
$\Theta_{1}$ and $\Theta_{2}$ are said to be cohomologous cocycles if
$\Theta_{1}-\Theta_{2}$ is a coboundary.
###### Theorem 3.7.
Let $(D,\dashv,\vdash,\alpha_{D},\beta_{D})$ be a BiHom-associative dialgebra,
$(M,\alpha_{M},\beta_{M})$ a BiHom-module,
$\theta_{1}:D\times D\rightarrow M\quad\mbox{and}\quad\theta_{2}:D\times
D\rightarrow M$
be bilinear maps. Let us set $D_{\Theta}=D\oplus M$, where
$\Theta=(\theta_{1},\theta_{2})$. For any $x,y\in D$, $v,w\in M$, let us
define
$\displaystyle(x+u)\triangleleft(y+v)=x\dashv
y+\theta_{1}(x,y)\quad\mbox{and}\quad(x+u)\triangleright(y+v)=x\vdash
y+\theta_{2}(x,y).$
Then,
$(D_{\Theta},\triangleleft,\triangleright,\alpha_{A}\otimes\alpha_{M},\beta_{A}\otimes\beta_{M})$
is a BiHom-associative dialgebra if and only if $\Theta$ is a $2$-cocycle.
###### Proof.
For any $x,y,z\in D,u,v,w\in M$, we have
$\displaystyle((x+v)\triangleleft(y+w))\triangleleft(\beta(z)+w)-(\alpha(x)+v)\triangleleft((y+w)\triangleleft(z+w))=$
$\displaystyle=$
$\displaystyle((x+v)\triangleleft(y+w))\triangleleft(\beta(z)+w)-(\alpha(x)+v)\triangleleft((y\dashv
z)+\theta_{1}(y,z))$ $\displaystyle=$ $\displaystyle((x\dashv
y)\dashv\beta(z))+\theta_{1}(x\dashv y,\beta(z))-(\alpha(x)\dashv(y\dashv
z))-\theta_{1}(\alpha(x),y\dashv z).$
The left hand vanishes by axioms (2.2) and (3.1). The other axioms are proved
analagously. ∎
###### Lemma 3.8.
Let $\Theta$ be a $2$-cocycle and $\Phi$ a $2$-coboundary. Then,
$D_{\Theta+\Phi}$ is a BiHom-associative dialgebra with
$\displaystyle(x+u)\unlhd(y+v)=x\dashv y+\varphi_{1}(x,y)+\theta_{1}(x,y),$
$\displaystyle(x+u)\unrhd(y+v)=x\vdash y+\varphi_{2}(x,y)+\theta_{2}(x,y).$
Moreover, $D_{\Theta}\cong D_{\Theta+\Phi}$.
###### Proof.
First, we have to shown that $D_{\Theta+\Phi}$ is a BiHom-associative
dialgebra. So, for any $x+u,\\\ y+v,z+w\in D\oplus M$,
$\displaystyle\qquad((x+u)\unlhd(y+v))\unlhd\beta(z+w)-\alpha(x+u)\unlhd((y+v))\unlhd(z+w))=$
$\displaystyle=(x\dashv
y+\varphi_{1}(x,y)+\theta_{1}(x,y))\unlhd(\beta(z)+\beta(w))-(\alpha(x)+\alpha(u))\unlhd(y\dashv
z+\varphi_{1}(y,z)+\theta_{1}(y,z))$ $\displaystyle=(x\dashv
y)\dashv\beta(z)+\varphi_{1}(x\dashv y,\beta(z))+\theta_{1}(x\dashv
y,\beta(z))-\alpha(x)\dashv y\dashv z-\varphi_{1}(\alpha(x),y\dashv
z)+\theta_{1}(\alpha(x),y\dashv z)$
The left hand side vanishes by (2.2) and (3.1). The proofs of the rest of
axioms are leaved to the reader.
Next, the isomorphism $f:D_{\Theta}\rightarrow D_{\Theta+\Phi}$ is given by
$f(x+v)=x+\nu(x)+v$. In fact, it is clear that $f$ is a bijective linear map
and
$\displaystyle f(\alpha_{D}+\alpha_{M})(x+v)$ $\displaystyle=$ $\displaystyle
f(\alpha_{D}(x)+\alpha_{M}(v))$ $\displaystyle=$
$\displaystyle\alpha_{D}(x)+\nu\alpha_{D}(x)+\alpha_{M}(v)$ $\displaystyle=$
$\displaystyle\alpha_{D}(x)+\alpha_{M}\nu(x)+\alpha_{M}(v)$ $\displaystyle=$
$\displaystyle(\alpha_{D}+\alpha_{M})(x+\nu(x)+v)$ $\displaystyle=$
$\displaystyle(\alpha_{D}+\alpha_{M})\circ f(x+v).$
Thus, $f$ commutes $\alpha_{D}+\alpha_{M}$, and similarly with
$\beta_{D}+\beta_{M}$.
Then,
$\displaystyle f((x+v)\triangleleft(y+w))$ $\displaystyle=$ $\displaystyle
f(x\dashv y+\theta_{1}(x,y))$ $\displaystyle=$ $\displaystyle f(x\dashv
y)+f(\theta_{1}(x,y))$ $\displaystyle=$ $\displaystyle x\dashv y+\nu(x\dashv
y)+\theta_{1}(x,y)$ $\displaystyle=$ $\displaystyle x\dashv
y+\varphi_{1}(x,y)+\theta_{1}(x,y).$
and
$\displaystyle f(x+v)\unlhd f(y+w)$ $\displaystyle=$
$\displaystyle(x+\nu(x)+v)\unlhd(y+\nu(y)+w)$ $\displaystyle=$
$\displaystyle(x\dashv y)+\varphi_{1}(x,y)+\theta_{1}(x,y).$
∎
###### Corollary 3.9.
Let $\Theta_{1},\Theta_{2}$ be two cohomologous $2$-cocycles on a BiHom-
associative dialgebra $D$, and $D_{1},D_{2}$ be the central extensions
constructed with these $2$-cocycles, respectively. The the central extensions
$D_{1}$ and $D_{2}$ are equivalent extensions. In particular a central
extension defined by a coboundary is equivalent with a trivial central
extension.
The following theorem is proved Mutatis Mutandis as ([25], Theorem 4.1). So we
omitted the proof.
###### Theorem 3.10.
There exists one to one correspondence between elements of $H^{2}(D,M)$ and
nonequivalents central extensions of associative dialgebra $D$ by $M$.
## 4 Classification
In this section, we give classification of BiHom-associative dialgebras in low
dimension.
Let $(D,\dashv,\vdash,\alpha,\beta)$ be an $n$-dimensional BiHom-associative
dialgebra, $\\{e_{i}\\}$ be a basis of $D$. For any $i,j\in\mathbb{N},1\leq
i,j\leq n$, let us put
$e_{i}\dashv e_{j}=\sum_{k=1}^{n}\gamma_{ij}^{k}e_{k},\quad e_{i}\vdash
e_{j}\sum_{k=1}^{n}\delta_{ij}^{k}e_{k},\quad\alpha(e_{j})=\sum_{k=1}^{n}\alpha_{kj}e_{k},\quad\beta(e_{j})=\sum_{k=1}^{n}\beta_{kj}e_{k}.$
The axioms in Definition 2.1 are respectively equivalent to
$\displaystyle\beta_{kj}\alpha_{pk}-\alpha_{ji}\beta_{pj}$ $\displaystyle=$
$\displaystyle 0,$ (4.1)
$\displaystyle\gamma_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\gamma_{jk}^{q}\gamma_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.2)
$\displaystyle\gamma_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\delta_{jk}^{q}\gamma_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.3)
$\displaystyle\gamma_{ij}^{p}\beta_{qk}\delta_{pq}^{r}-\alpha_{pi}\delta_{jk}^{q}\delta_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.4)
$\displaystyle\delta_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\gamma_{jk}^{q}\delta_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.5)
$\displaystyle\delta_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\delta_{jk}^{q}\delta_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0.$ (4.6)
### 4.1 One dimensional
There is only one $1$-dimensional BiHom-associative dialgebra ; the nul (or
trivial) BiHom-associative dialgebra.
### 4.2 Two dimensional
$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{1}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=ae_{1},\\\ e_{2}\dashv e_{1}=be_{1},\\\ e_{1}\vdash e_{2}=ce_{1},\\\ e_{2}\vdash e_{1}=de_{1},\\\ e_{2}\vdash e_{2}=fe_{1}.\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$
$\mathcal{A}lg_{2}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=ae_{1},\\\ e_{2}\dashv e_{1}=ae_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$
$\mathcal{A}lg_{3}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=ae_{1},\\\ e_{1}\vdash e_{2}=be_{1},\\\ e_{2}\vdash e_{1}=ce_{1},\\\ e_{2}\vdash e_{2}=de_{1}\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$
$\mathcal{A}lg_{4}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=ae_{1},\\\ e_{1}\vdash e_{2}=be_{1},\\\ e_{2}\vdash e_{1}=ce_{1},\\\ e_{2}\vdash e_{2}=de_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$
###### Remark 4.1.
In two dimensional, all of the BiHom-associative dialgebras are Hom-
associative dialgebras i.e. $\alpha=\beta$.
### 4.3 Three dimensional
$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{1}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=ae_{1},\\\ e_{2}\dashv e_{3}=be_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{2}=ce_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=de_{1},\\\ e_{3}\vdash e_{2}=fe_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{2}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ e_{3}\dashv e_{2}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
$\mathcal{A}lg_{3}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{,}\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ e_{3}\dashv e_{2}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{3}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
$\mathcal{A}lg_{4}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{3}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
$\mathcal{A}lg_{5}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{3}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
### 4.4 Four dimensional
$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{1}$ | $\begin{array}[]{ll}e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{1}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ e_{4}\dashv e_{4}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=ce_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{4}=de_{3},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=be_{2}\\\ \beta(e_{2})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{2}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{1}=ae_{4},\\\ e_{2}\dashv e_{3}=be_{4},\\\ e_{3}\dashv e_{1}=-ce_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=de_{4},\\\ e_{3}\vdash e_{3}=fe_{4},\\\ e_{3}\vdash e_{4}=e_{4},\\\ e_{4}\vdash e_{4}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \beta(e_{2})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{3}$ | $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=be_{4},\\\ e_{3}\dashv e_{1}=ce_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{2}=ce_{4},\\\ e_{3}\vdash e_{3}=de_{4},\\\ e_{4}\vdash e_{4}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{3})=e_{3}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{4}$ | $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=ae_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{1}=e_{4},\\\ e_{3}\dashv e_{2}=ce_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{4}\vdash e_{4}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{3})=e_{3}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{5}$ | $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{4}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{4}=e_{4},\\\ e_{1}\vdash e_{3}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{6}$ | $\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{4}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{3}=e_{4},\\\ e_{1}\vdash e_{4}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{1}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{4})=e_{4}\end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{7}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{4}=fe_{4},\\\ e_{3}\dashv e_{3}=-ge_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{4}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{1}=e_{4},\\\ e_{3}\vdash e_{2}=-he_{4},\\\ e_{3}\vdash e_{3}=ke_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{3})=e_{3},\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{8}$ | $\begin{array}[]{ll}e_{1}\dashv e_{3}=e_{4},\\\ e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{4}=e_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{1}=e_{4},\\\ e_{3}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{3})=e_{3}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{9}$ | $\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{1}+e_{4},\\\ e_{2}\dashv e_{3}=e_{1}+e_{4},\\\ e_{3}\dashv e_{2}=e_{1}+e_{4},\\\ e_{4}\dashv e_{2}=e_{1}+e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=-e_{1}+e_{4},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{3}\vdash e_{3}=e_{1}+e_{4},\\\ e_{4}\vdash e_{2}=e_{1}+e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \alpha(e_{3})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{3},\\\ \beta(e_{4})=e_{4},\\\ \end{array}$
$\mathcal{A}lg_{10}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{2}=e_{1}+e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{2}=e_{1},\\\ e_{3}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{4}\dashv e_{2}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{3}\vdash e_{3}=e_{1}+e_{4},\\\ e_{4}\vdash e_{2}=e_{1}+e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \alpha(e_{3})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{3},\\\ \beta(e_{4})=e_{4},\\\ \end{array}$
$\mathcal{A}lg_{11}$ | $\begin{array}[]{ll}e_{2}\dashv e_{2}=fe_{1}+ge_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{2}=e_{1}+e_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ e_{4}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=he_{1}-ke_{4},\\\ e_{3}\vdash e_{3}=e_{1}+e_{4},\\\ e_{4}\vdash e_{2}=e_{1}+e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \alpha(e_{3})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{3},\\\ \beta(e_{4})=e_{4},\\\ \end{array}$
$\mathcal{A}lg_{12}$ | $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=ae_{4},\\\ e_{2}\dashv e_{4}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=-be_{4},\\\ e_{3}\vdash e_{2}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{3})=e_{3}\\\ \alpha(e_{4})=e_{4}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
$\mathcal{A}lg_{13}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{1}\dashv e_{3}=e_{4},\\\ e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{1}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{3})=e_{3}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
$\mathcal{A}lg_{14}$ | $\begin{array}[]{ll}e_{1}\dashv e_{1}=e_{4},\\\ e_{1}\dashv e_{3}=-ce_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{1}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=-2ae_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=be_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{1})=e_{1}\\\ \alpha(e_{2})=e_{2}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{15}$ | $\begin{array}[]{ll}e_{1}\dashv e_{1}=-e_{4},\\\ e_{1}\dashv e_{2}=ae_{4},\\\ e_{2}\dashv e_{3}=be_{4},\\\ e_{3}\dashv e_{1}=ce_{4},\\\ e_{3}\dashv e_{2}=de_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=fe_{4},\\\ e_{1}\vdash e_{4}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{2}=ge_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{4}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
$\mathcal{A}lg_{16}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=ae_{4},\\\ e_{2}\dashv e_{4}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=be_{4},\\\ e_{2}\vdash e_{2}=ce_{4},\\\ e_{3}\vdash e_{2}=de_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{4}=e_{4},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{1})=ae_{1}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
## 5 Derivation of BiHom-associative dialgebras
In this section, we introduce and study derivations of BiHom-dendrifom, BiHom-
dialgebras.
###### Definition 5.1.
Let $(A,\mu,\alpha,\beta)$ be a BiHom-associative algebra. A linear map
${D}:A\longrightarrow A$ is called an $(\alpha^{s},\beta^{r})$-derivation of
$(A,\mu,\alpha,\beta)$, if it satisfies
${D}\circ\alpha=\alpha\circ{D}\quad\text{and}\quad{D}\circ\beta=\beta\circ{D}$
${D}\circ\mu(x,y)=\mu({D}(x),\alpha^{s}\beta^{r}(y))+\mu(\alpha^{s}\beta^{r}(x),{D}(y))$
###### Example 5.2.
We consider the $2$-dimensional BiHom-associative with a basis
$\left\\{e_{1},e_{2}\right\\}$. For
$\mu(e_{1},e_{1})=-e_{1},\quad\mu(e_{1},e_{2})=-e_{2},\quad\mu(e_{2},e_{1})=0,\quad\mu(e_{2},e_{2})=e_{2}$
and
$\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=-e_{2},\quad\beta(e_{1})=e_{1},\quad\beta(e_{2})=e_{2}.$
A direct computation gives that :
${D}(e_{1})=d_{22}e_{1},\quad{D}(e_{2})=d_{22}e_{2},$
$\alpha^{s}(e_{1})=\frac{\alpha_{21}\beta_{22}}{\beta_{21}}e_{1}+\frac{e_{2}}{2\beta_{21}},\quad\alpha^{s}(e_{2})=\alpha_{21}e_{1},\quad\beta^{r}(e_{1})=\frac{e_{2}}{2\beta_{21}}e_{2},\quad\beta^{r}(e_{2})=\beta_{21}e_{1}+\beta_{22}e_{2}$.
###### Definition 5.3.
Let $({D},\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra. A
linear map ${D}:{D}\rightarrow{D}$ is called an
$(\alpha^{k},\beta^{l})$-derivation of ${D}$ if it satisfies
1. $1.$
${D}\circ\alpha=\alpha\circ{D},\,{D}\circ\beta=\beta\circ{D}$;
2. $2.$
${D}(x\dashv
y)=\alpha^{k}\beta^{l}(x)\dashv{D}(y)+{D}(x)\dashv\alpha^{k}\beta^{l}(y);$
3. $3.$
${D}(x\vdash
y)=\alpha^{k}\beta^{l}(x)\vdash{D}(y)+{D}(x)\vdash\alpha^{k}\beta^{l}(y),$
for $x,y\in{D}.$
We denote by $Der({D}):=\displaystyle\bigoplus_{k\geq
0}\displaystyle\bigoplus_{l\geq 0}Der_{(\alpha^{k},\beta^{l})}({D})$, where
$Der_{(\alpha^{k},\beta^{l})}({D})$ is the set of all
$(\alpha^{k},\beta^{l})$-derivations of ${D}$.
###### Proposition 5.4.
For any ${D}\in Der_{(\alpha^{s},\beta^{r})}(A)$ and ${D}^{\prime}\in
Der_{(\alpha^{s^{\prime}},\beta^{r^{\prime}})}(A)$, we have
$\left[{D},{D^{\prime}}\right]\in
Der_{(\alpha^{s+s^{\prime}},\beta^{r+r^{\prime}})}(A)$.
###### Proof.
For $x,y\in A$, we have
$\begin{array}[]{ll}\left[{D},{D^{\prime}}\right]\circ\mu(x,y)&={D}\circ{D^{\prime}}\circ\mu(x,y)-{D^{\prime}}\circ{D}\circ\mu(x,y)\\\
&={D}(\mu({D}^{\prime}(x),\alpha^{s}\beta^{r}(y))+\mu(\alpha^{s}\beta^{r}(x),{D}^{\prime}(y)))\\\
&-{D}^{\prime}(\mu({D}(x),\alpha^{s}\beta^{r}(y))+\mu(\alpha^{s}\beta^{r}(x),{D}(y)))\\\
&=\mu({D}\circ{D^{\prime}}(x),\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(y))+\mu(\alpha^{s}\beta^{r}\circ{D^{\prime}}(x),{D}\circ\alpha^{s}\beta^{r}(y))\\\
&+\mu({D}\circ\alpha^{s}\beta^{r}(x),\alpha^{s}\beta^{r}\circ{D}^{\prime}(y))+\mu(\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(x),{D}\circ{D^{\prime}}(y))\\\
&-\mu({D^{\prime}}\circ{D}(x),\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(y))-\mu(\alpha^{s}\beta^{r}\circ{D}(x),{D^{\prime}}\circ\alpha^{s}\beta^{r}(y))\\\
&-\mu({D^{\prime}}\circ\alpha^{s}\beta^{r}(x),\alpha^{s}\beta^{r}{D}(y))-\mu(\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(x),{D^{\prime}}\circ{D}(y)).\end{array}$
Since ${D}$ and ${D}^{\prime}$ satisfy
${D}\circ\alpha=\alpha\circ{D},\,{D}^{\prime}\circ\alpha=\alpha\circ{D}^{\prime}$,
${D}\circ\beta=\beta\circ{D},\,{D}^{\prime}\circ\beta=\beta\circ{D}^{\prime}$.
We obtain
$\alpha^{s}\beta^{r}\circ{D^{\prime}}={D^{\prime}}\circ\alpha^{s}\beta^{r},\,{D}\circ\alpha^{s^{\prime}}\beta^{r^{\prime}}=\alpha^{s^{\prime}}\beta^{r^{\prime}}\circ{D}.$
Therefore, we arrive at
$\left[{D},{D^{\prime}}\right]\circ\mu(x,y)=\mu(\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(x),\left[{D},{D^{\prime}}\right](y))+\mu(\left[{D},{D^{\prime}}\right](x),\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(y)).$
Furthermore, it is straightforward to see that
$\begin{array}[]{ll}\left[{D},{D^{\prime}}\right]\circ\alpha&={D}\circ{D^{\prime}}\circ\alpha-{D^{\prime}}\circ{D}\circ\alpha\\\
&=\alpha\circ{D}\circ{D}^{\prime}-\alpha\circ{D}^{\prime}\circ{D}=\alpha\circ\left[{D},{D^{\prime}}\right].\end{array}$
$\begin{array}[]{ll}\left[{D},{D^{\prime}}\right]\circ\beta&={D}\circ{D^{\prime}}\circ\beta-{D^{\prime}}\circ{D}\circ\beta\\\
&=\beta\circ{D}\circ{D}^{\prime}-\beta\circ{D}^{\prime}\circ{D}=\beta\circ\left[{D},{D^{\prime}}\right]\end{array}$
which yields that $\left[{D},{D^{\prime}}\right]\in
Der_{(\alpha^{s+s^{\prime}},\beta^{r+r^{\prime}})}(A)$ with
$\mu=\dashv=\vdash.$ ∎
###### Proposition 5.5.
The space $Der_{(\alpha^{s},\beta^{r})}(A)$ is an invariant of the triple
BiHom-associative algebra A.
###### Proof.
Let
$\sigma:(A,\dashv_{A},\vdash_{A},\alpha^{s},\beta^{r})\longrightarrow(B,\dashv_{B},\vdash_{B},\alpha^{s},\beta^{r})$
be a triple BiHom-associative algebra isomorphism and let ${D}$ be a
$(\alpha^{s},\beta^{r})$-derivation of A. Then for any $x,y,z\in B$. We have :
$\begin{array}[]{ll}\sigma{D}\sigma^{-1}\circ(((x)\dashv_{B}(y))\dashv_{B}(z))&=\sigma{D}\circ((\sigma^{-1}(x)\dashv_{A}\sigma^{-1}(y))\dashv_{A}\sigma^{-1}(z))\\\
&=\sigma({D}\circ\sigma^{-1}(x)\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(y))\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(z))\\\
&+\sigma(\sigma^{-1}\circ\alpha^{s}\beta^{r}(x)\vdash_{A}{D}\circ\sigma^{-1}(y))\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(z))\\\
&+\sigma(\sigma^{-1}\circ\alpha^{s}\beta^{r}(x)\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(y)\vdash_{A}{D}\circ\sigma^{-1}(z))\\\
&=({D}\circ\sigma^{-1}(x)\dashv_{B}\alpha^{s}\beta^{r}(y))\dashv_{B}\alpha^{s}\beta^{r}(z))\\\
&+(\alpha^{s}\beta^{r}(x)\dashv_{B}\sigma\circ{D}\circ\sigma^{-1}(y))\dashv_{B}\alpha^{s}\beta^{r}(z))\\\
&+(\alpha^{s}\beta^{r}(x)\dashv_{B}\alpha^{s}\beta^{r}(y))\dashv_{B}{D}\circ\sigma^{-1}(z)).\end{array}$
Thus $\sigma\circ{D}\circ\sigma^{-1}$ is a $(\alpha^{s},\beta^{r})$-derivation
of $B$, hence the mapping.
$\psi:Der_{(\alpha^{s},\beta^{r})}(A)\longrightarrow
Der_{(\alpha^{s},\beta^{r})}(B)$,
${D}\longmapsto\sigma{D}\sigma^{-1}$ is an isomorphism of triple BiHom-
associative algebras.
In fact, it is easy to see that $\psi$ is linear. Moreover let
${D}_{1},{D}_{2},{D}_{3}$ be derivations of A :
$\begin{array}[]{ll}&\alpha^{s}\beta^{r}\circ\psi({D}_{1}\dashv_{tr}{D}_{2})\dashv_{tr}{D}_{3})=\\\
&=\alpha^{s}\beta^{r}\psi(tr({D}_{1})({D}_{2}\dashv{D}_{3}))+\alpha^{s}\beta^{r}\psi(tr({D}_{3})({D}_{1}{D}_{2}))+\alpha^{s}\beta^{r}\psi(tr({D}_{2})({D}_{3}\dashv{D}_{1}))\\\
&=\alpha^{s}\beta^{r}tr({D}_{1})\psi({D}_{2}\dashv{D}_{3})+\alpha^{s}\beta^{r}tr({D}_{3})\psi({D}_{1}\dashv{D}_{2})+\alpha^{s}\beta^{r}tr({D}_{2})\psi({D}_{3}\dashv{D}_{1})\\\
&=\alpha^{s}\beta^{r}tr(\psi({D}_{1}))(\psi({D}_{2})\dashv\psi({D}_{3}))+\alpha^{s}\beta^{r}tr(\psi({D}_{3}))(\psi({D}_{1})\dashv\psi({D}_{2}))\\\
&+\alpha^{s}\beta^{r}tr(\psi({D}_{2}))\psi((\psi({D}_{3})\dashv\psi({D}_{1})),\end{array}$
since $\psi$ is a morphism of the $Der_{(\alpha^{s},\beta^{r})}(A)$ and
$Der_{(\alpha^{s},\beta^{r})}(B)$, and
$tr({D})=tr(\sigma\circ{D}\circ\sigma^{-1}).$
Then
$\alpha^{s}\beta^{r}\psi(({D}_{1}\dashv_{tr}{D}_{2})\dashv_{tr}{D}_{3}))=\alpha^{s}\beta^{r}((\psi({D}_{1})\dashv_{tr}\psi({D}_{2}))\dashv_{tr}\psi({D}_{3})).$
∎
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# Aberrations in (3+1)D Bragg diffraction using pulsed Gaussian laser beams
A. Neumann<EMAIL_ADDRESS>Technical University of Darmstadt,
Institute of Applied Physics, Germany M. Gebbe ZARM, Universität Bremen,
Germany R. Walser Technical University of Darmstadt, Institute of Applied
Physics, Germany
(August 27, 2024)
###### Abstract
We analyze the transfer function of a three-dimensional atomic Bragg
beamsplitter formed by two counterpropagating pulsed Gaussian laser beams.
Even for ultracold atomic ensembles, the transfer efficiency depends
significantly on the residual velocity of the particles as well as on losses
into higher diffraction orders. Additional aberrations are caused by the
spatial intensity variation and wavefront curvature of the Gaussian beam
envelope, studied with (3+1)D numerical simulations. The temporal pulse shape
also affects the transfer efficiency significantly. Thus, we consider the
practically important rectangular-, Gaussian-, Blackman- and hyperbolic secant
pulses. For the latter, we can describe the time-dependent response
analytically with the Demkov-Kunike method. The experimentally observed
stretching of the $\pi$-pulse time is explained from a renormalization of the
simple Pendellösung frequency. Finally, we compare the analytical predictions
for the velocity-dependent transfer function with effective (1+1)D numerical
simulations for pulsed Gaussian beams, as well as experimental data and find
very good agreement, considering a mixture of Bose-Einstein condensate and
thermal cloud.
Bragg diffraction, atomic beamsplitter, atom optics, atom interferometer,
Bose-Einstein condensation
## I Introduction
Atoms represent the ultimate “abrasion free” quantum sensors for electro-
magnetic fields and gravitational forces. By a feat of nature, they occur with
bosonic or fermionic attributes, but are produced otherwise identically
without “manufacturing tolerance”. A beamsplitter based on Bragg diffraction
[1, 2, 3, 4] prepares superpositions of matter wavepackets by transferring
photon momentum from a laser to an atomic wave. Controlling the diffracted
populations, one can realize a beamsplitter and a mirror. These devices are
the central component of a matter-wave interferometer [1, 2, 3, 4, 5]. Due to
the well-defined properties of the atomic test masses and their precise
control by laser light, matter-wave interferometry can be used for high-
precision measurements of rotation and acceleration. Applications range from
tests of fundamental physics, like the equivalence principle [6, 7, 8, 9, 10,
11, 12, 13] or quantum electrodynamics [14, 15, 16], to inertial sensing [17,
18, 19, 20, 21]. Like all imaging systems, atom optics suffer from
imperfections and an accurate characterization is required in order to rectify
them. This is relevant for high-precision experiments, for instance gravimetry
[17, 22, 23] and extended free-fall experiments in large fountains, micro-
gravity and space [24, 25, 26, 27, 28, 29, 30]. Such challenging experiments
require realistic modeling and aberration studies, ideally hinting towards
rectification.
For ultra-sensitive atom interferometry a large and precise momentum transfer
is essential [31, 32, 33, 34]. Bragg scattering of atoms from a moving
standing light wave [35, 36, 37, 38], potentially in a retroreflective
geometry [39, 40], provides an efficient transfer of photon momentum without
changing the atomic internal state. In contrast, Raman scattering [41, 2]
couples different atomic internal states, enabling velocity filtering [42,
43]. While Raman pulses have lower demands on the atomic momentum distribution
[40, 44], Bragg pulses can be used for higher-order diffraction, also in
combination with Bloch oscillations [45, 46, 47, 48, 49, 32, 16, 34, 50].
The quasi-Bragg regime of atomic diffraction with smooth temporal pulse shapes
is optimal [51, 48, 31, 52, 32, 53, 54, 33, 16, 34]. It provides a high
diffraction efficiency with moderate velocity selectivity for relevant pulse
duration. However, losses into higher diffraction orders and the velocity
dispersion must be considered because atomic clouds do have a finite momentum
width.
The limit of the deep-Bragg regime with long interaction times and shallow
optical potentials gives a perfect on-resonance diffraction efficiency but
remains very narrow in momentum [2]. However, it is suitable to generate
velocity filters [26, 32]. In the opposite Raman-Nath limit short laser pulses
provide a vanishing velocity dispersion but the diffraction efficiency is very
low [2]. Despite their restrictions, both limits are popular as simple
analytical solutions can be given for rectangular pulse shapes and plane-wave
laser beams.
For smooth temporal envelopes there exist models based on adiabatic
elimination of the off-resonant coupled diffraction orders, solving the
effective two-level dynamics [51] and considering the velocity dispersion [55,
39]. The Bloch-band picture is suitable in the quasi-Bragg regime for
sufficiently slow (adiabatic) pulses [56]. An analytic theory for smooth
pulses based on the adiabatic theorem for single quasi-Bragg pulses is given
in [57]. Here, Doppler shifts are considered in terms of perturbation theory
to take finite atomic momentum widths into account.
Besides temporal envelopes, spatial envelopes also affect the beamsplitter
efficiency [58, 22, 26, 16], especially for large momentum transfer
interferometers. In particular, spatial variations due to three-dimensional
Gauss-Laguerre beams lead to aberrations.
FIG. 1: Bragg diffraction: energy diagram versus atomic wavenumber $k=p/\hbar$
in units of $k_{L}$ (2) in the lab frame $S$ (a) and an inertial frame
$S^{\prime}$ (b) moving with velocity $v_{g}$ (12). Ground- and excited state
eigen-frequencies of a free particle are $\omega_{g}(k)$, $\omega_{e}(k)$, the
two-photon and one-photon recoil frequencies $\omega_{2r}$ and $\omega_{r}$,
respectively. In frame $S$, we show that a deliberate detuning $\delta\omega$
(6) of the laser frequencies $\omega_{1},\omega_{2}$ leads to the same
fr6equency gap $\delta$ (7) (dashed-dotted arrows), as caused by a finite
initial momentum $p_{i}=\kappa\hbar k_{L}$ (5) (dotted arrows). In frame
$S^{\prime}$, the counterpropagating lasers have equal frequencies
$\omega_{1,2}^{\prime}=\omega_{L}$ (2) and link $p_{i}^{\prime}=-\hbar k_{L}$
with $p_{f}^{\prime}=\hbar k_{L}$. The velocity selectivity of Bragg
scattering leads to an incomplete transfer in the momentum ensembles (red,
shadowed). Odd momenta $\pm 3k_{L},\pm 5k_{L},\ldots$ are populated by higher
order diffraction.
In this article, we will revisit atomic beamsplitters in a moving frame in
Sec. II. We compare two common methods to solve the Schrödinger equation with
plane-wave laser beams in Sec. III. This is the Bloch-wave ansatz and an ad-
hoc ansatz, which leads to a more convenient extended zone scheme. In Sec. IV,
we study aberrations due to velocity selectivity, higher diffraction orders,
spatial variations of the beam intensities, wavefront curvatures and the
influence of four non-adiabatic temporal pulse envelopes in terms of the
complex transfer function and the fidelity. We introduce an explicitly
solvable Demkov-Kunike type model, which applies to hyperbolic sech pulses.
With the full (3+1)D simulations the effects of spatially Gauss-Laguerre laser
beams are studied. Finally, we gauge simulations and explicit models to
experimental data in Sec. V.
## II Matter-wave Bragg beamsplitter
### II.1 Conservation laws
The basic mechanism of an atomic beamsplitter is the stimulated absorption and
emission of two photons from bichromatic, counterpropagating laser beams [59,
1]. This process is depicted in Fig. 1a and satisfies energy and momentum
conservation
$\frac{p_{i}^{2}}{2M}+\hbar\omega_{1}=\frac{p_{f}^{2}}{2M}+\hbar\omega_{2},\qquad
p_{i}+\hbar k_{1}=p_{f}-\hbar k_{2}.$ (1)
Here, $p_{i,f}$ are the initial and final momenta of the particle with mass
$M$, $\pm\hbar k_{1,2}$ are photon momenta and $\omega_{1,2}$ are the laser
frequencies. We choose to work with positive wavenumbers $k_{1,2}>0$ and
emphasize the propagation directions with explicit signs, but retain the
directionality of $p_{i,f}$. Frequency and wavenumber are coupled by the
vacuum dispersion relation $\omega=ck$, with the speed of light $c$. One
chooses counterpropagating beams to maximize the momentum transfer
$p_{f}-p_{i}=2\hbar k_{L}$, introducing the average wavenumber and frequency
$k_{L}\equiv\frac{k_{1}+k_{2}}{2},\qquad\omega_{L}\equiv ck_{L}.$ (2)
Wave mechanics considers superpositions of momentum states $\ket{g,p_{i}}$ and
$\ket{g,p_{f}}$ in the internal atomic ground state $g$. For atoms initially
at rest $p_{i}=0$, energy and momentum conservation (1) requires laser
frequencies
$\omega_{1}=\omega_{2}+\omega_{2r}\approx\omega_{2}+\frac{\hbar(2k_{2})^{2}}{2M}.$
(3)
Due to the two-photon recoil, we need to introduce
$\omega_{2r}\equiv\frac{\hbar(2k_{L})^{2}}{2M}=4\omega_{r},$ (4)
as the two-photon frequency $\omega_{2r}$ in terms of the single photon
frequency $\omega_{r}$. The approximation (3) holds for non-relativistic
energies, just as the kinetic energy in (1).
### II.2 Off-resonant response
Releasing ultracold atomic ensembles from traps provides localized wavepackets
with a finite momentum dispersion. Therefore, one needs to study the response
of the Bragg beamsplitter with finite initial- and final momenta
$\bar{p}_{i}=\kappa\hbar k_{L}$, $\bar{p}_{f}=(2+\kappa)\hbar k_{L}$,
introducing a dimensionless momentum $\kappa$. This opens a frequency gap
$\mbox{$\delta$}\equiv\frac{\bar{p}_{f}^{2}}{2M\hbar}+\omega_{2}-\frac{\bar{p}_{i}^{2}}{2M\hbar}-\omega_{1}=\omega_{2r}\kappa,$
(5)
shown in Fig. 1 (a).
Alternatively, one can also probe the momentum response by a detuning of the
laser frequencies $\tilde{\omega}_{1,2}$ from the resonant values
$\omega_{1,2}$ in (3). Conveniently, this detuning is measured by
$\mbox{$\delta\omega$}\equiv\omega_{1}-\omega_{2}+\tilde{\omega}_{2}-\tilde{\omega}_{1}.$
(6)
Dash-dotted arrows mark the deviant frequencies in Fig. 1 (a). For a particle,
which is initially at rest $\tilde{p}_{i}=0$ and acquires a momentum
$\tilde{p}_{f}=\hbar(\tilde{k}_{1}+\tilde{k}_{2})$ after the momentum
transfer, one obtains a frequency gap
$\mbox{$\delta$}=\frac{\tilde{p}_{f}^{2}}{2M\hbar}+\tilde{\omega}_{2}-\tilde{\omega}_{1}\approx\mbox{$\delta\omega$}.$
(7)
The approximation holds for
$|\tilde{\omega}_{1,2}-\omega_{1,2}|\ll\omega_{L}$, which is satisfied very
well in the present context. Comparing Eqs. (5) and (7), one finds a linear
relation
$\mbox{$\delta\omega$}=\omega_{2r}\kappa,$ (8)
between laser-frequency mismatch $\delta\omega$ and the dimensionless initial
particle momentum $\kappa$. Therefore, both realizations are suitable to probe
the momentum response of Bragg diffraction and their results are related by
Eq. (8).
Experimentally, it is advantageous to modify the laser-frequencies (cf. Sec.
V) and to prepare atomic wavepackets initially at rest in the lab-frame $S$.
Theoretically, it is beneficial to emphasize the symmetries of the system.
Therefore, we will adopt a moving inertial frame $S^{\prime}$, wherein the
Doppler-shifted laser-frequencies coincide and the momentum coupled states
$p_{i}^{\prime}=-\hbar k_{L}$, $p_{f}^{\prime}=+\hbar k_{L}$ are distributed
symmetrically (cf. Sec. II.3, App. A). This is depicted in Fig. 1b.
### II.3 Counterpropagating, bichromatic fields
The superposition of two counterpropagating laser beams
$\bm{E}=\bm{E}_{1}+\bm{E}_{2}$, is defined by the constituent fields
$\bm{E}_{i}=\real[\bm{E}_{i}^{(+)}]$ with the positive frequency components
$\bm{E}_{i}^{(+)}(t,\bm{r})=\bm{\epsilon}_{i}e^{-i\phi_{i}(t,x)}\mathcal{E}_{i}(t,\bm{r}).$
(9)
Here, $\bm{\epsilon}_{i}$ denote the polarization vectors,
$\mathcal{E}_{i}(t,\bm{r})$ the slowly varying complex Gaussian envelopes and
$\phi_{1}(t,x)=\omega_{1}t-k_{1}x$, $\phi_{2}(t,x)=\omega_{2}t+k_{2}x$ are the
rapidly oscillating carrier phases for fields propagating along the
x-direction [60] (cf. App. A and B). From the superposition of two scalar
counterpropagating bichromatic fields
$\mathcal{E}=e^{-i\phi_{1}(t,x)}\mathcal{E}_{1}+e^{-i\phi_{2}(t,x)}\mathcal{E}_{2},$
(10)
one obtains a steady motion of the intensity pattern
$|\mathcal{E}|^{2}=|\mathcal{E}_{1}|^{2}+|\mathcal{E}_{2}|^{2}+2\real\left[\mathcal{E}_{2}^{\ast}\mathcal{E}_{1}^{\phantom{*}}e^{i(k_{1}+k_{2})(x-v_{g}t)}\right],$
(11)
where nodes move with the group velocity
$v_{g}=\frac{\omega_{1}-\omega_{2}}{\omega_{1}+\omega_{2}}c,\qquad|v_{g}|=\frac{\omega_{2r}}{2\omega_{L}}c\ll
c.$ (12)
If the lab frame $S$ has the coordinates $x$, then the moving interference
pattern defines another inertial frame $S^{\prime}$, where the grating is at
rest and the coordinates
$x^{\prime}=x-v_{g}t,$ (13)
are related to the lab frame coordinates $x$ by a passive Galilean
transformation.
### II.4 Interaction energy
The atom is represented by a ground $\ket{g}$ and an excited state $\ket{e}$.
These levels are separated by the transition frequency
$\omega_{0}=\omega_{e}-\omega_{g}$ and coupled by the electric dipole matrix
element $\bm{d}_{eg}=\bra{e}\hat{\bm{d}}\ket{g}$. To neglect spontaneous
emissions, the lasers are far-detuned from the atomic resonance frequencies
$|\omega_{0}-\omega_{i}|\gg\Gamma$, where $\Gamma$ is the natural linewidth of
the transition. In the lab frame $S$ the Hamilton operator of an atom with
mass $M$ reads
$\displaystyle\hat{H}(t)=$
$\displaystyle\frac{\hat{\bm{p}}^{2}}{2M}+\hbar\omega_{g}\hat{\sigma}_{g}+\hbar\omega_{e}\hat{\sigma}_{e}+V(t,\hat{\bm{r}}),$
(14) $\displaystyle V(t,\bm{r})=$
$\displaystyle\frac{\hbar}{2}\hat{\sigma}^{\dagger}\sum_{i=1}^{2}\Omega_{i}(t,\bm{r})e^{-i\phi_{i}(t,x)}+\text{h.c.},$
using the spin operators $\hat{\sigma}_{i=e,g}=\outerproduct{i}{i}$ and
$\hat{\sigma}=\outerproduct{g}{e}$. Here, we evaluate the electric dipole
interaction energy in the rotating-wave approximation and denote the Rabi
frequencies as
$\Omega_{i}(t,\bm{r})=-\bm{\varepsilon}_{i}\dotproduct\bm{d}_{ge}\,\mathcal{E}_{i}(t,\bm{r})/\hbar$
.
If we transform this Hamilton operator to the frame $S^{\prime}$, comoving
with the nodes of the interference pattern (13), and use a corotating internal
frame (96), it reads
$\begin{split}\hat{H}^{\prime\prime}(t)=&\frac{\hat{\bm{p}}^{2}}{2M}-\hbar\Delta\hat{\sigma}_{e}+\frac{\hbar}{2}\hat{\sigma}^{\dagger}\left[\tilde{\Omega}_{1}(t,\hat{\bm{r}})e^{ik_{L}\hat{x}}\right.\\\
&+\left.\tilde{\Omega}_{2}(t,\hat{\bm{r}})e^{-ik_{L}\hat{x}}\right]+\text{h.c.}\end{split}$
(15)
In this specific frame the atom responds only to a carrier wavenumber $k_{L}$.
We measure the laser detuning $\Delta\equiv\omega_{L}-\omega_{0}$ with respect
to the common Doppler-shifted frequency $\omega_{L}$. The Rabi frequencies
$\tilde{\Omega}_{i}\\!\left(\\!t,\bm{r}\\!\right)$ are given by the pulsed
Gauss-Laguerre beams of Eq. (107).
Dissipative processes are not an issue for large detunings, why we can resort
to the solution of the Schrödinger equation for $t>t_{i}$ and
$\ket{\psi}\equiv\ket{\psi^{\prime\prime}}$
$\ket{\psi(t)}=G(t,t_{i})\ket{\psi(t_{i})},$ (16)
with the propagator $G(t,t_{i})$ (125).
For the numerical solution of this two-component, (3+1)D problem, we use
Fourier methods with symplectic integrators [61] and operator disentangling
[62]. Analytical solutions are examined for rectangular pulses (Sec. IV.3) and
the hyperbolic secant pulse (Sec. IV.4).
### II.5 Ideal Bragg beamsplitter and mirror
The interaction of a two-state system with laser pulses can be understood
qualitatively by the “pulse area” [63]
$\theta(t)=\int_{-\infty}^{t}\text{d}t^{\prime}\,\Omega(t^{\prime}),$ (17)
which is rather a phase by dimension. In the context of ideal Bragg
scattering, the two states are the momentum states
$\\{\ket{-k_{L}}_{x},\ket{k_{L}}_{x}\\}$. One can visualize the evolution
during the action of the Bragg pulse as a motion on the Bloch sphere [64]. A
symmetrical 50:50 Bragg beamsplitter corresponds to a $\theta=\pi/2$ rotation
from the south pole to the equator at some longitude. This gives equal
probability to the outputs channels $\ket{\pm k_{L}}$. A $\theta=\pi$ rotation
from the south pole to the north pole reverses the momenta
$\ket{-k_{L}}\rightarrow\ket{k_{L}}$ and thus acts like a mirror. In the
following discussion, we will focus on the mirror configuration as it is most
susceptible to aberrations, due to the longer interaction time.
The polar decomposition of the transition amplitude
$\bra{\bm{k}^{\prime}}G(t,t_{i})\ket{\bm{k}}=\sqrt{\eta_{k^{\prime}k}}e^{i\phi_{k^{\prime}k}}$
(18)
between initial $\ket{\bm{k}}$ and final $\ket{\bm{k}^{\prime}}$ momentum
states characterizes the diffraction efficiency $0\leq\eta_{k^{\prime}k}\leq
1$. For atomic wavepackets, we use the phase sensitive fidelity
$F=|\langle\psi_{\text{ideal}}|\psi(t_{f})\rangle|^{2},\qquad\ket{\psi_{\text{ideal}}}=e^{2i\bm{k}_{L}\hat{\bm{x}}}|\psi_{i}\rangle,$
(19)
characterizing the overlap of the final state $\ket{\psi(t_{f})}$ of Eq. (16)
and the ideal final state $\ket{\psi_{\text{ideal}}}$. For an initial plane
wave, the fidelity is $F=\eta_{k^{\prime}k}$ with
$\bm{k}^{\prime}=\bm{k}+2\bm{k}_{L}$.
### II.6 Sources of aberrations
The velocity dispersion of Bragg diffraction [55] is significant and leads to
incomplete population transfer atomic wavepackets (cf. Fig. 1, Sec. IV.3.1).
Another cause for population loss is off-resonant coupling to higher
diffraction orders (cf. Sec. IV.3.2). This signals the crossover from the
deep-Bragg towards the Raman-Nath regime, referred to as quasi-Bragg regime
[51].
In general, smooth time-dependent laser pulses (cf. Sec. IV.1) lead to equally
smooth beamsplitter responses (cf. Sec. IV.4 and IV.6). In contrast, smooth
spatial envelopes lead to aberrations (cf. Sec. IV.7). Every Gauss-Laguerre
beam exhibits spatial inhomogeneity and wavefront curvature. This is relevant
for atomic clouds that are comparable in size to the laser beam waist, or for
clouds displaced from the symmetry axis. Static laser misalignment further
degrades the diffraction efficiency.
There are sundry other dynamical sources of aberrations, such as mechanical
vibrations of optical elements or stochastic laser noise [65]. The fundamental
process of spontaneous emission leads to decoherence and aberrations, too.
Fortunately, this can be suppressed by a detuning $|\Delta|\gg\Gamma$ much
larger than the linewidth $\Gamma$, as well as limiting the interaction time.
## III Plane-wave approximation
The basic mechanism of Bragg beamsplitters arises from the momentum transfer
of plane waves with a real, constant Rabi frequency
$\tilde{\Omega}_{1}(t,\bm{r})=\tilde{\Omega}_{2}(t,\bm{r})=\Omega_{0}$ within
the duration of a rectangular pulse. This model is the reference to gauge more
realistic calculations. Consequently, the two components
$\\{\psi_{e}(t,\bm{r}),\psi_{g}(t,\bm{r})\\}$ of the Schrödinger field evolve
according to
$\displaystyle
i\partial_{t}\psi_{e}=\left(-\frac{\hbar}{2M}\nabla^{2}-\Delta\right)\psi_{e}+\Omega_{0}\cos(k_{L}x)\psi_{g},$
(20a) $\displaystyle
i\partial_{t}\psi_{g}=-\frac{\hbar}{2M}\nabla^{2}\psi_{g}+\Omega_{0}^{\ast}\cos(k_{L}x)\psi_{e}.$
(20b)
using the Hamilton operator (15). Assuming the excited state is initially
empty, the atom’s kinetic energy is small and the lasers are far-detuned
$|\Delta|\gg\Gamma,\ \Omega_{0},\ \omega_{r}$, we can adiabatically eliminate
the excited state [66, 51]
$\psi_{e}\approx\frac{\Omega_{0}}{\Delta}\cos(k_{L}x)\psi_{g}.$ (21)
Then, the ground state Schrödinger equation reads
$i\partial_{t}\psi_{g}=\left(-\frac{\hbar}{2M}\bm{\nabla}^{2}+\mathcal{V}(x)\right)\psi_{g},$
(22)
with the dipole potential
$\mathcal{V}(x)=\cos^{2}\\!\left(\\!k_{L}x\\!\right)|\Omega_{0}|^{2}/\Delta$
[67]. Stationary solutions of the one-dimensional problem are Mathieu
functions [68]. Our goal is to formulate a suitable ansatz for the (3+1)
dimensional non-separable equation with time-dependent pulses.
### III.1 Bloch-wave ansatz
The Bloch picture is suitable for describing the velocity selective atomic
diffraction by a standing laser wave [1, 69, 70]. The characteristic
translation invariance of the Hamilton operator (22) by a displacement of
$a_{x}=\lambda_{L}/2$ defines a natural length scale. Its reciprocal is the
lattice vector $\mathfrak{q}_{x}=2\pi/a_{x}=2k_{L}$. It is convenient to embed
the total three-dimensional wavefunction in an orthorohmbic volume with
lengths $(N_{x}a_{x},a_{y},a_{z})$, with $N_{x}\in\mathbb{N}$ and to impose
periodic boundary conditions
$\psi_{g}(x+N_{x}a_{x},y+a_{y},z+a_{z})=\psi_{g}(x,y,z)$. Bragg scattering
involves at least two photons, one from each of the counterpropagating lasers.
Therefore, the two-photon recoil frequency $\omega_{2r}$ (4) emerges as the
frequency scale. In terms of the dimensionless length $\xi=\mathfrak{q}_{x}x$
and time $\tau=\omega_{2r}t$, the Schrödinger field
$\psi_{g}(t,\bm{r})=\sum_{r=-\left\lfloor\frac{N_{y}}{2}\right\rfloor}^{\left\lceil\frac{N_{y}}{2}\right\rceil-1}\sum_{s=-\left\lfloor\frac{N_{z}}{2}\right\rfloor}^{\left\lceil\frac{N_{z}}{2}\right\rceil-1}e^{i(r\mathfrak{q}_{y}y+s\mathfrak{q}_{z}z-\bar{\omega}_{r,s}\tau)}h^{(r,s)}(\tau,\xi),$
(23)
factorizes into one-dimensional fields $h^{(r,s)}(\tau,\xi)$ and two-
dimensional plane waves with the transversal lattice vectors
$\mathfrak{q}_{y,z}=2\pi/a_{y,z}$. The integers $N_{y,z}\in\mathbb{N}$ define
the maximal momentum resolution
$\mathfrak{q}_{i}^{\text{max}}=\mathfrak{q}_{i}\left\lfloor
N_{i}/2\right\rfloor$. Please note the use of the Gauss brackets rounding
towards the nearest integer at the “floor” $\left\lfloor\,\right\rfloor$ or
the “ceiling” $\left\lceil\,\right\rceil$. With a detuning dependent shift of
the frequency, introducing the two-photon Rabi frequency $\Omega$,
$\displaystyle\bar{\omega}_{r,s}$
$\displaystyle=\hbar\frac{r^{2}\mathfrak{q}_{y}^{2}+s^{2}\mathfrak{q}_{z}^{2}}{2M\omega_{2r}}+\Omega,$
$\displaystyle\Omega$
$\displaystyle=\frac{\Omega_{r}}{\omega_{2r}}=\frac{|\Omega_{0}|^{2}}{2\omega_{2r}\Delta},$
(24)
the Schrödinger equation for each amplitude simplifies to
$i\partial_{\tau}h(\tau,\xi)=\left(-\partial^{2}_{\xi}+\Omega\cos{\xi}\right)h(\tau,\xi).$
(25)
By construction, the potential is $2\pi$-periodic and the eigenfunctions
$h(\tau,\xi)=e^{-i\tau\omega^{(b)}(q)}h^{(b)}(\xi,q)$ are given by Bloch-waves
$h^{(b)}(\xi,q)$ [71, 72, 73, 74] with the lattice periodic function
$g^{(b)}(\xi,q)$ for momentum $q$ and band index $b$
$\displaystyle h^{(b)}(\xi,q)$ $\displaystyle=e^{iq\xi}g^{(b)}(\xi,q),$ (26)
$\displaystyle g^{(b)}(\xi+2\pi,q)$ $\displaystyle=g^{(b)}(\xi,q).$ (27)
From the periodic boundary conditions for the wavefunction $h^{(b)}(\xi+2\pi
N_{x},q)=h^{(b)}(\xi,q)$, one obtains a quantization of the wavenumber
$q_{n}=n/N_{x}$ with $n\in\mathbb{Z}$. The interval $-1/2\leq q_{n}<1/2$
defines the first Brillouin zone in the reduced zone scheme, whose extent
equals the _crystal momentum_ $Q=1$.
Bloch wavefunctions are also periodic in momentum space
$h^{(b)}(\xi,q+Q)=h^{(b)}(\xi,q)$, provided we define
$g^{(b)}(\xi,q)=\sum_{m=-\mathcal{N}}^{\mathcal{N}-1}e^{im\xi}g^{(b)}(m+q),$
(28)
by a Fourier series for a maximal diffraction order $\mathcal{N}\in\mathbb{N}$
with boundary
condition$g^{(b)}\left(q+\mathcal{N}\right)=g^{(b)}\left(q-\mathcal{N}\right)=0$.
From a superposition of these Bloch waves, one obtains the ansatz
$h(\tau,\xi)=\sum_{n=-\left\lfloor\frac{N_{x}}{2}\right\rfloor}^{\left\lceil\frac{N_{x}}{2}\right\rceil-1}\sum_{m=-\mathcal{N}}^{\mathcal{N}-1}e^{i(m+q_{n})\xi}g(\tau,m+q_{n}),$
(29)
for the time-dependent solution of Eq. (25), compatible with the Bloch theorem
and suitable for numerical computation. This ansatz transforms the partial
differential equation into the parametric difference equation
$i\partial_{\tau}g_{m}(\tau,q)=(m+q)^{2}g_{m}+\tfrac{\Omega}{2}(g_{m+1}+g_{m-1}).$
(30)
The $q$-dependence of the $m^{th}$-order scattering amplitude
$g_{m}(\tau,q)\equiv g(\tau,m+q)$ leads to the velocity dispersion of Bragg
diffraction. Assuming Dirichlet boundary conditions, one can use a
$(2\mathcal{N}-1)$-dimensional representation
$\bm{g}^{e}=(g_{-(\mathcal{N}-1)},\ldots,g_{\mathcal{N}-1}),$ to study the
initial value problem
$i\dot{\bm{g}}^{e}=H^{e}(q)\bm{g}^{e},\qquad H^{e}=D^{e}+L+L^{\dagger}.$ (31)
For the indices $1-\mathcal{N}\leq m\leq\mathcal{N}-1$, the Hamilton matrix
$H^{e}$ is formed by a diagonal matrix $D^{e}$ and a lower triangular matrix
$L$
$D^{e}_{m,n}=(m+q)^{2}\delta_{m,n},\qquad
L_{m,n}=\tfrac{\Omega}{2}\delta_{m,n+1}.$ (32)
In order to study the discrete Bloch energy bands $\omega^{(b)}(q)$, one has
to solve the eigenvalue problem
$\bm{g}^{e}(\tau,q)=e^{-i\tau\omega(q)}\bm{g}^{e}(q),\qquad\omega(q)\bm{g}^{e}=H^{e}(q)\bm{g}^{e}.$
(33)
In Fig. 2, we present the lowest few energy bands $\omega^{(b)}(q)$ versus the
lattice momentum $q$ in an extended momentum zone scheme. For reference, we
depict the quadratic dispersion relation of the empty lattice $\Omega\\!=\\!0$
and the dispersion relation for $\Omega\\!=\\!1$
($\Omega_{r}\\!=\\!\Omega\,\omega_{2r}\\!=\\!4\,\omega_{r}$), a moderately
deep lattice. Narrow momentum wavepackets $\psi(k)$ with $\sigma_{k}\ll k_{L}$
are ideal for beamsplitters. If they are located at the band edges
$k\\!=\\!q\mathfrak{q}_{x}\\!=\\!(\pm 1/2+m)2k_{L}$, the two-photon process
covers at least three Brillouin zones. For wavepackets at the center
$k\\!=\\!q\mathfrak{q}_{x}\\!=\\!2mk_{L}$, only two Brillouin zones are
coupled by a Bragg pulse.
FIG. 2: Energy bands $\omega^{(0,1,2)}(q)$ of a periodic lattice in the
extended zone scheme versus quasi-momentum $q$, with empty lattice $\Omega=0$
(dotted) and finite depth $\Omega=1$ (solid), where
$\Omega_{r}\\!=\\!\Omega\,\omega_{2r}\\!=\\!4\,\omega_{r}$. Initial
wavepackets with odd momenta $(2m+1)k_{L}$ are located at the edges
$q\\!=\\!\pm 1/2$ of the $1^{st}$ Brillouin zone, while even momenta $2mk_{L}$
are at the center $q=0$.
### III.2 Ad-hoc ansatz
There are alternatives formulations [51, 55] to the Bloch-wave ansatz, if we
define a Fourier series on the periodic lattice $h(x+N_{x}a_{x})=h(x)$ as
$h(x)=\sum_{l=-\infty}^{\infty}e^{i\frac{2\pi
l}{N_{x}a_{x}}x}g_{l},\qquad\frac{2\pi l}{N_{x}a_{x}}=\frac{2l}{N_{x}}k_{L}.$
(34)
By decomposing the index $l=N_{x}m+r$ into a quotient $m=\left\lfloor
l/N_{x}\right\rfloor$ and a remainder $0\leq r<N_{x}$, one obtains
$h(x)=\sum_{n=-\left\lfloor\frac{N_{x}}{2}\right\rfloor}^{\left\lceil\frac{N_{x}}{2}\right\rceil-1}\sum_{m=-\mathcal{N}}^{\mathcal{N}-1}g_{2m+1}(\kappa_{n})e^{ik_{2m+1}^{n}x},$
(35)
with $n=r-\left\lfloor N_{x}/2\right\rfloor$. In this series, we use a
momentum $k_{\mu}^{n}=(\mu+\kappa_{n})k_{L}$ and a quasimomentum $\kappa_{n}$
$-1\leq\kappa_{n}=\frac{2n}{N_{x}}-\frac{\left\lceil\frac{N_{x}}{2}\right\rceil-\left\lfloor\frac{N_{x}}{2}\right\rfloor}{N_{x}}<1,$
(36)
in an extended Brillouin zone. As the Schrödinger equation (25) has even
parity, parity is a conserved quantity. An ansatz with $\sin$ and $\cos$
functions would lead to a decoupling of (35) with respect to parity manifolds.
The decomposition of the index $l=N_{x}m+n$ is not unique, if we admit signed
integral remainders within the limits $-\left\lfloor N_{x}/2\right\rfloor\leq
n<\left\lceil N_{x}/2\right\rceil$. This implies a quotient
$m=\left\lfloor(l+\left\lfloor N_{x}/2\right\rfloor)/N_{x}\right\rfloor$. Now,
the Fourier series reads
$h(x)=\sum_{n=-\left\lfloor\frac{N_{x}}{2}\right\rfloor}^{\left\lceil\frac{N_{x}}{2}\right\rceil-1}\sum_{m=-\mathcal{N}}^{\mathcal{N}-1}g_{2m}(\kappa_{n})e^{ik_{2m}^{n}x},$
(37)
with the quasimomentum $\kappa_{n}$
$-1\leq\kappa_{n}=\frac{2n}{N_{x}}\leq 1-\frac{1}{N_{x}}.$ (38)
The definitions of the quasimomenta in Eqs. (36) and (38), agree exactly for
even number $N_{x}=2u$ of lattice sites or coincide asymptotically for
$N_{x}\rightarrow\infty$. The even/odd ambiguity of number of lattice sites
can not be of physical significance as the periodic boundary condition are
mere mathematical convenience. Therefore, assuming an even number of lattice
sites is no limitation.
Using time-dependent amplitudes $g_{\mu}(\tau,\kappa_{n})$ in the series (35)
and (37), transforms the Schrödinger equation (25) into a single difference
equation $\forall\mu\in\mathbb{Z}$
$i\partial_{\tau}g_{\mu}(\tau,\kappa)=\tfrac{1}{4}(\mu+\kappa)^{2}g_{\mu}+\tfrac{\Omega}{2}(g_{\mu+2}+g_{\mu-2}).$
(39)
Due to the two-photon transfer, there is no coupling between even and odd
solution manifolds. Consequently, it is advantageous to use Eq. (35) for
wavepackets located around odd multiples of $k_{L}$ or Eq. (37) for even
multiples of $k_{L}$ (cf. Fig. 2). As in the comoving frame $S^{\prime}$ (13)
mainly $\ket{-k_{L}}$ is coupled with $\ket{+k_{L}}$, we focus on the odd
solution manifold with $\mu=2m+1$. Therefore, Eq. (39) can be cast into a
tridiagonal system of linear differential equations
$i\dot{\bm{g}}^{o}=H^{o}\bm{g}^{o},\qquad H^{o}=D^{o}+L+L^{\dagger},$ (40)
for $\bm{g}^{o}=(g_{-2\mathcal{N}+1},g_{-2\mathcal{N}+3},\ldots
g_{2\mathcal{N}-1})$ with $L$ from (32) and a diagonal matrix
$D^{o}_{\mu,\nu}=\tfrac{1}{4}(\mu+\kappa)^{2}\delta_{\mu,\nu}\equiv
D_{\mu,\nu}+\varpi\delta_{\mu,\nu}.$ (41)
In the following, it will be prudent to adopt a rotating frame
$\bm{g}^{o}(\tau)=e^{-i\varpi\tau}\bm{g}(\tau)$ with a frequency offset
denoted by $\varpi=(-1+\kappa)^{2}/4$
$\displaystyle
i\dot{\bm{g}}=\mathcal{H}\bm{g},\qquad\mathcal{H}=D+L+L^{\dagger},$ (42)
$\displaystyle
D_{\mu,\nu}=\omega_{\mu}\delta_{\mu,\nu},\qquad\omega_{\mu}=\tfrac{1}{4}(\mu+\kappa)^{2}-\varpi.$
(43)
This grounds the frequency $\omega_{-1}=0$.
## IV Aberration analysis
Using the ad-hoc ansatz for Bragg scattering, we will successively consider
more realistic processes to assess their contribution to aberrations. We begin
with the plane-wave approximation and consider four temporal Bragg-pulse
shapes $f_{i}(\tau)$. We will analyze their influence on the velocity
dispersion as well as losses into higher diffraction orders. Finally, we will
add the spatial envelopes of the Gaussian-Laguerre beams and consider the
cumulative effect.
### IV.1 Bragg-pulse shapes
We examine temporal Gaussian- (G), rectangular- (R), hyperbolic secant- (S)
and Blackman- (B) Rabi pulses
$\Omega(\tau)=\Omega f_{j}(\tau),\quad j\in\\{G,R,S,B\\}.$ (44)
The shape functions $f_{j}$, depicted in Fig. 3, are all normalized to unity
at maximum and characterized by a window width $\tau_{j}$. Different Rabi
pulses (44) can be compared physically, if they cover the same pulse area (17)
$\displaystyle\theta$ $\displaystyle\equiv\theta(\tau=\infty)=\Omega T,$ (45a)
$\displaystyle T$ $\displaystyle\equiv T(-\infty,\infty),\quad
T(\tau_{i},\tau_{f})=\int_{\tau_{i}}^{\tau_{f}}\text{d}\tau\,f_{j}(\tau),\vspace{-7mm}$
(45b)
for equal nominal time $T=T_{G}=T_{R}=T_{B}=T_{S}$.
Rectangular pulses are popular in theory as they are constant during the
interaction time and lead to simple analytical approximations. They read
$f_{R}(|\tau|\leq\tau_{R})=1,\qquad T_{R}=2\tau_{R}$ (46)
and $f_{R}(|\tau|>\tau_{R})=0$, elsewhere.
Gaussian pulses are the standard shapes in pulsed laser experiments
$f_{G}(\tau)=e^{-\frac{\tau^{2}}{2\tau^{2}_{G}}},\qquad
T_{G}=\sqrt{2\pi}\,\tau_{G},$ (47)
with Gaussian width $\tau_{G}$.
Blackman pulses are characterized by a window function
$\displaystyle f_{B}(\tau)$
$\displaystyle=w_{B}\negthinspace\left(\frac{\tau}{\tau_{B}}\right),\qquad
T_{B}=\frac{21\pi}{25}\tau_{B},$ (48) $\displaystyle w_{B}(|\phi|\leq\pi)$
$\displaystyle=\tfrac{1}{50}[21+25\cos(\phi)+4\cos(2\phi)]$ (49)
and $w_{B}(|\phi|>\pi)=0$ elsewhere.
Hyperbolic secant pulses are defined with
$f_{S}(\tau)=\sech\left(\frac{\tau}{\tau_{S}}\right),\qquad
T_{S}=\pi\tau_{S}.$ (50)
They are amenable for analytical solutions [75, 76].
FIG. 3: Temporal envelopes $f(\tau)$ for rectangular-, Gaussian-, hyperbolic
secant- and Blackman pulses for equal nominal time $T=T_{j}$,
$j\in\\{G,R,S,B\\}$ and total pulse length $\Delta\tau=8\,\tau_{G}$. The
vertical lines indicates the pulse widths $\tau_{j}$.
### IV.2 Definition of $\pi$\- and $\frac{\pi}{2}$-pulses
The symmetrical 50:50 beamsplitter pulse and the 0:100 mirror pulse are the
two most relevant applications of atomic Bragg diffraction (cf. Sec. II.5).
Irrespective of the shape, a symmetrical beamsplitter pulse is defined by a
pulse area of $\theta=\pi/2$, while a complete specular reflection in momentum
space is achieved for $\theta=\pi$. This defines the nominal times
$T_{\pi}=\frac{\pi}{|\Omega|},\qquad T_{\pi/2}=\frac{T_{\pi}}{2}.$ (51)
In particular, the four pulse shapes yield mirror widths
$\tau_{G\pi}\\!=\\!\frac{\sqrt{\pi}}{\sqrt{2}|\Omega|},\
\tau_{R\pi}\\!=\\!\frac{\pi}{2|\Omega|},\
\tau_{B\pi}\\!=\\!\frac{25}{21|\Omega|},\
\tau_{S\pi}\\!=\\!\frac{1}{|\Omega|}.$ (52)
Due to the linearity, the symmetric beamsplitter width is just a half of the
mirror time i. e., $\tau_{\pi/2}=\tau_{\pi}/2$.
### IV.3 Diffraction efficiency of a rectangular pulse
FIG. 4: Fidelity $F$ versus two-photon intensity $\mathcal{I}=\Omega^{2}/16$,
respectively two-photon Rabi frequency $\Omega_{r}=\Omega\omega_{2r}$, and
inverse $\pi$-pulse stretching factor $\zeta^{-1}=\tau_{j\pi}/\tau_{j}$,
$j\in\\{G,B,S,R\\}$, for Gaussian (a, e), Blackman (b, f), sech (c, g) and
rectangular pulses (d, h). The initial state is a 1D Gaussian wavepacket (98),
initially centered at $(x,k_{x})=(0,-k_{L})$ with momentum width
$\sigma_{k}=0.01\,k_{L}$ (top), $\sigma_{k}=0.1\,k_{L}$ (bottom). The optimal
stretching factor $\zeta_{\pi}$ (60) (solid line) traverses the regions of
maximal fidelity. For the numerical (1+1)D integration (16) with pulse widths
$\zeta\tau_{j\pi}$, and total pulse length $\Delta\tau_{j}=8\zeta\tau_{G}$,
typical laser and atom parameters, used in experiments (Tab. 2), are applied.
#### IV.3.1 Velocity selective Pendellösung
In the deep-Bragg regime $\mathcal{N}=1$, off-resonant diffraction orders are
negligible. Thus, for first order diffraction $N=1$ the state vector in the
beamsplitter manifold
$k_{\pm}\equiv(\pm 1+\kappa)k_{L},$ (53)
simplifies to the amplitude tuple $\mathbf{g}_{\mp}(\tau)=(g_{-1},g_{+1})$
with $\mathbf{g}_{\mp}(\tau_{i})=(1,0)$. The well known Pendellösung [77, 78]
$\displaystyle\begin{split}g_{-1}(\tau)&=e^{-i\varphi}\left(\cos\vartheta-\frac{\kappa}{i\Omega_{\kappa}}\sin\vartheta\right),\\\
g_{+1}(\tau)&=e^{-i\varphi}\frac{\Omega}{i\Omega_{\kappa}}\sin\vartheta,\end{split}$
(54)
depends on $\varphi=\kappa(\tau-\tau_{i})/2$,
$\vartheta=\Omega_{\kappa}(\tau-\tau_{i})/2$ and the generalized two-photon
Rabi frequency $\Omega_{\kappa}=\sqrt{\kappa^{2}+\Omega^{2}}$. It follows from
(42) for the rectangular pulse shape (46)
$i\dot{\bm{g}}_{\mp}(\tau)=\mathcal{H}_{\mp}\bm{g}_{\mp},\qquad\mathcal{H}_{\mp}=\begin{pmatrix}0&\frac{\Omega}{2}\\\
\frac{\Omega}{2}&\kappa\end{pmatrix}.$ (55)
With this solution the mirror pulse width (52) can be generalized for
arbitrary $\kappa\neq 0$. Maximal efficiency
$\eta_{+-}(\tau_{R\pi})=|g_{+1}(\tau_{\pi})|^{2}$ is achieved for
$\vartheta=\pi/2$, which determines the mirror pulse width
$\tau_{R\pi}(\kappa)=\frac{\pi}{2\Omega_{\kappa}}.$ (56)
On resonance ($\kappa=0$), we recover Eq. (52). Finally, the diffraction
efficiency reads
$\eta_{+-}(\tau_{R\pi})=\frac{\Omega^{2}}{\Omega_{\kappa}^{2}}\sin^{2}{\vartheta_{\pi}},\qquad\vartheta_{\pi}=\frac{\pi}{2}\frac{\Omega_{\kappa}}{\Omega}.$
(57)
The relative phase of the transfer function (18), between the final $k_{-}$
and $k_{+}$ components is
$\Delta\phi\equiv\phi_{--}-\phi_{+-}=\arctan\\!\left(\\!\frac{\kappa}{\Omega_{\kappa}}\tan\vartheta\\!\right)\\!-\negmedspace\frac{\pi}{2}.$
(58)
For $\vartheta=\vartheta_{\pi}$, one obtains the phase shift after a mirror
pulse $\Delta\phi(\tau_{R\pi})$.
#### IV.3.2 Losses into higher diffraction orders
The transfer function $\bra{\bm{k}^{\prime}}G(t,t_{i})\ket{\bm{k}}$ (18)
exhibits resonances at $\bm{k}^{\prime}=\bm{k}+2N\bm{k}_{L}$. On the one hand,
resonances with $N\neq 1$ lead to a population loss from the $N=1$
beamsplitter manifold $\\{k_{\pm}\\}$ and reduce the diffraction efficiency.
On the other hand, they diminish the coupling strength within the beamsplitter
manifold. Consequently, this increases the optimal $\pi$-pulse time
$\tilde{\tau}_{\pi}>\tau_{\pi}$ of a Bragg mirror compared to the prediction
of the Pendellösung (52). Gochnauer _et al._ [56] have demonstrated this
effect experimentally for Gaussian pulses, proving that the effective coupling
strength is given by the energy bandgap in the quasimomentum space.
##### Renormalized $\pi$-pulse time
The influence of higher order resonances on the beamsplitter manifold can be
calculated perturbatively in terms of the generalized two-photon Rabi
frequency $\Omega_{\kappa}$. For $\Omega_{\kappa}\rightarrow 0$ all momentum
states are doubly degenerate with respect to their energies. We employ Kato’s
perturbation theory [79], as it can describe the generalized degenerate
eigenvalue problem (109). Remarkably, Kato’s 1st order perturbation theory
coincides with the Pendellösung (cf. App. C)
From a third order perturbation calculation
$\mathcal{O}(\Omega_{\kappa}^{4})$, we find the renormalized Rabi frequency
$\tilde{\Omega}=\sqrt{\kappa^{2}(1+2\mathcal{I})^{2}+\Omega^{2}(1-\mathcal{I})^{2}}\stackrel{{\scriptstyle\mathcal{I}\ll
1}}{{\longrightarrow}}\Omega_{\kappa}$ (59)
within the beamsplitter manifold using the abbreviation
$\mathcal{I}=\Omega^{2}/16$. For weak dressing $\mathcal{I}\ll 1$, it reduces
to the generalized Rabi frequency of the Pendellösung. From (52), one can
evaluate the $\pi$-pulse time stretching factor
$\zeta_{\pi}^{\kappa}\\!=\\!\frac{\tilde{\tau}_{R\pi}}{\tau_{R\pi}}\\!=\\!\frac{\Omega_{\kappa}}{\tilde{\Omega}},\qquad\zeta_{\pi}\equiv\zeta_{\pi}^{\kappa=0}\\!=\frac{1}{1-\mathcal{I}}\approx\\!1+\mathcal{I}.$
(60)
Figure 4 depicts a contour plot of the fidelity $F(\mathcal{I},\zeta)$ (19)
for a Bragg-mirror pulse versus the bare two-photon intensity $\mathcal{I}$
and the inverse pulse stretching factor $\zeta^{-1}=\tau_{j\pi}/\tau_{j}$.
This representation uncovers a linear relation. The numerically calculated
fidelity (19) considers four off-resonant diffraction orders
($\mathcal{N}=5$). As initial condition, we consider 1D Gaussian wavepackets
(98) centered at $k_{0}=-k_{L}$ with momentum width $\sigma_{k}$, localized in
the center of the laser beams $x_{0}=0$. Here, in the plane-wave
approximation, the results are independent of the expansion size. This size
$\sigma_{x}=(2\sigma_{k})^{-1}$ follows from the Heisenberg uncertainty.
Clearly, the $\pi$-pulse stretching factor $\zeta_{\pi}$ (60) traverses the
optimal fidelity regions for all pulse shapes and momentum widths, as a
universal rule, motivating the effective $\pi$-pulse widths
$\tilde{\tau}_{j\pi}=\zeta_{\pi}\tau_{j\pi},\quad j\in\\{G,R,B,S\\},$ (61)
with $\tau_{j\pi}$ from Eq. (52).
##### Renormalized $\pi$-pulse efficiency
In Fig. 5 the velocity dispersion of the response of an atomic mirror is
visualized for typical parameters used in experiments (cf. Tab. 2) and a two-
photon Rabi frequency $\Omega_{r}=\Omega\omega_{2r}=3\omega_{r}$.
FIG. 5: (a) Diffraction efficiency $\eta_{+-}$ ($\bm{\square}$) in the
beamsplitter manifold $N\\!=\\!1$, together with the relative phase shift
$\Delta\phi$ (58) ($\bm{\circ}$) and (b) losses into higher diffraction orders
$N\\!\neq\\!1$ versus detuning $\kappa$, after a rectangular mirror pulse. For
the numerical solution (solid), considering four off-resonant diffraction
orders (with $k=(-1+\kappa)k_{L}$ and $k^{\prime}\\!=\\!k+2Nk_{L}$) the
applied pulse width is $\tilde{\tau}_{R\pi}(\Omega)$ (61) and for the
Pendellösung (57), (58) (dotted), considering only the resonant diffraction
order, $\tau_{R\pi}(\Omega)$ (52) for
$\Omega_{r}\\!=\\!\Omega\,\omega_{2r}\\!=\\!3\,\mathrm{\omega_{r}}$. In (a),
the Pendellösung overestimates the efficiency and phase shift, while the Kato
corrections (121) (dashed) match the numerical results (solid) much better.
There are only deviations at the band edges, especially for $N\\!=\\!2$ (b).
The Pendellösung (54), valid in the deep-Bragg regime ($\mathcal{N}=1$),
applying the pulse width $\tau_{R\pi}(\Omega)$ (52), is compared to the
eigenvalue solution (42) with pulse width $\tilde{\tau}_{R\pi}(\Omega)$ (61).
Therefore, the diffraction efficiency $\eta_{k^{\prime}k}$ reveals the
velocity selectivity of the Bragg condition and the population loss into
higher diffraction orders, here in the quasi-Bragg regime ($\mathcal{N}=5$).
The phase difference $\Delta\phi$ (58) shows a $\pi$ jump at resonance. The
perturbative Kato solution (121) describes the beamsplitter response very
well, only at the band edges $\kappa\rightarrow\pm 1$, there are small
deviations. For weak coupling $\Omega$, the diffraction efficiency after a
mirror pulse of width $\tilde{\tau}_{\pi R}(\Omega)$ (61), exhibits a sinc
behavior [cf. Fig. 6 (a)]. It is the typical Fourier-response to a rectangular
pulse. Increasing the Rabi frequency $\Omega$, the response is power
broadened, in conjunction with a reduced efficiency. Simultaneously, the Kato
solution becomes less accurate for $|\kappa|>0$, while the resonant efficiency
$\eta_{0}\equiv\eta_{+-}(\kappa=0)=\eta_{k_{L},-k_{L}}$ can be approximated
further. This is also depicted in Fig. 6 (b), together with the efficiency’s
full width half maximum $\Delta\eta$ of the Bragg mirror. For an ideal mirror,
$\eta_{0}=1$ and $\Delta\eta\rightarrow\infty$ are desirable, but impossible.
FIG. 6: (a) Diffraction efficiency $\eta_{+-}$ after a mirror pulse of width
$\tilde{\tau}_{R\pi}(\Omega)$ (61) versus detuning $\kappa$ for different two-
photon Rabi frequencies $\Omega_{r}=\Omega\,\omega_{2r}$, numerical results
(solid) and Kato (121) solution (dashed). (b) Resonant transfer efficiency
$\eta_{0}$ ($\bm{\times}$) and efficiency width $\Delta\eta$
($\bm{\triangleright}$) versus $\Omega_{r}$. The numerically optimal
interaction time, for maximal efficiency (dash-dotted) is compared to the
approximations for the $\pi$-pulse width $\tau_{R\pi}$ (52) (dotted) and
$\tilde{\tau}_{R\pi}$ (61) (solid). The analytical Kato approximation
$\eta_{0}^{K}(\tilde{\tau}_{R\pi})$ (62) (dashed) provides meaningful
predictions.
In addition, we study the optimal interaction time in Fig. 6 (b). The
approximation $\tau_{R\pi}$ (52) for the deep-Bragg regime and
$\tilde{\tau}_{R\pi}$ (61) for the quasi-Bragg regime, considering higher
diffraction orders, are compared to the optimal interaction time, defined by
the maximum numerical transfer efficiency at resonance $\kappa=0$. With
increasing $\Omega$, in a regime where the losses into higher diffraction
orders are important, the approximation with $\tau_{R\pi}$ is less accurate,
while $\tilde{\tau}_{R\pi}$ can be used further. Please note that for the
maximized transfer efficiency the velocity acceptance $\Delta\eta$ is reduced,
while for $\tilde{\tau}_{R\pi}$ it remains larger, for increasing $\Omega$.
From the Kato solution (121) a simple analytic equation for the diffraction
efficiency on resonance, for the effective $\pi$-pulse time
$\tilde{\tau}_{R\pi}$ can be derived (cf. App. C) to
$\eta_{0}^{K}(\tilde{\tau}_{R\pi})=(1-2\mathcal{I})\left[1+|\Omega|\mathcal{I}\sin\left(\frac{2\pi}{|\Omega|}\frac{1+2\mathcal{I}}{1-\mathcal{I}}\right)\right],$
(62)
also depicted in Fig. 6 (b). This expression predicts losses into higher
diffraction orders within the convergence radius
$\Omega_{r}=\Omega\,\omega_{2r}<4\,\omega_{r}$ ($\mathcal{I}<0.0625$), very
well. The approximation remains positive for
$\Omega_{r}<8\sqrt{2}\,\omega_{r}$ $(\mathcal{I}=0.5)$.
### IV.4 Diffraction efficiency of a sech pulse
#### IV.4.1 Velocity selective Demkov-Kunike Pendellösung
For hyperbolic secant pulses $\Omega(\tau)=\Omega f_{S}(\tau)$ (50), one can
solve Eq. (55) also in a closed form [75, 76]. A decoupling of the first order
differential equation system with $g_{+1}=2i\Omega(\tau)^{-1}\dot{g}_{-1}$,
leads to Hill’s second order differential equation [68]
$0=\ddot{g}_{-1}-\biggl{(}\frac{\dot{\Omega}(\tau)}{\Omega(\tau)}-i\kappa\biggr{)}\,\dot{g}_{-1}+\frac{\Omega(\tau)^{2}}{4}g_{-1}.$
(63)
With the nonlinear map $z(\tau)=[1+\tanh{(\tau/\tau_{S})}]/2$, the
differential equation for $\gamma(z)\equiv g_{-1}(\tau)$ emerges as
$\displaystyle
z(1-z)\gamma^{\prime\prime}+\left[c-z(1+a+b)\right]\gamma^{\prime}-ab\gamma=0,$
(64)
with $a=\Omega\tau_{S}/2$, $b=-a$ and $c=(1+i\kappa\tau_{S})/2$. This is the
hypergeometric differential equation with solutions
$f_{1}={}_{2}F_{1}(a,b;c;z),f_{2}=z^{1-c}{}_{2}F_{1}(1+a-c,1+b-c;2-c;z)$ and
Wronski determinant $w=(1-z)^{c-1}z^{-c}$. Straightforward analysis (cf. App.
D) leads to the Demkov-Kunike (DK) solution with unitary propagator
$G_{\mp}(\tau,\tau_{i})$
$\bm{g}_{\mp}(\tau)=G_{\mp}(\tau,\tau_{i})\bm{g}_{\mp}(\tau_{i}),\qquad
G_{\mp}(\tau_{i},\tau_{i})=\mathds{1}.$ (65)
For the initial datum $\bm{g}_{\mp}(\tau_{i})=(1,0)$, one obtains
$g_{-1}(\tau)=[f_{1}(\tau)f^{\prime}_{2}(\tau_{i})-f_{2}(\tau)f^{\prime}_{1}(\tau_{i})]/w(\tau_{i}).$
(66)
For a pulse beginning in the remote past $\tau_{i}\ll-\tau_{S}$, this
simplifies to
$\displaystyle g_{-1}(\tau)$ $\displaystyle=\ _{2}F_{1}\left(a,-a;c,z\right),$
(67) $\displaystyle g_{+1}(\tau)$ $\displaystyle=\tfrac{a}{ic}\sqrt{z(1-z)}\
_{2}F_{1}\left(1-a,1+a;1+c,z\right).$ (68)
Now, the diffraction efficiency of a beamsplitter reads
$\eta_{+-}^{{DK}}(\kappa,\tau)=|g_{+1}(\tau)|^{2}=1-|g_{-1}(\tau)|^{2}.$ (69)
Furthermore, for very long pulse durations $\tau_{S}\ll\tau_{f},|\tau_{i}|$,
the diffraction efficiency simplifies to
$\eta_{+-}^{{DK}}(\kappa,\Omega,T)=\sech^{2}\\!\left(\frac{\kappa
T}{2}\right)\sin^{2}\\!\left(\frac{\Omega T}{2}\right)\\!,$ (70)
with the nominal time $T$ (45). In order to achieve full diffraction
efficiency $\eta_{0}^{{DK}}=\eta_{+-}^{{DK}}(\kappa=0)=1$, one should choose
the $\pi$-pulse width as $\tau_{S\pi}=|\Omega|^{-1}$, in agreement with the
pulse area (52). Waiting indefinitely long is hardly ever an option [80].
Therefore, the finite time approximation
$\eta_{0}^{{DK}}(\tau)\approx
z=\tfrac{1}{2}(1+\tanh{\Omega\tau})\vspace{-1mm}$ (71)
reveals the exponential convergence past several $\pi$-pulse times
$\tau\gg\tau_{S}$. It requires $\Omega_{r}=\Omega\,\omega_{2r}<3\,\omega_{r}$.
#### IV.4.2 Losses into higher diffraction orders
To consider losses into the higher diffraction orders, we use time-dependent
perturbation theory in Eq. (42)
$i\dot{\bm{g}}=\mathcal{H}(\tau)\bm{g},\qquad\mathcal{H}(\tau)=\mathcal{H}_{0}(\tau)+\mathcal{H}_{1}(\tau).\vspace{-1mm}$
(72)
The free evolution $\mathcal{H}_{0}(\tau)$ consist of a direct sum
$\displaystyle\mathcal{H}_{0}(\tau)$
$\displaystyle=\mathcal{H}_{\mp}(\tau)\bigoplus_{\begin{subarray}{c}\mu=-\mathcal{N}+1\\\
\mu\neq 0,1\end{subarray}}^{\mathcal{N}}\omega_{2\mu-1}\vspace{-1mm}$ (73)
of the DK-generator $\mathcal{H}_{\mp}(\tau)$ (55) in the beamsplitter
manifold and the unperturbed energies $\omega_{\mu}$ (43) in the higher
momentum states. The perturbation $\mathcal{H}_{1}(\tau)$ is simply the
complement of the complete Hamilton operator.
The free retarded propagator is defined for $\tau\geq\tau_{i}$ as
$G_{0}(\tau,\tau_{i})=G_{\mp}(\tau,\tau_{i})\bigoplus_{\begin{subarray}{c}\mu=-\mathcal{N}+1\\\
\mu\neq
0,1\end{subarray}}^{\mathcal{N}}e^{-i\omega_{2\mu-1}(\tau-\tau_{i})}\vspace{-1mm}$
(74)
and vanishes elsewhere (cf. App. D). It involves the DK-Pendellösung $G_{\mp}$
(65) and the free time evolution of off-resonant momentum states. The complete
solution
$\bm{g}(\tau)=G(\tau,\tau_{i})\bm{g}(\tau_{i})$ (75)
follows from the solution $G(\tau,\tau_{i})$ of the integral equation (127). A
second order approximation couples to the $\pm 3k_{L},\pm 5k_{L}$ momentum
states and shifts the frequencies of the beamsplitter manifold
$\displaystyle
G(\tau,\tau_{i})=G_{0}-i\int^{\infty}_{-\infty}\text{d}t\,G_{0}(\tau,t)\mathcal{H}_{1}(t)G_{0}(t,\tau_{i})$
(76)
$\displaystyle-\int^{\infty}_{-\infty}\text{d}t\text{d}t^{\prime}\,G_{0}(\tau,t)\mathcal{H}_{1}(t)G_{0}(t,t^{\prime})\mathcal{H}_{1}(t^{\prime})G_{0}(t^{\prime},\tau_{i}).$
This is required to observe the stretching of the $\pi$-pulse time. An
explicit analytical approximation can be obtained. It is numerically efficient
and useful for the interpretation, but remained unwieldy for display [81].
In Fig. 7, we compare the simple and the extended DK-model after a $\pi$
pulse, with the corresponding numerical (1+1)D simulations (16). The
diffraction efficiency is depicted in Fig. 7 (a) and the phase shift
$\Delta\phi$ between the coupled states in Fig. 7 (b). The simple DK-
Pendellösung (67) is valid for
$\Omega_{r}\\!=\\!\Omega\,\omega_{2r}\\!<\\!3\,\omega_{r}$. For
$\Omega_{r}\\!>\\!3\,\omega_{r}$, losses into higher diffraction orders are
significant, but the extended solution (76) still matches the numerical
solution.
FIG. 7: Velocity dispersion of (a) the diffraction efficiency $\eta_{+-}$ and
(b) the phase shift $\Delta\phi$ for sech-pulses with pulse width
$\tau_{S}=\tilde{\tau}_{S\pi}$ (61) and different Rabi frequencies
$\Omega_{r}=\Omega\,\omega_{2r}$. The DK-Pendellösung (67) (dotted) is
suitable for $\Omega_{r}<3\,\omega_{r}$ while the extended model (76) (dashed)
matches the numerical results (16) (solid) very well also for larger
$\Omega_{r}$.
##### Adiabaticity
The crossover from the deep- to the quasi-Bragg regime at $\Omega\approx
3\,\omega_{r}$ for atomic mirrors using $\tilde{\tau}_{j\pi}$ (61) is related
to the adiabaticity criterium [82]
$\max_{\tau\in[\tau_{i},\tau_{i}+\Delta\tau]}\left|\frac{d}{d\tau}\left(\frac{\bm{g}_{n}^{o}(\tau)^{\ast}\dot{\bm{g}}_{m}^{o}(\tau)}{\omega_{n}(\tau)-\omega_{m}(\tau)}\right)\right|\Delta\tau\ll
1,$ (77)
$\forall\,m\neq n$, with the eigenvalues $\omega_{m}(\tau)$ and eigenvectors
$\bm{g}_{m}^{o}(\tau)$ of $H^{o}$ (40). Equation (77) results in
$\Omega_{r}\\!=\\!\Omega\,\omega_{2r}\ll 4\,\omega_{r}$ for
$\tilde{\tau}_{S\pi}$ at $\kappa=0$. This is confirmed by the results of
Gochnauer et al. [56] and visible in Figs. 9 and 10. Therefore, while the DK-
Pendellösung (67) is valid in the adiabatic regime, the extended model (76)
can be even used for non-adiabatic pulses.
### IV.5 Diffraction efficiency of a Gaussian pulse in the deep-Bragg limit
Due to the similarity of the Gaussian- to the sech-pulses [cf. Eqs. (47) and
(50)], one can estimate the velocity selective diffraction efficiency for
infinitely long Gaussian pulses in the deep-Bragg regime. The different pulses
have equal nominal times (45). Therefore, approximating $\sech^{2}(a)$ from
Eq. (70), with a similar exponential form, providing the same integration area
as
$\int_{-\infty}^{\infty}\text{d}a\sech^{2}(a)=\int_{-\infty}^{\infty}\text{d}a\exp(-\pi
a^{2}/4)=2$, leads to
$\eta_{+-}^{G}(\kappa,\Omega,T)=\exp\left(\\!-\pi\Big{(}\frac{\kappa
T}{4}\Big{)}^{2}\right)\sin^{2}\left(\frac{T\Omega}{2}\right).$ (78)
The results are discussed in the next section.
### IV.6 Diffraction efficiency for all pulses in (1+1)D
FIG. 8: Velocity dependent diffraction efficiency $\eta_{+-}(\kappa)$ for a
Gaussian pulse ($j=G$, solid: numerical, dotted: deep-Bragg limit (78)) and
the $\sech$ pulse [$j=S$, dashed: analytical (76)]. A mirror pulse of width
$\tilde{\tau}_{j\pi}$ (61) with total pulse duration
$\Delta\tau=8\tilde{\tau}_{G\pi}$ is applied for three Rabi frequencies
$\Omega_{r}=\Omega\,\omega_{2r}$. FIG. 9: Comparison of the Bragg diffraction
for a mirror pulse width $\tilde{\tau}_{i\pi}$, for rectangular- (dash-dotted
$\bm{\times}$), Gaussian- (solid $\bm{\square}$), Blackman- (dotted
$\bm{\circ}$) and sech-pulses (dashed, numerical: $\bm{\triangledown}$, DK
(67) $\bm{\triangleleft}$, DK (76) $\bm{\triangleright}$). (a) Velocity
dispersion of the numerical diffraction efficiency $\eta_{+-}$ (without
plotmarkers) and phase shift $\Delta\phi$ (with plotmarkers) for
$\Omega_{r}=\Omega\,\omega_{2r}=3\,\mathrm{\omega_{r}}$. (b) On-resonance
diffraction efficiency $\eta_{0}$ and (c) width of the diffraction efficiency
$\Delta\eta$ versus $\Omega_{r}$. FIG. 10: Fidelity $F(\Omega_{r},\sigma_{k})$
after a mirror pulse of width $\tilde{\tau}_{i\pi}(\Omega)$ (61) versus the
two-photon Rabi frequency $\Omega_{r}\\!=\\!\Omega\,\omega_{2r}$ for different
initial atomic momentum widths
$\sigma_{k}\\!=\\!\\{0.01,0.05,0.1,0.2\\}k_{L}$, {$\bm{\times}$,
$\bm{\triangledown}$, $\bm{\square}$, $\bm{\circ}$}; for (a) Gaussian, (b)
Blackman, (c) sech and (d) rectangular pulses. The total interaction time is
$\Delta\tau=8\,\tilde{\tau}_{G\pi}$ (a-c) and
$\Delta\tau=2\tilde{\tau}_{R\pi}$ (d), cf. Eq. (61). The 1D initial Gaussian
wavepacket (98) is centered at $(x,k_{x})=(0,-k_{L})$. The DK-Pendellösung
(67) [dotted, (c)] matches the results of the numerical integration (16)
(solid) very well for $\Omega_{r}<3\,\omega_{r}$, considering population loss
to higher diffraction orders (76) (dashed) also for larger $\Omega_{r}$. The
Kato solution (121) (dashed) is depicted in (d), matching the numerical
results.
In beamsplitter experiments, Gaussian laser pulses are ubiquitous. There is a
good reason for it, as they are self-Fourier-transform functions. This is
evident in the numerical simulations of first order diffraction efficiency in
Fig. 8, which is free of the side lobes of rectangular pulses, seen in Fig. 6
(a). The diffraction efficiency becomes power-broadened for increasing Rabi
frequency. Beyond $\Omega_{r}>3\omega_{r}$, scattering into higher diffraction
order depletes the population in the beamsplitter manifold. However, in the
deep-Bragg regime, the approximation (78) matches the numerical solutions very
well. Sech-pulses [extended DK-model (76)] behave similarly, as shown in Fig.
8 and 9. The explicit solution for the sech-pulse [extended DK-model (76)]
deviates slightly from Gaussian- and Blackman-pulses, but provides very
detailed forecasts. Indeed, all smooth pulse shapes ($j=G,B,S$) with pulse
widths $\tilde{\tau}_{j\pi}$ are very similar and exhibit almost identical
phase shifts and efficiencies as depicted in Fig. 9. Here, for finite total
interaction times $\Delta\tau$, the $\pi$-pulse conditions are not met exactly
$\Omega T_{j}(-\Delta\tau/2,\Delta\tau/2)\approx\pi$ (45). One could adjust
the pulse width $\tilde{\tau}_{j\pi}$ for each pulse shape $j$ to obtain a
$\pi$ pulse individually $\Omega T_{j}(-\Delta\tau/2,\Delta\tau/2)=\pi$, but
this leads to unequal nominal times $T_{j}\neq T$ (45) and results in
significant phase differences. Thus, we consider the same $\pi$-pulse time
$\Delta\tau=8\tilde{\tau}_{G\pi}$ for all pulses and the widths
$\tau_{j}=\tilde{\tau}_{j\pi}$ connected via $T_{j}=T$, the resulting
differences in the pulse areas $\Omega T_{j}(-\Delta\tau/2,\Delta\tau/2)$ (45)
are negligible.
The phase sensitive fidelity (19) for different pulse shapes and momentum
widths $\sigma_{k}$ of an initial Gaussian wavepacket in 1D (98) are compared
in Fig. 10. For the smooth envelopes, an increasing $\sigma_{k}$ reduces the
range of admissible Rabi frequencies $\Omega_{r}\\!=\\!\Omega\,\omega_{2r}$,
which shifts the optimum to higher values. Evidently, the DK-Pendellösung (67)
matches numerical simulations for $\Omega_{r}<3\,\mathrm{\omega_{r}}$, while
the extended DK-model (76) remains further valid. The explicit Kato solution
(121) matches the results for rectangular pulses very well, demonstrating its
applicability for wavepackets with finite momentum width.
### IV.7 Diffraction efficiency for spatial Gauss-Laguerre modes with pulse
shapes in (3+1)D
#### IV.7.1 Gauss-Laguerre modes
The experimental beamsplitter beams are pulsed, bichromatic,
counterpropagating Gauss-Laguerre modes [83]. In the specific frame
$S^{\prime}$, comoving with the nodes of the interference pattern, there is
only a single wavenumber $k_{L}$ (cf. (15) and Apps. A, B). The slowly varying
amplitude of the electric field leads to Rabi frequencies
$\displaystyle\Omega_{j}(t,\bm{r})=\Omega_{j}(t,\varrho)e^{i\Phi(\varrho)},$
(79)
$\displaystyle\Omega_{j}(t,\varrho)=\Omega_{j}(t)\frac{w_{0}}{w_{j}}e^{-\frac{\varrho^{2}}{w_{j}^{2}}},\qquad\Phi(\varrho)=\frac{k_{L}\varrho^{2}}{2R_{j}}-\xi_{j}$
(80)
with beam parameters $w_{1,2}=w(\ell/2)$, $R_{1,2}=\pm R(\ell/2)$,
$\xi_{1,2}=\pm\xi(\ell/2)$ and the distance $\ell$ between both lasers beam
waists, as depicted in Fig. 11.
FIG. 11: Two counterpropagating, bichromatic Gauss-Laguerre beams form a
travelling, standing wave (12) with an intensity pattern in cylindrical
coordinates $(x,\varrho)$. The gray arrows are the local wavevectors, $w(x)$
is the local waist and $R(x)$ the local radius of curvature. The distance
between the two beam waists is $\ell$. The atomic cloud, generally localized
at $\bm{r}_{0}$ is indicated as red ellipse. FIG. 12: Fidelity
$F(\Omega_{r},\sigma_{k},\sigma_{x})$ after a mirror pulse versus two-photon
Rabi frequency $\Omega_{r}=\Omega\,\omega_{2r}$ for different atomic initial
momentum widths $\sigma_{k}=\\{0.05,\ 0.1,\ 0.2\\}\times k_{L}$, {solid blue,
dashed red, dashed-dotted green} of a 3D ballistically expanded Gaussian
wavepacket (100) for Gauss-Laguerre beams ($\bm{\circ}$) in comparison to
plane waves ($\bm{\square}$), using the (3+1)D numerical integration (16).
Gaussian temporal pulses of width $\tilde{\tau}_{G\pi}(\Omega)$ (61) and total
duration $\Delta\tau=8\,\tilde{\tau}_{G\pi}$ (61) are applied. Each column
represents a different ratio $\sigma_{x}/w_{0}$ between spatial width of the
initial state $\sigma_{x}$ and the beam waist $w_{0}$. In the bottom row the
atomic initial state is displaced in the radial direction of the Gaussian
beams to $\varrho_{0}=y_{0}=w_{0}/2$.
#### IV.7.2 Local plane-wave approximation
To isolate the momentum kick of the beamsplitter from the momentum imparted by
the dipole force, we consider a local plane-wave approximation of the Gauss-
Laguerre beam at the initial position $\bm{r}_{0}=(0,\bm{\varrho}_{0})$,
$\bm{\varrho}_{0}=(y_{0},z_{0})$ of the atomic cloud
$\Omega_{j}(t,\bm{r})\approx\Omega_{j}(t,\bm{r}_{0})=\Omega_{j}(t,\varrho_{0})e^{i\Phi(\varrho_{0})}.$
(81)
Thus, the atomic cloud feels only a reduced Rabi frequency but experiences no
spatial inhomogeneity. Therefore, simulations with plane waves must be
independent of the ratio $\sigma_{x}/w_{0}$ for $\sigma_{x}>\lambda$.
#### IV.7.3 Simulations
Beamsplitters perform best, if the atomic cloud (of size
$\sigma_{x}\sim$\mathrm{\SIUnitSymbolMicro m}$-$\mathrm{mm}$$) is well
localized within the beam waist $w_{0}$. For $w_{0}\sim$\mathrm{m}\mathrm{m}$$
and optical wavelengths $\lambda\sim$\mathrm{\SIUnitSymbolMicro m}$$ the
Rayleigh lengths $x_{R}$ are several meters, thus
$x_{R}\gg w_{0}>\sigma_{x}>\lambda.$ (82)
Therefore, one can expect that the transversal dipole forces will be stronger
than the forces along the propagation direction $x$. Small clouds centered at
the symmetry point $\bm{r}_{0}=(0,0,0)$ will feel the least degradation of the
beamsplitter fidelity [cf. Fig. 12 (a) and (b)] due to dipole forces. This
will be confirmed by displacing the initial cloud transversely to
$\bm{r}_{0}=(0,w_{0}/2,0)$, leading to larger aberrations [cf. Fig. 12
(e)-(h)].
In these simulations of a Bragg mirror, depicted in Fig. 12, we use the
effective $\pi$-pulse width $\tilde{\tau}_{G\pi}(\Omega(\bm{r}_{0}))$ (61) in
the local plane-wave approximation (81) for different Rabi frequencies
$\Omega_{r}=\Omega\,\omega_{2r}$ and a longitudinal laser displacement
$\ell=0.1\,x_{R}$, like in the experiment (cf. Tab. 2). As initial states of
the atomic cloud, we consider ballistically expanded 3D Gaussian wavepackets
(100) with different widths in real space $\sigma_{x}$ and in reciprocal space
$\sigma_{k}$.
For atoms located at the center of the Gaussian laser beams, the spatial
inhomogeneity (107) leads to significant aberrations only for large atomic
clouds [cf. Fig. 12 (c), (d)]. By contrast, even small displaced clouds [cf.
Fig. 12 (e), (f)] show a significant reduction of the fidelity in realistic
Gaussian beams compared to ideal plane waves. The latter uses a reduced Rabi
frequency according to the local plane-wave approximation (81). For large
clouds this reduction is detrimental [cf. Fig. 12 (g), (h)]. Please note that
we use parameters, where the simulation results for the fidelity only depend
on the ratio $\sigma_{x}/w_{0}<1$.
Besides the phase sensitive fidelity, the aberrations due to Gaussian beams
are already apparent in the diffraction efficiency. In Fig. 13 the momentum
density $\tilde{n}(k_{x},k_{y})$ is shown for the (3+1)D simulation with (a)
Gauss-Laguerre laser beams and (b) the idealized local plane-wave
approximation, after a mirror pulse with $\Omega_{r}=3\,\mathrm{\omega_{r}}$.
In the momentum space, the splitting is visible directly after the
$\pi$-pulse. We study a ballistically expanded Gaussian wavepacket (100) as
initial state with $\sigma_{k}=0.05\,\mathrm{k_{L}}$ and
$\sigma_{x}=1/25\,w_{0}$, located at $\bm{r}_{0}=(0,w_{0}/2,0)$. The
logarithmic scale highlights the imperfections of the Bragg diffraction, using
Gaussian laser beams. Even for the tiny momentum width, the diffraction
efficiency is reduced to $96.3\,\mathrm{\%}$ in comparison to
$97.8\,\mathrm{\%}$ for idealized plane waves. In addition, the dipole force
leads to a rogue, transversal momentum component
$\langle\hat{p}_{y}\rangle=0.012\,\mathrm{\hbar}k_{L}$. As opposed to the
diffraction efficiency and the fidelity, this momentum component depends not
only on the relation $\sigma_{x}/w_{0}$ but on the beam waist, here
$w_{0}=$62.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Further studies of the
mechanical light effects of the dipole force are subjects of our present
research.
Locating the initial state at the center $\bm{r}_{0}=(0,0,0)$ reduces the
aberrations due to Gauss-Laguerre laser beams. The diffraction efficiency of
$99.0\,\mathrm{\%}$ reaches almost the efficiency of idealized plane waves
with $99.1\,\mathrm{\%}$ and the transverse momentum component vanishes.
FIG. 13: Column integrated atom density in momentum space
$\tilde{n}(k_{x},k_{y})=\int\text{d}k_{z}\,|\tilde{\psi}(k_{x},k_{y},k_{z})|^{2}$
after a $\pi$ pulse for (a) Gaussian laser beams and (b) plane waves. The
initial state is a temporally evolved Gaussian wavepacket (100) located at
$\bm{r}_{0}=(0,w_{0}/2,0)$ with momentum width $\sigma_{k}=0.05\,k_{L}$ and
expansion size $\sigma_{x}=w_{0}/25=$2.5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$. Gaussian pulses with
$\Omega_{r}=\Omega\,\omega_{2r}=3\,\mathrm{\omega_{r}}$,
$\tau_{G}=\tilde{\tau}_{G\pi}(\Omega)$ (61),
$\Delta\tau=8\,\mathrm{\tilde{\tau}_{G\pi}}$ and beam waist
$w_{0}=$62.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ are applied. The final
momentum expectation value in $y$-direction
$\langle\hat{p}_{y}\rangle=0.012\,\mathrm{\hbar}k_{L}$ is highlighted with
grey lines.
## V Proving the Demkov-Kunike model experimentally
Experimentally, we employ an atom chip apparatus to Bose-condense 87Rb [84,
34] with a condensate fraction of $N^{c}=$(10\pm 1.)\text{\times}{10}^{3}$$
and a quantum depletion (thermal cloud) of $N^{t}=$(7\pm
1.)\text{\times}{10}^{3}$$ . After release from the trap (lab frame $S$), with
trap frequencies listed in Tab. 2, they expand ballistically and fall
vertically towards Nadir. The Bragg-laser beams are aligned horizontally. It
is sufficient to consider inertial motion during the short Bragg pulses
(<$\mathrm{m}\mathrm{s}$). After $10\text{\,}\mathrm{ms}$ time-of-flight
(TOF), at the beginning of the diffraction pulses, the temperature of the
thermal cloud is obtained from a bimodal fit [85] as $T=$20\pm
3\text{\,}\mathrm{nK}$$. So far, the cloud
$\sigma_{x}=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ is much smaller than
the beam waist $w_{0}=$1386\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
permits the plane-wave approximation.
Experimentally, the first order diffraction efficiency in the deep-Bragg limit
$\eta=\frac{N_{+}}{N_{-}+N_{+}}$ (83)
is obtained from the number of atoms $N_{+}$ diffracted into the first order
$k^{\prime}=k_{+}$ and the undiffracted atoms $N_{-}$ remaining in the initial
state $k^{\prime}=k_{-}$. The diffraction efficiency is either a function of
the detuning $\delta\omega$ (6) of the laser from the two-photon resonance
with atoms initially at rest $\langle\hat{p}_{x}(\tau_{i})\rangle=0$, or it is
the response for resonant lasers and an initial wavepacket centered at
$\langle\hat{p}_{x}(\tau_{i})\rangle=(-1+\bar{\kappa})\hbar
k_{L},\qquad\bar{\kappa}=\frac{\mbox{$\delta\omega$}}{\omega_{2r}},$ (84)
using Eq. (8) (cf. Sec. II.2).
Theoretically, we compute the diffraction efficiency (83) in the laser plane-
wave approximation from the number of diffracted atoms
$N_{\pm}(\bar{\kappa})=\int_{-1}^{1}\text{d}\kappa\,\eta_{\pm-}(\kappa)\,n(\kappa,\bar{\kappa}),$
(85)
following from a reaction equation derived in App. E, which completely
encloses the wavepacket with the effectively one-dimensional momentum density
$n(\kappa,\bar{\kappa})$ and the average initial momentum $\bar{\kappa}$.
Please note that for ideal plane matter-waves with wavenumber $\bar{\kappa}$
the diffraction efficiency (83) reduces to $\eta=\eta_{+-}(\bar{\kappa})$. In
the deep-Bragg regime, theoretically $N_{+}+N_{-}=N^{A}=N^{c}+N^{t}$ and the
diffraction efficiency simplifies to
$\eta=\frac{N_{+}(\bar{\kappa})}{N^{A}}=p^{c}\mathfrak{n}_{+}^{c}(\bar{\kappa})+p^{t}\mathfrak{n}_{+}^{t}(\bar{\kappa}),$
(86)
splitting into a condensate and a thermal cloud fraction with
$p^{c}=N^{c}/N^{A}$, $p^{t}=1-p^{c}$. Approximating the normalized initial
momentum distributions $\mathfrak{n}^{c}(\kappa,\bar{\kappa})$,
$\mathfrak{n}^{t}(\kappa,\bar{\kappa})$ by Gaussian functions (141) of widths
$\sigma_{k}^{c}=0.087\,k_{L}$ and $\sigma_{k}^{t}=(0.237\pm 0.015)\,k_{L}$
(cf. App. E.1) and using the Gaussian approximation (78) for the diffraction
efficiency $\eta_{\pm-}(\kappa)$, one obtains the analytical model
$\eta=\sin^{2}\left(\frac{\Omega
T}{2}\right)\sum_{a=\\{c,{t}\\}}\frac{p^{a}}{\tilde{\sigma}_{k}^{a}(\tilde{T})}e^{-\frac{(\bar{\kappa}\tilde{T})^{2}}{2\tilde{\sigma}_{k}^{a}(\tilde{T})^{2}}},$
(87)
with
$\tilde{\sigma}_{k}^{a}(\tilde{T})=\sqrt{1+(\tilde{T}\tilde{\sigma}_{k}^{a})^{2}}$,
$\tilde{T}=T\sqrt{\pi/8}$, $\sigma_{k}^{a}=\tilde{\sigma}_{k}^{a}k_{L}$.
FIG. 14: Experimental diffraction efficiency $\eta$ (83) for different laser
powers $P_{\bullet}=20\,\mathrm{mW}$ and $P_{\bm{\times}}=30\,\mathrm{mW}$ of
Gaussian pulses of width $\tau_{G}$ with numerical simulations (solid, blue)
and fits (87) (dashed, red) based on the DK-model. (a) Velocity selectivity
for $\Omega T_{G}=0.56\,\mathrm{\pi}$ pulses (45) versus detuning
$\bar{\kappa}$ of the initial central
momentum$\langle\hat{p}_{x}(\tau_{i})\rangle=(-1+\bar{\kappa}^{S}+\bar{\kappa})\hbar
k_{L}$, were $\bar{\kappa}^{S}k_{L}=0.12\,k_{L}$ is a small initial velocity
of the atoms in the lab frame $S$ and
$\bar{\kappa}=\mbox{$\delta\omega$}/\omega_{2r}$ (8). (b) Rabi oscillations of
the diffraction efficiency versus pulse width $\tau_{G}$, with total
interaction time $\Delta\tau=8\tau_{G}$ and highlighted pulse widths of (a).
Other parameters cf. Tab. 1, 2.
In Fig. 14, the diffraction efficiency (83) is depicted for two different
laser powers $P_{\bullet}=$20\text{\,}\mathrm{mW}$$,
$P_{\bm{\times}}=$30\text{\,}\mathrm{mW}$$ of a Gaussian pulse of width
$\tau_{G}$ (47) and total interaction time $\Delta\tau=8\tau_{G}$. In the
experiment, the atoms are displaced axially to $z_{0}=$1165\pm
50.\text{\,}\mathrm{\SIUnitSymbolMicro m}$=($0.84\pm 0.04$)w_{0}$, while
$x_{0}=y_{0}=$0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. This reduces the
effective Rabi frequency at the location of the atoms (81).
Fits using the model (87) describe the experimental data already very well and
provide starting parameters [$p_{c}$, $\Omega(\bm{r}_{0})$] for the effective
(1+1)D numerical simulations with Gaussian pulses, fully matching the
experimental data. The experimental, numerical and fit parameters are listed
in Tab. 1.
In Fig. 14 (a), the velocity dispersion of the diffraction efficiency uncovers
an initial motion $k_{x}^{S}=\bar{\kappa}^{S}k_{L}=0.12\,k_{L}$ of the atomic
cloud in the lab frame $S$. Considering this in
$\langle\hat{p}_{x}(\tau_{i})\rangle=(-1+\bar{\kappa}^{S}+\bar{\kappa})\hbar
k_{L}$ with $\bar{\kappa}=\mbox{$\delta\omega$}/\omega_{2r}$ leads to a very
good match of the fit model (87), the simulations and the experimental data.
In Fig. 14 (b), the diffraction efficiency displays damped Rabi oscillations
versus the pulse width $\tau_{G}$. This is a typical inhomogeneous line-
broadening caused by the momentum widths $\sigma_{k}^{c}$, $\sigma_{k}^{th}$,
the two-photon detuning $\mbox{$\delta\omega$}=\bar{\kappa}\omega_{2r}\neq 0$
and a residual velocity $\bar{\kappa}^{S}\neq 0$. It is also revealed by the
Gaussian approximation (87). The fit results for the two-photon Rabi frequency
are also optimal for the numerical simulations matching the experiment within
the error level.
Table 1: Parameters of Fig. 14 for the experiment (e), the numerical simulation (n) and the approximation (87) (a). | | | $P_{\bullet}\negmedspace=\negmedspace$20\pm 2.\text{\,}\mathrm{m}\mathrm{W}$$ | $P_{\bm{\times}}\negmedspace=\negmedspace$30\pm 3.\text{\,}\mathrm{m}\mathrm{W}$$
---|---|---|---|---
| e | $p_{c}$ | $0.59\pm 0.08$ | $0.59\pm 0.08$
| e | $\Omega$ | $6.60\pm 0.66\text{\,}\omega_{\mathrm{r}}$ | $9.89\pm 0.99\text{\,}\omega_{\mathrm{r}}$
| e | $\Omega(\bm{r}_{0})$ | $1.61\pm 0.27\text{\,}\omega_{\mathrm{r}}$ | $2.41\pm 0.40\text{\,}\omega_{\mathrm{r}}$
| e | $\bar{\kappa}^{S}k_{L}$ | $(0.12\pm 0.01)\,k_{L}$ | $(0.12\pm 0.01)\,k_{L}$
(a) | e | $\tau_{G}/\omega_{2r}$ | $147.45\,\mathrm{\mu s}$ | $98.3\,\mathrm{\mu s}$
| e | $\Omega(\bm{r}_{0})T_{G}$ (45) | $0.56\,\mathrm{\pi}$ | $0.56\,\mathrm{\pi}$
| a | $p_{c}$ | $0.59\pm 0.06$ | $0.59\pm 0.14$
| a | $\Omega(\bm{r}_{0})$ | $(1.74\pm 0.01)\,\omega_{r}$ | $(2.27\pm 0.01)\,\omega_{r}$
| n | $p_{c}$ | $0.59$ | $0.59$
| n | $\Omega(\bm{r}_{0})$ | $1.71\,\omega_{r}$ | $2.28\,\mathrm{\omega_{r}}$
(b) | e | $\mbox{$\delta\omega$}/2\pi$ | $-2\,\mathrm{kHz}$ | $-2.5\,\mathrm{kHz}$
| a | $p_{c}$ | $0.55\pm 0.03$ | $0.59\pm 0.04$
| a | $\Omega(\bm{r}_{0})$ | $(1.81\pm 0.01)\,\omega_{r}$ | $(2.30\pm 0.01)\,\omega_{r}$
| n | $p_{c}$ | $0.52$ | $0.52$
| n | $\Omega(\bm{r}_{0})$ | $1.81\,\omega_{r}$ | $2.30\,\omega_{r}$
It is worth mentioning that the velocity dispersion of the efficiency [Fig. 14
(a)] is less sensitive to the condensate ratio $p^{c}$ than the Rabi
oscillations [Fig. 14 (b)]. The Gaussian approximation (87) underestimates the
second maxima, but the fit of $p^{c}$ matches the experimental value within
its uncertainty. The numerical simulations predict a condensate ratio at the
lower bound of the experimental ratio, still within the uncertainty. The
reduction of condensate fraction $p^{c}$ in simulations and Gaussian
approximation is equivalent to increasing the momentum width of the condensate
or thermal cloud.
Thus, the Gaussian approximation (87) of the DK-model gives an unbiased
prediction of the experimental data. It assumes weak two-photon Rabi
frequencies $\Omega_{r}(\bm{r}_{0})<3\,\omega_{r}$, justifying the
Pendellösung (70) and small atomic clouds $\sigma_{x}\ll w_{0}$ to approximate
Gaussian beams by plane-waves.
## VI Conclusion
We present (3+1)D simulations and analytical models of a pulsed atomic Bragg
beamsplitter. Thereby, we characterize ubiquitous imperfections, like the
velocity dispersion and the population losses into higher diffraction orders.
We study the influence of four common temporal pulses (rectangular-,
Gaussian-, Blackman- and hyperbolic sech pulse). Clearly, the diffraction
efficiency and the fidelity benefit from Fourier-limited, smooth envelopes.
Much insight is gained from the analytical Demkov-Kunike model for a
hyperbolic secant pulse (67). It reveals the explicit dependence on the
multitude of physical parameters. Due to its similarity with a Gaussian pulse,
the diffraction efficiency (70) can also be used for it (78). For a large
parameter regime, the model is verified experimentally and matches the
velocity dispersion. The extended DK-model (76) matches also losses into
higher diffraction orders.
For a rectangular pulse, we have obtained explicit higher order diffraction
results from Kato degenerate perturbation theory, which provide insight in the
quasi-Bragg regime. Due to a renormalization of the effective Rabi frequency
in the beamsplitter manifold, one finds significant stretching of the optimal
$\pi$-pulse time, which has been seen experimentally [56]. We find this
stretching for all considered pulses in the quasi-Bragg regime and assume it
is universal.
Comparing Gauss-Laguerre beams with plane waves reduces the diffraction
efficiency and transfer fidelity, in general. The beam inhomogeneity becomes
relevant for $\sigma_{x}>w_{0}/10$. But even for smaller decentered clouds,
the fidelity suffers significantly. Currently, we investigate the aberrations
due to laser misalignment and transversal confinement, which will be reported
elsewhere.
###### Acknowledgements.
We like to thank Jan Teske for (3+1)D simulation of the initial Bose-Einstein-
condensate, Sven Abend and the members of the QUANTUS collaboration for
fruitful discussions. This work is supported by the DLR German Aerospace
Center with funds provided by the Federal Ministry for Economic Affairs and
Energy (BMWi) under Grant No. 50WM1957.
## Appendix A Comoving rotating frame
In quantum mechanics, a Galilean transformation is represented by the
displacement operator [86]
$\hat{G}(t)=e^{\frac{i}{\hbar}(\bm{\mathfrak{p}}\hat{\bm{r}}-\bm{\mathfrak{r}}(t)\hat{\bm{p}})}=e^{-\frac{i}{2\hbar}\bm{\mathfrak{p}}\bm{\mathfrak{r}}(t)}e^{\frac{i}{\hbar}\bm{\mathfrak{p}}\hat{\bm{r}}}e^{-\frac{i}{\hbar}\bm{\mathfrak{r}}(t)\hat{\bm{p}}}$
(88)
with a time-dependent coordinate
$\bm{\mathfrak{r}}(t)=\bm{\mathfrak{r}}_{0}+\bm{v}t$ and a momentum
$\bm{\mathfrak{p}}=m\bm{v}$. It transforms the corresponding Heisenberg
operators as
$\begin{pmatrix}\hat{\bm{r}}^{\prime}\\\
\hat{\bm{p}}^{\prime}\end{pmatrix}=\hat{G}\begin{pmatrix}\hat{\bm{r}}\\\
\hat{\bm{p}}\end{pmatrix}\hat{G}^{\dagger}=\begin{pmatrix}\hat{\bm{r}}-\bm{\mathfrak{r}}(t)\\\
\hat{\bm{p}}-\bm{\mathfrak{p}}\end{pmatrix}.$ (89)
In the Schrödinger picture, $\hat{G}(t)$ transforms the lab frame state
$\ket{\psi(t)}=\hat{G}(t)\ket{\psi^{\prime}(t)}$ into the state
$\ket{\psi^{\prime}(t)}$ of the comoving frame. Evaluating the comoving-frame
Hamilton operator $\hat{H}^{\prime}$ the Schrödinger equation reads
$\displaystyle
i\hbar\partial_{t}\ket{\psi^{\prime}}=\hat{H}\textquoteright\ket{\psi^{\prime}}=\hat{G}^{\dagger}(\hat{H}-i\hbar\partial_{t})\hat{G}\ket{\psi^{\prime}},$
(90)
$\displaystyle\hat{H}^{\prime}=\frac{\hat{\bm{p}}^{2}}{2M}+\hbar\omega_{g}\hat{\sigma}_{g}+\hbar\omega_{e}\hat{\sigma}_{e}+V(t,\hat{\bm{r}}+\bm{\mathfrak{r}}(t)).$
(91)
In the frame, moving with the group velocity $\bm{v}=v_{g}\bm{e}_{x}$ (12) in
the x-direction, the Doppler shifted laser phases
$\displaystyle\phi^{\prime}_{1}$
$\displaystyle=\omega_{1}t-k_{1}(\hat{x}+\mathfrak{x}_{0}+v_{g}t)=\omega_{L}t-k_{1}(\hat{x}+\mathfrak{x}_{0}),$
(92) $\displaystyle\phi^{\prime}_{2}$
$\displaystyle=\omega_{2}t+k_{2}(\hat{x}+\mathfrak{x}_{0}+v_{g}t)=\omega_{L}t+k_{2}(\hat{x}+\mathfrak{x}_{0})$
(93)
oscillate synchronously with
$\omega_{L}=\frac{\omega_{1}+\omega_{2}}{2}\left(1-\beta^{2}\right)\approx\frac{\omega_{1}+\omega_{2}}{2}.$
(94)
The second order correction in $\beta=v_{g}/c$ can be neglected safely in our
nonrelativistic scenario.
Another local frame transformation
$\ket{\psi^{\prime}}=\hat{F}\ket{\psi^{\prime\prime}}$, eliminates the rapid
temporal oscillations and establishes a single spatial period
$\lambda=2\pi/k_{L}$ of the optical potential
$\hat{F}(t)=e^{-i\omega_{g}t-i\omega_{L}t\hat{\sigma}_{e}+\frac{i}{2}[k_{12}(\hat{x}+\mathfrak{x}_{0})-\chi_{12}]\hat{\sigma}_{z}}.$
(95)
Now, the transformed Schrödinger equation reads
$\displaystyle\begin{split}&i\hbar\partial_{t}\ket{\psi^{\prime\prime}}=\hat{H}^{\prime\prime}\ket{\psi^{\prime\prime}},\end{split}$
(96)
$\displaystyle\begin{split}&\hat{H}^{\prime\prime}=\frac{(\hat{p}_{x}+\tfrac{1}{2}\hbar
k_{12}\hat{\sigma}_{z})^{2}}{2M}+\frac{\hat{p}^{2}_{y}+\hat{p}^{2}_{z}}{2M}-\hbar\Delta\hat{\sigma}_{e}\\\
&\phantom{+}+\frac{\hbar}{2}\hat{\sigma}^{\dagger}\left(\tilde{\Omega}_{1}(t,\hat{\bm{r}})e^{ik_{L}\hat{x}}+\tilde{\Omega}_{2}(t,\hat{\bm{r}})e^{-ik_{L}\hat{x}}\right)+\text{h.c.}\end{split}$
(97)
with a laser detuning $\Delta=\omega_{L}-\omega_{0}$, a common wavenumber
$k_{L}=(k_{1}+k_{2})/2$ and a relative wavenumber mismatch
$k_{12}=(k_{1}-k_{2})/2$. Global phases of the Rabi frequencies
$\Omega_{i}(t,\bm{r})=\tilde{\Omega}_{i}(t,\bm{r})e^{-i\chi_{i}}$ do vanish
with the proper gauge $\chi_{12}=(\chi_{1}+\chi_{2})/2$ and the shifted
coordinate origin $\mathfrak{x}_{0}=(\chi_{1}-\chi_{2})/2k_{L}$.
Please note,
$k_{12}=(\omega_{1}-\omega_{2})/c\sim$1\text{\times}{10}^{-10}\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$\sim$1\text{\times}{10}^{-11}\text{\,}\,$k_{L}$
is tiny in comparison to other relevant momenta. We will consider Bose-
Einstein condensates with Thomas-Fermi radii in the trap of a few microns (cf.
Sec. V and E.1), the momentum width can be approximated with the Heisenberg
width $\Delta
k_{\text{TF}}^{\text{H}}=3/2r_{\text{TF}}=$0.15\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$,
considering $r_{\text{TF}}=$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, while
the Rayleigh width gives $\Delta
k_{\text{TF}}^{\text{R}}=$0.51\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$
[87]. In our simulations, we consider atomic initial states as Gaussian
wavepackets with momentum widths $\sigma_{k}\in[0.01,0.05,0.1,0.2]\,k_{L}$,
with $k_{L}\approx$8\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$, to
compare $\Delta k_{\text{TF}}^{\text{R}}$ corresponds to
$\sigma_{k}^{\text{R}}\approx\Delta k_{\text{TF}}^{\text{R}}/3=0.02\,k_{L}$.
After release out of the trap the momentum width of the BEC increases. With
temperatures $T\leq$20\text{\,}\mathrm{nK}$$ this gives rise for momentum
widths of a thermal cloud
$\sigma_{k}=\sqrt{k_{B}TM}/\hbar\leq$0.23\text{\,}\,$k_{L}$. Therefore,
$k_{12}$ can be neglected safely.
## Appendix B Spreading Gaussian waves
##### Matter waves
Ballistically spreading Gaussian wavepackets are useful input states to test a
beamsplitter. Using different expansion times $t$, one can vary the position
width $\sigma_{x}$, while keeping the momentum width $\sigma_{k}$ constant. A
$n$-dimensional Gaussian unnormalized wavepacket is defined as
$\displaystyle\psi_{0}(\bm{r})=e^{i\bm{k}_{0}(\bm{r}-\bm{r}_{0})-\frac{1}{2}(\bm{r}-\bm{r}_{0})(2\Sigma_{0})^{-1}(\bm{r}-\bm{r}_{0})}$
(98)
$\displaystyle=\int\frac{\text{d}^{n}k}{(2\pi)^{\frac{n}{2}}}e^{i\bm{k}\bm{r}}\sqrt{|2\Sigma_{0}|}e^{-i\bm{k}\bm{r}_{0}-\frac{1}{2}(\bm{k}-\bm{k}_{0})(2\Sigma_{0})(\bm{k}-\bm{k}_{0})}$
and centered at
$(\bm{r}_{0},\bm{k}_{0})=(\langle\bm{r}\rangle,\langle-i\nabla\rangle)$. The
wavepacket is normalized to
$\int\text{d}^{n}r\,|\psi_{0}|^{2}=\sqrt{|2\pi\Sigma_{0}|}$ with the
covariance matrix
$\Sigma_{0}=\langle(\bm{r}-\bm{r}_{0})\otimes(\bm{r}-\bm{r}_{0})\rangle$. The
three dimensional free Schrödinger equation
$i\partial_{t}\psi(t,\bm{r})=-\frac{\alpha}{2}\Delta_{r}\psi,\qquad\alpha=\frac{\hbar}{M}$
(99)
describes the spreading of a matter-wave using the Fourier-transformed field
$\tilde{\psi}_{0}(\bm{k})$ implicitly defined in (98)
$\displaystyle\psi(t,\bm{r})=\int\frac{\text{d}^{n}k}{(2\pi)^{\frac{n}{2}}}e^{-it\frac{\alpha}{2}k^{2}}e^{i\bm{k}\bm{r}}\tilde{\psi}_{0}(\bm{k})$
(100)
$\displaystyle=\mathcal{A}(t)e^{-i\Theta(t)}e^{i\bm{k}_{0}[\bm{r}-\bm{r}_{0}]-\frac{1}{2}[\bm{r}-\bm{r}_{0}(t)][2\Sigma(t)]^{-1}[\bm{r}-\bm{r}_{0}(t)]}.$
The evolving center position $\bm{r}_{0}(t)$, spreading covariance
$\Sigma(t)$, dynamical phase $\Theta(t)$ and scale-factor $\mathcal{A}(t)$
read
$\displaystyle\Sigma(t)$ $\displaystyle=\Sigma_{0}+it\frac{\alpha}{2},$
$\displaystyle\bm{r}_{0}(t)$ $\displaystyle=\bm{r}_{0}+t\alpha\bm{k}_{0},$
(101) $\displaystyle\Theta(t)$ $\displaystyle=t\frac{\alpha k_{0}^{2}}{2},$
$\displaystyle\mathcal{A}(t)$
$\displaystyle=\sqrt{\frac{|\Sigma_{0}|}{|\Sigma(t)|}}.$ (102)
In the simulations, we assume an isotropic initial state with
$\Sigma_{ij}=\delta_{ij}\sigma_{x}^{2}$ and
$\sigma_{x}(t)=\sigma_{x}\sqrt{1+(t/t_{H})^{2}},$ (103)
with the Heisenberg time $t_{H}=2\sigma_{x}^{2}M/\hbar$.
##### Gaussian laser beams
The scalar mode of a circularly symmetric Gauss-Laguerre beam propagating
along the x-direction follows from the two-dimensional $n=2$ paraxial
approximation of the Helmholtz equation
$i\partial_{x}u(x,\bm{\varrho})=-\frac{\beta}{2}\Delta_{\varrho}u,\qquad\beta=k_{L}^{-1},\quad\bm{\varrho}=(y,z).$
(104)
The spatially evolved mode $u(x,\bm{\varrho})$ follows analogously from (100),
(101), substituting $(t,\alpha)\leftrightarrow(x,\beta)$
$\displaystyle
u(x,\bm{\varrho})=\frac{x_{R}}{iq(x)}e^{i\frac{k_{L}\varrho^{2}}{2q(x)}}=Ue^{i\Phi}$
(105) $\displaystyle
U(x,\varrho)=\frac{w_{0}}{w(x)}e^{-\frac{\varrho^{2}}{w(x)^{2}}},\quad\Phi(x,\varrho)=\frac{k_{L}\varrho^{2}}{2R(x)}-\xi(x),$
where $\varrho=\sqrt{y^{2}+z^{2}}$ is the normal distance to the symmetry axis
and $q(x)=x-ix_{R}$ is the complex beam parameter [83]. It is characterized by
the Rayleigh range $x_{R}=\pi w_{0}^{2}/\lambda$, the beam waist
$w(x)=w_{0}(1+(x/x_{R})^{2})^{1/2}$, the minimum waist $w_{0}=2\sigma$, the
radius of wavefront curvature $R(x)=x(1+(x_{R}/x)^{2})$, the Gouy phase
$\xi=\arctan(x/x_{R})$ and the wavelength $\lambda_{L}=2\pi/k_{L}$.
We consider two counterpropagating Gaussian laser beams, which are
symmetrically displaced with respect to their waists by a distance $\ell$.
Then, the dipole interaction energy in the comoving, rotating frame
(LABEL:eq:hamtrafoII), reads
$\hat{V}^{\prime\prime}=\frac{\hbar}{2}\hat{\sigma}^{\dagger}\left[\Omega_{1}(t,\bm{r})e^{ik_{L}x}+\Omega_{2}(t,\bm{r})e^{-ik_{L}x}\right]+\text{h.c.},$
(106)
with pulse amplitudes $\Omega_{j}(t)$ and spatial envelopes
$\Omega_{j}(t,\bm{r})=\Omega_{j}(t)U(x_{j},\varrho)e^{i\Phi(x_{j},\varrho)}.$
(107)
We use shifted coordinates $x_{1/2}=\pm(x+v_{g}t+\ell/2)$ and beam parameters
$w_{j}=w(x_{j})$, $R_{j}=R(x_{j})$ and $\xi_{j}=\xi(x_{j})$, which are slowly
varying for $x\ll x_{R}$. Beamsplitter pulses are typical short and one can
neglect the ballistic displacement $v_{g}t\sim$\mathrm{\SIUnitSymbolMicro
m}$\ll\ell,x_{R}$. For small atomic clouds $\sigma_{x}<w_{0}/3$, one can
approximate $x_{1}\approx-x_{2}\approx\ell/2$.
## Appendix C Degenerate perturbation theory
To rectify the Pendellösung (54) with contributions from higher order
diffraction, we employ Kato’s method for the stationary eigenvalue problem in
the presence of degeneracy [79]. All eigenvalues of the diagonal part
$D_{0}=D(\kappa=0)$ of the Bragg Hamilton operator (42) are doubly degenerated
$1\leq\alpha\leq 2$ on resonance. Therefore, we consider the flow of the
eigensystem
$\mathcal{H}(\lambda)\mathbf{v}_{i,\alpha}(\lambda)=\omega_{i,\alpha}(\lambda)\mathbf{v}_{i,\alpha}(\lambda)$
with
$\mathcal{H}=D_{0}+\lambda\mathcal{V},\qquad\mathcal{V}=D(\kappa)-D_{0}+L+L^{\dagger},$
(108)
for $0\leq\lambda\leq 1$ in the degenerate subspace $\mathcal{E}_{i}$. If we
denote the orthonormal eigenvectors of $D_{0}$ with
$\mathbf{v}_{i,\alpha}^{(0)}$ and their eigenvalues $\omega_{i}^{(0)}$, the
eigenvectors of the interacting Hamilton operator $\mathcal{H}_{i}(\lambda)$
restricted to the subspace $\mathcal{E}_{i}$, are
$\mathbf{v}_{i,\alpha}(\lambda)=P_{i}(\lambda)\mathbf{v}_{i,\alpha}^{(0)}$.
Now, all efforts are put in the perturbative evaluation of the projection
operator $P_{i}(\lambda)$, which evolves from the unperturbed projection
$P_{i}^{(0)}$. This results in the generalized eigenvalue problem
$\displaystyle\mathcal{H}_{i}\mathbf{v}_{i,\alpha}^{(0)}=\omega_{i,\alpha}K_{i}\mathbf{v}_{i,\alpha}^{(0)},$
(109)
$\displaystyle\mathcal{H}_{i}=P_{i}^{(0)}\mathcal{H}P_{i}P_{i}^{(0)},\quad
K_{i}=P_{i}^{(0)}P_{i}P_{i}^{(0)},$ (110)
with power series expressions for the operators
$\displaystyle P_{i}(\lambda)$
$\displaystyle=P_{i}^{(0)}+\sum_{n=1}^{\infty}\lambda^{n}A_{i}^{(n)},$ (111)
$\displaystyle A_{i}^{(n)}$
$\displaystyle=-\sum_{(n)}S_{i}^{(k_{1})}\mathcal{V}S_{i}^{(k_{2})}\mathcal{V}\dots\mathcal{V}S_{i}^{(k_{n+1})},$
(112) $\displaystyle\mathcal{H}P_{i}(\lambda)$
$\displaystyle=\omega^{(0)}_{i}P_{i}(\lambda)+\sum_{n=1}^{\infty}\lambda^{n}B_{i}^{(n)},$
(113) $\displaystyle B_{i}^{(n)}$
$\displaystyle=\sum_{(n-1)}S_{i}^{(k_{1})}\mathcal{V}S_{i}^{(k_{2})}\mathcal{V}\dots\mathcal{V}S_{i}^{(k_{n+1})}.$
(114)
Here $\sum_{(n)}$ denotes a sum over all combinations of integers
$k_{i}\in\mathbb{N}_{0}$ satisfying $k_{1}+k_{2}+\ldots+k_{n+1}=n$ and
$\vspace{-1mm}S_{i}^{(0)}=-P_{i}^{(0)},\
S_{i}^{(k>0)}=(S_{i})^{k},\,S_{i}=\frac{\mathds{1}-P_{i}^{(0)}}{\omega_{i}^{(0)}\mathds{1}-D_{0}}.$
(115)
It is straight forward to evaluate $\mathcal{H}_{i}$ and $\mathcal{K}_{i}$
from (110) for the ground-state manifold $i=1$ to order
$\mathcal{O}(\lambda^{n})$. We find that a third order truncation of the
series
$\displaystyle\mathcal{H}_{1}$
$\displaystyle=\begin{pmatrix}0&\frac{\Omega}{2}\\\
\frac{\Omega}{2}&\kappa\end{pmatrix}-2\mathcal{I}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-\mathcal{I}\begin{pmatrix}\kappa&\Omega\\\
\Omega&0\end{pmatrix},$ (116) $\displaystyle K_{1}$
$\displaystyle=(1-\mathcal{I})\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-\mathcal{I}\begin{pmatrix}\kappa&\frac{\Omega}{2}\\\
\frac{\Omega}{2}&-\kappa\end{pmatrix},\quad\mathcal{I}=\frac{\Omega^{2}}{16}$
agrees very good with the numerical results. The roots of the characteristic
equation $|\mathcal{H}_{1}-(\omega_{1}-\omega_{1}^{(0)})K_{1}|=0$, determine
the corrected eigenfrequencies of the Pendelösung. As the frequency shifts
$\omega_{1}(\lambda)-\omega_{1}^{(0)}$, are already $\mathcal{O}(\lambda)$, it
is consistent to use a lower approximation for $K_{1}$, which leads to better
results at the specified order. In particular, we have evaluated
$\tilde{\mathcal{H}}_{1}=K_{1}^{-1}\mathcal{H}_{1}$ and Taylor expanded it at
the specified order
$\tilde{\mathcal{H}}_{1}=\begin{pmatrix}-\mathcal{I}(2+\kappa)&\frac{\Omega}{2}(1-\mathcal{I})\\\
\frac{\Omega}{2}(1-\mathcal{I})&\kappa-\mathcal{I}(2-\kappa)\end{pmatrix}+\order{\lambda^{4}}.$
(117)
This leads to the succinct expression for the eigenvalues and -vectors
$\displaystyle\omega_{1,\pm}=\frac{\kappa}{2}-2\mathcal{I}\pm\frac{\tilde{\Omega}}{2},\quad\mathbf{v}_{1,\pm}^{(0)}=\begin{pmatrix}2(\mathcal{I}-1)\sqrt{\mathcal{I}}\\\
-\tfrac{1}{2}\kappa(1+2\mathcal{I})\pm\tilde{\Omega}\end{pmatrix},$
$\displaystyle\tilde{\Omega}=\sqrt{\kappa^{2}(1+2\mathcal{I})^{2}+\Omega^{2}(1-\mathcal{I})^{2}}$
(118)
in terms of a corrected Rabi frequency $\tilde{\Omega}$ (59).
Analogous to that the eigenvalues of the next subspace, coupling $\mu=\pm 3$
and representing the most important loss channel, can be calculated from
$\tilde{\mathcal{H}}_{3}=K_{3}^{-1}\mathcal{H}_{3}$
$\tilde{\mathcal{H}}_{3}=\begin{pmatrix}2(1+\mathcal{I})-\kappa&0\\\
0&2(1+\mathcal{I})+2\kappa\end{pmatrix}+\order{\lambda^{3}},$ (119)
skipping the $\lambda^{3}$ terms, which overestimate the losses into $\mu=\pm
3$. Including higher expansion orders would correct this, but we find that the
lower expansion (119) is sufficient. The eigenvalues and -vectors of
$\tilde{\mathcal{H}}_{3}$ are
$\omega_{3,\pm}=2(1+\mathcal{I})+\frac{\kappa}{2}\pm\frac{3\kappa}{2},\qquad(\mathbf{v}_{3,+}^{(0)},\mathbf{v}_{3,-}^{(0)})=\mathds{1}_{2}.$
(120)
With the eigenvectors $\mathbf{v}_{i,j}=P_{i}\mathbf{v}_{i,j}^{(0)}$, defined
by the projections (111), also expanded up to $\lambda^{3}$ for $\mu=\pm 1$
and $\lambda^{2}$ for $\mu=\pm 3$, the time-dependent solution of the Schrö-
dinger equation with the Hamiltonian (108) results in
$\displaystyle\bm{g}^{K}(\tau)$
$\displaystyle=\frac{\tilde{\bm{g}}^{K}(\tau)}{|\tilde{\bm{g}}^{K}(\tau)|^{2}},$
(121) $\displaystyle\tilde{\bm{g}}^{K}(\tau)$
$\displaystyle=\sum_{i=\\{1,3\\}}\sum_{j=\\{+,-\\}}c_{i,j}e^{-i\omega_{i,j}(\tau-\tau_{i})}\mathbf{v}_{i,j},$
(122)
where the integration constants $c_{i}^{(j)}$ are defined by the initial
condition $\tilde{\bm{g}}^{K}(\tau)=(0,1,0,0)$.
The population of the $\mu=1$ state is of special interest, because it defines
the diffraction efficiency $\eta_{+-}$. On resonance ($\kappa=0$), already
$\tilde{\bm{g}}^{K}(\tau)$ is approximately normalized. Therefore, it can be
approximated
$\displaystyle\eta_{0}^{K}(\tau)\approx|\left(\tilde{\bm{g}}^{K}(\tau)\right)_{3}|^{2}$
(123)
$\displaystyle=A\left(1+B\cos[4\tau^{\prime}(\mathcal{I}-1)\sqrt{\mathcal{I}}]+C\cos\theta_{+}+D\cos\theta_{-}\right)\\!,$
with
$\theta_{\pm}=2\tau^{\prime}(1\pm\sqrt{\mathcal{I}}+2\mathcal{I}-\mathcal{I}^{3/2})$,
$\tau^{\prime}=\tau-\tau_{i}$ and coefficients expanded up to the suited order
$\order{\mathcal{I}^{2}}$
$\displaystyle A$
$\displaystyle=\frac{1}{2}-\mathcal{I}-\frac{\mathcal{I}^{2}}{2}+\order{\mathcal{I}^{3}},\quad
B=-1+\order{\mathcal{I}^{3}},$ (124) $\displaystyle C$
$\displaystyle=-D=-4\mathcal{I}^{3/2}+\mathcal{O}(\mathcal{I}^{5/2}).$
After the effective $\pi$-pulse time
$\tilde{\tau}_{R\pi}=\pi/[2|\Omega|(1-\mathcal{I})]$ (61) the diffraction
efficiency (123) results in Eq. (62).
## Appendix D Demkov-Kunike model
The retarded Green’s function is defined as
$\displaystyle
G(\tau,\tau_{i})=\mathcal{T}e^{-i\int^{\tau}_{\tau_{i}}\text{d}t\,\mathcal{H}(t)}\theta(\tau-\tau_{i}),$
(125)
$\displaystyle[i\partial_{\tau}-\mathcal{H}(\tau)]G(\tau,\tau_{i})=i\delta(\tau-\tau_{i}),$
(126)
which hold equally for the free evolution $G_{0}(\tau,\tau_{i})$ by
substituting $\mathcal{H}\rightarrow\mathcal{H}_{0}$. This leads to the Dyson-
Schwinger integral equation
$G(\tau,\tau_{i})=G_{0}-i\int^{\infty}_{-\infty}\text{d}t\,G_{0}(\tau,t^{\prime})\mathcal{H}_{1}(t^{\prime})G(t^{\prime},\tau_{i}),$
(127)
which is central to time-dependent perturbation theory.
The two-dimensional Green’s function $G_{\mp}$ of the DK-model can be
expressed completely for $\Omega,\kappa\neq 0$ with the hypergeometric basis
functions $f_{1},f_{2}$ form Eq. (64)
$\displaystyle G_{\mp}(\tau,\tau_{i})=M(z)S(z)S^{-1}(z_{i})M^{-1}(z_{i}),$
(128) $\displaystyle M=\begin{pmatrix}1&0\\\
0&\frac{i}{a}\sqrt{z(1-z)}\end{pmatrix},\quad S=\begin{pmatrix}f_{1}&f_{2}\\\
f_{1}^{\prime}&f_{2}^{\prime}\end{pmatrix}.$ (129)
In the important case of exact resonance $\kappa=0$, further simplifications
are possible and lead to
$\displaystyle
G_{\mp}(\tau,\tau_{i})=\begin{pmatrix}\cos{\Delta\varphi}&-i\sin{\Delta\varphi}\\\
-i\sin{\Delta\varphi}&\cos{\Delta\varphi}\end{pmatrix},$ (130)
$\displaystyle\varphi(z)=\Omega\tau_{S}\arcsin{\sqrt{z}},\qquad\Delta\varphi=\varphi(z)-\varphi(z_{i}).$
(131)
The integrals (127) can be solved approximately analytically. However, the
expressions are bulky, why we forgo showing them [81].
## Appendix E Diffraction efficiency for partially coherent bosonic fields
The bosonic amplitude $\hat{a}_{g}(\mathbf{k})$ describes the ground state
atoms in momentum space and obeys the commutation relation
$[\hat{a}_{g}^{\phantom{\dagger}}(\mathbf{k}),\hat{a}_{g}^{\dagger}(\mathbf{q})]=\delta(\mathbf{k}-\mathbf{q})$.
For a Bose-condesed sample, the single-particle density matrix
$\rho(\mathbf{k},\mathbf{q})\equiv\langle\hat{a}_{g}^{\dagger}(\mathbf{q})\hat{a}_{g}(\mathbf{k})\rangle=\rho^{c}(\mathbf{k},\mathbf{q})+\rho^{t}(\mathbf{k},\mathbf{q}),$
(132)
separates into a condensate
$\rho^{c}(\mathbf{k},\mathbf{q})=\alpha^{\ast}(\mathbf{q})\alpha(\mathbf{k})$
and a quantum depletion $\rho^{t}(\mathbf{k},\mathbf{q})$. The momentum
density
$n(\mathbf{k})\equiv\rho(\mathbf{k},\mathbf{k})=N^{A}\left[p^{c}\mathfrak{n}^{c}(\mathbf{k})+p^{t}\mathfrak{n}^{t}(\mathbf{k})\right],$
(133)
is the observable in a beamsplitter. It is normalized to the total number of
$N^{A}=\int_{-\infty}^{\infty}\text{d}^{3}k\,n(\mathbf{k})=N^{c}+N^{t}$ atoms,
the densities $\mathfrak{n}^{c},\mathfrak{n}^{t}$ are probability normalized,
thus defining a condensate fraction $p^{c}=N^{c}/N^{A}$ and a thermal fraction
$p^{t}=N^{t}/N^{A}$. Dynamically, the classical field $\alpha(t)$ obeys the
Gross-Pitaevskii equation and extensions thereof for $\rho^{t}(t)$ [88, 89,
90].
During the short beamsplitter pulse ($<$1\text{\,}\mathrm{ms}$$), only single
particle dynamics (16) are relevant
$\rho(\tau)=G(\tau,\tau_{i})\rho(\tau_{i})G^{\dagger}(\tau,\tau_{i}),$ (134)
for the condensate and the thermal cloud. In the plane-wave approximation, the
three-dimensional Fourier propagator
$\mathcal{G}(\bm{k},\bm{q})=\mathcal{G}_{\parallel}\mathcal{G}_{\perp}$ (18)
factorizes into the transverse propagator
$\mathcal{G}_{\perp}(\tau,\mathbf{k}_{\perp},\mathbf{q}_{\perp})=e^{-i\frac{\hbar(k_{y}^{2}+k_{z}^{2})}{2M}\tau}\delta^{(2)}(\mathbf{k}_{\perp}-\mathbf{q}_{\perp}),$
(135)
and the longitudinal Greens function in x-direction
$G_{\parallel}(\tau,x,\xi)=\sum_{\mu,\nu,n}\tfrac{\mathcal{G}_{\mu,\nu}(\kappa_{n},\tau)}{N_{x}a_{x}}e^{i(k_{\mu}^{n}x-k_{\nu}^{n}\xi)},$
(136)
using definitions (35), (36). The discrete Greens-matrix
$\mathcal{G}_{\mu,\nu}(\tau,\kappa_{n})$ satisfies (40) with initial condition
$\mathcal{G}_{\mu,\nu}(0,\kappa_{n})=\delta_{\mu,\nu}$. In the continuum
limit, one uncovers the momentum conservation on a lattice with
$k_{x}=(\mu+\kappa)k_{L}$ and $q_{x}=(\nu+\kappa^{\prime})k_{L}$, from the
Fourier transformation
$\mathcal{G}_{\parallel}(\tau,k_{x},q_{x})=\delta(\kappa-\kappa^{\prime})\mathcal{G}_{\mu,\nu}(\tau,\kappa).$
(137)
All observables are along the x-direction. Thus, we average over the
transversal directions and introduce the marginal momentum densities at time
$\tau$
$n(\tau,k_{x})=\int_{-\infty}^{\infty}\text{d}k_{y}\text{d}k_{z}\,n(\tau,\mathbf{k}).$
(138)
We assume that the initial ensemble is well localized around
$k_{x}=(\nu+\kappa)k_{L}$ with $\nu=-1$, and denote the density by
$n_{i}(\kappa)=n(\tau_{i},k_{x})$. From the propagation equation (134), one
obtains the final density $n_{f}(\kappa)=n(\tau_{f},k_{x})$, with
$k_{x}=(\mu+\kappa)k_{L}$ at diffraction order $\mu$
$n_{f}(\mu,\kappa)=|\mathcal{G}_{\mu,-1}(\tau_{f},\kappa)|^{2}n_{i}(\kappa).$
(139)
Now, we can identify the diffraction efficiency as
$\eta_{+-}(\kappa)=|\mathcal{G}_{1,-1}(\tau_{f},\kappa)|^{2}$ and
$\eta_{--}(\kappa)=|\mathcal{G}_{-1,-1}(\tau_{f},\kappa)|^{2}$. Thus, for
atomic clouds with initial momentum
$\langle\hat{p}_{x}\rangle=(-1+\bar{\kappa})\hbar k_{L}$ (84), the number of
diffracted atoms read
$N_{\pm}(\bar{\kappa})=\int\limits_{-1}^{1}\text{d}\kappa\,\eta_{\pm-}(\kappa)n_{i}(\kappa,\bar{\kappa}),$
(140)
which are the observables in $1^{st}$ order diffraction theory.
### E.1 Initial momentum distribution
After release from the trap, the width of the BEC in momentum space increases
due to atomic mean-field interaction [91]. The momentum distribution is
determined by solving the (3+1)D Gross-Pitaevskii equation for the given
parameters of Tab. 2 and $10\text{\,}\mathrm{ms}$ time-of-flight before the
diffraction pulses. The result is confirmed by the scaling approach [92, 93,
94, 95] applied to the numerical Gross-Pitaevskii ground state. Finally, the
marginal, one-dimensional momentum density distribution of the BEC to begin of
the diffraction pulses $\mathfrak{n}_{i}^{c}\approx\tilde{\mathfrak{n}}^{c}$
(138), can be approximated with a Gaussian distribution
$\tilde{\mathfrak{n}}(\kappa,\bar{\kappa})=\frac{1}{\sqrt{2\pi}\tilde{\sigma}_{k}}e^{-\frac{(\kappa-\bar{\kappa})^{2}}{2(\tilde{\sigma}_{k})^{2}}},\quad\int_{-\infty}^{\infty}\negmedspace\text{d}\kappa\,\tilde{\mathfrak{n}}(\kappa,\bar{\kappa})=1,$
(141)
with the dimensionless momentum width $\tilde{\sigma}_{k}=\sigma_{k}/k_{L}$
and $\sigma_{k}^{c}=0.087\,k_{L}$, as depicted in Fig. 15.
The thermal cloud is also approximately a Gaussian distribution [85], where
the one-dimensional momentum width $\sigma_{k}^{t}=\sqrt{Mk_{B}T}/\hbar$
introduces a temperature $T$. Experimentally, time-of-flight measurements of
$\sigma_{x}(t)$ (103) lead to the momentum width $\sigma_{k}^{t}=(0.237\pm
0.015)\,k_{L}$ of $\mathfrak{n}^{t}$ (141) (cf. Fig. 15) and temperature
$T=$20\pm 3\text{\,}\mathrm{K}$$. The horizontal trap direction
$x^{\prime}=x\cos\phi$, $\phi=5.5^{\circ}\pm 1^{\circ}$ differs slightly from
the beamsplitter direction $x$. However, the resulting difference in the
momentum width $|\sigma_{k_{x}}-\sigma_{k_{x}^{\prime}}|=0.001\,k_{L}$ is
negligible within the uncertainty.
FIG. 15: One-dimensional density $\mathfrak{n}(\kappa)=p^{c}\mathfrak{n}^{c}+p^{t}\mathfrak{n}^{t}$ (133) ($p^{t}\\!=\\!0.51,p^{c}\\!=\\!0.49$) versus momentum detuning $\kappa$. The thermal cloud $\mathfrak{n}^{t}$ as well as the condensate $\mathfrak{n}^{c}$ obtained from (3+1)D GP simulation can be approximated with a Gaussian distribution $\mathfrak{n}^{a=\\{c,t\\}}\approx\tilde{\mathfrak{n}}^{a}$ (141). Table 2: Experimental parameters: On the $J=\nicefrac{{1}}{{2}}\rightarrow J^{\prime}=\nicefrac{{3}}{{2}}$ transition, far-detuned, linearly polarized light couples only to one component of the dipole operator. Therefore, the transition strength is reduced by $\sqrt{3}$. Quantity | | Symbol | | Value | | Reference
---|---|---|---|---|---|---
Atom
Number of atoms in condensate | | $N_{c}$ | | $(10\pm 1.)\text{\times}{10}^{3}$ | |
Number of atoms in thermal cloud | | $N_{t}$ | | $(7\pm 1.)\text{\times}{10}^{3}$ | |
Atomic mass | | $M$ | | 86.909 180 520(15) u | | [96]
Transition frequency Rb-87 D2 | | $\omega_{0}$ | | $2\pi\times 384.230\,484\,468\,5(62)\,$\mathrm{THz}$$ | | [97]
Lifetime | | $\tau$ | | $26.2348\pm 0.0077\text{\,}\mathrm{ns}$ | | [98]
Decay rate | | $\Gamma$ | | $2\pi\times$6.0666\pm 0.0018\text{\,}\mathrm{MHz}$$ | |
D2 ($5^{2}S_{\nicefrac{{1}}{{2}}}\rightarrow 5^{2}P_{\nicefrac{{3}}{{2}}}$) transition dipole matrix element | | $\mathcal{D}$ | | $3.58424(52)\times 10^{-29}\,$\mathrm{C}\text{\,}\mathrm{m}$$ | | [98]
Rabi-frequency | | $\Omega_{0}$ | | $\mathcal{E}_{0}\mathcal{D}/\hbar\sqrt{3}$ | |
Scattering length | | $a$ | | $98.96\,a_{0}$ | | [99]
Trap frequencies | | $[\omega_{x},\omega_{y},\omega_{z}]$ | | $2\pi\times[46\pm 2,18\pm 1,31\pm 1]\,$\mathrm{Hz}$$ | |
Thomas-Fermi radii inside trap | | $[r_{x},r_{y},r_{z}]$ | | $[4.2,10.8,6.2]\,$\mathrm{\SIUnitSymbolMicro m}$$ | |
Laser
Wavelength | | $\lambda_{L}$ | | $780.024\,500\,015\text{\,}\mathrm{nm}$ | |
Wavenumber | | $k_{L}$ | | $8.056\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$ | |
Detuning to atomic resonance | | $\Delta$ | | $97.875\text{\,}\mathrm{GHz}$ | |
Beam waist | | $w_{0}$ | | $1.386\text{\,}\mathrm{mm}$ | |
Rayleigh length | | $x_{R}$ | | $7.7\text{\,}\mathrm{m}$ | |
Total interaction time | | $\Delta t$ | | $(10^{2}...10^{3})$\mathrm{\SIUnitSymbolMicro s}$$ | |
Gaussian pulse width (47) | | $\tau_{G}$ | | $\Delta t/8$ | |
Distance between laser origins | | $\ell$ | | $0.1\,x_{R}$ | |
Total laser power | | $P$ | | $\mathcal{E}_{0}^{2}\epsilon_{0}\pi cw_{0}^{2}/4$ | |
Laser amplitude | | $\mathcal{E}_{0}$ | | | |
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|
# On Fuzzification of $n$-Lie Algebras
Shadi Shaqaqha Department of Mathematics, Yarmouk University, Irbid, Jordan
<EMAIL_ADDRESS>
###### Abstract.
The aim of this paper is to introduce the notion of intuitionistic fuzzy Lie
subalgebras and intutionistic fuzzy Lie ideals of $n$-Lie algebras. It is a
generalization of intuitionistic fuzzy Lie algebras. Then, we investigate some
of characteristics of intutionistic fuzzy Lie ideals (resp. subalgebras) of
$n$-Lie algeras. Finally, we define the image and the inverse image of
intuitionistic fuzzy Lie subalgebra under $n$-Lie algebra homomorphism. The
properties of intuitionistic fuzzy $n$-Lie subalgebras and intuitionistic
fuzzy Lie ideals under homomorphisms of $n$-Lie algebras are studied. Finally,
we define the intuitionistic fuzzy quotient $n$-Lie algebra by an
intuitionistic fuzzy ideal of n-Lie algebra and prove that it is a $n$-Lie
algebra.
###### Key words and phrases:
$n$-Lie algebras; $n$-Lie homomorphism; intutionistic fuzzy set;
intuitionistic fuzzy $n$-Lie subalgebra; intuitionistic fuzzy $n$-Lie ideal.
###### 2000 Mathematics Subject Classification:
08A72, 03E72, 20N25
## 1\. Introduction
The concept of $n$-Lie algebras was introduced by Filippov [15] in 1985. If
$n=2$, then we get Lie algebra structure which was introduced by Sophus Lie
(1842-1899) while he was attempting to classify certain smooth subgroups of
general linear groups that are now called Lie groups. The case where $n=3$ was
initially appeared in Nanmbu’s work [26] while he was attempting to describe
simultaneous classical dynamics of three particles. Takhtajan [29]
investigated the geometrical and algebraic aspects of the generalized Nambu
mechanics, and established the connection between the Nambu mechanics and
Filippov’s theory of $n$-Lie algebras. More applictions to the $n$-Lie
algebras can be found in [7, 16, 17, 23, 24, 25, 28, 34]. So, our results are
expected to be useful in various applications.
The notion of fuzzy sets was firstly introduced by Zadeh [37]. The fuzzy set
theory states that there are propositions with an infinite number of truth
values, assuming two extreme values, $1$ (totally true), $0$ (totally false)
and a continuum in between, that justify the term fuzzy. Applications of this
theory can be found, for example, in artificial intelligence, computer
science, control engineering, decision theory, logic and management science.
After introducing of fuzzy sets by Zadeh, many researches were conducted on
the generalizations of this fundamental concept. Among these generalizations
was the concept of intuitionistic fuzzy sets, which was introduced by
Attanasov [8] in 1986, is the most important and interesting one. The elements
of the intuitionistic fuzzy sets are distinguished by an additional degree
called the degree of uncertainty. There are numerous applications to this new
concept include computer science, mathematics, medicine, chemistry, economics,
astronomy etc.. Many mathematicians have involved in extending the concept of
intuitionistic fuzzy sets to border of abstract algebra. Biswas applied the
concepts of intuitionistic fuzzy sets to the theory of groups and studied
intuitionistic fuzzy subgroups in [10]. Also, Akram studied Lie algebra in
intuitionistic fuzzy sets and obtained some results in [3].
The fuzzy Lie subalgebras and fuzzy Lie ideals are considered in [21] by Kim
and Lee, and in [35, 36] by Yehia. They established the analogues of most of
the fundamental ground results involving Lie algebras in the fuzzy setting.
The study of fuzzy subalgebras (resp. ideals) of $n$-Lie algebras was
initiated by B. Davvaz and WA. Dudek [13]. Recently, the complex
(intuitionistic) fuzzy Lie algebras is studied in [32, 33] as a generalization
of (intutionistic) fuzzy Lie algebras.
In this paper we describe intuitionistic fuzzy n-Lie algebras. We will
introduce n-Lie algebras into intuitionistic fuzzy set. Our work will
generalize the theory of (intu- itionstic) fuzzy Lie algebras ([3, 4, 5, 6,
33, 35, 36]).
## 2\. Preliminaries
$n$-Lie algebras were originally introduced by Filippov [15] in 1985. They
generalize Lie algebras. In this article the ground field $F$ is arbitrary.
###### Definition 2.1.
Let $n\in\mathbb{N}$, $n\geq 2$. An $n$-Lie algebra is a pair $(L,[~{}])$
where $L$ is a vector space and
$[~{}]:L^{n}\rightarrow L;~{}(x_{1},\ldots,x_{n})\mapsto[x_{1},\ldots,x_{n}]$
is an $n$-linear map, called $n$-Lie bracket, that satisfies the following
identities for all $\sigma$ in the symmetric group $S_{n}$ and
$x_{1},\ldots,x_{n},y_{2},\ldots,y_{n}\in L$:
* (i)
Skew symmetry:
$[x_{\sigma(1)},\ldots,x_{\sigma(n)}]=\mathrm{sign}(\sigma)[x_{1},\ldots,x_{n}].$
* (ii)
The generalized Jacobi identity (called also the Filippov identity):
$[[x_{1},\ldots,x_{n}],y_{2},\ldots,y_{n}]=\sum_{i=1}^{n}[x_{1},\ldots,x_{i-1},[x_{i},y_{2},\ldots,y_{n}],x_{i},\ldots,x_{n}].$
Subalgebras of $n$-Lie algebras and homomorphisms (or isomorphisms) between
$n$-Lie algebras are defined as usual ([15]).
Throughout this paper, $L$ is a $n$-Lie algebra over field $F$.
The concept of intuitionistic fuzzy set was introduced by Atanassov [8], where
he added a new component (which determines the degree of non-membership) in
the definition of fuzzy set (FS) that was given by Zadeh. Let $X$ be a non-
empty set, and let
$A=(\mu_{A},\lambda_{A})=\\{(x,\mu_{A}(x),\lambda_{A}(x))~{}:~{}x\in X\\}$
where $\mu_{A}:X\rightarrow[0,1]$ and $\lambda_{A}:X\rightarrow[0,1]$ be
mappings such that $\mu_{A}(x)+\lambda_{A}(x)\leq 1$. Then $A$ is called an
intuitionistic fuzzy set. In this case the mappings $\mu_{A}$ and
$\lambda_{A}$ denote the degree of membership and the degree of non-membership
to $A$ respectively, for each element $x\in X$. The value
$\pi_{A}(x)=1-\mu_{A}(x)-\lambda_{A}(x)$ is called uncertainty or
intuitionistic index of the element $x\in X$ to the intuitionistic fuzzy set
$A$. It is obvious that each fuzzy set $A=\mu_{A}=\\{(x,\mu_{A}(x))~{}:~{}x\in
X\\}$ can be represented as an intuitionistic fuzzy set where
$A=\\{(x,\mu_{A}(x),1-\mu_{A}(x))~{}:~{}x\in X\\}$.
###### Definition 2.2.
Let $A=(\mu_{A},\lambda_{A})$ and $B=(\mu_{B},\lambda_{B})$ be two
intuitionistic fuzzy sets of a set $X$. Then
* (i)
The complement of $A$ is $\bar{A}=(\lambda_{A},\mu_{A})$,
* (ii)
$A\subseteq B$ if and only if $\mu_{A}(x)\leq\mu_{B}(x)$ and
$\lambda_{A}(x)\geq\lambda_{B}(x)$ for all $x\in X$,
* (iii)
the intersection of $A$ and $B$ is $A\cap
B=\\{(x,\mathrm{min}\\{\mu_{A}(x),\mu_{B}(x)\\},\mathrm{max}\\{\lambda_{A}(x),\lambda_{B}(x)\\}~{}|~{}x\in
X$,
* (iv)
the union of $A$ and $B$ is $A\cup
B=\\{(x,\mathrm{max}\\{\mu_{A}(x),\mu_{B}(x)\\},\mathrm{min}\\{\lambda_{A}(x),\lambda_{B}(x)\\}~{}|~{}x\in
X$,
* (v)
$\Box A=\\{(x,~{}\mu_{A}(x),~{}\mu_{A}^{\complement}(x)):~{}x\in X\\}$ where
$\mu_{A}^{\complement}(x)=(1-\mu_{A}(x))$ for all $x\in X$,
* (vi)
$\diamondsuit
A=\\{(x,~{}\lambda_{A}^{\complement}(x),~{}\lambda_{A}(x)):~{}x\in X\\}$ where
$\lambda_{A}^{\complement}(x)=(1-\lambda_{A}(x))$ for all $x\in X$.
Let $L$ be a $n$-Lie algebra over $F$. A fuzzy set
$A=\mu_{A}=\\{(x,~{}\mu_{A}(x)):x\in L\\}$ on $L$ is a fuzzy Lie subalgebra if
the following conditions are satisfied:
* (i)
$\mu_{A}(x+y)\geq\mathrm{min}\\{\mu_{A}(x),\mu_{A}(y)\\}$ for all $x,y\in L$,
* (ii)
$\mu_{A}(\alpha x)\geq\mu_{A}(x)$ for all $x\in L$ and $\alpha\in F$,
* (iii)
$\mu_{A}([x_{1},~{}x_{2},\ldots,x_{n}])\geq\mathrm{min}\\{\mu_{A}(x_{1}),\mu_{A}(x_{2}),\cdots,\mu_{A}(x_{n})\\}$
for all $x_{1},~{}x_{2},\ldots,~{}x_{n}\in L$.
It is called a fuzzy Lie ideal if the condition
$\mu_{A}([x_{1},~{}x_{2},\ldots,x_{n}])\geq\mathrm{min}\\{\mu_{A}(x_{1}),\mu_{A}(x_{2}),\cdots,\mu_{A}(x_{n})\\}$
is replaced by
$\mu_{A}([x_{1},~{}x_{2},\ldots,x_{n}])\geq\mathrm{max}\\{\mu_{A}(x_{1}),\mu_{A}(x_{2}),\cdots,\mu_{A}(x_{n})\\}$
([13]).
## 3\. Intuitionistic Fuzzy $n$-Lie Algebras
For the sake of simplicity, we shall use the symbols $a\wedge
b=\mathrm{min}\left\\{a,b\right\\}$ and $a\vee
b=\mathrm{max}\left\\{a,b\right\\}$.
###### Definition 3.1.
Let $A=(\mu_{A},\lambda_{A})=\\{(x,~{}\mu_{A}(x),~{}\lambda_{A}(x)):x\in L\\}$
be an intuitionistic fuzzy set of $L$. Then $A$ is called an intuitionistic
fuzzy Lie subalgebra of $L$ if the following conditions are satisfied:
* (i)
$\mu_{A}(x+y)\geq\mu_{A}(x)\wedge\mu_{A}(y)$ and
$\lambda_{A}(x+y)\leq\lambda_{A}(x)\vee\lambda_{A}(y)$ for all $x,y\in L$,
* (ii)
$\mu_{A}(\alpha x)\geq\mu_{A}(x)$ and $\lambda_{A}(\alpha
x)\leq\lambda_{A}(x)$ for all $x\in L$ and $\alpha\in F$,
* (iii)
$\mu_{A}([x_{1},~{}x_{2},\ldots,~{}x_{n}])\geq\mu_{A}(x_{1})\wedge\mu_{A}(x_{2})\wedge\cdots\wedge\mu_{A}(x_{n})$
and
$\lambda_{A}([x_{1},~{}x_{2},\ldots,~{}x_{n}])\leq\lambda_{A}(x_{1})\vee\lambda_{A}(x_{2})\vee\cdots\vee\lambda_{A}(x_{n})$
for all $x_{1},x_{2},\ldots,x_{n}\in L$.
An intuitionistic fuzzy set $A$ on $L$ is called an intuitionistic fuzzy Lie
ideal if the conditions $(i)$ and $(ii)$ are satisfied together with the
following addition condition:
$(iii)^{\prime}\mu_{A}([x_{1},~{}x_{2},\ldots,~{}x_{n}])\geq\mu_{A}(x_{1})\vee\mu_{A}(x_{2})\vee\cdots\vee\mu_{A}(x_{n})$
and
$\lambda_{A}([x_{1},~{}x_{2},\ldots,~{}x_{n}])\leq\lambda_{A}(x_{1})\wedge\lambda_{A}(x_{2})\wedge\cdots\wedge\lambda_{A}(x_{n})$
for all $x_{1},x_{2},\ldots,x_{n}\in L$.
In the special case that $n=2$, we obtain intuitionistic fuzzy Lie algebras
([3]). When first two conditions hold, we say that $A$ is an intuitionistic
fuzzy vector subspace of $L$. The second condition implies
$\mu_{A}(0)\geq\mu_{A}(x)$, $\mu_{A}(-x)\geq\mu_{A}(x)$,
$\lambda_{A}(0)\leq\lambda_{A}(x)$, and $\lambda_{A}(-x)\leq\lambda_{A}(x)$
for all $x\in L$. Also, every intuitionistic fuzzy Lie ideal of an $n$-Lie
algebra is an intuitionistic fuzzy Lie subalgebra, but the converse is not
necessary true ( see [32, Example 3.1]).
Let $\\{A_{i}=(\mu_{A_{i}},\lambda_{A_{i}})~{}|~{}i\in I\\}$ be a collection
of intuitionistic fuzzy sets on a nonempty set $X$. Then
$\bigcap_{i\in I}A_{i}=\\{(x,~{}\mu_{\bigcap_{i\in
I}A_{i}}(x),~{}\lambda_{\bigcap_{i\in I}A_{i}}(x)):~{}x\in X\\},$
where
$\mu_{\bigcap_{i\in I}A_{i}}(x)=\inf_{i\in I}\\{\mu_{A_{i}}(x)\\}$
and
$\lambda_{\bigcap_{i\in I}A_{i}}(x)=\sup_{i\in I}\\{\lambda_{A_{i}}(x)\\},$
is an intuitionistic fuzzy set too (see [33]). We shall give the proof of the
following theorem, established in [32] to the case of intuitionistic fuzzy Lie
algebras, which proves that the arbitrary intersection of intuitionistic fuzzy
Lie subalgebras (resp. ideals) of an $n$-Lie algebra $L$ is an intuitionistic
fuzzy Lie subalgebras (resp. ideal) of $L$ too. However it was proved in the
case that $L$ is a Lie (super)algebras and where the family is finite (see
[11]).
###### Theorem 3.1.
Let $\\{A_{i}\\}_{i\in I}$ $(A_{i}=(\mu_{A_{i}},\lambda_{A_{i}}),i\in I)$ be a
collection of intuitionistic fuzzy Lie subalgebras (resp. ideals) on $L$. Then
$\bigcap_{i\in I}A_{i}$ is an intuitionistic fuzzy Lie subalgebra (resp.
ideal) of $L$.
Proof. Here we will prove the case of intuitionistic fuzzy Lie subalgera. For
$x,~{}y\in L$ and $\alpha\in F$, we have
$\displaystyle\mu_{\bigcap_{i\in I}A_{i}}(x+y)=$ $\displaystyle\inf_{i\in
I}\\{\mu_{A_{i}}(x+y)\\}$ $\displaystyle\geq$ $\displaystyle\inf_{i\in
I}\\{\mu_{A_{i}}(x)\wedge\mu_{A_{i}}(y)\\}$ $\displaystyle=$
$\displaystyle\inf_{i\in I}\\{\mu_{A_{i}}(x)\\}\wedge\inf_{i\in
I}\\{\mu_{A_{i}}(y)\\}$ $\displaystyle=$ $\displaystyle\mu_{\bigcap_{i\in
I}A_{i}}(x)\wedge\mu_{\bigcap_{i\in I}A_{i}}(y).$
Also,
$\displaystyle\mu_{\bigcap_{i\in I}A_{i}}(\alpha x)=$ $\displaystyle\inf_{i\in
I}\\{\mu_{A_{i}}(\alpha x)\\}$ $\displaystyle\geq$ $\displaystyle\inf_{i\in
I}\\{\mu_{A_{i}}(x)\\}$ $\displaystyle=$ $\displaystyle\mu_{\bigcap_{i\in
I}A_{i}}(x).$
Similarly, we can prove that $\lambda_{\bigcap_{i\in
I}A_{i}}(x+y)\leq\lambda_{\bigcap_{i\in I}A_{i}}(x)\vee\lambda_{\bigcap_{i\in
I}A_{i}}(y)$ and $\lambda_{\bigcap_{i\in I}A_{i}}(\alpha
x)\leq\lambda_{\bigcap_{i\in I}A_{i}}(x)$. Next, if $x_{1},\ldots,x_{n}\in L$,
then
$\displaystyle\mu_{\bigcap_{i\in I}A_{i}}([x_{1},\ldots,x_{n}])=$
$\displaystyle\inf_{i\in I}\\{\mu_{A_{i}}([x_{1},\ldots,x_{n}])\\}$
$\displaystyle\geq$ $\displaystyle\inf_{i\in
I}\\{\mu_{A_{i}}(x_{1})\wedge\cdots\wedge\mu_{A_{i}}(x_{n})\\}$
$\displaystyle=$ $\displaystyle\inf_{i\in
I}\\{\mu_{A_{i}}(x)\\}\wedge\cdots\wedge\inf_{i\in I}\\{\mu_{A_{i}}(x_{n})\\}$
$\displaystyle=$ $\displaystyle\mu_{\bigcap_{i\in
I}A_{i}}(x_{1})\wedge\cdots\wedge\mu_{\bigcap_{i\in I}A_{i}}(x_{n}).$
In a similar way, one can show $\lambda_{\bigcap_{i\in
I}A_{i}}([x_{1},\ldots,x_{n}])\leq\lambda_{\bigcap_{i\in
I}A_{i}}(x)\vee\cdots\vee\lambda_{\bigcap_{i\in I}A_{i}}(x_{n})$. Therefore,
$\bigcap_{i\in I}A_{i}$ is an intuitionistic fuzzy Lie subalgebra. The proof
of the case of intuitionistic fuzzy Lie ideal is same, so we omit it. $\Box$
Let $A=\\{(x,~{}\mu_{A}(x),~{}\lambda_{A}(x)):~{}x\in X\\}$ be an
intuitionistic fuzzy set. For $s,t\in[0,~{}1]$ the set $A^{t}_{s}=\\{x\in
X:\mu_{A}(x)\geq s,~{}\lambda_{A}(x)\leq t\\}$ is called the upper level
subset of the intuitionistic fuzzy subset $A$. In particular if $t=1$, then we
get the upper $s$-level cut $A_{s}^{1}=U(\mu_{A};s)=\\{x\in
X~{}:~{}\mu_{A}(x)\geq s\\}$. Also, if $s=0$, then we get the lower $t$-level
cut $A_{0}^{t}=L(\lambda_{A};t)=\\{x\in X~{}:~{}\lambda_{A}(x)\leq t\\}$.
The following two theorems will show relations between intuitionistic fuzzy
Lie subalgebras of $L$ and Lie subalgebras of $L$. They are very similar to
the case that suggested by Kondo and Dudek in [22].
###### Theorem 3.2.
Let $A=(\mu_{A},~{}\lambda_{A})$ be an intuitionistic fuzzy set of an $n$-Lie
algebra $L$. Then A is an intuitionistic fuzzy Lie subalgebra of $L$ if and
only if the non-empty set $A^{t}_{s}$ is Lie subalgebra for all
$s,t\in[0,~{}1]$.
Proof. Let $A=\\{(x,~{}\mu_{A}(x),~{}\lambda_{A}(x)):~{}x\in L\\}$ be an
intuitionistic fuzzy Lie subalgebra. For $x,~{}y\in A^{s}_{t}$ and $\gamma\in
F$
* (i)
$\mu_{A}(x+y)\geq\mu_{A}(x)\wedge\mu_{A}(y)\geq s$ and
$\lambda_{A}(x+y)\leq\lambda_{A}(x)\vee\lambda_{A}(y)\leq t$,
* (ii)
$\mu_{A}(\gamma x)\geq\mu_{A}(x)\geq s$ and $\lambda_{A}(\gamma
x)\leq\lambda_{A}(x)\leq t$.
Thus $x+y,~{}\gamma x\in A_{s}^{t}$. Also for $x_{1},\ldots,x_{n}\in L$, we
have
$\mu_{A}([x_{1},,\ldots,x_{n}])\geq\mu_{A}(x_{1})\wedge\cdots\wedge\mu_{A}(x_{n}\geq
s$ and
$\lambda_{A}([x_{1},\ldots,x_{n}])\leq\lambda_{A}(x_{1})\vee\cdots\vee\lambda_{A}(x_{n})\leq
t$. That is $[x_{1},\ldots,x_{n}]\in A^{t}_{s}$. Therefore $A^{t}_{s}$ is Lie
subalgebra of $L$. Conversely, suppose that $A^{s}_{t}\neq\emptyset$ is a Lie
subalgebra of $L$ for every $s,t\in[0,~{}1]$. Let $x,y\in L$ and $\alpha\in
F$. Fix $s_{1}=\mu_{A}(x)\wedge\mu_{A}(x)$ and
$t_{1}=\lambda_{A}(x)\vee\lambda_{A}(y)$, so that $x,y\in A_{s_{1}}^{t_{1}}$.
Since $A_{s_{1}}^{t_{1}}$ is a subspace of $L$, we have $x+y$ and $\alpha x$
are in $A_{s_{1}}^{t_{1}}$, and so $\mu_{A}(x+y)\geq
s_{1}=\mu_{A}(x)\wedge\mu_{A}(y)$, $\lambda_{A}(x+y)\leq
t_{1}=\lambda_{A}(x)\vee\lambda_{A}(y)$. Also for $x\in L$, set
$s_{1}=\mu_{A}(x)$ and $t_{1}=\lambda_{A}(x)$. Then $x\in A_{s_{1}}^{t_{1}}$,
and so $\gamma x\in A^{t_{1}}_{s_{1}}$. Hence $\mu_{A}(\gamma
x)\geq\mu_{A}(x)$ and $\lambda_{A}(\gamma x)\leq\lambda_{A}(x)$. Finally let
$x_{1},x_{2},\ldots,x_{n}\in L$. Fix
$t=\mu_{A}(x_{1})\wedge\cdots\wedge\mu_{A}(x_{n})$ and
$s=\lambda_{A}(x_{1})\vee\cdots\vee\lambda_{A}(x_{n})$. Thus $x_{i}\in
A^{t}_{s}$ for all $i=1,\ldots,n$. Since $A^{t}_{s}$ is a subalgebra of $L$,
we have $[x_{1},\ldots,x_{n}]\in A^{t}_{s}$, so that
$\mu_{A}([x_{1},\ldots,x_{n}])\geq
s=\mu_{A}(x_{1})\wedge\cdots\wedge\mu_{A}(x_{2})$ and
$\lambda_{A}([x_{1},\ldots,x_{n}])\leq
t=\lambda_{A}(x_{1})\vee\cdots\vee\lambda_{A}(x_{n})$. The case of
intuitionistic fuzzy ideals is almost same. $\Box$
Let $A=\\{(x,~{}\mu_{A}(x),~{}\lambda_{A}(x)):~{}x\in X\\}$ be an
intuitionistic fuzzy set. For $s,t\in[0,~{}1]$ the set
$A^{t^{<}}_{s^{>}}=\\{x\in X:\mu_{A}(x)>s,~{}\lambda_{A}(x)<t\\}$ is called
the strong upper level subset of the intuitionistic fuzzy subset $A$.
###### Theorem 3.3.
Let $A=(\mu_{A},~{}\lambda_{A})$ be an intuitionistic fuzzy subset of $L$.
Then $A$ is an intuitionistic fuzzy Lie subalgebra of $L$ if and only if the
non empty set $A^{t^{<}}_{s^{>}}$ is Lie subalgebra, for all $s,t\in[0,~{}1]$.
Proof. The proof of the forward direction is almost identical to the proof in
Theorem 3.2. Conversely, suppose that the non-empty set $A^{t^{<}}_{s^{>}}$ is
a Lie subalgebra for all $s,t\in[0,~{}1]$. We need to show that the conditions
of Definition 3.1 are satisfied. Let $x,y\in L$. If $\mu_{A}(x)=0$ or
$\mu_{A}(y)=0$, then it is clear that
$\mu_{A}(x+y)\geq\mu_{A}(x)\wedge\mu_{A}(y)$, so we may assume that
$\mu_{A}(x)\neq 0$ and $\mu_{A}(y)\neq 0$. Let $s_{0}$ be the largest number
on the interval $[0,1]$ such that $s_{0}<\mu_{A}(x)\wedge\mu_{A}(y)$ and there
is no $a\in L$ satisfying $s_{0}<\mu_{A}(a)<\mu_{A}(x)\wedge\mu_{A}(y)$.
Having $x,y\in A_{s_{0}^{>}}^{t^{<}}$ where
$1>t>\lambda_{A}(x)\vee\lambda_{A}(y)$ (such $t$ exists because $\mu_{A}(x)$
and $\mu_{A}(y)$ are greater than $0$ in addition to
$\mu_{A}(x)+\lambda_{A}(x)$ and $\mu_{A}(y)+\lambda_{A}(y)$ are less than or
equals to $1$) implies that $x+y\in A^{t}_{s_{0}}$, and hence
$\mu_{A}(x+y)>s_{0}$. Since there exist no $a\in L$ with
$s_{0}<\mu_{A}(a)<\mu_{A}(x)\wedge\mu_{A}(y)$, it follows that
$\mu_{A}(x+y)\geq\mu_{A}(x)\wedge\mu_{A}(y)$. If $\lambda_{A}(x)=1$ or
$\lambda_{A}(y)=1$, then it is obvious that
$\lambda_{A}(x+y)\leq\lambda_{A}(x)\vee\lambda_{A}(y)$. Assume now that
$\lambda_{A}(x)\neq 1\neq\lambda_{A}(y)$. Let $t_{0}$ be the smallest number
on the interval $[0,1]$ such that $t_{0}>\lambda_{A}(x)\vee\lambda_{A}(y)$ and
there is no $a\in L$ with
$t_{0}>\lambda_{A}(a)>\lambda_{A}(x)\vee\lambda_{A}(y)$. Hence $x,y\in
A^{t_{0}^{<}}_{s^{>}}$ where $s<\mu_{A}(x)\wedge\mu_{A}(y)$, and so $x+y\in
A^{t_{0}^{<}}_{s^{>}}$. Therefore $\lambda_{A}(x+y)<t_{0}$. Since there is no
$a\in L$ such that $t_{0}>\lambda_{A}(a)>\lambda_{A}(x)\vee\lambda_{A}(y)$, it
follows that $\lambda_{A}(x+y)\leq\lambda_{A}(x)\vee\lambda_{A}(y)$ as
desired. In a similar way we can show that $\mu_{A}(\gamma x)\geq\mu_{A}(x)$
and $\lambda_{A}(\gamma x)\leq\lambda_{A}(x)$ for all $x\in L$ and $\gamma\in
F$. Next let $x_{1},\ldots,x_{n}\in L$. Again we may assume that neither of
$\mu_{A}(x_{1}),\ldots,\mu_{A}(x_{n})$ is $0$. Let $s_{1}$ be the greatest
number on the interval $[0,1]$ such that
$s_{1}<\mu_{A}(x)\wedge\cdots\wedge\mu_{A}(x_{n})$ and there is no $a\in L$
such that $s_{1}<\mu_{A}(a)<\mu_{A}(x_{1})\wedge\cdots\wedge\mu_{A}(x_{n})$.
Then $x_{i}\in A^{t^{<}}_{s_{1}^{>}}$, where
$1>t>\lambda_{A}(x_{1})\vee\cdots\vee\mu_{A}(x_{n})$, for each $i=1,\ldots,n$.
Since $A^{t^{<}}_{s_{1}^{>}}$ is subalgebra of $L$, we have
$[x_{1},\ldots,x_{n}]\in A^{t^{<}}_{s_{1}^{>}}$. Hence
$\mu_{A}(x_{1},\ldots,x_{n})\geq\mu_{A}(x_{1})\wedge\cdots\wedge\mu_{A}(x_{n})$.
In a similar fashion, we can prove that
$\lambda_{A}([x_{1},\ldots,x_{n}])\leq\lambda_{A}(x_{1})\vee\cdots\vee\lambda_{A}(x_{n})$
for all $x_{1},\ldots,x_{n}\in L$. $\Box$
From the proofs of Theorem 3.2 and Theorem 3.3, we can immediately obtain the
following result.
###### Corollary 3.1.
Let $A=(\mu_{A},\lambda_{A})$ be an intuitionistic fuzzy subset of an $n$-Lie
algebra $L$. The following statements are equivalent for every $s,t\in[0,1]$:
* (i)
$A$ is an intuitionistic fuzzy Lie subalgebra (resp. ideal) of $L$,
* (ii)
The nonempty subsets $A^{t}_{s}$ are $n$-Lie subalgebras (resp. ideals) of
$L$,
* (iii)
The nonempty subsets $A^{t^{<}}_{s}=\\{x\in L~{}|~{}\mu_{A}(x)\geq
s~{}\mathrm{and}~{}\lambda_{A}(x)<t\\}$ are $n$-Lie subalgebras (resp. ideals)
of $L$,
* (iv)
The nonempty subsets $A^{t}_{s^{>}}=\\{x\in
L~{}|~{}\mu_{A}(x)>s~{}\mathrm{and}~{}\lambda_{A}(x)\leq t\\}$ are $n$-Lie
subalgebras (resp. ideals) of $L$,
* (v)
The The nonempty subsets $A^{t^{<}}_{s^{>}}$ are $n$-Lie subalgebras (resp.
ideals) of $L$.
## 4\. Operations On Intuitionistic Fuzzy $n$-Lie ideals
We omit the proofs for the following two results because they are
straightforward.
###### Theorem 4.1.
Let $A=(\mu_{A},~{}\lambda_{A})$ be an intuitionistic fuzzy set in an $n$-Lie
algebra $L$. Then $A$ is an intuitionistic fuzzy Lie subalgebra (resp. ideal)
if and only if $\Box A$ and $\diamondsuit A$ are intuitionistic fuzzy Lie
subalgebras (resp. ideals).
Let $A=(\mu_{A},~{}\lambda_{A})$ and $B=(\mu_{B},~{}\lambda_{B})$ be two
intuitionistic fuzzy subsets on a $n$-Lie algebra $L$. Let us introduce the
following sum of $A$ and $B$, which was defined by Chen and Zhang [11] in the
case of intuitionistic fuzzy Lie superalgebras:
$A+B=(\mu_{A+B},~{}\lambda_{A+B}),$
where
$\mu_{A+B}(x)=\sup_{x=a+b}\\{\mu_{A}(a)\wedge\mu_{B}(b)\\},$
and
$\lambda_{A+B}(x)=\inf_{x=a+b}\\{\lambda_{A}(a)\vee\lambda_{B}(b)\\}.$
If $A=(\mu_{A},~{}\lambda_{A})$ and $B=(\mu_{B},~{}\lambda_{B})$ are
intuitionistic fuzzy sets on $L$, then $A+B$ is an intuitionistic fuzzy set of
$L$. Indeed if $x\in L$ with $\mu_{A+B}(x)+\lambda_{A+B}(x)>1$, then
$\sup_{x=a+b}\\{\mu_{A}(a)\wedge\mu_{B}(b)\\}+\inf_{x=a+b}\\{\lambda_{A}(a)\vee\lambda_{B}(b)\\}>1$,
and so there exist $a_{1},~{}b_{1}\in L$ such that $x=a_{1}+b_{1}$ with
$\mu_{A}(a_{1})\wedge\mu_{B}(b_{1})+\inf_{x=a+b}\\{\lambda_{A}(a)\vee\lambda_{B}(b)\\}>1$.
It follows that
$\mu_{A}(a_{1})\wedge\mu_{B}(b_{1})+\lambda_{A}(a_{1})\vee\lambda_{B}(b_{1})>1$.
Without loss of generality we may assume $\mu_{A}(a_{1})\leq\mu_{B}(b_{1})$.
If $\lambda_{A}(a_{1})\geq\lambda_{B}(b_{1})$, then it is a clear
contradiction. If $\lambda_{A}(a_{1})\leq\lambda_{B}(b_{1})$, then
$\mu_{A}(a_{1})+\lambda_{B}(b_{1})>1$. Hence
$\mu_{A}(a_{1})+1-\mu_{B}(b_{1})>1$, and so $\mu_{A}(a_{1})>\mu_{B}(b_{1})$.
Contradiction. Therefore, $A+B$ is an intuitionistic fuzzy set of $L$.
It is well known that if $A$ and $B$ are ideals of $L$, then $A+B$ is an ideal
of $L$ too. We are going to obtain an analogous result in the case of
intuitionistic fuzzy Lie ideals. Similar result was obtained for complex fuzzy
Lie subalgebra in [32], and for intuitionistic fuzzy Lie sub-superalgebras in
[11].
###### Theorem 4.2.
Let $A=(\mu_{A},~{}\lambda_{A})$ and $B=(\mu_{B},~{}\lambda_{B})$ be two
intuitionistic fuzzy $n$-Lie ideals on $L$. Then $A+B$ is an intuitionistic
fuzzy $n$-Lie ideal of $L$ too.
Proof. We proceed as in the proof of corresponding result on complex fuzzy Lie
algebras [33]. The only difference appears in the proof is to show that
$\mu_{A}([x_{1},\ldots,x_{n}])\geq\mu_{A}(x_{1})\vee\cdots\vee\mu_{A}(x_{n})$
and
$\lambda_{A}([x_{1},\ldots,x_{n}])\geq\lambda_{A}(x_{1})\wedge\mu_{A}(x_{n})$
for each $x_{1},\ldots,x_{n}\in L$. Let $x_{1},\ldots,x_{n}\in L$. Suppose
that
$\mu_{A+B}([x_{1},\ldots,x_{n}])<\mu_{A+B}(x_{1})\vee\cdots\vee\mu_{A+B}(x_{n})$.
Then there exists $t$ such that
$\mu_{A+B}([x_{1},\ldots,x_{n}])<t<\mu_{A+B}(x_{1})\vee\ldots\vee\mu_{A+B}(x_{n})$.
Without loss of generality we may assume
$\mu_{A+B}(x_{1})=\mu_{A+B}(x_{1})\vee\cdots\vee\mu_{A+B}(x_{n})$. Thus
$\mu_{A+B}([x_{1},\ldots,x_{n}])<t<\sup_{x_{1}=a+b}\\{\mu_{A}(a)\wedge\mu_{B}(b)\\}$,
and so there exist $a_{1},~{}b_{1}\in L$ such that
$\mu_{A+B}([x,~{}y])<t<\mu_{A}(a_{1})\wedge\mu_{B}(b_{1})$. Therefore
$\mu_{A}(a_{1})>t$ and $\mu_{B}(b_{1})>t$. Hence
$\displaystyle\mu_{A+B}([x_{1},\ldots,x_{n}])$ $\displaystyle=$
$\displaystyle\mu_{A+B}([a_{1}+b_{1},\ldots,x_{n}])$ $\displaystyle\geq$
$\displaystyle\sup_{[x_{1},\ldots,x_{n}]=[a,\ldots,x_{n}]+[b,\ldots,x_{n}]}\\{\mu_{A}([a,\ldots,x_{n}])\wedge\mu_{B}([b,\ldots,x_{n}])\\}$
$\displaystyle\geq$
$\displaystyle\mu_{A}([a_{1},\ldots,x_{n}])\wedge\mu_{B}([b_{1},\ldots,x_{n}])$
$\displaystyle\geq$
$\displaystyle(\mu_{A}(a_{1})\vee\cdots\vee\mu_{A}(x_{n}))\wedge(\mu_{B}(b_{1})\vee\cdots\vee\mu_{B}(x_{n}))$
$\displaystyle>$ $\displaystyle t$ $\displaystyle>$
$\displaystyle\mu_{A+B}([x_{1},\ldots,x_{n}]).$
Contradiction. Thus
$\mu_{A+B}([x_{1},\ldots,x_{n}]\geq\mu_{A+B}(x_{1})\vee\cdots\vee\mu_{A+B}(x_{n})$.
Finally, suppose
$\lambda_{A+B}([x_{1},\ldots,x_{n}])>\lambda_{A+B}(x_{1})\wedge\cdots\wedge\lambda_{A+B}(x_{n})$.
Then there exists $s$ such that
$\lambda_{A+B}([x_{1},\ldots,x_{n}])>s>\lambda_{A+B}(x_{1})\wedge\cdots\wedge\lambda_{A+B}(x_{n})$.
Without loss of generality we may assume
$\lambda_{A+B}(x_{1})=\lambda_{A+B}(x_{1})\wedge\cdots\wedge\mu_{A+B}(x_{n})$.
Thus
$\lambda_{A+B}([x_{1},\ldots,x_{n}])>s>\inf_{x_{1}=a+b}\\{\lambda_{A}(a)\vee\lambda_{B}(b)\\}$,
and so there exist $a_{1},~{}b_{1}\in L$ such that
$\lambda_{A+B}([x_{1},\ldots,x_{n}])>s>\lambda_{A}(a_{1})\vee\lambda_{B}(b_{1})$.
Therefore $\lambda_{A}(a_{1})<s$ and $\lambda_{B}(b_{1})<s$. Now
$\displaystyle\lambda_{A+B}([x_{1},\ldots,x_{n}])$ $\displaystyle=$
$\displaystyle\lambda_{A+B}([a_{1}+b_{1},\ldots,x_{n}])$ $\displaystyle\leq$
$\displaystyle\inf_{[x,~{}y]=[a,\ldots,x_{n}]+[b,\ldots,x_{n}]}\\{\lambda_{A}([a,\ldots,x_{n}])\vee\lambda_{B}([b,\ldots,x_{n}])\\}$
$\displaystyle\leq$
$\displaystyle\lambda_{A}([a_{1},\ldots,x_{n}])\vee\lambda_{B}([b_{1},\ldots,x_{n}])$
$\displaystyle\leq$
$\displaystyle(\lambda_{A}(a_{1})\wedge\cdots\wedge\lambda_{A}(x_{n}))\vee(\lambda_{B}(b_{1})\wedge\cdots\wedge\lambda_{B}(x_{n}))$
$\displaystyle<$ $\displaystyle s$ $\displaystyle<$
$\displaystyle\lambda_{A+B}([x_{1},\ldots,x_{n}]).$
Contradiction. Thus $A+B$ is a complex intuitionistic fuzzy $n$-Lie ideal of
$L$. $\Box$
In particular, if $A$ and $B$ are fuzzy Lie ideals of a Lie algebra $L$, then
we obtain a special case of Shaqaqha’s result [32, Theorem 3.5].
## 5\. Direct Product of Intuitionistic fuzzy Lie $n$-subalgebras
Let $A=(\mu_{A},\lambda_{A})$ and $B=(\mu_{B},\lambda_{B})$ be an
intuitionistic fuzzy subsets of $L_{1}$ and $L_{2}$, respectively. Then the
generalized cartesian product $A\times B$ is defined to be $A\times
B=(\mu_{A}\times\mu_{B},\lambda_{A},\lambda_{B})$ where
$\mu_{A}\times\mu_{B}:L\times
L\rightarrow[0,1];~{}(x,y)\mapsto\mu_{A}(x)\wedge\mu_{B}(y),$
and
$\lambda_{A}\times\lambda_{B}:L\times
L\rightarrow[0,1];~{}(x,y)\mapsto\lambda_{A}(x)\vee\lambda_{B}(y).$
We have the following result.
###### Theorem 5.1.
Let $A=(\mu_{A},\lambda_{A})$ and $B=(\mu_{B},\lambda_{B})$ be an
intuitionistic fuzzy $n$-Lie subalgebras of $L_{1}$ and $L_{2}$, respectively.
Then $A\times B$ is an intuitionistic fuzzy $n$-Lie subalgebra of $L_{1}\times
L_{2}$.
Proof. Note that $A\times B$ is an intuitionistic fuzzy subset of $L_{1}\times
L_{2}$. Indeed if $x\in L_{1}$ and $y\in L_{2}$ such that
$\mu_{A}(x)\leq\mu_{B}(y)$ and $\lambda_{B}(y)\geq\lambda_{B}(x)$, then
$(\mu_{A}\times\mu_{B})(x,y)+(\lambda_{A}\times\lambda_{B})(x,y)=\mu_{A}(x)+\lambda_{B}(y)\leq\mu_{B}(y)+\lambda_{B}(y)\leq
1.$
Similarly, if $\mu_{B}(y)\leq\mu_{A}(x)$ and
$\lambda_{A}(x)\geq\lambda_{B}(y)$. Let $(x_{1},y_{1}),(x_{2},y_{2})\in
L_{1}\times L_{2}$. Then the proofs for
$(\mu_{A}\times\mu_{B})((x_{1},y_{1})+(x_{2},y_{2}))\geq(\mu_{A}\times\mu_{B})((x_{1},y_{1}))\wedge(\mu_{A}\times\mu_{B})((x_{2},y_{2}))$,
$(\lambda_{A}\times\lambda_{B})((x_{1},y_{1})+(x_{2},y_{2}))\leq\lambda_{A}\times\lambda_{B}((x_{1},y_{1}))\vee\lambda_{A}\times\lambda_{B}((x_{2},y_{2}))$,
$(\mu_{A}\times\mu_{B})(c(x_{1},y_{1}))\geq(\mu_{A}\times\mu_{B})((x_{1},y_{1}))$,
and
$(\lambda_{A}\times\lambda_{B})(c(x_{1},y_{1}))\leq(\lambda_{A}\times\lambda_{B})((x_{1},y_{1}))$
are similar to the proof of [11, Lemma 2.1].
For $(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\in L\times L$, we have
$\displaystyle\mu_{A}([(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})])$
$\displaystyle=$
$\displaystyle(\mu_{A}\times\mu_{B})([x_{1},x_{2},\ldots,x_{n}],[y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle=$
$\displaystyle\mu_{A}([x_{1},x_{2},\ldots,x_{n}])\wedge\mu_{B}([y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle\geq$
$\displaystyle(\mu_{A}(x_{1})\wedge\cdots\wedge\mu_{A}(x_{n}))\wedge(\mu_{B}(y_{1})\wedge\cdots\wedge\mu_{B}(y_{n}))$
$\displaystyle=$
$\displaystyle(\mu_{A}\times\mu_{B})((x_{1},y_{1}))\wedge\cdots\wedge(\mu_{A}\times\mu_{B})((x_{n},y_{2})),$
and also
$\displaystyle\lambda_{A}([(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})])$
$\displaystyle=$
$\displaystyle(\lambda_{A}\times\lambda_{B})([x_{1},x_{2},\ldots,x_{n}],[y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle=$
$\displaystyle\lambda_{A}([x_{1},x_{2},\ldots,x_{n}])\vee\lambda_{B}([y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle\leq$
$\displaystyle(\lambda_{A}(x_{1})\vee\cdots\vee\lambda_{A}(x_{n}))\vee(\lambda_{B}(y_{1})\vee\cdots\vee\lambda_{B}(y_{n}))$
$\displaystyle=$
$\displaystyle(\lambda_{A}\times\lambda_{B})((x_{1},y_{1}))\vee\cdots\vee(\lambda_{A}\times\lambda_{B})((x_{n},y_{2})).$
$\Box$
However the direct product of two intuitionistic fuzzy Lie ideals of $n$-Lie
algebras $L_{1}$ and $L_{2}$ is not nesaccary to be an intuitionistic fuzzy
Lie ideal of the $n$-Lie algebra $L_{1}\times L_{2}$.
## 6\. On Lie Algebra Homomorphism of Intuitionistic Fuzzy $n$-Lie Algebras
Let $L_{1}$ and $L_{2}$ be $n$-Lie algebras, $A=(\mu_{A},\lambda_{A})$ be an
intuitionistic subset of $L_{1}$, and $f:L_{1}\rightarrow L_{2}$ be a
function. Then the intuitionistic fuzzy subset $f(A)$ of $f(L_{1})$ defined by
$f(A)=(\mu_{f(A)},\lambda_{f(A)})$ where $\mu_{f(A)}(y)=\sup_{x\in
f^{-1}(y)}\\{\mu_{A}(x)\\}$ and $\lambda_{f(A)}(y)=\inf_{x\in
f^{-1}(y)}\\{\lambda_{A}(x)\\}$ for $y\in f(L_{1})$ is called the image of $A$
under $f$. Similarly, if $B=(\mu_{B},\lambda_{B})$ is an intuitionistic fuzzy
subset of $L_{2}$, then the intuitionistic fuzzy set on $L_{1}$ is
$f^{-1}(B)=(\mu_{f^{-1}(B)},\lambda_{f^{-1}(B)})$ where
$\mu_{f^{-1}(B)}(x)=\mu_{B}(f(x))~{}\mathrm{and}~{}\lambda_{f^{-1}(B)}(x)=\lambda_{B}(f(x))$
(see for example [32]). The proofs of the following two results are omitted
because they are routine and parallel to the corresponding results on
intuitionistic fuzzy Lie algebras ([3, 33]).
###### Theorem 6.1.
Let $\varphi:L_{1}\rightarrow L_{2}$ be a $n$-Lie algebra homomorphism. If
$B=(\mu_{B},~{}\lambda_{B})$ is an intuitionistic fuzzy $n$-Lie subalgebra
(resp. ideal) on $L_{2}$, then the intuitionistic fuzzy set $\varphi^{-1}(B)$
is an intuitionistic fuzzy $n$-Lie subalgebra (rep. ideal) of $L_{1}$.
###### Theorem 6.2.
Let $\varphi:L_{1}\rightarrow L_{2}$ be a $n$-Lie algebra homomorphism. If
$A=(\mu_{A},~{}\lambda_{A})$ is an intuitionistic fuzzy $n$-Lie subalgebra
(resp. ideal) on $L_{1}$, then the intuitionistic fuzzy set $\varphi(A)$ is an
intuitionistic fuzzy $n$-Lie subalgebra (resp. ideal) of
$\mathrm{im}(\varphi)$.
## 7\. Intuitionistic Fuzzy Quotient $n$-Lie Algebras
Let $L$ be an $n$-Lie algebra and $I$ be an ideal of $L$. Then the factor
space $L/I=\\{x+I~{}:~{}x\in I\\}$ acquires an $n$-Lie algebra structure
(called a quotient $n$-Lie algebra) by setting
$[x_{1}+I,x_{2}+I,\ldots,x_{n}+I]=[x_{1},~{}x_{2},\ldots,x_{n}]+I$
for $x_{1},x_{2},\ldots,x_{n}\in L$. In this paper we define and study the
intuitionistic Fuzzy quotient $n$-Lie algebra by an intuitionistic fuzzy
$n$-Lie ideal.
###### Definition 7.1.
Let $A=(\mu_{A},\lambda_{A})$ be an intuitionistic fuzzy $n$-Lie ideal of an
$n$-Lie algebra $L$. Then for $x\in L$, the intuitionistic fuzzy subset
$x+A=(x+\mu_{A},x+\lambda_{A})$
where
$x+\mu_{A}:L\rightarrow[0,~{}1];~{}y\mapsto\mu_{A}(y-x)$
and
$x+\lambda_{A}:L\rightarrow[0,~{}1];~{}y\mapsto\lambda_{A}(y-x)$
is called a coset (determined by $x,\mu_{A}$, and $\lambda_{A}$) of the
intuitionistic fuzzy $n$-Lie ideal $A$.
The case where $n=2$ was introduced and studied by Chen [12] in 2010. The set
of all cosets of an intuitionistic fuzzy $n$-Lie ideal will be denoted by
$L/A$. The following lemma proves that a coset may have many different labels.
###### Lemma 1.
Let $A=(\mu_{A},~{}\lambda_{A})$ be an intuitionistic fuzzy $n$-Lie ideal of
an $n$-Lie algebra $L$, and let $x,y\in L$. The following statements are
equivalent:
* (i)
$x+A=y+A$,
* (ii)
$\mu_{A}(x-y)=\mu_{A}(0)$ and $\lambda_{A}(x-y)=\lambda_{A}(0)$.
Proof. If $x,~{}y\in L$ with $x+A=y+A$, then $x+\mu_{A}=y+\mu_{A}$ and
$x+\lambda_{A}=y+\lambda_{A}$. Consider $x+\mu_{A}=y+\mu_{A}$, then evaluating
both sides for $x$ implies that $\mu_{A}(0)=\mu_{A}(x-y)$. Similarly
$\lambda_{A}(x-y)=\lambda_{A}(0)$. Conversely, let $\mu_{A}(x-y)=\mu_{A}(0)$
and $\lambda_{A}(x-y)=\lambda_{A}(0)$. Then for any $z\in L$, we have
$(y+\mu_{A})(z)=\mu_{A}(z-y)\geq\mu_{A}(z-x)\wedge\mu_{A}(x-y)=\mu_{A}(z-x)\wedge\mu_{A}(0)=\mu_{A}(z-x)=(x+\mu_{A})(z)$.
Thus $y+\mu_{A}\geq x+\mu_{A}$. Also
$\mu_{A}(y-x)=\mu_{A}(-(x-y))=\mu_{A}(x-y)=\mu_{A}(0)$. So we can prove that
$x+\mu_{A}\geq y+\mu_{A}$ in the same way as above. Hence
$x+\mu_{A}=y+\mu_{A}$. By almost the same aegument we can prove that
$x+\lambda_{A}=y+\lambda_{A}$. $\Box$
###### Theorem 7.1.
Let $A=(\mu_{A},~{}\lambda_{A})$ be an intuitionistic fuzzy $n$-Lie ideal of
$L$ and $L/A$ be the set of all cosets of $L$ on $A$. Then the set $L/A$ is an
$n$-Lie algebra under the following operations:
* (i)
$(x+A)+(y+A)=(x+y)+A$ for all $x,y\in L$,
* (ii)
$\alpha(x+A)=\alpha x+A$ for all $x\in L$ and $\alpha\in F$,
* (iii)
$[(x_{1}+A),(x_{2}+A),\ldots,(x_{n}+A)]=[x_{1},x_{2},\ldots,x_{n}]+A$ for all
$x_{1},x_{2},\ldots,x_{n}\in L$.
Proof. First, we show that the operations are well defined. Let $x,~{}y,~{}u$
and $v$ be elements in $L$ such that $x+A=y+A$ and $u+A=v+A$. Consequently
$\mu_{A}(x+u-(y+v))=\mu_{A}((x-y)+(u-v))\geq\mu_{A}(x-y)\wedge\mu_{A}(u-v)=\mu_{A}(0)$
(because $\mu_{A}(x-y)=\mu_{A}(0)~{}\mathrm{and}~{}\mu_{A}(u-v)=\mu_{A}(0)$).
As $\mu_{A}(0)\geq\mu_{A}(x+u-(y+v))$, we have $\mu_{A}(x+u-y-v)=\mu_{A}(0)$.
Almost the same argument one can obtain that
$\lambda_{A}(x+u-y-v)=\lambda_{A}(0)$ Therefore $(x+u)+A=(y+v)+A$. Also,
$\mu_{A}(\alpha x-\alpha y)\geq\mu_{A}(x-y)=\mu_{A}(0)$ and
$\lambda_{A}(\alpha x-\alpha y)\leq\lambda_{A}(x-y)=\lambda_{A}(0)$. Hence
$\alpha(x+A)=\alpha(y+A)$. If $x_{1},x_{2},\ldots,x_{n}\in L$ such that
$x_{i}+A=y_{i}+A$ ($i=1,\ldots,n$), then
$\displaystyle\mu_{A}([x_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle=$ $\displaystyle\mu_{A}([x_{1}-y_{1},x_{2},\ldots,x_{n}]$
$\displaystyle+[y_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle\geq$ $\displaystyle\mu_{A}([x_{1}-y_{1},x_{2},\ldots,x_{n}])$
$\displaystyle\wedge\mu_{A}([y_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle\geq$
$\displaystyle\mu_{A}(0)\wedge\mu_{A}([y_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])$
Now,
$\displaystyle\mu_{A}([y_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])$
$\displaystyle=$ $\displaystyle\mu_{A}([y_{1},x_{2}-y_{2},x_{3},\ldots,x_{n}]$
$\displaystyle+[y_{1},y_{2},x_{3},\ldots,x_{2}]-[y_{1},y_{2},y_{3},\ldots,y_{n}])$
$\displaystyle\geq$
$\displaystyle\mu_{A}(y_{1})_{\vee}\mu_{A}(x_{2}-y_{2})\vee\mu_{A}(x_{3})\vee\cdots\mu_{A}(x_{n})$
$\displaystyle\wedge\mu_{A}([y_{1},y_{2},x_{3},\ldots,x_{2}]-[y_{1},y_{2},y_{3},\ldots,y_{n}])$
$\displaystyle\geq$
$\displaystyle\mu_{A}(0)\wedge\mu_{A}([y_{1},y_{2},x_{3},\ldots,x_{2}]-[y_{1},y_{2},y_{3},\ldots,y_{n}]).$
So
$\mu_{A}([x_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])\geq\mu_{A}(0)\wedge\mu_{A}(0)\wedge\mu_{A}([y_{1},y_{2},x_{3},\ldots,x_{2}]-[y_{1},y_{2},y_{3},\ldots,y_{n}]).$
Continuing in the same way, we obtain
$\mu_{A}([x_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])\geq\mu_{A}(0)\wedge\mu_{A}(0)\wedge\cdots\wedge\mu_{A}(0)~{}(n~{}\mathrm{times}).$
Hence
$\mu_{A}([x_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])=\mu_{A}(0)$.
Using the same method one can prove that
$\lambda_{A}([x_{1},x_{2},\ldots,x_{n}]-[y_{1},y_{2},\ldots,y_{n}])=\lambda_{A}(0)$.
Therefore $[x_{1},x_{2},\ldots,x_{n}]+A=[y_{1},y_{2},\ldots,y_{n}]+A$. Now it
is straightforward to see that the product on $L/A$ is an $n$-linear map
satisfying the Filippov identity. $\Box$
The $n$-Lie algebra constructed in Theorem 7.1 is called intuitionistic fuzzy
quotient $n$-Lie algebra of $L$ by $A$.
## References
* [1] K.S. Abdukhalikov, M.S. Tulenbaev, U.U. Umirbaev, On fuzzy subalgebras, Fuzzy Sets and Systems, 93 (1998), 257-262.
* [2] M. Akram, Fuzzy Lie algebras, Springer Nature Singapore Pte Ltd. (2018).
* [3] M. Akram, Intuitionistic fuzzy Lie subalgebras, Southeast Asian Bulletin of Mathematics 31 (2007), 843-855.
* [4] M. Akram, Intuitionistic (S,T)-fuzzy Lie ideals of Lie algebras, Quasigroups Relat. Systems 15 (2007), 201-215.
* [5] M. Akram, Intuitionistic fuzzy Lie ideals of Lie algebras, Int. Journal of Fuzzy Math., 6 (4) (2008), 991-1008.
* [6] M. Akram, Co-fuzzy Lie superalgebras over a co-fuzzy field, World Applied Sciences Journal, 7 (2009), 25-32.
* [7] D. Alekseevsky and P. Guha, On decomposability of Nambu-Poisson Tensor, Acta Mathematica Universitatis Comenianae, 65 (1996), 1-9
* [8] K.T Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87-96.
* [9] Y. Bahturin, Identical relations in Lie algebras, VNU Science Press, b.v., Utrecht, 1987.
* [10] R. Biswas, Intuitionistic fuzzy subgroups, Math. Forum 10 (1989), 37-46.
* [11] W. Chen, S. Zhang, Intuitionistic fuzzy Lie sub-superalgebra and intuitionistic fuzzy ideals, Computer and Mathematics with Applications 58 (2009), 1645-1661.
* [12] W. Chen, Intuitionistic fuzzy quotient Lie superalgebras, International Journal of Fuzzy Systems, 12 (4), (2010), 330-339.
* [13] B. Davvaz and WA. Dudek, Fuzzy $n$-Lie algebras, J Generalized Lie Theory Appl,11 (2017), 1-6.
* [14] W. A. Dudek, Fuzzifications of $n$-ary groupoids, Quasigroups and Related Systems,7 (2000), 45-66.
* [15] Filippov, $n$-ary Lie algebras, (Russian) Sibirsk Mat Zh, 26(6) (1985), 126–140.
* [16] P. Gautheron, Simple facts concerning Nambu algebras, Commun. Math. Phys., 195 (1998), 417-34
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* [22] M. Kondo and W. A. Dudek, On the transfer principal in fuzzy theorey, Mathware Soft Computing 12 (2005), 41-55.
* [23] G. Marmo, G. Vilasi and A. M. Vinogradov, The local structure of $n$-Poisson and $n$-Jacobi manifolds, J. Geom. Phys., 25 (1998), 141-82
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|
# Remark on an inequality for closed hypersurfaces in complete manifolds with
nonnegative Ricci curvature
Xiaodong Wang Department of Mathematics, Michigan State University, East
Lansing, MI 48824<EMAIL_ADDRESS>
###### Abstract.
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and
Mazzieri [AFM].
The following result was proved by Agostiniani, Fogagnolo and Mazzieri [AFM].
###### Theorem 1.
Let $\left(M^{n},g\right)$ ($n\geq 3$) be a complete Riemannian manifold with
nonnegative Ricci curvature and $\Omega\subset M$ a bounded open set with
smooth boundary. Then
(1)
$\int_{\partial\Omega}\left|\frac{H}{n-1}\right|^{n-1}d\sigma\geq\mathrm{AVR}\left(g\right)\left|\mathbb{S}^{n-1}\right|,$
where $H$ is the mean curvature of $\partial\Omega$ and
$\mathrm{AVR}\left(g\right)$ is the asymptotic volume ratio of $M$. Moreover,
if $\mathrm{AVR}\left(g\right)>0$, equality holds iff $M\backslash\Omega$ is
isometric to
$\left([r_{0},\infty)\times\partial\Omega,dr^{2}+\left(\frac{r}{r_{0}}\right)^{2}g_{\partial\Omega}\right)$
with
$r_{0}=\left(\frac{\left|\partial\Omega\right|}{\mathrm{AVR}\left(g\right)\left|\mathbb{S}^{n-1}\right|}\right)^{\frac{1}{n-1}}$
In particular, $\partial\Omega$ is a connected totally umbilic submanifold
with constant mean curvature.
The proof in [AFM] is highly nontrivial. It is based on the study of the
solution of the following problem
$\left\\{\begin{array}[c]{cc}\Delta u=0,&\text{on }M\backslash\Omega\\\
u=1&\text{on \ \ }\partial\Omega\\\ u\left(x\right)\rightarrow 0&\text{as
}x\rightarrow\infty,\end{array}\right.$
which exists when $\mathrm{AVR}\left(g\right)>0$. The key step consists of
showing that, with $\beta\geq\left(n-2\right)/\left(n-1\right)$
$U_{\beta}\left(t\right)=t^{-\beta\left(\frac{n-1}{n-2}\right)}\int_{u=t}\left|\nabla
u\right|^{\beta+1}d\sigma$
is monotone in $t\in(0,1]$. The geometric inequality (1) then follows by
analyzing the asymptotic behavior of $U_{\beta}\left(t\right)$ as
$t\rightarrow 0$. It is a beautiful argument.
In this short note, we show that this theorem can be proved by standard
comparison methods in Riemannian geometry.
To prove the inequality (1), we assume, without loss of generality, that
$\Omega$ has no hole, i.e. $M\backslash\Omega$ has no bounded component. In
the following we write $\Sigma=\partial\Omega$ and let $\nu$ be the outer unit
normal along $\Sigma$. For each $p\in\Sigma$ let
$\gamma_{p}\left(t\right)=\exp_{p}t\nu\left(p\right)$ be the normal geodesic
with initial velocity $\nu\left(p\right)$. We define
$\tau\left(p\right)=\sup\left\\{L>0:\gamma_{p}\text{ is minimizing on
}\left[0,L\right]\right\\}\in(0,\infty].$
It is well known that $\tau$ is a continuous function on $\Sigma$ and the
focus locus
$C\left(\Sigma\right)=\left\\{\exp_{p}\tau\left(p\right)\nu\left(p\right):\tau\left(p\right)<\infty\right\\}$
is a closed set of measure zero in $M$. Moreover the map
$\Phi\left(r,p\right)=\exp_{p}r\nu\left(p\right)$ is a diffeomorphism from
$E=\left\\{\left(r,p\right)\in\Sigma\times[0,\infty):r<\tau\left(p\right)\right\\}$
onto $\left(M\backslash\Omega\right)\backslash C\left(\Sigma\right)$. And on
$E$ the pull back of the volume form takes the form
$d\mu=\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)$. We will also
understand $r$ as the distance function to $\Sigma$ and it is smooth on
$M\backslash\Omega$ away from $C\left(\Sigma\right)$. By the Bochner formula
and nonnegative Ricci curvature condition
$\displaystyle 0$ $\displaystyle=\frac{1}{2}\Delta\left|\nabla
r\right|^{2}=\left|D^{2}r\right|^{2}+\left\langle\nabla r,\nabla\Delta
r\right\rangle+Ric\left(\nabla r,\nabla r\right)$
$\displaystyle\geq\frac{\left(\Delta
r\right)^{2}}{n-1}+\frac{\partial}{\partial r}\Delta r.$
In view of the initial condition $\Delta r|_{r=0}=H$, it is standard to deduce
from the above inequality $\tau\leq\frac{n-1}{H^{-}}$ and
$\frac{\mathcal{A}^{\prime}}{\mathcal{A}}=\Delta
r\leq\frac{\left(n-1\right)H}{n-1+Hr}$
This shows that the function
$\theta\left(r,p\right)=\frac{\mathcal{A}\left(r,p\right)}{\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}}$
is non-increasing in $r$ on $[0,\tau\left(p\right))$. As
$\theta\left(0,p\right)=1$, we obtain
$\mathcal{A}\left(r,p\right)\leq\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}.$
The above analysis is standard in Riemannian geometry. We also remark that
this argument involving the Bochner formula can be replaced by an argument
involving the index form along each individual geodesic $\gamma_{p}$. For more
details, cf. [P, S] or other books on Riemannian geometry.
Therefore for any $R>0$
$\displaystyle Vol\left\\{x\in M:d\left(x,\Omega\right)<R\right\\}=$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\left(1+\frac{H^{+}\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{R}\left(1+\frac{H^{+}\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle=$
$\displaystyle\left|\Omega\right|+\frac{R^{n}}{n}\int_{\Sigma}\left(\frac{H^{+}\left(p\right)}{n-1}\right)^{n-1}d\sigma\left(p\right)+O\left(R^{n-1}\right).$
Dividing both sides by
$\left|\mathbb{B}^{n}\right|R^{n}=\left|\mathbb{S}^{n-1}\right|R^{n}/n$ and
letting $R\rightarrow\infty$ yields
$\mathrm{AVR}\left(g\right)\leq\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma}\left(\frac{H^{+}}{n-1}\right)^{n-1}d\sigma,$
which implies (1).
We now analyze the equality case. Suppose
(2)
$\mathrm{AVR}\left(g\right)=\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma}\left(\frac{H^{+}}{n-1}\right)^{n-1}d\sigma>0.$
It is clear from the proof that $\tau\equiv\infty$ on the open set
$\Sigma^{+}=\left\\{p\in\Sigma:H\left(p\right)>0\right\\}$. For any
$R^{\prime}<R$ we have
$\displaystyle Vol\left\\{x\in M:d\left(x,\Omega\right)<R\right\\}$
$\displaystyle=$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\int_{0}^{R}\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)+\int_{\Sigma\backslash\Sigma^{+}}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\int_{0}^{R}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)+\int_{\Sigma\backslash\Sigma^{+}}\int_{0}^{R}drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\int_{R^{\prime}}^{R}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle+\int_{\Sigma^{+}}\int_{0}^{R^{\prime}}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)+O\left(R\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\theta\left(R^{\prime},p\right)\int_{R^{\prime}}^{R}\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle+\int_{\Sigma^{+}}\int_{0}^{R^{\prime}}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)+O\left(R\right).$
Dividing both sides by
$\left|\mathbb{B}^{n}\right|R^{n}=\left|\mathbb{S}^{n-1}\right|R^{n}/n$ and
letting $R\rightarrow\infty$ yields
$\mathrm{AVR}\left(g\right)\leq\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma^{+}}\left(\frac{H\left(p\right)}{n-1}\right)^{n-1}\theta\left(R^{\prime},p\right)d\sigma\left(p\right).$
Letting $R^{\prime}\rightarrow\infty$ yields
$\mathrm{AVR}\left(g\right)\leq\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma^{+}}\left(\frac{H}{n-1}\right)^{n-1}\theta_{\infty}d\sigma,$
where
$\theta_{\infty}\left(p\right)=\lim_{r\rightarrow\infty}\theta\left(r,p\right)\leq
1$. As we have equality (2) we must have $\theta_{\infty}\left(p\right)=1$ for
a.e. $p\in\Sigma^{+}$. It follows that
$\mathcal{A}\left(r,p\right)=\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}\text{
on }[0,\infty)$
for a.e. $p\in\Sigma^{+}$. By continuity the above identity holds for all
$p\in\Sigma^{+}$.
Inspecting the comparison argument, we must have on
$\Phi\left([0,\infty)\times\Sigma^{+}\right)$
$\displaystyle D^{2}r$ $\displaystyle=\frac{\Delta
r}{n-1}g=\frac{H}{n-1+Hr}g,$ $\displaystyle Ric\left(\nabla r,\nabla r\right)$
$\displaystyle=0.$
As $Ric\geq 0$, it follows that $Ric\left(\nabla r,\cdot\right)=0$. From the
1st equation above $\Sigma^{+}$ is an umbilic hypersurface, i.e. the 2nd
fundamental form $\Pi=\frac{H}{n-1}g_{\Sigma^{+}}$. Working with an
orthonormal frame $\left\\{e_{0}=\nu,e_{1},\cdots,e_{n-1}\right\\}$ along
$\Sigma^{+}$ we have by the Codazzi equation, with $1\leq i,j,k\leq n-1$
$R\left(e_{k},e_{j},e_{i},\nu\right)=\Pi_{ij,k}-\Pi_{ik,j}=\frac{1}{n-1}\left(H_{k}\delta_{ij}-H_{j}\delta_{ik}\right).$
Taking trace over $i$ and $k$ yields
$-\frac{n-2}{n-1}H_{j}=Ric\left(e_{j},\nu\right)=0.$
As a result $H$ is locally constant on $\Sigma^{+}$. Therefore $\Sigma^{+}$
must be the union of several components of $\Sigma$. We know that $\Phi$ is a
diffeomorphism form $[0,\infty)\times\Sigma^{+}$ onto its image and the
pullback metric $\Phi^{\ast}g$ takes the following form
$dr^{2}+h_{r},$
where $h_{r}$ is a $r$-dependent family of metrics on $\Sigma^{+}$ and
$h_{0}=g_{\Sigma^{+}}$. We have
$D^{2}r=\frac{H}{n-1+Hr}g.$
In terms of local coordinates $\left\\{x_{1},\cdots,x_{n-1}\right\\}$ on
$\Sigma^{+}$ the above equation implies
$\frac{1}{2}\frac{\partial}{\partial r}h_{ij}=\frac{H}{n-1+Hr}h_{ij}.$
Therefore $h_{r}=\left(1+\frac{H}{n-1}r\right)^{2}g_{\Sigma^{+}}$. This proves
that $\Phi\left([0,\infty)\times\Sigma^{+}\right)$ is isometric to
$\left([r_{0},\infty)\times\Sigma^{+},dr^{2}+\left(\frac{r}{r_{0}}\right)^{2}g_{\Sigma^{+}}\right)$,
where $r_{0}=\frac{n-1}{H}$.
Since $M$ has nonnegative Ricci curvature and Euclidean volume growth, it has
only one end by the Cheeger-Gromoll theorem. Therefore $\Sigma^{+}$ is
connected and if $\Sigma$ has other components besides $\Sigma^{+}$, they all
bound bounded components of $M\backslash\Omega$.
If we have the stronger identity
$\mathrm{AVR}\left(g\right)=\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma}\left|\frac{H}{n-1}\right|^{n-1}d\sigma>0,$
inspecting the proof of the inequality (1) shows that we must have $H\geq 0$
on $\Sigma$. Then $\overline{\Omega}$ is compact Riemannian manifold with mean
convex boundary. It is a classic fact that its boundary must be connected, see
[I, K] or [HW] for an analytic argument. Therefore $\Sigma=\Sigma^{+}$ is
connected and $M\backslash\Omega$ is isometric to
$\left([r_{0},\infty)\times\Sigma,dr^{2}+\left(\frac{r}{r_{0}}\right)^{2}g_{\Sigma}\right)$.
Acknowledgement. I would like to thank Fengbo Hang for helpful discussions.
## References
* [AFM] V. Agostiniani; M. Fogagnolo; L. Mazzieri. Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. Invent. Math. 222 (2020), no. 3, 1033-1101.
* [HW] F. Hang; X. Wang. Vanishing sectional curvature on the boundary and a conjecture of Schroeder and Strake, Pacific J. Math. 232 (2007), no. 2, 283-287.
* [I] R. Ichida. Riemannian manifolds with compact boundary. Yokohama Math. J. 29 (1981), no. 2, 169-177.
* [K] A. Kasue. Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary. J. Math. Soc. Japan 35 (1983), no. 1, 117-131.
* [P] P. Petersen. Riemannian Geometry. Third edition. Graduate Texts in Mathematics, 171. Springer, Cham, 2016.
* [S] T. Sakai. Riemannian geometry. Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, 1996.
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Analysis of Moral Judgement on Reddit
Nicholas Botzer, Shawn Gu, and Tim Weninger
Department of Computer Science and Engineering
University of Notre Dame
{nbotzer, sgu3<EMAIL_ADDRESS>
Moral outrage has become synonymous with social media in recent years. However, the preponderance of academic analysis on social media websites has focused on hate speech and misinformation. This paper focuses on analyzing moral judgements rendered on social media by capturing the moral judgements that are passed in the subreddit /r/AmITheAsshole on Reddit. Using the labels associated with each judgement we train a classifier that can take a comment and determine whether it judges the user who made the original post to have positive or negative moral valence. Then, we use this classifier to investigate an assortment of website traits surrounding moral judgements in ten other subreddits, including where negative moral users like to post and their posting patterns. Our findings also indicate that posts that are judged in a positive manner will score higher.
§ INTRODUCTION
How do people render moral judgements of others?
This question has been pondered for millennia. Aristotle, for example, considered morality in relation to the end or purpose for which a thing exists. Kant insisted that one's duty was paramount in determining what course of action might be good. Consequentialists argue that actions must be evaluated in relation to their effectiveness in bringing about a perceived good.
Regardless of the particular ethical frame that one ascribes to, the common practice of evaluating others' behavior in moral terms is widely regarded as important for the well-being of a community. Indeed, ethnographers and sociologists have documented how these kinds of moral judgements actually increase cooperation within a community by punishing those who commit wrongdoings and informing them of what they did wrong [3].
The process of rendering moral judgement has taken an interesting turn in the current era with the adoption of the Internet and social media in particular. Online social systems allow people to encounter and consider the lives of others from around the world.
At no other time in history have so many people been able to examine such a variety of cultures and viewpoints so readily. This increased sharing and mixing of viewpoints inevitably leads to online debates about various topics [29].
The content of these debates provides researchers with the opportunity to ask specific questions about argument, disagreement, moral evaluation, and judgement with the aid of new computational tools.
To that end, recent work has resulted in the creation of statistical models that can understand moral sentiment in text [25].
However, these models rely heavily on a gazette of words and topics and their alignment on moral axes.
The central motivation for these works are grounded in moral foundation theory [16] where studies also tend to investigate the use of morality as related to current events in the news such as politics or religion.
Despite their usefulness in understanding the moral valence of specific current events, the goal of the current work is to study moral judgements rendered on social media that apply to more common personal situations.
Example of (a) a post title and (b) a comment in the /r/AmItheAsshole subreddit. The NTA prefix and comment-score (not shown) indicates that the commenter judged the poster as “Not the Asshole”.
We focus on Reddit in particular, where users can create posts and have discussions in threaded comment-sections.
Although the details are complicated, users also perform curation of posts and comments through upvotes and downvotes based on their preference [12, 15].
This assigns each post and comment a score reflecting how others feel about the content.
Within Reddit there are a large number of subreddits, which are small communities dedicated to various topics.
The subreddit of interest for our question regarding moral judgements is called /r/AmItheAsshole.
Users of this subreddit post a description of a situation that they were involved in; they are also encouraged to explain details of the people involved and the final outcome of the situation.
Posters to /r/AmItheAsshole are typically looking to hear from other Reddit users whether or not they handled their personal situation in an ethically appropriate manner.
Other users then respond to the initial post with a moral judgement as to whether the original user was an asshole or not the asshole.
Figure <ref> shows an example of a typical post and one of its top responses.
One important rule of /r/AmItheAsshole is that top-level responses must categorize the behavior described in the original post to one of four categories: Not the Asshole (NTA), You're the Asshole (YTA), No assholes here (NAH), Everyone sucks here (ESH).
In addition to providing a categorical moral judgement, the responding user must also provide an explanation as to why they selected that choice.
Reddit's integrated voting system then allows other users to individually rate the judgements with which they most agree (upvote) or disagree (downvote). After some time has passed the competition among different judgements will settle, and one of the judgements will be rated highest. This top comment is then accepted as the judgement of the community.
This process of passing and rating moral judgement provides a unique view into our original question about how people make moral judgements.
Compared to other methodologies of computational evaluation of moral sentiments, collecting judgements from /r/AmItheAsshole (AITA) has some important benefits.
First, because posters and commenters are anonymous on Reddit, they are more likely to share their sensitive stories and frank judgements without fear of reprisal [21, 18].
Second, the voting mechanism of Reddit allows a large number of users to engage in an aggregated judgement in response to the original post [13]. However, the breadth and variety of this data does pose additional challenges. For instance, judgements are provided without an explicit moral-framing, and, similarly, Reddit-votes do not explicitly denote moral valence and are susceptible to path dependency effects [14].
In the present work we use data from AITA to investigate how users provide moral judgements of others. We then extract representative judgement-labels from each comment and use these labels and comments to train a classifier. This classifier is then broadly applied to infer the moral valence of other Reddit comments from ten different subreddits and used to answer the following research questions:
RQ1: What language is most closely associated with positive and negative moral valence?
RQ2: Is moral valence correlated with the score of a post?
RQ3: Do certain subreddit-communities attract users whose posts are typically classified by more negative or positive moral judgements?
RQ4: Are self-reported gender and age descriptions associated with positive or negative moral judgements?
In summary, we find that posts that are judged to have positive moral valence (, NTA label) typically score higher than posts with negative moral valence. We also find that certain subreddit-communities where users confess to something immoral (, such as /r/confessions) tend to attract users whose posts are characterized by negative moral valence. Among these immoral users we show that their posting habits tend towards three different types. Finally, we show that self-described male-users are more likely to be judged an asshole than female-users.
§ METHODOLOGY
We retrieve moral judgements by collecting posts and comments from the subreddit /r/AmItheAsshole, taken from the Pushshift data repository [1].
The questions raised in the present work are considered human subjects research, and relevant ethical consideration are present. We sought and received research approval from the Institution Review Board at the University of Notre Dame under protocol #20-01-5751.
Label Meaning # Comments
NTA Not the Asshole 717,006
YTA You're the Asshole 372,850
NAH No assholes here 91,903
ESH Everyone sucks here 79,059
The four judgements that users can pass on the subreddit /r/AmItheAsshole.
We restricted our data collections to posts submitted between January 1, 2017, and August 31, 2019. In order to assure that labels reflected the result of robust discussion we excluded those posts containing fewer than 50 comments. Subreddit rules require that top-level comments begin with one of four possible prefix-labels indicated in Tab. <ref>. Because of this rule, we further restrict our data collection to contain only top-level comments and their prefix-label. Comments with the INFO prefix, which indicates a request for more information, and comments with no prefix are also removed from consideration. This methodology resulted in a collection of 7,500 posts and 1,260,818 comments with explicit moral judgements. Posters and commenters appear to put a lot of thought and effort into these discussions. Each post contains 381 words on average and each comment contains 57 words on average.
§.§ Linguistic Analysis of Moral Judgement
Before we introduce our classifier, we first consider RQ1: what linguistic cues are associated with positive and negative moral judgement? To answer this question we split comments into two valence classes: positive and negative. The positive class contains comments that are labeled NTA or NAH; the negative class contains comments that are labeled YTA or ESH. Then we use the Allotaxonmeter system, which compares two Zipfian distributions using a scoring function called rank-turbulence divergence, to compare how terms are associated with these valence labels [9]. In our case, we constructed 1-gram multinomial distributions from each class, which, in the English language, is well known to exhibit a Zipfian distribution [22].
Negative Valence Positive Valence
[remember picture, overlay]
[fill=gray!20, anchor=east, rectangle, minimum width=3cm, text width=3cm, inner sep=.5,outer sep=0, align=right]Tyyou;
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[fill=blue!10, anchor=west, rectangle, minimum width=3cm, text width=3cm, inner sep=.5,outer sep=0, align=left]toTy;
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[fill=gray!20, anchor=east, rectangle, minimum width=7.5mm, text width=7.5mm, inner sep=.5,outer sep=0, align=right]quilt;
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[fill=blue!10, anchor=west, rectangle, minimum width=7.2mm, text width=7.2mm, inner sep=.5,outer sep=0, align=left]sheTy;
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[fill=white, anchor=east, rectangle, minimum width=7.4mm, text width=18.3mm, inner sep=.5,outer sep=0, align=right]beiause;
[fill=gray!20, anchor=east, rectangle, minimum width=7.3mm, text width=7.3mm, inner sep=.5,outer sep=0, align=right]causeT;
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[fill=blue!10, anchor=west, rectangle, minimum width=7.1mm, text width=7.1mm, inner sep=.5,outer sep=0, align=left]myTy;
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[fill=white, anchor=east, rectangle, minimum width=7.1mm, text width=18.3mm, inner sep=.5,outer sep=0, align=right]iTynn;
[fill=gray!20, anchor=east, rectangle, minimum width=7.1mm, text width=7.1mm, inner sep=.5,outer sep=0, align=right]nternTy;
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[fill=blue!10, anchor=west, rectangle, minimum width=7.0mm, text width=7.0mm, inner sep=.5,outer sep=0, align=left]cornellTy;
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[fill=gray!20, anchor=east, rectangle, minimum width=6.8mm, text width=6.8mm, inner sep=.5,outer sep=0, align=right]suckT;
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[fill=blue!10, anchor=west, rectangle, minimum width=6.6mm, text width=6.6mm, inner sep=.5,outer sep=0, align=left]theyTy;
Rank divergence scores of terms associated with positive and negative moral valences. Color bars indicate the relative contribution of terms, , contributes about 4 times as much to negative valence as .
Method Accuracy Precision Recall F1
Doc2Vec Embeddings 65.92 $\pm$ 0.04 61.22 $\pm$ 0.15 13.5 $\pm$ 0.09 22.1 $\pm$ 0.09
BERT Embeddings 70.10 $\pm$ 0.07 64.28 $\pm$ 0.2 36.96 $\pm$ 0.14 46.96 $\pm$ 0.08
Multinomial Naïve Bayes 72.12 $\pm$ 0.17 62.58 $\pm$ 0.17 55.22 $\pm$ 0.07 58.66 $\pm$ 0.05
Judge-BERT 89.03 $\pm$ 0.13 85.57 $\pm$ 0.18 83.48 $\pm$ 0.27 84.51 $\pm$ 0.17
Classification results on the AITA dataset. Results are mean-averages and standard deviations over five-fold cross-validation.
Table <ref> shows the terms that contain the largest divergence contribution for each valence class. Words with the highest negative valence include , , and in the top 5, but also , and within the top 10 (not shown). Simply put, these are the top words that are used when assigning negative moral judgement. Words associated with positive moral valence consist mainly of functional terms, but also include the names several ivy league schools in the top 50[A brief survey of these posts reveals that many posters ask if they are wrong to attend one of these schools even though their family member or partner was not admitted.].
§.§ Prediction Model
Given the dataset, a dataset with textual posts and textual comments labeled with positive or negative moral judgements, our goal is to predict whether an unlabeled comment assigns a positive (NTA or NAH) or negative (YTA or ESH) moral judgement to the user of the post.
It is important to note that this classifier is classifying the judgement of the commenter, not the morality of the poster.
We define our problem formally as follows.
Problem Definition
Given a top level comment $C$ with moral judgement $A \in \{+,-\}$ that responded to post $P$ we aim to find a predictive function $f$ such that
\begin{equation}
f : (C) \rightarrow A
\end{equation}
Formally, this takes the form of a text classification task where class inference denotes the valence of a moral judgement.
The choice of classification model $f$ is not particularly important, but we aim to train a model that performs well and generalizes to other datasets.
We selected four text classification models for use in the current work:
* Multinomial Naïve Bayes [19]: Uses word counts to learn a text classification model and has shown success in a wide variety of text classification problems.
* Doc2Vec [20]: Create comment-embeddings, which are input into a logistic regression classifier that calculates the class margin.
* BERT Embeddings [8]: Uses word embeddings from BERT, which are averaged together and input into a logistic regression classifier that calculates the class margin.
* Judge-BERT: We fine-tune the BERT-base model using the class labels. Specifically, we added a single dropout layer after BERT's final layer, followed by a final output layer that consists of our two classes. The model is trained using the Adam optimizer and uses the cross entropy loss function. We then trained for 3 epochs as recommended by Devlin et al [8].
§.§ Judgement Classification Results
We evaluate our four classifiers using accuracy, precision, recall, and F1 metrics. In this context a false positive is the instance when the classifier improperly assigns a negative (, asshole) label to a positive judgement. A false negative is the instance when the classifier improperly assigns a positive (, non-asshole) label to a negative judgement. We perform 5-fold cross-validation and, for each metric, report the mean-average and standard deviation over the 5 folds.
The results in Table <ref> indicate that the Doc2Vec, BERT, and Multinomial Naïve Bayes classifiers do not perform particularly well at this task. Fortunately, the fine-tuned Judge-BERT classifier performs relatively well, with an accuracy near 90% and where type 1 and type 2 errors are relatively similar. Overall, these results indicate that the Judge-BERT classifier is able to accurately classify moral judgements.
§ ANALYSIS OF MORAL JUDGEMENT
Using the Judge-BERT classifier, our next tasks are to better understand moral judgement across a variety of online social contexts and to analyze various trends in moral judgement. In order to minimize the transfer-error rate it is important to select subreddit-communities that are similar to the training dataset. In total we chose ten subreddits to explore in our initial analysis. The subreddits we chose can be broken into three main stylistic groups and are briefly described in Table <ref>.
Subreddit Description
64.3cmUsers pose questions in a scenario like the AITA dataset and receive advice or feedback on their situation.
54.3cmUsers confess to something that they have been keeping to themselves. Typically, confessions are about something immoral the poster has done.
64.8cmUsers engage in conversations with others to have a simple conversation or to here other opinions in order to change their worldview.
Subreddits used for analysis of moral judgement.
A post (in blue) made by a user along with the top response comment (white). The comment is then fed to our Judge-BERT classifier (green) to determine the moral valence of the post.
We applied the Judge-BERT classifier to the comments and posts of these ten subreddits. Specifically, given a post and its comment tree we identified the top-level comment with the highest score. This top-rated comment, which has received the most upvotes from the community, is considered to be the one passing judgement on the original poster. As illustrated in Fig. <ref>, this top-rated comment is then fed to our classifier and the resulting prediction is used to label the moral valence of the post and poster. It is important to be clear here: we are not predicting the moral valence of the comment itself, but rather the top-rated comment is used to pass judgement on the post.
Subreddit $\mu^+$ Score $\mu^-$ Score
/r/relationship_advice 44.32$\pm$1.77 23.90$\pm$1.81
/r/relationships 47.19$\pm$1.42 23.36$\pm$1.38
/r/dating_advice 43.49$\pm$2.51 33.45$\pm$3.03
/r/legaladvice 49.99$\pm$2.38 30.07$\pm$2.58
/r/dating 55.03$\pm$3.45 45.09$\pm$4.43
/r/offmychest 50.78$\pm$2.13 29.67$\pm$2.38
/r/TrueOffMyChest 47.93$\pm$1.40 23.95$\pm$1.38
/r/confessions 62.09$\pm$3.95 50.92$\pm$5.08
/r/CasualConversation 46.38$\pm$4.27 42.73$\pm$8.71
/r/changemyview 55.17$\pm$3.98 56.45$\pm$6.52
Mean scores and 95% confidence intervals of posts that were classified as having positive moral valence and negative moral valence. Positive posts are statistically significantly higher than negative posts ($p<0.001$).
§.§ Moral Valence and Popularity
Here we can begin to answer RQ2: Is moral valence correlated with the score of a post? In other words, do posts with positive moral valence score higher or lower than posts with negative moral valence. To answer this question, we extracted all posts and their highest scoring top-level comment from 2018 from each subreddit in Table <ref>.
Posts judged to have positive valence as a function of post score. Higher indicates more positive valence. Higher post scores are associated with more positive valence (Mann Whitney $\tau\in[0.395,0.41]$, $p<0.001$ two-tailed, Bonferroni corrected).
Popularity scores on Reddit exhibit a power-law distribution, so the mean-scores and their differences will certainly be misleading. Instead, we plot the ratio of comments judged to be positive against all comments as a function of the post score cumulatively in Fig. <ref>. Higher values in the plot indicate more positive valence. The results here are clear: post popularity is associated with positive moral valence. Most of the subreddits appear to have similar characteristics except for /r/CasualConversation, which has a much higher positive valence (on average) than the other subreddits. Mann-Whitney Tests for statistical significance on individual subreddits as well the aggregation of these tests with Bonferroni correction found that posts with positive valence have significantly higher scores than posts with negative valence ($\tau\in[0.395,0.41]$, $p<0.001$ two-tailed).
We take the additional step to argue that correlation does indeed imply causation in this particular case. Because posts are made before votes are cast, and because the text of a post is (typically) unchanged, and if we assume that scores are causally related to the text of the post, then the causal arrow can only point in one direction, , posts with positive moral valence result in higher scores than posts with negative moral valence.
These findings appear to conflict with other studies that have shown how negative posts elicit anger and encourage a negative feedback loop on social media [2, 7]. A further inspection of the posts indicated that posts classified as having positive moral valence often found users expressing that a moral norm had been breached. The difference in our results compared to others may be explained by perceived intent, that is, whether or not the moral violation occurred from an intentional agent towards a vulnerable agent, , dyadic morality [27]. Our inspection of comments expressing negative moral judgement confirms that the perceived intent of the poster is critical to the judgement rendered. These negative judgements typically highlight what the poster did wrong and advise the poster to reflect on their actions (or sometimes simply insult the poster). Conversely, we find that many posts judged to be positive clearly show that the poster is the vulnerable agent in the situation to some other intentional agent. The responses to these posts often display sympathy towards the poster and also outrage towards the other party in the scenario. These instances are perhaps best classified as examples of empathetic anger [17], which is anger expressed over the harm that has been done to another. We also note that some of the content labelled to have positive moral valence is simply devoid of a moral scenario. Examples of this can be primarily seen in /r/CasualConversation where the majority of posts are about innocuous topics.
Another possible explanation for our findings is that users on other online social media sites like Facebook and Twitter are more likely to like and share news headlines that elicit moral outrage; these social signals are then used by the site's algorithms to spread the headline further throughout the site [4, 15]. Furthermore, the content of the articles triggering these moral responses often covers current news events throughout the world. Our Reddit dataset, on the other hand, typically deal with personal stories and therefore tend to not have the same in-group/out-group reactions as those found on viral Facebook or Twitter posts.
§ ASSIGNING JUDGEMENTS TO USERS
Next we investigate RQ3: Do certain subreddit-communities attract users whose posts are typically classified by more negative or positive moral judgements? To answer this question we need to reconsider our unit of analysis. Rather than assigning moral valence to the individual post, in this analysis we consider the moral valence of the user who committed the post. To do this, we again find all posts and comments of the ten subreddits and find the highest scoring top-level comment; we classify whether that comment is judging the post to have positive or negative moral valence and then tally this for the posting user.
Of course, users are also able to post comments and sub-comments. So we expand this analysis to include judgements of users from throughout the entire comment tree. Each comment can have zero or more replies each with its own score. So, for each comment we identify the reply with the highest score and classify whether that reply is judging the comment to have positive or negative moral valence, and then tally this for the commenting user. We do this for each comment that has at least one reply at all levels in the comment tree.
By assigning moral valence scores to users we are able to capture all judgements across the ten subreddits and better-understand their behavior. It is important to remember that the classifier classifies the moral valence of text – with some amount of uncertainty – not the user specifically. So we emphasize that we do not label users as “good” or “bad” explicitly; rather, we identify users as having submitted posts and comments that were similar to comments that previously received positive or negative moral judgement.
Lorenz curve depicting the judgement inequality among users; Gini coefficient = 0.515
We include only users that were judged at least 50 times. Each user therefore has an associated count of positive and negative judgements. This begs an interesting question: are some users judged more positively or negatively than others? What does that distribution look like? To understand this breakdown we first plot a Lorenz curve in Fig. <ref>. We find that the distribution of moral valence is highly unequal: about 10% of users receive almost 40% of the observed negative judgements (Gini coefficient = 0.515).
This clearly indicates that there are a handful of users that receive the vast majority of negative judgements. To identify those users which receive a statistically significant proportion of negative judgements we perform a one-sided binomial test on each user. Simply put, this test emits a negativity probability, , the probability (p-value) that the negativity of a user is not due to chance.
Number of comments (normalized) as a the negativity threshold is raised. As the negativity threshold is raised the fraction of comments revealed tends towards 1. Higher lines indicate a higher concentration of negative users and vice versa.
Finally, we can illustrate the membership of each subreddit as a function of users' negativity probability. As expected, Fig. <ref> shows that as we increase the negativity threshold from almost certainly negative to uncertainty (from left to right) we begin to increase the fraction of comments observed. These curves therefore indicate the density of comments that are made from negative users (for varying levels of negativity); higher lines (especially on the left) indicate higher concentration of negativity. We find that /r/confessions, /r/changemyview, and /r/TrueOffMyChest contain a higher concentration of comments from more-negative users. On the opposite side of the spectrum, we find that /r/CasualConversation and /r/legaladvice have deep curves, which implies that these communities have fewer negative users than others.
§.§ Three Types of Negative Users
We select our group of users that have a statistically significant negative moral valence as those that were found to have a $p$-value less than 0.05 from our one-tailed binomial test.
Within this group we investigated into their posting habits to determine what types of posts they make to garner such a large number of negative judgements. From our analysis of these users we determined that they fall into three different stylistic groups.
* Explainer: These users will argue that what they did isn't that wrong.
* Stubborn Opinion: Users that do not acquiesce to the prevailing opinion of the responders.
* Returner: Users that repeatedly post the same situation hoping to elicit more-favorable responses.
The first type of user that we observe is the Explainer. The explainer typically makes a post and receives comments that condemn their immoral actions. In response to this judgement, the explainer will reply to many of the comments in an attempt to convince others that what they did was in fact moral. Often, this only serves to exacerbate the judgements made against them. This then leads to further negative judgements. In fact, we found that many of these users have only made a handful of posts that each have a large number of comments in self-defense. The large number of users that respond to these comments and with negative judgements is similar to the effect of online firestorms [24] but at a scale contained to only an individual post. For these types of posts we also note that some people do come to the defense of the poster, which follows similar findings that people show sympathy after a person has experienced a large amount of outrage [26].
A diagram showing the posting habits of a Returner. Posts are in the light blue boxes with blue arrows represent the order of posts. An example of a post response is shown in the white box with the red arrow representing the post it came from. Each post is prefaced with the overarching title, "Me and my partner are having a baby." followed by the current update on the situation. The response comments have also been condensed from their full length.
The second type of user we observe is the Stubborn Opinion user. These users are similar to but opposite from the Explainers. Rather than trying to change their perspective, the Stubborn Opinion user refuses to acquiesce to the prevailing opinion of the comment thread. For example, users posting to /r/changemyview that do not express a change of opinion despite the efforts and agreement of the commenting users often incur comments casting negative judgement. This back-and-forth sometimes becomes hostile. Many of these conversations end in personal attacks from one of the participants, which has also been shown in previous work on conversations derailing in /r/changemyview [5].
The third type of user is the Returner. The returner seeks repeated feedback from Reddit on the same subject. For example, when returners make posts seeking moral judgement, they will often engage in some of the discussion and may even agree with some of the critical responses. Some time later, the user will return and edit their original post or make another post providing an update about their situation. An example of a Returner is illustrated in Figure <ref>. In this case, a user continues to request advice after recently impregnating their partner. In these situations responding users often find previous posts on the same topic made by the same user and then use this information and highlight commentary from the previous post to build a stronger case against the user or highlight how the new post is nothing but a thinly-veiled attempt to shine a more-favorable light on their original situation. These attempts usually backfire and result in more negative judgments being cast against the user.
Positive Negative
Male 53,416 26,281
Female 57,126 20,714
Positive Negative
Male 139,163 74,384
Female 216,190 78,823
Contingency tables showing the number of positive and negative judgements for each self-reported gender. Moral valence has a small association with gender on /r/relationship_advice $\phi=0.07, p<0.01$ and /r/relationships $\phi=0.09, p<0.01$
§.§ Gender and Age Analysis
Our final task investigates RQ4: Are self-reported gender and age descriptions associated with positive or negative moral judgements? Recent studies on this topic have found that gender and moral judgements have a strong association [23]. Specifically, women are perceived to be victims more often than men and harsher punishments are sought for men. The rates at which men commit crimes tends to be higher than the rates of female crime and society generally views crimes as a moral violation [6]. If we apply these recent findings to our current research question we expect to find that male users will be judged negatively more often than females.
This task is not usually available on public social media services because gender and age are not typically revealed, while also allowing for anonymous posting. Fortunately, the posting guidelines of /r/relationships and /r/relationship_advice requires posters to indicate their age and gender in a structured manner directly in the post title. An example of this can be seen here:
where the poster uses [M27] to indicate that they identify as male aged 27 years and that their partner [F25] identifies as female aged 25 years. Using these conventions we are able to reliably extract users' self-reported age and gender.
We again apply our Judge-BERT model to assign a moral judgement to the post based on the top-scoring comment. In total we extracted judgements from 508,560 posts on /r/relationships and 157,537 posts on /r/relationship_advice. Because the posting age of a user may not be above the age of majority, we are careful to only collect data from users that are aged 18 and older via our research ethics guidelines. In general, the age breakdown appears to closely follow Reddit's age demographic. 90% of posters were between 18-30 years old.
Our first task is to determine if any association exists between moral judgement and gender. To answer this question we performed a $\chi^2$ test of independence. Contingency tables for this test are reported in Table <ref>. The $\chi^2$ test reports a significant association between gender and moral judgement in /r/relationships ($\chi^2 (1, 508,560) = 3874.6, p < .0001$) and in /r/relationship_advice ($\chi^2 (1, 157,537) = 762.2, p < .0001$). However, the $\chi^2$ test on such large sample sizes usually results in statistical significance; in fact, the $\chi^2$ test tends to find statistical significance for populations greater than 200 [28]. So we verify this association using $\phi$, which measures the strength of association controlled for population size. In this case, $\phi$ = 0.09 for /r/relationships and $\phi$ = 0.07 for /r/relationship_advice. These low values indicate that there is only a small association between gender and moral judgement.
Variable Coefficient p-value 95% CI
(Constant) -1.1575 $<$0.001 (-1.2093, -1.1058)
Gender 0.3076 $<$0.001 (0.2806, 0.3241)
Age 0.0059 $<$0.001 (0.0039, 0.0080)
Variable Coefficient p-value 95% CI
(Constant) -1.0923 $<$0.001 (-1.1214, -1.0631)
Gender 0.3814 $<$0.001 (0.3693, 0.3935)
Age 0.0034 $<$0.001 (0.0023, 0.0046)
Results for the logistic regression analysis for both subreddits.
§.§ Logistic Regression Analysis
Our second task is to determine if gender and age are associated with moral judgement. In other words, are young females, for instance, judged more positively than, say, old males?
To answer this question, we fit a two-variable logistic regression model where the binary-variable gender is encoded as 0 for female and 1 for male.
We report the findings from the logistic regressor for each subreddit in Table <ref>. These results indicate that males are judged more negatively than females. Specifically, in /r/relationship_advice being male is associated with a 35% increase in receiving a negative judgement. Similarly, in /r/relationships being male is associated with a 46% increase in receiving a negative judgement.
We also find that age has a relatively small effect on moral judgement; increased age is slightly correlated with negative judgement. Specifically, in /r/relationship_advice an increase in age by one year is associated with a 0.59% increase in receiving a negative judgement. In /r/relationships an increase in age by one year is associated with a 0.34% increase in receiving a negative judgement.
Simply put, those who are older and those who are male (independently) are statistically more likely to receive negative judgements from Reddit than those who are younger and female. Although gender is much more of a contributing factor than age and neither association is particularly strong.
An interesting avenue of work that has been investigated recently is the ability to preempt a conversation if you know further discussion will result in a personal attack <cit.>.
This differs from other frameworks that try to detect these personal attacks after they have already happened.
In this task two datasets have been used one from Wikipedia discussion pages and the other from the subreddit .
Each dataset consists of a back and forth conversation between users with some ending in a personal attack.
With this task in mind we wondered whether our asshole predictor could be transferred and used to predict these personal attacks as well.
The judgements user give on can be viewed as a form personal attacks when someone calls the original poster an asshole, since the user is expected to explain and give reasoning as to why this person is an asshole.
Due to this we wondered whether our classifier could be used on the conversations gone awry dataset to perform the same prediction task.
There are three different methods that have been used to solve this problem so far.
The first method uses a TF-IDF weighted bag-of-words representation extracted from the first comment-reply pair in the conversations.
The second method, called Awry, uses the first comment-reply pair of the conversation and extracts pragmatic features to use.
These pragmatic features include 38 different politeness strategies and 12 different prompt types.
The final model is called CRAFT (Conversational Recurrent Architecture for Forecasting) and is the current state of the art for this problem.
CRAFT consists of two parts, a pre-trained hierarchical recurrent encoder-decoder <cit.> and a prediction component that is just a multilayer perceptron with three fully-connected layers.
One major difference of the CRAFT model is that it works in an online fashion. This allows it to process comments in order and gain a context of what is happening in the conversation to signal it will de-reail.
Due to this factor the authors of CRAFT re-implemented the prior methods to also handle the comments in an online manner.
For our method we utilize our AITA classifier in both ways for this task. We first experiment with just using the second comment in each of the conversations. If our classifier predicts they are calling the original poster an asshole then we believe the conversation will de-rail. Otherwise, our prediction believes the conversation will proceed normally.
We also modified our model to stream comments in an online fashion as was done in the previous works.
Once our model makes a prediction that someone is an asshole we stop predicting and believe the conversation will end in a personal attack. Otherwise we force the model to continue down through all the comments and continue predicting.
We present our results in Table <ref>.
The results for our AITA classifier using just the second comment performs almost as well as the original Awry method on the Wikipedia talk pages. It does outperform a simple bag-of-words model. Part of the reason for it's performance is that it does not guess many of the conversations to de-rail. We can see a degradation in performance when we shift to the online model. In our online version our model ends up detecting too many of the posts as calling the other user an asshole to signal the derailment. This stands in contrast to all of the other methods that saw an increase in performance when shifting to an online version.
Overall our model does not seem to generalize to this other task very well. Our results do show that some of the language tendencies between calling someone an asshole and a conversation derailing may overlap.
5cWikipedia Talk Pages 5cReddit CMV
3-5 8-10
Model A P R FPR F1 A P R FPR F1
BoW 56.5 55.6 65.5 52.4 60.1 52.1 51.8 61.3 57.0 56.1
Awry 58.9 59.2 57.6 39.8 58.4 54.4 55.0 48.3 39.5 51.4
Cumul. BoW 60.6 57.7 79.3 58.1 66.8 59.9 58.8 65.9 46.2 62.1
Sliding Awry 60.6 60.2 62.4 41.2 61.3 56.8 56.6 58.2 44.6 57.4
CRAFT 66.5 63.7 77.1 44.1 69.8 63.4 60.4 77.5 50.7 67.9
AITA 57.3 57.9 57.3 14.4 57.5 53.3 53.3 53.3 22.4 53.3
AITA Online 54.8 58.6 54.8 66.8 56.6 52.1 55.9 52.1 44.1 53.9
Comparison of our AITA classifier used on the conversations gone awry dataset. Evaluation is done using accuracy, precision, recall, false positive rate, and F1 score.
The dynamics of discussion have been studied in many ways. One common way of doing so is analyzing the sentiment of the text in question. To do so, the text must first be embedded as a low dimensional feature vector. Then, given the sentiment of the text, e.g., positive or negative, one can train a classifier over the feature vectors, using the sentiment as the labels. As such, we discuss relevant text embedding methods, as follows.
Many text embedding methods focus on obtaining feature vectors for single words at a time <cit.>. However, it is also important to study groups of words (sentences, paragraphs) <cit.>. While simply combining feature vectors of each individual word in a sentence is one way to obtain group-level embeddings, this is often lacking. Doc2Vec [20] introduces an approach to embed groups of words at a time. It does so using a similar approach to its predecessor, word2vec <cit.>. However, Doc2Vec is also able to take into account the word order, and better embeds semantically similar words to each other. Several approaches have been proposed since, including many deep learning-based models <cit.>. However, BERT [8] is the current state-of-the-art. It uses a multilayer bidirectional Transformer encoder to learn the representation of a sequence of text, making it applicable to both words and groups of words.
The main drawback of sentiment analysis in the context of discussions, however, is that they do not account for how a given piece of text is responding to another. Presumably, the words people use in reaction to a situation depend on the morality of that situation. One approach does consider the response dynamic, though it does so in a general sense and not specifically for morality, described below.
Guimarães et al. <cit.> are interested in understanding different types of discussions that occur in the subreddits and . They propose four “conversational archetypes” that describe how a discussion progresses, and aim to test 10 hypotheses associated with them. For example, they hypothesize that “Harmonious” discussions, ones in which all the posts are not marked as “controversial” by Reddit standards, have high positive sentiment scores.
§ CONCLUSIONS
In this study, we show that it is possible to learn the language of moral judgements from text taken from /r/AmITheAsshole. We demonstrate that by extracting the labels and fine-tuning a BERT language model we can achieve good performance at predicting whether a user is rendering a positive or negative moral judgement. Using our trained classifier we then analyze a group of subreddits that are thematically similar to /r/AmITheAsshole for underlying trends. Our results showed that users prefer posts that have a positive moral valence rather than a negative moral valence. Another analysis revealed that a small portion of users are judged to have substantially more negative moral valence than others and they tend towards subreddits such as /r/confessions. We also show that these highly negative moral valence users fall into three different types based on their posting habits. Lastly, we demonstrate that age and gender have a minimal effect on whether a user is judged to be have positive or negative moral valence.
Although the Judge-BERT classifier enabled us to perform a variety of analysis it does pose some limitations. We are unable to verify if the classifier generalizes well to the other subreddits in our study. The test-subreddits do deviate from the types of moral analysis observed in the training data. Moral judgement is not the focus of /r/CasualConversation, for example.
In the future we hope to implement argument mining in order to gain a better understanding of the reasons for these judgements by extracting the underlying arguments given by users. Other works have done this by extracting rules of thumb through human annotation [11] but this limits the ability to perform a large scale analysis. Argument mining has shown success with extracting the persuasive arguments from subreddits like /r/changemyview [10] and would enable us to get a better understanding of moral judgements on social media. This would also allow us to aggregate the underlying themes from these judgements for further analysis.
§ ACKNOWLEDGEMENTS
We would like to thank Michael Yankoski and Meng Jiang for their help preparing this manuscript. This work is funded by the US Army Research Office (W911NF-17-1-0448) and the US Defense Advanced Research Projects Agency (DARPA W911NF-17-C-0094).
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|
# A duality operators/Banach spaces
Mikael de la Salle
(Date: )
###### Abstract.
Given a set $B$ of operators between subspaces of $L_{p}$ spaces, we
characterize the operators between subspaces of $L_{p}$ spaces that remain
bounded on the $X$-valued $L_{p}$ space for every Banach space on which
elements of the original class $B$ are bounded.
This is a form of the bipolar theorem for a duality between the class of
Banach spaces and the class of operators between subspaces of $L_{p}$ spaces,
essentially introduced by Pisier. The methods we introduce allow us to recover
also the other direction –characterizing the bipolar of a set of Banach
spaces–, which had been obtained by Hernandez in 1983.
## 1\. Introduction
All the Banach spaces appearing in this paper will be assumed to be separable,
and will be over the field $\mathbf{K}$ of real or complex numbers.
The _local theory of Banach spaces_ studies infinite dimensional Banach spaces
through their finite-dimensional subspaces. For example it cannot distinguish
between the (non linearly isomorphic if $p\neq 2$ [3, Theorem XII.3.8]) spaces
$L_{p}([0,1])$ and $\ell_{p}(\mathbf{N})$, as they can both be written as the
closure of an increasing sequence of subspaces isometric to
$\ell_{p}(\\{1,\dots,2^{n}\\})$ : the subspace of $L_{p}([0,1])$ made
functions that are constant on the intervals
$(\frac{k}{2^{n}},\frac{k+1}{2^{n}}]$, and the subspace of
$\ell_{p}(\mathbf{N})$ of sequences that vanish oustide of
$\\{0,\dots,2^{n}-1\\}$ respectively.
The relevant notions in the local theory of Banach spaces are the properties
of a Banach space that depend only on the collection of his finite dimensional
subspaces and not on the way they are organized. Said differently, the
properties that are inherited by finite representability. Such properties are
called _super-properties_. The central question is to understand whether one
super-property implies another, see Section 2 for terminology, details and
examples.
The main result is Theorem 1.6, where a theoretical criterion is obtained for
the implication of two super-properties which are moreover stable under
$\ell_{p}$-direct sums, for some $1\leq p<\infty$ which is fixed once and for
all. A result by Hernandez [12, 11] (Theorem 1.3 below) can be reformulated
as: a superproperty $P$ is stable under $\ell_{p}$-direct sums if and only if
it defined by $p$-homogeneous inequalities, _i.e._ if and only if there is an
operator $T$ between subspaces $\operatorname{dom}(T)$ and
$\operatorname{ran}(T)$ of $L_{p}$ spaces $L_{p}(\Omega_{1},m_{1})$ and
$L_{p}(\Omega_{2},m_{2})$ such that $X$ satisfies $P$ if and only if for every
$n$, every $f_{1},\dots,f_{n}$ in the domain of $T$ and every
$x_{1},\dots,x_{n}\in X$,
$\int_{\Omega_{2}}\|\sum_{i}(Tf_{i})(\omega_{2})x_{i}\|^{p}dm_{2}(\omega_{2})\leq\int_{\Omega_{1}}\|\sum_{i}f_{i}(\omega_{1})x_{i}\|^{p}dm_{1}(\omega_{1}).$
If one denotes by $\|T_{X}\|$ the (possibly infinite) norm of
$T\otimes\mathrm{id}_{X}$ between the subspaces $\operatorname{dom}(T)\otimes
X$ and $\operatorname{ran}(T)\otimes X$ of $L_{p}(\Omega_{i},m_{i};X)$, then
this condition can be shortly written as $\|T_{X}\|\leq 1$. So our result
characterizes, for two operators $S$ and $T$ between subspaces of $L_{p}$
spaces, when $\|T_{X}\|\leq 1$ implies $\|S_{X}\|\leq 1$.
This is a form of the bipolar theorem for a duality between the set
$\mathcal{X}$ of complex separable Banach spaces up to isometry and the set
$\mathcal{T}$ linear operators between subspaces of $L_{p}$ spaces defined by
the assignement $(T,X)\mapsto\|T_{X}\|$. Indeed, adapting the standard
terminology for locally convex topological vector spaces (see [4, II §6]), we
define :
###### Definition 1.1.
If $A\subset\mathcal{X}$ is a class of Banach spaces, then its _polar_
$A^{\circ}$ is the class of operators $T\in\mathcal{T}$ such that
$\|T_{X}\|\leq 1$ for every $X$ in $A$.
###### Definition 1.2.
If $B\subset\mathcal{T}$, then its polar ${}^{\circ}B$ is the class of Banach
spaces $X\in\mathcal{X}$ such that $\|T_{X}\|\leq 1$ for every $T$ in $B$.
This duality is a variant of the one considered in [27], where Pisier
restricts to operators between $L_{p}$ spaces (and not subspaces of $L_{p}$
spaces). If one is interested in the bipolar of a class of Banach spaces, the
two dualities are very different. On the other had, a description of the
bipolar for a class of operators for Pisier’s duality can be obtained from our
result, see Section 5 for details.
In a locally convex topological space, the bipolar theorem ([4, II §6]) states
that the bipolar of a set $C$ is equal to the closed convex hull of
$C\cup\\{0\\}$. The inclusion of the closed convex hull of $C\cup\\{0\\}$ in
the bipolar of $C$ is obvious; the content of the theorem is the other
inclusion, which follows from the Hahn-Banach theorem. The aim of this paper
is to state and prove a version of the bipolar theorem in this setting, for
the correct definition of “closed convex hull”. For the bipolar of a class of
Banach spaces, this is due to Hernandez. The methods we introduce allow us to
give a new proof of it (see Section 5 for the duality involving operators
between $L_{p}$ spaces).
###### Theorem 1.3.
([12]) The bipolar ${}^{\circ}(A^{\circ})$ of a class of Banach spaces
$A\subset\mathcal{X}$ is the class of Banach spaces finitely representable in
the class of all finite $\ell_{p}$-direct sums of elements in $A$.
There is also an isomorphic version of the previous result.
###### Theorem 1.4.
([12]) Let $A\subset\mathcal{X}$ and $X\in\mathcal{X}$. The following are
equivalent:
* •
$\|T_{X}\|<\infty$ for every $T\in A^{\circ}$.
* •
$X$ is isomorphic to a space finitely representable in the class of finite
$\ell_{p}$ direct sums of spaces in $A$, _i.e._ to a space in
${}^{\circ}A^{\circ}$.
In that case, the Banach-Mazur distance from $X$ to a space in
${}^{\circ}A^{\circ}$ is equal to $\sup_{T\in A^{\circ}}\|T_{X}\|$.
Our main result is the bipolar theorem for sets of operators. To state it we
have to introduce some definition.
###### Definition 1.5.
A _spatial isometry_ between finite dimensional subspaces of $L_{p}$ spaces is
a composition of isometries of the form:
* •
(Change of density) Restriction to a subspace of $L_{p}(\Omega,m)$ of the
multiplication by a nonvanishing measurable function
$h\colon\Omega\to\mathbf{K}^{*}$, _i.e._ $f\in L_{p}(\Omega,m)\mapsto hf\in
L_{p}(\Omega,|h|^{-p}m)$.
* •
(Equimeasurability outside of $0$) Maps of the form
$T\colon\operatorname{dom}(T)\subset L_{p}(\Omega,m)\to
L_{p}(\Omega^{\prime},m^{\prime})$ such that for every finite family
$f_{1},\dots,f_{n}\in\operatorname{dom}(T)$ and every Borel subset
$E\subset\mathbf{K}^{n}\setminus\\{0\\}$, $m(\\{x,(f_{1}(x),\dots,f_{n}(x))\in
E\\})=m^{\prime}(\\{x,(Tf_{1}(x),\dots,Tf_{n}(x))\in E\\})$.
It is not hard to prove (see Lemma 4.10 and Remark 4.11) that every spatial
isometry is of the form $C_{1}EC_{2}$ for $C_{1},C_{2}$ changes of phase and
measure and $E$ equimeasurable outside of $0$.
It is important that we require $0\notin E$, as we want for example that $f\in
L_{p}([0,1])\mapsto f\chi_{[0,1]}\in L_{p}([0,2])$ is a spatial isometry.
When $p$ is not an even integer, it is known that every isometry between
(separable) subspaces of $L_{p}$ spaces is a spatial isometry ([9]). The idea
developped in this article allows to recover this result, and to generalize it
to arbitrary $p$ : a linear map $T$ is a spatial isometry if and only if it is
a regular isometry, _i.e._ $\|T_{X}\|=\|T^{-1}_{X}\|=1$ for all $X$ (see
Remark A.2 and Corollary A.3).
We can now state the version of the bipolar theorem for sets of operators.
###### Theorem 1.6.
Let $B\subset\mathcal{T}$ and $T\colon\operatorname{dom}(T)\subset
L_{p}(\Omega_{1},m_{1})\to L_{p}(\Omega_{2},m_{2})$ be a linear map, and
$f_{1},f_{2},\dots,$ be a sequence generating a dense subspace of
$\operatorname{dom}(T)$. The following are equivalent :
* •
For every $X\in Banach$, $\sup_{S\in B}\|S_{X}\|\leq 1\implies\|T_{X}\|\leq
1$.
* •
For every $n$ and $\varepsilon>0$, there exist
* –
an operator $S=S_{0}\oplus S_{1}\oplus\dots\oplus S_{k}$ with $S_{0}$ of
regular norm $1$ and $S_{1}\dots,S_{k}\in B$,
* –
spatial isometries
$\displaystyle U\colon$ $\displaystyle\operatorname{dom}(U)\subset
L_{p}(\Omega_{1}\times[0,1])\oplus_{p}L_{p}([0,1])\to\operatorname{dom}(S),$
$\displaystyle V\colon$
$\displaystyle\operatorname{dom}(V)=S(\operatorname{ran}U)\to
L_{p}(\Omega_{2}\times[0,1])\oplus_{p}L_{p}([0,1]),$
* –
for every $i=1\dots,n$ there are $g_{i}\in L_{p}(\Omega_{1}\times[0,1])$,
$g^{\prime}_{i}\in L_{p}(\Omega_{2}\times[0,1])$ and $h_{i}\in L_{p}([0,1])$
such that $(g_{i},h_{i})\in\operatorname{dom}(U)$, $V\circ S\circ
U(g_{i},h_{i})=(g^{\prime}_{i},h_{i})$ and
$\displaystyle\left(\int_{\Omega_{1}\times[0,1]}|f_{i}(\omega)-g_{i}(\omega,s)|^{p}dm_{1}(\omega)ds\right)^{\frac{1}{p}}$
$\displaystyle\leq\varepsilon$
$\displaystyle\left(\int_{\Omega_{2}\times[0,1]}|(Tf_{i})(\omega)-g^{\prime}_{i}(\omega,s)|^{p}dm_{2}(\omega)ds\right)^{\frac{1}{p}}$
$\displaystyle\leq\varepsilon.$
It is instructing to work out explicitly a very simple case of this theorem,
namely for the obvious implication $\max(\|S_{X}\|,\|T_{X}\|)\leq
1\implies\|(T\circ S)_{X}\|\leq 1$. See Example 3.6.
In particular, we obtain the following characterization of the bipolar of a
set of operators.
###### Corollary 1.7.
The bipolar $({}^{\circ}B)^{\circ}$ of a class $B\subset\mathcal{T}$ is the
smallest class $B^{\prime}\subset\mathcal{T}$ containing $B$ and satisfying
the following properties :
1. (i)
$B^{\prime}$ contains $\\{T\in\mathcal{T},\sup_{X\in\mathcal{X}}\|T_{X}\|\leq
1\\}$.
2. (ii)
$B^{\prime}$ is stable under finite $\ell_{p}$-direct sums.
3. (iii)
If $T\in B^{\prime}$ and $U,V$ are spatial isometries then $U\circ T\circ V\in
B^{\prime}$.
4. (iv)
Let $T\in B^{\prime}$ such that $T\colon\operatorname{dom}(T)\subset
L_{p}(\Omega_{1},m_{1})\oplus L_{p}(\Omega,m)\to L_{p}(\Omega_{1},m_{1})\oplus
L_{p}(\Omega,m)$ is of the form $(f,g)\mapsto(Sf,g)$ for some
$S\in\mathcal{T}$ with domain equal to the image of $\operatorname{dom}(T)$ by
the first coordinate projection. Then $S\in B^{\prime}$.
5. (v)
If $T\in\mathcal{T}$ is an operator between subspaces of $L_{p}(\Omega,m)$ and
$L_{p}(\Omega^{\prime},m^{\prime})$ and if, for every finite family
$f_{1},\dots,f_{n}$ in the domain of $T$ and every $\varepsilon>0$, there is
$S\in B^{\prime}$ with domain contained in $L_{p}(\Omega,m)$ and range
contained in $L_{p}(\Omega^{\prime},m^{\prime})$ and elements
$g_{1},\dots,g_{n}\in\operatorname{dom}(S)$ such that
$\|f_{i}-g_{i}\|\leq\varepsilon$ and $\|Tf_{i}-Sg_{i}\|\leq\varepsilon$, then
$T\in B^{\prime}$.
Note however that Theorem 1.6 is a sense more precise than Corollary 1.7, as
it almost says that to obtain the bipolar of $B$ from $B$, it is enough to
apply the operations (i), (ii), (iii), (iv) and (v) only once, and in that
order. Almost because we obtain in this way all operators of the form
$T\otimes\mathrm{id_{L_{p}([0,1])}}$ with domain $\\{(\omega,s)\mapsto
f(\omega)\mid f\in\operatorname{dom}(T)\\}$ for $T\in({}^{\circ}B)^{\circ}$,
so one needs to apply one last time (iii) to obtain all of
$({}^{\circ}B)^{\circ}$. This improvement is not minor. For a long time, the
author was only able to prove Corollary 1.7, and actually expected that to
construct $({}^{\circ}B)^{\circ}$ out of $B$, it was necessary to iterate
these operations (and in particular (iv) and (v)) a large number of times
(even an arbitrarily large countable ordinal of times), and this ordinal
number was a measurement of the difficulty of computing the bipolar of a $B$.
This is closely related to the classical fact, essentially due to Banach,
that, to obtain the weak-* closure of a convex subset in the dual of a
separable Banach space, the number of times one needs to take limits of weak-*
convergent sequences can be an arbitrary countable ordinal. That this is not
the case will rely on a particularily strong form of the bipolar theorem (in
the linear setting) for the weak-* topology that we prove in Proposition 3.3.
See the discussion in subsection 3.1.
As for the usual bipolar theorem, the main content of the theorem is the
inclusion ${}^{\circ}B^{\circ}\subset B^{\prime}$. The reverse inclusion is
rather obvious because it is rather clear that ${}^{\circ}B^{\circ}$ contains
$B$ and satisfies all the properties (i-v).
So one can reformulate the non-trivial part of Corollary 1.7 as follows : if
$T\notin B^{\prime}$, then there is a Banach space $X$ such that
$X\in{}^{\circ}B$ but $\|T_{X}\|>1$. Constructing Banach spaces with
prescribed properties is a notoriously difficult problem in general. What
saves us here is that we do not construct $X$ explicitly, but we let the Hahn-
Banach theorem construct it for us. This is achieved by suitably encoding the
class of Banach spaces in a locally convex topological vector space $H$ and
the class of operators between subspaces of $L_{p}$ spaces in its dual
$H^{*}$, in such a way that the polarity between $\mathcal{X}$ and
$\mathcal{T}$ corresponds to the usual polarity in topological vector spaces.
So, once these two encodings are well understood, both Theorems 1.3 and 1.6
are just an application of the bipolar theorem in $H$ and $H^{*}$. When $X$ is
a Banach space of dimension $n$, $X$ will be encoded inside the real Banach
space $C(\mathbf{KP}^{n-1})$ of real-valued continuous functions on the
projective space of dimension $n-1$. Similarly an operator $T$ with a domain
of dimension $n$ will be encoded inside the dual of $C(\mathbf{KP}^{n-1})$.
The space $H$ evoked would then be the projective limit of a suitable system
of the spaces $C(\mathbf{KP}^{n-1})$. But since the study of the polarity
between $\mathcal{X}$ and $\mathcal{T}$ readily reduces to finite dimensional
Banach spaces and operators with finite dimensional domains, we prefer to work
directly with $C(\mathbf{KP}^{n-1})$ and never even formally introduce $H$.
### Notation
To avoid any set-theoretical problem (of $\mathcal{T}$ not being a set), all
the measure spaces appearing here will be standard measure spaces taken in
some fixed set containing $[0,1]$ with the Lebesgue measure and that is stable
by taking equivalent measures, measurable subsets with restriction of the
measure, and finite direct sums. By direct sum of a finite sequence
$(\Omega_{1},m_{1}),\dots,(\Omega_{n},m_{n})$ we mean the space
$(\Omega_{1}\cup\dots\cup\Omega_{n},m_{1}\oplus\dots\oplus m_{n})$ where
$\Omega_{1}\cup\dots\cup\Omega_{n}$ is the disjoint union and the measure is
$A\mapsto\sum_{i}m_{i}(A\cap\Omega_{i})$. None of the results depend on the
choice.
The $\ell_{p}$-direct sum of a finite family $T_{1},\dots,T_{n}$ of operators
from $\operatorname{dom}(T_{i})\subset L_{p}(\Omega_{i},m_{i})$ to
$\operatorname{ran}(T_{i})\subset L_{p}(\Omega^{\prime}_{i},m^{\prime}_{i})$
is the operator $T_{1}\oplus\dots\oplus T_{n}$ from
$\operatorname{dom}(T_{1})\oplus\dots\oplus\operatorname{dom}(T_{n})\subset
L_{p}(\Omega_{1}\cup\dots\cup\Omega_{n},m_{1}\oplus\dots\oplus m_{n})$ to
$\operatorname{ran}(T_{1})\oplus\dots\operatorname{ran}(T_{n})\subset
L_{p}(\Omega^{\prime}_{1}\cup\dots\cup\Omega^{\prime}_{n},m^{\prime}_{1}\oplus\dots\oplus
m^{\prime}_{n})$.
An operator $T\in\mathcal{T}$ is called regular if $\|T_{X}\|<\infty$ for
every Banach space $X$, or equivalently if $\|T_{\ell_{\infty}}\|<\infty$. In
that case the quantity $\|T_{\ell_{\infty}}\|=\sup_{X}\|T_{X}\|$ is called the
regular norm of $T$ and denoted $\|T\|_{r}$. We will denote by $REG$ the set
of operators $T\in\mathcal{T}$ such that $\|T\|_{r}\leq 1$.
### Organization of the paper
The first section presents some necessary background and some motivation for
studying this polarity. Section 3 contains various preliminaries, including
basic reminders on measure theory and on the linear bipolar theorem, as well
as one result on which the rest will rely: Proposition 3.3. Section 4 contains
the proof of the main theorem. It starts by defining the encoding of spaces
and operators in a linear duality, and then studies this encoding. Section 5
contains a discussion of variants of the duality presented in the
introduction. In an appendix we present a new proof and a generalization, in
the context of Section 4, of Hardin’s theorem [9]. Hardin’s theorem appears as
a direct corollary of the study of the invariant subspaces for some families
of representations of $\textrm{GL}_{n}(\mathbf{K})$ on $C(\mathbf{KP}^{n-1})$.
Some of the results have been announced in the report _Group actions on Banach
spaces and a duality spaces/operators_ [28, pp 2304–2307].
## 2\. Background and motivation
### 2.1. Reminders on Banach space geometry
If $n$ is an integer, the set $Q(n)$ of all $n$-dimensional normed space up to
isometry, equipped with the Banach-Mazur distance
$d(E,F)=\inf\\{\|u\|\|u^{-1}\|\mid u\colon E\to F\textrm{ linear
invertible}\\},$
becomes a compact metric space, _the Banach-Mazur compactum_. Beware that it
is not $d$ but $\log d$ which is a distance in the usual way ($d$ is
submultiplicative $d(E,G)\leq d(E,F)d(F,G)$ rather subadditive, and two
isometric spaces are at Banach-Mazur distance $1$), but following the
tradition we still call $d$ the Banach-Mazur distance.
We say that a Banach space $X$ is finitely representable in another Banach
space $Y$ if for every finite-dimensional space $E\subset X$ and every
$\varepsilon>0$ there is a subspace $F\subset Y$ of same dimension as $E$ such
that $d(E,F)\leq 1+\varepsilon$. In other words, if for every $n$, the closure
in $Q(n)$ of the space of $n$-dimensional subspaces of $X$ is contained in the
same closure but for $Y$. This is equivalent to $X$ being isometrically a
subspace of an ultraproduct of $Y$ [10].
More generally, we say that a Banach space $X$ is finitely representable in a
class $B$ of Banach spaces if for every finite-dimensional space $E\subset X$
and every $\varepsilon>0$ there is a subspace $F$ of a space in $B$ of same
dimension as $E$ such that $d(E,F)\leq 1+\varepsilon$. We can therefore define
_a class of Banach spaces up to finite representability_ as a collection
$A_{n}$ of closed subsets of $Q(n)$ such that for every $n>m$, every
$m$-dimensional subspace of every $E\in A_{n}$ belongs to $A_{m}$. In this
representation, finite representability corresponds to inclusion.
The $\ell_{p}$-direct sum of a finite family $X_{1},\dots,X_{n}$ of Banach
spaces is the space $X_{1}\oplus X_{2}\oplus\dots\oplus X_{n}$ for the norm
$\|(x_{1},\dots,x_{n})\|=(\|x_{1}\|^{p}+\dots+\|x_{n}\|^{p})^{\frac{1}{p}}$.
### 2.2. Motivation
Estimating $\|T_{X}\|$ in terms of the properties of $T$ and the geometric
properties of $X$ is a central aspect in the geometry of Banach spaces. Most
natural geometric classes of Banach spaces are characterized in terms of such
quantities, and most celebrated results can be expressed in the form “$T$
belongs to the bipolar of $B$” for specific $T$ and $B\subset\mathcal{T}$. We
list a few historical important examples for illustration. See [27, Section 4]
for other examples.
* •
Hilbert spaces are characterized by the parallelogram inequality, _i.e._ the
property $\|T_{X}\|\leq 1$ where $T\colon\ell_{2}^{2}\to\ell_{2}^{2}$ has
matrix $\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ -1&1\end{pmatrix}$.
* •
[6, 5] A Banach space has the UMD property (for Unconditional Martingale
Differences) if and only if the Hilbert transform $H\colon
L_{2}(\mathbf{R})\to L_{2}(\mathbf{R})$ satisfies $\|H_{X}\|<\infty$.
* •
Let $(\Omega,\mu)$ be a probability space and
$\varepsilon_{i}\colon\Omega\to\\{-1,1\\}$, $i\in\mathbf{N}$ be iid centered
(Bernoulli) random variables. A Banach space $X$ has type $p$ if there is a
constant $T_{p}$ such that
$\|\sum\varepsilon_{i}x_{i}\|_{L_{p}(\Omega;X)}\leq
T_{p}(\sum\|x_{i}\|^{p})^{\frac{1}{p}}$
for every $x_{i}\in X$. Equivalently if $\|T_{X}\|\leq 1$ where
$T\colon\operatorname{span}(\varepsilon_{i})\subset
L_{p}(\Omega)\to\ell_{p}(\mathbf{N})$ is the linear map sending
$\varepsilon_{i}$ to $\frac{1}{T_{p}}(1_{k=i})_{k\in\mathbf{N}}$.
* •
A Banach space $X$ has cotype $p$ if there is a constant $C_{p}$ such that
$\|\sum\varepsilon_{i}x_{i}\|_{L_{p}(\Omega;X)}\geq\frac{1}{C_{p}}(\sum\|x_{i}\|^{p})^{\frac{1}{p}}$
for every $x_{i}\in X$. Equivalently if $\|S_{X}\|\leq$ where
$S\colon\ell^{p}\to L_{p}(\Omega)$ is the linear map sending
$(a_{i})_{i\in\mathbf{N}}$ to $\frac{1}{C_{p}}\sum_{i}a_{i}\varepsilon_{i}$.
* •
A Banach space $X$ has type $>1$ if and only if it is K-convex [26]:
$\|T_{X}\|<\infty$, where $T\in B(L_{2}(\\{-1,1\\}^{\mathbf{N}}))$ is the
orthogonal projection on the space spanned by the coordinates
$\varepsilon_{i}\colon\omega=(\omega_{n})_{n\in\mathbf{N}}\mapsto\omega_{i}$.
* •
Denote by $d_{n}(X)$ the supremum over all $n$-dimensional subspaces $E$ of
$X$ subspaces of the Banach-Mazur distance from $U$ to $\ell_{2}^{n}$. Then
(this is due to Pisier but written in [14]) up to a factor $2$, $d_{n}(X)$ is
equal to $\sup\|T_{X}\|$, where the sup is taken over all $T\colon L_{2}\to
L_{2}$ of norm $1$ and rank $n$. We can therefore express the Milman-Wolfson
Theorem [20] as follows: a Banach space $X$ has type $p>1$ if and only if
$\|T_{X}\|=o(\|T\|\mathrm{rk}(T)^{\frac{1}{2}})$ as $\mathrm{rk}(T)\to\infty$.
We now move to a more detailed discussion of two of the author’s main
motivations.
### 2.3. Group representations on Banach spaces
Another motivation comes from the study of representations of groups on Banach
spaces. Let $G$ be a locally compact topological group with a fixed left Haar
measure. We recall that every strong-operator-topology (SOT) continuous
representation $\pi$ of $G$ on a Banach space $X$ extends to a representation
of the convolution algebra $C_{c}(G)$ of compactly supported continuous
functions on $G$ by setting $\pi(f)x=\int f(g)\pi(g)xdg$ for every $x\in X$.
For example, if $\lambda_{p}$ denotes the left-regular representation on
$L_{p}(G)$ $\lambda_{p}(g)f=f(g^{-1}\cdot)$, then $\lambda_{p}(f)$ is the
convolution operator $\xi\mapsto f\ast\xi$.
When $A$ is a class of Banach spaces, denote by $C_{A}(G)$ the completion of
$C_{c}(G)$ for the norm
$\|f\|_{C_{A}(G)}=\sup\|\pi(f)\|_{B(X)},$
where the supremum is over all SOT-continuous continuous isometric
representations $\pi$ of $G$ on a space $X$ in $A$.
The following result, which generalizes the classical fact that, for amenable
groups, the full and reduced $C^{*}$-algebras coincide, reduces the
understanding the representation theory of $G$ on a Banach space $X$ to the
understanding of $\|T_{X}\|$ for convolution operators $T$. This known fact
has already appeared in several unpublished texts (for example in the author’s
habilitation thesis), but seems to be missing from the published literature.
###### Proposition 2.1.
If $G$ is amenable and $\pi$ is an isometric representation of $G$ on a Banach
space $X$, then for every $f\in C_{c}(G)$,
$\|\pi(f)\|_{B(X)}\leq\|\lambda_{p}(f)_{X}\|.$
In particular, if a class of Banach spaces $A$ has the property that
$L_{p}(G;X)\in A$ for every $X\in A$, then
$\|f\|_{C_{A}(G)}=\sup_{X\in A}\|\lambda_{p}(f)_{X}\|.$
###### Proof.
Fix a norm $1$ element $\xi\in L_{p}(G)$ and define an isometric linear map
$\alpha\colon X\to L_{p}(G;X)$ by $\alpha(x)(g)=\xi(g)\pi(g^{-1})x$.
Then for $h\in G$,
$(\alpha(\pi(h)x)-\lambda(h)\alpha(x))(g)=(\xi(g)-\xi(h^{-1}g))\pi(g^{-1}h)x$,
and
$\|\alpha(\pi(h)x)-\lambda(h)\alpha(x)\|=\|x\|\|\xi-\lambda(h)\xi\|_{L_{p}(G)}.$
By the triangle inequality
$\|\alpha(\pi(f)x)-\lambda(f)\alpha(x)\|\leq\|f\|_{L_{1}(G)}\|x\|\sup_{h\in\mathrm{supp}(f)}\|\xi-\lambda(h)\xi\|_{L_{p}(G)},$
and using that $\alpha$ is isometric we obtain
$\|\pi(f)x\|\leq\|\lambda(f)_{X}\|\|x\|+\|f\|_{L_{1}(G)}\|x\|\sup_{h\in\mathrm{supp}(f)}\|\xi-\lambda(h)\xi\|_{L_{p}(G)}.$
We deduce
$\|\pi(f)\|\leq\|\lambda(f)_{X}\|+\|f\|_{L_{1}(G)}\sup_{h\in\mathrm{supp}(f)}\|\xi-\lambda(h)\xi\|_{L_{p}(G)}.$
When $G$ is amenable, the last term can be made arbitrarily small, which
proves the proposition. ∎
In the particular case of a compact group, this result lies at the heart of
the proofs of Lafforgue’s strong property (T) for higher-rank algebraic
groups. For example, thanks to the techniques of strong property (T), the
conjecture [2] that any action by isometries of a lattice in a connected
higher-rank simple Lie group on a super-reflexive Banach space has been
reduced to the following conjecture, see [16, 17, 13], see also [31]. Denote,
for any $\delta\in[-1,1]$, by $T_{\delta}$ the operator on
$L_{2}(\mathbf{S}^{2})$ mapping $f$ to the fonction $(T_{\delta}f)(x)=$ the
average of $f$ on the circle $\\{y\in\mathbf{S}^{2}\mid\langle
x,y\rangle=\delta\\}$. For any $\theta\in\mathbf{R}/2\pi\mathbf{Z}$, denote by
$S_{\theta}$ the operator on $L_{2}(\mathbf{S}^{3})$ mapping $f$ to the
fonction $(S_{\theta}f)(z)=$ the average of $f$ on the circle
$\\{\frac{1}{\sqrt{2}}(e^{i\theta}+e^{i\varphi}j)z\mid\varphi\in\mathbf{R}/2\pi\mathbf{Z}\\}$
(where we identify $\mathbf{S}^{3}$ with the norm $1$ quaternions in the usual
way). The conjecture is that for every super-reflexive Banach space, there
exist $\alpha>0$ and $C\in\mathbf{R}_{+}$ such that for every
$\delta\in[-1,1]$ and $\theta\in\mathbf{R}$,
$\|(T_{\delta}-T_{0})_{X}\|\leq C|\delta|^{\alpha}\textrm{ and
}\|(S_{\theta}-S_{\pi/2})_{X}\|\leq C|\theta-\pi/2|^{\alpha}.$
### 2.4. Super-expanders and embeddability of graphs in Banach spaces
Another motivation for studying the quantity $\|T_{X}\|$ is its well-known
connection with Poincaré inequalities and embeddability of expanders in $X$.
If $\mathcal{G}=(V,E)$ is a finite connected graph, we may define111There are
many small variants of the definition. But they do not matter for the
discussion here, though they do matter for other issues, see for example [15].
its $X$-valued $p$-Poincaré constant $\pi_{p,\mathcal{G}}(X)$ as the smallest
constant $\pi$ such that for every $f\colon V\to X$ satisfying $\sum_{v\in
V}\mathrm{deg}(v)f(v)=0$,
$\left(\sum_{v\in
V}\mathrm{deg}(v)\|f(v)\|^{p}\right)^{\frac{1}{p}}\leq\pi\left(\sum_{(v,w)\in
E}\|f(v)-f(w)\|^{p}\right)^{\frac{1}{p}}.$
Note that $\pi_{p,\mathcal{G}}(X)=\|T_{X}\|$ for $T$ the inverse of the linear
map $f\in\ell_{p}^{0}(V,\mathrm{deg})\mapsto(f(v)-f(w))_{(v,w)\in
E}\in\ell_{p}(E)$.
A sequence $\mathcal{G}_{n}=(V_{n},E_{n})$ of bounded degree graphs is called
_a sequence of expanders with respect to $X$_ if $\lim_{n}|V_{n}|=\infty$ and
$\sup_{n}\pi_{p,\mathcal{G}_{n}}(X)<\infty$. This does not depend on $p$ [21,
22, 7], see also [15, Proposition 3.9].
For example, if $p=2$ and $X=\mathbf{K}$ (or a Hilbert space), then
$\pi_{p,2}(\mathbf{K})$ is equal to $(2-2\lambda_{2})^{-\frac{1}{2}}$, for
$\lambda_{2}$ the second largest eigenvalue of the random walk operator on
$\mathcal{G}$. So being a sequence of expanders with respect to $\mathbf{K}$,
or to an $L^{p}$ space for some $p<\infty$, is the same as the usual
definition of expander graphs.
According to [19], a sequence $\mathcal{G}_{n}$ is called a sequence of super-
expanders if they are expanders with respect to all uniformly convex Banach
spaces. The existence of super-expanders is a difficult result. Essentially
two classes of examples have been obtained, by Lafforgue [17] and by Mendel
and Naor [19]. Lafforgue’s examples are even expanders with respect to all
Banach spaces of type $>1$. All these results are therefore results of the
norm “$T$ belongs to be bipolar of $S$”, where $S$ is any of the operators
quantifying the fact that a Banach space has nontrivial type or is super-
reflexive, and $T$ are correctly scaled operators in the definition of the
$p$-Poincaré constant. Many intriguing questions remain open, which can all be
formulated in the same way. For example,
###### Question 2.2.
[19] Are all expander sequences super-expanders? Expanders with respect to all
spaces of non-trivial type?
###### Question 2.3.
[19, 17] Does there exist a sequence of super-expanders of girth going to
infinity? And of logarithmic girth in the number of vertices? Are the
expanders coming from higher-rank simple Lie groups super-expanders?
###### Question 2.4.
[19, 17] Does there exist a sequence of expanders with respect to all Banach
spaces of nontrivial coptype?
A positive answer to this question is conjectured in [19], and Lafforgue even
suggests that the super-expanders coming from lattices in
$\mathrm{SL}_{3}(\mathbf{Q}_{p})$ (or other higher-rank simple algebraic
groups over non-archimedean local fields) as in [17] are such examples. But
this is wide open, as is the following.
###### Question 2.5.
[27] Are all expander sequences expanders with respect to all spaces of non-
trivial cotype?
One of the reasons for the interest in expanders with respect to Banach spaces
is the well-known fact, which essentially goes back to Gromov, that a sequence
of expanders with respect to $X$ does not coarsely embed into $X$. See for
example [27, Section 3]. Being an expander with respect to $X$ is much
stronger than non coarse embeddability (a striking example is given in [1]),
but by [34] (see also [24] for $L_{1}$ spaces) there is equivalence between
non-coarse embeddability into families of Banach spaces under closed finite
representability and $\ell_{p}$ direct sums and some _other_ forms of Poincaré
inequalities.
## 3\. Preliminaries
### 3.1. On the bipolar in a dual Banach space
In the whole paper, for a subset $C$ of a real Banach space $E$ with dual
$E^{*}$, we denote its polar
$C^{\circ}=\\{x^{*}\in E^{*},\langle x^{*},x\rangle\geq-1\textrm{ for all
}x\in C\\}.$
When $C\subset E$ is a cone (that is $x\in C$ implies $\\{tx\mid
t\in[0,\infty)\\}\subset C$), then its polar $C^{\circ}$ coincides with
$\\{x^{*}\in E^{*},\langle x^{*},x\rangle\geq 0\textrm{ for all }x\in C\\}$.
It is also a cone.
Similarily, when $C\subset E^{*}$ we denote its polar for the weak-* topology
by
${}^{\circ}C=\\{x\in E,\langle x^{*},x\rangle\geq-1\textrm{ for all }x^{*}\in
C\\}.$
Again, if $C$ is a cone, ${}^{\circ}C$ coincides with $\\{x\in E,\langle
x^{*},x\rangle\geq 0\textrm{ for all }x^{*}\in C\\}$ and is again a cone.
It should be always clear from the context whether the polarity is considered
in this linear setting of two vector spaces in duality or between
$\mathcal{X}$ and $\mathcal{T}$ as in Definition 1.1 and 1.2.
The classical bipolar theorems in this setting take the following forms:
###### Theorem 3.1.
Let $E$ be a real Banach space.
If $C\subset E$, then its bipolar ${}^{\circ}(C^{\circ})$ is equal to the norm
closure of the convex hull of $C\cup\\{0\\}$.
If $C\subset E^{*}$, then its bipolar $({}^{\circ}C)^{\circ}$ is equal to the
weak-* closure of the convex hull of $C\cup\\{0\\}$.
The second statement is not so useful for our purposes because taking the
weak-* closure can be quite complicated, as we shall soon recall. Fortunately,
there is an interesting consequence of the Krein-Smulian theorem [8, Theorem
V.12.1], which asserts that a convex subset of $E^{*}$ for a separable Banach
space $E$ is weak-* closed if and only if it is sequentially weak-* closed,
see [8, Theorem V.12.10]. This allows to significantly strengthen the result
for separable Banach spaces as follows.
If $C$ is a subset of a dual $E^{*}$, let us define an increasing family of
subsets $C_{\alpha}\subset E^{*}$ indexed by the ordinals $\alpha$ by letting
$C_{0}=C$, $C_{\alpha}$ be the set of all weak-* limits of sequences in
$C_{\alpha-1}$ if $\alpha$ is a successor and
$C_{\alpha}=\cup_{\beta<\alpha}C_{\beta}$ if $\alpha$ is a limit ordinal. The
smallest ordinal $\alpha$ such that $C_{\alpha}=C_{\alpha+1}$ (that is
$C_{\alpha}$ is weak-* sequentially closed) is sometimes called the order of
$C$. When $E$ is separable, the order of $C$ is countable, see for example the
argument in the proof of [8, Theorem V.12.10]. Moreover, if $C$ is convex,
then so is $C_{\alpha}$ for every $\alpha$. It follows from [8, Theorem
V.12.10] that, for the order of $C$, $C_{\alpha}$ coincides with the weak-*
closure of $C$. Let us summarize this discussion.
###### Proposition 3.2.
Let $E$ be a real separable Banach space and $C$ be a subset of $E^{*}$. There
is a countable ordinal $\alpha$ such that the bipolar of $C$ coincides with
$(\operatorname{conv}(C))_{\alpha}$.
The smallest ordinal $\alpha$ in the previous proposition measures the
difficulty to construct the bipolar of $C$ out of $C$.
The order has been more studied for linear subspaces $C$. It is known that for
many cases, every countable ordinal appears as the order of a linear subspace
of $E^{*}$. This was stated by Banach [18], and later examples such as
$E^{*}=\ell_{1}=(c_{0})^{*},\ell_{\infty},H_{\infty}$ were provided together
will full proofs [33, 32]. It is now known that this holds whenever $E$ is not
quasi-reflexive, that is when the canonical image of $E$ has infinite
codimension in its bidual [25]. See also the survey [23] for more information
on this.
It turns out that, for our applications, the order will always be equal to
$1$. This will follow from the following result.
###### Proposition 3.3.
Let $E$ be a real Banach space and $C\subset E^{*}$. Assume that there is a
convex subset $A\subset E$ such that $A\cap\\{x\in E\mid\|x\|\leq r\\}$ is
norm-compact for every $r>0$ and $A^{\circ}\subset C$.
Then the bipolar $({}^{\circ}C)^{\circ}$ of $C$ is equal to the norm closure
of the convex hull of $C$.
###### Proof.
Note that our assumptions implies that $0\in C$ (as $0\in A^{\circ}$). Let
$C^{\prime}$ be the norm closure of the convex hull of $C$. We know from the
bipolar theorem (Theorem 3.1) that $({}^{\circ}C)^{\circ}$ is equal to the
weak-* closure of $\mathrm{conv}(C)$, so the inlusion
$C^{\prime}\subset({}^{\circ}C)^{\circ}$ is obvious. To prove the converse
inclusion, consider $x\in E^{*}\setminus C^{\prime}$. We have to prove that
$x$ does not belong to the weak-* closure of the convex hull of $C$.
Let $j\colon E\to E^{**}$ be the canonical inclusion of $E$ in its bidual. By
the Hahn-Banach separation theorem in the Banach space $E^{*}$, there is
$\varphi\in E^{**}$ such that $\inf_{C}\varphi\geq-1$ and $\varphi(x)<-1$. In
particular, we have $\inf_{A^{\circ}}\varphi\geq-1$, that is
$\varphi\in(A^{\circ})^{\circ}=({}^{\circ}(j(A)))^{\circ}$. By Theorem 3.1
again, $({}^{\circ}(j(A)))^{\circ}$ is equal to the weak-* closure of (the
convex set) $j(A)$. But the assumption on $A$ implies that $j(A)$ is already
weak*-closed. Indeed, by the Krein-Smulian theorem, it is enough to show that
$j(A)\cap\overline{B}_{E^{**}}(0,r)$ is weak-* closed for every $r>0$. This is
true as $j(A)\cap\overline{B}_{E^{**}}(0,r)=j(A\cap\overline{B}_{E}(0,r))$ is
even norm-compact as a continuous image of a norm-compact set, and norm-
compact subsets of $E^{**}$ are weak-* closed. So $j(A)$ being weak-* closed,
we have proved that $\varphi\in j(A)$. In particular, $\varphi$ is
$\sigma(E^{*},E)$-continuous, and we obtain, as announced, that $x$ does not
belong to the weak-* closure of the convex hull of $C$. ∎
### 3.2. Reminders on the Jordan decomposition of measures
Recall that any signed measure $m$ on a Borel space has a unique decomposition
$m=m_{+}-m_{-}$ for two positive measures satisfying
$\|m\|=\|m_{+}\|+\|m_{-}\|$ (where the norm is the total variation norm). This
is the Jordan decomposition of $m$. If $m=m_{1}-m_{2}$ is any other
decomposition with $m_{1},m_{2}$ positive measures, then
$m_{1}-m_{+}=m_{2}-m_{-}$ is a positive measure. We will use the following
elementary fact.
###### Lemma 3.4.
Let $m$ and $m^{\prime}$ be any signed measure, and let $m_{1},m_{2}$ be any
positive finite measures such that $m=m_{1}-m_{2}$. There is a decomposition
$m^{\prime}=m^{\prime}_{1}-m^{\prime}_{2}$ with
$\|m_{1}-m^{\prime}_{1}\|+\|m_{2}-m^{\prime}_{2}\|=\|m-m^{\prime}\|.$
###### Proof.
Let $m=m_{+}-m_{-}$ and $m^{\prime}=m^{\prime}_{+}-m^{\prime}_{-}$ be the
Jordan decompositions. A small computation gives that
$\|m-m^{\prime}\|=\|m_{+}-m^{\prime}_{+}\|+\|m_{-}-m^{\prime}_{-}\|$.
By the property of the Jordan decomposition just recalled,
$m^{\prime\prime}:=m_{1}-m_{+}=m_{2}-m_{-}$ is a positive measure. Define
$m^{\prime}_{1}=m^{\prime}_{+}+m^{\prime\prime}$ and
$m^{\prime}_{2}=m^{\prime}_{2}+m^{\prime\prime}$, so that
$m^{\prime}=m^{\prime}_{1}-m^{\prime}_{2}$ and
$\|m_{1}-m^{\prime}_{1}\|+\|m_{2}-m^{\prime}_{2}\|=\|m_{+}-m^{\prime}_{+}\|+\|m_{-}-m^{\prime}_{-}\|=\|m-m^{\prime}\|.$
∎
### 3.3. On (iv) in Corollary 1.7
This short subsection is not needed anywhere else in the paper, but it
hopefully illustrates some basic things about Theorem 1.6 and Corollary 1.7.
We start by a lemma which clarifies in which situation an operator $T$ is of
the form (iv) in Corollary 1.7.
###### Lemma 3.5.
Let $T$ be a norm $\leq 1$ operator between subspaces
$\operatorname{dom}(T),\operatorname{ran}(T)\subset L_{p}(\Omega,m)$ and
$A\subset\Omega$ measurable. The following are equivalent.
* •
$Tf(x)=f(x)$ for almost every $x\in\Omega\setminus A$ and every
$f\in\operatorname{dom}(T)$.
* •
If we write $L_{p}(\Omega,m)=L_{p}(A,m)\oplus_{p}L_{p}(\Omega\setminus A,m)$,
then there is an operator $S$ with domain equal to the image of
$\operatorname{dom}(T)$ by the first coordinate projection such that
$T(f_{1},f_{2})=(Sf_{1},f_{2})$ for all
$(f_{1},f_{2})\in\operatorname{dom}(T)$.
In that case, $S$ is unique,
$\operatorname{dom}(S)=\\{f\left|{}_{A}\right.,f\in\operatorname{dom}(S)\\}$
and $S(f\left|{}_{A}\right.)=(Tf)\left|{}_{A}\right.$ for all
$f\in\operatorname{dom}(T)$.
###### Proof.
Clearly, the assumption that $Tf(x)=f(x)$ for almost every
$x\in\Omega\setminus A$ and every $f\in\operatorname{dom}(T)$ is equivalent to
the existence of a linear map $S\colon\operatorname{dom}(T)\to L_{p}(A,m)$
such that $T(f_{1},f_{2})=(S(f_{1},f_{2}),f_{2})$. So to prove the equivalence
stated in the lemma, we have to observe that, in this situation,
$S(f_{1},f_{2})$ depends only on $f_{1}$, _i.e._ (by linearity) that
$S(f_{1},f_{2})=0$ if $f_{1}=0$. For $(0,f_{2})\in\operatorname{dom}(T)$ we
have $\|T(0,f_{2})\|_{p}^{p}=\|S(0,f_{2})\|_{p}^{p}+\|f_{2}\|^{p}$, which (by
the assumption that $\|T\|\leq 1$) is less than $\|f_{2}\|^{p}$. This proves
that $\|S(0,f_{2})\|_{p}^{p}=0$, as requested.
The last assertion is a tautology. ∎
Finally, we provide an example that illustrates the main result.
###### Example 3.6.
The inequality $\|(T\circ S)_{X}\|\leq\|T_{X}\|\|S_{X}\|$ is clear for every
Banach space $X$ and every operators $T,S$ such that $T\circ S$ makes sense.
So it follows from Corollary 1.7 that, with the notation therein, if $S,T\in
B$ then $T\circ S$ belongs to $B^{\prime}$. We prove this directly, because it
illustrates the subtle property (iv).
So let $S,T\in B$ such that
$\operatorname{ran}(S)\subset\operatorname{dom}(T)$. By (ii) the operator
$S\oplus
T\colon\operatorname{dom}(S)\oplus\operatorname{dom}(T)\to\operatorname{ran}(S)\oplus\operatorname{ran}(T)$
belongs to $B^{\prime}$. By composing by the spatial isometry
$(f,g)\in\operatorname{ran}(S)\oplus\operatorname{ran}(T)\mapsto(g,f)\in\operatorname{ran}(T)\oplus\operatorname{ran}(S)$
(which is allowed by (iii)) and restricting to the subspace
$D=\\{(f,Sf)|f\in\operatorname{dom}(S)\\}\subset\operatorname{dom}(S)\oplus\operatorname{dom}(T)$
(which is allowed by (v)), we obtain that the map $(f,Sf)\in D\mapsto(T\circ
Sf,Sf)$ belongs to $B^{\prime}$. By (iv), we conclude that $T\circ S$ belongs
to $B^{\prime}$ as required.
## 4\. The space of degree $p$ homogeneous functions on $\mathbf{K}^{n}$
Let $n$ be a positive integer. Denote by $|z|$ the $\ell_{p}$-norm on
$\mathbf{K}^{n}$
$|z|=\left(|z_{1}|^{p}+\dots+|z_{n}|^{p}\right)^{\frac{1}{p}}.$
In the rare occasions when we want to insist on $p$, we write $|z|_{p}$ for
this quantity.
A function $\varphi\colon\mathbf{K}^{n}\to\mathbf{R}$ is called homogeneous of
degree $p$ if $\varphi(\lambda z)=|\lambda|^{p}\varphi(z)$ for all
$z\in\mathbf{K}^{n}$ and $\lambda\in\mathbf{K}$. The space $H_{n}$ of
continuous homogeneous of degree $p$ functions on $\mathbf{K}^{n}$ is a Banach
space over the field of real numbers for the topology of uniform convergence
on compact subsets on $\mathbf{K}^{n}$. A particular choice of norm is
$\|\varphi\|=\sup_{|z|\leq 1}|\varphi(z)|$, so that for this norm $H_{n}$ is
isometrically isomorphic to the space of real-valued continuous functions on
$\mathbf{KP}^{n-1}$ through the identification of $\varphi\in H_{n}$ with the
function $\mathbf{K}z\in\mathbf{KP}^{n-1}\mapsto\varphi(\frac{z}{|z|})$. An
equivalent definition of the norm of $\varphi\in H_{n}$ is the smallest number
such that for every $z\in\mathbf{K}^{n}$
(4.1) $|\varphi(z)|\leq(|z_{1}|^{p}+\dots+|z_{n}|^{p})\|\varphi\|.$
We encode a class $A\subset\mathcal{X}$ of Banach spaces by the cone
$N(A,n)\subset H_{n}$ (N for norms) of functions of the form
$z\mapsto\|\sum_{i=1}^{n}z_{i}x_{i}\|^{p}$ for $X\in A$ and
$x_{1},\dots,x_{n}\in X$.
When $(\Omega,m)$ is a measure space and $f=(f_{1},\dots,f_{n})$ is an
$n$-uple of elements of $L_{p}(\Omega,m)$, we can define a continuous linear
form $\mu_{f}$ on $H_{n}$ by
(4.2)
$\langle\mu_{f},\varphi\rangle=\int\varphi(f_{1}(\omega),\dots,f_{n}(\omega))dm(\omega).$
Indeed, it follows from (4.1) that the integral is well-defined and that
$\mu_{f}\in H_{n}^{*}$ with norm equal to
$\|f_{1}\|_{p}^{p}+\dots+\|f_{n}\|_{p}^{p}$ (the inequality $\leq$ is
immediate from (4.1), and the equality follows by evaluating $\mu_{f}$ at the
norm $1$ element $z\mapsto|z|^{p}$ in $H_{n}$).
We encode a class $B\subset\mathcal{T}$ of operators by the cone
$P(B,n)\subset H_{n}^{*}$
$P(B,n)=\\{\mu_{f}-\mu_{Tf},T\in B\textrm{ and
}f\in\operatorname{dom}(T)^{n}\\}$
where for $f=(f_{1},\dots,f_{n})\in\operatorname{dom}(T)^{n}$, we denote
$Tf=(Tf_{1},\dots,Tf_{n})$. It is a cone because for every $t\geq 0$,
$t(\mu_{f}-\mu_{Tf})=\mu_{t^{\frac{1}{p}}f}-\mu_{Tt^{\frac{1}{p}}f}$.
The crucial but obvious property motivating these definitions is that, if
$\varphi(z)=\|\sum_{i=1}^{n}z_{i}x_{i}\|_{X}^{p}$ for elements
$x_{1},\dots,x_{n}$ in a Banach space $X$, then
$\langle\mu_{f},\varphi\rangle=\|\sum_{i}f_{i}x_{i}\|_{L_{p}(\Omega,m;X)}^{p}$.
As a consequence,
$\langle\mu_{f}-\mu_{Tf},\varphi\rangle=\|\sum_{i}f_{i}x_{i}\|_{L_{p}(\Omega,m;X)}^{p}-\|\sum_{i}(Tf_{i})x_{i}\|_{L_{p}(\Omega,m;X)}^{p}.$
In particular, we have
###### Lemma 4.1.
Let $A\subset\mathcal{X}$ be a class of Banach spaces and
$B\subset\mathcal{T}$ a class of operators.
1. (1)
$B\subset A^{\circ}$ if and only if for every $n$, $P(B,n)\subset
N(A,n)^{\circ}$.
2. (2)
$A\subset{}^{\circ}B$ if and only if for every $n$,
$N(A,n)\subset{}^{\circ}P(B,n)$.
### 4.1. Polarity in $H_{n}$
We start by improving Lemma 4.1. The next result expresses that the polarity
in $\langle\mathcal{X},\mathcal{T}\rangle$ (see Definition 1.1 and 1.2) is
well encoded by the polarity $\langle H_{n},H_{n}^{*}\rangle$ (see Subsection
3.1). Recall that $REG$ the class of all operators $T\in\mathcal{T}$ with
regular norm $\|T\|_{r}:=\sup_{X\in\mathcal{X}}\|T_{X}\|\leq 1$.
###### Proposition 4.2.
Let $A\subset\mathcal{X}$ be a class of Banach spaces and
$B\subset\mathcal{T}$ a class of operators. Then
1. (1)
$P(A^{\circ},n)=N(A,n)^{\circ}$.
2. (2)
$N({}^{\circ}B,n)\subset{}^{\circ}P(B\cup REG,n)$.
In the proof, we need a description of the dual of $H_{n}$ :
###### Lemma 4.3.
Every continuous linear form $l$ on $H_{n}$ is of the form $\mu_{f}-\mu_{g}$
for some measure spaces $(\Omega,m)$ and $(\Omega^{\prime},m^{\prime})$ and
$n$-uples $f\in L_{p}(\Omega,m)^{n}$ and $g\in
L_{p}(\Omega^{\prime},m^{\prime})^{n}$. Moreover $\Omega,m,f$ and
$\Omega^{\prime},m^{\prime},g$ can be chosen so that $f$ and $g$ take almost
surely their values in $\\{z\in\mathbf{K}^{n},|z|=1\\}$ and so that
$m(\Omega)+m^{\prime}(\Omega^{\prime})$ is equal to the norm of $l$.
###### Proof.
By the identification of $H_{n}$ with $C(\mathbf{KP}^{n-1})$ and by the Riesz
representation theorem, every continuous linear form $l$ on $H_{n}$ is of the
form
$\varphi\mapsto\int_{\mathbf{KP}^{n-1}}\varphi\left(\frac{z}{|z|}\right)d\nu(\mathbf{K}z)$
for a unique signed measure $\nu$ on $\mathbf{KP}^{n-1}$, and the norm of $l$
is the total variation of $\nu$. Let $\nu=\nu_{+}-\nu_{-}$ be the Jordan
decomposition of $\nu$ and
$s\colon\mathbf{KP}^{n-1}\to\\{z\in\mathbf{K}^{n},|z|=1\\}$ a measurable
section. Define $(\Omega,m)=(\mathbf{KP}^{n-1},\nu_{+})$ and $f\in
L_{p}(\Omega,m)^{n}$ by $s(\omega)=(f_{1}(\omega),\dots,f_{n}(\omega))$.
Similarly define $(\Omega^{\prime},m^{\prime})=(\mathbf{KP}^{n-1},\nu_{-})$
and $g\in L_{p}(\Omega^{\prime},m^{\prime})^{n}$ by
$s(\omega)=(g_{1}(\omega),\dots,g_{n}(\omega))$. Then we have
$\int_{\mathbf{KP}^{n-1}}\varphi\left(\frac{z}{|z|}\right)d\nu(\mathbf{K}z)=\langle\mu_{f}-\mu_{g},\varphi\rangle.$
This proves the lemma, because by construction $f,g$ both take values in
$\\{z\in\mathbf{K}^{n},|z|=1\\}$ and
$m(\Omega)+m^{\prime}(\Omega)=(\nu_{+}+\nu_{-})(\mathbf{KP}^{n-1})$ is the
norm of $l$. ∎
###### Proof of Proposition 4.2.
We start by (1). If every space in $A$ is trivial (of dimension $0$), we have
$N(A,n)^{\circ}=H_{n}^{*}$, $A^{\circ}=\mathcal{T}$, and the result is easy.
We can therefore assume that $A$ contains a space of dimension $\geq 1$. Let
$f,g$ be $n$-uples in $L_{p}$ spaces. Note that if
$\varphi(z)=\|\sum_{i=1}^{n}z_{i}x_{i}\|^{p}$ then
$\langle\mu_{f}-\mu_{g},\varphi\rangle=\|\sum_{i}f_{i}x_{i}\|_{L_{p}(X)}^{p}-\|\sum_{i}g_{i}x_{i}\|_{L_{p}(X)}^{p}.$
So the linear form $\mu_{f}-\mu_{g}\in H_{n}^{*}$ belongs to $N(A,n)^{\circ}$
if and only if for every $X\in A$ and $x_{1},\dots,x_{n}\in X$, $\|\sum
f_{i}x_{i}\|_{L_{p}(X)}^{p}\geq\|\sum g_{i}x_{i}\|_{L_{p}(X)}^{p}$. Using that
there is a nonzero $X\in A$, this holds if and only if there is a linear map
$T$ sending $f_{i}$ to $g_{i}$ such that $T\in A^{\circ}$. This shows that
$\mu_{f}-\mu_{g}$ belongs to $N(A,n)^{\circ}$ if and only if it belongs to
$P(A^{\circ},n)$. By Lemma 4.3 every element of $H_{n}^{*}$ is of this form,
which proves (1).
We move to (2). Denote by $C_{n}$ the closed convex cone
$C_{n}=N(\mathcal{X},n)$. We first prove that
$N({}^{\circ}B,n)={}^{\circ}P(B,n)\cap C_{n}$. By definition
$N({}^{\circ}B,n)\subset C_{n}$. So we have to prove that for $\varphi\in
C_{n}$, $\varphi\in N({}^{\circ}B,n)$ if and only if
$\varphi\in{}^{\circ}P(B,n)$. But if
$\varphi(z)=\|\sum_{i=1}^{n}z_{i}x_{i}\|^{p}$ and
$X=\operatorname{span}(x_{1},\dots,x_{n})$, then we have that $\varphi\in
N(B^{\circ},n)$ if and only if $\|T\otimes id_{X}\|\leq 1$ for all $T\in B$,
if and only if for all $T\in B$ and
$f_{1},\dots,f_{n}\in\operatorname{dom}(T)$,
$\|\sum_{i}Tf_{i}x_{i}\|^{p}\leq\|\sum_{i}f_{i}x_{i}\|^{p}$, if and only if
$\varphi\in P(B,n)^{\circ}$.
We can now conclude with (2). By (1) for $A=\mathcal{X}$, we have
$C_{n}^{\circ}=P(REG,n)$. On the other hand, since $C_{n}$ is a closed convex
cone, the bipolar theorem implies that $C_{n}={}^{\circ}C_{n}^{\circ}$, and
hence $C_{n}={}^{\circ}P(REG,n)$. We therefore get
$\displaystyle N({}^{\circ}B,n)$
$\displaystyle={}^{\circ}P(B,n)\cap{}^{\circ}P(REG,n)$
$\displaystyle={}^{\circ}(P(B,n)\cup P(REG,n))$
$\displaystyle={}^{\circ}P(B\cup REG,n).$
This proves (2). ∎
By the bipolar theorem in $H_{n}$ and $H_{n}^{*}$, we obtain
###### Corollary 4.4.
Let $A\subset\mathcal{X}$ be a class of Banach spaces and
$B\subset\mathcal{T}$ a class of operators. Then
1. (1)
$N({}^{\circ}A^{\circ},n)=\overline{\mathrm{conv}}N(A,n)$.
2. (2)
$P({}^{\circ}B^{\circ},n)=\overline{\mathrm{conv}}^{w*}P(B\cup REG,n)$.
The rest of this section consists in understanding the closed convex hulls of
$N(A,n)$ and $P(B,n)$.
### 4.2. Understanding the encoding of Banach spaces in $H_{n}$
The following easy fact will be important later.
###### Lemma 4.5.
For every integer $n$, bounded subsets of $N(\mathcal{X},n)$ are relatively
norm-compact.
###### Proof.
By the Arzelà-Ascoli theorem, we have to prove that bounded subsets of
$N(\mathcal{X},n)$ are equicontinuous, seen in $C(\mathbf{KP}^{n-1})$. This
follows from the triangle inequality. For example for $p=1$ and
$\varphi(z)=\|\sum_{i}z_{i}x_{i}\|$, then we have
$|\varphi(z)-\varphi(z^{\prime})|\leq\|\sum_{i}(z_{i}-z^{\prime}_{i})x_{i}\|\leq\sum_{i}|z_{i}-z^{\prime}_{i}|\varphi(e_{i}).$
The case of arbitrary $p$ is similar. Alternatively, it follows from the case
$p=1$ by continuity of the map $t\mapsto t^{p}$. ∎
Lemma 4.5 allows to considerably strengthen the second statement in Corollary
4.4, replacing weak-* closure by norm closure.
###### Corollary 4.6.
Let $B\subset\mathcal{T}$ a class of operators. Then
$P({}^{\circ}B^{\circ},n)=\overline{\mathrm{conv}}^{\|\cdot\|}P(B\cup REG,n).$
###### Proof.
The set $N(\mathcal{X},n)$ is a closed convex cone in $H_{n}$, so by Lemma 4.5
$N(\mathcal{X},n)\cap\\{\varphi\in H_{n}\mid\|\varphi\|\leq r\\}$ is norm-
compact for every $r$. Moreover, we have that
$N(\mathcal{X},n)^{\circ}=P(REG,n)$ by Proposition 4.2. So, since $P(B\cup
REG,n)$ contains $N(\mathcal{X},n)^{\circ}$, Proposition 3.3 implies that its
bipolar is equal to the norm closure of its convex hull. ∎
Let us list elementary properties of $N$.
###### Lemma 4.7.
Let $A,A_{1},A_{2}\subset\mathcal{X}$ be classes of Banach spaces.
1. (1)
$N(A_{1},n)\subset N(A_{2},n)$ if and only if, for every $X\in A_{1}$, every
subspace of dimension $\leq n$ of $X$ is isometric to a subspace of a space in
$A_{2}$.
2. (2)
The convex hull of $N(A,n)$ is equal to $N(\oplus_{\ell_{p}}A,n)$, where
$\oplus_{\ell_{p}}A$ denotes the set of all finite $\ell_{p}$-direct sums of
Banach spaces in $A$.
3. (3)
The norm closure of $N(A,n)$ in $H_{n}$ coincides with $N(\overline{A},n)$
where $\overline{A}$ denotes the set of Banach spaces finitely represented in
$A$.
As a consequence of (1) and (3), if two classes of Banach spaces $A_{1},A_{2}$
are closed under finite representability, then $A_{1}=A_{2}$ if and only if
$N(A_{1},n)=N(A_{2},n)$ for all $n$.
###### Proof.
The first point is obvious from the following observation : if
$x_{1},\dots,x_{n}$ (respectively $y_{1},\dots,y_{n}$) are elements in a
Banach space $X$ (respectively in a Banach space $Y$), then the functions
$z\mapsto\|\sum_{i=1}^{n}z_{i}x_{i}\|^{p}$ and
$z\mapsto\|\sum_{i=1}^{n}z_{i}y_{i}\|^{p}$ coincide if and only if there is an
isometry from the linear span of $\\{x_{1},\dots,x_{n}\\}$ to the linear span
of $\\{y_{1},\dots,y_{n}\\}$ sending $x_{i}$ to $y_{i}$.
If $\varphi_{1},\dots,\varphi_{k}\in N(A,n)$ are given by
$\varphi_{j}(z)=\|\sum_{i=1}^{n}z_{i}x_{i}^{(j)}\|_{X_{j}}^{p}$ then by the
definition of the $\ell_{p}$-direct sum $X_{1}\oplus_{p}\dots\oplus_{p}X_{k}$
we can write
$\sum_{j=1}^{k}\varphi_{j}(z)=\|\sum_{i=1}^{n}z_{i}(x_{i}^{(j)})_{1\leq j\leq
k}\|_{X_{1}\oplus_{p}\dots\oplus_{p}X_{k}}^{p}.$
This shows that $N(\oplus_{\ell_{p}}A,n)$ coincides with
$\\{\varphi_{1}+\dots+\varphi_{k},k\in\mathbf{N},\varphi_{j}\in N(A,n)\\}.$
This is the convex hull of $N(A,n)$ because $N(A,n)$ is a cone.
We move to (3). If a sequence $\varphi_{k}\in N(A,n)$ converges uniformly on
compact subsets to $\varphi\in H_{n}$, then $\varphi^{\frac{1}{p}}$ is the
uniform limit on compact sets of the seminorms $\varphi_{k}^{\frac{1}{p}}$, so
it is a seminorm on $\mathbf{K}^{n}$. This means that there is a Banach space
$X\in\mathcal{X}$ and $x_{1},\dots,x_{n}$ spanning $X$ such that
$\varphi(z)=\|\sum_{i=1}^{n}z_{i}x_{i}\|^{p}$. The family $x_{1},\dots,x_{n}$
might not be linearly independant, so we extract from it a basis of $X$.
Without loss of generality we can assume that this basis is
$x_{1},\dots,x_{m}$ for some $m\leq n$. Write
$\varphi_{k}(z)=\|\sum_{i=1}^{n}z_{i}x_{i}^{(k)}\|_{X_{k}}^{p}$
for some $X_{k}\in A$ and $x_{1}^{(k)},\dots,x_{n}^{(k)}\in X_{k}$. From the
assumption that $\varphi_{k}$ converges uniformly on compacta to $\varphi$ and
the assumption that $x_{1},\dots,x_{m}$ is linearly independant, we get that
for every $\varepsilon>0$ there is $k$ such that
$(1-\varepsilon)\varphi(z,0)\leq\varphi_{k}(z,0)\leq(1+\varepsilon)\varphi(z,0)$
for all $z\in\mathbf{K}^{m}$. This means that the linear map $u\colon X\to
X_{k}$ sending $x_{i}$ to $(1-\varepsilon)^{-\frac{1}{p}}x_{i}^{(k)}$ for
$i\leq m$ satisfies
$\|x\|\leq\|u(x)\|\leq(\frac{1+\varepsilon}{1-\varepsilon})^{\frac{1}{p}}\|x\|\textrm{
for all }x\in X.$
Since $\varepsilon>0$ was arbitrary we have proved that $X$ is finitely
representable in $A$, _i.e._ that $\varphi\in N(\overline{A},n)$. This proves
that $\overline{N(A,n)}\subset N(\overline{A},n)$. The converse inclusion is
proved by reading the preceding argument backwards. ∎
We can conclude our proof of Hernandez’ theorem.
###### Proof of Theorem 1.3.
Let $A^{\prime}$ be the class of Banach spaces which are finitely
representable in the class of $\ell_{p}$-direct sums of spaces in $A$. It
follows from Corollary 4.4 and Lemma 4.7 that for every integer $n$,
$N({}^{\circ}A^{\circ},n)=N(A^{\prime},n).$
Since both ${}^{\circ}A^{\circ}$ and $A^{\prime}$ are closed under finite
representability, we get the equality ${}^{\circ}A^{\circ}=A^{\prime}$ by the
remark following Lemma 4.7. ∎
### 4.3. Proof of Theorem 1.4
If $X$ is at Banach-Mazur $\leq C$ from ${}^{\circ}A^{\circ}$; then the
inequality $\|T_{X}\|\leq C$ for every $T\in A^{\circ}$ is clear.
For the converse, we will need the following consequence of the Hahn-Banach
theorem.
###### Lemma 4.8.
Let $K$ be a compact Hausdorff topological space, and $C(K)$ the space of
real-valued continuous functions on $K$. Let $A$ be a closed convex cone in
the positive cone of $C(K)$ such that $A\cap B(0,1)$ is compact. Let $s\geq
1$. Then for every $\psi\in C(K)$, the following are equivalent
* •
$\exists\varphi\in A,\psi\leq\varphi\leq s\psi$.
* •
$\langle s\mu-\nu,\psi\rangle\geq 0$ for every positive measures $\mu,\nu$ on
$K$ such that $\langle\mu-\nu,\varphi\rangle\geq 0$ for all $\varphi\in A$.
###### Proof.
$\implies$ is easy because the inequality $\psi\leq\varphi\leq s\psi$ implies
$\langle s\mu-\nu,\psi\rangle\geq\langle\mu-\nu,\varphi\rangle$.
For the converse, since $A$ is a convex cone, the set $B$ of $\psi$ satisfying
$\exists\varphi\in A,\psi\leq\varphi\leq s\psi$ is a convex cone. Moreover,
the compactness assumption on $A$ implies that $B$ is also closed. Assume that
$\psi\notin B$. By Hahn-Banach there is a linear form on $C(K)$ which is
nonnegative on $B$ and negative at $\psi$. By the Riesz representation theorem
and the Hahn decomposition, this linear form can be written as $f\mapsto\int
fd(\mu-\nu)$ for positive measures $\mu,\nu$ such that there is a Baire
measurable subset $E\subset K$ satisfying $\nu(E)=0$ and $\mu(K\setminus
E)=0$.
Let $\varphi\in A$. Let $f_{n}\colon K\to[0,1]$ be a sequence of continuous
functions converging in $L^{1}(K,\mu+\nu)$ to the indicator function of $E$.
Then for every $n$, the function $(\frac{1}{s}+(1-\frac{1}{s})f_{n})\varphi$
belongs to $B$ so
$\langle\mu-\nu,(\frac{1}{s}+(1-\frac{1}{s})f_{n})\varphi\rangle>0$. By making
$n\to\infty$ we get $\langle\mu-\nu,\frac{1}{s}\varphi 1_{K\setminus
E}+\varphi 1_{E}\rangle\geq 0$, which can be written as
$\langle\frac{1}{s}\mu-\nu,\varphi\rangle\geq 0$.
So we have $\langle\frac{1}{s}\mu-\nu,\varphi\rangle\geq 0$ for every
$\varphi\in A$, whereas $\langle\mu-\nu,\psi\rangle<0$. This proves the lemma.
∎
We can now prove the converse implication in Theorem 1.4. Assume that
$\|T_{X}\|\leq C$ for every $T\in A^{\circ}$. Let $x_{1},\dots,x_{n}\in X$.
Define $\psi\in H_{n}$ by $\psi(z)=\|\sum_{i=1}^{n}z_{i}x_{i}\|^{p}$, and view
$\psi$ in $C(\mathbf{KP}^{n-1})$. The assumption that $\|T_{X}\|\leq C$ for
every $T\in A^{\circ}$ implies that $\langle C^{p}\mu-\nu,\psi\rangle\geq 0$
for every positive measures $\mu,\nu$ on $\mathbf{KP}^{n-1}$ such that
$\langle\mu-\nu,\varphi\rangle\geq 0$ for all $\varphi\in A$. By Lemma 4.8
(remember Lemma 4.5) this implies that there is $\varphi$ in the closed convex
hull of $N(A,n)$ such that $\psi\leq\varphi\leq C^{p}\psi$. By the proof of
Theorem 1.3, there is a space $Y\in{}^{\circ}A^{\circ}$ and
$y_{1},\dots,y_{n}\in Y$ such that $\varphi(z)=\|\sum_{i}z_{i}y_{i}\|^{p}$. By
taking the $1/p$-th power in the inequality $\psi\leq\varphi\leq C^{p}\psi$ we
get that $\|\sum z_{i}x_{i}\|\leq\|\sum z_{i}y_{i}\|\leq C\|\sum z_{i}x_{i}\|$
for every $y\in\mathbf{K}^{n}$. This means that the linear span of
$x_{1},\dots,x_{n}$ is at Banach-Mazur distance $\leq C$ from the linear span
on $\\{y_{1},\dots,y_{n}\\}$ and concludes the proof.
### 4.4. Understanding the encoding of operators in $H_{n}^{*}$
###### Lemma 4.9.
Let $f,g,\tilde{f},\tilde{g}$ be $n$-uples of elements of $L_{p}$ spaces. Then
$\mu_{f}-\mu_{g}=\mu_{\tilde{f}}-\mu_{\tilde{g}}$ if and only if there is
$h\in L_{p}(\Omega,m)^{n},\tilde{h}\in L_{p}(\tilde{\Omega},\tilde{m})^{n}$
such that $\mu_{(f_{i}\oplus
h_{i})_{i=1}^{n}}=\mu_{(\tilde{f}_{i}\oplus\tilde{h}_{i})_{i=1}^{n}}$ and
$\mu_{(g_{i}\oplus
h_{i})_{i=1}^{n}}=\mu_{(\tilde{g}_{i}\oplus\tilde{h}_{i})_{i=1}^{n}}$.
###### Proof.
The if direction is easy, because $\mu_{(f_{i}\oplus
h_{i})_{i=1}^{n}}=\mu_{f}+\mu_{h}$.
For the converse, assume that
$\mu_{f}-\mu_{g}=\mu_{\tilde{f}}-\mu_{\tilde{g}}$. Let $\nu_{f}$ be the
positive measure on $\mathbf{KP}^{n-1}$ such that, for every $\varphi\in
H_{n}$
(4.3)
$\langle\mu_{f},\varphi\rangle=\int_{\mathbf{KP}^{n-1}}\varphi\left(\frac{z}{|z|}\right)d\nu_{f}(\mathbf{K}z).$
Define similarly $\nu_{g},\nu_{\tilde{f}},\nu_{\tilde{g}}$. Then
$\nu_{f}-\nu_{g}=\nu_{\tilde{f}}-\nu_{\tilde{g}}$ is a signed measure on
$\mathbf{KP}^{n-1}$. Let $\nu_{+}-\nu_{-}$ be its Jordan decomposition. By the
properties of the Jordan decomposition, $\nu_{f}-\nu_{+}=\nu_{g}-\nu_{-}$ is a
positive measure on $\mathbf{KP}^{n-1}$, and therefore by the proof of Lemma
4.3 it is of the form to $\nu_{\tilde{h}}$ for some $n$-uple $\tilde{h}\in
L_{p}(\tilde{\Omega},\tilde{m})$. Similarly, there is a $h\in
L_{p}(\Omega,m)^{n}$ such that
$\nu_{\tilde{f}}-\nu_{+}=\nu_{\tilde{g}}-\nu_{-}=\nu_{h}$.
We can rewrite these equalities as
$\nu_{+}=\nu_{f}-\nu_{\tilde{h}}=\nu_{\tilde{f}}-\nu_{h}$
and
$\nu_{-}=\nu_{f}-\nu_{\tilde{h}}=\nu_{\tilde{f}}-\nu_{h}.$
This implies that $\mu_{f}+\mu_{h}=\mu_{\tilde{f}}+\mu_{\tilde{h}}$ and
$\mu_{g}+\mu_{h}=\mu_{\tilde{g}}+\mu_{\tilde{h}}$ and proves the lemma. ∎
###### Lemma 4.10.
For two families $f\in L_{p}(\Omega,m)^{n}$ and $g\in
L_{p}(\Omega^{\prime},m^{\prime})^{n}$, $\mu_{f}=\mu_{g}$ if and only if there
is a spatial isometry
$\operatorname{span}\\{f_{1},\dots,f_{n}\\}\to\operatorname{span}\\{g_{1},\dots,g_{n}\\}$
sending $f_{i}$ to $g_{i}$.
###### Proof.
The if direction is easy : firstly if there is a measurable function
$h\colon\Omega\to\mathbf{K}\setminus\\{0\\}$, if
$(\Omega^{\prime},m^{\prime})=(\Omega,|h|^{-p}m)$ and $g_{i}=hf_{i}$ for all
$i$, then for every $\varphi\in H_{n}$,
$\varphi(g_{1},\dots,g_{n})=|h|^{p}\varphi(f_{1},\dots,f_{n})$ and therefore
$\langle\mu_{g},\varphi\rangle=\langle\mu_{f},\varphi\rangle$. Secondly if
$f_{1},\dots,f_{n}$ and $g_{1},\dots,g_{n}$ are equimeasurable outside of $0$
in the sense of Definition 1.5, then
$\int\varphi(f_{1},\dots,f_{n})dm=\int\varphi(g_{1},\dots,g_{n})dm^{\prime}$
for every Borel function $\varphi$ vanishing at $0$ and such that the
integrals are defined. In particular $\mu_{f}=\mu_{g}$.
For the converse, assume that $\mu_{f}=\mu_{g}$. Take a measurable section
$s\colon\mathbf{KP}^{n-1}\to\mathbf{K}^{n}$ with values in
$\\{z\in\mathbf{K}^{n},|z|=1\\}$. Then there are measurable nonvanishing
functions $h\colon\Omega\to\mathbf{K}^{*}$ and
$h^{\prime}\colon\Omega^{\prime}\to\mathbf{K}^{*}$ such that
$f(\omega)=h(\omega)s(\mathbf{K}f(\omega))$ for every $\omega\in\Omega$ such
that $f(\omega)\neq 0$, and similarly
$g(\omega^{\prime})=h^{\prime}(\omega^{\prime})s(\mathbf{K}g(\omega^{\prime}))$
if $g(\omega^{\prime})\neq 0$. By replacing $m$ by $|h|^{-1/p}m$ and $f_{i}$
by $h_{i}f_{i}$ and similarly for $g$ we can assume that $h=1$ and
$h^{\prime}=1$, and we shall prove that $f$ and $g$ are equimeasurable outside
of $0$. By this we mean that for every Borel
$E\subset\mathbf{K}^{n}\setminus\\{0\\}$,
$m(\\{\omega,(f_{1}(\omega),\dots,f_{n}(\omega))\in
E\\})=m^{\prime}(\\{\omega^{\prime},(g_{1}(\omega^{\prime}),\dots,Tg_{n}(\omega^{\prime}))\in
E\\})$. It is clear that this implies that, for every matrix $A\in
M_{m,n}(\mathbf{K})$, $Af$ and $Ag$ are equimeasurable outside of $0$, and
therefore that the linear map sending $f_{i}$ to $g_{i}$ is well-defined and
is as in the definition of equimeasurability outside of $0$.
By the identification of $H_{n}$ with $C(\mathbf{KP}^{n-1})$, using that
$|f|\in\\{0,1\\}$ we have
$\int_{\Omega\setminus
f^{-1}(0)}\psi(\mathbf{K}f)dm=\int_{\Omega^{\prime}\setminus
g^{-1}(0)}\psi(\mathbf{K}g)dm^{\prime}$
for every continuous function $\psi\colon\mathbf{KP}^{n-1}\to\mathbf{K}$, and
therefore also for every bounded Borel function
$\varphi\colon\mathbf{KP}^{n-1}\to\mathbf{K}$. This implies, since $f$ and $g$
take values in $\\{0\\}\cup s(\mathbf{KP}^{n-1})$, that
$\int\varphi(f)dm=\int\varphi(g)dm^{\prime}$ for every Borel function
$\mathbf{K}^{n}\to\mathbf{K}$ vanishing at $0$. Equivalently, $f$ and $g$ are
equimeasurable outside of $0$. ∎
###### Remark 4.11.
The same proof shows actually a bit more : if a linear map $T\colon E\subset
L_{p}(\Omega,m)\to L_{p}(\Omega^{\prime},m^{\prime})$ satisfies
$\mu_{f}=\mu_{Tf}$ for every $n$ and every $f\in E^{n}$, then $T$ is a spatial
isometry. Indeed, since by our standing assumption $E$ (as every other space
considered in this paper) is separable, we can find a sequence $(f_{i})_{i\geq
0}$ generating a dense subspace of $E$ and satisfying
$\sum_{i}\|f_{i}\|^{p}<\infty$, and in particular $(f_{i}(\omega))_{i\geq 0}$
belongs to $\ell_{p}$ for almost every $\omega$. Then the same proof applies,
except that we replace $\mathbf{K}^{n}$ by $\ell_{p}$ and $\mathbf{KP}^{n-1}$
by its projectivization $\ell_{p}/\mathbf{K}^{*}$.
We shall also need the following variant :
###### Lemma 4.12.
For two families $f\in L_{p}(\Omega,m)^{n}$ and $g\in
L_{p}(\Omega^{\prime},m^{\prime})^{n}$ and $\varepsilon>0$,
$\|\mu_{f}-\mu_{g}\|<\varepsilon$ if and only if there are spatial isometries
$U\colon\operatorname{span}\\{f_{1},\dots,f_{n}\\}\to
L_{p}(\Omega^{\prime\prime},m^{\prime\prime})$ and
$V\colon\operatorname{span}\\{g_{1},\dots,g_{n}\\}\to
L_{p}(\Omega^{\prime\prime},m^{\prime\prime})$ such that
$\int_{\Omega^{\prime\prime}}(|Uf|^{p}+|Vg|^{p})\chi_{Uf\neq Vg}<\varepsilon.$
###### Proof.
We prove the slightly stronger statement with $<\varepsilon$ replaced by
$\leq\varepsilon$.
The if direction is easy : by Lemma 4.10 we have $\mu_{f}=\mu_{Uf}$ and
$\mu_{g}=\mu_{Vg}$, and therefore for every $\varphi\in H_{n}$,
$\displaystyle\langle\mu_{f}-\mu_{g},\varphi\rangle$
$\displaystyle=\int_{\Omega^{\prime\prime}}\varphi(Uf)-\varphi(g)$
$\displaystyle\leq\int_{\Omega^{\prime\prime}}(|\varphi(Uf)|+|\varphi(Vg)|)\chi_{f\neq
Ug}$
$\displaystyle\leq\int_{\Omega^{\prime\prime}}(|Uf|^{p}+|Vg|^{p})\|\varphi\|\leq\varepsilon\|\varphi\|.$
Taking the supremum over $\varphi$ we get
$\|\mu_{f}-\mu_{g}\|\leq\varepsilon$.
The converse follows from a coupling argument. Assume that
$\|\mu_{f}-\mu_{g}\|\leq\varepsilon$. Let $\nu_{f}$ and $\nu_{g}$ be the
measures on $\mathbf{KP}^{n-1}$ given by (4.3), so that the total variation
norm of $\nu_{f}-\nu_{g}$ is at most $\varepsilon$. This means that we can
decompose $\nu_{f}=\nu_{0}+\nu_{1}$ and $\nu_{g}=\nu_{0}+\nu_{2}$ for positive
measures with $(\nu_{1}+\nu_{2})(\mathbf{KP}^{n-1})\leq\varepsilon$. As in the
proof of Lemma 4.3, each $\nu_{k}$ corresponds by (4.3) to $\mu_{h^{k}}$ for
an $n$-uple $h^{k}\in L_{p}(\Omega_{k},m_{k})^{n}$ with
$\sum_{i}\|h^{k}_{i}\|_{p}^{p}=\nu_{k}(\mathbf{KP}^{n-1})$. In particular, we
have $\mu_{f}=\mu_{h^{0}}+\mu_{h^{1}}$ and $\mu_{g}=\mu_{h^{0}}+\mu_{h^{2}}$.
Let us define $\Omega^{\prime\prime}$ as the disjoint union
$\Omega_{0}\cup\Omega_{1}\cup\Omega_{2}$, $m^{\prime\prime}$ as
$m_{0}+m_{1}+m_{2}$, and $f^{\prime}=h^{0}\oplus h^{1}\oplus 0$ and
$g^{\prime}=h^{0}\oplus 0\oplus h^{1}$, so that
$\mu_{f^{\prime}}=\mu_{h^{0}}+\mu_{h^{1}}=\mu_{f}$ and
$\mu_{g^{\prime}}=\mu_{g}$. By Lemma 4.10, there are spatial isometries $U$
and $V$ sending $f$ to $f^{\prime}$ and $g$ to $g^{\prime}$ respectively, and
we have
$\int_{\Omega^{\prime\prime}}(|f^{\prime}|^{p}+|g^{\prime}|^{p})\chi_{f^{\prime}\neq
g^{\prime}}=\int_{\Omega_{1}}|h^{1}|^{p}+\int_{\Omega_{2}}|h^{2}|^{p}\leq\varepsilon.$
This proves the lemma. ∎
There is also an asymetric variant of the preceding lemma, that can be useful.
###### Remark 4.13.
In Lemma 4.12, we can moreover assume that
$(\Omega^{\prime\prime},m^{\prime\prime})=(\Omega\times[0,1],m\otimes
d\lambda)$ (for $\lambda$ the Lebesgue measure), and that the spatial isometry
$U$ is simply $U\xi(\omega,s)=\xi(\omega)$.
###### Proof.
Let $\mu_{f},\mu_{g},\nu_{f}=\nu_{0}+\nu_{1},\nu_{g}=\nu_{0}+\nu_{2}$ be as in
the proof of Lemma 4.12, where $\|\nu_{1}+\nu_{2}\|<\varepsilon$. We can even
assume that $\nu_{1}\neq 0$ (this is where the strict inequality
$<\varepsilon$ is used). Denote by
$\frac{d\nu_{0}}{d\nu_{f}}\colon\mathbf{KP}^{n-1}\to[0,1]$ the Radon-Nikodym
derivative. Define $A\subset\Omega\times[0,1]=\Omega^{\prime\prime}$ by
$A=\\{(\omega,s)\mid s\leq 0\leq
s\leq\frac{d\nu_{0}}{d\nu_{f}}(\mathbf{K}^{*}f(x))\\}$, so that
$\mu_{f\chi_{A}}=\nu_{0}$ and $\mu_{f\chi_{\Omega^{\prime\prime}\setminus
A}}=\nu_{1}$. In particular, $\Omega^{\prime\prime}\setminus A$ has positive
measure and is therefore an atomless standard measure space, and we can find
$h\in L_{p}(\Omega^{\prime\prime},m^{\prime\prime})^{n}$ that vanishes on $A$
such that $\mu_{h}$ corresponds to $\nu_{2}$. We then have
$\mu_{g}=\mu_{h}+\mu_{f\chi_{A}}=\mu_{h+f\chi_{A}}$. The last equality is
because $h$ and $f\chi_{A}$ are disjointly supported. By Lemma 4.10, there is
a spatial isometry $V$ sending $g$ to $h+f\chi_{A}$. Moreover, we have
$\int_{\Omega^{\prime\prime}}(|f|^{p}+|h+f\chi_{A}|^{p})\chi_{f\neq
h+f\chi_{A}}\leq\int_{\Omega^{\prime\prime}\setminus
A}(|f|^{p}+|h|^{p})<\varepsilon.$
∎
If $B\subset\mathcal{T}$, we define new (larger) classes as follows :
* •
$\Lambda_{1}(B)$ is the set of operators
$(T,\mathrm{id})\colon\operatorname{dom}(T)\oplus_{p}L_{p}(\Omega,\mu)\to\operatorname{ran}(T)\oplus
L_{p}(\Omega,\mu)$ for $T\in B$ and a measure space $(\Omega,\mu)$.
* •
$\Lambda_{2}(B)=\\{U\circ T\circ V\mid U,V\textrm{ spatial isometries, }T\in
B\\}$.
* •
$\Lambda_{3}(B)$ is the set of all $S\colon\operatorname{dom}(S)\subset
L_{p}(\Omega_{1},m_{1})\to L_{p}(\Omega_{2},m_{2})$ such that there is $T\in
B$ where $\operatorname{dom}(T)\subset L_{p}(\Omega_{1},m_{1})\oplus
L_{p}(\Omega,m)$, $\operatorname{ran}(T)\subset L_{p}(\Omega_{2},m_{2})\oplus
L_{p}(\Omega,m)$, $\operatorname{dom}(S)$ is the image of
$\operatorname{dom}(T)$ by the first coordinate projection and $T(f\oplus
g)=Sf\oplus g$ for every $f\oplus g\in\operatorname{dom}(T)$.
* •
$\Lambda_{4}(B)$ is the set of all $S\colon\operatorname{dom}(S)\subset
L_{p}(\Omega,m)\to L_{p}(\Omega^{\prime},m^{\prime})$ such that for every
finite family $f_{1},\dots,f_{n}$ in the domain of $T$ and every
$\varepsilon>0$, there is $T\in B$ with domain contained in $L_{p}(\Omega,m)$
and range contained in $L_{p}(\Omega^{\prime},m^{\prime})$ and elements
$g_{1},\dots,g_{n}\in D(S)$ such that $\|f_{i}-g_{i}\|\leq\varepsilon$ and
$\|Tf_{i}-Sg_{i}\|\leq\varepsilon$.
To save place, we denote
$\Lambda_{123}(B)=\Lambda_{3}(\Lambda_{2}(\Lambda_{1}(B)))$.
###### Corollary 4.14.
For every $T\in\mathcal{T}$ and $B\subset\mathcal{T}$, the following are
equivalent:
* •
for every $n$, $P(T,n)\subset P(B,n)$.
* •
The restriction of $T$ to every finite dimensional subspace of
$\operatorname{dom}(T)$ belongs to $\Lambda_{123}(B)$.
###### Proof.
Assume that, for a fixed $n$, $P(T,n)\subset P(B,n)$. This means that, for
every $f\in\operatorname{dom}(T)^{n}$, there is $S\in B$ and
$g\in\operatorname{dom}(S)^{n}$ such that
$\mu_{f}-\mu_{Tf}=\mu_{g}-\mu_{S_{g}}$. By Lemma 4.9 and Lemma 4.10, there are
$h\in L_{p}(\Omega,m)^{n}$ and $\overline{h}\in
L_{p}(\Omega^{\prime},m^{\prime})^{n}$ and spatial isometries
$U\colon\operatorname{span}\\{Sg_{i}\oplus\overline{h}_{i}\\}\to\operatorname{span}\\{Tf_{i}\oplus
h_{i}\\}$ sending $Sg_{i}\oplus\overline{h}_{i}$ to $Tf_{i}\oplus h_{i}$ and
$V\colon\operatorname{span}\\{f_{i}\oplus h_{i}\to
g_{i}\oplus\overline{h}_{i}\\}$ sending $f_{i}\oplus h_{i}$ to
$g_{i}\oplus\overline{h}_{i}$. The operator $S_{1}=(S,\mathrm{id})$ on
$\operatorname{dom}(T)\oplus L_{p}(\Omega^{\prime},m^{\prime})$ belongs to
$\Lambda_{1}(B)$, so the operator $S_{2}=U\circ S_{1}\circ V$, which sends
$f_{i}\oplus h_{i}$ to $Tf_{i}\oplus h_{i}$ belongs to
$\Lambda_{2}(\Lambda_{1}(B))$, and therefore the restriction of $T$ to
$\operatorname{span}\\{f_{1},\dots,f_{n}\\}$ belongs to
$\Lambda_{3}(\Lambda_{2}(\Lambda_{1}(B)))$. This proves one direction.
The converse is simpler: it follows from the easy directions in Lemma 4.9 and
Lemma 4.10 that $P(\Lambda_{i}(B),n)=P(B,n)$ for $i=1,2,3$. In particular, if
the restriction of $T$ to every $\leq n$-dimensional subspace of
$\operatorname{dom}(T)$ belongs to $\Lambda_{123}(B)$, then $P(T,n)\subset
P(B,n)$. ∎
### 4.5. Convergences in $H_{n}^{*}$
This section is devoted to the understanding of the encoding of both weak-*
sequential convergence and norm convergence in $H_{n}^{*}$. Our first result
asserts that weak-* convergence of sequences corresponds to the operation
$\Lambda_{4}$ we just defined.
###### Proposition 4.15.
Let $B\subset\mathcal{T}$. The smallest class containing $B$ and stable by all
operations $\Lambda_{1},\Lambda_{2},\Lambda_{3},\Lambda_{4}$ coincides with
the set of $T\in\mathcal{T}$ such that for every $n$, $P(T,n)$ is contained in
the sequential weak-* closure of $P(B,n)$.
###### Proof.
We define by transfinite induction, for every ordinal $\alpha$, a class
$B_{\alpha}$ as follows. $B_{0}$ is $\Lambda_{123}(B)$. If $\alpha$ is a
successor ordinal, $B_{\alpha}=\Lambda_{123}(\Lambda_{4}(B_{\alpha-1}))$. If
$\alpha$ is a limit ordinal we set $B_{\alpha}=\cup_{\beta<\alpha}B_{\beta}$.
Similarly, we define, for every integer $n$ and every ordinal $\alpha$, a
subset $C^{n}_{\alpha}\subset H_{n}^{*}$ by $C^{n}_{0}=P(B,n)$, for a
successor ordinal $C^{n}_{\alpha}$ is the set of all limits of weak-*
converging sequences of elements of $C^{n}_{\alpha-1}$. If $\alpha$ is a limit
ordinal we set $C^{n}_{\alpha}=\cup_{\beta<\alpha}C^{n}_{\beta}$.
We claim that, for every $T\in\mathcal{T}$ with $\operatorname{dom}(T)$
finite-dimensional, $P(T,n)\subset C^{n}_{\alpha}$ for every $n$ if and only
if $T$ belongs to $B_{\alpha}$. We prove it by transfinite induction. If
$\alpha=0$, this is Corollary 4.14. Let $\alpha>0$ and assume that the claim
holds for all $\beta<\alpha$. If $\alpha$ is a limit ordinal, the claim is
clear.
So assume that $\alpha$ is a successor. Assume first that $P(T,n)\subset
C^{n}_{\alpha}$ for every $n$. Let $n$ be the dimension of
$\operatorname{dom}(T)$ and $f=(f_{1},\dots,f_{n})$ a basis. Then
$\mu_{f}-\mu_{Tf}$ is a limit of a weak-* converging sequence $\nu_{k}$ of
elements of $C^{n}_{\alpha-1}$. By Lemma 4.3 there are $f^{(k)}\in
L_{p}(\Omega_{k},m_{k})^{n}$ and $g^{(k)}\in
L_{p}(\Omega^{\prime}_{k},m^{\prime}_{k})^{n}$ with values in
$\\{z\in\mathbf{K}^{n},|z|=1\\}$ such that
$\nu_{k}=\mu_{f^{(k)}}-\mu_{g^{(k)}}$ and
$m_{k}(\Omega_{k})+m^{\prime}_{k}(\Omega^{\prime}_{k})$ is the norm of the
corresponding linear form, which is bounded by Banach-Steinhaus. For
simplicity of the exposition assume that
$m_{k}(\Omega_{k})+m^{\prime}_{k}(\Omega^{\prime}_{k})\leq 1$. By the
induction hypothesis, there is an operator $S_{k}\in B_{\alpha-1}$ such that
$f^{(k)}\in D(S_{k})^{n}$ and $S_{k}f^{(k)}=g^{(k)}$. We have two sequences of
probability measures, $f^{(k)}_{*}m_{k}+(1-m_{k}(\Omega_{k}))\delta_{0}$ and
$g^{(k)}_{*}m^{\prime}_{k}+(1-m^{\prime}_{k}(\Omega^{\prime}_{k}))\delta_{0}$,
on $\\{0\\}\cup\\{z\in\mathbf{K},|z|=1\\}\subset\mathbf{K}^{n}$. By
compactness, up to an extraction we can assume that both sequences converge
weak-*, and by Skorohod’s representation theorem we can assume that
$(\Omega_{k},m_{k})$ does not depend on $k$ and that $f^{(k)}$ converges
almost surely to some $f^{(\infty)}\in L_{p}^{n}$ and similarly $g^{(k)}$
converges almost surely, and in particular in $L_{p}$, to $g^{(\infty)}$ (this
modifies the operators $S_{k}$, but they still satisfy
$\nu_{k}=\mu_{f^{(k)}}-\mu_{Sf^{(k)}}$ and therefore still belong to
$B_{\alpha-1}$). In particular, the operator $S^{(\infty)}$ from
$\operatorname{dom}(S^{(\infty)})=\operatorname{span}\\{f^{(\infty)}_{1},\dots,f^{(\infty)}_{n}\\}\to\operatorname{span}\\{g^{(\infty)}_{1},\dots,g^{(\infty)}_{n}\\}$
sending $f^{(\infty)}_{i}$ to $g^{(\infty)}_{i}$ belongs to
$\Lambda_{4}(B_{\alpha-1})$ and it satisfies
$\mu_{f^{(\infty)}}-\mu_{S^{(\infty)}f^{(\infty)}}=\lim_{k}\mu_{f^{(k)}}-\mu_{S^{(k)}f^{(k)}}=\mu_{f}-\mu_{Tf}.$
By Corollary 4.14 again, the restriction of $T$ to
$\operatorname{span}\\{f_{1},\dots,f_{n}\\}=E$ belongs to
$\Lambda_{123}(S^{(\infty)})\subset B_{\alpha}$. This concludes the proof that
$P(T,n)\subset C^{n}_{\alpha}$ for all $n$ implies that the restriction of $T$
belongs to $B_{\alpha}$. The converse is similar but easier and left to the
reader. ∎
Similarly, norm convergence is well encoded.
###### Lemma 4.16.
Let $B\subset\mathcal{T}$, $T\colon\operatorname{dom}(T)\subset
L_{p}(\Omega_{1},m_{1})\to L_{p}(\Omega_{2},m_{2})$ a linear map with domain
of finite dimension $n$, and $(f_{1},\dots,f_{n})$ be a basis of
$\operatorname{dom}(T)$. Then $P(T,n)$ is contained in the norm-closure of
$P(B,n)$ if and only if for every $\varepsilon>0$, there is
$S\in\Lambda_{123}(B)$ with $\operatorname{dom}(S)\subset
L_{p}(\Omega_{1}\times[0,1],m_{1}\otimes d\lambda)$ and
$\operatorname{ran}(S)\subset L_{p}(\Omega_{2}\times[0,1],m_{2}\otimes
d\lambda)$, there are $g_{1},\dots,g_{n}\in\operatorname{dom}(S)$ such that
$\int_{\Omega_{1}\times[0,1]}(|f(\omega)|^{p}+|g(\omega,s)|^{p})\chi_{f(\omega)\neq
g(\omega,s)}dm_{1}(\omega)ds\leq\varepsilon$
and
$\int_{\Omega_{2}\times[0,1]}(|Tf(\omega)|^{p}+|Sg(\omega,s)|^{p})^{\frac{1}{p}}\chi_{Tf(\omega)\neq
Sg(\omega,s)}dm_{2}(\omega)ds\leq\varepsilon.$
###### Proof.
Assume that $P(T,n)$ is contained in the norm closure of $P(B,n)$. This means
that for every $\varepsilon>0$, there $\mu^{\prime}\in P(B,n)$ such that
$\|\mu_{f}-\mu_{Tf}-\mu^{\prime}\|<\varepsilon$. By Lemma 3.4, we can write
$\mu^{\prime}=\mu_{g}-\mu_{h}$ for $n$-uples of elements of $L^{p}$ spaces
$g,h$ where $\|\mu_{f}-\mu_{g}\|+\|\mu_{Tf}-\mu_{h}\|<\varepsilon$. Since we
have some room $(<\varepsilon$), we can even assume that
$\\{g_{1},\dots,g_{n}\\}$ are linearly independant, so that we can define a
linear map $S$ sending $g_{i}$ to $h_{i}$. By Corollary 4.14, $S$ belongs to
$\Lambda_{123}(B)$, and so does $S$ composed with any spatial isometry. So the
only if direction follows from Lemma 4.12 and its improvement in Remark 4.13 .
The converse is proved the same way. ∎
### 4.6. Proof of the main Theorem
We are also ready to prove our main Theorem 1.6. Before we do so, we only need
to understand the operation of taking convex hulls.
###### Lemma 4.17.
Let $B\subset\mathcal{T}$ and $n\in\mathbf{N}$. The convex hull of $P(B,n)$ is
equal to $P(\oplus_{\ell_{p}}(B),n)$ where $\oplus_{\ell_{p}}(B)$ is the class
of all finite $\ell_{p}$-direct sums of operators in $B$.
###### Proof.
This is clear: if $T_{1},\dots,T_{k}\in B$ and $f^{(j)}\in D(T_{j})^{n}$ for
all $j$, then
$\sum_{j}\mu_{f^{(j)}}-\mu_{Tf^{(j)}}=\mu_{f}-\mu_{(T_{1}\oplus\dots\oplus
T_{k})f}$
where $f_{i}=f_{i}^{(1)}\oplus\dots\oplus f_{i}^{(k)}\in D(T_{1})\oplus\dots
D(T_{k})$ and $f=(f_{1},\dots,f_{n})\in(D(T_{1})\oplus\dots D(T_{k}))^{n}$. ∎
We can conclude.
###### Proof of Theorem 1.6.
We start with the easy direction. Assume that for every $n$ and $\varepsilon$,
the assumption in the second bullet point holds. Let $X$ be Banach space such
that $\sup_{S\in B}\|S_{X}\|\leq 1$. We have to prove that $\|T_{X}\|\leq 1$.
That is, for every integer $n$ and every $x_{1},\dots,x_{n}$,
(4.4)
$\|\sum_{i}(Tf_{i})x_{i}\|_{L_{p}(\Omega_{2};X)}\leq\|\sum_{i}f_{i}x_{i}\|_{L_{p}(\Omega_{1},X)}.$
Let $\varepsilon>0$, and $S=S_{0}\oplus S_{1}\dots S_{k}$, $U$, $V$,
$g_{i},g^{\prime}_{i},h_{i}$ given by the assumption. In the following
computation, we view $Tf_{i}$ as an element of $L_{p}(\Omega_{2}\times[0,1])$
that does not depend on the second variable in $[0,1]$, and similarly for
$f_{i}$. We denote simply by $\|\cdot\|_{p}$ the norm in
$L_{p}(\Omega_{i}\times[0,1];X)$ or $L_{p}(\Omega_{i}\times[0,1])$. We can
bound
$\displaystyle\|\sum_{i}(Tf_{i})x_{i}\|_{L_{p}(\Omega_{2};X)}$
$\displaystyle\leq\sum_{i}\|Tf_{i}-g^{\prime}_{i}\|_{p}\|x_{i}\|+\|\sum_{i}g^{\prime}_{i}x_{i}\|_{p}$
$\displaystyle\leq\varepsilon\sum_{i}\|x_{i}\|+\|\sum_{i}g^{\prime}_{i}x_{i}\|_{p}$
$\displaystyle=\varepsilon\sum_{i}\|x_{i}\|+\left(\|\sum_{i}(g^{\prime}_{i},h_{i})x_{i}\|_{p}^{p}-\|\sum_{i}h_{i}x_{i}\|_{p}^{p}\right)^{\frac{1}{p}}.$
The quantity inside the parenthesis is equal to
$\|\sum_{i}S(g_{i},h_{i})x_{i}\|_{p}^{p}-\|\sum_{i}h_{i}x_{i}\|_{p}^{p},$
so using that $\|S_{X}\|=\max_{0\leq i\leq k}\|(S_{i})_{X}\|\leq 1$, we obtain
that it is bounded above by
$\|\sum_{i}(g_{i},h_{i})x_{i}\|_{p}^{p}-\|\sum_{i}h_{i}x_{i}\|_{p}^{p}=\|\sum_{i}g_{i}x_{i}\|_{p}^{p}.$
We can therefore go on with our computation and get
$\displaystyle\|\sum_{i}(Tf_{i})x_{i}\|_{L_{p}(\Omega_{2};X)}$
$\displaystyle\leq\varepsilon\sum_{i}\|x_{i}\|+\|\sum_{i}g_{i}x_{i}\|_{p}$
$\displaystyle\leq\varepsilon\sum_{i}\|x_{i}\|+\sum_{i}\|f_{i}-g_{i}\|_{p}\|x_{i}\|+\|\sum_{i}f_{i}x_{i}\|_{p}$
$\displaystyle\leq 2\varepsilon\sum_{i}\|x_{i}\|+\|\sum_{i}f_{i}x_{i}\|_{p}.$
Making $\varepsilon\to 0$, we obtain (4.4) as required.
The converse direction relies on everything we have obtained so far. Assume
that $T\in({}^{\circ}B)^{\circ}$. We know from Corollary 4.6 that for every
integer $n$, $P(T,n)\subset\overline{\mathrm{conv}}^{\|\cdot\|}P(B\cup
REG,n)$, which is the same as the norm-closure of $P(\oplus_{\ell_{p}}(B\cup
REG))$ by Lemma 4.17. So by Lemma 4.16, for every $\varepsilon>0$ there is
$S\in\Lambda_{123}(B\cup REG)$ with $\operatorname{dom}(S)\subset
L_{p}(\Omega_{1}\times[0,1],m_{1}\otimes d\lambda)$ and
$\operatorname{ran}(S)\subset L_{p}(\Omega_{2}\times[0,1],m_{2}\otimes
d\lambda)$, there are $g_{1},\dots,g_{n}\in\operatorname{dom}(S)$ such that
$\int_{\Omega_{1}\times[0,1]}(|f(\omega)|^{p}+|g(\omega,s)|^{p})\chi_{f(\omega)\neq
g(\omega,s)}dm_{1}(\omega)ds\leq\varepsilon^{p}$
and
$\int_{\Omega_{2}\times[0,1]}(|Tf(\omega)|^{p}+|Sg(\omega,s)|^{p})^{\frac{1}{p}}\chi_{Tf(\omega)\neq
Sg(\omega,s)}dm_{2}(\omega)ds\leq\varepsilon^{p}.$
In particular, using that $\int(|a|^{p}+|b|^{p})\chi_{a\neq
b}\geq\int|a-b|^{p}=\sum_{i}\int|a_{i}-b_{i}|^{p}$ for every
$a,b\in(L_{p})^{n}$, we have for every $i$,
$\left(\int_{\Omega_{1}\times[0,1]}|f_{i}(\omega)-g_{i}(\omega,s)|^{p}dm_{1}(\omega)ds\right)^{\frac{1}{p}}\leq\varepsilon$
and
$\left(\int_{\Omega_{2}\times[0,1]}|Tf_{i}(\omega)-Sg_{i}(\omega,s)|^{p}dm_{2}(\omega)ds\right)^{\frac{1}{p}}\leq\varepsilon.$
Also, using that $REG$ contains the identity and is stable by
$\ell_{p}$-direct sums, $\Lambda_{1}(\oplus_{\ell_{p}}(B\cup REG))$ is the set
of all operators of the form $S_{0}\oplus S_{1}\oplus\dots\oplus S_{k}$ for
$S_{0}\in REG$ and $S_{1},\dots,S_{k}\in B$. So the fact that $S$ belongs to
$\Lambda_{123}(B\cup REG)$ means that there exist $S_{0}\in REG$,
$S_{1},\dots,S_{k}\in B$, spatial isometries $U,V$ and a measure space
$(\Omega,m)$ such that $V\circ(S_{0}\oplus\dots\oplus S_{k})\circ V$ contains
elements of the form $(g_{i},h_{i})$ in its domain for all $1\leq i\leq n$ and
some $h_{i}\in L_{p}(\Omega,m)$ and so that $V\circ(S_{0}\oplus\dots\oplus
S_{k})\circ V(g_{i},h_{i})=(Sg_{i},h_{i})$. We can of course replace
$(\Omega,m)$ by any standard measure space as this amounts to conjugating by
another spatial isometry. So we have obtained the conclusion of the theorem. ∎
## 5\. Variants of the duality
The duality defined in the Introduction was implicit in many early works on
the geometry of Banach spaces, and was essentially present in [27], where
Pisier explicitly considered a duality that is very close to ours. He defines
the polars by the same formulas as in Definition 1.1 and 1.2, but with
different classes of operators instead of $\mathcal{T}$.
###### Definition 5.1.
Denote by $\mathcal{T}_{f}$ the class of all linear operators between (full,
as opposed to subspace of) $L_{p}$ spaces. If $n$ is an integer, denote
$\mathcal{T}_{f,n}$ the class of all linear operators $\ell_{p}^{n}\to L_{p}$
and $\mathcal{T}_{f,<\infty}=\cup_{n\in\mathbf{N}}\mathcal{T}_{f,n}$.
In particular, for the duality between $\mathcal{X}$ and $\mathcal{T}_{f}$,
the polar of a set $B$ of Banach spaces is smaller than for the duality
between $\mathcal{X}$ and $\mathcal{T}$, and therefore its bipolar is larger.
Hernandez also obtained a description of the bipolar $B$ for this duality: it
is the set of Banach spaces that are finitely representable in subspaces of
quotients of finite $\ell_{p}$ direct sums of spaces in $B$, see Theorem 5.5.
This is quite different from Theorem 1.3. For example for the duality
considered here, every Banach space in the bipolar of $\ell_{1}$ has cotype
$\max(p,2)$ (this is immediate from Hernandez’s Theorem 1.3 and the fact that
$\ell_{p}(\ell_{1})$ has cotype $\max(p,2)$), whereas for the duality in [27],
the bipolar of $\ell_{1}$ contains every space finitely representable in a
quotient of $\ell_{1}$, _i.e._ every Banach space.
However, as far as the bipolar of a set of operators is concerned, the two
dualities are very related : if $B\subset\mathcal{T}_{f}$, then its bipolar
for the polarity between $\mathcal{X}$ and $\mathcal{T}_{f}$ is the set of
operators between $L_{p}$ spaces which belong to ${}^{\circ}B^{\circ}$ (for
the polarity between $\mathcal{X}$ and $\mathcal{T}$). So our Theorem 1.6 also
provides an answer to [27, Problem 4.1].
The dualities discussed so far are isometric variants of two other isomorphic
forms of the duality in [27], where $A^{\circ}$ is the class of operators such
that $\|T_{X}\|<\infty$ for all $X\in A$, and ${}^{\circ}B$ is the class of
Banach space such such $\|T_{X}\|<\infty$ for all $T\in B$. But, if $B$ is
finite, the bipolar of $B$ for this “isomorphic” duality coincides with
$\cup_{R>0}R{}^{\circ}(R^{-1}B)^{\circ}$. If $B$ is infinite, the “isomorphic”
bipolar of $B$ is
$\cup_{R>0}\cup_{B^{\prime}}R({}^{\circ}B^{\prime})^{\circ}$, where
$B^{\prime}=\\{\\{c_{T}T\mid T\in B\\}\mid c\in(0,1]^{B}\\}$. So our bipolar
Theorem 1.6 also allows to describe the bipolar for the isomorphic forms of
the duality.
It turns out that the methods of this paper also allow us to recover
Hernandez’s characterization of the bipolar of a set of Banach spaces for the
duality between $\mathcal{X}$ and $\mathcal{T}_{f}$. Let us start with an easy
fact, which allows us to resctrict our attention to $\mathcal{T}_{f,n}$.
###### Lemma 5.2.
The bipolar of a subset $A\subset\mathcal{X}$ for the duality between
$\mathcal{X}$ and $\mathcal{T}_{f}$ coincides with its bipolar for the duality
between $\mathcal{X}$ and $\mathcal{T}_{f,<\infty}$.
###### Proof.
Any $L_{p}$ space can be written as the closure of an increasing net of finite
dimensional $L_{p}$ spaces, which are isomorphic to $\ell_{p}^{n}$. ∎
Denote by $e=(e_{1},\dots,e_{n})$ the standard basis of $\ell_{p}^{n}$. We
encode a subset $B\subset\mathcal{T}_{f,n}$, as the cone
$\widetilde{P}(B,n)\subset H_{n}^{*}$
$\widetilde{P}(B,n)=\\{s(\mu_{e}-\mu_{Te})\mid T\in B,s>0\\}.$
We warn the reader that $\widetilde{P}(B,n)$ is strictly smaller than $P(B,n)$
even for $B\subset\mathcal{T}_{f,n}$. Note however that the duality is still
efficiently encoded. The following result is the analogue of Lemma 4.1 and is
proved identically.
###### Lemma 5.3.
Let $n\in\mathbf{N}$ be an integer, $A\subset\mathcal{X}$ be a class of Banach
spaces and $B\subset\mathcal{T}_{f,n}$ a class of operators $\ell_{p}^{n}\to
L_{p}$.
1. (1)
$B\subset A^{\circ}$ if and only if $\widetilde{P}(B,n)\subset
N(A,n)^{\circ}$.
2. (2)
$A\subset{}^{\circ}B$ if and only if
$N(A,n)\subset{}^{\circ}\widetilde{P}(B,n)$.
Moreover, we have
###### Lemma 5.4.
The subset $\widetilde{P}(\mathcal{T}_{f,n},n)\subset H_{n}^{*}$ is a weak-*
closed convex cone. Its polar is
$C_{n}:=\\{\varphi\in H_{n}\mid\varphi\leq
0,\varphi(e_{1})=\dots=\varphi(e_{n})=0\\}.$
###### Proof.
The convexity of the cone $\widetilde{P}(\mathcal{T}_{f,n},n)$ is clear, as
$\theta(\mu_{e}-\mu_{Te})+(1-\theta)(\mu_{e}-\mu_{Se})$ can be written as
$\mu_{e}-\mu_{Re}$ for the linear map $R\colon\ell_{p}^{n}\to L_{p}\oplus
L_{p}$ given in matrix form by $R=\begin{pmatrix}\theta^{\frac{1}{p}}T\\\
(1-\theta)^{\frac{1}{p}}s\end{pmatrix}$. For the weak-* closedness, by the
Krein-Smulian theorem [8, Theorem V.12.1], we have to show that the
intersection of $\widetilde{P}(\mathcal{T}_{f,n},n)$ with the closed unit ball
$B_{H_{n}^{*}}$ is sequentially weak-* closed. Recall that every element of
$H_{n}^{*}$ can be regarded as a signed measure on $\mathbf{KP}^{n-1}$. If it
belongs to $\widetilde{P}(\mathcal{T}_{f,n},n)\cap B_{H_{n}^{*}}$, then its
positive part in the Jordan decomposition has total mass $\leq 1$ and has
support contained in $\\{\mathbf{K}e_{1},\dots,\mathbf{K}e_{n}\\}$. In
particular, it is less than $\mu_{e}$. It follows that it can be written as
$\mu_{e}-\mu_{f}$ for some $f\in(L_{p})^{n}$. So we are reduced to showing
that $\\{\mu_{e}-\mu_{f}\mid f\in L_{p}^{n}\\}$ is weak-* closed in
$H_{n}^{*}$, which is clear.
It remains to identify the polar of $\widetilde{P}(\mathcal{T}_{f,n},n)$. If
$\varphi\in C_{n}$, and $T\in\mathcal{T}_{f,n}$, we have
$\langle\mu_{e},\varphi\rangle=\sum_{i}\varphi(e_{i})=0,$
so
$\langle\mu_{e}-\mu_{Te},\varphi\rangle=-\langle\mu_{Te},\varphi\rangle\geq
0.$
This shows the inclusion
$C_{n}\subset{}^{\circ}\widetilde{P}(\mathcal{T}_{f,n},n)$.
For the converse inclusion, consider
$\varphi\in{}^{\circ}\widetilde{P}(\mathcal{T}_{f,n},n)$. Then for every
$f\in(L_{p})^{n}$, we have
(5.1) $\langle\mu_{e}-\mu_{f},\varphi\rangle\geq 0.$
In particular, replacing $f$ by $sf$ and making $s\to\infty$, we obtain
$-\langle\mu_{f},\varphi\rangle\geq 0$
for every $f\in(L_{p})^{n}$. Taking for $f$ a constant $z$, this forces
$\varphi(z)\leq 0$ for every $z\in\mathbf{K}^{n}$. Taking $f=0$ in (5.1) leads
to
$\sum_{i}\varphi(e_{i})=\langle\mu_{e},\varphi\rangle\geq 0.$
This implies that $\varphi(e_{i})=0$ for every $i$, and that $\varphi$ belongs
to $C_{n}$. This concludes the proof of the inclusion
${}^{\circ}\widetilde{P}(\mathcal{T}_{f,n},n)\subset C_{n}$ and of the lemma.
∎
We can now reprove Hernandez’ Theorem.
###### Theorem 5.5.
([12]) Let $A\subset\mathcal{X}$ and $X\in\mathcal{X}$. The following are
equivalent.
1. (1)
For every operator $T\colon L_{p}\to L_{p}$, $\|T_{X}\|\leq\sup_{Y\in
A}\|T_{Y}\|$.
2. (2)
$X$ is finitely representable in the class of all quotients of finite
$\ell_{p}$-direct sums of elements in $A$.
###### Proof.
The interesting direction is (1)$\implies$(2). So assume that (1) holds.
Equivalently by Lemma 5.2 $X$ belongs to the bipolar of $A$ for the duality
between $\mathcal{X}$ and $\mathcal{T}_{f,<\infty}$. By Lemma 5.3, this holds
if and only if for every $n$,
$N(X,n)\subset{}^{\circ}(\widetilde{P}(\mathcal{T}_{f,n},n)\cap
N(Y,n)^{\circ}).$
By Lemma 5.4 and the bipolar theorem, $\widetilde{P}(\mathcal{T}_{f,n},n)$
coincides with $C_{n}^{\circ}$, so the previous inclusion becomes
$N(X,n)\subset{}^{\circ}((C_{n}\cup N(Y,n))^{\circ}).$
By the bipolar theorem again, we obtain that $N(X,n)$ belongs to the closed
convex hull of $C_{n}\cup N(Y,n)$, which is nothing but
$C_{n}+\overline{\mathrm{conv}}(N(Y,n))$ (use compactness as in Lemma 4.5 to
see that $C_{n}+\overline{\mathrm{conv}}(N(Y,n))$ is closed). By Lemma 4.7, we
obtain that for every $n\in\mathbf{N}$ and every $x_{1},\dots,x_{n}\in X$,
there is a space $Y$ finitely representable in the finite $\ell_{p}$-direct
sums of elements in $A$ and elements $y_{1},\dots,y_{n}$ spanning $Y$ such
that
$\forall i,\|x_{i}\|=\|y_{i}\|\textrm{ and }\forall
z\in\mathbf{K}^{n},\|\sum_{i}z_{i}x_{i}\|_{X}\leq\|\sum_{i}z_{i}y_{i}\|_{Y}.$
Now if $E$ is any finite dimensional subspace of $X$, and $\varepsilon>0$, we
can pick a finite family $x_{1},\dots,x_{n}$ in its unit sphere whose convex
hull contains the ball of radius $(1+\varepsilon)^{-1}$. Applying the
preceding to these $x_{i}$’s, we obtain a space $Y$ finitely representable in
$\oplus_{\ell_{p}}A$ and a linear map $u\colon Y\to E$ of norm $1$ such that
the image of the unit ball contains the ball of radius $(1+\varepsilon)^{-1}$
of $E$. In other words, $E$ is at Banach-Mazur distance $\leq 1+\varepsilon$
from a quotient of $Y$. But a subspace of a quotient is the same as a quotient
of a subspace, so we have obtained that $X$ is finitely representable in the
quotients of spaces in $\oplus_{\ell_{p}}A$. This is (2).
The converse implication (2)$\implies$(1) can be proved using the same
arguments, but it is easy and classical to check it directly. The point that
perhaps deserves a small justification is why $\|T_{X}\|\leq\|T_{Y}\|$ if $X$
is a quotient of $Y$ and $T\colon L_{p}\to L_{p}$ is an operator. One argument
is by duality. Indeed, $X^{*}$ identifies then as a subspace of $Y^{*}$, and
if $T^{*}\colon L_{q}\to L_{q}$ denotes the dual of $T$ (for
$\frac{1}{q}+\frac{1}{p}=1$), then
$\|T_{X}\|=\|T^{*}_{X^{*}}\|\leq\|T^{*}_{Y^{*}}\|=\|T_{Y}\|.$
∎
## Appendix A On the $\textrm{GL}(n,\mathbf{K})$ invariant subspaces of the
space of homogeneous functions
Let $0<p<\infty$. We recall some definition that already appeared in the body
of the paper for $p\geq 1$.
Let $n$ be a positive integer. Denote by $|z|$ the $\ell_{p}$-”norm” on
$\mathbf{K}^{n}$
$|z|=\left(|z_{1}|^{p}+\dots+|z_{n}|^{p}\right)^{\frac{1}{p}}.$
A function $\varphi\colon\mathbf{K}^{n}\to\mathbf{R}$ is called homogeneous of
degree $p$ if $\varphi(\lambda z)=|\lambda|^{p}\varphi(z)$ for all
$z\in\mathbf{K}^{n}$ and $\lambda\in\mathbf{K}$. The space $H_{n}$ of real
continuous homogeneous of degree $p$ functions on $\mathbf{K}^{n}$ is a Banach
space for the topology of uniform convergence on compact subsets on
$\mathbf{K}^{n}$. A particular choice of norm is $\|\varphi\|=\sup_{|z|\leq
1}|\varphi(z)|$, so that for this norm $H_{n}$ is isometrically isomorphic to
the space of continuous functions on $\mathbf{KP}^{n-1}$ through the
identification of $\varphi\in H_{n}$ with the function
$\mathbf{K}z\in\mathbf{KP}^{n-1}\mapsto\varphi\left(\frac{z}{|z|}\right)$. For
this identification, the natural action of $\textrm{GL}_{n}(\mathbf{K})$ on
$H_{n}$ corresponds to the action of $\textrm{GL}_{n}(\mathbf{K})$ on
$C(\mathbf{KP}^{n-1})$ given by
$A\cdot\varphi(\mathbf{K}z)=\frac{|A^{-1}z|^{p}}{|z|^{p}}\varphi(A^{-1}\mathbf{K}z).$
###### Theorem A.1.
The $\textrm{GL}_{n}(\mathbf{K})$-invariant closed subspaces of the Banach
space $H_{n}$ of continuous $p$-homogeneous functions
$\mathbf{K}^{n}\to\mathbf{R}$ are
* •
$\\{0\\}$ and $H_{n}$ if $p$ is not an even integer.
* •
$\\{0\\}$, $H_{n}$ and the subspace of degree $p$ homogeneous polynomials if
$p$ is an even integer.
###### Remark A.2.
This theorem allows to reprove the result [9] that if $p$ is not an even
integer, then every isometry between subspaces of $L_{p}$ spaces is a spatial
isometry. Indeed, if $T$ is such an isometry, $n$ is an integer, $f\in
D(T)^{n}$ and $\varphi(z)=|z_{1}|^{p}$, then we get for every
$A\in\textrm{GL}_{n}(\mathbf{K})$ (with the notation of (4.2))
$\langle\mu_{f}-\mu_{Tf},\varphi\circ
A\rangle=\|\sum_{j}a_{1,j}f_{j}\|^{p}-\|\sum_{j}a_{1,j}Tf_{j}\|^{p}=0.$
The linear form $\mu_{f}-\mu_{Tf}$ therefore vanishes on the
$\textrm{GL}_{n}(\mathbf{K})$-invariant subspace spanned by $\\{\varphi\circ
A,A\in\textrm{GL}_{n}(\mathbf{K})\\}$. By Theorem A.1 this subspace is dense,
which implies that $\mu_{f}-\mu_{Tf}=0$. One concludes by Remark 4.11 that $T$
is a spatial isometry.
When $p$ is an even integer, the same argument shows that if $X$ is a Banach
space and $x,y\in X$ are so that $(z_{1},z_{2})\mapsto\|z_{1}x+z_{2}y\|^{p}$
is not a polynomial in $z_{1},z_{2},\overline{z}_{1},\overline{z}_{2}$ (for
example if $X=\mathbf{K}^{2}$ with the $\ell_{q}$ norm for $q$ which is not an
even divisor of $p$), then every operator $T$ between subspaces of $L_{p}$
spaces such that $\|T_{X}\|=\|T^{-1}_{X}\|=1$ is a spatial isometry. In
particular we have:
###### Corollary A.3.
For any $0<p<\infty$ (even integer or not) a linear map $T$ between subspaces
of $L_{p}$ spaces is a spatial isometry if and only if $T$ is a regular
isometry.
Rudin’s proof in [29] relied on the Wiener Tauberian theorem. In the proof of
Theorem A.1, we shall need the following variant.
###### Proposition A.4.
Let $f,g\colon\mathbf{R}^{d}\to\mathbf{C}$ be two measurable functions and
$C>0$ such that $|f(x)|\leq C(1+|x|)^{p}$ and $|g(x)|\leq C(1+|x|)^{-p-d-1})$
for all $x\in\mathbf{R}^{d}$. Assume that $g\ast f=0$. Then the support of the
tempered distribution $\hat{f}$ is contained in
$\\{\xi\in\mathbf{R}^{d},\hat{g}(\xi)=0\\}$.
###### Proof.
First observe that the assumption on $g$ implies that $g\in
L_{1}(\mathbf{R}^{d})$.
If $g$ belongs to $\mathcal{D}(\mathbf{R}^{d})$ (the space of compactly
supported $C^{\infty}$ functions), then the proposition is easy : by taking
Fourier transform we have $\hat{g}\hat{f}=0$ (multiplication of a distribution
by a $C^{\infty}$ function), from which the conclusion follows. The strategy
will be to approximate $g$ by compactly supported $C^{\infty}$ functions.
We have to prove that for every $\xi\in\mathbf{R}^{d}$ with $\hat{g}(\xi)\neq
0$, there is a neighbourhood $V$ of $\xi$ such that
$\langle\hat{f},\varphi\rangle=0$ for every $\varphi\in\mathcal{D}(V)$. By
standard translation/convolution/dilation arguments, we can assume that
$\xi=0$, $g$ is $C^{\infty}$, and that $\hat{g}$ does not vanish on the
closure of $B(0,1)$. We will prove that $\langle\hat{f},\varphi\rangle=0$ for
every $\varphi\in\mathcal{D}(B(0,1))$.
Let $\rho\colon\mathbf{R}^{d}\to[0,1]$ be a compactly supported $C^{\infty}$
function, equal to $1$ on $B(0,1)$, and define a sequence of functions
$g_{n}\in\mathcal{D}(\mathbf{R}^{d})$ by $g_{n}(x)=g(x)\rho(\frac{x}{n})$. By
the dominated convergence theorem, $\|g_{n}-g\|_{L_{1}(\mathbf{R}^{d})}\to 0$,
and so $\|\hat{g}_{n}-\hat{g}\|_{L_{\infty}}\to 0$. In particular there exists
$n_{0}$ such that $\hat{g}_{n}$ does not vanish on $B(0,1)$ for all $n\geq
n_{0}$.
Let $\varphi\in\mathcal{D}(B(0,1))$. Then $\frac{\varphi}{\hat{g}_{n}}$
belongs to $\mathcal{D}(B(0,1))$, so we can write
$\langle\hat{f},\varphi\rangle=\langle\hat{g}_{n}\hat{f},\frac{\varphi}{\hat{g}_{n}}\rangle=\langle
g_{n}\ast f,\mathcal{F}^{-1}(\frac{\varphi}{\hat{g}_{n}})\rangle$
where $\mathcal{F}^{-1}$ is the inverse Fourier transform. Using that
$g_{n}\ast f(x)=(g_{n}-g)\ast f(x)=O(\frac{1}{n}(1+\frac{|x|}{n})^{p})$ (this
inequality will be explained below), we get
(A.1)
$|\langle\hat{f},\varphi\rangle|\leq\frac{C}{n}\int(1+\frac{|x|}{n})^{p}|\mathcal{F}^{-1}(\frac{\varphi}{\hat{g}_{n}})|dx.$
To justify to domination of $(g_{n}-g)\ast f(x)=\int(g_{n}-g)(y)f(x-y)dy$, use
that
$|(g_{n}-g)(y)|\lesssim(1+|y|)^{-p-d-1}1_{|y|>n}$
and
$|f(x-y)|\lesssim(1+|x-y|)^{p}\lesssim(1+\max(|x|,|y|))^{p}$
to obtain
$|f\ast(g-g_{n})(x)|\lesssim\int_{|y|>n}(1+|y|)^{-p-d-1}(1+\max(|x|,|y|))^{p}dy.$
If $|x|\leq n$, then the preceding inequality becomes
$|f\ast(g-g_{n})(x)|\lesssim\int_{|y|>n}(1+|y|)^{-d-1}dy\lesssim\frac{1}{n}.$
If $|x|\geq n$, then we cut the integral as
$\int_{n<|y|\leq|x|}+\int_{|x|<|y|}$ and get
$\displaystyle|f\ast(g-g_{n})(x)|$ $\displaystyle\lesssim$
$\displaystyle\int_{n<|y|\leq|x|}\frac{(1+|x|)^{p}}{(1+|y|)^{p+d+1}}dy+\int_{|y|>|x|}(1+|y|)^{-d-1}dy$
$\displaystyle\lesssim$
$\displaystyle|x|^{p}\frac{1}{n^{p+1}}+\frac{1}{|x|}\lesssim\frac{|x|^{p}}{n^{p+1}}.$
This proves the announced inequality.
In view of (A.1), we see that our goal is to prove good integrability
properties on the function $\mathcal{F}^{-1}(\frac{\varphi}{\hat{g}_{n}})$,
_i.e._ good regularity properties of its Fourier transform
$\frac{\varphi}{\hat{g}_{n}}$. To achieve this, we denote by
$A(\mathbf{R}^{d})$ the Fourier algebra of $\mathbf{R}^{d}$, _i.e._ the Banach
space $\mathcal{F}(L_{1}(\mathbf{R}^{d}))$ for the norm
$\|h\|_{A(\mathbf{R}^{d})}=\|\mathcal{F}^{-1}h\|_{L_{1}(\mathbf{R}^{d})}$. The
inequality
(A.2)
$\|h_{1}h_{2}\|_{A(\mathbf{R}^{d})}\leq\|h_{1}\|_{A(\mathbf{R}^{d})}\|h_{2}\|_{A(\mathbf{R}^{d})}$
is the reason for the term “algebra” and is clear from the usual properties of
convolution and Fourier transform. We have the following lemmas.
###### Lemma A.5.
For every $\varphi\in\mathcal{D}(B(0,1))$, there is a constant $C=C(\varphi)$
such that $\frac{\varphi}{\hat{g}_{n}}$ belongs to $A(\mathbf{R}^{d})$ with
norm $\leq C$ for all $n\geq n_{0}$.
###### Lemma A.6.
There is a constant $C^{\prime}$ such that $D^{\alpha}\hat{g}_{n}$ belongs to
$A(\mathbf{R}^{d})$ with norm $\leq C^{\prime}$ for all $n\in\mathbf{N}$ and
$\alpha\in\mathbf{N}^{d}$, $|\alpha|<p+1$.
These two lemmas, together with the Leibniz derivation rule and the fact that
$A(\mathbf{R}^{d})$ is a Banach algebra (A.2), imply that, for every
$\varphi\in\mathcal{D}(B(0,1))$, there is a constant $C$ such that
$D^{\alpha}\frac{\varphi}{\hat{g}_{n}}$ belongs to $A(\mathbf{R}^{d})$ with
norm less than $C$ for all $n\geq n_{0}$ and $\alpha\in\mathbf{N}^{d}$,
$|\alpha|<p+1$. Therefore, for every such $n$ and $\alpha$ we have
$\int|x^{\alpha}\mathcal{F}^{-1}(\frac{\varphi}{\hat{g}_{n}})|dx\leq C.$
This implies that, if $k$ is the unique integer in the interval $[p,p+1)$,
then for every $n\geq n_{0}$
$\int(1+|x|)^{p}\mathcal{F}^{-1}(\frac{\varphi}{\hat{g}_{n}})|dx\leq\int(1+|x|)^{k}\mathcal{F}^{-1}(\frac{\varphi}{\hat{g}_{n}})|dx\leq
C^{\prime}.$
A fortiori, by (A.1) we have
$|\langle\hat{f},\varphi\rangle|\leq\frac{C^{\prime}}{n},$
so making $n\to\infty$ we obtain $\langle\hat{f},\varphi\rangle=0$. This
concludes the proof. ∎
We have to prove the two lemmas used above.
###### Proof of Lemma A.5.
Let $\rho\in D(B(0,1))$ which is equal to $1$ on the support of $\varphi$. The
fact that $\frac{\rho}{\hat{g}}$ (and $\frac{\rho}{\hat{g}_{n}}$ for every
$n\geq n_{0}$) belongs to $A(\mathbf{R}^{d})$ is essentially the Wiener
tauberian theorem. Indeed, the proof in [30, Theorem 9.3] shows that for every
$x\in\mathbf{C}$ such that $\hat{g}(x)\neq 0$, there is $\varepsilon>0$ such
that $\frac{\rho}{\hat{g}}\in A(\mathbf{R}^{d})$ for every $\rho\in
D(B(x,\varepsilon))$. The claimed result follows by a partition of unity
argument. To obtain a bound on $\frac{\rho}{\hat{g}_{n}}$ independant from
$n$, we write
$\frac{\varphi}{\hat{g}_{n}}=\frac{\varphi}{\hat{g}}\frac{1}{1-\frac{\rho}{\hat{g}}(\hat{g}-\hat{g}_{n})}.$
Since $\frac{\rho}{\hat{g}}$ belongs to $A(\mathbf{R}^{d})$ and
$\|\hat{g}-\hat{g}_{n}\|_{A(\mathbf{R}^{d})}=\|g-g_{n}\|_{L_{1}(\mathbf{R}^{d})}\to
0$, there is $n_{1}\geq n_{0}$ such that
$\frac{\rho}{\hat{g}}(\hat{g}-\hat{g}_{n})$ has $A(\mathbf{R}^{d})$-norm less
than $\frac{1}{2}$ for all $n\geq n_{1}$. This implies that for $n\geq n_{1}$
$\frac{\varphi}{\hat{g}_{n}}=\sum_{k\geq
0}\frac{\varphi}{\hat{g}}\left(\frac{\rho}{\hat{g}}(\hat{g}-\hat{g}_{n})\right)^{k}$
belongs to $A(\mathbf{R}^{d})$ with norm less than
$2\|\frac{\varphi}{\hat{g}}\|_{A(\mathbf{R}^{d})}$. The lemma follows with
$C=\max(2\|\frac{\varphi}{\hat{g}}\|_{A(\mathbf{R}^{d})},\max_{n_{0}\leq
n<n_{1}}\|\frac{\varphi}{\hat{g}_{n}}\|_{A(\mathbf{R}^{d})})).$
∎
###### Proof of Lemma A.6.
We have
$\|D^{\alpha}\hat{g}_{n}\|_{A(\mathbf{R}^{d})}=\|x^{\alpha}g_{n}\|_{L_{1}(\mathbf{R}^{d})}\leq\|x^{\alpha}g\|_{L_{1}(\mathbf{R}^{d})}$
because $g_{n}(x)=g(x)\rho(\frac{x}{n})$ and $0\leq\rho\leq 1$. The quantity
$\|x^{\alpha}g\|_{L_{1}(\mathbf{R}^{d})}$ is finite because
$g(x)=O(|x|^{-p-d-1})$ and $|\alpha|<p+1$. ∎
We can now prove the main result on $\textrm{GL}_{n}(\mathbf{K})$-invariant
subspaces of $H_{n}$.
###### Proof of Theorem A.1.
For simplicity we write the proof for $\mathbf{K}=\mathbf{C}$. The real case
is similar, see Remark A.9. Let $f_{0}\in H_{n}$ be a nonzero function such
that the space spanned by the functions $f_{0}\circ A$ for
$A\in\textrm{GL}_{n}(\mathbf{C})$ is not dense in $H_{n}$. We will prove that
$p$ is an even integer and that $f_{0}$ is a homogeneous polynomial. By the
Hahn-Banach theorem, there is a nonzero linear form $\varphi$ on $H_{n}$ which
vanishes on $f_{0}\circ A$ for all $A$. By the Riesz representation theorem,
there is a unique nonzero signed measure $\mu$ on $\mathbf{CP}^{n-1}$ such
that $\varphi(f)=\int f(\frac{z}{|z|})d\mu(\mathbf{C}z)$. We can assume that
$\mu$ is absolutely continuous with respect to the Lebesgue measure (=the
unique $\textrm{U}(n)$-invariant probability measure) on $\mathbf{CP}^{n-1}$,
with a $C^{\infty}$ Radon-Nykodym derivative. Indeed, if $\rho$ is a
$C^{\infty}$ function on $\textrm{U}(n)$, then the measure
$\rho\ast\mu=\int(u_{*}\mu)\rho(u)du$ is absolutely continuous with respect to
the Lebesgue measure on $\mathbf{CP}^{n-1}$, has a $C^{\infty}$ density, and
still satisfies $\int f_{0}\circ A(\frac{z}{|z|})d(h\ast\mu)(\mathbf{C}z)=0$
for every $A\in\textrm{GL}_{n}(\mathbf{C})$. Moreover if $\rho\geq 0$ has a
support which is a small enough neighbourhoud of the identity, then
$\rho\ast\mu\neq 0$.
So in particular, $\mu$ has a nonzero bounded Radon-Nykodym derivative $h$
with respect to the Lebesgue measure. By Lemma A.8 we can write
$\int_{\mathbf{CP}^{n-1}}F(\mathbf{C}z)d\mu(z)=\int_{\mathbf{C}^{n-1}}(Fh)(\mathbf{C}(1,z))\frac{c}{(1+|z_{1}|^{2}+\dots|z_{n-1}|^{2})^{n}}dz.$
Taking $F(\mathbf{C}z)=(f_{0}\circ A)(\frac{z}{|z|})$, we get
$F(\mathbf{C}(1,z))=\frac{f_{0}\circ A(1,z)}{1+|z|^{p}}$ and
(A.3) $0=\int_{\mathbf{C}^{n-1}}f_{0}\circ A(1,z)g(z)dz$
for the nonzero function
$g(z)=\frac{1}{1+|z|^{p}}\frac{d\mu}{d\lambda}(\mathbf{C}(1,z))\frac{c}{(1+|z|_{2}^{2})^{n}}$,
which satisfies.
(A.4) $g(z)=O((1+|z|)^{-p-2n}).$
Now if we take for $A=\begin{pmatrix}1&0\\\ b&-A^{\prime-1}\end{pmatrix}$ for
$A^{\prime}\in\textrm{GL}_{n-1}(\mathbf{C})$, then (A.3) becomes
$0=\int f_{0}(1,b-A^{\prime-1}z)g(z)dz=|detA^{\prime}|\int(g\circ
A^{\prime})(z)f_{0}(1,b-z)dz.$
The second equality is a change of variable. In other words, if
$f\colon\mathbf{C}^{n-1}\to\mathbf{R}$ is the function $f(z)=f_{0}(1,z)$, then
$f$ is a continuous function satisfying $f(z)=O(1+|z|^{p})$ as $z\to\infty$,
and such that $(g\circ A^{\prime})\ast f=0$ for every
$A^{\prime}\in\textrm{GL}_{n-1}(\mathbf{C})$.
Viewing $\mathbf{C}^{n-1}$ as a real vector space $\mathbf{R}^{d}$ with
$d=2n-2$, we see that we are in the setting of Proposition A.4 ( (A.4) indeed
implies that $(g\circ A^{\prime})(z)=O((1+|z|)^{-p-d-2})$). So the proposition
implies that the support of $\hat{f}$ is contained in
$\\{\xi\in\mathbf{C}^{n-1},\mathcal{F}(g\circ A^{\prime})(\xi)\neq 0\\}$. But,
$g$ being nonzero, there exists $\xi\neq 0$ such that $\hat{g}(\xi)\neq 0$.
Since $\textrm{GL}_{n-1}(\mathbf{C})$ acts transitively on
$\mathbf{C}^{n-1}\setminus\\{0\\}$, we get that the support of $\hat{f}$ is
contained in $\\{0\\}$. This implies that $f$ is a polynomial function in
$z,\overline{z}$. So we have proved that (A.3) implies that the function
$z\mapsto f_{0}(1,z)$ is a polynomial function in $z,\overline{z}$. But since
(A.3) for $f_{0}$ clearly implies (A.3) for $f_{0}\circ A$ for every
$A\in\textrm{GL}_{n}(\mathbf{C})$, we get that $z\mapsto f_{0}\circ A(1,z)$ is
a polynomial for every $A$. This implies that $p$ is an even integer and that
$f_{0}$ is a homogeneous polynomial, see Lemma A.7.
This shows that if $p$ is not an even integer, then $\\{0\\}$ and $H_{n}$ are
the only closed $\textrm{GL}_{n}(\mathbf{C})$-invariant subspaces of $H_{n}$,
and that otherwise all other invariant closed subspaces are contained in the
space of degree $p$ homogeneous polynomials. It remains to show that for every
nonzero degree $p$ homogeneous polynomial, every other such polynomial belongs
to the linear space spanned by its $\textrm{GL}_{n}(\mathbf{C})$ orbit. This
is not difficult. ∎
###### Lemma A.7.
Let $f_{0}\in H_{n}$ be a nonzero function such that, for every
$A\in\textrm{GL}_{n}(\mathbf{C})$, $z\in\mathbf{C}^{n-1}\mapsto f_{0}\circ
A(1,z)$ is a polynomial in $z,\overline{z}$. Then $p$ is an even integer and
$f_{0}$ is a homogeneous polynomial of degree $p$.
###### Proof.
Let $P\in\mathbf{C}[X_{1},\dots,X_{2n-2}]$ such that
$f_{0}(1,z)=P(z,\overline{z})$. Using that $f_{0}\in H_{n}$, we have that
$|P(z,\overline{z})|=O((1+|z|)^{p})$, and in particular $\mathrm{deg}(P)\leq
p$, so we can write
$P(z,\overline{z})=\sum_{\alpha,\beta\in\mathbf{N}^{d},|\alpha|+|\beta|\leq
p}a_{\alpha,\beta}z^{\alpha}\overline{z}^{\beta}.$
Let $c\in\mathbf{C}^{n-1}$ and $A=\begin{pmatrix}1&c^{*}\\\ 0&1\end{pmatrix}$.
Similarly there is $P_{c}\in\mathbf{C}[X_{1},\dots,X_{2n-2}]$ of degree $\leq
p$ such that $f\circ A(1,z)=P_{c}(z)$. Then
$P_{c}(z)=|1+\langle z,c\rangle|^{p}f(1,\frac{z}{1+\langle
z,c\rangle})=|1+\langle z,c\rangle|^{p}P(\frac{z}{1+\langle
z,c\rangle},\frac{\overline{z}}{1+\overline{\langle z,c\rangle}}).$
We can rewrite this quantity as
$\sum_{\alpha,\beta}a_{\alpha,\beta}(1+\langle
z,c\rangle)^{\frac{p}{2}-|\alpha|}(1+\overline{\langle
z,c\rangle})^{\frac{p}{2}-|\beta|}z^{\alpha}\overline{z}^{\beta}.$
By expanding $(1+t)^{l}=\sum_{n\geq 0}\binom{l}{n}t^{n}$, for small $z$ the
preceding sum is
$\sum_{\alpha,\beta,n,m}a_{\alpha,\beta}\binom{\frac{p}{2}-|\alpha|}{n}\binom{\frac{p}{2}-|\beta|}{m}\langle
z,c\rangle^{n}\overline{\langle
z,c\rangle}^{m}z^{\alpha}\overline{z}^{\beta}.$
Since $P_{c}$ is a polynomial of degree $\leq p$, we get that for every $N>p$,
$\sum_{|\alpha|+|\beta|+n+m=N}a_{\alpha,\beta}\binom{\frac{p}{2}-|\alpha|}{n}\binom{\frac{p}{2}-|\beta|}{m}\langle
z,c\rangle^{n}\overline{\langle
z,c\rangle}^{m}z^{\alpha}\overline{z}^{\beta}=0.$
Since this is valid for every $c$, we get
$a_{\alpha,\beta}\binom{\frac{p}{2}-|\alpha|}{n}\binom{\frac{p}{2}-|\beta|}{m}=0$
for every $\alpha,\beta\in\mathbf{N}^{d}$ and $n,n\in\mathbf{N}$ such that
$|\alpha|+|\beta|+n+m>p$.
Let $\alpha,\beta$ such that $a_{\alpha,\beta}\neq 0$ (such $\alpha,\beta$
exist by the assumption that $f_{0}$ is nonzero). Then taking $n=0$ and $m$
very large, we find that $\binom{\frac{p}{2}-|\beta|}{m}=0$, which implies
that $\frac{p}{2}-|\beta|$ is a nonnegative integer. Similarly
$\frac{p}{2}-|\alpha|$ is a nonnegative integer. This proves that $p$ is an
even integer and
$f_{0}(1,z)=\sum_{|\alpha|,|\beta|\leq\frac{p}{2}}a_{\alpha,\beta}z^{\alpha}\overline{z}^{\beta}.$
By homogeneity we get
$f_{0}(z_{1},z)=\sum_{|\alpha|,|\beta|\leq\frac{p}{2}}z_{1}^{\frac{p}{2}-|\alpha|}z^{\alpha}\overline{z}_{1}^{\frac{p}{2}-|\beta|}\overline{z}^{\beta}.$
This is the lemma. ∎
###### Lemma A.8.
The Lebesgue measure $\lambda$ on $\mathbf{CP}^{n-1}$ is given by
$\int_{\mathbf{CP}^{n-1}}F(\mathbf{C}z)d\lambda(z)=c\int_{\mathbf{C}^{n-1}}F(\mathbf{C}(1,z))\frac{1}{(1+|z_{1}|^{2}+\dots+|z_{n-1}|^{2})^{n}}dz$
for some number $c>0$.
###### Proof.
It is a change of variable to compute that the finite measure
$F\in
C(\mathbf{CP}^{n-1})\mapsto\int_{\mathbf{C}^{n-1}}F(\mathbf{C}(1,z))\frac{1}{(1+|z_{1}|^{2}+\dots+|z_{n-1}|^{2})^{n}}dz$
is invariant by $\textrm{U}(n)$. ∎
###### Remark A.9.
We did not use the full strength of Proposition A.4 for
$\mathbf{K}=\mathbf{C}$, as we used it for a function $g$ satisfying
$g(z)=O((1+|z|)^{-p-d-2})$, which is strictly stronger that the required
$g(z)=O((1+|z|)^{-p-d-1})$. The reason for this $2$ is that the real dimension
drops by $2$ between $\mathbf{C}^{n}$ and $\mathbf{CP}^{n-1}$. In the real
case, the dimension drops by $1$, and the same proof (using all the
assumptions of Proposition A.4 this time) leads to the following.
The $\textrm{GL}_{n}(\mathbf{R})$-invariant closed subspace of the Banach
space $H_{n,\mathbf{R}}$ of continuous $p$-homogeneous functions
$\mathbf{R}^{n}\to\mathbf{R}$ are (1) $\\{0\\}$ and $H_{n,\mathbf{R}}$ if $p$
is not an even integer (2) $\\{0\\}$, $H_{n,\mathbf{R}}$ and the space of
homogeneous degree $p$ polynomials if $p$ is an even integer.
As a consequence, the conclusion of remark A.2 holds also over $\mathbf{R}$.
### Acknowledgements
The author thanks Jean-Christophe Mourrat for interesting discussions that
lead to the proof of Proposition A.4, and Alexandros Eskenazis, Mikhail
Ostrovskii and Ignacio Vergara for useful comments, suggestions and
corrections. Special thanks are due to Gilles Pisier for all that, and also
for encouraging the author to think about the content of Section 5.
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|
A framework to compare music generative models using automatic evaluation
metrics extended to rhythm.
Sebastian Garcia-Valenciaa,b, Alejandro Betancourtc and Juan G. Lalinde-
Pulidoa
aComputer Science Department, Universidad EAFIT, Medellin, Colombia
bResearch and Development, AIVA Technologies, Luxembourg City, Luxembourg
cDigital Department, Ecopetrol, Bogota, Colombia
###### Abstract
To train a machine learning model is necessary to take numerous decisions
about many options for each process involved, in the field of sequence
generation and more specifically of music composition, the nature of the
problem helps to narrow the options but at the same time, some other options
appear for specific challenges. This paper takes the framework proposed in a
previous research that did not consider rhythm to make a series of design
decisions, then, rhythm support is added to evaluate the performance of two
RNN memory cells in the creation of monophonic music. The model considers the
handling of music transposition and the framework evaluates the quality of the
generated pieces using automatic quantitative metrics based on geometry which
have rhythm support added as well.
## keywords
Automatic Evaluation; Rhythm; Framework; Monophonic Music; Generative Model;
Recurrent Neural Network
## Introduction
Artificial intelligence is a term appearing everywhere in the last years, the
variety of fields it is transforming is huge and the examples of breaking
advances in everything it is applied to seems not to stop. Music composition
is not the exception, every day the level of compositions generated by AI gets
only better, and more and more composers begin to integrate AI-powered systems
in their arsenal of tools.
With the development of the field, comes the fact that there are too many
options for each aspect of the development of a machine learning model, i.e.
datasets, representations, objective functions, hyperparameters, etc.
Therefore the importance of a way to formally narrow these options.
We can characterize the challenges for this research in two categories. On one
hand, we have the algorithmic challenges that any machine learning problem has
(which data to use, which algorithmic architecture, how to train the model).
On the other hand, we must consider some further challenges specific to the
field of music composition (how to represent the music, how to keep long term
consistency, how to deal with transposition, and how to evaluate the quality
of the compositions)
In previous research done in (Garcia-Valencia et al. 2020), we proposed a
framework to progressively compare and justify a good set of combinations for
these options in the case of a sequence generator of melodies without rhythm
(all notes are quarter notes). Here we will apply the same framework adding
support for rhythm as follows, getting a whole monophonic music generator.
Training Data: Music datasets are usually the result of scraping music
repositories. The most popular container in recent years is the MuseScore
sheet music archive (Froment and Bonte 2002), because there is a significant
community supporting and contributing to it, an example of a dataset of
monophonic music with pieces of musescore is the mono-musicxml-dataset (van
der Wel and Ullrich 2017). We use an updated version of the mono-midi-
transposition-dataset already used in the previous research but with support
for rhythm.
Algorithmic Architecture:
Just like in the previous research, we use an architecture based on Recurrent
Neural Networks with memory mechanisms but defined and implemented from
scratch. It has a multicell RNN layer where n units of a certain type of RNN
cell can be set, the election of this number and type is not trivial.
Objective Function: Concerning the optimization of the model in training,
usually, cross-entropy is a popular choice for classification and generative
models (Johnson 2017; Hutchings and McCormack 2017; Whorley and Conklin 2016;
Kawthekar et al. 2017). The behavior of this metric is used as criteria to
choose the number of units in the multicell RNN since it was already tested
that optimization of cross-entropy corresponds with improvements in the music
generation (Garcia-Valencia 2020a).
Respect to the musical aspects:
1. 1.
Music Representation: The notes are transformed into tuples (p,d), where p is
the pitch as midi value and d is the duration, then they are encoded as
embeddings to add semantic meaning as tested in (Garcia-Valencia 2020b).
2. 2.
Long Term Consistency: A common problem in sequence generation is that, after
some iterations, the generated sequence loses sense and becomes random data.
To address this problem we analyze the performance of two different types of
memory cells:
1. (a)
LSTM (Hochreiter and Urgen Schmidhuber 1997): The Long-Short-Term Memory NN is
composed of 4 elements: a cell state which determines the current context, a
forget gate which decides which information remove, an input gate to insert
new information, and an output gate to decide the output (Olah 2015).
2. (b)
GRU (Cho et al. 2014): The Gated Recurrent Unit is very similar to the LSTM.
It has two gates, the update gate which is a combination of the forget and
input gates in the LSTM, and the reset gate which decides the rate of past
information kept. Training a GRU network is usually faster because it has
fewer tensors than LSTM.
The authors in Chung et al. (2014) compare LSTM and GRU and conclude that GRU
performs better in 3 out of 4 datasets, however, there is no consensus about
which is better and it is normal to try both to find which fits better the use
case.
3. 3.
Music Transposition: Music transposition makes two pieces of music to be
perceived in essence the same song despite having almost all notes different.
The updated version of the dataset with time support has the same 3 strategies
variations to address music transposition: i) A control case where the
sequences are the notes, ii) A modified case storing the pitch intervals iii)
The last case on which each sequence is augmented to 12.
4. 4.
Music Evaluation: According to the recent bibliography, music evaluation can
be divided into two groups. The first one uses mathematical formulations
(Colombo et al. 2017; Jaques et al. 2016; Tymoczko 2011), while the second one
uses musical experiments to consider human judgments about the quality of the
composition. To quantify the quality of the compositions, we add rhythm
support to the three quantitative metrics proposed in the previous research
(Conjunct Melodic Motion, Limited Macroharmony and Centricity) which considers
indications of tonality based on music theory and geometry. The proposed
metrics are used to compare different combinations of hyperparameters and the
impact of the Music Transposition strategies. See section "Description of the
Quantitative Evaluation" for a further description of the metrics.
The Framework: In summary, coming from the conclusions of the previous
research (Garcia-Valencia et al. 2020) we already chose the mono-midi-
transposition-dataset for training purposes, the general architecture based in
a multi-cell RNN, cross-entropy as objective function, tuples as music
representation, embeddings as encoding and quantitative metrics for quality
evaluation. we still have 3 options of dataset strategy for transposition, 2
types of memory cells and n numbers of units to put in the multicell RNN for
long-term consistency. First, we will use the cross-entropy convergence of the
models in training and validation to choose the number of units, and then the
quantitative metrics to evaluate the datasets and memory cell types.
The remaining part of this paper is organised as follows: Chapter "Generating
melodies", introduces our approach for the RNN architecture, the training
phase, and the music evaluation. Chapter "Experiments", presents the
experiments to understand the selected architectures and finally tune the
hyperparameters of the proposed networks (e.g., Number of units, Memory Cell).
Chapter "Results", analyses the models and the generated melodies from the
quantitative and musical perspective. Finally, chapter "Conclusions and future
research", concludes and provides some future research lines of this work.
## Generating melodies
Following the ideas from section "Introduction", our goal is to develop a
network to generate melodies. Section "RNN Architecture", describes the
architecture and training of the RNN. Section "Mono-midi-transposition-
dataset" and "Description of the Quantitative Evaluation", illustrates
respectively the updates to the dataset and quantitative metrics respect to
the previous research. The result of applying these metrics to the dataset is
used as the baseline for the performance analysis of section "Analysis of
Metrics in the Models".
### RNN Architecture
This model with time support was done from scratch and do not reuse the one of
the previous paper, however, its high-level logic is very similar.
The workflow through the network varies depending on the use, namely training
mode (fig. 1) or sampling mode (fig. 2). For training, the starting point is
the dataset. First, every sample goes to the embedding component to be encoded
as a vector.
Figure 1: RNN Architecture training mode
The embedding goes to the multicell RNN, which can contain an array of cells
of any type (LSTM or GRU). Once the embedding flows through the multicell RNN,
the internal state of the cells changes and goes as feedback input for the
next iteration. After that, there is a densely connected layer.
Using the output of the dense layer and the encoded Y, the network calculates
the cross-entropy. Using an Adam optimiser, this last layer back-propagates
the error through all the layers (light lines in fig. 1).
In the case of sampling mode, the starting point is the seed, which is the
sequence that the network will use to initiate its internal state and then
generate new notes. From here, the sampling mode has two phases. In the first
one (light lines in fig. 2), each sample of the seed flows through the
embedding encoder to become its vector representation and then change the
internal state of the cells in multicell RNN. Once it finishes with all the
samples in the seed, phase 2 begins (dark lines in fig. 2). The output of the
multicell flows through the dense layer and a multinomial distribution layer
which outputs the most probable next embedding. This output goes back as an
input to the multicell RNN in the next iteration, and simultaneously it is
decoded and added to the generated melody. This process repeats n times.
Figure 2: RNN Architecture sampling mode
### Mono-midi-transposition-dataset
The dataset111The general dataset is available in
https://sebasgverde.github.io/mono-midi-transposition-dataset/ used in this
paper is an updated version of the dataset used in the previous research. The
process consists of the same 3 steps222All the code, files and final datasets
to replicate the results of the whole paper are available in
https://sebasgverde.github.io/rnn-time-music-paper/ than the previous paper (
i) Scraping, ii) Preprossessing and iii) Cleaning) with the difference that in
the preprocessing, instead of just an array containing the sequence of notes,
each midi is transformed into an array containing a list of tuples (p,d),
where p is the pitch as midi value and d is the duration using a bar
resolution of 16, this means that a whole note is represented with a value of
16 and a 16th note with a value of 1. Also, in the cleaning, only pieces with
at least 12 notes and with durations of max 16 are kept.
The final list of arrays is used as the base to create 3 datasets (fig 3.4 and
3.5):
Figure 3: Dataset workflow
Control Dataset: This is the base case, on which X is the concatenated array
of the original songs and Y is X shifted by one position.
DB12 Dataset: To construct this dataset, each song is transposed 12 times (on
each degree of the chromatic scale).
Intervals Dataset: In this case, we do not have a sequence of notes, but a
sequence of relative changes.
### Description of the Quantitative Evaluation
Here we define the adaption of the 3 metrics (i.e. Conjunct Melodic Motion,
Limited Macroharmony and Centricity) proposed in Tymoczko (2011) and
implemented in the previous research (Garcia-Valencia et al. 2020) to evaluate
automatically the quality of the generated pieces measuring the tonality.
Since we need to support rhythm, the calculation of the span changes respect
to the previous research.
Let’s convert the melody into an array of resolution 16 (16 elements per bar),
where the pitch is in the position of the onsets and all other elements are
zero. Let’s define the $span_{i}$ as a subset of song from $X_{i}$ to
$X_{i+n}$, where n = $|$span$|$ is the size of the span (in this case the
constant 32, the size of two bars), m is the size of the step we use to move
through the array (in this case 4, the size of a quarter note) and $Sq$ the
quantity of spans given by equation 1.
$Sq=\begin{cases}1,&\text{if }|song|\leq n\\\
\frac{|song|-n}{m}+1,&\text{otherwise}\end{cases}$ (1)
1. 1.
Conjunct Melodic Motion (CMM): CMM looks for smooth transitions through the
melody, in other words, the changes between notes must have short intervals.
2. 2.
Limited Macroharmony (LM): Macroharmony measures the diversity of notes in
melodies. This is captured by measuring the number of different notes in short
slices of time (spans).
3. 3.
Centricity (CENTR): Centricity claims that in short slices of time (spans),
there must be a note which appears with more frequency than the others.
#### Geometry descriptive statistics for dataset
We apply now these metrics to the database to have a baseline to compare. As
table 1 shows, for the CMM and the LM the average is above the perfect score
of 1 by 1.23 and 1.05 respectively, this evidence that a score above 1 in the
generated pieces is not necessarily a bad result. In the case of centricity,
we can conclude that a tonal melody should have a note that is at least the
27% of the melody.
CMM | LM | CENTR
---|---|---
2.23 $\pm$ 0.98 | 2.05 $\pm$ 1.18 | 0.27 $\pm$ 0.14
Table 1: Dataset evaluation
## Experiments
This section compares different types of RNN cells (2), with different
quantities of units in the multicell RNN layer (5) and different transposition
strategies (3). The combination of these decisions gives 30 different models
to be analysed. This section defines the optimal number of units, for every
data-set and memory cell, reducing the number of models to 6 (3 datasets and 2
memory cells). Looking for simplicity, the experiments are grouped by data-
set, we try powers of two number of units beginning with 128 (i.e. 128, 256,
512, 1024, 2048). The convergence of the cross-entropy is used as evaluation
metric looking for a balance between a good training and validation loss
value.
Regarding the training conditions: i) In all the cases the batch size is 64
and the sequence length is 100, which means that every training iteration uses
64 sequences of 100 notes simultaneously. ii) For the control and the interval
dataset, the model trains 200 epochs, which means that uses the complete
dataset 200 times. For the DB12 dataset, which is 12 times bigger than the
other 2, it trains only 90 epochs.
### Control dataset experiment
The optimal number of units for the LSTM model is 2048, after epoch 60 the
training loss stops improving (Fig. 4(c)) while the validation loss continues
getting worse (Fig. 4(a)), this is a clear insight of overfitting, however, in
the context of sequence generation, overfitting doesn’t have the same meaning
that it has in other tasks like classification, and therefore even if is a
good criteria to choose a model and train step, it is not the definitive one.
In the case of LSTM, the 2048 model has a stable behavior and the minimal
training loss, we chose this model around epoch 60, where the validation loss
is minimal.
Even if the GRU model with 2048 units has the minimal training loss, it is too
unstable (Fig. 4(d)), therefore we use the 1024 units one around epoch 50
which is more stable and shows less overfitting (Fig. 4(b))
(a) LSTM validation
(b) GRU validation
(c) LSTM training
(d) GRU training
Figure 4: Control Dataset Experiment Learning Curves
### Interval dataset Experiment
For the interval dataset case we use the same reasoning for both memory cells,
the minimal training loss in a train step of minimal validation loss. In the
case of LSTM is the 2048 units model around epoch 60, even if the 1042 units
one is close (Fig. 5(c)) the validation loss shows a worse behavior (Fig.
5(a)).
The behavior of the GRU model for this dataset shows interesting results, the
2048 units model never reaches the minimum value as it did with the control
dataset where it did it at least for a while before becoming unstable, in this
case, 512 and 1024 are better for the training (Fig. 5(d)), we can notice a
tendency of the GRU cells to have some unexpected behaviors.
(a) LSTM validation
(b) GRU validation
(c) LSTM training
(d) GRU training
Figure 5: Interval dataset Experiment Learning Curves
### DB12 Dataset Experiment
In the LSTM experiment (Fig. 6(c)), results are similar to the other two
datasets, with correspondence between training loss and units count, the 2048
unit model around epoch 50 is the chosen in this case.
GRU DB12 model has a bad performance in this experiment as well, showing a
general difference in stability for dense models between LSTM and GRU. The
chosen model, in this case, is the 512 units one in the minimal training loss
epoch, this is because it is very close to the 1024 units model (Fig 6(d)) but
shows more stability and less overfitting in the validation loss (Fig. 6(b)).
(a) LSTM validation
(b) GRU validation
(c) LSTM training
(d) GRU training
Figure 6: DB12 Dataset Experiment Learning Curves
### Analysing the Network Depths
Table 2 summarises the optimal number of units for each dataset and memory
cell. The following part of this section analyses the advantages and
disadvantages of each dataset (rows) and the performance of each memory-
mechanism (columns). The remaining part of this paper uses the number of units
defined in Table 2.
| LSTM | GRU
---|---|---
CONTROL | 2048 | 1024
INTERVAL | 2048 | 1024
DB12 | 2048 | 512
Table 2: Optimal number of units per model
Regarding the datasets, it is interesting, that the DB12 dataset is the only
one with 512 units as the best option, it is as well, the only dataset where
none of the models reach a training loss below 1 within the first 50 epochs,
this suggests that models take longer to converge with this dataset, but this
can be explained by the fact that the DB12 dataset is 12 times bigger than the
other two.
Regarding the type of memory cell, the LSTM shows a very stable behavior
across the datasets, always having a direct correspondence between unit count
and loss convergence, as well as a general stable behavior of the curves. The
GRU model, on the other hand, shows instability in the curves, especially for
dense models like the ones with 1024 and 2048 units.
## Results
This chapter introduces the evaluation and analysis of the 6 models of table
2. In section "Analysis of Metrics in the Models", the 3 metrics described in
"Description of the Quantitative Evaluation", test 100 songs for each of the 6
models to quantify their composition capabilities. Section "Generated Melodies
Musical Analysis", visualises the 6 most representative songs, and provide
analysis from the musical viewpoint.
### Analysis of Metrics in the Models
To evaluate the 6 models from table 2, each model generates 100 songs adding
30 new notes to a seed of a D4 half note followed by an E4 quarter note and an
F4 quarter note. Table 3 shows the average and standard deviation for each
metric and model.
| | LSTM | GRU
---|---|---|---
CMM | control | 2.06 $\pm$ 0.66 | 2.01 $\pm$ 0.53
interval | 1.65 $\pm$ 0.61 | 2.21 $\pm$ 0.73
| db12 | 2.30 $\pm$ 0.89 | 1.66 $\pm$ 0.65
LM | control | 1.97 $\pm$ 0.60 | 1.84 $\pm$ 0.55
interval | 1.67 $\pm$ 0.70 | 1.99 $\pm$ 0.70
db12 | 2.25 $\pm$ 1.02 | 2.20 $\pm$ 0.87
CENTR | control | 0.30 $\pm$ 0.10 | 0.28 $\pm$ 0.10
interval | 0.24 $\pm$ 0.11 | 0.27 $\pm$ 0.11
db12 | 0.30 $\pm$ 0.14 | 0.34 $\pm$ 0.17
Table 3: Means and standard deviation for the 100 songs generated by the final
models with the 3 metrics.
According to the Conjunct Melodic Motion (CMM) the closest model to 1 is the
INTERVAL-LSTM (1.65) outperforming the average of the dataset (2.23), close to
it is the DB12-GRU(1.66) but with a higher standard deviation. The DB12-LSTM
model shows a significantly worse result, and the largest standard deviation,
positioning it as the least reliable model according to the CMM.
In the case of LM the DB12 dataset has the worse performance in general for
this metric with both values above the dataset average (2.05). The model with
the best LM is the INTERVAL-LSTM with 1.67, which also outperforms the 2.05 of
the dataset (table 1).
Regarding the centricity, The model which shows higher centricity is the
DB12-GRU. Notice that this model has poor LM, this makes sense, while
centricity measures the prevalence of a note, the LM, makes the opposite. The
bad performance of this model in the LM metric indicates a strong repetition
of notes
In general, INTERVAL models show a low centricity (tends to use more notes),
which is evident in the good results for LM. Between the 2 strategies for
transposition learning, the interval representation is the most stable.
With respect to the best model for the 3 metrics, the CONTROL-GRU model shows
the best trade-off for the three metrics. It is the only model that
outperforms the centricity of the dataset without affecting the LM
considerably. In summary, concerning only the quantitative analysis, the best
models are i) INTERVAL-LSTM for CMM and LM, ii) DB12-GRU for CENTR and iii)
CONTROL-GRU for general tonality.
### Generated Melodies Musical Analysis
Using the average metrics for the 100 songs generated by each of the 6 models
(Table 3), it is possible to select a representative song per model using the
Euclidean Distance to the average as selection criteria. Table 4 summarises
the metrics for the selected songs.
| LSTM | GRU
---|---|---
| CMM | LM | CENTR | CMM | LM | CENTR
CONTROL | 1.94 | 2.00 | 0.30 | 2.00 | 1.80 | 0.26
INTERVAL | 1.62 | 1.67 | 0.32 | 2.16 | 2.07 | 0.24
DB12 | 2.22 | 2.33 | 0.36 | 1.62 | 2.14 | 0.29
Table 4: The 3 metrics for the most representative of the 100 songs generated
by the final models in each case.
Figures 7, 8 and 9 shows the musical sheet of the 6 melodies from table 4.
Both CONTROL model songs have only natural notes and an evident high use of
quarter notes, in the GRU case (Fig. 7(b)) the LM has relatively good results
with a good balance of total pitches.
The DB12-GRU model (Fig. 8(b)) also shows no alterations and especially a good
CMM with smooth changes across the whole piece and more variation in the
rhythm using some half and eighth notes, however, the LM is the second worse
between the models, in this case, because almost all the spans are below the
minimum number of pitches (5), having as result a song with few pitches.
(a) LSTM Model
(b) GRU Model
Figure 7: Generated Melodies Control Dataset
The DB12-LSTM (Fig. 8(a)) has, in this case, the higher centricity, the G4
quarter note repeated 3 times in bar 7 is a good example of why. This model
also has good use of rhythms and phrase structure, you can describe the melody
with an individual motif of 4 quarter notes followed by a whole note, and just
some variations of this, the problem in this song is the distance in some
changes like the jumps from a G4 to a D5 in bar 2, 8 and 9. This explains the
score for the CMM, although this value is close to the expected one in the
dataset evaluation (table 1).
(a) LSTM Model
(b) GRU Model
Figure 8: Generated Melodies DB12 Dataset
In the case of the INTERVAL models, as expected by the CMM and LM metrics
(Tables 2 and 3 ), The INTERVAL-LSTM (Fig. 9(a)) song has no abrupt changes
from note to note and the range of pitches is well balanced. The INTERVAL-GRU
song (Fig. 9(b)) has almost no quarter notes, and show some rhythmic and
Melodic patterns even when this model didn’t highlight in the metrics, in this
model we can see also some half notes crossing bars in bars 2, 3 and 4.
(a) LSTM Model
(b) GRU Model
Figure 9: Generated Melodies Interval Dataset
## Conclusions and future research
About the 2 cell types, it is interesting that the representative melody of
the LSTM model, when trained with the DB12 dataset, show a good learning
capacity of phrase patterns. The GRU models are in general less stable than
the LSTM ones (Sec Experiments), especially with the densest models. LSTM
models have a more expected behavior, with a clear correspondence between the
network depth and learning convergence, it is important to say though, that
sections "Generated Melodies Musical Analysis" and "Analysis of Metrics in the
Models" suggest that GRU models tend to explore more rhythmic patterns in the
generation.
As found in the previous research, the depth of the network is not a trivial
problem of using so many units as possible, section "Experiments" shows many
examples where models with 512 and 1024 units have better results for the cost
function than the ones with 2048.
Finally, concerning the dataset variations, the CONTROL dataset seems to have
less exploratory models since they use only natural notes and a lot of quarter
notes in the representative melodies. The DB12 dataset has the highest scores
for the CENTR metric and in general shows more interesting musical patterns
than the other representation alternative as intervals.
As future work. It would be interesting to see the performance of our optimal
six models when used to generate sequences in other areas. Also, The
implementation of the automatic metrics is a good starting point to make them
more robust. These metrics could be used to train models using reinforcement
learning.
## Acknowledgement
Special thanks to the Center of Excellence and Appropriation on Big Data and
Data Analytics (Alianza CAOBA).
## Supplementary Materials
As support for the paper, there is a web page in the URL:
https://sebasgverde.github.io/rnn-time-music-paper/ with all the material
necessary for research replication, which includes:
1. 1.
Datasets
2. 2.
Network Weights
3. 3.
Midi songs of the final models
4. 4.
RNN model
5. 5.
Library for music evaluation
6. 6.
Detailed instructions to run the scripts
Scripts available include:
1. 1.
Environment setting
2. 2.
Datasets and weights downloading
3. 3.
Number of units experiment
4. 4.
Generating songs with trained models
5. 5.
Music Evaluation of models, songs and datasets
## References
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* Colombo et al. (2017) Colombo, F., A. Seeholzer, and W. Gerstner. 2017. “Deep Artificial Composer: A Creative Neural Network Model for Automated Melody Generation.” In J. Correia, V. Ciesielski, and A. Liapis, (editors) _Computational Intelligence in Music, Sound, Art and Design: 6th International Conference, EvoMUSART 2017, Amsterdam, The Netherlands, April 19–21, 2017, Proceedings_. Cham: Springer International Publishing, pp. 81–96. URL http://dx.doi.org/10.1007/978-3-319-55750-2{_}6.
* Froment and Bonte (2002) Froment, N., and T. Bonte. 2002. “MuseScore.” URL https://musescore.org.
* Garcia-Valencia (2020a) Garcia-Valencia, S. 2020a. “Cross entropy as objective function for music generative models.” _ArXiv e-prints_ abs/2006.0. URL https://arxiv.org/abs/2006.02217.
* Garcia-Valencia (2020b) Garcia-Valencia, S. 2020b. “Embeddings as representation for symbolic music.” _ArXiv e-prints_ abs/2005.0. URL https://arxiv.org/abs/2005.09406.
* Garcia-Valencia et al. (2020) Garcia-Valencia, S., A. Betancourt, and J. G. Lalinde-Pulido. 2020. “Sequence Generation using Deep Recurrent Networks and Embeddings: A study case in music.” _ArXiv e-prints_ abs/2012.0. URL https://arxiv.org/abs/2012.01231.
* Hochreiter and Urgen Schmidhuber (1997) Hochreiter, S., and J. Urgen Schmidhuber. 1997. “LONG SHORT-TERM MEMORY.” _Neural Computation_ 9(8):1735–1780. URL http://www7.informatik.tu-muenchen.de/{~}hochreithttp://www.idsia.ch/{~}juergen.
* Hutchings and McCormack (2017) Hutchings, P., and J. McCormack. 2017. “Using Autonomous Agents to Improvise Music Compositions in Real-Time.” In J. Correia, V. Ciesielski, and A. Liapis, (editors) _Computational Intelligence in Music, Sound, Art and Design: 6th International Conference, EvoMUSART 2017, Amsterdam, The Netherlands, April 19–21, 2017, Proceedings_. Cham: Springer International Publishing, pp. 114–127. URL http://dx.doi.org/10.1007/978-3-319-55750-2{_}8.
* Jaques et al. (2016) Jaques, N., S. Gu, R. E. Turner, and D. Eck. 2016. “Tuning Recurrent Neural Networks with Reinforcement Learning.” _Thesis_ :410–420URL http://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2008112111/2/E-Diss839{%}7B{_}{%}7Dthesis.pdf{%}5Cnhttp://arxiv.org/abs/1611.02796.
* Johnson (2017) Johnson, D. D. 2017. “Generating Polyphonic Music Using Tied Parallel Networks.” In J. Correia, V. Ciesielski, and A. Liapis, (editors) _Computational Intelligence in Music, Sound, Art and Design: 6th International Conference, EvoMUSART 2017, Amsterdam, The Netherlands, April 19–21, 2017, Proceedings_. Cham: Springer International Publishing, pp. 128–143. URL http://dx.doi.org/10.1007/978-3-319-55750-2{_}9.
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* Whorley and Conklin (2016) Whorley, R. P., and D. Conklin. 2016. “Music Generation from Statistical Models of Harmony.” _Journal of New Music Research_ 45(2):160–183. URL https://doi.org/10.1080/09298215.2016.1173708.
|
# Finite transverse conductance in topological insulators under an applied in-
plane magnetic field
Dhavala Suri Tata Institute of Fundamental Research, Hyderabad-500046, India
Abhiram Soori Corresponding author<EMAIL_ADDRESS>School of Physics,
University of Hyderabad, C. R. Rao Road, Gachibowli, Hyderabad-500046, India
###### Abstract
Recently, in topological insulators (TIs) the phenomenon of planar Hall effect
(PHE) wherein a current driven in presence an in-plane magnetic field
generates a transverse voltage has been experimentally witnessed. There have
been a couple of theoretical explanations of this phenomenon. We investigate
this phenomenon based on scattering theory on a normal metal-TI-normal metal
hybrid structure and calculate the conductances in longitudinal and transverse
directions to the applied bias. The transverse conductance depends on the
spatial location between the two NM-TI junctions where it is calculated. It is
zero in the drain electrode when the chemical potentials of the top and the
bottom TI surfaces ($\mu_{t}$ and $\mu_{b}$ respectively) are equal. The
longitudinal conductance is $\pi$-periodic in $\phi$-the angle between the
bias direction and the direction of the in-plane magnetic field. The
transverse conductance is $\pi$-periodic in $\phi$ when $\mu_{t}=\mu_{b}$
whereas it is $2\pi$-periodic in $\phi$ when $\mu_{t}\neq\mu_{b}$. As a
function of the magnetic field, the magnitude of transverse conductance
increases initially and peaks. At higher magnetic fields, it decays for angles
$\phi$ closer to $0,\pi$ whereas oscillates for angles $\phi$ close to
$\pi/2$. The conductances oscillate with the length of the TI region. A finite
width of the system makes the transport separate into finitely many channels.
The features of the conductances are similar to those in the limit of
infinitely wide system except when the width is so small that only one channel
participates in the transport. When only one channel participates in
transport, the transverse conductance in the region $0<x<L$ is zero for
$\mu_{t}=\mu_{b}$ and the transverse conductance in the region $x>L$ is zero
even for the case $\mu_{t}\neq\mu_{b}$. We understand the features in the
obtained results.
## I Introduction
In the last few decades, novel materials such as topological insulators (TIs)
and Weyl semimetals which exhibit nontrivial electrical properties stemming
from the topology of their bandstructures were predicted and realized Qi and
Zhang (2011); Hasan and Kane (2010); Yan and Felser (2017); Armitage _et al._
(2018). Under an external magnetic field, a current driven results in
development of a voltage transverse to the current in the plane of magnetic
field and current, and this phenomenon is called planar Hall effect (PHE). PHE
along with negative longitudinal magnetoresistance has been seen as a direct
signature of chiral anomaly in Weyl semimetals Burkov (2017); Nandy _et al._
(2017); Kumar _et al._ (2018). PHE has also been observed in TIs Taskin _et
al._ (2017); Rakhmilevich _et al._ (2018); He _et al._ (2019); Bhardwaj _et
al._ (2021) and its origin is ascribed to spin-flip scattering of surface
electrons from impurities. Another explanation of PHE comes from the tilting
of the Dirac cone that describes the surface states of the TIs Zheng _et al._
(2020). Also there has been an attempt at explaining PHE emanating from the
bulk states of the TI Nandy _et al._ (2018). It is interesting to note that
PHE in TIs was predicted by considering scattering at junction of TIs with a
ferromagnet in proximity to one part of the TI surface Scharf _et al._
(2016), without the need of either the scattering from impurities or the
titling of the Dirac cones due to magnetic field. But a TI has two surfaces-
one on top and another at bottom, as a result, it is not clear whether the
transverse deflections of the incident electrons will cancel from the two
surfaces. Motivated by these developments, we examine transport in a system of
in-plane magnetic field applied to top and bottom surfaces of a TI connected
to two-dimensional normal metal (NM) leads on either sides. We follow Landuer-
Büttiker approach Landauer (1957); Büttiker _et al._ (1985); Datta (1995) and
calculate currents in transverse and longitudinal directions in response to a
bias applied in the longitudinal direction. This is in contrast to the
experiments where a current is driven in longitudinal direction and voltages
developed in transverse and longitudinal directions are measured in Hall bar
geometry. Also, we study the effect of unequal chemical potentials on the top
and the bottom surfaces of TI which can be achieved in experiments by applying
different gate voltages to the two surfaces. Finally, we study the case of
finite width of the sample.
Figure 1: Schematic diagram of the setup: the topological insulator (TI) is
connected to normal metal (NM) leads on either sides. The two NM’s and the TI
are taken to be infinitely long along $y$. Both the NM leads are semi-infinite
along $x$. A voltage $V$ applied from left NM to the right NM results in a
current $I$. Planar Hall effect is when the current $I$ has a nonzero
component along $y$ direction. Such a transverse current could be either in
the TI region or in the right NM region.
The paper is organized as follows. In sec. II, the system under consideration
and details of the calculation comprising of the Hamiltonian, the boundary
conditions and the formulae for the longitudinal and the transverse
conductances are discussed. In sec. III, the results are presented and
analyzed. In sec. IV, we discuss the implications of our results and conclude.
## II Details of calculation
The setup under study is a NM-TI-NM junction, with the TI in the middle having
a top surface and a bottom surface as shown in the Fig. 1. We shall take both
the NMs and the TI to be of length $L_{y}$ along $y$. The NM lead on the left
extends all the way from $x=-\infty$ to $x=0$ and makes a junction with both
the surfaces of TI along $x=0$. TI extends from $x=0$ to $x=L$ and makes a
junction with the NM on the right along the line $x=L$. From now on, we shall
denote the coordinates of the top (bottom) surface of the TI with a subscript
$t$ ($b$). The in-plane magnetic field applied is present only in the TI
region. The NM lead on the right extends from $x=L$ to $x=\infty$. The
Hamiltonian describing the system being investigated is
$\displaystyle H$ $\displaystyle=$
$\displaystyle\Big{[}-\frac{\hbar^{2}}{2m}\Big{(}\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\Big{)}-\mu_{N}\Big{]}\sigma_{0},{\rm~{}~{}for~{}}x<0{~{}\rm
and~{}}x>L,$ (1) $\displaystyle=$ $\displaystyle i\hbar
v_{F}\Big{(}\sigma_{y}\frac{\partial}{\partial
x_{t}}-\sigma_{x}\frac{\partial}{\partial
y_{t}}\Big{)}+(b_{x}\cdot\sigma_{x}+b_{y}\cdot\sigma_{y})-\mu_{t}\sigma_{0},{\rm~{}~{}for~{}}0<x_{t}<L,$
$\displaystyle=$ $\displaystyle-i\hbar
v_{F}\Big{(}\sigma_{y}\frac{\partial}{\partial
x_{b}}-\sigma_{x}\frac{\partial}{\partial
y_{b}}\Big{)}+(b_{x}\cdot\sigma_{x}+b_{y}\cdot\sigma_{y})-\mu_{b}\sigma_{0},{\rm~{}~{}for~{}}0<x_{b}<L.$
Here, $\mu_{N}$ is the chemical potential of the NM leads, $\mu_{t/b}$ is the
chemical potential on the top/bottom surface of the TI which can be controlled
by applied gate voltages, $(b_{x},b_{y})=b(\cos\phi,\sin\phi)$ where $\phi$ is
the angle the in-plane magnetic field makes with $x$-axis (we refer to the
Zeeman energy $b$ as magnetic field), $\sigma_{0}$ is identity matrix and
$\sigma_{x,y}$ are Pauli spin matrices. The Hamiltonians for the top and the
bottom surfaces have a relative minus sign for the following reason. The two
surfaces are part of the same TI and are connected at the boundaries. If
$L_{y}$ is the length of the TI in $y$-direction, we can think of the top and
the bottom surfaces of TI as a single TI surface described by the Hamiltonian
$i\hbar v_{F}(\sigma_{y}\partial_{x}-\sigma_{x}\partial_{y})$ along with the
periodic boundary conditions: $x=0\equiv 2L$ and $y=0\equiv 2L_{y}$. The
coordinates of the top and the bottom surfaces are $x_{t}=x$ in the range
$0<x<L$, $x_{b}=2L-x$ in the range $L<x<2L$, $y_{t}=y$ in the range
$0<y<L_{y}$ and $y_{b}=2L_{y}-y$ in the range $L_{y}<y<2L_{y}$, which imply
$\partial_{x_{b}}=-\partial_{x}$ and $\partial_{y_{b}}=-\partial_{y}$ leading
to the relative minus sign. This can also be shown starting from the bulk four
band Hamiltonian Udupa _et al._ (2018). Though the bulk TI Hamiltonian has
four bands, two coming from spin and another two coming from bipartite nature
of the underlying lattice, the magnetic field couples only to the spin through
Zeeman coupling resulting in the term $(b_{x}\sigma_{x}+b_{y}\sigma_{y})$. We
have chosen the gauge for the vector potential so that it is zero in $(x,y)$
plane: $\vec{A}=(0,0,b_{x}y-b_{y}x)$. The in-plane magnetic field shifts the
Dirac point of the top (bottom) surface to $\vec{k}=\pm(b_{y},-b_{x})/\hbar
v_{F}$ respectively. The dispersion relations for the top and the bottom TI
surfaces are respectively
$\displaystyle E$ $\displaystyle=$ $\displaystyle-\mu_{t}\pm\sqrt{(\hbar
v_{F}k_{x}-b_{y})^{2}+(\hbar v_{F}k_{y}+b_{x})^{2}},~{}~{}~{}~{}$ (2)
$\displaystyle E$ $\displaystyle=$ $\displaystyle-\mu_{b}\pm\sqrt{(\hbar
v_{F}k_{x}+b_{y})^{2}+(\hbar v_{F}k_{y}-b_{x})^{2}}~{}.~{}~{}~{}$ (3)
To solve the scattering problem, boundary conditions need to be specified at
$x=0$ and $x=L$. Boundary conditions at NM-TI junctions have been discussed in
literature Modak _et al._ (2012); Soori _et al._ (2013); Soori (2020). The
probability current operators for the top and bottom surfaces can be shown to
be $\hat{\vec{j}}_{t}=(-v_{F}\sigma_{y},v_{F}\sigma_{x})$ and
$\hat{\vec{j}}_{b}=(v_{F}\sigma_{y},-v_{F}\sigma_{x})$ respectively. So, the
conservation of current along $x$-direction between NM and TI surfaces reads
$\displaystyle\frac{\hbar~{}{\rm
Im}[\psi^{\dagger}_{N}\partial_{x}\psi_{N}]_{x=x_{0}}}{mv_{F}}$
$\displaystyle=$
$\displaystyle-\psi^{\dagger}_{t}\sigma_{y}\psi_{t}|_{x_{t}=x_{0}}+\psi^{\dagger}_{b}\sigma_{y}\psi_{b}|_{x_{b}=x_{0}},~{}$
at both the junctions located at $x_{0}=0,~{}L$, where $\psi_{N}$ is the
wavefunction on the NM side and $\psi_{t/b}$ is the wavefunction on the
top/bottom surface of the TI. The most general boundary condition satisfying
the current conservation eq. (LABEL:eq:current-cons) is
$\displaystyle\psi_{N}$ $\displaystyle=$ $\displaystyle
c[M(-\chi_{t})\psi_{t}+M(\chi_{b})\psi_{b}],$
$\displaystyle\frac{\hbar}{mv_{F}}\partial_{x}\psi_{N}-\chi_{N}\psi_{N}$
$\displaystyle=$
$\displaystyle\frac{i}{c}\sigma_{y}\cdot\Big{[}-M(-\chi_{t})\psi_{t}+M(\chi_{b})\psi_{b}\Big{]},$
where all the wavefunctions and $\partial_{x}\psi_{N}$ are evaluated at the
junction at $x=0$. Here, $M(\chi)={\exp}[i\chi\sigma_{y}]$. We shall soon see
that the dimensionless parameters $\chi_{N}$, $\chi_{t}$ and $\chi_{b}$
quantify the strengths of the delta-function barriers infinitesimally close to
the junction from the NM-, top TI- and bottom TI- sides respectively Modak
_et al._ (2012); Sen and Deb (2012a); *sen12err. The boundary conditions at
the junction located at $x=L$ is same as eq. (LABEL:eq:bc), except that the
dimensionless parameters $\chi_{N}$, $\chi_{t}$ and $\chi_{b}$ acquire
opposite signs. A delta function barrier on the NM side of a junction results
in a wavefunction which continuous at the location of the barrier, accompanied
by a discontinuity in $\partial_{x}\psi$ proportional to the strength of the
barrier multiplied by $\psi$. Hence, $\chi_{N}$ is the strength of the barrier
on the NM side made dimensionless. On a TI surface described by the
Hamiltonian $H_{TI}=i\hbar
v_{F}(\sigma_{y}\partial_{x}-\sigma_{x}\partial_{y})+V_{0}\Delta_{l}(x-x_{0})$
(where $\Delta_{l}(x-x_{0})$ is $1$ for $x_{0}<x<x_{0}+l$ and $0$ elsewhere),
the wavefunction in the region $x_{0}<x<x_{0}+l$ obeys $i\hbar
v_{F}\sigma_{y}\partial_{x}\psi=-V_{0}\psi$ for large $V_{0}$ and has a
solution of the form $\psi(x)=\exp[iV_{0}x\sigma_{y}/(\hbar
v_{F})]\psi(x_{0})$. The delta function limit is $l\to 0$ and $V_{0}\to\infty$
so that $V_{0}l$ is a finite constant. In this limit,
$\psi(x_{0}^{+})=\exp(i\chi_{0}\sigma_{y})\psi(x_{0}^{-})$, where
$\chi_{0}=V_{0}l/(\hbar v_{F})$. So, the wavefunction of the top/bottom TI
surface across a delta function barrier is related by
$\psi(x_{0}^{+})=\exp(\pm i\chi_{0}\sigma_{y})\psi(x_{0}^{-})$. This justifies
the introduction of parameters $\chi_{t}$ and $\chi_{b}$ in the boundary
conditions eq. (LABEL:eq:bc). We shall set all the dimensionless barrier
strengths close to the junction to zero at both the junctions to allow maximal
transmission. We shall set $c=(mv_{F}/\sqrt{2m\mu_{N}})^{1/2}$ so that
transmission of normally incident electron at the junction is perfect at zero
energy in absence of a magnetic field Soori (2020).
Due to translational invariance of the system in $y$-direction, the momentum
$\hbar k_{y}$ along $y$ can be taken to be equal in all the four regions. The
component of the current along $x$ is conserved and is same anywhere. But
along $y$, $k_{y}$ is same in all the regions and the component of current
along $y$ need not be same at all $x$.
### II.1 Limit as $L_{y}\to\infty$
The wavefunction of a spin-$\sigma$ electron incident from the left NM with
energy $E$, making an angle $\theta$ with $x$-axis has the following form in
different regions (except for a multiplicative factor of $e^{ik_{y}y}$):
$\displaystyle\psi_{N}(x)$ $\displaystyle=$
$\displaystyle(e^{ik_{x}x}+r_{\sigma,\sigma}e^{-ik_{x}x})|\sigma\rangle+r_{\overline{\sigma},\sigma}e^{-ik_{x}x}|\overline{\sigma}\rangle,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{~{}\rm
for~{}}x<0,$ $\displaystyle\psi_{p}(x_{p})$ $\displaystyle=$ $\displaystyle
s_{\sigma,p,+}e^{ik_{x,p,+}x_{p}}|k_{p,+}\rangle+s_{\sigma,p,-}e^{ik_{x,p,-}x_{p}}|k_{p,-}\rangle,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{~{}\rm
for~{}}0<x_{p}<L,~{}{\rm and~{}}p=t,b,$ $\displaystyle\psi_{N}(x)$
$\displaystyle=$ $\displaystyle
t_{\uparrow,\sigma}e^{ik_{x}x}|\uparrow\rangle+t_{\downarrow,\sigma}e^{ik_{x}x}|\downarrow\rangle,~{}{\rm
for~{}}x>L,~{}$ (6)
where $\sigma=\uparrow,\downarrow$, $\overline{\sigma}$ is the spin opposite
to $\sigma$, $|\uparrow\rangle=[1,0]^{T}$, $|\downarrow\rangle=[0,1]^{T}$,
$k_{x}=\sqrt{2m(\mu_{N}+E)}\cos{\theta}/\hbar$,
$k_{y}=\sqrt{2m(\mu_{N}+E)}\sin{\theta}/\hbar$, $k_{x,p,s}$’s for $s=+,-$
correspond to the two roots for $x$-wavenumber obtained from the dispersion in
the $p$-TI surface ($p=t,b$ stand for top, bottom surfaces) as a function of
$E$ and $k_{y}$, $|k_{p,s}\rangle$ is the spinor on $p$-TI surface for
electron with wavenumber $(k_{x,p,s},k_{y})$ which can be found from the
Hamiltonian for the TI and the coefficients $r_{\sigma^{\prime},\sigma}$,
$s_{\sigma,p,s}$, $t_{\sigma^{\prime},\sigma}$ are to be determined by
matching the boundary conditions in eq. (LABEL:eq:bc) at $x_{0}=0,L$.
If $\psi_{p,\sigma}(x)$ is the wavefunction due to an $\sigma$-spin electron
incident at an angle $\theta$ at energy $E$ on $p$-TI surface at $x_{p}=x$ in
the range $0\leq x_{p}\leq L$, the current along $y$ at the location $x$ from
this wavefunction will be
$I_{\sigma,y}(E,\theta,x)=ev_{F}\sum_{p=t,b}\psi_{p,\sigma}(x)^{\dagger}\sigma_{p}\sigma_{x}\psi_{p,\sigma}(x)$,
where $e$ is electron charge, $\sigma_{t}=1$ and $\sigma_{b}=-1$. If
$[I_{x},I_{y}(x)]$ is the current flowing at $x$ in response to a voltage bias
$V$ in the bias window $(0,eV)$, the longitudinal and transverse differential
conductances are defined as $G_{xx}=dI_{x}/dV$ and $G_{yx}(x)=dI_{y}(x)/dV$
respectively. These are given by the expressions
$\displaystyle G_{xx}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2m(\mu_{N}+eV)}}{mv_{F}}G_{0}\sum_{\sigma,\sigma^{\prime}=\uparrow,\downarrow}\int_{-\pi/2}^{\pi/2}d\theta\cos{\theta}|t_{\sigma^{\prime},\sigma}|^{2},$
$\displaystyle G_{yx}(x)$ $\displaystyle=$
$\displaystyle\frac{G_{0}}{ev_{F}}\sum_{\sigma=\uparrow,\downarrow}\int_{-\pi/2}^{\pi/2}I_{\sigma,y}(eV,\theta,x)d\theta,~{}{\rm
for~{}}0<x<L,$
where $G_{0}=(e^{2}/h)\cdot(mv_{F}L_{y}/h)$ and $L_{y}$ is the length of the
system in $y$-direction. The current deflected in the transverse direction in
the right NM is same at all locations $x>L$ and the transverse differential
conductance due to this current is given by
$\displaystyle G_{yx}(x>L)$ (8) $\displaystyle=$
$\displaystyle\frac{\sqrt{2m(\mu_{N}+eV)}}{mv_{F}}G_{0}\sum_{\sigma,\sigma^{\prime}=\uparrow,\downarrow}\int_{-\pi/2}^{\pi/2}d\theta\sin{\theta}|t_{\sigma^{\prime},\sigma}|^{2}~{}.$
### II.2 Finite $L_{y}$
For a finite $L_{y}$, we take the same Hamiltonian as in eq. (1), make the
length along $y$-direction in all the regions $L_{y}$ finite and apply
periodic boundary conditions along $y$. This makes $k_{y}$ take values:
$k_{y}=n2\pi/L_{y}$, for integer $n$. The scattering problem becomes one-
dimensional and separated in channels described by $n$. At a given energy $E$,
there are a finite number of channels participating in the transport given by
$(2N+1)$, where $N=[\sqrt{2m(E+\mu_{N})}L_{y}/h]$, $[x]$ being the largest
integer less than $x$. For a given $k_{y}=n2\pi/L_{y}$ at energy $E$,
$k_{x}=\sqrt{2m(E+\mu_{N})/\hbar^{2}-k_{y}^{2}}$ and the wavefunction is given
by eq. (6) except that the wavefunction and the scattering coefficients carry
an additional channel index $n$. Transverse current $I_{\sigma,y,n}$ carried
by the channel indexed by $n$ due to an incident spin $\sigma$ electron is
$I_{\sigma,y,n}=ev_{F}(\psi_{t,n}^{\dagger}\sigma_{x}\psi_{t,n}-\psi_{b,n}^{\dagger}\sigma_{x}\psi_{b,n})$.
The longitudinal and the transverse conductances are given by
$\displaystyle G_{xx}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{h}\sum_{n=-N}^{N}\sum_{\sigma,\sigma^{\prime}}|t_{\sigma^{\prime},\sigma,n}|^{2}$
$\displaystyle G_{yx}(x)$ $\displaystyle=$ $\displaystyle
G_{0}\frac{2\pi}{L_{y}}\sum_{n=-N}^{N}\sum_{\sigma}\frac{I_{\sigma,y,n}}{ev_{F}},~{}~{}{\rm
for~{}}0<x<L,$ $\displaystyle G_{yx}(x>L)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{h}\sum_{n=-N}^{N}\sum_{\sigma,\sigma^{\prime}}\frac{k_{y}}{k_{x}}|t_{\sigma^{\prime},\sigma,n}|^{2}~{}.$
(9)
## III Results and Analysis
To obtain numerical results, we shall fix $\mu_{N}$ and $v_{F}$, and choose
other parameters as combinations of these parameters. The mass $m$ decides the
size of the Fermi wavenumber. We choose $m=0.025\mu_{N}/v_{F}^{2}$ so that the
wavenumbers on NM and TI at energy $-0.2\mu_{N}$ are equal when
$\mu_{t}=\mu_{b}=0$ and $b=0$. The length of the TI is chosen to be $L=5\hbar
v_{F}/\mu_{N}$. These are the values of the parameters unless otherwise
stated. First we will discuss the results for the case $\lim{L_{y}\to\infty}$
and deliberate upon the effect of finite $L_{y}$ at the end.
### III.1 $\mu_{t}=\mu_{b}$
Figure 2: (a) $G_{xx}$ and (b) $G_{yx}$ as functions of bias for different
angles $\phi$ made by the in-plane magnetic field with $x$-axis with
$b=0.4\mu_{N}$. The values shown in the legend for (a) and (b) are the
respective values of $\phi$ for which $G_{xx}$ and $G_{yx}(L/2)$ are plotted.
(c) $G_{xx}$ and (d) $G_{yx}$ at zero bias as functions of $\phi$ at different
values of magnetic field $b$ mentioned within the plot legend. (e) $G_{xx}$
and (f) $G_{yx}$ at zero bias as functions of magnetic field $b$ for different
$\phi$ specified in the plot legends. Parameters: $L=5\hbar v_{F}/\mu_{N}$,
$\mu_{t}=\mu_{b}=0.1\mu_{N}$ and $m=0.025\mu_{N}/v_{F}^{2}$.
First, we set $\mu_{t}=\mu_{b}=0.1\mu_{N}$ and study the dependence of
$G_{xx}$ and $G_{yx}(L/2)$ on the bias at different angles $\phi$ when the
magnitude of the magnetic field is fixed at $b=0.4\mu_{N}$ in Fig 2(a) and
Fig. 2(b) respectively. In Fig. 2(c) and Fig. 2(d) we show the dependence of
the longitudinal and the transverse conductances respectively at zero bias on
$\phi$. The slow increase in $G_{xx}$ with bias is due to increase in density
of states of incident electrons with bias. For an angle $\phi$ between
$x$-axis and the magnetic field, the Dirac cones on the TI surfaces are
displaced in $y$-direction by an amount $|b\cos{\phi}/(\hbar v_{F})|$ thereby
making the wavenumbers $k_{x,p,s}$’s in the TI region complex (when
$\cos{\phi}\neq 0$) for a range of angle of incidence $\theta$. This reduces
the transmission probabilities $|t_{\sigma,\sigma^{\prime}}|^{2}$ for larger
values of $|\cos{\phi}|$ and for larger values of $|b|$ which agrees with the
observed features of $G_{xx}$ in Fig. 2(a,c). In Fig. 2(b,d), we find that
$G_{yx}(L/2)$ is exactly zero at $\phi=0,\pm\pi/2,\pi$. When $\phi=\pm\pi/2$,
the Dirac cone is shifted along $x$-direction and the system is symmetric
under $y\to-y$ thereby giving zero total current along $y$. When $\phi=0,\pi$,
the Dirac cones in the top and bottom surfaces are displaced exactly along
$\pm y$ directions, and the currents deflected along $y$ in the top and the
bottom surfaces are equal in magnitude and opposite in sign thus giving zero
$G_{yx}$. Now, we address the question why there is a nonzero $G_{yx}(L/2)$
for a nonzero $b$ in a direction $\phi$ other than $0,\pm\pi/2,\pi$. Under a
magnetic field $(b_{x},b_{y})$, the Dirac points of the top and the bottom
surfaces are shifted to $\pm(b_{y},-b_{x})/(\hbar v_{F})$. The currents in
$y$-direction carried by the electrons incident at angles $\theta$ and
$-\theta$ on one surface do not cancel due to a finite shift of the Dirac cone
in $y$-direction. At the same time, the net current in $y$-direction carried
by the top surface and the bottom surface do not cancel despite the opposite
shifts of the two Dirac cones because the wavenumbers in $x$-direction of the
corresponding surfaces $k_{x,t,s}$ and $k_{x,b,s}$ are different. From Fig.
2(b), it can be seen that at $eV=-\mu_{t}=-\mu_{b}$, the transverse
conductance is exactly zero implying that the net current in the transverse
direction carried by the evanescent waves in the TI region is zero. The
transverse conductance $G_{yx}(x)$ is $\pi$-periodic in $\phi$ and
$G_{yx}(x>L)$ is exactly zero for the case $\mu_{t}=\mu_{b}$.
Turning to the dependence of the two conductances on $b$, in Fig. 2(e), we
find monotonic dependence of $G_{xx}$ on $|b|$ for $|\cos{\phi}|$ close to $1$
and oscillatory dependence of $G_{xx}$ on $|b|$ for $|\cos{\phi}|$ small
compared to $1$. This is because, the displacement of TI Dirac cones on in $y$
direction is by an amount proportional to $\cos{\phi}$. Nearly normal
incidences with $\theta$ close to zero contribute the most to $G_{xx}$. When
$|\cos{\phi}|$ is large, for angles of incidences $\theta$ close to zero, the
transport in TI region is diffusive characterized by a complex $k_{x,p,s}$
whose imaginary part grows in magnitude with $b$. When $|\cos{\phi}|$ is small
compared to $1$, the displacement of the TI Dirac cones along $y$-direction is
minimal. Nearly normal incidences from NM will find a real $k_{x,p,s}$ in the
TI and the transport is ballistic except for scatterings at the interfaces
which leads to interference between the forward moving and the backward moving
waves. This is the reason for oscillatory behavior of $G_{xx}$ with $b$. Under
the transformation $\phi\to\phi+\pi$, the transmission probabilities
$|t_{\sigma,\sigma^{\prime}}|^{2}$ for angles of incidence $\theta$ and
$-\theta$ get interchanged thereby making $G_{xx}$ even in $b$. In Fig. 2(f),
we plot $G_{yx}(L/2)$ versus $b$. The nonzero values of the transverse
conductance $G_{yx}$ at certain values of $\phi$ increases in magnitude with
$|b|$ for small $|b|$ since increasing value of $|b|$ gives scope for higher
asymmetry between scatterings at angles of incidence $\theta$ and $-\theta$.
But beyond a value of $|b|$, the displacement of the Dirac cone in
$y$-direction causes the wavefunction to decay into the TI (which is
particularly the case for $|\cos{\phi}|$ close to $1$), resulting in decrease
in magnitude of $G_{yx}$ with $|b|$. For values of $\phi$ such that
$|\cos{\phi}|$ is small compared to $1$, the scattering from angles of
incidence away from normal incidence centered around $\pm\theta_{b}$ which
depend on $|b|$ contribute dominantly to $G_{yx}$. The Fabry-Pérot type
interference Soori _et al._ (2012) of these modes results in oscillations in
$G_{yx}$ with $|b|$. Under $\phi\to\phi+\pi$, the displacement of each of the
Dirac cones is opposite to that before the transformation. This hints at the
reversal of sign of $G_{yx}$ upon $b\to-b$. But, since $b_{y}\to-b_{y}$ the
displacement of each of the Dirac cones along $x$ is opposite to that before
the transformation making the surface dominantly contributing to $G_{yx}$
switch under the transformation $\phi\to\phi+\pi$. Hence the displacement of
the Dirac cone along $y$-direction for the surface dominantly contributing to
$G_{yx}$ is shifted in the same direction for both choices of magnetic field
directions $\phi$ and $\phi+\pi$, making $G_{yx}$ $\pi$-periodic. Further, the
transmission probability at angle of incidence $\theta$ for $\phi$ is equal to
the transmission probability at angle $-\theta$ for $\phi+\pi$ since under
these transformations, the top and bottom surface Hamiltonians and the
respective $k_{x,p,s}$’s get interchanged [see eq.s (3)&(2)] leaving the
transport problem along $x$ unchanged. Hence, it can be seen from eq. (8) that
transverse conductance in the NM region $G_{yx}(x>L)$ reverses sign under
$\phi\to\phi+\pi$. This combined with $\pi$-periodicity of $G_{yx}$ implies
$G_{yx}(x>L)$ is zero when the two chemical potentials are equal.
Figure 3: Dependence of the zero bias transverse differential conductance on
the location in the TI region for different angles $\phi$ mentioned in the
legend with the choice of parameters: $L=5\hbar v_{F}/\mu_{N}$,
$b=0.2\mu_{N}$, $\mu_{t}=\mu_{b}=0.1\mu_{N}$ and $m=0.025\mu_{N}/v_{F}^{2}$.
To study the dependence of the transverse conductance on the location, we plot
$G_{yx}(x)$ versus $x$ in Fig. 3. We find that the magnitude of the transverse
conductance is peaked at $x=L/2$ for this choice of parameters.
### III.2 $\mu_{t}\neq\mu_{b}$
To study the conductances in this case, we choose the same set of parameters
as in the Fig. 2 except when mentioned otherwise. We choose
$\mu_{t}=-\mu_{b}=0.1\mu_{N}$. The longitudinal differential conductance shows
characteristics very similar to the ones in Fig. 2(c) except for a change in
the numerical value. Even in the case $\mu_{t}\neq\mu_{b}$, the longitudinal
conductance is still $\pi$-periodic in $\phi$. The $\pi$-periodic behavior of
longitudinal conductance can be understood as follows. Under the
transformation $\phi\to(\phi+\pi)$, $(b_{x},b_{y})\to-(b_{x},b_{y})$ and the
Dirac cones of the TI-surfaces get displaced exactly by the same magnitude but
in opposite direction away from the origin in the $(k_{x},k_{y})$ plane. The
transverse shift in opposite direction does not alter the longitudinal
conductance. Furthermore, the longitudinal shift of Dirac cones in the
opposite direction to the same extent does not alter the longitudinal
conductance because, this is minus of the longitudinal conductance when the
same bias is applied in the opposite direction before reversing the magnetic
field and the net longitudinal conductance at zero applied bias is exactly
zero. In Fig.4, we plot the transverse differential conductances at $x=L/2$
and at $x>L$ versus $\phi$.
Figure 4: Transverse differential conductance $G_{yx}(x)$ at (a) $x=L/2$ and
at (b) $x>L$ as functions of $\phi$ for different values of bias mentioned in
the legend. $\mu_{t}=-\mu_{b}=0.1\mu_{N}$, $b=0.2\mu_{N}$, $L=5\hbar
v_{F}/\mu_{N}$, and $m=0.025\mu_{N}/v_{F}^{2}$.
It is interesting to see that $G_{yx}(x)$ is $2\pi$-periodic for
$\mu_{t}=-\mu_{b}$. Also, $G_{yx}(x>L)$ is nonzero generically except at zero
bias. Also, interestingly both $G_{yx}(L/2)$ and $G_{yx}(x>L)$ are nonzero at
$\phi=0$ for this case. This is because of the displacement of the two Dirac
cones in $\pm y$-direction equally but in opposite directions does not lead to
cancellation of transverse currents at nonzero bias due to broken perfect
antisymmetry of the top-bottom surface dispersions under $y\to-y$. For a fixed
bias, the values of $G_{yx}(x)$ for $\phi$ and $\pi-\phi$ are equal in
magnitude and opposite in sign since the transverse shift of the Dirac cones
is exactly opposite for $\phi\to(\pi-\phi)$. The transverse conductance is
$\pi$-periodic only when the chemical potentials of the top and bottom
surfaces are the same since under $\phi\to\phi+\pi$, the Dirac cones of the
top and the bottom surfaces get interchanged, whereas when
$\mu_{t}\neq\mu_{b}$, under $\phi\to\phi+\pi$ the top and the bottom Dirac
cones do not get interchanged.
Figure 5: Zero bias transverse conductance as a function of the location for
different choices $\phi$ indicated in the legend for (a)
$\mu_{t}=\mu_{b}=0.5\mu_{N}$ and (b) $\mu_{t}=-\mu_{b}=0.5\mu_{N}$. Other
parameters: $L=20\hbar v_{F}/\mu_{N}$, $b=0.2\mu_{N}$ and
$m=0.025\mu_{N}/v_{F}^{2}$.
In Fig. 5, we plot $G_{yx}(x)$ versus $x$ for a longer TI region with
$L=20\hbar v_{F}/\mu_{N}$ for (a) $\mu_{t}=\mu_{b}=0.5\mu_{N}$ and (b)
$\mu_{t}=-\mu_{b}=0.5\mu_{N}$ for different choices of $\phi$ to show the
dependence of the transverse conductance on spatial location. Compared to Fig.
3, the transverse conductance oscillates more as a function of $x$ in the
range $0\leq x\leq L$ due to Fabry-Pérot interference of the modes in TI. The
relatively higher magnitude of transverse conductance in Fig. 5(a) in
comparison with that in Fig.3 is because of a higher value of
$\mu_{t}=\mu_{b}$. Further, we find that $G_{yx}(x)=G_{yx}(L-x)$ in the range
$0\leq x\leq L$ when $\mu_{t}=\mu_{b}$, whereas $G_{yx}(L^{-})=0$ always.
Let us reason out analytically why $G_{yx}(L^{-})=0$. The wavefunctions on the
two TI surfaces and on the NM side at the location $x=L$ are related by the
boundary condition eq. (LABEL:eq:bc) simplified to
$\psi_{t}+\psi_{b}=\psi_{N}/c$ and
$\psi_{t}-\psi_{b}=-dk_{x}\sigma_{y}\psi_{N}$, where $d=\hbar c/(mv_{F})$.
From these two equations, $\psi_{N}$ can be eliminated resulting in a relation
between $\psi_{t}$ and $\psi_{b}$ from which it can be shown that
$\psi_{t}^{\dagger}\sigma_{x}\psi_{t}=\psi_{b}^{\dagger}\sigma_{x}\psi_{b}$.
This means the net transverse current
$I_{\sigma,y}=ev_{F}(\psi_{t}^{\dagger}\sigma_{x}\psi_{t}-\psi_{b}^{\dagger}\sigma_{x}\psi_{b})$
due to the two surfaces at $x=L^{-}$ is zero.
As a function of length $L$, the value of the transverse current
$I_{\sigma,y}$ at given values of energy, angle of incidence $\theta$ and
spatial location (for instance at $x=L/2$) oscillates periodically due to
Fabry-Pérot type interference. But the transverse conductance which is
obtained by integrating $I_{\sigma,y}$ over $\theta$ need not be periodic in
$L$ since the periods for different $\theta$ will be different. However,
$G_{yx}$ evaluated at $x=L/2$ oscillates about $0$ as a function of length $L$
as can be seen in Fig. 6.
Figure 6: Zero bias transverse conductance evaluated at $x=L/2$ as a function
of length $L$ for $\phi=0.2\pi$, $b=0.2\mu_{N}$, $\mu_{t}=\mu_{b}=0.5\mu_{N}$
and $m=0.25\mu_{N}/v_{F}^{2}$.
### III.3 Finite $L_{y}$
Now, we turn to the case of finite $L_{y}$. The longitudinal and the
transverse conductances for this case are given by eq. (9). From these
formulae, it can be seen that as $L_{y}$ increases, the conductances draw
contributions from more number of channels and hence at large $L_{y}$ the
conductances are proportional to $L_{y}$. Hence we plot the conductances in
units of $G_{0}$ which is proportional to $L_{y}$. We first choose the
parameters: $L=5\hbar v_{F}/\mu_{N}$, $\mu_{t}=\mu_{b}=0.1\mu_{N}$,
$b=0.2\mu_{N}$, $m=0.025\mu_{N}/v_{F}^{2}$, $\phi=0.25\pi$ and study the
dependence of $G_{xx}/G_{0}$ and $G_{yx}(x=L/2)/G_{0}$ as functions of $L_{y}$
in Fig. 7. For $\mu_{t}=\mu_{b}$, the transverse conductance $G_{yx}(x>L)$ in
the region $x>L$ is zero. We can see that for large lengths, the two
conductances saturate to the respective values in the limit of
$L_{y}\to\infty$ that can be read from Fig. 2 (c,d). For
$\log_{10}[L_{y}\mu_{N}/(\hbar v_{F})]<1.461$, there is only one channel
participating in the transport and $G_{xx}/G_{0}$ increases as $L_{y}$
decreases since $G_{0}\propto L_{y}$. For the case of single channel, the
contribution to transverse current from the top and the bottom surfaces in the
region $0<x<L$ are equal and opposite when $\mu_{t}=\mu_{b}$ and hence
$G_{yx}(x)$ in this region is zero.
Figure 7: Dependence of zero bias- (a) longitudinal and (b) transverse
conductances on $L_{y}$ for the choice of parameters: $L=5\hbar
v_{F}/\mu_{N}$, $\mu_{t}=\mu_{b}=0.1\mu_{N}$, $b=0.2\mu_{N}$,
$m=0.025\mu_{N}/v_{F}^{2}$, $\phi=0.25\pi$. The saturation values of the
conductances are mentioned in the figure.
Now, we turn to the case $\mu_{t}\neq\mu_{b}$. The longitudinal conductance
shows features similar to the case $\mu_{t}=\mu_{b}$. But, the transverse
conductance in the region $x>L$ is nonzero typically. For the choice of
parameters: $L=5\hbar v_{F}/\mu_{N}$, $\mu_{t}=-\mu_{b}=0.1\mu_{N}$,
$b=0.2\mu_{N}$, $m=0.025\mu_{N}/v_{F}^{2}$, $\phi=0.25\pi$, $E=0.2\mu_{N}$, we
plot the transverse conductances at $x=L/2$ and $x>L$ as functions of $L_{y}$
in Fig. 8. A contrasting feature in this case compared to the case of
$\mu_{t}=\mu_{b}$ is that the transverse conductance at $x=L/2$ for the values
of $L_{y}$ corresponding to single channel is non-zero here. This can be
understood by the following argument. If $k_{xt},~{}k_{xb}$ are the
$x$-components of wavenumbers on top and bottom TI surfaces, their
Hamiltonians for single channel case ($k_{y}=0$) are $[(-\hbar
v_{F}k_{xt}+b_{y})\sigma_{y}+b_{x}\sigma_{x}-\mu_{t}\sigma_{0}]$ and $[(\hbar
v_{F}k_{xb}+b_{y})\sigma_{y}+b_{x}\sigma_{x}-\mu_{b}\sigma_{0}]$ respectively.
It can be seen from here that when $\mu_{t}=\mu_{b}$,
$\psi_{t}^{\dagger}\sigma_{x}\psi_{t}=\psi_{b}^{\dagger}\sigma_{x}\psi_{b}$
implying the transverse current
$I_{\sigma,y}=ev_{F}(\psi_{t}^{\dagger}\sigma_{x}\psi_{t}-\psi_{b}^{\dagger}\sigma_{x}\psi_{b})=0$.
However, when $\mu_{t}\neq\mu_{b}$, the expectation values of $\sigma_{x}$ on
the top and the bottom surfaces are not equal due to the terms
$-\mu_{t/b}\sigma_{0}$ in the two Hamiltonians leading to a nonzero value of
$I_{\sigma,y}=ev_{F}(\psi_{t}^{\dagger}\sigma_{x}\psi_{t}-\psi_{b}^{\dagger}\sigma_{x}\psi_{b})$.
Another feature of unequal chemical potentials on the two TI surfaces is that
$G_{yx}(x>L)$ is generically nonzero. But for value of $L_{y}$ corresponding
to the single channel case, $G_{yx}(x>L)=0$ since $G_{yx}(x>L)\propto k_{y}$
from eq. (9) and $k_{y}=0$ for the only channel.
Figure 8: The dependence of transverse conductances at (a) $x=L/2$ and (b)
$x>L$ on $L_{y}$ for the choice of parameters: $L=5\hbar v_{F}/\mu_{N}$,
$\mu_{t}=-\mu_{b}=0.1\mu_{N}$, $b=0.2\mu_{N}$, $m=0.025\mu_{N}/v_{F}^{2}$,
$\phi=0.25\pi$ and $E=0.2\mu_{N}$. The saturation values of the conductances
are shown in the figure.
The typical dependences of the conductances on $\phi$ for finite $L_{y}$ is
similar to those observed for the case $\lim L_{y}\to\infty$, except for a
difference in exact numerical values.
## IV Discussion and Conclusion
We have essentially studied the phenomenon of PHE in TIs with the scattering
theory approach when TI is connected to NM leads on either sides. We use the
boundary condition for the NM-TI junction obtained by demanding the current
conservation. The longitudinal and the transverse conductances are
$\pi$-periodic in $\phi$ when $\mu_{t}=\mu_{b}$. For angles $\phi$ close to
$0$ or $\pi$, the longitudinal conductance decays with magnetic field whereas
for angles $\phi$ close to $\pm\pi/2$, the longitudinal conductance decays
with the magnetic field much slowly showing a slight periodic behavior with
magnetic field strength at $\phi=\pm\pi/2$. Magnitude of the transverse
conductance first increases with magnetic field, peaks and then decreases for
angles $\phi$ close to but not equal to $0$ or $\pi$ whereas oscillates after
an initial monotonic increase for angles close to $\pm\pi/2$. Such
oscillations are rooted in Fabry-Pérot type interference of the modes in the
TI region between the two NM-TI junctions. The transverse conductance depends
on the spatial location and is zero in the right NM lead when
$\mu_{t}=\mu_{b}$. When $\mu_{t}=-\mu_{b}$, the transverse conductance is
nonzero though small in magnitude in the right NM region. We also find that
when the width of the system $L_{y}$ is finite the scattering problem reduces
to a one dimensional problem separated into a finite number of channels and
the conductances depend on the width of the system. The transverse conductance
in the limit of small $L_{y}$ corresponding to a single channel, is zero at
$x>L$ always whereas is zero in the region $0<x<L$ when $\mu_{t}=\mu_{b}$. The
differential gating of the top and the bottom surfaces of the TI can be
experimentally achieved which means $\mu_{t}$ and $\mu_{b}$ can be separately
controlled Taskin _et al._ (2017). While many features in our results
qualitatively agree with the experimental findings of Taskin et al. Taskin
_et al._ (2017), the angular dependence of the transverse resistance for the
case of differentially gated top and bottom surfaces of the TI, the dependence
of the conductances on the magnetic field strength and the dependence on width
of the sample $L_{y}$ need to be probed experimentally.
###### Acknowledgements.
Authors thank Bertrand Halperin, Karthik V. Raman and Archit Bhardwaj for
useful discussions. AS thanks DST-INSPIRE Faculty Award (Faculty Reg. No. :
IFA17-PH190) for financial support.
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|
# Edge-Featured Graph Attention Network
Jun Chen
School of Software
Shanghai Jiao Tong University
Shanghai, China
<EMAIL_ADDRESS>
Haopeng Chen
School of Software
Shanghai Jiao Tong University
Shanghai, China
<EMAIL_ADDRESS>
###### Abstract
Lots of neural network architectures have been proposed to deal with learning
tasks on graph-structured data. However, most of these models concentrate on
only node features during the learning process. The edge features, which
usually play a similarly important role as the nodes, are often ignored or
simplified by these models. In this paper, we present edge-featured graph
attention networks, namely EGATs, to extend the use of graph neural networks
to those tasks learning on graphs with both node and edge features. These
models can be regarded as extensions of graph attention networks (GATs). By
reforming the model structure and the learning process, the new models can
accept node and edge features as inputs, incorporate the edge information into
feature representations, and iterate both node and edge features in a parallel
but mutual way. The results demonstrate that our work is highly competitive
against other node classification approaches, and can be well applied in edge-
featured graph learning tasks.
## 1 Introduction
In many real-world applications, data are best constructed as graphs to
analyze and display. The graph is such a natural structure whose nodes and
edges can be used to characterize the entities and their inner-relationships
among data. Recently, several works have defined neural networks on
graphszhou2018graph ; zhang2018deep . Kipf et al.kipf2016semi proposed graph
convolutional networks, namely GCNs, based on spectral graph theory.
Veličković et al.velivckovic2017graph presented graph attention networks
(GATs), which aggregate features following a self-attention strategy. Though
such graph neural networks have been proven successful in some of node
classification tasks, they still have obvious shortcomings. One of these
networks’ major problems is that the edge features are not incorporated into
the models. In fact, most of the current state-of-the-art graph neural
networks have not consider the edge features.
However, edges with their features play an essential role in many real-world
node classification tasks. For example, in a trading network, the node labels
may be highly relevant to the transactions. In such a case, the information
contained in edges may have a more significant contribution to the
classification accuracy compared with node features. Actually, different
graphs have different preferences for features of nodes and edges, whereas all
existing GNNs ignore or shun this fact.
In this paper, we proposed edge-featured graph attention networks (EGATs) to
address the above challenges. This work can be regarded as an extension of
GATs. To exploit the edge features effectively, we enhance the original
attention mechanism; thus, the edge information can be an important factor in
attention-weight computing. Further, the structures and learning processes of
traditional attention models are also redesigned in our work, so the models
can accept both node and edge features and iterate them individually. The
updating of edge features is necessary and should be node-equivalent, because
an iterative consistency between nodes and edges should be kept during the
learning. Besides, a multi-scale merge strategy, which concatenates features
from different iterations, is also adopted in our work. All node and edge
features will be gathered in the final layer so that the model can learn the
necessary characteristics which benefit the classification from various
scales.
To our best knowledge, we are the first to incorporate edges into GATs as
equivalent entities like nodes, point out graphs’ different different
preferences for features, and handle them spontaneously within the models. Our
models can be applied to graphs with discrete and continuous features for both
nodes and edges, which can satisfy the demands of many real-world node
classification tasks.
## 2 Related work
Graph neural networks (GNNs)scarselli2008graph first extended neural network
architectures to graphs. As a result, many related works were sprung up
ensuingly. Spectral approaches, established on spectral graph theory, are
among the most critical parts of these works. Bruna et al.bruna2013spectral
first defined convolution operations on Fourier domain. However, the filters
this model computed are non-spatially localized, so Henaff et
al.henaff2015deep improved it by generating spatially localized filters. Kipf
et al.kipf2016semi proposed graph convolutional networks (GCNs), which
further simplified above methods by using a first-order approximation on the
Chebyshev polynomials. Unlike GCNs, Veličković et al.velivckovic2017graph
proposed graph attention networks (GATs) to dynamically aggregate node
features. Numerous variants have been derived from such design. Wang et
al.wang2019heterogeneous introduced heterogeneous graph attention networks
(HANs) to process various heterogeneous graphs. Ma et al.ma2019disentangled
put forward disentangled graph convolutional networks using a routing
algorithm. Several other works also made efforts to learn graph
representations. GraphSAGEhamilton2017inductive generates node embeddings by
aggregating node features using several pre-defined aggregate operations.
Inspired by RNNs like LSTMhochreiter1997long and GRUcho2014learning , gated
graph neural networksli2015gated , namely GGNNs, were proposed. Furthermore,
Xu et al.xu2018representation explored jumping knowledge networks, in which
layer aggregation is adopted to acquire multi-scale features.
However, all these graph neural networks have a common characteristic: focus
on node rather than edge features. Only very few works have tried to integrate
edge features into GNN architecture. Schlichtkrull et
al.schlichtkrull2018modeling proposed an extension architecture of GCNs named
R-GCNs. Gong et al.gong2019exploiting presented a framework that augments
GCNs and GATs with edges. However, such approaches are somewhat not as
reasonable as they are. We will highlight their limitations in the next
section.
## 3 Motivation
The demands of processing edge-featured graphs are quite common in real-world
tasks. For example, if there is a need to find users who may have illegal
behaviors in a trading network, it is better to use some node classification
approaches to pick them up. Evidently, one user is suspicious or not is highly
relevant to the amount he paid or received some time. In other words, the edge
features are likely to have a more significant impact on classification than
node features under such a situation. However, traditional GNNs cannot handle
these graphs in a direct, elegant, and reasonable way. There may be some
doubts that if it is possible to convert graphs to which current models can
readily accept. Obviously, ignoring edge features is unacceptable. Using pre-
defined aggregate functions to integrate edge features into nodes may be a
better solution. We do not deny it may perform well on certain graphs, whereas
it is not a panacea suitable for every condition, for the selection of
function is highly dependent on graphs’ traits. It is more like feature
engineering, rather than a universal approach.
Only a few works exploit edge features in graph neural networks, and all of
them have obvious limitations. Schlichtkrull et al.schlichtkrull2018modeling
proposed R-GCNs to process modeling relational data. However, the models can
accept graphs only when their edges are labeled, which indicates that the
edges cannot include continuous attributes. Gong et al.gong2019exploiting
presented a framework enhances GCNskipf2016semi and GATsvelivckovic2017graph
. This framework can accept continuous attributes of edges, whereas it merely
regards them as weights between different node pairs. In most cases, it is
somewhat unreasonable. For example, a special graph can be constructed, with
the same features for nodes and different features for edges. If we consider
edge features as weights and all weights of each node sum as one, it is
interesting to find that no matter how the node features update, it will
remain unchanged during the learning process.
The above phenomenon shows a fact, which has never been discussed in recent
researches, that different graphs may have different preferences for node and
edge features. To those graphs that edges possess a great impact, it is
infelicitous to treat edge features as weights or labels. However, all
existing works have ignored such a key fact. Our work’s motivation is not only
to integrate edge features into GATs but also to propose general models that
spontaneously handle such preferences. To our best knowledge, we are the first
to try to solve such problems. It should be emphasized that we do not want to
present disparate models against state-of-the-art approaches. Since the
attention mechanism has proved itself in lots of tasks, it is unnecessary to
propose a completely new one. Our work is an extension of
GATsvelivckovic2017graph , and all the improvements we made are served for our
motivation.
## 4 The proposed model
### 4.1 EGAT layer overview
A single EGAT layer contains two different blocks: node attention block and
edge attention block. Each EGAT layer is designed in a symmetrical scheme;
thus, the node and edge features can update themselves in a parallel and
equivalent way. Figure 1 (a) gives an illustration of the EGAT layer.
Each EGAT layer accepts a set of node features,
$\textbf{H}=\\{\vec{h}_{1},\vec{h}_{2}\dots,\vec{h}_{N}\\}$,
$\vec{h}_{i}\in\mathbb{R}^{F_{H}}$, as well as a set of edge features,
$\textbf{E}=\\{\vec{e}_{1},\vec{e}_{2}\dots,\vec{e}_{M}\\}$,
$\vec{e}_{p}\in\mathbb{R}^{F_{E}}$, as inputs. $N$ and $M$ represent the
number of nodes and edges, while $F_{H}$ and $F_{E}$ symbolize the number of
their respective features. After processing, the layer will produce high-level
outputs, which include a new set of node features,
$\textbf{H}^{\prime}=\\{\vec{h}_{1}^{\prime},\vec{h}_{2}^{\prime}\dots,\vec{h}_{N}^{\prime}\\}$,
$\vec{h}_{i}^{\prime}$ $\in\mathbb{R}^{F_{H}^{\prime}}$, and a new set of edge
features,
$\textbf{E}^{\prime}=\\{\vec{e}_{1}^{\prime},\vec{e}_{2}^{\prime}\dots,\vec{e}_{M}^{\prime}\\}$,
$\vec{e}_{p}^{\prime}$ $\in\mathbb{R}^{F_{E}^{\prime}}$.
The cardinality of $F$ and $F^{\prime}$ may be different (whether the $F$ is
$F_{H}$ or $F_{E}$), since the linear transformations performed on the node
and edge features are not same. We use two learnable matrices,
$\textbf{W}_{H}\in\mathbb{R}^{F_{H}\times F_{H}^{\prime}}$, and
$\textbf{W}_{E}\in\mathbb{R}^{F_{E}\times F_{E}^{\prime}}$, to achieve such
transformations. For each node $i$ and edge $p$, their transformed features
can be computed by $\vec{h}_{i}^{*}=\mathbf{W}_{H}\vec{h}_{i}$, and
$\vec{e}_{p}^{*}=\mathbf{W}_{E}\vec{e}_{p}$, respectively. Then, both of them
will be fed into the node attention block and the edge attention block, which
individually producing the new sets of node and edge features. Moreover, the
adjacency and mapping matrices of nodes and edges will also be injected into
the two blocks for ancillary computation. For simplicity, we will re-use some
_symbols_ , which include H, E, $\vec{h}_{i}$, and $\vec{e}_{p}$. These
symbols will have new meanings that characterize the features transformed by
linear transformations in the rest of the paper.
Figure 1: (a) An illustration of one EGAT layer. It accepts H and E as inputs,
and produces two sets of new features. $\textit{A}_{H}$ and $\textit{A}_{E}$
are adjacency matrices, while $\textit{M}_{H}$ and $\textit{M}_{E}$ are
mapping matrices for nodes and edges, respectively. The $\textit{H}_{m}$ is
edge-integrated node features generated by node attention block, which will
only be used in the merge layer; (b) The architecture of EGATs. The model is
constructed by several EGAT layers and a merge layer. The node and edge
features generated from each iteration will be concatenated in the merge layer
to achieve a multi-scale feature fusion. For convenience, the adjacency and
mapping matrices are not shown in this figure, as well as the multi-head
attention.
### 4.2 Node attention block
The node attention block accepts H, a set of node features, and E, a set of
edge features, and produces $\textbf{H}^{\prime}$, a new set of node features.
In E, the edge features are ranked in a preset order, so it is hard to find
the relations between edges and their adjacent nodes. Thus, a mapping
transformation will be first applied to E in the block, to re-organize it into
another common form $\textbf{E}^{*}$. Every element in $\textbf{E}^{*}$ can be
represented as $\vec{e}_{ij}$, while $i$ and $j$ denote the nodes on each end
of an edge. The transformation from E to $\textbf{E}^{*}$ can be realized by
matrix multiplication using an edge mapping matrix $\textbf{M}_{E}$, which is
an $N\times N\times M$ tensor. Compared with the adjacency matrix, it expands
the third dimension to indicate where each edge should be placed. Figure 2 (b)
gives a simple example about the mapping process.
Before the multiplication, the edge mapping matrix should first be reshaped
into $N^{2}\times M$ so that $\textbf{M}_{E}$ and E could have the same
dimension of 2. Eventually, the multiplication result needs to reshape back
with a size of $N\times N\times F_{E}^{\prime}$, transforming the edge set E
into the adjacency form. The edge mapping matrix is unique for a particular
graph structure with determining orders of nodes and edges so that it can be
constructed in a pre-processing step before the learning process.
Thanks to the adjacent form, the model can quickly seek out the edge between
two specified nodes. Based on that, an edge-integrated attention mechanism can
be performed on each node, generating the attention weights of its neighbors
includes not only the features of the two nodes but also the edge connecting
them. For each node $i$, the weight $w_{ij}$ will be computed for every
$j\in\mathcal{N}_{i}$, where the $\mathcal{N}_{i}$ is the set including the
first-order neighbors of node $i$ as well as the node $i$ itself. During the
process, features will be concatenated, parameterized by a weight vector
$\vec{a}$, and applied LeakyReLU as the activation function. Normalization
will also be performed on these weights across all choices of node $j$, where
$j\in\mathcal{N}_{i}$, by using a softmax function. The whole process can be
formulated as follows:
$\alpha_{ij}=\frac{{\rm exp}({\rm
LeakyReLU}(\vec{\textbf{a}}^{T}[\vec{h}_{i}\|\vec{h}_{j}\|\vec{e}_{ij}]))}{\sum_{k\in\mathcal{N}_{i}}{\rm
exp}({\rm
LeakyReLU}(\vec{\textbf{a}}^{T}[\vec{h}_{i}\|\vec{h}_{k}\|\vec{e}_{ik}]))}$
(1)
It is interesting to note that, for each node, the aggregated features include
not only the neighbors’ but also the ones of itself. Without edge features,
the problem can be solved by adding an identity matrix to the adjacency
matrix. However, the introduction of edge features makes it more difficult. In
our work, we use a tricky method by adding virtual featured self-loops to the
graph. If a node does not have an edge that connected itself, a virtual self-
loop will be attached to it. In particular, for every virtual self-loop, its
features will be computed as an average of all its adjacent edges’ features in
each dimension as a compromise. All these operations should be done before
being fed to the model.
Figure 2: Left: (a) An illustration of a regular graph; (b) An example of the
mapping transformation performed on (a). ${M}_{E}$ is the edge mapping matrix
in size of ${N}\times{N}\times{M}$, with its last dimension encoded in a one-
hot scheme. For simplicity, we draw the matrix by replacing one-hot encoding
vectors with non-zero indices. $E$ is the edge feature set with a size of
${M}\times{F}_{E}^{\prime}$. ${M}_{E}$ should first be reshaped to
${N}^{2}\times{M}$, and recover to ${N}\times{N}\times{F}_{E}^{\prime}$ after
multiplication. Right: (c) An example of graph transformation. The nodes and
edges’ roles are inversed in the new graph; (d) The adjacency matrix ${A}_{E}$
of the new graph, namely edge adjacency matrix. An identity matrix has been
added to the matrix to pre-build the self-adjacent relations; (e) The mapping
matrix ${M}_{H}$ of the new graph.
After acquiring the normalized attention weights for each neighborhood, we can
perform a weighted sum on these neighbor node features. In addition, a non-
linearity $\sigma$ will be applied to these summation results. The final
results, which is also the outputs of this node attention block, can be
expressed as:
$\vec{h}_{i}^{\prime}=\sigma(\sum\limits_{j\in\mathcal{N}_{i}}\alpha_{ij}\vec{h}_{j})$
(2)
It should be noticed that we only aggregate the node features to generate the
new set of node features. The edge features only play a part in weight
computing but not a part of the new node features. It is for the clarity and
symmetry of the model that we design such a strategy. If we merge edge
features into nodes in each iteration, all the features may tangle up together
and make the network more complicated and confusing. In fact, we also produce
the set of edge-integrated node features $\textbf{H}_{m}$ in the node
attention block. For each node $i$, we generate its new edge-integrated
features as follows:
$\vec{m}_{i}=\sigma(\sum\limits_{j\in\mathcal{N}_{i}}\alpha_{ij}(\vec{h}_{j}\|\vec{e}_{ij}))$
(3)
However, these features will only be used in the last-level merge layer to
achieve a multi-scale concatenation. They will never be passed to the next
EGAT layer as the inputs.
### 4.3 Edge Attention Block
The node features can update themselves periodically in node attention blocks
to acquire high-level features, whereas it is unreasonable to reuse the
original low-level edge features during the weight computation. Besides, we
also need high-level edge features to keep a balance of importance between
nodes and edges. Thus, we proposed edge attention blocks, each of which
accepts a set of node features, H, and a set of edge features, E, and produces
$\textbf{E}^{\prime}$, a new set of edge features.
A natural idea to realize such blocks is to update each edge’s features by
aggregating adjacent edges’ features. In undirected graphs, we consider two
edges are adjacent only if two edges have at least one common vertex. To
achieve the aggregation, we adopt a tricky approach in our work by first
switching the roles of nodes and edges in the graph. A similar concept on
directed graphs has been proposed by Chen et al.chen2017supervised for
community detection. To achieve this, we create a new graph based on the
original graph, whose nodes and edges are the edges and nodes of the original
one, respectively. The transformation of the graph and the matrices structured
by us are illustrated in Figure 2 (right).
The inputs of node and edge features are organized in the same sequential
form. Thanks to the symmetric design, we can easily perform the attention
mechanism on the new graph because the node feature set can be converted into
the adjacency form by using $\textbf{M}_{H}$, the node mapping matrix, with no
difficulty. For each edge $p$, the normalized attention weight of edge $q$ can
be expressed as:
$\beta_{pq}=\frac{{\rm exp}({\rm
LeakyReLU}(\vec{\textbf{b}}^{T}[\vec{e}_{p}\|\vec{e}_{q}\|\vec{h}_{pq}]))}{\sum_{k\in\mathcal{N}_{p}}{\rm
exp}({\rm
LeakyReLU}(\vec{\textbf{b}}^{T}[\vec{e}_{p}\|\vec{e}_{k}\|\vec{h}_{pk}]))}$
(4)
where $\mathcal{N}_{p}$ is the first-order neighbor set of edge $p$ (including
$p$), and $\vec{\textbf{b}}$ is a weight vector with a size of
$\mathbb{R}^{2F_{E}^{\prime}+F_{H}^{\prime}}$. One noteworthy point is that,
when we compute the attention weight of an arbitrary edge and the edge itself,
it is no middle node between the two edges. In our experiments, we logically
create an empty node between the two edges, by padding all the features of
this virtual node as zeros. Like node features, the computing of new set of
edge features can be represented as:
$\vec{e}_{p}^{\prime}=\sigma(\sum\limits_{q\in\mathcal{N}_{p}}\beta_{pq}\vec{e}_{q})$
(5)
### 4.4 EGAT Architecture
In this section, we present EGATs, which are constructed by stacking several
EGAT layers and appending a merge layer at the tail. The architecture of EGATs
is illustrated by Figure 1 (b).
Multi-scale strategies have been widely used to aggregate hierarchical feature
maps in CNN models. Xu et al.xu2018representation first introduced such
strategies into GNNs and proposed jumping knowledge networks, which further
improve the accuracy. Inspired by such works, we adopt a multi-scale merge
strategy by adding a merge layer in EGATs. Unlike jump knowledge networks, we
collect not only node features but also edge features. Edge features will be
integrated into nodes in each EGAT layer, result in $\textbf{H}_{m}$, which we
mentioned in 4.2. All $\textbf{H}_{m}$ generated from different iterations
will aggregate together using a concatenation operation. Besides, we adopt the
_multi-head attention_ in the merge layer to further stabilize the attention
mechanism. Unlike GATsvelivckovic2017graph , our multi-head attention is
performed on the unity of all EGAT layers rather than a single layer. $K$
independent multi-scale edge-integrated features would be computed and merged,
resulting in the feature representation as follows:
$\vec{h}_{i}^{*}=\bigparallel_{k=1}^{K}(\bigparallel_{l=1}^{L}m_{i}^{l,k})$
(6)
where $L$ indicates the number of the EGAT layers, and $m_{i}^{l,k}$
represents the edge-integrated node features of node $i$ produced in iteration
$l$ of the group $k$. To obtain a more refined representation, we apply a one-
dimensional convolution to the results as a linear transformation and a non-
linearity. For node classification tasks, a softmax function will be applied
in the end to generate predicted labels.
## 5 Experiments
We experientially assess the efficiency of EGAT by performing comparative
evaluation against state-of-the-art approaches on several node classification
datasets, which include both node-sensitive and edge-sensitive graphs.
Besides, some additional analyses are also included in this section.
### 5.1 Datasets
We conduct our experiments on five node classification tasks containing both
node-sensitive and edge-sensitive datasets. The former are graphs whose node
features highly correlate with node labels, while the latter are those whose
edges possess a dominant position. Such a division is somewhat relative, and
it does not mean the features in a weak status have no contributions to the
final results.
Node-Sensitive Graphs. Three real-world node classification datasets, which
include _Cora_ , _Citeseer_ , and _Pubmed_sen2008collective , are utilized in
our experiments for node-sensitive graph learning. Such datasets are citation
networks and have been widely used in graph learning research works as
standard benchmarks. Notably, these datasets are undirected and do not have
edge features within the graphs. For a fair comparison, in our work, we adopt
the same dataset splits used in papers of GCNskipf2016semi and
GATsvelivckovic2017graph .
Edge-Sensitive Graphs. We derive two trading networks, _Trade-B_ and _Trade-M_
, to test the effectiveness of our models on edge-sensitive graphs. The two
datasets are financial-collaborative and refer to real-world trading records.
For confidentiality, we cleaned and extracted some distinct patterns of
abnormal behaviors from the original data provided by a bank and regenerated
them as new datasets. In these datasets, each node represents a customer, with
an attribute indicating the risk level of it. The edges, however, represent
the relations among customers, whose features contain the number and total
amount of recent transactions. _Trade-B_ is a binary classification dataset,
which possesses 3907 nodes (97 of them are labeled) and 4394 edges. _Trade-M_
, however, is ternary classified, with 4431 nodes (139 of them are labeled)
and 4900 edges. For both datasets, we separated the labeled nodes into three
parts, for training, validation, and test, with a ratio of 3:1:1. The two
datasets are directed initially; however, to make it suitable for EGATs, we
converted them into an undirected form.
### 5.2 Experimental Setup
For all the experiments, we implement EGATs based on the Pytorch
frameworkpaszke2019pytorch . Because of the large memory usage for both
adjacency and mapping matrices, we convert them into sparse forms to reduce
the memory requirement and computational complexity during the learning
process. The experimental setup for node-sensitive and edge-sensitive graphs
are described as follows.
Node-Sensitive Graphs. Because all three citation networks do not possess even
an edge feature, we generate one weak topological feature for each edge, by
enumerating the numbers of its adjacent edges. In those experiments, we adopt
an EGAT model with $L$ = 2 and $K$ = 8, where $L$ and $K$ represent the number
of the EGAT layers and the attention heads. For simplicity, we use the same
numbers of the output features for every EGAT layer, where $F_{H}^{\prime}$ =
8 and $F_{E}^{\prime}$ = 4, for nodes and edges separately. A one-dimensional
convolution operation are performed in the merge layer to produce C features
(where C is the number of classes), followed by a softmax function. To improve
accuracy, some techniques like dropoutsrivastava2014dropout and $L_{2}$
regularization are also used in EGATs. All these experiments, but _Pubmed_ ,
were run on a machine with two GPUs of Geforce RTX 1080 Ti. Because of a
larger requirement on video memory, _Pubmed_ was run on Tesla V100 instead.
Edge-Sensitive Graphs. _Trade-B_ and _Trade-M_ are the two edge-sensitive
benchmarks used in our experiments. As we mentioned above, some virtual self-
loops are added to the graphs before the training. For those datasets, we
apply an EGAT model whose $L$ = 2 and $K$ = 8, with the same output feature
dimension for each EGAT layer. Differing from the node experiments, we use
three kinds of combinations of $F_{H}^{\prime}$ and $F_{E}^{\prime}$ here with
different ratios, which can be listed as 8:4, 6:6, 4:8, respectively. Besides,
all other details of this model are similar to those used for node-sensitive
graph learning. All these experiments were run on a machine with two GPUs of
Geforce RTX 1080 Ti.
### 5.3 Results
For the node-sensitive tasks, we report the classification accuracy on the
test nodes after 10 runs, which are listed in Table 1. We compare our results
against several strong baselines and state-of-the-art approaches proposed in
previous works. In particular, we re-implement a two-layer GAT
modelvelivckovic2017graph by PyTorch, namely SP-GAT*, with $F^{\prime}$, the
number of hidden units, equals to 8. For a fair comparison, SP-GAT* accepts
the same sparse representations of matrices used in our model.
The results show that EGATs are highly competitive against the state-of-the-
art models on such node-sensitive graphs. We notice that there is a slight
decrease for both _Cora_ and _Citeseer_ compared with SP-GAT*, which may be
caused by the introduction of edge features. Since we generate the feature for
each edge with the number of its adjacent edges, some interference may occur
if these features are kind of useless. However, those negative effects are
quite insignificant. Thanks to the symmetrical design, EGATs can adjust
themselves during the learning and put more concentration on these useful
features and produce acceptable results. In a word, EGATs can achieve high
accuracy in node-sensitive classification tasks, surpassing the performance of
most state-of-the-art approaches.
Table 1: Summary of the results on node classification accuracy, for Cora, Citeseer and Pubmed. SP-GAT* corresponds to the best result of GAT implemented by us with a sparse form. Method | Cora | Citeseer | Pubmed
---|---|---|---
MLP | 55.1% | 46.5% | 71.4%
ManiRegbelkin2006manifold | 59.5% | 60.1% | 70.7%
SemeiEmbweston2012deep | 59.0% | 59.6% | 71.7%
LPzhu2003semi | 68.0% | 45.3% | 63.0%
DeepWalkperozzi2014deepwalk | 67.2% | 43.2% | 65.3%
ICAlu2003link | 75.1% | 69.1% | 73.9%
Planetoidyang2016revisiting | 75.7% | 64.7% | 77.2%
Chebyshevdefferrard2016convolutional | 81.2% | 69.8% | 74.4%
GCNkipf2016semi | 81.5% | 70.3% | 79.0%
Monetmonti2017geometric | 81.7% | - | 78.8%
SP-GAT* | 82.5$\pm$0.4% | 70.8$\pm$0.5% | 78.1$\pm$0.4%
EGAT (ours) | 82.1$\pm$0.7% | 70.3$\pm$0.5% | 78.1$\pm$0.4%
Table 2: Summary of the results on node classification accuracy, for Trade-B and Trade-M. The hyper-parameters $h$ and $e$ represent $F_{H}^{\prime}$ and $F_{E}^{\prime}$ used in our EGAT model, respectively. Method | Trade-B | Trade-M
---|---|---
SP-GAT* | 65.0% | 46.4%
SP-GAT-sum* | 85.0% | 51.1%
SP-GAT-avg* | 78.0% | 71.4%
SP-GAT-max* | 81.5% | 65.7%
EGAT ($h$ = 8, $e$ = 4) | 87.5% | 84.3%
EGAT ($h$ = 6, $e$ = 6) | 88.0% | 85.4%
EGAT ($h$ = 4, $e$ = 8) | 92.0% | 78.2%
For the edge-sensitive tasks, we report the mean classification accuracy on
test nodes after 10 runs, and apply SP-GAT* and its variants to the benchmarks
as comparisons. For SP-GAT*, we only feed the original node features as
inputs. To ensure fairness, we further create three variants of SP-GAT*, by
aggregating edge features into nodes as node features in advance using
different functions, including sum, average, and max pooling. Besides, we
evaluate the accuracy by comparing three EGATs with different ratios of
$F_{H}^{\prime}$ and $F_{E}^{\prime}$. The comparative results are listed in
Table 2. EGATs show an incredible performance from the table, which is streets
ahead of other approaches on both two datasets. For _Trade-B_ and _Trade-M_ ,
the best classification accuracy of EGATs can reach 92.0% and 85.4%. It is
also interesting to observe that different datasets may possess different
characteristics. For example, the edge features within _Trade-B_ can be better
expressed by summing up together. On the contrary, the average operation may
be more applicable to representing the edge features in _Trade-M_. Despite
their traits, EGATs can achieve high accuracy against these baselines on all
these datasets, which means that EGATs can learn these characteristics of
graphs spontaneously. To our best knowledge, there are no existing approaches
that can process these kinds of graphs effectively.
We also investigate the effects of the ratio of $F_{H}^{\prime}$ and
$F_{E}^{\prime}$ on accuracy. According to the results, if the edge features
may play a more important role than node ones, we recommend choosing a small
or balance value of $F_{H}^{\prime}:F_{E}^{\prime}$ so that edges will have a
higher chance to show themselves. However, there may be exceptions in some
cases. For example, the accuracy decreased to 78.2% when we select a high
$F_{E}^{\prime}$ in _Trade-M_. Due to the mutual effect of the features of
nodes and edges, the model becomes complex, and it is hard to consider both
features separately. So, if better performance is demanded, it is better to
adjust these hyper-parameters several times to choose the most suitable ones.
### 5.4 Complexity Analysis
The complexity analysis of EGATs is given in this subsection. Since the
constructions of adjacency and mapping matrices occur in a pre-processing step
rather than the critical path, we merely ignore them and concentrate on the
learning process. In EGATs, the matrix multiplication is the most time-
consuming operation, which can be regarded as the entry point. Assume that now
we have a graph with $N$ nodes and $E$ edges. In each node attention block, we
introduce the edge features by applying a mapping transformation, which is
actually matrix multiplication. It can be easily proved that the computation
complexity of multiplying an $m\times n$ sparse matrix $A$ and an $n\times p$
dense matrix $B$ can be reduced to $O(cp)$, where $c$ denotes to the number of
non-zero elements in $A$. Thus, the complexity of such a transformation is
$O(E)$, for the reason that $p$, the number of features, can be seen as a
constant. Because the complexity of GATs is no less than O(E) in each
iteration, so the introduction of edges in node attention blocks will not
significantly increase the complexity.
Things may be a little different occurring in edge attention blocks. Consider
a graph with one central node and $K$ neighbors. Because every two edges are
neighbors, when we switch the roles of nodes and edges, the number of edges in
the new graph, $E^{*}$, is on the order of $O(K^{2})$. When we extend this
conclusion to a regular graph with $N$ nodes and $E$ edges, the number of non-
zero elements in the converted mapping matrix can be represented as
$O(\sum_{i=1}^{N}{d_{i}^{2}})$, where $d_{i}$ indicate the degree of each node
in the graph. Thus, the complexity of edge attention block is in the same
order. However, based on our experimental results, the delay of EGATs is quite
acceptable on the benchmarks and those graphs with a similar scale. Besides,
all the test datasets except _Pubmed_ can run on Geforce RTX 1080 Ti without
exceeding the memory limit. If someone has higher performance requirements,
some modifications could be made in edge attention block. For example, each
edge can regard the two adjacent nodes as virtual edges and only aggregate the
edge part of $\textbf{H}_{m}$ during the learning.
## 6 Conclusions
We proposed edge-featured graph attention networks (EGATs), novel edge-
integrated graph neural networks that performed on graphs with node and edge
features. We incorporate edge features into GNNs and present a symmetrical
approach to exploit them. To our best knowledge, we are the first to
incorporate edges as node-equivalent entities, point out graphs’ different
preferences for features, and handle them spontaneously. The results
demonstrate that EGATs have successfully achieved state-of-the-art performance
on node classification tasks, especially for edge-sensitive datasets.
There are some potential improvements to EGATs that could be addressed as
future work. In EGATs, we update each edge’s features by aggregating its
neighbors’ information. However, the number of neighbors of each edge may be
huge. Despite transforming matrices into sparse forms, the models still need a
large memory usage when it performed on large-scale graphs. Thus, it is better
to find an improved way to reduce the models’ memory requirement. Besides,
EGATs do not naturally support directed graphs as well as multi-graphs. We
intend to achieve these extensions further on.
## Broader Impact
In this work, edge-featured graph attention networks (EGATs), novel edge-
integrated graph neural networks, were proposed to perform node classification
on those graphs with node and edge features. This work has the following
potential positive impact on society. First, the models proposed in this paper
are kind of versatile and have a broad application foreground in many fields.
For example, the models can be applied in the financial sector, as an aid to
finding those people who may have a suspicion in financial fraud, money
laundering, etc. Second, given the absence of edge features in traditional GNN
approaches, our work may attract the attention of other researchers, spawning
a series of related research, to enhance further the basic framework of graph
neural networks from a theoretical level. At the same time, this works may
have some negative consequences with a small probability. Because there are
very few works trying to exploit edge features in graph neural networks, this
field is still kind of immature and receives little attention. Therefore, the
negative impact of our models on society are quite unclear and need further
exploration. Besides, we should be cautious about the result of the failure of
the system. It should be noticed that the prediction results of our models
should only be regarded as an auxiliary reference rather than a definite
truth. Users of the models should perform a second manual verification by
their own to ensure the authenticity of the results. We will not be
responsible for the negative effects caused by the wrong prediction of EGATs.
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|
# The Six Hug Commandments: Design and Evaluation of a Human-Sized Hugging
Robot with Visual and Haptic Perception
Alexis E. Block 0000-0001-9841-0769 MPI-IS and ETH ZürichStuttgartGermany ,
Sammy Christen 0000-0002-3511-8565 ETH ZürichZürichSwitzerland , Roger
Gassert 0000-0002-6373-8518 ETH ZürichZürichSwitzerland , Otmar Hilliges
0000-0002-5068-3474 ETH ZürichZürichSwitzerland and Katherine J. Kuchenbecker
0000-0002-5004-0313 MPI for Intelligent SystemsStuttgartGermany
(2021)
###### Abstract.
Receiving a hug is one of the best ways to feel socially supported, and the
lack of social touch can have severe negative effects on an individual’s well-
being. Based on previous research both within and outside of HRI, we propose
six tenets (“commandments”) of natural and enjoyable robotic hugging: a
hugging robot should be soft, be warm, be human sized, visually perceive its
user, adjust its embrace to the user’s size and position, and reliably release
when the user wants to end the hug. Prior work validated the first two tenets,
and the final four are new. We followed all six tenets to create a new robotic
platform, HuggieBot 2.0, that has a soft, warm, inflated body (HuggieChest)
and uses visual and haptic sensing to deliver closed-loop hugging. We first
verified the outward appeal of this platform in comparison to the previous
PR2-based HuggieBot 1.0 via an online video-watching study involving 117
users. We then conducted an in-person experiment in which 32 users each
exchanged eight hugs with HuggieBot 2.0, experiencing all combinations of
visual hug initiation, haptic sizing, and haptic releasing. The results show
that adding haptic reactivity definitively improves user perception a hugging
robot, largely verifying our four new tenets and illuminating several
interesting opportunities for further improvement.
††journalyear: 2021††copyright: acmlicensed††doi:
10.1145/3434073.3444656††conference: Proceedings of the 2021 ACM/IEEE
International Conference on Human-Robot Interaction; March 8–11, 2021;
Boulder, CO, USA††booktitle: Proceedings of the 2021 ACM/IEEE International
Conference on Human-Robot Interaction (HRI ’21), March 8–11, 2021, Boulder,
CO, USA††price: 15.00††isbn: 978-1-4503-8289-2/21/03††submissionid:
hrifp1096††ccs: Computer systems organization Robotics
## 1\. Introduction
Figure 1. A user hugging HuggieBot 2.0.
Hugging has significant social and physical health benefits for humans. Not
only does a hug help lower blood pressure, alleviate stress and anxiety, and
increase the body’s levels of oxytocin, but it also provides social support,
increases trust, and fosters a sense of community and belonging (Cohen et al.,
2015). Social touch in a broader sense is also vital for maintaining many
kinds of relationships among humans and primates alike (Suvilehto et al.,
2015); hugs seem to be a basic evolutionary need. They are therefore highly
popular! An online study conducted in 2020 polled 1,204,986 people to find out
“what is the best thing?” Hugs earned fifth place out of 8,850 things, behind
only sleep, electricity, the Earth’s magnetic field, and gravity (Scott,
2020). The absence of social touch can have detrimental effects on child
development (Cascio et al., 2019). Unfortunately, ever more interactions are
happening remotely and online, especially during this unprecedented time of
physical distancing due to COVID-19. An increasing number of people are
suffering from loneliness and depression due to increased workload and
population aging (Neira and Barber, 2014; Morrison and Gore, 2010). Our long-
term research goal is to determine the extent to which we can close the gap
between the virtual and physical worlds via hugging robots that provide high-
quality social touch.
Making a good hugging robot is difficult because it must understand the user’s
nonverbal cues, realistically replicate a human hug, and ensure user safety.
We believe that such robots need multi-modal perception to satisfy all three
of these goals, a target no previous system has reached. Some approaches focus
primarily on safety, providing the user with the sensation of being hugged
without being able to actively reciprocate the hugging motion (Tsetserukou,
2010; Teh et al., 2008; Duvall et al., 2016). Conversely, other researchers
focus on providing the user with an item to hug, but that item cannot hug the
user back (Sumioka et al., 2013; Stiehl et al., 2005; DiSalvo et al., 2003).
Other robotic solutions safely replicate a hug, but they are teleoperated,
meaning they have no perception of their user and require an additional person
to control the robot any time a user wants a hug (Hedayati et al., 2019;
Shiomi et al., 2017a; Yamane et al., 2017). Finally, some robots have basic
levels of perception but are not fully autonomous or comfortable (Block and
Kuchenbecker, 2019; Miyashita and Ishiguro, 2004). Section 2 further details
prior research in this domain.
To tackle the aforementioned goal of safely delivering pleasant hugs, we
propose the six tenets (“commandments”) of robotic hugging: a hugging robot
should be soft, warm, and sized similar to an adult human, and it should see
and react to an approaching user, adjust automatically to that user’s size and
position while hugging, and reliably respond when the user releases the
embrace. After presenting these tenets and our accompanying hypotheses in
Section 3, we use the tenets to inform the creation of HuggieBot 2.0, a novel
humanoid robot for close social-physical interaction, as seen in Fig. 1 and
described in Section 4. HuggieBot 2.0 uses computer vision to detect an
approaching user and automatically initiate a hug based on their distance to
the robot. It also uniquely models hugging after robot grasping, using two
slender padded arms, an inflated torso, and haptic sensing to automatically
adjust to the user’s body and detect user hug initiation and termination.
HuggieBot 2.0 is the first human-sized hugging robot with visual and haptic
perception for closed-loop hugging.
We then seek to validate the four new tenets by conducting two experiments
with HuggieBot 2.0. First, we confirmed user preference for the created
platform’s physical size, visual appearance, and movements through a
comparative online study, as described in Section 5. We then conducted an in-
person study (Section 6) to understand how HuggieBot 2.0 and its three new
perceptual capabilities (vision, sizing, and release detection) affect user
opinions. Section 7 discusses the study results, which show that the six
tenets significantly improve user perception of hugging robots. Section 8
discusses the limitations of our approach and concludes the paper.
## 2\. Related Work
### Using Vision for Person Detection
One challenge of accurate and safe robotic hugging is detecting a user’s
desire for a hug. Many researchers solve this problem by using a remote
operator to activate the hug (DiSalvo et al., 2003; Yamane et al., 2017;
Yamazaki et al., 2016; Sumioka et al., 2013). This form of telehug is not a
universal approach because it requires a hugging partner to be available at
the exact moment a user would like the comfort of a hug. Having the user press
a button is a simpler alternative but differs greatly from human-human
hugging. One method that could allow robots to provide hugs autonomously is to
detect an approaching user via computer vision. Human detection has long been
of interest in many research fields, including autonomous driving (Yurtsever
et al., 2020), surveillance and security (Paul et al., 2013), computer vision
(Viola and Jones, 2004), and human-robot interaction (Vo et al., 2014). Early
works focus mostly on finding a representative feature set that distinguishes
the humans in the scene from other objects. Different methods for the feature
extraction have been proposed, such as using Haar wavelets (Oren et al.,
1997), histograms of oriented gradients (HOG) (Dalal et al., 2006), and
covariance matrices (Tuzel et al., 2007). Several approaches try to combine
multiple cues for person detection; for example, Choi et al. (Choi et al.,
2011) and Vo et al. (Vo et al., 2014) combine the Viola-Jones face detector
(Viola and Jones, 2001) and an upper-body detector based on HOG features
(Dalal et al., 2006). In computer vision, deep-learning-based systems relying
on convolutional neural networks (CNNs) are often used for person detection.
However, the computational cost of these detection pipelines is very high, so
they are not suited for use on a real-time human-robot interaction platform.
The recent decrease of the computational cost of improved models, e.g., (Liu
et al., 2016), has facilitated their adaptation to robot platforms. Hence, we
integrate such a model into our pipeline so that HuggieBot 2.0 can recognize
an approaching user to initiate a hug with minimal on-board computational
load.
### Hugging as a Form of Grasping
Once the user arrives, safely delivering a hug is challenging for robots
because users come in widely varying shapes and sizes and have different
preferences for hug duration and tightness. No existing hugging robots are
equipped to hug people adaptively. We propose looking to the robotics research
community to find a solution. Grasping objects of varying shape, size, and
mechanical properties is a common and well-studied problem, e.g., (Romano et
al., 2011; Ma et al., 2016; Costanzo et al., 2020; Barber et al., 1986).
Therefore, we look at hugging as a large-scale two-fingered grasping problem,
where the item to be grasped is a human body. For example, the BarrettHand, a
commercially available three-fingered gripper, automatically adjusts to
securely grasp widely varying objects by closing all finger joints
simultaneously and then stopping each joint individually when that joint’s
torque exceeds a threshold (Townsend, 2000).The robot arms used for HuggieBot
2.0 have torque sensors at every joint, making this torque-thresholded
grasping method an ideal way to achieve a secure embrace that neither leaves
air gaps nor applies excessive pressure to the user’s body. Torque sensors can
also enable the robot to feel when the user wishes to leave the embrace.
### Previous Hugging Robots
In recent years, there has been a dramatic increase in the number of
researchers developing and studying hugging robots. High press coverage shows
this topic is of great interest to the general public. Interestingly,
researchers are taking many different approaches to create robotic systems
that can receive and/or give hugs. Smaller hugging robots have been created to
provide comfort, but they typically cannot actively hug the user back. The
Huggable is a small teddy bear to accompany children during long stays in the
hospital (Stiehl et al., 2005). The Hug is a pillow whose shape mimics a child
wrapping his or her limbs around an adult (DiSalvo et al., 2003). Hugvie is
also a pillow users can hug; a cellphone inside lets the user talk to a
partner while he or she hugs the pillow (Sumioka et al., 2013; Yamazaki et
al., 2016). Their small size makes these huggable systems inherently safer
than larger devices, but it also prevents them from providing the benefits of
social touch to the user because they cannot administer deep touch pressure
therapy (Edelson et al., 1999).
Teleoperated hugging robots have also been created, and they can be closer to
human size. Some research groups focus on non-anthropomorphic solutions like
using a large panda or teddy bear stuffed animal to hide the mechanical
components (Hedayati et al., 2019; Shiomi et al., 2017a, b). These robots all
require that either an operator or partner is available at the time any user
wants a hug. They also may not be very comfortable for the user because the
robots are unable to stand on their own; the user must crouch or crawl to get
a hug from the robot. Disney also patented a teleoperated hugging robot to be
similar to its famous character Baymax (Yamane et al., 2017), though a
physical version has not been reported. Negatively, none of these robots seem
to have the ability to adjust their embrace to the size or location of the
user. In addition to creating an appropriately sized hugging robot, all of
these researchers covered the robot’s rigid components with soft materials to
create an enjoyable contact experience for their users. In contrast, Miyashita
and Ishiguro (2004) previously created a robot that measures the distance
between the robot and user with range sensors to initiate the hug sequence;
this robot has a hard surface that may be uncomfortable to touch, and its
inverted pendulum design appears to use the human to balance during the hug.
Block and Kuchenbecker added padding, heating pads, and a soft tactile sensor
to a Willow Garage Personal Robot 2 (PR2) to enable it to exchange hugs with
human users (Block and Kuchenbecker, 2019, 2018). The experimenter manually
adjusted the robot to match the height and size of each user, a process that
takes time and prevents spontaneous hugging. She also initiated every hug for
the user. The PR2 was successful in adapting to the user’s desired hug
duration through the use of the tactile sensor, but the user had to place his
or her hand in a specific location on the PR2’s back, which was not natural
for all users, and the user had to press this sensor to tell the robot to
release them. Some users also criticized the size and shape of this robot as
being awkward to hug. On the positive side, this study showed that both
softness and warmth are important for a robot to deliver good hugging
experiences; we therefore incorporate these already validated elements as our
first two tenets. From Trovato et al. (2016) we learned that softness alone is
not enough for a hugging robot; people are more receptive to hugging a robot
that is wearing clothing, so we placed suitable clothes on HuggieBot 2.0.
## 3\. Hugging Tenets and Hypotheses
We propose six tenets to guide the creation of future hugging robots. The
first two were validated by Block and Kuchenbecker (Block and Kuchenbecker,
2019), and the other four are proposed and validated in this paper. We believe
that a hugging robot should:
* T1.
be soft,
* T2.
be warm,
* T3.
be sized similar to an adult human,
* T4.
visually perceive and react to an approaching user, rather than requiring a
triggering action such as a button press by the user, an experimenter, or a
remote hugging partner,
* T5.
autonomously adapt its embrace to the size and position of the user’s body,
rather than hug in a constant manner, and
* T6.
reliably detect and react to a user’s desire to be released from a hug
regardless of his or her arm positions.
Building off the previously described research, this project seeks to evaluate
the extent to which the six tenets benefit user perception of hugging robots.
Specifically, we aim to test the following three hypotheses:
* H1.
When viewing from a distance, potential users will prefer the embodiment and
movement of a hugging robot that incorporates our four new tenets over a
state-of-the-art hugging robot that violates the tenets.
* H2.
Obeying the four new tenets during physical user interactions will
significantly increase the perceived safety, naturalness, enjoyability,
intelligence, and friendliness of a hugging robot.
* H3.
Repeated hugs with a robot that follows all six tenets will improve user
opinions about robots in general.
## 4\. System Design and Engineering
We introduce a new human-sized hugging robot with visual and haptic
perception. This platform is designed to have a friendlier and more
comfortable appearance than previous state-of-the-art hugging robots. Building
off feedback from users of prior robots (Section 2), we focused on six areas
for the design of this new, self-contained robot: the frame, arms, inflated
sensing torso, head and face, visual person detection, and software
architecture.
### Frame
HuggieBot 2.0’s core consists of a custom stainless steel frame with a
v-shaped base. The robot’s height can be manually adjusted if needed, and the
shape of the base allows users to come very close to the robot, as seen in
Fig. 1. The user does not need to lean over a large base to receive a hug,
unlike the PR2-based hugging robot (Block and Kuchenbecker, 2019). The
v-shaped base also increases the safety and stability of the robot by acting
to counteract any leaning force imparted by a user. This large base and
counterweight ensure that even a large user approaching at a high speed
intending to make an incorrect impact with the robot will not flip it over,
inflict injury upon themselves, or cause damage to the robot.
### Arms
Two 6-DOF Kinova JACO arms are horizontally mounted to a custom stainless
steel bracket attached to the top of the metal frame. To create a more
approachable appearance, the grippers of the JACO arms were removed, and large
padded mittens were placed over the wrist joints that terminate each arm. The
arms are controlled by commanding target joint angles; movement is quiet, and
the joints are not easily backdrivable when powered. The torque sensors at
each joint are continually monitored so that hugs can be automatically
adjusted to each user’s size and position. The second joint (shoulder pan) and
third joint (elbow flex) on each arm stop closing individually when they
surpass a torque threshold, which we empirically set to 10 Nm and 5 Nm,
respectively. The joint torques are also used to detect when a user is pushing
back against the arms with a torque higher than 20 Nm, indicating his/her
desire to be released from a hug. To create a comfortable and enjoyable
tactile experience for the user, we covered the arms in soft foam and a
sweatshirt.
### HuggieChest: Inflatable Torso
We developed a simple and inherently soft inflatable haptic sensor to serve as
the torso of our hugging robot, as pictured in Fig. 2. The torso was
constructed by both heat sealing and gluing (with HH-66 vinyl cement) two
sheets of PVC vinyl to create an airtight seal. This chest has one chamber
located in the front and another in the back. There is no airflow between the
two chambers. Each chamber has a valve from an inflatable swim armband to
inflate, seal, and deflate the chamber. Inside the chamber located on the back
of the robot is an Adafruit BME680 barometric pressure sensor and an Adafruit
electret microphone amplifier MAX4466 with adjustable gain. Both sensors were
secured in the center of the chest on the inner wall of the chamber. The two
sensors are connected to a single Arduino Uno micro-controller outside the
chamber. The microphone and pressure sensor are sampled at 45 Hz, and the
readings are sent over serial to the HuggieBot 2.0 computer for real-time
processing. We originally tested with the same sensing capabilities in both
chambers but did not find the information from the front chamber to be very
useful; thus, no data are collected from the front chamber. This novel
inflatable haptic sensor is called the HuggieChest.
Figure 2. The inflated HuggieChest when lying flat. The two air chambers form
the front and back of the robot’s torso.
The HuggieChest’s shape was created by following a pattern of a padded vest
that goes over the wearer’s head and is secured with a belt around the waist.
Because the HuggieChest is heat-sealed at the shoulders to stop airflow
between the chambers and allow the chest to bend once inflated, the robot is
unable to feel contacts in these locations. The HuggieChest is placed directly
on top of the metal frame of HuggieBot 2.0. On top of the inflatable torso, we
put two Thermophore MaxHeat Deep-Heat Therapy heating pads (35.6 cm $\times$
68.6 cm), which are attached together at one short edge with two shoulder
straps. The sweatshirt is placed on top of the heating pads to create the
final robot torso.
### Head and Face
We designed and 3D-printed a head to house a Dell OptiPlex 7050 minicomputer
that controls the robot, a face screen, an RGB-D camera, the Arduino from the
HuggieChest, a wireless JBL speaker, and cables. The head splits into two
halves with a rectangular plate on each side that can be removed to access the
inside. The final piece of the head is the front-facing frame, which secures
the face screen and camera. The face screen is an LG LP101WH1 Display 10.1”
LCD screen with a 1366 $\times$ 768 resolution in portrait orientation. The
screen displays faces based on designs created and validated for the Baxter
robot (Fitter and Kuchenbecker, 2016).
### Vision and Person Detection
We use a commercially available Intel Realsense RGB-D camera with custom
software to recognize an approaching person and initiate a hug. To this end,
we integrate a deep-learning-based person detection module into our pipeline.
The module consists of two parts. First, our software recognizes an
approaching person using an open-source Robot Operating System (ROS)
integration (Odabasi, 2017) of Tensorflow’s object detection library, which is
based on the SSD mobilenet model (Liu et al., 2016). In the next step, we
utilize the camera’s depth sensor to estimate the distance of the person to
the robot. We use a sliding window to ensure a person is actively approaching
the robot; we observe the distance measured from the depth sensor, which can
be noisy, and check whether the mean distance decreases. This strategy
prevents undesired hugs in case a person walks away from the robot. Once the
person is actively approaching the robot, we initiate a hug as soon as a tuned
distance threshold of 2.45 m is passed. This threshold was selected as it
informs the robot the person is attempting to move from the social space into
the robot’s personal space (Hall et al., 1968).
### Robot Software Architecture
The robot is controlled via ROS Kinetic. Each robot arm joint has both angle
and torque sensors. A PID controller is used to control each joint angle over
time. The robot arms begin by moving to a home position. The camera module
starts and waits for an approaching user. Upon detection, the robot asks the
user, “Can I have a hug, please?” as in (Block and Kuchenbecker, 2018), while
the robot’s face changes to an opening and closing mouth. The specific hug it
is supposed to run (with or without haptic sizing and release) is executed by
commanding each joint to move at a fixed angular velocity toward a
predetermined goal pose. For hugs without haptic sizing, the robot hugs in a
one-size-fits-most manner, where the robot’s second and third joints each
close by 20∘. This pose was large enough such that it did not apply high
forces to the bodies of any of our users; it was not adjusted for different
subjects. For hugs with haptic sizing, the robot arms move toward a pose sized
for a small user; we continually monitor each joint torque and stop a joint’s
movement if it exceeds the pre-set torque threshold. This method leads to
automatic adjustment to the user’s size (T5).
The Arduino communicates the microphone and pressure sensor data from inside
the back chamber of the HuggieChest to ROS over serial at 45 Hz. This data
stream is analyzed in real time. The program first determines the ambient
pressure and noise in the chamber by averaging the first 20 samples to create
a baseline that accounts for different levels of inflation and noise. The
chamber detects the user beginning to hug when the chamber’s pressure
increases by 50 kPa above the baseline pressure. Contact is determined to be
broken, thus indicating that the user wants to be released, when the pressure
returns to 10 kPa above the baseline pressure. The measured torques from the
robot’s shoulder pan and elbow flex joints are monitored continually during a
hug. A haptic release is also triggered when any of these torques surpasses a
threshold limit of 20 Nm. For a timed hug, rather than detecting the instant
at which the user wants to be released, the robot waits 1 second after the
arms fully close before releasing the user, so it is apparent to the user they
are not in control of the duration of the hug. Overall, our proposed method of
closed-loop hugging works on a higher level of abstraction than the low-level
control, i.e., including both visual and haptic perception in the loop of the
hugging process. The robot’s haptic perception is two-fold: adjusting to the
size of the user and sensing when he/she wants to be released.
## 5\. Online User Study
We ran an online study to get feedback from a broad audience on the embodiment
and movement of our robot as part of our user-centered design process, and to
compare it to the PR2-based HuggieBot 1.0 (Block and Kuchenbecker, 2019). The
main stimuli were two videos from (Block and Kuchenbecker, 2019) along with
two newly recorded videos of people hugging HuggieBot 2.0 with matched gender,
enthusiasm, and timing; these videos are included as supplementary material
for this paper. This study was approved by the Max Planck Society ethics
council under the HI framework.
### Participants
All participants for the online study were non-compensated English-speaking
volunteers recruited via emails and social media. A total of 117 subjects took
part in the online survey: 42.7% male, 56.4% female, and 0.9% who identify as
other. The participants ranged in age from 20 to 86 (M = 37.5, SD = 16.75).
The majority of respondents indicated they had little (30.7%) or no experience
(43.6%) interacting physically with robots.
### Procedures
After someone reached out to the experimenter and indicated interest in
participating in the online study, the experimenter sent the subject an
informed consent document by email. The participant read it thoroughly, asked
any questions, and signed it and sent it back to the experimenter only if they
wanted to participate. At this point, the user was assigned a subject number
and sent a unique link to the online survey.
Table 1. The questions participants answered after viewing or experiencing
robot hugs. This hug made the robot seem (unfriendly – friendly)
---
This robot behavior seemed (unsafe – safe)
This hug made the robot seem socially (stupid – intelligent)
This hug interaction felt (awkward – enjoyable)
This robot behavior seemed (unnatural – natural)
First, participants filled out their demographic information. Next, they were
shown two videos of an adult (one male and one female) hugging a robot labeled
“Robot A”. Robot A was HuggieBot 2.0 for half of the participants and Block
and Kuchenbecker’s HuggieBot 1.0 (Block and Kuchenbecker, 2019) for the other
half. Users could watch these videos as many times as they liked before
answering several questions. Users described their first impressions of the
robot they saw. Then, they answered the questions shown in Table 1 on a
5-point Likert scale. Afterwards, there was an optional space for additional
comments. Next, they were shown two videos of people hugging the other robot
labeled “Robot B” under the same conditions as the first videos. Users
answered the same questions for Robot B as they did for Robot A. Finally,
since the videos were shot from behind the robot, users were shown frontal
images of both robots posed in a similar manner. Participants were then asked
“In what ways is Robot A better than Robot B?” and “In what ways is Robot B
better than Robot A?” Then, they were asked to select which robot they would
prefer to hug, Robot A or B, and why.
After the first 40 participants, we noticed that several users were commenting
on the purple fuzzy appearance of the PR2, rather than on the robots
themselves. Therefore, we added a new section at the end of the survey, which
was completed by the remaining 77 participants. This new section showed
photographs of Robot A and Robot B plus two additional photographs showing
HuggieBot 2.0 in different clothing. In one of the new photographs, HuggieBot
2.0 wore a fuzzy purple robe similar to the fabric cover on the PR2, and in
the other it wore its gray sweatshirt over this same fuzzy purple robe.
Participants were asked which of these four robots they would prefer to hug
and why.
Figure 3. The responses to the five questions asked after users watched two
videos of people hugging HuggieBot 2.0 (HB2) and HuggieBot 1.0 (HB1).
### Results
The responses to the five Likert-style questions from Table 1 can be seen in
Fig. 3. For all statistical analyses, we applied a Bonferroni alpha correction
to $\alpha$ = 0.05 to determine significance and to account for the multiple
comparisons. Because the data from these questions were non-parametric, we
conducted a Wilcoxon signed-rank test. No statistically significant
differences were found between the responses to any of these questions for the
two robots.
The responses to the first and second rounds of voting for which robot users
would prefer to hug can be found in Fig. 4. We ran a Wilcoxon signed-rank test
for the first round of voting and determined users would significantly prefer
to hug our new robot over HuggieBot 1.0 (p ¡ 0.001). In the second voting
round, no significant preference was found between any of the four options,
indicating HuggieBot 2.0 was preferred over HuggieBot 1.0 approximately 3:1.
We ran several one-way analyses of variance (ANOVA) to see if participant
gender, robot presentation order, or participant level of extroversion had a
significant effect on which robot the user selected. No significance was found
in any of these cases.
Figure 4. The breakdown of preferences when users had two choices (top) and
four choices (bottom), with the associated images. The colors of the second
plot show which robot the user preferred in the first selection round.
### Changes to Platform
After analyzing the results of the online study and reviewing the user
feedback, we found several areas to improve our hugging robot. Some users
found the initial HuggieBot 2.0 voice off-putting, so we changed the robot’s
voice to sound less robotic. We made the purple robe and the sweatshirt the
robot’s permanent outfit, as it had the highest number of votes and received
many positive comments. We changed the color of the robot’s face from its
initial green to purple to match the robe and create a more polished look.
Several users commented on the slow speed of HuggieBot 2.0’s arms. Since the
arm joints cannot move faster than the maximum angular velocity specified by
the manufacturer, we instead moved their starting position closer to the goal
position to reduce the time they need to close.
## 6\. In-Person User Study
The goal of the in-person study was to evaluate our updated robotic platform
and the four new hugging tenets that drove its design. This study was also
approved by the Max Planck Society ethics council under the HI framework.
### Participants
The recruitment methods for the in-person study were the same as for the
online study. Participants not employed by the Max Planck Society were
compensated 12 euros. A total of 32 subjects participated in the in-person
study: 37.5% male and 62.5% female. Our participants ranged in age from 21 to
60 (M = 30, SD = 7) and came from 13 different countries. We took significant
precautions beyond government regulations to protect participant health when
running this study during the COVID-19 pandemic.
### Procedures
After confirming their eligibility given the exclusion criteria, users
scheduled an appointment for a 1.5-hour-long session with HuggieBot 2.0. Upon
arrival, the experimenter explained the study, and the potential subject read
the informed consent document and asked questions. If he/she still wanted to
participate, the subject signed the consent form and the video release form,
at which point the video cameras were turned on to record the experiment.
Users began by filling out a demographics survey on a computer. Next, the
investigator introduced the robot as the personality “HuggieBot” and explained
its key features, including the emergency stop. The experimenter explained how
the trials would work and how the subject should be prepared to move. She also
explained the two different ways to initiate a hug (walking, key press) and
the three different ways to be released from a hug (release hands, lean back,
wait until robot releases). At this point the subject filled out an opening
survey to document his/her initial impressions of the robot; participants
rated on a sliding scale from 0 (disagree) to 10 (agree) how much they agreed
with the statements found in Table 2. Note that these questions were asked
before the user had any experience physically hugging the robot. Next, the
user performed practice hugs with the robot to acclimate to the hug initiation
methods and the timing of the robot’s arms closing. A participant was allowed
to perform as many practice hugs as desired, verbally indicating to the
experimenter when they were ready to begin the experiment. On average users
did 2 or 3 practice hugs, but users taller than the robot (1.75 m) averaged 5
or 6 practice hugs because it took more time for them to find the most
comfortable arm positions. The eight hugging conditions that made up this
experiment are all three possible pairwise combinations of our three binary
factors (vision, sizing, and release detection). The video associated with
this paper shows a hugging trial from each of the eight conditions.
Table 2. The fifteen questions asked in the opening and closing
questionnaires. I feel understood by the robot
---
I trust the robot
Robots would be nice to hug
I like the presence of the robot
I think using the robot is a good idea
I am afraid to break something while using the robot
People would be impressed if I had such a robot
I could cooperate with the robot
I think the robot is easy to use
I could do activities with this robot
I feel threatened by the robot
This robot would be useful for me
This robot could help me
This robot could support me
I consider this robot to be a social agent
We used an $8\times 8$ Latin square to counter-balance any effects of
presentation order (Grant, 1948) and recruited 32 participants to have
complete Latin squares. After each hug, the participant returned to the
computer and answered six questions. The first question was a free-response
asking for the user’s “first impressions of this interaction.” Then, the
participant used a sliding scale from 0 to 10 to answer the five questions
found in Table 1, which were the same questions as in the online study. A
subject could request to experience the same hug again if needed. After
experiencing all eight hug conditions, the participants experienced an average
of 16 more hugs, during which they contacted the robot’s back in different
ways and received various robot responses. Data were collected for these
additional hugs, but they will not be analyzed in this paper due to space
constraints. At the end of the experiment, the subject answered the same
questions from the beginning of the study (Table 2). Finally, users could
provide additional comments at the end.
All slider-type questions in the survey were based on previous surveys in HRI
research and typical Unified Theory of Acceptance and Use of Technology
(UTAUT) questionnaires (Weiss et al., 2008; Heerink et al., 2009). The free-
response questions were designed to give the investigators any other
information the participant wanted to share about the experience. A within-
subjects study was selected for this experiment because we were most
interested in the differences between the conditions, rather than the overall
response levels to a robot hug. We also preferred this design for its higher
statistical power given the same number of participants compared to a between-
subjects study (de Winter and Dodou, 2017).
Figure 5. A comparison of the responses to the opening (blue) and closing
(red) surveys. The top and bottom of the box represent the 25th and 75th
percentile responses, respectively, while the line in the center represents
the median, and the triangle indicates the mean. The lines extending past the
boxes show the farthest data points not considered outliers. The + marks
indicate outliers. The black lines with stars at the top of the graph indicate
statistically significant differences.
### Results
This in-person study was the first robustness test of the fully integrated
HuggieBot 2.0 system. Each subject experienced a minimum of 24 hugs during the
study, plus practice hugs. With 32 total participants, the robot executed more
than 850 successful hugs over the course of the entire study, sometimes giving
200 hugs in one day.
Figure 6. A comparison of the responses to the survey questions after each
hug, grouped by factors. The purple colors represent without vision (v) and
with vision (V), the pink colors represent without sizing (s) and with sizing
(S), and the green colors represent timed release (r) and haptic release (R).
For all statistical analyses, we applied a Bonferroni alpha correction (to
account for 15 multiple comparisons) to $\alpha$ = 0.05 to determine
significance. We use Pearson’s linear correlation coefficient, $\rho$, to
report effect size. Box plots of the responses to the opening and closing
survey questions from Table 2 are shown in Fig. 5. In this study, answers were
submitted on a continuous sliding scale, so a paired t-test comparison of the
opening and closing survey was conducted. We found that users felt understood
by (p = 0.0025, $\rho$ = 0.57) and trusted the robot more (p ¡ 0.001, $\rho$ =
0.70) after participating in the experiment. Users also felt that robots were
nicer to hug (p ¡ 0.001, $\rho$ = 0.76 )
The responses to the five questions asked after each hug can be seen grouped
by the presence and absence of each of the three tested factors (vision,
sizing, and release) in Fig. 6. These responses were analyzed using three-way
repeated measures analysis of variance via the built-in MATLAB function
ranova; our data satisfy all assumptions of this analytical approach. No
significant improvements were noticed in the perceived safety of the robot in
any of the tested conditions, as the robot was consistently rated highly safe.
The automatic size adjustment significantly increased users’ impressions of
the naturalness of the robot’s movement (F(1,31) = 25.192, p ¡ 0.001, $\rho$ =
0.4158). Users found the robot’s hug significantly more enjoyable when it
adjusted to their size (F(1,31) = 70.553, p ¡ 0.001, $\rho$ = 0.3610).
Automatic size adjustment to the user caused a significant increase (F(1,31) =
25.102, p ¡ 0.001, $\rho$ =0.4258) in the perceived intelligence of the robot.
Finally, the robot was considered significantly friendlier when it adjusted to
the size of the user (F(1,31) = 84.925, p ¡ 0.001, $\rho$ = 0.3205). In
summary, haptic hug sizing significantly affected every aspect except safety.
Trials that included visual perception trended slightly positive but were not
significantly different for any five of the investigated questions; small
positive trends for haptic release were also not significant.
Eight users (25%) verbally stated and wrote about their preference for not
having to push a button to activate a hug. Out of the 256 distinct hug
response surveys (32 users, 8 surveys per user), the physical warmth of the
robot was positively mentioned 100 times (39%), further validating that
physical warmth is critical to pleasant robot hugs (T2) (Block and
Kuchenbecker, 2019). These positive comments were most commonly seen in the
conditions with automatic hug sizing, presumably because the increased contact
with the robot torso made the heat more apparent. Additionally, we observed
that our participants used a mixture of pressure release and torque release to
indicate their desire to end the hugs in the study. 17 users (53%) voiced
their preference for the haptic release hugs, saying when the robot released
before they were ready (in hugs with a timed release), “he didn’t want to hug
me!” or that the hug was “too short!” Interestingly, the hug condition where
all three perceptual factors were present had the highest number of positive
comments. 31 out of 32 users (96.8%) commented that this condition was the
most “pleasant interaction,” “natural,” “friendly,” or “fun.”
## 7\. Discussion
Our three hypotheses were largely supported by the results. First, H1
hypothesized that when viewing from a distance, potential users will prefer
the embodiment and movement of a hugging robot that incorporates our four new
tenets over a state-of-the-art hugging robot that violates these tenets. The
online study found that users significantly preferred HuggieBot 2.0 over
HuggieBot 1.0; we believe our new robot was preferred because it obeys all six
tenets. Written comments from the online community mention the “large,”
“hulking,” and “over-powering” PR2 robot as unnerving when compared to the
size of the user. In comparison, our robot is considered “nice” and
“friendlier.” Several users also wrote comments on how the people in the PR2
videos had to push a button on the robot’s back, which seemed “unnatural”,
whereas the HuggieBot 2.0 release seemed more “intuitive”. We did our best to
match the videos of our new robot to the pre-existing videos of the PR2 so as
not to bias the online viewers. These videos included but could not showcase
visual hug initiation and haptic size adjustment. Based on the strong
preference for our hugging robot in our carefully controlled online study, we
conclude that users do prefer the embodiment and movement of a hugging robot
that obeys our four new tenets over a state-of-the-art hugging robot that
violates most of them.
H2 conjectured that obeying the four new tenets during physical interactions
with users will significantly increase the perceived safety, naturalness,
enjoyability, intelligence, and friendliness of a hugging robot. We found that
the haptic perception tenets had the greatest effects on these aspects of the
robot, with haptic sizing positively affecting many responses and both haptic
sizing and haptic release garnering positive comments. The lack of significant
effects of visual initiation does not match the comments that users prefer the
interaction when the robot recognizes their approach, rather than them having
to push a button to initiate the hug. It is possible that users might not have
included the button pushing in their rating of the hug as we simply asked
users to “rate their experience with this hug” and did not explicitly tell
them to include the hug initiation. Users might also have been confused that
they had to walk towards the robot to initiate a hug, and then the robot would
ask “Can I have a hug, please?” We chose to have the robot say the same phrase
for both initiation methods to minimize variables, but as the user was
initiating the hug, it may have made more sense for the robot to say something
else or not speak.
We also believe there is room for improvement of the visual perception of our
robot, which could contribute to higher ratings of the five questions.
Currently, our perception of the user is based solely on his/her approach. To
take perception even further, we believe hugs would be more comfortable if the
robot could adjust its arm poses to match the approaching user’s height and
arm positions. Our taller users found the robot hugged them too low, and our
shorter users found the opposite. Adjusting to user height would more fully
obey T4 (visual perception) and therefore should be more acceptable to users.
While the torso of the robot and dual release methods ensure our robot follows
T6 of reliably releasing the user, a robot that could adjust its arm positions
to the reciprocating pose of the user could greatly strengthen the user’s
impression of the robot’s visual perception and improve the user opinion of
the robot. We concede that our robot’s rudimentary visual perception
contributed to our lack of finding significant differences in the areas we
investigated when testing with and without this factor.
Finally, H3 hypothesized that repeated hugs with a robot that follows all six
tenets will improve user opinions about robots in general. We asked the same
opening and closing survey questions as Block and Kuchenbecker (2019), with
similar results. Both studies’ users felt more understood by, trusted, and
thought that robots were nicer to hug after participating. HuggieBot 1.0’s
users also liked the presence of the robot more afterwards, found the robot
easier to use, and viewed it as more of a social agent after the experiment,
although these findings were reported without any statistical correction for
multiple comparisons. The PR2 robot used in that experiment is significantly
larger than an adult human, which violates T3. This domineering physical
presence, therefore, contributed to lower initial ratings for users liking the
presence of the robot, their perceived ease of use of the robot, and viewing
the robot as a social agent. Our new robot, whose physical stature obeys the
first three tenets, received higher initial ratings in these categories.
HuggieBot 2.0 appeared as a friendly social agent from the beginning, and
prolonged interaction with it confirmed these high initial impressions, which
is why we did not find any significant differences for these questions in our
study. As first impressions are often critical to determine whether a user
will interact with a robot, here we see that it is important to obey the tenet
prescribing the physical size of a robot. Therefore, we conclude that a robot
that follows the six tenets does indeed improve user opinions about robots in
general. A positive impression of the robot is crucial because it will make
users more willing to receive a robot hug, and thus more likely to receive
these health benefits when they cannot receive them from other people.
## 8\. Limitations and Conclusion
This study represents an important step in understanding intimate social-
physical human-robot interactions, but it certainly has limitations. Due to
COVID-19, the first study relied solely on videos and images, rather than the
participants physically interacting with robots. Since we do not have access
to a PR2, this online study enabled a fair comparison between our new platform
and a previously well-rated hugging robot. Watching other people hug a robot
is how users will decide if they also want to interact with a hugging robot in
the wild.
One weakness of our new platform, HuggieBot 2.0, was the slow speed of the
Kinova JACO arms. We selected these arms because of their inherent safety
features; however, the distance the arms had to travel made their slow speed
obvious and caused a long delay after hug initiation. When users started the
hug with a button press, they could wait before walking to the robot for
better timing. With visual hug initiation, the users were required to walk
before the arms began moving, which resulted in many awkwardly waiting in
front of the robot for the arms to close. Related to this limitation is the
size of the room in which we conducted the experiment. A larger room would
have let us set the threshold distance farther back to accommodate the speed
of the arms. Both of these limitations could have contributed to the lack of
significant differences between the hugs with and without visual perception.
Another limitation is the self-selection bias of our participants. For
transparency, we advertised the experiment as a hugging robot study. While we
succeeded in recruiting a diverse and largely non-technical audience to make
our results as applicable to the general public as possible, we nevertheless
acknowledge that users who chose to participate in the study were interested
in robots. Because we did not hide the nature of our study, we did not have
any participants who refused to hug the robot, as might occur in a more
natural in-the-wild study design.
This project took a critical look at state-of-the-art hugging robots, improved
upon their flaws, built upon their successes, and created a new hugging robot,
HuggieBot 2.0. We also propose to the HRI community six tenets of hugging that
future designers should consider to improve user acceptance of hugging robots.
During times of social distancing, the consequences of lack of physical
contact with others can be more damaging and prevalent than ever. If we cannot
seek comfort from other people due to physical distance or health or safety
concerns, it is important that we seek other opportunities to reap the
benefits of this helpful interaction.
###### Acknowledgements.
This work is partially supported by the Max Planck ETH Center for Learning
Systems and the IEEE Technical Committee on Haptics. The authors thank Bernard
Javot, Michaela Wieland, Hasti Seifi, Felix Grüninger, Joey Burns, Ilona
Jacobi, Ravali Gourishetti, Ben Richardson, Meike Pech, Nati Egana, Mayumi
Mohan, and Kinova Robotics for supporting various aspects of this research
project.
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|
# Hyperspectral Image Denoising via Multi-modal and Double-weighted Tensor
Nuclear Norm
Sheng Liu, Xiaozhen Xie, and Wenfeng Kong Sheng Liu, Xiaozhen Xie, and Wenfeng
Kong are with College of Science, Northwest A&F University, Yangling 712100,
China (e-mail<EMAIL_ADDRESS>e-mail<EMAIL_ADDRESS>(Corresponding author); e-mail: [email protected]).
###### Abstract
Hyperspectral images (HSIs) usually suffer from different types of pollution.
This severely reduces the quality of HSIs and limits the accuracy of
subsequent processing tasks. HSI denoising can be modeled as a low-rank tensor
denoising problem. Tensor nuclear norm (TNN) induced by tensor singular value
decomposition plays an important role in this problem. In this letter, we
first reconsider three inconspicuous but crucial phenomenons in TNN. In the
Fourier transform domain of HSIs, different frequency slices (FS) contain
different information; different singular values (SVs) of each FS also
represent different information. The two physical phenomenons lie not only in
the spectral mode but also in the spatial modes. Then based on them, we
propose a multi-modal and double-weighted TNN. It can adaptively shrink the FS
and SVs according to their physical meanings in all modes of HSIs. In the
framework of the alternating direction method of multipliers, we design an
effective alternating iterative strategy to optimize our proposed model.
Denoised experiments on both synthetic and real HSI datasets demonstrate their
superiority against related methods.
###### Index Terms:
Hyperspectral image, tensor nuclear norm, double weighting, frequency slices,
multi-modal.
## I Introduction
Hyperspectral image (HSI) has been widely used in many fields [1] due to its
wealthy spatial and spectral information of a real scene. However, the
observed HSIs are usually corrupted by different noises, e.g., Gaussian noise,
impulse noise, deadlines, stripes and their mixtures. This seriously affects
the subsequent applications of HSI, such as unmixing, target detection, and so
on. Therefore, HSI denoising , as a preprocessing step to remove mixed noise
for various subsequent applications, is a valuable and active research topic.
In HSIs, different spectral bands are images from the same scene under
different wavelengths. It means the global correlation or low-rank prior lies
in HSIs [2]. Based on this prior, how to measure the HSI low-rankness becomes
the key to denoising tasks. As HSIs can be treated as 3rd-order tensor, this
problem is turned into a 3rd-order tensor low-rank problem. Due to the
nonunique definitions of the tensor rank, different tensor decompositions and
their corresponding tensor ranks are proposed, such as the Tucker
decomposition [3, 4], PARAFAC decomposition [5], and tensor singular value
decomposition (t-SVD) [6, 7], to exploit the low-rankness of HSIs.
Among them, the tensor tubal rank induced by t-SVD can characterize the low-
rank structure of HSIs very well.Its convex relaxation is the tensor nuclear
norm (TNN) [8]. TNN is effective to keep the intrinsic structure of HSIs.
Hence, TNN has attracted extensive attention for HSI denoising problems in
recent years [7, 9, 10]. However, during the definition of TNN, there are
three kinds of prior knowledge that are underutilized for further exploiting
the low-rankness in HSIs. Firstly, in the Fourier transform domain of HSIs,
the low-frequency slices carry the profile information of HSIs, while the
high-frequency slices mainly carry the noise information of HSIs. Secondly, in
each frequency slices, bigger singular values mainly contain information on
clean data and smaller singular values mainly contain information on noise.
Thirdly, low-rankness not only exists in the spectral dimension but also lies
in the spatial dimensions [3]. The classical TNN only takes the Fourier
transform to connect the spatial dimensions with the spectral dimension and
lacks flexibility for handling different correlations along with different
modes of HSIs [7].
In this letter, to take full advantage of the above prior knowledge and
improve the capability and flexibility of TNN, we propose a multi-modal and
double-weighted TNN for HSI denoising tasks. The merits of our model are four-
fold. First, according to information types in different frequency slices in
the Fourier transform domain, we adaptively assign bigger weights to slices
that mainly contain noise information and smaller weights to slices that
mainly contain profile information, which can depress noise more and
simultaneously preserve the profile information of clean HSIs better. Second,
in each frequency slice, we use the partial sum of singular values to only
shrink small singular values, which can better protect the clean data
information contained in big singular values. Third, we apply the double-
weighted TNN in all modes of HSIs, which can achieve a more flexible and
accurate characterization of HSI low-rankness. Finally, we develop an
alternating direction method of multiplier based algorithm to efficiently
solve the proposed model, Compared with various competing HSI denoising
methods, the best-denoised performances are obtained by our method synthetic
and real HSI datasets.
## II Preliminaries
### II-A Notations
In this letter, matrix and tensor are denoted as bold upper-case letter
$\mathbf{X}$ and calligraphic letter $\mathcal{X}$, respectively. For a 3rd-
order tensor $\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$, its
$(i,j,k)$-th component is represented as $\mathcal{X}(i,j,k)$. For
$\mathcal{X}$, $\mathcal{Y}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$,
their inner product is defined as
$<\mathcal{X},\mathcal{Y}>=\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}\sum_{k=1}^{n_{3}}x_{ijk}y_{ijk}$.
Then the Frobenius norm of a tensor $\mathcal{X}$ is defined as
$\|\mathcal{X}\|_{F}=\sqrt{<\mathcal{X},\mathcal{X}>}$. The $k$-th frontal
slice of $\mathcal{X}$ is denoted as $\mathbf{X}^{(k)}=\mathcal{X}(:,:,k)$.
The fast Fourier transform along the third mode of $\mathcal{X}$ is
represented as $\bar{\mathcal{X}}=\texttt{fft}(\mathcal{X},[],3)$ and its
inverse operation is $\mathcal{X}=\texttt{ifft}(\bar{\mathcal{X}},[],3)$. The
mode-$p$ permutation of $\mathcal{X}$ is defined as
$\mathcal{X}_{p}=\texttt{permute}(\mathcal{X},p)$, $p=1,2,3$, where the $m$-th
mode-3 slice of $\mathcal{X}_{p}$ is the $m$-th mode-$p$ slice of
$\mathcal{X}$, i.e.,
$\mathcal{X}(i,j,k)=$$\mathcal{X}_{1}(j,k,i)=$$\mathcal{X}_{2}(k,i,j)=$$\mathcal{X}_{3}(i,j,k)$.
Also, its inverse operation is
$\mathcal{X}=\texttt{ipermute}(\mathcal{X}_{p},p)$.
### II-B Problem Formulation
An ideal HSI can be viewed as a 3rd-order tensor
$\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ and usually is
assumed to be low-rank. Corrupted by mixed noise, its observed version can be
modeled as
$\mathcal{Y}=\mathcal{X}+\mathcal{S}+\mathcal{N},$ (1)
where $\mathcal{Y},\mathcal{S},\mathcal{N}\in\mathbb{R}^{n_{1}\times
n_{2}\times n_{3}}$; $\mathcal{S}$ denotes the sparse noise; $\mathcal{N}$
denotes the Gaussian white noise.
HSI denoising aims to recover the ideal HSI $\mathcal{X}$ from the observed
HSI $\mathcal{Y}$ in (1). Under the framework of regularization theory, it can
briefly be formulated as
$\displaystyle\arg\min_{\mathcal{X},\mathcal{S},\mathcal{N}}$
$\displaystyle\texttt{Rank}(\mathcal{X})+\lambda\|\mathcal{S}\|_{1}+\tau\|\mathcal{N}\|_{F}^{2},$
(2) $\displaystyle s.t.$
$\displaystyle\mathcal{Y}=\mathcal{X}+\mathcal{S}+\mathcal{N},$
where $\|\cdot\|_{1}$ is $L_{1}$ norm to detect the sparse noise;
$\|\cdot\|_{F}$ describes the Gaussian noise; $\texttt{Rank}(\cdot)$
represents the rank of unknown ideal HSI; $\lambda$ and $\tau$ are non-
negative parameters.
In model (2), regularization term Rank is approximated by different
relaxations. As mentioned above, TNN is widely used convex relaxation, which
can be defined as
$\|\mathcal{X}\|_{*}:=\frac{1}{n_{3}}\sum_{k=1}^{n_{3}}\|\bar{\textbf{X}}^{(k)}\|_{*}.$
(3)
## III The Weighted TNN
### III-A Frequency-Weighted TNN
In (3), the frontal slice of $\bar{\mathcal{X}}$ corresponds to the frequency
component of $\mathcal{X}$ [11]. Specifically, for $\mathcal{X}$, its profile
information is contained in the low-frequency frontal slices, while its
detailed information is contained in the high-frequency ones. When
$\mathcal{X}$ is distorted by outliers, the effects on high-frequency frontal
slices are more severe. However, different frequency slices of
$\bar{\mathcal{X}}$ have the same impact on TNN in (3), which is obviously
inconsistent with the physics meaning of frequency components. Therefore, we
improve TNN in (3) by assigning different weights for different frequency
slices, and propose the frequency-weighted TNN as follows:
$\|\mathcal{X}\|_{w*}:=\sum_{k=1}^{n_{3}}w_{k}(\bar{\textbf{X}}^{(k)})\|\bar{\textbf{X}}^{(k)}\|_{*},$
(4)
where $w_{k}(\bar{\textbf{X}}^{(k)})$ is the $k$-th weight parameter. For HSI
denoising problems, the lower the frequencies are, the less the corresponding
frequency slices should be punished. By amounts of data simulations, the
weights $w_{k}$ approximatively consist with the frequencies and are inversely
proportionate to $\|\bar{\textbf{X}}^{(k)}\|_{F}$. We let
$w_{k}(\bar{\textbf{X}}^{(k)})=\frac{c_{1}}{\log(\|\bar{\textbf{X}}^{(k)}\|_{F}^{2})+\varepsilon}+c_{2},$
(5)
where $\varepsilon=10^{-10}$ is to avoid dividing by zero; $c_{1}$ is the
scaling factor after frequency normalization; $c_{2}$ is a constant. The
proposed FWTNN is different from the frequency-filtered tensor nuclear norm
(FTNN) [11]. Our weights are data dependent, but FTNN’s weights are pre-
weight.
### III-B Double-Weighted TNN
For $\bar{\textbf{X}}^{(k)}$ in (3), the matrix nuclear norm is used as the
tightest convex surrogate for rank. However, it has limitation in the accuracy
of approximation due to its convexity. Recently, a series of improvement
methods are proposed for better approximation [12, 13]. To differently treat
singular values of $\bar{\textbf{X}}^{(k)}$, we choose partial sum of singular
values (PSSV) to only punish the smaller singular values which mainly contain
the noise information of HSIs. Then, a double-weighted TNN is proposed by
replacing the matrix nuclear norm in (4) with the PSSV of
$\bar{\textbf{X}}^{(k)}$, which is defined as
$\|\mathcal{X}\|_{dw*}:=\frac{1}{n_{3}}\sum_{k=1}^{n_{3}}w_{k}(\bar{\textbf{X}}^{(k)})\|\bar{\textbf{X}}^{(k)}\|_{\textrm{PSSV}},$
(6)
where
$\|\bar{\textbf{X}}^{(k)}\|_{\textrm{PSSV}}=\sum_{r=R+1}^{\min\\{n_{1},n_{2}\\}}\sigma_{r}(\bar{\textbf{X}}^{(k)})$;
$\sigma_{r}(\bar{\textbf{X}}^{(k)})$ is the $r$-th biggest singular value of
matrix $\bar{\textbf{X}}^{(k)}$; $R$ is a parameter indicating the number of
main singular values. The double-weighted TNN minimization problem can be
solved by following theorem.
###### Theorem 1.
Assuming that $\tau>0$, $\mathcal{X},\mathcal{Y}\in$ $\mathbb{R}^{n_{1}\times
n_{2}\times n_{3}}$, for the minimization problem
$\mathcal{X}^{*}=\arg\min_{\mathcal{X}}\tau\lVert\mathcal{X}\rVert_{dw*}+\frac{1}{2}\lVert\mathcal{X}-\mathcal{Y}\rVert_{F}^{2},$
(7)
its solution is
$\begin{array}[]{rl}\mathcal{X}^{*}=\mathcal{D}\mathcal{W}^{w,R,\tau}\left(\mathcal{Y}\right)=\texttt{ifft}(\bar{\mathcal{U}}\cdot\bar{\mathcal{S}}_{dw*}\cdot\bar{\mathcal{V}}^{T},[],3),\end{array}$
(8)
where
$\bar{\mathcal{Y}}=\bar{\mathcal{U}}\cdot\bar{\mathcal{S}}\cdot\bar{\mathcal{V}}^{T}$;
$\bar{\mathcal{S}}_{dw*}(r,r,k)=\max(\bar{\mathcal{S}}(r,r,k)-\tau
w_{r}w_{k},0);$, $R=TW(k)$
$w_{r}=\left\\{\begin{array}[]{l}\begin{matrix}0,&r\leq R\\\ \end{matrix}\\\
\begin{matrix}1,&r>R\\\ \end{matrix}\\\ \end{array}\right.$;
$w_{k}=\frac{c_{1}}{\log(\lVert\boldsymbol{\bar{\textbf{X}}}^{(k)}\rVert_{F}^{2}+\varepsilon)}+c_{2}$.
###### Lemma 1.
(PSVT [12]). Let $\tau>0$, $l=\min(m,n)$ and
$\mathbf{X},\mathbf{Y}\in\mathbb{R}^{m\times n}$ which can be decomposed by
SVD. $\mathbf{Y}$ can be considered as the sum of two matrices,
$\mathbf{Y}=\mathbf{Y}_{1}+\mathbf{Y}_{2}=\mathbf{U}_{Y1}\mathbf{D}_{Y1}\mathbf{V}_{Y1}^{\top}+\mathbf{U}_{Y2}\mathbf{D}_{Y2}\mathbf{V}_{Y2}^{\top}$
, where $\mathbf{U}_{Y1},\mathbf{V}_{Y1}$ are the singular vector matrices
corresponding to the $R$ largest singular values by S V D , and
$\mathbf{U}_{Y2}$, $\mathbf{V}_{Y2}$ from the $(R+1)$ -th to the last singular
values. Define a minimization problem for the PSSV as
$\underset{\mathbf{X}}{\arg\min}\frac{1}{2}\|\mathbf{X}-\mathbf{Y}\|_{F}^{2}+\tau\|\mathbf{X}\|_{\textrm{PSSV}}$
(9)
Then, the optimal solution of Eq.(9) can be expressed by the PSVT operator
defined as:
$\displaystyle\mathbb{P}_{R,\tau}[\mathbf{Y}]$
$\displaystyle=\mathbf{U}_{Y}\left(\mathbf{D}_{Y1}+\mathcal{S}_{\tau}\left[\mathbf{D}_{Y2}\right]\right)\mathbf{V}_{Y}^{\top}$
(10)
$\displaystyle=\mathbf{Y}_{1}+\mathbf{U}_{Y2}\mathcal{S}_{\tau}\left[\mathbf{D}_{Y2}\right]\mathbf{V}_{Y2}^{\top}$
where
$\begin{array}[]{l}\mathbf{D}_{Y1}=\operatorname{diag}\left(\sigma_{1},\cdots,\sigma_{R},0,\cdots,0\right)\\\
\mathbf{D}_{Y2}=\operatorname{diag}\left(0,\cdots,0,\sigma_{R+1},\cdots,\sigma_{l}\right)\end{array}$
(11)
and $\mathcal{S}_{\tau}[x]=\max(|x|-\tau,0)$ is the soft-thresholding operator
[14].
###### Proof.
Let $\tau>0$, $\mathcal{X},\mathcal{Y}\in$ $\mathbb{R}^{n_{1}\times
n_{2}\times n_{3}}$, $MR\in\mathbb{R}^{n_{3}}$ is multi-rank of $\mathcal{X}$,
and $c_{1},c_{2}$ are given constants. According to the definition of DWTNN
(6) and the properties of tensors in the Fourier domain, the Eq. (7) is
equivalent to
$\displaystyle\mathcal{X}^{*}=\arg\min_{\mathcal{X}}\tau\|X\|_{dw*}+\frac{1}{2}\|\mathcal{X}-\mathcal{Y}\|_{F}^{2},$
(12)
$\displaystyle=\arg\min_{\mathcal{X}}\tau\frac{1}{n_{3}}\sum_{k=1}^{n_{3}}w_{k}\left(\overline{\mathbf{X}}^{(k)}\right)\left\|\overline{\mathbf{X}}^{(k)}\right\|_{\textrm{PSSV}}+\frac{1}{2n_{3}}\|\overline{\mathcal{X}}-\overline{\mathcal{Y}}\|_{F}^{2}$
$\displaystyle=\arg\min_{\mathcal{X}}\frac{1}{n_{3}}\sum_{k=1}^{n_{3}}\tau
w_{k}\left(\overline{\mathbf{X}}^{(k)}\right)\left\|\overline{\mathbf{X}}^{(k)}\right\|_{\textrm{PSSV}}+\frac{1}{2}\left\|\overline{\mathbf{X}}^{(k)}-\overline{\mathbf{Y}}^{(k)}\right\|_{F}^{2}.$
Therefore, the Eq. (12) can be divided into $n_{3}$ subproblems as follows:
$\arg\min_{\mathcal{X}}\tau
w_{k}\left(\overline{\mathbf{X}}^{(k)}\right)\|\overline{\mathbf{X}}\|_{p=R}+\frac{1}{2}\|\overline{\mathbf{X}}-\overline{\mathbf{Y}}\|_{F}^{2},$
(13)
the Eq.(13) can be solved by Lemma 1. Let
$w_{r}=\left\\{\begin{array}[]{ll}0,&i\leqslant R\\\ 1,&i>R\end{array}\right.$
(14)
Base on Eq.(14), Eq.(10) in Lemma 1 is equivalent to
$\displaystyle\mathbb{P}_{R,\tau,w_{k}}\left[\overline{\mathbf{Y}}\right]=\overline{\mathbf{U}}_{Y}\left(\mathcal{S}_{R,\tau,w_{k}}\left[\overline{\mathbf{D}}\right]\right)\overline{\mathbf{V}}_{Y}^{\top}$
(15)
where
$\displaystyle\mathcal{S}_{R,\tau,w_{k}}\left[\overline{\mathbf{D}}_{i}\right]$
$\displaystyle=\begin{cases}\sigma_{i}\left(\overline{\mathbf{Y}}\right)&\text{if}\
i<R+1\\\ \max\left(\sigma_{i}\left(\overline{\mathbf{Y}}\right)-\tau
w_{k},0\right)&\text{otherwise}\\\ \end{cases}$ (16)
$\displaystyle=\max\left(\sigma_{i}\left(\overline{\mathbf{Y}}\right)-\tau
w_{k}w_{r},0\right)$
Therefore, the solution of Eq. (7) is
$\begin{array}[]{rl}\mathcal{X}^{*}=\mathcal{D}\mathcal{W}^{w,R,\tau}\left(\mathcal{Y}\right)=\texttt{ifft}(\bar{\mathcal{U}}\cdot\bar{\mathcal{S}}_{dw*}\cdot\bar{\mathcal{V}}^{T},[],3),\end{array}$
(17)
where
$\bar{\mathcal{Y}}=\bar{\mathcal{U}}\cdot\bar{\mathcal{S}}\cdot\bar{\mathcal{V}}^{T}$;
$\bar{\mathcal{S}}_{dw*}(r,r,k)=\max(\bar{\mathcal{S}}(r,r,k)-\tau
w_{r}w_{k},0);$, $R=TW(k)$
$w_{r}=\left\\{\begin{array}[]{l}\begin{matrix}0,&r\leq R\\\ \end{matrix}\\\
\begin{matrix}1,&r>R\\\ \end{matrix}\\\ \end{array}\right.$;
$w_{k}=\frac{c_{1}}{\log(\lVert\boldsymbol{\bar{\textbf{X}}}^{(k)}\rVert_{F}^{2}+\varepsilon)}+c_{2}$.
∎
###### Remark 1.
In theorem 1, there are two key weights that need to be given in advance.
Frequency weights $w_{k}$ can be calculated by Eq. (5). Truncation weight
$w_{r}$ can be calculated by Algorithm 1. It refers to the way multi-rank is
calculated in reference [15]. In Algorithm 1, the ratio of the maximum value
of the singular values is chosen, which can also be replaced by the ratio of
the sum of the singular values.
Algorithm 1 Calculate truncation weight $TW\in\mathbb{R}^{n_{3}}$.
Input: A tensor $\mathcal{Y}$; ratio $\eta$.
Output:Truncation weight $TW\in\mathbb{R}^{n_{3}}$.
1: Compute $\overline{\mathcal{Y}}=\operatorname{fft}(\boldsymbol{Y},[],3)$ .
2: $\text{for}\ i=1,\cdots,n_{3}\ \text{do}$
$[\overline{\mathbf{U}}^{(i)},\overline{\mathbf{S}}^{(i)},\overline{\mathbf{V}}^{(i)}]$
$s=\text{diag}(\overline{\mathbf{S}}^{(i)})$
$TW(i)=\text{length}(s(s>max(s)\times\eta))$.
end for
### III-C Multi-modal and Double-Weighted TNN
TNN in (3) only approximates the correlations connected by mode-3 Fourier
transform in the spatial dimensions with the spectral dimension. It lacks of
flexibility for describing low-rankness in all modes of HSIs. To connect the
$p$-th mode with other two modes, we can define the double-weighted TNN for
each mode-$p$ permutation of HSIs, i.e., $\|\mathcal{X}_{p}\|_{dw*},p=1,2,3.$
As $\|\mathcal{X}_{p}\|_{dw*}$ are different according to different modes, we
use the weighted average of double-weighted TNNs along all modes to
approximate the tensor rank of HSIs. Finally, the multi-modal and double-
weighted TNN (MDWTNN) is proposed as follows:
$\displaystyle\|\mathcal{X}\|_{mdw*}:=\sum_{p=1}^{3}\alpha_{p}\|\mathcal{X}_{p}\|_{dw*}=\sum_{p=1}^{3}\sum_{k=1}^{n_{p}}\alpha_{p}w_{k}^{p}\|\bar{\textbf{X}}^{(k)}_{p}\|_{\textrm{PSSV}},$
(18)
where $\bar{\mathcal{X}}_{p}=\texttt{fft}(\mathcal{X}_{p},[],3)$;
$\bar{\textbf{X}}^{(k)}_{p}$ is the $k$-th frontal slice of
$\bar{\mathcal{X}}_{p}$ and its assigned weight is $w_{k}^{p}$; $\alpha_{p}>0$
and $\sum_{p=1}^{3}\alpha_{p}=1$. Fig. 1 shows the schematic diagram of
MDWTNN.
Figure 1: MDWTNN Schematic diagram. (I) Multi-modal permutations and Fourier
transforms. (II) Double-weighted TNN. (a) mode-3 Fourier transform of
$\mathcal{X}_{p},p=1,2,3$. Due to the repeatability of the operation, II only
represents one of $\mathcal{X}_{1}$, $\mathcal{X}_{2}$, $\mathcal{X}_{3}$, and
the other two only need to repeat the operation represented by II. (b)
different frequency components. (c) PSSV relaxation on each frequency slice
$\\{\bar{\mathcal{X}}_{i}\\}$. (d) The singular values of frequency slices.
(e) Frequency weights. The top is the sum of singular values in each frequency
slice, and the bottom is the weight of each frequency slice. (f) Doubel-
weighted TNN minimization with respect to $\mathcal{X}_{p}$. (III) Synthesis
from multi-modal denoised results.
## IV HSI denoising via MDWTNN Minimization
MDWTNN in (18) takes full advantage of physical meanings in frequency
components, singular values, and modes of HSIs, which can provide a better
approximation to the tensor rank. Then we use MDWTNN to replace the
regularization term Rank in (2) and propose the HSI denoising model as
follows:
$\begin{array}[]{rl}\arg\min_{\mathcal{X},\mathcal{S},\mathcal{N}}&\|\mathcal{X}\|_{mdw*}+\lambda\|\mathcal{S}\|_{1}+\tau\|\mathcal{N}\|_{F}^{2},\\\
s.t.&\mathcal{Y}=\mathcal{X}+\mathcal{S}+\mathcal{N}.\end{array}$ (19)
Introducing auxiliary variables, model (19) is equivalent to
$\begin{array}[]{rl}\displaystyle\arg\min_{\mathcal{X},\mathcal{S},\mathcal{N}}\sum_{p=1}^{3}\alpha_{p}\|\mathcal{Z}_{p}\|_{dw*}+\lambda\lVert\mathcal{S}\rVert_{1}+\tau\lVert\mathcal{N}\rVert_{F}^{2},\\\
\displaystyle s.t.\ \mathcal{Y}=\mathcal{X}+\mathcal{S}+\mathcal{N},\
\mathcal{Z}_{p}=\mathcal{X}_{p},\ p=1,2,3.\\\ \end{array}$ (20)
By augmented Lagrangian multiplier method, the Lagrangian function of model
(20) can be written as
$\begin{array}[]{l}L_{\mu_{p},\beta}\left(\mathcal{X},\mathcal{Z}_{p},\mathcal{N},\mathcal{S},\Gamma_{p},\Lambda\right)=\lambda\lVert\mathcal{S}\rVert_{1}+\tau\lVert\mathcal{N}\rVert_{F}^{2}\\\
+<\mathcal{Y}-\left(\mathcal{X}+\mathcal{S}+\mathcal{N}\right),\Lambda>+\frac{\beta}{2}\lVert\mathcal{Y}-\left(\mathcal{X}+\mathcal{S}+\mathcal{N}\right)\rVert_{F}^{2},\\\
\displaystyle+\sum_{p=1}^{3}{\left\\{\alpha_{p}\lVert\mathcal{X}_{p}\rVert_{dw*}+<\mathcal{X}_{p}-\mathcal{Z}_{p},\Gamma_{p}>+\frac{\mu_{p}}{2}\lVert\mathcal{X}_{p}-\mathcal{Z}_{p}\rVert_{F}^{2}\right\\}},\\\
\end{array}$
where $\Lambda$ and $\Gamma_{p}$ are the Lagrangian multipliers; $\beta$ and
$\mu_{p}$ are the Lagrange penalty parameters. Its minimization problem can be
efficiently solved in the framework of ADMM [16]. At the $(n+1)$-th iteration,
each variable in the Lagrangian function can be updated by solving its
corresponding subproblem respectively when other variables are fixed at the
$n$-th iteration.
For $\mathcal{Z}_{p}$, $p=1,2,3$, their corresponding subproblems can be
written as
$\arg\min_{\mathcal{Z}_{p}}\alpha_{p}\lVert\mathcal{Z}_{p}\rVert_{dw*}+\frac{\mu_{p}}{2}\left\|\mathcal{Z}_{p}-\left(\mathcal{X}_{p}^{n}+\frac{\Gamma_{p}^{n}}{\mu_{p}}\right)\right\|_{F}^{2}.$
(21)
The closed-form solution of (21) obtained from theorem 1 are as follows:
$\begin{array}[]{rl}\mathcal{Z}_{p}^{n+1}=\mathcal{D}\mathcal{W}^{w\left(\mathcal{X}_{p}^{n}\right),R,\frac{\alpha_{p}}{\mu_{p}}}\left(\mathcal{X}_{p}^{n}+\frac{\Gamma_{p}^{n}}{\mu_{p}}\right).\\\
\end{array}$ (22)
For $\mathcal{X}$, its corresponding subproblem can be written as
$\displaystyle\mathcal{X}^{n+1}=$
$\displaystyle\arg\min_{\mathcal{X}}\sum_{p=1}^{3}{\frac{\mu_{p}}{2}}\lVert\mathcal{X}-\mathcal{Z}_{p}^{n+1}+\frac{\Gamma_{p}^{n}}{\mu_{p}}\rVert_{F}^{2}$
(23)
$\displaystyle+\frac{\beta}{2}\lVert\mathcal{Y}-\left(\mathcal{X}+\mathcal{S}^{n}+\mathcal{N}^{n}\right)+\frac{\Lambda^{n}}{\beta}\rVert_{F}^{2}.$
It has the closed-form solution as follows:
$\begin{array}[]{rl}\mathcal{X}^{n+1}=\frac{\sum_{p=1}^{3}{\mu_{p}}\left(\mathcal{Z}_{p}^{n+1}-\frac{\Gamma_{p}^{n}}{\mu_{p}}\right)+\beta\left(\mathcal{Y}-\mathcal{S}^{n}-\mathcal{N}^{n}+\frac{\Lambda^{n}}{\beta}\right)}{1+\beta}.\\\
\end{array}$ (24)
For $\mathcal{S}$, its corresponding subproblem can be written as
$\arg\min_{\mathcal{S}}\lambda\lVert\mathcal{S}\rVert_{1}+\frac{\beta}{2}\lVert\mathcal{Y}-\left(\mathcal{X}^{n+1}+\mathcal{S}+\mathcal{N}^{n}\right)+\frac{\Lambda^{n}}{\beta}\rVert_{F}^{2}.$
(25)
It can be solved by the soft-thresholding operator [14] as:
$\begin{array}[]{rl}\mathcal{S}^{n+1}=\texttt{shrink}\left(\mathcal{Y}-\mathcal{X}^{n+1}-\mathcal{N}^{n+1}+\frac{\Lambda^{n}}{\beta},\frac{\lambda}{\beta}\right).\end{array}$
(26)
For $\mathcal{N}$, its corresponding subproblem can be written as
$\arg\min_{\mathcal{N}}\tau\|\mathcal{N}\|_{F}^{2}+\frac{\beta}{2}\|\mathcal{Y}-\left(\mathcal{X}^{n+1}+\mathcal{S}^{n+1}+\mathcal{N}\right)+\frac{\Lambda^{n}}{\beta}\|_{F}^{2}.$
(27)
It has the closed-form solution as follows :
$\begin{array}[]{rl}\mathcal{N}^{n+1}=\frac{\beta\left(\mathcal{Y}-\mathcal{X}^{n+1}-\mathcal{S}^{n}+\frac{\Lambda^{n}}{\beta}\right)}{2\tau+\beta}.\end{array}$
(28)
For multipliers $\Gamma_{p}$ and $\Lambda$, they can be updated as follows:
$\left\\{\begin{array}[]{l}\Gamma_{p}^{n+1}=\Gamma_{p}^{n}+\mu_{p}\left(\mathcal{Z}_{p}^{n+1}-\mathcal{X}^{n+1}\right),p=1,2,3\\\
\Lambda^{n+1}=\Lambda^{n}+\beta\left(\mathcal{Y}-\mathcal{X}^{n+1}-\mathcal{S}^{n+1}-\mathcal{N}^{n+1}\right).\\\
\end{array}\right.$ (29)
The proposed algorithm for our HSI denoising model is summarized in Algorithm
2.
Algorithm 2 HSI denoising via the MDWTNN minimization
Input: The observed tensor $\mathcal{Y}$; weight parameters $c_{1}$, $c_{2}$,
ratio $\eta$; regularization parameters $\lambda$, $\tau$; and stopping
criterion $\epsilon$.
Output: Denoised image $\mathcal{X}$.
1: Initialize:
$\mathcal{Y}$=$\mathcal{X}$=$\mathcal{S}$=$\mathcal{N}$=$\mathcal{Z}_{p}$;
$\Gamma_{p}=\Lambda=0$; $\mu_{p}$=$\beta$=$10^{-3}$;
$p=1,2,3$; $\mu_{max}=10^{10}$; $\rho=1.2$ and $n=0$.
2: Repeat until convergence:
3\. Update
$\mathcal{Z}_{p}$,$\mathcal{X}$,$\mathcal{S}$,$\mathcal{N}$,$\mathcal{X}$
$\Gamma_{p},\Lambda$ by (22), (24), (26), (28), (29)
Update $\mu_{p}=\rho\mu_{p}$, $\beta=\rho\beta$, $w_{k}$ by (5)
4: Check the convergence condition.
## V Experiments
To verify the effectiveness of our MDWTNN based HSI denoising model, various
experiments were performed on two challenging simulated datasets and two real
HSI datasets. For comparison, four state-of-the-art HSI denoising methodes
were employed as the benchmark in the experiments, i.e., BM4D [17], LRMR [18],
LRTDTV [3] and 3DTNN [7]. Since the BM4D method was only suitable to remove
Gaussian noise, we implemented it on HSIs which were preprocessed by the RPCA
denoising method [6].
### V-A Simulated Data Experiments
In the simulation experiments, we selected two datasets. From the Washington
DC Mall dataset111http://lesun.weebly.com/hyperspectral-data-set.html, we
chose a sub-block with the size of $256\times 256\times 191$ as a simulation
dataset. From the Pavia City Center
dataset222http://www.ehu.eus/ccwintco/index.php/, we chose a sub-block with
the size of $200\times 200\times 80$ as a simulation dataset. The hybrids of
white Gaussian and impulse noises with 5 different intensity levels were added
to the simulation datasets band by band. Let $G$ and $P$ denoted the variance
of Gaussian white noise and percentage of impulse noise, respectively. In
noise case 1-3, the same intensity noise was added to all the bands. In noise
case 1, $G$=0.1 and $P$=0.2; In noise case 2, $G$=0.2 and $P$=0.2; In noise
case 3, $G$=0.1 and $P$=0.4; In noise case 4 and 5, the noise intensities were
different for different bands. In noise case 4, $G$ was randomly selected from
0.1 to 0.2 and $P$=0.2; In noise case 5, $G$=0.1 and $P$ was randomly selected
from 0.2 to 0.4.
For quantitatively evaluating the denoised results of all the test methods,
the CPU times and the means of PSNR, SSIM and SAM in each band, i.e., MPSNR,
MSSIM and MSAM, were listed in Table I. Although the CPU times of our model
were not the shortest, one could update $\mathcal{Z}_{p}$ by (22) in parallel
to further shorten the CPU times of our model. For visual evaluation in Fig.
2, we showed the denoised results of the Pavia City Center dataset in Case 1.
TABLE I: Quantitative comparison and time of all competing methods under
different levels of noises on simulate dataset.
Dataset | Noise case | Index | Noise | BM4D | LRMR | LRTDTV | 3DTNN | Our
---|---|---|---|---|---|---|---|---
Washington DC Mall | Case1 | PSNR | 11.068 | 31.014 | 31.567 | 32.800 | 34.270 | 36.095
SSIM | 0.085 | 0.893 | 0.867 | 0.896 | 0.936 | 0.950
MSAM | 43.139 | 4.576 | 5.042 | 4.327 | 3.481 | 2.937
time | - | 547.005 | 378.898 | 538.160 | 270.211 | 334.656
Case2 | PSNR | 10.216 | 27.192 | 27.642 | 29.489 | 29.055 | 32.525
SSIM | 0.061 | 0.791 | 0.743 | 0.807 | 0.801 | 0.894
MSAM | 45.297 | 6.830 | 7.859 | 6.382 | 6.726 | 4.393
time | - | 529.243 | 395.461 | 584.461 | 309.977 | 378.728
Case3 | PSNR | 8.305 | 29.691 | 29.183 | 30.270 | 29.572 | 34.185
SSIM | 0.037 | 0.866 | 0.798 | 0.844 | 0.784 | 0.928
MSAM | 50.092 | 5.264 | 6.579 | 6.182 | 6.542 | 3.604
time | - | 528.440 | 394.272 | 582.318 | 316.547 | 375.719
Case4 | PSNR | 10.648 | 28.970 | 29.518 | 31.073 | 31.852 | 34.379
SSIM | 0.073 | 0.848 | 0.810 | 0.859 | 0.887 | 0.930
MSAM | 44.265 | 5.695 | 6.439 | 5.468 | 4.873 | 3.579
time | - | 541.960 | 396.478 | 537.374 | 273.466 | 340.113
Case5 | PSNR | 9.669 | 30.419 | 30.412 | 31.560 | 32.783 | 35.180
SSIM | 0.060 | 0.883 | 0.836 | 0.874 | 0.902 | 0.942
MSAM | 47.270 | 4.865 | 5.761 | 5.405 | 4.311 | 3.220
time | - | 546.378 | 397.474 | 541.124 | 277.123 | 340.162
Pavia City Center | Case1 | MPSNR | 11.122 | 29.701 | 31.259 | 32.297 | 31.696 | 33.951
MSSIM | 0.105 | 0.920 | 0.905 | 0.914 | 0.924 | 0.942
MSAM | 45.712 | 5.84 | 6.824 | 4.93 | 4.844 | 4.426
time | - | 112.235 | 113.723 | 131.91 | 54.461 | 72.224
Case2 | MPSNR | 10.265 | 25.619 | 27.321 | 28.605 | 27.009 | 30.124
MSSIM | 0.074 | 0.835 | 0.791 | 0.821 | 0.799 | 0.879
MSAM | 46.978 | 7.170 | 8.385 | 6.685 | 6.923 | 5.666
time | - | 113.246 | 113.115 | 129.155 | 56.610 | 71.322
Case3 | MPSNR | 8.384 | 27.745 | 28.937 | 29.741 | 27.696 | 31.924
MSSIM | 0.043 | 0.892 | 0.848 | 0.873 | 0.790 | 0.916
MSAM | 48.580 | 6.745 | 7.74 | 6.007 | 9.875 | 5.095
time | - | 115.908 | 116.961 | 127.981 | 54.448 | 71.435
Case4 | MPSNR | 10.659 | 27.361 | 29.094 | 30.251 | 28.934 | 32.041
MSSIM | 0.088 | 0.877 | 0.853 | 0.870 | 0.860 | 0.915
MSAM | 46.457 | 6.6 | 7.796 | 5.877 | 6.389 | 5.048
time | - | 115.684 | 133.241 | 129.088 | 51.669 | 68.251
Case5 | MPSNR | 9.565 | 28.674 | 30.044 | 30.977 | 30.055 | 33.038
MSSIM | 0.070 | 0.907 | 0.878 | 0.893 | 0.879 | 0.932
MSAM | 47.885 | 6.293 | 7.333 | 5.493 | 6.795 | 4.682
time | - | 133.811 | 81.989 | 120.418 | 61.742 | 77.520
(a) Original image (b) Noise image (c) BM4D(24.53dB) (d) LRMR(27.84dB)
(e) LRTDTV (27.13dB) (f) 3DTNN(27.55dB) (g) Our(28.08dB)
Figure 2: The denoised results of the Pavia City Center dataset in Case 1.
Pseudocolor image with bands (78, 58, 14).
### V-B Real Data Experiments
In the real experiments, we selected two datasets. From the Indian Pines
dataset333https://engineering.purdue.edu/∼biehl/MultiSpec/hyperspectral, we
chose a sub-block with the size of $145\times 145\times 224$. From the
Australian dataset444http://remote-
sensing.nci.org.au/u39/public/html/index.shtml, we chose a sub-block with the
size of $200\times 200\times 150$. In Fig. 3, we listed all denoised results
of the Indian Pines and Australian datasets. For BM4D and LRTDTV, although
they could remove more noise, they also lost more details. This made the
denoised result too smooth. For LRMR and 3DTNN, they mainly used the low-rank
information of HSI. Although they retained more details, they also retained
more noise. Compared with them, our proposed model could remove more noise
while retaining more details. Fig. 4 showed that the vertical mean profiles of
band 218 before and after denoising. Here, one could see that the curve of the
proposed MDWTNN method was most stable.
(a) Real band (b) BM4D (c) LRMR (d) LRTDTV
(e) 3DTNN (f) Our
(g) Real band (h) BM4D (i) LRMR (j) LRTDTV
(k) 3DTNN (l) Our
Figure 3: All denoised results for the Indian Pines and Australian dataset.
(a)-(f) are the denoised results of the 150 band of the Indian Pines dataset.
(g)-(l) are the denoised results of the 48 band of the Australian dataset.
(a) Real dataset (b) BM4D (c) LRMR (d) LRTDTV
(e) 3DTNN (f) Our
Figure 4: Vertical mean profiles of band 218 by all denoised results for the
Indian Pines.
## VI Conclusion
In this letter, we propose a multi-modal and double-weighted TNN for HSI
denoising tasks. The proposed TNN can efficiently characterize the physical
meanings of the frequency components, singular values, and orientations
ignored by the standard TNN. And the weight parameters also can be obtained
adaptively. They powerfully improve capability and flexibility for describing
low-rankness in HSIs. The experiments conducted with both simulate and real
HSI datasets show that our MDWTNN based HSI denoising model is a competitive
method to remove the hybrid noise. Besides, our proposed MDWTNN regularization
term can also be applied to other low-rankness based tasks, i.e.,
hyperspectral imagery classification, tensor completion, MRI reconstruction.
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|
11institutetext: Department of Physics, DSB Campus, Kumaun University,
Nainital – 263 001, India
11email<EMAIL_ADDRESS>22institutetext: LESIA, Observatoire de
Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place
Jules Janssen, F-92190 Meudon, France 33institutetext: Laboratoire Cogitamus,
1 3/4 rue Descartes, 75005 Paris, France 44institutetext: Centre for
Mathematical Plasma Astrophysics, Dept. of Mathematics, KU Leuven, 3001
Leuven, Belgium 55institutetext: Udaipur Solar Observatory, Physical Research
Laboratory, Udaipur 313004, India
# Observations of a prominence eruption and loop contraction
Pooja Devi 11 Pascal Démoulin 2233 Ramesh Chandra 11 Reetika Joshi 11 Brigitte
Schmieder 2244 Bhuwan Joshi 55
###### Abstract
Context. Prominence eruptions provide key observations to understand the
launch of coronal mass ejections as their cold plasma traces a part of the
unstable magnetic configuration.
Aims. We select a well observed case to derive observational constraints for
eruption models.
Methods. We analyze the prominence eruption and loop expansion and contraction
observed on 02 March 2015 associated with a GOES M3.7 class flare
(SOL2015-03-02T15:27) using the data from Atmospheric Imaging Assembly (AIA)
and the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). We
study the prominence eruption and the evolution of loops using the time-
distance techniques.
Results. The source region is a decaying bipolar active region where magnetic
flux cancellation is present for several days before the eruption. AIA
observations locate the erupting prominence within a flux rope viewed along
its local axis direction. We identify and quantify the motion of loops in
contraction and expansion located on the side of the erupting flux rope.
Finally, RHESSI hard X-ray observations identify the loop top and two foot-
point sources.
Conclusions. Both AIA and RHESSI observations support the standard model of
eruptive flares. The contraction occurs 19 minutes after the start of the
prominence eruption indicating that this contraction is not associated with
the eruption driver. Rather, this prominence eruption is compatible with an
unstable flux rope where the contraction and expansion of the lateral loop is
the consequence of a side vortex developing after the flux rope is launched.
###### Key Words.:
Sun: filaments, prominences – Sun: magnetic fields – Sun: flares
## 1 Introduction
Solar prominence, or filament, eruptions are one of the violent illustrations
of solar activity. Because of their associations with coronal mass ejections
(CMEs) and interplanetary coronal mass ejections (ICMEs), their study is
important for the space-weather point of view (for example see: Schwenn, 2006;
Verbanac et al., 2011; Schmieder et al., 2020). In the standard flare model,
also called the CSHKP model (Carmichael, 1964; Sturrock, 1966; Hirayama, 1974;
Kopp & Pneuman, 1976), the ejection of a flux rope (FR), possibly containing a
filament, drives the magnetic reconnection behind it which implies a two-
ribbon flare. In the FR, the magnetic field lines have helical structures and
the dense as well as cold prominence plasma material is concentrated in
magnetic dips. Therefore, the existence of prominence can be considered as an
indicator of the magnetic FR in solar corona (Schmieder et al., 2013; Filippov
et al., 2015). The relationship between the FR rise and associated radiative
signatures has been a topic of considerable interest (e.g., Alexander et al.,
2006; Liu et al., 2009b; Joshi et al., 2013, 2016; Mitra & Joshi, 2019). In
particular, comparisons of the location, timing, and strength of high-energy
emissions (e.g., temporal and spatial evolution of HXR sources) with respect
to the dynamical evolution of the prominence provide critical clues to help
understand the characteristics of the underlying energy release phenomena,
such as the expected site of magnetic reconnection, particle acceleration, and
heating.
Two- and three-dimensional models have been proposed to explain prominence and
filament eruptions and their associated activities such as the spatial
location and separation of flare ribbons. Models have also been proposed to
explain the triggering mechanism of the eruptions, for example, the magnetic
breakout (Antiochos, 1998; Antiochos et al., 1999), the tether cutting (Moore
et al., 2001; Moore & Sterling, 2006), the kink instability (Sakurai, 1976;
Török & Kliem, 2005), and the torus instability (Forbes & Isenberg, 1991;
Kliem & Török, 2006) models. The instability of an FR, modeled by a line
current in equilibrium in a bipolar potential field, was first proposed by van
Tend & Kuperus (1978) to model solar eruptions. This model was further
developed toward an MHD model (e.g., Démoulin et al., 1991; Forbes & Isenberg,
1991; Lin & Forbes, 2000; Priest & Forbes, 2002; Forbes et al., 2006; Kliem &
Török, 2006; Aulanier et al., 2010). It is presently known as the torus
instability model, or equivalently the catastrophe model (Démoulin & Aulanier,
2010). This catastrophic instability occurs in a bipolar magnetic field
configuration with an FR in equilibrium above the photospheric inversion line
(PIL) of the magnetic field vertical component. This model is tied to the
decrease in the magnetic field strength with height that implies a decrease in
the downward and stabilizing force as the FR is forced to evolve to a larger
height. This evolution is typically generated by the photospheric evolution
(e.g., new magnetic flux emergence, shearing, and/or converging motions around
the PIL). At some critical height, the FR become unstable and erupts.
During the FR eruption, the arcade-like magnetic field lines, passing above
the FR, are stretched upward, while their bottom parts are pushed against each
other below the FR in order to fill the region where the FR was present
earlier on. As a result, a current sheet is created behind the FR. This
induces magnetic reconnection, which results in the formation of closed field
lines at low heights (flare loops) as well as a further build-up of the
erupting FR, by creating new twisted field lines wrapping around the original
FR. This reconnection is crucial as it allows the erupting FR to be ejected
toward the interplanetary space as a CME. However, even with the fastest
possible reconnection rate, the entire incoming magnetic flux below the FR
cannot be reconnected, thus a long and thin current sheet was predicted behind
an erupting FR (Lin & Forbes, 2000). Indeed, pieces of evidence of the current
sheet formation during solar eruptions were reported (e.g., Takasao et al.,
2012; Innes et al., 2015; Scott et al., 2016; Cheng et al., 2018; Lee et al.,
2020).
Observations reveal another aspect of eruptions with contracting and expanding
loops that were observed during eruptive solar flares (Veronig et al., 2006;
Joshi et al., 2007; Liu et al., 2009a, 2012; Simões et al., 2013; Kushwaha et
al., 2014; Petrie, 2016; Dudík et al., 2016, 2017, 2019, and references cited
therein). Liu et al. (2009a) and Joshi et al. (2009) found that the coronal
loop contraction is associated with the converging motion of the conjugate
hard X-ray (HXR) footpoints and the downward motion of the HXR loop top
sources. The contracting and expending coronal loops are located at the
periphery of the legs of the erupting FR, then these loops are different than
the flare loops formed by reconnection behind the erupting FR. Also, the loops
which are located near the active region (AR) contract first, whereas the
loops located far away contract later on (Gosain, 2012; Simões et al., 2013;
Shen et al., 2014). Both Shen et al. (2014) and Kushwaha et al. (2015)
observed the contraction of loops in the pre-flare phase for a duration of
about half an hour. Dudík et al. (2016) also reported the expansion and
contraction of loops during an X-class flare. They interpreted this as the
result of growing FR that subsequently erupts. Next, they presented the
observations of loop expansion and contraction for two eruptive flares (one
major GOES X-class and another small GOES C-class). In these events, the
expanding and contracting loops coexist for a period of more than half an
hour. The observed speeds varies from 1.5 to 39 km s-1. Wang et al. (2018)
considered four events with loop contraction. In the first event, the
contraction occurred during the impulsive phase. For the second event, the
prominence was already erupting before the contraction (see their movie).
Finally, for the two last events, no evidence of ejection was present.
In earlier papers, the loop contraction was associated with the conjecture
proposed by Hudson (2000). According to this conjecture in low plasma $\beta$
and with negligible gravity, the magnetic energy released in a solar eruption
must originate in a “magnetic implosion”. More specifically, some portion of
solar corona needs to implode in order to decrease the total magnetic energy,
then to a power a solar eruption. Shen et al. (2014) and Russell et al. (2015)
instead interpreted the loop contraction as a modification to the equilibrium
of nearby loops due to the balance between the magnetic pressure and the
magnetic tension. Liu & Wang (2009) also reported the expansion and
contraction of some coronal loops. They found the expansion and contraction of
the overlying coronal loops during the eruption, such that when the filament
started to rise, the loops were also pushed upward, and as soon as the
filament rose explosively, they began to contract. In the slow phase of the
eruption, loops expanded at a slow speed ($\sim$8 km s-1). This expansion
became fast in the impulsive phase, with a speed roughly equal to the speed of
the eruption ($\sim$56 km s-1). When the filament erupted out, the loops began
to contract inward with a speed ranging from 60 km s-1 to 140 km s-1.
In contrast, Dudík et al. (2016) proposed that the apparent implosion is a
result of the large-scale dynamics involving the FR eruption in three
dimensions. Next, Dudík et al. (2017) explained their observations of two
eruptive flares on the basis of the three-dimensional (3D) MHD model of
erupting FR proposed by Zuccarello et al. (2017). This model was derived from
the analysis of 3D line-tied visco-resistive MHD simulations realized with the
OHM-MPI code (Zuccarello et al., 2015). In this model, the eruption is
triggered by the torus instability and the corona is treated as a zero $\beta$
plasma without gravity. According to these numerical simulations, the coronal
loop expansion and contraction during the FR eruption are the result of
hydromagnetic effects related to the generation of a vortex on each side of
the erupting FR. These vortices lead to advection of closed coronal loops.
Outward flows and returning flows are in the vortex part close and further
away from the erupting FR, respectively. This implies the possible
simultaneous presence of loops in expansion and contraction.
In this paper, we analyze the observations of the prominence eruption on 02
March 2015. This event shows the erupting FR and the associated flare loops in
various EUV and X-ray wavelengths. Moreover, coronal loop expansion and
contraction are well observed on one side of the eruption. The paper is
organized as follows: Sect. 2 presents the observational data, with a
description of the AR magnetic field evolution on the solar disk then of the
temporal and spatial evolution of the prominence eruption. Next, the time-
distance analysis of loop expansion and contraction is presented in Sect. 3,
followed by a theoretical analysis. Finally, we summarize our results and
conclude in Sect. 4.
Figure 1: Spatial evolution of the prominence eruption observed in GONG
H$\alpha$ (a – e), AIA 304 (f – j), 171 (k – o), and 193 (p – t) Å wavelengths
(see Electronic Supplementary Materials for the movies). The onset of the
eruption is $\approx$ 14:40 UT. The northern and southern legs of the filament
are labeled as the “Northern Leg” and “Southern Leg” in panels (h) and (i),
respectively. RHESSI X-ray contours of 6 – 12 (pink), 25 – 50 (blue), and 50 –
100 keV (red) energy ranges were over-plotted on AIA 193 Å images in (q – t).
The contour levels were set to 50% and 80% of the peak flux of the X-ray flux
peak. The integration time for RHESSI images is 20 sec. Figure 2: Spatial
evolution of the eruption using the running difference method (a – e) and with
MGN processed images of the prominence eruption in AIA 304 (f – j) and 171 (k
– o) Å wavelengths (see Sect. 2.1). The northern and southern legs of the
filament are indicated by arrows in panels (h) and (i), respectively. The
flare loops represented by “hot loops” are shown in panel (j). The black and
cyan arrows in panel (h) show the outer and inner quasi circular features
which is the FR configuration.
## 2 Observations
On 02 March 2015, an M3.7 class flare, associated with a prominence eruption,
occurred between 15:10 UT and 15:40 UT in the AR NOAA 12290, close to the
western limb of the Sun (N21W86). For our present study, we have analyzed the
data from the following instruments:
Firstly, the _Atmospheric Imaging Assembly_ (AIA: Lemen et al., 2012) on board
the _Solar Dynamics Observatory_ (SDO: Pesnell et al., 2012) observes the
different layers of the Sun in seven EUV channels, two UV channels, and one
white-light channel with a cadence of 12 seconds, 24 seconds, and 3600
seconds, respectively. The pixel size of AIA data is 0′′.6. In this paper, we
have analyzed AIA images in 304 Å, 171 Å, and 193 Å wavelengths to study the
prominence eruption, flare, and loop expansion and contraction during the
eruption. The AR magnetic configuration was analyzed with the _Helioseismic
Magnetic Imager_ (HMI: Schou et al., 2012). The cadence of HMI data is 45
seconds and the pixel resolution is 0′′.5.
Secondly, the _Reuven Ramaty High Energy Solar Spectroscopic Imager_ (RHESSI:
Lin et al., 2002) observed this flare from the rise phase at 15:15 UT until
its decay phase at 15:35 UT. RHESSI observes the full Sun with an
unprecedented combination of spatial resolution (as fine as $\approx$ 2′′.3)
and energy resolution (1 – 5 keV) in the energy range from 3 keV to 17 MeV.
Thirdly, the eruption was also observed by the _Global Oscillation Network
Group_ (GONG: Harvey et al., 2011) in the H$\alpha$ center. GONG continuously
observes the full Sun with observatories spread around Earth. The temporal and
the pixel resolution of this data are 1 minute and 1′′, respectively.
The morphology and the temporal evolution of the prominence eruption and of
the associated flare are observed in H$\alpha$, EUV, and X-rays. They are
described in the following subsections.
Figure 3: (a) AIA 304 Å image showing the ejected plasma and the position of
the selected slice. (b) Time-distance plot along the selected slice. When the
emission is less saturated and the spatial extension large enough, the
erupting plasma has a range of velocities as displayed with over-plotted
straight lines approximating the mean plasma blob trajectories (velocities are
still increasing with time). This outlines the global expansion of the FR.
### 2.1 Morphology of the prominence eruption
Figure 1 displays the evolution of the prominence eruption observed in
H$\alpha$, AIA 304, 171, and 193 Å. The onset of the eruption is $\approx$
14:40 UT and almost simultaneous in all wavelengths (see the movie attached to
Fig. 1 and the analysis in Sect. 3.1). The major part of the prominence erupts
non-radially in the northwest direction. The extension of the erupting
structure is better outlined in 304 Å with a distribution of emitting plasma
showing a quasi circular type of feature. The filament height grows with time
and a void of emitting plasma in 304 Å develops, outlining the core of the FR
(Fig. 1i,j). The quasi circular pattern and the central void are indications
that the line of sight is nearly aligned with the local direction of the FR
axis. This event is comparable to the one studied by Shen et al. (2014);
however, it is more clearly structured than the other event, likely because it
was observed more along the FR axis direction.
During the eruption, some part of the prominence is diverted toward the west
side (Fig. 1g – j and the attached movie). We interpret this as prominence
plasma which is initially located southward and within a part of the FR more
inclined on the line of sight. Then, the FR axis changes along the
configuration which is a 3D writhed configuration. The presence of a twisted
field is much less obvious in this southern part as the FR is seen partly from
the side.
During the uplifting of the filament, the coronal plasma shows a twisted
structure in AIA 171 and 193 Å (Fig. 1 bottom rows). However, we cannot define
a sign for the twist because we cannot determine which structures are in the
foreground and background with the optically thin coronal emission. The
presence of a prominence before the eruption is compatible with the FR since
an upward curvature of magnetic field lines is needed to support the dense
prominence plasma against gravity. However, a sheared magnetic arcade can also
possess magnetic dips and hence be capable of supporting a filament against
gravity (see the review of Mackay et al., 2010).
In H$\alpha$ observations, the upper part of the prominence is first seen to
get detached from a remaining part staying above the chromosphere (Fig. 1a,b).
Later on, the erupting prominence splits into two parts (Fig. 1c,d). The
superposition with the co-aligned 304 Å data shows that one part is located
below the FR center, which is as expected in the gravitationally stable
configuration with upward curvature of the field lines, while the other part
is located well above the FR axis which is not expected. This implies that
enough kinetic energy should be injected into this plasma during the earlier
phase of eruption in order to bring it up in the FR. This observation could be
related to the results of Lepri & Zurbuchen (2010) at 1 AU, which were
obtained using the Solar Wind Ion Composition Spectrometer (SWICS)
measurements. They found that cold plasma could be present in a fraction of
the extension of some magnetic clouds, with no specific location, in
particular not necessarily in the rear part.
As the prominence erupted, a straight and emitting structure is formed behind
the erupting FR (Fig. 1h). This is most likely the northern filament leg that
is stretched thin and long by the erupting magnetic configuration. Next, this
structure gets broader and less coherent with time (Fig. 1i,j). The southern
filament leg is also visible in Fig. 1i. In the later phase of the eruption,
flare loops formed which are seen in projection below the northern leg of the
filament (see Fig. 1j), although we cannot say it is actually below the leg of
the filament or not because of the line-of-sight confusion on the limb.
For a better understanding of the eruption, we created images using the
running difference method (Fig. 2, top row). These images were obtained by the
subtraction of the image obtained one minute before. These images outline the
development of the filament eruption in AIA 304 Å with a contrast better than
the original images as the method emphasizes the changes. The FR is better
seen in the difference images at 15:19 and 15:21 UT with dark gray and quasi
circular-like structures (Fig. 2c,d).
A limitation of the running difference method is that we do not see the loops
well. Then, we also applied the Multi-Gaussian Normalization (MGN) method
developed by Morgan & Druckmüller (2014). The evolution of the MGN images of
AIA 304 and 171 Å are presented in the middle and bottom rows of Fig. 2,
respectively. The MGN method defines a normalized image by using the local
mean and standard deviation computed with a local Gaussian function (called a
kernel). The observed image was first convolved with the kernel to compute the
local mean and standard deviation. By subtracting the local mean and dividing
by the local standard deviation, the MGN method defines the normalized image.
Then, this normalized image is transformed by the arctan function. This
process was repeated with different kernel widths, and the final image is a
weighted combination of the normalized components.
Figure 4: Evolution of the AR magnetic field from 25 to 28 February 2015. The
locations of the flux cancellation are shown by green ellipses. The location
of the H$\alpha$ filament on 28 February 2015 is displayed in panel (d) with a
red contour. A movie of magnetic field during 25 – 28 is available in the
Electronic Supplementary Materials.
Compared to Fig. 1, the MGN method emphasizes the northern and southern legs
of the filament as pointed in Fig. 2h,i. The internal structure of the
erupting FR is also better seen than in the original images. When well
identified with the MGN method, most of the structures can also be identified
in the original images, albeit less precisely. In particular, parts of
circular structures are also present inside the quasi circular structure
identified in Fig. 1 and marked with arrows in Fig. 2h. With a likely 3D
writhed configuration outlined by only a few loops that are dense enough, it
is indeed remarkable to be able to see the traces of an FR in part of the
erupting configuration. Next, similar erupting structures are present in the
different AIA filters. This indicates an important plasma component within the
range of transition region temperatures because this is the temperature range
shared by the temperature response functions of the three AIA EUV filters.
The MGN method also allows one to see the loops surrounding the erupting FR,
since they are barely seen in the original data. These loops are not
necessarily near the eruption as the integration length along the line of
sight is large at the solar limb. For example, the eruption is seen to
progress on a foreground or background of undisturbed large-scale loops in AIA
171 Å, while other loops, of a similar height as the eruption in Fig. 2l,m,
are evolving; the optically thin emission did not allow us to determine which
structure is in front. The full dynamic of the coronal loops and of the
eruption is shown in the associated movies.
The prominence plasma is nearly at a constant location before 14:20 UT and
later on it progressively accelerates outward with a significant inclination
on the local vertical (Fig. 3). The early part of the eruption, before 15:15
UT, is too saturated in 304 Å to see more than a global upward motion. Later
on, an increasing spreading of the erupting plasma started to develop with
different plasma blobs moving at different speeds. The average speeds range
from 110 to 240 km s-1 within the FR. These speeds were calculated using a
time distance analysis of AIA 304 Å data sets. In Fig. 3b, we outline the main
structures with straight dashed lines. Plasma in different parts of the FR
move at a different speed. We interpret the range of speed to be due to the
expansion of the FR. The cone shape defined by these lines indicates that the
FR expands nearly self-similarly with a linear rescaling with time of its
spatial configuration (i.e., the FR is rescaled globally with a scalar factor
depending linearly on time). The expansion speed of the FR border, relative to
the FR center, is $\approx$ 65 km s-1, which is slightly more than one-third
of the speed of the FR center ($\approx$ 175 km s-1), then we conclude that
the expansion is significant.
Later on, this prominence eruption was accompanied by a CME observed by the
Large Angle and Spectrometric Coronagraph (LASCO, Brueckner et al. (1995))
onboard the SOHO satellite and it is given in the SOHO/LASCO CME catalog at
https://cdaw.gsfc.nasa.gov/CME$\\_$list/ (Yashiro et al., 2004). The CME first
appears in the LASCO C2 field-of-view at $\approx$ 15:45 UT. The speed of the
CME leading edge is 452 km s-1 so it is less than twice its velocity in the
low corona (Fig. 3b). At the distances observed by C2, the CME was already
decelerating with a mean value of deceleration of 12 m s-2.
Figure 5: Evolution of the corona in AIA 171 Å from 6 to 1 day before the
studied eruption. A sheared loop built up in the AR core from 24 to 28
February (green and blue arrows in panels (a – e). The purple arrows point to
large scale loops of the AR (a – e) which become open, or at least large scale
in panel (f). The location of the filament is displayed in panel (d, e) with a
red contour.
### 2.2 Photospheric magnetic field
We analyze the AR magnetic configuration with the HMI magnetic field data.
Since the studied eruption was at the limb on 02 March 2015, the photospheric
magnetic field is not available. Therefore, we follow its associated AR
evolution when it was on the solar disk. AR NOAA 12290 was on the eastern limb
on 18 February 2015. One day later, it started to show on the solar disk as a
small bipolar AR with a negative leading and a positive following polarity.
During its disk crossing, it progressively grew in size as its magnetic field
got dispersed by convective motions. Then, AR 12290 was well in its decay
phase when it reached the western limb.
Figure 4 shows four magnetograms of the longitudinal field component with a
time cadence of one day. The AR was located at $\approx$ 15∘ of longitude
after the central meridian on 25 February 2015 in panel (a), and at $\approx$
55∘ of longitude on 28 February 2015 in panel (d). The magnetograms are
spatially de-rotated to the central meridian so that the spatial extension of
the polarities could be compared between the different panels. However, we
kept the longitudinal component so projection effects appeared as the AR
approached the limb (mostly on the limb side of the leading polarity where
fake positive polarities appear). The magnetic field mostly disperses with
time in each polarity so that the polarities become more extended. Then, the
global bipolar magnetic configuration is progressively larger in size, which
is expected to induce a global expansion of the coronal loops. This dispersion
also implies that the polarities of opposite signs become into contact at the
PIL. This induces cancellations of the magnetic flux, for example, at the
locations surrounded by a green ellipse in Fig. 4a,c. These cancellations of
flux are best seen in the movie associated with Fig. 4. Such cancellations
imply a progressive buildup of an FR above the PIL (van Ballegooijen &
Martens, 1989; Amari et al., 2003; Green et al., 2018). Finally, the FR
becomes unstable leading to an eruption (Sect. 2.1).
The above analysis shows that AR 12290 is mainly a decaying bipolar AR with
cancellations occurring at the PIL. This evolution is expected to continue
slowly in the following days as typically observed in decaying ARs (see the
review of van Driel-Gesztelyi & Green, 2015). Therefore, it is justified to
take the closest possible on-disk magnetic field of the western limb to
approximate the photospheric field distribution during the eruption. The last
magnetogram shown (Fig. 4d) is about 2 days before the studied eruption at the
limb. The location of the filament on 28 February 2015 is over-plotted with a
red contour.
Next, AR 12290 has the typical configuration of ARs with a more dispersed
following polarity than the preceding one (van Driel-Gesztelyi & Green, 2015).
This may lead to the non-radial eruption as present in numerical simulations
(Aulanier et al., 2010). However, the asymmetry seen in Fig. 4 is moderate and
the tilt of the AR bipole on the solar equator is small while the observed
eruption is tilted well away from the radial direction (Fig. 1).
Figure 6: GOES and RHESSI X-ray time profiles of the M3.7 flare. For RHESSI,
profiles were plotted in the energy bands of 6 – 12 keV (pink), 12 – 25 keV
(green), 25 – 50 keV (blue), and 50 – 100 keV (red). For clarity of
presentation, we scaled RHESSI count rates by factors of 1, 1/2, 1, and 1/5
for 6 – 12 keV, 12 – 25 keV, 25 – 50 keV, and 50 – 100 keV energy bands,
respectively. RHESSI light curves were corrected for change in attenuator
states during flare observations. Horizontal bars at the bottom represent the
RHESSI observing state (N: night; F: flare).
Fig. 4, and more precisely the associated movie, shows the convergence of
polarities of both signs at the PIL. This induces a magnetic shear and indeed
positively sheared loops are observed to develop from 24 to 28 February with
AIA 171 Å observations (Fig. 5a – e). The polarity cancellation is expected to
transform these sheared loops in an FR. This cancellation happens, in
particular, at the northern part of the PIL (Fig. 4a – c). A C3.7 flare starts
after 05:00 UT on 01 March 2015. This is the largest event observed in AR
12290 since the beginning of its disk transit. This event is eruptive and
associated with a slow CME with a speed $\approx$ 191 km s-1. This event
changes the coronal configuration drastically by opening the magnetic field on
the northern part of the AR. This is seen after the flare in AIA 171 Å
observations by the disappearance of the coronal loops, and the creation of a
region of low emission embedded in open, or at least large-scale, loops (Fig.
5f). Then, we interpret the non-radial motion of the filament as a consequence
of the transformation generated by the previous event, which occurred around
05:00 UT on 01 March 2015. This event opens, or at least creates larger
scales, within the northern coronal field of the AR. This new configuration is
expected to channel the later filament eruption, which is then diverted from
the central AR part toward its northern side (Fig. 1).
### 2.3 Temporal and spatial evolution of X-ray sources
The physical processes during a flare associated with the prominence eruption,
in particular the transformation of magnetic to kinetic energy (acceleration
of particles), are best characterized by X-ray emissions. The temporal
evolution of the flare in HXR and associated sources yield insights about the
eruption phenomena (heating and nonthermal emissions) in the source region.
For this purpose, we compare GOES and RHESSI light curves for the M3.7 flare
associated with the prominence eruption in Fig. 6. The GOES 1 – 8 Å time
profile shows a typical temporal behavior of a long duration event during
$\approx$ 15:00 – 15:40 UT with a distinct peak at $\approx$ 15:28 UT. The
peak in a higher energy light curve of GOES (i.e., 0.5 – 4.0 Å) occurs
slightly earlier at $\approx$ 15:27 UT, which is expected. RHESSI profiles at
energies $\leq 25$ keV indicate a continuous rise in the X-ray count rate
until $\approx$ 15:26 UT and they decline thereafter. From these RHESSI light
curves, we note the rise phase to be more gradual than the decline phase which
is rather unusual.
RHESSI 25 – 50 and 50 – 100 keV light curves have a maximum at $\approx$ 15:25
UT. Notably, in the 25 – 50 keV profile, we observe significant fluctuations
with a few pronounced sub-peaks during the rise phase. Furthermore, a
prominent sub-peak around 15:21 UT can also be seen at the high energy channel
of 50 – 100 keV (indicated by left vertical line). These sub-peaks at $>$25
keV represent distinct episodes of particle acceleration as the erupting FR
forces magnetic reconnection events underneath it. Apart from two brief
intervals (around the two vertical lines), the flare does not show flux
enhancement in the 50 – 100 keV energy band observations.
Figure 7: A few representative RHESSI intensity isocontours showing relative
positions and spatial evolution of X-ray sources in 6 – 12 keV (pink), 12 – 25
keV (green), 25 – 50 keV (blue), and 50 – 100 keV (red) during the M3.7 flare.
The contour levels were set at 55%, 70%, 80%, and 95% of the peak flux of each
image. The integration time for each image is 20 sec.
Limb events provide us with a unique opportunity to distinguish the HXR
emission that originated at the coronal loop tops from that of their foot-
points. To discuss this in detail and to study the morphological evolution of
HXR sources, we utilized the imaging capabilities of RHESSI. We reconstructed
RHESSI images by the CLEAN algorithm with the natural weighing scheme (Hurford
et al., 2002) using front detector segments 3 – 8 (excluding 7). The images
are produced at 6 – 12, 12 – 25, 25 – 50, and 50 – 100 keV energy bands. In
Fig. 7, we present a series of co-temporal RHESSI images in different energy
channels. The counts statistics during the selected times, as seen from the
time profiles, together with the integration time of 20 sec ensure the
reliability of the HXR source structures. To examine the spatial location of
HXR sources with respect to erupting prominence, we plotted HXR contours over
AIA 193 Å images in Fig. 1q – t.
Comparisons of RHESSI images at different channels show a complex structure
and evolution of the X-ray sources. At the beginning, the X-ray sources, which
were observed up to 50 keV energies, are co-spatial (Fig. 7a). This X-ray
emitting region lies in the lower corona where the initiation of the plasma
eruption occurred. Next, we note the onset of the X-ray emission from a new
location which lies northeast of the initial X-ray emitting region (Figs. 1r
and 7b,c). Around the first HXR peak ($\approx$ 15:21 UT), the X-ray sources
at 25 – 50 keV exhibit an extended structure with a distinct appearance of a
coronal source beside another one at lower heights (Fig. 7e). In the
subsequent phases, the X-ray emission is only observed from the latter, which
developed in the northeastern X-ray emitting site (Fig. 7f – l). It is
noteworthy that, although the 50 – 100 keV sources appear very briefly during
the flare maximum at 15:25 UT, they exhibit an extended structure from lower
to higher coronal heights with the strongest source extending onto the solar
disk (Fig. 7h). These sources probably are a composite emission from the
coronal and foot-point regions (see also Fig. 1t). Finally, the coronal
emission continued during the decline phase at energies up to 25 keV (Fig. 7i
– l).
The temporal and imaging analyses of HXR emission suggest a large variability
in the flux for all sources is present, especially in the hardest channels.
These observations are in agreement with the standard model of the formation
of a coronal source above the flare loops observed in EUV and two foot-point
sources in the magnetically connected foot-points. These X-ray sources are
expected to be formed by the energetic particles that accelerate at the base
of the reconnecting region or in the newly formed flare loops, which are
typically in contraction, and thus a favorable configuration for particle
acceleration (e.g., see the review by Benz, 2017).
Figure 8: Location of slices S1 and S2 (top row) chosen for the time-distance
analysis of AIA 304 (a), 171 (b), and 193 (c) Å. The time-distance plots for
slice S1 and S2 are presented in the middle and the bottom rows, respectively,
in AIA 304, 171, and 193 Å. The blue dashed line in panels (d-f) is the fit of
a combination of a linear and exponential function to the trajectory shown in
the time slice data. The vertical white dashed line in the time-distance plots
indicates the onset time of the prominence eruption and also the time of the
images shown in the top row. Slice S1 was selected in the direction of the
prominence eruption so that the erupting plasma would be clearly visible in
the time-distance image (middle row). Slice S2 was selected on the opposite
side of the AR from the eruption region to explore the extension of the
eruption effects. A movie of AIA 304, 171, and 193 Å is available in the
Electronic Supplementary Materials.
## 3 Loop expansion and contraction
### 3.1 Observational evidences
In the present study, we found observations of coronal loop expansion and
contraction at various EUV wavelengths of the AIA telescope during the
prominence eruption on 02 March 2015. To understand this phenomena, we created
time-distance plots in several directions and finalized four slices. We name
these slices S1, S2, S3, and S4. The measured speeds for all the selected loop
systems and the time ranges for the expansion and contraction are tabulated in
Appendix A with Table 1.
Slice S1: The top panels of Fig. 8 depict the location of slice S1 and S2.
Abscissa $s_{1}$ and $s_{2}$ are defined along the cuts, and the associated
time-distance plots are shown in the middle and the bottom panels,
respectively. Slice S1 was set in the direction of prominence eruption. From
panels (e) and (f) in Fig. 8, we inferred that the eruption is initiated
around 14:40 UT when a significant upward displacement of the emitting plasma
could be detected after a phase with a nearly constant abscissa $s_{1}$. Later
on, the prominence plasma progressively accelerated with an exponential
behavior. Then, we used a combination of an exponential and linear increase of
height as a function of time to perform a least square fit of the data (Cheng
et al., 2020). The fitted function is in the form of $f(t)=a\,e^{t/b}+c\,t+d$,
where a, b, c, and d are the coefficients determined by the fit to the data.
This fitted function nicely describes the prominence plasma evolution observed
in 171 and 193 Å with no significant difference between the two filters
(panels (e) and (f) of Fig. 8).
Slice S1 of AIA 304 Å mostly images the plasma located at a low height around
$s_{1}\approx$25′′ at 14:00 UT (Fig. 8d). This plasma becomes dark around
14:40 UT, which is the starting time of the eruption, then it becomes very
bright, saturating the detector. Later on, the emitting plasma accelerated
upward, up to a velocity around 200 km s-1 (Fig. 3b). After the prominence
eruption, the falling back of erupting prominence plasma was observed (see
attached movie). The function fitted to 171 and 193 Å data is also compatible
with the 304 Å observations which are less constraining at lower $s_{1}$
values (Fig. 8d).
We label ”a” the loop system located well above the erupting prominence in the
range of $s_{1}\approx$120 – 180′′ at 14:40 UT (Fig. 8b). In both 171 and 193
Å, the loop system ”a” is in global expansion before the prominence eruption.
The expansion velocity of this loop system in AIA 171 Å varies from 7 to 9 km
s-1, and in AIA 193 Å from 9 to 10 km s-1 with a tendency of a slightly larger
velocity at lower height. This expansion becomes much faster when the
prominence eruption approaches $s_{1}\approx$100$\arcsec$, around 15:10 UT;
the prominence plasma is clearly seen in 304 Å in panel (d), and at the same
locations in 171 and 193 Å in panels (e) and (f).
We also find some loops in both 171 and 193 Å which are stationary before and
during the prominence eruption process (e.g., $s_{1}\approx 15,\,115,$ and
$145\arcsec$ at 14:00 UT in panels (e, f)). These stationary structures are
almost vertical (radial), and more precisely they trace the legs of large-
scale loops in panels (b, c). With the projection effect at the solar limb, we
observe them closer to the eruption than they really are. These stationary
loops are not likely rooted in the vicinity of the eruption site, thus they
are not disturbed during the eruption process.
Figure 9: Location of slices S3 and S4 (top row) chosen for the time-distance
analysis of AIA 304, 171, and 193 Å presented in the panels (a), (b), and (c),
respectively. These slices were selected for the analysis of loop expansion
and contraction. Different expanding and contracting loops are shown by dashed
lines in the middle and bottom rows. The white vertical dashed line indicates
the onset time of the eruption and the time of the top row images. The loop
systems in the direction of slices S3 and S4 are labeled in panels (e) and
(h), respectively. A movie of AIA 304 Å is available in the Electronic
Supplementary Materials.
Slice S2: The bottom panels of Fig. 8 display the time-distance plots related
to slice S2 in AIA 304, 171, and 193 Å. We label ”f” the main loops in this
direction (Fig. 8b). This slit tests if the eruption affects the side of the
AR opposite to the eruption location. In both 171 Å and 193 Å wavelengths, the
expansion speed before eruption is about 2 km s-1, which is slow (Fig. 8h,i).
This expansion mostly continues for some loops during and after the eruption
in 193 Å, while there is some evidences of contraction for others. The
observations are clearer in 171 Å with a contraction starting at $\approx$
15:10 UT, so during the eruption, for example, at $s_{2}\approx$ 40$\arcsec$.
This is followed by a relatively fast expansion $\approx$ 6 km s-1 starting at
$\approx$ 15:40 UT.
Slice S3: The top panels of Fig. 9 show the location of slice S3 and S4. The
associated time-distance plots are shown in the middle and bottom panels,
respectively. Before the eruption, AIA 304 Å only detected plasma at
$s_{3}\approx$18′′ (Fig. 9d). After $\approx$ 14:30 UT, this plasma moved up
with a global expansion speed of about 11 km s-1. Then, at $\approx$ 15:02 UT,
the motion changed to a downward motion at a similar speed. At larger heights,
$s_{3}>50\arcsec$, part of the ejected plasma crossed S3.
The observations in hotter temperature ranges have different and complementary
information on three main sets of coronal loops. The loops in AIA 171 and 193
Å crossing S3 are labeled ”b” ($s_{3}$ in 25 – 60$\arcsec$), ”d” ($s_{3}$ in
70 – 85$\arcsec$), and ”e” ($s_{3}>$ in 90$\arcsec$) at 14:40 UT (Fig. 9b).
These loops are in global expansion with a speed in the range of 3 to 6 km
s-1, with the speed decreasing with height. After this global expansion, these
loops suddenly contracted at $\approx$ 15:18 UT in phase at different heights.
Later on, around 15:40 UT traces of ejected plasma were present in both 171
and 193 Å while much less visible than in AIA 304 Å (Fig. 9d).
A closer analysis shows that the loop systems have some differences in their
evolution. The loop system labeled ”b,” which is between $s_{3}=25$ and
$60\arcsec$ at 14:40 UT, contracts earlier, as it starts at 14:59 UT (Figs.
9e,f). The speed of this earlier contraction is in the range of 9 – 22 km s-1.
Next, the loop, labeled ”d” (Fig. 9b), is likely a set of unresolved loops. It
has a different shape and likely also different photospheric connections than
loop systems ”b” and ”e.” Loop ”d” started to contract later than ”b,” around
15:16 UT, in phase with all the loops located at larger heights (system ”e”).
The speed of this contraction varies from 9 to 34 km s-1 with a tendency of a
larger speed at a larger height. The magnitude of these speeds is comparable
to the speed of the earlier contraction of ”b,” while higher than the speed of
the earlier expansion. The above speed values are coherent with the expansion
and contraction speed found previously, from a few (1 – 2) km s-1 to 39 km
s-1, for the eruption of 05 March 2012 and 19 June, 2013 (Dudík et al., 2017).
Slice S4: The time-distance plots for slice S4 in AIA 304, 171, and 193 Å are
presented in panels (g), (h), and (i), respectively, in Fig. 9. In 304 Å, an
upward motion, followed by a downward one, of loop system ”b” is observed as
was previously observed for slice S3. Indeed, the whole loop system ”b” moves
in phase (see the associate movie in 304 Å). In 171 and 193 Å, the loops
encountered with growing $s_{4}$ abscissa are labeled as ”c,” ”b,” ”d,” ”b,”
and ”e.” We observed a similar expansion before the eruption, followed by a
contraction during the eruption as in the case of S3 for the loops ”b,” ”d,”
and ”e.” Again, these loops move in phase all along their length. An earlier
contraction of loops ”b” and ”d” is also present for $s_{4}<90\arcsec$,
starting at 14:59 UT just as for S3 . The speed of this earlier contraction
varies from 6 to 22 km s-1.
### 3.2 Theoretical interpretation
As the analyzed eruption is at the limb, it has less background and foreground
than when an eruption is observed on the solar disk. Then, the early coronal
evolution of this event can be observed better. There is a weak upward motion
of the prominence before the event in the time range of 14:00 – 14:30 UT in
Fig. 8e,f. This evolution is likely driven by the photospheric cancellation of
the magnetic field at the PIL, which could only be observed days before an
eruption. (Sect. 2.2). Then, the prominence plasma seen in H$\alpha$ and in
AIA filters starts to progressively move up faster with an exponential growth,
which rapidly dominates in magnitude the earlier linear evolution of its
position. The nonlinear increase in s1 abscissa starts to be significant
around 14:40 UT. This prominence plasma traces a part of the magnetic
configuration and the start of its exponential evolution indicates that the
magnetic field becomes unstable. This defines the physical starting time of
the eruption.
Around the prominence eruption starting time, 14:40 UT, GOES fluxes weakly
evolve with time. This small evolution continues even much later on as both
GOES channels start to increase in flux only at $\approx$ 15:05 UT, so 25
minutes later (Fig. 6). The increase in EUV flux is also weak during that time
period and even in localized regions of the AR. We interpret these data with
the model of Lin & Forbes (2000). The time interval between $\approx$ 14:40
and 15:05 UT corresponds to an almost ideal-MHD evolution of the FR which is
unstable. In the model, a current sheet forms behind the FR with a
reconnection rate that is too slow to process a significant amount of the
magnetic field. The small amount of magnetic energy that is liberated is
mostly transformed to macroscopic kinetic energy and it builds the current
sheet up; we did not succeed identify this in the present observations. Then,
the small amount of magnetic flux reconnected per unit of time, with its
associated magnetic energy release, is expected to have a low contribution to
the coronal emissions, as observed.
We interpret the increase after $\approx$ 15:05 UT of the X-ray flux measured
by GOES as an indication of significant magnetic reconnection, which
accelerates energetic particles and then heats the plasma; this results in the
evaporation of part of the chromospheric plasma, then an increase in the
coronal density, and finally an enhancement to the EUV emission. This
reconnection further decreases the stabilizing effect of the overlying
magnetic field. Then, a strong increase in the prominence speed (Fig. 8d) is a
logical consequence since there is positive feedback on the dynamic of the
reconnected field with a decrease in the downward magnetic tension (Welsch,
2018). Further, a sudden transition in the kinematic evolution of the
prominence from its slow to fast ascent together with the simultaneity in the
increase of the HXR flux point toward the feedback process between the early
dynamics of the eruption and the strength of the flare magnetic reconnection
(Temmer et al., 2008; Vršnak, 2016).
The upward motion of coronal loops occurred before a significant upward motion
of the prominence could be detected (at about 14:40 UT, middle and bottom
panels of Figs. 8 and 9). The upward motion of loops has been observed in
other eruptive events and interpreted as the upward progression of the
magnetic reconnection site into the corona due to the effect of the erupting
magnetic structure, frequently with a filament present within, on the observed
loops (Liu & Wang, 2009; Gosain, 2012; Simões et al., 2013; Wang et al.,
2018). HXR coronal sources together with conjugate foot-point sources are a
natural consequence of the coronal magnetic reconnection. In this process, the
strongest sources are flare loop foot-point sources due to thick-target
bremsstrahlung in or near the chromosphere (Aschwanden et al., 2002). As the
erupting configuration goes on one side of the loops, they could be pushed
sideways, then they finally retract as a consequence of a lower magnetic
pressure in the region emptied by the erupting configuration.
For the present event, the earlier contraction of some coronal loops started
at $\approx$ 14:59 UT (Fig. 9), so at least about 19 minutes after the start
of the prominence eruption when the exponential amplitude was large enough to
be detectable. This result is in contrast to the expectation of Hudson (2000);
since the eruption is not associated with any evidence of “magnetic implosion”
for 19 minutes, then there is no evidence that the magnetic energy released is
powered by such an implosion. These observations are rather compatible with
the theoretical models involving an ideal instability (e.g., Lin & Forbes,
2000, and related models summarized in Sect. 1). To put it simply, unstable
magnetic configurations, for example, with an unstable FR, have a decreasing
magnetic energy while they are in expansion. Furthermore, an instability can
occur in models with an invariance along the PIL. These models do not have any
“magnetic implosion” while the FR is erupting and expanding.
The earlier contraction of some coronal loops is closer to the increase in the
X-ray flux measured by GOES, and most of the observed contractions are in the
time range of the X-ray flux enhancement observed by both GOES and RHESSI
(15:10 – 15:40 UT, Fig. 6). This is in agreement with some earlier studies
which found the loop contraction to be associated to the impulsive phase of
some eruptive flares (Liu et al., 2009a; Simões et al., 2013; Wang et al.,
2018). In the present studied event, this time period also corresponds to the
higher speed regime of the prominence (on the order of 200 km s-1).
Next, we notice a propagation of the contraction with height. This result
agrees with previous observations of other eruptions (e.g., Gosain, 2012;
Simões et al., 2013; Shen et al., 2014). In addition to an expected longer
delay with an increasing distance to the launch site, a gradient of the Alfvén
and fast mode speeds may be needed to explain the observations, as follows.
Both speeds have similar values in the low plasma $\beta$ of the corona. A
lower Alfvén speed at a lower height allows one to see the propagation of the
contraction, while a much higher Alfvén speed at larger heights implies a
propagation that is too fast in order to be detected with AIA time cadence (12
s). Furthermore, this difference in speed could not only be due to a
difference in height, but also to a difference in location in the AR (along
the line of sight) of the short and large loops. Indeed, the projection along
the line of sight still allows one to infer different loop shapes, and then
different magnetic connectivities.
With a different view point, Fig. 9 shows that, at the same time, loops expand
or contract at different locations of the coronal configuration. This is in
agreement with previous results obtained for other eruptions (Dudík et al.,
2016, 2017, 2019). This evolution variety could be interpreted in the context
of the vortex model derived from MHD simulations (Zuccarello et al., 2017).
The studied AR is suited for a comparison to these simulations because the
large-scale distribution of the radial magnetic field component, at the
photospheric level, is comparable to the set-up of the numerical simulations
(i.e., bipolar with a moderately more disperse following magnetic polarity). A
first difference between the numerical simulations and observations is an
expected magnetic Reynolds larger by orders of magnitude in the corona than in
simulations, which is due to the limitations of current computer resources. A
second difference is the earlier eruption observed on 01 March. This eruption
opened, or at least made the loops large scale, in the northern side of the AR
magnetic field (Fig. 5). The comparison of the present observations to the MHD
simulations is limited by the few observed loops which are dense enough to be
clearly visible, so their motion can be derived. Then, present observations do
not allow us to visualize if a vortex is forming at the difference of MHD
simulations where as many plasma blobs as needed could be followed in time.
Then, we only claim that the studied loop evolution is compatible with the
development of a lateral vortex, while more observations, in particular with a
larger number of dense loops, are needed to confirm or refute this conclusion.
## 4 Conclusions
We studied a prominence eruption which occurred at the solar western limb in
AR 12290 on 02 March 2015. It is associated with an M3.7 GOES class flare. In
the field of view of AIA, the initially stable prominence progressively
accelerates, with an exponential behavior, to speeds in the range of 110 – 240
km s-1. This range corresponds to different plasma blobs and it traces the
expansion of the magnetic configuration in the low corona. The prominence
erupted away from the local vertical toward the northeast direction.
The source AR, which was observed during its disk transit, is a simple bipolar
region in the decaying stage. For about one week before the eruption, several
episodes of magnetic flux cancellation were observed in between the polarities
in the region where the filament and prominence was observed. Sheared coronal
loops were also observed with AIA. The AR magnetic configuration is asymmetric
with a following polarity more extended than the leading one. However, the
coronal loop observations indicate that this does not create a significant
asymmetry in the coronal connections. An earlier eruption created an open
field- and high-lying loops configuration in the northern side of the AR about
one day before the studied prominence eruption. We conclude that this open-
and high-lying field provides an easier channel for the prominence eruption,
which implies a non-radial eruption.
AIA observations show that the prominence is embedded in a coronal structure
which has the typical shape of an FR viewed along its local axis direction in
its northern part. As the eruption develops, the FR-shaped emission increases
in size and plasma following backward toward the chromosphere is observed at
both prominence legs. Finally, RHESSI observations observed the foot-points
and coronal sources of the flare loops. We conclude that the event of 02 March
2015 has the main characteristics of an erupting FR.
The cold material emission of the prominence, observed in H$\alpha$, were
split into two main blobs during the eruption. The co-alignment with AIA
observations show that one blob is at the rear side while the other one is at
the front side of the erupting FR. The blob at the rear side is at the
expected location for dense plasma to be supported in the concave-up part of
the FR. The detection of the cold plasma in the front part of the FR is
unusual. This plasma is away of its equilibrium position in the magnetic dips.
Then, during the eruption, significant kinetic energy was inputed to drive it
up to the top of the FR field lines. Such cold plasma present in the front is
found in 35% of the cases of interplanetary CMEs where cold plasma is detected
in situ at 1 AU (Lepri & Zurbuchen, 2010). Then, our observations provide an
alternative explanation to the one proposed by Manchester et al. (2014), where
the strong deceleration of a fast CME by the interaction with the surrounding
solar wind implies a drift of the dense and cold plasma toward the front of
the CME.
The prominence eruption first initiates the expansion, and then the
contraction of different sets of coronal loops located in the southern side of
the eruption. The expansion and contraction speeds of coronal loops are in the
range of 2 km s-1 to 34 km s-1. Such results are coherent with the results
obtained in other eruptions (Sect. 1). Here, with the analyzed eruption being
at the limb, we take advantage of the fact that there is less background and
foreground than with the previous analyzed eruptions observed on the solar
disk. This event also provides a different and useful view point directed
almost along the FR axis in its northern part.
Our results show many similarities with previous studies of loop contraction
in eruptions, see Sect. 1, and especially with the studies of Shen et al.
(2014) and Dudík et al. (2016, 2017). Similar to the latter studies, we
observe a slow expansion of coronal loops before the onset of a filament and
prominence eruption. Significantly after the prominence eruption starts, loop
contraction was detected on the southern side of the erupting FR. This delay
between prominence eruption and loop contraction onsets, of about 19 minutes,
is well within the range of these three studies ($\approx$ 4, 7, 50 minutes).
Also in agreement with Dudík et al. (2016, 2017), we simultaneously observe,
at different distances from the erupting FR, sets of loops that are either in
contraction or expansion. The expansion and contraction speed magnitudes are
also comparable.
The present studied eruption, as well as the observations cited in the
previous paragraph could be interpreted as follows. First, due to the FR
uplifting, the coronal loops were pushed upward. Afterwards, the loops began
to contract in order to reach a new equilibrium state. Another explanation is
provided by the numerical simulations of Aulanier et al. (2010) and Zuccarello
et al. (2017). In these simulations, the FR dynamics, linked to its
instability, is the driver of the eruption. Both in the present observations
and these simulations, the upward motion of the FR starts before the loop
contraction which occurs around the time when the FR strongly accelerates
upward. Hydrodynamic vortexes are generated which drive coronal loops upward
or downward depending on their spatial locations within the time-dependent
vortexes. A conclusion to further tests on a broader set of events where the
dynamic of the magnetic configuration could be inferred from the beginning of
the prominence and filament eruption, which is typically earlier than the
associated flare. For that purpose, the presence of dense plasma in the
largest possible fraction of the magnetic configuration is an important clue.
###### Acknowledgements.
We would like to thank the referee for the constructive comments and
suggestions. We recognize the collaborative and open nature of knowledge
creation and dissemination, under the control of the academic community as
expressed by Camille Noûs at http://www.cogitamus.fr/indexen.html. SDO is a
mission for NASA′s Living With a Star (LWS) Program. SDO data are courtesy of
the NASA/SDO science team. RHESSI is a NASA Small Explorer Mission. PD
acknowledges the support from CSIR, New Delhi. The work of RC is supported by
the Bulgarian Science Fund under Indo-Bulgarian bilateral project. RJ thanks
the Department of Science and Technology (DST), Government of India for an
INSPIRE fellowship and to CEFIPRA for a Raman Charpak Fellowship at
Observatoire de Paris, Meudon, France. BS would like to thank Ramesh Chandra
for her stay in Nainital in October 2019 where this work was initiated.
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## Appendix A Measured velocities
The velocities are deduced from the time-distance analysis of the AIA
observation shown in Figs. 8 and 9. These velocities are summarized in the two
most right columns of Table 1. They correspond to the loop systems ”a” to ”f”
defined in Figs. 8b and 9b. The mean velocities were computed from the slope
found in the time-distance plot of panels (e) and (h) of Figs. 8 and 9 within
the time range indicated in the third and fourth columns.
Table 1: Expansion and contraction velocities of loops observed with AIA 304, 171, and 193 Å filters. Loops | Wavelength | Time (UT) | Velocity (km s-1)
---|---|---|---
| (Å) | Expansion | Contraction | Expansion | Contraction
a | 171 | 14:00 – 15:12 | – | 7 | –
| | 14:00 – 15:16 | – | 8 | –
| | 14:00 – 15:10 | – | 9 | –
| 193 | 14:00 – 15:16 | – | 9 | –
| | 14:22 – 15:10 | – | 10 | –
b | 304 | 14:08 – 14:56 | 15:04 – 15:42 | 8 | 11
| | 14:36 – 15:03 | 14:56 – 15:12 | 9 | 14
| | 14:28 – 15:04 | 15:03 – 15:34 | 11 | 19
| 171 | – | 15:10 – 15:34 | – | 9
| | – | 15:15 – 15:37 | – | 12
| | – | 15:03 – 15:41 | – | 12
| | – | 15:15 – 15:34 | – | 13
| | – | 15:04 – 15:12 | – | 18
| | – | 14:53 – 15:15 | – | 20
| | – | 14:55 – 15:06 | – | 22
| | – | 15:18 – 15:37 | – | 22
| | – | 15:00 – 15:24 | – | 22
| 193 | – | 14:52 – 15:24 | – | 9
| | – | 15:11 – 15:24 | – | 9
| | – | 14:44 – 15:27 | – | 12
| | – | 15:18 – 15:50 | – | 21
c | 171 | – | 15:14 – 15:28 | – | 6
| 193 | – | 15:18 – 15:34 | – | 6
d | 171 | 14:13 – 15:12 | 15:18 – 15:30 | 6 | 23
| 193 | 14:20 – 15:14 | 15:14 – 15:28 | 6 | 17
e | 171 | 14:00 – 15:08 | 15:13 – 15:32 | 4 | 7
| | 14:00 – 15:06 | 15:12 – 15:32 | 5 | 9
| | – | 15:14 – 15:28 | – | 23
| | – | 15:18 – 15:38 | – | 24
| | – | 15:18 – 15:32 | – | 31
| 193 | 14:26 – 15:06 | 15:20 – 15:28 | 3 | 11
| | 14:40 – 15:08 | 15:20 – 15:32 | 4 | 14
| | 14:16 – 15:08 | 15:27 – 16:00 | 5 | 14
| | 14:54 – 15:16 | 15:14 – 15:30 | 5 | 24
| | – | 15:14 – 15:23 | – | 30
| | – | 15:18 – 15:30 | – | 34
f | 171 | 14:00 – 14:56 | – | 2 | –
| | 14:00 – 15:10 | – | 2 | –
| 193 | 14:00 – 16:20 | – | 2 | –
|
11institutetext: Dipartimento di Fisica Enrico Fermi, Università di Pisa,
Largo B. Pontecorvo 3, Pisa, Italy 22institutetext: INFN Sez. di Pisa, Largo
B. Pontecorvo 3, Pisa, Italy 33institutetext: CNR-SPIN, Complesso
Universitario di Monte Sant’Angelo, via Cintia, I-80126 Napoli, Italy
44institutetext: INFN Sez. di Napoli, Complesso Universitario di Monte
Sant’Angelo, via Cintia, I-80126 Napoli, Italy
# Effects of temperature variations in high sensitivity Sagnac gyroscope
Andrea Basti 1122 Nicolò Beverini 1122 Filippo Bosi 22 Giorgio Carelli 1122
Donatella Ciampini 1122 Angela D.V. Di Virgilio 22 Francesco Fuso 1122 Umberto
Giacomelli 1122 Enrico Maccioni 1122 Paolo Marsili 1122 Giuseppe Passeggio 33
Alberto Porzio 3344 Andrea Simonelli 22 Giuseppe Terreni 22
(Received: date / Revised version: date)
###### Abstract
GINGERINO is one of the most sensitive Sagnac laser-gyroscope based on an
heterolithic mechanical structure. It is a prototype for GINGER, the laser
gyroscopes array proposed to reconstruct the Earth rotation vector and in this
way to measure General Relativity effects. Many factors affect the final
sensitivity of laser gyroscopes, in particular, when they are used in long
term measurements, slow varying environmental parameters come into play. To
understand the role of different terms allows to design more effective
mechanical as well as optical layouts, while a proper model of the dynamics
affecting long term (low frequency) signals would increase the effectiveness
of the data analysis for improving the overall sensitivity. In this
contribution we focus our concerns on the effects of room temperature and
pressure aiming at further improving mechanical design and long term stability
of the apparatus. Our data are compatible with a local orientation changes of
the Gran Sasso site below $\mu$rad as predicted by geodetic models. This value
is, consistent with the requirements for GINGER and the installation of an
high sensitivity Sagnac gyroscope oriented at the maximum signal, i.e. along
the Earth rotation axes.
###### pacs:
## 1 Introduction
The family of inertial angular rotation sensors based on the Sagnac effect is
rather large anderson1994 . They rely on the Sagnac effect that appears as a
difference in the optical path between waves (no matter if atomic or e.m.
waves) propagating in opposite direction in a rotating closed loop. This
effect can be observed, like in the case of fibre gyroscopes Veliko2012 , as a
phase difference between the two counter-propagating beams or, when the loop
is a resonant cavity, as a frequency difference between them. In this last
case, there are two possible strategies: to interrogate the cavities through
external laser sources (passive ring cavity, PRC Liu19 ) or to observe the
beat note between the radiation emitted in the two directions by a laser
medium (active ring cavity, usually called ring-laser-gyro, RLG RSIUlli ). In
principle a ring could operate both in active or in passive configuration, and
this is a very interesting feature for very high sensitivity measurements to
get rid of systematic that affects differently active and passive devices.
Eq. (1) reports the general relation connecting the Sagnac frequency $f_{s}$,
the quantity measured for any RLG, and the modulus of the local angular
rotation rate $\Omega$:
$f_{s}=4\frac{A}{P\lambda}\Omega\cdot\cos(\theta)$ (1)
where $A$ is the area enclosed by the optical path, $P$ its perimeter,
$\lambda$ the wavelength and $\theta$ the angle between the area vector and
the local rotational axis. Since for a RLG rigidly connected to the ground,
the Earth rotation velocity is by far the dominant component of $\Omega$, we
can approximate $\theta$ with the angle between the area vector and the Earth
rotational axis and define the RLG Scale Factor (SF) as
$4A/(P\lambda)\cos(\theta)$. Any change in $f_{s}$ can be ascribed to
different sources and, in particular for high sensitivity measurements, it is
in principle very hard to discriminate between spurious rotations (affecting
$\Omega$) and changes in the scale factor, due to geometrical modifications
and/or orientation changes, as they will produce the same effective change
into $f_{s}$. For this reason, the details of the experimental apparatus
matter in the final sensitivity, and even more in its long time response.
Up to now, the most sensitive RLG was built with a so called monolithic
design, a block of thermally stable material with optically contacted mirrors.
This scheme is very expensive and, once installed, cannot be further enlarged
or oriented at will RSIUlli . Since RLG sensitivity increases with its
dimension, a very large area, 833 m2, heterolithic (HL) RLG was built, whose
mirrors were hold inside metallic boxes fixed to the floor Hurst2009 . New HL
apparatuses composed on a rigid monument supporting the mirrors metallic boxes
have been developed in the last decade Belfi2012 ; 90day ; beverini_2020 ;
igel2021 , (see fig. 1). Large frame high-performance HL passive optical
gyroscopes have been also recently reported Martynov19 ; Zhang20 ; Liu19 ;
Liu20 ; korth2016 .
Figure 1: GINGERINO at the time of assembling. In the foreground the mirror
box and the pipes connecting the boxes can be seen, one of the Mitutoyo screw
used to orientate the mirrors is visible beyond the box. It’s a steel
mechanical structure attached to a cross shaped granite monument, which
provide stability to the ring perimeter.
In HL structures, the whole light path is inside a single vacuum chamber where
the mirror corner boxes are connected by pipes. Mirror holders are also
provided with mechanical and PZT driven tools to have a fine control on their
position and orientation. This control can be made active for geometry
stabilization Santagata2015 . External disturbances can in principle produce
spurious rotations and changes in the orientation of the area vector;
moreover, couplings between the different mirrors are surely present, since
there is a continuous mechanical connection, generating spurious signals
GINGERINO is a highly sensitive HL RLG prototype continuously running,
unattended and without any active control. It has been assembled inside the
underground INFN Gran Sasso laboratories (LNGS) in order to probe the site
noise level and to study the potential sensitivity of HL RLG in the
perspective of the GINGER project, (Gyroscopes IN GEneral Relativity), which
aims at measuring the Lense-Thirring effect on the Earth with $1\%$ accuracy
or even better prd2011 ; angela2017 . Since the main goal of the GINGER
project is to measure General Relativity and geodetic effects that produce DC
or periodic signals, with periods ranging from few days to years, the
investigation of the role of slow varying environmental perturbations, such as
temperature and pressure, becomes of great importance. The GINGER project is
based on an array of RLG whose relative orientation can be chosen in order to
optimally reconstruct the angular rotation. In particular, it is convenient to
orientate one of the RLG at the maximum Sagnac frequency, i.e. with the area
vector parallel to the polar axis. In this conditions a variation of the
absolute inclination of the RLG affects its scale factor only at the second
order (see eq. 1), and it is possible to reconstruct the orientation with
respect to the total angular rotation axis of the other RLGs angela2017 ;
angelaFrontiers to the micro-rad. This accuracy is required to measure the
Lense-Thirring effect at the 1$\%$ level. In this contest, the stability of
the underneath bedrock matters and should be further investigated.
GINGERINO has clearly shown that HL mechanical structure can operate in the
LNGS site with high sensitivity and long term stability even if, in absence of
an active control, mode jumps and split mode operation occasionally take
place. These failures, however, affect the instrument duty cycle but not the
sensitivity. It has been already proved that this perturbation typically
affects $5\%$ of the data RSI2016 , thanks to the LNGS low environment noise.
However, it appears clearly that there is large room for improvements.
The effects of temperature and pressure on the apparatus must be investigated
in order to further improve the design in view of GINGER. Clearly, data and
findings should be handled with care since, as mentioned above, any change is
totally equivalent to a fluctuation of the angular velocity modulus.
In the present paper the data analysis results are used to find bounds on the
effect of temperature and pressure on the instrument sensitivity in order to
assess the environmental constrains to be fulfilled by the GINGER design.
It is well established that known geophysical signals induce local rotations
and tilts that affect in a well defined manner high sensitivity RLG. These
signal are for us a very useful for assessing the reliability of our
instrument and check the effectiveness of data analysis.
It is important to note that, while we draw specific conclusion in view of the
GINGER design, the analysis and its procedure are more general and applies
also to PRC.
The paper is organised as follows. In Sect. 2 we review the elements of the
quite complex data analysis we usually run on GINGERINO data. In particular,
we focus on the role of temperature and its interplay with local tilts as
disturbances on the Sagnac signal. Then, in Sect. 3 conclusions are
summarised.
## 2 Data analysis
This paper aims at relating data coming from the long time observation of the
environmental parameters with the Sagnac signal, We investigated the effects
of temperature, pressure, air flow speed and local tilts whose probes are all
co-located with the RLG. The GINGERINO apparatus is contained inside a closed
box made of thermal and acoustic shielding walls, floor and roof, and it is
composed by the vacuum chamber, whose corner are rigidly screwed to a granite
structure attached to the bedrock through a central reinforced concrete
support. Temperature and pressure probes record the environmental data inside
the box. A 2-channels tilt-meter is placed on the top of the granite monument
to look for local tilts. An anemometer is also installed in the tunnel outside
the box to measure air-flow speed. All environmental data are sampled at 1 Hz.
The analysis has been applied to three time series (30 days in June 2018, 70
days in Autumn 2019, and 103 days in Winter 2020). The RLG interference signal
was elaborated following the technique described in details in Refs.
Angela2019 ; Angela_epj2020 ; PRR2020 ; divirgilio2021 . There, we take into
account the laser dynamic and reconstruct the true Sagnac frequency $f_{s}$,
(in the following often expressed as the angular Sagnac frequency
$\omega_{s}=2\pi f_{s}$). So far, we have structured the analysis in order to
recover from the data the global Earth rotation by taking into account laser
dynamics and local disturbances coming from known geophysical signals. Here,
we aim at gain deeper inside the role of local environmental conditions, whose
slow motion variations are indeed low frequency noise affecting the Sagnac
signal. In particular, we look at correlation between environmental time
series with the angular velocity resulting from the main data analysis. In our
analysis we consider $\omega_{s}$ as
$\omega_{s}=\omega_{geo}+\omega_{local}$ (2)
with $\omega_{geo}$ given by $2\pi\cdot SF\cdot\Omega_{geo}$, where
$SF=A\cos\theta/(P\lambda)$ is the instrumental scale factor and
$\Omega_{geo}$ is the global Earth rotation rate routinely measured and
elaborated by the International Earth Rotation and Reference Systems Service
(IERS). Moreovoer, $\omega_{local}=2\pi\cdot SF\cdot\Omega_{local}$. In
particular, $\Omega_{local}$ combines the rotational contribution of local
geophysical origin as tides, ocean loading, etc. with instrumental spurious
rotations that may be result from changes in the environmental operating
condition. In principle, RLG data alone do not allow to discriminate between
these two contributions. It is then important to identify any kind of
disturbances on the apparatus coming from the environment by using the data
from environmental probes, in order to improve the sensitivity and possibly
the accuracy of future experimental apparatuses.
### 2.1 Pressure and air–flow contribution
We found no clear correlation of the gyroscope data with pressure and
anemometer signals, notwithstanding the fact that GINGERINO box is not
pressure isolated, by tight doors, from the tunnel as it is usual for these
kind of high sensitivity instrumentation. As a matter of fact, while air flows
variations are themselves negligible, pressure excursions are $\sim 4\%$. The
lack of evidence of pressure fingerprint on the data does not rule out the
possibility of an influence on the Sagnac signal. Surely, at low frequency,
these variations does make sensible effect at the present stage. The pressure
time line shows rather fast slopes most probably due to human activities that
are not transferred to GINGERINO. However, pressure variations may be seen by
the cave as a single mechanical structure with its own resonances, so that a
tight isolation is certainly wise for GINGER that will occupy a larger cave
volume.
### 2.2 Tilts and local rotation
Figure 2: Time behaviour of the two tilt-meter channels during the analysed 30
days.
A different finding came out for the two-axis tilt-meter and temperature data.
Any local rotation, due to known geophysical signals and/or instrumental and
environmental noise results in changing the effective SF that includes the
orientation of the ring plane with respect to the actual rotation axis and the
geometry of the cavity. Preliminary we note that tilt-meter data give a clear
signal of the Earth solid tides. The tides show-up as tiny bi-daily
oscillation in both time series of Fig. 2. The period of such oscillations can
be better estimated by looking at the amplitude spectral density (ASD) of
tilt-meter reading for the two axis (N–S and E–W respectively) shown in Fig.3.
These plot refer to the data collected in the 103 days long run in 2020. Such
a long observation period allows a much better spectral resolution of low
frequency signal such as solid tides.
Figure 3: Amplitude spectral density of the two tilt-meter channels during the
first 103 days of 2020.
Indeed, the ASD of tilt-meter data shows a peak at at 1.932 cpd (corresponding
to a period of 12.42 hours), that corresponds to the frequency of the main
tide component. Before going into details of the interplay between temperature
and tilts, we stress that our tilt meter signals are quite similar to the
typical geophysical signal recorded elsewhere by other high sensitive
instrument. For instance the records of the clinometer Marussi, installed in
the cave of Bus de la Genziana (Friuli, Italy) Devoti ; devoti2019 , show that
most of the tilts of the monument can be assumed to have geophysical origin.
We will see that long time drift may be ascribed to temperature changes. We
hereby stress that known geophysical signals play the role of a “test signal”
for our apparatus giving us the possibility of evaluating its reliability in
view of GINGER.
### 2.3 Temperature Effects
As above said the analysis evaluates $\Omega_{local}$, the total local
disturbance, using these information and the environmental monitors to recover
at our best the Earth rotation velocity. In this contest, known geophysical
signals are of great importance since they can be used to increase the
instrument accuracy and sensitivity. Any GR effects is contained into the
Earth rotation signal once it’s cleared from any local perturbation, that’s
why the study of room temperature effects on the apparatus is a key point. The
temperature variations are very slow, for this reason they have a great
importance in the frequency region where the GR terms should appear.
Figure 4: Temperature in ∘C (blue line), and $\Omega_{local}$, in rad/s (red
line) from the 2018 30 days data set
A typical temporal behavior of temperature and $\Omega_{local}$ is shown in
Fig. 4 that compares the time series obtained from the 30 days 2018 data set.
It is clear that temperature variations may affect the apparatus in many
different ways. They change the geometry of the gyroscope, but also deform the
monument, generating spurious rotation. It is not straightforward to separate
one effect from the other. Aim the present analysis is then to understand the
effective role of as many as possible thermal contribution to the instrument
signals. June 2018 data set has a nice clean temperature behaviour with the
maximum excursion limited to less than 0.03 ∘C. Then we will use these data to
investigate residual contribution of the temperature in $\Omega_{local}$.
As clearly visible in Fig.4, $\Omega_{local}$ on quite long time scales
follows the temperature behavior. This is a clear indication of a possible
linear relation between the two variables that can be interpreted as a
consequence of the bare instrument expansion due to thermal drifts. From Eq.
(1) we see that the SF depends linearly on the ring side length. We can, then,
assume that for long term operations, the relative effect of a temperature
variation $\Delta T$ on the Sagnac frequency is given by $\rho\cdot\Delta T$,
where $\rho$ is the thermal expansion coefficient. In our case, the expansion
coefficient is the granite one $\rho_{g}=6.5\cdot 10^{-6}/\,^{\circ}$C and the
average Sagnac frequency is $f_{s}=280$ Hz. The expected effect is of the
order of $f_{s}\cdot\rho\cdot\Delta T$ Hz. A linear regression carried on the
2018 data set, after separating local and global signals, gives a temperature
coefficient of $0.7\cdot 10^{-3}$ Hz/∘C, to be compared to the expected value
of 1.8$\cdot 10^{-3}$ Hz/∘C.
Another approach is to scatter plot the two variables, $\Omega_{local}$ and
$T$ in a region where the temperature varies linearly. This can be done
selecting the data relative to the upward slope of the first $\sim 3$ days.
Fig. 5 shows in a scatter plot $\Omega_{local}$ vs. T, the linear relation in
this case is of the order of 0.45 Hz/∘C, as obtained by a linear fit on the
plotted data. This may indicate that temperature induces perturbations in the
apparatus much larger than the geometrical scale factor change thus confirming
that temperature effects are multifaceted. Eventually, we note that the same
approach is not worthy for the remaining of the data of the 2018 run. As a
matter of fact, faster change in the temperature suppresses the relative
scattering of $\omega_{local}$ but, once the temperature get more quite, the
scatter around the mean relatively increases.
Figure 5: Relationship between $\frac{\omega_{local}}{2\pi}$ and temperature
variations.
### 2.4 Interplay between temperature and tilts
To further understand the role of the temperature we have investigated the
relation between temperature and tilt signals. In this case, as it is evident
comparing the time series in Figs. 4 and 2 there is no evidence of a simple
relation. However, it is plausible that temperature changes induce some
spurious rotation of the apparatus due to any anisotropy of the mechanical
structure and/or of the underneath concrete monument. To appreciate whether or
not an effect arises in the tilts we assumed a third order polynomial in the
temperature variable as the driving for tilts. Consequently, we have, at
first, calculated the maximum tilt direction for each time point, then fitted
these tilt values with a polynomial fit of the third order leaving the
temperature as explanatory variable.
These procedure has a twofold aim. On one hand, modelling the contribution to
tilts coming from the temperature due to instrumental deformation allows to
better use tilt-meter data in calculating the final instrument accuracy. On
the other hand, the standard deviation of the residuals give an evaluation of
the orientation stability of the cave itself. Standard deviations of the
residuals, indeed, gives the fraction of inclination not related to
temperature in this model.
We found a residual standard deviation of 0.726 $\mu$rad for the 2018 30 days
series (third order fit r-squared=0.707 with 0.01 ∘C thermal stability, and
$\pm 2.5\ \mu$rad inclination range and STD 1.3 $\ \mu$rad) while it goes down
to 0.25$\ \mu$rad in the case of 103 days 2020 (third order fit
r-squared=0.891 with 0.1 ∘C thermal stability, $\pm 1.5\,\mu$rad range, and
STD 0.73 $\mu$rad). From this result we can infer that the orientation of the
underneath bedrock is stable at the level of microradian.
The relation between tilts and temperature in some portion of data has been
estimated of the order of 500 $\mu$rad/∘C. In GINGER, the RLG at maximum
signal requires a long term stability of the monument of the order 1 $\mu$rad
angela2017 . If we assume in GINGER a long term temperature stability of 0.01
∘C, we need to improve at least a factor 5 the orientation stability of the
monument. We expect that in GINGERINO the changes of orientation are mostly
related to the non uniformity of the basement which is connected to the
bedrock through a reinforced concrete block, whose homogeneity was not cured.
We have also investigated possible effects of the temperature derivative
$dT/dt$. It could deform the shape of the HL mechanical structure, producing a
spurious rotation of the ring mirrors. This derivative, however, is a very
small quantity, at the limit of the measurement noise level. To be sure that,
at the present sensitivity, GINGERINO does not see any effect of this term we
looked at first 10 days data of June 2018, showing larger temperature drift
being acquired soon after a closing of the box. There we had for $dT/dt$ a
maximum value of the order of $10^{-7}$ ∘C/s but we did not find a clear
relation between the gyroscope signal and $dT/dt$. In any case, we expect that
this effect is very small, and can be further reduced by an active control of
the ring geometry and by improving the isolation of the apparatus from
external perturbation.
Typical long term (some weeks) temperature fluctuations are of the order of
$10^{-2}$ ∘C, but in the 2020 long run we observed only $\sim 0.1$ ∘C
stability, ten times worse. This is probably due to deterioration of the
protection box, which we will repair as soon as possible. 111Access to the
experiment is presently limited due CoVid pandemic. The whole GINGERINO
experiment is enclosed in a box, to isolate from the cave. The box is wormed
up by infrared lamps in order to keep the relative humidity below $60\%$
### 2.5 Coupling between the rotation of the HL RLG and the inclination of
the monument
GINGERINO is based on one of the first mechanical model for heterolithic RLG,
usually called GEOSENSOR, developed for application in seismology. The mirrors
can be easily aligned using mechanical levers in air, which act on the mirror
boxes. This smart and convenient solution has the drawback that all the
mechanical parts form a continuous object, and basically very small rotation
of one part can effectively make the cavity rotate. This means that any tilt
and/or mechanical stress due to temperature may induce a spurious, tiny and
slow, rotation of the optical cavity. Under this assumption we expect to
observe a phase variation $\phi$ associated to the tilt. This variation can be
reconstructed integrating in time $\omega_{local}$, at this purpose we remind
that the data are acquired at 600s rate, and before integration, interpolated
in order to fill the gaps due to the missed points, associated with mode jumps
and split mode operation. In Fig. 6 we report the change of the absolute value
of inclination $\Delta Tilt$ versus the phase $\phi$ for the 2020 data, the
longer data set. The plot shows a clear correlation between the two variable
indicating that whenever the ring structure tilts we see a rotation of the
optical cavity.
Figure 6: Scatter plot of the absolute value of the change of inclination
$\Delta_{Tilt}$ versus the phase variation $\phi$ associated to the tilt.
Similar behaviour is found using the other data sets. To have a quantitative
estimation of the connection between tilt and rotation we have run a linear
fit on the plotted data. The fit indicates that there is a linear relationship
of the phase with the inclination of the monument, of the order of $550\pm 5$
rad per rad of inclination (r-squared = 0.93). This connection between tilts
and instrumental rotation will be reduced by the new mechanical design we are
developing for GINGER. In GINGER each mirror will be uncoupled from the rest
of the mechanics. Active controls, absent in GINGERINO, on mirrors position
and tilt should be designed in order to avoid couplings to the entire
structure. Moreover, in order to control and eventually subtract this
contribution, it is possible to develop system to monitor the position of each
mirror with respect to the granite support.
## 3 Conclusions
GINGERINO is the first underground HL RLG operative in a continuous basis,
with sensitivity better than prad/s. RLG signal is the sum of a global
contribution coming from the Earth rotation and a local one that contains
geophysical signals and local and instrument disturbances. We have
investigated the fingerprints of environmental parameters, such as pressure,
temperature and local tilts, in the local contribution to the Sagnac frequency
as measured by GINGERINO. Our study proves the very close relation between
temperature and Sagnac signal. It is not only given by the thermal expansion
of the granite support, which changes the perimeter of the cavity and
accordingly the geometrical scale factor of the Sagnac gyroscope. However, the
coupling of temperature variations seems to be more complicate affecting
mainly the support structure and the HL mechanical structure of GINGERINO. By
looking at the local tilts, measured by a two-axis tilt-meter attached to the
granite support, we found evidence of a coupling between this degrees of
freedom with the temperature variations. It most probably comes from the
reinforced concrete interface between the granite and the underneath bedrock
that is not homogeneous and so its orientation changes with temperature
variation. The study of the local disturbances $\omega_{local}$ shows that the
RLG cavity rotates when the monument tilts, this is associated with the HL
mechanical design of GINGERINO, in which the mirrors are not fixed to the
monument. Cross checking the temperature behaviour and the tilt-meter signal
we also proved that the residual orientation stability at the level of
$\mu$rad at LNGS is suited for the construction of GINGER the RLG array able
to measure General Relativity effects. To reach the required sensitivity and
accuracy improvement in the mechanics, especially for the structure underneath
the ring, are necessary. The present study indicate the path to follow toward
GINGER: increase the instrument isolation; improve the holding structure
homogeneity; reduce the coupling between tilt and cavity rotation. All of
these have a feasible solution.
###### Acknowledgements.
We thank the Gran Sasso staff in support of the experiments, particularly
Stefano Gazzana, Nazzareno Taborgna and Stefano Stalio . We are thankful for
technical assistance to Alessio Sardelli and Alessandro Soldani of INFN
Sezione di Pisa and Francesco Francesconi of Dipartimento di Fisica. A special
thank to Gaetano De Luca of Istituto Nazionale di Geofisica e Vulcanologia for
regularly checking Gingerino operation status.
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# Efficient Mining of Frequent Subgraphs with Two-Vertex Exploration
Peng Jiang The University of Iowa<EMAIL_ADDRESS>, Rujia Wang
Illinois Institute of Technology<EMAIL_ADDRESS>and Bo Wu Colorado School
of Mines<EMAIL_ADDRESS>
###### Abstract.
Frequent Subgraph Mining (FSM) is the key task in many graph mining and
machine learning applications. Numerous systems have been proposed for FSM in
the past decade. Although these systems show good performance for small
patterns (with no more than four vertices), we found that they have difficulty
in mining larger patterns. In this work, we propose a novel two-vertex
exploration strategy to accelerate the mining process. Compared with the
single-vertex exploration adopted by previous systems, our two-vertex
exploration avoids the large memory consumption issue and significantly
reduces the memory access overhead. We further enhance the performance through
an index-based quick pattern technique that reduces the overhead of
isomorphism checks, and a subgraph sampling technique that mitigates the issue
of subgraph explosion. The experimental results show that our system achieves
significant speedups against the state-of-the-art graph pattern mining systems
and supports larger pattern mining tasks that none of the existing systems can
handle.
## 1\. Introduction
Frequent Subgraph Mining (FSM) is an important operation on graphs and is
widely used in various application domains, including bioinformatics (Milo et
al., 2002; Vazquez et al., 2004), computer vision (Chu and Tsai, 2012), and
social network analysis (Ugander et al., 2013). The task is to discover
frequently occurring subgraph patterns from an input graph. Different from
graph pattern matching problems where a query pattern is given, FSM needs to
find the important patterns based on a support measure and thus has a much
larger exploration space.
Since the patterns of interest are unknown, most systems for FSM take an
explore-aggregate-filter approach (Teixeira et al., 2015; Dias et al., 2019;
Wang et al., 2018; Chen et al., 2020). The principle is to explore all the
subgraphs, aggregate the subgraphs according to their patterns, and filter out
the subgraphs that are redundant or are not of interest. The exploration
happens in a vertex-by-vertex manner where smaller subgraphs are iteratively
extended based on the connections in the graph. There are mainly two ways for
exploration: breadth-first and depth-first. Starting from all vertices in the
graph, breadth-first exploration stores all subgraphs of size $l$ and extends
them with one more vertex to find subgraphs of size $l+1$. The main problem of
breadth-first exploration is that the intermediate data can easily exceeds the
memory capacity as the subgraph size grows. With depth-first exploration, a
subgraph of size $l$ is immediately extended to a subgraph of size $l+1$
without seeing other subgraphs of size $l$. It needs not save the intermediate
subgraphs and thus can explore larger patterns. However, depth-first
exploration cannot exploit the anti-monotone property to prune the search
space (Dias et al., 2019), resulting in a lot of unnecessary computation.
Some recent graph mining systems take a pattern-based approach (Mawhirter and
Wu, 2019; Jamshidi et al., 2020). The idea is to enumerate the (unlabeled)
subgraph patterns and then perform pattern matching on the graph. Because the
pre-generated patterns guide the exploration, these systems need not store any
intermediate data, and the aggregation overhead can be reduced as the topology
of the subgraphs is given. However, this approach only works well for small
patterns because when the pattern is larger (more than 6), listing all
subgraph patterns itself becomes a hard problem (McKay et al., 1981; McKay and
Piperno, [n.d.]; ng, [n.d.]). It is also difficult for the pattern-based
systems to exploit the anti-monotone property to prune the search space.
Peregrine (Jamshidi et al., 2020) maintains a list of frequent patterns,
extend the patterns with one vertex or edge, and then re-match the extended
patterns on the graph. It prunes the search space without storing the
intermediate subgraphs, but the re-matching incurs a lot of redundant
computation. These issues have impeded the existing graph mining systems from
supporting FSM for large patterns. In fact, most of the prior work only
reports experimental results for FSM with no more than 4 vertices.
To enable large pattern mining, we propose a novel two-vertex exploration
method in this work. Our key observation is that vertex-by-vertex exploration
is not necessary for pattern mining. Instead, we can perform two-vertex
exploration that joins size-($n-2$) subgraphs with size-$3$ subgraphs on a
common vertex to obtain subgraphs of size-$n$. The new exploration method
significantly accelerates the exploration process and reduces the memory
access overhead in the join operation. It also allows us the exploit the anti-
monotone property to prune the exploration space without storing the
intermediate subgraphs or re-matching the patterns.
To further accelerate the mining process, we propose two new techniques to
overcome the performance bottlenecks. One performance bottleneck is due to the
expensive isomorphism checks in the aggregation step. To aggregate the
subgraphs based on their patterns, we need to generate a canonical form for
each subgraph such that the subgraphs with the same canonical form are
isomorphic. Unfortunately, the best known algorithms for generating such
canonical forms have exponential complexity (Shang et al., 2008; Babai et al.,
1983; Xifeng Yan and Jiawei Han, 2002). Therefore, we want to perform
isomorphism check for as few subgraphs as possible. Previous work has employed
a quick pattern technique to reduce the number of isomorphism checks (Teixeira
et al., 2015; Wang et al., 2018). The main is to first group the subgraphs
based on an easily computed pattern (e.g., a list of all edges). Since
subgraphs in the same group must be isomorphic, only one isomorphism check is
needed for each group. We improve on this idea by proposing an index-based
quick pattern technique. It assigns an index to each pattern and uses the
indices to compute a quick pattern for the joined subgraph. Compared with the
quick pattern technique used in prior work, our quick pattern encodes the
information of sub-patterns and achieves more accurate grouping of the
subgraphs, leading to a significant reduction of isomorphism checks.
Another more fundamental challenge of mining large patterns on graphs is due
to the exponential growth of the exploration space. For example, in a median
size graph, MiCo (Elseidy et al., 2014), which has $9\times 10^{4}$ vertices
and $10^{6}$ edges, there are more than $10^{12}$ size-5 subgraphs. When the
pattern size increases to 7, the estimated number of subgraphs is in the order
of $10^{17}$ for which exhaustive enumeration becomes infeasible. To mitigate
this issue, we propose a subgraph sampling technique. The idea is that we
sample a small subset of size-3 subgraphs for exploring larger subgraphs
during the joining and/or the matching phase. Since the subgraphs of frequent
patterns are more likely to be sampled, we are able to discover frequent
patterns with only a small number of sampled subgraphs. Compared with previous
works that apply edge or neighbor sampling to FSM (Iyer et al., 2018;
Mawhirter et al., 2018), we can discover more frequent patterns with the same
or less computation. This is because subgraph samples preserve more structural
information of the graph than edge samples.
We perform extensive evaluation of our system and compare with three state-of-
the-art graph mining systems: AutoMine (Mawhirter and Wu, 2019), Peregrine
(Jamshidi et al., 2020), and Pangolin (Chen et al., 2020). The results show
that without using sampling our system achieves 1.8x to 8.4x speedups on tasks
for which the compared systems can return. By using sampling, our system can
discover larger patterns that none of the existing systems can handle.
## 2\. Background
This section introduces the graph related concepts that are important to our
discussion and formally defines the frequent subgraph mining problem.
### 2.1. Graph Basics
A graph $G$ is defined as $G=(V,E,L)$ consisting of a set of vertices $V$, a
set of edges $E$ and a labeling function $L$ that assigns labels to the
vertices and edges. A graph $G^{\prime}=(V^{\prime},E^{\prime},L^{\prime})$ is
a subgraph of graph $G=(V,E,L)$ if $V^{\prime}\subseteq V$,
$E^{\prime}\subseteq E$ and $L^{\prime}(v)=L(v),\forall v\in V^{\prime}$. A
subgraph $G^{\prime}=(V^{\prime},E^{\prime},L^{\prime})$ is vertex-induced if
all the edges in $E$ that connect the vertices in $V^{\prime}$ are included
$E^{\prime}$. A subgraph is edge-induced if it is connected and is not vertex-
induced.
###### Definition 0 (Isomorphism).
Two graphs $G_{a}=(V_{a},E_{a},L_{a})$ and $G_{b}=(V_{b},E_{b},L_{b})$ are
isomorphic if there is a bijective function $f:V_{a}\Rightarrow V_{b}$ such
that $(v_{i},v_{j})\in E_{a}$ if and only if $(f(v_{i}),f(v_{j}))\in E_{b}$.
We say two (sub)graphs have the same pattern if they are isomorphic. The
pattern is a template for the isomorphic subgraphs, and a subgraph is an
instance (also called embedding) of its pattern. To determine the pattern of a
subgraph, a canonical form for each subgraph can be computed, and the
subgraphs with the same canonical form are isomorphic. There are various tools
and algorithms available for graph isomorphism check (McKay et al., 1981;
Junttila and Kaski, 2007; Xifeng Yan and Jiawei Han, 2002). All of these
algorithms have exponential complexity. We use bliss (Junttila and Kaski,
2007) for isomorphism check in our system as it is fast in practice and is
widely used in graph mining systems (Teixeira et al., 2015; Wang et al., 2018;
Jamshidi et al., 2020). A related concept is automorphism check which checks
if two subgraphs are identical, even though they might have different
orderings of vertices and edges.
### 2.2. Frequent Subgraph Mining
The task of Frequent Subgraph Mining (FSM) is to obtain all frequent subgraph
patterns from a labeled input graph. A pattern is considered frequent if it
has a support above a threshold. While the definition of the support measure
can vary across applications, the support usually needs to satisfy an anti-
monotone property, i.e., the support of a pattern should be no greater than
the support of its sub-patterns (Meng and Tu, 2017).
###### Definition 0 (MNI Support).
Given a pattern $P=(V_{p},E_{p},L_{p})$ and an input graph $G=(V,E,L)$, if $P$
has $m$ embeddings $\\{f_{1},f_{2},\ldots,f_{m}\\}$ in $G$, the minimum image
based (MNI) support of $P$ in $G$ is defined as
$\sigma_{MNI}(P,G)=\min_{v\in V_{p}}|\\{f_{i}(v):i=1,2,\ldots,m\\}|.$
Other support measures include maximum independent set based (MIS) support,
minimum instance based (MI) support, and maximum vertex cover based (MVC)
support. All these support measures are anti-monotone. MNI support is the most
commonly used one because it has linear computation complexity while achieving
a good accuracy in measuring the ‘frequency’ of patterns in a graph. The
readers are refered to (Meng and Tu, 2017) for detailed descriptions and
computation complexity of different support measures. We adopt the MNI support
for our experiments, although our proposed techniques are applicable to any
support measure with the anti-monotone property.
With a support measure $\sigma$, the frequent subgraph mining problem is
defined as finding all patterns $\\{P_{i}=(V_{i},E_{i},L_{i})\\}$ in a graph
$G$ such that $|V_{i}|=s$ and $\sigma(P_{i},G)\geq t$ where $s$ is the given
pattern size and $t$ is the given support threshold. The support can be
calculated with either vertex-induced subgraphs or edge-induced subgraphs. Our
proposed techniques work for both cases. We use edge-induced subgraphs for
experiments, as it is the common setting in prior work (Wang et al., 2018;
Dias et al., 2019; Jamshidi et al., 2020).
## 3\. Illustration of the Idea
(a) An example graph
(b) Size-3 subgraphs
(c) Join the size-3 subgraphs on every column
Figure 1. An example of finding size-5 subgraphs by joining size-3 subgraphs.
Before getting into technical details, we describe our idea of two-vertex
exploration with an example. We use an unlabeled graph for simple
illustration.
Suppose our task is to discover size-5 patterns in an input graph (as shown in
Figure 1a). We can first find all size-3 subgraphs and join them on a common
vertex to obtain size-5 subgraphs. In this example, we first apply a matching
algorithm to obtain all the embeddings of size-3 patterns (i.e., wedge and
triangle) as listed in Figure 1b. Each pattern is assigned an index (0 for
wedge and 1 for triangle in this example), and the index is stored with each
embedding during the pattern matching.
Next, we calculate the MNI support for each size-3 pattern and prune the
patterns with support less than the threshold. In this example, the supports
for both wedge and triangle is 3. Suppose we set the support threshold to 3.
Neither of the patterns will be pruned. After we obtain the pruned size-3
subgraphs, we perform binary join on every pair of the columns (i.e., $\langle
1,1\rangle$, $\langle 1,2\rangle$,$\langle 1,3\rangle$,$\langle
2,1\rangle$,$\langle 2,2\rangle$,$\langle 2,3\rangle$, $\langle
3,1\rangle$,$\langle 3,2\rangle$,$\langle 3,3\rangle$) to explore size-5
subgraphs. Figure 1c shows how we can obtain four size-5 subgraphs by joining
column $\langle 1,1\rangle$ on key $3$. Every pair of subgraphs with key $3$
are tested (i.e., $\langle$‘342’, ‘342’$\rangle$, $\langle$‘342,
‘352’$\rangle$, …, $\langle$‘387’, ‘385’$\rangle$, $\langle$‘387’,
‘387’$\rangle$). If two subgraphs have one and only one common vertex, they
compose a valid size-5 subgraph. In this example, ‘342’ and ‘375’ make up a
valid size-5 subgraph ‘34275’; ‘342’ and ‘387’ make up ‘34287’; ‘352’ and
‘387’ make up ‘35287’; and ‘342’ and ‘385’ make up ‘34285’. These valid joins
are marked with connected arrows in Figure 1c. We can see that, through the
join operation, we grow the pattern size from 3 to 5 in one exploration step.
We will show that such two-vertex exploration is exhaustive for subgraph
exploration in §4.1.
One may notice that the result of joining ‘374’ with ‘385’ (‘37485’) is not
included in Figure 1c. This is because our system performs an automorphism
check when generating the join results to remove redundancy. We propose a
smallest-vertex first dissection method that ensures only the results that are
obtained by joining the subgraph of the smallest spanning vertex indices are
saved. In this case, the ‘37485’ subgraph will be generated when we join the
third column of ‘543’ and the first column of ‘387’. More details on the
automorphism check and redundancy removal are explained in §4.3.
The above procedure can be extended to explore larger subgraphs by joining
multiple subgraph lists. For example, a 3-way join of two size-3 subgraphs and
one size-2 subgraphs (i.e. edges) will explore all size-6 subgraphs. A 3-way
join of size-3 subgraphs will explore all size-7 subgraphs. Given an input
graph $G$, a pattern size $s$, and a support threshold $t$, the workflow of
our frequent subgraph mining algorithm is summarized as follows:
1. Step1: Obtain all size-3 subgraphs by matching.
2. Step2: Calculate the support for each size-3 and size-2 pattern, and remove patterns with support smaller than $t$ along with their subgraphs.
3. Step3: Perform multi-way join of size-3 subgraphs and/or edges to obtain subgraphs of size s: if $s==2n+1$, join $n$ size-3 subgraph lists; if $s==2n$, join the edge list with $n-1$ size-3 subgraph lists.
4. Step4: Calculate support for each size-$s$ patterns and remove patterns with support smaller than $t$.
For Step1, any matching algorithm will work; we use AutoMine (Mawhirter and
Wu, 2019) in our implementation. Step2 and Step4 are straightforward based on
the definition of the support measure. Step3 is the most important step in the
algorithm. We will detail this step in the next section.
## 4\. Subgraph Exploration Process
All the current graph mining systems based on the explore-aggregate-filter
approach use single-vertex exploration because it ensures that all the
size-$n$ subgraphs can be found by extending the size-($n-1$) subgraphs with
an edge. We find that limiting the step size to 1 is not a must to find all
patterns. This section describes our two-vertex exploration idea and explains
its advantage over single-vertex exploration.
### 4.1. Two-Vertex Exploration
We propose to explore the size-$n$ subgraphs by joining the size-($n-2$)
subgraphs with the size-3 subgraphs (i.e., wedges and triangles). The
completeness of this two-vertex exploration method is summarized as follows.
###### Theorem 1.
All of the size-$n$ subgraphs can be discovered by joining the size-($n-2$)
subgraphs with the size-$3$ subgraphs on a common vertex.
###### Proof.
Our goal is to show that any size-$n$ subgraph can be dissected into a
connected size-($n-2$) subgraph and a connected size-$3$ subgraph on one
vertex. Because we join all size-($n-2$) and size-$3$ subgraphs in all
possible ways, if a dissection exists for a size-$n$ subgraph, it will be
discovered by the join operation. Suppose any size-$n$ subgraph can be
dissected into a size-$(n-2)$ and a size-$3$ subgraph. There are only two way
a size-($n+1$) subgraph can be constructed from a size-$n$ subgraph: 1) the
new vertex is connected with the size-$(n-2)$ subgraph, and in this case, the
size-($n+1$) subgraph can be dissected in the same way as the size-$n$
subgraph; 2) if the new vertex is only connected with the size-$3$ subgraph,
it is easy to verify that for either of the two cases (wedge or triangle), we
can always pick three connected vertices as the new dissection. As the base
case, all the six size-$4$ patterns can be dissected into a size-$3$ subgraph
and an edge. The proof finishes by induction. ∎
Note that multi-vertex exploration is not complete with more than two
vertices. For example, a seven-vertex three-pronged star graph with two
vertices in each prong cannot be obtained by joining any two size-4 subgraphs.
Therefore, we cannot explore more than two vertices in each step.
Two-vertex exploration can be either vertex-induced or edges induced. For
vertex-induced exploration, we add all the connecting edges between the two
joining subgraphs to the resulting subgraph. For edge-induced exploration, we
enumerate all possible combinations of the connecting edges between the
joining subgraphs and generate a resulting subgraph for each combination.
### 4.2. Depth-First Multi-Way Join of Subgraphs
Figure 2. Code of multi-way join.
To avoid the large memory consumption, we implement the exploration process as
a depth-first multi-way join. Suppose we want to join $t$ subgraph lists
$SL_{1},SL_{2},...,SL_{t}$ and the subgraph in $SL_{i}$ has $l_{i}$ vertices.
For each subgraph list $SL_{i}$, we group the subgraphs by each of its $l_{i}$
columns, create $l_{i}$ hash tables and store the hash tables in $H_{i}$. For
example, the size-$3$ subgraphs in Figure 1c are grouped by the vertex indices
in the first column. Once the hash tables are created, the multi-way join
operation is simply a nested loop that iterates over all possible combinations
of subgraphs in different hash tables, as shown in Figure 2. We first
enumerate all possible combinations of columns in different subgraph lists by
iterating over all hash tables of each subgraph list. Then, we identify the
matching keys $k_{1}$ in the first two hash tables and try to combine the
subgraphs ($s_{1}$ and $t_{2}$) on the key. If the two subgraphs make up a
valid larger subgraph $s_{2}$, we iterate over all the vertices of $s_{2}$ and
look up each vertex $k_{2}$ in the third hash table. For every subgraph
$t_{3}$ with key $k_{2}$ in the third hash table, we combine $s_{2}$ with
$t_{3}$ to obtain a larger subgraph. For joining more subgraph lists, the code
simply repeats the for loop.
Depth-first join is also used in Fractal (Dias et al., 2019) for single-vertex
exploration. The main issue is that it incurs a huge amount of redundant
memory accesses. Our two-vertex exploration mitigates this issue as it
requires fewer join steps to enumerate subgraphs of a certain size. To see
this, let us consider the exploration of size-5 subgraphs. With single-vertex
exploration, it requires a 4-way join of the edges in the graph. The first
join operation of the edge list does not incur any redundant memory accesses
as each neighbor list is accessed only once in the two hash tables. However,
when we join the intermediate size-3 subgraphs with the edge list, we need to
query the edge list for each intermediate subgraph. For non-consecutive size-3
subgraphs of the same key, each neighbor list will be accessed multiple times
during the join process. The same problem exists when joining size-4 subgraphs
with the edge list. In contrast, two-vertex exploration obtains size-5
subgraphs by performing a binary join of size-3 subgraphs which incurs no
redundant memory accesses. Our experimental results also validate this point.
### 4.3. Redundancy Removal through Smallest-Vertex-First Dissection
Input : $\;\;\;\text{subgraph }s;\text{subgraph }t;\text{joining key }k$
Output : $\;\;\;\text{combined subgraph }s^{\prime}$
1
2 func _dissect(_$s^{\prime}$ , $n$_)_:
3 foreach _$v$ in $s^{\prime}$ in ascending order_ do
4 $l=$ the first $n$ vertices visited by starting from $v$ and spanning to the
smallest vertex at each step;
5 $r^{\prime}=$ the unvisited vertices in $s^{\prime}$;
6 foreach _$v^{\prime}$ in $l$ in ascending order_ do
7 $r=r^{\prime}\cup v^{\prime}$;
8 if _$r$ is connected_ then return $l,r$ ;
9
10
11
12
13if _$s$ and $t$ have identical vertices other than $k$_ then return
$\emptyset$;
// $s^{\prime}$ is a valid subgraph joined by $s$ and $t$
14 $s^{\prime}=s\cup t$;
// find the smallest dissection of $s^{\prime}$
15 $l,r$ = dissect($s^{\prime}$, $t.size$);
// if the two joining subgraphs correspond to the smallest dissection, return
$s^{\prime}$
16 if _$l==t$ and $r==s$_ then return $s^{\prime}$;
17 else return $\emptyset$;
Algorithm 1 Combine two subgraphs and check for automorphism.
Combing small subgraphs in different ways can lead to identical results. As we
briefly mentioned in Figure 1c, joining subgraph ‘$342$’ and ‘$375$’ generates
the same subgraph as joining ‘$352$’ and ‘$274$’. These redundant subgraphs
incur redundant computation, and the redundancy can accumulate over the
exploration steps. To eliminate the redundant subgraphs, we perform an
automorphism check when a subgraph is generated. The previous automorphism
check technique for single-vertex exploration is based on the concept of the
canonicality of the subgraphs (Teixeira et al., 2015). This canonicality check
does not work for multi-vertex exploration because the small subgraphs are
generated by a matching algorithm and may not have the canonicality property.
We propose a smallest-vertex-first dissection method that enables the
redundancy removal for multi-vertex exploration.
Our method is based on the following observation: for any subgraph, there is
only one way to divide it into two smaller subgraphs with both subgraphs being
connected and one of them having the smallest spanning vertex indices. Thus,
we can eliminate redundancy by producing a subgraph $s^{\prime}$ only if the
two joining subgraphs correspond to this unique dissection of $s^{\prime}$.
The automorphism check is performed each time we combine two subgraphs (i.e.,
in the $combine$ function in Figure 2). Algorithm 1 shows the procedure of the
$combine$ function. For a pair of input subgraphs $s$ and $t$ ($t$ is usually
a size-$3$ subgraph), we first check if there are any other identical vertices
except for the joining vertex $k$. If yes, $s$ and $t$ cannot form a valid
subgraph, and the function return an empty set. If no, we give the combined
subgraph to a dissection procedure that divides the subgraph into two small
subgraphs $l$ and $r$. From the vertex with the smallest index, the dissection
procedure finds the smallest $n$ vertices and store them in $l$ where $n$ is
the size of $t$. Next, the algorithm checks if the remaining vertices can
constitute a connected subgraph $r$ with any of the vertices in $l$. If yes,
the dissection procedure stops and returns $l$ and $r$. The algorithm returns
as soon as the first dissection is found, and it will always return because of
Theorem 1. Once we have the smallest dissection $l$ and $r$, we check if they
are the same as $t$ and $s$. If yes, the $combine$ function returns the
combined subgraph; otherwise, it returns an empty set.
Example: The smallest-vertex-first dissection of the subgraph ‘$34257$’ in
Figure 1a can be obtained by spanning from vertex $2$. The two adjacent
vertices of $2$ are $3$ and $8$. Because $3$ is smaller, we take $3$ in the
first step, and the visited set contains vertex $2$ and $3$. The vertices that
are adjacent to the two visited vertices are $4,5,7,8$. Because $4$ is the
smallest, we take $4$ in the next step, and we have three vertices $2,3,4$ in
$l$. The unvisited vertices are $5$ and $7$. We check if any of $2,3,4$ can
form a connected graph with $5,7$, and we find $3$ is the smallest vertex that
connects 5 and 7. The algorithm stops and returns $l=\\{2,3,4\\}$ and
$r=\\{3,5,7\\}$. When joining the two subgraph lists in Figure 1c, our system
generates ‘$34275$’ (by combining ‘$342$’ and ‘$375$’) instead of ‘$35274$’
(by combing ‘$352$’ and ‘$274$’). For the same reason, ‘$37485$’ is not
generated by combing ‘$374$’ and ‘$385$’ as the smallest dissection of
‘$37485$’ is ‘$345$’ and ‘$387$’.
The worst cases complexity of the algorithm is $O(|s^{\prime}|^{3})$. Although
it is higher than the linear complexity of the automorphism check for single-
vertex exploration (Teixeira et al., 2015; Wang et al., 2018), the actual
number of instructions does not increase much because $s^{\prime}$ is small
and the algorithm usually returns early at line 7.
### 4.4. Pattern Aggregation with Index-based Quick Pattern
Next, we need to aggregate the subgraphs according to their patterns. This is
done by computing the canonical form of each subgraph. The subgraphs with the
same canonical form are isomorphic and will be put in the same group. As
pointed out in §2, computing the canonical form is expensive, especially for
large patterns. Previous work has used a quick pattern technique to reduce the
canonical form computation. However, their quick patterns encode little
topological information of the subgraphs, resulting in a lot of quick pattern
groups of isomorphic subgraphs.
We propose an index-based quick pattern technique that can achieve more
accurate grouping of subgraphs and reduce the overhead of canonical form
computation. The idea is to assign an index to each pattern in a subgraph list
and use the indices for computing the quick pattern of the combined subgraph.
If a subgraph list is generated by the matching algorithm, we simply index the
input patterns and store the indices with each subgraph. As shown in Figure
1b, the size-$3$ subgraphs are obtained by matching the two size-$3$ patterns.
We store the $pattern\\_idx$ with each of its embeddings. When two subgraphs
are combined, we construct a 4-tuple as the quick pattern for the combined
subgraph. The first two elements in the 4-tuple are the pattern indices of the
two joining subgraphs. The third element represents the position of the
joining vertex in the two subgraphs. Suppose the two joining subgraphs $s_{1}$
and $s_{2}$ are of size $n_{1}$ and $n_{2}$. If the joining vertex is the
$i$th vertex in $s_{1}$ and the $j$th vertex in $s_{2}$, then the value of the
third element is ($i\times n_{2}+j$). The last element is a bitarray
representing connections between the two subgraphs. If the $i$th vertex in
$s_{1}$ is connected with the $j$th vertex in $s_{2}$, then the ($i\times
n_{2}+j$)th bit in the bitarray is set.
Example: In Figure 1c, the resulting subgraph ‘$34275$’ is obtained by
joining $s_{1}=`342$’ and $s_{2}=`375$’, and its quick pattern is $\langle
0,0,0,32\rangle$. The first two elements are the pattern index of ‘$342$’ and
‘$375$’. The third element is 0 because the joining vertex is at position 0 in
both subgraphs. The last element is $32$ because the $s_{1}[1]=4$ is connected
with $s_{2}[2]=5$ in the graph and the $(1\times 3+2)$th bit is set in the
bitarray. Similarly, the quick pattern of both ‘$34287$’ and ‘$35287$’ is
$\langle 0,0,0,128\rangle$, and the quick pattern of ‘$34285$’ is $\langle
0,0,0,272\rangle$.
By encoding the sub-pattern information, our quick pattern achieves more
accurate grouping of the subgraphs and thus reduces the canonical form
computation. The computation is further reduced by multi-vertex exploration as
larger subgraphs contains more accurate sub-pattern information. To see this
point, let us consider the number of possible size-4 unlabeled patterns. We
have known that any size-4 subgraph can be obtained by joining a size-3
subgraph and an edge. The total number of possible 4-tuples with our index-
based quick pattern is 48 ($=2\times 1\times 6\times 4$) where $2$ represents
there are two types of size-3 subgraphs (i.e., triangle and wedge), $6$ is the
number of possible joining positions, and $4$ is the number of possible values
of the last element in the 4-tuple. In comparison, if we use the edge list as
the quick pattern as in previous work (Teixeira et al., 2015; Wang et al.,
2018), the fully-connected size-4 graph alone has 624 ($=6!-4\times 4!$)
possible quick patterns where $6!$ represents all possible permutations of the
six edges and $4\times 4!$ represents the permutations that do not have
adjacent edges. This indicates that our index-based technique has much fewer
possible patterns compared with the technique used in previous work, leading
to fewer groups for isomorphism check.
The quick pattern is computed after every $combine$ function in Figure 2. If
the $combine$ function returns a valid subgraph, we compute its quick pattern
and look for the quick pattern in a global dictionary. The dictionary keeps a
mapping from quick patterns to their indices. If the quick pattern exists, we
store its index with the subgraph. If a quick pattern is not found in the
dictionary, we increase the global index number and insert a new pair of quick
pattern and its index. In our implementation, we parallelize the for-loop that
iterates over all keys in the first joining hash table. To avoid
synchronization among threads, we store a quick pattern dictionary for each
thread.
### 4.5. Exploration Space Pruning
Figure 3. An example of exploration space pruning.
An optimization that most graph mining systems adopt for frequent subgraph
mining is to filter out the subgraphs of infrequent patterns so as to reduce
the subgraph exploration space (Teixeira et al., 2015; Wang et al., 2018; Chen
et al., 2020; Jamshidi et al., 2020). All of the existing systems achieve this
optimization with breadth-first exploration. They either store all
intermediate subgraphs (e.g., RStream (Wang et al., 2018), Pangolin (Chen et
al., 2020)) or maintain a list of frequent patterns and re-match these pattern
(e.g., Peregrine (Jamshidi et al., 2020)) in each exploration step. The
problem with the first approach is that it takes a lot memory and needs to
aggregate the subgraphs in each step. The problem with the second approach is
that it needs to perform redundant matching in each step, and it only works
for support measures that can be computed without storing all the embeddings
(e.g., MNI). If the user wants to use more accurate support measures (e.g.,
MIS, MVC (Meng and Tu, 2017)), the second approach will not work. An advantage
of two-vertex exploration is that it enables exploration space pruning without
storing intermediate results or re-matching.
Our main idea is that, instead of checking the support of the combined
pattern, we check whether the vertices around the joining point form any
subgraphs of smaller infrequent patterns. If an infrequent subgraph is found,
then the combined subgraph must be infrequent and should be discarded. Figure
3 shows an example of this method. When the system tries to join two subgraph
‘012’ and ‘234’ at vertex 2, if finds that there is an edge connecting vertex
0 and 3 and an edge connecting vertex 1 and 4. This forms two triangles ‘023’
and ‘124’. While triangle ‘124’ is frequent, triangle ‘023’ is not, according
to the list of frequent size-3 patterns. Due to the anti-monotone property of
the support measure, a frequent pattern cannot contain infrequent subpatterns.
Thus, the combined subgraph ‘01234’ must be infrequent and should not be used
for further exploration.
The above pruning procedure is done in the $combine$ function (line 9 in
Algorithm 1) when we check the connectivity among vertices of the two joining
subgraphs. For any size-3 subgraph ‘abc’ with ‘a’ being the joining vertex and
‘b’, ‘c’ from different subgraphs, if the subgraph is not in the list of
frequent size-3 patterns, the $combine$ function returns an empty set
immediately.
## 5\. Subgraph Sampling for Faster Exploration
In real applications, we may not need to find all frequent patterns, and
exhaustive exploration is unnecessary (Iyer et al., 2018). Thus, we propose a
subgraph sampling technique to accelerate the exploration process.
Sampling during Joining: The idea is to add a sampling operation each time we
iterate over the joining subgraphs, i.e., before each for-loop in the dotted
boxes in Figure 2. Because the MNI support measures the frequency of a pattern
as the number of distinct matching vertices, we sample a fixed number of
iterations in each of the boxed for-loops in Figure 2, in order to achieve a
more even distribution of subgraphs over all vertices. If a loop has fewer
iterations than the sampling threshold, we execute all of them; if a loop has
more iterations than the threshold, we sample the iterations uniformly to the
threshold number. This subgraph sampling during the joining phase can be
considered as a generalization of the neighbor sampling technique in ASAP
(Iyer et al., 2018). ASAP samples a subset of the edges when it extends the
matched subgraph from one vertex to its neighbors. We sample the neighboring
size-3 subgraphs instead. Intuitively, our subgraph sampling is more accurate
than neighbor sampling because size-3 subgraphs preserve more graph structures
than edges.
Sampling during Matching: For very large graphs, we may not be able to store
all size-3 subgraphs in memory or even on disk. To achieve fast mining, we can
sample the subgraphs during the matching phase and only store the sampled
subgraphs. Similar to the sampling in the joining phase, we sample a fixed
number of subgraphs around each vertex in order to have subgraphs evenly
distributed over all vertices. More specifically, we permute the vertex list
at each inner loop of the nested for-loop generated by AutoMine (Mawhirter and
Wu, 2019). The execution continues to the next iteration of the outermost loop
if $t$ subgraphs have been matched in the current iteration. This will give us
$t$ subgraphs sampled from each vertex. We set $t$ to a number such that all
the sampled subgraphs can be stored in memory. These sampled size-3 subgraphs
are then given to the join procedure to explore larger subgraphs. This
subgraph sampling during the matching phase can be considered as a
generalization of the edge sampling technique for approximate graph processing
(Agarwal et al., 2013; Zou and Holder, 2010). Previous work has shown that
edge sampling does not work well for graph mining tasks (Iyer et al., 2018).
Our subgraph sampling is much more robust than edge sampling for graph pattern
mining as it preserves more structures of the graph. Our experiments also
validate this point.
## 6\. Experimental Results
This section presents our experimental setup and performance comparison with
the existing graph mining systems and methods.
### 6.1. Experimental Setup
Platform: We run all the experiments on a workstation with an Intel Xeon
W-3225 CPU containing 8 physical cores (16 logical cores with hyper-
threading), 196GB memory, and a 4TB SSD. We use GCC 7.3.1 for compilation with
optimization level O2 enabled. All the systems are configured to run with 16
threads. We use OpenMP to parallelize the for-loop that iterates over all keys
in the first joining hash table.
Table 1. Graph datasets Graphs | #vertices | #edges | Description
---|---|---|---
CiteSeer (CI) (Elseidy et al., 2014) | 3264 | 4536 | Publication citation
MiCo (MI) (Elseidy et al., 2014) | 100K | 1.1M | Co-authorship
Orkut (OK) (ork, [n.d.]) | 3.1M | 117.2M | Social network
UK-2005 (UK) (you, [n.d.]) | 39M | 936M | Social network
Friendster (FR) (Yang and Leskovec, 2015) | 65M | 1.8B | Social network
Datasets: We test on five graphs as listed in Table. 1. These graphs are
commonly used for evaluating performance of graph mining systems. CiteSeer and
MiCo are labeled, and the other four are unlabeled. For the unlabeled graphs,
we randomly assign $30$ labels to the vertices.
Settings: We compare our system with three state-of-the-art graph mining
systems: Peregrine (Jamshidi et al., 2020) and AutoMine (Mawhirter and Wu,
2019) which represent the pattern-based systems, and Pangolin (Chen et al.,
2020) which represents the explore-aggregate-filter systems. We run edge-
induced FSM since it is more commonly evaluated by the existing graph mining
systems (Wang et al., 2018; Dias et al., 2019; Jamshidi et al., 2020). The
original code of AutoMine only supports vertex-induced FSM (which has much
less computation than edge-induced FSM) and uses number of embeddings as the
support measure (which is not anti-monotone). We adapt the code to support
edge-induced FSM with MNI support, and we use it to find all size-3 subgraphs
for our two-vertex exploration.
For most graphs, we set the MNI support threshold $t=0.001n$, $0.005n$,
$0.01n$ and $0.05n$ where $n$ is the number of nodes in the graph. The reason
we use proportional thresholds is that the MNI support measures frequency as
the number of distinct vertices (Meng and Tu, 2017). The threshold means that
if every vertex in a pattern maps to at least $t$ different vertices in the
graph, we consider the pattern frequent. For UK and FR, because $n$ is large,
there are few patterns that can meet threshold $0.001n$. Therefore, we test
with $0.0001n$ and $0.0005n$ on UK and FR.
### 6.2. Performance without Sampling
Table 2. Execution times in seconds. Systems: Two-Vertex exploration (TV), Peregrine (PR), AutoMine (AM) and Pangolin (PG). ‘T’ represents timeout after 24 hours of execution. ‘F’ execution failure due to insufficient memory or disk space. Size | Support | Gr. | TV | PR | AM | PG
---|---|---|---|---|---|---
4-FSM | 0.001 | CI | 0.99 | 5.4 | 5.1 | 5.5
0.005 | 0.89 | 4.8 | 4.8
0.01 | 0.81 | 3.4 | 3.7
0.05 | 0.61 | 1.1 | 2.8
4-FSM | 0.001 | MI | 41645 | F | 78244 | F
0.005 | 32763
0.01 | 29698
0.05 | 25682
5-FSM | 0.001 | CI | 25.2 | F | 68.2 | F
0.005 | 22.1
0.01 | 21.5
0.05 | 16.9
6-FSM | 0.001 | CI | 615 | F | 1924 | F
0.005 | 597
0.01 | 564
0.05 | 416
7-FSM | 0.001 | CI | 26760 | F | 63362 | F
0.005 | 24644
0.01 | 23697
0.05 | 16257
Since none of the compared systems supports sampling, we first run our
algorithm without sampling to compare the performance. Table 2 summarizes the
execution time of FSM for which at least one of the compared systems can
return result within 24 hours. The execution time of our system reported here
is the time of Step 2,3,4 as described in Section 3. We do not include the
time for Step1 because 1) it is negligible on these two graphs (0.08 seconds
on CI and 102 seconds on MI) compared with the joining time, and 2) Step1 can
be considered as preprocessing. We find that Peregrine and Pangolin abort for
most tasks. In fact, Peregrine paper (Jamshidi et al., 2020) only reports
results of 3-FSM. Pangolin (Chen et al., 2020) reports results mostly for
3-FSM. It reports 4-FSM for only one graph using large support thresholds, but
it fails to give result for MI. For the only one testcase (4-FSM on CI) that
Peregrine and Pangolin do return, our system is 1.8x to 5.6x faster. AutoMine
is able to return results for these tasks. However, because it matches the
patterns in a depth-first order, it cannot benefit from the anti-monotone
property (i.e., it does not run faster for larger support thresholds). Our
system is 1.9x to 8.4x faster than AutoMine for these tasks.
Figure 4. Performance of two-vertex exploration, single-vertex exploration,
and two-vertex exploration without our index-based quick pattern. The
Execution times are normalized for each task with the execution time of two-
vertex exploration in Table 2. Figure 5. Total memory access size in multi-
way join with two-vertex and single-vertex exploration for different FSM tasks
(MNI support threshold $0.001n$).
Advantage over Single-Vertex Exploration: As discussed in Section 4.2, one
advantage of two-vertex exploration over single-vertex exploration is that it
reduces the memory access overhead in depth-first multi-way join. To show the
advantage, we configure our system to run single-vertex exploration. The
single-vertex version still uses our index-based quick patterns, but it does
not support exploration space pruning since the size-3 subgraphs are not
computed. The execution times of single-vertex exploration are shown in Figure
4. We can see that single-vertex exploration is 1.02x to 1.52x slower than
two-vertex exploration. We also collect the total memory access sizes to the
hash tables with two-vertex exploration and single-vertex exploration
(assuming every query to the hash tables is a cache miss). As shown in Figure
5, two-vertex exploration reduces the memory access overhead by 5x to 189x.
The results are collected with support threshold $0.001n$. Other support
thresholds show a similar pattern.
Figure 6. Number of isomorphism checks for different FSM tasks (MNI support
threshold $0.001n$) with and without our index-based quick pattern technique.
Benefit of Index-Based Quick Pattern: To show the benefit of our index-based
quick pattern technique, we disable our index-based quick pattern and use the
quick pattern technique in previous work instead (i.e., a list of edges with
labels of adjacent nodes). Figure 4 shows the execution times of two-vertex
exploration without our index-based quick pattern. We can see that it leads to
1.75x to 2.78x slowdown. To further verify the advantage, we collect the
number of invocations to the bliss function (Junttila and Kaski, 2007) for
computing the canonical forms of subgraphs. As shown in Figure 6, our index-
based quick pattern reduces the number of isomorphism checks by 31x to 564x
for different tasks, which explains the speedups.
### 6.3. Performance with Sampling
Next, we evaluate the effectiveness of our sampling methods. Since all the
size-3 subgraphs of CI and MI can be stored in memory, we only perform
sampling during the joining phase for these two graphs.
Figure 7. Number of discovered size-4 frequent patterns on MI graph with
different support thresholds and different sampling thresholds. For single-
vertex exploration, ST$x$ means in every join step only $x$ edges are sampled
in each neighbor list. For two-vertex exploration, it means that $x$ edges and
$x^{2}$ size-3 subgraphs are sampled in each key group when we join the edge
list with the size-3 subgraphs. Figure 8. Number of discovered size-7
frequent patterns on CI graph with different support thresholds and different
sampling thresholds. For single-vertex exploration, ST$x$ means in every join
step only $x$ edges are sampled in each neighbor list. For two-vertex
exploration, it means that in every join step only $x^{2}$ size-3 subgraphs
are sampled in each key group.
Figure 7 shows the number of size-4 frequent patterns we can find on MI graph
with different support thresholds and different sampling thresholds. The
execution time of different runs are labeled on top of the bars. When the
support threshold is set to $0.001n$, there are 215025 frequent patterns in
total, and to discover all these patterns precisely our system needs to run
for 41645 seconds (as shown in Table 2). If we sample $10$ edges and $100$
size-3 subgraphs in each key group when we join the edge list and the size-3
subgraphs (ST10 in the figure), our two-vertex exploration returns 49% of the
frequent patterns in 671 seconds. The execution time is reduced by 62x. When
the support threshold is set to $0.005n$, we can find 41% of the total
frequent patterns within 524 seconds, which is 1/63 of the total execution
time. When the support threshold is set to $0.05n$, there are only 8 frequent
patterns, and our two-vertex exploration with ST6 sampling can find 7 of them
in 58 seconds, which leads to a 443x speedup compared with the accurate
execution. The figure also shows that the larger sampling thresholds we use
the more frequent patterns we can find.
Figure 8 shows the number of size-7 frequent patterns found on CI graph with
different support thresholds and different sampling thresholds. When the
support threshold is $0.001n$, our two-vertex exploration can find 86% of the
frequent patterns in 1284 seconds with ST10 sampling. Compared with the time
of accurate execution in Table 2, sampling achieves a 21x speedup. When the
support threshold is set to $0.005n$ and $0.01n$, there are fewer frequent
patterns, and our two-vertex exploration with ST10 sampling can find more than
90% of the frequent patterns with less than 1/21 of the total execution time.
When the support threshold is $0.05n$, there are only 22 frequent patterns,
and our two-vertex exploration with ST6 sampling can find all of them within
170 seconds, which is 1/96 of the accurate execution time.
Table 3. Results of 9-FSM on CI graph with sampling threshold 4 for two-vertex
exploration (TV) and sampling threshold 2 for single-vertex exploration (SV).
Support | 0.001 | 0.005 | 0.01 | 0.05
---|---|---|---|---
TV | 63941 | 6050 | 1770 | 16
SV | 402 | 0 | 0 | 0
(a) Number of discovered patterns
0.001 | 0.005 | 0.01 | 0.05
---|---|---|---
73 | 71 | 54 | 45
75 | 75 | 63 | 46
(b) Execution time (sec)
Advantage over Single-Vertex Exploration: As discussed in Section 5, another
advantage of two-vertex exploration over single-vertex exploration is that it
leads to more accurate sampling. To verify this, we configure our system to
run sampled single-vertex exploration. Since single-vertex exploration needs
twice join steps as two-vertex exploration, we set its sampling threshold to
the square root of the threshold for two-vertex exploration in order to
achieve a similar size of overall exploration space. As shown in Figure 7 and
8, if we do not include the matching time, two-vertex exploration has a
slightly shorter execution time than single-vertex exploration when they use
the corresponding sampling thresholds. Even if we add the time for matching
size-3 subgraphs (102 seconds on MI and 0.08 seconds on CI), the total
execution time is close to that of single-vertex exploration. For 4-FSM on MI,
two-vertex exploration finds 6% to 57% more frequent patterns than single-
vertex exploration. For 7-FSM on CI, the number of frequent patterns found by
two-vertex exploration is 1.18x to 4x that of single-vertex exploration. Table
9b shows the number of size-9 frequent patterns found on CI graph with
different support thresholds. We can see that with a similar execution time
two-vertex exploration discovers much more frequent patterns than single-
vertex exploration. It is worth noting that none of the previous systems can
return results for 9-FSM even on a small graph like CI. AutoMine cannot even
enumerate all the size-9 unlabeled patterns in 24 hours.
Table 4. Results of 5-FSM on OK graph with support threshold $0.001n$ and
different sampling thresholds. ‘M. ST’ stands for Matching Sampling Threshold,
‘M. Time’ stands for Matching Time, ‘J. ST’ stands for Joining Sampling
Threshold, ‘J. Time’ stands for Joining Time.
M. ST | M. Time (sec) | J. ST | J. Time (sec) | # of Patterns
---|---|---|---|---
2 | 119 | 4 | 157 | 18
16 | 168 | 18
4 | 119 | 4 | 207 | 20
16 | 216 | 20
8 | 120 | 4 | 268 | 20
16 | 269 | 20
16 | 122 | 4 | 299 | 20
16 | 318 | 20
(c) Two-vertex exploration
J. ST | J. Time | # of Patterns
---|---|---
1 | 1184 | 15
2 | 6780 | 18
(d) Single-vertex exploration
Table 5. Results of 5-FSM on UK and FR graph with different sampling
thresholds and support thresholds. ‘M. ST’ stands for Matching Sampling
Threshold, ‘M. Time’ stands for Matching Time, ‘J. ST’ stands for Joining
Sampling Threshold, ‘J. Time’ stands for Joining Time.
Support | M. ST | M. Time (sec) | J. ST | J. Time (sec) | # of Patterns
---|---|---|---|---|---
0.0001 | 2 | 2254 | 4 | 140 | 76
16 | 288 | 143
4 | 2530 | 4 | 185 | 105
16 | 505 | 188
0.0005 | 2 | 2254 | 4 | 105 | 1
16 | 235 | 3
4 | 2530 | 4 | 138 | 8
16 | 421 | 14
(e) Two-vertex exploration on UK
Support | M. ST | M. Time (sec) | J. ST | J. Time (sec) | # of Patterns
---|---|---|---|---|---
0.0001 | 2 | 6010 | 4 | 499 | 1
16 | 7620 | 1
4 | 6657 | 4 | 547 | 4
16 | 9955 | 8
0.0005 | 2 | 6010 | 4 | 315 | 1
16 | 4637 | 1
4 | 6657 | 4 | 377 | 1
16 | 5417 | 1
(f) Two-vertex exploration on FR
Results on Large Graphs: Since the size-3 subgraphs of OK, UK and FR cannot
be entirely stored in memory, we perform sampling during the matching phase
and only store the sampled size-3 subgraphs. Table 9c shows the number of
size-5 frequent patterns with support larger than $0.001n$ found on OK graph.
In the table, a matching sampling threshold $x$ means that $x$ subgraphs are
sampled from each vertex during the matching phase. A larger matching sampling
threshold results in longer matching time, although they are not proportional
– the matching time is mainly determined by the number of subgraph groups that
need isomorphism checks. We find that the number of discovered patterns does
not increase much with larger sampling thresholds. This is because $0.001n$ is
a relatively large support threshold for this graph, and there are not many
frequent patterns. To show the advantage of two-vertex exploration, we run
single-vertex exploration for the same task. Since single-vertex exploration
needs not match the size-3 subgraphs, we only perform sampling during the
joining phase with threshold 1 and 2. The results are shown in Table 9d. We
can see that single-vertex exploration takes a longer time and finds fewer
patterns.
The results of 5-FSM on UK and FR graph are shown in Table 5. We can see that
matching takes a large proportion of the total execution time. This is because
there are a lot of size-3 subgraphs and patterns in these two large graphs.
However, if we consider matching as preprocessing and store the sampled size-3
subgraphs in memory, the joining procedure is fast. As shown in Table 9e, our
system can find frequent patterns on UK within a few minutes, and more
patterns can be found by using larger joining sampling thresholds. Table 9f
shows the results of 5-FSM on FR graph. Again, the matching procedure is
expensive. Once the size-3 subgraph are sampled, we can find size-5 frequent
patterns in a relatively short time with sampled join. For comparison, we also
run single-vertex exploration with sampling threshold 2 on these two graphs.
It cannot finish execution within 24 hours, so we print out the found patterns
after 24 hours of execution. For UK, it returns 5 frequent patterns when
support threshold is set to $0.0001n$, and 0 frequent pattern when support
threshold is $0.0005n$. For FR, single-vertex exploration with sampling
threshold 2 cannot return any frequent pattern within 24 hours.
## 7\. Related Work
This section summarizes the graph pattern mining systems that are most related
to our work.
Exploration-based Systems: Arabesque (Teixeira et al., 2015) is a distributed
graph pattern mining system. It enumerates all possible embeddings in multiple
rounds and uses a filter-process model to generate the results. It first
propose the quick pattern technique for reducing isomorphism checks. RStream
(Wang et al., 2018) is the first single-machine, out-of-core graph mining
system. It supports a rich programming model that exposes relational algebra
for developers to express various mining tasks and a runtime engine that can
efficiently compute the relational operations. Pangolin (Chen et al., 2020)
also targets single-machine but provides GPU programming interface for
acceleration. DistGraph (Talukder and Zaki, 2016), ScaleMine (Abdelhamid et
al., 2016) and G-miner (Chen et al., 2018) are all distributed graph mining
systems that adopt breadth-first exploration. DistGraph focuses on reducing
the communication of distributed computing when each node can only have a
portion of the graph. ScaleMine proposes a two-phase mining approach to
achieve good load balance and reduce communication in distributed computing.
G-miner proposes a block-based graph partitioning technique and uses work
stealing to achieve good load balance. Because these systems use breadth-first
exploration and need to store all intermediate results, they are not able to
mine large patterns on large graphs. Fractal (Dias et al., 2019) is also
exploration-based, but it supports depth-first exploration to reduce the
memory consumption. All of the existing systems adopt single-vertex
exploration. Our system is the first to adopt multi-vertex exploration for
mining larger pattern in graphs.
Pattern-based Systems: AutoMine (Mawhirter and Wu, 2019) is a single-machine
graph mining system that features compiler-based optimizations. Their main
idea is to enumerate all the unlabeled patterns of a particular size and match
them one-by-one on a graph. Because the patterns are given, AutoMine is able
to search an optimal matching strategy and combine the matching procedures of
multiple patterns. Because of its depth-first matching order, AutoMine is hard
to benefit from the anti-monotone property of FSM. Also, when the pattern size
is more than 7, enumerating the patterns becomes difficult. Peregrine
(Jamshidi et al., 2020) is another pattern-based system. Instead of
enumerating all the patterns before matching, it discovers patterns based on
the subgraphs it has explored and maintains a list of the patterns. The main
issue with Peregrine is that it needs to rematch the frequent patterns in each
step, which leads to redundant computation. DwavesGraph (Chen and Qian, 2020)
is a recently proposed pattern-based graph mining system. It is based on the
idea that the task of matching a large pattern can be divided into smaller
tasks of matching the subpatterns. Similar to AutoMine, it needs to know all
the unlabeled patterns in advance. Thus, it cannot discover large patterns. In
fact, DwavesGraph paper only reports result for 3-FSM.
Approximate Pattern Mining: Sampling has been proposed by earlier works (Al
Hasan and Zaki, 2009; Saha and Al Hasan, 2015) to accelerate FSM in a database
of graphs. In this setting, a pattern is considered frequent if it exists in
more than a certain amount of graphs. The main idea of these works is to
perform random walk in the space of all patterns. Every time it walks from one
pattern to another, it calculates a probability distribution of all candidate
patterns. By carefully setting the sampling probability at each step, they
ensure that patterns of higher supports are more likely to be sampled (Al
Hasan and Zaki, 2009). More recent works consider FSM on a single graph since
it is more commonly used in real applications and is more general (a list of
graphs can be considered as a single graph with disconnected components)
(Elseidy et al., 2014). Sampling has also proposed to accelerate pattern-based
graph mining in this setting (Iyer et al., 2018; Mawhirter et al., 2018; Pavan
et al., 2013). The main idea is to sample edges in the graph based on the
given patterns and estimate the actual results with the sampled results. These
methods need to know the patterns in advance. It is not obvious how they can
be applied to the exploration-based systems. We fill this gap and show that
FSM can be accelerated by sampling the subgraphs in each key group of the join
operation.
## 8\. Conclusion
In this work, we propose a novel two-vertex exploration method to accelerate
frequent subgraph mining. Based on two-vertex exploration, we further improve
the performance through an index-based quick pattern technique and subgraph
sampling. The experiments show that our method outperforms other state-of-the-
art graph mining systems for FSM on various input graphs and pattern sizes.
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|
Saarland University and Max Planck Institute for Informatics, Saarland
Informatics Campus, Saarbrücken<EMAIL_ADDRESS>work
is part of the project TIPEA that has received funding from the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation programme (grant agreement No. 850979). Saarbrücken Graduate School
of Computer Science and Max Planck Institute for Informatics, Saarland
Informatics Campus, Saarbrücken<EMAIL_ADDRESS>of this
work was done at BARC, Copenhagen University, supported by the VILLUM
Foundation grant 16582. Karl Bringmann and André Nusser [500]Theory of
computation Problems, reductions and completeness
# Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff
Distance Under Translation
Karl Bringmann André Nusser
###### Abstract
Computing the similarity of two point sets is a ubiquitous task in medical
imaging, geometric shape comparison, trajectory analysis, and many more
settings. Arguably the most basic distance measure for this task is the
Hausdorff distance, which assigns to each point from one set the closest point
in the other set and then evaluates the maximum distance of any assigned pair.
A drawback is that this distance measure is not translational invariant, that
is, comparing two objects just according to their shape while disregarding
their position in space is impossible.
Fortunately, there is a canonical translational invariant version, the
Hausdorff distance under translation, which minimizes the Hausdorff distance
over all translations of one of the point sets. For point sets of size $n$ and
$m$, the Hausdorff distance under translation can be computed in time
$\tilde{\mathcal{O}}(nm)$ for the $L_{1}$ and $L_{\infty}$ norm [Chew, Kedem
SWAT’92] and $\tilde{\mathcal{O}}(nm(n+m))$ for the $L_{2}$ norm
[Huttenlocher, Kedem, Sharir DCG’93].
As these bounds have not been improved for over 25 years, in this paper we
approach the Hausdorff distance under translation from the perspective of
fine-grained complexity theory. We show (i) a matching lower bound of
$(nm)^{1-o(1)}$ for $L_{1}$ and $L_{\infty}$ (and all other $L_{p}$ norms)
assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of
$n^{2-o(1)}$ for $L_{2}$ in the imbalanced case of $m=\mathcal{O}(1)$ assuming
the 3SUM Hypothesis.
###### keywords:
Hausdorff Distance Under Translation, Fine-Grained Complexity Theory, Lower
Bounds
## 1 Introduction
As data sets become larger and larger, the requirement for faster algorithms
to handle such amounts of data becomes increasingly necessary. One very common
type of data that is created during measurements is point sets in the plane,
for example when recording GPS trajectories or describing shapes of objects,
in medical image analysis, and in various data science applications.
A fundamental algorithmic tool for analyzing point sets is to compute the
similarity of two given sets of points. There are several different measures
of similarity in this setting, for example Hausdorff distance [21], geometric
bottleneck matching [18], Fréchet distance [3], and Dynamic Time Warping [25].
Among these measures, the Hausdorff distance is arguably the most basic and
intuitive: It assigns to each point from one set the closest point in the
other set and then evaluates the maximum distance of all assigned pairs of
points.111There is a directed and an undirected variant of the Hausdorff
distance, see Section 2. In this introduction, we do not differentiate between
these two, since all our statements hold for both variants. For a discussion
of the other previously mentioned distance measures, see Section 1.1.
While these similarity measures are of great practical relevance, for some
applications it is a drawback that they are not translational invariant, i.e.,
when translating one of the point sets, the distance can – and in most cases
will – change. This is unfavorable in applications that ask for comparing the
shape of two objects, meaning that the absolute position of an object is
irrelevant. Examples of this task arise for example in 2D object shape
similarity, medical image analysis [19], classification of handwritten
characters [10], movement patterns of animals [12], and sports analysis [17].
Fortunately, any point set similarity measure has a canonical translational
invariant version, by minimizing the similarity measure over all translations
of the two given point sets. For the Hausdorff distance this variant is known
as the _Hausdorff distance under translation_ , see Section 2 for a formal
definition. Given two point sets in the plane of size $n$ and $m$, the
Hausdorff distance under translation can be computed in time
$\mathcal{O}(nm\log^{2}nm)$ for the $L_{1}$ and $L_{\infty}$ norm [16], and in
time $\mathcal{O}(nm(n+m)\log nm)$ for the $L_{2}$ norm [22]. We are not aware
of any lower bounds for this problem, not even conditional on a plausible
hypothesis. The only results in this direction are $\Omega(n^{3})$ lower
bounds on the arrangement size [16] and on the number of connected components
of the feasible translations [28] (for the decision problem on points in the
plane with $n=m$). However, these bounds also hold for $L_{1}$ and
$L_{\infty}$, where they are “broken” by the $\mathcal{O}(nm\log^{2}nm)$-time
algorithm [16], so apparently these bounds are irrelevant for the running time
complexity.
In this paper, we approach the Hausdorff distance under translation from the
viewpoint of fine-grained complexity theory [29]. For two problem settings, we
show that the known algorithms are optimal up to lower order factors assuming
standard hypotheses:
1. 1.
We show an $(nm)^{1-o(1)}$ lower bound for all $L_{p}$ norms — and in
particular $L_{1}$ and $L_{\infty}$, matching the
$\mathcal{O}(nm\log^{2}nm)$-time algorithm from [16] up to lower order
factors, see Section 3.
This result holds conditional on the Orthogonal Vectors Hypothesis, which
states that finding two orthogonal vectors among two given sets of $n$ binary
vectors in $d$ dimensions cannot be done in time
$\mathcal{O}(n^{2-\varepsilon}\textrm{poly}(d))$ for any $\varepsilon>0$. It
is well-known that the Orthogonal Vectors Hypothesis is implied by the Strong
Exponential Time Hypothesis [30], and thus our lower bound also holds assuming
the latter [23]. These two hypotheses are the most standard assumptions used
in fine-grained complexity theory in the last decade [29].
2. 2.
We show an $n^{2-o(1)}$ lower bound for $L_{2}$ in the imbalanced case
$m=\mathcal{O}(1)$, matching the $\mathcal{O}(nm(n+m)\log nm)$-time algorithm
from [16] up to lower order factors, see Section 4. Previously, an
$n^{2-o(1)}$ lower bound was only known for the more general problem of
computing the Hausdorff distance under translation of sets of _segments_ in
the case that both sets have size $n$ (a problem for which the best known
algorithm runs in time222By $\tilde{\mathcal{O}}$-notation we ignore
logarithmic factors in $n$ and $m$. $\tilde{\mathcal{O}}(n^{4})$) [6].
Our result holds conditional on the 3SUM Hypothesis, which states that
deciding whether, among $n$ given integers, there are three that sum up to 0
requires time $n^{2-o(1)}$. This hypothesis was introduced by Gajentaan and
Overmars [20], is a standard assumption in computational geometry [24], and
has also found a wealth of applications beyond geometry (see, e.g., [1, 2, 4,
26]).
Our lower bounds close gaps that have not seen any progress over 25 years.
Furthermore, note that our second lower bound shows a separation between the
$L_{2}$ norm and the $L_{1}$ and $L_{\infty}$ norms, as in the imbalanced case
$m=\mathcal{O}(1)$ the latter admits a $\tilde{\mathcal{O}}(n)$-time algorithm
[16] while the former requires time $n^{2-o(1)}$ assuming the 3SUM Hypothesis.
We leave it as an open problem whether for $L_{2}$ the balanced case $n=m$
requires time $n^{3-o(1)}$.
### 1.1 Related work
Our work continues a line of research on fine-grained lower bounds in
computational geometry, which had early success with the 3SUM Hypothesis [20]
and recently got a new impulse with the Orthogonal Vectors Hypothesis (or
Strong Exponential Time Hypothesis) and resulting lower bounds for the Fréchet
distance [7], see also [13, 11]. Continuing this line of research is getting
increasingly difficult, although there are still many classical problems from
computational geometry without matching lower bounds. In this paper we obtain
such bounds for two settings of the classical Hausdorff distance under
translation.
Besides Hausdorff distance, there are several other distance measures on point
sets, including geometric bottleneck matching [18], Fréchet distance [3], and
Dynamic Time Warping [25]. The geometric bottleneck matching minimizes the
maximal distance in a perfect matching between the two given point sets.
Fréchet distance and Dynamic Time Warping additionally take the order of the
input points into account. They both consider the same class of _traversals_
of the input points, and the Fréchet distance minimizes the _maximal_ distance
that occurs during the traversal, while Dynamic Time Warping minimizes the
_sum_ of distances.
Let us discuss the canonical translational invariant versions of these
distance measures. For geometric bottleneck matching under translation, Efrat
et al. designed an $\tilde{\mathcal{O}}(n^{5})$ algorithm [18]. The discrete
Fréchet distance under translation has an
$\tilde{\mathcal{O}}(n^{4.66\dots})$-time algorithm and a conditional lower
bound of $n^{4-o(1)}$ [9], see also [10] for algorithm engineering work on
this topic. While Dynamic Time Warping is a very popular measure (in
particular for video and speech processing), no exact algorithm for its
canonical translational invariant version is known in $L_{2}$ since it
contains the geometric median problem as a special case [5].
Further work on the Hausdorff distance under translation includes an
$\mathcal{O}((n+m)\log nm)$-time algorithm for point sets in one dimension
[27]. For generalizations to dimensions $d>2$ see [16, 15].
## 2 Preliminaries
In this paper we consider finite point sets which lie in $\mathbb{R}^{2}$. For
any $p\in\mathbb{R}^{2}$, we use $p_{x}$ and $p_{y}$ to refer to its first and
second component, respectively. For a point set $A\subset\mathbb{R}^{2}$ and a
translation $\tau\in\mathbb{R}^{2}$, we define $A+\tau\coloneqq\\{a+\tau\mid
a\in A\\}$. To denote index sets, we often use $[n]\coloneqq\\{1,\dots,n\\}$.
Given a point $q\in\mathbb{R}^{2}$, its $p$-norm is defined as
$\lVert q\rVert_{p}\coloneqq\left(\lvert q_{x}\rvert^{p}+\lvert
q_{y}\rvert^{p}\right)^{\frac{1}{p}}.$
We now introduce several distance measures, which are all versions of the
famous Hausdorff distance. First, let us define the most basic version. Let
$A,B\subset\mathbb{R}^{2}$ be two point sets. The _directed Hausdorff
distance_ is defined as
$\delta_{\vec{H}}(A,B)\coloneqq\max_{a\in A}\min_{b\in B}\lVert
a-b\rVert_{p}.$
Note that, intuitively, the directed Hausdorff distance measures the distance
from $A$ to $B$ but not from $B$ to $A$, and it is not symmetric. A symmetric
variant of the Hausdorff distance, the _undirected Hausdorff distance_ , is
defined as
$\delta_{H}(A,B)\coloneqq\max\\{\delta_{\vec{H}}(A,B),\delta_{\vec{H}}(B,A)\\}.$
Note that, by definition, $\delta_{\vec{H}}(A,B)\leq\delta_{H}(A,B)$. Both of
the above distance measures can be modified to a version which is invariant
under translation. The _directed Hausdorff distance under translation_ is
defined as
$\delta_{\vec{H}}^{T}(A,B)\coloneqq\min_{\tau\in\mathbb{R}^{2}}\delta_{\vec{H}}(A,B+\tau),$
and the _undirected Hausdorff distance under translation_ is defined as
$\delta_{H}^{T}(A,B)\coloneqq\min_{\tau\in\mathbb{R}^{2}}\delta_{H}(A,B+\tau).$
Again, it holds that $\delta_{\vec{H}}^{T}(A,B)\leq\delta_{H}^{T}(A,B)$.
Naturally, for all of the above distance measures, the decision problem is
defined such that we are given two point sets $A,B$ and a threshold distance
$\delta$, and ask if the distance of $A,B$ is at most $\delta$.
For the Hausdorff distance on point sets (without translation) the undirected
distance is at most as hard as the directed distance, because the undirected
distance can be calculated using two calls to an algorithm computing the
directed distance.333Actually, the directed Hausdorff distance is also at most
as hard as the undirected Hausdorff distance (thus, they are equally hard), as
$\delta_{\vec{H}}(A,B)=\delta_{H}(A\cup B,B)$. However, note that for the
Hausdorff distance under translation, we cannot just compute the directed
distance twice and then obtain the undirected distance as we have to take the
maximum for the same translation.
## 3 OV based $(mn)^{1-o(1)}$ lower bound for $L_{p}$
We now present a conditional lower bound of $(mn)^{1-o(1)}$ for the Hausdorff
distance under translation — first for $L_{1}$ and $L_{\infty}$, and then we
discuss how to generalize this bound to $L_{p}$. We present the first lower
bound only for the $L_{1}$ case, as the same construction carries over to the
$L_{\infty}$ case via a rotation of the input sets by $\tfrac{\pi}{4}$. Our
lower bound is based on the hypothesized hardness of the Orthogonal Vectors
problem.
###### Definition 3.1 (Orthogonal Vectors Problem (OV)).
Given two sets $X,Y\subset\\{0,1\\}^{d}$ with $|X|=m,|Y|=n$, decide whether
there exist $x\in X$ and $y\in Y$ with $x\cdot y=0$.
A popular hypothesis from fine-grained complexity theory is as follows.
###### Definition 3.2 (Orthogonal Vectors Hypothesis (OVH)).
The Orthogonal Vectors problem cannot be solved in time
$\mathcal{O}((nm)^{1-\varepsilon}\text{poly}(d))$ for any $\varepsilon>0$.
This hypothesis is typically stated and used for the balanced case $n=m$.
However, it is known that the hypothesis for the balanced case is equivalent
to the hypothesis for any unbalanced case $n=m^{\alpha}$ for any fixed
constant $\alpha>0$, see, e.g, [8, Lemma 2.1].
We now describe a reduction from Orthogonal Vectors to Hausdorff distance
under translation. To this end, we are given two sets of $d$-dimensional
binary vectors $X=\\{x_{1},\dots,x_{m}\\}$ and $Y=\\{y_{1},\dots,y_{n}\\}$
with $\lvert X\rvert=m$ and $\lvert Y\rvert=n$, and we construct an instance
of the undirected Hausdorff distance under translation defined by point sets
$A$ and $B$ and a decision distance $\delta=1$. First, we describe the high-
level structure of our reduction. The point set $A$ consists only of Vector
Gadgets, which encode the vectors of $X$ using $2md$ points. The point set $B$
consists of three types of gadgets:
* •
_Vector Gadgets:_ They encode the vectors from $Y$, very similarly to the
Vector Gadgets of $A$.
* •
_Translation Gadget:_ It restricts the possible translations of the point set
$B$.
* •
_Undirected Gadget:_ It makes our reduction work for the undirected Hausdorff
distance under translation by ensuring that the maximum over the directed
Hausdorff distances is always attained by $\delta_{\vec{H}}(B+\tau,A)$.
See Figure 1 for an overview of the reduction. Intuitively, the first
dimension of the translation chooses the vector $y\in Y$ while the second
dimension of the translation chooses the vector $x\in X$. An alignment of the
Vector Gadgets within distance 1 is then possible if and only if $x$ and $y$
are orthogonal. Alignments that can circumvent this orthogonality check are
not possible as we restrict the translations to a small set of candidates by
placing dummy Vector Gadgets on the right side and by including a Translation
Gadget.
### 3.1 Gadgets
Figure 1: Sketch of the reduction from OV to the undirected Hausdorff distance
under translation.
We now describe the gadgets in detail. Let $\varepsilon>0$ be a sufficiently
small constant, e.g., $\varepsilon=\frac{1}{20mnd}$. Recall that the distance
for which we want to solve the decision problem is $\delta=1$. Furthermore, we
denote the $i$th component of a vector $v$ by $v[i]$ and we use $0^{d}$ and
$1^{d}$ to denote the $d$-dimensional all-zeros and all-ones vector,
respectively.
##### Vector Gadget
We define a general Vector Gadget, which we then use at several places by
translating it. Given a vector $v\in\\{0,1\\}^{d}$, the Vector Gadget consists
of the points $p_{1},\dots,p_{d}\in\mathbb{R}^{2}$:
$p_{i}=\begin{cases}(\varepsilon^{2},i\varepsilon),&\text{if }v[i]=0\\\
(0,i\varepsilon),&\text{if }v[i]=1\\\ \end{cases}$
We denote the Vector Gadget created from vector $v$ by $V(v)$. Additionally,
we define a mirrored version of the gadget $V$ as
$\overline{V}(v)\coloneqq V(\bar{v}),$
where $\bar{v}$ is the inversion of $v$, i.e., each bit is flipped.
###### Lemma 3.3.
Given two vectors $v_{1},v_{2}\in\\{0,1\\}^{d}$ and corresponding Vector
Gadgets $V_{1}=V(v_{1})$ and $V_{2}=\overline{V}(v_{2})+(1,0)$, we have
$\delta_{H}(V_{1},V_{2})\leq 1$ if and only if $v_{1}\cdot v_{2}=0$.
###### Proof 3.4.
Let the points of $V_{1}$ (resp. $V_{2}$) be denoted as $p_{1},\dots,p_{d}$
(resp. $q_{1},\dots,q_{d}$). First, note that $\lVert
p_{i}-q_{j}\rVert_{1}=1+\lvert
i-j\rvert\varepsilon+(v_{1}[i]+v_{2}[j]-1)\varepsilon^{2}>1$ for $i\neq j$.
Thus, for the Hausdorff distance to be at most $1$, we have to match $p_{i}$
to $q_{i}$ for all $i\in[d]$. This is possible if and only if $v_{1}[i]=0$ or
$v_{2}[i]=0$, as $p_{i}$ and $q_{i}$ are only at distance larger than 1 for
$v_{1}[i]=1$ and $v_{2}[i]=1$.
See Figure 2 for an example. Note that if we swap both gadgets and invert both
vectors (i.e., flip all their bits), the Hausdorff distance does not change
and thus an analogous version of Lemma 3.3 holds in this case, as we are just
performing a double inversion.
###### Lemma 3.5.
Given two vectors $v_{1},v_{2}\in\\{0,1\\}^{d}$ and corresponding Vector
Gadgets $V_{1}=\overline{V}(v_{1})$ and $V_{2}=V(v_{2})+(1,0)$, we have
$\delta_{H}(V_{1},V_{2})\leq 1$ if and only if
$\bar{v}_{1}\cdot\bar{v}_{2}=0$, where $\bar{v}_{1},\bar{v}_{2}$ are the
inversions of $v_{1},v_{2}$.
Figure 2: A depiction of the two types of Vector Gadgets and how they are
placed to check for orthogonality.
For any $x,y,D\in\mathbb{R}$, we call Vector Gadgets $V_{1}=V(v_{1})+(x,y)$
and $V_{2}=\overline{V}(v_{2})+(x+D,y)$ _vertically aligned_ , or more
precisely, _vertically aligned at distance $D$_.
##### Translation Gadget
To ensure that $B$ cannot be translated arbitrarily, we introduce a gadget to
restrict the translations to a restricted set of candidates. The Translation
Gadget $T$ consists of two translated Vector Gadgets of the zero vector:
$T\coloneqq(\overline{V}(1^{d})-(2-n\varepsilon,0))\cup(\overline{V}(0^{d})+(2+2\varepsilon,0)).$
We show that restricting the coordinates of the points of the other set
involved in the Hausdorff distance under translation instance, already
restricts the feasible translations significantly.
###### Lemma 3.6.
Let
$P\subset[-1-\frac{1}{2}\varepsilon,1+\frac{1}{2}\varepsilon]\times\mathbb{R}$
be a point set and $T$ the Translation Gadget. If
$\delta_{\vec{H}}^{T}(T,P)\leq 1$, then
$\tau_{x}^{*}\in[-(n+\frac{1}{2})\varepsilon-\varepsilon^{2},-\frac{3}{2}\varepsilon]$,
where $\tau^{*}$ is any translation satisfying
$\delta_{\vec{H}}(T,P+\tau^{*})\leq 1$.
###### Proof 3.7.
We show the contrapositive. Therefore, assume the converse, i.e., that
$\tau_{x}^{*}$ is not contained in
$[-(n+\frac{1}{2})\varepsilon-\varepsilon^{2},-\frac{3}{2}\varepsilon]$. If
$\tau_{x}^{*}<-(n+\frac{1}{2})\varepsilon-\varepsilon^{2}$, then
$-1-\frac{1}{2}\varepsilon-(-2+n\varepsilon+\varepsilon^{2}+\tau_{x}^{*})>1$
and thus the left part of $T$ cannot contain any point of $P$ at distance at
most $1$. If $\tau_{x}^{*}>-\frac{3}{2}\varepsilon$, then
$2+2\varepsilon+\tau_{x}^{*}-(1+\frac{1}{2}\varepsilon)>1$ and thus the right
part of $T$ cannot contain any point of $P$ at distance at most $1$. Thus,
$\delta_{\vec{H}}^{T}(T,P)>1$.
##### Undirected Gadget
To ensure that each point in $A$ can be matched to a point in $B$ within
distance $1$, we add auxiliary points to $B$. The Undirected Gadget is defined
by the point set
$U\coloneqq\\{(-\frac{1}{2},0),(\frac{1}{2},0)\\}.$
###### Lemma 3.8.
Given a set of points
$P\subset[-1-\frac{1}{2}\varepsilon,1+\frac{1}{2}\varepsilon]\times[-\frac{1}{8},\frac{1}{8}]$,
it holds that $\delta_{\vec{H}}(P,U+\tau)\leq 1$ for any
$\tau\in[-(n+\frac{1}{2})\varepsilon-\varepsilon^{2},(n+\frac{1}{2})\varepsilon+\varepsilon^{2}]\times[-\frac{1}{8},\frac{1}{8}]$.
###### Proof 3.9.
By symmetry, we can restrict to proving that the distance of the point set
$P^{\prime}=P\cap[0,1+\frac{1}{2}\varepsilon]\times[-\frac{1}{8},\frac{1}{8}]$
to $(\frac{1}{2},0)+\tau$ is at most $1$. For any $p^{\prime}\in P^{\prime}$,
we have $\lvert
p^{\prime}_{x}-(\frac{1}{2}+\tau_{x})\rvert\leq\frac{1}{2}+(n+\frac{1}{2})\varepsilon+\varepsilon^{2}\leq\frac{1}{2}+\frac{1}{10}$,
where the last inequality follows from plugging in
$\varepsilon=\frac{1}{20mnd}$, and also $\lvert
p^{\prime}_{y}-\tau_{y}\rvert\leq\frac{1}{4}$. Thus, $\lVert
p^{\prime}-((\frac{1}{2},0)+\tau)\rVert_{1}\leq\frac{3}{4}+\frac{1}{10}<1$.
### 3.2 Reduction and correctness
We now describe the reduction and prove its correctness. We construct the
point sets of our Hausdorff distance under translation instance as follows.
The first set, i.e., set $A$, consists only of Vector Gadgets:
$A\coloneqq\left(\bigcup_{i\in[m]}V(x_{i})+(-1-\frac{1}{2}\varepsilon,i\cdot
2d\varepsilon)\right)\cup\left(\bigcup_{i\in[m]}V(1^{d})+(1+\frac{1}{2}\varepsilon,i\cdot
2d\varepsilon)\right)$
The second set, i.e., set $B$, consists of Vector Gadgets, the Translation
Gadget, and the Undirected Gadget:
$B\coloneqq\left(\bigcup_{j\in[n]}\overline{V}(y_{j})+(j\varepsilon,0)\right)\cup
T\cup U$
See Figure 1 for a sketch of the above construction. To reference the vector
gadgets as they are used in the reduction, we use the notation
$V_{r}(x_{i})\coloneqq V(x_{i})+(-1-\frac{1}{2}\varepsilon,i\cdot
2d\varepsilon)\quad\text{and}\quad\overline{V_{r}}(y_{j})\coloneqq\overline{V}(y_{j})+(j\varepsilon,0).$
We can now prove correctness of our reduction. In the reduction, we return
some canonical positive instance, if the $0^{d}$ vector is contained in any of
the two OV sets. This allows us to drop all $1^{d}$ vectors from the input, as
they cannot be orthogonal to any other vector. Thus, we can assume that all
vectors in our input contain at least one 0-entry and at least one 1-entry.
###### Theorem 3.10.
Computing the directed or undirected Hausdorff distance under translation in
$L_{1}$ or $L_{\infty}$ for two point sets of size $n$ and $m$ in the plane
cannot be solved in time $\mathcal{O}((mn)^{1-\gamma})$ for any $\gamma>0$,
unless the Orthogonal Vectors Hypothesis fails.
###### Proof 3.11.
Recall that we only have to consider the $L_{1}$ case. We first prove that
there is a pair of orthogonal vectors $x\in X$ and $y\in Y$ if and only if
$\delta_{H}^{T}(A,B)\leq 1$. To prove the theorem for the directed and
undirected Hausdorff distance under translation at the same time, it suffices
to show ?$\Rightarrow$? for the undirected version and ?$\Leftarrow$? for the
directed version.
$\mathbf{\Rightarrow}$:
Assume that there exist $x_{i}\in X$, $y_{j}\in Y$ with $x_{i}\cdot y_{j}=0$.
Then consider the translation $\tau=(-(j+\frac{1}{2})\varepsilon,i\cdot
2d\varepsilon)$ which vertically aligns the Vector Gadgets $V_{r}(x_{i})$ and
$\overline{V_{r}}(y_{j})+\tau$ at distance $1$. As $x_{i}$ and $y_{j}$ are
orthogonal, it follows from Lemma 3.3 that
$\delta_{\vec{H}}(\overline{V_{r}}(y_{j})+\tau,A)\leq 1$. We now show that all
of the remaining points of $B+\tau$ have a point of $A$ at distance at most
$1$. The Vector Gadgets $\overline{V_{r}}(y_{j^{\prime}})+\tau$ with
$j^{\prime}<j$ are strictly to the left of $\overline{V_{r}}(y_{j})+\tau$ and
are thus also in Hausdorff distance at most $1$ from $V_{r}(x_{i})$. If $j=n$,
then we are done with the Vector Gadgets. Otherwise, consider the Vector
Gadget $\overline{V_{r}}(y_{j+1})+\tau$. We claim that each point of it is at
distance at most $1$ from $V(1^{d})+(1+\frac{1}{2}\varepsilon,i\cdot
2d\varepsilon)$. As the two gadgets are vertically aligned, we just have to
check their horizontal distance, which is
$1+\frac{1}{2}\varepsilon-((j+1)\varepsilon-(j+\frac{1}{2})\varepsilon)=1.$
Thus, by Lemma 3.3, we have
$\delta_{\vec{H}}(\overline{V_{r}}(y_{j+1})+\tau,A)\leq 1$. Now, by the same
argument as above, all gadgets $\overline{V_{r}}(y_{j^{\prime}})+\tau$ with
$j^{\prime}>j+1$ are in directed Hausdorff distance at most $1$ from $A$.
As the points of the Undirected Gadget $U+\tau$ are closer by a distance of
almost $\frac{1}{2}$ to $A$ than the Vector Gadgets in $B+\tau$, also
$\delta_{\vec{H}}(U+\tau,A)\leq 1$ holds. Finally, we have to show that the
Translation Gadget $T+\tau$ is at distance at most $1$ from $A$. As the left
part of $T$ and $V_{r}(x_{i})$ are aligned vertically, we only have to check
the horizontal distance. The horizontal distance is
$-1-\frac{1}{2}\varepsilon-(-2+n\varepsilon-(j+\frac{1}{2})\varepsilon)=1-(n-j)\varepsilon\leq
1$
for any $j\in[n]$. Similarly, the distance of the right part of the
Translation Gadget from the vertically aligned $V(1^{d})$ in $A$ is
$2+2\varepsilon-(j+\frac{1}{2})\varepsilon-(1+\frac{1}{2}\varepsilon)=1-(j-1)\varepsilon\leq
1$
for any $j\in[n]$. Thus, by Lemma 3.3 and Lemma 3.5, it holds that
$\delta_{\vec{H}}(T+\tau,A)\leq 1$. As
$\tau\in[-(n+\frac{1}{2})\varepsilon-\varepsilon^{2},-\frac{3}{2}\varepsilon]\times[-\frac{1}{8},\frac{1}{8}]$,
we know by Lemma 3.8 that $\delta_{\vec{H}}(A,B+\tau)\leq 1$ and thus also
$\delta_{H}^{T}(A,B)\leq 1$.
$\mathbf{\Leftarrow}$:
Now, assume that $\delta_{H}^{T}(A,B)\leq 1$ and let $\tau$ be any translation
for which $\delta_{\vec{H}}(B+\tau,A)\leq 1$. Note that we used the directed
Hausdorff distance in the previous statement on purpose, as we prove hardness
for both versions. Lemma 3.6 implies that
$\tau_{x}\in[-(n+\frac{1}{2})\varepsilon-\varepsilon^{2},-\frac{3}{2}\varepsilon]$.
Let $\overline{V_{r}}(y_{j})+\tau,\overline{V_{r}}(y_{j+1})+\tau$ be the
Vector Gadgets such that $\overline{V_{r}}(y_{j})+\tau$ has directed Hausdorff
distance at most $1$ to the left Vector Gadgets of $A$ and
$\overline{V_{r}}(y_{j+1})+\tau$ has directed Hausdorff distance at most $1$
to the right Vector Gadgets of $A$. This is well-defined as the left Vector
Gadgets of $A$ and the right Vector Gadgets of $A$ are at distance at least
$2+\varepsilon-\varepsilon^{2}$ from each other, and thus no Vector Gadget of
$B+\tau$ can be at distance at most $1$ from both sides. Furthermore, as
$\tau_{x}\leq-\frac{3}{2}\varepsilon$, the Vector Gadget
$\overline{V_{r}}(y_{j})+\tau$ has directed Hausdorff distance at most $1$ to
the left Vector Gadgets of $A$, as
$j\varepsilon-\frac{3}{2}\varepsilon-(-1-\frac{1}{2}\varepsilon)=1+(j-1)\varepsilon\leq
1$
for $j=1$. If $j=n$, then $\overline{V_{r}}(y_{j+1})+\tau$ is undefined.
As $\delta_{\vec{H}}(B+\tau,A)\leq 1$, we know that
$\overline{V_{r}}(y_{j})+\tau$ has directed Hausdorff distance at most $1$ to
a gadget $V_{r}(x)$ for some $x\in X$. We claim that this distance cannot be
closer than $1$ as $\overline{V_{r}}(y_{j+1})+\tau$ must have a directed
Hausdorff distance at most $1$ from the right side of $A$ or, in case $j=n$,
due to the restrictions imposed by the Translation Gadget. Let us consider the
case $j\neq n$ first. Any translation $\tau^{\prime}$ which places
$\overline{V_{r}}(y_{j+1})+\tau^{\prime}$ in directed Hausdorff distance at
most $1$ from the right side of $A$ needs to fulfill
$1+\frac{1}{2}\varepsilon-((j+1)\varepsilon+\tau^{\prime}_{x})\leq 1$
and thus $\tau^{\prime}_{x}\geq-(j+\frac{1}{2})\varepsilon$, using the fact
that each vector in $Y$ contains at least one $0$-entry. This, on the other
hand, implies that $\overline{V_{r}}(y_{j})+\tau^{\prime}$ is in Hausdorff
distance at least
$j\varepsilon-(j+\frac{1}{2})\varepsilon-(-1-\frac{1}{2}\varepsilon)=1$
from $V_{r}(x)$. Now consider the case $j=n$. As by Lemma 3.6 we have
$\tau_{x}\geq-(n+\frac{1}{2})\varepsilon-\varepsilon^{2}$, it follows that
$\overline{V_{r}}(y_{n})+\tau$ is in Hausdorff distance at least
$n\varepsilon-(n+\frac{1}{2})\varepsilon-(-1-\frac{1}{2}\varepsilon)=1$
from $V_{r}(x)$, using the fact that each vector in $Y$ contains at least one
$0$-entry (this is the reason why the $\varepsilon^{2}$ disappears).
By the arguments above, the two gadgets $\overline{V_{r}}(y_{j})+\tau$ and
$V_{r}(x)$ have to be horizontally aligned as required by Lemma 3.3. They also
have to be vertically aligned as a vertical deviation would incur a Hausdorff
distance larger than $1$ for the pair of points in the two gadgets that are in
horizontal distance $1$. Then, applying Lemma 3.3, it follows that $x$ and
$y_{j}$ are orthogonal.
It remains to argue why the above reduction implies the lower bound stated in
the theorem. Assume we have an algorithm that computes the Hausdorff distance
under translation for $L_{1}$ or $L_{\infty}$ in time $(mn)^{1-\gamma}$ for
some $\gamma>0$. Then, given an Orthogonal Vectors instance $X,Y$ with $\lvert
X\rvert=m$ and $\lvert Y\rvert=n$, we can use the described reduction to
obtain an equivalent Hausdorff under translation instance with point sets
$A,B$ of size $\lvert A\rvert=\mathcal{O}(md)$ and $\lvert
B\rvert=\mathcal{O}(nd)$ and solve it in time
$\mathcal{O}((mn)^{1-\gamma}\text{poly}(d))$, contradicting the Orthogonal
Vectors Hypothesis.
### 3.3 Generalization to $L_{p}$
We can extend the above construction such that it works for all $L_{p}$ norms
with $p\neq\infty$ by changing the spacing between $0$ and $1$ points of the
Vector Gadgets and also set $\varepsilon$ accordingly. More precisely, we can
set $\varepsilon=\frac{1}{40pmnd}$ (instead of $\frac{1}{20mnd}$) and use
$\varepsilon^{2p}$ as spacing (instead of $\varepsilon^{2}$), i.e., the Vector
Gadget for a vector $v\in\\{0,1\\}^{d}$ then consists of the points
$p_{1},\dots,p_{d}\in\mathbb{R}^{2}$:
$p_{i}=\begin{cases}(\varepsilon^{2p},i\varepsilon),&\text{if }v[i]=0\\\
(0,i\varepsilon),&\text{if }v[i]=1\\\ \end{cases}$
We prove that these modifications suffice in the remainder of this section.
To this end, first note that in the proof of Theorem 3.10, the proof for
?$\Rightarrow$? for $L_{p}$ already follows from the $L_{1}$ case as the
$L_{1}$ norm is an upper bound on all $L_{p}$ norms. Thus, we only have to
modify the proof of ?$\Leftarrow$?. To show ?$\Leftarrow$?, note that the only
place where we use the $L_{1}$ norm in the proof is in the invocation of Lemma
3.3. Otherwise, we only argue via distances with respect to a single
dimension, which carries over to $L_{p}$ as $\|(x,0)\|_{p}=|x|$. Thus, we now
prove Lemma 3.3 for the general $L_{p}$ case.
###### Proof 3.12 (Proof of Lemma 3.3 for $L_{p}$).
To adapt the proof of Lemma 3.3 to the $L_{p}$ case, we only have to argue
that we cannot match any $p_{i},q_{j}$ for $i\neq j$, as the remaining
arguments merely argue about distances in a single dimension. We have that
$\lVert p_{i}-q_{j}\rVert_{p}=\left((\lvert
i-j\rvert\varepsilon)^{p}+(1-(v_{1}[i]+v_{2}[j]-1)\varepsilon^{2p})^{p}\right)^{1/p}\geq\left(\varepsilon^{p}+(1-\varepsilon^{2p})^{p}\right)^{1/p},$
which is greater than 1 if $\varepsilon^{p}+(1-\varepsilon^{2p})^{p}>1$, which
we obtain by using Bernoulli’s inequality:
$\varepsilon^{p}+(1-\varepsilon^{2p})^{p}\geq\varepsilon^{p}+1-p\varepsilon^{2p}\geq
1+\left(\frac{1}{40pmnd}\right)^{p}-p\left(\frac{1}{40pmnd}\right)^{2p}>1.$
The remainder of the proof is analogous to the remainder of the proof of Lemma
3.3.
By all of the above arguments, the following theorem follows.
###### Theorem 3.13 (Theorem 3.10 for $L_{p}$).
Computing the directed or undirected Hausdorff distance under translation in
$L_{p}$ for two point sets of size $n$ and $m$ in the plane cannot be solved
in time $\mathcal{O}((mn)^{1-\gamma})$ for any $\gamma>0$, unless the
Orthogonal Vectors Hypothesis fails.
## 4 3Sum based $n^{2-o(1)}$ lower bound for $m\in\mathcal{O}(1)$
We now present a hardness result for the unbalanced case of the directed and
undirected Hausdorff distance under translation. We base our hardness on
another popular hypothesis of fined-grained complexity theory: the 3Sum
Hypothesis. Before stating the hypothesis, let us first introduce the 3Sum
problem.444Note that we do not explicitly restrict the universe of the
integers here. In the WordRAM model, we use the standard assumption that each
integer in the input has bit complexity $\mathcal{O}(\log n)$. In the RealRAM
model, we can perform the common arithmetic operations on reals in constant
time, so there is no need to restrict the universe. With these conventions,
our reduction works in both models.
###### Definition 4.1 (3Sum).
Given three sets of positive integers $X,Y,Z$ all of size $n$, do there exist
$x\in X,y\in Y,z\in Z$ such that $x+y=z$?
The corresponding hardness assumption is the 3Sum Hypothesis.
###### Definition 4.2 (3Sum Hypothesis).
There is no $\mathcal{O}(n^{2-\varepsilon})$ algorithm for 3Sum for any
$\varepsilon>0$.
There are several equivalent variants of the 3Sum problem. Most important for
us is the convolution 3Sum problem, abbreviated as Conv3Sum [26, 14].
###### Definition 4.3 (Conv3SUM).
Given a sequence of positive integers $X=(x_{0},\dots,x_{n-1})$ of size $n$,
do there exist $i,j$ such that $x_{i}+x_{j}=x_{i+j}$?
This problem has a trivial $\mathcal{O}(n^{2})$ algorithm and, assuming the
3Sum Hypothesis, this is also optimal up to lower order factors. As 3Sum and
Conv3Sum are equivalent, a lower bound conditional on Conv3Sum implies a lower
bound conditional on 3Sum.
Figure 3: Sketch of the reduction from Conv3Sum to the directed and undirected
Hausdorff distance under translation in the Euclidean plane.
Therefore, given a Conv3Sum instance defined by the sequence of integers $X$
with $\lvert X\rvert=n$, we create an equivalent instance of the directed
Hausdorff distance under translation for $L_{2}$ by constructing two sets of
points $A$ and $B$ with $\lvert A\rvert=\mathcal{O}(n)$ and $\lvert
B\rvert=\mathcal{O}(1)$ and providing a decision distance $\delta$. We provide
some intuition for the reduction in the following. See Figure 3 for an
overview. Intuitively, we define a low-level gadget from which we build three
separate high-level gadgets by rotation and scaling. Recall that in the
Conv3Sum problem we have to find values $i,j$ which fulfill the equation
$x_{i}+x_{j}=x_{i+j}$. Intuitively, we encode the choice of these two values
into the two dimensions of the translation: the horizontal translation chooses
the pair $(i,x_{i})$ in the first high-level gadget and the vertical
translation chooses the pair $(j,x_{j})$ in the second high-level gadget. The
third high-level gadget then allows for a Hausdorff distance below the
threshold iff the chosen $i$ and $j$ fulfill the Conv3Sum constraint
$x_{i}+x_{j}=x_{i+j}$. To make this construction also work for the directed
Hausdorff distance under translation, we add a simple gadget that restricts
translations. In the remainder of this section, we present the details of our
reduction and prove that it implies the claimed lower bound.
### 4.1 Construction
Given a Conv3Sum instance with $X\subset[M]$ where $n=\lvert X\rvert$, we now
describe the construction of the Hausdorff distance under translation instance
with point sets $A,B$ and threshold distance $\delta$. We use a small enough
$\varepsilon$, e.g., $\varepsilon=(4Mn^{2})^{-4}$, as value for
microtranslations. Furthermore, we set $\delta=1+4n^{2}\varepsilon^{2}$. The
additional $4n^{2}\varepsilon^{2}$ term compensates for the small variations
in distance that occur on microtranslations due to the curvature of the
$L_{2}$-ball.
#### 4.1.1 Low-level gadget
Figure 4: The $A$ set of the low-level gadget of the 3Sum reduction, which is
used to build the high-level gadgets. We just show the leftmost part of the
gadget, but the remainder is similar.
We use a single low-level gadget, which is then scaled and rotated to obtain
high-level gadgets. This gadget consists of two point sets $A_{l}$ and
$B_{l}$. The point set $A_{l}$ contains what we call _number points_
$p_{i}^{1},p_{i}^{2}$ and _filling points_ $q_{i}$ for $0\leq i<n$. The set
$B_{l}$ just contains two points: $r_{1}$ and $r_{2}$. The number points
$p_{i}^{1},p_{i}^{2}$ encode the number $x_{i}$, while the filling points make
sure that no other translations than the desired ones are possible. See Figure
4 for an overview. All of the points in this gadget are of the form $(x,0)$.
The number points are
$p_{i}^{1}=\left(2i\varepsilon+x_{i}\varepsilon^{1.5},0\right),\quad
p_{i}^{2}=p_{i}^{1}+\left(\varepsilon,0\right)$
for $0\leq i<n$. The filling points are
$q_{i}=\left(\left(2i+\frac{3}{2}\right)\varepsilon,0\right)$
for $0\leq i<n$.
The points in $B_{l}$ should introduce a gap to only allow alignment of the
number gadgets such that the microtranslations (i.e., those in the order of
$\varepsilon^{1.5}$) correspond to the number of the gap in the number gadget.
To this end, $B_{l}$ contains the points
$r_{1}=(-1,0),\quad r_{2}=(1+\varepsilon,0).$
Before we prove properties of the low-level gadget, we first prove that the
error due to the curvature of the $L_{2}$-ball is small.
###### Lemma 4.4.
Let $(p_{x},p_{y}),(q_{x},q_{y})\in\mathbb{R}^{2}$ be two points with $\lvert
p_{x}-q_{x}\rvert\in[\frac{1}{2},2]$ and $p_{y}=q_{y}$. For any
$\tau\in[0,(2n-1)\varepsilon]^{2}$, we have
$\lvert p_{x}-(q_{x}+\tau_{x})\rvert\leq\lVert p-(q+\tau)\rVert_{2}\leq\lvert
p_{x}-(q_{x}+\tau_{x})\rvert+4n^{2}\varepsilon^{2}.$
###### Proof 4.5.
As each component is a lower bound for the $L_{2}$ norm, the first inequality
follows. Thus, let us prove the second inequality. We first transform
$\lVert
p-(q+\tau)\rVert_{2}=\sqrt{(p_{x}-(q_{x}+\tau_{x}))^{2}+\tau_{y}^{2}}=\lvert
p_{x}-(q_{x}-\tau_{x})\rvert\sqrt{1+\tau_{y}^{2}/(p_{x}-(q_{x}+\tau_{x}))^{2}}.$
As $\sqrt{1+x}\leq 1+\frac{x}{2}$ for any $x\geq 0$, we have
$\lVert p-(q+\tau)\rVert_{2}\leq\lvert
p_{x}-(q_{x}-\tau_{x})\rvert+\tau_{y}^{2}/(2\lvert
p_{x}-(q_{x}-\tau_{x})\rvert).$
As $\tau_{y}\leq 2(n-1)\varepsilon$ and $\lvert
p_{x}-(q_{x}-\tau_{x})\rvert\geq\frac{1}{2}$, we obtain the desired upper
bound.
An analogous statement holds when swapping the $x$ and $y$ coordinates. Note
that the $4n^{2}\varepsilon^{2}$ term also occurs in the value of $\delta$
that we chose, as this is how we compensate for these errors in our
construction. While we have to consider this error in the following arguments,
it should already be conceivable that it will be insignificant due to its
magnitude.
To this end, we use a compact notation to denote a value being in a certain
range around a value. More concretely, for any $y,r\in\mathbb{R}$, let $x=y\pm
r$ denote $x\in[y-r,y+r]$. We now state two lemmas which show how the
Hausdorff distance under translation decision problem is related to the
structure of the low-level gadget.
###### Lemma 4.6.
Given a low-level gadget $A_{l},B_{l}$ as constructed above and the
translation being restricted to $\tau\in[0,(2n-1)\varepsilon]^{2}$, it holds
that if $\delta_{\vec{H}}(A_{l},B_{l}+\tau)\leq\delta$, then
$\exists i\in\mathbb{N}:\tau_{x}=2i\varepsilon+x_{i}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2}.$
###### Proof 4.7.
Let $\tau\in[0,(2n-1)\varepsilon]^{2}$ and assume
$\delta_{\vec{H}}(A_{l},B_{l}+\tau)\leq\delta$. Then all points in $A_{l}$ are
at distance at most $\delta$ from one of the two points in $B_{l}$.
Furthermore, both points in $B_{l}+\tau$ also have at least one close point in
$A_{l}$, as
$\lVert r_{1}+\tau-p_{0}^{1}\rVert_{2}\leq
1-\tau_{x}+4n^{2}\varepsilon^{2}\leq\delta\quad\text{and}\quad\lVert
r_{2}+\tau-q_{n-1}\rVert_{2}\leq
1+\tau_{x}-(2n-\frac{3}{2})\varepsilon+4n^{2}\varepsilon^{2}<\delta,$
using that $n\geq 1$ and Lemma 4.4.
The gaps between neighboring points in $A_{l}$ either have width close to
$\frac{1}{2}\varepsilon$, if the gap is between a number point and a filling
point ($p_{i}^{1}$ and $q_{i-1}$, or $p_{i}^{2}$ and $q_{i}$), or they have a
width of $\varepsilon$, if the gap is between two number points ($p_{i}^{1}$
and $p_{i}^{2}$). Furthermore, the two points in $B_{l}$ have distance
$2+\varepsilon$, so there is an $\varepsilon-8n^{2}\varepsilon^{2}$ gap
between their $\delta$-balls. Thus, there is an $i$ such that $p_{i}^{1}$ has
distance at most $\delta$ to $r_{1}$, and $p_{i}^{2}$ has distance at most
$\delta$ to $r_{2}$. This alignment of the gadgets can only be realized by a
translation $\tau$ for which
$\tau_{x}=2i\varepsilon+x_{i}\varepsilon^{1.5}\pm 4n^{2}\varepsilon^{2},$
which completes the proof.
###### Lemma 4.8.
Given a low-level gadget $A_{l},B_{l}$ as constructed above and the
translation being restricted to $\tau\in[0,(2n-1)\varepsilon]^{2}$, it holds
that if
$\exists i\in\mathbb{N}:\tau_{x}=2i\varepsilon+x_{i}\varepsilon^{1.5},$
then $\delta_{H}(A_{l},B_{l}+\tau)\leq\delta$.
###### Proof 4.9.
Let $i\in\mathbb{N}$ and let $\tau_{x}=2i\varepsilon+x_{i}\varepsilon^{1.5}$.
Consider any translations $\tau\in\\{\tau_{x}\\}\times[0,2(n-1)\varepsilon]$.
Due to the restricted translation and Lemma 4.4, we can disregard the error
terms that arise from the vertical translation $\tau_{y}$ as they are
compensated for by $\delta$. Then all the points in $A_{l}$ before and
including $p_{i}^{1}$ are at distance at most $\delta$ from $r_{1}\in
B_{l}+\tau$ and all the points afterwards are at distance at most $\delta$
from $r_{2}\in B_{l}+\tau$. Clearly, both points in $B_{l}+\tau$ then also
have points from $A_{l}$ at distance $\delta$, and thus
$\delta_{H}(A_{l},B_{l}+\tau)\leq\delta$.
#### 4.1.2 High-level gadgets
This construction is inspired by the hard instance that was given in [28]. We
want to obtain a grid of translations of spacing $\varepsilon$ with some
microtranslations in the $\mathcal{O}(\varepsilon^{1.5})$ range. We already
defined the low-level gadget above, and we now define the high-level gadgets.
##### Column Gadget
The column gadget induces columns in translational space, i.e., it enforces
that valid translations have to lie on one of these columns. The column gadget
is actually the low-level gadget we already described above. You can see a
sketch of this gadget in Figure 5(a). To semantically distinguish it from the
low-level gadget, we refer to the point sets of the column gadget as $A_{c}$
and $B_{c}$.
##### Row Gadget
The row gadget induces rows in translational space, i.e., it enforces that
valid translations have to lie on one of these rows. We obtain the row gadget
by rotating all points in the low-level gadget around the origin by $\pi/2$
counterclockwise. You can see a sketch of this gadget in Figure 5(b). We call
the point sets of the row gadget $A_{r}$ and $B_{r}$.
##### Diagonal Gadget
The diagonal gadget induces diagonals in translational space, i.e., it
enforces that valid translations have to lie on one of these diagonals. As
opposed to the column and row gadget, the diagonal gadget also has to be
scaled. We scale the sets $A_{l}$ and $B_{l}$ separately. We scale $A_{l}$
such that the gap between the number point pairs $p_{i}^{1},p_{i}^{2}$ becomes
$\frac{1}{\sqrt{2}}\varepsilon$. And we scale $B_{l}$ such that the gap
between the points becomes $2+\frac{1}{\sqrt{2}}\varepsilon$. After scaling,
we rotate the points counterclockwise around the origin by $\pi/4$. You can
see a sketch of this gadget in Figure 5(c). We call the point sets of the
diagonal gadget $A_{d}$ and $B_{d}$.
##### Translation Gadget
To restrict the translations for the directed Hausdorff distance under
translation, we introduce another gadget. The first set of points $A_{t}$
contains
$z_{l}\coloneqq(-1+(2n-1)\varepsilon,0),\quad z_{r}\coloneqq(1,0),\quad
z_{b}\coloneqq(0,-1+(2n-1)\varepsilon),\quad z_{t}\coloneqq(0,1).$
The second point set $B_{t}$ only contains the origin $z_{c}\coloneqq(0,0)$.
We want to make sure that this gadget behaves well in a certain range.
###### Lemma 4.10.
Given $\tau\in[0,(2n-1)\varepsilon]^{2}$, it holds that
$\delta_{H}(A_{t},B_{t}+\tau)\leq\delta$.
###### Proof 4.11.
As $B_{t}$ has a point on all sides, clearly
$\delta_{\vec{H}}(B_{t}+\tau,A_{t})\leq\delta$. Furthermore,
$\lVert z_{l}-(z_{c}+\tau)\rVert_{2}\leq
1+4n^{2}\varepsilon^{2}\leq\delta\quad\text{and}\quad\lVert
z_{r}-(z_{c}+\tau)\rVert_{2}\leq\delta,$
using Lemma 4.4. Analogous statements hold for $z_{b}$ and $z_{t}$. Thus, also
$\delta_{\vec{H}}(A_{t},B_{t}+\tau)\leq\delta$.
(a) Column Gadget
(b) Row Gadget
(c) Diagonal Gadget
Figure 5: Three of the high-level gadgets. The points of $A$ are all in the
low-level gadgets, while the points in $B$ are explicitly shown including
their $\delta$-ball.
#### 4.1.3 Complete construction
To obtain the final sets of the reduction, we now place all four described
high-level gadgets (i.e., column gadget, row gadget, diagonal gadget, and
translation gadget) far enough apart. More explicitly, the point sets $A,B$ of
the Hausdorff distance under translation instance are defined as
$A\coloneqq A_{c}\cup(A_{r}+(10,0))\cup(A_{d}+(20,0))\cup(A_{t}+(30,0))$
and
$B\coloneqq B_{c}\cup(B_{r}+(10,0))\cup(B_{d}+(20,0))\cup(B_{t}+(30,0)).$
The far placement ensures that the two point sets of the respective gadgets
have to be matched to each other when the Hausdorff distance under translation
is at most delta $\delta$.
### 4.2 Proof of correctness
First, we want to ensure that everything relevant happens in a very small
range of translations.
###### Lemma 4.12.
Let $\tau\in\mathbb{R}^{2}$. If $\delta_{\vec{H}}(A,B+\tau)\leq\delta$, then
$\tau\in[0,(2n-1)\varepsilon]^{2}$.
###### Proof 4.13.
Note that for a Hausdorff distance at most $\delta$, the sets $A_{c}$ and
$B_{c}$ have to matched to each other and analogously for $A_{r},B_{r}$, and
$A_{d},B_{d}$, and $A_{t},B_{t}$. To show the contrapositive, assume
$\tau\notin[0,(2n-1)\varepsilon]^{2}$. For simplicity, we refer to the points
in the high-level gadgets with the notation of the low-level gadget. Due to
the translation gadget, we have
$\lVert
z_{l}-(z_{c}+\tau)\rVert_{2}>\delta\quad\text{for}\quad\tau_{x}>(2n-1)\varepsilon+4n^{2}\varepsilon^{2},$
and
$\lVert
z_{r}-(z_{c}+\tau)\rVert_{2}>\delta\quad\text{for}\quad\tau_{x}<-4n^{2}\varepsilon^{2}.$
We now show that under these restricted translations and as
$\delta_{\vec{H}}(A,B+\tau)\leq\delta$, both points $r_{1},r_{2}$ in $B_{c}$
have at least one point of $A_{c}$ at distance $\delta$. In the column gadget
for $\tau_{x}\in[-4n^{2}\varepsilon^{2},0)$, we have
$\lVert(r_{1}+\tau)-p_{0}^{1}\rVert_{2}\geq\lvert-1-(p_{0}^{1})_{x}+\tau_{x}\rvert>\delta\quad\text{and}\quad\lVert(r_{2}+\tau)-p_{0}^{1}\rVert_{2}\geq
1+\varepsilon-\mathcal{O}(\varepsilon^{1.5})>\delta$
for small enough $\varepsilon$ and as $x_{0}>0$ and thus there is a component
of order $\varepsilon^{1.5}$. On the other hand, for
$\tau_{x}\in((2n-1)\varepsilon,(2n-1)\varepsilon+4n^{2}\varepsilon^{2}]$, we
have
$\lVert r_{2}+\tau-p_{n-1}^{2}\rVert_{2}\geq
1+\varepsilon+\tau_{x}-(2n-1)\varepsilon>\delta\quad\text{and}\quad\lVert
r_{1}+\tau-p_{n-1}^{2}\rVert_{2}\geq
1+\mathcal{O}(\varepsilon^{1.5})-4n^{2}\varepsilon^{2}>\delta$
for small enough $\varepsilon$. An analogous argument holds for the row gadget
and $\tau_{y}$, as the row gadget is just a rotated version of the column
gadget and the translation gadget is symmetric with respect to these gadgets.
We can now prove the main result of this section.
###### Theorem 4.14.
Computing the directed or undirected Hausdorff distance under translation in
$L_{2}$ for two sets of size $n$ and $7$ cannot be solved in time
$\mathcal{O}(n^{2-\gamma})$ for any $\gamma>0$, unless the 3Sum Hypothesis
fails.
###### Proof 4.15.
We construct a Hausdorff under translation instance in this proof from a
Conv3Sum instance as described previously in this section, and then show that
they are equivalent. We first consider how to apply Lemma 4.6 and Lemma 4.8 to
the diagonal gadget. More precisely, we consider which translations align the
gaps of $A_{d}$ and $B_{d}$ as is used in these two lemmas. Consider the
constraint $\tau_{x}=2k\varepsilon+x_{k}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2}$ that is encoded by the low-level gadget. Recall that we
scale this gadget by $\frac{1}{\sqrt{2}}$ and rotate it by $\frac{\pi}{4}$,
i.e., we apply the transformation matrix
$\frac{1}{\sqrt{2}}\cdot\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}\cdot\begin{pmatrix}\frac{1}{\sqrt{2}}&0\\\
0&\frac{1}{\sqrt{2}}\\\ \end{pmatrix}=\frac{1}{2}\cdot\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}$
to the right side of the constraint. Thus, for any
$\alpha\in[0,(2n-1)\varepsilon]$, the diagonal gadget encodes the constraints
$\begin{pmatrix}\tau_{x}\\\
\tau_{y}\end{pmatrix}=\frac{1}{2}\cdot\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}\cdot\begin{pmatrix}2k\varepsilon+x_{k}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2}\\\
\alpha\end{pmatrix}=\frac{1}{2}\cdot\begin{pmatrix}2k\varepsilon+x_{k}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2}&-\alpha\\\ 2k\varepsilon+x_{k}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2}&\alpha\end{pmatrix}.$
By adding up the two constraints, we obtain
$\tau_{x}+\tau_{y}=2k\varepsilon+x_{k}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2}.$
We now show correctness of the reduction.
$\mathbf{\Leftarrow}$:
Assume $X$ is a positive Conv3Sum instance. Then there exist $x_{i},x_{j}$
such that $x_{i}+x_{j}=x_{i+j}$. Consider
$\tau=(2i\varepsilon+x_{i}\varepsilon^{1.5},2j\varepsilon+x_{j}\varepsilon^{1.5})$
as translation. Due to Lemma 4.8, we have that
$\delta_{H}(A_{c},B_{c}+\tau)\leq\delta$ and analogously
$\delta_{H}(A_{r},B_{r}+\tau)\leq\delta$. By the initial observation, we can
also apply Lemma 4.8 to the diagonal gadget, and thus
$\delta_{H}(A_{d},B_{d}+\tau)\leq\delta$. Finally, by Lemma 4.10, we also have
that $\delta_{H}(A_{t},B_{t}+\tau)\leq\delta$ for the given $\tau$.
$\mathbf{\Rightarrow}$:
Assume $\delta_{\vec{H}}^{T}(A,B)\leq\delta$. From Lemma 4.12, it follows that
$\tau\in[0,(2n-1)\varepsilon]^{2}$. Then, due to Lemma 4.6 and the initial
observation about the diagonal gadget, we have that there exist $i,j,k$ that
fulfill
$\displaystyle\tau_{x}$ $\displaystyle=2i\varepsilon+x_{i}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2},$ $\displaystyle\tau_{y}$
$\displaystyle=2j\varepsilon+x_{j}\varepsilon^{1.5}\pm 4n^{2}\varepsilon^{2},$
$\displaystyle\tau_{x}+\tau_{y}$
$\displaystyle=2k\varepsilon+x_{k}\varepsilon^{1.5}\pm 4n^{2}\varepsilon^{2}.$
It follows that
$2i\varepsilon+x_{i}\varepsilon^{1.5}+2j\varepsilon+x_{j}\varepsilon^{1.5}\pm
8n^{2}\varepsilon^{2}=2k\varepsilon+x_{k}\varepsilon^{1.5}\pm
4n^{2}\varepsilon^{2},$
and thus $i+j=k$ and $x_{i}+x_{j}=x_{k}$.
It remains to argue why the above reduction implies the lower bound stated in
the theorem. Assume we have an algorithm that computes the Hausdorff distance
under translation in $L_{2}$ in time $\mathcal{O}(n^{2-\gamma})$ for some
$\gamma>0$. Then, given a Conv3Sum instance $X$ with $\lvert X\rvert=n$, we
can use the described reduction to obtain an equivalent Hausdorff under
translation instance with point sets $A,B$ of size $\lvert
A\rvert=\mathcal{O}(n)$ and $\lvert B\rvert=7$ and solve it in time
$\mathcal{O}(n^{2-\gamma})$, contradicting the 3Sum Hypothesis.
## 5 Conclusion
In this work, we provide matching lower bounds for the running time of two
important cases of the fundamental distance measure Hausdorff distance under
translation. These lower bounds are based on popular standard hypotheses from
fine-grained complexity theory. Interestingly, we use two different hypotheses
to show hardness. For the Hausdorff distance under translation for $L_{p}$, we
show a lower bound of $(nm)^{1-o(1)}$ using the Orthogonal Vectors Hypothesis,
while for the imbalanced case of $m=\mathcal{O}(1)$ in $L_{2}$, we show an
$n^{2-o(1)}$ lower bound using the 3Sum Hypothesis. We leave it as an open
problem whether Hausdorff distance under translation for the balanced case
admits a strongly subcubic algorithm or if conditional hardness can be shown.
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|
# Proof Automation in the Theory of Finite Sets and Finite Set Relation
Algebra
Maximiliano Cristiá Universidad Nacional de Rosario and CIFASIS, Argentina
<EMAIL_ADDRESS>Ricardo D. Katz CIFASIS-CONICET, Argentina
<EMAIL_ADDRESS>Gianfranco Rossi Università di Parma, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
$\\{log\\}$ (‘setlog’) is a satisfiability solver for formulas of the theory
of finite sets and finite set relation algebra (FS&RA). As such, it can be
used as an automated theorem prover (ATP) for this theory. $\\{log\\}$ is able
to automatically prove a number of FS&RA theorems, but not all of them.
Nevertheless, we have observed that many theorems that $\\{log\\}$ cannot
automatically prove can be divided into a few subgoals automatically
dischargeable by $\\{log\\}$. The purpose of this work is to present a
prototype interactive theorem prover (ITP), called $\\{log\\}$-ITP, providing
evidence that a proper integration of $\\{log\\}$ into world-class ITP’s can
deliver a great deal of proof automation concerning FS&RA. An empirical
evaluation based on 210 theorems from the TPTP and Coq’s SSReflect libraries
shows a noticeable reduction in the size and complexity of the proofs with
respect to Coq.
###### keywords:
$\\{log\\}$; Set theory; Automated proofs; Constraint logic programming
## 1 Introduction
Interactive theorem proving (ITP) [1] is increasingly used in the
formalization and proof of results of mathematics and logic, and also a widely
used approach to formal verification of hardware and software. ITP’s such as
Coq [2], Isabelle/HOL [3] and HOL Light [4] vary in the level of expressivity
and automation, but typically support rich specification languages including
higher-order logic or dependent type theory. Since interactive theorem proving
is labor intensive, thus costly and limited, much research has been devoted to
the development of automated reasoning.
SMT solvers [5] implement decision procedures for the satisfiability problem
of formulas of specific theories. Since they have significantly improved their
power in the last decades, it has become increasingly common for ITP’s the use
of SMT solvers as efficient automated theorem provers (ATP) for the
corresponding theories. As a consequence, users of mainstream ITP’s can call
ATP’s [6, 7] and SMT solvers [8, 9] to automatically advance their proofs.
These ad-ons exploit the idea of mixing interactive and automated proof steps.
Our proposal fits in this line of work as we propose to integrate $\\{log\\}$
as a special purpose ATP into ITP’s.
$\\{log\\}$ is a satisfiability solver accepting an input language at least as
expressive as the class of full set relation algebras on finite sets (FS&RA)
[10]. FS&RA essentially corresponds to the first-order fragment of formal
notations such as Alloy [11], B [12] and Z [13] restricted to finite sets. In
consequence, this input language can be used as a specification language for a
large class of software systems and $\\{log\\}$ as a tool to reason about
them.
$\\{log\\}$ can automatically prove111By an abuse of language, from now on we
will say that $\\{log\\}$ can or cannot prove a theorem to mean that it can
decide or not the satisfiability of its negation (see Remark 3.2 below). 97%
of the theorems on Boolean algebra (BOO), relation algebra (REL) and set
theory (SET) gathered in the TPTP library [14] that can be expressed in its
input language (in .3 s each in average), see [15]. Since the equational
theory of FS&RA is undecidable [16], $\\{log\\}$ cannot decide the
satisfiability of all the formulas it accepts. Moreover, $\\{log\\}$ can take
too long to decide the satisfiability of some formulas. For example, it takes
a long time to prove the following result:
$\begin{split}f\in{}&A\rightarrow
B{}\land{}\mathop{\mathrm{ran}}f=B{}\land{}g\in B\rightarrow C\\\
{}\land{}&h\in B\rightarrow C\land f\circ g=f\circ h\implies g=h\end{split}$
(T1)
where $A$, $B$ and $C$ denote any finite sets, $A\rightarrow B$ denotes the
set of all (finite) functions from $A$ to $B$ (in this context a function is a
binary relation where no two ordered pairs have the same first component) and
$\mathop{\mathrm{ran}}f$ is the range of $f$. Nevertheless, since $g=h\iff
g\subseteq h\land h\subseteq g$, the proof of (T1) can be reduced to the
proofs of the following two implications, on which $\\{log\\}$ spends a few
seconds:
$\displaystyle\begin{split}f\in{}&\star\rightarrow\star{}\land{}\mathop{\mathrm{ran}}f=B{}\land{}g\in
B\rightarrow\star\\\ {}\land{}&h\in\star\rightarrow\star\land f\circ g=f\circ
h\implies g\subseteq h\end{split}$
$\displaystyle\begin{split}f\in{}&\star\rightarrow\star{}\land{}\mathop{\mathrm{ran}}f=B{}\land{}g\in\star\rightarrow\star\\\
{}\land{}&h\in B\rightarrow\star\land f\circ g=f\circ h\implies h\subseteq
g\end{split}$
Here $\star$ means that the corresponding hypotheses can be dropped (for
example, $f\in\star\rightarrow\star$ says that it does not matter what the
domain and range of $f$ are). Thus, by _dividing_ the proof of (T1) into
subgoals and by _dropping_ the appropriate hypotheses in each, $\\{log\\}$ can
automatically do the rest.
We have noticed that in practice the approach above succeeds on proving many
results that $\\{log\\}$ cannot automatically prove. As a consequence, in this
paper we present $\\{log\\}$-ITP, a prototype interactive theorem prover where
users can enter any $\\{log\\}$ formula and interactively prove it. More
precisely, in $\\{log\\}$-ITP users can divide the proof into subgoals, drop
hypotheses and call $\\{log\\}$ to perform the actual mathematical steps. This
follows the way other tools, such as Atelier B [17] and the Coq’s why3 [18]
tactic, work. We point out that $\\{log\\}$-ITP is just a vehicle to provide
evidence that a proper integration of $\\{log\\}$’s rewriting system into
world-class ITP’s can deliver a great deal of proof automation concerning
FS&RA.
In order to validate our proposal in practice we perform a number of proofs
with $\\{log\\}$-ITP and Coq. On these theorems $\\{log\\}$ either does not
terminate or takes a very long time to do it. This comparison shows promising
results as Coq proofs are harder and longer than $\\{log\\}$-ITP’s (see
Section 4 for details).
As SMT solvers, $\\{log\\}$ generates a solution when it determines that a
formula is satisfiable. Indeed, $\\{log\\}$ provides a finite representation
of all the (possibly infinitely many) solutions [10]. In the context of
integrating $\\{log\\}$ into an ITP, this means that if $\\{log\\}$ is called
to advance a proof but it happens that the goal is not provable from the
premises, a counterexample is generated. This counterexample might help the
user to adjust the theorem or the theory which contains it. QuickChick has
been proposed as a tool to decrease the number of failed proof attempts in Coq
by generating counterexamples before a proof is attempted [19]. This tool
relies on a random counterexample generator. Although $\\{log\\}$
counterexamples are generated in a deductive fashion, it is also more limited
as it works only for a specific theory.
Let us finally mention that traditional ATP’s such as E prover [20] and
Vampire [21] can automate proofs of FS&RA. Since they can efficiently solve
many FOL problems, they work by encoding set theory in (most often) untyped
first-order logic. One of the simplest encodings applies extensionality and
rewrites away all definitions, thus arriving at formulas based on set
membership. However, these encodings must deal with typing information when
sets do not have the same set support. The easiest way to deal with this issue
is to omit all type information, but this approach is unsound. Another way to
deal with types is to annotate terms with type tags or guards. This
considerably increases the size of the problems passed to generic ATP, with a
dramatic impact on their performance [22, 23]. Another approach to proof
automation in set theory is to use polymorphic provers. Our work follows this
approach as $\\{log\\}$ can be seen as a specialized prover for polymorphic
set theory. The empirical assessment presented in this paper confirms the
results reported by other polymorphic provers [23, 24, 25, 26, 27].
This paper is structured as follows. The logic language supported by
$\\{log\\}$ and some of its main features are presented in Section 2. Section
3 describes $\\{log\\}$-ITP which is empirically evaluated in Section 4. We
give our final conclusions in Section 5.
## 2 $\\{log\\}$: a satisfiability solver for finite sets and relations
In this section we provide a brief, informal introduction to the $\\{log\\}$
system [29]. A formal presentation of $\\{log\\}$’s language can be found in
A; deeper presentations can be found elsewhere [10, 15, 28].
### 2.1 The $\\{log\\}$ language
$\\{log\\}$ [29] is a satisfiability solver implemented in Prolog whose input
language is denoted $\mathcal{L}_{\mathcal{BR}}$. This is a multi-sorted
first-order predicate language with two distinct sorts: the sort
$\mathsf{Set}$ of all the terms which denote sets (including binary relations)
and the sort $\mathsf{O}$ of all the other terms. Thus, we do not introduce
distinct sorts for sets and binary relations. Binary relations are just sets
of ordered pairs. This allows sets and relations to be freely mixed; in
particular all set operators are directly applicable to binary relations.
###### Note 1 (Background on binary relations).
Let $R$ and $S$ be binary relations, and $A$ a set. Then, we define following
relational operators: relational composition, $R\circ S=\\{(x,z)\mid\exists
y((x,y)\in R\land(y,z)\in S)\\}$; converse (or inverse) of $R$,
$R^{\smallsmile}=\\{(y,x)\mid(x,y)\in R\\}$; the identity relation on $A$,
$\mathop{\mathrm{id}}A=\\{(x,x)\mid x\in A\\}$; domain of $R$,
$\mathop{\mathrm{dom}}R=\\{x\mid\exists y((x,y)\in R)\\}$; range of $R$,
$\mathop{\mathrm{ran}}R=\mathop{\mathrm{dom}}R^{\smallsmile}$; domain
restriction of $R$ on $A$, $A\dres R=\mathop{\mathrm{id}}(A)\circ R$; range
restriction of $R$ on $A$, $R\rres A=R\circ\mathop{\mathrm{id}}(A)$; domain
anti-restriction of $R$ on $A$, $A\mathbin{\hbox to0.0pt{\raise
0.21529pt\hbox{$-$}\hss}{\dres}}R=R\setminus(A\dres R)$; range anti-
restriction of $R$ on $A$, $R\mathbin{\hbox to0.0pt{\raise
0.21529pt\hbox{$-$}\hss}{\rres}}A=R\setminus(R\rres A)$; relational image of
$A$ through $R$, $R[A]=\mathop{\mathrm{ran}}(A\dres R)$; and relational
overriding of $R$ by $S$, $R\oplus S=(\mathop{\mathrm{dom}}S\mathbin{\hbox
to0.0pt{\raise 0.21529pt\hbox{$-$}\hss}{\dres}}R)\cup S$. A binary relation is
a (partial) function if no two of its ordered pairs have the same first
component. Given that functions are just binary relations, all relational
operators can be applied to functions. ∎
In $\mathcal{L}_{\mathcal{BR}}$ set operators are encoded as constraints over
the domain of finite hybrid sets. For example: $\mathit{un}(A,B,C)$ is a
constraint interpreted as $C=A\cup B$. $\\{log\\}$ implements a wide range of
set and relational operators (cf. Note 2). For instance, $\in$ is a constraint
interpreted as set membership; $=$ is set equality; $\mathit{dom}(R,D)$
corresponds to $\mathop{\mathrm{dom}}R=D$; $A\subseteq B$ corresponds to the
subset relation; $\mathit{comp}(R,S,T)$ is interpreted as $T=R\circ S$; and
$\mathit{apply}(F,X,Y)$ is equivalent to $\mathit{pfun}(F)\land[X,Y]\in F$,
where $\mathit{pfun}(F)$ constrains $F$ to be a (partial) function. Formulas
in $\\{log\\}$ are conjunctions ($\land$) and disjunctions ($\lor$) of
constraints. Negation is introduced by means of so-called _negated
constraints_. For example $\mathit{nun}(A,B,C)$ is interpreted as $C\neq A\cup
B$ and $\notin$ as the negation of $\notin$. In general, if $\pi$ is a
constraint, $n\pi$ corresponds to its negated form. For formulas to fit inside
the decision procedures implemented in $\\{log\\}$, users must only use this
form of negation.
In turn, terms can be either uninterpreted Herbrand terms (as in Prolog) or
set terms, i.e., terms with the following form and interpretation: $\emptyset$
to denote the empty set; $\\{x\mathbin{\scriptstyle\sqcup}A\\}$, called
_extensional set_ , which is interpreted as $\\{x\\}\cup A$, where $A$ is a
set term; and $A\times B$ to represent the Cartesian product between the sets
denoted by set terms $A$ and $B$.
In $\\{log\\}$ sets are always finite and untyped and they are allowed as set
elements (i.e., sets can be nested). As the second argument of an extensional
set can be a variable, sets in $\\{log\\}$ can be unbounded.
###### Note 2 (Binary relations and expressivenes).
$\mathcal{L}_{\mathcal{BR}}$ turns out to be at least as expressive as _the
class of full set relation algebras on finite sets_. A _full set relation
algebra_ [16] over a base set $A$, denoted $\mathfrak{R}(A)$, is the relation
algebra where relations are subsets of $A\times A$. A mapping from formulas of
a $\mathfrak{R}(A)$ with $A$ finite to $\mathcal{L}_{\mathcal{BR}}$ formulas
can be easily defined [10]. The class of full set relation algebras on finite
sets is the class of relation algebras $\mathfrak{R}(A)$ where $A$ is a finite
set. Further, $\mathcal{L}_{\mathcal{BR}}$ allows for the definition of the
class of full set heterogeneous relation algebras on finite sets. An
heterogeneous relation algebra deals with relations between two arbitrary sets
$A$ and $B$, i.e. with subsets of $A\times B$ [33]. We use the acronym FS&RA
to denote such a class, for any $A$ and $B$ finite.
The class of full set heterogeneous relation algebras roughly corresponds to
the first-order fragment of formal notations such as Alloy [11], B [12] and Z
[13]. A large class of programs can be specified within this fragment. The
limitation of $\mathcal{L}_{\mathcal{BR}}$ to finite sets is not so severe as
many programs operate only on finite data structures. Therefore,
$\mathcal{L}_{\mathcal{BR}}$ can be used as a specification language for a
large class of software systems and $\\{log\\}$ as a tool to reason about them
[31, 34, 35, 30, 32]. ∎
### 2.2 A rewriting system for $\mathcal{L}_{\mathcal{BR}}$
$\\{log\\}$ implements a rewriting system for $\mathcal{L}_{\mathcal{BR}}$
formulas, called $\mathit{SAT}_{\mathcal{BR}}$, whose global organization is
shown in Algorithm 1. Basically, $\mathit{SAT}_{\mathcal{BR}}$ uses two
procedures: sort_infer and STEP.
$\Phi\leftarrow\textsf{sort\\_infer}(\Phi)$;
repeat
$\Phi^{\prime}\leftarrow\Phi$;
$\Phi\leftarrow\textsf{STEP}(\Phi)$
until $\Phi=\Phi^{\prime}$;
return $\Phi$
Algorithm 1 The solver $\mathit{SAT}_{\mathcal{BR}}$. $\Phi$ is the input
formula.
$\mathcal{L}_{\mathcal{BR}}$ does not provide variable declarations. For this
reason, sort_infer($\Phi$) automatically adds either a $\mathit{set}$ or a
$\mathit{rel}$ constraint for each variable $X$ in $\Phi$ which is required to
represent, respectively, either a set or a binary relation according to the
intended interpretation of the terms or constraints where $X$ occurs.
STEP applies specialized rewriting procedures to the current formula $\Phi$
and returns either $\mathit{false}$ or a modified formula. Each rewriting
procedure applies a few non-deterministic rewrite rules which reduce the
syntactic complexity of $\mathcal{L}_{\mathcal{BR}}$ constraints of one kind.
The execution of STEP is iterated until a fixpoint is reached, i.e., the
formula is irreducible. STEP returns $\mathit{false}$ whenever (at least) one
of the procedures in it rewrites $\Phi$ to $\mathit{false}$.
The rewriting procedures implemented in $\\{log\\}$ can be divided into two
classes: those for set constraints and those for relational constraints. The
former were introduced in [36] and extensively discussed from then on. They
constitute the base for a decision procedure for finite sets based on set
unification and set constraint solving. The latter were introduced more
recently [10, 15]. Roughly, there are 50 rewriting procedures adding up 175
rewrite rules.
$\displaystyle\begin{split}\\{x{}\mathbin{\scriptstyle\sqcup}{}&A\\}=\\{y\mathbin{\scriptstyle\sqcup}B\\}\rightarrow\\\
&x=y\land A=B\\\ &\lor x=y\land\\{x\mathbin{\scriptstyle\sqcup}A\\}=B\\\ &\lor
x=y\land A=\\{y\mathbin{\scriptstyle\sqcup}B\\}\\\ &\lor
A=\\{y\mathbin{\scriptstyle\sqcup}N\\}\land\\{x\mathbin{\scriptstyle\sqcup}N\\}=B\end{split}$
(1)
$\displaystyle\begin{split}\mathit{un}&(A,B,\\{t\mathbin{\scriptstyle\sqcup}C\\})\rightarrow\\\
&\\{t\mathbin{\scriptstyle\sqcup}C\\}=\\{t\mathbin{\scriptstyle\sqcup}N\\}\land
t\notin N\\\
&\land(A=\\{t\mathbin{\scriptstyle\sqcup}N_{1}\\}\land\mathit{un}(N_{1},B,N)\\\
&{}\qquad\lor
B=\\{t\mathbin{\scriptstyle\sqcup}N_{1}\\}\land\mathit{un}(A,N_{1},N)\\\
&{}\qquad\begin{split}{}\lor A&=\\{t\mathbin{\scriptstyle\sqcup}N_{1}\\}\\\
&\land
B=\\{t\mathbin{\scriptstyle\sqcup}N_{2}\\}\land\mathit{un}(N_{1},N_{2},N))\end{split}\\\
\end{split}$ (2)
$\displaystyle\begin{split}\mathit{inv}&(R,\\{(y,x)\mathbin{\scriptstyle\sqcup}S\\})\rightarrow\\\
&R=\\{(x,y)\mathbin{\scriptstyle\sqcup}N\\}\land\mathit{inv}(N,S)\end{split}$
(3) Figure 1: Representative rewriting rules
Here we just show some of the most representative rewrite rules in Figure 1
(the reader can find a comprehensive list online [37]). Note that these rules
are recursive. Rule (1) finds all possible solutions for the equality between
two non-empty extensional sets. The second and third disjuncts take care of
duplicates in the right- and left-hand side terms, respectively, while the
last disjunct takes care of permutativity of the set constructor
$\\{\cdot\mathbin{\scriptstyle\sqcup}\cdot\\}$. Specifically, the last
disjunct can be read as ‘$y$ must belong to $A$, $x$ must belong to $B$ and
there exists a set $N$ containing the remaining elements of both $A$ and $B$’.
In turn, rule (2) finds all possible solutions of a set union operation when
the result is a non-empty extensional set, where $N$, $N_{1}$ and $N_{2}$ are
new variables (implicitly existentially quantified). Note that set unification
is used to avoid possible repetitions of $t$ in $C$. Also observe that the
disjunction captures the three possible solutions: $t$ belongs to $A$, $t$
belongs to $B$ and $t$ belongs to $A$ and $B$. Finally, rule (3) finds the
binary relation whose converse is a non-empty extensional relation in a very
simple way.
The rewriting system implemented by $\\{log\\}$ has been proved to be a semi-
decision procedure [10]. More precisely, it has been proved that: a) when
Algorithm 1 terminates, the returned formula preserves the set of solutions of
the input formula; b) the returned formula is $\mathit{false}$ if and only if
the input formula is unsatisfiable; and c) if the returned formula is not
$\mathit{false}$ it is trivial to calculate one of its solutions (basically by
substituting all set variables by the empty set). In this context, a
‘solution’ is an assignment of values to all the free variables of the
formula. Furthermore, when $\\{log\\}$ terminates, it has the ability to
produce a finite representation of all the (possibly infinitely many)
solutions of the input formula, in the form of a finite disjunction of
$\mathcal{L}_{\mathcal{BR}}$ formulas. In other words, whenever $\\{log\\}$
terminates, it either produces a proof of unsatisfiability or a finite
representation of all the solutions.
### 2.3 Using $\\{log\\}$
Users interact with $\\{log\\}$ by simply entering a formula; there are no
user commands. If the formula is unsatisfiable $\\{log\\}$ will simply return
$\mathit{false}$ and if it is satisfiable it will return a finite
representation of all its solutions.
###### Example 2.1.
For example, the following is a satisfiable formula (note that binary
relations can be freely combined with extensional sets, and set operators can
take relations as arguments):
$\begin{split}\mathit{un}&(A,B,\\{(1,1),(h,3)\mathbin{\scriptstyle\sqcup}C\times
D\\})\\\ &\land\mathit{id}(E,A)\land\mathit{inv}(B,B)\land 1\notin
E\end{split}$ (4)
The relevant part of a solution returned by $\\{log\\}$ is:
$\displaystyle
A=\\{(3,3)\mathbin{\scriptstyle\sqcup}N_{3}\\},B=\\{(1,1)\mathbin{\scriptstyle\sqcup}N_{2}\\},$
$\displaystyle h=3,E=\\{3\mathbin{\scriptstyle\sqcup}N_{1}\\}$
$\displaystyle\textsf{Constraint: }3\notin C,un(N_{2},N_{3},C\times
D),id(N_{1},N_{3}),$ $\displaystyle\qquad\qquad\;\;\;inv(N_{2},N_{2}),\dots$
where $N_{i}$ are fresh variables. That is, each solution is composed of a
(possibly empty) conjunction of equalities between variables and terms and a
(possibly empty) conjunction of constraints. The conjunction of constraints is
guaranteed to be trivially satisfiable. ∎
In this context, $\\{log\\}$ can be used as a set-based, constraint-based
programming language. Users can give values to what they consider to be input
variables in the formula and $\\{log\\}$ will return values for the remaining
variables. For instance, if (4) is thought as a program where $A$ and $B$ are
inputs and the user enters (4) conjoined with $A=\\{(2,2),(3,3)\\}\land
B=\\{(1,1),(1,2),(2,1)\\}$, the answer will be
$h=3,C=D=\\{1,2\\},E=\\{2,3\\}$. In $\\{log\\}$, _formulas are programs_.
Given that $\\{log\\}$ is a satisfiability solver we can use it also as an
automated theorem prover. To prove that formula $\Phi$ is a theorem,
$\\{log\\}$ has to be called on $\lnot\Phi$ waiting an $\mathit{false}$
answer, meaning that $\lnot\Phi$ is unsatisfiable (and thus $\Phi$ is a
theorem).
###### Example 2.2.
We can prove that set intersection is commutative by asking $\\{log\\}$ to
prove the following formula is unsatisfiable:
$\mathit{inters}(A,B,C)\land\mathit{inters}(B,A,D)\land C\neq D$
As there are no finite sets satisfying this formula, $\\{log\\}$ returns
$\mathit{false}$. The formula can also be written as:
$\mathit{inters}(A,B,C)\land\mathit{ninters}(B,A,C)\hfill\qed$
All these properties along with programming facilities not discussed in this
paper [29], make $\\{log\\}$ a versatile verification tool [31, 34, 35, 30,
32].
## 3 Automating complex FS&RA proofs
As we have pointed out, $\\{log\\}$ may not terminate or may take a very long
time when it is used to prove some theorems of FS&RA. However, we have
observed that in practice the proofs of many of such theorems can be divided
into a few subgoals each of which can be automatically and quickly discharged
by $\\{log\\}$. In fact, these proofs follow a recurring pattern: divide the
proof by introducing some assumptions, drop zero or more hypotheses in each
subgoal and call $\\{log\\}$ to do the hard, annoying mathematical work. This
would imply that complex theorems of FS&RA can be easily proved by calling
$\\{log\\}$ at the right points.
In order to provide evidence of these observations, we have developed a proof-
of-concept ITP on top of $\\{log\\}$ that we call $\\{log\\}$-ITP. This is a
freely available +500 LOC Prolog program [38] that allows users to declare a
$\\{log\\}$ formula as a theorem and attempt to prove it interactively through
some proof commands. First, we will show a typical proof using
$\\{log\\}$-ITP, and then we will give technical details about the proof
system.
###### Remark 3.1 (Limitations).
It is important to bear in mind that $\\{log\\}$-ITP is intended to be an ITP
_only_ for theorems of FS&RA and _only_ to empirically validate our proposal.
This means that it cannot be compared in no way with general-purpose ITP’s
such as Coq or Isabelle/HOL.
### 3.1 A typical proof
[$\mathsf{rewrite}$] [$\mathsf{drop}$] [$\mathsf{prove}$]$\\{log\\}$Γ_1 ⊢g ⊆h
Γ⊢g ⊆h & [$\mathsf{drop}$] [$\mathsf{prove}$]$\\{log\\}$Γ_2 ⊢g ⊆h Γ⊢h ⊆g Γ⊢g =
h
Figure 2: $\\{log\\}$-ITP proof of theorem $\mathsf{T1}$ or (T1)
(T1) is declared as a $\\{log\\}$-ITP theorem with the $\mathsf{theorem}$
command:
$\begin{split}\mathsf{theorem}(&\mathsf{T1},\\\ &\mathit{pfun}(f)\land
f\subseteq A\times B\land\mathit{dom}(f,A)\land\mathit{ran}(f,B)\\\
&\land\mathit{pfun}(g)\land g\subseteq B\times C\land\mathit{dom}(g,B)\\\
&\land\mathit{pfun}(h)\land h\subseteq B\times C\\\
&\land\mathit{dom}(h,B)\land\mathit{comp}(f,g,N)\land\mathit{comp}(f,h,N),\\\
&g=h)\end{split}$
where the first parameter is just a name for the theorem, the second one is a
(possibly empty) conjunction of hypotheses and the third one is the thesis,
both entered as $\\{log\\}$ formulas. Note that $f\in A\rightarrow B$ is
encoded in $\\{log\\}$ as $\mathit{pfun}(f)\land f\subseteq A\times
B\land\mathit{dom}(f,A)$ and that $f\circ g=f\circ h$ is encoded as two
$\mathit{comp}$ constraints yielding the same result ($N$).
As we have said, an automated proof of (T1) would take a long time. However,
the interactive proof shown in Figure 2 takes only a few seconds of computing
time. There, $\Gamma$ represents the hypotheses of (T1). The proof starts with
the $\mathsf{rewrite}$ command which splits the proof into the two subgoals
shown in Figure 2. Attempting to use $\\{log\\}$ to prove these subgoals by
means of the command $\mathsf{prove}$ would consume as much time as the proof
of the initial goal because they are essentially the same. As the proof of
these two subgoals is symmetric, we will explain in detail only the first one.
In this case the user can use the following $\mathsf{drop}$ command, which
expects a list of $\\{log\\}$ constraints:
$\begin{split}\mathsf{drop}([&f\subseteq A\times B,\mathit{dom}(f,A),\\\
&g\subseteq B\times C,h\subseteq B\times C,\mathit{dom}(h,B)])\end{split}$ (5)
These constraints are expected to be part of $\Gamma$ in which case they are
removed from it, thus yielding the following hypotheses:
$\begin{split}&\mathit{pfun}(f)\land\mathit{ran}(f,B)\land\mathit{pfun}(g)\land\mathit{dom}(g,B)\\\
&\land\mathit{pfun}(h)\land\mathit{comp}(f,g,N)\land\mathit{comp}(f,h,N)\end{split}$
called $\Gamma_{1}$ in Figure 2. Now, $\mathsf{prove}$ discharges the current
subgoal in a few seconds. Since in this case $\\{log\\}$ succeeds in proving
the goal, the system shows to the user the remaining subgoal. A similar course
of action is taken to discharge the remaining subgoal, where a different list
of constraints is passed in to the $\mathsf{drop}$ command.
In Section 3.3 we discuss some aspects of this proof and present the complete
proof script in (6).
### 3.2 Proof commands
Γ⊢φΓ⊢Δ | Γ⊢ξ,ΔΓ⊢φ∧ξ, Δ | [$\mathsf{drop}(\varphi)$]Γ⊢ΔΓ,φ⊢Δ
---|---|---
[$\mathsf{define}(\pi)$] Γ, π(…,n) ⊢Δ Γ⊢Δ | [$\mathsf{prove}$] setlog(Γ⟹Δ) Γ⊢Δ
Γ, (w = u)[n/x] ⊢ξ[n/x]Γ⊢(w = u ⟹ξ)[n/x] Γ⊢∀x:w = u ⟹ξ Γ⊢(∀x:v = t ⟹φ) ∧(∀x:w
= u ⟹ξ) Γ⊢π(v,w)
Figure 3: Main $\\{log\\}$-ITP proof commands as inference rules
In this section we present in detail the main proof commands of
$\\{log\\}$-ITP (see Figure 3). Some of them are direct implementations of
well-known inference rules while others implement a few such rules in a single
proof step.
###### Remark 3.2 (Interfacing with $\\{log\\}$).
As we already said, as $\\{log\\}$ is a satisfiability solver, it can be used
as an ATP. Indeed, if $\\{log\\}$ finds that formula $\varphi$ is
unsatisfiable in FS&RA, then $\lnot\varphi$ is a theorem (in FS&RA). Thus,
when $\\{log\\}$ is used as a back-end system for an ITP, formulas must be
negated before sending them from the ITP to $\\{log\\}$. However, with the
intention to simplify the presentation, we are not going to mention this
negation process in the remaining of the paper. This implies that, for
example, when in the command called $\mathsf{prove}$ (Figure 3) we say that
$\Gamma\implies\Delta$ is sent to $\\{log\\}$ it actually means that its
negation $\Gamma\land\lnot\Delta$ is sent to it. ∎
Concerning Figure 3, the $\mathsf{assume}$ command can be seen as the
implementation of a special case of the Cut rule; $\mathsf{cases}$ corresponds
to conjunction introduction; and $\mathsf{drop}$ is a specialized version of
the Weakening rule where the antecedent is weakened in order to deliver to
$\\{log\\}$ exactly the necessary hypotheses that yield the consequent.
$\mathsf{define}$ waits for a constraint
$\pi\in\\{\mathit{un},\mathit{inv},\mathit{id},\mathit{comp}\\}$. The last
argument of $\pi$ is expected to be a new variable and all the others must be
variables in the current scope (of the proper sort). For instance, if in the
current scope $R$ is a binary relation then the user can issue
$\mathsf{define}(\mathit{inv}(R,S))$, where $S$ is a new variable, in which
case $\mathit{inv}(R,S)$ is added as a new hypothesis. This is sound because
what we are doing is no more than asserting that the converse of $R$ is called
$S$. Now the user can make assumptions on $S$. For example,
$\mathsf{assume}(\mathit{pfun}(S))$, which means that $S$ (i.e., the converse
of $R$) is a function. Without such a command it would be impossible to
consider assumptions on expressions assembled from variables in the current
scope, as in $\\{log\\}$-ITP set and relational operations are represented as
predicates.
The $\mathsf{prove}$ command simply calls $\\{log\\}$ on the current subgoal.
In this case, there are three possible behaviors: _a_) $\\{log\\}$ answers
that the current subgoal is indeed valid and so it is proved and the next one
(if any) is shown to the user; _b_) $\\{log\\}$ answers that there is a
counterexample (for instance if too many hypotheses have been dropped) in
which case a proper error message is printed; and _c_) $\\{log\\}$ takes too
long and the user decides to interrupt the command. In _b_ and _c_ the proof
is unchanged. In _b_ users can execute command $\mathsf{counterex}$ to get a
counterexample witnessing why the goal failed (recall the discussion on the
generation of counterexamples in the introduction).
$\mathsf{rewrite}$ calls $\\{log\\}$ to rewrite the thesis; that is
$\\{log\\}$ is called to apply a rewrite rule to the thesis (eg. one of the
rules of Figure 1) thus generating one or more new goals to prove—this last
case occurs when the rewrite rule is nondeterministic. The thesis is assumed
to be a single constraint. Each of these new goals should be simpler to prove
than the orginal one. For each new goal generated by the rewrite rule,
$\mathsf{rewrite}$ performs three proof steps in one (see Figure 3):
1. 1.
It applies conjunction introduction to the goal. If the subgoal is a
conjunction of constraints, then the user will prove one after the other.
2. 2.
It applies universal introduction on each subgoal generated in step 1. As the
new goal will in general be a universally quantified formula, these quantified
variables are ‘introduced’.
3. 3.
It applies conditional introduction on each subgoal generated in step 1. As in
the previous step, the new goal will in general be a conditional and so
hypotheses are ‘introduced’ as well.
For example, if $\Gamma\vdash\mathit{pfun}(f)$ is the current goal, the net
effect of $\mathsf{rewrite}$ is shown in Figure 4, where $x$, $y$, $z$, $v$
and $N$ are new variables. If, for instance, $\\{log\\}$ does not terminate on
$\Gamma\vdash\mathit{pfun}(f)$, after $\mathsf{rewrite}$ the user has two
simpler goals to prove. In particular, the one on the right most often can be
automatically and quickly discharged with $\mathsf{prove}$. In Figure 4
$\mathsf{rewrite}$ produces those two subgoals because in
$\mathcal{L}_{\mathcal{BR}}$ we have:
$\begin{split}&\mathit{pfun}(f)\Leftrightarrow\\\ &\qquad(\forall v:v\in
f\Rightarrow\mathit{pair}(v))\\\ &\qquad\land(\forall x,y,z:(x,y)\in
f\land(x,z)\in f\Rightarrow y=z)\end{split}$
where $v\in f$ is equivalent to $f=\\{v\mathbin{\scriptstyle\sqcup}N\\}$ for
some new variable $N$, which yields the equalities seen in Figure 4. Then,
when conjunction, universal quantification and implication are introduced as
in Figure 3, we get the result shown in Figure 4.
$\quad\quad\quad\quad\quad\quad\quad\inference{\Gamma,f=\\{(x,y),(x,z)\mathbin{\scriptstyle\sqcup}N\\}\vdash
y=z&\Gamma,f=\\{v\mathbin{\scriptstyle\sqcup}N\\}\vdash\mathit{pair}(v)}{\Gamma\vdash\mathit{pfun}(f)}$
Figure 4: Example of a rewrite step
Actually, the inference rule given for $\mathsf{rewrite}$ in Figure 3 is a
simplification of the real rule. Here we assume that the current thesis is a
constraint $\pi$ depending on two variables ($v$ and $w$), which when
rewritten by $\\{log\\}$ delivers a conjunction of two universal formulas (in
the real case it can be any number of them). These formulas have all the same
form $\mathit{equalities}\implies\mathit{predicate}$, where
$\mathit{equalities}$ is a (possibly empty) conjunction of equalities of the
form $var=term$ where $var$ is one of the variables on which $\pi$ depends on;
and $\mathit{predicate}$ is a (possibly empty) conjunction of
$\mathcal{L}_{\mathcal{BR}}$ constraints. In Figure 3 we assume that the
$\mathit{equalities}$ in each conjunct have exactly one equality.
It is important to remark that $\\{log\\}$-ITP does not need a command, for
instance, to perform equality substitution because this is performed by
$\\{log\\}$ when $\mathsf{prove}$ is executed. In effect, if $\mathsf{prove}$
is issued, for example, on
$\Gamma,f=\\{v\mathbin{\scriptstyle\sqcup}N\\}\vdash\mathit{pair}(v)$, then
$\\{log\\}$ will substitute $f$ by $\\{v\mathbin{\scriptstyle\sqcup}N\\}$ in
$\Gamma$.
### 3.3 Discussion
As can be seen in Figure 3, many $\\{log\\}$-ITP’s proof commands correspond
to standard inference rules present in ITP’s. Then, they can be easily
replaced by the proof commands present in a particular ITP.
As concerns proof automation, the $\mathsf{drop}$ command plays a central
role. In effect, it allows to call $\\{log\\}$ with the minimal set of
hypotheses as to prove the goal. This implies a reduction of the proof term
and consequently of the computing time. As opposed to the usual fact that the
more hypotheses are available during an interactive proof, the better,
dropping hypotheses is decisive to the success of our approach. Indeed, when
an ATP is called to perform a proof step, unnecessary hypotheses may make it
walk through many actually useless proof paths (and, in the case of tools like
$\\{log\\}$, that might not terminate in some cases, they can take an infinite
proof path). Hence, by dropping unnecessary hypotheses the prover has fewer
proof paths to walk through, thus augmenting the chances to end the proof and
to do it faster.
On the downside, the key role of $\mathsf{drop}$ sensibly changes the proof
style as now the user must determine which hypotheses are superfluous to prove
a subgoal instead of using them to prove it.
We also note that our approach tends to reduce the influence of a good _lemma
engineering_. That is, users usually plan which lemmas go first and which
follow, so as to use the former in the proofs of the latter. For example, the
Coq proof of (T1) is the following222The reader does not need to understand
the proofs, just to have an idea of their length and complexity.:
⬇
move=>is_function_F is_function_G is_function_H range_F_eq_B rel_comp_eq.
apply/eqP; rewrite eqEsubset; apply/andP; split.
- exact: (auxT is_function_F is_function_G range_F_eq_B rel_comp_eq).
- symmetry in rel_comp_eq.
exact: (auxT is_function_F is_function_H range_F_eq_B rel_comp_eq).
where `auxT` is a helper lemma whose importance was made evident after the
first proof attempt (T1). Indeed, `auxT` states that $g\subseteq h$ holds if
the hypotheses of (T1) are satisfied. Its proof is the following:
⬇
move=>[fun_cond_F _ _] [_ domain_G_eq_B _] range_F_eq_domain_G rel_comp_eq;
apply/subsetP => p p_in_G.
have: p.1 \in range F.
rewrite range_F_eq_domain_G -domain_G_eq_B.
apply/in_domainP; exists p.2.
by rewrite -[(p.1,p.2)]surjective_pairing.
move=> /in_range_restP [a [_ pair_in_F]].
have: (a,p.2) \in rel_comp F H.
by rewrite -rel_comp_eq;apply/in_rel_compP; exists p.1; split;
[exact: pair_in_F |
rewrite -[(p.1,p.2)]surjective_pairing].
move=> /in_rel_compP [b [in_F in_H]].
have p1_eq_b: p.1 = b by apply:
((((fun_cond_F (a,p.1)) (a,b)) pair_in_F) in_F).
by rewrite [p]surjective_pairing p1_eq_b.
Now, compare Coq’s proof of (T1) with $\\{log\\}$-ITP’s (cf. Section 3.1):
$\displaystyle\mathsf{rewrite}.$ (6)
$\displaystyle\begin{split}-\;&\mathsf{drop}([f\subseteq A\times
B,\mathit{dom}(f,A),\\\ &\qquad g\subseteq B\times
C,\mathit{dom}(g,B),h\subseteq B\times C]),\\\ &\mathsf{prove}.\end{split}$
$\displaystyle\begin{split}-\;&\mathsf{drop}([f\subseteq A\times
B,\mathit{dom}(f,A),\\\ &\qquad g\subseteq B\times C,h\subseteq B\times
C,\mathit{dom}(h,B)]),\\\ &\mathsf{prove}.\end{split}$
where no helper lemma is necessary and the complex proof of `auxT` is replaced
by dropping hypotheses and then calling $\\{log\\}$. On the other hand, as in
Coq, the user still needs to guide the proof by splitting the initial goal
into $g\subseteq h$ and $h\subseteq g$ and the symmetry used in the Coq proof
(i.e., `symmetry`) is still present in the $\\{log\\}$-ITP proof, when
symmetric $\mathsf{drop}$ commands are executed. Hence, were $\\{log\\}$
available in Coq the proof of (T1) would not need the helper lemma and it
would still be compact and semi-automatic.
However, there is still room for further automation. CoqHammer [6] uses
external ATPs to automate Coq proofs. CoqHammer helps in automating the proof
of (T1):
⬇
move=> is_function_F is_function_G is_function_H range_F_eq_B rel_comp_eq.
apply/eqP; rewrite eqEsubset; apply/andP; split.
- hammer.
- hammer.
where `hammer` needs lemma `auxT` to prove both subgoals. _However,_ `hammer`
_cannot prove_ `auxT` _automatically_. Then, were CoqHammer _and_ $\\{log\\}$
available, the Coq proof of (T1) could be _almost_ automatic: first use
$\mathsf{drop}$ and $\mathsf{prove}$ to prove `auxT`; and then use `hammer` to
prove (T1) as above.
CoqHammer depends on a good lemma engineering. Conversely, $\\{log\\}$ does
not depend on such engineering but it can only prove results of FS&RA. A
combination between a general tool like CoqHammer with specialized provers
such as $\\{log\\}$, seems to be a promising strategy towards proof
automation.
## 4 Empirical assessment
As we have said, our intention is to provide evidence that integrating
$\\{log\\}$’s rewriting system into mainstream ITP’s will yield a noticeable
increment in proof automation concerning FS&RA. This is $\\{log\\}$-ITP’s
single purpose. To this end, we performed 210 proofs with $\\{log\\}$-ITP and
Coq in order to compare their complexity and length. In an attempt to avoid as
much as possible a bias towards $\\{log\\}$-ITP, 21 proofs correspond to
problems listed in the REL and SET collections of the TPTP library ((T1) is an
example) which satisfy that $\\{log\\}$ either does not terminate or takes a
very long time to do it when it is applied to prove them333$\\{log\\}$
automatically and quickly proves all the other problems of TPTP.SET and
TPTP.REL expressible in its input language [10].. The remaining 189 proofs
correspond to lemmas included in Coq’s SSReflect finite set library,
`finset`444The remaining 147 lemmas of finset are not expressible in the input
language of $\\{log\\}$ as they include operators such as generalized union or
powerset. [39]. We chose `finset` for three reasons: _a_) it has been designed
as to make it easy to prove those lemmas in Coq; _b_) a fragment of `finset`’s
set theory keeps a clear relationship with respect to $\\{log\\}$’s; and _c_)
it would provide evidence that proof automation of real Coq results can be
achieved with our proposal. Finally, the Coq proofs were performed by one of
the authors (with experience in working with SSReflect), while
$\\{log\\}$-ITP’s were done by another author.
As concerns the TPTP problems, they are encoded in Coq by extending
SSReflect’s `finset`. SSReflect is a proof language extending Coq with
additional tactics oriented to support long mathematical proofs. `finset`
defines a type for sets over a finite type. It includes definitions such as
set membership, union, Cartesian product, etc. However, it does not include
set relation algebra definitions such as the identity relation, converse (or
inverse), composition, and (partial) function. Since these are necessary to
reason about $\mathcal{L}_{\mathcal{BR}}$ formulas, we defined them in Coq.
For example, our Coq set-based definition of function from $A$ to $B$ (i.e.,
$f\in A\rightarrow B$) is the following:
⬇
Definition is_function_from_to
(S1 S2:finType) (R:{set (S1 * S2)})
(A:{set S1}) (B:{set S2}) :=
[/\ is_function R, domain R = A
& (range R) \subset B].
where `is_function`, `domain` and `range` are defined in a similar fashion,
but where `\subset` is part of SSReflect’s finite type interface (on which
`finset` is based on). We also give a set-based definition of the relational
composition of two binary relations:
⬇
Definition rel_comp (S1 S2 S3:finType)
(R1:{set (S1 * S2)}) (R2:{set (S2 * S3)}) :=
[set p | [ exists u, ((p.1, u) \in R1)
&& ((u, p.2) \in R2)]].
These extensions to `finset` lead to the following encoding of theorem (T1):
⬇
Theorem T1 A B C (F:{set (S1 * S2)})
(G H:{set (S2 * S3)}) :
is_function_from_to F A B ->
is_function_from_to G B C ->
is_function_from_to H B C ->
range F = B ->
rel_comp F G = rel_comp F H -> G = H.
A similar encoding was used to state the 189 lemmas of `finset`, as
$\\{log\\}$-ITP theorems. For example, the following `finset` lemma:
⬇
Lemma setCU A B :
~: (A :|: B) = ~: A :&: ~: B
where `~:` is complement (-), `:|:` is $\cup$ and `:&:` is $\cap$, is encoded
as the following $\\{log\\}$-ITP theorem:
$\begin{split}\mathsf{theorem}(&\text{{setCU}},\\\ &A\subseteq T\land
B\subseteq T\land\mathit{un}(A,B,M_{1})\\\
&\land\mathit{un}(M_{1},M_{2},T)\land M_{1}\parallel
M_{2}\land\mathit{un}(A,M_{3},T)\\\ &\land A\parallel
M_{3}\land\mathit{un}(B,M_{4},T)\land B\parallel M_{4},\\\
&\mathit{inters}(M_{3},M_{4},M_{2}))\end{split}$
where $T$ corresponds to the variable `T` of type `finType` declared in the
section containing Lemma setCU (i.e., all sets of this section are subsets of
`T`).
Table 1 summarizes the results of the evaluation. As we have said, $\\{log\\}$
is unable to automatically prove any of the 21 TPTP theorems (thus, for the
TPTP theorems, the Auto entry is set to zero). On the other hand, Coq needs
690 proof commands to prove them, while $\\{log\\}$-ITP can do it with 219, of
which only 146 are other than the $\mathsf{prove}$ command. This is a
reduction of about 68% in the number of proof commands (and about 79% if
$\mathsf{prove}$ is not counted). By ‘proof command’ we understand all the
characters accommodated in the same line (which intend to represent basic
mathematical proof steps). For example, for us, this is a single Coq proof
command: by move=> a; apply/setP=> x; rewrite inE; case: eqP => ->. In this
sense, $\\{log\\}$-ITP proof commands tend to be simpler than Coq’s. Actually,
the 690 proof commands used in Coq to prove the TPTP problems are composed of
about 1030 applications of SSReflect tactics, which represent 35,014
characters while those used in $\\{log\\}$-ITP just 3,662 characters. Then,
apart from the gain in the number of proof commands, there is notable gain in
their complexity ($\approx$ 90%). Finally, collectively all the
$\mathsf{prove}$ commands consume 55 s of computing time which means that the
automated part of each theorem is executed in 2.6 s in average. On the other
hand, the completion of all the Coq proofs took around 10 man-hour; while
completing all the $\\{log\\}$-ITP took around 1 man-hour.
Collection | # | Auto | % | Commands | Computing Time | Avg Computing Time
---|---|---|---|---|---|---
| | | | Coq | $\\{log\\}$ | non-$\mathsf{prove}$ | |
TPTP | 21 | 0 | 0 | 690 | 219 | 146 | 55 s | 2.6 s
SSReflect finset | 189 | 183 | 97 | 223 | 195 | 6 | 182 s | 1 s
Summary | 210 | 183 | 87 | 913 | 435 | 158 | 237 s | 1.1 s
Table 1: Summary of the empirical evaluation
As concerns the problems taken from `finset`, Table 1 indicates that
$\\{log\\}$-ITP automatically proves 97% of them (in 1 s in average). In this
case the gain in the number of proof commands is minimal. However, it should
be noted that in $\\{log\\}$-ITP 183 problems are proved via the _same_
command ($\mathsf{prove}$), while the Coq proofs require, roughly, 214
_different_ commands. In other words, a Coq user needs to reason on how to
prove 183 theorems, while a $\\{log\\}$-ITP user does not. This fact can be
quantified if only the non-$\mathsf{prove}$ commands are considered, as they
amount to only 2% of the Coq commands.
The Coq proofs present in these experiments are the result of some degree of
lemma engineering. For instance, in the TPTP Coq proofs we first proved 21
helper lemmas and then the 21 theorems used as experiment. The helper lemmas
correspond to properties that are used several times in the proofs of the 21
theorems. These can be simple properties, such as the characterization of the
fact that an ordered pair belongs to the relational composition of two binary
relations, or more complex ones, such as the helper lemma `auxT` described in
Section 3.3 (which is applied twice in the proof of (T1) thanks to the use of
symmetry). Without this lemma engineering, proofs would be longer and more
complex. As usual, many of these helper lemmas make themselves evident after
some of the main theorems are proved. In $\\{log\\}$-ITP no lemma engineering
was used (actually, there is no way to use or apply a previous lemma in the
current proof). This is another indication of a gain in simplicity when
$\\{log\\}$-ITP is used.
This evaluation suggests that an integration of $\\{log\\}$ into Coq would
produce more fully automated FS&RA proofs and would semi-automate many others.
The full data set of our experiments can be found online at:
https://www.dropbox.com/s/c6z45thxlvr1q1h/setlogITP.zip?dl=0. They were
performed on a Dell Latitude E7470 (06DC) laptop with a 4 core Intel(R) Core™
i7-6600U CPU at 2.60GHz with 8 Gb of main memory, running Linux Ubuntu 18.04.2
LTS 64-bit with kernel 4.15.0-56-generic. The following software versions were
used: $\\{log\\}$ 4.9.6-5d over SWI-Prolog (multi-threaded, 64 bits, version
7.6.4); Coq 8.9.0; CoqHammer 1.1 using Vampire 4.2.2, E prover 2.3, Z3 4.8.4.0
and CVC4 1.6.
## 5 Conclusions
We have proposed $\\{log\\}$ as a special purpose ATP for the theory of finite
sets and finite set relation algebra, that can be integrated into ITPs to
semi-automate proofs of this theory. We have also empirically evaluated the
approach by implementing a prototype ITP where users can call $\\{log\\}$ as a
proof command. The prototype was assessed on 210 theorems and compared with
Coq. The assessment shows good results in: _a_) the number of automated
proofs; _b_) the computing times needed to complete them; and _c_) the
reduction in the length and complexity of interactive proofs that call
$\\{log\\}$ to discharge subgoals.
Concerning future work, in the case of Coq we see two possible integration
strategies. The most obvious one is to use $\\{log\\}$ as an external ATP,
along the lines of SMTCoq [8] or CoqHammer [6]. In this case proof
reconstruction might be difficult or infeasible. A second integration strategy
may consist in implementing $\\{log\\}$’s rewriting system as a Coq tactic.
The first steps of this approach have already been done by Dubois and Weppe
[40]. Depending on the way this is done, as a side effect, this approach might
yield a formal verification of $\\{log\\}$. It remains as open issues, though,
whether the size of the proof term produced by the new tactic will be more
manageable than in the first strategy and whether or not the tactic will be
fast enough as to be worth it.
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## Appendix A Syntax and semantics of $\mathcal{L}_{\mathcal{BR}}$
In this appendix we provide a formal, detailed introduction of the syntax and
semantics of $\mathcal{L}_{\mathcal{BR}}$.
The input constraint language accepted by $\\{log\\}$,
$\mathcal{L}_{\mathcal{BR}}$, is a first-order predicate language with terms
of two sorts: terms designating sets (including binary relations), and terms
designating ur-elements. Terms of either sort are allowed to enter in the
formation of set terms (in this sense, the designated sets are hybrid), no
nesting restrictions being enforced (in particular, membership chains of any
finite length can be modeled). In a term which is not a variable and
designates an ur-element, the main functor (be it a constant or a function
symbol) will act as a free (‘uninterpreted’) Herbrand constructor; a special
set constructor, and a handful of reserved predicate symbols endowed with a
pre-designated set-theoretic meaning, are also available. Formulas are built
in the usual way by using conjunction, disjunction and negation of atomic
predicates. A number of complex operators (in the form of predicates) are
defined as $\mathcal{L}_{\mathcal{BR}}$ formulas, thus making it simpler for
the user to write complex formulas.
### A.1 Syntax
The syntax of $\mathcal{L}_{\mathcal{BR}}$ is defined primarily by giving the
signature upon which terms and formulas are built.
###### Definition A.1.
The signature $\Sigma_{\cal BR}$ of $\mathcal{L}_{\mathcal{BR}}$ is a tuple
$\langle\mathcal{F},\Pi,\mathsf{Set},\mathsf{O},\mathcal{V}\rangle$ where:
* •
$\mathcal{F}$ is the set of function symbols partitioned as
$\mathcal{F}\mathrel{\widehat{=}}\mathcal{F}_{S}\cup\mathcal{F}_{\mathcal{X}}$,
where
$\mathcal{F}_{S}\mathrel{\widehat{=}}\\{\emptyset,\\{\cdot\mathbin{\scriptstyle\sqcup}\cdot\\},\cdot\times\cdot\\}$
and $\mathcal{F}_{\mathcal{X}}$ is a set of uninterpreted constant and
function symbols, including at least the binary function symbol
$(\cdot,\cdot)$.
* •
$\Pi$ is the set of predicate symbols partitioned as
$\Pi\mathrel{\widehat{=}}\Pi_{\mathit{S}}\cup\Pi_{T}\cup\Pi_{\mathit{R}}$,
where
$\Pi_{\mathit{S}}\mathrel{\widehat{=}}\\{=,\neq,\in,\notin,\mathit{un},\parallel\\}$,
$\Pi_{T}\mathrel{\widehat{=}}\\{\mathit{set},\mathit{nset},\mathit{rel},\mathit{nrel},\mathit{pair},\mathit{npair}\\}$
and
$\Pi_{\mathit{R}}\mathrel{\widehat{=}}\\{\mathit{id},\mathit{comp},\mathit{inv}\\}$.
* •
$\\{\mathsf{Set},\mathsf{O}\\}$ is the set of sorts.
* •
$\mathcal{V}$ is a denumerable set of variables partitioned as
$\mathcal{V}\mathrel{\widehat{=}}\mathcal{V}_{S}\cup\mathcal{V}_{O}$, where
$\mathcal{V}_{S}$ and $\mathcal{V}_{O}$ contain variables of sort
$\mathsf{Set}$ and $\mathsf{O}$, respectively. ∎
To complete the definition of $\mathcal{L}_{\mathcal{BR}}$, in addition to the
signature it is necessary to specify the sorts of function and predicate
symbols: if $f\in\mathcal{F}$ (resp., $\pi\in\Pi$) is of arity $n$, then its
sort is an $n+1$-tuple $\langle s_{1},\ldots,s_{n+1}\rangle$ (resp., an
$n$-tuple $\langle s_{1},\ldots,s_{n}\rangle$) of non-empty subsets of the set
of sorts $\\{\mathsf{Set},\mathsf{O}\\}$. This notion is denoted by $f:\langle
s_{1},\ldots,s_{n+1}\rangle$ (resp., by $\pi:\langle
s_{1},\ldots,s_{n}\rangle$).
###### Definition A.2.
The sorts of the function symbols in $\mathcal{F}$ are as follows:
* •
$\emptyset:\langle\\{\mathsf{Set}\\}\rangle$;
* •
$\mathsf{\\{\cdot\mathbin{\scriptstyle\sqcup}\cdot\\}:\langle\\{\mathsf{Set},\mathsf{O}\\},\\{\mathsf{Set}\\},\\{\mathsf{Set}\\}\rangle}$;
* •
$\cdot\times\cdot:\langle\\{\mathsf{Set}\\},\\{\mathsf{Set}\\},\\{\mathsf{Set}\\}\rangle$;
* •
$(\cdot,\cdot):\langle\\{\mathsf{Set},\mathsf{O}\\},\\{\mathsf{Set},\mathsf{O}\\},\\{\mathsf{O}\\}\rangle$;
* •
$f:\langle\\{\mathsf{O}\\},\dots,\\{\mathsf{O}\\}\rangle\in(\\{\mathsf{O}\\})^{n+1}$
if $f\in\mathcal{F}_{\mathcal{X}}$ is of arity $n$.
The sorts of the predicate symbols in $\Pi$ are as follows (symbols $=$,
$\neq$, $\in$, $\notin$ and $\parallel$ are infix; all other symbols in $\Pi$
are prefix):
* •
$\mathit{pair},\mathit{npair},\mathit{set},\mathit{nset}:\langle\\{\mathsf{Set},\mathsf{O}\\}\rangle$;
* •
$=,\neq:\langle\\{\mathsf{Set},\mathsf{O})\\},\\{\mathsf{Set},\mathsf{O}\\}\rangle$;
* •
$\in,\notin:\langle\\{\mathsf{Set},\mathsf{O}\\},\\{\mathsf{Set}\\}\rangle$;
* •
$\mathit{un},\mathit{comp}:\langle\\{\mathsf{Set}\\},\\{\mathsf{Set}\\},\\{\mathsf{Set}\\}\rangle$;
* •
$\parallel,\mathit{id},\mathit{inv}:\langle\\{\mathsf{Set}\\},\\{\mathsf{Set}\\}\rangle$;
* •
$\mathit{rel},\mathit{nrel}:\langle\\{\mathsf{Set}\\}\rangle$. ∎
We can now define the set of admissible (i.e., well-sorted)
$\mathcal{L}_{\mathcal{BR}}$ terms.
###### Definition A.3.
All $\mathcal{BR}$-terms and their sorts are build inductively as follows:
* •
each variable $v\in V$ is a $\mathcal{BR}$-term of sort
$\langle\\{\mathsf{Set}\\}\rangle$ if $v\in\mathcal{V}_{S}$ or sort
$\langle\\{\mathsf{O}\\}\rangle$ if $v\in\mathcal{V}_{O}$.
* •
if $f\in\mathcal{F}$ is a function symbol of sort $\langle
s_{1},\ldots,s_{n+1}\rangle$, and for each $i=1,\ldots,n$, $t_{i}$ is a
$\mathcal{BR}$-term of sort $\langle s^{\prime}_{i}\rangle$ with
$s^{\prime}_{i}\subseteq s_{i}$, then $f(t_{1},\ldots,t_{n})$ is a
$\mathcal{BR}$-term of sort $\langle s_{n+1}\rangle$. ∎
Note that the sort of any $\mathcal{BR}$-term $t$ is always of the form
$\langle\\{\mathsf{Set}\\}\rangle$ or $\langle\\{\mathsf{O}\\}\rangle$. In the
former case we simply say that $t$ is of sort $\mathsf{Set}$, or a set term,
and in the latter case that $t$ is of sort $\mathsf{O}$. In particular,
$\mathcal{BR}$-terms of the form
$\\{\cdot\mathbin{\scriptstyle\sqcup}\cdot\\}$ are called _extensional_ set
terms. The first parameter of an extensional set term is called _element part_
and the second is called _set part_. Observe that one can write terms
representing sets which are nested at any level.
The following notation is introduced to make reading of set terms simpler:
$\\{t_{1},t_{2},\dots,t_{n}\mathbin{\scriptstyle\sqcup}t\\}$ as a shorthand
for
$\\{t_{1}\mathbin{\scriptstyle\sqcup}\\{t_{2}\,\mathbin{\scriptstyle\sqcup}\,\cdots\\{t_{n}\mathbin{\scriptstyle\sqcup}t\\}\cdots\\}\\}$
and the notation $\\{t_{1},t_{2},\dots,t_{n}\\}$ as a shorthand for
$\\{t_{1},t_{2},\dots,t_{n}\mathbin{\scriptstyle\sqcup}\emptyset\\}$.
###### Example A.4.
The following are set terms:
* -
$\emptyset$
* -
$\\{a,(b,c)\\}$, i.e.,
$\\{a\mathbin{\scriptstyle\sqcup}\\{(b,c)\mathbin{\scriptstyle\sqcup}\emptyset\\}\\}$,
where $a$, $b$ and $c$ are constants of sort $\mathsf{O}$
* -
$\\{x\mathbin{\scriptstyle\sqcup}A\times\\{y\mathbin{\scriptstyle\sqcup}B\\}\\}$,
where $x,y$ are variables of sort $\mathsf{Set}$ or $\mathsf{O}$, and $A,B$
are variables of sort $\mathsf{Set}$.
* -
$\\{x\mathbin{\scriptstyle\sqcup}A\\}$, where $x$ is a variable of sort
$\mathsf{Set}$ or $\mathsf{O}$, and $A$ is a variable of sort $\mathsf{Set}$.
On the opposite, $\\{x\mathbin{\scriptstyle\sqcup}(a,b)\\}$ is not a set term
because $(a,b)$ is not of sort $\mathsf{Set}$. ∎
Finally, from $\mathcal{L}_{\mathcal{BR}}$ terms, we define
$\mathcal{L}_{\mathcal{BR}}$ formulas.
###### Definition A.5.
All $\mathcal{BR}$-formulas are build inductively as follows:
* •
if $\pi\in\Pi$ is a predicate symbol of sort $\langle
s_{1},\ldots,s_{n}\rangle$, and for each $i=1,\ldots,n$, $t_{i}$ is a
$\mathcal{BR}$-term of sort $\langle s^{\prime}_{i}\rangle$ with
$s^{\prime}_{i}\subseteq s_{i}$, then $\pi(t_{1},\ldots,t_{n})$ is a
$\mathcal{BR}$-constraint, a particular case of $\mathcal{BR}$-formula.
* •
if $\alpha$ and $\beta$ are $\mathcal{BR}$-formulas, then so are
$\alpha\land\beta$ and $\alpha\lor\beta$. ∎
###### Example A.6.
The following are $\mathcal{BR}$-formulas:
$\displaystyle a\in A\land a\notin B\land\mathit{un}(A,B,C)\land C=\\{x\\}$
$\displaystyle\mathit{un}(A,B,C)\land A\parallel C\land\mathit{inv}(R,A)\land
R\neq\emptyset$
where $a$ is a constant of sort $\mathsf{O}$, $x$ is a variable of sort
$\mathsf{Set}$ or $\mathsf{O}$, and $A$, $B$, $C$ and $R$ are variables of
sort $\mathsf{Set}$. On the contrary, $\mathit{un}(A,B,(x,y))$ is not a
$\mathcal{BR}$-formula because $\mathit{un}(A,B,(x,y))$ is not a
$\mathcal{BR}$-constraint ($(x,y)$ is not of sort $\mathsf{Set}$ as required
by the sort of $\mathit{un}$). ∎
### A.2 Semantics
Semantics of $\mathcal{BR}$-formulas is given by defining a suitable
interpretation structure for $\mathcal{L}_{\mathcal{BR}}$.
Sorts and symbols in $\Sigma_{\cal BR}$ are interpreted according to the
interpretation structure $\mathcal{R}\mathrel{\widehat{=}}\langle
D,(\cdot)^{\mathcal{R}}\rangle$, where $D$ and $(\cdot)^{\mathcal{R}}$ are
defined as follows.
###### Definition A.7 (Interpretation domain).
The interpretation domain $D$, of the interpretation structure $\mathcal{R}$,
is partitioned as $D\mathrel{\widehat{=}}D_{\mathsf{Set}}\cup D_{\mathsf{O}}$
where:
* •
$D_{\mathsf{Set}}$ is the collection of all hereditarily finite hybrid sets
built from elements in $D$; and
* •
$D_{\mathsf{O}}$ is a collection of other objects, including ordered pairs of
elements in $D$. ∎
Hereditarily finite sets are those sets that admit (hereditarily finite) sets
as their elements. Note that, finite binary relations and functions, as
defined in Note 2, belong to $D_{\mathsf{Set}}$.
###### Definition A.8 (Interpretation function).
The interpretation function $(\cdot)^{\mathcal{R}}$, of the interpretation
structure $\mathcal{R}$, is defined as follows.
* •
Each sort $\mathsf{S}\in\\{\mathsf{Set},\mathsf{O}\\}$ is mapped to the domain
$D_{\mathsf{S}}$.
* •
For each sort $\mathsf{S}\in\\{\mathsf{Set},\mathsf{O}\\}$, each variable $x$
of sort $\mathsf{S}$ is mapped to an element $x^{\mathcal{R}}$ in
$D_{\mathsf{S}}$.
The constant and function symbols in $\mathcal{F}_{S}$ are interpreted as
follows:
* •
$\emptyset$ as the empty set;
* •
$\\{x\mathbin{\scriptstyle\sqcup}A\\}$ as the set $\\{x^{\mathcal{R}}\\}\cup
A^{\mathcal{R}}$; and
* •
$A\times B$ as the set $A^{\mathcal{R}}\times B^{\mathcal{R}}$.
The predicate symbols in $\Pi$ are interpreted as follows:
* •
$x=y$ as $x^{\mathcal{R}}=y^{\mathcal{R}}$;
* •
$x\in A$ as $x^{\mathcal{R}}\in A^{\mathcal{R}}$;
* •
$\mathit{un}(A,B,C)$ as $C^{\mathcal{R}}=A^{\mathcal{R}}\cup B^{\mathcal{R}}$;
* •
$A\parallel B$ as $A^{\mathcal{R}}\cap B^{\mathcal{R}}=\emptyset$;
* •
$\mathit{set}(x)$ as $x^{\mathcal{R}}\in D_{\mathsf{Set}}$;
* •
$\mathit{pair}(x)$ as $x^{\mathcal{R}}\in\\{(a,b):a,b\in D\\}$;
* •
$\mathit{rel}(R)$ as $R^{\mathcal{R}}\subset\\{(a,b):a,b\in D\\}$;
* •
$\mathit{id}(A,R)$ as $R^{\mathcal{R}}=\mathop{\mathrm{id}}A^{\mathcal{R}}$;
* •
$\mathit{inv}(R,S)$ as $S^{\mathcal{R}}=(R^{\mathcal{R}})^{\smallsmile}$;
* •
$\mathit{comp}(R,S,T)$ as $T^{\mathcal{R}}=R^{\mathcal{R}}\circ
S^{\mathcal{R}}$; and
* •
any symbol $\pi^{\prime}$ in
$\\{\neq,\notin,\mathit{nrel},\mathit{nset},\mathit{npair}\\}$ is interpreted
as $\lnot\pi$ for the corresponding symbol $\pi$ in
$\\{=,\in,\mathit{rel},\mathit{set},\mathit{pair}\\}$, where $\lnot$ is
logical negation. ∎
The interpretation structure $\mathcal{R}$ is used to evaluate each
$\mathcal{RIS}$-formula $\Phi$ into a truth value
$\Phi^{\mathcal{R}}=\\{\mathit{true},\mathit{false}\\}$ in the following way:
$\mathcal{RIS}$-constraints are evaluated by $(\cdot)^{\mathcal{R}}$ according
to the meaning of the corresponding predicates in set theory as defined above;
$\mathcal{RIS}$-formulas are evaluated by $(\cdot)^{\mathcal{R}}$ according to
the rules of propositional logic.
In particular, observe that equality between two set terms is interpreted as
the equality in $D_{\mathsf{Set}}$; that is, as set equality between
hereditarily finite hybrid sets. Such equality is regulated by the standard
_extensionality axiom_ , which has been proved to be equivalent, for
hereditarily finite sets, to the following equational axioms [36]:
$\displaystyle\\{x,x\mathbin{\scriptstyle\sqcup}A\\}=\\{x\mathbin{\scriptstyle\sqcup}A\\}$
$\displaystyle\\{x,y\mathbin{\scriptstyle\sqcup}A\\}=\\{y,x\mathbin{\scriptstyle\sqcup}A\\}.$
###### Note A.9.
$\mathcal{L}_{\mathcal{BR}}$ can be extended to support other set and
relational operators definable by means of suitable
$\mathcal{L}_{\mathcal{BR}}$ formulas. Dovier et al. [36] proved that symbols
in $\Pi_{\mathit{S}}$ are sufficient to define constraints implementing the
set operators $\cap$, $\subseteq$ and $\setminus$. Cristiá and Rossi extend
that result in [10] showing that symbols in
$\Pi_{\mathit{S}}\cup\Pi_{\mathit{R}}$ are sufficient to define constraints
implementing all the operators defined in Note 2.
As any of these constraints can be replaced by its definition, we can
completely ignore the presence of them in $\mathcal{L}_{\mathcal{BR}}$
formulas. ∎
###### Note A.10 (Negation).
The negated versions of both set and relational constraints can be introduced
as $\mathcal{L}_{\mathcal{BR}}$ formulas [36, 10]. For example, $\lnot
R=S^{\smallsmile}$ is introduced as the following ${\cal BR}$-formula:
$\begin{split}((x&,y)\in R\land(y,x)\notin S){}\lor{}((x,y)\notin
R\land(y,x)\in S)\\\ &{}\lor{}\mathit{nrel}(R)\lor\mathit{nrel}(S)\end{split}$
($x$ and $y$ are implicitly existentially quantified). Thanks to the
availability of negative constraints, (general) logical negation is not
strictly necessary in $\mathcal{L}_{\mathcal{BR}}$. ∎
Data Availability Statement The data underlying this article are available in
Dropbox at https://www.dropbox.com/s/c6z45thxlvr1q1h/setlogITP.zip?dl=0, and
can be accessed with the URL just given.
|
# High-Energy Neutrino Production in Clusters of Galaxies
Saqib Hussain, ${}^{\href https://orcid.org/0000-0002-0458-0490~{}^{1}}$
Rafael Alves Batista, ${}^{\href
https://orcid.org/0000-0003-2656-064X~{}^{2}}$ Elisabete M. de Gouveia Dal
Pino, ${}^{\href https://orcid.org/0000-0001-8058-4752~{}^{3}}$ and Klaus
Dolag, ${}^{\href https://orcid.org/0000-0003-1750-286X~{}^{4,5}}$
1,3 Institute of Astronomy, Geophysics and Atmospheric Sciences (IAG),
University of São Paulo (USP), São Paulo, Brazil
2 Radboud University Nijmegen, Department of Astrophysics/IMAPP, 6500 GL
Nijmegen, The Netherlands
4University Observatory Munich, Scheinerstr. 1, 81679 Munchen, Germay
5Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str 1, 85741
Garching, Germany E-mail<EMAIL_ADDRESS>(SH)E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Clusters of galaxies can potentially produce cosmic rays (CRs) up to very-high
energies via large-scale shocks and turbulent acceleration. Due to their
unique magnetic-field configuration, CRs with energy $\leq 10^{17}$ eV can be
trapped within these structures over cosmological time scales, and generate
secondary particles, including neutrinos and gamma rays, through interactions
with the background gas and photons. In this work we compute the contribution
from clusters of galaxies to the diffuse neutrino background. We employ three-
dimensional cosmological magnetohydrodynamical simulations of structure
formation to model the turbulent intergalactic medium. We use the distribution
of clusters within this cosmological volume to extract the properties of this
population, including mass, magnetic field, temperature, and density. We
propagate CRs in this environment using multi-dimensional Monte Carlo
simulations across different redshifts (from $z\sim 5$ to $z=0$), considering
all relevant photohadronic, photonuclear, and hadronuclear interaction
processes. We find that, for CRs injected with a spectral index
$\alpha=1.5-2.7$ and cutoff energy $E_{\text{max}}=10^{16}-5\times
10^{17}\;\text{eV}$, clusters contribute to a sizeable fraction to the diffuse
flux observed by the IceCube Neutrino Observatory, but most of the
contribution comes from clusters with $M\gtrsim 10^{14}\;M_{\odot}$ and
redshift $z\lesssim 0.3$. If we include the cosmological evolution of the CR
sources, this flux can be even higher.
###### keywords:
galaxies: clusters: intracluster medium, neutrinos, magnetic fields
††pubyear: 2020††pagerange: High-Energy Neutrino Production in Clusters of
Galaxies–LABEL:lastpage
## 1 Introduction
The IceCube Neutrino Observatory reported evidence of an isotropic
distribution of neutrinos with $\sim$ PeV energies (Aartsen et al., 2017,
2020). Their origin is not known yet, but the isotropy of the distribution
suggests that they are predominantly of extragalactic origin. They might come
from various types of sources, such as galaxy clusters (Murase et al., 2013;
Hussain et al., 2019), starbursts galaxies, galaxy mergers, AGNs (Murase et
al., 2013; Kashiyama & Mészáros, 2014; Anchordoqui et al., 2014; Khiali & de
Gouveia Dal Pino, 2016; Fang & Murase, 2018), supernova remnants (Chakraborty
& Izaguirre, 2015; Senno et al., 2015), gamma-ray bursts (Hümmer et al., 2012;
Liu & Wang, 2013). Since neutrinos can reach the Earth without being deflected
by magnetic fields or attenuated due to any sort of interaction, they can help
to unveil the sources of ultra-high-energy cosmic rays (UHECRs) that produce
them.
Their origin and that of the diffuse gamma-ray emission are among the major
mysteries in astroparticle physics. The fact that the observed energy fluxes
of UHECRs, high-energy neutrinos, and gamma rays are all comparable suggests
that these messengers may have some connection with each other (Ahlers &
Halzen, 2018; Alves Batista et al., 2019a; Ackermann et al., 2019). The three
fluxes could, in principle, be explained by a single class of sources (Fang &
Murase, 2018), like starburst galaxies or galaxy clusters (e.g., Murase et
al., 2008; Kotera et al., 2009; Alves Batista et al., 2019a, for reviews ).
Clusters of galaxies form in the universe possibly through violent processes,
like accretion and merging of smaller structures into larger ones (Voit,
2005). These processes release large amounts of energy, of the order of the
gravitational binding energy of the clusters ($\sim
10^{61}-10^{64}~{}\text{erg}$). Part of this energy is depleted via shock
waves and turbulence through the intracluster medium (ICM), which accelerate
CRs to relativistic energies. These can be also re-accelerated by similar
processes in more diffuse regions of the ICM, including relics, halos,
filaments, and cluster mergers (e.g., Brunetti & Jones, 2014; Brunetti &
Vazza, 2020, for reviews). Furthermore, clusters of galaxies are attractive
candidates for UHECR production due to their extended sizes ($\simeq$ Mpc) and
suitable magnetic field strength ($\sim 1\;\mu\text{G}$) (e.g., Fang & Murase,
2018; Kim et al., 2019). Those with energies $E>7\times 10^{18}$ eV have most
likely an extragalactic origin (e.g., Aab et al., 2018; Alves Batista et al.,
2019b), and those with $E\lesssim 10^{17}$ eV are believed to have Galactic
origin (see e.g., Blasi, 2013; Amato & Blasi, 2018), although the exact
transition between galactic and extragalactic CRs is not clear yet (see e. g.,
Aloisio et al., 2012; Parizot, 2014; Giacinti et al., 2015; Thoudam et al.,
2016; Kachelriess, 2019).
CRs with $E\lesssim 10^{17}$ eV can be confined within clusters for a time
comparable to the age of the universe (e.g. Hussain et al., 2019). This
confinement makes clusters efficient sites for the production of secondary
particles including, electron-positron pairs, neutrinos and gamma rays due to
their interaction with the thermal protons and photon fields (e.g. Berezinsky
et al., 1997; Rordorf et al., 2004; Kotera et al., 2009). Non-thermal radio to
gamma-ray and neutrino observations are, therefore, the most direct ways of
constraining the properties of CRs in clusters (Berezinsky et al., 1997; Wolfe
& Melia, 2008; Yoast-Hull et al., 2013; Zandanel et al., 2015). Conversely,
the diffuse flux of gamma rays and neutrinos depend on the energy budget of CR
protons in the ICM. Clusters also naturally can introduce a spectral softening
due to the fast escape of high-energy CRs from the magnetized environment
which might explain the second knee that appears around $\sim 10^{17}$ eV, in
the CR spectrum (Apel et al., 2013).
To calculate the fluxes of CRs and secondary particles from clusters, there
are many analytical and semi-analytical works (Berezinsky et al., 1997; Wolfe
& Melia, 2008; Murase et al., 2013), but in most of the approaches, the ICM
model is overly simplified by assuming, for instance, uniform magnetic field
and gas distribution. There are more realistic numerical approaches in Rordorf
et al. (2004) and Kotera et al. (2009) exploring the three-dimensional (3D)
magnetic fields of clusters. More recently, Fang & Olinto (2016) estimated the
flux of neutrinos from these objects assuming an injected CR spectrum $\propto
E^{-1.5}$, an isothermal gas distribution, a radial profile for the total
matter (baryonic and dark) density profile, and a Kolmogorov turbulent
magnetic field with coherence length $\sim 100\;\text{kpc}$. They found these
estimates to be comparable to IceCube measurements. Here we revisit these
analyses by employing a more rigorous numerical approach. We take into account
the non-uniformity of the gas density and magnetic field distributions in
clusters, as obtained from MHD simulations. We consider additional factors
such as the location of CR sources within a given cluster, and the obvious
mass dependence of the physical properties of clusters. This last
consideration is important because massive clusters ($\gtrsim
10^{15}\;M_{\odot}$) are much less common than lower-mass ones ($\lesssim
10^{13}\;M_{\odot}$). Consequently, clusters that can confine CRs of energy
above PeV for longer are probably more relevant for detection of high-energy
neutrinos.
Our main goal is to derive the contribution of clusters to the diffuse flux of
high-energy neutrinos. To this end, we follow the propagation and cascading of
CRs and their by-products in the cosmological background simulations by Dolag
et al. (2005). We use the Monte Carlo code CRPropa (Alves Batista et al.,
2016) that accounts for all relevant photohadronic, photonuclear, and
hadronuclear interaction processes. Ultimately, we obtain the CR and neutrino
fluxes that emerge from the clusters.
This paper is organized as follows: in section 2 we describe the numerical
setup for both the cosmological background simulations and for CR propagation
through this environment; in section 3 we characterize the 3D-MHD simulations
and present our results for the fluxes of CRs and neutrinos; in section 4 we
discuss our results; finally, in section 5 we draw our conclusions.
## 2 Numerical Method
### 2.1 Background MHD Simulation
To study the propagation of CRs in the ICM we consider the large scale
cosmological 3D-MHD simulations performed by Dolag et al. (2005), who employed
the Lagrangian smoothed particle hydrodynamics (SPH) code GADGET (Springel et
al., 2001; Springel, 2005). These simulations capture the essential features
of the mass, temperature, density, and magnetic field distributions in galaxy
clusters, filaments and voids.
Figure 1: This figure shows the temperature (upper panel) and magnetic field
(lower panel) for one of the eight regions of our background 3D-MHD
cosmological simulation at redshift $z=0.01$, with dimension
$240~{}\text{Mpc}^{3}$, performed by Dolag et al. (2005).
We consider here seven snapshots of these simulations with redshifts
$z=0.01;\;0.05;\;0.2;\;0.5;\;0.9;\;1.5;\;5.0$, each having the same volume
$(240\;\text{Mpc})^{3}$. We have divided the domain of each snapshot into
eight regions. Fig. 1 shows the temperature and magnetic-field distributions
for one of the regions, at redshift $z=0.01$.
The filaments in Fig. 1 are populated with galaxy clusters and have dimensions
$\sim 50\;\text{Mpc}^{3}$, while the voids have dimensions of the same order,
which are compatible with observations (e.g., Govoni et al., 2019; Gouin et
al., 2020). In this simulation, the comoving intensity of the seed magnetic
field was chosen to be $B=2\times 10^{-12}\;\text{G}$, which leads to a quite
reasonable match with the field strength observed in different clusters of
galaxies today. Feedback and star formation were not included in these
cosmological simulations. The background cosmological parameters assumed are
$h\equiv H_{0}/(100\;\text{km~{}s}^{-1}~{}\text{Mpc}^{-1})=0.7$,
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and the baryonic fraction
$\Omega_{b}/\Omega_{m}=14\;\%$.
### 2.2 Simulation Setup for Cosmic Rays
The simulations described in the previous section provide the background
magnetic field, gas density and temperature distributions of the ICM. In order
to study the CR propagation in this environment, we employ the CRPropa 3 code
(Alves Batista et al., 2016), with stochastic differential equations (Merten
et al., 2017).
In these simulations, we assume that CRs are composed only by protons. We
consider all relevant interactions during their propagation including
photohadronic, photonuclear, and hadronuclear processes, namely photopion
production, photodisintegration, nuclear decay, proton-proton (pp)
interactions, and adiabatic losses due to the expansion of the universe. The
cosmic microwave background radiation (CMB) and the extragalactic background
light (EBL) are two essential ingredients, but other contributions comes from
the hot gas component of the ICM, of temperatures between $\sim 10^{6}-10^{8}$
K, that produces bremsstrahlung radiation (Rybicki & Lightman, 2008) and
serves as target for pp-interactions. This is calculated in section 3.1.
#### 2.2.1 Cosmic-ray Propagation
To investigate the flux of different particle species and the change of their
energy spectrum, we use the Parker transport equation, which is a simplified
version of the Fokker-Planck equation. It gives a good description of the
transport of CRs for an isotropic distribution in the diffuse regime. It is
given by:
$\frac{\partial n}{\partial t}+\vec{u}.\nabla n=\nabla.(\hat{\kappa}\nabla
n)+\frac{1}{p^{2}}\frac{\partial}{\partial
p}\left(p^{2}\kappa_{pp}\frac{\partial n}{\partial
p}\right)+\frac{1}{3}(\nabla\vec{u})\frac{\partial n}{\partial\ln
p}+S(\vec{x},p,t).$ (1)
Here $\vec{u}$ is the advection speed, $\hat{\kappa}$ is the spatial diffusion
tensor, $p$ is the absolute momentum, $\kappa_{pp}$ is the diffusion
coefficient of momentum used to describe the reacceleration, n is the particle
density, $\vec{x}$ gives position and $S(\vec{x},p,t)$ is the source of CRs
(distribution of CRs at the source).
Propagation of CRs can be diffusive or semi-diffusive, depending on the Larmor
radius ($r_{\text{L}}=1.08E_{15}/B_{\mu\text{G}}$ pc) of the particles and the
magnetic field of the ICM. The diffusive regime corresponds to
$r_{\text{L}}\ll R_{\text{cluster}}$, and the semi-diffusive is for
$r_{\text{L}}\gtrsim R_{\text{cluster}}$, wherein $R_{\text{cluster}}$ is the
radius of the cluster, typically $\sim 1\;\text{Mpc}$. Because
$B\sim\mu\text{G}$, for the energy range of interest
($10^{14}-10^{19}\;\text{eV}$), $r_{\text{L}}\ll R_{\text{cluster}}$, so we
are in the diffusive regime. CRs in this energy range would be confined
completely by the magnetic field of the clusters for a time longer than the
Hubble time ($t_{\text{H}}\sim 14$ Gyr) (e.g. Fang & Murase, 2018). For
instance, a CR with energy $\sim 10^{17}$ eV in a cluster of mass $\sim
10^{14}\;M_{\odot}$ with central magnetic field strength $\sim
10^{-6}\;\mu\text{G}$ has $r_{\text{L}}\sim 0.1$ kpc much smaller than the
size of the cluster ($\sim 2$ Mpc) and the trajectory length of this CRs
inside the cluster is $\sim 10^{3}$ Mpc. The confinement time for this CR can
be calculated as $t_{\text{con}}\sim 1000~{}\text{Mpc}/c\;\sim t_{\text{H}}$
(e.g. Hussain et al., 2019). Hence, CRs with energy $E>10^{17}\;\text{eV}$
have more chances to escape the magnetized cluster environment. The flux of
CRs that can escape a cluster depends on its mass and magnetic-field profile,
with the latter directly correlated with the density distribution, being
larger in denser regions.
## 3 Results
### 3.1 Cosmological Background
Our background simulation includes seven snapshots in the redshift range
$0.01<z<5.0$. We have identified clusters in the densest regions of the
isocontour maps of the whole volume, in each snapshot (see Fig. 1). We then
selected five clusters with distinct masses ranging from $10^{12}$ to
$10^{16}\;M_{\odot}$, which we assumed to be representative of all the
clusters in the corresponding snapshot. Finally, we injected CRs in each of
these clusters to study their propagation and production of secondary
particles. As an example, Fig. 2 illustrates relevant properties for two of
these clusters with masses $\sim 10^{14}\;M_{\odot}$ (left panel) and $\sim
10^{15}\;M_{\odot}$ (right panel) at redshift $z=0.01$. To estimate the total
mass of a cluster from the simulations, we integrated the baryonic and dark
matter densities within a volume of $2$ Mpc, assuming an approximate spherical
volume. We note that this specific evaluation is not much affected by the
deviations from spherical symmetry that we detect in Fig. 2.
Figure 2: Maps of gas density (left column), magnetic field (middle column),
temperature (right column) of two clusters of masses $\sim 10^{14}M_{\odot}$
(upper panels) and $\sim 10^{15}M_{\odot}$ (bottom panels), at redshift
$z=0.01$.
To illustrate general average properties of the simulated clusters, we
converted the Cartesian into spherical coordinates and divided the cluster in
$10$ concentric spherical shells of different radii ($R_{\text{shell}}$).
Starting from the center of the cluster, the shells were first divided in
intervals of $100\;\text{kpc}$, then between $300\;\text{kpc}$ and
$1500\;\text{kpc}$, they were divided in intervals of $200\;\text{kpc}$, and
the last shell in the outskirts was taken between
$1500\;\text{kpc}<r<2000\;\text{kpc}$.
Fig. 3 depicts volume-averaged profiles of different quantities as a function
of the radial distance for a cluster of mass $\sim 10^{15}\;M_{\odot}$ at four
different redshifts. The overdensity in Fig. 3 (bottom-right panel) is defined
as $\Delta=\rho(r)/\rho_{\text{bary}}$, where $\rho(r)$ is the total density
at a given point and $\rho_{\text{bary}}$ is the mean baryonic density,
$\rho_{\text{bary}}=\Omega_{\text{bary}}\times\rho_{\text{crit}}$,
$\rho_{\text{crit}}=3H^{2}/8\pi\text{G}$. We see that, in general, these
radial profiles are very similar across the cosmological time, except for the
temperature that varies non-linearly with time by about four orders of
magnitude in the inner regions of the cluster. Fig. 4 shows profiles for the
temperature, gas density, magnetic field and overdensity for a cluster of mass
$\sim 10^{15}\;M_{\odot}$, as a function of the azimuthal ($\phi$) angle for
different latitudes ($\theta$), within a radial distance of
$R=300\;\text{kpc}$, at a redshift $z=0.01$. We see that there are substantial
variations in the angular distributions of all the quantities. These
variations characterize a deviation from spherical symmetry that may affect
the emission pattern of the CRs and consequently secondary gamma rays and
neutrinos.
We also found that the magnetic field strength of a cluster depends on its
mass: the heavier the cluster, the stronger the average magnetic field is, due
to the larger extension of denser regions (see middle column of Fig. 2 and
Fig. 5). Inside all clusters, magnetic fields vary in the range
$10^{-8}<B/\text{G}<10^{-5}$ (see also Dolag et al., 2005; Ferrari et al.,
2008; Xu et al., 2009; Brunetti & Jones, 2014; Brunetti et al., 2017; Brunetti
& Vazza, 2020).
Figure 3: Volume-averaged profiles as a function of the radial distance from
the center for a cluster of mass $M\sim 10^{15}\;M_{\odot}$, at four different
redshifts. The quantities shown are: dark-matter mass (top left); gas number
density (top center); gas mass (top right); magnetic field (bottom left);
temperature (bottom-center) and overdensity (bottom right). Figure 4:
Volume-averaged profiles as a function of the azimuthal ($\phi$) angle for
different latitudes ($\theta$), within a radial distance $R=300$ kpc from the
center, for a cluster of mass $M\sim 10^{15}\;M_{\odot}$. From top left to
bottom right clockwise, temperature, gas number density, overdensity and
magnetic field.
Figure 5: Upper panel shows the whole volume-averaged value of the magnetic
field as a function of the cluster mass. Lower panel compares the volume-
averaged magnetic field as a function of the radial distance for clusters of
different masses.
In the upper panel of Fig. 6, we compare the radial density profile of our
simulated cluster of mass $10^{15}\;M_{\odot}$ with the model used by Fang &
Olinto (2016). We see that both profiles look similar up to $\sim
10^{3}\;\text{kpc}$. Above this scale, the density distribution of our
simulated clusters decays much faster than the assumed distribution in Fang &
Olinto (2016).
Figure 6: Comparison of the density profile of a cluster of mass
$10^{15}\;M_{\odot}$, from our simulation with the model used by Fang & Olinto
(2016), given in the upper panel. The lower panel shows the number of clusters
per mass interval in our background simulation for different redshifts.
To estimate the total flux of CRs and neutrinos, we need to evaluate the total
number of clusters in our background simulations as a function of their mass,
at different redshifts. From the entire simulated volume,
($240\;\text{Mpc})^{3}$, we selected $20$ sub-samples of
$(20\;\text{Mpc})^{3}$ from different regions, as representative of the whole
background. We then calculated the average number of clusters per mass
interval in each of these sub-samples ($dN_{\text{clusters, avg}}/dM$),
between $10^{12}\;M_{\odot}$ and $10^{16}\;M_{\odot}$. To obtain the total
number of clusters per mas interval we multiplied this quantity by the number
of intervals $N=(240\;\text{Mpc})^{3}/(20\;\text{Mpc})^{3}$ in which the whole
volume was divided. So, the total number of clusters per mass interval was
calculated as $(dN_{\text{clusters,\; avg}}/dM)\times N$. Since we have seven
redshifts in our cosmological background simulations,
$z=0.01,\;0.05,\;0.2,\;0.5,\;0.9,\;1.5,\;5.0$, we then have repeated the
calculation above for each snapshot to obtain the number of clusters per mass
interval at different redshifts. This is shown in the lower panel of Fig. 6
for different redshifts.
To calculate the photon field of the ICM, we assume that the clusters are
filled with photons from Bremsstrahlung radiation of the hot, rarefied ICM gas
(see Figs. 1 to 4). For typical temperatures and densities, we can further
assume an optically thin gas. Taking a photon density ($n_{\text{ph}}$)
distribution with approximately spherical symmetric within the cluster, we
have the following relations for an optically thin gas (Rybicki & Lightman,
2008):
$\frac{dn_{\text{ph}}}{d\epsilon}=\frac{4\pi
I_{\nu}}{ch\epsilon},\;\;\;\;\;I_{\nu}=R_{\text{shell}}~{}J_{\nu}^{\text{ff}}.$
(2)
where $I_{\nu}$ is the specific intensity of the emission, $c$ is the speed of
light, $h$ is the Planck constant, $\epsilon$ is the photon energy,
$R_{\text{shell}}$ is the radius of concentric spherical shells, and
$J_{\nu}^{\text{ff}}$ is related with the Bremsstrahlung emission coefficient:
$4\pi J_{\nu}^{\text{ff}}=\epsilon_{\nu}^{\text{ff}}(\nu,n,T)=6.8\times
10^{-38}Z^{2}n_{e}n_{i}T^{-1/2}e^{-h\nu/k_{B}T},$ (3)
which is given in units of
$\text{erg}\;\text{cm}^{-3}\;\text{s}^{-1}\;\text{Hz}^{-1}$.
In Fig. 7 we compare the radiation fields for two EBL models (Gilmore et al.,
2012; Dominguez et al., 2011) with the Bremsstrahlung photon fields of two
clusters of masses $\sim 10^{15}\;M_{\odot}$ (cluster$\;1$) and $\sim
10^{14}\;M_{\odot}$ (cluster$\;2$). For both clusters, we calculated the
internal photon field at the center ($R<100\;\text{kpc}$) and for the
($700<R\;/\;\text{kpc}<900$). It can be seen that the Bremsstrahlung photon
field is dominant at X-rays, but only near the center of the clusters, while
the EBL dominates at infrared and optical wavelengths mainly.
Figure 7: Comparison of EBL with the Bremsstrahlung radiation of the ICM as a
function of the photon energy. The Bremsstrahlung is calculated for two
clusters at different radial distance intervals. Cluster$\;1$ has mass
$10^{15}\;M_{\odot}$, and Cluster$\;2$ , $10^{14}\;M_{\odot}$.
The interaction rates of CRs with the Bremsstrahlung photon fields in each
shell were also calculated (see Fig. 8 and appendix A) and implemented in
CRPropa. We note that though the assumption of spherical symmetry for
evaluating the Bremsstrahlung radiation and its interaction rate with CRs
seems to be in contradiction with the results of Fig. 4, our computation of
these quantities in CRPropa have revealed no significant contribution of the
Bremsstrhalung photons to neutrinos production. Indeed, the upper panel of
Fig. 8 indicates that the $\lambda$ for these interactions is larger than the
Hubble horizon. Thus deviations from spherical symmetry for this photon field
will not be relevant in this study.
We have also implemented the proton-proton (pp) interactions using the spatial
dependent density field extracted directly from the background cosmological
simulations, using the same procedure described by Rodríguez-Ramírez et al.
(2019). We further notice that, for the computation of the CR fluxes, the
magnetic field distribution has been also extracted directly from the
background simulations, without considering any kind of space symmetry.
### 3.2 Mean free paths for different CR interactions
CRPropa 3 employs a Monte Carlo method for particle propagation and previously
loaded tables of the interaction rates in order to calculate the interaction
of CRs with photons along their trajectories. We implemented the spatially-
dependent interaction rates into the code, based on the gas and photon density
distributions for the clusters of different masses. The mean free paths
($\lambda$) for the different interactions of CRs are described in appendix A.
The values of $\lambda$ for all the interactions of CRs with the background
photon fields and the gas, are plotted in the upper panel of Fig. 8. For
photopion production, we compare $\lambda$ due to interactions with the photon
fields (i.e., the Bremsstrahlung radiation, red solid line) of a cluster of
mass $10^{15}\;M_{\odot}$ with the EBL (red dotted line) and the CMB (red
dashed line). For the Bremsstrahlung radiation, we considered only the photons
within a sphere of radius $100\;\text{kpc}$ around the center of the cluster
(i.e., the densest region, which is shown in Figs. 2 & 4). High-energy CR
interactions with CMB photons is a well-understood process that limits the
distance from which CRs can reach Earth leading to the GZK cutoff. The upper
panel of Fig. 8 shows that $\lambda$ for this interaction is much smaller than
that for the EBL and Bremsstrahlung. So, CR interactions with CMB photons
dominate at energies $E\gtrsim 10^{17}\;\text{eV}$. We also see that $\lambda$
for Bremsstrahlung is greater than the size of the universe ($\sim 10^{6}$
Mpc), and for EBL, it is $\sim 10^{3}\;\text{Mpc}$. The $\lambda$ for pp-
interactions (green line) is much less than the Hubble horizon. Therefore,
this kind of interaction is more likely to occur than photopion production
specially at energies $<10^{17}$ eV. Upper panel of Fig. 8 also shows that we
can neglect the CR interactions with the local Bremsstrahlung photon field, as
well as the interaction of high-energy gamma rays with the local gas of the
ICM (yellow) in photopion production.
The lower panel of Fig. 8 shows the distribution of the trajectory lengths
(total distance travelled by a CR inside the cluster up to the observation
time), for different energy bins of CRs. There is a substantial number of
events with trajectory length greater than $D\gtrsim 10^{3}$ Mpc for each
energy bin. Thus, the trajectory lengths of CRs are comparable to the mean
free paths of pp-interactions and photopion production in the CMB and EBL
case, so that these interactions can produce secondary particles including
gamma rays and neutrinos.
Figure 8: The upper panel shows the mean-free path $\lambda$ for CR
interactions which produce neutrinos. It is shown $\lambda$ for photopion
production in the bremsstrahung photon field (red solid line), CMB (red dashed
line) and EBL (red dotted line). Also shown is $\lambda$ for pp-interactions
(green) calculated within a sphere of radius $r=100$ kpc around the center of
a massive cluster (with mass $10^{15}\;M_{\odot}$ and shown in Fig. 2). The
$\lambda$ for the interaction of high-energy gamma rays with the local gas of
the ICM (yellow) is also depicted. The thick black line represents the Hubble
horizon in the upper panel. The lower panel shows the distribution of the
total trajectory length of CRs inside the cluster as a function of their
energy bins.
### 3.3 CR Flux Calculation
To study the propagation of CRs in the diffuse ICM, we used the transport
equation as implemented in CRPropa 3 by (Merten et al., 2017, see also
equation 1). There are three possible scenarios in CRPropa3 for each particle
until its detection: the particle reaches the detector within a Hubble time;
the energy of the particle becomes smaller than a given threshold; or the
trajectory length of a CR exceeds the maximum propagation distance allowed.
We inject CRs isotropically with a power-law energy distribution with spectral
index $\alpha$ and exponential cut-off energy $E_{\text{max}}$ which follows
the relation $dN_{\text{CR},E}/dE\propto
E_{i}^{-\alpha}\exp(-E_{i}/E_{\text{max}})$ (see Appendix B). We take
different values for $\alpha\simeq 1.5-2.7$, and for $E_{\text{max}}=5\times
10^{15}-10^{18}$ eV (e.g. Brunetti & Jones, 2014; Fang & Olinto, 2016;
Brunetti et al., 2017; Hussain et al., 2019, for review).
As stressed, the lower and upper limits of the mass of the galaxy clusters are
taken to be $10^{12}\ M_{\odot}$ and $10^{16}\;M_{\odot}$, respectively. This
is because for $10^{14}\lesssim E/\text{eV}\lesssim 10^{19}$, clusters with
mass $M<10^{12}\;M_{\odot}$ barely contribute to the total flux of neutrino,
due to low gas density, while there are few clusters with $M\gtrsim
10^{15}\;M_{\odot}$ at high redshifts ($z>1.5$) (Komatsu et al., 2009; Ade et
al., 2014). The closest galaxy clusters are located at $z\sim 0.01$, so we
consider the redshift range $0.01\leq z\leq 5.0$.
The amount of power of the clusters that goes into CR production is left as a
free parameter to be regulated by the observations (e.g. Gonzalez et al.,
2013; Brunetti & Jones, 2014; Fang & Olinto, 2016, for reviews). We here
assume that about $0.5-3\;\%$ of the cluster luminosity is available for
particle acceleration.
We did not consider the feedback from active galactic nuclei (AGN) or star
formation rate (SFR) in our background cosmological simulations (as performed
e.g. in Barai et al., 2016; Barai & de Gouveia Dal Pino, 2019). AGN are
believed to be the most promising CR accelerators inside clusters of galaxies
and star-forming galaxies contain many supernova remnants that can also
accelerate CRs up to very-high energies ($E\gtrsim 100\;\text{PeV}$) (He et
al., 2013). AGN are more powerful and more numerous at higher redshifts
(Hasinger et al., 2005; Khiali & de Gouveia Dal Pino, 2016; D’Amato et al.,
2020), and their luminosity density evolves more strongly for $z\gtrsim 1$.
Also, supernovae are more common at high redshifts (He et al., 2013; Moriya et
al., 2019). Therefore, it is reasonable to expect that, if high energy cosmic
ray (HECR) sources have a cosmological evolution similar to AGN or following
the star-formation rate (SFR), then the flux of neutrinos may be higher at
high redshifts due to the larger CR output from these objects.
For the evolution of AGN sources (Hopkins & Beacom, 2006; Heinze et al., 2016)
and SFR (Yüksel et al., 2008; Wang et al., 2011; Gelmini et al., 2012) we
consider the following parametrization:
$\psi_{\text{SFR}}(z)=\frac{1}{B}\begin{cases}(1+z)^{3.4}\;\;\text{if}\;z<1\\\
(1+z)^{-0.3}\;\;\text{if}\;1<z<4\\\ (1+z)^{-3.5}\;\;\text{if}\;z>4\\\
\end{cases}$ (4)
$\psi_{\text{AGN}}(z)=\frac{(1+z)^{m}}{A}\begin{cases}(1+z)^{3.44}\;\;\text{if}\;z<0.97\\\
10^{1.09}(1+z)^{-0.26}\;\;\text{if}\;0.97<z<4.48\\\
10^{6.66}(1+z)^{-7.8}\;\;\text{if}\;z>4.48\\\ \end{cases}$ (5)
where $A=360.6$ and $B=6.66$ are normalization constants in equations (5) and
(4), respectively. For AGN evolution $\psi_{AGN}(z)\propto(1+z)^{5}$, for low
redshift $z<1$ (Gelmini et al., 2012; Alves Batista et al., 2019b) and also
according to (Gelmini et al., 2012; Heinze et al., 2016), in equation (5),
$m>1.5$ for AGN, so we consider $m=1.7$. Typically, the luminosity of AGNs
ranges from $10^{42}$ to $10^{47}\;\text{erg/s}$ and their evolution depends
on their luminosities. The AGNs with luminosities $\sim
10^{44}-10^{46}\;\text{erg/s}$ are more important as they are more numerous
and believed to be able to accelerate particles to ultra-high energies (e.g.
Waxman, 2004; Khiali & de Gouveia Dal Pino, 2016). AGNs with luminosities
greater than $10^{46}\;\text{erg/s}$ are less numerous (Hasinger et al., 2005)
and their evolution function ($\psi_{\text{AGN}}(z)$) is different from
equation (5). For no source evolution, $\psi(z)=1$.
The total flux of CRs is estimated from the entire population of clusters. The
number of clusters per mass interval $dN/dM$ at redshift $z$ is given in the
lower panel of Fig. 6, which was obtained from our cosmological simulations.
It is related to the flux through:
$E^{2}\Phi(E)=\int\limits_{z_{\text{min}}}^{z_{\text{max}}}dz\int\limits_{M_{\text{min}}}^{M_{\text{max}}}dM\dfrac{dN}{dM}~{}E^{2}\dfrac{d\dot{N}(E/(1+z),M,z)}{dE}\left(\dfrac{\psi_{\text{ev}}(z)}{4\pi
d_{L}^{2}(z)}\right)$ (6)
where $\psi_{\text{ev}}(z)$ stands for, $\psi_{\text{SFR}}(z)$ and
$\psi_{\text{AGN}}(z)$, $\dot{N}$ is the number of CRs per time interval $dt$
with energies between $E$ and $E+dE$ that reaches the observer. The quantity
$E^{2}\;d\dot{N}/dE$ in equation (6) is the power of CRs calculated from our
propagation simulation and is several orders of magnitude smaller than the
luminosity of observed clusters (e.g., Brunetti & Jones, 2014).
In order to convert the code units of the CR simulation to physical units, we
have used a normalization factor (Norm). To calculate Norm, we first evaluate
the X-ray luminosity of the cluster using the empirical relation
$L_{\text{X}}\propto f_{g}^{2}\;M_{\text{vir}}$ (Schneider, 2014), where
$f_{g}=M_{g}/M_{\text{vir}}$ denotes the gas mass ($M_{g}$) fraction with
respect to the total mass of the cluster within the Virial radius
($M_{\text{vir}}$) and then, since we are assuming that $(0.5-3)\;\%$ of this
luminosity goes into CRs, this implies that
$\text{Norm}\sim(0.5-3)\;\%~{}L_{\text{X}}/L_{\text{CRsim}}$ and
$L_{\text{CRsim}}$ is the luminosity of the simulated CRs. Therefore, the CR
power that reaches the observer (at the Earth) is $\sim
E^{2}~{}d\dot{N}/dE\times\text{Norm}$. In equation (6) $d_{L}$ is the
luminosity distance, given by:
$d_{L}=(1+z)\dfrac{c}{H_{0}}\int\limits_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})},$
(7)
with
$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}=\frac{H(z)}{H_{0}},$ (8)
where the Hubble constant, as well as the matter ($\Omega_{m}$) and dark-
energy ($\Omega_{\Lambda}$) densities are defined in section 2.1, assuming a
flat $\Lambda$CDM universe.
We selected different injection points inside the clusters of different masses
in order to study the spectral dependence with the position, which may
correspond to different scenarios of acceleration of CRs. For instance, the
larger concentration of galaxies near the center must favor more efficient
acceleration, but compressed regions by shocks in the outskirts may also
accelerate CRs. The schematic diagram of the simulation of CRs propagation is
shown in Fig. 9. CRs are injected at three different positions within each
selected cluster denoted by $R_{\text{Offset}}$. The spectra of CRs have been
collected by an observer in a sphere of $2~{}\text{Mpc}$ radius
($R_{\text{Obs}}$), centred at the cluster, with a redshift window ($-0.1\leq
z\leq 0.1$) for all the injection points of CRs. All-flavour neutrino fluxes
are also computed at the same observer (see Section-3.4 below).
Figure 9: Scheme of the CR simulation geometry. They are injected at three
different positions inside each cluster represented by $R_{\text{Offset}}$,
and $R_{\text{Obs}}$ is the radius of the observer.
The spectrum of CRs obtained from our simulations is shown in Figs. 10 & 11.
Its dependence on the position where the CR source is located within the
cluster for $z=0.01$ is shown for three clusters of different masses in Fig.
10. Particles injected at $1~{}\text{Mpc}$ distance away from the clusters
center can leave them in short time, with almost no interaction, as both the
magnetic field and the gas number density are very low compared to the central
regions. On the other hand, CRs injected at the center or at
$300~{}\text{kpc}$ away from the cluster center can be easily deflected by the
magnetic field and trapped in dense regions. This explains the higher CR flux
for the injection point at $1~{}\text{Mpc}$ in Fig. 10. Also, because the
confinement of CRs in the central regions of the clusters is comparable to a
Hubble time, and because of the value of $\lambda$ for the relevant
interactions, the production of secondary particles including neutrinos and
gamma rays in the clusters is substantial, as we will see in section 3.4.
Figure 10: This figure shows the CR flux of individual clusters of distinct
masses, $M\sim 10^{15}$ (red); $10^{14}$ (green); and $M\sim
10^{13}~{}M_{\odot}$ (blue color). This diagram shows the flux of CRs, for
sources located at the center of the cluster (solid), at $300$ kpc (dashed),
and at $1$ Mpc (dash-dotted lines) away from the centre. The flux is computed
at the edge of the clusters. The spectral parameters are $\alpha=2$ and
$E_{\text{max}}=5\times 10^{17}$ eV, and it is assumed that $2\%$ of the
luminosity of the clusters is converted into CRs.
In Fig. 11 we show the CR spectrum of all the clusters at different redshifts
integrated up to the Earth. Although the spectra in this diagram have been
integrated up to the Earth, we have not considered any interactions of the CRs
with the background photon and magnetic fields during their propagation from
the edge of the clusters to the Earth. Though not quantitatively realistic, it
provides important qualitative information. One obvious result is that most of
the contribution in the CR flux comes from clusters at low redshifts. Moreover
there is a significant suppression in the flux of CRs at $\gtrsim 10^{17}$ eV,
which indicates the trapping of lower-energy CRs within the clusters (Alves
Batista et al., 2018).
Figure 11: This figure shows the total CR flux (at the Earth distance) from
all the clusters distributed in different redshifts: $z=0.01$ (blue); $z=0.05$
(orange); $z=0.2$ (green). The total CRs flux for the redshift range $0.01\leq
z\leq 0.3$ is given by the red dotted line.
### 3.4 Flux of Neutrinos
To calculate the neutrino flux, the CRPropa 3 code integrates a relation
similar to equation (6) for neutrino species, and the procedure is the same as
described in Section 3.3 .
In general, neutrino production occurs mainly due to photopion production and
pp-interactions. In Fig. 8, where we show $\lambda$ for different
interactions, we see that protons with energies $E<10^{17}$ eV produce
neutrinos principally due to pp-interactions, while for $E>10^{17}$ eV, they
produce neutrinos both, by pp-interactions and photopion process. We have also
seen in Fig. 8 (lower panel) that the total trajectory length of CRs inside a
cluster is comparable or larger than $\lambda$ for these interactions and
thus, neutrino production is inevitable.
In Fig. 12 we show the dependence of the neutrino flux with the position of
the corresponding CR source within clusters of different masses. As in the
case of the CR flux, it can be seen that there is less neutrino production for
the injection position at $1$ Mpc away from the center of the cluster.
Furthermore, massive clusters produce more neutrinos than the light ones. In
Fig. 13 we present the redshift distribution of neutrinos as a function of
their energy, as observed at a distance of $2$ Mpc from the center of
individual clusters with different masses.
Figure 12: This figure shows the neutrino flux of individual clusters of
distinct masses: $M\sim 10^{15}$ (red); $10^{14}$ (green) and
$10^{13}~{}M_{\odot}$ (blue color). The CR sources are located at the center
of the cluster (solid lines), at $300$ kpc (dashed lines), and at $1$ Mpc away
from the center (dash-dotted lines). The flux is computed at the edge of
clusters. The CR injection follows $dN/dE\propto E^{-2}$,
$E_{\text{max}}=5\times 10^{16}$ eV, and it is assumed that $2~{}\%$ of the
luminosity of the clusters is converted to CRs. Figure 13: Redshift
distribution of the neutrinos as a function of their energy, as observed at
$2$ Mpc away from the center of clusters with different masses.
In Fig. 14 & 15, we present the total flux of neutrinos from the whole
population of clusters, as measured at Earth, integrated over the entire
redshift range within the Hubble time (solid brown curve in the panels). In
the left panel of Fig. 14 and in Fig. 15, the injected CR spectrum is assumed
to follow $E^{-1.5}$, with an exponential cut-off $E_{\text{max}}=5\times
10^{16}\;\text{eV}$. Also, we assumed in these cases that $0.5\%$ of the
kinetic energy of the clusters is converted to the CRs. Besides the total
flux, this panel also shows the flux of neutrinos for several cluster mass
intervals. The softening effect at higher energies is due to the shorter
diffusion time of the CRs, and to the mass distribution of the clusters, as
higher flux reflects lower population of massive clusters. In Fig. 15 we
present the integrated flux in different redshift intervals and it can also be
seen that the clusters at high redshift contribute less to the total flux of
neutrinos. Those at $z>1$ barely contribute to the flux due to the low
population of massive clusters and their large distances. Fig. 14 and 15 also
compares our results with the IceCube observations. We see that for the
assumed scenario for CRs injection in left panel of Fig. 14 and in Fig. 15,
they can reproduce the IceCube observations for $E>20$ TeV. In right panel of
Fig. 14, instead, we have assumed that $2\;\%$ of the kinetic energy of the
clusters is converted into CRs, with a CR energy power-law spectrum $E^{-2}$,
with $E_{\text{max}}$ following the dependence below with the cluster mass and
magnetic field:
$E_{\text{max}}=2.8\times
10^{18}\left(\dfrac{M_{\text{cluster}}}{{10^{15}M_{\odot}}}\right)^{2/3}\left(\dfrac{B_{\text{cluster}}~{}\text{G}}{10^{-6}~{}\text{G}}\right)~{}\text{eV},$
(9)
which is similar to Fang & Olinto (2016). In this scenario we find that the
clusters contribution to the neutrino flux is smaller than IceCube
measurements.
For all diagrams of Fig. 14 & 15, we also compare our results with those of
Fang & Olinto (2016)) (blue lines). The total fluxes in both are similar, in
general.Moreover, we see that in both cases, the largest contribution to the
flux of neutrinos comes from the cluster mass group
$10^{14}~{}M_{\odot}<M<10^{15}~{}M_{\odot}$. However, the contribution from
the mass group $10^{12}~{}M_{\odot}<M<10^{14}~{}M_{\odot}$ in our results is a
factor twice larger than that of Fang & Olinto (2016), and smaller by the same
factor for the mass group $M>10^{15}~{}M_{\odot}$, at energies $E>0.01$ PeV
(left panel of Fig. 14).
A striking difference between the two results is that, according to Fang &
Olinto (2016), the redshift range $0.3\leq z\leq 1$ amounts for the largest
contribution to neutrino production, but in our case the redshift range
$0.01\leq z\leq 0.3$ provides a more significant contribution (see Fig. 15).
Besides, there is a difference of factor $\sim 2$ to $\sim 3$ between ours and
their results at these redshift ranges. This difference may be due to the more
simplified modeling of the background distribution of clusters in their case
specially for the lower mass group
($10^{12}~{}M_{\odot}<M<10^{14}~{}M_{\odot}$) at high redshifts ($z>1$).
Figure 14: Neutrino spectrum at Earth obtained using our simulations (brown
lines), compared with the IceCube data (markers), and Fang & Olinto (2016)
results (blue lines). The panels show the total flux integrated over all
clusters and redshifts between $0.01\leq z\leq 5$ (solid thick lines). The
left panel shows the neutrino spectra (thin blue and brown lines) for cluster
mass ranges of: $10^{12}~{}M_{\odot}<M<10^{14}~{}M_{\odot}$ (dash-dotted),
$10^{14}~{}M_{\odot}<M<10^{15}~{}M_{\odot}$ (dashed), and
$M>10^{15}~{}M_{\odot}$ (dotted lines). The left panel corresponds to the case
with $\alpha=1.5$ and $E_{\text{max}}=5\times 10^{16}$ eV, whereas in the
right panel $\alpha={-2}$ and $E_{\text{max}}$ follows equation (9). These
diagrams do not include the redshift evolution of the CR sources,
$\psi_{ev}=1$ in equation equation (6). Figure 15: This figure shows the
neutrino spectrum for different redshift ranges: $z<0.3$ (dotted lines),
$0.3<z<1.0$ (dashed), and $1.0<z<5.0$ (dash-dotted lines). The solid blue and
brown lines correspond to the total spectrum in Fang & Olinto (2016), and in
this work, respectively. The CR injection in this figure follows $dN/dE\propto
E^{-1.5}$, and $E_{\text{max}}=5\times 10^{16}\;\text{eV}$. This figure does
not include the redshift evolution of the CR sources, $\psi_{ev}=1$ in
equation equation (6).
In Fig. 16, we present the total neutrino spectra calculated for different
spectral indices of the injected CRs, while in Fig. 17 we show the total
neutrino spectra calculated for several cut-off energies. In order to try to
fit the observed IceCube data, we have considered a $3\;\%$ conversion of the
kinetic energy of the cluster into CRs in Figs. 16 & 17.
Figure 16: Total spectrum of neutrinos for different injected CR spectra,
$\sim E^{-\alpha}$, with $\alpha=1.5$ (blue), $1.9$ (orange), $2.3$ (green),
$2.7$ (red). We consider $E_{\text{max}}=5\times 10^{17}~{}\text{eV}$. This
figure does not include the redshift evolution of the CR sources,
$\psi_{ev}=1$ in equation equation (6).
Figure 17: Total neutrino spectrum for different cutoff energies i.e.,
$E_{max}=5\times 10^{15}$ (red), $10^{16}$ (green), $10^{17}$ (orange), and
$5\times 10^{17}~{}\text{eV}$ (blue). In the upper panel the spectral index is
$\alpha=2$, and in lower panel $\alpha=1.5$. This figure does not include the
redshift evolution of the CR sources, $\psi_{ev}=1$ in equation equation (6).
So far, we have computed the CR and neutrino fluxes from the clusters,
considering no evolution function with redshift for both CR sources, AGN and
SFR, i.e. we assumed $\psi_{\text{ev}}(z)=1$ in equation (6). In Fig. 18, we
have included these contributions and plotted the flux of neutrinos for the
redshift ranges: $z<0.3,\;0.3<z<1.0$, and $1.0<z<5.0$. The flux is obtained
for spectral index $\alpha=2$ and cutoff energy $E_{\text{max}}=5\times
10^{17}\;\text{eV}$.
Clusters can directly accelerate CRs through shocks, but any type of
astrophysical object that can produce HECRs can also contribute to the diffuse
neutrino flux. In the former case, the sources evolve only according to the
background MHD simulations, dubbed here “no evolution”, whereas in the latter
some assumptions have to be made regarding the CR sources. In Fig. 18 we
illustrate the impact of the source evolution. We consider, in addition to the
case wherein sources do not evolve, SFR and AGN-like evolutions (see equations
5 and 4 and accompanying discussion). Our results suggest that, while the
neutrino fluxes for the AGN and the SFR evolutions are relatively close to
each other, the case without evolution contributes slightly less to the total
flux. Moreover, at high redshifts ($1.0<z<5.0$), AGNs in clusters produce more
neutrinos than sources with SFR-like evolutions, whereas the same is not true
for $z\lesssim 1$.
Figure 18: Neutrino spectrum for different assumptions on the evolution of the
CR sources: SFR (blue), AGN (green), and no evolution (brown). The fluxes are
shown for different redshift ranges: $z<0.3$ (dotted lines), $0.3<z<1.0$
(dashed), and $1.0<z<5.0$ (dash-dotted lines). The CR injection spectrum has
parameters $\alpha=2$ and $E_{\text{max}}=5\times 10^{17}\;\text{eV}$.
In Fig. 19, we plotted the flux for different combinations of spectral index
$\alpha$ and $E_{max}$, with different source evolution assumptions as in Fig.
18. In both panels all the combinations of $\alpha$ and $E_{\text{max}}$ are
roughly matching with IceCube data, except $\alpha=1.5$, and $E_{max}=5\times
10^{17}\;\text{eV}$ in the upper panel as it overshoots the IceCube points.
Figure 19: Flux of neutrinos for different assumptions on the evolution of the
CR sources: no evolution (solid lines), SFR (dashed lines), AGN (dotted lines)
and AGN $+$ SFR (dash-dotted lines). In upper panel green and red lines
represent $\alpha=1.5$ for $E_{\text{max}}=10^{16}$ and $5\times
10^{17}\;\text{eV}$ respectively. In lower panel orange and blue lines
correspond to $\alpha=2$ for $E_{\text{max}}=10^{16}$ and $5\times
10^{17}\;\text{eV}$, respectively.
## 4 Discussion
In our simulations, the central magnetic field strength and gas number density
of the ICM are $\sim 10~{}\mu\text{G}$ and $\sim 10^{-2}~{}\text{cm}^{-3}$,
respectively, for a cluster with mass $10^{15}~{}M_{\odot}$ at $z=0.01$, and
both decrease toward the outskirts of the cluster. These quantities depend on
the mass of the clusters, being smaller for less massive clusters (see Fig. 2
& 5). Thus, high-energy CRs will escape with a higher probability without much
interactions in the case of less massive clusters. Lower-energy CRs, on the
other hand, contribute less to the production of high-energy neutrinos.
Therefore, we have a lower neutrino flux from less massive clusters. In
contrast, for massive clusters, higher magnetic field and gas density produce
higher neutrino flux due to the longer confinement time, as we see in Fig. 12.
We tested several injection CRs spectral indices ($\alpha\simeq 1.5-2.7$),
cut-off energies ($E_{\text{max}}=5\times 10^{15}-10^{18}$ eV), and source
evolution (AGN, SFR, no evolution), in order to try to interpret the IceCube
data (see Figs. 14, 15, 16, 17, 18 and 19). Overall, our results indicate that
galaxy clusters can contribute to a considerable fraction of the diffuse
neutrino flux measured by IceCube at energies between $100\;\text{TeV}$ and
$10\;\text{PeV}$, or even all of it, provided that that protons compose most
of the CRs.
Our results also look, in principle, similar to those of Fang & Olinto (2016)
with no source evolution, who considered essentially the same redshift
interval, but employed semi-analytical profiles to describe the cluster
properties. In particular, in both cases, the largest contribution to the flux
of neutrinos comes from the cluster mass group
$10^{14}<M<10^{15}~{}M_{\odot}$. However, they did not consider the
interactions of CRs with CMB and EBL background as they considered it
subdominant compared to the hadronic background following Kotera et al.
(2009). But, it can be seen from the upper panel of Fig. 8 that $\lambda$ for
pp-interaction and photopion production in the CMB are comparable for CRs of
energy $\gtrsim 10^{17}$ eV. Therefore, the neutrino production due to CR
interactions with the CMB is not negligible. Perhaps the most relevant
difference between our results and theirs is that, in their case, the redshift
range $0.3\leq z\leq 1$ makes the largest contribution to neutrino production,
while in our case this comes from the redshift range $z\lesssim 0.3$, when
considering no source evolution (see Fig 15).
When including source evolution, there is also a dominance in the neutrino
flux from the redshift range $z\lesssim 0.3$, though the contribution due to
the evolution of star forming galaxies (SFR) from redshifts $0.3\leq z\leq 1$
is also important. Overall, the inclusion of source evolution can increase the
diffuse neutrino flux by a factor of $\sim 3$ (when considering the separate
contributions of AGN or SFR) to $\sim 5$ (when considering both contributions
concomitantly) in the cases we studied, compared to the case with no
evolution, which is in agreement with (Murase & Waxman, 2016). Also, our
results agree with the IceCube measurements for $E\gtrsim 10^{14}\;\text{eV}$
and are in rough accordance with (Murase, 2017; Fang & Murase, 2018).
Nevertheless, since there are uncertainties related to the choice of specific
populations for the CR sources, obtaining a full picture of the diffuse high-
energy neutrino emission by clusters is not a straightforward task.
It is also worth comparing our results with Zandanel et al. (2015), who
evaluated the neutrino spectrum based on estimations of the radio to gamma-ray
luminosities of the clusters in the universe. Although our work has assumed an
entirely different approach, both results are consistent, especially for a CR
spectral index $\alpha\simeq 2$. High-energy ($E>10^{17}$ eV) CRs can escape
easily from clusters, effectively leading to a spectral steepening that was
not considered by Zandanel et al. (2015). However, not all the clusters are
expected to produce hadronic emission (Zandanel et al., 2015, 2014). In fact,
we observe less hadronic interactions in the case of low-mass clusters
($M\lesssim 10^{14}\;M_{\odot}$), which could further limit the neutrino
contribution from clusters.
The cluster scenario may get strong backing due to anisotropy detections above
PeV energies. Recently, only a few sources of high-energy neutrinos have been
observed (Aartsen et al., 2013, 2015; Albert et al., 2018; Ansoldi et al.,
2018; Aartsen et al., 2020), but there are also expectations to increase the
observations with future instruments like IceCube-Gen2 (The IceCube-Gen2
Collaboration, 2020), KM3NeT (Adrián-Martínez et al., 2016), and the Giant
Radio Array for Neutrino Detection (GRAND) (Álvarez-Muñiz et al., 2020).
Specifically, neutrinos from clusters are more likely to be observed if the
flux of cosmogenic neutrinos is low, which might contaminate the signal, as
discussed by Alves Batista et al. (2019b).
## 5 Conclusions
We considered a cosmological background based on 3D-MHD simulations to model
the cluster population of the entire universe, and a multidimensional Monte
Carlo technique to study the propagation of CRs in this environment and obtain
the flux of neutrinos they produce. Our results can be summarized as follows:
* •
We found that CRs with energy $E\lesssim 10^{17}$ eV cannot escape from the
innermost regions of the clusters, due to interactions with the background
gas, thermal photons and magnetic fields. Massive clusters ($M\gtrsim
10^{14}\;M_{\odot}$) have stronger magnetic fields which can confine these
high-energy CRs for a time comparable to the age of the universe.
* •
Our simulations predict that the neutrino flux above PeV energies comes from
the most massive clusters because the CR interactions with the gas of the ICM
are rare for clusters with $M<10^{14}\;M_{\odot}$.
* •
Most of the neutrino flux comes from nearby clusters in the redshift range
$z\lesssim 0.3$. The high-redshif clusters contribute less to the total flux
of neutrinos compared to the low-redshift ones, as the population of massive
clusters at high redshifts is low.
* •
The total integrated neutrino flux obtained from the interactions of CRs with
the ICM gas and CMB during their propagation in the turbulent magnetic field
can account for sizeable percentage of the IceCube observations, especially,
between energy $100\;\text{TeV}$ and $10\;\text{PeV}$.
* •
Our results also indicate that the redshift evolution of CR sources like AGN
and SFR, enhance the flux of neutrinos.
Finally, more realistic studies considering cosmological simulations that
account for AGN and star formation feedback from galaxies ( e.g. Barai et al.,
2016; Barai & de Gouveia Dal Pino, 2019) will allow to constrain better the
redshift evolution of the CR sources in the computation of the total neutrino
flux from clusters. Furthermore, in the future, IceCube will have detected
more events. Then, combined with diffuse gamma-ray searches by the forthcoming
CTA (Cherenkov Telescope Array Consortium et al., 2019), it will be possible
to better assess the contribution of galaxy clusters to the total
extragalactic neutrino flux.
## Acknowledgements
Saqib Hussain acknowledges support from the Brazilian funding agency CNPq.
EMdGDP is also grateful for the support of the Brazilian agencies FAPESP
(grant 2013/10559-5) and CNPq (grant 308643/2017-8). RAB is currently funded
by the Radboud Excellence Initiative, and received support from FAPESP in the
early stages of this work (grant 17/12828-4). KD acknowledges support by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy – EXC-2094 – 39078331 and by the funding for the
COMPLEX project from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation program grant agreement
ERC-2019-AdG 860744. The numerical simulations presented here were performed
in the cluster of the Group of Plasmas and High-Energy Astrophysics (GAPAE),
acquired with support from FAPESP (grant 2013/10559-5). This work also made
use of the computing facilities of the Laboratory of Astroinformatics
(IAG/USP, NAT/Unicsul), whose purchase was also made possible by a FAPESP
(grant 2009/54006-4). We also acknowledge very useful comments from K. Murase
on an earlier version of this manuscript.
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## Appendix A Mean Free Paths
The mean free path $\lambda$ for different CR interactions in the ICM are
defined below.
For a CR proton with Lorentz factor $\gamma_{p}$ traversing an isotropic
photon field, one obtains the rate $\lambda_{p\gamma}^{-1}(E_{p})$
(Schlickeiser, 2002)
$\displaystyle\lambda_{p\gamma}^{-1}(E_{p})$
$\displaystyle=\frac{1}{2\gamma_{p}}\int\limits_{\epsilon_{\text{th}}/2\gamma_{p}}^{\infty}d\epsilon\frac{n_{\text{ph}}(\epsilon,r_{i})}{\epsilon^{2}}\int\limits_{\epsilon_{\text{th}}}^{2\gamma_{p}\epsilon}d\epsilon^{\prime}\epsilon^{\prime}\sigma_{p\gamma}(\epsilon^{\prime})K_{p}(\epsilon^{\prime}),$
(10) $\displaystyle\epsilon_{\text{th}}$
$\displaystyle=Km_{\pi}c^{2}\left[1+\frac{Km_{\pi}}{2m_{p}}\right]=145\;\text{MeV}.$
(11)
Where $n_{\text{ph}}(\epsilon,r_{i})$ denotes the number density of photons of
energy $\epsilon$ at a given distance $r_{i}$ from the center of the cluster
and $\sigma_{p\gamma}$ is the cross section of the interaction of CRs with
background photons. The threshold energy for the production of $K$ pions is
given by equation (11), so that for the production of a single ($K=1$) pion
the rest system threshold energy is $\epsilon_{\text{th}}=145~{}\text{MeV}$
(Schlickeiser, 2002).
To calculate the rate for the interactions of high-energy photons (produced
during the propagation of CRs inside a cluster) with the local protons in the
ICM, we can use equation (10) with the following modification in the center-
of-mass (CM) energy. The energy $E$ and 3-momentum ${\bf p}$ of a particle of
mass $m$ form a 4-vector $p=(E,p)$ whose square $p^{2}=(E/c)^{2}-{\bf
p}^{2}=m^{2}c^{4}$. The velocity of the particle is $\beta c={\bf v}/c={\bf
p}/E$. In the collision of two particles of masses $m_{1}$ and $m_{2}$, the
total CM energy can be expressed in the Lorentz-invariant form as
$\displaystyle\epsilon_{CM}$
$\displaystyle=\left[\frac{1}{c^{2}}(E_{1}+E_{2})^{2}-({\bf p_{1}}+{\bf
p_{2}})c^{2}\right]^{1/2}$ (12)
$\displaystyle=\left[m_{1}^{2}c^{4}+m_{2}^{2}c^{4}+\frac{2E_{1}E_{2}}{c^{2}}(1-{\bf\beta_{1}\beta_{2}}\cos\theta)\right]^{1/2},$
(13)
where $\theta$ is the angle between the particles that we can consider zero.
In the frame where one particle (of mass $m_{2}$) is at rest (lab frame) then,
$\epsilon_{CM}=(m_{1}^{2}c^{4}+m_{2}^{2}c^{4}+2E_{1}m_{2}c^{2})^{1/2}.$ (14)
If we consider $m_{2}$ is proton and $m_{1}$ is photon, then the above
relation becomes
$\epsilon_{CM}=(m_{2}^{2}c^{4}+2E_{1}m_{2}c^{2})^{1/2}.$ (15)
$\lambda_{\gamma
p}^{-1}(\epsilon_{\text{ph}})=\frac{\epsilon_{p}}{2\epsilon_{\text{ph}}}\int\limits_{\epsilon_{\text{th}}/(2\epsilon_{\text{ph}}/\epsilon_{p})}^{\infty}d\epsilon\frac{n_{p}(\epsilon,r_{i})}{\epsilon_{p}^{2}}\int\limits_{\epsilon_{\text{th}}}^{(2\epsilon_{\text{ph}}/\epsilon_{p})\epsilon}d\epsilon^{\prime}\epsilon^{\prime}\sigma_{\gamma
p}(\epsilon^{\prime}),$ (16)
so that the rest frame is in the local protons. We used equation (15) for the
energy of the CM in equation (16). In equation (16), $n_{p}(\epsilon,r_{i})$
is number density of local protons with energy $\epsilon_{p}=m_{p}c^{2}\sim 1$
GeV at a given distance $r_{i}$ from the center of a cluster and decreases
toward the outskirt, $\epsilon_{\text{th}}\sim 1.4\times 10^{8}$ eV is the
threshold energy for this interaction and the cross section $\sigma_{\gamma
p}(\epsilon^{\prime})$ is of the order $\sim 10^{-37}~{}(\text{cm}^{2})$. With
these values used in equation (16) we solve this integral to calculate
$\lambda$ for $\gamma$-proton interaction. We calculated $\lambda$ from
equations 10-16 with some modifications to include the information of the
spatially dependent Bremsstrahlung photon field of the clusters
$n_{\text{ph}}(\epsilon,r)$.
For proton-proton (pp) interaction, the rate is given by
$\lambda^{-1}_{\text{pp}}(E_{p},r_{i})=K_{\text{pp}}~{}\sigma_{\text{pp}}(E_{p})~{}n_{i}(r_{i})$
(17)
Where $K_{\text{pp}}=0.5$ is the inelasticity factor, $n_{i}(r_{i})$ denotes
the number density of proton at a given distance $r_{i}$ from the center of
the cluster and $E_{p}$ is the energy of the protons.
To obtain the proton number density, we consider that the background plasma
consists of electrons and protons in near balancing. Since the abundance is
mostly of H and this is mostly ionized in the hot ICM, this is a reasonable
assumption. Thus $n_{p}\simeq n_{e}$, and
$\rho_{\text{gas}}=n_{p}m_{p}+n_{e}m_{e}\sim n_{p}m_{p}$, so that $n_{i}\simeq
n_{e}\simeq\rho_{\text{gas}}/m_{p}$, where $m_{p}$ is the proton mass and
$\rho_{\text{gas}}$ is the gas mass density in the system.
For $\sigma_{\text{pp}}=70~{}\text{mb}$
($1\text{barn}=10^{-29}\;\text{m}^{2}$), we have for the cross section
(Kafexhiu et al., 2014):
$\sigma_{\text{pp}}=\left[30.7-0.96\log\left(\frac{E_{p}}{E_{p}^{\text{th}}}\right)+0.18\log\left(\frac{E_{p}}{E_{p}^{\text{th}}}\right)\right]\left[1-\left(\frac{E_{p}^{\text{th}}}{E_{p}}\right)^{1.9}\right]^{3}~{}\text{mb},$
(18)
where $E_{p}$ is the energy of the proton and $E_{p}^{\text{th}}$ is the
threshold kinetic energy $E_{p}^{\text{th}}=2m_{\pi}+m_{\pi}/m_{p}\approx
0.2797$ GeV. We used equations 17 and (18) to calculate $\lambda_{\text{pp}}$.
## Appendix B Spectral Index
To calculate the flux of neutrinos corresponding to injected CRs with an
arbitrary power-law spectrum with power law index $\alpha$,
$dN_{\text{CR},~{}E}/dE\propto
E_{i}^{-\alpha}\exp\\{-E_{i}/E_{\text{max}}\\}$, we can normalize the spectrum
as follows:
$J(\alpha)=\dfrac{\ln(E_{\text{CR},~{}\text{max}}/E_{\text{min}})}{\int\limits_{E_{\text{min}}}^{E_{\text{CR,
max}}}E_{i}^{1-\alpha}\exp\left(-\frac{E_{i}}{E_{\text{max}}}\right)dE}E_{i}^{1-\alpha}\exp\left(-\frac{E_{i}}{E_{\text{max}}}\right)$
(19)
Where, $E_{i}$ is the injection energy of the simulated CRs, $E_{\text{max}}$
is the exponential cut-off energy, and $E_{\text{CR, max}}$ is the maximum
injection energy of the CRs.
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# Club Stationary Reflection and the Special Aronszajn Tree Property
Omer Ben-Neria and Thomas Gilton Hebrew University of Jerusalem The Edmond J.
Safra Campus - Givat Ram, Jerusalem, Israel, 9190401<EMAIL_ADDRESS>University of Pittsburgh Department of Mathematics. The Dietrich School of
Arts and Sciences, 301 Thackeray Hall, Pittsburgh, PA 15260, United States
<EMAIL_ADDRESS>
###### Abstract.
We prove that it is consistent that _Club Stationary Reflection_ and the
_Special Aronszajn Tree Property_ simultaneously hold on $\omega_{2}$, thereby
contributing to the study of the tension between compactness and incompactness
in set theory. The poset which produces the final model follows the collapse
of an ineffable cardinal first with an iteration of club adding (with
anticipation) and second with an iteration specializing Aronszajn trees.
In the first part of the paper, we prove a general theorem about specializing
Aronszajn trees on $\omega_{2}$ after forcing with what we call
$\cal{F}$_-Strongly Proper_ posets, where $\cal{F}$ is either the weakly
compact filter or the filter dual to the ineffability ideal. This type of
poset, of which the Levy collapse is a degenerate example, uses systems of
exact residue functions to create many strongly generic conditions. We prove a
new result about stationary set preservation by quotients of this kind of
poset; as a corollary, we show that the original Laver-Shelah model, which
starts from a weakly compact cardinal, satisfies a strong stationary
reflection principle, though it fails to satisfy the full Club Stationary
Reflection. In the second part, we show that the composition of collapsing and
club adding (with anticipation) is an $\cal{F}$-Strongly Proper poset. After
proving a new result about Aronszajn tree preservation, we show how to obtain
the final model.
###### Key words and phrases:
forcing, Aronszajn Trees, stationary reflection, compactness, specialization
###### 2010 Mathematics Subject Classification:
Primary 03E05, 03E35
The first author was partially supported by the Israel Science Foundation
(Grant 1832/19).
## 1\. Introduction
This work is a contribution to the study of the tension between compactness
and incompactness principles in set theory. We focus on the second uncountable
cardinal, $\omega_{2}$, and consider the strong compactness principle of Club
Stationary Reflection and the strong incompactness principle known as the
Special Aronszajn Tree Property (these are defined below).
The two properties have been shown to be consistent separately by Magidor [34]
and Laver and Shelah [33], respectively. Since the properties represent strong
forms of opposing phenomena (compactness and incompactness) it is natural to
suspect that they are jointly inconsistent. The main result of this paper
shows, on the contrary, that the conjunction of the two principles is
consistent. More precisely, we prove:
###### Theorem 1.1.
It is consistent relative to the existence of an ineffable cardinal that Club
Stationary Reflection and the Special Aronszajn Tree Property simultaneously
hold at $\omega_{2}$.
Our work also shows that a weaker version of stationary reflection holds in
the original Laver-Shelah model (which uses a weakly compact):
###### Theorem 1.2.
In the original Laver-Shelah model the following stationary reflection
principle holds: for every sequence $\langle
S_{\alpha}\mid\alpha<\omega_{2}\rangle$ of stationary subsets of
$\omega_{2}\cap\operatorname{cof}(\omega)$ there is $\beta<\omega_{2}$ so that
$S_{\alpha}\cap\beta$ is stationary in $\beta$, for every $\alpha<\beta$.
However, Club Stationary Reflection at $\omega_{2}$ fails.
We proceed to define the relevant terms and contextualize our result. If $\nu$
is a regular cardinal, we use $\operatorname{cof}(\nu)$ denote the class of
ordinals with cofinality $\nu$. We recall that if
$\operatorname{cf}(\alpha)>\omega$, then $S\subseteq\alpha$ is _stationary_ if
$S\cap C\neq\emptyset$ for each club $C\subseteq\alpha$. We say that $S$
_reflects_ if there is some $\beta<\alpha$ with
$\operatorname{cf}(\beta)>\omega$ so that $S\cap\beta$ is stationary in
$\beta$. If $\kappa$ is regular, we say that _stationary reflection_ holds at
$\kappa^{++}$ if every stationary
$S\subseteq\kappa^{++}\cap\operatorname{cof}(\leq\kappa)$ reflects.
Baumgartner originally showed ([5]) that stationary reflection at $\omega_{2}$
is consistent from a weakly compact cardinal. Harrington and Shelah ([21])
later improved this, showing that the optimal assumption of a Mahlo cardinal
suffices. One obtains stronger principles by requiring that multiple
stationary sets reflect simultaneously. Recall that a collection $\\{S_{i}\mid
i<\tau\\}$ of $\tau<\alpha$ stationary subsets of $\alpha$ is said to reflect
_simultaneously_ if there is some $\beta<\alpha$ with
$\operatorname{cf}(\beta)>\omega$ so that $S_{i}\cap\beta$ is stationary in
$\beta$ for every $i<\tau$. Magidor ([34]) has shown that the consistency
strength of “any two stationary subsets of
$\omega_{2}\cap\operatorname{cof}(\omega)$ simultaneous reflect” implies the
consistency of a weakly compact cardinal. One may also consider stronger
diagonal versions of the above, defined in the natural way.
We are interested in the following very strong form of stationary reflection
which implies all of the above:
###### Definition 1.3.
Suppose that $\kappa$ is regular. We say that Club Stationary Reflection holds
at $\kappa^{++}$ if for any stationary
$S\subseteq\kappa^{++}\cap\operatorname{cof}(\leq\kappa)$, there exists a club
$C\subseteq\kappa^{++}$ so that for all $\beta\in
C\cap\operatorname{cof}(\kappa^{+})$, $S$ reflects at $\beta$. We write
$\mathsf{CSR}(\kappa^{++})$.
We will concern ourselves with the case $\kappa=\omega$, i.e., with stationary
subsets of $\omega_{2}\cap\operatorname{cof}(\omega)$. Most relevant for us,
Magidor ([34]) showed that $\mathsf{CSR}(\omega_{2})$ is consistent from a
weakly compact cardinal; by the above remarks, this is the optimal hypotheses.
Extensions of $\mathsf{CSR}$ to other cardinals have been shown to have
limitations. For example, Jech and Shelah [25] proved that for every
$n<\omega$, if every stationary subset of
$\omega_{n+3}\cap\operatorname{cof}(\omega_{n+1})$ reflects then
$\mathsf{CSR}(\omega_{n+2})$ fails. However, Jech and Shelah [25], and later
Cummings and Wylie [14], proved the consistency of certain best possible
variations of club stationary reflection below $\aleph_{\omega}$.
Limitations on stationary reflection emerge from incompactness principles. One
of the most prominent of these is Jensen’s $\square_{\kappa}$. In [26], Jensen
showed that $\square_{\kappa}$ holds in $L$ for all $\kappa>\omega$ and that
$\Box_{\kappa}$ implies the existence of many nonreflecting stationary subsets
of $\kappa^{+}$.
Further studies showed that variations of $\square_{\kappa}$ place limitations
on the cofinality of reflection points, as well as the amount of simultaneous
reflection. For instance, in [41], Schimmerling introduced the hierarchy of
square principles, $\square_{\kappa,\lambda}$, $1\leq\lambda\leq\kappa^{+}$.
As $\lambda$ increases, this hierarchy is strictly decreasing in strength; see
Jensen [27] for $\kappa$ regular and [11] for $\kappa$ singular. For a regular
cardinal $\kappa$ and $\lambda\leq\kappa$, Schimmerling and independently
Foreman and Magidor have observed that if $\kappa^{<\lambda}=\kappa$ and
$\square_{\kappa,<\lambda}$ holds then every stationary subset of $\kappa^{+}$
has a stationary subset which does not reflect at any point of cofinality
$\geq\lambda$; see [13]. In particular, $\kappa^{<\kappa}=\kappa$ and
$\square_{\kappa,<\kappa}$ imply that every stationary subset of $\kappa^{+}$
has a stationary subset which does not reflect at any point in
$\kappa^{+}\cap\operatorname{cof}(\kappa)$. In [11], Cummings, Foreman, and
Magidor extended these results and developed the theory for $\kappa$ singular.
Other notable weakenings of $\square_{\kappa}$ were introduced and developed
by Todorčević ([43]). These principles, denoted $\square(\kappa^{+})$ and
$\square(\kappa^{+},\lambda)$, place refined limitations on the extent of
stationary reflection. See [15], [23], and [39].
The weakest nontrivial form of square studied by Jensen is the so-called Weak
Square, denoted $\square_{\kappa}^{*}$, Remarkably, $\square_{\kappa}^{*}$ is
equivalent to a key incompactness phenomenon, the existence of a special
$\kappa^{+}$-Aronszajn tree. Let us recall the relevant definitions. A _tree_
is a partially ordered set $(T,\leq_{T})$ so that for each $x\in T$, the set
of $\leq_{T}$-predecessors of $x$ is well-ordered; we refer to the _height_ of
$x$ in $T$ as the ordertype of this set. If $\alpha$ is an ordinal, we use
Lev${}_{\alpha}(T)$ to denote all $x\in T$ of height $\alpha$. The _height_ of
$T$ is the least ordinal $\alpha$ so that $T$ has no elements of height
$\alpha$. A _branch_ through $T$ is a linearly ordered subset of $T$, and a
_cofinal branch_ is a branch which intersects every level below the height of
$T$.
Let $\kappa$ be regular. A _$\kappa$ -tree_ is a tree $T$ of height $\kappa$
so that each level has size $<\kappa$; we will always assume that for each
such tree, each node in the tree has incompatible extensions to all higher
levels. $\kappa$ is said to have the _tree property_ if every $\kappa$-tree
has a cofinal branch. K$\ddot{\text{o}}$nig showed ([28]) that $\omega$ has
the tree property, while Aronszjan has shown that the tree property fails at
$\omega_{1}$ (the result was communicated in [32]). The extent of the tree
property on cardinals greater than $\omega_{1}$, a famous question of
Magidor’s, is independent of $\mathsf{ZFC}$. A watershed in our understanding
is due to Mitchell and Silver ([37]) who showed that the tree property at
$\omega_{2}$ is consistent from a weakly compact cardinal.
A tree which witnesses the failure of the tree property is said to be
_Aronszajn_ (i.e., a $\kappa$-tree which has no cofinal branches); the
existence of such a tree is an instance of incompactness. A particularly
strong witness that a tree is Aronszajn is given by a _specializing function_
: in the case that $\kappa=\lambda^{+}$, a specializing function is an
$f:T\longrightarrow\lambda$ so that if $x<_{T}y$, then $f(x)\neq f(y)$. (For
an exploration of these concepts at an arbitrarily regular cardinal, see
[30].) Having a specializing function is a particularly strong witness to
being Aronszajn since the function witnesses that $T$ remains Aronszajn in any
extension of that model in which $\kappa$ is still a cardinal. $T$ is said to
be a _special Aronszajn tree_ if there is a specializing function for $T$. The
property of interest to us is the following:
###### Definition 1.4.
Let $\kappa$ be regular. We say that $\kappa^{+}$ has the Special Aronszajn
Tree Property if there are Aronszajn trees on $\kappa^{+}$, and if every
Aronszajn tree on $\kappa^{+}$ is special. We denote this property by
$\mathsf{SATP}(\kappa^{+})$.
By a result of Specker ([42]), if $\kappa^{<\kappa}$ holds, then there is a
special Aronszajn tree on $\kappa^{+}$; in particular, if the $\mathsf{CH}$
holds, then there is a special Aronszajn tree on $\omega_{2}$. Jensen ([26])
later showed that the principle $\Box^{*}_{\kappa}$ holds if and only if there
is a special Aronszajn tree on $\kappa^{+}$; since $\kappa^{<\kappa}$ implies
$\Box^{*}_{\kappa}$, this strengthens Specker’s result.
With regards to constructing specializing functions by forcing, Baumgartner,
Malitz, and Reinhardt showed ([8]) that $\mathsf{MA}+\neg\mathsf{CH}$ implies
$\mathsf{SATP}(\omega_{1})$. Later, Laver and Shelah showed ([33]) that
$\mathsf{SATP}(\omega_{2})$ is consistent from a weakly compact cardinal.
Generalizing this further, Golshani and Hayut have recently shown ([20]),
using posets which specialize with anticipation, that it is consistent that,
simultaneously, for every regular cardinal $\kappa$,
$\mathsf{SATP}(\kappa^{+})$ holds. Krueger has generalized the result of
Laver-Shelah (and also Abraham-Shelah, [2]) in a different direction ([31]),
showing that it is consistent with the $\mathsf{CH}$ that any two countably
closed Aronszjan trees on $\omega_{2}$ are club isomorphic. And finally,
Asperó and Golshani ([3]) have announced a positive solution to the question
of whether $\mathsf{SATP}(\omega_{2})$ is consistent with the $\mathsf{GCH}$.
This work continues the study of the tension between different manifestations
of compactness and incompactness phenomena in set theory, which together with
the study of tension with other fundamental principles such as approximation
principles (e.g., [9], [19]) and cardinal arithmetic (e.g., [16], [40]) is
central to our understanding of their extent and limitations.
We proceed to describe our result in general terms and highlight the
challenges that appear in the process. Let $\kappa$ be a cardinal which is
either ineffable or weakly compact in a ground model $V$ of $\mathsf{GCH}$; we
will specify later (Definition 2.14) exactly when $\kappa$ is ineffable or
weakly compact. We obtain the model which witnesses Theorem 1.1 by first
defining, in the extension by
$\mathbb{P}=\operatorname{Col}(\omega_{1},<\kappa)$, a $\kappa^{+}$-length
iteration
$\mathbb{C}_{\kappa^{+}}=\langle\mathbb{C}_{\tau},\mathbb{C}(\tau)\mid\tau<\kappa^{+}\rangle$
of adding clubs which will eventually witnesses $\mathsf{CSR}(\omega_{2})$.
After forcing with $\mathbb{C}_{\kappa^{+}}$, we then force with a
$\kappa^{+}$-iteration
$\mathbb{S}_{\kappa^{+}}=\langle\mathbb{S}_{\tau},\mathbb{S}(\tau)\mid\tau<\kappa^{+}\rangle$
specializing the desired Aronszajn trees $\dot{T}_{\tau}$, i.e.,
$\mathbb{S}(\tau)=\mathbb{S}(\dot{T_{\tau}})$ (see Section 1.1 for precise
definitions of the posets).
To make this strategy work, we need, among other things, that all stationary
subsets of $\omega_{2}\cap\operatorname{cof}(\omega)$ which appear in the
final generic extension by
$\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}}*\dot{\mathbb{S}}_{\kappa^{+}}$
reflect as in the definition of $\mathsf{CSR}(\omega_{2})$. Consequently, the
club adding posets must anticipate names for stationary sets added by the
later specializing iteration. In order to carry this through, we define the
names $\dot{\mathbb{C}}_{\tau}$ and $\dot{\mathbb{S}}_{\tau}$, for
$\tau<\kappa^{+}$, simultaneously. More precisely, for each $\tau<\kappa^{+}$,
given that the $\mathbb{P}$-name $\dot{\mathbb{C}}_{\tau}$ and the
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\tau})$-name $\dot{\mathbb{S}}_{\tau}$ have
been defined, we use a bookkeeping function to pick the
$(\mathbb{P}*\dot{\mathbb{C}}_{\tau}*\dot{\mathbb{S}}_{\tau})$-name
$\dot{S_{\tau}}$ of a stationary subset of
$\kappa\cap\operatorname{cof}(\omega)$, and we set $\dot{\mathbb{C}}(\tau)$ to
be the $(\mathbb{P}\ast\dot{\mathbb{C}}_{\tau})$-name for the poset to add,
with $\dot{\mathbb{S}}_{\tau}$-anticipation, the desired club. Then we select
the $(\mathbb{P}*\dot{\mathbb{C}}_{\tau+1}*\dot{\mathbb{S}}_{\tau})$-name
$\dot{T}_{\tau}$ for an Aronszajn tree on $\kappa$.
As expected, the tension between compactness and incompactness gives rise to
tension between the different parts of the forcing construction. We list three
notable manifestations:
(1) Working with Intermediate Generic Extensions. A central property of the
Laver-Shelah forcing ([33]) is the existence of intermediate forcing
extensions in which regular cardinals $\alpha<\kappa$ become $\omega_{2}$ and
the relevant portion $\dot{T}_{\tau}\cap(\alpha\times\omega_{1})$ of the trees
are Aronszajn trees on $\alpha$. Accompanying this is machinery for projecting
conditions of $\mathbb{P}*\dot{\mathbb{S}}_{\tau}$ to those intermediate
extensions. In [33] the existence of such intermediate extensions is secured
by the weak compactness of $\kappa$, and the fact that
$\mathbb{P}*\dot{\mathbb{S}}_{\tau}$ is $\kappa$-c.c. However, in our case,
the presence of the poset $\dot{\mathbb{C}}_{\tau}$ prevents the initial
segment $\mathbb{P}*\dot{\mathbb{C}}_{\tau}$ from being $\kappa$-c.c. To
overcome this difficulty, we use the fact that the full collapse poset
$\mathbb{P}$ absorbs many restricted subforcings of
$\mathbb{P}*\dot{\mathbb{C}}_{\tau}$, which allows us to place upper bounds on
various generic filters of the restricted poset. We then couple this in
Section 6 with a generalization of a result of Abraham’s ([1]) that (stated in
current language) if $\dot{\mathbb{Q}}$ is an
$\operatorname{Add}(\omega,\omega_{1})$-name for an $\omega_{1}$-closed poset,
then $\operatorname{Add}(\omega,\omega_{1})\ast\dot{\mathbb{Q}}$ is strongly
proper. This secures the existence of sufficiently many strongly generic
conditions (and in turn, the existence of intermediate extensions).
(2) Preservation of Stationary Sets by Quotients. The ability to add a closed
unbounded set through the reflection points $\alpha<\kappa$ of a stationary
set $\dot{S_{\tau}}\subseteq\kappa\cap\operatorname{cof}(\omega)$ hinges upon
the fact that many such points exist. The ineffability (in fact, just weak
compactness) of $\kappa$ guarantees that for many $\alpha<\kappa$,
$\dot{S_{\tau}}\cap\alpha$ is a stationary subset of $\alpha$ in the
restricted generic extension where $\alpha=\omega_{2}$. The forcing
construction of [34] uses the fact that the related quotient of
$\mathbb{P}*\dot{\mathbb{C}}_{\tau}$ by its initial segment is
$\sigma$-closed, and an argument of Baumgartner’s ([5]) shows that
$\sigma$-closed posets preserve the stationarity of stationary sets of
countable cofinality ordinals. By contrast, for us the stationary sets
$\dot{S}_{\tau}$ further rely on the specializing poset
$\dot{\mathbb{S}}_{\tau}$, and although the poset
$\mathbb{P}*\dot{\mathbb{C}}_{\tau}*\dot{\mathbb{S}}_{\tau}$ is,
$\sigma$-closed, it does not in general admit $\sigma$-closed quotients by its
natural restrictions to heights $\alpha<\kappa$. Nevertheless, in Section 4 we
analyze the Laver-Shelah iteration $\dot{\mathbb{S}}_{\tau}$ to prove that the
relevant quotients preserve the stationary of $\dot{S_{\tau}}\cap\alpha$ for
many suitable $\alpha<\kappa$.
(3) Preservation of Aronszajn Trees. The organization of the posets
$\dot{\mathbb{C}}_{\tau}$ and $\dot{\mathbb{S}}_{\tau}$, described above,
guarantees that for each $\tau<\kappa^{+}$, $\dot{T}_{\tau}$ is a
$(\mathbb{P}*\dot{\mathbb{C}}_{\tau+1}*\dot{\mathbb{S}}_{\tau})$-name of an
Aronszajn tree on $\kappa$, which is specialized by
$\mathbb{P}*\dot{\mathbb{C}}_{\tau+1}*\dot{\mathbb{S}}_{\tau+1}$.
However, in the final forcing construction, $\mathbb{S}_{\tau+1}$ follows the
extended iteration $\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}}$, and on its
face, $\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}}*\dot{\mathbb{S}}_{\tau}$ might
introduce a cofinal branch to $T_{\tau}$, causing the specializing poset
$\mathbb{S}(\tau)$ to collapse $\kappa$. To guarantee that this cannot occur,
an Aronszajn preservation theorem is required for the quotient of
$\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}}*\dot{\mathbb{S}}_{\tau}$ by
$\mathbb{P}*\dot{\mathbb{C}}_{\tau+1}*\dot{\mathbb{S}}_{\tau}$. The fact that
no new reals are added during the iteration, and that
$\dot{\mathbb{C}}_{\kappa^{+}}$ is not $\kappa$-closed, prevents us from using
known preservation arguments (for instance those of [44]). Therefore, in
Section 6 we develop an alternative preservation argument which fits the
properties of the poset $\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}}$, and we
apply them in Section 7 to show that the tree $T_{\tau}$ remains Aronszajn.
Structure of this work: In the rest of this section, we review relevant
preliminaries regarding forcing as well as ineffable and weakly compact
cardinals. The first part of the work consists of Sections 2 through 4. In
Section 2 we develop the notion and fundamental properties of posets which are
strongly proper with respect to the filter $\cal{F}$ on $\kappa$, which is
either the weakly compact filter on $\kappa$ or the filter dual to the
ineffability ideal on $\kappa$ (depending on whether $\kappa$ is weakly
compact or ineffable, respectively). We will later verify that initial
segments of the form $\mathbb{P}*\dot{\mathbb{C}}_{\tau}$, for
$\tau<\kappa^{+}$, are members this class. Section 3 studies an iteration of
specializing posets $\mathbb{S}_{\tau}$, following an $\cal{F}$-strongly
proper poset $\mathbb{P}^{*}$. We prove that the main results of the Laver-
Shelah analysis apply in this context as well. In the case when
$\mathbb{P}^{*}$ is just the Levy collapse to make $\kappa$ become
$\omega_{2}$, we only need a weakly compact (and $\cal{F}$ is the weakly
compact filter). This is just the Laver-Shelah argument. However, when
$\mathbb{P}^{*}$ becomes a more complicated poset, we needed to use the
stronger assumption that $\kappa$ is ineffable (and $\cal{F}$ is the filter
dual to the ineffability ideal). Nevertheless, we only need the ineffability
for the case when $\mathbb{P}^{*}$ is not just the collapse, and in this case,
only for the proof of Proposition 3.22 and the corollaries of that
proposition. Section 4 is devoted to showing that suitable quotients of
specializing iterations of the form $\mathbb{P}^{*}*\dot{\mathbb{S}}_{\tau}$,
where $\mathbb{P}^{*}$ is $\cal{F}$-strongly proper (and in either case for
$\cal{F}$) preserve stationary subsets of countable cofinality ordinals.
In Part 2 of the paper, we construct specific posets playing the role of
$\mathbb{P}^{*}$ above, and we prove our theorem. In Section 5, we introduce
the complementary notion of posets which are completely proper with respect to
$\cal{F}$, and later we apply this analysis to $\dot{\mathbb{C}}_{\tau}$,
$\tau<\kappa^{+}$. We show that the composition of the Levy collapse and a
poset which is completely proper with respect to $\cal{F}$ is strongly proper
with respect to $\cal{F}$. Section 6 develops the main properties of the club
adding iteration $\mathbb{C}_{\tau}$. And finally, we combine the results of
the previous sections in Section 7 to prove Theorem 1.1.
### 1.1. Forcing
In this subsection, we review our conventions about forcing and provide
explicit definitions of posets which we will use throughout the paper.
To begin, in order to anticipate working with iterations later, we will work
with pre-orderings (i.e., relations which are transitive and reflexive) rather
than partial orders. Moreover, we will use the Jerusalem convention for
forcing. Thus we view a forcing poset as a triple
$(\mathbb{Q},\leq_{\mathbb{Q}},0_{\mathbb{Q}})$, where $\leq_{\mathbb{Q}}$ is
a pre-ordering and where $0_{\mathbb{Q}}$ is a smallest element; for
conditions $p,q\in\mathbb{Q}$, we will write $p\geq_{\mathbb{Q}}q$ to indicate
that $p$ is an extension of $q$. When context is clear, we will drop explicit
mention of $\mathbb{Q}$ in $0_{\mathbb{Q}}$ and $\leq_{\mathbb{Q}}$. Given
that we are only working with pre-orderings rather than partial orders, we
will often have conditions $p,q\in\mathbb{Q}$ so that $p\leq_{\mathbb{Q}}q$
and $q\leq_{\mathbb{Q}}p$ but $q$ and $p$ are not literally equal as sets. In
this case, we will write $p=^{*}_{\mathbb{Q}}q$, or simply $p=^{*}q$ if
$\mathbb{Q}$ is clear from context.
If $\mathbb{Q}$ is a poset, we say that $\mathbb{Q}$ is $\omega_{1}$-closed
with sups if for any increasing sequence $\langle q_{n}:n\in\omega\rangle$ of
conditions in $\mathbb{Q}$, there exists a $\leq_{\mathbb{Q}}$-least upper
bound $q$ of the sequence. Any such $q$ is referred to as a sup of the
sequence. Note that this does not say that any two compatible conditions in
$\mathbb{Q}$ have a sup. Moreover, it also does not require that a sup of an
increasing $\omega$-sequence is unique. However, if $q_{1}$ and $q_{2}$ are
two sups of such a sequence, then $q_{1}=^{*}_{\mathbb{Q}}q_{2}$. These
observations will be important later when we deal with iterations of posets
with this property.
If $\mathbb{Q}$ is a poset and $q\in\mathbb{Q}$, then we use $\mathbb{Q}/q$ to
denote all conditions in $\mathbb{Q}$ which extend $q$.
Let $M\prec H(\theta)$ be an elementary substructure and $\mathbb{U}\in M$ a
poset. A condition $u$ in $\mathbb{U}$ is $(M,\mathbb{U})$-completely generic,
if the set $\left\\{\bar{u}\in\mathbb{U}\cap M:u\geq\bar{u}\right\\}$ of
weaker conditions in $M$ meets all dense subsets $D\subseteq\mathbb{U}$ which
belong to $M$, and thus forms a $(M,\mathbb{U})$-generic filter.
For the remainder of the paper, we fix a cardinal $\kappa$ which will be
either weakly compact or ineffable (we specify in Definition 2.14 exactly when
$\kappa$ is ineffable or weakly compact). In the next subsection, we will
review facts about ineffability and weak compactness.
Throughout the paper, we will use $\mathbb{P}$ to denote the Levy collapse
$\operatorname{Col}(\omega_{1},<\kappa)$. If $\alpha<\kappa$ is inaccessible,
we use $\mathbb{P}\upharpoonright\alpha$ to denote the collapse
$\operatorname{Col}(\omega_{1},<\alpha)$. We view conditions in
$\operatorname{Col}(\omega_{1},<\kappa)$ as countable functions $p$ so that
$\operatorname{dom}(p)\subseteq\kappa$ and so that for each
$\nu\in\operatorname{dom}(p)$, $p(\nu)$ is a countable, partial function from
$\omega_{1}$ to $\nu$. If $G$ is a $V$-generic filter over $\mathbb{P}$, then
we use $G\upharpoonright\alpha$ to denote the $V$-generic filter
$\left\\{p\upharpoonright\alpha:p\in G\right\\}$ over
$\mathbb{P}\upharpoonright\alpha$.
For adding clubs, we generalize the club-adding poset of Magidor ([34]) by
incorporating anticipation; we only state the definition in the generality
needed for our paper. Recall that if $S$ is stationary in $\alpha$, then the
_trace_ of $S$, denoted $\operatorname{tr}(S)$, consists of all $\beta<\alpha$
so that $S$ reflects at $\beta$.
###### Definition 1.5.
Let $\mathbb{S}$ be a cardinal-preserving poset in some model $W$, and let
$\dot{S}$ be an $\mathbb{S}$-name for a stationary subset of
$\omega_{2}\cap\operatorname{cof}(\omega)$. We let
$\mathsf{CU}(\dot{S},\mathbb{S})$ denote the poset, defined in $W$, where
conditions are closed, bounded subsets $c$ of $\omega_{2}$ so that
$\Vdash_{\mathbb{S}}\check{c}\subseteq\operatorname{tr}(\dot{S})\cup\left(\omega_{2}\cap\operatorname{cof}(\omega)\right).$
The ordering is end-extension.
We emphasize that in order to be a condition in
$\mathsf{CU}(\dot{S},\mathbb{S})$, a given closed, bounded subset of
$\omega_{2}$ must be outright forced by $\mathbb{S}$ to be contained, mod
cofinality $\omega$ points, in $\operatorname{tr}(\dot{S})$. Since any
condition $c$ in $\mathsf{CU}(\dot{S},\mathbb{S})$ can be extended by placing
an ordinal of cofinality $\omega$ above $\max(c)$, we see that
$\mathsf{CU}(\dot{S},\mathbb{S})$ does add a club subset of $\omega_{2}$ of
the model. Moreover, the poset is trivially $\omega_{1}$-closed, so preserves
$\omega_{1}$. However preservation of $\omega_{2}$ is a non-trivial matter.
We now review the definition of the poset which we will use to specialize
Aronszajn trees on $\omega_{2}$. The poset itself will decompose such a tree
into a union of $\omega_{1}$-many antichains, which in this case is equivalent
to having a specializing function.
###### Definition 1.6.
Suppose that $T$ is an Aronszajn tree on $\omega_{2}$. Let $\mathbb{S}(T)$
denote the poset where conditions are functions $f$ with countable domain
$\operatorname{dom}(f)\subseteq\omega_{1}$, and where for each
$\alpha\in\operatorname{dom}(f)$, $f(\alpha)\subseteq T$ is a countable
antichain in $<_{T}$. Recalling that we are using the Jerusalem convention for
forcing, we say that $g$ extends $f$, written $f\leq g$, if
$\operatorname{dom}(f)\subseteq\operatorname{dom}(g)$ and if for all
$\alpha\in\operatorname{dom}(f)$, $f(\alpha)\subseteq g(\alpha)$.
It is clear that $\mathbb{S}(T)$ is $\omega_{1}$-closed. Moreover, if a tree
$T^{\prime}$ is not Aronszajn, then the analogously defined poset
$\mathbb{S}(T^{\prime})$ will collapse $\omega_{2}$.
### 1.2. Weak Compactness
In this final subsection, we review facts about the ineffability of $\kappa$.
However, we will also need various facts about the weak compactness of
$\kappa$, and so we begin with these.
###### Definition 1.7.
$\mathcal{F}_{\operatorname{WC}}$ is the filter generated by subsets $A$ of
$\kappa$ for which there is some $U\subseteq V_{\kappa}$ and a
$\Pi^{1}_{1}$-statement $\Phi$, satisfied by $(V_{\kappa},\in,U)$, so that
$A=\\{\alpha<\kappa\mid\alpha\text{ is regular, and }(V_{\alpha},\in,U\cap
V_{\alpha})\models\Phi\\}.$
The filter $\mathcal{F}_{\operatorname{WC}}$ is $\kappa$-closed as well as
normal. It will be helpful at later parts in our argument to phrase membership
in the weakly compact filter in terms of embeddings. The idea will be that for
a subset $B$ of $\kappa$, where $B$ is a member of a $\kappa$-model $M$,
$B\in\mathcal{F}_{\operatorname{WC}}$ iff for all $M$-normal ultrafilters $U$,
$\kappa\in j_{U}(B)$, where $j_{U}$ is the ultrapower embedding. We make this
precise in the following few items.
###### Definition 1.8.
Suppose that $\alpha$ is an inaccessible cardinal. We say that a transitive
set $M$ is an $\alpha$-model if $M\models\mathsf{ZFC}^{-}$, $|M|=\alpha$,
$\alpha\in M$, and $\,{}^{<\alpha}M\subseteq M$.
Weak compactness is naturally associated to various embedding properties; here
we mention the following result from [22]:
###### Proposition 1.9.
For any $\kappa$-model $M$, there exist a $\kappa$-model $N$ and an elementary
embedding $j:M\longrightarrow N$ so that $\operatorname{crit}(j)=\kappa$ and
$j,M\in N$.
However, we are mostly interested a different case, namely, when $j$ is the
elementary embeddings associated with an $M$-normal ultrafilter on $\kappa$
and where $N$ is the ultrapower of $M$. A filter
$U\subseteq\mathcal{P}(\kappa)\cap M$ is an $M$-normal ultrafilter if $U$ is
an $M$-ultrafilter, and for every $A\in U$ and regressive function
$f:A\to\kappa$ in $M$ there exists some $A^{\prime}\subseteq A$ in $U$ so that
$f\upharpoonright A^{\prime}$ is constant. We note that being $M$-normal
implies that $U$ is closed under intersections of $<\kappa$-sequences in $M$
consisting of sets in $U$.
It is routine to verify that each elementary embedding $j:M\to N$ as in the
proposition above gives rise to an $M$-normal ultrafilter
$U_{j}=\\{A\in\mathcal{P}(\kappa)\cap M\mid\kappa\in j(A)\\}$. Conversely, we
can associate to each $M$-normal ultrafilter $U$ its ultrapower embedding
$j_{U}:M\to N\cong\operatorname{Ult}(M,U)$.
###### Proposition 1.10.
Let $M$ be a $\kappa$-model and $B\in M$ a subset of $\kappa$. If $B\in U$ for
every $M$-normal ultrafilter $U$ on $\kappa$, then
$B\in\mathcal{F}_{\operatorname{WC}}$.
###### Proof.
Let $M,B$ be as in the statement of the proposition, and fix a subset
$E_{M}\subseteq\kappa\times\kappa$ so that $(\kappa,E_{M})$ is isomorphic to
$M$. To prove that $B\in\mathcal{F}_{\operatorname{WC}}$, it suffices to show
that there is a $\Pi^{1}_{1}$ statement $\Psi$ satisfied by
$(V_{\kappa},\in,E_{M},B)$ so that the set $\\{\alpha<\kappa:\alpha\text{ is
regular and }(V_{\alpha},\in,E_{M}\cap V_{\alpha},B\cap\alpha)\models\Psi\\}$
is contained in $B$.
To begin, we observe that the assertions that “$E_{M}$ is well-founded”, that
“$(\kappa,E_{M})$ is isomorphic to a transitive $\kappa$-model”, and that “$B$
is represented in $(\kappa,E_{M})$ by $b\in\kappa$”, are all within the class
of $\Pi^{1}_{1}$ formulas over $(V_{\kappa},\in,E_{M},B)$ (see [29], Section 2
for details). Let $\Phi_{0}$ denote their conjunction.
We also note that for a subset $U_{M}\subseteq\kappa$, the assertion “$U_{M}$
codes a subset of $M\cap\mathcal{P}(\kappa)$ which is an $M$-normal
ultrafilter,” is $\Sigma^{0}_{\omega}$. Therefore the assertion $\Phi_{1}$
stating that
$``\forall\,U_{M}\subseteq\kappa\text{, if }U_{M}\text{ codes an
}M\text{-normal ultrafilter, then }B\in U_{M}"$
is $\Pi^{1}_{1}$. Let $\Phi$ be the conjunction $\Phi_{0}\wedge\Phi_{1}$, a
$\Pi^{1}_{1}$ formula satisfied in $(V_{\kappa},\in,E_{M},B)$.
Define $X:=\\{\alpha<\kappa:\alpha\text{ is regular, and
}(V_{\alpha},\in,E_{M}\cap V_{\alpha},B\cap\alpha)\models\Phi\\}$, and we show
that $X\subseteq B$. Fix some $\alpha\in X$. Then the relation
$E_{M}\upharpoonright\alpha=E_{M}\cap V_{\alpha}\subseteq\alpha\times\alpha$
is well-founded, and it codes an $\alpha$-model $M_{\alpha}$ with
$B\cap\alpha$ represented in $(\alpha,E_{M}\upharpoonright\alpha)$ by the same
element $b\in\kappa$ which represents $B$ in $(\kappa,E_{M})$. Let
$i_{\alpha}:M_{\alpha}\to M_{\kappa}$ be the elementary embedding resulting
from the identifications
$M_{\alpha}\cong(\alpha,E_{M}\upharpoonright\alpha)\prec(\kappa,E_{M})\cong
M$. It is straightforward to verify that
$\operatorname{cp}(i_{\alpha})=\alpha$, $i_{\alpha}(\alpha)=\kappa$, and
$i_{\alpha}(B\cap\alpha)=B$.
It follows that $U_{\alpha}=\\{A\subseteq\alpha:\alpha\in i_{\alpha}(A)\\}$ is
an $M_{\alpha}$-normal ultrafilter. Let $j_{\alpha}:M_{\alpha}\to N_{\alpha}$
be the induced ultrapower embedding, and let $k_{\alpha}:N_{\alpha}\to M$ be
the factor map given by $k_{\alpha}([f]_{U_{\alpha}})=i_{\alpha}(f)(\alpha)$.
We note that ${\operatorname{cp}}(k_{\alpha})>\alpha$. Since
$(V_{\alpha},\in,E_{M}\upharpoonright\alpha,B\cap\alpha)$ satisfies $\Phi$,
$B\cap\alpha\in U_{\alpha}$. The last implies that $\alpha\in
j_{\alpha}(B\cap\alpha)$, which in turn implies that
$\alpha=k_{\alpha}(\alpha)\in k_{\alpha}\circ
j_{\alpha}(B\cap\alpha)=i_{\alpha}(B\cap\alpha)=B$. ∎
We now review the relevant facts about ineffable cardinals; see [6] for the
details. A cardinal $\lambda$ is ineffable if for any sequence
$\vec{A}=\langle A_{\nu}:\nu<\lambda\rangle$ so that $A_{\nu}\subseteq\nu$ for
all $\nu<\lambda$, there is an $A\subseteq\lambda$ so that
$\left\\{\nu<\lambda:A_{\nu}=A\cap\nu\right\\}$ is stationary in $\lambda$.
Such a set $A$ is said to be coherent for $\vec{A}$.
###### Notation 1.11.
In the case that $\kappa$ is ineffable, we will denote the ineffability ideal
on $\kappa$ by $\cal{I}_{in}$ throughout the paper, and we will let
$\cal{F}_{in}$ denote the filter on $\kappa$ which is dual to $\cal{I}$.
$\cal{I}_{in}$ consists of all $S\subseteq\kappa$ so that for some sequence
$\vec{A}$ as above, no stationary subset of $S$ is coherent for $\vec{A}$. In
the case that $\kappa$ is ineffable, $\cal{I}_{in}$ is a proper, normal ideal
on $\kappa$. Finally, we mention the following theorem of Baumgartner’s ([6],
Theorem 7.2); see [24] for more information.
###### Theorem 1.12.
(Baumgartner) $F_{WC}$ is contained in $\cal{F}_{in}$.
In fact, Baumgartner showed that $\cal{F}_{WC}$ is contained in the filter
dual to the _weak_ ineffability ideal on $\kappa$.
###### Notation 1.13.
$\cal{F}$ will denote either $\cal{F}_{WC}$ or $\cal{F}_{in}$ throughout this
paper depending on whether $\kappa$ is weakly compact or ineffable
(respectively). $\cal{I}$ denotes the ideal dual to $\cal{F}$. We will specify
in Definition 2.14 in the next section exactly when $\cal{F}$ is equal to
$\cal{F}_{WC}$ or equal to $\cal{F}_{in}$.
###### Remark 1.14.
1. (1)
In our paper, we only use the ineffability of $\kappa$ to prove Proposition
3.22 and in the corollaries of this proposition. All other results in our
paper can be carried out assuming only that $\kappa$ is weakly compact.
2. (2)
As is immediate from Theorem 1.12 and the fact that $\cal{F}$ is either
$\cal{F}_{WC}$ or $\cal{F}_{in}$, all $\cal{F}$-positive sets are also
$F_{WC}$-positive.
3. (3)
Thus, if $B\in\cal{F}^{+}$ and if $M$ is a $\kappa$-model with $B\in M$, then
by Proposition 1.10 there is some $M$-normal ultrafilter $U$ on $\kappa$ so
that $B\in U$. We will use this fact throughout the paper.
As an illustration of the type of argument in (3) above, we prove the
following fact which we will need in the proof of Proposition 4.4.
###### Lemma 1.15.
Suppose that $B\in\cal{F}^{+}$. Then
$B\backslash\operatorname{tr}(B)\in\cal{I}$.
###### Proof.
Suppose otherwise, for a contradiction. Then
$B\backslash\operatorname{tr}(B)\in\cal{F}^{+}$. Let $M^{*}$ be a
$\kappa$-model containing $B$, and hence $B\backslash\operatorname{tr}(B)$. By
Proposition 1.10, there is an $M^{*}$-normal ultrafilter $U$ so that, letting
$j:M^{*}\longrightarrow N$ be the ultrapower embedding, $\kappa\in
j(B\backslash\operatorname{tr}(B))$. However, $B$ is a stationary subset of
$\kappa$, since $B\in\cal{F}^{+}$. Thus $\kappa\in j(\operatorname{tr}(B))$.
Since $\kappa\in j(B)$ also, we have $\kappa\in j(B)\cap
j(\operatorname{tr}(B))=j(B\cap\operatorname{tr}(B))$, which is a
contradiction. ∎
## 2\. $\cal{F}$-strongly proper posets
In this section we transition into the main body of the paper. After briefly
reviewing some important facts about strong genericity in Subsection 2.1, we
then define, in Subsection 2.2, the class of $\cal{F}$-strongly proper posets
(see Notation 1.13 for some information about $\cal{F}$ and see Definition
2.14 for the exact definition). This class, which includes the collapse
$\mathbb{P}$, consists of posets for which we may build various residue
systems and thereby obtain strongly generic conditions for models of interest.
### 2.1. Review of Strongly Generic Conditions
Here we review the definition and basic properties of strongly generic
conditions. Much of this material was originally developed by Mitchell [37].
Parts of our exposition here summarize the exposition in [31], Section 1, to
which we refer the reader for proofs.
###### Definition 2.1.
Let $N\prec H(\theta)$, where $\theta$ is regular. Let $\mathbb{Q}\in N$ be a
poset. A condition $q\in\mathbb{Q}$ is said to be a strongly
$(N,\mathbb{Q})$-generic condition if for any set $D$ which is dense in
$\mathbb{Q}\cap N$, $D$ is predense above $q$ in $\mathbb{Q}$.
###### Remark 2.2.
Note that if $\mathbb{Q}\in N$, with $N$ as above, then any strongly
$(N,\mathbb{Q})$-generic condition is also an $(N,\mathbb{Q})$-generic
condition. Moreover, $q$ is a strongly $(N,\mathbb{Q})$-generic condition iff
$q\Vdash_{\mathbb{Q}}\dot{G}_{\mathbb{Q}}\cap N$ is a $V$-generic filter over
$\mathbb{Q}\cap N$.
We now review a combinatorial characterization of strongly generic conditions,
implicit in [37], Proposition 2.15, in terms of the existence of residue
functions.
###### Definition 2.3.
Suppose that $\mathbb{Q}\in N\prec H(\theta)$, $q\in\mathbb{Q}$, and
$s\in\mathbb{Q}\cap N$. $s$ is said to be a residue of $q$ to $N$ if for all
$t\geq_{\mathbb{Q}\cap N}s$, $t$ and $q$ are compatible in $\mathbb{Q}$.
A residue function for $N$ above $q$ is a function $f_{N}$ defined on
$\mathbb{Q}/q$ so that for each $r\in\mathbb{Q}/q$, $f_{N}(r)$ is a residue of
$r$ to $N$.
Finally, if $q,r\in\mathbb{Q}$ and $s\in\mathbb{Q}\cap N$, we say that $s$ is
a dual residue of $q$ and $r$ to $N$ if $s$ is a residue for both $q$ and $r$
to $N$.
###### Lemma 2.4.
$q\in\mathbb{Q}$ is $(N,\mathbb{Q})$-strongly generic iff there is a residue
function for $N$ above $q$.
In the next subsection, we will isolate further properties of residue
functions of interest. For now, we review the process by which strongly
generic conditions allow us to break apart the forcing $\mathbb{Q}$ into a
two-step iteration.
###### Notation 2.5.
Let $\mathbb{Q}$ be a poset and $q\in\mathbb{Q}$. Suppose $\mathbb{Q}\in
N\prec H(\theta)$ and $q$ is a strongly $(N,\mathbb{Q})$-generic condition.
Fix a $V$-generic filter $\bar{G}$ over $\mathbb{Q}\cap N$. In $V[\bar{G}]$,
let $(\mathbb{Q}/q)/\bar{G}$ denote the poset where conditions are all
$r\in(\mathbb{Q}/q)$ which are $\mathbb{Q}$-compatible with every condition in
$\bar{G}$. The ordering is the same as in $\mathbb{Q}$.
The following two results originate in [36]; our formulation of them follows
[31].
###### Lemma 2.6.
Suppose $\mathbb{Q}\in N\prec H(\theta)$ and $q$ is a strongly
$(N,\mathbb{Q})$-generic condition. Then for all $r\geq q$ and
$s\in\mathbb{Q}\cap N$, $s$ is a residue of $r$ to $N$ iff
$s\Vdash_{\mathbb{Q}\cap N}r\in(\mathbb{Q}/q)/\dot{G}_{\mathbb{Q}\cap N}$.
###### Lemma 2.7.
Suppose $\mathbb{Q}\in N\prec H(\theta)$ and $q$ is a strongly
$(N,\mathbb{Q})$-generic condition. Then
1. (1)
if $r\geq q$, $s\in\mathbb{Q}\cap N$, and $r$ and $s$ are
$\mathbb{Q}$-compatible, then there exists $t\geq_{\mathbb{Q}\cap N}s$ so that
$t$ is a residue of $r$ to $N$;
2. (2)
if $D\subseteq\mathbb{Q}$ is dense above $q$, then $\mathbb{Q}\cap N$ forces
that $D\cap(\mathbb{Q}/q)/\dot{G}_{\mathbb{Q}\cap N}$ is dense in
$(\mathbb{Q}/q)/\dot{G}_{\mathbb{Q}\cap N}$.
### 2.2. Exact Residue Functions and $\cal{F}$-Strong Properness
Following Neeman ([18]), we next isolate the properties of residue functions
(see Definition 2.3) of interest. We will apply this in our work to the
iteration $\mathbb{P}*\dot{\mathbb{C}}$, consisting of the collapse poset
$\mathbb{P}$ followed by a Magidor-style, club-adding iteration
$\dot{\mathbb{C}}$. Neeman also connected this with countable closure of the
quotient forcing ([18], subsection 2.2). However, we were not able to apply
this analysis to the final poset, which also includes the specializing
iteration, as we do not know if the quotients involving the specializing
iteration are even strategically closed. This will lead us later, in Section
4, to an ad-hoc proof that the quotients of the final poset preserve
stationary sets consisting of countable cofinality ordinals, without having
$\omega_{1}$-closed quotients.
Recalling that we are working with pre-orders (in anticipation of working with
iterations later), we begin our discussion with the following definition.
###### Definition 2.8.
Let $\mathbb{Q}$ be a poset which is $\omega_{1}$-closed with sups.
$D\subseteq\mathbb{Q}$ is said to be countably $=^{*}$-closed if
1. (a)
for each $q\in D$ and $r\in\mathbb{Q}$, if $r=^{*}q$, then $r\in D$;
2. (b)
if $\langle q_{n}:n\in\omega\rangle$ is an increasing sequence of conditions
all of which are in $D$ and if $q^{*}$ is a sup of the sequence, then
$q^{*}\in D$.
Such sets $D$ will arise later as the domains of _exact_ (see below) residue
functions, whose domains need not in general be all of the poset under
consideration, but only a dense, $=^{*}$-closed subset. We will construct such
functions in Proposition 5.6.
The following is Neeman’s notion of an exact strong residue function for $N$
with dense domain above $q$ ([18], Definitions 1.6, 2.10), but with the
requirement of strategic continuity strengthened to continuity.
###### Definition 2.9.
Let $\mathbb{Q}$ be a poset which is $\omega_{1}$-closed with sups, and fix
$N$ with $\mathbb{Q}\in N\prec H(\theta)$. Let $q\in\mathbb{Q}$.
A partial function $f:\mathbb{Q}/q\rightharpoonup\mathbb{Q}\cap N$ is said to
be an exact, strong residue function for $N$ above $q$ if it satisfies the
following properties:
1. (1)
(dense domain) the domain of $f$ is a dense, countably $=^{*}$-closed subset
$D$ of $\mathbb{Q}/q$;
2. (2)
(projection) $r\geq f(r)$ for all $r\in D$;
3. (3)
(order preservation) for all $r^{*},r\in D$, if $r^{*}\geq r$, then
$f(r^{*})\geq f(r)$;
4. (4)
(strong residue) for any $r\in D$ and any $u\in\mathbb{Q}\cap N$ so that
$u\geq f(r)$, there exists $r^{*}\geq r$ with $r^{*}\in D$ so that
$f(r^{*})\geq u$;
5. (5)
(countable continuity) if $\langle r_{n}:n\in\omega\rangle$ is an increasing
sequence of conditions in $D$ with a sup $r^{*}$, then $f(r^{*})$ is a sup of
$\langle f(r_{n}):n\in\omega\rangle$.111Note that $r^{*}$ is in $D$ by (1) and
also that the sequence $\langle f(r_{n}):n\in\omega\rangle$ is increasing by
(3).
We call such a pair $\langle q,f\rangle$ a residue pair for $(N,\mathbb{Q})$,
or just a residue pair for $N$ if $\mathbb{Q}$ is clear from context.
The following appears in [18] (Lemma 2.11).
###### Lemma 2.10.
Suppose that $\mathbb{Q}$ is separative, $q\in\mathbb{Q}$, and that
$f:\mathbb{Q}/q\longrightarrow\mathbb{Q}\cap N$ is a function satisfying
properties (2) and (4) of Definition 2.9. Then $f$ is order-preserving on its
domain.
###### Example 2.11.
Let $\alpha<\kappa$ be inaccessible. Then the function
$f:\mathbb{P}\longrightarrow\mathbb{P}\upharpoonright\alpha$ given by
$f(p)=p\upharpoonright\alpha$ is an exact, strong residue function for any
$M\prec H(\theta)$ with $M\cap\kappa=\alpha$ above the condition $\emptyset$
and has all of $\mathbb{P}$ as its domain.
Our next task is to isolate the models which for us will play the role of
“$N$” in Definition 2.9. First some notation which we will fix for the
remainder of the paper.
###### Notation 2.12.
Let $\lhd$ be a fixed well-order of $H(\kappa^{+})$.
In the following definitions and claims we make a standard use of continuous
sequences of elementary substructures $M_{\alpha}$, where
$|M_{\alpha}|=\alpha<\kappa$ and $M_{\alpha}\cap\kappa=\alpha$, to form
natural restrictions $\mathbb{P}^{*}\cap M_{\alpha}$ of posets
$\mathbb{P}^{*}$ which are members of the models on the chain. For ease of
notation in describing such chains, we use terminology similar to [31] and
introduce the notion of a $P$-suitable sequence, for a parameter $P$.
###### Definition 2.13.
Let $P\in H(\kappa^{+})$ be a parameter. We say that a sequence $\langle
M_{\alpha}:\alpha\in A\rangle$ is $P$-suitable if
1. (1)
$A\in{\cal{F}^{+}}$;
2. (2)
for each $\alpha\in A$, $\alpha$ is inaccessible,
$M_{\alpha}\cap\kappa=\alpha$, $\,{}^{<\alpha}M_{\alpha}\subseteq M_{\alpha}$,
and $|M_{\alpha}|=\alpha$;
3. (3)
for each $\alpha\in A$, $M_{\alpha}\prec(H(\kappa^{+}),\in,\lhd)$ and $P\in
M_{\alpha}$;
4. (4)
if $\alpha<\beta$ are in $A$, then $M_{\alpha}\in M_{\beta}$, and also if
$\gamma\in A\cap\lim(A)$, then $M_{\gamma}=\bigcup\left\\{M_{\delta}:\delta\in
A\cap\gamma\right\\}$.
We refer to a single model $M$ satisfying (2) and (3) as a $P$-suitable model.
It is clear from the definition that if $\vec{M}$ is $P$-suitable and
$B\subseteq\operatorname{dom}(\vec{M})$ is in $\cal{F}^{+}$, then $\langle
M_{\alpha}:\alpha\in B\rangle$ is also $P$-suitable. It is also clear that for
any $P\in H(\kappa^{+})$, there exists a $P$-suitable sequence.
The next definition is the main item of this section; it specifies a class of
posets which contains the Levy collapse $\mathbb{P}$ and each of which can
play the role of a preparatory forcing for a $\aleph_{2}$-c.c. iteration
specializing Aronszajn trees. (The work in Section 3 is devoted to showing
this.) It is helpful to recall Notation 1.11.
###### Definition 2.14.
Let $\mathbb{P}^{*}$ be a poset in $H(\kappa^{+})$ which is
$\omega_{1}$-closed with sups and which collapses all cardinals in the
interval $(\omega_{1},\kappa)$. In the case that $\mathbb{P}^{*}=\mathbb{P}$
(recall that $\mathbb{P}=\operatorname{Col}(\omega_{1},<\kappa)$), we assume
that $\kappa$ is weakly compact, and in the case that $\mathbb{P}^{*}$ is not
$\mathbb{P}$, we assume that $\kappa$ is ineffable. Let $\cal{F}$ denote the
following filter on $\kappa$:
${\cal{F}=}\begin{cases}{\cal{F}_{WC}}&{\text{if
}\mathbb{P}^{*}=\mathbb{P}}\\\ {\cal{F}_{in}}&{\text{otherwise,}}\end{cases}$
and let $\cal{I}$ denote the ideal dual to $\cal{F}$.
We say that $\mathbb{P}^{*}$ is $\cal{F}$-strongly proper if for any
$\mathbb{P}^{*}$-suitable sequence $\vec{M}$ there exist an
$A\subseteq\operatorname{dom}(\vec{M})$ with
$\operatorname{dom}(\vec{M})\backslash A\in\cal{I}$, a sequence $\langle
p^{*}(M_{\alpha}):\alpha\in A\rangle$ of conditions in $\mathbb{P}^{*}$, and a
sequence $\langle\varphi^{M_{\alpha}}:\alpha\in A\rangle$ of functions
satisfying the following properties, for each $\alpha\in A$:
1. (1)
$\varphi^{M_{\alpha}}$ is an exact, strong residue function for
$(M_{\alpha},\mathbb{P}^{*})$ above $p^{*}(M_{\alpha})$ and
$\varphi^{M_{\alpha}}(p^{*}(M_{\alpha}))=0_{\mathbb{P}^{*}}$;
2. (2)
if $\beta\in A$ is greater than $\alpha$, then $p^{*}(M_{\alpha})$ and
$\varphi^{M_{\alpha}}$ are members of $M_{\beta}$.
We will refer to the sequence of pairs $\langle\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle:\alpha\in A\rangle$ as a residue
system for $\vec{M}\upharpoonright A$ and $\mathbb{P}^{*}$.
###### Remark 2.15.
1. (1)
A corollary of (1) of the above definition is that $p^{*}(M_{\alpha})$ is
compatible with every condition in $\mathbb{P}^{*}\cap M_{\alpha}$. Such
conditions were called _universal_ in [10].
2. (2)
Note that any $\cal{F}$-strongly proper poset has size exactly $\kappa$. It
has size at least $\kappa$ since the $\mathsf{GCH}$ holds and it collapses all
cardinals in the interval $(\omega_{1},\kappa)$ and has size no more than
$\kappa$ since it is a member of $H(\kappa^{+})$.
###### Example 2.16.
The Levy collapse poset $\mathbb{P}$ is an example of a $\cal{F}$-strongly
proper poset (noting that $\cal{F}=\cal{F}_{WC}$ in this case). Indeed,
letting $\vec{M}$ be any suitable sequence, we may take the set $A$ in
Definition 2.14 to just be $\operatorname{dom}(\vec{M})$. Then we define
$p(M_{\alpha}):=\emptyset$ for all $\alpha\in A$ and define
$\varphi^{M_{\alpha}}$ on the entire poset $\mathbb{P}$ by
$\varphi^{M_{\alpha}}(p)=p\upharpoonright\alpha$. As stated in Example 2.11,
each $\varphi^{M_{\alpha}}$ is an exact, strong residue function for
$M_{\alpha}$ above $\emptyset$. The remaining properties of Definition 2.14
are trivial.
In our intended applications, the posets playing the role of $\mathbb{P}^{*}$
in Definition 2.14 will be of the form $\mathbb{P}\ast\dot{\mathbb{C}}$, where
$\dot{\mathbb{C}}$ is a $\mathbb{P}$-name for an iteration of club-adding with
anticipation.
We now check, by a standard argument, that forcing with an $\cal{F}$-strongly
proper poset preserves $\kappa$.
###### Lemma 2.17.
Suppose that $\mathbb{P}^{*}$ is $\cal{F}$-strongly proper. Then forcing with
$\mathbb{P}^{*}$ preserves $\kappa$.
###### Proof.
By Definition 2.14, we know that $\mathbb{P}^{*}$ preserves $\omega_{1}$ and
collapses all cardinals in the interval $(\omega_{1},\kappa)$. Thus if
$\mathbb{P}^{*}$ does not preserve $\kappa$, then we may find a condition
$p\in\mathbb{P}^{*}$ and a $\mathbb{P}^{*}$-name $\dot{f}$ for a function with
domain $\omega_{1}$ which $p$ forces is cofinal in $\kappa$. Since
$\mathbb{P}^{*}$ has size $\kappa$, we may assume that the name $\dot{f}$ is a
member of $H(\kappa^{+})$. Let $\vec{M}$ be a
$\left\\{\dot{f},p,\mathbb{P}^{*}\right\\}$-suitable sequence. Also let
$A\subseteq\operatorname{dom}(\vec{M})$ witness Definition 2.14, and let $N$
denote the least model on the sequence $\vec{M}\upharpoonright A$. By
Definition 2.14, we may find a condition $p^{*}(N)$ and an exact, strong
residue function for $(N,\mathbb{P}^{*})$ above $p^{*}(N)$. Since $p\in N$,
Definition 2.14(1) implies that $p^{*}(N)$ and $p$ are compatible. So let $q$
be an extension of them both. Then since $q$ is an
$(N,\mathbb{P}^{*})$-strongly generic condition and $\dot{f}\in N$, $q$ forces
that $\operatorname{ran}(\dot{f})\subseteq N\cap\kappa<\kappa$. But this
contradicts the fact that $p$ forces that $\dot{f}$ is unbounded in $\kappa$.
∎
The remainder of the subsection is dedicated to proving lemmas about how
suitable sequences interact with the weak compactness of $\kappa$.
###### Lemma 2.18.
Let $P\in H(\kappa^{+})$ and $\vec{M}$ be $P$-suitable. Then
$P\subseteq\bigcup_{\alpha\in\operatorname{dom}(\vec{M})}M_{\alpha}$.
###### Proof.
By definition of a suitable sequence, each model on the sequence is elementary
with respect to the fixed well-order $\lhd$ on $H(\kappa^{+})$, and therefore
each model contains the $\lhd$-least surjection $\psi$ from $\kappa$ onto $P$.
Then
$P=\psi[\kappa]=\bigcup_{\alpha\in\operatorname{dom}(\vec{M})}\psi[\alpha]\subseteq\bigcup_{\alpha\in\operatorname{dom}(\vec{M})}M_{\alpha}.$
∎
###### Lemma 2.19.
Let $P\in H(\kappa^{+})$, and let $\vec{M}$ be $P$-suitable. Suppose that
$M^{*}$ is a $\kappa$-model containing $\vec{M}$ and that $U$ is an
$M^{*}$-normal ultrafilter on $\kappa$ so that $\operatorname{dom}(\vec{M})\in
U$. Let $j:M^{*}\longrightarrow N$ be the ultrapower embedding. Then
1. (1)
$\kappa\in\operatorname{dom}(j(\vec{M}))$, and
$j(\vec{M})(\kappa)=\bigcup_{\alpha\in\operatorname{dom}(\vec{M})}j[M_{\alpha}]$;
2. (2)
$j(P)\cap j(\vec{M})(\kappa)=j[P];$
3. (3)
$\bigcup_{\alpha\in\operatorname{dom}(\vec{M})}M_{\alpha}$ is transitive, and
$j^{-1}\upharpoonright M_{\kappa}$ is the transitive collapse of $M_{\kappa}$.
###### Proof.
Let $B:=\operatorname{dom}(\vec{M})$. The first part of item (1) follows since
$B\in U=\left\\{X\in\cal{P}(\kappa)\cap M^{*}:\kappa\in j(X)\right\\}$. Thus
$\kappa\in\operatorname{dom}(j(\vec{M}))$, and so we may let
$M_{\kappa}:=j(\vec{M})(\kappa)$. Additionally, $j(B)\cap\kappa=B$, and so by
Definition 2.13(4), $M_{\kappa}=\bigcup_{\alpha\in B}j(M_{\alpha})$. But
$|M_{\alpha}|=\alpha<\kappa$ for each $\alpha\in B$, and hence
$j(M_{\alpha})=j[M_{\alpha}]$. Thus
$M_{\kappa}=\bigcup_{\alpha\in B}j[M_{\alpha}],$
completing the proof of (1). (2) follows immediately.
For (3), observe that if $x\in
M:=\bigcup_{\alpha\in\operatorname{dom}(\vec{M})}M_{\alpha}$, then a tail of
the sequence $\vec{M}$ is $x$-suitable, and so $x\subseteq M$ by Lemma 2.18.
Thus $M$ is transitive. Since $j^{-1}\upharpoonright M_{\kappa}$ is an
$\in$-isomorphism whose range (namely $M$) is transitive, $j^{-1}$ is the
transitive collapse. ∎
###### Lemma 2.20.
Suppose that $\vec{M}$ is $\mathbb{P}^{*}$-suitable and that $\langle\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle:\alpha\in\operatorname{dom}(\vec{M})\rangle$
is a residue system for $\vec{M}$ and $\mathbb{P}^{*}$. Then we may find some
$A\subseteq\operatorname{dom}(\vec{M})$ with
$\operatorname{dom}(\vec{M})\backslash A\in\cal{I}$ so that for all $\alpha\in
A$, $\mathbb{P}^{*}\cap M_{\alpha}\Vdash\check{\alpha}=\dot{\aleph}_{2}$.
###### Proof.
Suppose that $\vec{M}$ and $\langle\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle:\alpha\in\operatorname{dom}(\vec{M})\rangle$
are as in the statement of the lemma. For a contradiction, assume that
${B:=\left\\{\alpha\in\operatorname{dom}(\vec{M}):\mathbb{P}^{*}\cap
M_{\alpha}\not\Vdash\check{\alpha}=\dot{\aleph}_{2}\right\\}\in\cal{F}^{+}.}$
Since $\cal{F}_{WC}\subseteq\cal{F}$, $B$ is also in $\cal{F}_{WC}^{+}$.
Let $M^{*}$ be a $\kappa$-model containing $\vec{M}$, $B$, $\mathbb{P}^{*}$,
and the sequence $\langle\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle:\alpha\in\operatorname{dom}(\vec{M})\rangle$.
By Proposition 1.10, since $\kappa\backslash B\notin\cal{F}_{WC}$, we may find
some $M^{*}$-normal measure $U$ so that, letting $j:M^{*}\longrightarrow N$ be
the associated ultrapower embedding, $\kappa\in j(B)$. Let
$M_{\kappa}=j(\vec{M})(\kappa)$. Then $N$ satisfies that
$j(\mathbb{P}^{*})\cap M_{\kappa}$ does not force that
$\check{\kappa}=\dot{\aleph}_{2}$. On the other hand, by the previous lemma,
we know that $j(\mathbb{P}^{*})\cap M_{\kappa}=j[\mathbb{P}^{*}]$. Since
$j[\mathbb{P}^{*}]$ is isomorphic to $\mathbb{P}^{*}$ and $\mathbb{P}^{*}$
forces that $\kappa$ becomes $\aleph_{2}$ (by Lemma 2.17), this implies that
$j[\mathbb{P}^{*}]$ forces that $\kappa=\dot{\aleph}_{2}$. Thus $N$ also
satisfies that $j[\mathbb{P}^{*}]=j(\mathbb{P}^{*})\cap M_{\kappa}$ forces
that $\check{\kappa}=\dot{\aleph}_{2}$, a contradiction.
∎
We recall that in Definition 2.14(1), the condition $p^{*}(M_{\alpha})$ is
required to be compatible with every condition in $\mathbb{P}^{*}\cap
M_{\alpha}$. A practical corollary of this is that any generic for
$\mathbb{P}^{*}$ contains plenty of conditions of the form
$p^{*}(M_{\alpha})$.
###### Lemma 2.21.
Suppose that $\vec{M}$ is $\mathbb{P}^{*}$-suitable, that $\langle\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle:\alpha\in\operatorname{dom}(\vec{M})\rangle$
is a residue system for $\vec{M}$ and $\mathbb{P}^{*}$, and that
$B\subseteq\operatorname{dom}(\vec{M})$ is in $\cal{F}^{+}$. Suppose that
there is a condition $\bar{p}\in\mathbb{P}^{*}$ satisfying that for each
$\alpha\in B$, there is a condition
$p_{\alpha}\in\operatorname{dom}(\varphi^{M_{\alpha}})$ so that
$\bar{p}=^{*}\varphi^{M_{\alpha}}(p_{\alpha})$. Then
$\bar{p}\Vdash\dot{X}:=\left\\{\alpha<\kappa:p_{\alpha}\in\dot{G}_{\mathbb{P}^{*}}\right\\}\text{
is unbounded in }\kappa.$
In particular, taking $p_{\alpha}:=p^{*}(M_{\alpha})$ with $\bar{p}$ the
trivial condition, and recalling Definition 2.14(1),
$\mathbb{P}^{*}\Vdash\left\\{\alpha<\kappa:p^{*}(M_{\alpha})\in\dot{G}_{\mathbb{P}^{*}}\right\\}\text{
is unbounded in }\kappa.$
Moreover, letting $E$ abbreviate $\operatorname{dom}(\vec{M})$, if $\alpha\in
E\cap\lim(E)$, then
$\Vdash_{\mathbb{P}^{*}\cap
M_{\alpha}}\left\\{\xi<\alpha:p^{*}(M_{\xi})\in\dot{G}_{\mathbb{P}^{*}\cap
M_{\alpha}}\right\\}\text{ is unbounded in }\alpha.$
###### Proof.
Let $p\in\mathbb{P}^{*}$ be a condition extending $\bar{p}$, and let
$\gamma<\kappa$. We find an extension of $p$ which forces that
$\dot{X}\backslash\gamma\neq\emptyset$. Since $B$ is unbounded in $\kappa$ and
$\vec{M}$ is $\mathbb{P}^{*}$-suitable, Lemma 2.18 implies that $p\in
M_{\beta}$ for some $\beta\in B\backslash(\gamma+1)$. By definition of an
exact, strong residue function, $p_{\beta}$ is compatible with
$p\geq\bar{p}=^{*}\varphi^{M_{\beta}}(p_{\beta})$. Therefore, let $p^{*}$ be a
common extension; then $p^{*}\Vdash\beta\in\dot{X}\backslash\gamma$.
The proof in the case $\alpha\in E\cap\lim(E)$ is identical, using the fact
that, by Definition 2.13, $M_{\alpha}=\bigcup_{\xi\in B\cap\alpha}M_{\xi}$ in
this case. ∎
## 3\. $\cal{F}$-Strongly Proper Posets and Specializing Aronszajn Trees on
$\omega_{2}$
In this section we will prove that if $\mathbb{P}^{*}$ is an
$\cal{F}$-strongly proper poset, then we can iterate to specialize Aronszajn
trees on $\kappa$ in the extension by $\mathbb{P}^{*}$ (see Definition 2.14
for the definition of $\cal{F}$). Recall from Lemma 2.17 that $\mathbb{P}^{*}$
forces that $\kappa$ becomes $\aleph_{2}$ and also preserves the
$\mathsf{CH}$; thus there are in fact Aronszajn trees on $\kappa$ in any
$\mathbb{P}^{*}$-extension. By Example 2.16, the collapse poset $\mathbb{P}$
is $\cal{F}$-strongly proper, and therefore, in the case that
$\mathbb{P}^{*}=\mathbb{P}$, our results here generalize those of [33] (recall
that $\cal{F}=\cal{F}_{WC}$ when $\mathbb{P}^{*}=\mathbb{P}$).
We consider a countable support iteration
$\dot{\mathbb{S}}=\langle\dot{\mathbb{S}}_{\xi},\dot{\mathbb{S}}(\xi):\xi<\kappa^{+}\rangle$
of length $\kappa^{+}$, specializing Aronszajn trees on $\kappa$ in the
$\mathbb{P}^{*}$-extension. More precisely, $\dot{\mathbb{S}}$ is a
$\mathbb{P}^{*}$-name for an iteration with countable support so that for any
$\xi<\kappa^{+}$, $\dot{\mathbb{S}}(\xi)$ is an $\dot{\mathbb{S}}_{\xi}$-name
for the poset $\dot{\mathbb{S}}(\dot{T}_{\xi})$, where $\dot{T}_{\xi}$ is a
nice $\dot{\mathbb{S}}_{\xi}$-name for an Aronszajn tree on $\kappa$; see
Definition 1.6 for the exact definition of posets of the form $\mathbb{S}(T)$.
The selection of the names $\dot{T}_{\xi}$ \- and hence the definition of the
iteration - is determined by using the fixed well-order $\lhd$ of
$H(\kappa^{+})$ from Notation 2.12 as a bookkeeping function. In particular,
for each $\xi<\kappa^{+}$, the name $\dot{\mathbb{S}}_{\xi}$ is definable in
$(H(\kappa^{+}),\in,\lhd)$ from $\mathbb{P}^{*}$ and $\xi$, and consequently
it is a member of any model which is suitable with respect to $\mathbb{P}^{*}$
and $\xi$. We will use $\mathbb{R}_{\xi}$ to abbreviate
$\mathbb{P}^{*}\ast\dot{\mathbb{S}}_{\xi}$ for each $\xi<\kappa^{+}$.
Since the poset $\mathbb{R}_{\xi}$ is $\omega_{1}$-closed, it is
straightforward to see that $\mathbb{R}_{\xi}$ has a dense set of determined
conditions, i.e., conditions $(p,\dot{f})$ for which there is some function
$f$ in $V$ so that $p\Vdash_{\mathbb{P}^{*}}\dot{f}=\check{f}$. The dense set
of determined conditions is also closed under sups of countable increasing
sequences. Thus we will assume that all future conditions are determined.
###### Notation 3.1.
Strictly speaking, the domain of a (determined) condition $f$ in
$\mathbb{R}_{\xi}$ is a countable subset of $\xi$, and for each
$\zeta\in\operatorname{dom}(f)$, $f(\zeta)$ is itself a function whose domain
is a countable subset of $\omega_{1}$. However, we will often make an abuse of
notation and write $f(\zeta,\nu)$ to mean the countable set of tree nodes
$f(\zeta)(\nu)$.
The main goal of this section is to prove that $\dot{\mathbb{S}}$ is forced to
be $\kappa$-c.c. For this it suffices to prove the following:
###### Theorem 3.2.
For every $\rho<\kappa^{+}$ it is forced by the trivial condition of
$\mathbb{P}^{*}$ that $\dot{\mathbb{S}}_{\rho}$ is $\kappa$-c.c.
We will prove Theorem 3.2 by induction on $\rho$. Doing so will require two
induction hypotheses, the first of which is the following:
Inductive Hypothesis I: For each $\xi<\rho$,
$\mathbb{P}^{*}\Vdash\dot{\mathbb{S}}_{\xi}$ is $\kappa$-c.c.
We will assume Inductive Hypothesis I throughout the entire section. Later in
the section, after developing more of the theory, we will introduce a second,
more technical inductive hypothesis; we state this after Remark 3.15. Though
we assume the first inductive hypothesis throughout, we will only use the
second inductive hypothesis once it is introduced, and the results prior to
the statement thereof do not require it.
The rest of the section will proceed as follows. We will first establish, in
Proposition 3.4, that for all $\xi<\rho$, there are plenty of intermediate
generic extensions between $V$ and the full $\mathbb{R}_{\xi}$-extension in
which various restrictions of Aronszajn trees are Aronszajn in the
intermediate model. In light of this, we will define analogues of the
“hashtag” and “star” principles from [33]; the former will say that two
conditions have the same restriction to a given model, whereas the latter says
that two conditions have a dual residue to a given model. Afterwards, we
define the notion of a splitting pair of conditions, a notion which will play
a key role in later amalgamation arguments. Next, we will state our second
induction hypothesis, which describes the interplay between the star and
hashtag principles. Using the second induction hypothesis, we will prove that
splitting pairs exist and isolate sufficient conditions under which they can
be amalgamated (see Lemma 3.18). Finally, we show that
$\dot{\mathbb{S}}_{\rho}$ is forced to be $\kappa$-c.c., and we verify that
the second induction hypothesis holds at $\rho$. As mentioned in Remark
1.14(1), the only substantial use of the ineffability of $\kappa$ is in
verifying that the second induction hypothesis holds at $\kappa$.
###### Definition 3.3.
Let $G^{*}$ be $V$-generic over $\mathbb{P}^{*}$. If $\xi\leq\rho$,
$f\in\mathbb{S}_{\xi}$, and $\bar{f}$ is a function (not necessarily a
condition), we write $f\geq\bar{f}$ to mean that
$\operatorname{dom}(\bar{f})\subseteq\operatorname{dom}(f)$ and for all
$\langle\zeta,\nu\rangle\in\operatorname{dom}(\bar{f})$,
$\bar{f}(\zeta,\nu)\subseteq f(\zeta,\nu)$.
The next item establishes the existence of the desired intermediate generic
extensions between $V$ and $V[\mathbb{R}_{\xi}]$ for $\xi<\rho$, and in turn
the existence of plenty of residues. We recommend recalling Lemma 2.19 and
Notation 1.11 before reading the proof. In the statement of the following
proposition, we will assume that the various names are nice names for subsets
of $H(\kappa^{+})$. Thus, in light of our discussion about determined
conditions, the name $\dot{\mathbb{S}}_{\zeta}$ will be viewed as a union of
sets of the form $\left\\{f\right\\}\times A_{f}$, where
$f:\zeta\times\omega_{1}\rightharpoonup\kappa\times\omega_{1}$ is a countable
partial function and $A_{f}\subseteq\mathbb{P}^{*}$ is an antichain. This will
ensure, for instance, that $\dot{\mathbb{S}}_{\zeta}\cap M_{\alpha}$ is really
a $(\mathbb{P}^{*}\cap M_{\alpha})$-name. Similar considerations apply to the
$\mathbb{R}_{\zeta}$-name $\dot{T}_{\zeta}$.
###### Proposition 3.4.
Suppose that $\vec{M}$ is $\mathbb{R}_{\rho}$-suitable. Then there exists
$B^{*}\subseteq\operatorname{dom}(\vec{M})$ with
$\operatorname{dom}(\vec{M})\backslash B^{*}\in\cal{I}$ so that for any
$\alpha\in B^{*}$, for any residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$, and for any $\zeta\in M_{\alpha}\cap\rho$, the
following are true:
1. (1)
$(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}})$ forces that
$\dot{G}_{\mathbb{R}_{\zeta}}\cap{M_{\alpha}}$ is a $V$-generic filter for
$\mathbb{R}_{\zeta}\cap M_{\alpha}$;
2. (2)
$(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}})$ forces that
$\dot{T}_{\zeta}\cap(\check{\omega}_{1}\times\check{\alpha})=(\dot{T}_{\zeta}\cap
M_{\alpha})[\dot{G}_{\mathbb{R}_{\zeta}}\cap M_{\alpha}]$;
3. (3)
$(\mathbb{P}^{*}\cap M_{\alpha})\Vdash(\dot{\mathbb{S}}_{\zeta}\cap
M_{\alpha})$ is $\alpha$-c.c. Furthermore, $(\mathbb{R}_{\zeta}\cap
M_{\alpha})\Vdash\dot{T}_{\zeta}\cap{M_{\alpha}}$ is an Aronszajn tree on
$\alpha=\dot{\aleph}_{2}$.
###### Proof.
Fix an $\mathbb{R}_{\rho}$-suitable sequence $\vec{M}$, and let
$B:=\operatorname{dom}(\vec{M})$ so that $B\in\cal{F}^{+}$ by Definition 2.13.
Since $\mathbb{P}^{*}$ is $\cal{F}$-strongly proper, we may assume that $B$
satisfies the conclusion of Definition 2.14, by removing an $\cal{I}$-null set
if necessary. Our goal is to show that each of (1)-(3) fail only on a set in
$\cal{I}$. Thus we define $B_{1}$ to be the set of $\alpha\in B$ so that for
some $\zeta\in M_{\alpha}$ and some residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$, (1) fails for these objects;
we define $B_{2}$ to be the set of $\alpha\in B$ so that for some $\zeta\in
M_{\alpha}\cap\rho$ and some residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$, (2) fails for $M_{\alpha}$ and
$\zeta$; and we define $B_{3}$ similarly. We show that each of these is in
$\cal{I}$.
Suppose for a contradiction that $B_{i}\in\cal{F}^{+}$ for some
$i\in\left\\{1,2,3\right\\}$. Then $B_{i}\in\cal{F}_{WC}^{+}$ since
$\cal{F}_{WC}\subseteq\cal{F}$. Let $M^{*}$ be a $\kappa$ model containing
$B_{i}$ as well as the sequence $\vec{M}$. Applying Proposition 1.10, we may
fix an $M^{*}$-normal ultrafilter $U$ containing $B_{i}$, and we let
$j:M^{*}\longrightarrow N$ be the induced ultrapower embedding. In particular
$\kappa\in j(B_{i})$.
Let $M_{\kappa}:=j(\vec{M})(\kappa)$. Fix $\zeta^{*}\in M_{\kappa}\cap
j(\rho)$ for the remainder of the proof which witnesses the relevant failure
of (1), (2), or (3) on the $j$-side. Since
$M_{\kappa}=j\left[\bigcup_{\alpha\in B}M_{\alpha}\right]$, by Lemma 2.19, we
have that $\zeta^{*}\in\operatorname{ran}(j)$, and so $\zeta^{*}=j(\zeta)$ for
some $\zeta<\rho$. Moreover, $M_{\kappa}\cap
j(\mathbb{P}^{*})=j[\mathbb{P}^{*}]$.
Case 1: $i=1$ Since $\kappa\in j(B_{1})$, we may fix a residue pair $\langle
p^{*}(M_{\kappa}),\varphi^{M_{\kappa}}\rangle$ for
$(M_{\kappa},j(\mathbb{P}^{*}))$ so that
$(p^{*}(M_{\kappa}),0_{j(\dot{\mathbb{S}}_{\zeta})})$ does not force that
$\dot{G}_{j(\mathbb{R}_{\zeta})}\cap M_{\kappa}$ is generic over
$j(\mathbb{R}_{\zeta})\cap M_{\alpha}$.
To obtain our contradiction, we show that
$(p^{*}(M_{\kappa}),0_{j(\dot{\mathbb{S}}_{\zeta})})$ in fact _does_ force
that $\dot{G}_{j(\mathbb{R}_{\zeta})}\cap M_{\kappa}$ is generic over
$j(\mathbb{R}_{\zeta})\cap M_{\kappa}$. Thus fix an extension
$(q^{*},\dot{g})$ of $(p^{*}(M_{\kappa}),0_{j(\dot{\mathbb{S}}_{\zeta})})$ in
$j(\mathbb{R}_{\zeta})$. Let $A^{*}\in N$ be a maximal antichain of
$j(\mathbb{R}_{\zeta})\cap M_{\kappa}=j[\mathbb{R}_{\zeta}]$, and we will find
some extension of $(q^{*},\dot{g})$ which forces that
$A^{*}\cap\dot{G}_{j(\mathbb{R}_{\zeta})}\neq\emptyset$. Since $q^{*}$ extends
$p^{*}(M_{\kappa})$ in $j(\mathbb{P}^{*})$ and $\varphi^{M_{\kappa}}$ is an
exact, strong residue function, we may extend and relabel, if necessary, to
assume that $q^{*}\in\operatorname{dom}(\varphi^{M_{\kappa}})$. Then
$\varphi^{M_{\kappa}}(q^{*})\in j(\mathbb{P}^{*})\cap
M_{\kappa}=j[\mathbb{P}^{*}]$. Therefore $\varphi^{M_{\kappa}}(q^{*})=j(q)$
for some $q\in\mathbb{P}^{*}$.
Now let $A:=j^{-1}[A^{*}]$; since $A^{*}$ is a maximal antichain in
$j[\mathbb{R}_{\zeta}]$, $A$ is a maximal antichain in $\mathbb{R}_{\zeta}$.
However, note that since $\mathbb{R}_{\zeta}$ is not necessarily
$\kappa$-c.c., $A$ could very well have size $\kappa$, and therefore we cannot
assume that it is an element of $M^{*}$. Until after the proof of the next
claim, we will work in $V$, not $M^{*}$. Let $\dot{A}(1)$ be the
$\mathbb{P}^{*}$-name for $\left\\{f\in\dot{\mathbb{S}}_{\zeta}:(\exists
p\in\dot{G}_{\mathbb{P}^{*}})\;\;(p,f)\in A\right\\}.$ Then
$q\Vdash\dot{A}(1)$ is a maximal antichain in $\dot{\mathbb{S}}_{\zeta}$.
Since $\mathbb{P}^{*}\Vdash\dot{\mathbb{S}}_{\zeta}$ is $\kappa$-c.c., by
Inductive Hypothesis I, we may extend $q$ in $\mathbb{P}^{*}$ to some
condition $q^{\prime}$ and find an ordinal $\beta<\kappa$ and a sequence
$\langle\dot{f}_{\gamma}:\gamma<\beta\rangle$ of $\mathbb{P}^{*}$-names so
that
$q^{\prime}\Vdash^{V}_{\mathbb{P}^{*}}\dot{A}(1)=\left\\{\dot{f}_{\gamma}:\gamma<\beta\right\\}.$
Since a given $\dot{f}_{\gamma}$ needn’t be a member of $M^{*}$, we show how
to replace these names with ones that are in $M^{*}$, up to extending
$q^{\prime}$.
###### Claim 3.5.
There exist a condition $u\geq_{\mathbb{P}^{*}}q^{\prime}$ and a sequence of
$\mathbb{P}^{*}$-names $\langle\dot{h}_{\gamma}:\gamma<\beta\rangle$ in
$M^{*}$ so that
$u\Vdash^{V}(\forall\gamma<\beta)\,\dot{h}_{\gamma}=\dot{f}_{\gamma}$.
_Proof._ To find $u$, let $G^{*}$ be a $V$-generic filter over
$\mathbb{P}^{*}$ containing $q^{\prime}$. Then
$\mathbb{S}_{\zeta}:=\dot{\mathbb{S}}_{\zeta}[G^{*}]\subseteq\bigcup_{\eta\in
B}M_{\eta}[G^{*}]$, since $\mathbb{R}_{\zeta}\subseteq\bigcup_{\eta\in
B}M_{\eta}$. Since $\kappa=\aleph_{2}^{V[G^{*}]}$ and $\beta<\kappa$, there
exists some $\eta\in B$ so that for all $\gamma<\beta$,
$\dot{f}_{\gamma}[G^{*}]\in M_{\eta}[G^{*}]$. By Lemma 2.21, there exists a
$\delta\geq\eta$ so that $p^{*}(M_{\delta})\in G^{*}$. Now let $u\in G^{*}$ be
an extension of $p^{*}(M_{\delta})$ and $q^{\prime}$ so that
$u\Vdash(\forall\gamma<\beta)\,\dot{f}_{\gamma}\in
M_{\delta}[\dot{G}_{\mathbb{P}^{*}}]$.
Back in $V$ we define new names $\dot{h}_{\gamma}$ for each $\gamma<\beta$;
recalling Notation 3.1, we view conditions in $\dot{\mathbb{S}}_{\zeta}$ as
having a domain which is a countable subset of $\zeta\times\omega_{1}$ so that
each element in the range is a countable subset of $\kappa\times\omega_{1}$.
For each $\gamma<\beta$, each $\bar{\zeta}\in\zeta\cap M_{\delta}$ (an
iteration stage), each $\nu<\omega_{1}$ (corresponding to the $\nu$th tree
antichain), and each $\theta\in\delta\times\omega_{1}$ (a node with height
below $\delta$), let $A(\gamma,\bar{\zeta},\nu,\theta)$ be a maximal antichain
in $\mathbb{P}^{*}\cap M_{\delta}$ of conditions $p$ which decide whether or
not $\theta$ is a member of $\dot{f}_{\gamma}(\bar{\zeta},\nu)$. Let
$\dot{h}_{\gamma}$ be the $\mathbb{P}^{*}$-name which is interpreted in an
arbitrary generic extension via some $G^{*}$ as follows:
$\theta\in{h}_{\gamma}(\bar{\zeta},\nu)$ iff there is some $p\in
A(\gamma,\bar{\zeta},\nu,\theta)\cap G^{*}$ which forces that
$\theta\in\dot{f}_{\gamma}(\bar{\zeta},\nu)$. Otherwise $h_{\gamma}$ is
undefined.
We claim that
$u\Vdash^{V}(\forall\gamma<\beta)\,\dot{h}_{\gamma}=\dot{f}_{\gamma}$. To see
this, let $G^{*}$ be a $V$-generic filter containing $u$. Fix
$\bar{\zeta}<\zeta$ and $\nu<\omega_{1}$, and we verify that
$f_{\gamma}(\bar{\zeta},\nu)=h_{\gamma}(\bar{\zeta},\nu)$. On the one hand, if
$\theta\in h_{\gamma}(\bar{\zeta},\nu)$, then by definition
$f_{\gamma}(\bar{\zeta},\nu)$ is defined and also contains the node $\theta$.
On the other hand, if $\tau\in f_{\gamma}(\bar{\zeta},\nu)$ is a node, then
since $f_{\gamma}\in M_{\delta}[G^{*}]$ has a countable domain,
$\bar{\zeta}\in M_{\delta}[G^{*}]$, and since $f_{\gamma}(\bar{\zeta},\nu)\in
M_{\delta}[G^{*}]$ is countable, $\tau\in M_{\delta}[G^{*}]$ too. But
$M_{\delta}[G^{*}]\cap V=M_{\delta}$, since $u\in G^{*}$ is a (strongly)
$(M_{\delta},\mathbb{P}^{*})$-generic condition (as it extends
$p^{*}(M_{\delta})$). Hence $\bar{\zeta},\tau\in M_{\delta}$. Thus
$\bar{\zeta}\in M_{\delta}\cap\zeta$ and $\tau$ has height below
$M_{\delta}\cap\kappa=\delta$. Next, since $u$ is a strongly
$(M_{\delta},\mathbb{P}^{*})$-generic condition which is in $G^{*}$, and since
$A(\gamma,\bar{\zeta},\nu,\tau)$ is a maximal antichain $\mathbb{P}^{*}\cap
M_{\delta}$, we know that $A(\gamma,\bar{\zeta},\nu,\tau)\cap
G^{*}\neq\emptyset$, say with $\bar{u}$ in the intersection. But as $\tau\in
f_{\gamma}(\bar{\zeta},\nu)$, we must have that $\bar{u}$ forces that
$\tau\in\dot{f}_{\gamma}(\bar{\zeta},\nu)$, and hence $\tau\in
h_{\gamma}(\bar{\zeta},\nu)$. This completes the proof that
$u\Vdash\dot{f}_{\gamma}=\dot{h}_{\gamma}$ for each $\gamma<\beta$.
Finally, since $M^{*}$ is $<\kappa$-closed, the sequence of antichains
$\langle A(\gamma,\bar{\zeta},\nu,\theta):\gamma<\beta,\bar{\zeta}\in
M_{\delta}\cap\zeta,\nu<\omega_{1},\theta\in(\delta\times\omega_{1})\rangle$
is a member of $M^{*}$. Therefore, the sequence
$\langle\dot{h}_{\gamma}:\gamma<\beta\rangle$ is a member of $M^{*}$ too.
∎(Claim 3.5)
Continuing with the main body of the argument, let
$u\geq_{\mathbb{P}^{*}}q^{\prime}$ and
$\langle\dot{h}_{\gamma}:\gamma<\beta\rangle$ witness the above claim. Since
$q^{\prime}\Vdash_{\mathbb{P}^{*}}\dot{A}(1)=\left\\{\dot{f}_{\gamma}:\gamma<\beta\right\\}$
and $u\geq q^{\prime}$, we have
$(*)\;\;u\Vdash_{\mathbb{P}^{*}}\left\\{\dot{h}_{\gamma}:\gamma<\beta\right\\}\;\text{
is a maximal antichain in }\;\dot{\mathbb{S}}_{\zeta}.$
Since $\langle\dot{h}_{\gamma}:\gamma<\beta\rangle$ and $u$ are in $M^{*}$,
$(*)$ is satisfied in $M^{*}$. Applying $j$,
$j(u)\Vdash^{N}_{j(\mathbb{P}^{*})}\left\\{j(\dot{h}_{\gamma}):\gamma<\beta\right\\}\text{
is a maximal antichain in }j(\dot{\mathbb{S}}_{\zeta}).$
Next, $u\geq_{\mathbb{P}^{*}}q^{\prime}\geq_{\mathbb{P}^{*}}q$ so
$j(u)\geq_{j(\mathbb{P}^{*})}j(q)=\varphi^{M_{\kappa}}(q^{*})$, and $j(u)\in
j[\mathbb{P}^{*}]\subseteq M_{\kappa}$ so $j(u)$ and $q^{*}$ are compatible in
$j(\mathbb{P}^{*})$. Let $q^{**}$ be a condition extending both of them with
$\varphi^{M_{\kappa}}(q^{**})\geq j(u)$. Since $q^{**}$ extends $q^{*}$, which
forces that $\dot{g}$ is a condition in $j(\dot{\mathbb{S}}_{\zeta})$,
$q^{**}$ forces this too. As $q^{**}\geq j(u)$ also forces that
$\left\\{j(\dot{h}_{\gamma}):\gamma<\beta\right\\}$ is a maximal antichain in
$j(\dot{\mathbb{S}}_{\zeta})$, we may find an extension $r^{*}$ of $q^{**}$, a
$j(\mathbb{P}^{*})$-name $\dot{g}^{*}$, and an ordinal $\gamma<\beta$ so that
$r^{*}\Vdash\dot{g}^{*}\geq\dot{g},j(\dot{h}_{\gamma}).$
We may also extend, if necessary, to assume that
$r^{*}\in\operatorname{dom}(\varphi^{M_{\kappa}})$, since $r^{*}\geq q^{*}\geq
p^{*}(M_{\kappa})$. Let $r\in\mathbb{P}^{*}$ so that
$\varphi^{M_{\kappa}}(r^{*})=j(r)$.
Now $r^{*}\geq_{j(\mathbb{P}^{*})}q^{**}$ are both in
$\operatorname{dom}(\varphi^{M_{\kappa}})$. Since $\varphi^{M_{\kappa}}$ is
order-preserving,
$j(r)=\varphi^{M_{\kappa}}(r^{*})\geq\varphi^{M_{\kappa}}(q^{**})\geq j(u)$.
Then $r\geq u$. As a result,
$r\Vdash^{V}\dot{h}_{\gamma}=\dot{f}_{\gamma}\in\dot{A}(1)$. By definition of
$\dot{A}(1)$, we may find some $\mathbb{P}^{*}$-extension $r^{\prime}$ of $r$
so that $(r^{\prime},\dot{h}_{\gamma})$ extends some element
$(r^{\prime}_{0},\dot{f})$ of $A$. Then $j(r^{\prime}_{0},\dot{f})\in A^{*}$,
as $A=j^{-1}[A^{*}]$. Since $r^{\prime}$ extends $r$ in $\mathbb{P}^{*}$, we
get that
$j(r^{\prime})\geq_{j(\mathbb{P}^{*})}j(r)=\varphi^{M_{\kappa}}(r^{*})$. So
$j(r^{\prime})$ and $r^{*}$ are compatible in $j(\mathbb{P}^{*})$. Let
$r^{**}$ be a condition extending both of them. Then $(r^{**},\dot{g}^{*})$
extends $j(r^{\prime}_{0},\dot{f})$. Indeed, $r^{**}$ extends $j(r^{\prime})$
which extends $j(r^{\prime}_{0})$. Furthermore, $r^{**}$ extends $r^{*}$ which
forces that $\dot{g}^{*}\geq j(\dot{h}_{\gamma})$, and $r^{*}$ extends
$j(r^{\prime})$ which forces that $j(\dot{h}_{\gamma})\geq j(\dot{f})$. Thus
$(r^{**},\dot{g}^{*})$ extends $j(r^{\prime}_{0},\dot{f})$, and therefore
$(r^{**},\dot{g}^{*})\Vdash_{j(\mathbb{R}_{\zeta})}j(r^{\prime}_{0},\dot{f})\in
A^{*}\cap\dot{G}_{j(\mathbb{R}_{\zeta})}\neq\emptyset.$
However, $(r^{**},\dot{g}^{*})$ also extends the starting condition
$(q^{*},\dot{g})$. This finishes the proof that $\kappa$ is not a member of
$j(B_{1})$, which contradicts our initial case assumption otherwise.
Case 2: $i=2$ In this case, we are assuming that $\kappa\in j(B_{2})$, and we
will derive a contradiction. Let $\langle
p^{*}(M_{\kappa}),\varphi^{M_{\kappa}}\rangle$ be a residue pair for
$(M_{\kappa},j(\mathbb{P}^{*}))$ so that
$(p^{*}(M_{\kappa}),0_{j(\dot{\mathbb{S}}_{\zeta})})$ does not force the
desired equality. We will show, however, that this residue pair does in fact
force the desired equality.
Towards this end, let $G^{*}$ be $V$-generic for $j(\mathbb{R}_{\zeta})$
containing $(p^{*}(M_{\kappa}),0_{j(\dot{\mathbb{S}}_{\zeta})})$, and let
$\bar{G}^{*}:=G^{*}\cap M_{\kappa}$ which, by item (1), is a $V$-generic
filter over $j(\mathbb{R}_{\zeta})\cap M_{\kappa}=j[\mathbb{R}_{\zeta}]$. Let
$G:=j^{-1}[\bar{G}^{*}]$, which is $V$-generic over $\mathbb{R}_{\zeta}$. Let
$j(T_{\zeta})$ denote $j(\dot{T}_{\zeta})[G^{*}]$, let
$T_{\zeta}:=\dot{T}_{\zeta}[G]$, and let
$T^{\prime}_{\zeta}:=(j(\dot{T}_{\zeta})\cap
M_{\kappa})[\bar{G}^{*}]=j[\dot{T}_{\zeta}][\bar{G}^{*}]$.
Since $\dot{T}_{\zeta}$ is a nice $\mathbb{R}_{\zeta}$-name for a tree order
on $\kappa$, for each $\tau,\theta\in\kappa\times\omega_{1}$, there exists an
antichain $B_{\theta,\tau}$ of $\mathbb{R}_{\zeta}$ so that, letting
$\operatorname{op}(\theta,\tau)$ denote the canonical name for
$\langle\theta,\tau\rangle$,
$<_{\dot{T}_{\zeta}}=\bigcup\left\\{\left\\{\operatorname{op}(\theta,\tau)\right\\}\times
B_{\theta,\tau}:\theta,\tau<\kappa\right\\}.$
It is straightforward to see from this that $T_{\zeta}=T^{\prime}_{\zeta}$. So
we will show that $T_{\zeta}$ equals the restriction of $j(T_{\zeta})$ to
$\kappa\times\omega_{1}$. However, we know that $j:M^{*}\longrightarrow N$
lifts to $j:M^{*}[G]\longrightarrow N[G^{*}]$, since
$j[G]=\bar{G}^{*}\subseteq G^{*}$ and since each of the filters is generic
over the appropriate models. From this it follows that
$T_{\zeta}=j(T_{\zeta})\cap(\kappa\times\omega_{1})$. Therefore, the equality
in (2) is in fact satisfied, which contradicts the assumption that $\kappa\in
j(B_{2})$.
Case 3: $i=3$ Let $j(q)\in j[\mathbb{P}^{*}]=j(\mathbb{P}^{*})\cap M_{\kappa}$
be a condition forcing that $\dot{A}$ is a name for a $\kappa$-sized antichain
in $j(\dot{\mathbb{S}}_{\zeta})\cap M_{\kappa}$. Since $B$ satisfies
Definition 2.14, and since $B_{1}\subseteq B$ is in $U$, we may fix a residue
pair $(p^{*}(M_{\kappa}),\varphi^{M_{\kappa}})$ for
$(M_{\kappa},j(\mathbb{P}^{*}))$. Let $q^{*}\geq q,p^{*}(M_{\kappa})$ be a
condition, and let $G^{*}$ be a $V$-generic filter over $j(\mathbb{P}^{*})$
containing $q^{*}$. Then $\bar{G}^{*}$ is a $V$-generic filter over
$j(\mathbb{P}^{*})\cap M_{\kappa}=j[\mathbb{P}^{*}]$. We recall that
$j^{-1}:M_{\kappa}\longrightarrow\bigcup_{\alpha\in B_{1}}M_{\alpha}$ is the
transitive collapse, and that $j^{-1}$ lifts in the standard way from
$M_{\kappa}[G^{*}]$ to $(\bigcup_{\alpha\in B_{1}}M_{\alpha})[G]$. Now let
$A:=\dot{A}[\bar{G}^{*}]$ so that $A$ is an antichain in
$(j(\dot{\mathbb{S}}_{\zeta})\cap
M_{\kappa})[\bar{G}^{*}]=j[\dot{\mathbb{S}}_{\zeta}][\bar{G}^{*}]=j(\dot{\mathbb{S}}_{\zeta}[G^{*}])\cap
M_{\kappa}[G^{*}]$. Also, $A$ is a member of $V[\bar{G}^{*}]$ and has size
$\kappa$ there. Applying the elementarity of $j^{-1}$, we have that
$j^{-1}[A]=:\bar{A}$ is an antichain in $\dot{\mathbb{S}}_{\zeta}[G]$, where
$G:=j^{-1}[\bar{G}^{*}]$ is $V$-generic over $\mathbb{P}^{*}$. Finally,
$\bar{A}\in V[G]$ since $j^{-1}[\dot{A}]$ is a $\mathbb{P}^{*}$-name in $V$
and $\bar{A}=j^{-1}[\dot{A}][G]$. Since $\bar{A}$ has size $\kappa$ in
$V[G]=V[\bar{G}^{*}]$, this contradicts the assumption that
$\mathbb{P}^{*}\Vdash\dot{\mathbb{S}}_{\zeta}$ is $\kappa$-c.c.
For the “furthermore” part of (3), suppose now that $\dot{b}^{\prime}$ is a
$j(\mathbb{R}_{\zeta})\cap M_{\kappa}=j[\mathbb{R}_{\zeta}]$-name which $j(q)$
forces is a branch through $j(\dot{T}_{\zeta})\cap M_{\kappa}$. Fix $q^{*}$
and $G^{*}$ as in the previous paragraph. Then by (2), we see that
$j(\dot{T}_{\zeta})[G^{*}]\cap(\kappa\times\omega_{1})=\dot{T}_{\zeta}[G]$.
Moreover,
$j(\dot{T}_{\zeta})[G^{*}]\cap(\kappa\times\omega_{1})=j(\dot{T}_{\zeta})[G^{*}]\cap
M_{\kappa}[G^{*}]=(j(\dot{T}_{\zeta})\cap M_{\kappa})[\bar{G}^{*}]$. Thus
$\dot{b}^{\prime}[\bar{G}^{*}]$ is a branch through
$T_{\zeta}:=\dot{T}_{\zeta}[G]$. But $\dot{b}^{\prime}[\bar{G}^{*}]$ is a
member of $V[\bar{G}^{*}]$ and $V[\bar{G}^{*}]=V[G]$. Thus $T_{\zeta}$ is not
Aronszajn in $V[G]$, a contradiction.
Thus we see that $\kappa$ cannot be a member of $j(B_{3})$, completing Case 3
and thereby the proof. ∎
In both the previous result and Definition 2.14, there was the apparent
necessity of refining the domain of a suitable sequence so that various
desired behavior obtains on each level of the refined sequence. The next item
amalgamates this into one definition which we use frequently throughout.
###### Definition 3.6.
Let $\vec{M}$ be an $\mathbb{R}_{\rho}$-suitable sequence. We say that
$\vec{M}$ is in pre-splitting configuration up to $\rho$ if there is a residue
system $\langle\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle:\alpha\in\operatorname{dom}(\vec{M})\rangle$
satisfying items (1) and (2) of Definition 2.14 (with respect to
$\mathbb{P}^{*}$) as well as items (1)-(3) of Proposition 3.4 for all
$\alpha\in\operatorname{dom}(\vec{M})$ (with respect to $\mathbb{R}_{\rho}$).
###### Definition 3.7.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$,
$\xi\leq\rho$, and $\alpha\in\operatorname{dom}(\vec{M})$ so that $\xi\in
M_{\alpha}$. Fix a residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$.
1. (1)
For a (determined) condition $(p,f)$, we define $f\upharpoonright M_{\alpha}$
to be the function $\bar{f}$ with domain $\operatorname{dom}(f)\cap
M_{\alpha}$ so that for each
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f)\cap M_{\alpha}$,
$\bar{f}(\zeta,\nu)=f(\zeta,\nu)\cap M_{\alpha}.$
2. (2)
We define $D(\varphi^{M_{\alpha}},\xi)$ to be the set of conditions
$(p,f)\in\mathbb{R}_{\xi}$ so that
$p\in\operatorname{dom}(\varphi^{M_{\alpha}})$ and
$(\varphi^{M_{\alpha}}(p),f\upharpoonright M_{\alpha})$ is a condition in
$\mathbb{R}_{\xi}$ (and hence in $\mathbb{R}_{\xi}\cap M_{\alpha}$).
3. (3)
If $(p,f)\in D(\varphi^{M_{\alpha}},\xi)$, we make a slight abuse of notation
and define $(p,f)\upharpoonright M_{\alpha}$ to be the pair
$(\varphi^{M_{\alpha}}(p),f\upharpoonright M_{\alpha})$, when
$\varphi^{M_{\alpha}}$ is clear from context.
We observe that in general, for $(p,f)\in D(\varphi^{M_{\alpha}},\xi)$,
although $(p,f)\upharpoonright M_{\alpha}\in\mathbb{R}_{\xi}\cap M_{\alpha}$
is a condition, it need not be a residue of $(p,f)$ to $M_{\alpha}$ in the
sense that it is possible for some
$(p^{\prime},f^{\prime})\in\mathbb{R}_{\xi}\cap M_{\alpha}$ which extends
$(p,f)\upharpoonright M_{\alpha}$ to not be compatible with $(p,f)$. However,
$(p,f)\upharpoonright M_{\alpha}$ must have _some_ extension
$(\bar{p},\bar{f})\in\mathbb{R}_{\xi}\cap M_{\alpha}$ which is a residue of
$(p,f)$. This is because if $G\subseteq\mathbb{R}_{\xi}$ is $V$-generic and
contains $(p,f)$, then by Proposition 3.4 and the definition of pre-splitting
configuration, $\bar{G}:=G\cap M_{\alpha}$ is $V$-generic over
$\mathbb{R}_{\xi}\cap M_{\alpha}$ and $(p,f)\upharpoonright
M_{\alpha}\in\bar{G}$. Since $(p,f)\in G$ is compatible with every condition
in $\bar{G}$, there must be some $(\bar{p},\bar{f})\in\bar{G}$ which extends
$(p,f)\upharpoonright M_{\alpha}$ and is a residue of $(p,f)$, by Lemma 2.6.
###### Lemma 3.8.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$. Then
for each $\alpha\in\operatorname{dom}(\vec{M})$, each residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$, and each $\xi\leq\rho$ with $\xi\in
M_{\alpha}$, $D(\varphi^{M_{\alpha}},\xi)$ is dense and $=^{*}$-countably
closed in $(\mathbb{P}^{*}/p^{*}(M_{\alpha}))\ast\dot{\mathbb{S}}_{\xi}$.
###### Proof.
We will first prove the result for $\xi<\rho$ and then use this to prove the
result at $\rho$. Note that the $=^{*}$-countable closure of
$D(\varphi^{M_{\alpha}},\xi)$, in either case for $\xi$, follows from the
continuity of $\varphi^{M_{\alpha}}$ and the $=^{*}$-countable closure of the
posets. We therefore concentrate on showing density.
Let $(p_{0},f_{0})\in\mathbb{R}_{\xi}$ be given with $p_{0}\geq
p^{*}(M_{\alpha})$. By the observations following Definition 3.7, we may build
increasing sequences $\langle(p_{n},f_{n}):n\in\omega\rangle$ and
$\langle(\bar{p}_{n},\bar{f}_{n}):n\in\omega\rangle$ so that
1. (i)
$(\bar{p}_{n},\bar{f}_{n})$ is a residue of $(p_{n},f_{n})$ to $M_{\alpha}$
with $\bar{p}_{n}$ extending $\varphi^{M_{\alpha}}(p_{n})$ and with
$\bar{f}_{n}$ extending the function $f_{n}\upharpoonright M_{\alpha}$ in the
sense of Definition 3.3 (note that $f_{n}\upharpoonright M_{\alpha}$ needn’t
be forced by $\varphi^{M_{\alpha}}(p_{n})$ to be a condition);
2. (ii)
$(p_{n+1},f_{n+1})$ extends both $(p_{n},f_{n})$ and
$(\bar{p}_{n},\bar{f}_{n})$;
3. (iii)
$p_{n+1}\in\operatorname{dom}(\varphi^{M_{\alpha}})$ and
$\varphi^{M_{\alpha}}(p_{n+1})\geq\bar{p}_{n}$.
Now let $(p^{*},f^{*})$ be a sup of $\langle(p_{n},f_{n}):n\in\omega\rangle$.
Note that $p^{*}\in\operatorname{dom}(\varphi^{M_{\alpha}})$ since
$p_{n}\in\operatorname{dom}(\varphi^{M_{\alpha}})$ for each $n$ and also that
$\varphi^{M_{\alpha}}(p^{*})$ is a sup of
$\langle\varphi^{M_{\alpha}}(p_{n}):n\in\omega\rangle$. We claim that
$(\varphi^{M_{\alpha}}(p^{*}),f^{*}\upharpoonright M_{\alpha})$ is a
condition. Now
$\varphi^{M_{\alpha}}(p^{*})\geq\varphi^{M_{\alpha}}(p_{n+1})\geq\bar{p}_{n}$
for each $n$, so $\varphi^{M_{\alpha}}(p^{*})$ forces that
$\langle\bar{f}_{n}:n\in\omega\rangle$ is an increasing sequence of conditions
in $\dot{\mathbb{S}}_{\xi}$, and therefore forces that
$\bigcup_{n}\bar{f}_{n}$ is a condition too. However, $\bar{f}_{n+1}$ extends
(in the sense of Definition 3.3) $f_{n+1}\upharpoonright M_{\alpha}$ which
extends $\bar{f}_{n}$ for all $n$. Therefore
$\bigcup_{n}\bar{f}_{n}=\bigcup_{n}(f_{n}\upharpoonright
M_{\alpha})=(\bigcup_{n}f_{n})\upharpoonright M_{\alpha}=f^{*}\upharpoonright
M_{\alpha}$, which finishes the claim.
Now we show that the lemma holds for $\xi=\rho$. We deal with $\rho$ limit
first. If $\operatorname{cf}(\rho)>\omega$, then the result holds since any
$(p,f)\in\mathbb{R}_{\rho}$ is in $\mathbb{R}_{\xi}$ for some $\xi<\rho$. On
the other hand, if $\operatorname{cf}(\rho)=\omega$, then let
$\langle\xi_{n}:n\in\omega\rangle$ be an increasing sequence of ordinals in
$M_{\alpha}$ which is cofinal in $\rho$. By applying the lemma below $\rho$,
we define an increasing sequence of extensions
$\langle(p_{n},f_{n}):n\in\omega\rangle$ of $(p,f)$ so that
$f_{n}\upharpoonright[\xi_{n},\rho)=f\upharpoonright[\xi_{n},\rho)$ and so
that $(p_{n},f_{n}\upharpoonright\xi_{n})\in D(\varphi^{M_{\alpha}},\xi_{n})$.
Now let $p^{*}$ be a sup of $\langle p_{n}:n\in\omega\rangle$ and
$f^{*}:=\bigcup_{n}f_{n}$. Then $(p^{*},f^{*})$ extends $(p,f)$ and is a
member of $D(\varphi^{M_{\alpha}},\rho)$.
Finally, assume that $\rho=\rho_{0}+1$ is a successor, and let
$(p,h)\in\mathbb{R}_{\rho}$ be given. By the remarks after Definition 3.7, we
may find a residue $(\bar{p},\bar{h}_{0})$ of $(p,h\upharpoonright\rho_{0})$
to $M_{\alpha}$ with respect to the poset $\mathbb{R}_{\rho_{0}}$. Recalling
Notation 3.1, we use $h(\rho_{0})\cap M_{\alpha}$ in what follows as an abuse
of notation for $\langle h(\rho_{0})(\nu)\cap
M_{\alpha}:\nu\in\operatorname{dom}(h(\rho_{0}))\rangle$. By the elementarity
of $M_{\alpha}$ and the fact that $h(\rho_{0})\cap M_{\alpha}$ is a member of
$M_{\alpha}$, we may find an extension of $(\bar{p},\bar{h}_{0})$, say
$(\bar{p}^{\prime},\bar{h}^{\prime}_{0})$, which either forces that
$h(\rho_{0})\cap M_{\alpha}\in\dot{\mathbb{S}}(\rho_{0})$ or forces that
$h(\rho_{0})\cap M_{\alpha}\notin\dot{\mathbb{S}}(\rho_{0})$. Since
$(\bar{p}^{\prime},\bar{h}^{\prime}_{0})$ extends $(\bar{p},\bar{h}_{0})$, it
is compatible with $(p,h\upharpoonright\rho_{0})$, and hence it must force
that $h(\rho_{0})\cap M_{\alpha}\in\dot{\mathbb{S}}(\rho_{0})$. Finally, since
$D(\varphi^{M_{\alpha}},\rho_{0})$ is dense and $\rho_{0}\in M_{\alpha}$, we
may find an extension $(q,g_{0})$ of $(\bar{p}^{\prime},\bar{h}^{\prime}_{0})$
and $(p,h\upharpoonright\rho_{0})$ which is in
$D(\varphi^{M_{\alpha}},\rho_{0})$ and which satisfies that
$\varphi^{M_{\alpha}}(q)\geq\bar{p}^{\prime}$. Then $(q,g_{0})\upharpoonright
M_{\alpha}$ is a condition which forces that $h(\rho_{0})\cap M_{\alpha}$ is a
condition in $\dot{\mathbb{S}}(\rho_{0})$. Thus
$(\varphi^{M_{\alpha}}(q),(g_{0}\upharpoonright M_{\alpha})^{\frown}\langle
h(\rho_{0})\cap M_{\alpha}\rangle)$ is a condition and equals
$(q,g_{0}^{\frown}\langle h(\rho_{0})\rangle)\upharpoonright M_{\alpha}$. ∎
###### Notation 3.9.
We will often find it useful to denote conditions in $\mathbb{R}_{\xi}$ by the
letters $u,v$ and $w$. If $u\in\mathbb{R}_{\xi}$, we write $p_{u}$ and $f_{u}$
to denote the objects so that $u=(p_{u},f_{u})$. Furthermore, if
$\zeta\leq\xi$, then we write $u\upharpoonright\zeta$ to denote the pair
$(p_{u},f_{u}\upharpoonright\zeta)$, which restricts the length. This should
not be confused with $u\upharpoonright
M_{\alpha}=(\varphi^{M_{\alpha}}(p),f\upharpoonright M_{\alpha})$ from
Definition 3.7, which restricts the height.
The following definitions of $\\#$ and $*$ are taken from [33] and modified to
the current presentation. The dual residue property defined in (2) of the
upcoming definition is the natural translation into the current situation of
the statement that “$\\#$ implies $*$” at $\alpha$ from [33].
###### Definition 3.10.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$, that
$\alpha\in\operatorname{dom}(\vec{M})$, and that $\zeta\leq\rho$ is in
$M_{\alpha}$.
1. (1)
Fix conditions $u,v\in\mathbb{R}_{\zeta}$ and $w\in\mathbb{R}_{\zeta}\cap
M_{\alpha}$. Fix a residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$.
1. (a)
We say that
$\\#^{\zeta}_{\varphi^{M_{\alpha}}}(u,v,w)$
holds if $u,v\in D(\varphi^{M_{\alpha}},\zeta)$ and $u\upharpoonright
M_{\alpha}=^{*}w=^{*}v\upharpoonright M_{\alpha}$.222Note that if $(p,f)$ and
$(q,g)$ are (determined) conditions with $(p,f)=^{*}(q,g)$, then $f=g$.
2. (b)
We say that
$*^{\zeta}_{\varphi^{M_{\alpha}}}(u,v,w)$
holds if $u,v\in D(\varphi^{M_{\alpha}},\zeta)$ and if $w\geq u\upharpoonright
M_{\alpha},v\upharpoonright M_{\alpha}$ is a dual residue for $u$ and $v$ (see
Definition 2.3).333One can in fact argue that if $w$ is a dual residue, then
it follows that $w\geq u\upharpoonright M_{\alpha}$ and $w\geq
v\upharpoonright M_{\alpha}$; however, we don’t need this fact.
2. (2)
We say that $\mathbb{R}_{\zeta}$ satisfies the dual residue property at
$M_{\alpha}$ if for any residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$ and any conditions $u,v,w$ so that
$\\#^{\zeta}_{\varphi^{M_{\alpha}}}(u,v,w)$ holds, there exists
$w^{*}\geq_{\mathbb{R}_{\zeta}\cap M_{\alpha}}w$ so that
$*^{\zeta}_{\varphi^{M_{\alpha}}}(u,v,w^{*})$ holds.
###### Lemma 3.11.
Suppose that $*^{\zeta}_{\varphi^{M_{\alpha}}}(u,v,w)$ holds and that $D$ is
dense and countably $=^{*}$-closed in
$\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}})$. Then:
1. (1)
there exist $u^{\prime}\geq u$, $v^{\prime}\geq v$ with
$u^{\prime},v^{\prime}\in D$ and there exists $w^{\prime}\geq w$ so that
$u^{\prime}\upharpoonright M_{\alpha}\geq w$, $v^{\prime}\upharpoonright
M_{\alpha}\geq w$, and
$*^{\zeta}_{\varphi^{M_{\alpha}}}(u^{\prime},v^{\prime},w^{\prime})$ hold;
2. (2)
there exist $u^{*}\geq u$ and $v^{*}\geq v$ with $u^{*},v^{*}\in D$, and there
exists $w^{*}\geq w$ so that
$\\#^{\zeta}_{\varphi^{M_{\alpha}}}(u^{*},v^{*},w^{*})$ holds.
###### Proof.
First define $E$ to be the set of conditions $s$ in
$\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}})$ so that
$s\in D(\varphi^{M_{\alpha}},\zeta)\cap D$ and so that either
$s\upharpoonright M_{\alpha}\geq w$ or $s\upharpoonright M_{\alpha}$ is
incompatible with $w$; then $E$ is dense in
$\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}})$. Now fix
a $V$-generic filter $\bar{G}$ over $\mathbb{R}_{\zeta}\cap M_{\alpha}$
containing $w$, and note that $u$ and $v$ are in
$(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),{0_{\dot{\mathbb{S}}_{\zeta}})})/\bar{G}$.
By Lemma 2.7(2), we can find $u^{\prime}\geq u$ and $v^{\prime}\geq v$ so that
$u^{\prime},v^{\prime}$ are in $E$ as well as in
$(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}}))/\bar{G}$.
We next observe that $u^{\prime}\upharpoonright M_{\alpha}\in\bar{G}$. Indeed,
since $u^{\prime}\in D(\varphi^{M_{\alpha}},\zeta)$,
$u^{\prime}\upharpoonright M_{\alpha}$ is a condition in
$\mathbb{R}_{\zeta}\cap M_{\alpha}$. Additionally, since
$u^{\prime}\in(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}}))/\bar{G}$
and $u^{\prime}\geq u^{\prime}\upharpoonright M_{\alpha}$, we have that
$u^{\prime}\upharpoonright M_{\alpha}$ (a condition) is compatible with every
condition in $\bar{G}$. Thus $u^{\prime}\upharpoonright M_{\alpha}\in\bar{G}$.
However, by definition of $E$, and since $w\in\bar{G}$,
$u^{\prime}\upharpoonright M_{\alpha}$ must extend $w$. A symmetric argument
shows that $v^{\prime}\upharpoonright M_{\alpha}\geq w$.
Now let $w^{\prime}\in\bar{G}$ be a condition extending $w$ which forces that
$u^{\prime},v^{\prime}$ are in
$(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}}))/\dot{\bar{G}}$.
By Lemma 2.6, we have that
$*^{\zeta}_{\varphi^{M_{\alpha}}}(u^{\prime},v^{\prime},w^{\prime})$ holds.
Since $u^{\prime}\upharpoonright M_{\alpha}$ and $v^{\prime}\upharpoonright
M_{\alpha}$ both extend $w$, this completes the proof of (1).
For (2), suppose that we are given conditions $u_{0},v_{0}$, and $w_{0}$ so
that $*^{\zeta}_{\varphi^{M_{\alpha}}}(u_{0},v_{0},w_{0})$ holds. By
repeatedly applying (1), we may define a coordinate-wise increasing sequence
$\langle\langle u_{n},v_{n},w_{n}\rangle:n\in\omega\rangle$ so that for all
$n\in\omega$, $u_{n+1}\geq u_{n}$ and $v_{n+1}\geq v_{n}$;
$u_{n+1}\upharpoonright M_{\alpha}\geq w_{n}$ and $v_{n+1}\upharpoonright
M_{\alpha}\geq w_{n}$; and
$*^{\zeta}_{\varphi^{M_{\alpha}}}(u_{n},v_{n},w_{n})$ holds. Let $u^{*}$ be a
sup of $\langle u_{n}:n\in\omega\rangle$, and let $v^{*}$ and $w^{*}$ be
defined similarly. Since $*^{\zeta}_{\varphi^{M_{\alpha}}}(u_{n},v_{n},w_{n})$
holds for each $n$, by definition we have that $w_{n}\geq u_{n}\upharpoonright
M_{\alpha},v_{n}\upharpoonright M_{\alpha}$. Therefore the sequences $\langle
u_{n}\upharpoonright M_{\alpha}:n\in\omega\rangle$ and $\langle
v_{n}\upharpoonright M_{\alpha}:n\in\omega\rangle$ are each intertwined with
$\langle w_{n}:n\in\omega\rangle$, and consequently, they have suprema which
are $=^{*}$-related. It follows by the continuity of $\varphi^{M_{\alpha}}$
that $\\#^{\zeta}_{\varphi^{M_{\alpha}}}(u^{*},v^{*},w^{*})$ holds. ∎
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$, that
$\alpha\in\operatorname{dom}(\vec{M})$ and that $\zeta\in M_{\alpha}\cap\rho$.
Fix $\theta\in(\kappa\setminus\alpha)\times\omega_{1}$, a node in the tree
$\dot{T}_{\zeta}$ of level greater than or equal to $\alpha$.
Let $\dot{b}_{\zeta}(\theta,\alpha)$ denote the $\mathbb{R}_{\zeta}$-name for
$\left\\{\bar{\theta}\in\alpha\times\omega_{1}:\bar{\theta}<_{\dot{T}_{\zeta}}\theta\right\\}$.
By Proposition 3.4(2), the condition
$(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}})$ forces that
$\dot{b}_{\zeta}(\theta,\alpha)$ is a cofinal branch through
$(\dot{T}_{\zeta}\cap M_{\alpha})[\dot{G}_{\mathbb{R}_{\zeta}\cap
M_{\alpha}}]$. Note that by Proposition 3.4 and the definition of pre-
splitting configuration, $(\dot{T}_{\zeta}\cap
M_{\alpha})[G_{\mathbb{R}_{\zeta}\cap M_{\alpha}}]$ is an Aronsjzan tree on
$\alpha$ in the $V$-generic extension $V[G_{\mathbb{R}_{\zeta}\cap
M_{\alpha}}]$ over $\mathbb{R}_{\zeta}\cap M_{\alpha}$. In light of this, we
make the following definition:
###### Definition 3.12.
Let $\zeta<\rho$, $\alpha<\kappa$, and $\langle\theta,\tau\rangle$ be a pair
of tree nodes (possibly equal) at or above level $\alpha$, which we view as
nodes in the tree $\dot{T}_{\zeta}$. We say that two conditions $u$ and $v$ in
$\mathbb{R}_{\zeta}$ split $\langle\theta,\tau\rangle$ below $\alpha$ in
$\dot{T}_{\zeta}$ if there exist a level $\bar{\alpha}<\alpha$ and _distinct_
nodes $\bar{\theta},\bar{\tau}$ on level $\bar{\alpha}$ so that
$u\Vdash\bar{\theta}<_{\dot{T}_{\zeta}}\theta$ and
$v\Vdash\bar{\tau}<_{\dot{T}_{\zeta}}\tau$. More generally, if
$\zeta\leq\xi<\rho$ and $u^{\prime},v^{\prime}\in\mathbb{R}_{\xi}$, then we
say that $u^{\prime}$ and $v^{\prime}$ split $\langle\theta,\tau\rangle$ below
$\alpha$ in $\dot{T}_{\zeta}$ if $u=u^{\prime}\upharpoonright\zeta$ and
$v=v^{\prime}\upharpoonright\zeta$ do.
###### Lemma 3.13.
Suppose that $\zeta\leq\xi<\rho$ and $\mathbb{R}_{\xi}$ satisfies the dual
residue property at some $M_{\alpha}$, where $\zeta\in M_{\alpha}$ (see
Definition 3.10). Fix $u,v\in\mathbb{R}_{\xi}$ so that for some
$w\in\mathbb{R}_{\xi}\cap M_{\alpha}$,
$\\#^{\xi}_{\varphi^{M_{\alpha}}}(u,v,w)$. Let $\langle\theta,\tau\rangle$ be
a pair of tree nodes (possibly equal) each of which is at or above level
$\alpha$. Then there exist extensions $u^{*}\geq u$, $v^{*}\geq v$, and
$w^{*}\geq w$ so that $\\#^{\xi}_{\varphi^{M_{\alpha}}}(u^{*},v^{*},w^{*})$
and so that $u^{*}$ and $v^{*}$ split $\langle\theta,\tau\rangle$ below
$\alpha$ in $\dot{T}_{\zeta}$.
###### Proof.
Since $\mathbb{R}_{\xi}$ satisfies the dual residue property at $M_{\alpha}$,
$\mathbb{R}_{\zeta}$ does too, and so we may find some
$w^{\prime}\in\mathbb{R}_{\zeta}\cap M_{\alpha}$ so that
$w^{\prime}\geq_{\mathbb{R}_{\zeta}}w\upharpoonright\zeta$ and
$*^{\zeta}_{\varphi^{M_{\alpha}}}(u\upharpoonright\zeta,v\upharpoonright\zeta,w^{\prime})$.
Fix a $V$-generic filter $\bar{G}$ over $\mathbb{R}_{\zeta}\cap M_{\alpha}$
containing $w^{\prime}$. As a result, $u\upharpoonright\zeta$ and
$v\upharpoonright\zeta$ are in
$(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}}))/\bar{G}$.
By the discussion preceding Definition 3.12, we know that
$u\upharpoonright\zeta$ forces in the quotient that
$\dot{b}_{\zeta}(\theta,\alpha)$ is a cofinal branch through
$\bar{T}:=(\dot{T}_{\zeta}\cap M_{\alpha})[\bar{G}]$, which by Proposition 3.4
is an Aronszajn tree on $\alpha$ in $V[\bar{G}]$. Consequently, we may find
two conditions $u_{0},u_{1}$ which extend $u\upharpoonright\zeta$ in
$(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}}))/\bar{G}$,
some level $\bar{\alpha}<\alpha$, and two _distinct_ nodes
$\theta_{0},\theta_{1}$ on level $\bar{\alpha}$ of $\bar{T}$ so that $u_{i}$
forces in
$(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}}))/\bar{G}$
that $\theta_{i}<_{\dot{T}_{\zeta}}\theta$. Since $v\upharpoonright\zeta$ also
forces that $\dot{b}_{\zeta}(\tau,\alpha)$ is a cofinal branch through
$\bar{T}$, we may find some extension $v_{0}$ of $v\upharpoonright\zeta$ in
the quotient so that $v_{0}$ decides the $<_{\dot{T}_{\zeta}}$-predecessor,
say $\bar{\tau}$, of $\tau$ on level $\bar{\alpha}$ of $\bar{T}$. As
$\theta_{0}\neq\theta_{1}$, there exists some $i\in 2$ so that
$\theta_{i}\neq\bar{\tau}$. Set $\bar{\theta}=\theta_{i}$.
Now fix an extension $w^{\prime\prime}$ of $w^{\prime}$ in $\bar{G}$ so that
$w^{\prime\prime}$ forces the following statements: (i) $u_{i},v_{0}$ are in
the quotient; (ii) $u_{i}$ forces in the quotient that
$\bar{\theta}<_{\dot{T}_{\zeta}}\theta$; (iii) $v_{0}$ forces in the quotient
that $\bar{\tau}<_{\dot{T}_{\zeta}}\tau$.
By two applications of Lemma 2.7(3), we may find conditions $\bar{u},\bar{v}$
in the quotient so that $\bar{u}$ extends $u_{i}$ and $w^{\prime\prime}$, so
that $\bar{v}$ extends $v_{0}$ and $w^{\prime\prime}$, and so that
$\bar{u},\bar{v}\in D(\varphi^{M_{\alpha}},\zeta)$.
We now see that
$\bar{u}\Vdash_{\mathbb{R}_{\zeta}}\bar{\theta}<_{\dot{T}_{\zeta}}\theta$,
since
$w^{\prime\prime}\Vdash_{\mathbb{R}_{\zeta}\cap
M_{\alpha}}\left(\;u_{i}\Vdash_{(\mathbb{R}_{\zeta}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\zeta}}))/\dot{\bar{G}}}\bar{\theta}<_{\dot{T}_{\zeta}}\theta\right)$
and since $\bar{u}\geq w^{\prime\prime},u_{i}$. Similarly,
$\bar{v}\Vdash_{\mathbb{R}_{\zeta}}\bar{\tau}<_{\dot{T}_{\zeta}}\tau$.
Finally, let $\bar{w}\geq w^{\prime\prime}$ be a condition in $\bar{G}$ which
forces that $\bar{u}$ and $\bar{v}$ are in the quotient, so that by Lemma 2.6,
$*^{\zeta}_{\varphi^{M_{\alpha}}}(\bar{u},\bar{v},\bar{w})$ holds. By Lemma
3.11, we can find some $\bar{u}^{*}\geq\bar{u}$, $\bar{v}^{*}\geq\bar{v}$, and
$\bar{w}^{*}\geq w$ so that
$\\#^{\zeta}_{\varphi^{M_{\alpha}}}(\bar{u}^{*},\bar{v}^{*},\bar{w}^{*})$
holds. Now let $u^{*}$ be the condition where $p_{u^{*}}=p_{\bar{u}^{*}}$, and
where $f_{u^{*}}=f_{\bar{u}^{*}}\,^{\frown}f_{u}\upharpoonright[\zeta,\xi)$.
Let $v^{*}$ and $w^{*}$ be defined similarly. Then
$\\#^{\xi}_{\varphi^{M_{\alpha}}}(u^{*},v^{*},w^{*})$, and $u^{*}$ and $v^{*}$
split $\langle\theta,\tau\rangle$ below $\alpha$.
∎
One of the most important uses of the dual residue property is to obtain
splitting pairs of conditions. Obtaining such conditions will also crucially
use the “exactness” conditions of Definition 2.14.
###### Definition 3.14.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$.
1. (1)
Let $\alpha\in\operatorname{dom}(\vec{M})$ and $\xi\in M_{\alpha}\cap\rho$.
Fix a residue pair $\langle p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$ and conditions $u,v\in\mathbb{R}_{\xi}$. We say
that $u$ and $v$ are a splitting pair for $(\varphi^{M_{\alpha}},\xi)$ if
* •
for some $w\in\mathbb{R}_{\xi}\cap M_{\alpha}$,
$\\#^{\xi}_{\varphi^{M_{\alpha}}}(u,v,w)$;
* •
for any
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f_{u})\cap\operatorname{dom}(f_{v})\cap
M_{\alpha}$ and any
$\langle\theta,\tau\rangle\in(f_{u}(\zeta,\nu))\times(f_{v}(\zeta,\nu)),$ both
at or above level $\alpha$, $u$ and $v$ split $\langle\theta,\tau\rangle$
below $\alpha$ in $\dot{T}_{\zeta}$.
2. (2)
Given fixed enumerations
$f_{u}(\zeta,\nu)\backslash(\alpha\times\omega_{1})=\\{\theta_{n}\mid
n<\omega\\}$ and
$f_{v}(\zeta,\nu)\backslash(\alpha\times\omega_{1})=\\{\tau_{m}\mid
m<\omega\\}$ (possibly with repetitions in the case the sets are finite,
nonempty), we define a splitting function to be a function $\Sigma$ with
domain444Recall our convention from Notation 3.1 regarding conditions $f_{u}$
and their domains.
$\operatorname{dom}(\Sigma)=(\operatorname{dom}(f_{u})\cap\operatorname{dom}(f_{v})\cap
M_{\alpha})\times\omega\times\omega,$
so that for any $\langle\zeta,\nu,m,n\rangle\in\operatorname{dom}(\Sigma)$,
$\Sigma(\zeta,\nu,m,n)$ is a pair $\langle\bar{\theta},\bar{\tau}\rangle$ of
tree nodes satisfying Definition 3.12 with respect to
$\langle\theta_{m},\tau_{n}\rangle$. We will denote $\bar{\theta}$ by
$\Sigma(\zeta,\nu,m,n)(L)$ and $\bar{\tau}$ by $\Sigma(\zeta,\nu,m,n)(R)$.
###### Remark 3.15.
Let $\Sigma$ be as in Definition 3.14.
1. (1)
We emphasize the fact that if
$\langle\zeta,\nu,m,n\rangle\in\operatorname{dom}(\Sigma)$, then
$\Sigma(\zeta,\nu,m,n)(L)\neq\Sigma(\zeta,\nu,m,n)(R)$
are two _distinct_ tree nodes _on the same level_. We will usually suppress
explicit mention of the level.
2. (2)
Any splitting function $\Sigma$ is a member of $M_{\alpha}$ since $M_{\alpha}$
is countably closed and since $\Sigma$ maps from a countable subset of
$M_{\alpha}$ into $M_{\alpha}$.
3. (3)
We only require the splitting pair in Definition 3.14 to split nodes coming
from coordinates which are members of $M_{\alpha}$. As we will see from Lemma
3.18 and later thinning out arguments, this is sufficient for our purposes.
Now we are ready to state our second inductive hypothesis, the point of which
is to provide plenty of instances of the dual residue property. We will assume
the second inductive hypothesis for the rest of the section. We again recall
the filter $\cal{F}$ and its dual ideal $\cal{I}$ from Definition 2.14.
Inductive Hypothesis II: Let $\xi<\rho$, and suppose that $\vec{M}$ is
$\mathbb{R}_{\xi}$-suitable. Then there is an
$A\subseteq\operatorname{dom}(\vec{M})$ with $A\in\cal{I}$ so that for all
$\alpha\in\operatorname{dom}(\vec{M})\backslash A$, $\mathbb{R}_{\xi}$
satisfies the dual residue property at $M_{\alpha}$.
Now we move to the main part of the proof that $\mathbb{P}^{*}$ satisfies
inductive hypotheses I and II with respect to $\rho$. After a bit more set-up,
we will verify inductive hypothesis I for $\rho$ and then use this to verify
inductive hypothesis II at $\rho$.
The following lemma amalgamates instance of the second induction hypothesis
below $\rho$, stating that if $\vec{M}$ is $\rho$-suitable, then for almost
all $\alpha\in\operatorname{dom}(\vec{M})$ and all $\xi\in
M_{\alpha}\cap\rho$, $\mathbb{R}_{\xi}$ satisfies the dual residue property at
$M_{\alpha}$. However, we note that the following lemma is far from showing
that the second induction hypothesis holds at $\rho$ itself, only asserting
(roughly) that it holds up to $\rho$. The proof involves a standard diagonal
union, using the normality of $\cal{I}$.
###### Lemma 3.16.
Suppose that $\vec{M}$ is $\mathbb{R}_{\rho}$-suitable. Then there is an
$A\subseteq\operatorname{dom}(\vec{M})$ with $A\in\cal{I}$ so that for all
$\alpha\in\operatorname{dom}(\vec{M})\backslash A$ and all $\xi\in
M_{\alpha}\cap\rho$, $\mathbb{R}_{\xi}$ satisfies the dual residue property at
$M_{\alpha}$.
###### Proof.
Fix $\vec{M}$ which is $\mathbb{R}_{\rho}$-suitable. If $\rho<\kappa$, then
the lemma follows by taking the union of $<\kappa$-many sets in $\cal{I}$
which witness the second induction hypothesis below $\rho$. Suppose, then,
that $\rho\geq\kappa$, and let $h:\kappa\longrightarrow\rho$ be the
$\lhd$-least bijection from $\kappa$ onto $\rho$. Note that $h$ is in
$M_{\alpha}$ for all $\alpha\in\operatorname{dom}(\vec{M})$ (since $\rho$ is
an element of $M_{\alpha}$) and that for each such $\alpha$,
$M_{\alpha}\cap\rho=h[M_{\alpha}\cap\kappa]$. Next observe that for all
$\xi<\rho$, a tail of the sequence $\vec{M}$ is $\mathbb{R}_{\xi}$-suitable,
since a tail of this sequence contains $\xi$ as an element and since each
model on $\vec{M}$ is $\mathbb{R}_{\rho}$-suitable.
For each $\xi<\rho$, we may then find $A_{\xi}\in\cal{I}$ so that for all
$\alpha\in\operatorname{dom}(\vec{M})\backslash A_{\xi}$, $M_{\alpha}$ is
$\mathbb{R}_{\xi}$-suitable and so that $\mathbb{R}_{\xi}$ satisfies the dual
residue property at $M_{\alpha}$. For each $\nu<\kappa$, let
$B_{\nu}:=A_{h(\nu)}$, and let
$B:=\nabla_{\nu<\kappa}B_{\nu}=\left\\{\beta<\kappa:(\exists\nu<\beta)\,[\beta\in
B_{\nu}]\right\\}$. $B\in\cal{I}$ since $\cal{I}$ is a normal ideal.
We now claim that for all $\alpha\in\operatorname{dom}(\vec{M})$ and all
$\xi\in M_{\alpha}\cap\rho$, $M_{\alpha}$ is $\mathbb{R}_{\xi}$-suitable and
$\mathbb{R}_{\xi}$ satisfies the dual residue property at $M_{\alpha}$. So let
such $\alpha$ and $\xi$ be given. $\xi\in M_{\alpha}\cap\rho$, and hence
$\xi=h(\bar{\nu})$ for some $\bar{\nu}<\alpha$. However, $\alpha\notin B$, and
therefore for all $\nu<\alpha$, $\alpha\notin B_{\nu}=A_{h(\nu)}$. In
particular, $\alpha\notin A_{h(\bar{\nu})}=A_{\xi}$. Thus by choice of
$A_{\xi}$, $M_{\alpha}$ is $\mathbb{R}_{\xi}$-suitable, and $\mathbb{R}_{\xi}$
satisfies the dual residue property at $M_{\alpha}$. ∎
At important parts of the following proofs we will need to understand the
circumstances under which we can amalgamate conditions in $\mathbb{R}_{\rho}$,
and in particular, in $\dot{\mathbb{S}}_{\rho}.$ We will often be interested
in the following strong sense of amalgamation:
###### Definition 3.17.
Let $u,v\in\mathbb{R}_{\rho}$ so that $p_{u}$ and $p_{v}$ are compatible in
$\mathbb{P}^{*}$. We say that $f_{u}$ and $f_{v}$ are strongly compatible over
$p_{u}$ and $p_{v}$ if for any condition $q\in\mathbb{P}^{*}$ which extends
$p_{u}$ and $p_{v}$, $q$ forces that
$\check{f}_{u}\cup\check{f}_{v}\in\dot{\mathbb{S}}_{\rho}$.
The next lemma gives sufficient conditions under which we may amalgamate
conditions in $\mathbb{R}_{\rho}$ whose specializing parts are strongly
compatible as above.
###### Lemma 3.18.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$, that
$\alpha<\beta$ are in $\operatorname{dom}(\vec{M})$, and that $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ and $\langle
p^{*}(M_{\beta}),\varphi^{M_{\beta}}\rangle$ are residue pairs for
$(M_{\alpha},\mathbb{P}^{*})$ and $(M_{\beta},\mathbb{P}^{*})$, respectively.
Let $\langle u_{\alpha},v_{\alpha}\rangle$ and $\langle
u_{\beta},v_{\beta}\rangle$ be two pairs of conditions in $\mathbb{R}_{\rho}$
which satisfy the following:
1. (1)
$\langle u_{\alpha},v_{\alpha}\rangle$ is a splitting pair for
$(\varphi^{M_{\alpha}},\rho)$ with splitting function $\Sigma_{\alpha}$;
2. (2)
$\langle u_{\beta},v_{\beta}\rangle$ is a splitting pair for
$(\varphi^{M_{\beta}},\rho)$ with splitting function $\Sigma_{\beta}$;
3. (3)
$\Sigma_{\alpha}=\Sigma_{\beta}$;
4. (4)
there exists $w\in M_{\alpha}$ so that
$\\#^{\rho}_{\varphi^{M_{\alpha}}}(u_{\alpha},v_{\alpha},w)$ and
$\\#^{\rho}_{\varphi^{M_{\beta}}}(u_{\beta},v_{\beta},w)$ both hold; and
5. (5)
$u_{\alpha},v_{\alpha}\in M_{\beta}$.
Then $u_{\alpha}$ and $v_{\beta}$ are compatible in $\mathbb{R}_{\rho}$; in
fact, $f_{u_{\alpha}}$ is strongly compatible with $f_{v_{\beta}}$ over
$p_{u_{\alpha}}$ and $p_{v_{\beta}}$.
###### Proof.
We first observe that $p_{u_{\alpha}}$ and $p_{v_{\beta}}$ are compatible in
$\mathbb{P}^{*}$. Indeed, by (4),
$\varphi^{M_{\alpha}}(p_{u_{\alpha}})=^{*}p_{w}=^{*}\varphi^{M_{\beta}}(p_{v_{\beta}})$,
and by (5), $p_{u_{\alpha}}\in M_{\beta}$. Thus as
$p_{u_{\alpha}}\geq\varphi^{M_{\alpha}}(p_{u_{\alpha}})\geq\varphi^{M_{\beta}}(p_{v_{\beta}})$,
and as $\varphi^{M_{\beta}}$ is a residue function, $p_{u_{\alpha}}$ is
compatible with $p_{v_{\beta}}$.
Now let $q\in\mathbb{P}^{*}$ be any common extension of $p_{u_{\alpha}}$ and
$p_{v_{\beta}}$. We will argue by induction on $\zeta\leq\rho$ that
$q\Vdash(\check{f}_{u_{\alpha}}\cup\check{f}_{v_{\beta}})\upharpoonright\zeta\in\dot{\mathbb{S}}_{\zeta}$.
Limit stages are immediate. For the successor stage, suppose that
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f_{u_{\alpha}})\cap\operatorname{dom}(f_{v_{\beta}})$
and that we have proven that
$q\Vdash(\check{f}_{u_{\alpha}}\cup\check{f}_{v_{\beta}})\upharpoonright\zeta\in\dot{\mathbb{S}}_{\zeta}$.
Since $f_{u_{\alpha}}\in M_{\beta}$ by (5), $\langle\zeta,\nu\rangle\in
M_{\beta}$. Thus
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f_{v_{\beta}})\cap
M_{\beta}=\operatorname{dom}(f_{w})$, since $w=^{*}v_{\beta}\upharpoonright
M_{\beta}$. Since we also have that $w=^{*}u_{\beta}\upharpoonright
M_{\beta}$, it follows that
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f_{u_{\beta}})$. Thus
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f_{u_{\beta}})\cap\operatorname{dom}(f_{v_{\beta}})\cap
M_{\beta}=\operatorname{dom}(f_{u_{\alpha}})\cap\operatorname{dom}(f_{v_{\alpha}})\cap
M_{\alpha}$, with equality holding by (3) and the definition of a splitting
function. Moreover, $\zeta\in M_{\alpha}$ since
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f_{w})\subseteq M_{\alpha}$.
Now pick a pair of distinct nodes $\langle\theta,\tau\rangle\in
f_{u_{\alpha}}(\zeta,\nu)\times f_{v_{\beta}}(\zeta,\nu)$, and we will show
that $(q,(f_{u_{\alpha}}\cup f_{v_{\beta}})\upharpoonright\zeta)$ forces in
$\mathbb{R}_{\zeta}$ that $\theta$ and $\tau$ are
$\dot{T}_{\zeta}$-incompatible. If $\theta$ is below level $\alpha$, then
$\theta\in(f_{u_{\alpha}}\upharpoonright
M_{\alpha})(\zeta,\nu)=f_{w}(\zeta,\nu)\subseteq f_{v_{\beta}}(\zeta,\nu)$.
Thus $(q,f_{v_{\beta}}\upharpoonright\zeta)\Vdash\theta,\tau$ are
$\dot{T}_{\zeta}$-incompatible, and so $(q,(f_{u_{\alpha}}\cup
f_{v_{\beta}})\upharpoonright\zeta)$ forces this too. A similar argument
applies if $\tau$ is below level $\beta$.
We therefore assume that $\theta$ is at or above level $\alpha$ and $\tau$ is
at or above level $\beta$. Let $m$ and $n$ be chosen so that $\theta$ is the
$m$th node in $f_{u_{\alpha}}(\zeta,\nu)\backslash(\alpha\times\omega_{1})$
and $\tau$ is the $n$th node in
$f_{v_{\beta}}(\zeta,\nu)\backslash(\beta\times\omega_{1})$. By assumption
(3), letting $\Sigma:=\Sigma_{\alpha}=\Sigma_{\beta}$, we know that
$\Sigma(\zeta,\nu,m,n)(L)$ and $\Sigma(\zeta,\nu,m,n)(R)$ are two distinct
nodes on the same level and also that
$(p_{u_{\alpha}},f_{u_{\alpha}}\upharpoonright\zeta)\Vdash\Sigma(\zeta,\nu,m,n)(L)<_{\dot{T}_{\zeta}}\theta$
and
$(p_{v_{\beta}},f_{v_{\beta}}\upharpoonright\zeta)\Vdash\Sigma(\zeta,\nu,m,n)(R)<_{\dot{T}_{\zeta}}\tau.$
Therefore $(q,(f_{u_{\alpha}}\cup f_{v_{\beta}})\upharpoonright\zeta)$ forces
that $\tau$ and $\theta$ are incompatible in $\dot{T}_{\zeta}$, as we intended
to show. ∎
The following item shows how we can obtain the desired splitting pairs of
conditions.
###### Lemma 3.19.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$ and that
$\operatorname{dom}(\vec{M})$ satisfies the conclusion of Lemma 3.16. Fix
$\alpha\in\operatorname{dom}(\vec{M})$, and suppose that $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ is a residue pair for
$(M_{\alpha},\mathbb{P}^{*})$. Finally, fix $u,v,w$ so that
$\\#^{\rho}_{\varphi^{M_{\alpha}}}(u,v,w)$. Then there exist extensions
$u^{*}\geq u$ and $v^{*}\geq v$ so that $u^{*},v^{*}$ are a splitting pair for
$(\varphi^{M_{\alpha}},\rho)$.
###### Proof.
Fix $u,v,w$ as in the statement of the lemma. We define a coordinate-wise
increasing sequence of triples $\langle\langle
u_{n},v_{n},w_{n}\rangle:n\in\omega\rangle$ of conditions and a sequence
$\langle\langle\zeta_{n},\nu_{n},\theta_{n},\tau_{n}\rangle:n\in\omega\rangle$
of tuples of ordinals and tree nodes so that $\langle
u_{0},v_{0},w_{0}\rangle=\langle u,v,w\rangle$ and so that for each $n$,
* •
$\\#^{\rho}_{\varphi^{M_{\alpha}}}(u_{n},v_{n},w_{n})$ holds;
* •
$\langle\zeta_{n},\nu_{n}\rangle\in\operatorname{dom}(f_{u_{n}})\cap\operatorname{dom}(f_{v_{n}})\cap
M_{\alpha}$ and
$\langle\theta_{n},\tau_{n}\rangle\in(f_{u_{n}}(\zeta_{n},\nu_{n})\backslash(\alpha\times\omega_{1}))\times(f_{v_{n}}(\zeta_{n},\nu_{n})\backslash(\alpha\times\omega_{1}))$;
and
* •
$u_{n+1}$ and $v_{n+1}$ split $\langle\theta_{n},\tau_{n}\rangle$ below
$\alpha$.
This is done with respect to some bookkeeping device in such a way that if
$u^{*}$ is a sup of $\langle u_{n}:n\in\omega\rangle$ (and similarly for
$v^{*}$), then for each
$\langle\zeta,\nu\rangle\in\operatorname{dom}(f_{u^{*}})\cap\operatorname{dom}(f_{v^{*}})\cap
M_{\alpha}$ and each
$\langle\theta,\tau\rangle\in(f_{u^{*}}(\zeta,\nu)\backslash(\alpha\times\omega_{1}))\times(f_{v^{*}}(\zeta,\nu)\backslash(\alpha\times\omega_{1}))$,
$\langle\zeta,\nu,\theta,\tau\rangle$ appears as the $n$th tuple for some $n$.
To show the successor step, suppose that $u_{n},v_{n}$ and $w_{n}$ are given,
and consider $\langle\zeta_{n},\nu_{n},\theta_{n},\tau_{n}\rangle$. Note that
$\\#^{\zeta_{n}}_{\varphi^{M_{\alpha}}}(u_{n}\upharpoonright\zeta_{n},v_{n}\upharpoonright\zeta_{n},w_{n}\upharpoonright\zeta_{n})$
also holds. Then Lemma 3.13 applies since $\zeta_{n}\in M_{\alpha}$ and since
$\mathbb{R}_{\eta}$ satisfies the dual residue property at $M_{\alpha}$ for
all $\eta\in M_{\alpha}\cap\rho$. Thus we may find conditions
$u^{\prime}_{n}\geq u_{n}\upharpoonright\zeta_{n}$, $v^{\prime}_{n}\geq
v_{n}\upharpoonright\zeta_{n}$, and $w^{\prime}_{n}\geq
w_{n}\upharpoonright\zeta_{n}$ so that
$\\#^{\zeta_{n}}_{\varphi^{M_{\alpha}}}(u^{\prime}_{n},v^{\prime}_{n},w^{\prime}_{n})$
and so that $u^{\prime}_{n}$ and $v^{\prime}_{n}$ split
$\langle\theta_{n},\tau_{n}\rangle$ below $\alpha$. Now define $f_{u_{n+1}}$
to be the function which equals $f_{u^{\prime}_{n}}$ on $\zeta_{n}$ and which
equals $f_{u_{n}}$ on $[\zeta_{n},\rho)$. Also, let $u_{n+1}$ be the pair
$(p_{u^{\prime}_{n}},f_{u_{n+1}})$. Let $v_{n+1}$ and $w_{n+1}$ be defined
similarly. Then $\\#^{\rho}_{\varphi^{M_{\alpha}}}(u_{n+1},v_{n+1},w_{n+1})$
holds and $u_{n+1}$ and $v_{n+1}$ split $\langle\theta_{n},\tau_{n}\rangle$
below $\alpha$.
This completes the construction of the sequence. Fix sups $u^{*},v^{*},w^{*}$.
Since $\\#^{\rho}_{\varphi^{M_{\alpha}}}(u_{n},v_{n},w_{n})$ holds for all
$n$, $\\#^{\rho}_{\varphi^{M_{\alpha}}}(u^{*},v^{*},w^{*})$ also holds. By the
choice of bookkeeping, $u^{*},v^{*}$ is a splitting pair for
$(\varphi^{M_{\alpha}},\rho)$, completing the proof. ∎
###### Lemma 3.20.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$. Suppose
that for each $\alpha\in\operatorname{dom}(\vec{M})$, there exist
$u_{\alpha},v_{\alpha}$ which are a splitting pair for
$(\varphi^{M_{\alpha}},\rho)$, where $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ is a residue pair for
$(M_{\alpha},\mathbb{P}^{*})$. Then there exists
$B\subseteq\operatorname{dom}(\vec{M})$ in $\cal{F}^{+}$ so that for any
$\alpha<\beta$ in $B$, $u_{\alpha},v_{\alpha}\in M_{\beta}$,
$u_{\alpha}\upharpoonright M_{\alpha}=^{*}v_{\beta}\upharpoonright M_{\beta}$,
and $u_{\alpha}$ is compatible with $v_{\beta}$.
###### Proof.
Suppose that for each $\alpha\in\operatorname{dom}(\vec{M})$, we have a
splitting pair $u_{\alpha},v_{\alpha}$ for $(\varphi^{M_{\alpha}},\rho)$; we
also let $w_{\alpha}\in\mathbb{R}_{\rho}\cap M_{\alpha}$ be a condition
witnessing
$\\#^{\rho}_{\varphi^{M_{\alpha}}}(u_{\alpha},v_{\alpha},w_{\alpha})$. Let
$\Sigma_{\alpha}$ be a splitting function for $(u_{\alpha},v_{\alpha})$ with
respect to $M_{\alpha}$, as in Definition 3.14. By Remark 3.15,
$\Sigma_{\alpha}\in M_{\alpha}$. Now the function on
$\operatorname{dom}(\vec{M})$ defined by $\alpha\mapsto\langle
w_{\alpha},\Sigma_{\alpha}\rangle$ is regressive (since the pair can be coded
by an ordinal below $\alpha$). Since
$\operatorname{dom}(\vec{M})\in{\cal{F}^{+}}$ and ${\cal{F}}$ is normal, there
exists some ${B}\subseteq\operatorname{dom}(\vec{M})$ which is also in
${\cal{F}^{+}}$ on which that function takes a constant value, say
$\langle\bar{w},\Sigma\rangle$. Moreover, by intersecting with a club and
relabelling if necessary, we may assume that if $\alpha<\beta$ are in ${B}$,
then $u_{\alpha},v_{\alpha}\in M_{\beta}$. But then for any $\alpha<\beta$ in
$B$, we have that $u_{\alpha}\upharpoonright
M_{\alpha}=^{*}\bar{w}=^{*}v_{\beta}\upharpoonright M_{\beta}$. Therefore, for
all $\alpha<\beta$ in ${B}$, the assumptions of Lemma 3.18 are satisfied, and
consequently $u_{\alpha}$ and $v_{\beta}$ are compatible. ∎
###### Proposition 3.21.
$\mathbb{P}^{*}\Vdash\dot{\mathbb{S}}_{\rho}$ is $\kappa$-c.c.
###### Proof.
Let $p\in\mathbb{P}^{*}$ be a condition, and suppose that
$p\Vdash\langle\dot{f}_{\gamma}:\gamma<\kappa\rangle$ is a sequence of
conditions in $\dot{\mathbb{S}}_{\rho}$. We will find some extension $p^{*}$
of $p$ which forces that this sequence does not enumerate an antichain.
Let $\vec{M}$ be a sequence which is suitable with respect to the three
parameters $\mathbb{R}_{\rho}$, $p$ and
$\langle\dot{f}_{\gamma}:\gamma<\kappa\rangle$, and which is in pre-splitting
configuration up to $\rho$. By removing an $\cal{I}$-null set, we may assume
that $\operatorname{dom}(\vec{M})$ satisfies the conclusion of Lemma 3.16.
Let $B:=\operatorname{dom}(\vec{M})$. Since $\vec{M}$ is in pre-splitting
configuration up to $\rho$, let $\langle\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle:\alpha\in B\rangle$ be a residue
system. For each $\alpha\in B$, $p\in\mathbb{P}^{*}\cap M_{\alpha}$, and
therefore we may find some extension $p_{\alpha}$ of $p$ so that
$p_{\alpha}\in\operatorname{dom}(\varphi^{M_{\alpha}})$. We may also assume
that for some function $f_{\alpha}$ in $V$,
$p_{\alpha}\Vdash_{\mathbb{P}^{*}}\dot{f}_{\alpha}=\check{f}_{\alpha}$. Now
extend $\langle p_{\alpha},f_{\alpha}\rangle$ to a condition $u_{\alpha}$ in
$D(\varphi^{M_{\alpha}},\rho)$. By Lemma 3.19, we may further extend
$u_{\alpha}$ to a splitting pair $\langle
u^{*}_{\alpha},v^{*}_{\alpha}\rangle$ for $(\varphi^{M_{\alpha}},\rho)$. By
Lemma 3.20, we may find some $B^{*}\subseteq B$ with $B^{*}\in{\cal{F}^{+}}$
so that for all $\alpha<\beta$ in $B^{*}$, $u^{*}_{\alpha}$ and
$v^{*}_{\beta}$ are compatible. Let $w$ be a condition extending them both.
Then $p_{w}$ forces that $\check{f}_{w}$ extends both
$\check{f}_{u^{*}_{\alpha}}$ and $\check{f}_{v^{*}_{\beta}}$ and hence extends
$\dot{f}_{\alpha}$ and $\dot{f}_{\beta}$. Therefore $p_{w}$ forces that
$\dot{f}_{\alpha}$ and $\dot{f}_{\beta}$ are compatible in
$\dot{\mathbb{S}}_{\rho}$. ∎
We are now ready to verify that the second induction hypothesis holds at
$\rho$. We again remark that this proposition (and the later results which
build off of it) is the only place in our work where we need the ineffability
of $\kappa$. In all other cases, the weak compactness of $\kappa$ suffices.
###### Proposition 3.22.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$. Then
there is an $A\in\cal{I}$ so that for all
$\alpha\in\operatorname{dom}(\vec{M})\backslash A$, $\mathbb{R}_{\rho}$
satisfies the dual residue property at $M_{\alpha}$.
###### Proof.
We only deal with the case when $\mathbb{P}^{*}$ is not just the collapse
poset $\mathbb{P}$ (and hence we’re in the case where $\kappa$ is ineffable,
and $\cal{F}=\cal{F}_{in}$). The case when $\mathbb{P}^{*}$ is the collapse
$\mathbb{P}$ is simpler and taken care of in [33].
Suppose otherwise, for a contradiction. Then
$B:=\left\\{\alpha\in\operatorname{dom}(\vec{M}):\mathbb{R}_{\rho}\text{ does
not satisfy the dual residue property at }M_{\alpha}\right\\}$
is in ${\cal{F}^{+}}$. Moreover, by removing an $\cal{I}$-null set if
necessary, we may assume that $B$ satisfies the conclusion of Lemma 3.16. We
will derive our contradiction by creating a $\kappa$-sized antichain in
$\mathbb{R}_{\rho}$ for which we can amalgamate many of the
$\mathbb{P}^{*}$-parts. This will then lead to a $\kappa$-sized antichain in
$\mathbb{S}_{\rho}$ in some $V$-generic extension over $\mathbb{P}^{*}$.
For each $\alpha\in B$, we fix a residue pair $\langle
p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$ and a triple $\langle
u_{\alpha},v_{\alpha},w_{\alpha}\rangle$ which witnesses that
$\mathbb{R}_{\rho}$ does not satisfy the dual residue property at
$M_{\alpha}$. Thus
$\\#^{\rho}_{\varphi^{M_{\alpha}}}(u_{\alpha},v_{\alpha},w_{\alpha})$ holds,
but for any $w^{*}\geq_{\mathbb{R}_{\rho}\cap M_{\alpha}}w_{\alpha}$, either
$w^{*}$ is not a residue for $u_{\alpha}$ to $M_{\alpha}$ or $w^{*}$ is not a
residue for $v_{\alpha}$ to $M_{\alpha}$. In particular, for each such
$w^{*}$, we may find a further extension in $\mathbb{R}_{\rho}\cap M_{\alpha}$
which is either incompatible with $u_{\alpha}$ or incompatible with
$v_{\alpha}$ in $\mathbb{R}_{\rho}$.
By Lemma 3.19, we may extend $\langle u_{\alpha},v_{\alpha},w_{\alpha}\rangle$
to another triple $\langle
u^{*}_{\alpha},v^{*}_{\alpha},w^{*}_{\alpha}\rangle$ so that $u^{*}_{\alpha}$
and $v^{*}_{\alpha}$ are a splitting pair for $M_{\alpha}$. By Lemma 3.20, we
may find some $B^{*}\subseteq B$ with $B^{*}\in\cal{F}^{+}$ so that for all
$\alpha,\beta\in B^{*}$ with $\alpha<\beta$,
$w^{*}_{\alpha}=^{*}w^{*}_{\beta}$, $u^{*}_{\alpha}$ and $v^{*}_{\alpha}$ are
in $M_{\beta}$, and $u_{\alpha}^{*}$ is compatible with $v_{\beta}^{*}$. We
let $\bar{w}^{*}$ denote a condition which is $=^{*}$ equal to
$w^{*}_{\alpha}$ for $\alpha\in B^{*}$.
Next, for each $\alpha<\beta$ both in $B^{*}$, we define a condition
$w^{*}_{\alpha,\beta}$. Fix such $\alpha$ and $\beta$. Since
$u^{*}_{\alpha}\in M_{\beta}$ is compatible with $v^{*}_{\beta}$, there is an
extension $w^{*}_{\alpha,\beta}$ of $u^{*}_{\alpha}$ in $\mathbb{R}_{\rho}\cap
M_{\beta}$ which is a residue for $v_{\beta}^{*}$ to $\mathbb{R}_{\rho}\cap
M_{\beta}$. Since $\beta\in B$ and since $w^{*}_{\alpha,\beta}$ is a residue
for $v^{*}_{\beta}$, we may further extend (and relabel if necessary) to
assume that $w^{*}_{\alpha,\beta}$ is incompatible with $u^{*}_{\beta}$. Since
$p_{w^{*}_{\alpha,\beta}}\geq p_{u^{*}_{\alpha}}\geq p^{*}(M_{\alpha})$, we
may further assume that $p_{w^{*}_{\alpha,\beta}}$ is in the domain of
$\varphi^{M_{\alpha}}$.
We now set up an application of the ineffability of $\kappa$. For each
$\beta\in B^{*}$, we define the function $E_{\beta}$ by
${E_{\beta}:=\left\\{(\alpha,w^{*}_{\alpha,\beta}):\alpha\in
B^{*}\cap\beta\right\\}\subseteq\beta\times(M_{\beta}\cap\mathbb{R}_{\rho}).}$
Formally, we ought to apply the ineffability of $\kappa$ to a sequence
$\vec{A}$ where the $\alpha$-th element on the sequence is a subset of
$\alpha$. However, we will work with the sets $E_{\beta}$; this poses no loss
of generality since, by using the $\lhd$-least bijection from
$\mathbb{R}_{\rho}$ onto $\kappa$ and the Gödel pairing function, we can code
$E_{\beta}$ as a subset of $\beta$.
Since $\kappa$ is ineffable and $B^{*}\in\cal{F}^{+}$, we can find a subset
$E$ of $\kappa\times\mathbb{R}_{\rho}$ and a stationary $S\subseteq B^{*}$ so
that for all $\beta\in S$,
$E\cap(\beta\times(M_{\beta}\cap\mathbb{R}_{\rho}))=E_{\beta}$. We observe
that $E$ is a function: if $(\alpha,w)$ and $(\alpha,w^{\prime})$ are both in
$E$, fix some $\beta\in S$ large enough so that $w,w^{\prime}\in
M_{\beta}\cap\mathbb{R}_{\rho}$. Then $(\alpha,w)$ and $(\alpha,w^{\prime})$
are in $E\cap(\beta\times(M_{\beta}\cap\mathbb{R}_{\rho}))=E_{\beta}$. Since
$E_{\beta}$ is a function, $w=w^{\prime}$. We can now rephrase the coherence
as follows: if $\beta\in S$, then $E\upharpoonright\beta=E_{\beta}$, since
$E\upharpoonright\beta$ and $E_{\beta}$ are both functions with domain
$B^{*}\cap\beta$ and $E_{\beta}\subseteq E\upharpoonright\beta$.
Next, $B^{*}\subseteq\operatorname{dom}(E)$. Indeed, for each $\beta\in S$,
$E\upharpoonright\beta=E_{\beta}$, and the domain of $E_{\beta}$ is
$B^{*}\cap\beta$. Since $S$ is unbounded (in fact stationary) in $\kappa$,
there are unboundedly-many $\beta$ so that $E\upharpoonright\beta=E_{\beta}$,
from which the conclusion follows. And finally, if $\alpha\in B^{*}$ then for
any $\beta\in S\backslash(\alpha+1)$, $E(\alpha)=w^{*}_{\alpha,\beta}$, since
$E(\alpha)=E_{\beta}(\alpha)=w^{*}_{\alpha,\beta}$.
Now we press down residues for conditions indexed by $S$. Since $S\subseteq
B^{*}=\operatorname{dom}(E)$, $E(\alpha)$ is defined for each $\alpha\in S$.
And moreover, $p_{E(\alpha)}$ is in the domain of $\varphi^{M_{\alpha}}$ since
it equals $w^{*}_{\alpha,\beta}$ for some/any $\beta\in
S\backslash(\alpha+1)$, and since $w^{*}_{\alpha,\beta}$ is in the domain of
$\varphi^{M_{\alpha}}$. Then $\varphi^{M_{\alpha}}(p_{E(\alpha)})$ is a
condition in $M_{\alpha}\cap\mathbb{R}_{\rho}$, and each such condition can be
coded by an element of $\alpha$, using the $\lhd$-least bijection from
$\mathbb{R}_{\rho}$ onto $\kappa$. Thus the function
$\alpha\mapsto\varphi^{M_{\alpha}}(p_{E(\alpha)})$ on $S$ is regressive, and
so we can find a stationary $S^{*}\subseteq S$ so that it has a constant
value, say the condition $p^{**}$.
Now fix $\alpha<\beta$ in $S^{*}$, and we will show that $p_{E(\alpha)}$ and
$p_{E(\beta)}$ are compatible in $\mathbb{P}^{*}$. Indeed,
$E(\alpha)=w^{*}_{\alpha,\beta}$ is an element of
$M_{\beta}\cap\mathbb{R}_{\rho}$. Additionally,
$p_{w^{*}_{\alpha,\beta}}\geq\varphi^{M_{\alpha}}(p_{w^{*}_{\alpha,\beta}})=p^{**}=\varphi^{M_{\beta}}(p_{E(\beta)})$.
Thus $p_{E(\alpha)}=p_{w^{*}_{\alpha,\beta}}$ extends, inside of $M_{\beta}$,
the residue of $p_{E(\beta)}$ to $M_{\beta}$. $p_{E(\alpha)}$ and
$p_{E(\beta)}$ are therefore compatible in $\mathbb{P}^{*}$.
However, for such $\alpha<\beta$, we also know that $E(\alpha)$ and $E(\beta)$
are incompatible conditions in $\mathbb{R}_{\rho}$, since $E(\beta)$ extends
$u^{*}_{\beta}$ and since $E(\alpha)=w^{*}_{\alpha,\beta}$ is incompatible
with $u^{*}_{\beta}$. Thus if $q$ is any condition which extends
$p_{E(\alpha)}$ and $p_{E(\beta)}$ in $\mathbb{P}^{*}$, then $q$ must force
that $f_{E(\alpha)}$ and $f_{E(\beta)}$ are incompatible conditions in
$\dot{\mathbb{S}}_{\rho}$.
Now we can create our $\kappa$-sized antichain of specializing conditions. Let
$G^{*}$ be a $V$-generic filter over $\mathbb{P}^{*}$ which contains the
condition $p^{**}$, and recall that
$p^{**}=\varphi^{M_{\alpha}}(p_{E(\alpha)})$ for all $\alpha\in S^{*}$. By
Lemma 2.21, the set
${X:=\left\\{\alpha\in S^{*}:p_{E(\alpha)}\in G^{*}\right\\}}$
is unbounded in $\kappa$. Therefore if $\alpha<\beta$ are in $X$, then
$f_{E(\alpha)}$ and $f_{E(\beta)}$ are incompatible conditions in
$\dot{\mathbb{S}}_{\rho}[G]$. Since $\kappa$ is a cardinal after forcing with
$\mathbb{P}^{*}$ and since $X$ has size $\kappa$, this gives a $\kappa$-sized
antichain in $\dot{\mathbb{S}}_{\rho}[G]$. This contradicts Proposition 3.21
and completes the proof. ∎
Here we comment on the use of the ineffability of $\kappa$. In the original
Laver-Shelah argument, $\mathbb{P}^{*}$ is just the collapse forcing. Thus
their entire forcing is $\kappa$-c.c. However, in our set-up, $\mathbb{P}^{*}$
will in general fail to be $\kappa$-c.c., and consequently it is not enough to
find a $\kappa$-sized antichain in
$\mathbb{P}^{*}\ast\dot{\mathbb{S}}_{\rho}=\mathbb{R}_{\rho}$. Rather, we need
to arrange that there is a $\kappa$-sized antichain in $\mathbb{R}_{\rho}$ for
which we can amalgamate plenty of the $\mathbb{P}^{*}$-parts of the
conditions. This in turn requires that we be able to press down on the
residues of the $\mathbb{P}^{*}$-parts. Considering the array $\langle
w^{*}_{\alpha,\beta}:\alpha,\beta\in B^{*}\wedge\alpha<\beta\rangle$ from the
proof of the previous result, we need to find a stationary $S\subseteq B^{*}$
on which, for each $\alpha\in S$, the function on $S\backslash(\alpha+1)$
taking $\beta$ to $w^{*}_{\alpha,\beta}$ is independent of $\beta$, say taking
value $w^{**}_{\alpha}$. Then using the stationarity of $S$, we pressed down
on the residue of $w^{**}_{\alpha}$. The ineffability of $\kappa$ allowed us
to create a function, namely $E$, out of the above array with
$\operatorname{dom}(E)$ containing a stationary set on which the
approximations (the $E_{\beta}$) cohere. We were not able to create this
function and set up an application of pressing down just assuming that
$\kappa$ is weakly compact.
However, in the case that $\mathbb{P}^{*}$ is just the collapse, then a weakly
compact cardinal suffices for the entirety of the argument, since the entire
poset $\mathbb{P}\ast\dot{\mathbb{S}}_{\rho}$ is then $\kappa$-c.c. In this
case, we only need to create an unbounded $Z\subseteq B^{*}$ on which the
function $\beta\mapsto w^{*}_{\alpha,\beta}$ is independent of $\beta$, for
each $\alpha\in Z$ and $\beta\in Z\backslash(\alpha+1)$; this is because
$\langle w^{**}_{\alpha}:\alpha\in Z\rangle$ would then be a $\kappa$-sized
antichain in $\mathbb{P}\ast\dot{\mathbb{S}}_{\rho}$, a contradiction. $Z$ can
be constructed by working inside a $\kappa$-model $M^{*}$ containing all of
the relevant information, for which there exists an $M^{*}$-normal ultrafilter
containing $B$ as an element.
We have now completed the proof of Theorem 3.2. We conclude with a corollary
which adds to that theorem an additional clause about the dual residue
property; this will be useful later.
###### Corollary 3.23.
Suppose that $\mathbb{P}^{*}$ is ${\cal{F}}$-strongly proper and that
$\dot{\mathbb{S}}_{\kappa^{+}}$ is a $\mathbb{P}^{*}$-name for a
$\kappa^{+}$-length, countable support iteration specializing Aronszajn trees
on $\kappa$. Then for all $\rho<\kappa^{+}$,
1. (1)
$\mathbb{P}^{*}$ forces that $\dot{\mathbb{S}}_{\rho}$ is $\kappa$-c.c.; and
2. (2)
if $\vec{M}$ is in pre-splitting configuration up to $\rho$, then there is
some $A\subseteq\operatorname{dom}(\vec{M})$ with $A\in\cal{I}$ so that for
all $\alpha\in\operatorname{dom}(\vec{M})\backslash A$, $\mathbb{R}_{\rho}$
satisfies the dual residue property at $M_{\alpha}$. Hence, for all $\zeta\in
M_{\alpha}\cap(\rho+1)$, $\mathbb{R}_{\zeta}$ satisfies the dual residue
property at $M_{\alpha}$.
###### Proof.
If the corollary is false, let $\rho$ be the least such that it fails at
$\rho$. Then Induction Hypotheses I and II hold below $\rho$, so Propositions
3.21 and 3.22 show that (1) and (2) hold at $\rho$, a contradiction. ∎
## 4\. ${\cal{F}}$-Strongly Proper Posets and Preserving Stationary Sets
In this section, we will prove that the appropriate quotients preserve
stationary sets of cofinality $\omega$ ordinals. We will apply this result in
Section 6 when we show that our intended club-adding iteration is
${\cal{F}}$-completely proper (see Definition 5.8). In the first part of this
section, we will prove some helpful lemmas which we use in the second part to
complete proof of the preservation of the relevant stationary sets.
For the remainder of this section, we fix a ${\cal{F}}$-strongly proper poset
$\mathbb{P}^{*}$ and an iteration $\dot{\mathbb{S}}_{\rho}$ of length
$\rho<\kappa^{+}$ specializing Aronszajn trees in the extension by
$\mathbb{P}^{*}$; see the beginning of Section 3 for a more precise definition
and relevant notation. Note that the conclusions of Corollary 3.23 hold.
We first prove two lemmas which describe how the residue functions with
respect to two models on a suitable sequence interact. More precisely, suppose
we have a suitable sequence $\vec{M}$, where $\alpha<\beta$ are both in
$\operatorname{dom}(\vec{M})$ and $M_{\alpha}$ and $M_{\beta}$ have respective
residue pairs $\langle p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ and
$\langle p^{*}(M_{\beta}),\varphi^{M_{\beta}}\rangle$ with respect to
$\mathbb{P}^{*}$. A natural question is whether, on a dense set,
$\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(q))=^{*}\varphi^{M_{\alpha}}(q)$,
i.e., whether the $M_{\alpha}$-residue of the $M_{\beta}$-residue is
equivalent to the $M_{\alpha}$-residue. Proposition 4.2 below shows that this
is the case.
###### Lemma 4.1.
Suppose that $\vec{M}$ is $\mathbb{P}^{*}$-suitable with residue system
$\langle\langle
p^{*}(M_{\gamma}),\varphi^{M_{\gamma}}\rangle:\gamma\in\operatorname{dom}(\vec{M})\rangle$
and that $\alpha<\beta$ are in $\operatorname{dom}(\vec{M})$.
1. (1)
For every $p\in\mathbb{P}^{*}$ that extends both $p^{*}(M_{\alpha})$ and
$p^{*}(M_{\beta})$, there is an extension $p^{*}\geq_{\mathbb{P}^{*}}p$ with
$p^{*}\in\operatorname{dom}(\varphi^{M_{\alpha}})\cap\operatorname{dom}(\varphi^{M_{\beta}})$.
2. (2)
$D_{0}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}}):=\left\\{q\in\operatorname{dom}(\varphi^{M_{\alpha}})\cap\operatorname{dom}(\varphi^{M_{\beta}}):\varphi^{M_{\beta}}(q)\in\operatorname{dom}(\varphi^{M_{\alpha}})\right\\}$
is $=^{*}$-countably closed and dense in
$\mathbb{P}^{*}/\left\\{p^{*}(M_{\alpha}),p^{*}(M_{\beta})\right\\}$.555This
denotes the set of $r\in\mathbb{P}^{*}$ which extend both $p^{*}(M_{\alpha})$
and $p^{*}(M_{\beta})$.
###### Proof.
For (1), we apply a dovetailing construction using the properties of the
residue functions. Define, by recursion, an increasing sequence $\langle
p_{n}:n\in\omega\rangle$ of extensions of $p$ so that
$p_{2n+1}\in\operatorname{dom}(\varphi^{M_{\alpha}})$ and, for $n>0$,
$p_{2n}\in\operatorname{dom}(\varphi^{M_{\beta}})$. Let $p^{*}$ be a sup of
$\langle p_{n}:n\in\omega\rangle$. Then
$p^{*}\in\operatorname{dom}(\varphi^{M_{\alpha}})$ since it is also a sup of
$\langle p_{2n+1}:n\in\omega\rangle$, and
$p^{*}\in\operatorname{dom}(\varphi^{M_{\beta}})$ since it is a sup of
$\langle p_{2n}:n>0\rangle$.
For (2), fix a condition $q_{-1}\in\mathbb{P}^{*}$ which extends both
$p^{*}(M_{\alpha})$ and $p^{*}(M_{\beta})$, where by (1) we may assume that
$q_{-1}\in\operatorname{dom}(\varphi^{M_{\alpha}})\cap\operatorname{dom}(\varphi^{M_{\beta}})$.
We first make a cosmetic improvement to $q_{-1}$ before the main construction.
Since $q_{-1}$ extends both $\varphi^{M_{\beta}}(q_{-1})$ and
$p^{*}(M_{\alpha})$ and since both of these conditions are in $M_{\beta}$
(using Definition 2.14(2) to see that $p^{*}(M_{\alpha})\in M_{\beta}$), we
may apply the elementarity of $M_{\beta}$ to find a condition $s_{-1}\in
M_{\beta}$ which extends $\varphi^{M_{\beta}}(q_{-1})$ and
$p^{*}(M_{\alpha})$. Now find an extension $q_{0}\geq q_{-1}$ so that
$\varphi^{M_{\beta}}(q_{0})\geq s_{-1}$, noting that we may assume that
$q_{0}\in\operatorname{dom}(\varphi^{M_{\alpha}})\cap\operatorname{dom}(\varphi^{M_{\beta}})$.
Having completed this modification, we now define, by recursion, an increasing
sequence of conditions $\langle q_{n}:n\in\omega\rangle$ in
$\operatorname{dom}(\varphi^{M_{\alpha}})\cap\operatorname{dom}(\varphi^{M_{\beta}})$
and an increasing sequence $\langle s_{n}:n\in\omega\rangle$ of conditions in
$M_{\beta}\cap\operatorname{dom}(\varphi^{M_{\alpha}})$ so that for each $n$,
$\varphi^{M_{\beta}}(q_{n+1})\geq s_{n}\geq\varphi^{M_{\beta}}(q_{n})$. So
assume that $q_{n}$ is defined. Since $\varphi^{M_{\beta}}(q_{n})\geq
p^{*}(M_{\alpha})$ (using the previous paragraph for the case $n=0$), we may
find an extension $s_{n}$ of $\varphi^{M_{\beta}}(q_{n})$ which is a member of
$M_{\beta}\cap\operatorname{dom}(\varphi^{M_{\alpha}})$. Then let $q_{n+1}\geq
q_{n}$ be a condition with $\varphi^{M_{\beta}}(q_{n+1})\geq s_{n}$. Finally,
let $q^{*}$ be a sup of $\langle q_{n}:n\in\omega\rangle$, and let $s^{*}$ be
a sup of $\langle s_{n}:n\in\omega\rangle$, noting by the intertwined
construction that $s^{*}$ is also a sup of
$\langle\varphi^{M_{\beta}}(q_{n}):n\in\omega\rangle$. By the countable
continuity of the residue functions, we have
$\varphi^{M_{\beta}}(q^{*})=^{*}s^{*}$. But
$s^{*}\in\operatorname{dom}(\varphi^{M_{\alpha}})$, since it is the sup of the
increasing sequence $\langle s_{n}:n\in\omega\rangle$ of conditions in
$\operatorname{dom}(\varphi^{M_{\alpha}})$. Consequently,
$\varphi^{M_{\beta}}(q^{*})$ is also in
$\operatorname{dom}(\varphi^{M_{\alpha}})$. Since $q_{-1}$ in
$\mathbb{P}^{*}/\left\\{p^{*}(M_{\alpha}),p^{*}(M_{\beta})\right\\}$ was
arbitrary, this completes the proof of (2). ∎
###### Proposition 4.2.
Suppose that $\vec{M}$ is $\mathbb{P}^{*}$-suitable with residue system
$\langle\langle
p^{*}(M_{\gamma}),\varphi^{M_{\gamma}}\rangle:\gamma\in\operatorname{dom}(\vec{M})\rangle,$
and let $\alpha<\beta$ be in $\operatorname{dom}(\vec{M})$. Then
$E(\varphi^{M_{\alpha}},\varphi^{M_{\beta}}):=\left\\{p\in\mathbb{P}^{*}:\varphi^{M_{\beta}}(p)\in\operatorname{dom}(\varphi^{M_{\alpha}})\;\wedge\;\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(p))=^{*}\varphi^{M_{\alpha}}(p)\right\\}$
is $=^{*}$-countably closed and dense in
$\mathbb{P}^{*}/\left\\{p^{*}(M_{\alpha}),p^{*}(M_{\beta})\right\\}$.
###### Proof.
We begin by observing that if $q\in
D_{0}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$, then
$\varphi^{M_{\alpha}}(q)$ extends
$\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(q))$. Indeed, since
$q\in\operatorname{dom}(\varphi^{M_{\beta}})$, $q\geq\varphi^{M_{\beta}}(q)$,
and since $\varphi^{M_{\alpha}}$ is order-preserving and both $q$ and
$\varphi^{M_{\beta}}(q)$ are in $\operatorname{dom}(\varphi^{M_{\alpha}})$, we
conclude that
$\varphi^{M_{\alpha}}(q)\geq\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(q))$.
With this observation in mind, let $p\in\mathbb{P}^{*}$ extend both
$p^{*}(M_{\alpha})$ and $p^{*}(M_{\beta})$, and by extending further if
necessary, we may assume that $p$ is in
$D_{0}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$. We will define by recursion
an increasing sequence of conditions $\langle p_{n}:n\in\omega\rangle$ in
$D_{0}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$ with $p_{0}=p$ so that for
all $n$,
$\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(p_{n+1}))\geq\varphi^{M_{\alpha}}(p_{n})\geq\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(p_{n}));$
note that all of the above items are defined, by definition of
$D_{0}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$.
Suppose we are given $p_{n}$. As observed earlier, since $p_{n}\in
D_{0}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$, we have
$\varphi^{M_{\alpha}}(p_{n})\geq\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(p_{n}))$.
Since $\varphi^{M_{\alpha}}(p_{n})$ extends, in $M_{\alpha}$, the residue of
$\varphi^{M_{\beta}}(p_{n})$ to $M_{\alpha}$, we may find a condition $q\in
M_{\beta}$ extending $\varphi^{M_{\beta}}(p_{n})$ so that
$\varphi^{M_{\alpha}}(q)\geq\varphi^{M_{\alpha}}(p_{n})$. Since $q\in
M_{\beta}$ extends $\varphi^{M_{\beta}}(p_{n})$, there is an $r\geq p_{n}$ so
that $\varphi^{M_{\beta}}(r)\geq q$. Finally, let $p_{n+1}\geq r$ be a
condition in $D_{0}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$. Then
$\varphi^{M_{\beta}}(p_{n+1})\geq\varphi^{M_{\beta}}(r)\geq q$, and hence
$\varphi^{M_{\alpha}}(\varphi^{M_{\beta}}(p_{n+1}))\geq\varphi^{M_{\alpha}}(q)\geq\varphi^{M_{\alpha}}(p_{n})$.
This completes the construction of the desired sequence.
Let $p^{*}$ be a sup of $\langle p_{n}:n\in\omega\rangle$. It is
straightforward to verify that it witnesses the lemma. ∎
The last lemma that we will need before turning to the main result of this
section is a technical refinement of Lemma 3.19 which isolates circumstances
in which for $\alpha<\beta<\kappa$ as above, we can find splitting pairs $u,v$
for $(M_{\alpha},\rho)$ with the additional property that $u\upharpoonright
M_{\beta}$ and $v\upharpoonright M_{\beta}$ also form a splitting pair for
$(M_{\alpha},\rho)$. Moreover, $u\upharpoonright M_{\beta}$ and
$v\upharpoonright M_{\beta}$ will split the nodes on levels between $\alpha$
and $\beta$ in the same way that $u$ and $v$ do. For the statement of the next
result, recall the way we denote restriction of (iteration) length
$p\upharpoonright\xi$, and restriction in the poset height, $p\upharpoonright
M_{\alpha}$, from Notation 3.9.
###### Lemma 4.3.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$ and that
$\operatorname{dom}(\vec{M})$ satisfies the conclusion of Corollary 3.23(2).
Suppose $\alpha<\beta$ are both in $\operatorname{dom}(\vec{M})$ and that
$\langle p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ and $\langle
p^{*}(M_{\beta}),\varphi^{M_{\beta}}\rangle$ are residue pairs for
$(M_{\alpha},\mathbb{P}^{*})$ and $(M_{\beta},\mathbb{P}^{*})$ respectively.
Finally, fix a condition $u\in\mathbb{R}_{\rho}$ with
$p_{u}\in\operatorname{dom}(\varphi^{M_{\alpha}})\cap\operatorname{dom}(\varphi^{M_{\beta}})$.
Then there exist a splitting pair $(u^{*},v^{*})$ for
$(\varphi^{M_{\alpha}},\rho)$ extending $u$ and a splitting function $\Sigma$
satisfying the following:
1. (1)
$p_{u^{*}}$ and $p_{v^{*}}$ are both in
$E(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$ (see Proposition 4.2);
2. (2)
$(u^{*}\upharpoonright M_{\beta},v^{*}\upharpoonright M_{\beta})$ is also an
$(M_{\alpha},\rho)$-splitting pair, and for any tuple
$(\zeta,\nu,m,n)\in\operatorname{dom}(\Sigma)$ so that the $m$-th node of
$f_{u^{*}}(\zeta,\nu)\backslash(\alpha\times\omega_{1})$ and the $n$-th node
of $f_{v^{*}}(\zeta,\nu)\backslash(\alpha\times\omega_{1})$ are both in
$M_{\beta}$,
$(\varphi^{M_{\beta}}(p_{u^{*}}),(f_{u^{*}}\upharpoonright
M_{\beta})\upharpoonright\zeta)\Vdash_{\mathbb{R}_{\zeta}}\Sigma(\zeta,\nu,m,n)(L)<_{\dot{T}_{\zeta}}\theta$
and
$(\varphi^{M_{\beta}}(p_{v^{*}}),(f_{v^{*}}\upharpoonright
M_{\beta})\upharpoonright\zeta)\Vdash_{\mathbb{R}_{\zeta}}\Sigma(\zeta,\nu,m,n)(R)<_{\dot{T}_{\zeta}}\tau.$
###### Proof.
By Lemma 3.8, we know that $D(\varphi^{M_{\alpha}},\rho)\cap
D(\varphi^{M_{\beta}},\rho)$ is dense and $=^{*}$-countably closed in
$\mathbb{R}_{\rho}/\left\\{p^{*}(M_{\alpha}),p^{*}(M_{\beta})\right\\}$.
Moreover, by Proposition 4.2 (with the notation from the statement thereof),
$E(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$ is dense and countably
$=^{*}$-closed in
$\mathbb{P}^{*}/\left\\{p^{*}(M_{\alpha}),p^{*}(M_{\beta})\right\\}$.
Consequently,
$E^{*}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}},\rho):=\left\\{v\in\mathbb{R}_{\rho}:v\in
D(\varphi^{M_{\alpha}},\rho)\cap D(\varphi^{M_{\beta}},\rho)\wedge p_{v}\in
E(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})\right\\}$
is dense and countably $=^{*}$-closed in
$\mathbb{R}_{\rho}/\left\\{p^{*}(M_{\alpha}),p^{*}(M_{\beta})\right\\}$. For
use later, we also let $E^{*}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}},\zeta)$
be defined similarly, with $\zeta$ replacing $\rho$ in the above definition.
Let $u$ be as in the assumption of the current lemma. We may extend and
relabel if necessary to assume that $u\in
E^{*}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}},\rho)$. We set
$u_{0}:=v_{0}:=u$ and $w_{0}:=u\upharpoonright M_{\alpha}$. We will now define
a coordinate-wise increasing sequence of triples $\langle\langle
u_{n},v_{n},w_{n}\rangle:n\in\omega\rangle$ and a sequence of tuples
$\langle\langle\zeta_{n},\nu_{n},\theta_{n},\tau_{n}\rangle:n\in\omega\rangle$
(with respect to some bookkeeping device) of tree nodes and ordinals so that
the following conditions are satisfied for all $n$:
1. (1)
$\\#^{\rho}_{\varphi^{M_{\alpha}}}(u_{n},v_{n},w_{n})$;
2. (2)
$u_{n},v_{n}\in D(\varphi^{M_{\beta}},\rho)$;
3. (3)
$\langle\zeta_{n},\nu_{n}\rangle\in\operatorname{dom}(f_{u_{n}})\cap\operatorname{dom}(f_{v_{n}})\cap
M_{\alpha}$, and
$\langle\theta_{n},\tau_{n}\rangle\in(f_{u_{n}}(\zeta_{n},\nu_{n})\backslash(\alpha\times\omega_{1}))\times(f_{v_{n}}(\zeta_{n},v_{n})\backslash(\alpha\times\omega_{1}))$;
4. (4)
$u_{n+1}$ and $v_{n+1}$ split $\langle\theta_{n},\tau_{n}\rangle$ below
$\alpha$ in $\dot{T}_{\zeta_{n}}$, and if $\theta_{n}$ and $\tau_{n}$ are both
below level $\beta$, then in fact $u_{n+1}\upharpoonright M_{\beta}$ and
$v_{n+1}\upharpoonright M_{\beta}$ split $\langle\theta_{n},\tau_{n}\rangle$
below $\alpha$ in $\dot{T}_{\zeta_{n}}$. Moreover, in this case, there is a
pair of nodes $\langle\bar{\theta}_{n},\bar{\tau}_{n}\rangle$ below level
$\alpha$ which witnesses the splitting for both $u_{n+1}$ and $v_{n+1}$ as
well as their restrictions to $M_{\beta}$;
5. (5)
$p_{u_{n}}$ and $p_{v_{n}}$ are in
$E(\varphi^{M_{\alpha}},\varphi^{M_{\beta}})$.
For $n=0$, we have that (1), (2), and (5) hold because $u\in
E^{*}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}},\rho)$. (3) holds by definition
and (4) is vacuous.
Suppose, then, that we have defined $u_{n},v_{n}$, and $w_{n}$. By Lemma 3.13,
we may find extensions $u^{\prime}_{n}\geq u_{n}$, $v^{\prime}_{n}\geq v_{n}$,
and $w^{\prime}_{n}\geq w_{n}$ so that
$\\#^{\rho}_{\varphi^{M_{\alpha}}}(u^{\prime}_{n},v^{\prime}_{n},w^{\prime}_{n})$
holds and so that $u^{\prime}_{n}$ and $v^{\prime}_{n}$ split
$\langle\theta_{n},\tau_{n}\rangle$ below $\alpha$ in $\dot{T}_{\zeta_{n}}$.
Let $\bar{\theta}_{n}$ and $\bar{\tau}_{n}$ be nodes below level $\alpha$
which witness the splitting.
We now define conditions $u^{***}_{n}$, $v^{***}_{n}$, and $w^{***}_{n}$ (the
superscript for later notational purposes) which extend, respectively,
$u^{\prime}_{n}\upharpoonright\zeta_{n}$,
$v^{\prime}_{n}\upharpoonright\zeta_{n}$, and
$w^{\prime}_{n}\upharpoonright\zeta_{n}$. If either $\theta_{n}$ or $\tau_{n}$
are at or above level $\beta$ (namely, outside of $M_{\beta}$), then we simply
set $u^{***}_{n}:=u^{\prime}\upharpoonright\zeta_{n}$,
$v^{***}_{n}:=u^{\prime}\upharpoonright\zeta_{n}$. Since
$\operatorname{dom}(\vec{M})$ satisfies the conclusion of Corollary 3.23 (2),
and since $\zeta_{n}\in M_{\alpha}$, we may find a dual residue $w^{***}_{n}$
of $u^{***}_{n}$ and $v^{***}_{n}$ to $M_{\alpha}$. This completes the
definition of the triple $(u^{***}_{n},v^{***}_{n},w^{***}_{n})$ in the case
that either $\theta_{n}$ or $\tau_{n}$ are at or above level $\beta$.
Suppose on the other hand that $\theta_{n}$ and $\tau_{n}$ are both below
level $\beta$ and therefore are in $M_{\beta}$. Since
$\operatorname{dom}(\vec{M})$ satisfies the conclusion of Corollary 3.23 (2),
and since $\zeta_{n}\in M_{\alpha}\cap\rho$ and
$\\#^{\zeta_{n}}_{\varphi^{M_{\alpha}}}(u^{\prime}_{n}\upharpoonright\zeta_{n},v^{\prime}_{n}\upharpoonright\zeta_{n},w^{\prime}_{n}\upharpoonright\zeta_{n})$,
we may find a condition $w^{*}_{n}\in M_{\alpha}\cap\mathbb{R}_{\zeta_{n}}$
which is a dual residue of $u^{\prime}_{n}\upharpoonright\zeta_{n}$ and
$v^{\prime}_{n}\upharpoonright\zeta_{n}$ to $M_{\alpha}$. We next extend
$u^{\prime}_{n}\upharpoonright\zeta_{n}$ to a condition which extends not only
$w^{*}_{n}$ but also some residue to $M_{\beta}$.
Let $u^{**}_{n}$ be an extension in $D(\varphi^{M_{\beta}},\zeta_{n})$ of
$u^{\prime}_{n}\upharpoonright\zeta_{n}$ and $w^{*}_{n}$. By the remarks
before Lemma 3.8, we may let $\bar{u}^{**}_{n}\in M_{\beta}$ be a residue of
$u^{**}_{n}$ to $M_{\beta}$ in $\mathbb{R}_{\zeta_{n}}$. Finally, let
$u^{***}_{n}$ be a condition in $D(\varphi^{M_{\beta}},\zeta_{n})\cap
D(\varphi^{M_{\alpha}},\zeta_{n})$ which extends $u_{n}^{**}$ and
$\bar{u}^{**}_{n}$ and which satisfies that $u^{***}_{n}\upharpoonright
M_{\beta}\geq_{\mathbb{R}_{\zeta_{n}}}\bar{u}^{**}_{n}$.
By definition of $\bar{\theta}_{n}$ above, we know that
$u^{\prime}_{n}\upharpoonright\zeta_{n}\Vdash_{\mathbb{R}_{\zeta_{n}}}\bar{\theta}_{n}<_{\dot{T}_{\zeta_{n}}}\theta_{n}$,
and hence the extension $u^{**}_{n}$ of $u^{\prime}\upharpoonright\zeta_{n}$
forces this too. Since $\bar{u}^{**}_{n}$ is a residue of $u^{**}_{n}$ to
$M_{\beta}$ in $\mathbb{R}_{\zeta_{n}}$ and $\bar{\theta}_{n}$, $\theta_{n}$
are nodes in $M_{\beta}$, we conclude that $\bar{u}^{**}_{n}$ also forces that
$\bar{\theta}_{n}<_{\dot{T}_{\zeta_{n}}}\theta_{n}$. Finally, since
$u^{***}_{n}\upharpoonright M_{\beta}$ is a condition (because $u^{***}_{n}\in
D(\varphi^{M_{\beta}},\zeta_{n})$) which extends $\bar{u}^{**}_{n}$, we
conclude that $u^{***}_{n}\upharpoonright M_{\beta}$ forces that
$\bar{\theta}_{n}<_{\dot{T}_{\zeta_{n}}}\theta_{n}$. This completes the first
round of extensions of $u^{\prime}_{n}\upharpoonright\zeta_{n}$.
We now turn to extending $v^{\prime}_{n}\upharpoonright\zeta_{n}$. Since
$u_{n}^{***}\in D(\varphi^{M_{\alpha}},\zeta_{n})$, we may let $w^{**}_{n}$ be
a residue of $u^{***}_{n}$ to $M_{\alpha}$ which extends $w_{n}^{*}$. Since
$w^{**}_{n}\geq w_{n}^{*}$ and $w_{n}^{*}$ is a residue of
$v^{\prime}_{n}\upharpoonright\zeta_{n}$ to $M_{\alpha}$, $w^{**}_{n}$ is also
a residue of $v^{\prime}_{n}\upharpoonright\zeta_{n}$ to $M_{\alpha}$.
Applying the same argument as in the previous two paragraphs to
$v^{\prime}_{n}\upharpoonright\zeta_{n}$ with $w^{**}_{n}$ playing the role of
$w^{*}_{n}$ and with $\bar{\tau}_{n}$ and $\tau_{n}$ playing the respective
roles of $\bar{\theta}_{n}$ and $\theta_{n}$, we may find an extension
$v^{***}_{n}$ of $v^{\prime}_{n}\upharpoonright\zeta_{n}$ in
$D(\varphi^{M_{\beta}},\zeta_{n})\cap D(\varphi^{M_{\alpha}},\zeta_{n})$ and a
condition $w^{***}_{n}$ so that $v^{***}_{n}\upharpoonright M_{\beta}$ forces
$\bar{\tau}_{n}<_{\dot{T}_{\zeta_{n}}}\tau_{n}$ and so that $w^{***}_{n}$ is a
residue of $v^{***}_{n}$ to $M_{\alpha}$ in $\mathbb{R}_{\zeta_{n}}$ which
extends $w^{**}_{n}$. Note that since $w^{***}_{n}\geq w^{**}_{n}$,
$w^{***}_{n}$ is also a residue of $u^{***}_{n}$ to $M_{\alpha}$ in
$\mathbb{R}_{\zeta_{n}}$.
To summarize, we now have extensions $u^{***}_{n}$ and $v^{***}_{n}$ of
$u^{\prime}_{n}\upharpoonright\zeta_{n}$ and
$v^{\prime}_{n}\upharpoonright\zeta_{n}$ respectively which are both in
$D(\varphi^{M_{\beta}},\rho)$ and which satisfy that
$u^{***}_{n}\upharpoonright M_{\beta}$ and $v^{***}_{n}\upharpoonright
M_{\beta}$ split $\langle\theta_{n},\tau_{n}\rangle$ as witnessed by
$\langle\bar{\theta}_{n},\bar{\tau}_{n}\rangle$. Moreover, $w^{***}_{n}$ is a
dual residue of $u^{***}_{n}$ and $v^{***}_{n}$ to $M_{\alpha}$ in
$\mathbb{R}_{\zeta_{n}}$, i.e.,
$*^{\zeta_{n}}_{\varphi^{M_{\alpha}}}(u^{***}_{n},v^{***}_{n},w^{***}_{n})$.
This completes the definition of the triple
$(u^{***}_{n},v^{***}_{n},w^{***}_{n})$ in the case that $\theta_{n}$ and
$\tau_{n}$ are below level $\beta$.
We now apply Lemma 3.11, to find $u_{n+1}\upharpoonright\zeta_{n}\geq
u^{***}_{n}$, $v_{n+1}\upharpoonright\zeta_{n}\geq v^{***}_{n}$ and
$w_{n+1}\upharpoonright\zeta_{n}\geq w^{***}_{n}$ so that
$\\#^{\zeta_{n}}_{\varphi^{M_{\alpha}}}(u_{n+1}\upharpoonright\zeta_{n},v_{n+1}\upharpoonright\zeta_{n},w_{n+1}\upharpoonright\zeta_{n})$
and so that
$u_{n+1}\upharpoonright\zeta_{n},v_{n+1}\upharpoonright\zeta_{n}\in
E^{*}(\varphi^{M_{\alpha}},\varphi^{M_{\beta}},\zeta_{n})$. Define
$u_{n+1}:=(u_{n+1}\upharpoonright\zeta_{n})^{\frown}(u^{\prime}_{n}\upharpoonright[\zeta_{n},\rho))$,
with $v_{n+1}$ and $w_{n+1}$ defined similarly. $u_{n+1},v_{n+1}$, and
$w_{n+1}$ then satisfy (1)-(5), completing the successor step of the
construction.
If we now let $u^{*},v^{*},w^{*}$ be sups of their respective sequences, it is
straightforward to see that they satisfy the lemma, using (4) to secure the
desired splitting function. ∎
Having laid the groundwork in the previous results, we next turn to analyzing
when quotients of $\mathbb{R}_{\rho}$ preserve stationary sets of cofinality
$\omega$ ordinals. We will prove the following proposition:
###### Proposition 4.4.
Suppose that $\vec{M}$ is in pre-splitting configuration up to $\rho$ and that
$\operatorname{dom}(\vec{M})$ satisfies Corollary 3.23(2). Then there exists
some $B^{*}\subseteq\operatorname{dom}(\vec{M})$ with
$\operatorname{dom}(\vec{M})\backslash B^{*}\in\cal{I}$ so that for any
$\alpha\in B^{*}$, any $(\mathbb{R}_{\rho}\cap M_{\alpha})$-name $\dot{S}$ for
a stationary subset of $\alpha\cap\operatorname{cof}(\omega)$, and any residue
pair $\langle p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$, the poset
$\mathbb{R}_{\rho}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\rho}})$ forces
that $\dot{S}$ remains stationary.
Thus the quotient forcing of $\mathbb{R}_{\rho}$ above the condition
$(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\rho}})$ preserves the stationarity
of $\dot{S}$. The remainder of the section is devoted to the proof.
###### Proof.
To begin, we define the set $B^{*}:=\operatorname{tr}(B)\cap B$, where
$B=\operatorname{dom}(\vec{M})$. Since $B\in\cal{F}^{+}$, Lemma 1.15 implies
that $B\backslash B^{*}\in\cal{I}$.
Now fix, for the rest of the proof, an ordinal $\alpha\in B^{*}$ and a residue
pair $\langle p^{*}(M_{\alpha}),\varphi^{M_{\alpha}}\rangle$ for
$(M_{\alpha},\mathbb{P}^{*})$; since $\vec{M}$ is in pre-splitting
configuration up to $\rho$, we may also fix, for each $\gamma\in B\cap\alpha$,
a residue pair $\langle p^{*}(M_{\gamma}),\varphi^{M_{\gamma}}\rangle$ for
$(M_{\gamma},\mathbb{P}^{*})$.
Next, fix a condition $(p,f)$ in
$\mathbb{R}_{\rho}/(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\rho}})$ and an
$\mathbb{R}_{\rho}$-name $\dot{C}$ for a closed unbounded subset of $\alpha$.
We will find some extension $(p^{*},f^{*})$ of $(p,f)$ which forces in
$\mathbb{R}_{\rho}$ that $\dot{C}\cap\dot{S}\neq\emptyset$. By Lemma 3.8, we
may assume that $(p,f)\in D(\varphi^{M_{\alpha}},\rho)$.
In $V$, let $\theta>\kappa^{+}$ be a large enough regular cardinal, and let
$K\prec H(\theta)$ be chosen so that $|K|=\kappa$, $\,{}^{<\kappa}K\subseteq
K$, and so that $K$ has the following parameters as elements:
1. (i)
the sequences $\vec{M}$ and $\langle\langle
p^{*}(M_{\gamma}),\varphi^{M_{\gamma}}\rangle:\gamma\in
B\cap(\alpha+1)\rangle$, the set $B^{*}$, the poset $\mathbb{R}_{\rho}$, the
$\mathbb{R}_{\rho}$-condition $(p,f)$, the $\mathbb{R}_{\rho}$-name $\dot{C}$,
and the $(\mathbb{R}_{\rho}\cap M_{\alpha})$-name $\dot{S}$;
2. (ii)
the fixed well-order $\lhd$ of $H(\kappa^{+})$ from Notation 2.12.
Finally, let $\mathbb{K}$ denote the tuple
$(K,\in,\alpha,\vec{M},B^{*},\mathbb{R}_{\rho},(p,f),\dot{C},\dot{S},\lhd).$
Define $E_{0}$ to be the club of $\beta<\alpha$ so that
$\operatorname{Sk}^{\mathbb{K}}(\beta)\cap\alpha=\beta$. Since $\alpha\in
B^{*}$, we know that $B\cap\alpha$ is stationary in $\alpha$. Thus
$B\cap\lim(B)\cap\alpha$ is stationary in $\alpha$, and therefore
$E:=\lim(E_{0}\cap B)$ is a club in $\alpha$.
Recalling that $(p,f)\in D(\varphi^{M_{\alpha}},\rho)$, we can find a residue
$(\bar{p},\bar{f})$ of $(p,f)$ to $M_{\alpha}$ which extends the condition
$(\varphi^{M_{\alpha}}(p),f\upharpoonright M_{\alpha})$. Let $\dot{X}$ be the
$(\mathbb{R}_{\rho}\cap M_{\alpha})$-name for
$\left\\{\beta\in E_{0}\cap
B{\cap\lim(B)}\cap\alpha:(p^{*}(M_{\beta}),0_{\dot{\mathbb{S}}_{\rho}})\in\dot{G}_{\mathbb{R}_{\rho}\cap
M_{\alpha}}\right\\};$
by Lemma 2.21 we know that $\dot{X}$ is forced by $\mathbb{R}_{\rho}\cap
M_{\alpha}$ to be unbounded in $\alpha$. Since $\dot{S}$ is an
$(\mathbb{R}_{\rho}\cap M_{\alpha})$-name of a stationary subset of
$\alpha\cap\operatorname{cof}(\omega)$, $\dot{S}$ is forced to contain a limit
point of $\dot{X}$. This, combined with the fact that $\mathbb{R}_{\rho}\cap
M_{\alpha}$ does not add new $\omega$-sequences, implies that we can find an
extension $(q,g)\geq_{\mathbb{R}_{\rho}\cap M_{\alpha}}(\bar{p},\bar{f})$ and
an increasing sequence $\langle\beta_{n}:n\in\omega\rangle$ in $E_{0}\cap
B{\cap\lim(B)}$ with $\sup_{n}\beta_{n}=\nu\in
E\cap\operatorname{cof}(\omega)$, so that (i)
$(q,g)\Vdash_{\mathbb{R}_{\rho}\cap M_{\alpha}}\nu\in\dot{S}$, and (ii) for
all $n\in\omega$, $q\geq_{\mathbb{P}^{*}}p^{*}(M_{\beta_{n}})$.
For the rest of the proof, we will fix $(q,g)\in\mathbb{R}_{\rho}\cap
M_{\alpha}$, $\langle\beta_{n}\mid n<\omega\rangle$, and $\nu$ with the above
properties. Define $K_{\beta_{n}}:=\operatorname{Sk}^{\mathbb{K}}(\beta_{n}),$
noting that $K_{\beta_{n}}\cap\alpha=\beta_{n}$ because $\beta_{n}\in E_{0}$.
It will be helpful for later to see that $M_{\beta_{n}}\subseteq
K_{\beta_{n}}$ for each $n$. Indeed, $\beta_{n}\in B\cap\lim(B)$ which implies
that $M_{\beta_{n}}=\bigcup_{\eta\in B\cap\beta_{n}}M_{\eta}$. Moreover,
$\beta_{n}\subseteq K_{\beta_{n}}$, and therefore applying the elementarity of
$K_{\beta_{n}}$, we see that for all $\eta\in B\cap\beta_{n}$, $M_{\eta}\in
K_{\beta_{n}}$. Since $\beta_{n}\subseteq K_{\beta_{n}}$, $\eta\subseteq
K_{\beta_{n}}$ too. Thus $M_{\eta}\subseteq K_{\beta_{n}}$ since
$K_{\beta_{n}}$ sees a bijection between $M_{\eta}$ and $\eta$. Combining all
of this, we see that $M_{\beta_{n}}=\bigcup_{\eta\in
B\cap\beta_{n}}M_{\eta}\subseteq K_{\beta_{n}}$.
We proceed to find an extension $(p^{*},f^{*})$ of $(p,f)$ which is compatible
with $(q,g)$ and forces that $\nu\in\dot{C}$. We will secure this by building
two increasing $\omega$-sequences of conditions, one above $(p,f)$ and another
above $(q,g)$, in such a way that the limits of each sequence can be
amalgamated; the resulting condition will then force $\nu$ into
$\dot{S}\cap\dot{C}$. Let $(p_{0},f_{0}):=(p,f)$ and $(q_{0},g_{0}):=(q,g)$.
###### Claim 4.5.
There exists an increasing sequence $\langle(p_{n},f_{n}):n\in\omega\rangle$
of conditions in $\mathbb{R}_{\rho}$ and an increasing sequence
$\langle(q_{n},g_{n}):n\in\omega\rangle$ of conditions in
$\mathbb{R}_{\rho}\cap M_{\alpha}$ so that for each $n\in\omega$,
1. (1)
$(p_{n},f_{n})\in K_{\beta_{n}}$;
2. (2)
$(p_{n+1},f_{n+1})\Vdash_{\mathbb{R}_{\rho}}\dot{C}\cap(\beta_{n},\nu)\neq\emptyset$;
3. (3)
$(p_{n},f_{n})\in D(\varphi^{M_{\alpha}},\rho)$;
4. (4)
$q_{n}\geq_{{\mathbb{P}^{*}}\cap M_{\alpha}}\varphi^{M_{\alpha}}(p_{n})$;
5. (5)
$f_{n+1}$ and $g_{n+1}$ are strongly compatible (Definition 3.17) over
$p_{n+1}$ and $q_{n+1}$.
Before we prove this claim, we show that proving it suffices to obtain the
desired condition $(p^{*},f^{*})$. So suppose that Claim 4.5 is true. Let
$(p^{*},f^{*})$ be a sup of $\langle(p_{n},f_{n}):n\in\omega\rangle$, and let
$(q^{*},g^{*})$ be a sup of $\langle(q_{n},g_{n}):n\in\omega\rangle$.
Observe that by item (2) of Claim 4.5 and the fact that the sequence
$\langle\beta_{n}:n\in\omega\rangle$ is cofinal in $\nu$, we have that
$(p^{*},f^{*})\Vdash_{\mathbb{R}_{\rho}}\nu\in\lim(\dot{C})$ and hence forces
that $\nu\in\dot{C}$ as $\dot{C}$ names a club. Also, since
$(q^{*},g^{*})\geq(q_{0},g_{0})$ and since $(q_{0},g_{0})=(q,g)$ forces that
$\nu\in\dot{S}$, $(q^{*},g^{*})$ forces that $\nu\in\dot{S}$ too. We claim
that $p^{*}$ and $q^{*}$ are compatible in $\mathbb{P}^{*}$, from which it
follows by item (5) of Claim 4.5 that $f^{*}$ and $g^{*}$ are strongly
compatible over $p^{*}$ and $q^{*}$. Indeed, $(p^{*},f^{*})\in
D(\varphi^{M_{\alpha}},\rho)$ since this set is closed under sups of
increasing $\omega$-sequences by Lemma 3.8. Furthermore, by the countable
continuity of $\varphi^{M_{\alpha}}$, $\varphi^{M_{\alpha}}(p^{*})$ is a sup
of the increasing sequence
$\langle\varphi^{M_{\alpha}}(p_{n}):n\in\omega\rangle$. Thus to show that
$p^{*}$ and $q^{*}$ are compatible, since $q^{*}\in\mathbb{P}^{*}\cap
M_{\alpha}$, it suffices to show that $q^{*}\geq_{\mathbb{P}^{*}\cap
M_{\alpha}}\varphi^{M_{\alpha}}(p^{*})$. However, we know that $q^{*}\geq
q_{n}$ for all $n$ and so by (4) of Claim 4.5,
$q^{*}\geq\varphi^{M_{\alpha}}(p_{n})$ for all $n$. Therefore $q^{*}$ extends
$\varphi^{M_{\alpha}}(p^{*})$, by definition of a supremum.
Now let $(p^{**},f^{**})$ be a condition in $\mathbb{R}_{\rho}$ above both
$(p^{*},f^{*})$ and $(q^{*},g^{*})$. Then because $(p^{**},f^{**})$ extends
$(p^{*}(M_{\alpha}),0_{\dot{\mathbb{S}}_{\rho}})$ as well as $(q^{*},g^{*})$,
which in turn forces in $\mathbb{R}_{\rho}\cap M_{\alpha}$ that
$\nu\in\dot{S}$, we have that
$(p^{**},f^{**})\Vdash_{\mathbb{R}_{\rho}}\nu\in\dot{S}$. And finally, as
$(p^{**},f^{**})$ extends $(p^{*},f^{*})$ which forces in $\mathbb{R}_{\rho}$
that $\nu\in\dot{C}$, we conclude that
$(p^{**},f^{**})\Vdash_{\mathbb{R}_{\rho}}\nu\in\dot{S}\cap\dot{C}$. Thus it
suffices to prove Claim 4.5 in order to finish the proof of Proposition 4.4.
_Proof._ (of Claim 4.5) We will construct the sequences satisfying (1)-(5) of
Claim 4.5 recursively. For the base case $n=0$, items (2) and (5) hold
vacuously. For item (1), we have that $(p_{0},f_{0})=(p,f)\in K_{\beta_{0}}$
as $(p,f)=(p_{0},f_{0})$ was chosen to be definable by a constant in the
language of $\mathbb{K}$. We also ensured that $(p_{0},f_{0})\in
D(\varphi^{M_{\alpha}},\rho)$, which establishes (3). Finally,
$q_{0}\geq_{\mathbb{P}^{*}\cap M_{\alpha}}\bar{p}\geq_{\mathbb{P}^{*}\cap
M_{\alpha}}\varphi^{M_{\alpha}}(p_{0}),$ which establishes (4).
Suppose, then, that we have defined $(p_{n},f_{n})$ and $(q_{n},g_{n})$
satisfying (1)-(5). We first observe that $(p_{n},f_{n})$ and $(q_{n},g_{n})$
are compatible. If $n=0$, this holds since $(q_{0},g_{0})$ is in
$\mathbb{R}_{\rho}\cap M_{\alpha}$ and extends $(\bar{p},\bar{f})$, which is a
residue of $(p_{0},f_{0})$ to $\mathbb{R}_{\rho}\cap M_{\alpha}$. If $n>0$,
then we have that $q_{n}\geq_{\mathbb{P}^{*}\cap
M_{\alpha}}\varphi^{M_{\alpha}}(p_{n})$, and therefore $p_{n}$ and $q_{n}$ are
$\mathbb{P}^{*}$-compatible. Moreover, $f_{n}$ and $g_{n}$ are strongly
compatible over the compatible conditions $p_{n}$ and $q_{n}$, and therefore
$(p_{n},f_{n})$ and $(q_{n},g_{n})$ are compatible in $\mathbb{R}_{\rho}$.
Next choose some condition $(r,h)$ in $\mathbb{R}_{\rho}$ which extends
$(p_{n},f_{n})$ and $(q_{n},g_{n})$, and by extending if necessary, we may
assume that there is some ordinal $\mu>\beta_{n}$ so that
$(r,h)\Vdash_{\mathbb{R}_{\rho}}\mu\in\dot{C}\backslash(\beta_{n}+1)$. Since
$r\geq p_{n}\geq p^{*}(M_{\alpha})$ and since $r\geq q_{n}\geq q_{0}\geq
p^{*}(M_{\beta_{n+1}})$, we may also extend if necessary to assume, by Lemma
4.1(1), that
$r\in\operatorname{dom}(\varphi^{M_{\beta_{n+1}}})\cap\operatorname{dom}(\varphi^{M_{\alpha}})$
and also that $\varphi^{M_{\alpha}}(r)\geq q_{n}$.
We now apply Lemma 4.3, with $\alpha$ and $\beta_{n+1}$ playing the respective
roles of “$\beta$” and “$\alpha$” in the statement thereof, to find extensions
$(r_{L},h_{L})$ and $(r_{R},h_{R})$ of $(r,h)$ which satisfy the conclusion of
that lemma. We let $(\bar{r},\bar{h})$ be a condition so that
$\\#^{\rho}_{\varphi^{M_{\beta_{n+1}}}}((r_{L},h_{L}),(r_{R},h_{R}),(\bar{r},\bar{h}))$.
Let $\Sigma$ be a splitting function for $(r_{L},h_{L})$ and $(r_{R},h_{R})$
with respect to the model $M_{\beta_{n+1}}$ which satisfies Lemma 4.3. For
$Z\in\left\\{L,R\right\\}$, set
$x_{Z}:=\operatorname{dom}(h_{Z})\cap M_{\beta_{n+1}};$
this is a countable subset of $M_{\beta_{n+1}}$ and therefore is a member of
$M_{\beta_{n+1}}$. Since $M_{\beta_{n+1}}\subseteq K_{\beta_{n+1}}$, as shown
earlier, $x_{Z}$ is also an element of $K_{\beta_{n+1}}$.
We are now in a position to reflect into the model $K_{\beta_{n+1}}$. We
observe that in $H(\theta)$ the following statement is true in the following
parameters
$\beta_{n},\Sigma,\bar{r},\bar{h},\mathbb{R}_{\rho},(p_{n},f_{n}),\alpha$,
$B,\dot{C},x_{L}$, $x_{R}$, $\vec{M}$, and $\langle\langle
p^{*}(M_{\gamma}),\varphi^{M_{\gamma}}\rangle:\gamma\in
B\cap(\alpha+1)\rangle$, all of which are in $K_{\beta_{n+1}}$: there exists a
condition $(r^{*},h^{*})$ in $\mathbb{R}_{\rho}$ and a pair
$(r^{*}_{Z},h^{*}_{Z})_{Z\in\left\\{L,R\right\\}}$ of conditions above
$(r^{*},h^{*})$ in $\mathbb{R}_{\rho}$ as well as ordinals $\mu^{*},\eta$, so
that
1. (i)
$(r^{*},h^{*})\geq_{\mathbb{R}_{\rho}}(p_{n},f_{n})$;
2. (ii)
$\eta\in B$;
3. (iii)
$\\#^{\rho}_{\varphi^{M_{\eta}}}((r^{*}_{L},h^{*}_{L}),(r^{*}_{R},h^{*}_{R}),(\bar{r},\bar{h}))$;
4. (iv)
for each $Z\in\left\\{L,R\right\\}$, $(r^{*}_{Z},h^{*}_{Z})$ and
$(r^{*},h^{*})$ are in $E^{*}(\varphi^{M_{\eta}},\varphi^{M_{\alpha}})$ (see
Proposition 4.2);
5. (v)
$(r^{*},h^{*})\Vdash_{\mathbb{R}_{\rho}}\mu^{*}\in\dot{C}\backslash(\beta_{n}+1)$;
6. (vi)
$\operatorname{dom}(h^{*}_{Z})\cap M_{\eta}=x_{Z}$;
7. (vii)
$(r^{*}_{L},h^{*}_{L})$ and $(r^{*}_{R},h^{*}_{R})$ are an
$(M_{\eta},\rho)$-splitting pair, and $\Sigma$ is a splitting function for
$(r^{*}_{L},h^{*}_{L})$ and $(r^{*}_{R},h^{*}_{R})$ with respect to the model
$M_{\eta}$.
This statement is true in $H(\theta)$ as witnessed by the conditions
$(r_{Z},h_{Z})_{Z\in\left\\{L,R\right\\}}$ and $(r,h)$, the ordinal $\mu$
playing the role of $\mu^{*}$, and the ordinal $\beta_{n+1}$ playing the role
of $\eta$. Since the parameters of this statement are in $K_{\beta_{n+1}}$, we
may therefore find, in $K_{\beta_{n+1}}$, conditions
$(r^{*}_{Z},h^{*}_{Z})_{Z\in\left\\{L,R\right\\}}$ extending some
$(r^{*},h^{*})\geq(p_{n},f_{n})$, an ordinal $\mu^{*}$, and an ordinal
$\eta\in B$ so that (i)-(vii) above are satisfied of these objects.
We now define $(p_{n+1},f_{n+1}):=(r^{*}_{L},h^{*}_{L})$. We need to extend
the condition $(\varphi^{M_{\alpha}}(r_{R}),h_{R}\upharpoonright M_{\alpha})$
a bit more before defining $(q_{n+1},g_{n+1})$. The following claim will help
us do this:
###### Subclaim 4.6.
$\varphi^{M_{\alpha}}(p_{n+1})$ and $\varphi^{M_{\alpha}}(r_{R})$ are
compatible in $\mathbb{P}^{*}\cap M_{\alpha}$.
_Proof._ Both the condition $p_{n+1}$ and the function $\varphi^{M_{\alpha}}$
are members of $K_{\beta_{n+1}}$. Therefore $\varphi^{M_{\alpha}}(p_{n+1})\in
K_{\beta_{n+1}}\cap M_{\alpha}\cap\mathbb{P}^{*}$. Recall that $\mathbb{K}$
contained the fixed well-order $\lhd$ of $H(\kappa^{+})$ and that all suitable
models are elementary in $H(\kappa^{+})$ with respect to $\lhd$. Thus if we
let $e^{\mathbb{P}^{*}}$ denote the $\lhd$-least bijection from $\kappa$ onto
$\mathbb{P}^{*}$, then we have that $e^{\mathbb{P}^{*}}$ is in $M_{\alpha}$
and in $K_{\beta_{n+1}}$. Since $M_{\alpha}$ is elementary and contains
$e^{\mathbb{P}^{*}}$, we see that
$\varphi^{M_{\alpha}}(p_{n+1})=e^{\mathbb{P}^{*}}(\zeta)$ for some
$\zeta<\alpha$. But then by the elementarity of $K_{\beta_{n+1}}$, we see that
$\zeta\in K_{\beta_{n+1}}\cap\alpha=\beta_{n+1}$. Therefore
$\varphi^{M_{\alpha}}(p_{n+1})=e^{\mathbb{P}^{*}}(\zeta)\in M_{\beta_{n+1}}$.
Furthermore, we know that
$\bar{r}=^{*}\varphi^{M_{\beta_{n+1}}}(r_{R})=^{*}\varphi^{M_{\beta_{n+1}}}(\varphi^{M_{\alpha}}(r_{R}))$
where the first equality holds by definition of $\bar{r}$ and the second
because $r_{R}$ satisfies Lemma 4.3. Applying (iii) and (iv) above we also
have that,
$\bar{r}=^{*}\varphi^{M_{\eta}}(p_{n+1})=^{*}\varphi^{M_{\eta}}(\varphi^{M_{\alpha}}(p_{n+1})).$
Additionally, since $\varphi^{M_{\eta}}$ is an exact, strong residue function
and $\varphi^{M_{\alpha}}(p_{n+1})\in\operatorname{dom}(\varphi^{M_{\eta}})$,
we know that
$\varphi^{M_{\alpha}}(p_{n+1})\geq\varphi^{M_{\eta}}(\varphi^{M_{\alpha}}(p_{n+1}))=^{*}\bar{r}.$
Therefore, as $\varphi^{M_{\alpha}}(p_{n+1})\in M_{\beta_{n+1}}$ extends
$\bar{r}$, which is a residue of $\varphi^{M_{\alpha}}(r_{R})$ to
$M_{\beta_{n+1}}$, we conclude that $\varphi^{M_{\alpha}}(p_{n+1})$ is
compatible with $\varphi^{M_{\alpha}}(r_{R})$ in $\mathbb{P}^{*}\cap
M_{\alpha}$. ∎(Subclaim 4.6)
Using Subclaim 4.6, we may fix some condition $q_{n+1}$ in $\mathbb{P}^{*}\cap
M_{\alpha}$ which is above both $\varphi^{M_{\alpha}}(p_{n+1})$ and
$\varphi^{M_{\alpha}}(r_{R})$. We finally set $g_{n+1}:=h_{R}\upharpoonright
M_{\alpha}$, noting that $(q_{n+1},g_{n+1})\in M_{\alpha}$.
We next verify that items (1)-(5) of Claim 4.5 hold for $n+1$. We have that
$(p_{n+1},f_{n+1})\geq(p_{n},f_{n})$ by (i) of the reflection, more precisely,
since
$(p_{n+1},f_{n+1})=(r^{*}_{L},h^{*}_{L})\geq(r^{*},h^{*})\geq(p_{n},f_{n}).$
Additionally, because
$q_{n+1}\geq\varphi^{M_{\alpha}}(r_{R})\geq\varphi^{M_{\alpha}}(r)\geq q_{n},$
we have that $q_{n+1}\geq q_{n}$. Moreover, $g_{n+1}$ extends $g_{n}$ as a
function: $g_{n+1}=h_{R}\upharpoonright M_{\alpha}$, $g_{n}\in M_{\alpha}$,
and $(r_{R},h_{R})\geq(r,h)\geq(q_{n},g_{n})$. Thus $(q_{n+1},g_{n+1})$
extends $(q_{n},g_{n})$. (1) of Claim 4.5 holds because we found the witnesses
in the model $K_{\beta_{n+1}}$. For (2), $\mu^{*}\in
K_{\beta_{n+1}}\cap\alpha=\beta_{n+1}\subseteq\nu$, and since
$\mu^{*}>\beta_{n}$, we have that
$(p_{n+1},f_{n+1})\Vdash\mu^{*}\in\dot{C}\cap(\beta_{n},\nu)$. For (3), we
have $(p_{n+1},f_{n+1})\in D(\varphi^{M_{\alpha}},\rho)$ by (iii) and the
definition of $\\#^{\rho}_{\varphi^{M_{\eta}}}$ (see Definition 3.10). For
(4), we have that $q_{n+1}\geq\varphi^{M_{\alpha}}(p_{n+1})$ by choice of
$q_{n+1}$.
It remains therefore to check that item (5) of Claim 4.5 holds. Since
$q_{n+1}\geq_{\mathbb{P}^{*}\cap M_{\alpha}}\varphi^{M_{\alpha}}(p_{n+1})$, we
know that $p_{n+1}$ and $q_{n+1}$ are compatible in $\mathbb{P}^{*}$; let
$p^{*}\in\mathbb{P}^{*}$ be any condition extending both. We claim that
$p^{*}\Vdash_{\mathbb{P}^{*}}\check{f}_{n+1}\cup\check{g}_{n+1}\in\dot{\mathbb{S}}_{\rho}.$
Suppose by induction on $\delta<\rho$ that
$\langle\delta,\nu\rangle\in\operatorname{dom}(f_{n+1})\cap\operatorname{dom}(g_{n+1})$
and that $p^{*}$ forces that the union of
$\check{f}_{n+1}\upharpoonright\delta$ and
$\check{g}_{n+1}\upharpoonright\delta$ is a condition in
$\dot{\mathbb{S}}_{\delta}$. Again using the fixed well-order $\lhd$ of
$H(\kappa^{+})$, we may let $\psi$ be the $\lhd$-least bijection from $\kappa$
onto $\rho$, so that $\psi$ is a member of $K_{\beta_{n+1}}$ as well as every
model on the $\mathbb{R}_{\rho}$-suitable sequence $\vec{M}$. Since
$\langle\delta,\nu\rangle\in\operatorname{dom}(g_{n+1})$ and $g_{n+1}\in
M_{\alpha}$, $\delta\in M_{\alpha}\cap\rho=\psi[\alpha]$. Furthermore, since
$\langle\delta,\nu\rangle\in\operatorname{dom}(f_{n+1})$ and
$f_{n+1}=h^{*}_{L}\in K_{\beta_{n+1}}$, we have that $\delta\in
K_{\beta_{n+1}}$. Thus
$\delta\in\psi[\alpha]\cap
K_{\beta_{n+1}}=\psi[K_{\beta_{n+1}}\cap\alpha]=\psi[\beta_{n+1}]\subseteq
M_{\beta_{n+1}}.$
Therefore (recalling that $g_{n+1}=h_{R}\upharpoonright M_{\alpha}$),
$\langle\delta,\nu\rangle\in\operatorname{dom}(h_{R})\cap
M_{\beta_{n+1}}=x_{R}=\operatorname{dom}(h^{*}_{R})\cap M_{\eta}.$
Continuing, fix a pair
$\langle\theta,\tau\rangle\in f_{n+1}(\delta,\nu)\times g_{n+1}(\delta,\nu)$
with $\theta\neq\tau$. We need to show that $\theta$ and $\tau$ are forced to
be incompatible nodes in the tree $\dot{T}_{\delta}$ by the condition
$\big{(}p^{*},(f_{n+1}\cup g_{n+1})\upharpoonright\delta\big{)}$. Recall,
going forward, that $(\bar{r},\bar{h})$ equals both
$(r_{R},h_{R})\upharpoonright M_{\beta_{n+1}}$ and
$(p_{n+1},f_{n+1})\upharpoonright M_{\eta}$; in particular, $f_{n+1}$ and
$h_{R}\upharpoonright M_{\alpha}=g_{n+1}$ both extend $\bar{h}$. Continuing,
if $\tau$ is below level $\beta_{n+1}$, then
$\tau\in\bar{h}(\delta,\nu)\subseteq f_{n+1}(\delta,\nu)$ and we are done.
Furthermore, if $\theta$ is below level $\eta$, then
$\theta\in\bar{h}(\delta,\nu)\subseteq g_{n+1}(\delta,\nu)$ and we are done in
this case too. Thus we assume that $\theta$ is at or above level $\eta$ and
that $\tau$ is at or above level $\beta_{n+1}$. With respect to the fixed
enumerations, let $k$ and $m$ be chosen so that $\theta$ is the $k$th element
of $f_{n+1}(\delta,\nu)\backslash(\eta\times\omega_{1})$ and $\tau$ is the
$m$th element of $h_{R}(\delta,\nu)\backslash(\beta_{n+1}\times\omega_{1})$.
Then because the function $\Sigma$ is the same for both pairs of splitting
conditions, we know that
$(p_{n+1},f_{n+1}\upharpoonright\delta)\Vdash_{\mathbb{R}_{\delta}}\Sigma(\delta,\nu,k,m)(L)<_{\dot{T}_{\delta}}\theta$
and that
$(r_{R},h_{R}\upharpoonright\delta)\Vdash_{\mathbb{R}_{\delta}}\Sigma(\delta,\nu,k,m)(R)<_{\dot{T}_{\delta}}\tau.$
However, $\tau\in g_{n+1}(\delta,\nu)=(h_{R}\upharpoonright
M_{\alpha})(\delta,\nu)$, and therefore $\tau$ is below level $\alpha$ of the
tree $\dot{T}_{\delta}$. Therefore by Lemma 4.3, we have that
$\Big{(}\varphi^{M_{\alpha}}(r_{R}),(h_{R}\upharpoonright
M_{\alpha})\upharpoonright\delta\Big{)}\Vdash_{\mathbb{R}_{\delta}}\Sigma(\delta,\nu,k,m)(R)<_{\dot{T}_{\delta}}\tau.$
Since $q_{n+1}\geq\varphi^{M_{\alpha}}(r_{R})$ and
$g_{n+1}=h_{R}\upharpoonright M_{\alpha}$, we conclude that
$(q_{n+1},g_{n+1}\upharpoonright\delta)\Vdash_{\mathbb{R}_{\delta}}\Sigma(\delta,\nu,k,m)(R)<_{\dot{T}_{\delta}}\tau.$
Finally, since $\big{(}p^{*},(f_{n+1}\cup
g_{n+1})\upharpoonright\delta\big{)}$ is above both
$(p_{n+1},f_{n+1}\upharpoonright\delta)$ and
$(q_{n+1},g_{n+1}\upharpoonright\delta)$, it follows that
$\big{(}p^{*},(f_{n+1}\cup
g_{n+1})\upharpoonright\delta\big{)}\Vdash\Sigma(\delta,\nu,k,m)(R)<_{\dot{T}_{\delta}}\tau\wedge\Sigma(\delta,\nu,k,m)(L)<_{\dot{T}_{\delta}}\theta.$
Since the distinct nodes $\Sigma(\delta,\nu,k,m)(L)$ and
$\Sigma(\delta,\nu,k,m)(R)$ are on the same level, $\big{(}p^{*},(f_{n+1}\cup
g_{n+1})\upharpoonright\delta\big{)}$ therefore forces that $\theta$ and
$\tau$ are incompatible nodes in the tree $\dot{T}_{\delta}$.
This completes the proof that $f_{n+1}$ and $g_{n+1}$ are strongly compatible
over $p_{n+1}$ and $q_{n+1}$. Therefore the proof of Claim 4.5 is now
complete. ∎(Claim 4.5)
As remarked earlier, this completes the proof of Proposition 4.4. ∎
###### Remark 4.7.
As we’ve noted before, if $\mathbb{P}^{*}$ is just equal to the collapse poset
$\mathbb{P}$, then the results from Section 3 which are needed for this
section hold only assuming that $\kappa$ is weakly compact (since then
$\cal{F}=\cal{F}_{WC}$; see Definition 2.14). We then see that if
$\mathbb{P}^{*}$ is just $\mathbb{P}$, then the arguments in this section can
also be carried out only using a weakly compact.
As a corollary of Proposition 4.4, we can now prove Theorem 1.2.
###### Proof.
Recall that the Laver-Shelah model $V[G*F]$ is obtained by starting from a
ground model $V$ with a weakly compact cardinal $\kappa$, and forcing with the
Levy collase $\mathbb{P}$ followed by a countable support iteration
$\mathbb{S}=\langle\mathbb{S}_{\tau},\mathbb{S}(\tau)\mid\tau<\kappa^{+}\rangle$
of specializing posets $\mathbb{S}(\tau)=\mathbb{S}(\dot{T_{\tau}})$ of
Aronszajn trees on $\kappa$, chosen by a bookkeeping function. Since
$\mathbb{S}$ satisfies the $\kappa$-c.c, every sequence of stationary sets
$\langle S_{\alpha}\mid\alpha<\kappa\rangle$ as in the statement of Theorem
1.2, belongs to an intermediate extension $V[G*F_{\tau}]$, where
$F_{\tau}:=F\cap\mathbb{S}_{\tau}$, for some $\tau<\kappa^{+}$. Now work in
$V$, and take a $(\mathbb{P}*\dot{\mathbb{S}}_{\tau})$-name
$\langle\dot{S}_{\alpha}\mid\alpha<\kappa\rangle$ for the sequence of
stationary sets. Let $\vec{M}$ be suitable with respect to these parameters.
By the weak compactness of $\kappa$, let $\beta\in\operatorname{dom}(\vec{M})$
be such that $\langle\dot{S}_{\alpha}\cap V_{\beta}\mid\alpha<\beta\rangle$
are names for stationary subsets of $\beta$ in the restricted poset
$(\mathbb{P}*\dot{\mathbb{S}}_{\tau})\cap M_{\beta}$. Recalling Remark 4.7, we
see that Proposition 4.4 can be applied to $\mathbb{P}$, and in this case,
$\cal{F}=\cal{F}_{WC}$. We conclude that each $S_{\alpha}\cap\beta$ remains
stationary in the full $\mathbb{P}*\dot{\mathbb{S}}_{\tau}$ generic extension
$V[G*F_{\tau}]$ and hence in $V[G\ast F]$.
To see that $\mathsf{CSR}(\omega_{2})$ fails in the Laver-Shelah model,
observe that in the ground model, there exist stationary sets
$S\subseteq\kappa\cap\operatorname{cof}(\omega)$ and
$T\subseteq\kappa\cap\operatorname{cof}(\omega_{1})$ so that $S$ does not
reflect at any point in $T$ (see Proposition 1.1 of [25]). The stationarity of
$S$ and $T$ are preserved by the $\kappa$-c.c. forcing of Laver and Shelah,
and since $\omega_{1}$ is also preserved, we have that $S$ and $T$ witness the
failure of $\mathsf{CSR}(\omega_{2})$ in the final model. ∎
## 5\. $\cal{F}$-completely proper posets
In this section, we will specify what $\mathbb{P}$-names $\dot{\mathbb{C}}$
for posets are such that $\mathbb{P}\ast\dot{\mathbb{C}}$ is
${\cal{F}}$-strongly proper, and we will draw some conclusions from this.
Since $\mathbb{P}\ast\dot{\mathbb{C}}$ will be playing the role of
$\mathbb{P}^{*}$ in Definition 2.14, and since
$\mathbb{P}\ast\dot{\mathbb{C}}$ is not merely the collapse, we are in the
case when $\kappa$ is ineffable and $\cal{F}=\cal{F}_{in}$. However, we only
use the ineffability of $\kappa$ in the following section when applying
Proposition 3.22 through its use in Proposition 4.4. Recall Definition 2.14
for the definition of ${\cal{F}}$, and also recall that $\mathbb{P}$ denotes
the Levy collapse poset $\operatorname{Col}(\omega_{1},<\kappa)$, where
$\kappa$ is either ineffable or weakly compact. Two main ideas come into play
in this section. The first is an axiomatization of various properties of the
iterated club adding $\dot{\mathbb{C}}_{\text{Magidor}}$ from [34], which will
allow us to place upper bounds on various “local” filters added by the Levy
collapse. We then couple this axiomatization with a generalization of a result
of Abraham’s ([1]) that, in current language, if $\dot{\mathbb{Q}}$ is an
$\operatorname{Add}(\omega,\omega_{1})$-name for an $\omega_{1}$-closed poset,
then $\operatorname{Add}(\omega,\omega_{1})\ast\dot{\mathbb{Q}}$ is strongly
proper; see [18] for a proof of this fact as stated here. The strong
properness results from using so-called “guiding reals.”
We recall that in [34], to show that an iteration
$\dot{\mathbb{C}}_{\text{Magidor}}$ of length $<\kappa^{+}$ adding the desired
clubs is $\kappa$-distributive, Magidor argued, in part, as follows: let
$j:M\longrightarrow N$ be a weakly compact embedding, where $M$ has the
relevant parameters. Let $G^{*}$ be $N$-generic over $j(\mathbb{P})$ and
$G:=G^{*}\cap\mathbb{P}$, so that in $N[G^{*}]$, we may construct an
$M[G]$-generic filter $H$ for $\mathbb{C}_{\text{Magidor}}$. Moreover, $j[H]$
has a least upper bound in $j(\mathbb{C}_{\text{Magidor}})$, namely, the
function obtained by placing $\kappa$ on top of each coordinate in the domain
of $j[H]$; by the closure of the quotient (which implies the preservation of
the stationary sets appearing along the way in the definition of
$\mathbb{C}_{\text{Magidor}}$), this is indeed a condition.
The property of $\dot{\mathbb{C}}_{\text{Magidor}}$ which we will axiomatize
is a reflection of the above to an $\cal{F}$-positive set of $\alpha<\kappa$.
Roughly, we want to say that for many $\alpha$, if you “cut off”
$\mathbb{P}\ast\dot{\mathbb{C}}$ at $\alpha$, then many generics added by the
tail of the collapse for “$\dot{\mathbb{C}}$ cut off at $\alpha$” have upper
bounds in the full poset $\dot{\mathbb{C}}$. More precisely, given a
$\mathbb{P}$-name $\dot{\mathbb{C}}$ in $H(\kappa^{+})$ for a poset which is
$\omega_{1}$-closed with sups and given a $\dot{\mathbb{C}}$-suitable model
(see Definition 2.13) $M$, say with $M\cap\kappa=\alpha<\kappa$, we consider
the poset $\pi_{M}(\dot{\mathbb{C}})$, where $\pi_{M}$ denotes the transitive
collapse of $M$ to $\bar{M}$. An easy absoluteness argument shows that
$\pi_{M}(\dot{\mathbb{C}})$ is a name in
$\pi_{M}(\mathbb{P})=\mathbb{P}\upharpoonright\alpha$. Appealing to the
closure of $\bar{M}$ under $<\alpha$-sequences, and hence $\omega$-sequences,
we see that $\pi_{M}(\dot{\mathbb{C}})$ is forced by
$\mathbb{P}\upharpoonright\alpha$ to be $\omega_{1}$-closed with sups. The
desired condition on $\mathbb{P}$-names $\dot{\mathbb{C}}$ can now be stated a
bit more precisely: we will demand that after forcing with $\mathbb{P}$, say
to add the generic $G$, for many $\alpha$ as above and many
$V[G_{\mathbb{P}\upharpoonright\alpha}]$-generics $H$ for
$\pi_{M}(\dot{\mathbb{C}})[G_{\mathbb{P}\upharpoonright\alpha}]$ in $V[G]$,
$\pi^{-1}_{M[G]}[H]$ has an upper bound in $\dot{\mathbb{C}}[G]$. Note that we
are implicitly appealing to the properness of $\mathbb{P}$ with respect to $M$
to see that $\pi_{M}:M\longrightarrow\bar{M}$ lifts to
$\pi_{M[G]}:M[G]\longrightarrow\bar{M}[G_{\mathbb{P}\upharpoonright\alpha}]$;
we discuss this more later.
The first step to making this work is to isolate exactly which filters we will
use; for reasons related to building strong, exact residue functions later, we
will not consider all filters added by the tail of the collapse for
$\pi_{M}(\dot{\mathbb{C}})$. The definition is meant to capture the behavior
of filters generated by using the generic surjections to guide choices of
conditions, similar to how Abraham used guiding reals in [1].
### 5.1. Residue Functions from Local Filters
###### Definition 5.1.
Let $\alpha<\kappa$ be inaccessible, and let $\dot{\mathbb{Q}}$ be a
$(\mathbb{P}\upharpoonright\alpha)$-name for a poset of size $\alpha$ which is
$\omega_{1}$-closed with sups. Since $\mathbb{P}\upharpoonright\alpha$ is
$\alpha$-c.c there exists a list $\langle\dot{\gamma}_{i}:i<\alpha\rangle$ of
$(\mathbb{P}\upharpoonright\alpha)$-names, which is forced to enumerate all
conditions in $\dot{\mathbb{Q}}$. We say that a sequence
$\dot{s}=\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ of $\mathbb{P}$-names
(not $(\mathbb{P}\upharpoonright\alpha)$-names) for conditions in
$\dot{\mathbb{Q}}$ is guided by the collapse at $\alpha$ for
$\dot{\mathbb{Q}}$ if the following conditions are satisfied:
1. (1)
$\Vdash_{\mathbb{P}}\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ is
$\leq_{\dot{\mathbb{Q}}}$-increasing, $\dot{d}_{0}$ is the weakest condition
in $\dot{\mathbb{Q}}$, and if $\nu$ is limit, then $\dot{d}_{\nu}$ is a sup of
$\langle\dot{d}_{\mu}:\mu<\nu\rangle$;
2. (2)
if $p\in\mathbb{P}$ and $\operatorname{dom}(p(\alpha))$ is an ordinal
$\nu<\omega_{1}$, then there exists $p^{\prime}\geq p$ with
$p^{\prime}\upharpoonright[\alpha,\kappa)=p\upharpoonright[\alpha,\kappa)$ and
a sequence $\langle\beta(\mu):\mu\leq\nu\rangle$ of ordinals in $V$ so that
$p^{\prime}\Vdash\dot{d}_{\mu}=^{*}\dot{\gamma}_{\beta(\mu)}$ for all
$\mu\leq\nu$. In this case we will say that $p^{\prime}$ determines an initial
segment of $\dot{s}$;
3. (3)
if $p^{\prime}$ as in (2) determines an initial segment of $\dot{s}$ and if
$\dot{\gamma}$ is a $(\mathbb{P}\upharpoonright\alpha)$-name for a
$\dot{\mathbb{Q}}$-extension of $\dot{\gamma}_{\beta(\nu)}$, then there exists
$p^{*}\geq p^{\prime}$ so that $p^{*}\Vdash\dot{d}_{\nu+1}\geq\dot{\gamma}$.
###### Lemma 5.2.
Suppose that $\dot{s}:=\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ is guided
by the collapse at $\alpha$. Let $H(\dot{s})$ be the $\mathbb{P}$-name for the
filter on $\dot{\mathbb{Q}}$ generated by $\dot{s}$. Then $\mathbb{P}$ forces
that $H(\dot{s})$ is $V[\dot{G}\upharpoonright\alpha]$-generic over
$\dot{\mathbb{Q}}$.
###### Proof.
Fix a condition $p\in\mathbb{P}$ and a $\mathbb{P}$-name $\dot{D}$ for a dense
subset of $\dot{\mathbb{Q}}$ which is a member of
$V[\dot{G}\upharpoonright\alpha]$. We find an extension of $p$ which forces
that $\dot{D}\cap H(\dot{s})\neq\emptyset$. By extending $p$ and applying (2)
of Definition 5.1 if necessary, we may assume the following:
1. (1)
there is a $(\mathbb{P}\upharpoonright\alpha)$-name $\dot{D}_{0}$ for a dense
subset of $\dot{\mathbb{Q}}$ so that $p\Vdash\dot{D}=\dot{D}_{0}$;
2. (2)
$\operatorname{dom}(p(\alpha))$ is an ordinal $\nu$, and there is a sequence
$\langle\beta(\mu):\mu\leq\nu\rangle$ of ordinals in $V$ so that
$p\Vdash\dot{d}_{\mu}=^{*}\dot{\gamma}_{\beta(\mu)}$ for all $\mu\leq\nu$.
Let $\dot{\gamma}$ be a $(\mathbb{P}\upharpoonright\alpha)$-name for a
condition in $\dot{D}_{0}$ forced to extend $\dot{\gamma}_{\beta(\nu)}$. By
item (3) of Definition 5.1, we may find an extension $p^{*}$ of $p$ so that
$p^{*}\Vdash\dot{d}_{\nu+1}\geq\dot{\gamma}$. Then
$p^{*}\Vdash\dot{\gamma}\in\dot{D}\cap H(\dot{s})$, finishing the proof. ∎
###### Definition 5.3.
Let $\dot{H}$ be a $\mathbb{P}$-name for a filter on $\dot{\mathbb{Q}}$. We
say that $\dot{H}$ is guided by the collapse at $\alpha$ if there is a
sequence $\dot{s}=\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ of conditions
guided by the collapse at $\alpha$ so that $\dot{H}=H(\dot{s})$.
Suppose that $M$ is a suitable model. Let $\alpha:=M\cap\kappa$, and let
$\pi_{M}:M\to\bar{M}$ be the transitive collapse map of $M$. Let
$G\subseteq\mathbb{P}$ be generic over $V$, and set
$G_{\alpha}=G\cap(\mathbb{P}\upharpoonright\alpha)$. We have that
$\mathbb{P}\upharpoonright\alpha=\pi_{M}(\mathbb{P})\in\bar{M}$, and
$G_{\alpha}\subset\pi_{M}(\mathbb{P})$ is generic for $\bar{M}$. Moreover,
setting $M[G]=\\{\dot{x}[G]\mid\dot{x}\in M\text{ is a
}{\mathbb{P}}\text{-name}\\}$, we have that $\bar{M}[G_{\alpha}]$ is the
transitive collapse of $M[G]$, with the transitive collapse map $\pi_{M[G]}$
being the natural extension of $\pi_{M}$, given by
$\pi_{M[G]}(\dot{x}[G])=\pi_{M}(\dot{x})[G_{\alpha}]$.
###### Lemma 5.4.
Suppose that $M$ is a $(\mathbb{P}\ast\dot{\mathbb{C}})$-suitable model, where
$\dot{\mathbb{C}}$ is a $\mathbb{P}$-name for a poset on $\kappa$ which is
$\omega_{1}$-closed with sups. Let $\alpha:=M\cap\kappa$ and $\pi_{M}$ be the
transitive collapse map of $M$. Suppose that $\dot{H}$ is a $\mathbb{P}$-name
for a subset of $\pi_{M}(\dot{\mathbb{C}})$ which is guided by the collapse at
$\alpha$ for $\pi_{M}(\dot{\mathbb{C}})$, and further suppose that there is a
$\mathbb{P}$-name $\dot{c}$ for a condition in $\dot{\mathbb{C}}$ which is
forced to be an upper bound for $\pi^{-1}_{M[\dot{G}]}[\dot{H}]$. Then
$\mathbb{P}$ forces that $\dot{c}$ is an
$(M[\dot{G}],\dot{\mathbb{C}})$-completely-generic condition.
###### Proof.
Fix $G$. To see that $c={\dot{c}[G]}$ is $(M[G],\mathbb{C})$-completely
generic, fix a dense, open $E\subseteq\mathbb{C}$ with $E\in M[G]$. We show
that $c$ extends some condition in $E$.
By the elementarity of $\pi_{M[G]}:M[G]\to\bar{M}[G_{\alpha}]$, we know that
$\pi_{M[G]}(E)$ is dense in
$\pi_{M[G]}(\mathbb{C})=\pi_{M}(\dot{\mathbb{C}})[G_{\alpha}]$. Since
$H:=\dot{H}[G]$ is a $V[{G_{\alpha}}]$-generic filter, by Lemma 5.2,
$H\cap\pi_{M[G]}(E)\neq\emptyset$, and thus $\pi^{-1}_{M[G]}[H]\cap
E\neq\emptyset$. ∎
There are two particularly useful properties of this class of names for
generic filters. On the one hand, filters in this class will allow us to
generate strong, exact residue functions by isolating the information which a
given conditions determines about the filters. On the other hand, the class of
such filters for one poset, such as a two-step iteration, often projects to
the class of such filters for another poset, such as the first step in a two-
step iteration. This property will be particularly useful in Section 6 when we
want to show, by induction, that our club adding poset is well-behaved.
It is straightforward to verify that the notion of $\mathbb{P}$-names of
filters, which are guided by the collapse at a given cardinal, factor well in
iterations.
###### Lemma 5.5.
Suppose that $\alpha<\kappa$ is inaccessible and that
$\dot{\mathbb{Q}_{0}}\ast\dot{\mathbb{Q}}_{1}$ is a
$(\mathbb{P}\upharpoonright\alpha)$-name for a two-step poset of size $\alpha$
which is $\omega_{1}$-closed with sups. Let
$\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ be a sequence of
$\mathbb{P}$-names which is guided by the collapse at $\alpha$ for
$\dot{\mathbb{Q}}_{0}\ast\dot{\mathbb{Q}}_{1}$. Then the sequence
$\langle\dot{d}_{\nu}(0):\nu<\omega_{1}\rangle$ of $\mathbb{P}$-names of
conditions in $\dot{\mathbb{Q}}_{0}$ is guided by the collapse at $\alpha$ for
$\dot{\mathbb{Q}}_{0}$.
The following proposition shows how to generate exact, strong residue
functions from the filters discussed above.
###### Proposition 5.6.
Suppose that $\dot{\mathbb{C}}$ is a $\mathbb{P}$-name in $H(\kappa^{+})$ for
a poset of size $\kappa$ which is $\omega_{1}$-closed with sups, and set
$\mathbb{P}^{*}:=\mathbb{P}\ast\dot{\mathbb{C}}$. Let $M$ be a
$(\mathbb{P}\ast\dot{\mathbb{C}})$-suitable model, say with
$\alpha:=M\cap\kappa<\kappa$, and let $\pi_{M}$ denote the transitive collapse
of $M$. Also let $\dot{s}=\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ be a
sequence of $\mathbb{P}$-names guided by the collapse at $\alpha$ for
$\pi_{M}(\dot{\mathbb{C}})$. Additionally, suppose that there is a
$\mathbb{P}$-name $\dot{d}^{*}$ for a condition in $\dot{\mathbb{C}}$ which is
forced to be an upper bound for the sequence $\pi_{M[\dot{G}]}^{-1}[\dot{s}]$.
Define $p^{*}(M)$ to be the condition
$p^{*}(M):=(0_{\mathbb{P}},\dot{d}^{*}),$
and let
$D(M):=\left\\{(p,\dot{d})\geq p^{*}(M):p\emph{ determines an initial segment
of }\dot{s}\right\\}.$
Finally, define $\varphi^{M}$ on $D(M)$ by
$\varphi^{M}(p,\dot{d})=(p\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}_{\beta})),$
where $\beta<\alpha$ is the least so that
$p\Vdash\dot{d}_{\operatorname{dom}(p(\alpha))}=^{*}\dot{\gamma}_{\beta}$
($\beta$ exists by definition of “$p$ determines an initial segment of
$\dot{s}$”). Then
1. (a)
$D(M)$ is a dense, countably $=^{*}$-closed subset of
$\mathbb{P}^{*}/p^{*}(M)$;
2. (b)
$p^{*}(M)$ is compatible with every condition in $\mathbb{P}^{*}\cap M$; and
3. (c)
$\varphi^{M}$ is an exact, strong residue function from $D(M)$ to
$M\cap\mathbb{P}^{*}$.
###### Proof.
Let $\langle\dot{\gamma}_{i}:i<\alpha\rangle$ be a sequence of
$(\mathbb{P}\upharpoonright\alpha)$-names which is forced to enumerate all
conditions in $\pi_{M}(\dot{\mathbb{C}})$, and with $\dot{d}^{*}$, satisfies
Definition 5.1, witnessing that
$\dot{s}=\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ is guided by the collapse
at $\alpha$ for $\pi_{M}(\dot{\mathbb{C}})$.
We first prove item (a). Given a condition $(p,\dot{d})$ in
$\mathbb{P}^{*}/p^{*}(M)$, by item (2) of Definition 5.1, we may find an
extension $p^{\prime}$ of $p$ so that $p$ determines an initial segment of
$\dot{s}$. Then $(p^{\prime},\dot{d})\geq(p,\dot{d})$ is in $D(M)$, proving
density. Similarly, $D(M)$ is $=^{*}$-closed: if $(p_{1},\dot{d}_{1})\in D(M)$
and $(p_{1},\dot{d}_{1})=^{*}(p_{2},\dot{d}_{2})$, then $p_{2}$ determines an
initial segment of $\dot{s}$, because $p_{2}=p_{1}$ (recall these are collapse
conditions) and because
$(p_{2},\dot{d}_{2})\geq(p_{1},\dot{d}_{1})\geq(0_{\mathbb{P}},\dot{d}^{*})$.
To see that $D(M)$ is closed under sups of increasing $\omega$-sequences,
suppose that $\langle(p_{n},\dot{c}_{n}):n\in\omega\rangle$ is an increasing
sequence of conditions in $D(M)$, and let $(p^{*},\dot{c}^{*})$ be a sup. Set
$\nu_{n}:=\operatorname{dom}(p_{n}(\alpha))$ and
$\nu^{*}:=\operatorname{dom}(p^{*}(\alpha))$. If $\nu^{*}=\nu_{m}$ for some
$m\in\omega$, then because $p_{m}$ determines $\dot{s}$ up to $\nu_{m}$, we
have that $p^{*}$ determines $\dot{s}$ up to $\nu^{*}=\nu_{m}$. Thus $p^{*}$
determines an initial segment of $\dot{s}$ in this case. So consider the case
that $\nu^{*}>\nu_{m}$ for all $m$; in particular, $\nu^{*}$ is a limit
ordinal. Since for all $n\in\omega$, $p_{n}$ determines an initial segment of
$\dot{s}$ and $p^{*}\geq p_{n}$, we may find a sequence
$\langle\beta(\mu):\mu<\nu^{*}\rangle$ in $V$ so that
$p^{*}\Vdash\dot{d}_{\mu}=^{*}\dot{\gamma}_{\beta(\mu)}$ for all
$\mu<\nu^{*}$. Now let $\beta(\nu^{*})$ be chosen so that
$\dot{\gamma}_{\beta(\nu^{*})}$ is forced to be a sup of
$\langle\dot{\gamma}_{\beta(\mu)}:\mu<\nu\rangle$, if this sequence is
increasing, and equals the trivial condition otherwise. Since $\nu^{*}$ is a
limit, $\Vdash_{\mathbb{P}}\dot{d}_{\nu^{*}}$ is a sup of
$\langle\dot{d}_{\mu}:\mu<\nu^{*}\rangle$. But $p^{*}$ forces that
$\dot{d}_{\mu}=^{*}\dot{\gamma}_{\beta(\mu)}$ for all $\mu<\nu^{*}$, and
therefore $p^{*}$ forces that
$\dot{d}_{\nu^{*}}=^{*}\dot{\gamma}_{\beta(\nu^{*})}$. Thus, in either case,
$p^{*}$ determines an initial segment of $\dot{s}$, which finishes the proof
of (a).
Now we verify item (b). Fix a condition $(u,\dot{c}_{0})$ in
$\mathbb{P}^{*}\cap M$, and we will show that it is compatible with
$p^{*}(M)$. We observe that, trivially, $u$ determines an initial segment of
$\dot{s}$ since $\operatorname{dom}(u(\alpha))=0$ and
$\Vdash_{\mathbb{P}}\dot{d}_{0}$ is the trivial condition in
$\pi_{M}(\dot{\mathbb{C}})$, by (1) of Definition 5.1. By (3) of the same
definition, we may find an extension $p\geq u$ s.t.
$p\Vdash\dot{d}_{1}\geq\pi_{M}(\dot{c}_{0})$. Then $(p,\dot{d}^{*})$ extends
$(u,\dot{c}_{0})$ since $p$ forces that $\dot{d}^{*}$ is an upper bound for
the sequence $\pi_{M}^{-1}[\dot{s}]$ and that
$\pi^{-1}_{M}(\dot{d}_{1})\geq\dot{c}_{0}$.
It therefore remains to verify that $\varphi^{M}$ is an exact, strong residue
function. Condition (1) of Definition 2.9 holds since, by (a), $D(M)$ is dense
and countably $=^{*}$-closed in $\mathbb{P}^{*}/p^{*}(M)$. For the projection
condition of Definition 2.9, fix $(p,\dot{c})\in D(M)$, and let $\dot{\gamma}$
be the $(\mathbb{P}\upharpoonright\alpha)$-name so that
$\varphi^{M}(p,\dot{c})=(p\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}))$.
Since $(p,\dot{c})\in D(M)$, $p$ determines an initial segment of $\dot{s}$,
and therefore
$p\Vdash\dot{d}_{\operatorname{dom}(p(\alpha))}=^{*}\dot{\gamma}$. Since $p$
also forces that $\dot{c}\geq\dot{d}^{*}$, $p$ forces that $\dot{c}$ is an
upper bound for $\pi_{M}^{-1}[\dot{s}]$ and therefore that $\dot{c}$ extends
$\pi_{M}^{-1}(\dot{d}_{\operatorname{dom}(p(\alpha))})=^{*}\pi_{M}^{-1}(\dot{\gamma})$.
Therefore
$(p,\dot{c})\geq(p\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}))=\varphi^{M}(p,\dot{c})$.
It is straightforward to verify that $\varphi^{M}$ is order preserving, i.e.,
condition (3) of Definition 2.9. So we prove that $\varphi^{M}$ has the strong
residue property (condition (4) of Definition 2.9). Thus fix $(p,\dot{c})\in
D(M)$, where we let $\nu:=\operatorname{dom}(p(\alpha))$ and $\dot{\gamma}$ so
that
$\varphi^{M}(p,\dot{c})=(p\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}))$.
Fix a condition $(u,\dot{\delta})$ in $\mathbb{P}^{*}\cap M$ with
$(u,\dot{\delta})\geq(p\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}))$, and
we will verify that $(u,\dot{\delta})$ is compatible with $(p,\dot{c})$. Let
$p^{\prime}:=u\cup p$, a condition in $\mathbb{P}$, and observe that
$p^{\prime}$ still determines an initial segment of $\dot{s}$ and
$\nu=\operatorname{dom}(p^{\prime}(\alpha))$. By item (3) of Definition 5.1,
we may find some $p^{*}\geq p^{\prime}$ so that
$p^{*}\Vdash\dot{d}_{\nu+1}\geq\pi_{M}(\dot{\delta})$. Then $(p^{*},\dot{c})$
extends both $(p,\dot{c})$ and $(u,\dot{\delta})$.
We now check that $\varphi^{M}$ is $\omega$-continuous, which will finish the
proof of (c) and thereby the proof of the proposition. Fix an increasing
sequence of conditions $\langle(p_{n},\dot{c}_{n}):n\in\omega\rangle$ in
$D(M)$, and let $(p^{*},\dot{c}^{*})$ be a supremum of this sequence. Then
$p^{*}:=\bigcup_{n}p_{n}$, and $\dot{c}^{*}$ is forced by $p^{*}$ to be a sup
of $\langle\dot{c}_{n}:n\in\omega\rangle$. By item (a) of the proposition,
$(p^{*},\dot{c}^{*})\in D(M)$. We need to show that
$\varphi^{M}(p^{*},\dot{c}^{*})$ is a sup of
$\langle\varphi^{M}(p_{n},\dot{c}_{n}):n\in\omega\rangle$. For each
$n<\omega$, set $\nu_{n}:=\operatorname{dom}(p_{n}(\alpha))$, and also set
$\nu^{*}:=\operatorname{dom}(p^{*}(\alpha))$. Additionally, for each $n$, fix
the least ordinal $\beta(\nu_{n})$ with
$p_{n}\Vdash\dot{d}_{\nu_{n}}=^{*}\dot{\gamma}_{\beta(\nu_{n})}$, so that
$\varphi^{M}(p_{n},\dot{c}_{n})=(p_{n}\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}_{\beta(\nu_{n})}))$.
Finally let $\beta(\nu^{*})$ be least so that
$\varphi^{M}(p^{*},\dot{c}^{*})=(p^{*}\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}_{\beta(\nu^{*})}))$.
We claim that $p^{*}\Vdash\dot{\gamma}_{\beta(\nu^{*})}$ is a sup of
$\langle\dot{\gamma}_{\beta(\nu_{n})}:n\in\omega\rangle$. Note that proving
this claim suffices: indeed, then $p^{*}\upharpoonright\alpha$ forces that
$\dot{\gamma}_{\beta(\nu^{*})}$ is a sup of
$\langle\dot{\gamma}_{\beta(\nu_{n})}:n\in\omega\rangle$, and as a result
$\varphi^{M}(p^{*},\dot{c}^{*})=(p^{*}\upharpoonright\alpha,\pi_{M}^{-1}(\dot{\gamma}_{\beta(\nu^{*})}))$
is a sup of $\langle\varphi^{M}(p_{n},\dot{c}_{n}):n\in\omega\rangle$.
To prove the claim, we have two cases on $\nu^{*}$. Since
$\nu^{*}=\sup_{m}\nu_{m}$, either $\nu^{*}>\nu_{m}$ for all $m$, or
$\nu^{*}=\nu_{m}$ for almost all $m$. In the first case, $\nu^{*}$ is a limit,
and so $\mathbb{P}$ forces that $\dot{d}_{\nu^{*}}$ is a sup of
$\langle\dot{d}_{\nu}:\nu<\nu^{*}\rangle$. Since
$p_{n}\Vdash\dot{d}_{\nu_{n}}=^{*}\dot{\gamma}_{\beta(\nu_{n})}$ for each $n$,
$p^{*}$ forces that $\langle\dot{\gamma}_{\beta(\nu_{n})}:n\in\omega\rangle$
is cofinal in $\langle\dot{d}_{\nu}:\nu<\nu^{*}\rangle$. Therefore $p^{*}$
forces that these two sequences have the same sups. Consequently, $p^{*}$
forces that $\dot{\gamma}_{\beta(\nu^{*})}=^{*}\dot{d}_{\nu^{*}}$ is a sup of
$\langle\dot{\gamma}_{\beta(\nu_{n})}:n\in\omega\rangle$. For the second case,
$\nu^{*}=\nu_{m}$ for all $m$ above some $k$. Then $\mathbb{P}$ forces that
$\langle\dot{d}_{\nu_{n}}:n\in\omega\rangle$ is eventually equal to
$\dot{d}_{\nu^{*}}$. Because
$p_{n}\Vdash\dot{d}_{\nu_{n}}=^{*}\dot{\gamma}_{\beta(\nu_{n})}$ for all $n$
and $p^{*}\geq p_{n}$, we have that
$p^{*}\Vdash\langle\dot{\gamma}_{\beta(\nu_{n})}:n\in\omega\rangle$ is
eventually equal to $\dot{d}_{\nu^{*}}$. Finally,
$p^{*}\Vdash\dot{\gamma}_{\beta(\nu^{*})}=^{*}\dot{d}_{\nu^{*}}$, and
therefore $p^{*}$ forces that
$\langle\dot{\gamma}_{\beta(\nu_{n})}:n\in\omega\rangle$ is eventually
$=^{*}$-equal to $\dot{\gamma}_{\beta(\nu^{*})}$, and therefore that
$\dot{\gamma}_{\beta(\nu^{*})}$ is a sup. This finishes the proof of the claim
and thereby the proof that $\varphi^{M}$ is $\omega$-continuous. ∎
The final result in this subsection shows that we can create filters which are
guided by the collapse at $\alpha$ by using the generic surjection from
$\omega_{1}$ onto $\alpha$ to guide extensions in the second coordinate. This
combines ideas of collapse absorption with, as previously mentioned, Abraham’s
use of guiding reals.
###### Lemma 5.7.
Suppose that $\dot{\mathbb{Q}}$ is a $(\mathbb{P}\upharpoonright\alpha)$-name
for a poset of size $\alpha$, which is $\omega_{1}$-closed with sups, and let
$\langle\dot{\gamma}_{i}:i<\alpha\rangle$ be forced to enumerate all
conditions in $\dot{\mathbb{Q}}$. Let $\dot{f}_{\alpha}$ be the
$\mathbb{P}$-name for the standard surjection added from $\omega_{1}$ onto
$\alpha$.
Suppose that $\dot{s}=\langle\dot{d}_{\nu}:\nu<\omega_{1}\rangle$ is a
sequence of $\mathbb{P}$-names for conditions in $\dot{\mathbb{Q}}$ forced by
$\mathbb{P}$ to satisfy the following properties:
1. (1)
$\dot{s}$ is $\leq_{\dot{\mathbb{Q}}}$-increasing, $\dot{d}_{0}$ names the
trivial condition, and if $\nu$ is a limit, then $\dot{d}_{\nu}$ is a
$\leq_{\dot{\mathbb{Q}}}$-sup of $\langle\dot{d}_{\mu}:\mu<\nu\rangle$;
2. (2)
if $\nu<\omega_{1}$ and $\dot{\gamma}_{\dot{f}_{\alpha}(\nu)}$ extends
$\dot{d}_{\nu}$ in $\dot{\mathbb{Q}}$, then $\dot{d}_{\nu+1}$ extends
$\dot{\gamma}_{\dot{f}_{\alpha}(\nu)}$;
3. (3)
for each $\nu<\omega_{1}$, the sequence
$\langle\dot{d}_{\mu}:\mu\leq\nu\rangle$ is definable in
$V[\dot{G}\upharpoonright\alpha]$ from $\dot{f}_{\alpha}\upharpoonright\nu$.
Then $\dot{s}$ is guided by the collapse at $\alpha$ for $\dot{\mathbb{Q}}$.
In particular, there exists a $\mathbb{P}$-name for a sequence which is guided
by the collapse at $\alpha$ for $\dot{\mathbb{Q}}$.
###### Proof.
We will verify that items (1)-(3) of Definition 5.1 hold. Item (1) of the
definition is immediate from assumption (1) of the lemma.
For item (2) of Definition 5.1, suppose that $p\in\mathbb{P}$ is a condition
where $\operatorname{dom}(p(\alpha))$ is an ordinal $\nu<\omega_{1}$. Let $G$
be $V$-generic over $\mathbb{P}$ containing $p$, and let
$\bar{G}:=G\cap(\mathbb{P}\upharpoonright\alpha)$. For each $\mu\leq\nu$, let
$d_{\mu}:=\dot{d}_{\mu}[G]$, and let $\beta(\mu)$ be an ordinal $<\alpha$ so
that $d_{\mu}=\dot{\gamma}_{\beta(\mu)}[\bar{G}]$. By assumption (3) of the
lemma, the sequence $\langle d_{\mu}:\mu\leq\nu\rangle$ is definable in
$V[\bar{G}]$ from $f_{\alpha}\upharpoonright\nu=p(\alpha)$. Therefore, there
exists a condition $\bar{p}\geq p\upharpoonright\alpha$ with
$\bar{p}\in\bar{G}$ so that $\bar{p}$ forces that
$\langle\dot{\gamma}_{\beta(\mu)}:\mu\leq\nu\rangle$ satisfies the definition
with respect to $p(\alpha)$. Then $p^{\prime}:=p\cup\bar{p}$ witnesses item
(2) of Definition 5.1, since $p^{\prime}$ forces that
$\langle\dot{\gamma}_{\beta(\mu)}:\mu\leq\nu\rangle$ and
$\langle\dot{d}_{\mu}:\mu\leq\nu\rangle$ both satisfy the same definition in
$V[\dot{\bar{G}}]$ with the parameter
$\dot{f}_{\alpha}\upharpoonright\nu=p(\alpha)$.
Turning to item (3) of Definition 5.1, let $p^{\prime}$ be a condition as in
the previous paragraph. Fix a $(\mathbb{P}\upharpoonright\alpha)$-name
$\dot{\gamma}$ for a $\dot{\mathbb{Q}}$-extension of
$\dot{\gamma}_{\beta(\nu)}$, and let $\delta<\alpha$ and $\bar{p}^{*}\geq
p^{\prime}\upharpoonright\alpha$ so that
$\bar{p}^{*}\Vdash_{\mathbb{P}\upharpoonright\alpha}\dot{\gamma}=\dot{\gamma}_{\delta}$.
Define $p^{*}$ to be the minimal extension of $\bar{p}^{*}\cup
p^{\prime}\upharpoonright[\alpha,\kappa)$ so that
$p^{*}\Vdash\dot{f}_{\alpha}(\nu)=\delta$. Then
$p^{*}\Vdash\dot{\gamma}_{\dot{f}_{\alpha}(\nu)}\geq\dot{\gamma}_{\beta(\nu)}=^{*}\dot{d}_{\nu}$,
so there exists an extension $p^{**}$ of $p^{*}$ so that $p^{**}$ forces
$\dot{d}_{\nu+1}\geq\dot{\gamma}_{\dot{f}_{\alpha}(\mu)}=\dot{\gamma}$.
For the “in particular” claim of the lemma, define a sequence $\dot{s}$ by
recursion so that it satisfies (1) and so that if $\nu<\omega_{1}$, then
$\dot{d}_{\nu+1}$ is forced to be equal to $\gamma_{\dot{f}_{\alpha}(\nu)}$ if
this extends $\dot{d}_{\nu}$, and otherwise equals $\dot{d}_{\nu}$. Then (2)
and (3) are also satisfied, so $\dot{s}$ is guided by the collapse at $\alpha$
for $\dot{\mathbb{Q}}$. ∎
### 5.2. ${\cal{F}}$-Complete Properness
We are now ready to isolate a sufficient condition on names $\dot{\mathbb{C}}$
so that $\mathbb{P}\ast\dot{\mathbb{C}}$ is ${\cal{F}}$-strongly proper.
###### Definition 5.8.
Let $\dot{\mathbb{C}}$ be a $\mathbb{P}$-name in $H(\kappa^{+})$ for a poset
forced to be $\omega_{1}$-closed with sups. We say that $\dot{\mathbb{C}}$ is
${\cal{F}}$-Completely Proper if for any
$(\mathbb{P}\ast\dot{\mathbb{C}})$-suitable sequence $\vec{M}$ there is some
$A\subseteq\operatorname{dom}(\vec{M})$ with $A\in\cal{I}$ so that for each
$\alpha\in\operatorname{dom}(\vec{M})\backslash A$ and each $\mathbb{P}$-name
$\dot{H}$ for filter over $\pi_{M_{\alpha}}(\dot{\mathbb{C}})$ which is guided
by the collapse at $\alpha$, there exists a $\mathbb{P}$-name
$\dot{c}_{\dot{H}}$ for a condition in $\dot{\mathbb{C}}$ which is forced to
be a least upper bound for
$\pi^{-1}_{M_{\alpha}[\dot{G}_{\mathbb{P}}]}[\dot{H}]$.
We recall that a poset $\mathbb{U}$ is _$\lambda$ -distributive_ if forcing
with $\mathbb{U}$ adds no sequences of ordinals of length less than $\lambda$.
A sufficient condition to guarantee this is that the intersection of fewer
than $\lambda$-many dense, open subsets of $\mathbb{U}$ is dense, open. They
are equivalent if $\mathbb{U}$ is separative.
###### Lemma 5.9.
Suppose that $\dot{\mathbb{C}}$ is ${\cal{F}}$-completely proper. Then
$\mathbb{P}$ forces that the intersection of fewer than $\kappa$-many dense,
open subsets of $\dot{\mathbb{C}}$ is dense, open. Hence $\dot{\mathbb{C}}$ is
forced to be $\kappa$-distributive.
###### Proof.
Let $\vec{D}=\langle\dot{D}_{i}:i<\omega_{1}\rangle$ be a sequence of
$\mathbb{P}$-names for dense, open subsets of $\dot{\mathbb{C}}$. We show that
the intersection is forced to be non-empty. Fix a sequence $\vec{M}$ which is
suitable with respect to $\mathbb{P}\ast\dot{\mathbb{C}}$ and $\vec{D}$ so
that $\operatorname{dom}(\vec{M})$ satisfies the conclusion of Definition 5.8.
Let $\alpha\in\operatorname{dom}(\vec{M})$. By Lemma 5.7, there exists a
$\mathbb{P}$-name $\dot{H}$ for a filter which is guided by the collapse at
$\alpha$ for $\pi_{M_{\alpha}}(\dot{\mathbb{C}})$. Since
$\operatorname{dom}(\vec{M})$ satisfies Definition 5.8, we may find a
$\mathbb{P}$-name $\dot{d}$ for a condition which is forced to be an upper
bound in $\dot{\mathbb{C}}$ for
$\pi^{-1}_{M_{\alpha}[\dot{G}_{\mathbb{P}}]}[\dot{H}]$. Then, by Lemma 5.4,
$\dot{d}$ is forced to be an
$(M_{\alpha}[\dot{G}],\dot{\mathbb{C}})$-_completely_ generic condition. But
$\dot{D}_{i}\in M_{\alpha}$ for each $i<\omega_{1}$ and is dense, open.
Therefore it is forced that $\dot{d}\in\bigcap_{i\in\omega_{1}}\dot{D}_{i}$. ∎
By combining Lemma 5.7, Proposition 5.6, and Lemma 2.20, we conclude with the
following key result:
###### Proposition 5.10.
Suppose that $\dot{\mathbb{C}}$ is ${\cal{F}}$-completely proper. Then
$\mathbb{P}\ast\dot{\mathbb{C}}$ is ${\cal{F}}$-strongly proper.
## 6\. Properties of the Club-Adding Poset
In this section, we have two main tasks. In the first subsection, we will
prove that our intended club adding iteration, as well as useful variants
thereof, are ${\cal{F}}$-completely proper, and in the second subsection, we
will prove that our intended club adding iteration does not add branches
through various Aronszajn trees. Each of these results will be used as part of
a larger inductive argument in the final section in which we prove Theorem
1.1. Again we comment that, in this case, $\kappa$ is ineffable, but we are
only using the ineffability of $\kappa$ when we apply Proposition 4.4, since
this requires Proposition 3.22.
### 6.1. Adding Clubs is ${\cal{F}}$-Completely Proper
In order to anticipate arguments in the next subsection, where we show that
appropriate ${\cal{F}}$-completely proper posets do not add branches through
certain Aronszajn trees, we will need to not only show that our club adding
poset is ${\cal{F}}$-completely proper, but also show that variants of it have
this property. These variants are created by iterating the process of taking
an initial segment of the iteration followed by products of finitely-many
copies of the tail.
The following iteration follows Magidor’s work [34] on adding clubs through
reflection points of stationary subsets of a weakly compact cardinal $\kappa$,
which has been collapsed to become $\omega_{2}$.
Let $\rho<\kappa^{+}$, and suppose that we have defined a $\mathbb{P}$-name
for an iteration
$\langle\dot{\mathbb{C}}_{\sigma},\dot{\mathbb{C}}(\eta):\sigma\leq\rho,\eta<\rho\rangle$
and a $(\mathbb{P}\ast\dot{\mathbb{C}}_{\rho})$-name
$\langle\dot{\mathbb{S}}_{\sigma},\dot{\mathbb{S}}(\eta):\sigma\leq\rho,\eta<\rho\rangle$
for an iteration so that for all $\sigma<\rho$ the following assumptions are
satisfied:
1. (1)
$\mathbb{P}$ forces that the iteration
$\langle\dot{\mathbb{C}}_{\sigma},\dot{\mathbb{C}}(\eta):\sigma\leq\rho,\eta<\rho\rangle$
has $<\kappa$-support, and $\mathbb{P}\ast\dot{\mathbb{C}}_{\rho}$ forces that
$\langle\dot{\mathbb{S}}_{\sigma},\dot{\mathbb{S}}(\eta):\sigma\leq\rho,\eta<\rho\rangle$
has countable support. Furthermore, $\dot{\mathbb{S}}_{\sigma}$ is a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma})$-name;
2. (2)
$\dot{\mathbb{C}}(\sigma)$ is a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma})$-name for
$\mathsf{CU}(\dot{S}_{\sigma},\dot{\mathbb{S}}_{\sigma})$ (see Definition
1.5), where $\dot{S}_{\sigma}$ is a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma}\ast\dot{\mathbb{S}}_{\sigma})$-name
for a stationary subset of $\kappa\cap\operatorname{cof}(\omega)$ and
$\dot{\mathbb{C}}_{\sigma+1}=\dot{\mathbb{C}}_{\sigma}\ast\dot{\mathbb{C}}(\sigma)$;
3. (3)
$\dot{\mathbb{S}}(\sigma)$ is a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma+1}\ast\dot{\mathbb{S}}_{\sigma})$-name
for $\mathbb{S}(\dot{T}_{\sigma})$ (see Definition 1.6), where
$\dot{T}_{\sigma}$ is a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma+1}\ast\dot{\mathbb{S}}_{\sigma})$-name
for an Aronszajn tree on $\kappa$;
4. (4)
$\dot{\mathbb{C}}_{\sigma}$ is ${\cal{F}}$-completely proper (and hence
$\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma}$ is $\cal{F}$-strongly proper, by
Proposition 5.10), and $\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma}$ forces that
$\dot{\mathbb{S}}_{\sigma}$ is a countable support iteration specializing
Aronszajn trees, as defined in Section 3.
Working in an arbitrary generic extension by $\mathbb{P}$, we now define the
variations of $\dot{\mathbb{C}}_{\rho}$ mentioned above; we call these
_Doubling Tail Products_. These will be the posets
$\mathbb{C}_{\rho}(\vec{\delta})$, where
$\vec{\delta}=\langle\delta_{0},\delta_{1},\dots,\delta_{n-1}\rangle\in[\rho]^{n}$
is a strictly decreasing sequence of ordinals. We use
$[\rho]^{<\omega}_{\text{dec}}$ to denote the set of all finite, strictly
decreasing tuples from $\rho$; $[\rho]^{n}_{\text{dec}}$ is defined similarly.
We first introduce an auxiliary name for a poset
$\dot{\mathbb{C}}_{\delta^{*},\rho}(\vec{\delta})$, where $\vec{\delta}$ is as
above and $\delta^{*}\leq\delta_{n-1}$ is an additional ordinal. This is done
by recursion on $n=|\vec{\delta}|$ as follows:
* •
For $n=0$ (i.e., $\vec{\delta}=\emptyset$) we define
$\dot{\mathbb{C}}_{\delta^{*},\rho}(\emptyset)=\dot{\mathbb{C}}_{\delta^{*},\rho}$
to be the tail-segment of the iteration $\mathbb{C}_{\rho}$, starting from
stage $\delta^{*}$.
* •
For $n\geq 1$, $\vec{\delta}\in[\rho]^{n}$, and $\delta^{*}\leq\delta_{n-1}$,
the poset $\mathbb{C}_{\delta^{*},\rho}(\vec{\delta})$ is given by-
$\dot{\mathbb{C}}_{\delta^{*},\rho}(\vec{\delta})=\dot{\mathbb{C}}_{\delta^{*},\delta_{n-1}}*(\dot{\mathbb{C}}_{\delta_{n-1},\rho}(\vec{\delta}\upharpoonright
n-1))^{2},$
where $\dot{\mathbb{C}}_{\delta^{*},\delta_{n-1}}$ is the segment of the
iteration $\mathbb{C}_{\rho}$ starting from (and including) stage $\delta^{*}$
to stage $\delta_{n-1}$, and
$\vec{\delta}\upharpoonright(n-1)=\langle\delta_{0},\dots,\delta_{n-2}\rangle$.
We can now define $\mathbb{C}_{\rho}(\vec{\delta})$.
###### Definition 6.1.
For $\vec{\delta}\in[\rho]^{<\omega}_{\text{dec}}$, define
$\mathbb{C}_{\rho}(\vec{\delta})=\mathbb{C}_{0,\rho}(\vec{\delta})$ (i.e., as
$\mathbb{C}_{\delta^{*},\rho}(\vec{\delta})$ with $\delta^{*}=0$).
For example, if $\vec{\delta}=\langle\delta_{0}\rangle$ is a singleton, then
$\mathbb{C}_{\rho}(\langle\delta_{0}\rangle)=\mathbb{C}_{\delta_{0}}*\dot{\mathbb{C}}_{\delta_{0},\rho}^{2}$.
Similarly, if $\vec{\delta}=\langle\delta_{0},\delta_{1}\rangle$ has two
elements $\delta_{0}>\delta_{1}$ then
$\mathbb{C}_{\rho}(\langle\delta_{0},\delta_{1}\rangle)=\mathbb{C}_{\delta_{1}}*\dot{\mathbb{C}}_{\delta_{1},\rho}^{2}(\langle\delta_{0}\rangle)=\mathbb{C}_{\delta_{1}}*\left(\dot{\mathbb{C}}_{\delta_{1},\delta_{0}}*\dot{\mathbb{C}}^{2}_{\delta_{0},\rho}\right)^{2}$
We refer to posets $\mathbb{C}_{\rho}(\vec{\delta})$ as the doubling tail
products of $\mathbb{C}_{\rho}$. We now return to working in $V$, in
particular with the statement of the next item.
###### Proposition 6.2.
(Given assumptions (1)-(4) stated at the beginning of the subsection) For each
$\rho<\kappa^{+}$ and
$\vec{\delta}=\langle\delta_{0},\dots,\delta_{n-1}\rangle\in[\rho]^{<\omega}_{\text{dec}}$,
the doubling tail product $\dot{\mathbb{C}}_{\rho}(\vec{\delta})$ is
${\cal{F}}$-completely proper. In particular, $\dot{\mathbb{C}}_{\rho}$ is
${\cal{F}}$-completely proper.
We will explain the necessity of proving Proposition 6.2 for the doubling tail
products of $\dot{\mathbb{C}}_{\rho}$ in the final section of the paper.
###### Proof.
We will first work with $\dot{\mathbb{C}}_{\rho}$, rather than with the
doubling tail products, in order to establish that a certain statement $(*)$
(see below) holds. We will then show that this statement $(*)$ can be used to
prove the desired result for the doubling tail products. We use
$\mathbb{R}_{\sigma}$, for $\sigma\leq\rho$, to denote
$\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma}*\dot{\mathbb{S}}_{\sigma}$.
To begin, we fix an $\mathbb{R}_{\rho}$-suitable sequence $\vec{M}$; by
removing a set in $\cal{I}$, we may assume that $\vec{M}$ is in pre-splitting
configuration up to $\rho$. By removing a further $\cal{I}$-null set, we may
assume that $\vec{M}$ satisfies the conclusion of Proposition 4.4.
Let $M$ be a $\kappa$-model so that $M$ contains the relevant parameters,
including $\vec{M}$. Since $\operatorname{dom}(\vec{M})$ is in $\cal{F}^{+}$,
we may apply Proposition 1.10 to find an $M$-normal ultrafilter $U$ so that,
letting $j:M\longrightarrow N$ be the corresponding ultrapower map, $\kappa\in
j(\operatorname{dom}(\vec{M}))$. Let $M_{\kappa}$ be the $\kappa$-th model on
the sequence $j(\vec{M})$.
Fix a $V$-generic filter $G^{*}$ over $j(\mathbb{P})$, and let
$G:=G^{*}\cap\mathbb{P}$. For notational simplicity, we continue using $j$ to
denote the lifted map $j:M[G]\longrightarrow N[G^{*}]$. Recall by Lemma 2.19
that $j^{-1}\upharpoonright M_{\kappa}$ is the transitive collapse map of
$M_{\kappa}$ and that $j^{-1}$ lifts in the standard way to
$M_{\kappa}[G^{*}]$.
Suppose that $\sigma\leq\rho$ and that $\dot{H}$ is a $j(\mathbb{P})$-name in
$N$ for a generic filter over
$\pi_{M_{\kappa}[\dot{G}^{*}]}(j(\dot{\mathbb{C}}_{\sigma}))=\dot{\mathbb{C}}_{\sigma}$.
We define the $\kappa$-flat function for (the pull-back of) $\dot{H}$ to be
the $j(\mathbb{P})$-name for the function with domain $j[\sigma]$,666We note
that $j[\sigma]\in N$: $M_{\kappa}\cap j(\rho)=j[\rho]$, and so $j[\rho]$ is
in $N$. Then intersect with $j(\sigma)$. so that for each $\eta<\sigma$,
$\dot{r}(j(\eta))$ is forced to be equal to
$\left(\bigcup\dot{H}(\eta)\right)\cup\left\\{\kappa\right\\}$.
We will prove the following proposition $(*)$ by induction:
1. $(*)$
for any $\sigma\leq\rho$, if $\dot{H}$ is a $j(\mathbb{P})$-name in $N$ for a
generic filter over
$\dot{\mathbb{C}}_{\sigma}=\pi_{M_{\kappa}[G^{*}]}(j(\dot{\mathbb{C}}_{\sigma}))$
which is guided by the collapse at $\kappa$ (see Definition 5.3), then it is
forced by $j(\mathbb{P})$ that the $\kappa$-flat function for $\dot{H}$ is a
condition in $j(\dot{\mathbb{C}}_{\sigma})$.
We first consider the case that $\sigma\leq\rho$ is limit. Suppose that we
know the result for all $\eta<\sigma$. We use throughout the fact that
$j^{-1}$ equals the transitive collapse map of $M_{\kappa}[G^{*}]$.
Let $H\in N[G^{*}]$ be a filter over $\mathbb{C}_{\sigma}$ which is guided by
the collapse at $\kappa$, and let $r$ be the $\kappa$-flat function for $H$.
Since $|\operatorname{dom}(r)|^{N}<j(\kappa)$ and $j(\mathbb{C}_{\sigma})$ is
taken with $<j(\kappa)$-supports, in order to see that $r\in
j(\mathbb{C}_{\sigma})$, it suffices to show that for all $\eta<\sigma$,
$r\upharpoonright j(\eta)\in j(\mathbb{C}_{\eta}).$ So let $\eta<\sigma$ be
fixed. Since
$\mathbb{C}_{\sigma}\cong\mathbb{C}_{\eta}\ast\dot{\mathbb{C}}_{{\eta},\sigma}$
and since $H$ is guided by the collapse at $\kappa$ over
$\mathbb{C}_{\sigma}$, we have by Lemma 5.5 that
$H\upharpoonright\mathbb{C}_{\eta}$ is also guided by the collapse at $\kappa$
over $\mathbb{C}_{\eta}$. By induction, this implies that the $\kappa$-flat
function for $H\upharpoonright\mathbb{C}_{\eta}$, namely $r\upharpoonright
j(\eta)$, is a condition in $j(\mathbb{C}_{\eta})$. This completes the proof
of $(*)$ in the limit case.
Now suppose that $\sigma+1\leq\rho$ and that we know that $(*)$ holds at
$\sigma$. Let $H\in N[G^{*}]$ be a filter over $\mathbb{C}_{\sigma+1}$ which
is guided by the collapse at $\kappa$, and let $H_{\sigma}$ denote the
restriction of $H$ to $\mathbb{C}_{\sigma}$. Again appealing to Lemma 5.5, we
know that $H_{\sigma}$ is guided by the collapse at $\kappa$.
Let $r$ be the $\kappa$-flat function for $H$, and let $\bar{r}$ denote
$r\upharpoonright j(\sigma)$, the $\kappa$-flat function for $H_{\sigma}$.
Since $H_{\sigma}$ is guided by the collapse at $\kappa$, we may apply the
induction hypothesis to conclude that $\bar{r}$ is a condition in
$j(\mathbb{C}_{\sigma})$. By Proposition 5.6, since $H_{\sigma}$ is guided by
the collapse at $\kappa$, we know that in $N$ we may find a residue pair
$\langle(0_{j(\mathbb{P})},\dot{\bar{r}}),\varphi^{M_{\kappa}}\rangle$ for the
pair $(M_{\kappa},j(\mathbb{P}^{*}))$, where $\dot{\bar{r}}$ is a
$j(\mathbb{P})$-name in $N$ for $\bar{r}$. We use $p^{*}(M_{\kappa})$ to
denote $(0_{j(\mathbb{P})},\dot{\bar{r}})$.
Since $\bar{r}$ is an upper bound for
$\pi_{M_{\kappa}[G^{*}]}^{-1}[H_{\sigma}]=j[H_{\sigma}]$, we conclude that
$\bar{r}$ forces in $j(\mathbb{C}_{\sigma})$ over $N[G^{*}]$ that $\bigcup
j[H(\sigma)]{=\bigcup H(\sigma)}$ is club in $\kappa$ (equality follows since
$j$ is the identity on bounded subsets of $\kappa$). Therefore, to see that
$r\in j(\mathbb{C}_{\sigma+1})$ (which finishes the proof of $(*)$ in the
successor case), it suffices to show that
$\bar{r}\Vdash^{N[G^{*}]}_{j(\mathbb{C}_{\sigma})}\left(\bigcup
H(\sigma)\cup\left\\{\kappa\right\\}\right)\in j(\dot{\mathbb{C}}(\sigma)).$
Since $j(\dot{\mathbb{C}}(\sigma))$ is a
$j(\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma})$-name for
$\mathsf{CU}(j(\dot{S}_{\sigma}),j(\dot{\mathbb{S}}_{\sigma}))$, the above
holds if and only if
$(\bar{r},0_{j(\dot{\mathbb{S}}_{\sigma})})\Vdash^{N[G^{*}]}_{j(\mathbb{C}_{\sigma}\ast\dot{\mathbb{S}}_{\sigma})}\left(\bigcup
H(\sigma)\cup\left\\{\kappa\right\\}\right)\subseteq\left(\operatorname{tr}\left(j(\dot{S}_{\sigma})\right)\cup\left(j(\kappa)\cap\operatorname{cof}(\omega)\right)\right).$
By the elementarity of $j$ and since $\bar{r}$ is an upper bound for
$j[H_{\sigma}]$, we see that
$(\bar{r},0_{j(\dot{\mathbb{S}}_{\sigma})})\Vdash^{N[G^{*}]}_{j(\mathbb{C}_{\sigma}\ast\dot{\mathbb{S}}_{\sigma})}\bigcup
H(\sigma)\subseteq\left(\operatorname{tr}\left(j(\dot{S}_{\sigma})\right)\cup\left(j(\kappa)\cap\operatorname{cof}(\omega)\right)\right).$
Since $\kappa$ has cofinality $\omega_{1}$ after forcing with
$j(\mathbb{C}_{\sigma}\ast\dot{\mathbb{S}}_{\sigma})$, it therefore suffices
to show that
$(\bar{r},0_{j(\dot{\mathbb{S}}_{\sigma})})\Vdash^{N[G^{*}]}_{j(\mathbb{C}_{\sigma}\ast\dot{\mathbb{S}}_{\sigma})}(j(\dot{S}_{\sigma})\cap\kappa)\text{
is stationary in }\kappa.$
Before continuing, we recall that $\mathbb{R}_{\sigma}$ denotes
$\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma}\ast\dot{\mathbb{S}}_{\sigma}$. By
Proposition 3.4, we know that
$(p^{*}(M_{\kappa}),0_{j(\dot{\mathbb{S}}_{\sigma})})\Vdash^{N}_{j(\mathbb{R}_{\sigma})}j(\dot{S}_{\sigma})\cap\kappa=(j(\dot{S}_{\sigma})\cap
M_{\kappa})[\dot{G}_{j(\mathbb{R}_{\sigma})}\cap M_{\kappa}].$
However, $j(\mathbb{R}_{\sigma})\cap M_{\kappa}=j[\mathbb{R}_{\sigma}]$ is
isomorphic to $\mathbb{R}_{\sigma}$, and $j(\dot{S}_{\sigma})\cap
M_{\kappa}=j[\dot{S}_{\sigma}]$. Since $\dot{S}_{\sigma}$ is a nice
$\mathbb{R}_{\sigma}$-name for a stationary subset of
$\kappa\cap\operatorname{cof}(\omega)$, $j[\dot{S}_{\sigma}]$ is therefore a
$j(\mathbb{R}_{\sigma})\cap M_{\kappa}$-name for a stationary subset of
$\kappa\cap\operatorname{cof}(\omega)$.
We now apply Proposition 4.4 in $N$, recalling that $\vec{M}$ satisfies the
conclusion of that proposition and that $\kappa\in
j(\operatorname{dom}(\vec{M}))$. Thus, applying the observations in the
previous paragraph, we conclude that
$(p^{*}(M_{\kappa}),0_{j(\dot{\mathbb{S}}_{\sigma})})$ forces over $N$ that
$j[\dot{S}_{\sigma}]$ is stationary in $\kappa$. Recalling that
$p^{*}(M_{\kappa})=(0_{j(\mathbb{P})},\dot{\bar{r}})$, we now conclude that
$(\bar{r},0_{j(\dot{\mathbb{S}}_{\sigma})})\Vdash^{N[G^{*}]}_{j(\mathbb{C}_{\sigma}\ast\dot{\mathbb{S}}_{\sigma})}j(\dot{S}_{\sigma})\cap\kappa=j[\dot{S}_{\sigma}][\dot{G}_{j(\mathbb{R}_{\sigma})}\cap
M_{\kappa}]\text{ is stationary}.$
This completes the proof that $r$, the $\kappa$-flat function for $H$, is a
condition in $j(\mathbb{C}_{\sigma})$ and also finishes the proof that $(*)$
holds.
To finish the proof of Proposition 6.2, we prove by induction on $k<\omega$
that for any
$\vec{\delta}=\langle\delta_{0},\dots,\delta_{k-1}\rangle\in[\rho]^{k}_{\text{dec}}$,
the poset $\mathbb{C}_{\rho}(\vec{\delta})$ is $\cal{F}$-completely proper.
Working by contradiction, let $k\in\omega$ be the least so that for some
(empty in case $k=0$) $\vec{\delta}\in[\rho]^{k}_{\text{dec}}$, the
proposition fails for $\mathbb{C}_{\rho}(\vec{\delta})$. Let $\vec{M}$ be a
$\vec{\delta}$-suitable sequence which witnesses that
$\mathbb{C}_{\rho}(\vec{\delta})$ is not $\cal{F}$-suitable. Since $\vec{M}$
is $\vec{\delta}$-suitable it is also
$\mathbb{P}\ast\dot{\mathbb{C}}_{\rho}(\vec{\delta})$-suitable. By removing a
$\cal{I}$-null set, we may assume that $\vec{M}$ satisfies the conclusion of
Proposition 4.4.
Let $M$ be a $\kappa$-model containing the relevant parameters, including
$\vec{M}$. Since $\operatorname{dom}(\vec{M})\in{\cal{F}^{+}}$, we may find
some $M$-normal ultrafilter $U$ so that, letting $j:M\longrightarrow N$ be the
ultrapower embedding, $\kappa\in j(\operatorname{dom}(\vec{M}))$.
First we deal with the case $k=0$. Since $\vec{M}$ witnesses that
$\dot{\mathbb{C}}_{\rho}$ is not $\cal{F}$-completely proper, $N$ satisfies
that the conclusion of Definition 5.8 fails at $\kappa$ with respect to
$j(\mathbb{P})$ and $j(\dot{\mathbb{C}}_{\rho})$. However, this directly
contradicts $(*)$, which we showed holds in this set-up.
Now we assume that $k=l+1$ is a successor. Then we may find an $N$-generic
filter $G^{*}$ over $j(\mathbb{P})$ so that in $N[G^{*}]$ there is a filter
$H^{*}$ over
$\pi_{{M_{\kappa}}[G^{*}]}(j(\mathbb{C}_{\rho}(\vec{\delta})))=\mathbb{C}_{\rho}(\vec{\delta})$
so that no condition in $j(\mathbb{C}_{\rho}(\vec{\delta}))$ is a least upper
bound for $\pi^{-1}_{M_{\kappa}[G^{*}]}[H^{*}]$. We will show, on the
contrary, that there is such a least upper bound for the pull-back of $H^{*}$.
Write $\vec{\delta}=\langle\delta_{0},\dots,\delta_{k-1}\rangle$. For
simplicity of notation, we also write
$\mathbb{C}_{\rho}(\vec{\delta}\upharpoonright
k-1)=\mathbb{C}_{\delta_{k-2}}\ast\dot{\mathbb{D}}$ so that
$\mathbb{C}_{\rho}(\vec{\delta})=\mathbb{C}_{\delta_{k-1}}\ast\left(\dot{\mathbb{C}}_{[\delta_{k-1},\delta_{k-2})}\ast\dot{\mathbb{D}}\right)^{2}$.
Note in the case $k=1$, we just have
$\mathbb{C}_{\rho}(\vec{\delta})=\mathbb{C}_{\delta_{0}}\ast(\dot{\mathbb{C}}_{[\delta_{0},\rho)})^{2}$.
The filter $H^{*}$ adds generics $J_{0},J_{1}$ over
$\mathbb{C}_{\rho}(\vec{\delta}\upharpoonright k-1)$ so that $J_{0}$ and
$J_{1}$ agree on $\mathbb{C}_{\delta_{k-1}}$ (recall that
$\delta_{k-1}<\delta_{k-2}$) but are mutually generic afterwards. Since
$H^{*}$ is guided by the collapse at $\kappa$, both $J_{0}$ and $J_{1}$ are
guided by the collapse at $\kappa$. Hence, our inductive assumption implies
that for each $i\in 2$, $\pi^{-1}_{M_{\kappa}[G^{*}]}[J_{i}]$ has a sup
$r_{i}$ in $j(\mathbb{C}_{\rho}(\vec{\delta}\upharpoonright k-1))$. By the
agreement between $J_{0}$ and $J_{1}$ up to stage $\delta_{k-1}<\delta_{k-2}$,
we know that $r_{0}\upharpoonright
j(\mathbb{C}_{\delta_{k-1}})=r_{1}\upharpoonright
j(\mathbb{C}_{\delta_{k-1}})$. We let $\bar{r}$ denote the common value.
Finally, we define $r^{*}$ to be the function $\bar{r}\,^{\frown}\langle
r_{i}\upharpoonright j(\mathbb{C}_{\rho}(\vec{\delta}\upharpoonright
k-1)):i<2\rangle$. Then $r^{*}$ is a condition in
$j(\mathbb{C}_{\rho}(\vec{\delta}))$ which is a sup of
$\pi^{-1}_{M_{\kappa}[G^{*}]}[H^{*}]$.
This completes the inductive step and the proof of the proposition.
∎
### 6.2. No New Branches
In this subsection, we will show that various ${\cal{F}}$-completely proper
posets do not add branches through Aronszajn trees of interest. We will use
this general result to show, in particular, that tails of the club adding
poset $\mathbb{C}_{\rho}$ do not add any branches to trees $\dot{T}$ which are
Aronszajn trees in an intermediate extension obtained by forcing with
$\mathbb{C}_{\sigma}{\ast\dot{\mathbb{S}}_{\sigma}}$, for $\sigma<\rho$. This
will ensure that the tree specializing iteration of length $\rho$ above is in
fact an iteration of specializing _Aronszajn_ trees in the extension by
$\mathbb{C}_{\rho}$, a conclusion which is essential in order to see that the
specializing iteration does not collapse $\kappa$.
Arguments for securing that certain posets do not add new cofinal branches to
trees play a crucial role in consistency results concerning the tree property,
going back to the work of Mitchell and Silver ([37]), and Magidor and Shelah
([35]). Lemma 6 of Unger [44] provides such an argument with respect to closed
posets and trees named by posets with reasonable chain condition, given
constraints on the continuum function. Here, we prove a version of these
results, in which the relevant posets (which in practice are variants of the
club-adding poset) are $\cal{F}$-completely proper (and thus
$\kappa$-distributive) but not $\kappa$-closed.
The statement of the following Proposition involves (names of) posets
$\dot{\mathbb{Q}}_{1},\dot{\mathbb{Q}}_{2}$, and $\dot{\mathbb{S}}$. To relate
the statement to our scenario, we suggest keeping in mind the following
assignments of the posets: Fixing $\rho<\rho^{*}<\kappa^{+}$,
$\dot{\mathbb{Q}}_{1}=\dot{\mathbb{C}}_{\rho}$ is the $(\mathbb{P}$-name) of
the first $\rho$ steps of the club adding iteration,
$\dot{\mathbb{S}}=\dot{\mathbb{S}}_{\rho}$ is the
$\mathbb{P}*\dot{\mathbb{C}}_{\rho}$-name of the first $\rho$ steps of the
iteration specializing trees, and
$\dot{\mathbb{Q}}_{2}=\dot{\mathbb{C}}_{[\rho,\rho^{*})}$ is the
$\mathbb{P}*\dot{\mathbb{C}}_{\rho}$-name of the segment of the final
iteration from (and including) stage $\rho$ to stage $\rho^{*}$ (i.e.
$\dot{\mathbb{Q}}_{1}*\dot{\mathbb{Q}}_{2}=\dot{\mathbb{C}}_{\rho^{*}}$).
###### Proposition 6.3.
Suppose that $\dot{\mathbb{Q}}_{1}$ is a $\mathbb{P}$-name and that
$\dot{\mathbb{Q}}_{2}$ and $\dot{\mathbb{S}}$ are
$(\mathbb{P}\ast\dot{\mathbb{Q}}_{1})$-names so that
$\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{Q}}^{2}_{2}$ is ${\cal{F}}$-completely
proper and so that
$\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{Q}}^{2}_{2}$ forces that
$\dot{\mathbb{S}}$ is $\kappa$-c.c. Let $\dot{T}$ be a
$(\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{S}})$-name for an
Aronszajn tree on $\kappa$. Then
$\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast(\dot{\mathbb{Q}}_{2}\times\dot{\mathbb{S}})$
forces that $\dot{T}$ is an Aronszajn tree.
That is to say, forcing with $\dot{\mathbb{Q}}_{2}$ after
$\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{S}}$ does not add branches
to $\dot{T}$. To show this, we will follow the standard approach and show that
if $\dot{\mathbb{Q}}_{2}$ were to add such a branch, then we can find some
model in which a level of the tree has too many nodes.
For the rest of this subsection, we suppose for a contradiction that $\dot{b}$
is
$(\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast(\dot{\mathbb{Q}}_{2}\times\dot{\mathbb{S}}))$-name
for a branch through $\dot{T}$, where $\dot{T}$ is a
$(\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{S}})$-name for an
Aronszajn tree on $\kappa$. In the context of working with the forcing
$\mathbb{R}^{*}:=\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast(\dot{\mathbb{Q}}^{2}_{2}\times\dot{\mathbb{S}})$,
for which a typical generic looks like $G\ast Q_{1}\ast(Q_{2}^{L}\times
Q_{2}^{R}\times F)$, we will use $\dot{b}_{L}$ to denote the
$(\mathbb{P}\ast\dot{\mathbb{Q}}_{1}\ast(\dot{\mathbb{Q}}_{2}\times\dot{\mathbb{S}}))$-name
for $\dot{b}[\dot{G}\ast{\dot{Q}}_{1}\ast(\dot{Q}_{2}^{L}\times\dot{F})]$,
i.e., the interpretation of $\dot{b}$ using the left generic filter added by
$\dot{\mathbb{Q}}^{2}_{2}$. $\dot{b}_{R}$ is defined similarly.
The next lemma will be used as a successor step in obtaining a tree of
conditions forcing incompatible information about a branch.
###### Lemma 6.4.
(Under the assumptions of Proposition 6.3) $\mathbb{P}$ forces that for each
$\dot{\mathbb{Q}}_{1}$-name $\dot{d}$ for a condition in
$\dot{\mathbb{Q}}_{2}$, there is a dense, open set of $c$ in
$\dot{\mathbb{Q}}_{1}$ satisfying the following property: there exist names
$\dot{d}_{L},\dot{d}_{R}$ for conditions in $\dot{\mathbb{Q}}_{2}$ and an
ordinal $\xi<\kappa$ so that
1. (1)
$c\Vdash\dot{d}_{Z}\geq\dot{d}$ for each $Z\in\left\\{L,R\right\\}$;
2. (2)
$\langle
c,\dot{d}_{L},\dot{d}_{R},0_{\dot{\mathbb{S}}}\rangle\Vdash^{V[\dot{G}]}_{\dot{\mathbb{Q}}_{1}\ast(\dot{\mathbb{Q}}_{2}\times\dot{\mathbb{Q}}_{2}\times\dot{\mathbb{S}})}\dot{b}_{L}(\xi)\neq\dot{b}_{R}(\xi)$.
###### Proof.
We work in $V[G]$. Fix $c\in\mathbb{Q}_{1}$ and a $\mathbb{Q}_{1}$-name
$\dot{d}$ for a condition in $\dot{\mathbb{Q}}_{2}$. Let $Q_{1}$ be
$V[G]$-generic over $\mathbb{Q}_{1}$ containing $c$, and let $Q_{2}^{L}\times
Q_{2}^{R}$ be $V[G\ast Q_{1}]$-generic over $\mathbb{Q}_{2}^{2}$ containing
$(d,d)$.
We first claim that $0_{\mathbb{S}}$ forces over $V[G\ast
Q_{1}\ast(Q_{2}^{L}\times Q_{2}^{R})]$ that $\dot{b}_{L}\neq\dot{b}_{R}$. Thus
let $F$ be an arbitrary $V[G\ast Q_{1}\ast(Q_{2}^{L}\times
Q_{2}^{R})]$-generic filter for $\mathbb{S}$. Since $\mathbb{S}$ and
$\mathbb{Q}_{2}^{2}$ both live in $V[G\ast Q_{1}]$, the product lemma implies
that $Q_{2}^{L}\times Q_{2}^{R}$ is $V[G\ast Q_{1}\ast F]$-generic over
$\mathbb{Q}_{2}^{2}$. Since $Q_{2}^{L}$ and $Q_{2}^{R}$ are mutually $V[G\ast
Q_{1}\ast F]$-generic filters over $\mathbb{Q}_{2}$, we conclude that
$V[G\ast Q_{1}\ast F]=V[G\ast Q_{1}\ast(F\times Q_{2}^{L})]\cap V[G\ast
Q_{1}\ast(F\times Q_{2}^{R})].$
Therefore, if $b:=b_{L}=b_{R}$, then $b$ is in $V[G\ast Q_{1}\ast F]$, and
therefore $T$ is not an Aronszajn tree in that model, a contradiction.
We now claim that there is an ordinal $\xi$ so that $0_{\mathbb{S}}$ forces
over $V[G\ast Q_{1}\ast(Q_{2}^{L}\times Q_{2}^{R})]$ that
$\dot{b}_{L}(\xi)\neq\dot{b}_{R}(\xi)$. Let $A\subseteq\mathbb{S}$ be a
maximal antichain in $V[G\ast Q_{1}\ast(Q_{2}^{L}\times Q_{2}^{R})]$
consisting of conditions $g\in\mathbb{S}$ so that for some $\zeta_{g}<\kappa$,
$g\Vdash^{V[G\ast Q_{1}\ast(Q_{2}^{L}\times
Q_{2}^{R})]}_{\mathbb{S}}\dot{b}_{L}(\zeta_{g})\neq\dot{b}_{R}(\zeta_{g}).$
Because $\dot{b}_{L}$ and $\dot{b}_{R}$ name branches in $\dot{T}$, we see
that for any $g\in A$ and $\zeta\geq\zeta_{g}$, $g$ forces that
$\dot{b}_{L}(\zeta)\neq\dot{b}_{R}(\zeta)$. Since $\mathbb{S}$ is still
$\kappa$-c.c. after forcing to add $Q_{2}^{L}\times Q_{2}^{R}$, we know that
$A$ has size $<\kappa$ in $V[G\ast Q_{1}\ast(Q_{2}^{L}\times Q_{2}^{R})]$.
Therefore, letting $\xi:=\sup_{g\in A}\zeta_{g}$, $\xi<\kappa$. Then $\xi$
witnesses the claim: indeed, if $f\in\mathbb{S}$ is any condition, we may
extend it to $f^{*}$ so that $f^{*}$ is above some $g\in A$. By the remarks
above and since $\xi\geq\zeta_{g}$, we know that
$f^{*}\Vdash\dot{b}_{L}(\xi)\neq\dot{b}_{R}(\xi)$, completing the proof of the
second claim.
Since $(c,\dot{d},\dot{d})\in Q_{1}\ast(Q_{2}^{L}\times Q_{2}^{R})$, we may
find an extension $(c^{*},\dot{d}_{L},\dot{d}_{R})$ of $(c,\dot{d},\dot{d})$
as well as an ordinal $\xi<\kappa$ so that $(c^{*},\dot{d}_{L},\dot{d}_{R})$
forces that $0_{\dot{\mathbb{S}}}$ forces that
$\dot{b}_{L}(\xi)\neq\dot{b}_{R}(\xi)$. Then $c^{*}\geq c$ is in the desired
dense set. ∎
For the rest of the subsection, let $\vec{M}=\langle M_{\alpha}:\alpha\in
B\rangle$ be a sequence which is suitable with respect to all parameters of
interest. Since $\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{Q}}^{2}_{2}$ is
$\cal{F}$-completely proper, which implies that
$\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{Q}}_{2}$ is $\cal{F}$-completely proper,
we may assume that $\operatorname{dom}(\vec{M})$ satisfies the conclusion of
Definition 5.8 with respect to
$\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{Q}}^{2}_{2}$ and with respect to
$\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{Q}}_{2}$.
Fix $M^{*}\prec H(\kappa^{++})$, where $M^{*}$ has size $\kappa$, is closed
under $<\kappa$-sequences, and contains $\vec{M}$ as well as $\lhd$ from
Notation 2.12 as an element. Let $M$ denote the transitive collapse of
$M^{*}$, so that $M$ is a $\kappa$-model. Since
$\operatorname{dom}(\vec{M})\in\cal{F}^{+}$, we may find an $M$-normal
ultrafilter $U$ so that, letting $j:M\longrightarrow N$ be the associated
ultrapower map, $\kappa\in j(\operatorname{dom}(\vec{M}))$. As usual, we use
$M_{\kappa}$ to denote $j(\vec{M})(\kappa)$. The following claim shows that we
can build the desired tree of conditions forcing incompatible information
about the branch.
###### Claim 6.5.
$j(\mathbb{P})$ forces over $N$ that there exist sequences
$\langle\dot{c}_{\nu}:\nu<\omega_{1}\rangle$ and $\langle\dot{d}_{s}:s\in
2^{<\omega_{1}}\rangle$ so that the following properties hold:
1. (1)
for each $f\in(2^{\omega_{1}})^{N[\dot{G}_{j(\mathbb{P})}]}$,
$\langle(\dot{c}_{\nu},\dot{d}_{f\upharpoonright\nu}):\nu<\omega_{1}\rangle$
is an increasing sequence of conditions in
$\dot{\mathbb{Q}}_{1}\ast\dot{\mathbb{Q}}_{2}$ which is guided by the collapse
at $\kappa$ (see Definition 5.3);
2. (2)
if $s\neq t$ are in $2^{\nu}$ for some $\nu<\omega_{1}$, then
$j(\langle\dot{c}_{\nu},\dot{d}_{s},\dot{d}_{t},0_{\dot{\mathbb{S}}}\rangle)\Vdash^{N[\dot{G}_{j(\mathbb{P})}]}_{j\left(\dot{\mathbb{Q}}_{1}\ast(\dot{\mathbb{Q}}_{2}\times\dot{\mathbb{Q}}_{2}\times\dot{\mathbb{S}})\right)}j(\dot{b})_{L}\upharpoonright\kappa\neq
j(\dot{b})_{R}\upharpoonright\kappa.$
###### Proof.
The definition is by recursion. Let $G^{*}$ be an arbitrary $V$-generic over
$j(\mathbb{P})$, and let $G=G^{*}\cap\mathbb{P}$. Let $(c_{0},\dot{d}_{0})$ be
the trivial condition in $\mathbb{Q}_{1}\ast\dot{\mathbb{Q}}_{2}$. In order to
show that the desired sequences generate filters which are guided by the
collapse at $\kappa$ (which in turn will guarantee that they have upper
bounds), we will show that (1)-(3) of Lemma 5.7 are satisfied. In particular,
to secure (3) of that lemma, throughout the construction we will select
objects which are minimal according to the fixed well-order $\lhd$ of
$H(\kappa^{+})$. We remark that the entire construction takes place in
$N[G^{*}]$, but the proper initial segments can be carried out in $M[G]$ using
a proper initial segment of $f_{\kappa}$, the standard surjection from
$\kappa$ onto $\omega_{1}$ added by $G^{*}$.
Suppose that $\nu$ is a limit and that for all $\mu<\nu$ and all $s\in
2^{\mu}$, we have defined $c_{\mu}$ and $\dot{d}_{s}$. Then we let
$\dot{c}_{\nu}$ be the $\lhd$-least $\mathbb{P}$-name for a condition in
$\dot{\mathbb{Q}}_{1}$ so that $c_{\nu}:=\dot{c}_{\nu}[G]$ is a sup of
$\langle c_{\mu}:\mu<\nu\rangle$. Similarly, for $s\in 2^{\nu}$, we let
$\dot{d}_{s}$ be a $\mathbb{Q}_{1}$-name forced to be a sup of
$\langle\dot{d}_{s\upharpoonright\mu}:\mu<\nu\rangle$ so that a
$\mathbb{P}$-name for $\dot{d}_{s}$ is $\lhd$-minimal. Note that item (2) in
the claim still holds since if $s\neq t$ are in $2^{\nu}$, then there exists
some $\mu<\nu$ so that $s\upharpoonright\mu\neq t\upharpoonright\mu$. So
$j(\langle
c_{\mu},\dot{d}_{s\upharpoonright\mu},\dot{d}_{t\upharpoonright\mu},0_{\dot{\mathbb{S}}}\rangle)$
forces that $j(\dot{b})_{L}\upharpoonright\kappa\neq
j(\dot{b})_{R}\upharpoonright\kappa$. Hence the extension $j(\langle
c_{\nu},\dot{d}_{s},\dot{d}_{t},0_{\dot{\mathbb{S}}}\rangle)$ also forces
this.
Now for the successor step. Suppose that we have defined $c_{\nu}$ and
$\dot{d}_{s}$ for all $s\in 2^{\nu}$. In order to ensure that the assumptions
of Lemma 5.7 are satisfied, and thereby ensure that the sequences are guided
by the collapse at $\kappa$ (which in turn will guarantee they have an upper
bound), we will first define an auxiliary extension $c^{*}_{\nu}\geq c_{\nu}$
and for each $s\in 2^{\nu}$, a $\mathbb{Q}_{1}$-name $\dot{d}^{*}_{s}$ forced
by $c^{*}_{\nu}$ to extend $\dot{d}_{s}$. Towards this end, let
$\gamma:=f_{\kappa}(\nu)$, where $f_{\kappa}$ is the standard surjection added
by $G^{*}$ from $\omega_{1}$ onto $\kappa$. Let $u_{\gamma}$ be the
$\gamma$-th condition in $\mathbb{Q}_{1}\ast\dot{\mathbb{Q}}_{2}$, and write
$u_{\gamma}$ as $\langle c^{\gamma},\dot{d}^{\gamma}\rangle$. If $c^{\gamma}$
does not extend $c_{\nu}$ in $\mathbb{Q}_{1}$, set $c^{*}_{\nu}=c_{\nu}$ and
$\dot{d}^{*}_{s}=\dot{d}_{s}$. On the other hand, if $c^{\gamma}\geq c_{\nu}$,
we set $c^{*}_{\nu}=c^{\gamma}$. Then, given $s\in 2^{\nu}$, if
$c^{*}_{\nu}\Vdash\dot{d}^{\gamma}\geq\dot{d}_{s}$, we set
$\dot{d}^{*}_{s}=\dot{d}^{\gamma}$, and otherwise we set
$\dot{d}^{*}_{s}=\dot{d}_{s}$. Note that there is at most one $s$ that falls
into the first of these, since $c^{*}_{\nu}\Vdash\left\\{\dot{d}_{s}:s\in
2^{\nu}\right\\}$ is an antichain in $\dot{\mathbb{Q}}_{2}$.
Now we move to defining $c_{\nu+1}$ and $\dot{d}_{t}$ for all $t\in
2^{\nu+1}$. By Lemma 6.4, for each $s\in 2^{\nu}$, the set $D_{s}$ of all
$c\in{\mathbb{Q}_{1}}$ for which there exist names $\dot{d}_{L}$ and
$\dot{d}_{R}$ and an ordinal $\xi<\kappa$ satisfying
1. (i)
$c\Vdash\dot{d}_{Z}\geq\dot{d}_{s}$ for each $Z\in\left\\{L,R\right\\}$; and
2. (ii)
$\langle
c,\dot{d}_{L},\dot{d}_{R},0_{\dot{\mathbb{S}}}\rangle\Vdash{\dot{b}_{L}(\xi)\neq\dot{b}_{R}(\xi)}$
is dense, open in $\mathbb{Q}_{1}$. By Lemma 5.9 applied to $\mathbb{Q}_{1}$,
there exists an extension of $c_{\nu}$ inside $\bigcap_{s\in 2^{\nu}}D_{s}$.
Let $\dot{c}_{\nu+1}$ be the $\lhd$-minimal $\mathbb{P}$-name so that
$c_{\nu+1}:=\dot{c}_{\nu+1}[G]$ is such an extension. For each $s\in 2^{\nu}$,
we may find an ordinal $\xi_{s}<\kappa$ and $\mathbb{Q}_{1}$-names
$\dot{d}_{s^{\frown}\langle 0\rangle}$ and $\dot{d}_{s^{\frown}\langle
1\rangle}$ so that $c_{\nu+1}\Vdash\dot{d}_{s^{\frown}\langle
i\rangle}\geq\dot{d}_{s}$, so that $\langle
c_{\nu+1},\dot{d}_{s^{\frown}\langle 0\rangle},\dot{d}_{s^{\frown}\langle
1\rangle},0_{\dot{\mathbb{S}}}\rangle$ forces that
$\dot{b}_{L}(\xi_{s})\neq\dot{b}_{R}(\xi_{s})$, and so that $\mathbb{P}$-names
for $\dot{d}_{s^{\frown}\langle 0\rangle}$ and $\dot{d}_{s^{\frown}\langle
1\rangle}$ are $\lhd$-minimal. Applying $j$ to this statement, we secure (2)
of the claim. This completes the proof. ∎
Now that we have proven the above claim, we can finish the proof of
Proposition 6.3.
###### Proof.
(of Proposition 6.3) Let $G^{*}$ be $N$-generic over $j(\mathbb{P})$, let
$G:=G^{*}\cap\mathbb{P}$, and fix sequences $\langle
c^{{0}}_{\nu}:\nu<\omega_{1}\rangle$ and $\langle\dot{d}^{{0}}_{s}:s\in
2^{<\omega_{1}}\rangle$ satisfying Claim 6.5. Let $\langle
c_{\nu}:\nu<\omega_{1}\rangle$ and $\langle\dot{d}_{s}:s\in
2^{<\omega_{1}}\rangle$ denote the sequences of their $j$-images. For each
$f\in(2^{\omega_{1}})^{N[G^{*}]}$,
$\langle{(c^{0}_{\nu},\dot{d}^{0}_{f\upharpoonright\nu})}:\nu<\omega_{1}\rangle$
is guided by the collapse at $\kappa$. Moreover, $\operatorname{dom}(\vec{M})$
satisfies Definition 5.8, and $\kappa\in j(\operatorname{dom}(\vec{M}))$.
Since $j(\mathbb{Q}_{1}\ast\dot{\mathbb{Q}}_{2})$ is $j(\cal{F})$-completely
proper, we may find a condition $(c^{*},\dot{d}_{f})$ which is a sup in
$j(\mathbb{Q}_{1}\ast\dot{\mathbb{Q}}_{2})$ of
$\langle(c_{\nu},\dot{d}_{f\upharpoonright\nu}):\nu<\omega_{1}\rangle$ (note
that $c^{*}$ is independent of $f$ since any two sups are $=^{*}$-equal). By
item (2) of the previous claim, we know that if $f\neq g$ are in
$(2^{\omega_{1}})\cap N[G^{*}]$, then $\langle
c^{*},\dot{d}_{f},\dot{d}_{g},0_{j(\dot{\mathbb{S}})}\rangle$ forces that
$j(\dot{b})_{L}\upharpoonright\kappa\neq j(\dot{b})_{R}\upharpoonright\kappa$.
Now let $Q_{1}^{*}\ast F^{*}$ be $N[G^{*}]$-generic over
$j(\mathbb{Q}_{1}\ast\dot{\mathbb{S}})$ with $Q_{1}^{*}$ containing $c^{*}$.
Applying item (2) of the previous claim again, we know that if $f\neq g$ are
in $(2^{\omega_{1}})\cap N[G^{*}]$, then $\langle d_{f},d_{g}\rangle$ forces
in $j(\mathbb{Q}^{2}_{2})$ that $j(\dot{b})_{L}\upharpoonright\kappa\neq
j(\dot{b})_{R}\upharpoonright\kappa$. We note here that the tree of interest,
namely $T^{*}:=j(\dot{T})[G^{*}\ast Q_{1}^{*}\ast F^{*}]$, is a member of
$N[G^{*}\ast Q_{1}^{*}\ast F^{*}]$, i.e., exists prior to forcing with
$j(\mathbb{Q}_{2}^{2})$.
For each $f\in(2^{\omega_{1}})\cap N[G^{*}]$, let $d^{*}_{f}$ be an extension
of $d_{f}$ in $j(\mathbb{Q}_{2})$ which decides the value of
$j(\dot{b})(\kappa)$, say as $\alpha_{f}$. We claim that if $f\neq g$ are in
$(2^{\omega_{1}})\cap N[G^{*}]$, then $\alpha_{f}\neq\alpha_{g}$. Indeed,
suppose for a contradiction that there were $f\neq g$ with
$\alpha_{f}=\alpha_{g}$. Then force in $j(\mathbb{Q}^{2}_{2})$ above the
condition $\langle d^{*}_{f},d^{*}_{g}\rangle$ to obtain a pair
$\bar{Q}_{2}^{L}\times\bar{Q}_{2}^{R}$ of mutually generic filters for
$j(\mathbb{Q}_{2})$. Since $\alpha_{f}=\alpha_{g}$, the branch of $T^{*}$
below $\alpha_{f}$ is the same as the branch of $T^{*}$ below $\alpha_{g}$.
But the branch of $T^{*}$ below $\alpha_{f}$ equals
$(j(\dot{b})[\bar{Q}_{2}^{L}])\upharpoonright\kappa$ and the branch of $T^{*}$
below $\alpha_{g}$ equals
$(j(\dot{b})[\bar{Q}_{2}^{R}])\upharpoonright\kappa$, contradicting the fact
that $\langle d^{*}_{f},d^{*}_{g}\rangle$ forces that the interpretations
diverge below $\kappa$.
Since $(2^{\omega_{1}})\cap N[G^{*}]$ has size $j(\kappa)$ in $N[G^{*}]$ and
$j(\mathbb{Q}_{2}\ast\dot{\mathbb{S}})$ preserves $j(\kappa)$, this set still
has size $j(\kappa)$ in $N[G^{*}\ast Q_{1}^{*}\ast F^{*}]$. Thus in the model
$N[G^{*}\ast Q_{1}^{*}\ast F^{*}]$, the function taking
$f\in(2^{\omega_{1}})\cap N[G^{*}]$ to $\alpha_{f}$ is an injection. Therefore
level $\kappa$ of $T^{*}$ has size $j(\kappa)$ which contradicts the fact that
$j(\kappa)$ is $\aleph_{2}$ in $N[G^{*}\ast Q_{1}^{*}\ast F^{*}]$ and that
$T^{*}$ is an Aronszajn tree on $j(\kappa)$. ∎
## 7\. Putting it all Together
Up to this point in the paper, we have worked to establish a number of
isolated results. In this section, we will now define the poset which will
witness Theorem 1.1. Each of the previous sections will function as a
component in the inductive verification that this poset has the desired
properties.
We recall that $\mathbb{P}$ denotes $\operatorname{Col}(\omega_{1},<\kappa)$,
the Levy collapse of the ineffable cardinal $\kappa$. We define a
$\mathbb{P}$-name $\dot{\mathbb{C}}_{\kappa^{+}}$ for a $\kappa^{+}$-length
iteration adding clubs, and we also define a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\kappa^{+}})$-name
$\dot{\mathbb{S}}_{\kappa^{+}}$ for an iteration which specializes Aronszajn
trees. This is done in such a way that for all $\rho<\kappa^{+}$, the
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\kappa^{+}})$-name $\dot{\mathbb{S}}_{\rho}$
for the first $\rho$-stages of $\dot{\mathbb{S}}_{\kappa^{+}}$ is actually a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\rho})$-name.
More precisely, we define by recursion on $\rho\leq\kappa^{+}$ the names
$\dot{\mathbb{C}}_{\rho}$ and $\dot{\mathbb{S}}_{\rho}$. Suppose that
$\rho=\rho_{0}+1$ is a successor and that $\dot{\mathbb{C}}_{\rho_{0}}$ and
$\dot{\mathbb{S}}_{\rho_{0}}$ are both defined. Using the fixed well-order
$\lhd$ from Notation 2.12 as a bookkeeping device, we select a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\rho_{0}}\ast\dot{\mathbb{S}}_{\rho_{0}})$-name
$\dot{S}_{\rho_{0}}$ for a stationary subset of
$\kappa\cap\operatorname{cof}(\omega)$, and we define
$\dot{\mathbb{C}}_{\rho}:=\dot{\mathbb{C}}_{\rho_{0}}\ast\mathsf{CU}(\dot{S}_{\rho_{0}},\dot{\mathbb{S}}_{\rho_{0}})$;
see Definition 1.5. Next, we use $\lhd$ to select a
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\rho}\ast\dot{\mathbb{S}}_{\rho_{0}})$-name
$\dot{T}_{\rho_{0}}$ for an Aronszajn tree on $\kappa$, and we set
$\dot{\mathbb{S}}_{\rho}$ to be the
$(\mathbb{P}\ast\dot{\mathbb{C}}_{\rho})$-name for
$\dot{\mathbb{S}}_{\rho_{0}}\ast\mathbb{S}(\dot{T}_{\rho_{0}})$; see
Definition 1.6.
Now suppose that $\rho$ is a limit and that we have defined the sequences
$\langle\dot{\mathbb{C}}_{\xi},\dot{\mathbb{C}}(\xi):\xi<\rho\rangle$ and
$\langle\dot{\mathbb{S}}_{\xi},\dot{\mathbb{S}}(\xi):\xi<\rho\rangle$. We
first let $\dot{\mathbb{C}}_{\rho}$ be the $<\kappa$-support limit of
$\langle\dot{\mathbb{C}}_{\xi},\dot{\mathbb{C}}(\xi):\xi<\rho\rangle$. Second,
we see that $\mathbb{P}\ast\dot{\mathbb{C}}_{\rho}$ forces that
$\langle\dot{\mathbb{S}}_{\xi},\dot{\mathbb{S}}(\xi):\xi<\rho\rangle$ names an
iteration with countable support: by induction, if $\xi<\rho$ is a limit, then
$\dot{\mathbb{S}}_{\xi}$ is the $(\mathbb{P}\ast\dot{\mathbb{C}}_{\xi})$-name
for the countable support limit of
$\langle\dot{\mathbb{S}}_{\zeta},\dot{\mathbb{S}}(\zeta):\zeta<\xi\rangle$.
But $\dot{\mathbb{C}}_{\rho}$ is $\omega_{1}$-closed, and consequently, the
countable support limit of
$\langle\dot{\mathbb{S}}_{\zeta},\dot{\mathbb{S}}(\zeta):\zeta<\xi\rangle$ is
the same in both the extension by $\mathbb{P}\ast\dot{\mathbb{C}}_{\xi}$ and
the extension by $\mathbb{P}\ast\dot{\mathbb{C}}_{\rho}$. In light of this, we
let $\dot{\mathbb{S}}_{\rho}$ denote the countable support limit of
$\langle\dot{\mathbb{S}}_{\xi},\dot{\mathbb{S}}(\xi):\xi<\rho\rangle$, noting
that this is an $\omega_{1}$-closed poset in the extension by
$\mathbb{P}\ast\dot{\mathbb{C}}_{\rho}$. This completes the definitions of the
names. We may now define
$\mathbb{R}^{*}:=\mathbb{P}\ast\dot{\mathbb{C}}_{\kappa^{+}}\ast\dot{\mathbb{S}}_{\kappa^{+}}$.
We begin our analysis of $\mathbb{R}^{*}$ with some simple remarks. First,
$\mathbb{R}^{*}$ is $\omega_{1}$-closed, since all the posets under
consideration are (and since our iterations were taken with supports which are
at least countable). Additionally, $\mathbb{R}^{*}$ is $\kappa^{+}$-c.c.
Indeed, $\mathbb{P}$ trivially is. Furthermore, $\kappa^{<\kappa}=\kappa$
after forcing with $\mathbb{P}$, and so if $\beta<\kappa^{+}$,
$\dot{\mathbb{C}}_{\beta}$ is forced to be a poset of size $\kappa$. Since
direct limits in the iteration $\dot{\mathbb{C}}_{\kappa^{+}}$ are taken at
all stages in $\kappa^{+}\cap\operatorname{cof}(\kappa)$, standard arguments
(e.g., see [7]) show that $\dot{\mathbb{C}}_{\kappa^{+}}$ is $\kappa^{+}$-c.c.
Finally, since for every $\beta<\kappa^{+}$, $\dot{\mathbb{S}}_{\beta}$ is
forced to have size $\kappa$ by $\mathbb{P}\ast\dot{\mathbb{C}}_{\kappa^{+}}$,
and since $\dot{\mathbb{S}}_{\kappa^{+}}$ is taken with countable supports, a
standard $\Delta$-System argument shows that $\dot{\mathbb{S}}_{\kappa^{+}}$
is $\kappa^{+}$-c.c.
We next claim that if $\mathbb{R}^{*}$ preserves $\kappa$, then it forces all
Aronszajn trees on $\kappa$ are special, that such trees exist, and that every
stationary subset $S\subseteq\kappa\cap\operatorname{cof}(\omega)$ reflects on
every ordinal of cofinality $\omega_{1}$ in some closed unbounded subset on
$\kappa$. First, suppose that $\dot{T}$ is an $\mathbb{R}^{*}$-name for an
Aronszajn tree on $\kappa$. Because $\mathbb{R}^{*}$ is $\kappa^{+}$-c.c.,
$\dot{T}$ is an
$(\mathbb{P}*\dot{\mathbb{C}}_{\gamma}*\dot{\mathbb{S}}_{\gamma})$-name for
some $\gamma<\kappa^{+}$, and hence names an Aronszajn tree in any extension
between that given by
$\mathbb{P}*\dot{\mathbb{C}}_{\gamma}*\dot{\mathbb{S}}_{\gamma}$ and the full
$\mathbb{R}^{*}$-extension. By our bookkeeping device, there is some
$\delta\geq\gamma$ so that $\dot{\mathbb{S}}(\delta)$ is forced by
$\mathbb{P}*\dot{\mathbb{C}}_{\delta+1}*\dot{\mathbb{S}}_{\delta}$ to equal
$\dot{\mathbb{S}}(\dot{T})$. Hence $\mathbb{R}^{*}$ forces that $\dot{T}$ is
special. Similarly, if $\dot{S}$ is an $\mathbb{R}^{*}$-name for a stationary
subset of $\kappa\cap\operatorname{cof}(\omega)$, then there is some
$\alpha<\kappa^{+}$ so that $\dot{S}$ is a
$(\mathbb{P}*\dot{\mathbb{C}}_{\alpha}*\dot{\mathbb{S}}_{\alpha})$-name, and
hence there is some $\beta\geq\alpha$ so that $\dot{\mathbb{C}}(\beta)$ is
forced by $\mathbb{P}*\dot{\mathbb{C}}_{\beta}$ to equal
$\mathsf{CU}(\dot{S},\dot{\mathbb{S}}_{\beta})$. Thus in the extension by
$\mathbb{P}*\dot{\mathbb{C}}_{\beta+1}*\dot{\mathbb{S}}_{\beta}$, $\dot{S}$
reflects almost everywhere, and since the forcing to complete
$\mathbb{P}*\dot{\mathbb{C}}_{\beta+1}*\dot{\mathbb{S}}_{\beta}$ to
$\mathbb{R}^{*}$ is $\omega_{1}$-closed, $\dot{S}$ still reflects almost
everywhere in the full $\mathbb{R}^{*}$-extension.
As a result of the previous discussion, we see that in order to show that
$\mathbb{R}^{*}$ witnesses Theorem 1.1, we need to prove that $\mathbb{R}^{*}$
preserves $\kappa$. To achieve this we verify that (i) $\mathbb{P}$ forces
$\dot{\mathbb{C}}_{\kappa^{+}}$ is $\kappa$-distributive, and that (ii)
$\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}}$ forces that
$\dot{\mathbb{S}}_{\kappa^{+}}$ is $\kappa$-c.c.
To this end, we consider the following simplifications. First, concerning (i),
we note that since $\dot{\mathbb{C}}_{\kappa^{+}}$ is forced to be
$\kappa^{+}$-c.c, it is sufficient to verify that $\dot{\mathbb{C}}_{\rho}$ is
forced to be $\kappa$-distributive for all $\rho<\kappa^{+}$ to show that (i)
holds. We will use Lemma 5.9 to verify this, by proving that for every
$\rho<\kappa^{+}$, $\dot{\mathbb{C}}_{\rho}$ is $\cal{F}$-completely proper.
Second, concerning (ii), since $\dot{\mathbb{S}}_{\kappa^{+}}$ names a
countable support iteration, any $\kappa$-sized antichain would witness that
some proper initial segment $\dot{\mathbb{S}}_{\gamma}$ is not $\kappa$-c.c.
Therefore, it is sufficient to verify that
$\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}}$ forces $\dot{\mathbb{S}}_{\gamma}$
is $\kappa$-c.c for every $\gamma<\kappa^{+}$. Howover, as
$\dot{\mathbb{C}}_{\kappa^{+}}$ names a $\kappa^{+}$-cc poset, every
$(\mathbb{P}*\dot{\mathbb{C}}_{\kappa^{+}})$-name for a subset of
$\dot{\mathbb{S}}_{\gamma}$ of size $\kappa$ is equivalent to a
$(\mathbb{P}*\dot{\mathbb{C}}_{\rho})$-name, for some $\rho\geq\gamma$.
Clearly, if $\mathbb{P}*\dot{\mathbb{C}}_{\rho}$ forces
$\dot{\mathbb{S}}_{\gamma}$ fails to satisfy the $\kappa$-c.c for some
$\gamma\leq\rho$, then it forces $\dot{\mathbb{S}}_{\rho}$ is not
$\kappa$-c.c. We conclude that (ii) follows from the assertion that for every
$\rho<\kappa^{+}$, $\mathbb{P}*\dot{\mathbb{C}}_{\rho}$ forces that
$\dot{\mathbb{S}}_{\rho}$ is $\kappa$-c.c.
Combining the two simplifications, it remains to prove the next claim.
###### Claim 7.1.
The following holds for every $\rho<\kappa^{+}$:
1. (1)
$\dot{\mathbb{C}}_{\rho}$ is $\cal{F}$-completely proper, and
2. (2)
$\mathbb{P}*\dot{\mathbb{C}}_{\rho}$ forces $\dot{\mathbb{S}}_{\rho}$ is
$\kappa$-c.c.
We prove the claim by induction on $\rho<\kappa^{+}$. Let $\rho<\kappa^{+}$,
and suppose that the claim holds for every $\sigma<\rho$. In particular, if
$\sigma<\rho$, then since $\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma}$ forces
that $\dot{\mathbb{S}}_{\sigma}$ is a countable support iteration specializing
trees and since (2) holds at $\sigma$, we must have that
$\mathbb{P}\ast\dot{\mathbb{C}}_{\sigma}$ forces that
$\dot{\mathbb{S}}_{\sigma}$ is a countable support iteration specializing
_Aronszajn_ trees. Hence we see that assumptions (1)-(4) from the beginning of
section 6 hold. Applying Proposition 6.2, it follows that
$\dot{\mathbb{C}}_{\rho}$ is $\cal{F}$-completely proper, and moreover, so is
$\mathbb{C}_{\rho}(\vec{\delta})$ for every finite, decreasing sequence
$\vec{\delta}$ of ordinals below $\rho$. We use this to prove that (2) of the
claim holds at $\rho$.
We aim to apply Corollary 3.23 to the $\cal{F}$-strongly proper poset
$\mathbb{P}^{*}:=\mathbb{P}\ast\dot{\mathbb{C}}_{\rho}$, and to do so, we need
to verify that for each $\sigma<\rho$,
$\mathbb{P}^{*}\ast\dot{\mathbb{S}}_{\sigma}\Vdash\dot{T}_{\sigma}$ is an
Aronszjan tree. This is where the doubling tail products come into play.
Indeed, we consider a slightly more general statement, which would allow us to
use Proposition 6.3. For each decreasing sequence
$\vec{\delta}=\langle\delta_{0},\dots,\delta_{n-1}\rangle$ in $\rho$, we let
$\star_{\rho}(\vec{\delta})$ be the statement that
$\mathbb{P}*\dot{\mathbb{C}}_{\rho}(\vec{\delta})$ forces
$\dot{\mathbb{S}}_{\delta_{n-1}}$ is $\kappa$-c.c. (note that if
$\vec{\delta}=\emptyset$ then the statement holds vacuously). We prove by
induction on the reverse lexicographic order $<_{\operatorname{rLex}}$ on
$[\rho]^{<\omega}_{\text{dec}}$ that $\star_{\rho}(\vec{\delta})$ holds. The
base case of the induction, where $\vec{\delta}=\emptyset$, trivially holds,
as mentioned above. For the induction step, fix
$\vec{\delta}=\langle\delta_{{0}},\dots,\delta_{{n-1}}\rangle\in[\rho]^{<\omega}_{\text{dec}}$
and suppose that $\star_{\rho}(\vec{\gamma})$ holds for every
$\vec{\gamma}<_{\operatorname{rLex}}\vec{\delta}$. To show that
$\star_{\rho}(\vec{\delta})$ holds, we need to verify that
$\mathbb{P}*\dot{\mathbb{C}}_{\rho}(\vec{\delta})$ forces that
$\dot{\mathbb{S}}_{\delta_{n-1}}$ is $\kappa$-c.c. For this in turn, by
Corollary 3.23, it is sufficient to verify that for every
$\gamma<\delta_{n-1}$,
$\mathbb{P}*\dot{\mathbb{C}}_{\rho}(\vec{\delta})\ast\dot{\mathbb{S}}_{\gamma}$
forces that $\dot{T}_{\gamma}$ is an Aronszajn tree on $\kappa$. If
$\delta_{n-1}=0$ there is nothing to prove. Otherwise, let
$\gamma<\delta_{n-1}$, and set
$\vec{\gamma}:=\vec{\delta}^{\frown}\langle\gamma\rangle$. By Proposition 6.3
and the definition of $\dot{\mathbb{C}}_{\rho}(\vec{\gamma})$, it suffices to
verify that $\mathbb{P}*\dot{\mathbb{C}}_{\rho}(\vec{\gamma})$ forces that
$\dot{\mathbb{S}}_{\gamma}$ is $\kappa$-c.c, to conclude that
$\mathbb{P}*\dot{\mathbb{C}}_{\rho}(\vec{\delta})\ast\dot{\mathbb{S}}_{\gamma}$
forces that $\dot{T}_{\gamma}$ is Aronszajn. However, the last is just
$\star_{\rho}(\vec{\gamma})$, which holds by our inductive assumption and the
fact that $\vec{\gamma}<_{\operatorname{rLex}}\vec{\delta}$. This concludes
the proof of Claim 7.1, which in turn finishes the proof of Theorem 1.1.
We conclude the paper with two questions:
###### Question 7.2.
Is an ineffable cardinal necessary for proving Theorem 1.1? Is a weakly
compact cardinal sufficient?
As we’ve remarked throughout the paper, we only use ineffability in the proof
of Proposition 3.22 (and thus in the corollaries of this proposition). If one
could prove this proposition assuming only a weakly compact, then that would
suffice to show that a weakly compact is optimal.
We also mention the following long-standing question:
###### Question 7.3.
Is a weakly compact cardinal needed for
$\mathsf{SATP}(\omega_{2})+2^{\omega_{1}}=\omega_{3}$?
We recall that in the Laver-Shelah model of $\mathsf{SATP}(\omega_{2})$,
$2^{\omega_{1}}=\omega_{3}$. Moreover, Rinot has shown ([38]) that if the
$\mathsf{GCH}$ and $\mathsf{SATP}(\omega_{2})$ both hold, then $\omega_{2}$ is
weakly compact in $L$.
## 8\. Acknowledgements
The authors would like to thank Itay Neeman, for helpful comments and
corrections, and also the referee, for a thorough reading of the manuscript
and many valuable suggestions and comments.
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|
# Communication-Efficient Sampling for Distributed Training of Graph
Convolutional Networks
Peng Jiang
Department of Computer Science
The University of Iowa
Iowa City, IA 52242, USA
<EMAIL_ADDRESS>
&Masuma Akter Rumi
Department of Computer Science
The University of Iowa
Iowa City, IA 52242, USA
<EMAIL_ADDRESS>
###### Abstract
Training Graph Convolutional Networks (GCNs) is expensive as it needs to
aggregate data recursively from neighboring nodes. To reduce the computation
overhead, previous works have proposed various neighbor sampling methods that
estimate the aggregation result based on a small number of sampled neighbors.
Although these methods have successfully accelerated the training, they mainly
focus on the single-machine setting. As real-world graphs are large, training
GCNs in distributed systems is desirable. However, we found that the existing
neighbor sampling methods do not work well in a distributed setting.
Specifically, a naive implementation may incur a huge amount of communication
of feature vectors among different machines. To address this problem, we
propose a communication-efficient neighbor sampling method in this work. Our
main idea is to assign higher sampling probabilities to the local nodes so
that remote nodes are accessed less frequently. We present an algorithm that
determines the local sampling probabilities and makes sure our skewed neighbor
sampling does not affect much to the convergence of the training. Our
experiments with node classification benchmarks show that our method
significantly reduces the communication overhead for distributed GCN training
with little accuracy loss.
## 1 Introduction
Graph Convolutional Networks (GCNs) are powerful models for learning
representations of attributed graphs. They have achieved great success in
graph-based learning tasks such as node classification Kipf and Welling
(2017); Duran and Niepert (2017), link prediction Zhang and Chen (2017, 2018),
and graph classification Ying et al. (2018b); Gilmer et al. (2017).
Despite the success of GCNs, training a deep GCN on large-scale graphs is
challenging. To compute the embedding of a node, GCN needs to recursively
aggregate the embeddings of the neighboring nodes. The number of nodes needed
for computing a single sample can grow exponentially with respect to the
number of layers. This has made mini-batch sampling ineffective to achieve
efficient training of GCNs. To alleviate the computational burden, various
neighbor sampling methods have been proposed Hamilton et al. (2017); Ying et
al. (2018a); Chen et al. (2018b); Zou et al. (2019); Li et al. (2018); Chiang
et al. (2019); Zeng et al. (2020). The idea is that, instead of aggregating
the embeddings of all neighbors, they compute an unbiased estimation of the
result based on a sampled subset of neighbors.
Although the existing neighbor sampling methods can effectively reduce the
computation overhead of training GCNs, most of them assume a single-machine
setting. The existing distributed GCN systems either perform neighbor sampling
for each machine/GPU independently (e.g., PinSage Ying et al. (2018a),
AliGraph Zhu et al. (2019), DGL Wang et al. (2019)) or perform a distributed
neighbor sampling for all machines/GPUs (e.g., AGL Zhang et al. (2020)). If
the sampled neighbors on a machine include nodes stored on other machines, the
system needs to transfer the feature vectors of the neighboring nodes across
the machines. This incurs a huge communication overhead. None of the existing
sampling methods or the distributed GCN systems have taken this communication
overhead into consideration.
In this work, we propose a communication-efficient neighbor sampling method
for distributed training of GCNs. Our main idea is to assign higher sampling
probabilities for local nodes so that remote nodes will be accessed less
frequently. By discounting the embeddings with the sampling probability, we
make sure that the estimation is unbiased. We present an algorithm to generate
the sampling probability that ensures the convergence of training. To validate
our sampling method, we conduct experiments with node classification
benchmarks on different graphs. The experimental results show that our method
significantly reduces the communication overhead with little accuracy loss.
## 2 Related Work
The idea of applying convolution operation to the graph domain is first
proposed by Bruna et al. (2013). Later, Kipf and Welling (2017) and Defferrard
et al. (2016) simplify the convolution computation with localized filters.
Most of the recent GCN models (e.g., GAT Velickovic et al. (2018), GraphSAGE
Hamilton et al. (2017), GIN Xu et al. (2019)) are based on the GCN in Kipf and
Welling (2017) where the information is only from 1-hop neighbors in each
layer of the neural network. In Kipf and Welling (2017), the authors only
apply their GCN to small graphs and use full batch for training. This has been
the major limitation of the original GCN model as full batch training is
expensive and infeasible for large graphs. Mini-batch training does not help
much since the number of nodes needed for computing a single sample can grow
exponentially as the GCN goes deeper. To overcome this limitation, various
neighbor sampling methods have been proposed to reduce the computation
complexity of GCN training.
Node-wise Neighbor Sampling: GraphSAGE Hamilton et al. (2017) proposes to
reduce the receptive field size of each node by sampling a fixed number of its
neighbors in the previous layer. PinSAGE Ying et al. (2018a) adopts this node-
wise sampling technique and enhances it by introducing an importance score to
each neighbor. It leads to less information loss due to weighted aggregation.
VR-GCN Chen et al. (2018a) further restricts the neighbor sampling size to two
and uses the historical activation of the previous layer to reduce variance.
Although it achieves comparable convergence to GraphSAGE,VR-GCN incurs
additional computation overhead for convolution operations on historical
activation which can outweigh the benefit of reduced number of sampled
neighbors. The problem with node-wise sampling is that, due to the recursive
aggregation, it may still need to gather the information of a large number of
nodes to compute the embeddings of a mini-batch.
Layer-wise Importance Sampling: To further reduce the sample complexity,
FastGCN Chen et al. (2018b) proposes layer-wise importance sampling. Instead
of fixing the number of sampled neighbors for each node, it fixes the number
of sampled nodes in each layer. Since the sampling is conduced independently
in each layer, it requires a large sample size to guarantee the connectivity
between layers. To improve the sample density and reduce the sample size,
Huang et al. (2018) and Zou et al. (2019) propose to restrict the sampling
space to the neighbors of nodes sampled in the previous layer.
Subgraph Sampling: Layer-wise sampling needs to maintain a list of neighbors
and calculate a new sampling distribution for each layer. It incurs an
overhead that can sometime deny the benefit of sampling, especially for small
graphs. GraphSAINT Zeng et al. (2020) proposes to simplify the sampling
procedure by sampling a subgraph and performing full convolution on the
subgraph. Similarly, ClusterGCN Chiang et al. (2019) pre-partitions a graph
into small clusters and constructs mini-batches by randomly selecting subsets
of clusters during the training.
All of the existing neighbor sampling methods assume a single-machine setting.
As we will show in the next section, a straightforward adoption of these
methods to a distributed setting can lead to a large communication overhead.
## 3 Background and Motivation
In a $M$-layer GCN, the $l$-th convolution layer is defined as
$H^{(l)}=P\sigma(H^{(l-1)})W^{(l)}$ where $H^{(l)}$ represents the embeddings
of all nodes at layer $l$ before activation, $H^{(0)}=X$ represents the
feature vectors, $\sigma$ is the activation function, $P$ is the normalized
Laplacian matrix of the graph, and $W^{(l)}$ is the learnable weights at layer
$l$. The multiple convolution layers in the GCN can be represented as
$H^{(M)}=P\sigma(H^{(l-1)}(...\sigma(\underbrace{PXW^{(1)}}_{H^{(1)}})...))W^{(M)}.$
(1)
The output embedding $H^{(M)}$ is given to some loss function $F$ for
downstream learning tasks such as node classification or link prediction.
GCN as Multi-level Stochastic Compositional Optimization: As pointed out by
Cong et al. (2020), training a GCN with neighbor sampling can be considered as
a multi-level stochastic compositional optimization (SCO) problem (although
their description is not accurate). Here, we give a more precise connection
between GCN training and multi-level SCO. Since the convergence property of
algorithms for multi-level SCO has been extensively studied Yang et al.
(2019); Zhang and Xiao (2019); Chen et al. (2020), this connection will allow
us to study the convergence of GCN training with different neighbor sampling
methods. We can define the graph convolution at layer $l\in[1,M]$ as a
function $f^{(l)}=P\sigma(H^{(l-1)})W^{(l)}$. The embedding approximation with
neighbor sampling can be considered as a stochastic function
$f_{\omega_{l}}^{(l)}=\tilde{P}^{(l)}\sigma(H^{(l-1)})W^{(l)}$ where
$\tilde{P}^{(l)}$ is a stochastic matrix with
$\mathbb{E}_{\omega_{l}}[\tilde{P}^{(l)}]=P$. Therefore, we have
$f^{(l)}=\mathbb{E}_{\omega_{l}}[f_{\omega_{l}}^{(l)}]$. The loss function of
the GCN can be written as
$\mathcal{L}(\theta)=\mathbb{E}_{\omega_{(M+1)}}\left[f_{\omega_{(M+1)}}^{(M+1)}\left(\mathbb{E}_{\omega_{M}}\left[f_{\omega_{M}}^{(M)}\left(...{E}_{\omega_{1}}[f^{(1)}(\theta)]...\right)\right]\right)\right].$
(2)
Here, $\theta$ is the set of learnable weights at all layers
$\\{W^{(1)},...,W^{(M)}\\}$, $f^{(M+1)}=F(H^{(M)})$, and the stochastic
function $f_{\omega_{(M+1)}}^{(M+1)}$ corresponds to the mini-batch sampling.
Distributed Training of GCN: As the real-world graphs are large and the
compute/memory capacity of a single machine is limited, it is always desirable
to perform distributed training of GCNs. A possible scenario would be that we
train a GCN on a multi-GPU system. The global memory of a single GPU cannot
accommodate the feature vectors of all nodes in the graph. It will be
inefficient to store the feature vectors on the CPU main memory and move the
feature vectors to GPU in each iteration of the training process because the
data movement incurs a large overhead. We want to split the feature vectors
and store them on multiple GPUs so that each GPU can perform calculation on
its local data. Another possible scenario would be that we have a large graph
with rich features which cannot be store on a single machine. For example, the
e-commerce graphs considered in AliGraph Zhu et al. (2019) can ‘contain tens
of billions of nodes and hundreds of billions of edges with storage cost over
10TB easily’. Such graphs need to be partitioned and stored on different
machines in a distributed system. Figure 1 shows an example of training a two-
layer GCN on four GPUs. Suppose full neighbor convolution is used and each GPU
computes the embeddings of its local nodes. GPU-0 needs to compute the
embeddings of node A and B and obtain a stochastic gradient $\tilde{g_{0}}$
based on the loss function. GPU-1 needs to compute the embeddings of node C
and D and obtain a stochastic gradient $\tilde{g_{1}}$. Similarly, GPU-2 and
GPU-3 compute the embeddings of their local nodes and obtain stochastic
gradient $\tilde{g_{2}}$ and $\tilde{g_{3}}$. The stochastic gradients
obtained on different GPUs are then averaged and used to update the model
parameters.
Figure 1: An example of distributed GCN training. Left: A graph with 8 nodes
are divided into four parts and stored on four GPUs. Right: For a two-layer
GCN, to compute the embedding of node A, we need the feature vectors of node
A, B, C, E and F; to compute the embedding of node B, we need the feature
vectors of node B, C, D, E, F, G. Nodes that are not on the same GPU need to
be transferred through the GPU connections.
Communication of Feature Vectors: As shown in Figure 1, the computation of a
node’s embedding may involve reading the feature vector of a remote node. To
compute the embedding of node A on GPU-0, we need the intermediate embeddings
of node B and E, which in turn need the feature vectors of node A, B, C, E and
F (Note that the feature vector of node E itself is needed to compute its
intermediate embedding; the same for node B). Since node C, E, F are not on
GPU-0, we need to send the feature vectors of node C from GPU-1 and node E, F
from GPU-2. Similarly, to compute the embedding of node B on GPU-0, we need
feature vectors of node B, C, D, E, F and G, which means that GPU-0 needs to
obtain data from all of the other three GPUs. This apparently incurs a large
communication overhead. Even with neighbor sampling, the communication of the
feature vectors among the GPUs are unavoidable. In fact, in our experiments on
a four-GPU workstation, the communication can take more than 60% of the total
execution time with a naive adoption of neighbor sampling. The problem is
expected to be more severe on distributed systems with multiple machines.
Therefore, reducing the communication overhead for feature vectors is critical
to the performance of distributed training of GCNs.
## 4 Communication-Efficient Neighbor Sampling
To reduce the communication overhead of feature vectors, a straightforward
idea is to skew the probability distribution for neighbor sampling so that
local nodes are more likely to be sampled. More specifically, to estimate the
aggregated embedding of node $i$’s neighbors (i.e., $\sum_{j\in
N(i)}w_{ij}x_{j}$ where $N(i)$ denotes the neighbors of node $i$, $x_{j}$ is
the embedding of node $j$, and $w_{ij}$ is its weight), we can define a
sequence of random variables $\xi_{j}\sim$ Bernoulli($p_{j}$) where $p_{j}$ is
the probability that node $j$ in the neighbor list is sampled. We have an
unbiased estimate of the result as
$\sum_{j\in N(i)}\frac{1}{p_{j}}\xi_{j}w_{ij}x_{j}.$ (3)
The expected communication overhead with this sampling strategy is
$comm\\_overhead\propto\mathbb{E}\left[\sum_{j\in R}\xi_{j}\right]=\sum_{j\in
R}p_{j}$ (4)
where $R$ is the set of remote nodes. Suppose we have a sampling budget $B$
and we denote all the local nodes as $L$. We can let $\sum_{j\in
N(i)}p_{j}=\sum_{j\in L}p_{j}+\sum_{j\in R}p_{j}=B$ so that $B$ neighbors are
sampled on average. It is apparent that, if we increase the sampling
probability of local nodes (i.e., $\sum_{j\in L}p_{j}$), the expected
communication overhead will be reduced. However, the local sampling
probability cannot be increased arbitrarily. As an extreme case, if we let
$\sum_{j\in L}p_{j}=B$, only the local nodes will be sampled, but we will not
be able to obtain an unbiased estimate of the result, which can lead to poor
convergence of the training algorithm. We need a sampling strategy that can
reduce the communication overhead while maintaining an unbiased approximation
with small variance.
### 4.1 Variance of Embedding Approximation
Consider the neighbor sampling at layer $l+1$. Suppose $S_{l}$ is the set of
sampled nodes at layer $l$. We sample from all the neighbors of nodes in
$S_{l}$ and estimate the result for each of the node using (3). The total
estimation variance is
$V=\mathbb{E}\left[\sum_{i\in S_{l}}\left\lVert\sum_{j\in
N(S_{l})}\frac{1}{p_{j}}\xi_{j}w_{ij}x_{j}-\sum_{j\in
N(S_{l})}w_{ij}x_{j}\right\rVert^{2}\right]=\sum_{j\in
N(S_{l})}\left(\frac{1}{p_{j}}-1\right)\left\lVert
w_{*j}\right\rVert^{2}\left\lVert x_{j}\right\rVert^{2}.$ (5)
Here $\left\lVert w_{*j}\right\rVert^{2}=\sum_{i\in S_{l}}w_{ij}^{2}$ is the
sum of squared weights of edges from nodes in $S_{l}$ to node $j$. Clearly,
the smallest variance is achieved when $p_{j}=1,\forall j$, and it corresponds
to full computation. Since we are given a sampling budget, we want to minimize
$V$ under the constraint $\sum_{j\in N(S_{l})}p_{j}\leq B$. The optimization
problem is infeasible because the real value of $\left\lVert
x_{j}\right\rVert^{2}$ is unknown during the sampling phase. Some prior work
uses $\left\lVert x_{j}\right\rVert^{2}$ from the previous iteration of the
training loop to obtain an optimal sampling probability distribution (e.g.,
Cong et al. (2020)). This however incurs an extra overhead for storing $x_{j}$
for all the layers. A more commonly used approach is to consider $\left\lVert
x_{j}\right\rVert^{2}$ as bounded by a constant $C$ and minimize the upper
bound of $V$ Chen et al. (2018b); Zou et al. (2019). The problem can be
written as a constrained optimization problem:
$\displaystyle\min\;\;\;\;\;\;\;\;$ $\displaystyle\sum_{j\in
N(S_{l})}\left(\frac{1}{p_{j}}-1\right)\left\lVert w_{*j}\right\rVert^{2}C$
(6) subject to $\displaystyle\sum_{j\in N(S_{l})}p_{j}\leq B.$ $\displaystyle
0<p_{j}\leq 1.$
The Sampling Method Used in Previous Works: Although the solution to the
above problem can be obtained, it requires expensive computations. For
example, Cong et al. (2020) give an algorithm that needs to sort $\left\lVert
w_{*j}\right\rVert^{2}$ and searches for the solution. As neighbor sampling is
performed at each layer of the GCN and in each iteration of the training
algorithm, finding the exact solution to (6) may significantly slowdown the
training procedure. Chen et al. (2018b) and Zou et al. (2019) adopt a simpler
sampling method. They define a discrete probability distribution over all
nodes in $N(S_{l})$ and assign the probability of returning node $j$ as
$q_{j}=\frac{\left\lVert w_{*j}\right\rVert^{2}}{\sum_{k\in
N(S_{l})}\left\lVert w_{*k}\right\rVert^{2}}.$ (7)
They run the sampling for $B$ times (without replacement) to obtain $B$
neighbors. We call this sampling method linear weighted sampling. Intuitively,
if a node is in the neighbor list of many nodes in $S_{j}$ (i.e., $\left\lVert
w_{*j}\right\rVert$ is large), it has a high probability of being sampled.
More precisely, the probability of node $j$ being sampled is
$p_{j}=1-(1-q_{j})^{B}\leq q_{j}B.$ (8)
Plugging (7) into (8) and (6), we can obtain an upper bound of the variance of
embedding approximation with this linear weighted sampling method as
$V_{lnr}=\left(\frac{|N(S_{l})|}{B}-1\right)\sum_{k\in N(S_{l})}\left\lVert
w_{*k}\right\rVert^{2}C$ (9)
Due to its easy calculation, we adopt this sampling strategy in our work, and
we skew the sampling probability distribution to the local nodes so that the
communication overhead can be reduced.
### 4.2 Skewed Linear Weighted Sampling
Our idea is to scale the sampling weights of local nodes by a factor $s>1$.
More specifically, we divide the nodes in $N(S_{l})$ into the local nodes $L$
and the remote nodes $R$, and we define the sampling probability distribution
as
$q_{j}=\begin{cases}\frac{s\left\lVert w_{*j}\right\rVert^{2}}{\sum_{k\in
L}s\left\lVert w_{*k}\right\rVert^{2}+\sum_{k\in R}\left\lVert
w_{*k}\right\rVert^{2}}&\text{if $j\in L$}\\\ \frac{\left\lVert
w_{*j}\right\rVert^{2}}{\sum_{k\in L}s\left\lVert
w_{*k}\right\rVert^{2}+\sum_{k\in R}\left\lVert
w_{*k}\right\rVert^{2}}&\text{if $j\in R$}.\end{cases}$ (10)
Compared with (7), (10) achieves a smaller communication overhead because
$\sum_{j\in R}p_{j}$ is smaller. We call our sampling method skewed linear
weighted sampling. Clearly, the larger $s$ we use, the more communication we
save. Our next task is to find $s$ that can ensure the convergence of the
training.
We start by studying the approximation variance with our sampling method.
Plugging (10) into (8) and (6), we can obtain an upper bound of the variance
as
$\displaystyle V_{skewed}$
$\displaystyle=\left(\left(\frac{|L|}{sB}+\frac{|R|}{B}\right)\left(\sum_{k\in
L}s\left\lVert w_{*k}\right\rVert^{2}+\sum_{k\in R}\left\lVert
w_{*k}\right\rVert^{2}\right)-\sum_{k\in N(S_{l})}\left\lVert
w_{*k}\right\rVert^{2}\right)C$ (11)
$\displaystyle=V_{lnr}+\frac{(s-1)|R|}{B}\sum_{k\in L}\left\lVert
w_{*k}\right\rVert^{2}C+\frac{(1-s)|L|}{sB}\sum_{k\in R}\left\lVert
w_{*k}\right\rVert^{2}C.$
Note that the variance does not necessarily increase with the skewed neighbor
sampling (the last term of (11) is negative).
Since GCN training is equivalent to multi-level SCO as explained in the
Background section, we can use the convergence analysis of multi-level SCO to
study the convergence of GCN training with our skewed neighbor sampling.
Although different algorithms for multi-level SCO achieve different
convergence rates Yang et al. (2019); Zhang and Xiao (2019); Chen et al.
(2020), for general non-convex objective function $\mathcal{L}$, all of these
algorithms have the optimality error
$\left(\mathcal{L}(x_{k+1})-\mathcal{L}^{*}\right)$ or
$\nabla\mathcal{L}(x_{k+1})$ bounded by some terms that are linear to the
upper bound of the approximation variance at each level. This means that if we
can make sure $V_{skewed}=\Theta(V_{lnr})$, our skewed neighbor sampling will
not affect the convergence of the training. In light of this, we have the
following theorem for determining the scaler $s$ in (10).
###### Theorem 1.
When the number of remote nodes $|R|>0$, with some small constant $D$, if we
set
$s=\left(\frac{D(|N(S_{l})|-B)}{|R|}+\frac{1}{2}\right),$ (12)
the training algorithm using our sampling probability in (10) will achieve the
same convergence rate as using the linear weighted sampling probability in
(7).
###### Proof.
If we set $V_{skewed}\leq D_{1}V_{lnr}$ with some constant $D_{1}$, we can
calculate an exact upper bound of $s$ by solving the equations with (7) and
(10). The upper bound is
$\frac{T_{1}+T_{2}}{2}+\frac{\sqrt{(T_{1}+T_{2})^{2}-4T_{3}}}{2}$ where
$T_{1}=\frac{(D_{1}-1)(|L|+|R|-B)}{|R|}+1$,
$T_{2}=\frac{((D_{1}-1)(|L|+|R|-B)+|L|)\sum_{k\in R}\left\lVert
w_{*k}\right\rVert^{2}}{|R|\sum_{k\in L}\left\lVert w_{*k}\right\rVert^{2}}$,
$T_{3}=\frac{|L|\sum_{k\in R}\left\lVert w_{*k}\right\rVert^{2}}{|R|\sum_{k\in
L}\left\lVert w_{*k}\right\rVert^{2}}$. $|L|$ is the number of local nodes,
$|R|$ is the number of remote nodes, and $B$ is the sampling budget. For
simple computation, we ignore all the terms dependant on $\sum_{k\in
L}\left\lVert w_{*k}\right\rVert^{2}$ and $\sum_{k\in R}\left\lVert
w_{*k}\right\rVert^{2}$, and it gives us a feasible solution
$s=\frac{T_{1}}{2}=\left(\frac{D(|N(S_{l})|-B)}{|R|}+\frac{1}{2}\right)$ where
$D=\frac{D_{1}-1}{2}$. ∎
Intuitively, if there are few remote nodes (i.e., $\frac{|N(S_{l})|}{|R|}$ is
large), we can sample the local nodes more frequently, and (12) gives us a
larger $s$. If we have a large sampling budget $B$, the estimation variance of
the linear weighted sampling (9) is small. We will need to sample enough
remote nodes to keep the variance small, and (12) gives us a smaller $s$.
## 5 Experimental Results
We evaluate our communication-efficient sampling method in this section.
### 5.1 Experimental Setup
Platform: We conducted our experiments on two platforms: a workstation with
four Nvidia RTX 2080 Ti GPUs, and eight machines each with an Nvidia P100 GPU
in a HPC cluster. The four GPUs in the workstation are connected through PCIe
3.0 x16 slot. The nodes in the HPC cluster are connected with 100Gbps
InfiniBand based on a fat-tree topology. Our code is implemented with PyTorch
1.6. We use CUDA-aware MPI for communication among the GPUs. To enable the
send/recv primitive in PyTorch distributed library, we compile PyTorch from
source with OpenMPI 4.0.5.
tableGraph datasets. Graph #Nodes #Edges Cora 2.7K 10.5K CiteSeer 3.3K 9.2K
Reddit 233K 58M YouTube 1.1M 6.1M Amazon 1.6M 132M
Datasets: We conduct our experiments on five graphs as listed in Table 5.1.
Cora and CiteSeer are two small graphs that are widely used in previous works
for evaluating GCN performance Zou et al. (2019); Chen et al. (2018b); Zeng et
al. (2020). Reddit is a medium-size graph with 233K nodes and an average
degree of 492. Amazon is a large graph with more than 1.6M nodes and 132M
edges Zeng et al. (2020). We use the same configurations of training set,
validation set, and test set for the graphs as in previous works Zou et al.
(2019); Chen et al. (2018b); Zeng et al. (2020). Youtube is a large graph with
more than 1M nodes Mislove et al. (2007). Each node represents a user, and the
edges represent the links among users. The graph does not have feature vector
or label information given. We generate the labels and feature vectors based
on the group information of the nodes. More specifically, we choose the 64
largest groups in the graph as labels. The label of each node is a vector of
length 64 with element value of 0 or 1 depending on whether the node belongs
to the group. Only the nodes that belong to at least one group are labeled.
For feature vector, we randomly select 2048 from the 4096 largest groups. If a
node does not belong to any group, its feature vector is 0. We use 90% of the
labeled nodes for training and the remaining 10% for testing.
Benchmark and Settings: We use a 5-layer standard GCN (as in Formula (1)) to
perform node classification tasks on the graphs. For Cora, CiteSeer, Reddit
and Amazon whose labels are single value, we use the conventional cross-
entropy loss to perform multi-class classification. For YouTube, since the
nodes’ labels are vectors, we use binary cross entropy loss with a rescaling
weight of 50 to perform multi-label classification. The dimension of the
hidden state is set to 256 for Cora, CiteSeer and Reddit. The dimension of the
hidden state is set to 512 for YouTube and Amazon. For distributed training,
we divide the nodes evenly among different GPUs. Each GPU runs sampling-based
training independently, with the gradients averaged among GPUs in each
iteration.
We incorporate our skewed sampling technique into LADIES Zou et al. (2019) and
GraphSAINT Zeng et al. (2020), both of which use linear weighted sampling as
in (7). The only difference is that LADIES uses all neighbors of nodes at
layer $l$ as $N(S_{l})$ when it samples nodes at layer $l+1$, while GraphSAINT
considers all training nodes as $N(S_{l})$ when it samples the subgraph.
We compare the performance of three versions. The first version (Full) uses
the original linear weighted sampling and transfers sampled neighbors among
GPUs. The second version (Local) also uses linear weighted sampling but only
aggregates neighboring nodes on the same GPU. The third version (Our) uses our
skewed sampling and transfers sampled neighbors among GPUs.
### 5.2 Results on Single-Machine with Multiple GPUs
We use LADIES to train the GCN with Cora, CiteSeer, Reddit and YouTube graph
on the four-GPU workstation. The batch size on each GPU is set to 512, and the
number of neighbor samples in each intermediate layer is also set to 512.
Table 1: Comparison of our sampling method with a naive adoption of the LADIES sampler on Cora and CiteSeer. Graph | Sampling Method | F1-Score (%) | Communication Data Size (#nodes)
---|---|---|---
Cora | | Full
---
Our (D=4)
Our (D=8)
Our (D=16)
Our (D=32)
| $74.46\pm 1.36$
---
$74.82\pm 0.97$
$75.84\pm 1.05$
$75.80\pm 1.69$
$74.96\pm 1.15$
| $402582291\pm 410933$
---
$316248197\pm 165161$
$299891935\pm 267364$
$284031348\pm 295241$
$270444788\pm 290565$
CiteSeer | | Full
---
Our (D=4)
Our (D=8)
Our (D=16)
Our (D=32)
| $66.54\pm 1.80$
---
$65.58\pm 2.52$
$65.50\pm 2.43$
$65.36\pm 2.48$
$65.64\pm 2.51$
| $900858804\pm 595144$
---
$722616380\pm 588753$
$704497231\pm 518062$
$689739665\pm 438101$
$679202038\pm 413854$
Cora and CiteSeer Results: Although these two graphs are small and can be
easily trained on a single GPU, we apply distributed training to these two
graphs and measure the total communication data size to show the benefit of
our sampling method. Table 1 shows the best test accuracy and the total
communication data size in 10 epochs of training. The results are collected in
10 different runs and we report the mean and deviation of the numbers.
Compared with the full-communication version, our sampling method does not
cause any accuracy loss for Cora with $D$ (in Formula (12)) set to
$4,8,16,32$. For CiteSeer, the mean accuracy in different runs decreases by
about 1% with our sampling method. However, the best accuracy in different
runs matches the best accuracy of full-communication version. Figure 2 shows
the training loss over epochs with different sampling methods on Cora. We can
see that local aggregation leads to poor convergence, due to the loss of
information in the edges across GPUs. The other versions have the model
converge to optimal after 3 epochs. The training loss on CiteSeer follows a
similar pattern. The results indicate that our sampling method does not impair
the convergence of training.
Figure 2: Training loss over epochs on Cora.
The execution times of different versions are almost the same on these two
small graphs because the communication overhead is small and reducing the
communication has little effect to the overall performance. Therefore, instead
of reporting the execution time, we show the communication data size of
different versions. The numbers in Table 1 are the numbers of nodes whose
feature vectors are transferred among GPUs during the entire training process.
We can see that the communication is indeed reduced. When $D=32$, our sampling
method saves the communication overhead by 1.48x on Cora and 1.32x on
CiteSeer.
(a) Training Loss
(b) Validation Accuracy
(c) Execution Time
Figure 3: Results on Reddit graph.
(a) Training Loss
(b) Validation Accuracy
(c) Execution Time
Figure 4: Results on YouTube graph.
(a) Training Loss
(b) Validation Accuracy
(c) Execution Time
Figure 5: Results on Amazon graph.
Reddit and YouTube Results: These are two larger graphs for which
communication overhead is more critical to the performance of distributed
training. Figure 3 shows the results on Reddit graph. We run the training for
50 epochs and compare the training loss, validation accuracy and execution
time of different versions. The breakdown execution time is shown in Figure
3c. We can see that communication takes more than 60% of the total execution
time if we naively adopt the linear weighted sampling method. Our sampling
method reduces the communication time by 1.4x, 2.5x and 3.5x with $D$ set to
4, 8, 16, respectively. The actual communication data size is reduced by 2.4x,
3.6x and 5.2x. From Figure 3a, we can see that our sampling method converges
at almost the same rate as the full-communication version when $D$ is set to 4
or 8. The training converges slower when $D=16$, due to the large
approximation variance. Figure 3b shows the validation accuracy of different
versions. The best accuracy achieved by full-communication version is 93.0%.
Our sampling method achieves accuracy of 94.3%, 92.4%, 92.2% with $D$ set to
4, 8, 16, respectively. The figures also show that training with local
aggregation leads to a significant accuracy loss.
Figure 4 shows the results on YouTube graph. We run the training for 300
epochs. As shown in Figure 4c, the communication takes more than 70% of the
total execution time in the full-communication version. Our sampling method
effectively reduces the communication time. The larger $D$ we use, the more
communication time we save. The actual communication data size is reduced by
3.3x, 4.6x, 6.7x with $D$ set to 4, 8, 16. Despite the communication
reduction, our sampling method achieves almost the same convergence as full-
communication version, as shown in Figure 4a and 4b. The full-communication
version achieves a best accuracy of 34.0%, while our sampling method achieves
best accuracy of 33.4%, 36.0%, 33.2% with $D$ set to 4, 8, 16. In contrast,
local aggregation leads to a noticeable accuracy loss. The best accuracy it
achieves is 28.5%.
tableExecution time of the full-communication version with distributed
features on GPUs (Time-Distr) and a centralized version with all features
stored on CPU and copied to GPU in each iteration (Time-Centr). Graph Time-
Distr (sec) Time-Centr (sec) Cora 4.8 58.1 Citeseer 4.6 59.8 Reddit 2763.1
6622.0 YouTube 5852.2 7028.9
Comparison with Centralized Training: As we described in Section 3, the
motivation of partitioning the graph and splitting the nodes’ features among
GPUs in a single machine is that each GPU cannot hold the feature vectors of
the entire graph. As oppose to this distributed approach, an alternative
implementation is to store all the features on CPU and copy the feature
vectors of the sampled nodes to different GPUs in each iteration. For example,
PinSage Ying et al. (2018a) adopts this approach for training on large graphs.
To justify our distributed storage of the feature vectors, we collect the
performance results of GCN training with this centralized storage of feature
vectors on CPU. Table 5.2 lists the training time for different graphs. Even
compared with the full-communication baseline in Figure 3c and 4c, this
centralized approach is 1.2x to 13x slower. This is because the centralized
approach incurs a large data movement overhead for copying data from CPU to
GPU. The results suggest that, for training large graphs on a single-machine
with multiple GPUs, distributing the feature vectors onto different GPUs is
more efficient than storing the graph on CPU.
### 5.3 Results on Multiple Machines
We incorporate our sampling method to GraphSAINT and train a GCN with Amazon
graph on eight machines in a HPC cluster. The subgraph size is set to 4500.
With our skewed sampling method, a subgraph is more likely to contain nodes on
the same machine, and thus incurs less communication among machines. We run
the training for 20 epochs. As shown in Figure 5c, the communication takes
more than 75% of the total execution time in the full-communication version.
Our sampling method effectively reduces the communication overhead. The
overall speedup is 1.7x, 2.8x, 4.2x with $D$ set to 4, 8, 16. As shown in
Figure 5a and 5b, our sampling method achieves almost the same convergence as
full-communication version. The full-communication version achieves a best
accuracy of 79.31%, while our sampling method achieves best accuracy of 79.5%,
79.19%, 79.29% with $D$ set to 4, 8, 16. Although the training seems to
converge with local aggregation on this dataset, there is a clear gap between
the line of local aggregation and the lines of other versions. The best
accuracy achieved by local aggregation is 76.12%.
## 6 Conclusion
In this work, we study the training of GCNs in a distributed setting. We find
that the training performance is bottlenecked by the communication of feature
vectors among different machines/GPUs. Based on the observation, we propose
the first communication-efficient sampling method for distributed GCN
training. The experimental results show that our sampling method effectively
reduces the communication overhead while maintaining a good accuracy.
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22 2125
# Resource Bisimilarity in Petri Nets is Decidable
Irina A. Lomazova
<EMAIL_ADDRESS>Address for correspondence<EMAIL_ADDRESS>Vladimir
A. Bashkin Petr Jančar
Dept. of Computer Science Faculty of Science
Palacký University Olomouc Czechia
<EMAIL_ADDRESS>
###### Abstract
Petri nets are a popular formalism for modeling and analyzing distributed
systems. Tokens in Petri net models can represent the control flow state or
resources produced/consumed by transition firings. We define a resource as a
part (a submultiset) of Petri net markings and call two resources equivalent
when replacing one of them with another in any marking does not change the
observable Petri net behavior. We consider resource similarity and resource
bisimilarity, two congruent restrictions of bisimulation equivalence on Petri
net markings. Previously it was proved that resource similarity (the largest
congruence included in bisimulation equivalence) is undecidable. Here we
present an algorithm for checking resource bisimilarity, thereby proving that
this relation (the largest congruence included in bisimulation equivalence
that is a bisimulation) is decidable. We also give an example of two resources
in a Petri net that are similar but not bisimilar.
###### keywords:
labeled Petri nets, bisimulation, congruence, decidability, resource
bisimilarity
††volume: 186††issue: 1-4
Resource Bisimilarity in Petri Nets is Decidable
## 1 Introduction
The concept of process equivalence can be formalized in many different ways
[1]. One of the most important is _bisimulation equivalence_ [2, 3], which
captures the main features of the observable behavior of a system. Generally,
bisimulation equivalence, also called _bisimilarity_ , is defined as a
relation on sets of states of labeled transition systems (LTS). Two states are
bisimilar if their behavior cannot be distinguished by an external observer.
Petri nets are a popular formalism for modeling and analyzing distributed
systems, on which many notations used in practice are based. In Petri nets,
states are represented as markings — multisets of tokens residing in Petri net
places. The interleaving semantics associates a labeled Petri net (in which
transitions are labeled with actions) and its initial marking with the
corresponding LTS that describes the behavior of the net. For Petri nets, many
important behavioral properties, e.g. reachability, are decidable, but
behavioral equivalences like bisimilarity are undecidable [4].
As a mathematical formalism, Petri nets are equivalent to the subclass
(P,P)-PRS of Process Rewrite Systems (PRS) introduced by R. Mayr [5]. PRS
allow a (possibly infinite) LTS to be represented by a finite set of rewrite
rules using algebraic operations of sequential and parallel composition.
Several well-studied classes of infinite-state systems are included in the PRS
hierarchy, which is based on operations used in rewrite rules to define the
specific PRS: basic process algebras (BPA) are (1,S)-PRS, basic parallel
processes (BPP) are (1,P)-PRS, pushdown automata (PDA) are (S,S)-PRS, Petri
nets are (P,P)-PRS and process algebras (PA) are (1,G)-PRS.
The PRS hierarchy is strict (regarding expressivity) and sets a number of
interesting decidability borders, in particular for bisimilarity. While
bisimilarity is undecidable for (P,P)-PRS, corresponding to Petri nets, it is
decidable for the class (1,P)-PRS [6], corresponding to labeled communication-
free Petri nets, in which each transition has no more than one input place.
Due to the restriction on the structure of communication-free Petri nets,
bisimilarity for them is a congruence, and this property was crucial for the
decidability proof in [6]. (This property is also implicitly used in the later
proof [7].)
In Petri net models, places can be interpreted as resource repositories, and
tokens represent resource availability or resources themselves. A transition
firing, in turn, can be thought of as resource handling, relocation, deletion,
and creation. Hence, Petri nets have a clear resource perspective and are
widely used to model and analyze various types of resource-oriented systems
such as production systems, resource allocation systems, etc.
Since bisimulation equivalence cannot be effectively checked for Petri nets,
it is natural either to restrict the model (for example, to consider
communication-free or one-counter Petri nets), or to strengthen the
equivalence relation. In this paper we explore the second option. We consider
resources as parts of Petri net markings (multisets of tokens residing in
Petri net places) and study the possibility of replacing one resource with
another in any marking without changing the observable behavior of the net.
_Related work._
Numerous studies have been devoted to various aspects of resources in Petri
nets. We name just a few of them.
In open nets special resource places are used for modeling a resource
interface of the system [8, 9, 10, 11]. In workflow nets resource places
represent resources that can be consumed and produced during the business
process execution. Obviously, resource places not only demonstrate a resource
flow, but may substantially influence the system control flow [12, 13].
As a mathematical formalism, Petri nets are closely connected with Girard’s
linear logic [14], which also has a nice resource interpretation, and for
different classes of Petri nets it is possible to express a Petri net as a
linear logic formula [15, 16].
We have already mentioned bisimulation equivalence, which is defined as the
largest bisimulation in a given LTS. In [17], an equivalence on places of a
Petri net was defined which when lifted to the reachable markings preserves
their token game and distribution over the places. This structural equivalence
is called “strong bisimulation” in [17]111Not related to the more common use
of terms “strong bisimulation” and “weak bisimulation” for fundamental state
bisimulation and state bisimulation in LTS with invisible (silent)
transitions., because it is a non-trivial subrelation of bisimilarity.
The term _place bisimulation_ comes from [18] where Olderog’s concept was
improved and used for a reduction of nets. In a Petri net, two bisimilar
places can be fused without changing the observable net behavior.
In [19], a polynomial algorithm for computing the largest place bisimulation
was presented. Equivalences on sets of places were further explored in [20,
21, 22]. In particular, it was proven (using the technique from [4]) that a
weaker relation, called the _largest correct place fusion_ , is undecidable.
Places $p$ and $q$ can be correctly fused if for any marking $M$ the two
markings $M+p$ and $M+q$ are bisimilar.
Independently, in [23] a similar equivalence relation, called _structural
bisimilarity_ , was studied for labeled Place-Transition Nets. This relation
is defined on the set of all vertices of a Petri net (both places and
transitions) and also uses a kind of weak transfer property.
In [24, 25], a notion of _team bisimulation_ was defined for communication-
free Petri nets. Two distributed systems, each composed of a team of
sequential non-cooperating agents (tokens in a communication-free net), are
equivalent if it is possible to match each sequential component of one system
with a team-bisimilar sequential component of another system (as in sports,
where competing teams have the same number of equivalent player positions).
For Petri nets, the team bisimulation coincides with the place bisimulation
from [18].
In [26], a new equivalence on Petri net resources, called _resource
similarity_ was proposed. As already mentioned, a resource in a Petri net is a
part of its marking, and two resources are similar if replacing one of them by
another in any marking does not change the observable net behavior; in our
case it means that the behavior remains in the same class of bisimulation
equivalence. It was shown that the correct place fusion equivalence from [22]
is a special case of resource similarity, namely the place fusion is resource
similarity for one-token resources. Hence the undecidability result for the
place fusion extends to resource similarity. On the other hand, resource
similarity is a congruence and thus can be generated by a finite basis. A
special type of minimal basis, called the _ground basis_ , was presented in
[26].
In [26] also a stronger equivalence of resources in a Petri net, called
_resource bisimilarity_ 222Not related to the notion of “resource
bisimulation” from [27]. was defined; an equivalence of resources was called a
resource bisimulation if its additive and transitive closure is a
bisimulation. Properties of the mentioned resource equivalences were studied
in [28, 29]. In particular, it was shown that resource bisimulation is defined
by a weak transfer property, similar to the weak transfer property of place
bisimulation. It was proven that there exists the largest resource
bisimulation, and it coincides with resource bisimilarity. The questions of
whether the resource bisimilarity is decidable and whether it is strictly
stronger than resource similarity remained open in [28, 29, 30].
_Our contribution._
In this paper we give answers to these open questions. Based on the well-known
“tableau method” (see, e.g., [31]), we construct an algorithm for checking
resource bisimilarity in Petri nets. Thus, it is proved that resource
bisimilarity is decidable in Petri nets and is strictly stronger than resource
similarity. We also give an example of a labeled Petri net in which two
similar resources are not resource bisimilar.
Interestingly, for resource similarity and resource bisimilarity we have that
both are congruences, and thus both have finite bases, but one of them is
decidable, and the other is not.
_Organization of the paper._ In Section 2 we recall basic notions and
notations, including the result on finitely-based congruences. Section 3
clarifies how resource bisimilarity, resource similarity, and bisimilarity are
related. Section 4 shows the announced algorithm deciding resource
bisimilarity, and Section 5 provides conclusions and additional remarks.
## 2 Preliminaries
By $\mathord{\mathbb{N}}$ and $\mathord{\mathbb{N}}_{+}$ we denote the sets of
non-negative integers and of positive integers, respectively.
#### Multisets.
A _multiset_ $m$ over a set $S$ is a mapping
$m:S\rightarrow\mathord{\mathbb{N}}$; hence a multiset may contain several
copies of the same element. By $\mathcal{M}(S)$ we denote the set of multisets
over $S$ (which could be also denoted by $\mathord{\mathbb{N}}^{S}$). The
inclusion $\subseteq$ on $\mathcal{M}(S)$ coincides with the component-wise
order $\leq$ : we put $m\subseteq m^{\prime}$, or $m\leq m^{\prime}$, if
$\forall s\in S:m(s)\leq m^{\prime}(s)$. By $m=\emptyset$ we mean that
$m(s)=0$ for all $s\in S$.
Given $m,m^{\prime}\in\mathcal{M}(S)$, the union $m\cup m^{\prime}$ is
generally different than the sum $m+m^{\prime}$: for each $s\in S$ we have
$(m\cup m^{\prime})(s)=\max\,\\{m(s),m^{\prime}(s)\\}$ and
$(m+m^{\prime})(s)=m(s)+m^{\prime}(s)$. We also define the multiset
subtraction $m-m^{\prime}$: for each $s\in S$ we have
$(m-m^{\prime})(s)=\max\,\\{\,m(s)-m^{\prime}(s),0\,\\}$.
In this paper we only deal with multisets over finite sets $S$. In this case
the cardinality $|m|$ of each $m\in\mathcal{M}(S)$ is finite: we have
$|m|=\sum_{s\in S}m(s)$. We note that the partial order $\leq$ on
$\mathcal{M}(S)$ can be naturally extended to a total order, the _cardinality-
lexicographic_ order $\sqsubseteq$ :
we put $m\sqsubseteq m^{\prime}$ if $|m|<|m^{\prime}|$, or $|m|=|m^{\prime}|$
and $m\leq_{\textsc{lex}}m^{\prime}$,
where $\leq_{\textsc{lex}}$ is a lexicographic order. More precisely, the
_lexicographic order_ $\leq_{\textsc{lex}}$ on $\mathcal{M}(S)$ assumes that
the elements of $S$ are ordered, in which case $m\in\mathcal{M}(S)$ can be
also viewed as a vector $m=((m)_{1},(m)_{2},\dots,(m)_{|S|})$ in
$\mathord{\mathbb{N}}^{|S|}$; we put $m\leq_{\textsc{lex}}m^{\prime}$ if
$m=m^{\prime}$ or $(m)_{i}<(m^{\prime})_{i}$ for the least $i$ for which
$(m)_{i}$ and $(m^{\prime})_{i}$ differ.
#### Labeled Petri nets.
A _Petri net_ is a tuple $N=(P,T,W)$ where $P$ and $T$ are finite disjoint
sets of _places_ and _transitions_ , respectively, and $W:(P\times
T)\cup(T\times P)\to\mathord{\mathbb{N}}$ is an _arc-weight function_. A
_labeled Petri net_ is a tuple $N=(P,T,W,l)$ where $(P,T,W)$ is a Petri net
and $l:T\to\mathord{\textit{Act}}$ is a labeling function, mapping the
transitions to _actions_ (observed events) from a set
$\mathord{\textit{Act}}$. Figure 1 shows an example of a labeled Petri net,
with three transitions depicted by boxes, all being labeled with the same
action $b$; the places are depicted by circles, and the arc-weight function is
presented by (multiple) directed arcs (there is no arc from $p\in P$ to $t\in
T$ if $W(p,t)=0$, and similarly no arc from $t$ to $p$ if $W(t,p)=0$).
A _marking_ in a Petri net $N=(P,T,W)$ is a function
$M:P\to\mathord{\mathbb{N}}$, hence a multiset $M\in{\cal M}(P)$; it gives the
number of _tokens_ in each place. Tokens residing in a place are often
interpreted as resources of some type, which are consumed or produced by
transition firings (as defined below).
For a transition $t\in T$, the _preset_ ${\rm Phys.~{}Rev.~{}E}{t}$ and the
_postset_ $t^{\bullet}$ are defined as the multisets over $P$ such that ${\rm
Phys.~{}Rev.~{}E}{t}(p)=W(p,t)$ and $t^{\bullet}(p)=W(t,p)$ for each $p\in P$.
A _transition_ $t\in T$ _is enabled in a marking_ $M$ if ${\rm
Phys.~{}Rev.~{}E}{t}\leq M$; such a _transition may fire in_ $M$, yielding the
marking $M^{\prime}=(M-{\rm Phys.~{}Rev.~{}E}{t})+t^{\bullet}$ (hence
$M^{\prime}(p)=M(p)-W(p,t)+W(t,p)$ for each $p\in P$). We denote this firing
by $M\stackrel{{\scriptstyle t}}{{\to}}M^{\prime}$.
#### Bisimulation equivalence (in labeled Petri nets).
Informally speaking, two markings (states) in a labeled Petri net are
considered equivalent if they generate the same observable behavior. Finding
equivalent states may be very helpful for reducing the state space when
analyzing behavioral properties of a given net. A classical behavioral
equivalence is bisimulation equivalence, also called bisimilarity [3]. We
recall its definition in the framework of labeled Petri nets; below we thus
implicitly assume a fixed labeled Petri net $N=(P,T,W,l)$.
We say that a _relation_ $R\subseteq{\cal M}(P)\times{\cal M}(P)$ _satisfies_
the _transfer property w.r.t. a relation_ $R^{\prime}\subseteq{\cal
M}(P)\times{\cal M}(P)$ if for each pair $(M_{1},M_{2})\in R$ and for each
firing $M_{1}\stackrel{{\scriptstyle t}}{{\to}}M_{1}^{\prime}$ there exists an
(“imitating”) firing $M_{2}\stackrel{{\scriptstyle u}}{{\to}}M_{2}^{\prime}$
such that $l(u)=l(t)$ and $(M_{1}^{\prime},M_{2}^{\prime})\in R^{\prime}$. We
can illustrate this property by the following diagram.
$\qquad\qquad\qquad\qquad\qquad\qquad M_{1}\qquad R\qquad M_{2}$
$\qquad\qquad\qquad\qquad\qquad\qquad~{}\downarrow
t\quad\qquad\qquad\downarrow(\exists)u:l(u)=l(t)$
$\qquad\qquad\qquad\qquad\qquad\qquad M_{1}^{\prime}\qquad R^{\prime}\qquad
M_{2}^{\prime}$
By saying that $R$ _has the transfer property_ we mean that $R$ satisfies the
transfer property w.r.t. itself (hence $R^{\prime}=R$ in the diagram).
A relation $R\subseteq{\cal M}(P)\times{\cal M}(P)$ is a _bisimulation_ if $R$
has the transfer property and also $R^{-1}$ has the transfer property.
Markings $M_{1}$ and $M_{2}$ are _bisimilar_ (or _bisimulation equivalent_),
written $M_{1}\sim M_{2}$, if there exists a bisimulation $R$ such that
$(M_{1},M_{2})\in R$; the relation $\sim$, called _bisimilarity_ (or
_bisimulation equivalence_), is thus the union of all bisimulations. It is
easy to check that bisimilarity is an equivalence (a reflexive, symmetric, and
transitive relation), and that it is itself a bisimulation. Hence the relation
$\sim$ is the largest bisimulation (related to our fixed labeled Petri net
$N=(P,T,W,l)$).
It is also standard to define _stratified bisimilarity relations_ [32]
$\sim_{i}\subseteq{\cal M}(P)\times{\cal M}(P)$, for
$i\in\mathord{\mathbb{N}}$:
* •
$\sim_{0}\mathop{=}{\cal M}(P)\times{\cal M}(P)$ (hence $M_{1}\sim_{0}M_{2}$
for all $M_{1},M_{2}\in{\cal M}(P)$);
* •
$\sim_{i+1}$ is the largest symmetric relation that satisfies the transfer
property w.r.t. $\sim_{i}$ (hence $M_{1}\sim_{i+1}M_{2}$ iff for each
$M_{1}\stackrel{{\scriptstyle t}}{{\to}}M_{1}^{\prime}$ there is
$M_{2}\stackrel{{\scriptstyle u}}{{\to}}M_{2}^{\prime}$ where $l(u)=l(t)$ and
$M_{1}^{\prime}\sim_{i}M_{2}^{\prime}$ and for each
$M_{2}\stackrel{{\scriptstyle t}}{{\to}}M_{2}^{\prime}$ there is
$M_{1}\stackrel{{\scriptstyle u}}{{\to}}M_{1}^{\prime}$ where $l(u)=l(t)$ and
$M_{1}^{\prime}\sim_{i}M_{2}^{\prime}$).
It is easy to note that $\sim_{i}$ are equivalences (for all
$i\in\mathord{\mathbb{N}}$),
$\sim_{0}\mathop{\supseteq}\sim_{1}\mathop{\supseteq}\sim_{2}\mathop{\supseteq}\cdots$,
and $\sim\mathop{=}\bigcap_{i\in\mathord{\mathbb{N}}}\sim_{i}$; hence
$M_{1}\sim M_{2}$ iff $M_{1}\sim_{i}M_{2}$ for all $i\in\mathord{\mathbb{N}}$
(since labeled Petri nets generate image-finite labeled transition systems
[32]).
While bisimilarity is undecidable for labeled Petri nets [4], we note that it
is decidable for so called _communication-free labeled Petri nets_ [6] (also
know as (1,P)-systems [5], or Basic Parallel Processes [33]). The
communication-free Petri nets $(P,T,W)$ satisfy $|{\rm
Phys.~{}Rev.~{}E}{t}|\leq 1$ for all $t\in T$, and it is thus easy to observe
that in this case bisimulation equivalence is a congruence w.r.t. the marking
addition. For our aims it is useful to look at congruences on ${\cal M}(P)$
for finite sets $P$ in general; we do this in the next paragraph, using rather
symbols $r,s$ instead of $M$ for elements of ${\cal M}(P)$, to stress that it
is not important here whether or not $P$ is the set of places of a Petri net.
#### Congruences on ${\cal M}(P)$.
Given a finite set $P$ (which can be the set of places of a Petri net), an
equivalence relation $\rho$ on ${\cal M}(P)$ is a _congruence_ if
$r_{1}\mathop{\rho}r_{2}$ implies $(r_{1}+s)\mathop{\rho}\,(r_{2}+s)$ for all
$s\in{\cal M}(P)$.
(Hence $r_{1}\mathop{\rho}r_{2}$ and
$r^{\prime}_{1}\mathop{\rho}r^{\prime}_{2}$ implies
$(r_{1}+r^{\prime}_{1})\mathop{\rho}\,(r_{2}+r^{\prime}_{1})$ and
$(r^{\prime}_{1}+r_{2})\mathop{\rho}\,(r^{\prime}_{2}+r_{2})$, and thus also
$(r_{1}+r^{\prime}_{1})\mathop{\rho}\,(r_{2}+r^{\prime}_{2})$.)
An important known fact is captured by the next proposition, which says that
each congruence on ${\cal M}(P)$ has a finite basis (by which the congruence
is generated); this was an important ingredient in the decidability proof in
[6]. We can refer, e.g., to [34, 35], but we provide a self-contained proof
for convenience.
###### Proposition 2.1
Each congruence $\rho$ on ${\cal M}(P)$, where $P$ is a finite set, is
generated by a finite subset of $\rho$, in particular by the set
$\rho_{\textsc{b}}$ (called a _basis of_ $\rho$) consisting of the minimal
elements of the set $\\{(r,s)\in\rho\mid r\neq s\\}$ w.r.t. the partial order
$\leq$, where we put $(r,s)\leq(r^{\prime},s^{\prime})$ if $r\leq r^{\prime}$
and $s\leq s^{\prime}$. (Finiteness of $\rho_{\textsc{b}}$ follows by
Dickson’s lemma.)
###### Proof 2.2
For the sake of contradiction, we assume that the least congruence
$\rho^{\prime}$ containing $\rho_{\textsc{b}}$ is a proper subset of $\rho$.
We choose a pair $(r_{0},s_{0})\in\rho\smallsetminus\rho^{\prime}$ such that
$r_{0}+s_{0}$ is the least in the set
$\\{r+s\mid(r,s)\in\rho\smallsetminus\rho^{\prime}\\}$ w.r.t. the cardinality-
lexicographic order $\sqsubseteq$ on ${\cal M}(P)$; we note that the pair
$(s_{0},r_{0})$ also has this property. Since
$(r_{0},s_{0})\in\rho\smallsetminus\rho^{\prime}$, we have $r_{0}\neq s_{0}$
and $(r,s)\leq(r_{0},s_{0})$ for some
$(r,s)\in\rho_{\textsc{b}}\subseteq\rho^{\prime}$; w.l.o.g. we assume
$r\sqsubseteq s$ (since otherwise we would consider the pairs
$(s,r)\leq(s_{0},r_{0})$, where $(s,r)\in\rho^{\prime}$ and
$(s_{0},r_{0})\in\rho\smallsetminus\rho^{\prime}$).
We derive $(r+(s_{0}-s),s+(s_{0}-s))\in\rho^{\prime}$, i.e.,
$(r+(s_{0}-s),s_{0})\in\rho^{\prime}\subseteq\rho$; since
$(r_{0},s_{0})\in\rho$, we deduce $(r_{0},r+(s_{0}-s))\in\rho$. Since
$r\sqsubseteq s$ and $r\neq s$, we have either $|r|<|s|$, or $|r|=|s|$ and
$r<_{\textsc{lex}}s$; this entails that either
$|r_{0}+(r+(s_{0}-s))|<|r_{0}+s_{0}|$, or
$|r_{0}+(r+(s_{0}-s))|=|r_{0}+s_{0}|$ and
$r_{0}+(r+(s_{0}-s))<_{\textsc{lex}}r_{0}+s_{0}$. Due to our choice of
$(r_{0},s_{0})$ we deduce that $(r_{0},r+(s_{0}-s))\in\rho^{\prime}$; together
with the above fact $(r+(s_{0}-s),s_{0}))\in\rho^{\prime}$ this entails
$(r_{0},s_{0})\in\rho^{\prime}$ – a contradiction.
Remark.
We might also note that the above defined basis $\rho_{\textsc{b}}$ could be
made smaller by including just one of symmetric pairs $(r,s)$ and $(s,r)$.
We note that bisimilarity, i.e. the equivalence $\sim$, in labeled Petri nets
is not a congruence in general (unlike the case of communication-free labeled
Petri nets); we can use the simple example in Figure 2, where
$\\{X\\}\sim\\{Y\\}$, but $(\\{X\\}+\\{X\\})\not\sim(\\{Y\\}+\\{X\\})$ (since
the firing $\\{X,X\\}\stackrel{{\scriptstyle t}}{{\to}}\emptyset$, where $t$
is labeled with $b$, has no imitating firing from $\\{X,Y\\}$, where no
$b$-labeled transition is enabled).
Instead of looking at subclasses where $\sim$ is a congruence, we will look at
natural equivalences refining $\sim$ that are congruences for the whole class
of labeled Petri nets. The finite-basis result (Proposition 2.1) might give
some hope regarding the decidability of such congruences.
## 3 Resource similarity and resource bisimilarity
In Section 3.1 we look at the largest congruence that refines bisimilarity
(the equivalence $\sim$) in labeled Petri nets. This relation is known as
resource similarity (denoted here by $\approx$), and it is also known to be
undecidable (which entails that its finite basis is not effectively
computable). In Section 3.2 we then recall another congruence, known as
resource bisimilarity (denoted here by $\simeq$). In fact, it is the largest
congruence refining $\sim$ that is a bisimulation. In Section 4 we show the
decidability of resource bisimilarity (thus answering a question that was left
open in the literature); nevertheless, the effective computability of its
finite basis is not established by this result. (We add further comments in
Section 5.)
10 centShopBought5 cent$b$$b$$b$$t_{1}$$t_{2}$$t_{3}$ Figure 1: Buying goods
for 20 cents
### 3.1 Resource similarity
The relation of _resource similarity_ , introduced in [26] for labeled Petri
nets, is a congruent strengthening of bisimulation equivalence. Intuitively, a
resource in a Petri net can be viewed as a part of its marking; formally it is
also a multiset of places, hence its definition does not differ from the
definition of a marking. The notions “markings” and “resources” are viewed as
different only due to their different interpretation in particular contexts.
Resources are parts of markings that may or may not provide this or that kind
of net behavior; e. g., in the Petri net example in Fig. 1 from [30], two ten-
cent coins form a resource — enough to buy an item of goods.
Two resources are viewed as similar for a given labeled Petri net if replacing
one of them by another in any marking (i.e., in any context) does not change
the observable behavior of the net. In this sense, resource similarity is a
congruence that preserves visible behaviors up to bisimilarity. Now we capture
this description formally. We use symbols $r,s,w$ for elements of ${\cal
M}(P)$ (rather than $M$) to stress our “resource motivation” here.
###### Definition 3.1
Let $N=(P,T,W,l)$ be a labeled Petri net. A _resource_ $r$ in $N$ is a
multiset over the set $P$ of places; hence $r\in{\cal M}(P)$. Two resources
$r$ and $s$ in $N$ are called _similar_ , which is denoted by $r\approx s$, if
for all $w\in{\cal M}(P)$ we have $(r+w)\sim(s+w)$, where $\sim$ is
bisimulation equivalence. The relation $\approx$ is called _resource
similarity_.
###### Proposition 3.2
For any labeled Petri net $N=(P,T,W,l)$, the relation $\approx$ (resource
similarity) is the largest congruence included in $\sim$ (bisimilarity).
###### Proof 3.3
It is straightforward to check that $\approx$ is an equivalence relation
(reflexive, symmetric, and transitive), since $\sim$ is an equivalence. (In
particular, if $r_{1}\approx r_{2}$ and $r_{2}\approx r_{3}$, then for each
$w\in{\cal M}(P)$ we have $(r_{1}+w)\sim(r_{2}+w)$ and
$(r_{2}+w)\sim(r_{3}+w)$, and thus $(r_{1}+w)\sim(r_{3}+w)$; hence
$r_{1}\approx r_{3}$.) Moreover, $\approx$ is a congruence w.r.t. multiset
addition, i.e., $r\approx s$ implies $(r+w)\approx(s+w)$; indeed, for any
$w^{\prime}$ we have $((r+w)+w^{\prime})\sim((s+w)+w^{\prime})$, since
$r\approx s$ entails $(r+(w+w^{\prime}))\sim(s+(w+w^{\prime}))$. We trivially
have $\approx\mathop{\subseteq}\sim$. Moreover, each congruence
$\rho\mathop{\subseteq}\sim$ satisfies that $r\mathop{\rho}s$ implies
$(r+w)\mathop{\rho}\,(s+w)$, and thus $(r+w)\sim(s+w)$, for all $w\in{\cal
M}(P)$; this entails $\rho\mathop{\subseteq}\approx$.
Since resource similarity is a congruence on ${\cal M}(P)$, it has a finite
basis by Proposition 2.1. However, the place fusion (studied in [21, 22]) is a
special case of resource similarity for resources of capacity one. Place
fusion has been proven to be undecidable, and this implies undecidability of
resource similarity; it also entails that the finite basis of $\approx$ is not
effectively computable.
Remark.
The problem of deciding non-bisimilarity, i.e. the relation $\not\sim$, is
semidecidable (for labeled Petri nets); this follows from the fact that the
equivalences $\sim_{i}$ are decidable for all $i\in\mathord{\mathbb{N}}$, and
$\sim\mathop{=}\bigcap_{i\in\mathord{\mathbb{N}}}\sim_{i}$. On the other hand,
the halting problem for Minsky machines can be reduced to deciding $\not\sim$,
which entails that $\sim$ is not semidecidable. Since $r\not\approx s$ means
that $(r+w)\not\sim(s+w)$ for some $w$, we easily derive the semidecidability
of $\not\approx$. Moreover, the mentioned reduction of the halting problem to
deciding $\not\sim$ can be adjusted so that it reduces to deciding
$\not\approx$. This entails that $\approx$ is not semidecidable, and thus the
finite basis of $\approx$ cannot be effectively computable.
We note that resource similarity is not a bisimulation in general; the
relation $\approx$ may not have the transfer property. This is exemplified by
the labeled Petri net in Figure 3: we shall later show that
$\\{X_{1}\\}\approx\\{Y_{1}\\}$, but the firing
$\\{X_{1}\\}\stackrel{{\scriptstyle t_{4}}}{{\to}}\\{X_{3}\\}$ has no
“imitating” firing from $\\{Y_{1}\\}$; indeed, both candidates
$\\{Y_{1}\\}\stackrel{{\scriptstyle u_{1}}}{{\to}}\emptyset$ and
$\\{Y_{1}\\}\stackrel{{\scriptstyle u_{2}}}{{\to}}\\{Y_{2}\\}$ fail, since
$\\{X_{3}\\}\not\approx\emptyset$ (because
$(\\{X_{3}\\}+\\{Z\\})\not\sim_{1}(\emptyset+\\{Z\\})$) and
$\\{X_{3}\\}\not\approx\\{Y_{2}\\}$ (because
$(\\{X_{3}\\}+\emptyset)\not\sim_{1}(\\{Y_{2}\\}+\emptyset)$).
### 3.2 Resource bisimilarity
In view of the above facts, that the largest congruence $\approx$ refining
$\sim$ is undecidable and is not a bisimulation, it is natural to look at the
largest congruence $\simeq$ refining $\sim$ that _is_ a bisimulation (whence
we have $\simeq\mathop{\subseteq}\approx\mathop{\subseteq}\sim$). In fact, the
relation $\simeq$, named _resource bisimilarity_ , was already defined in
[26], though technically slightly differently than we do this below. We also
use a new term, namely _the resource transfer property_ ; this property is
stronger than _the transfer property_ defined for relations on ${\cal M}(P)$
in Section 2. An analogous notion was called _the weak transfer property_ in
[26], due to a different viewpoint (related to the variant of the transfer
property for place bisimulation defined in [19]).
###### Definition 3.4
Let $N=(P,T,W,l)$ be a labeled Petri net. We say that a _relation_
$R\subseteq{\cal M}(P)\times{\cal M}(P)$ _satisfies_ the _resource transfer
property w.r.t. a relation_ $R^{\prime}\subseteq{\cal M}(P)\times{\cal M}(P)$
if for each pair $(r,s)\in R$ and for each firing $(r+({\rm
Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle t}}{{\to}}r^{\prime}$ there
exists an (“imitating”) firing $(s+({\rm
Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle u}}{{\to}}s^{\prime}$ such
that $l(u)=l(t)$ and $(r^{\prime},s^{\prime})\in R^{\prime}$. We can
illustrate this property by the following diagram.
$r\qquad\quad R\qquad\quad s$ $\qquad r+({\rm
Phys.~{}Rev.~{}E}{t}-r)\qquad\qquad s+({\rm Phys.~{}Rev.~{}E}{t}-r)$
$\qquad\qquad\qquad\quad\quad\downarrow
t\qquad\qquad\qquad\downarrow(\exists)u:\ l(u)=l(t)$ $r^{\prime}\qquad\quad
R^{\prime}\qquad\quad s^{\prime}$
(By our multiset definitions, $r+({\rm Phys.~{}Rev.~{}E}{t}-r)=r\cup{\rm
Phys.~{}Rev.~{}E}{t}$. We also note that ${\rm Phys.~{}Rev.~{}E}{t}\leq r$
entails that ${\rm Phys.~{}Rev.~{}E}{t}-r=\emptyset$, hence $r+({\rm
Phys.~{}Rev.~{}E}{t}-r)=r$ and $s+({\rm Phys.~{}Rev.~{}E}{t}-r)=s$; therefore
the resource transfer property implies the transfer property.)
By saying that $R$ _has the resource transfer property_ we mean that $R$
satisfies the resource transfer property w.r.t. itself (hence $R^{\prime}=R$
in the diagram).
A relation $R\subseteq{\cal M}(P)\times{\cal M}(P)$ is a _resource
bisimulation_ if $R$ has the resource transfer property and also $R^{-1}$ has
the resource transfer property. Resources (or markings) $r,s\in{\cal M}(P)$
are _resource bisimilar_ , written $r\simeq s$, if there exists a resource
bisimulation $R$ such that $(r,s)\in R$; the relation $\simeq$, called
_resource bisimilarity_ , is thus the union of all resource bisimulations.
###### Example 3.5
In the net in Figure 1 we can present any multiset $r$ over the set of places
by the vector
$(r(\texttt{10cent}),r(\texttt{Shop}),r(\texttt{5cent}),r(\texttt{Bought}))$.
We can show that $r_{0}\simeq s_{0}$ where $r_{0}=(1,0,0,0)$ and
$s_{0}=(0,0,2,0)$, by verifying that the relation
$\\{\big{(}(1,0,0,k),(0,0,2,k)\big{)}\mid k\geq
0)\\}\cup\\{\big{(}(0,0,0,k),(0,0,0,k)\big{)}\mid k\geq 1\\}$ is a resource
bisimulation.
E.g., ${\rm Phys.~{}Rev.~{}E}{t_{1}}-r_{0}=(1,1,0,0)$, and we have
$r_{0}+({\rm Phys.~{}Rev.~{}E}{t_{1}}-r_{0})=(2,1,0,0)\stackrel{{\scriptstyle
t_{1}}}{{\to}}(0,0,0,1)$. This firing can be imitated from $s_{0}+({\rm
Phys.~{}Rev.~{}E}{t_{1}}-r_{0})=(1,1,2,0)$ by the firing
$(1,1,2,0)\stackrel{{\scriptstyle t_{2}}}{{\to}}(0,0,0,1)$ (since both $t_{1}$
and $t_{2}$ have the same label $b$). It is a routine to finish this
verification.
###### Proposition 3.6
For any labeled Petri net $N=(P,T,W,l)$, the relation $\simeq$ (resource
bisimilarity) is the largest congruence included in $\sim$ that is a
bisimulation; hence we have
$\simeq\mathop{\subseteq}\approx\mathop{\subseteq}\sim$ .
###### Proof 3.7
Since the relation $\simeq$ is the union of all resource bisimulations, i.e.
$\simeq\mathop{=}\bigcup\,\\{\,R\mathop{\subseteq}{\cal M}(P)\times{\cal
M}(P)\mid R$ and $R^{-1}$ have the resource transfer property$\,\\}$,
it is obvious that also $\simeq$ and $\simeq^{-1}$ have the resource transfer
property; hence $\simeq$ is the largest resource bisimulation (related to the
considered labeled Petri net $N=(P,T,W,l)$). Since the resource transfer
property is stronger than the transfer property (if $R$ has the resource
transfer property, then $R$ also has the transfer property), each resource
bisimulation is a (standard) bisimulation; hence $\simeq$ is a bisimulation,
and we have $\simeq\mathop{\subseteq}\sim$.
The reflexivity and symmetry of $\simeq$ is straightforward; the next two
points show that $\simeq$ is a congruence:
1. 1.
We prove that $r\simeq s$ implies $(r+w)\simeq(s+w)$ by showing that the set
$R=\\{(r+w,s+w)\mid r,s,w\in{\cal M}(P),r\simeq s\\}$
is a resource bisimulation (hence $R\mathop{=}\simeq$). To this aim we fix
some $(r+w,s+w)\in R$, where $r\simeq s$, and consider a firing $(r+w+({\rm
Phys.~{}Rev.~{}E}{t}-(r+w)))\stackrel{{\scriptstyle t}}{{\to}}r^{\prime}$; we
look for an imitating firing from $s+w+({\rm Phys.~{}Rev.~{}E}{t}-(r+w))$. We
note that $w+({\rm Phys.~{}Rev.~{}E}{t}-(r+w))=({\rm
Phys.~{}Rev.~{}E}{t}-r)+w^{\prime}$ for a (maybe nonempty) multiset
$w^{\prime}\in{\cal M}(P)$; we thus have $(r+({\rm
Phys.~{}Rev.~{}E}{t}-r)+w^{\prime})\stackrel{{\scriptstyle
t}}{{\to}}r^{\prime}$, and look for an imitating firing from $s+({\rm
Phys.~{}Rev.~{}E}{t}-r)+w^{\prime}$. We have $(r+({\rm
Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle t}}{{\to}}r^{\prime\prime}$,
and by $r\simeq s$ we deduce that there is an imitating firing $(s+({\rm
Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle
t^{\prime}}}{{\to}}s^{\prime\prime}$, where $r^{\prime\prime}\simeq
s^{\prime\prime}$. Since $(r+({\rm
Phys.~{}Rev.~{}E}{t}-r)+w^{\prime})\stackrel{{\scriptstyle
t}}{{\to}}r^{\prime\prime}+w^{\prime}$ (for the above $r^{\prime}$ we have
$r^{\prime}=r^{\prime\prime}+w^{\prime}$), the firing $(s+({\rm
Phys.~{}Rev.~{}E}{t}-r)+w^{\prime})\stackrel{{\scriptstyle
t^{\prime}}}{{\to}}s^{\prime\prime}+w^{\prime}$ satisfies our need:
$(r^{\prime\prime}+w^{\prime},s^{\prime\prime}+w^{\prime})\in R$.
2. 2.
Transitivity of $\simeq$ (i.e., $r_{1}\simeq r_{2}$ and $r_{2}\simeq r_{3}$
entails $r_{1}\simeq r_{3}$) is demonstrated by showing that the relation
$R=\\{(r_{1},r_{3})\mid r_{1}\simeq r_{2}$ and $r_{2}\simeq r_{3}$ for some
$r_{2}\in{\cal M}(P)\\}$
is a resource bisimulation. Let us consider a pair $(r_{1},r_{3})\in R$, where
$r_{1}\simeq r_{2}$ and $r_{2}\simeq r_{3}$, and a firing $(r_{1}+({\rm
Phys.~{}Rev.~{}E}{t_{1}}-r_{1}))\stackrel{{\scriptstyle
t_{1}}}{{\to}}r^{\prime}_{1}$. Since $r_{1}\simeq r_{2}$, there is an
imitating firing $(r_{2}+({\rm
Phys.~{}Rev.~{}E}{t_{1}}-r_{1}))\stackrel{{\scriptstyle
t_{2}}}{{\to}}r^{\prime}_{2}$, where $r^{\prime}_{1}\simeq r^{\prime}_{2}$. By
Point $1$, $r_{2}\simeq r_{3}$ entails $(r_{2}+({\rm
Phys.~{}Rev.~{}E}{t_{1}}-r_{1}))\simeq(r_{3}+({\rm
Phys.~{}Rev.~{}E}{t_{1}}-r_{1}))$, hence $(r_{2}+({\rm
Phys.~{}Rev.~{}E}{t_{1}}-r_{1}))\stackrel{{\scriptstyle
t_{2}}}{{\to}}r^{\prime}_{2}$ (where ${\rm
Phys.~{}Rev.~{}E}{t_{2}}-(r_{2}+({\rm
Phys.~{}Rev.~{}E}{t_{1}}-r_{1}))=\emptyset$) has an imitating firing
$(r_{3}+({\rm Phys.~{}Rev.~{}E}{t_{1}}-r_{1}))\stackrel{{\scriptstyle
t_{3}}}{{\to}}r^{\prime}_{3}$, where $r^{\prime}_{2}\simeq r^{\prime}_{3}$.
Since $(r^{\prime}_{1},r^{\prime}_{3})\in R$, we are done.
Finally we note that each congruence $\rho$ on ${\cal M}(P)$ that refines
$\sim$ (i.e., $\rho\mathop{\subseteq}\sim$) and is a (standard) bisimulation
is also a resource bisimulation (since $r\mathop{\rho}s$ implies $(r+({\rm
Phys.~{}Rev.~{}E}{t}-r))\mathop{\rho}\,(s+({\rm Phys.~{}Rev.~{}E}{t}-r))$);
hence we have $\rho\mathop{\subseteq}\simeq$ for all such congruences.
Similarly as for (standard) bisimilarity, we define the stratified versions
for resource bisimilarity; we also observe their properties that underpin our
algorithm in Section 4.
###### Definition 3.8 (of equivalences $\simeq_{i}$ and equivalence-levels
$\textsc{EqLev}(r,s)$)
Assuming a labeled Petri net $N=(P,T,W,l)$, we define the relations
$\simeq_{i}$, $i\in\mathord{\mathbb{N}}$, as follows:
* •
$\simeq_{0}\mathop{=}{\cal M}(P)\times{\cal M}(P)$;
* •
$\simeq_{i+1}$ is the largest symmetric relation that satisfies the resource
transfer property w.r.t. $\simeq_{i}$.
For $r,s\in{\cal M}(P)$, by $\textsc{EqLev}(r,s)$ we denote the largest $i$
such that $r\simeq_{i}s$, stipulating $\textsc{EqLev}(r,s)=\omega$ (where
$i<\omega$ for all $i\in\mathord{\mathbb{N}}$) if $r\simeq_{i}s$ for all
$i\in\mathord{\mathbb{N}}$.
###### Proposition 3.9
1. 1.
For all $i\in\mathord{\mathbb{N}}$, $\simeq_{i}$ are congruences on ${\cal
M}(P)$.
2. 2.
$\simeq_{0}\mathop{\supseteq}\simeq_{1}\mathop{\supseteq}\simeq_{2}\mathop{\supseteq}\cdots$,
and $\simeq\mathop{=}\bigcap_{i\in\mathord{\mathbb{N}}}\simeq_{i}$.
3. 3.
If $\textsc{EqLev}(s,s^{\prime})>\textsc{EqLev}(r,s)$, then
$\textsc{EqLev}(r,s^{\prime})=\textsc{EqLev}(r,s)$.
###### Proof 3.10
1) We can proceed by induction on $i$, and analogously as in the proof of
Proposition 3.6.
2) The fact $\simeq_{i}\mathop{\supseteq}\simeq_{i+1}$ and the inclusion
$\simeq\mathop{\subseteq}\,(\,\bigcap_{i\in\mathord{\mathbb{N}}}\simeq_{i})$
are obvious. Now we observe that the relation
$\bigcap_{i\in\mathord{\mathbb{N}}}\simeq_{i}$ is a resource bisimulation
(which entails
$(\bigcap_{i\in\mathord{\mathbb{N}}}\simeq_{i})\mathop{\subseteq}\simeq$);
here we use image-finiteness, i.e., the fact that each firing has only
finitely many candidates for imitating firings (with the same action-label).
Hence if $(r,s)\in\bigcap_{i\in\mathord{\mathbb{N}}}\simeq_{i}$, and $(r+({\rm
Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle t}}{{\to}}r^{\prime}$, then
there is an imitating firing $(s+({\rm
Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle t^{\prime}}}{{\to}}s^{\prime}$
(with $l(t^{\prime})=l(t)$) such that $r^{\prime}\simeq_{i}s^{\prime}$ for
infinitely many $i\in\mathord{\mathbb{N}}$; but this entails that
$(r^{\prime},s^{\prime})\in\bigcap_{i\in\mathord{\mathbb{N}}}\simeq_{i}$.
3) Let $\textsc{EqLev}(s,s^{\prime})>\textsc{EqLev}(r,s)=i$; hence
$r\simeq_{i}s$, $r\not\simeq_{i+1}s$, and $s\simeq_{i+1}s^{\prime}$ (and
$s\simeq_{i}s^{\prime}$). Since $\simeq_{i}$ and $\simeq_{i+1}$ are
equivalences, we have $r\simeq_{i}s^{\prime}$ and
$r\not\simeq_{i+1}s^{\prime}$ (since $r\simeq_{i+1}s^{\prime}$ would entail
$r\simeq_{i+1}s$).
The next theorem summarizes the previously derived facts on the equivalences
$\simeq$ (resource bisimilarity), $\approx$ (resource similarity), $\sim$
(bisimilarity), and adds that these equivalences really differ in general.
###### Theorem 3.11 (Mutual relations of resource bisimilarity, resource
similarity, and bisimilarity)
1. 1.
For each labeled Petri net we have
$\simeq\mathop{\subseteq}\approx\mathop{\subseteq}\sim$; moreover, $\approx$
is the largest congruence included in $\sim$, and $\simeq$ is the largest
congruence included in $\sim$ that is a bisimulation.
2. 2.
There exists a labeled Petri net for which $\approx\mathop{\neq}\sim$.
3. 3.
There exists a labeled Petri net for which $\simeq\mathop{\neq}\approx$.
4. 4.
For each labeled communication-free Petri net we have
$\simeq\mathop{=}\approx\mathop{=}\sim$.
###### Proof 3.12
Point $1$ summarizes Propositions 3.2 and 3.6.
Point $4$ follows by the fact that $\sim$ is a congruence for labeled
communication-free Petri nets.
$a$$a$$b$$X$$Y$
Figure 2: Resource similarity does not coincide with bisimilarity:
$\\{X\\}\sim\\{Y\\}$, but $\\{X\\}\not\approx\\{Y\\}$.
Point $2$ is shown by the example depicted in Fig. 2. We have
$\\{X\\}\sim\\{Y\\}$, since the relation
$\\{(\\{X\\},\\{Y\\}),(\emptyset,\emptyset)\\}$ is a bisimulation; but
$\\{X\\}\not\approx\\{Y\\}$, since
$(\\{X\\}+\\{X\\})\not\sim(\\{Y\\}+\\{X\\})$ (we even have
$\\{X,X\\}\not\sim_{1}\\{X,Y\\}$ due to the action $b$ that is enabled just in
one of these multisets).
$b$$b$$a$$a$$a$$X_{3}$$X_{2}$$X_{1}$$Z$$t_{1}$$t_{3}$$t_{5}$$t_{2}$$t_{4}$$b$$a$$a$$Y_{2}$$Y_{1}$$u_{1}$$u_{3}$$u_{2}$
Figure 3: Resource similarity does not coincide with resource bisimilarity:
$\\{X_{1}\\}\approx\\{Y_{1}\\}$, but $\\{X_{1}\\}\not\simeq\\{Y_{1}\\}$.
Point $3$ is shown by the example depicted in Fig. 3. At the end of Section
3.1 we have, in fact, already shown that $\\{X_{1}\\}\not\simeq\\{Y_{1}\\}$:
since the pair $(\\{X_{1}\\},\\{Y_{1}\\})$ (viewed as a relation with a single
element) does not satisfy the transfer property w.r.t. $\approx$, it surely
does not satisfy the resource transfer property w.r.t. $\simeq$ (since
$\simeq\mathop{\subseteq}\approx$, and the resource transfer property is
stronger than the transfer property). It thus remains to show that
$\\{X_{1}\\}\approx\\{Y_{1}\\}$, i.e. that
$(\\{X_{1}\\}+w)\sim(\\{Y_{1}\\}+w)$ for each multiset $w$ over the set
$P=\\{X_{1},X_{2},X_{3},Y_{1},Y_{2},Z\\}$.
We show this by verifying that the following relation $R$ on ${\cal M}(P)$ is
a bisimulation; $R$ contains the pairs $(r,s)$ for which one of the following
conditions is satisfied (where the condition $1$ guarantees that
$(\\{X_{1}\\}+w,\\{Y_{1}\\}+w)\in R$):
1. 1.
$r(X_{1})-s(X_{1})=1$, $s(Y_{1})-r(Y_{1})=1$, and $r,s$ coincide on the set
$\\{X_{2},X_{3},Y_{2},Z\\}$;
2. 2.
$r(X_{1})=s(X_{1})$, $r(Y_{1})=s(Y_{1})$, and
$r(X_{2})+\min\\{r(X_{3}),r(Z)\\}+r(Y_{2})=s(X_{2})+\min\\{s(X_{3}),s(Z)\\}+s(Y_{2})$.
It is a routine to check that both $R$ and $R^{-1}$ have the transfer
property. The only interesting cases are the firings from one side of
$(r,s)\in R$ that cannot be matched (“imitated”) by firing the same transition
from the other side. Such cases are described in what follows:
* •
$(r,s)$ satisfies $2$:
Here any $b$-transition (i.e., $t_{3},t_{5},u_{3}$) fired from $r$ (or from
$s$) can be matched by firing any $b$-transition from $s$ (or from $r$,
respectively); the resulting pair $(r^{\prime},s^{\prime})$ obviously
satisfies $2$ as well.
For the firing $r\stackrel{{\scriptstyle t_{4}}}{{\to}}r^{\prime}$ ($t_{4}$ is
an $a$-transition) we have
* –
either $\min\\{r^{\prime}(X_{3}),r^{\prime}(Z)\\}=\min\\{r(X_{3}),r(Z)\\}+1$
(when $r(X_{3})<r(Z)$), in which case the firing $r\stackrel{{\scriptstyle
t_{4}}}{{\to}}r^{\prime}$ is matched by $s\stackrel{{\scriptstyle
t_{2}}}{{\to}}s^{\prime}$,
* –
or $\min\\{r^{\prime}(X_{3}),r^{\prime}(Z)\\}=\min\\{r(X_{3}),r(Z))\\}$ (when
$r(X_{3})\geq r(Z)$), in which case the firing $r\stackrel{{\scriptstyle
t_{4}}}{{\to}}r^{\prime}$ is matched by $s\stackrel{{\scriptstyle
t_{1}}}{{\to}}s^{\prime}$.
In both cases $(r^{\prime},s^{\prime})$ satisfies $2$. The firing
$s\stackrel{{\scriptstyle t_{4}}}{{\to}}s^{\prime}$ is matched analogously.
* •
$(r,s)$ satisfies $1$ and $r(Y_{1})=0$ or $s(X_{1})=0$:
If $r(Y_{1})=0$ (hence $s(Y_{1})=1$), then the firing
$s\stackrel{{\scriptstyle u_{1}}}{{\to}}s^{\prime}$ (or
$s\stackrel{{\scriptstyle u_{2}}}{{\to}}s^{\prime}$) is matched by
$r\stackrel{{\scriptstyle t_{1}}}{{\to}}r^{\prime}$ (or
$r\stackrel{{\scriptstyle t_{2}}}{{\to}}r^{\prime}$, respectively); in both
cases $(r^{\prime},s^{\prime})$ satisfies $2$.
If $r(X_{1})=1$ (hence $s(X_{1})=0$), then the firing
$r\stackrel{{\scriptstyle t_{1}}}{{\to}}r^{\prime}$ (or
$r\stackrel{{\scriptstyle t_{2}}}{{\to}}r^{\prime}$) is matched by
$s\stackrel{{\scriptstyle u_{1}}}{{\to}}s^{\prime}$ (or
$s\stackrel{{\scriptstyle u_{2}}}{{\to}}s^{\prime}$, respectively).
The firing $r\stackrel{{\scriptstyle t_{4}}}{{\to}}r^{\prime}$ is matched
similarly as discussed above:
* –
if $\min\\{r^{\prime}(X_{3}),r^{\prime}(Z)\\}=\min\\{r(X_{3}),r(Z)\\}+1$, then
$r\stackrel{{\scriptstyle t_{4}}}{{\to}}r^{\prime}$ is matched by
$s\stackrel{{\scriptstyle u_{2}}}{{\to}}s^{\prime}$;
* –
if $\min\\{r^{\prime}(X_{3}),r^{\prime}(Z)\\}=\min\\{r(X_{3}),r(Z)\\}$, then
$r\stackrel{{\scriptstyle t_{4}}}{{\to}}r^{\prime}$ is matched by
$s\stackrel{{\scriptstyle u_{1}}}{{\to}}s^{\prime}$.
In both cases $(r^{\prime},s^{\prime})$ satisfies $2$.
As we already mentioned, bisimilarity (the relation $\sim$) and resource
similarity (the relation $\approx$) are undecidable. In the next section
(Section 4) we show that resource bisimilarity (the relation $\simeq$) is
decidable. Here we still show that the decidability does not immediately
follow from the fact that
$\simeq=\bigcap_{i\in\mathord{\mathbb{N}}}\simeq_{i}$ (where
$\simeq_{0}\mathop{\supseteq}\simeq_{1}\mathop{\supseteq}\simeq_{2}\mathop{\supseteq}\cdots$),
even though all $\simeq_{i}$, as well as $\simeq$, are congruences and thus
have finite bases (by Proposition 2.1). We could even easily derive that
finite bases for $\simeq_{i}$, $i\in\mathord{\mathbb{N}}$, are effectively
computable; nevertheless, it is not immediately clear how they “converge” to a
finite basis for $\sim$. The following simple example aims to illustrate this,
even for the case of a labeled communication-free Petri net (where $\simeq$
coincides with $\sim$).
$a$$b$$a$$b$$X$$Y$$Z$ Figure 4: A labeled communication-free Petri net
###### Example 3.13
Let us consider the labeled Petri net in Fig. 4; it is a communication-free
net, since the preset of each transition is a (multi)set containing a single
place. Here bisimilarity (the relation $\sim$) is a congruence (thus
coinciding with $\simeq$), and it is not difficult to derive that $\sim$ is
the identity relation (we can thus take the empty set as its finite basis).
We now look at finite bases of congruences $\simeq_{i}$. To this aim we
present multisets $r$ over $\\{X,Y,Z\\}$ as the vectors $(r(X),r(Y),r(Z))$.
For $\simeq_{0}$, a basis is the set
$\\{\big{(}(0,0,0),(0,0,1)\big{)},\big{(}(0,0,0),(0,1,0)\big{)},\big{(}(0,0,0),(1,0,0)\big{)}\\}$
(where we do not include the symmetric pairs).
For $\simeq_{1}$, a basis contains elements like
$\big{(}(0,0,1),(0,0,2)\big{)},\big{(}(0,0,1),(0,1,1)\big{)},\big{(}(0,0,1),(1,0,1)\big{)},\big{(}(0,0,1),(1,1,0)\big{)},\dots$
For $\simeq_{2}$, a basis can not contain, e.g.,
$\big{(}(0,0,1),(0,0,2)\big{)}$, but “instead” it contains elements like
$\big{(}(0,0,2),(0,0,3)\big{)}$, $\big{(}(1,1,1),(1,1,2)\big{)}$, $\dots$.
It seems to be a subtle problem in general, to derive a basis for $\simeq$ by
constructing a row of bases for $\simeq_{1}$, $\simeq_{2}$, $\dots$.
Nevertheless, for labeled _communication-free_ Petri nets, a finite basis for
$\sim$ (coinciding with $\simeq$) can be effectively computed; further remarks
about this are in Section 5.
## 4 Algorithm for checking resource bisimilarity
The problem:
Given a labeled Petri net $N=(P,T,W,l)$ and two resources $r_{0},s_{0}\in{\cal
M}(P)$, we need to check whether they are resource bisimilar, i.e. whether
$r_{0}\simeq s_{0}$. We assume that $P\neq\emptyset$ and $T\neq\emptyset$
(otherwise the problem is trivial).
The algorithm ALG we present here to solve this problem uses the well-known
tableau technique (see e. g., [31]), adapted to the resource bisimilarity
relation and its resource transfer property.
Below we give a pseudocode of ALG (with some comments inside $(*$ $*)$). In
its computation, ALG will be also comparing the cardinalities $|r|$, $|s|$ of
multisets $r,s\in{\cal M}(P)$, and in the case $|r|=|s|$ it will use the
_lexicographic order_ $r\leq_{\textsc{lex}}s$. In other words, it will use the
cardinality-lexicographic order $\sqsubseteq$ that was defined in Section 2
for multisets. Besides that, ALG will also use the standard component-wise
order $r\leq s$, extended to the pairs of multisets as in Proposition 2.1: we
put $(r,s)\leq(r^{\prime},s^{\prime})$ if $r\leq r^{\prime}$ and $s\leq
s^{\prime}$.
> Nondeterministic algorithm ALG deciding resource bisimilarity
>
> _Input:_ A labeled Petri net $N=(P,T,W,l)$ and two resources
> $r_{0},s_{0}\in{\cal M}(P)$.
>
> _Output:_ If $r_{0}\simeq s_{0}$, then at least one computation returns YES;
> if $r_{0}\not\simeq s_{0}$, then all computations return NO.
>
> Procedure:
>
> $(*$ It stepwise constructs a tree (a _proof tree_ , also called a
> _tableau_), which is stored in a “program variable” CT (Current Tree); its
> nodes are labeled with pairs of resources. A node of CT is called an
> _identity-node_ if its label is of the form $(r,r)$; each such node will be
> a _successful leaf_ of the constructed tree. The outcome YES will be
> returned iff a _successful tree_ , i.e. a tree whose all leaves are
> successful, will be constructed. $*)$
>
> 1. 1.
>
> Create the root labeled with the input pair $(r_{0},s_{0})$; this node
> constitutes the initial current tree, hence the initial value of CT.
>
> 2. 2.
>
> while there is a non-identity leaf in CT do
>
> begin
>
> In CT choose a leaf n labeled with $(r,s)$ where $r\neq s$, and process it
> as follows:
>
> * •
>
> if the rule REDUCE $(*$ described below $*)$ is applicable to n
> then apply it; by doing this, n gets exactly one child-node
> $\bar{\textsc{n}}$
>
> $(*$ $\bar{\textsc{n}}$ becomes a new leaf in (the extended) CT, and is
> labeled with some $(\bar{r},\bar{s})$ where $\bar{r}+\bar{s}$ is strictly
> smaller than $r+s$ in the cardinality-lexicographic order $\sqsubseteq$
> $*)$;
>
> * •
>
> otherwise $(*$ when REDUCE is not applicable to n $*)$, apply EXPAND to n
>
> $(*$ which fails if $r\not\simeq_{1}s$, in which case the computation
> returns NO, and otherwise creates (at most) $2\cdot|T|$ children of n $*)$.
>
> end
>
> RETURN YES $(*$ here CT is successful, since all leaves are identity-nodes
> $*)$.
>
>
To formulate the rule EXPAND, we introduce the following definitions,
referring to the underlying labeled Petri net $N=(P,T,W,l)$. For $t\in T$, and
$r,s,r^{\prime},s^{\prime}\in{\cal M}(P)$, the pair $(r^{\prime},s^{\prime})$
is a _$t$ -child of_ $(r,s)$ if there is $u\in T$ such that $l(u)=l(t)$,
$(r+({\rm Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle
t}}{{\to}}r^{\prime}$, and $(s+({\rm
Phys.~{}Rev.~{}E}{t}-r))\stackrel{{\scriptstyle u}}{{\to}}s^{\prime}$. (Recall
the diagram in Definition 3.4.) By $\mathord{\textit{next}}_{t}(r,s)$ we
denote the set of all $t$-children of $(r,s)$.
We observe a trivial fact that shows the soundness of the following
description of EXPAND.
###### Claim 1
We have $r\not\simeq_{1}s$ iff $\mathord{\textit{next}}_{t}(r,s)$ or
$\mathord{\textit{next}}_{t}(s,r)$ is empty for some $t\in T$.
> Application of EXPAND to a node n labeled with $(r,s)$ (where $r\neq s$):
>
> if $r\not\simeq_{1}s$ then RETURN NO
>
> $(*$ hence the algorithm ALG invoking EXPAND returns NO in this case $*)$,
>
> otherwise for each $t\in T$ select (nondeterministically) exactly one pair
> of resources from $\mathord{\textit{next}}_{t}(r,s)$ and exactly one pair of
> resources from $\mathord{\textit{next}}_{t}(s,r)$, and create (at most)
> $2\cdot|T|$ children of n whose labels are precisely the pairs selected for
> all $t\in T$.
>
> $(*$ We recall that $T\neq\emptyset$; hence if $r\simeq_{1}s$, then the set
> of children of n is nonempty. $*)$
We note some simple facts regarding EXPAND (that are used later, in the proof
of Theorem 4.3):
###### Claim 2 (Properties of EXPAND)
Let n be a leaf of CT, labeled with $(r,s)$ where $r\neq s$. Then we have:
1. 1.
If $r\simeq s$, then there is at least one application of EXPAND to n such
that each arising child of n is labeled with an equivalent pair, i.e. with
some $(r^{\prime},s^{\prime})$ where $r^{\prime}\simeq s^{\prime}$.
2. 2.
If $\textsc{EqLev}(r,s)=k\in\mathord{\mathbb{N}}_{+}$ (hence $r\simeq_{k}s$,
$r\not\simeq_{k+1}s$, $k\geq 1$), then each application of EXPAND to n gives
rise to at least one child of n that is labeled with some
$(r^{\prime},s^{\prime})$ where $\textsc{EqLev}(r^{\prime},s^{\prime})<k$.
###### Proof 4.1
The claims easily follow from the definitions of relations $\simeq$ and
$\simeq_{i}$.
Now we describe the rule REDUCE, and also note its useful properties.
> Application of REDUCE to a node n labeled with $(r,s)$ (where $r\neq s$), in
> a given current tree CT:
>
> If there a node $\textsc{n}^{\prime}\neq\textsc{n}$ on the path from the
> root to n in CT that is labeled with $(r^{\prime},s^{\prime})$ where
> $(r^{\prime},s^{\prime})\leq(r,s)$, then the rule REDUCE is applicable;
> otherwise it is not applicable.
>
> If the rule is applicable, then $(r^{\prime},s^{\prime})\leq(r,s)$ related
> to one such node $\textsc{n}^{\prime}$ is selected, and n gets precisely one
> child, namely $\bar{\textsc{n}}$ labeled with $(\bar{r},\bar{s})$ where we
> have:
>
> * •
>
> if $|r^{\prime}|<|s^{\prime}|$, or $|r^{\prime}|=|s^{\prime}|$ and
> $r^{\prime}<_{\textsc{lex}}s^{\prime}$, then
> $(\bar{r},\bar{s})=(r,(s-s^{\prime})+r^{\prime})$, and
>
> * •
>
> if $|s^{\prime}|<|r^{\prime}|$, or $|r^{\prime}|=|s^{\prime}|$ and
> $s^{\prime}<_{\textsc{lex}}r^{\prime}$, then
> $(\bar{r},\bar{s})=((r-r^{\prime})+s^{\prime},s)$.
>
>
>
> $(*$ Since identity-nodes are leaves, we have $r^{\prime}\neq s^{\prime}$.
> But we note that the case $(r^{\prime},s^{\prime})=(r,s)$ is not excluded;
> in this case the child $\bar{\textsc{n}}$ is an identity-node, labeled with
> $(r,r)$ or $(s,s)$. $*)$
###### Claim 3 (Properties of REDUCE)
Let us consider a current tree CT and an application of REDUCE to a node n in
CT, labeled with $(r,s)$, where the application is based on a node
$\textsc{n}^{\prime}$ labeled with $(r^{\prime},s^{\prime})$ (where
$(r^{\prime},s^{\prime})\leq(r,s)$); let the resulting child
$\bar{\textsc{n}}$ of n be labeled with $(\bar{r},\bar{s})$. We then have:
1. 1.
$|\bar{r}+\bar{s}|<|r+s|$, or $|\bar{r}+\bar{s}|=|r+s|$ and
$\bar{r}+\bar{s}<_{\textsc{lex}}r+s$;
2. 2.
the edges on the path from $\textsc{n}^{\prime}$ to n could not be created by
applications of REDUCE only (i. e., at least one edge has been created by
applying EXPAND);
3. 3.
if $r^{\prime}\simeq s^{\prime}$ and $r\simeq s$, then $\bar{r}\simeq\bar{s}$;
4. 4.
if $\textsc{EqLev}(r^{\prime},s^{\prime})>\textsc{EqLev}(r,s)$, then
$\textsc{EqLev}(\bar{r},\bar{s})=\textsc{EqLev}(r,s)$.
###### Proof 4.2
1) If $|r^{\prime}|<|s^{\prime}|$, then
$|\bar{s}|=|(s-s^{\prime})+r^{\prime}|<|s|$ (since $s^{\prime}\leq s$); hence
$|\bar{r}+\bar{s}|=|r+\bar{s}|<|r+s|$.
If $r^{\prime}<_{\textsc{lex}}s^{\prime}$, then
$(s-s^{\prime})+r^{\prime}<_{\textsc{lex}}s$ (due to the first component $i$
in which $r^{\prime}$ and $s^{\prime}$ differ; hence
$(r^{\prime})_{i}<(s^{\prime})_{i}$, and therefore
$((s-s^{\prime})+r^{\prime})_{i}<(s)_{i}$); this entails that
$r+((s-s^{\prime})+r^{\prime})<_{\textsc{lex}}r+s$. Hence if
$|r^{\prime}|=|s^{\prime}|$ and $r^{\prime}<_{\textsc{lex}}s^{\prime}$, then
$\bar{r}+\bar{s}<_{\textsc{lex}}r+s$ (since $\bar{r}=r$ and
$\bar{s}=(s-s^{\prime})+r^{\prime}$). The case $|r^{\prime}|>|s^{\prime}|$, or
$|r^{\prime}|=|s^{\prime}|$ and $s^{\prime}<_{\textsc{lex}}r^{\prime}$ is
analogous.
2) Since $(r^{\prime},s^{\prime})\leq(r,s)$, we have either
$|r^{\prime}+s^{\prime}|<|r+s|$, or $(r^{\prime},s^{\prime})=(r,s)$. By 1) it
is thus obvious that the edges on the path from $\textsc{n}^{\prime}$ to n
could not be all created by applications of REDUCE.
3) Since $\simeq$ is a congruence, $r^{\prime}\simeq s^{\prime}$ entails
$r^{\prime}+(s-s^{\prime})\simeq s^{\prime}+(s-s^{\prime})$, hence
$r^{\prime}+(s-s^{\prime})\simeq s$. Then $r\simeq s$ entails $r\simeq
r^{\prime}+(s-s^{\prime})$ (by symmetry and transitivity of $\simeq$). The
claim is thus clear.
4) We note that
$\textsc{EqLev}(r^{\prime},s^{\prime})\leq\textsc{EqLev}(r^{\prime}+(s-s^{\prime}),s^{\prime}+(s-s^{\prime}))$,
since $\simeq_{i}$ are congruences (by Proposition 3.9(1)). Hence
$\textsc{EqLev}(r^{\prime},s^{\prime})>\textsc{EqLev}(r,s)$ entails
$\textsc{EqLev}(r^{\prime}+(s-s^{\prime}),s)>\textsc{EqLev}(r,s)$, and thus
$\textsc{EqLev}(r,r^{\prime}+(s-s^{\prime}))=\textsc{EqLev}(r,s)$, by
Proposition 3.9(3).
The following theorem asserts the termination and correctness of the algorithm
ALG.
###### Theorem 4.3 (ALG decides resource bisimilarity)
Let $N=(P,T,W,l)$ be a labeled Petri net and $r_{0},s_{0}\in{\cal M}(P)$ be
its resources. Then:
1. 1.
For the input $N,r_{0},s_{0}$ there are only finitely many computations of the
above nondeterministic algorithm ALG, and each computation finishes by
constructing a finite tree $\mathcal{T}$ whose nodes are labeled with pairs of
resources; moreover, either one leaf of $\mathcal{T}$ is _unsuccessful_ , i.e.
labeled with $(r,s)$ where $r\not\simeq_{1}s$ (in which case the computation
returns NO), or $\mathcal{T}$ is successful, i.e. all leaves of $\mathcal{T}$
are identity-nodes (in which case the computation returns YES).
2. 2.
We have $r_{0}\simeq s_{0}$ if, and only if, at least one computation (of ALG
on $N,r_{0},s_{0}$) constructs a successful tree.
###### Proof 4.4
1) Any constructed tree is finitely branching, since each node has at most
$2\cdot|T|$ children. If there was an infinite computation, it would construct
a tree with an infinite branch (by König’s lemma); let us fix such a branch b
for the sake of contradiction. Each edge on b results by an application of
EXPAND or REDUCE. Claim 3(1) entails that infinitely many edges in b have
arisen by using EXPAND (since we cannot have an infinite row of REDUCE
applications). Hence we get an infinite sequence $(r_{1},s_{1})$,
$(r_{2},s_{2})$, $\dots$ of labels related to nodes
$\textsc{n}_{1},\textsc{n}_{2},\dots$ in b where EXPAND was used, and thus
REDUCE was not applicable. But Dickson’s lemma contradicts this, since there
must exist some $i<j$ such that $(r_{i},s_{i})\leq(r_{j},s_{j})$ (and thus
REDUCE would be applicable to $\textsc{n}_{j}$).
2) We analyze the respective two cases:
* •
Let $r_{0}\simeq s_{0}$. We consider a computation which keeps the property
that all nodes are labeled with equivalent pairs (each node is labeled with
some $(r,s)$ where $r\simeq s$); there is such a computation by Claim 2(1) and
Claim 3(3). All leaves of the constructed tree are then successful (being
identity-nodes).
* •
Let $r_{0}\not\simeq s_{0}$ (hence
$\textsc{EqLev}(r_{0},s_{0})=k\in\mathord{\mathbb{N}}$), and let us consider
the tree $\mathcal{T}$ constructed by an arbitrarily chosen computation. By
recalling Claim 2(2) and Claim 3(2,4) we deduce that there must be a branch of
$\mathcal{T}$ along which the equivalence level never increases (it drops
along each edge related to EXPAND, and remains the same along each edge
related to REDUCE). Hence the leaf of this branch must be unsuccessful
(labeled with some $(r,s)$ where $r\not\simeq_{1}s$).
## 5 Conclusions and additional remarks
In this paper we have investigated the decidability issues for two congruent
restrictions of bisimulation equivalence (denoted by $\sim$) in classical
labeled Petri nets (P/T-nets).
These congruences are resource similarity (denoted by $\approx$) and resource
bisimilarity (denoted by $\simeq$), as defined in [26]; both try to clarify
when replacing a resource (submarking) in a Petri net marking with a similar
resource does not change the observable system behavior.
Here we have stressed that $\approx$ is the largest congruence included in
$\sim$, and that $\simeq$ is the largest congruence included in $\sim$ that is
a bisimulation; we thus also have
$\simeq\mathop{\subseteq}\approx\mathop{\subseteq}\sim$. Besides a
straightforward fact that $\approx$ is a _strict_ refinement of $\sim$ in
general (i.e., $\approx\mathop{\subsetneq}\sim$ for some labeled Petri nets),
we have shown here also that $\simeq$ strictly refines $\approx$
($\simeq\mathop{\subsetneq}\approx$); the latter fact answered a question that
was left open in [26, 30]. We can also notice that deciding the relations
$\approx$ and $\simeq$ can be used for deducing bisimilar states, and thus
help to increase the efficiency of verification by reducing the state space of
the relevant systems. For another application of resource equivalences we can
refer to a Petri net reduction, in [36].
Since resource similarity and resource bisimilarity are congruences (w.r.t.
addition), they are finitely-based; more specifically, they are generated by
their minimal non-identity elements. The existence of such finite bases for
$\approx$ and $\simeq$ gives some hope at least for semidecidability of these
relations; the decidability proof in [6] for labeled communication-free Petri
nets (under the name Basic Parallel Processes), where we have
$\simeq\mathop{=}\approx\mathop{=}\sim$, was based on such semidecidability.
Nevertheless, the previous research [26, 30] already clarified that resource
similarity is undecidable (as well as bisimilarity), while decidability of
resource bisimilarity remained open.
In this paper we have shown that resource bisimilarity is decidable for
labeled Petri nets. In principle, we could proceed along the lines of two
semidecision procedures like [6] but we have chosen to present a tableau-based
algorithm deciding the problem. Regarding the computational complexity, we
only mention the known PSPACE-completeness of $\sim$ for Basic Parallel
Processes [7], where $\sim\mathop{=}\simeq$. (The PSPACE-lower bound was shown
in [37].)
An interesting question that we have left open here is if the mentioned finite
basis of $\simeq$ is computable (which does not follow from the decidability
of $\simeq$). In the case of labeled communication-free Petri nets (i.e.,
Basic Parallel Processes) the answer is positive: it follows by the fact that
[7] shows that a Presburger-arithmetic description of the relation $\sim$ can
be computed in this case. We note that for resource similarity we know that
the finite basis is not computable, since the relation $\approx$ is
undecidable.
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|
TTK-21-01, P3H-21-001
Modelling of top quark decays in $t\overline{t}\gamma$ production at the LHC
Malgorzata Worek 111Work supported in part by the German Research Foundation
(DFG) Collaborative Research Centre/Transregio project CRC/TRR 257: P3H -
Particle Physics Phenomenology after the Higgs Discovery.
Institute for Theoretical Particle Physics and Cosmology
RWTH Aachen University
D-52056 Aachen, Germany
> In this proceedings we briefly report on the state-of-the-art NLO QCD
> computation for the $pp\to t\overline{t}\gamma$ process in the di-lepton
> channel. We describe higher-order corrections to the
> $e^{+}\nu_{e}\,\mu^{-}\overline{\nu}_{\mu}\,b\overline{b}\gamma$ final state
> at the LHC with $\sqrt{s}=13$ TeV. Off-shell top quarks, double-, single- as
> well as non-resonant top-quark contributions along with all interference
> effects are consistently incorporated at the matrix element level. Results
> are presented in the form of fiducial integrated and differential cross
> sections. The impact of top quark modelling is scrutinised by an explicit
> comparison with the results in the narrow-width approximation. Both types of
> predictions are now available in the Helac-Nlo framework.
> PRESENTED AT
>
>
>
>
> $13^{\mathrm{th}}$ International Workshop on Top Quark Physics
> Durham, UK (videoconference), 14–18 September, 2020
## 1 Introduction
Top quark pair production in association with an additional photon provides
particularly promising means for a direct measurement of the top quark
electric charge, which governs the coupling strength of the
$t\overline{t}\gamma$ interaction. The $t\overline{t}\gamma$ process can
probe, however, not only the strength but also the structure of the
$t\overline{t}\gamma$ vertex. A generic way for the parameterisation of new
physics effects is provided by an effective field theory (EFT). The Lagrangian
of the EFT is expressed in terms of the Standard Model (SM) Lagrangian and a
non-SM part, which is dominated by the contributions due to operators of
dimension six. Such contributions can be translated into top quark magnetic
and electric dipole moments [1, 2] and measured at the LHC. Since the top
quark is the heaviest quark, effects of physics beyond the SM on its couplings
are larger than for any other fermion, thus, deviations with respect to SM
predictions for this process should be easier to detect. Furthermore,
$t\overline{t}\gamma$ production can be used to obtain predictions for
integrated and differential $t\overline{t}\gamma/t\overline{t}$ cross section
ratios [3]. Such cross section ratios are more stable against radiative
corrections and have reduced scale dependence. Consequently, they have
enhanced predictive power. They can, therefore, be used to study the
$t\overline{t}\gamma$ process with a higher precision. Finally, the integrated
and differential top quark charge asymmetries and lepton charge asymmetry can
be investigated in $t\overline{t}\gamma$ production at the LHC. They provide
complementary information to the measured charge asymmetries in
$t\overline{t}$ production [4, 5, 6, 7].
For an accurate comparison with LHC data, reliable theoretical predictions,
which include higher order effects, are mandatory. NLO QCD corrections for the
$t\overline{t}\gamma$ process have been calculated in Ref. [8, 5], whereas NLO
electroweak corrections have been considered in Ref. [9]. In both cases, top
quarks are treated as stable particles. For more realistic studies, however,
top quark decays are required. First attempts in this direction have been
carried out in Ref. [10], where NLO QCD predictions for $t\overline{t}\gamma$
have been matched to parton shower programs. In this study top quark decays
are treated in the parton shower approximation omitting spin correlations and
photon emission in parton shower evolution. Fully realistic theoretical
predictions for $t\overline{t}\gamma$ at NLO in QCD have been presented in
Ref. [11]. In this case, top quark decays are included using the narrow width
approximation (NWA). Consequently, spin correlations of final state particles
are maintained at the NLO level and photon radiation off top quark decay
products is incorporated. Finally, in Ref. [12] a complete description of
$pp\to t\overline{t}\gamma$ in the di-lepton top quark decay channel has been
presented. In this case, all resonant and non-resonant diagrams,
interferences, and off-shell effects of the top quarks and the $W$ gauge
bosons are consistently taken into account. These state-of-the-art theoretical
predictions for $t\overline{t}\gamma$ have been compared to the NWA case in
Ref. [13] using the Helac-Nlo common framework [14].
In this proceedings we briefly summarise the state-of-the-art theoretical
predictions for $t\overline{t}\gamma$ production at the LHC and analyse the
applicability of the NWA approach for this process.
## 2 Results
Modelling Approach | $\sigma^{\rm LO}$ [fb] | $\sigma^{\rm NLO}$ [fb]
---|---|---
full off-shell | ${7.32}^{+2.45\,(33\%)}_{-1.71\,(23\%)}$ | ${7.50}^{+0.11\,(1.0\%)}_{-0.45\,(6.0\%)}$
NWA | ${7.18}^{+2.39\,(33\%)}_{-1.68\,(23\%)}$ | ${7.33}_{-0.24\,(3.3\%)}^{-0.43\,(5.9\%)}$
NWAγ-prod | ${3.85}^{+1.29\,(33\%)}_{-0.90\,(23\%)}$ | ${4.15}_{-0.21\,(5.1\%)}^{-0.12\,(2.3\%)}$
NWAγ-decay | ${3.33}^{+1.10\,(33\%)}_{-0.77\,(23\%)}$ | ${3.18}^{-0.31\,(9.7\%)}_{-0.03\,(0.9\%)}$
NWALOdecay | | ${4.63}^{+0.44\,(9.5\%)}_{-0.52\,(11\%)}$
Table 1: LO and NLO integrated cross sections for various approaches to the
modelling of top quark decays. We additionally provide theoretical
uncertainties as obtained from the scale dependence. All results are provided
for $\mu_{0}=H_{T}/4$.
We present results for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\overline{\nu}_{\mu}\,b\overline{b}\gamma$ process for
the LHC Run II energy of $\sqrt{s}=13$ TeV. Our calculation uses CT14 parton
distribution functions (PDFs) [15] and employs the following SM parameters:
$G_{\mu}=1.166378\times 10^{-5}\,{\rm GeV}^{-2}$, $m_{W}=80.385$ GeV,
$\Gamma_{W}=2.0988$ GeV, $m_{Z}=91.1876$ GeV and $\Gamma_{Z}=2.50782$ GeV. The
electroweak coupling is derived from the Fermi constant. For the emission of
the isolated photon, however, $\alpha_{\rm QED}=1/137$ is used instead. The
top quark mass is set to $m_{t}=173.2$ GeV. The top quark width is calculated
using formulas from Ref. [16]. All other QCD partons including $b$ quarks as
well as leptons are treated as massless. The final state jets are constructed
using the IR-safe anti-$k_{T}$ jet algorithm [17] with $R=0.4$. We require at
least two jets for our process, of which exactly two must be bottom flavoured
jets. Furthermore, we request two charged leptons, missing transverse momentum
and an isolated hard photon. For the latter, we use the Frixione photon
isolation prescription, which is based on a modified cone approach [18]. We
apply basic selection cuts to these final states to ensure that they are
observed inside the detector and are well separated from each other, see [13]
for more details. We utilise the following scale
$\mu_{R}=\mu_{F}=\mu_{0}=H_{T}/4$ where $H_{T}$ is defined as
$H_{T}=p_{T}(e^{+})+p_{T}(\mu^{-})+p_{T}^{miss}+p_{T}(b_{1})+p_{T}(b_{2})+p_{T}(\gamma)\,,$
(1)
with $p_{T}(b_{1})$ and $p_{T}(b_{2})$ being bottom-flavoured jets and
$p_{T}^{miss}$ the missing transverse momentum from the two neutrinos. The
scale systematics is evaluated by varying $\mu_{R}$ and $\mu_{F}$
independently in the range between $\mu_{0}/2$ and $2\mu_{0}$. For
comparisons, in the case of the $t\overline{t}\gamma$ cross section results in
the NWA also the fixed scale $\mu_{0}=m_{t}/2$ is used. Our results for the
integrated fiducial cross sections for $t\overline{t}\gamma$ production are
summarised in Table 1. We compare the full off-shell results with the
calculations in the NWA. For the NWA case two versions are examined: the full
NWA and the NWALOdecay. The former comprises NLO QCD corrections to the
production and top quark decays as well as photon radiation from all charged
particles. The latter includes the NWA predictions with LO decays of top
quarks and photon radiation in the production stage only. In Table 1 we
additionally quote results for the LO and NLO QCD cross sections where photon
radiation occurs either in the production or in the decay stage.
A few comments can be made here. First, we observe that the NLO QCD
corrections are small of the order of a few percent only. We can also assess
the size of the non-factorizable top quark corrections for our setup. These
effects, which imply a cross-talk between production and decays of top quarks,
change the NLO cross section by less than $3\%$. For both the full off-shell
and NWA case, theoretical uncertainties due to the scale dependence are
consistently at the $6\%$ when higher order effects are incorporated. Using
results from Table 1, we can additionally see that $57\%$ of all isolated
photons are emitted in the production stage either from the initial state
light quarks or off-shell top quarks that afterwards go on-shell. Thus, $43\%$
are emitted in the decay stage, either from on-shell top quarks or its
(charged) decay products. Not only is the contribution from photon emission in
top quark decays substantial but NLO QCD corrections to decays are also
relevant. Therefore, it is not surprising that NWALOdecay results can not
reproduce the correct normalisation. The discrepancy with respect to the full
NWA approach amounts to almost $60\%$. NLO QCD corrections to top quark decays
are negative and at the level of $12\%$ when $\mu_{0}=H_{T}/4$ is employed.
Furthermore, theoretical uncertainties increase up to $11\%$ in this case.
Figure 1: Differential cross section distributions as a function of the
minimum invariant mass of the positron and bottom-jet and the (averaged)
transverse momentum of the $b$-jet. NLO QCD results for various approaches to
the modelling of top quark production and decays are shown.
Figure 2: Differential cross section distributions in function of $H_{T}$ and
the transverse momentum of the isolated photon. NLO QCD predictions in the
full NWA are shown together with fractions of events originating from photon
radiation in the production and in decays.
The situation changes for the full NWA case once differential cross sections
are examined instead. In Figure 1, we present two examples of differential
fiducial cross section distributions where off-shell effects are particularly
sizeable. Specifically, we show the minimum invariant mass of the positron and
bottom-jet and the (averaged) transverse momentum of the $b$-jet. The upper
plots show absolute NLO QCD predictions for the same three different
theoretical descriptions. The ratios to the full off-shell result including
its scale uncertainty band is plotted in the middle and bottom plots. When the
full off-shell and NWA results are compared, non-factorizable top quark
corrections up to $50\%-60\%$ can be noticed in the specific phase space
regions. At the same time, we can see that the NWALOdecay predictions are
unable to correctly describe these observables in the whole plotted range.
Among all observables that we have examined, only dimensionful observables are
sensitive to non-factorizable top quark corrections. They can be divided into
two categories: observables with kinematical thresholds or edges and
observables in high $p_{T}$ regions.
We have also studied the composition of photon emissions at the differential
level. As an example, in Figure 2 differential cross section distributions in
function of $H_{T}$ and $p_{T}(\gamma)$ are given at NLO in QCD. We present
predictions in the full NWA together with fractions of events originating from
photon radiation in the production and in decays. For low values of the
transverse momentum, the differential distributions are dominated by photon
emission in the decay stage. However, once the high $p_{T}$ regions of these
observables are probed, photon emission from the production part of the
$t\overline{t}\gamma$ process dominates completely the full results. Based on
our findings, selection criteria can be developed to reduce such contributions
that constitute a background for the measurement of the anomalous couplings in
the $t\overline{t}\gamma$ vertex.
Last but not least, our state-of-the-art theoretical predictions for the
$t\overline{t}\gamma$ process in the $e\mu$ channel at $13$ TeV have already
been compared to the LHC data [19].
## 3 Summary
In this proceedings, we have briefly described a comparison between the
complete off-shell and NWA calculations for the
$e^{+}\nu_{e}\,\mu^{-}\overline{\nu}_{\mu}\,b\overline{b}\gamma$ final state
at the LHC. We underlined the importance of the higher order corrections and
photon emission in top quark decays. Furthermore, we have shown that
dimensionful observables are sensitive to the non-factorizable top quark
corrections in the high $p_{T}$ regions and close to kinematical thresholds or
edges. This shows that proper modelling of the top quark production and decays
is essential already now in the presence of inclusive cuts and it will become
even more important in the presence of more exclusive cuts and in the high
luminosity phase of the LHC.
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|
# A survey on shape-constraint deep learning for medical image segmentation
Simon Bohlender
Department of Computer Science
TU Darmstadt
Darmstadt, Germany
Ilkay Oksuz
Computer Engineering Department
Istanbul Technical University
Istanbul, Turkey
School of Biomedical Engineering Imaging Sciences
King's College
London, U.K.
Anirban Mukhopadhyay
Department of Computer Science
TU Darmstadt
Darmstadt, Germany
###### Abstract
Since the advent of U-Net, fully convolutional deep neural networks and its
many variants have completely changed the modern landscape of deep learning
based medical image segmentation. However, the over dependence of these
methods on pixel level classification and regression has been identified early
on as a problem. Especially when trained on medical databases with sparse
available annotation, these methods are prone to generate segmentation
artifacts such as fragmented structures, topological inconsistencies and
islands of pixel. These artefacts are especially problematic in medical
imaging since segmentation is almost always a pre-processing step for some
downstream evaluation. The range of possible downstream evaluations is rather
big, for example surgical planning, visualization, shape analysis, prognosis,
treatment planning etc. However, one common thread across all these downstream
tasks is the demand of anatomical consistency. To ensure the segmentation
result is anatomically consistent, approaches based on Markov/ Conditional
Random Fields, Statistical Shape Models are becoming increasingly popular over
the past 5 years. In this review paper, a broad overview of recent literature
on bringing anatomical constraints for medical image segmentation is given,
the shortcomings and opportunities of the proposed methods are thoroughly
discussed and potential future work is elaborated. We review the most relevant
papers published until the submission date. For quick access, important
details such as the underlying method, datasets and performance are tabulated.
_K_ eywords Medical Image Segmentation $\cdot$ Shape Priors $\cdot$ Shape
Models $\cdot$ CRF $\cdot$ MRF $\cdot$ Active Contours
## 1 Introduction
Semantic segmentation is the task of predicting the category of individual
pixels in the image which has been one of the key problems in the field of
image understanding and computer vision for a long time. It has a vast range
of applications such as autonomous driving (detecting road signs, pedestrians
and other road users), land use and land cover classification, image search
engines, medical field (detecting and localizing the surgical instruments,
describing the brain tumors, identifying organs in different image
modalities). This problem has been tackled by a combination of machine
learning and computer vision, approaches in the past. Despite their popularity
and success, deep learning era changed main trends. Many of the problems in
computer vision - semantic segmentation among them - have been solved with
convolutional neural networks (CNNs) .
Incorporating prior knowledge into traditional image segmentation algorithms
has proven useful for obtaining more accurate and plausible results. The
highly constrained nature of anatomical objects can be well captured with
learning based techniques. However, in most recent and promising techniques
such as CNN based segmentation it is not obvious how to incorporate such prior
knowledge. Segmenting images that suffer from low-quality and low signal-to-
noise ratio without any shape constraint remains problematic even for CNNs.
Though it has been shown that incorporation of shape prior information
significantly improves the performance of the segmentation algorithms,
incorporation of such prior knowledge is a tricky practical problem. In this
work, we provide an overview of efforts of shape prior usage in deep learning
frameworks.
### 1.1 Yet another review paper
There already appeared a variety of review papers about shape modelling and
deep learning for medical image segmentation in the recent past. McInerney and
Terzopoulos (1996) presents various approaches that apply deformable models.
Peng et al. (2013) deals with different categories of graph-based models where
meaningful objects are represented by sub-graphs. The review by Heimann and
Meinzer (2009) is about statistical shape models and concentrates especially
on landmark-based shape representations. Elnakib et al. (2011) also reviews
different shape feature based models, that include statistical shape models,
as well as deformable models. A more recent review by Nosrati and Hamarneh
(2016) provides insights into segmentation models that incorporate shape
information as prior knowledge. Later surveys of Litjens et al. (2017), Razzak
et al. (2017), Rizwan I Haque and Neubert (2020) and Lei et al. (2020) shift
their focus to deep learning approaches. Hesamian et al. (2019) and Taghanaki
et al. (2019) present different network architectures and training techniques,
whereas Jurdi et al. (2020) take it a step further and reviews prior-based
loss functions in neural networks.
Since deep learning became the method of choice for many computer vision
tasks, including medical image segmentation, we focus our review on models
that combine neural networks with explicit shape models in order to
incorporate shape knowledge into the segmentation process. Segmentation models
solely based on neural networks usually do not incorporate any form of shape
knowledge. They are based on traditional loss functions that only regard
objects at pixel level and do not evaluate global structures. The papers we
present in this review improve these networks by combining them with
additional models that are especially built with shape in mind. This is also
the point that delimits this review from existing surveys which either focus
mostly deep learning approaches or on traditional shape and deformable model
methods, but not on the combination of both.
The explicit models applied in this review can be divided into three main
categories as shown in Figure 1: 1) Conditional or Markov Field models that
establish connections between different pixel regions 2) Active/Statistical
Shape Models that learn a special representation for valid shapes 3) Active
Contour Models or snakes that use deformable splines for shape detection.
These models are either applied as pre-processing steps to create initial
segmentations, post-processing steps to refine the neural network
segmentations, or used in multi-step models consisting of various models along
a specific pipeline.
We are aware that the field is heavily shifting from explicit ways of modeling
shape to more implicit approaches where networks are trained in an end-to-end
way.Up and coming Works propose more intelligent loss functions that no longer
require additional explicit shape modelling, but only consist of a single
neural network. Zhang et al. (2020a) proposed a new geometric loss for lesion
segmentation. Other examples are Mohagheghi and Foruzan (2020) and Han et al.
(2020) where the loss contains shape priors. Li et al. (2020) introduces a
spatially encoded loss with a special shape attention mechanism. Clough et al.
(2019b) uses a topology based loss function.
However the overwhelming majority of articles combine neural networks and
explicit models to introduce shape knowledge. This combination often stems
from a rather principled engineering design choice (as shown in Figure 1)
which is not detailed in any of the previous review articles. This review
focuses on this overarching design principle of shape constraint which, along
with being a quick access guide to explicit approaches, will work as a
research catalyzer of implicit constraints.
Figure 1: Overview of related work approaches
## 2 CRF / MRF approaches
Markov Random Fields (MRF) Li (1994) belong to the domain of graphical models
and model relationships between pixels or high-level features with a
neighborhood system. The label probability of a single pixel is thereby
conditioned on all neighboring pixels which allow to model contextual
constraints. The maximum a posteriori probability (MAP) can then be calculated
by applying the Bayes rule. Conditional Random Fields (CRF) Lafferty et al.
(2001) are an extension of MRFs and allow to incorporate arbitrary global
features over regions of pixels. For medical image segmentation this means
that they generate smooth edges by using this global knowledge about
surrounding regions which is a reason why the are often applied alongside
neural networks to perform medical image segmentation.
#### CRFs used for postprocessing
The largest category of methods that utilize CRFs or MRFs apply them as a pos-
tprocessing step. A large portion of papers focus on the straight-forward
approach where the CNN generates initial segmentations maps which are directly
passed to a CRF or MRF model as inputs for further refinements. These
approaches are evaluated on a variety of anatomies and mostly differ in the
utilized network architectures but follow the same idea. They are applied to
lung nodules (Yaguchi et al. (2019), Gao et al. (2016)), retinal vessel (Fu et
al. (2016b)), brain tumor (Zhao et al. (2016), Li et al. (2017a)), cervical
nuclei (Liu et al. (2018)), eye sclera (Mesbah et al. (2017)), melanoma (Luo
and Yang (2018)), ocular structure (Nguyen et al. (2018)), left atrial
appendage (Jin et al. (2018)), lymph node (Nogues et al. (2016)), liver (Dou
et al. (2016)) and prostate cancer lesion (Cao et al. (2019)) segmentation
tasks. A slightly different approach for skin lesion detection by Qiu et al.
(2020) is based on the same idea, but uses not just a single CNN network, but
an ensemble of seven or fifteen which are combined inside the CRF. Two other
approaches to highlight here for brain region (Zhai and Li (2019)) and optical
discs in fundus image (Bhatkalkar et al. (2020)) segmentation integrate a
special attention mechanism into their networks with the motivation to improve
the segmentations by detecting and exploiting salient deep features. Another
special version that operates on weakly segmented bounding box images for
fetal brain & lung segmentation is introduced by Rajchl et al. (2017). Given
the initial weak segmentations, the model iteratively optimizes the pixel
predictions with a CNN followed by a CRF to obtain the final segmentation
maps.
Instead of CRFs, Shakeri et al. (2016) use a MRF to impose volumetric
homogenity on the outputs of a CNN for subcortical region segmentation. MRFs
are also utilized in the approach shown by Xia et al. (2019) for kidney
segmentation where the MRF is integrated into a SIFT-Flow model.
Besides these classical approaches, another method that came up focused on
cascading CNN networks that generate segmentations in a coarse-to-fine
fashion. Wachinger et al. (2018) use this strategy with a first network that
segments fore- from background pixels in brain MRIs and a second one that
classifies the actual brain regions. The same method is also used by Shen and
Zhang (2017) for brain tumor segmentation, by Dou et al. (2017) for liver and
whole heart segmentation, and by Christ et al. (2016) for liver-based lesion
segmentation.
A somewhat different cascading structure, for brain tumor segmentation, is
introduced by Hu et al. (2019) where multiple subsequent CNNs are used to
extract more discriminative multi-scale features and to capture dependencies.
Feng et al. (2020) extend this version on the task of brain tumor segmentation
with the introduction of residual connections that improve the overall
performance. Similar to the cascading methods, there are CNNs with two
pathways that combine two parallel networks on different resolution levels
that aim for capturing larger 3D contexts. The approach was originally
introduced by Alansary et al. (2016) for placenta segmentation, but was also
applied in Cai et al. (2017) to the task of pancreas segmentation. Kamnitsas
et al. (2017) proposes another related approach where two parallel networks, a
FCN that extracts a rough mask and a HED that outputs a contour, are fused
inside a CRF. In the approach by Shen et al. (2018) that deals with brain
tumor segmentation, a third path is added where in total three concurrent FCNs
are trained based on different filtered (gaussian, mean, median) input images.
After each network an individual CRF is applied and their results are fused in
a linear regression model.
Figure 2: Overview of relevant papers per year for each category
#### Training CNN and CRF models end-to-end
The idea of integrating CRF models directly into neural networks origins from
the task of semantic image segmentation and was introduced by Zheng et al.
(2015). They combine the strengths of both models into a unified framework
that allows end-to-end training. Broken down, the basic task of CRFs is to
minimize an energy term with an iterative mean field approximation. Since CRFs
are graphical models, each iteration step can be formulated as a stack of CNN
layers. Multiple iterations can then be implemented by repeatedly executing
this stack or alternatively as an equivalent Recurrent Neural Network (RNN).
The resulting network is denoted as a CRF-RNN and can be applied on top of any
CNN architecture. Fu et al. (2016a) are the first to transfer this method to
medical image segmentations with a model called DeepVessel for the task of
retinal vessel segmentation. For the same task Luo et al. (2017) achieve
similar results by using a slightly deeper base CNN network with more
convolution layers. Besides retinal vessel, CRF-RNN approaches are applied to
a variety of other anatomical structures. Zhao et al. (2016) applies them to
brain tumor segmentation and extend it with some additional pre- and post-
processing steps later on Zhao et al. (2018b). Xu et al. (2018) uses a V-Net
combined with CRF-RNN for bladder segmentation and in Monteiro et al. (2018)
they are also applied on brain tumor as well as prostate segmentation with 3D
multi-modal images. Analogues Chen and de Bruijne (2018) utilizes a U-Net as
their base-network to deal with white matter lesion segmentation. On the same
idea as CRF-RNN Deng et al. (2020) uses a CRF-Recurrent Regression based
Neural Network (CRF-RRNN) integrated with a heterogeneous CNN for brain tumor
segmentation where the combined network can also be trained end-to-end.
Instead of using a full RNN, Zhang et al. (2020d) propose a method where MRF
is integrated into the segmentation network as a block of local and global
convolution layers that take the CNN output as unary potentials to calculate
the corresponding pairwise potentials.
CNNs combined with CRF / MRF models | | |
---|---|---|---
Authors | Anatomy | Title | Method
CRF / MRF used for post-processing | | |
Li et al. (2017a) | Brain Tumor | Low-Grade Glioma Segmentation Based on CNN with Fully Connected CRF | CRF refines CNN segmentation
Wachinger et al. (2018) | Brain Region | DeepNAT: Deep convolutional neural network for segmenting neuroanatomy | CRF refines hierarchical CNN segmentations
Hu et al. (2019) | Brain Tumor | Brain Tumor Segmentation Using Multi-Cascaded Convolutional Neural Networks and Conditional Random Field | FC-CRF refines segmentations of three CNNs
Shen and Zhang (2017) | Brain Tumor | Fully connected CRF with data-driven prior for multi-class brain tumor segmentation | Multiple FC-CRFs
Kamnitsas et al. (2017) | Brain Lesion | Efficient Multi-Scale 3D CNN with Fully Connected CRF for Accurate Brain Lesion Segmentation | FC-CRF refines two-pathway CNN
Alansary et al. (2016) | Placenta | Fast Fully Automatic Segmentation of the Human Placenta from Motion Corrupted MRI | FC-CRF refines two-pathway CNN
Shakeri et al. (2016) | Sub-cortical regions | Sub-cortical brain structure segmentation using F-CNN’s | MRF refines FCNN segmentation
Zhai and Li (2019) | Brain region | An Improved Full Convolutional Network Combined with Conditional Random Fields for Brain MR Image Segmentation Algorithm and its 3D Visualization Analysis | FC-CRF refines CNN with attention
Dou et al. (2016) | Liver | 3D Deeply Supervised Network for Automatic Liver Segmentation from CT Volumes | FC-CRF refines 3D FCNN with 3D supervision mechanism
Dou et al. (2017) | Heart | 3D Deeply Supervised Network for Automated Segmentation of Volumetric Medical Images | FC-CRF refines cascading U-Nets
Christ et al. (2016) | Liver | Automatic Liver and Lesion Segmentation in CT Using Cascaded Fully Convolutional Neural Networks and 3D Conditional Random Fields | FC-CRF refines cascaded FCNs
Fu et al. (2016b) | Retinal Vessel | Retinal vessel segmentation via deep learning network and fully-connected conditional random fields | FC-CRF refines CNN with side-outputs
Jin et al. (2018) | Left atrial appendage | Left Atrial Appendage Segmentation Using Fully Convolutional Neural Networks and Modified Three-Dimensional Conditional Random Fields | FC-CRF combines slices of FCN
Cai et al. (2017) | Pancreas | Pancreas Segmentation in MRI using Graph-Based Decision Fusion on Convolutional Neural Networks | CRF refines results from FCN and HED network
Xia et al. (2019) | Kidney | Deep Semantic Segmentation of Kidney and Space-Occupying Lesion Area Based on SCNN and ResNet Models Combined with SIFT-Flow Algorithm | MRF refines combined ResNet and SCNN
Rajchl et al. (2017) | Fetal Brain / Lung | DeepCut: Object Segmentation from Bounding Box Annotations using Convolutional Neural Networks | Iterative CRF and CNN
Nogues et al. (2016) | Lymph Node | Automatic Lymph Node Cluster Segmentation Using Holistically-Nested Neural Networks and Structured Optimization in CT Images | CRF refines HNN (FCN + DSN) segmentations
Yaguchi et al. (2019) | Lung Nodules | 3D fully convolutional network-based segmentation of lung nodules in CT images with a clinically inspired data synthesis method | CRF refines 3D FCN segmentations
Gao et al. (2016) | Lung | Segmentation label propagation using deep convolutional neural networks and dense conditional random field | CRF refines CNN segmentations
Feng et al. (2020) | Brain Tumor | Study on MRI Medical Image Segmentation Technology Based on CNN-CRF Model | CRF refines DCNN segmentations
Liu et al. (2018) | Cervical Nuclei | Automatic segmentation of cervical nuclei based on deep learning and a conditional random field | Locally FC-CRF refines Mask-RCNN segmentation
Shen et al. (2018) | Brain Tumor | Brain tumor segmentation using concurrent fully convolutional networks and conditional random fields | Concurrent FCN refined by FC-CRF
Mesbah et al. (2017) | Eye Sclera | Conditional random fields incorporate convolutional neural networks for human eye sclera semantic segmentation | Initial CNN boundaries refined by CRF
Luo and Yang (2018) | Melanoma | Fast skin lesion segmentation via fully convolutional network with residual architecture and CRF | CRF refines FCN segmentations
Bhatkalkar et al. (2020) | Fundus Optic Disk | Improving the Performance of Convolutional Neural Network for the Segmentation of Optic Disc in Fundus Images Using Attention Gates and Conditional Random Fields | FC-CRF refines CNN segmentations
Qiu et al. (2020) | Skin Lesion | Inferring Skin Lesion Segmentation With Fully Connected CRFs Based on Multiple Deep Convolutional Neural Networks | CRF refines segmentations of DCNN ensemble
Nguyen et al. (2018) | Ocular structures | Ocular structures segmentation from multi-sequences mri using 3d unet with fully connected crfs | FC-CRF refines CNN segmentations
Cao et al. (2019) | Prostate cancer lesions | Prostate Cancer Detection and Segmentation in Multi-parametric MRI via CNN and Conditional Random Field | Selective Dense CRF refines CNN segmentations
CNN and CRF trained end-to-end
Zhao et al. (2018b) | Brain Tumor | A deep learning model integrating FCNNs and CRFs for brain tumor segmentation. | Combination of FCNN and CRF-RNN
Monteiro et al. (2018) | Prostate / Brain Tumor | Conditional Random Fields as Recurrent Neural Networks for 3D Medical Imaging Segmentation | Combination of FCNN and CRF-RNN
Fu et al. (2016a) | Retinal Vessel | DeepVessel: Retinal Vessel Segmentation via Deep Learning and Conditional Random Field | Combination of CNN and CRF-RNN layers
Chen and de Bruijne (2018) | White matter hyperintensities | An End-to-end Approach to Semantic Segmentation with 3D CNN and Posterior-CRF in Medical Images | Combination of U-Net and FC-CRF
Xu et al. (2018) | Bladder | Automatic bladder segmentation from CT images using deep CNN and 3D fully connected CRF-RNN | Combination of CNN and CRF-RNN
Deng et al. (2020) | Brain Tumor | Deep Learning-Based HCNN and CRF-RRNN Model for Brain Tumor Segmentation | Combination of HCNN and CRF-RRNN
Zhang et al. (2020d) | Prostate | ARPM-net: A novel CNN-based adversarial method with Markov Random Field enhancement for prostate and organs at risk segmentation in pelvic CT images | CNN combined with MRF block
Zhao et al. (2016) | Brain tumor | Brain tumor segmentation using a fully convolutional neural network with conditional random fields | CRF integrated into FCNN
Luo et al. (2017) | Retinal Vessel | Efficient CNN-CRF network for retinal image segmentation | Combination of CNN and CRF
## 3 Shape model based approaches
The second category of model assumptions often combined with CNNs are active
shape models (ASM) Cootes et al. (1995) or probabilistic active shape models
(PASM). ASMs require a training set with a fixed number of manually annotated
landmark points of the segmented object. Each point represents a particular
part of the object and has to be in the same position over all images. These
annotated shapes are then iteratively matched and a mean shape is derived. The
landmark points show different variabilities that are modeled by a Point
Distribution Model (PDM). Performing a principal component analysis (PCA) and
weighting the eigenvectors allows creating new shapes in the allowed
variability range. For detecting an object in an unknown image an algorithm is
used that updates pose and shape parameters iteratively to improve the match
until convergence. An extension to this approach are probabilistic ASMs (PASM)
Wimmer et al. (2009). They impose a weaker constraint on shapes which allows
more flexible contours with more variations from the mean shape. This is
achieved by introducing a probabilistic energy function which is minimized in
order to fit a shape to a given image. The model’s ability to generalize is
thereby improved and the segmentation results outperform standard ASMs.
#### Shape Models for post-processing
Though CNN based segmentation models yield good segmentation results, they
tend to produce anatomically implausible segmentation maps that can contain
detached islands or holes at parts where they do not occur in reality. Since
shape models represent valid and anatomically plausible shapes, it makes sense
to apply them in post-processing steps to regularize initial CNN segmentations
and transform them into a valid shape domain. Xing et al. (2016) take up this
idea and apply it to nucleus segmentation. The initial segmentations are
generated by a CNN and the post-processing step includes a sparse selection-
based shape model for top-down shape inference, which is more insensitive to
object occlusions compared to PCA-based shape models, and an additional
deformable model for bottom-up shape deformation. Also Hsu (2019) follows this
strategy for segmentation and tracking of the left ventricle. They swap out
the CNN for a Faster-RCNN and use an improved ASM that allows to obtain
matching points in greater ranges. Fauser et al. (2019) continue on improving
the ASM by using a probabilistic ASM that is more flexible and allows leaving
the shape space. The segmentation of the left ventricle is performed by
combining the results of three CNN-PASM models for each dimension. Another
modified ASM is proposed by Medley et al. (2020). The authors use Expectation-
Maximization to deal with outliers during optimizing the ASM. They also
evaluate different ASM features and conclude that a CNN that learns the input
feature maps for the EM-ASM performs best. Besides improving on the ASM a
different approach by Karimi et al. (2019) aims for generating better
predictions with an ensemble of U-Net like CNN models with different filters
and parameters. In their approach a SSM model, based on the thresholded
segmentations from all individual models, is only applied if the disagreement
between the ensemble models becomes to high. Instead of using the CNN for
generating segmentation maps, it is also sufficient to only predict bounding
boxes as initializations for ASMs. Such an approach is applied by Tabrizi et
al. (2018) on kidney segmentation where a fuzzy-ASM produces the final
segmentations. Li et al. (2018) also uses a CNN for bounding box prediction,
but adds an intermediate step before utilizing a statistical shape model for
myocardial segmentation, in which a random forest classifier builds
probability maps from the given bounding boxes. Another tree model, more
specific an adaptive feature learning probability boosting tree (AFL-PBT) is
also utilized by He et al. (2018) as an initial step to classify voxels for
prostate segmentation. A subsequent CNN then extracts boundary probability
maps and a three-level ASM is employed to generate final segmentations.
#### Shape Models for prior knowledge
In this second paragraph we present some papers where the shape models are
applied pre-hoc before any deep learning network. Two straight forward models
for this category are proposed by Cheng et al. (2016) and Fan et al. (2020).
In Fan et al. (2020) a 3D U-Net-like CNN segments Itra-Cholear anatomy based
on initial segmentations from an ASM and the original CT images. Cheng et al.
(2016) on the other hand use a CNN for refining initial segmentations from an
Active Appearance Model (AAM) that produces only coarse prostate
segmentations. The AAM is basically an extended shape model that adds an
additional texture model for better fitting capabilities. The other two models
already introduce some pipeline-like approaches, but use both a shape model as
prior knowledge. The pipeline for subcortical region segmentation in Duy et
al. (2018) starts with a pre-processing SVM that classifies sagittal slices
into groups of similar shape. The prior ASM then creates rough segmentations
for each group which are finalized by a CNN. Further the authors propose an
optional CRF model for post-processing. Nguyen et al. (2019) introduce the ASM
as a more traditional prior for uveal melanoma segmentation where it is used
as a constraining term for a CRF model that is based on Grad-CAM (class
activation maps) heatmaps. The final segmentations are again generated with a
U-Net that combines the CRF with original input CTs.
Figure 3: Overview of anatomical structures examined in the relevant papers
#### Pipeline approaches with multiple CNN and ASM models
The last category for combining shape models and neural networks contains all
approaches that consist of different models arranged along pipelines. The
motivation is to process input images stage-wise or in a coarse-to-fine way
that allows to capture more information and hence result in more accurate
segmentation maps. In the models by Tack et al. (2018) for knee menisci,
Ambellan et al. (2019) for knee bone & cartilage, and Brusini et al. (2020)
for hippocampus segmentation, the pipelines combine multiple CNNs and SSMs.
All three start with initial 2D U-Nets regularized by SSMs which are used to
extract smaller 3D subvolumes. Tack et al. (2018) and Ambellan et al. (2019)
apply an additional 3D U-Net afterwards, whereas Brusini et al. (2020) uses
three U-Nets and averages their predictions to obtain final segmentations.
Ambellan et al. (2019) further continues after this step and utilizes a second
3D SSM model to obtain the knee bone segmentations and even applies a third
U-Net to segment the cartilage afterwards. Besides these typical pipelines,
there are also some hybrid approaches we count to this category that integrate
shape models and neural networks. They use special CNNs that directly predicts
the parameters of an SSM, which are the shape coefficients (weights for the
modes of variations), the pose parameters. Qin et al. (2020) use such a SSM-
Net inside a small pipeline for prostate segmentation. They propose an
inception-based network that directly predicts parameters of the SSM which can
be back-translated into a prostate contour prediction. Parallel to this, a
residual U-Net generates probability maps from the inputs. The final
segmentations are generated by averaging the outputs of both models. The
method of Tilborghs et al. (2020) for left ventricle segmentation is based on
the same idea, but removes the small pipeline. Instead they modify the CNN and
add a third output which is an actual distance map. A special loss function is
used to train the network toward optimizing the segmentation map alongside the
SSM parameters. A nearly identical approach by Karimi et al. (2018) is applied
to prostate segmentation. Their CNN predicts center position of the prostate,
the shape model parameters, and a rotation vector which are passed to a final
layer that outputs the coordinates of the landmark points which resemble the a
final segmentation map. Schock et al. (2020) relies on the same method for
knee bone & cartilage segmentation, but extend it with additional pre- and
post-processing steps. They add a preprocessing 2D U-Net that detects initial
bone positions and crop the volume into subvolumes which only contain the
femur or tibia bone. Afterwards their SSM-Net comes into place that predicts
the SSM parameters and the actual landmarks in a subsequent PCA layer. An
additional fine-tuning step then generates the cartilage segmentations with a
3D U-Net based on subvolumes centered at the bones’ landmark points. Rather
than integrating the SSM and CNN, Ma et al. (2018) introduces a Bayesian model
that integrates both, the CNN and a robust kernel SSM (RKSSM) for the task of
pancreas segmentation. At first the RKSSM is initialized to fit the detected
ROI of a Dense U-Net. A Gaussian Mixture Model afterwards guides the shape
adaption and iteratively projects the adapted shape onto the RKSSM until
convergence which results in the final segmentation map.
CNNs combined with Active Shape Models | | |
---|---|---|---
Authors | Anatomy | Title | Method
ASM for post-processing | | |
Xing et al. (2016) | Nucleus | An Automatic Learning-Based Framework for Robust Nucleus Segmentation | Shape Model refines CNN segmentation
He et al. (2018) | Prostate | Automatic Magnetic Resonance Image Prostate Segmentation Based on Adaptive Feature Learning Probability Boosting Tree Initialization and CNN-ASM Refinement | Three-level-ASM refines segmentations of CNN
Fauser et al. (2019) | Temporal Bone | Toward an automatic preoperative pipeline for image-guided temporal bone surgery | Probabilistic ASM refines 2D U-Net segmentation
Li et al. (2018) | Myocardial | Fully Automatic Myocardial Segmentation of Contrast Echocardiography Sequence Using Random Forests Guided by Shape Model | ASM refines random-forest segmentations initialized by a CNN
Medley et al. (2020) | Left Ventricle | Deep Active Shape Model for Robust Object Fitting | ASM initialized with CNN generated features maps
Karimi et al. (2019) | Prostate | Accurate and robust deep learning-based segmentation of the prostate clinical target volume in ultrasound images | SSM refines segmentations from ensemble of CNNs
Tabrizi et al. (2018) | Kidney | Automatic kidney segmentation in 3D pediatric ultrasound images using deep neural networks and weighted fuzzy active shape model | Fuzzy ASM segmentations based on DNN generated bounding boxes
Hsu (2019) | Left Ventricle | Automatic Left Ventricle Recognition, Segmentation and Tracking in Cardiac Ultrasound Image Sequences | ASM improves R-CNN segmentations for detection and tracking
ASM as prior-knowledge
Duy et al. (2018) | Brain Region | Accurate brain extraction using Active Shape Model and Convolutional Neural Networks | CNN refines ASM segmentations
Cheng et al. (2016) | Prostate | Active appearance model and deep learning for more accurate prostate segmentation on MRI | 2D-CNN refines segmentations from an Active Appearance Model
Fan et al. (2020) | Intra-Cholear Anatomy | Combining model- and deep-learning-based methods for the accurate and robust segmentation of the intra-cochlear anatomy in clinical head CT images | U-Net refines ASM segmentations
Nguyen et al. (2019) | Uveal Melanoma | A novel segmentation framework for uveal melanoma based on magnetic resonance imaging and class activation maps | U-Net segmentations based on a CRF that uses ASM as prior knowledge
Pipelines with multiple ASM and CNN models & Hybrid approaches
Ambellan et al. (2019) | Knee Bone / Cartilage | Automated Segmentation of Knee Bone and Cartilage combining Statistical Shape Knowledge and Convolutional Neural Networks: Data from the Osteoarthritis Initiative | Three CNN and two SSM models
Tack et al. (2018) | Knee Menisci | Knee Menisci Segmentation using Convolutional Neural Networks: Data from the Osteoarthritis Initiative | 3D CNN and SSM initialized by 2D models
Brusini et al. (2020) | Hippocampus | Shape Information Improves the Cross-Cohort Performance of Deep Learning-Based Segmentation of the Hippocampus | ASM as input for CNN
Ma et al. (2018) | Pancreas | A novel bayesian model incorporating deep neural network and statistical shape model for pancreas segmentation | U-Net and SSM segmentations combined within Bayesian model
Qin et al. (2020) | Prostate | A weakly supervised registration-based framework for prostate segmentation via the combination of statistical shape model and CNN | Segmentations combined of U-Net and SSM-Net predictions
Tilborghs et al. (2020) | Left Ventricle | Shape Constrained CNN for Cardiac MR Segmentation with Simultaneous Prediction of Shape and Pose Parameters | Hybrid approach where CNN generates segmentations and ASM parameters
Karimi et al. (2018) | Prostate | Prostate segmentation in MRI using a convolutional neural network architecture and training strategy based on statistical shape models | CNN predicts segmentations and 3D-ASM parameters
Schock et al. (2020) | Knee Bone & Cartilage | A Method for Semantic Knee Bone and Cartilage Segmentation with Deep 3D Shape Fitting Using Data from the Osteoarthritis Initiative | CNN that predicts segmentations and 3D-ASM parameters is refined by U-Net
## 4 Active contour approaches
A last type of models that often combined with deep learning models to
incorporate shape knowledge are Active Contour Models (ACM) Kass et al. (1988)
, also known as snakes. A snake is a deformable controlled continuity spline
that is pushed towards edges or contours by minimizing an energy function
under the influence of different forces and constraints. It consists of an
internal energy that keeps the contour continuous and smooth, an image energy
that attracts it to contours, and an external constraint force that adds user-
imposed guidance. A similar approach are level set functions (LSF) introduced
by Andrew (2000) and firstly applied to image segmentation by Malladi et al.
(1995). An LSF is a higher dimensional function where a contour is defined as
its zero level set. With a speed function, derived from the image, that
controls the evolution of the surface over time, a Hamilton-Jacobi partial
differential equation can be obtained.
#### ACM models for post-processing
Since ACM models are based on the idea of evolving a contour, it makes sense
to apply them as a post-processing step to improve an initial segmentation
map. An early model by Middleton and Damper (2004) uses only a simple
multilayer perceptron (MLP) that creates binary pixel-wise boundary
predictions for lung segmentation. Since these are very rough and contain
misclassifications the ASM is used to improve and close the contour. Salimi et
al. (2018) is also based on an MLP, but adds an vector field convolution to
the ACM to make it more robust for prostate segmentation
However, the more recent ACM post-processing models are exclusively based on
different CNN architectures and are applied to a variety of anatomies. Li et
al. (2017b) use a FCN that is refined by a classic ACM for left ventricle
segmentation. The same approach is taken by Guo et al. (2019) for liver
segmentation and Zhao et al. (2018a) utilize it for nucleus segmentation. In
the approaches by Xu et al. (2019) the ACM refinements are not yet the final
steps and additional adaptive ellipse fitting is used to segment breast
nuclei. Hu et al. (2018) and Fang et al. (2019) transfer the basic refinement
method to breast tumor detection with a phase-based ACM that improves over
multiple iterations. A different slightly modified ACM post-processing method
is based on geodesic computations and is further used by Ma and Yang (2019)
for dental root segmentation and Nunes et al. (2020) for lung segmentation.
Zhang et al. (2020b) also introduces a special ACM that integrates a fourth-
order partial differential equation and segments plaque based on an initial
R-CNN segmentation. Instead of just refining an initial CNN predicted per-
pixel segmentation map, da Silva et al. (2020) use a Chan-Vese ACM to generate
prostate segmentation on DCNN coarsely classified superpixels which only
represent rough initialization for the contour model. The authors of Kot et
al. (2020) further separate the two models where the CNN masks bone tissue
which is removed for the ACM to segment brain tumors. The last special
approach in the ACM category by Zhang et al. (2020c) is a hybrid model that
integrates an ACM into a U-Net. The resulting deep active contour network
(DACN) is end-to-end trainable with a special ACM based loss function and
automatically segments cervical cells and skin lesions. Besides ACM, another
large number of approaches rely on level set functions (LSF). Same as before a
CNN is used for generating initial segmentation maps which are then refined by
the LSF. Hatamizadeh et al. (2019) uses this for brain, liver, lung
segmentation, Gong et al. (2019) for pancreas segmentation, Carbajal-Degante
et al. (2020) for ventricle and liver segmentation, and Xie et al. (2020) for
left ventricle segmentation. Some extra processing is made in Yang et al.
(2021) for dental pulp segmentation where the initial CNN segmentations are
used to calculate elliptic curves which are used to guide the LSF. In general,
for the LSF it is often sufficient to initialize them only with a rough
bounding boxes or region of interest annotations. So, Liu et al. (2019) use a
Faster RCNN to generate location boxes of left atriums which serve as input
for the LSF after Otsu thresholding. Avendi et al. (2016) inserts an
additional step between CNN ROI detection and LSF segmentation where the
initial left-ventricle shape is inferred with an stacked auto-encoder. In
comparison to these two approaches, in Cha et al. (2016) the CNN is not used
to predict ROI, but to classify if an ROI is part of the bladder. The outputs
are then refined by three different 3D LSF and a final 2D LSF afterwards.
Another idea is to use recurrent pipelines where the segmentations are refined
iteratively. Such an approach is introduced by Tang et al. (2017) where both
models are integrated into a FCN-LSF. The method is used for left ventricle
and liver segmentation with semi-supervised training where the LSF gradually
refines the segmentation and backpropagates a loss to improve the FCN. Hoogi
et al. (2017) proposed a different iterative process. Hereby the CNN estimates
if the zero level set is inside, outside or near the lesion boundary. Based on
these the LSF parameters are calculated and the contour is evolved. The
process then repeats until convergence.
#### Using a CNN to refine ACM segmentations
Besides the majority of approaches that use ACMs for post-processing, there
are also methods where ACMs are used to obtain the initial segmentations or
are guided by CNNs. The earliest of these approaches by Ahmed et al. (2009)
uses an ACM to remove skull tissue from images and applies a simple artificial
neural network to classify the remaining brain regions. Rupprecht et al.
(2016) introduce an approach where the ACM is guided by the CNN. The ACM
generated rough segmentations of the left ventricle. A CNN then predicts
vectors on patches around each pixel of this initial contour that point
towards closes object boundary points and are used to further evolve the
contour. The latest method for this category by Kasinathan et al. (2019) also
uses the ACM to generate initial segmentations, more specific it segments all
lung nodules. A post-processing CNN afterwards classifies them or removes
false positives.
CNNs combined with Active Contour Models | | |
---|---|---|---
Authors | Anatomy | Title | Method
ACM for post-processing | | |
Middleton and Damper (2004) | Lung | Segmentation of magnetic resonance images using a combination of neural networks and active contour models | ACM refines MLP segmentation
Salimi et al. (2018) | Prostate | Fully automatic prostate segmentation in MR images using a new hybrid active contour-based approach | ACM refines MLP segmentation
Li et al. (2017b) | Left Ventricle | Left ventricle segmentation by combining convolution neural network with active contour model and tensor voting in short-axis MRI | ACM refines FCN segmentation
Hu et al. (2018) | Breast Tumor | Automatic tumor segmentation in breast ultrasound images using a dilated fully convolutional network combined with an active contour model | Phase-based ACM refines dilated FCN segmentation
Guo et al. (2019) | Liver | Automatic liver segmentation by integrating fully convolutional networks into active contour models | ACM refines multi-branch FCN segmentation
Zhao et al. (2018a) | Nucleus | Improved Nuclear Segmentation on Histopathology Images Using a Combination of Deep Learning and Active Contour Model | Hybrid ACM refines multi-branch FCN segmentation
Hatamizadeh et al. (2019) | Liver / Brain Lesion / Lung | Deep Active Lesion Segmentation | ACM refines signed distance maps from FC-CNN
Tang et al. (2017) | Liver / Left Ventricle | A Deep Level Set Method for Image Segmentation | Level-set ACM refines FCN segmentations iteratively
Cha et al. (2016) | Bladder | Urinary bladder segmentation in CT urography using deep-learning convolutional neural network and level sets | Multiple level-set functions segment CNN output ROIs
Hoogi et al. (2017) | Liver Lesion | Adaptive Estimation of Active Contour Parameters Using Convolutional Neural Networks and Texture Analysis | Level-set function iteratively improves CNN segmentation
Fang et al. (2019) | Breast Tumor | Combining a Fully Convolutional Network and an Active Contour Model for Automatic 2D Breast Tumor Segmentation from Ultrasound Images | Phase-based ACM refines initial contours from dilated FCNN
Xu et al. (2019) | Breast Cancer Nuclei | Convolutional neural network initialized active contour model with adaptive ellipse fitting for nuclear segmentation on breast histopathological images | ACM refines CNN segmentations
Ma and Yang (2019) | Teeth | Automatic dental root CBCT image segmentation based on CNN and level set method | ACM refines CNN segmentations
Carbajal-Degante et al. (2020) | Ventricles | Active contours for multi-region segmentation with a convolutional neural network initialization | Phase level-set function refines CNN segmentations
Liu et al. (2019) | Left Atrium | A Framework for Left Atrium Segmentation on CT Images with Combined Detection Network and Level Set Model | 3D level-set model initialized by Faster RCNN
Yang et al. (2021) | Teeth | Accurate and automatic tooth image segmentation model with deep convolutional neural networks and level set method | Level-set based on contours derived from U-Net predictions
Nunes et al. (2020) | Lung | Adaptive Level Set with region analysis via Mask R-CNN: A comparison against classical methods | ACM improves Mask R-CNN segmentations
Xie et al. (2020) | Left Ventricle | Automatic left ventricle segmentation in short-axis MRI using deep convolutional neural networks and central-line guided level set approach | Level-set model improves CNN initialization
Gong et al. (2019) | Pancreas | Convolutional Neural Networks Based Level Set Framework for Pancreas Segmentation from CT Images | Level-set model based on initial contour from CNN
Zhang et al. (2020c) | Cervical Cell / Skin Lesion | Deep Active Contour Network for Medical Image Segmentation | ACM integrated into CNN that learns initial parameters (end-to-end)
Zhang et al. (2020b) | Plaque | Faster R-CNN, fourth-order partial differential equation and global-local active contour model (FPDE-GLACM) for plaque segmentation in IV-OCT image | ACM initialized with bounding box from R-CNN
da Silva et al. (2020) | Prostate | Superpixel-based deep convolutional neural networks and active contour model for automatic prostate segmentation on 3D MRI scans | ACM refines DCNN segmentations
Kot et al. (2020) | Brain Tumor | U-Net and Active Contour Methods for Brain Tumour Segmentation and Visualization | ACM refines U-Net segmentations
Avendi et al. (2016) | Left Ventricle | A combined deep-learning and deformable-model approach to fully automatic segmentation of the left ventricle in cardiac MRI | CNN and AE initialize level set function
CNN refines ACM
Kasinathan et al. (2019) | Lung Tumor / Nodule | Automated 3-D Lung Tumor Detection and Classification by an Active Contour Model and CNN Classifier | CNN refines multiple ACM segmentations
Rupprecht et al. (2016) | Left ventricular cavity | Deep Active Contour | CNN refines ACM
Ahmed et al. (2009) | Brain | A Hybrid Approach for Segmenting and Validating T1-Weighted Normal Brain MR Images by Employing ACM and ANN | ANN based on ACM preprocessed images
## 5 Topology based Approaches
An alternative approach to integrating shape priors into network-based
segmentation was presented in Lee et al. (2019). Here, the segmentation
started with a candidate shape which was topologically correct (and
approximately correct in terms of its shape), and the network was trained to
provide the appropriate deformation to this shape such that it maximally
overlapped with the ground truth segmentation.
Such methods can be considered to have a ‘hard prior’ rather than the ‘soft-
prior’ of the methods presented above in the sense that the end result can be
guaranteed to have the correct shape. However, this approach may be limited by
a requirement that the initial candidate shape be very close to an acceptable
answer such that only small shape deformations are needed. A further potential
issue is that the deformation field provided by the network may need to be
restricted to prevent the shape from overlapping itself and consequently
changing its topology.
The differentiable properties of persistent homology Edelsbrunner et al.
(2000) make it a promising candidate for the integration of topological
information into the training of neural networks. The key idea is that it
measures the presence of topological features as some threshold or length
scale changes. Persistent features are those which exist for a wide range of
filtration values, and this persistence is differentiable with respect to the
original data. There have recently been a number of approaches suggested for
the integration of PH and deep learning, which we briefly review here.
In Chen et al. (2018) a classification task was considered, and PH was used to
regularise the decision boundary. Typical regularisation of a decision
boundary might encourage it to be smooth or to be far from the data. Here, the
boundary was encouraged to be simple from a topological point of view, meaning
that topological complexities such as loops and handles in the decision
boundary were discouraged. Rieck et al. (2018) proposed a measure of the
complexity of a neural network using PH. This measure of ‘neural persistence’
was evaluated as a measure of structural complexity at each layer of the
network, and was shown to increase during network training as well as being
useful as a stopping criterion.
PH is applied to image segmentation, but the PH calculation has typically been
applied to the input image and used as a way to generate features which can
then be used by another algorithm. Applications have included tumour
segmentation Qaiser et al. (2016), cell segmentation Assaf et al. (2017) and
cardiac segmentation from computed tomography (CT) imaging Gao et al. (2013).
Recently Clough et al. (2019a) proposed to use PH not to the input image being
segmented, but rather to the candidate segmentation provided by the network.
In an extended work Clough et al. Clough et al. (2020) the topological
information found by the PH calculation can be used to provide a training
signal to the network, allowing an differentiable loss function to compare the
topological features present in a proposed segmentation, with those specified
to exist by some prior knowledge.
## 6 Discussion
As the deep learning research effort for medical image segmentation is
consolidating towards incorporating shape constraints to ensure downstream
analysis, certain patterns are emerging as well. In the next few subsections,
we discuss such clear patterns and emerging questions relevant for the
progress of research in this direction.
### 6.1 End-to-End vs post/pre-hoc
With the maturity of research, this field is clearly moving beyond post-/pre-
hoc setting towards more systematic end-to-end training approaches. This
effect is depicted in Figure 4 where the paper counts are aggregated from this
work and Jurdi et al. (2020). The maturity of deep learning frameworks
(especially PyTorch), novel architectures (especially generative modeling) and
automatic differentiation make it possible to incorporate complex shape-based
loss functions during training. With the availability of these tools, large
models can be trained with tailored shape streams in the model architecture to
incorporate shape information.
Figure 4: Temporal trend towards end-to-end approaches
### 6.2 Semi-supervised segmentation
The ability to incorporate additional information using shape as a prior can
aid in reducing the total number of necessary annotations in achieving a good
segmentation. The shape priors can useful in generating controlled data
augmentations for the medical image analysis task in hand and reduce the
number of unrealistic augmentations. This would be instrumental in particular
in the case of rare diseases, where there is not enough of data and manual
annotations to train a neural network. The shape priors that are giving clues
about the expected pathology in such cases can lead to better segmentation
accuracy in the final output.
### 6.3 Effectiveness in pathological cases
One common theme identified by last few decades worth research on shape
modeling is the difficulty in representing the pathological shapes. While the
"typical shapes" i.e. normal shapes lie in a low-dimensional sub-manifold, the
pathological cases have a long tail in the distribution (e.g. congenital heart
diseases). That is normal shapes are self-similar but pathological cases
contain atypical shapes along with typical pathologies. Traditional linearized
shape modeling had trouble addressing this issue whereas the non-linear
modeling of shape statistics had its issue in terms of intractable numerics.
Whether a neural approach can address this overarching problem of encoding
pathological shapes is an open problem. Unfortunately, from our literature
search, we have not found any clear direction to address this perennial issue
of shape modeling.
### 6.4 Evaluation
While the shape constraints are becoming increasingly commonplace for medical
image segmentation, we believe the visual perception and human comprehension
plays a significant role behind the interest of the community. The more
general question of real world effectiveness of these methods are not often
studied. For example, how effective these shape constraints are under noisy
annotation is an open question? While the segmentation quality is most often
measured by the Dice metric, Maier-Hein et al. (2018) has already prescribed
to move beyond Dice to evaluate the segmentation quality. Topological accuracy
of anatomical structures is increasingly used as an evaluation metric to
address the shortcomings of classical image segmentation evaluation metric in
medical image analysis Byrne et al. (2020). Finally, segmentation is typically
a mean to an end. As such, the effectiveness of these segmentation techniques
should be measured quantitatively for downstream evaluation tasks such as
visualization, planning Fauser et al. (2019) etc.
## 7 Conclusion
Bringing prior knowledge about the shape of the anatomy for semantic
segmentation is a rather well-trodden idea. The community is devising new ways
to incorporate such prior knowledge in deep learning models trained with
frequentist approach. While the Bayesian interpretation of deep learning
segmentation networks is an upcoming trend, it is already shown that under
careful considerations, prior knowledge about the shape can be incorporated
even in frequentist approaches with significant success.
We see the future research concentrating more on end-to-end networks with the
overarching theme of learning using Analysis-by-synthesis. Early work has
demonstrated the effectiveness of shape constraints in federated learning and
this will be a major direction in the coming years.
We believe the community needs to address the issues discussed in Section 6
before shape constrained segmentation can be considered as a trustworthy
technology in practical medical image analysis. To this end, we can think of
shape constrained segmentation as a technical building block within a bigger
image analysis pipeline rather than a stand-alone piece of technology. For
example, in the case of surgical planning and navigation pipeline, such shape
constraints can be meaningful provided the performance is thoroughly validated
under pathological cases with multiple quality metrics. Important steps have
already been taken in this direction. In short, along with exciting results,
shape constrained deep learning for segmentation opens up many possible
research questions for the next few years. Proper understanding and answering
those hold the key to their successful deployment in the real clinical
scenario.
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# Aalto-1, multi-payload CubeSat: In-orbit results and lessons learned
M. Rizwan Mughal111M. Rizwan Mughal is also associated with Electrical
Engineering Department, Institute of Space Technology, Islamabad, Pakistan,
Correspondence<EMAIL_ADDRESS>J. Praks R. Vainio P. Janhunen J.
Envall A. Näsilä P. Oleynik P. Niemelä A. Slavinskis J. Gieseler N.
Jovanovic B. Riwanto P. Toivanen H. Leppinen T. Tikka A. Punkkinen R.
Punkkinen H.-P. Hedman J.-O. Lill J.M.K. Slotte Department of Electronics
and Nanoengineering, Aalto University School of Electrical Engineering, 02150
Espoo, Finland Department of Physics and Astronomy, University of Turku,
20014 Turku, Finland Finnish Meteorological Institute, Space and Earth
Observation Centre, Helsinki, Finland VTT Technical Research Centre of
Finland Ltd, Espoo, Finland Tartu Observatory, University of Tartu,
Observatooriumi 1, 61602 Tõravere, Estonia Department of Future Technologies,
University of Turku, 20014 Turku, Finland Accelerator Laboratory, Turku PET
Centre, Åbo Akademi University, 20500 Turku, Finland Physics, Faculty of
Science and Technology, Åbo Akademi University, 20500 Turku, Finland
###### Abstract
The in-orbit results and lessons learned of the first Finnish satellite
Aalto-1 are briefly presented in this paper. Aalto-1, a three-unit CubeSat
which was launched in June 2017, performed Aalto Spectral Imager (AaSI),
Radiation Monitor (RADMON) and Electrostatic Plasma Brake (EPB) missions. The
satellite partly fulfilled its mission objectives and allowed to either
perform or attempt the experiments. Although attitude control was partially
functional, AaSI and RADMON were able to acquire valuable measurements. EPB
was successfully commissioned but the tether deployment was not successful.
In this paper, we present the intended mission, in-orbit experience in
operating and troubleshooting the satellite, an overview of experiment
results, as well as lessons learned that will be used in future missions.
###### keywords:
Aalto-1 , CubeSat , In-orbit results , Lessons learned , Aalto Spectral Imager
, Radiation Monitor , Electrostatic Plasma Brake
††journal: Acta Astronautica
## 1 Introduction
There has been a significant increase in the design, development, launch and
operation of nano and micro satellites since last two decades. A large number
countries initiated their space activities and a large number of Newspace
companies emerged as an outcome. A number of innovative platform subsystems,
payloads and missions have been proposed, designed and launched by
universities and small industry thanks to significantly reduction of
development and launch costs [1, 2, 3, 4, 5, 6, 7, 8, 9]. This has been made
possible due to availability of Commercial Off The Shelf (COTS), technology
miniaturization and affordable rides. The CubeSat standard, initially
perceived for educational purposes only, was defined by Stanford and
California Polytechnic State Universities in 1999 [10]. Since the launch of
first CubeSat in 2003, this standard has revolutionized the space industry by
playing an increasingly important role in technology demonstrations, remote
sensing, Earth observation and education [11, 12]. More recently, the CubeSats
have started to increasingly exploit the scientific and commercial use cases
[12, 13]. Being small in size, they have transformed the traditional design
approach of space systems by providing low-cost access to space [14, 15, 16,
17]. A single ride of launch vehicle can carry hundreds of CubeSat-class
satellites. Many universities are effectively using CubeSats as hands-on tools
to teach the challenging engineering concepts about the design and development
of complex interdisciplinary systems. The launch and operation phase provides
a unique learning experience to university teams enabling them to learn
essential skills in mission design and operations [18]. Now a day’s university
CubeSat missions aim at real science and technology demonstration while also
ensuring the educational objectives. It is important for CubeSat community to
share the knowledge, in orbit experiences, lessons learned and mission details
which will consequently help other teams to gain valuable experience and not
repeat the same mistakes.
The current small satellite literature lacks the whole life cycle: i.e. all
aspects relating to mission planning, design, launch, operations and lessons
learned. The teams either report very specific technical information of the
design, or come up with mission descriptions and in-orbit results. One can
barely find information in the current literature about complete life cycle
covering a wide range of aspects. In order to provide the CubeSat community
with the sufficient details on complete aspects in terms of technology
development, technology demonstration and key experiences, we present the
design, development and in-orbit experience of Aalto-1, the first satellite of
Aalto University, Finland. We present our findings in two papers: the first
one covering the technology development aspects [19] whereas the present paper
covers the in-orbit results and lessons learned.
## 2 Mission overview
Aalto-1, shown in Fig. 1, is a 3U CubeSat designed and developed by Aalto
university and partner organizations. The spacecraft was launched in June 2017
and hosted three payloads: AaSI, RADMON and EPB.
Figure 1: Overview of Aalto 1 subsystems and photograph of FM.The highlighted
subsystems are: 1) Radiation Monitor (RADMON), 2) Electrostatic Plasma Brake
(EPB) 3) Global Positioning System’s (GPS’s) antenna and stack interface
board, 4) Attitude Determination and Control System (ADCS), 5) GPS and S-band
radio, 6) Aalto Spectral Imager (AaSI), 7) Electrical Power System (EPS), 8)
On-Board Computer (OBC), 9) Ultra High Frequency (UHF) radios, 10) solar
panels, 11) electron guns for EPB, 12) S-band antenna, 13) debug connector
AaSI is the first hyperspectral imaging system compatible with nanosatellites,
based on a piezo-actuated tunable Fabry–Pérot Interferometer (FPI) which
allows for an unprecedented miniaturization [20]. The instrument fits in a
half of CubeSat unit and, within a few seconds, can acquire spectral images in
tens of freely programmable channels. The filter works in the spectral range
of 500–900 nm where each channel is 10–20 nm wide. A 512$\times$512-pixel
sensor with a 10∘ field of view provides a ground resolution better than 200 m
per pixel.
RADMON, fitting within 0.4 CubeSat units, is one of the smallest particle
detectors, which has proven itself capable of taking scientific measurements
[21, 22]. It measures electron energies in the $>$1.5 MeV range and protons in
the $>$10 MeV range.
EPB is novel deorbiting technology which employs the coulomb drag between the
ionospheric plasma and a long charged tether [23, 24]. The tether is deployed
using a centrifugal force and it is estimated that a 100-m tether (such as on-
board Aalto-1) could decrease an altitude by 100 km of a three-unit CubeSat
within 600 days [25]. A similar experiment was carried on-board ESTCube-1 [26,
27] where tether deployment was not successful [28]. While Aalto-1 EPB
experiment was improved based on ESTCube-1 ground test results, yet the
deployment of EPB was not successful. This is due to the fact that Aalto-1
flight hardware had to be delivered soon after the ESTCube-1 experiment was
carried out and, therefore, the team did not have time and resources to
redesign the EPB module, as it is being done for the FORESAIL-1 mission [25].
In this paper, section 3 briefly introduces mission timeline representing the
launch and operations. Section 4 presents the in-orbit results and lessons
learned of all the payloads. RADMON in-orbit results are introduced briefly
based on previously published results [21, 22]. EPB in-orbit results and
lessons learned are discussed in detail, especially the possible reasons of
tether deployment failure. AaSI detailed in-orbit results are presented here
for the first time. Furthermore, Section 5 introduces in-orbit experience of
platform’s subsystems. Section 6 discusses the results and concludes the
paper.
## 3 Mission timeline
The spacecraft was launched aboard PSLV-C38 launch vehicle at 05:59 Eastern
European Time (EET) and the first beacon was recorded by a Software Defined
Radio (SDR) located in South Africa at approximately 08:30 EET. The first
contact with the Aalto University ground station was established during the
first pass at 10:07 EET.
During the consequent passes, several responses were recorded, but were not
decoded due to an unidentified problem in the ground station reception chain.
Later on the problem was troubleshooted to be in mast pre-amplifier. While
powering it off provided a directional link with the CubeSat, it came at a
cost – a loss in the signal strength.
The mission wise timeline on the commissioning and operations of each
experiment is presented in Fig. 2. During Launch & Early Operations Phase
(LEOP), the first AaSI picture was downloaded and RADMON commissioning phase
was started. As part of Aalto-1 operations, multiple AaSI campaigns have been
completed. RADMON operations resulted in a useful data set during nominal
conditions and also during a solar storm. EPB campaign resulted in partial
success in commissioning phase but failure in tether deployment.
Figure 2: Mission timeline
## 4 Mission payloads
This section describes the in orbit performance of RADMON, EPB and AaSI
payloads. The thorough design approach, selection and implementation has been
presented in accompanying paper [19].
### 4.1 Radiation monitor mission
The RADMON is a small (4$\times$9$\times$10 $\text{cm}^{3}$, 360 g) low-power
(1 W) radiation monitor [29, 19]. The monitor detects protons and electrons
employing a regular E – E analysis to distinguish between particle species.
The detectors of the instrument are a 2.1$\times$2.1$\times$0.35
$\text{mm}^{3}$ silicon detector and a 10$\times$10$\times$10 $\text{mm}^{3}$
CsI(Tl) scintillation detector placed inside a brass envelope (see Figure 3).
The envelope of the detector compartment is opaque for protons below 50 MeV
and electrons below 8 MeV. The envelope has a 280 $\mathrm{\mu m}$ aluminum
entrance window that stops low energy photons and low energy charged
particles. A particle must hit both detectors to be registered. Therefore, the
thicknesses of the entrance window and the silicon detector set the lower
energy threshold for protons to about 10 MeV and electrons to about 1.5 MeV. A
detailed description of the instrument calibration is presented in [22].
Figure 3: The RADMON radiation monitor cross section. The arrow on the picture
shows a particle that is incident within the instrument aperture. The brass
case is light-brown. The silicon detector is light blue, surrounded by a blue
passive silicon area, which is fixed on a printed circuit board (PCB) shown as
dark gray. The CsI(Tl) scintillator is shown in green. Under the scintillator
there is a photodiode shown in dark blue. White structure on the bottom is an
alumina case of the photodiode.
#### 4.1.1 In-orbit results
RADMON in-orbit calibration campaign was carried out in September 2017. It was
discovered that the gain of the scintillator did not match the value obtained
from ground calibrations, but was about 30% lower. The reason could not be
positively determined, but the deterioration of the optical contact between
the CsI(Tl) crystal and the photodiode during launch vibrations could
potentially be responsible for this decay of performance. A successful in-
flight calibration was, however, achieved using data obtained in a dedicated
calibration mode, which allows raw data from detectors to be down-linked. The
in-flight calibration is discussed in detail in [22].
The first observational campaign of RADMON started on 10 October 2017 and
lasted until 2 May 2018. Using these data, it has been demonstrated in [21]
that the instrument is able to measure the integral intensities of electrons
above 1.5 MeV and protons above 10 MeV in Low Earth Orbit (LEO), reflecting
the dynamic environment of the radiation belts. Fig. 4 shows the temporal
evolution of daily electron intensities from October to December 2017 with
respect to McIlwain $L$ parameter [30] (indicating the equatorial distance of
drift shells) together with the $Dst$ (disturbance storm time) index as a
measure for geomagnetic storm intensity [31, 32]. The two observed moderate
geomagnetic storms result in strong enhancements of the outer radiation belt,
while periods following small storms are characterized by reduced electron
intensities in the outer belt.
Figure 4: From top to bottom: Time series of $Dst$ index and four histograms
of integral intensities with respect to $L$ parameter obtained by the
different RADMON electron channels from 10 Oct 2017 to 21 Dec 2017. The z-axis
gives color-coded arithmetic daily mean of intensity per bin – note that the
color scale is different for all panels in order to enhance the details of all
channels which have different sensitivities. Figure adapted from [21] by
permission of Elsevier, ©2019 COSPAR.
Figure 4 also illustrates the contamination of all electron measurements by
higher energy protons: the constantly increased intensities in the $L$ range
below 2 correspond to the proton-dominated inner radiation belt. Further
comparisons with electron spectra observed in a similar but slightly higher
orbit (820 km) by the Energetic Particle Telescope (EPT) onboard the ESA
minisatellite (volume $<$1 m3) PROBA-V (PRoject for OnBoard Autonomy-
Vegetation) showed a good agreement for the $>$1.5 MeV electron channel of
RADMON [21].
Next observation effort was made late in 2019 to check if the instrument
functions well. We have ensured that the instrument is in a good shape, but
the satellite lacks power for continuous operations of RADMON. A compromise
was found to keep RADMON operating for every 12 hours with a 3-hour break to
ensure recharge of the satellite battery. A new set of calibration data from
the end of 2019 confirmed that the calibration of the detectors had not
changed during the 2.5 years in space and that no visible signs of detector
degradation could be identified.
#### 4.1.2 Lessons learned
RADMON is a successful space experiment and, certainly, it can be improved.
Minimization of the contamination of electron channels by high-energy protons
would be the most valuable improvement for the instrument. The collimator
geometry should also be streamlined to achieve an optimal instrument aperture.
The current design is such that particles enter the instrument within a
$\approx 20\degree$ half-width cone defined by an opening in the brass
container. The opening is manufactured as a right-angle shaft sufficiently
larger than the dimension of the silicon detector (see Fig. 3).
An incident particle may, therefore, hit a side of the silicon detector in a
way that it deposits energy into its active area and its passive area in an
arbitrary proportion. Further, it hits the scintillation detector. This effect
leads to an underestimation of energy deposited in the E detector.
Subsequently, such a particle is misclassified. A silicon detector with two
concentric active areas would contribute to better particle classification and
reduce contamination of electron channels by protons. The detector should
trigger on the central dot and add the energy deposited in the encircling area
to its output signal. One of the possible geometries could be a ”sandwich”
detector with a thinner layer carrying the central spot and a thicker layer
beneath. In this case, it is even easier to get the correct E signal since the
energy loss in the thinner layer would not be needed for the pulse height
analysis. Any signal above the threshold would gate the particle detection by
the E – E detectors below. The thickness of the top detector can be about
100–150 $\mathrm{\mu}$m and should be optimized for scientific requirements. A
thicker detector would show more edge effects than a thin one, but could have
a better signal-to-noise ratio. The thickness of the entrance window should be
adjusted as well during the optimization.
Another issue is that the current geometry allows a gradual increase in the
angle of the acceptance cone. The collimator should be designed as a conical
opening in the shielding container so that it becomes transparent at sharper
energy threshold. It would improve the flatness of the particle response at
moderate energies.
A simulation of the suggested layered design carried out within the Geant4
[33, 34] framework is compared to a simulation of the current design in Fig.
5. The ”sandwich” has a thin silicon detector right on top of the E silicon
detector. Both detectors are square and of the same size. The instrument
container has a tantalum front wall, which can be optimized further to a
tantalum lining of the container opening. This reduction is possible since the
upper thin silicon detector sets the accepted solid angle for a particle to be
detected. High energy protons coming within the aperture are still detected as
electrons. Nevertheless, limiting the solid angle of the instrument acceptance
for such protons improves the quality of the observational data. In a proton-
rich environment, such as South Atlantic geomagnetic anomaly, contamination of
electron channels could be used as a secondary proxy on the proton population.
Figure 5: The contamination of electron channels (e3 and e5 are chosen as
examples) by high energy protons in comparison to a proposed ”sandwich”
design.
As a positive takeaway from the experiment, the successful RADMON re-
calibration using in-flight data showed that a dedicated mode allowing the
full pulse height data to be downloaded also from space can render a RADMON-
like instrument to a self-calibrating device. Thus, an expensive full
calibration campaign in high-energy beam facilities, reaching hundreds of MeVs
in proton energies, can be avoided using this approach.
### 4.2 Electrostatic Plasma Brake
The key components of the EPB payload are those of the tether reeling
mechanism as shown in Fig 6. These include the tether reel, reel motor (not
visible), tether chamber, tether tip mass, tip mass launch lock (Kaiku), and
reel launch lock (Kieku). The reel motor (vacuum qualified piezo motor) is
nested inside the reel. The control electronics underneath the tether reeling
Printed Circuit Board (PCB), separate high voltage PCB, and electron emitters
are similar to those of ESTCube-1 as described earlier in the literature [27].
Only changes introduced were related to the revised launch locks and
additional diagnostics. The high voltage converter was changed to double the
voltage from $\pm$ 500 kV to $\pm$ 1000 kV which caused some minor changes in
the electron emitters.
The revised launch locks and additional diagnostics included the following
components. The reel lock was newly designed and diagnostics was added. Behind
the spring loaded lock shaft is an optoport (black component next to the lock
in the left panel of Fig. 6. When the lock was burned the state of the
optoport was designed to change from open to close. To monitor the tip mass
before and after the tip mass lauch lock was released, a pair of
phototransistor (Kyylä) and IR LED (Soihtu) was mounted in the tether chamber
opposite to the opening tube of the tip mass and tether (two holes in the top
right corner of the tether chamber in the right panel of Fig. 6). The IR LED
can be used to healty check the phototransistor prior to the tip mass release.
After the release, if the tether should be damaged, the phototransistor
observed light freely entering to the tether chamber.
Figure 6: EPB Mechanical parts on the PCB (left) and key tether reeling
components: reel, tip mass, tether chamber, and tip mass launch lock (right).
The reel lock can be seen on the left panel left side of the tether chamber.
#### 4.2.1 In-orbit results
The in-orbit tests of the plasma brake started with a commission phase, in
which the On Board Computer (OBC) sent EPB a number of commands with the goal
of verifying its operational state. This list included essentially all the
commands which were safe to run without any risk of hazard. This restriction
ruled out e.g. the commands that would initiate physical changes in the
payload’s status (launch lock burns, motor activation) or the ones not usable
at this point of the mission (high voltage or electron gun activation). The
commands that were run all worked as designed, returning some housekeeping
data for analysis. Most importantly at this point, the data showed that all
systems were at nominal state and that the launch locks had kept the tether
reel and the tip mass intact.
The second step in in-orbit tests was to open the two launch locks that had
locked the tether reel and the tip mass during the launch. Each lock was
opened by applying a 150 mA current, which would melt the dyneema string
keeping the spring loaded lock at closed state. The tether reel lock, named
Kieku, had an integrated optical diagnostics system whose state could be read
by the OBC at any time. During the burn sequence the system’s state switched
from “locked” to “deployed” after about 12 seconds of burning as shown in Fig.
7. Similar diagnostics were not available for the tip mass lock Kaiku. The
duration of the burn current for Kaiku was chosen long enough to ensure a
proper deployment.
Figure 7: Flight data showing the deployment of the tether reel lock Kieku.
The final verification of the operational readiness before attempting tether
deployment was performed with the help of the photosensor Kyylä. Kyylä is a
simple phototransistor placed inside the tether reel chamber. It has the
backside of the tip mass in the center of its field of view. If the tip mass
had been ejected from its nest prematurely, the light (e.g. from Earth albedo)
entering the chamber could easily be detected in Kyylä’s signal. An example
data plot from an early Kyylä scan is shown in Fig. 8. The extremely narrow
width of the peaks indicate that even though light is able to enter the
chamber, it is able to do so over a very narrow angle only, as the satellite
is spinning. This may be explained as follows. The tip mass, roughly
cylinderic, remains like a plug in the tether opening tube. The tip mass is
not tightly in the tube but held to its place by the launch lock. Thus there
is a tiny gap between the tip mass and the tube walls that provides the light
with a passage of the narrow angle. If the tip mass was completely removed,
the shape of the peaks would be considerably wider. Another piece of
information obtained from these tests was the confirmation of the satellite’s
spin rate. An approximate seven second periodicity of the peaks coincided
precisely to the angular velocity data of the Attitude Determination & Control
Subsystem (ADCS). Simultaneously it provided proof that Kyylä was indeed
measuring real phenomena of its surroundings and not some arbitrary electrical
disturbances.
Figure 8: Flight data from the phototransistor Kyylä. The periodicity of the
signal corresponds to the satellite’s spin rate at the time.
After the successful initial tests and preparations it was time to attempt
tether deployment. A controlled spin-up of the satellite could not be
performed due to the shortcomings of the satellite’s ADCS which are described
in detail in section5. The satellite was nonetheless spinning through natural
causes and its spin axis and angular velocity ($\approx$50 degrees per second)
were, by chance, suitable for taking a shot at deployment.
The spin rate for EPB deployment was verified by magnetometer and gyroscope
telemetry data. Figures 9 and 10 present the high resolution measurement data
in time and frequency domains respectively and Fig. 11 presents the calibrated
gyroscope data during the EPB deployment campaign.
Figure 9: Magnetometer high resolution data during EPB deployment campaign
Figure 10: Magnetometer high resolution flight data during EPB deployment
campaign in frequency domain confirming the spin rate around the spin axis
Figure 11: High resolution gyroscope calibrated flight data during EPB
deployment campaign
Despite having achieved the desired spin rate around tether deployment axis,
the deployment attempts all failed, unfortunately. In each attempt the motor
was commanded to make a turn that is relatively small but still easily
detectable. We couldn’t observe any changes in the tether reel rotary
position. The vacuum qualified piezo motor has an in-built potentiometer based
rotary encoder. Fig. 12 shows the values measured by this encoder throughout
the tether deployment trials. The peak-to-peak variation of the values
corresponds to a 1.4° turn, or 0.4 mm on the perimeter of the reel. The
conclusion must be that no detectable motor movement has taken place. If the
motor had worked nominally, the turn angle would have been tens of degrees.
Figure 12: Measured values of the rotary encoder of the tether reel motor.
These values were recorded over several tether deployment attempts. The peak-
to-peak variation of the values corresponds to a turn of 1.4 degrees. The pre-
launch value recorded in the last ground tests was 421.
#### 4.2.2 Lessons learned
The most noticeable result of the EPB mission is obviously the failure of the
tether deployment hardware. It is somewhat unclear why this happened, even
though several clues exist. Figure 12 shows examples of the measured motor
voltage during two tether deployment attempts. In normal operation the motor
voltage would remain in its nominal value of approximately 40 volts. As the
plots show, the voltage is cut off and starts a rather rapid decay as soon as
it has been switched on. The motor voltage is generated within the EPB control
electronics with the help of a boost converter. In Fig. 12 the voltage appears
to saturate at the level of the boost converter’s input voltage. This would
indicate that the faulty operation of the boost converter is the source of all
grief.
Figure 13: Two data sets of the motor voltage during the tether deployment
attempts. Notice the saturation at the level of the input voltage ($\approx$11
volts) of the boost converter. In each set the last data point was recorded
after the input voltage had been switched off.
Not all went haywire, though. Several newly developed systems, some including
moving parts, worked exactly as planned. Especially the completely renewed
design for the tether reel lock Kieku proved to be a reliable work horse in
space. At this point it is important to introduce the reader to the launch
history of the EPB payload. A very similar payload was first launched on-board
ESTCube-1 [26, 27]. Its fate was identical to that of Aalto-1 EPB. It is
important to note that the timelines of the two satellite missions overlapped
in a most unfortunate way. Once the in-orbit results of ESTCube-1 were ready
and verified, the delivery date of the Aalto-1 flight model hardware was only
four months away. Also, due to the lack of proper on-board diagnostics, the
reasons for the failure were mostly unknown. Therefore the EPB team was
lacking both the proper time and the accurate knowledge of the problem in
order to make fundamental changes in the motor hardware and control
electronics. Instead, a number of features were added to gather all the
information possible, in order to at least see what is happening in case of
repeated failure. All these diagnostics tools described above (Kyylä, Kieku’s
optical feedback, motor’s position encoder) worked as planned. This allowed
the EPB team to have an instant view of the situation in orbit and finally get
valuable clues of what happened on-board ESTCube-1 as well. The last minute
changes could not help the Aalto-1 EPB to complete its mission, but at the
very least they helped in compiling a road-map towards more successful
missions in the future. A small step for Coulomb drag industry, but a step
forward nonetheless.
### 4.3 Aalto-1 Spectral Imager AaSI
#### 4.3.1 In-orbit results
After establishing communications, the VIS camera was first powered on the 3rd
of July, 2017. The first housekeeping data from the camera indicated nominal
behaviour, and the instrument temperature was ca. $-5^{\circ}$C. The first
image, as shown in Fig. 14, was taken on the 5th of July, while the satellite
was still tumbling. During image acquisition, the satellite was located over
Norway with the field of view pointed to the southern direction towards
Denmark. Based on visual analysis, the image quality is good, and no visible
de-focusing or new aberrations are present.
Figure 14: The first image downlinked from Aalto-1. The image is taken with
the VIS camera on 5th of July, 2017 and it shows the coastline of Denmark
together with Earth’s horizon.
The spectral camera was first powered on the 25th of July, and the instrument
housekeeping data was nominal. The temperature was around $-5^{\circ}$C, and
the piezo voltages for the FPI were between 26 V and 28 V, which indicated
perfect health for the FPI unit. When compared to piezo voltages measured on
ground prior to launch, there was approximately 10 V difference in one of the
channels. This was expected as there is a temperature difference between the
measurements (+22∘C at the pre-launch check vs. $-5^{\circ}$C in orbit) and
the water absorbed by the piezo actuators has evaporated at the time of taking
in-orbit measurements.
Figure 15: False color composite of the first spectral image captured by AaSI.
The approximate wavelengths in the image are R=710 nm, G=535 nm and B=510 nm.
The tumbling of the satellite is clearly visible as the frames do not contain
much overlap. The bottom part of the image shows as yellow, as the wavelengths
535 nm and 710 nm are extracted from the same raw image.
First images were taken with the spectral camera on the 3rd of August. From
these images, the functionality of the camera optics was verified. The imaged
scene was covered by clouds, and in the false color composite as shown in Fig.
15, one can see spectral variation in the clouds. During this time, the
satellite was still tumbling quite rapidly, so the imaged area is moving
significantly between the spectral frames.
After the performance of optics was verified, the on-board spectral
calibration method was tested. The calibration is based on measuring a bright
target (e.g. cloud or desert) and scanning the spectral filter over the cutoff
wavelength of the 900 nm short pass filter and taking an average of the pixel
values. The sequential images are recorded with very small wavelength
increment. When the spectral transmission peak passes over the short pass
filter, the signal level will drop. When the signal is plotted as a function
of FPI set point voltage, the drop in signal level is visible. The location
where the slope is steepest corresponds to the cutoff wavelength of the
shortpass filter. Successful calibration measurement was performed on the 5th
of September which is shown in Fig. 16. When compared to measurements done on
ground, it can be seen that the spectral behaviour is similar, but due to the
cold temperature ($-16^{\circ}$C) and different illumination conditions the
shape of the calibration spectrum is different.
Figure 16: Calibration measurement comparison. In the top figure, the signal
level is plotted as function of FPI set point voltage. Signal derivative is
plotted in the bottom figure. The position with the steepest slope corresponds
to the filter cutoff wavelength. The filter cutoff position is visible in both
cases, but the measurement performed in orbit is distorted. This is mainly due
to the cold temperature, which is outside the instrument’s operation
temperature.
The satellite was de-tumbled in June 2018 and the imaging campaign was
continued immediately after de-tumbling. During this campaign, an image mosaic
was created from VIS images, and finally on August 6, 2018 the first cloud-
free images of land targets were acquired. The imaging sequence started at the
equator above Congo, and continued for about six minutes while the satellite
was travelling south toward South Africa. The images of six different
wavelengths were acquired, and, from the resulting spectrum, the red-edge of
vegetation is clearly visible, as shown in Fig. 17.
Figure 17: The first cloud-free spectral image of a land target (top). The
image is centered on Tshuapa River near Mbandaka. The false color image is
constructed from R=752 nm, G=671 nm, B=565 nm. The bottom figure shows the
spectrum of the central area of the image measured at 6 wavelenghts.
An image compression program was uploaded to the satellite during the spring
of 2018. This was first tested around the midsummer of 2018, and several
series of images were taken. In order to downlink the image mosaics, image
compression was required. After compression, the images were successfully
downlinked. The stiched mosaic is shown in Fig. 18. The slow tumbling of the
satellite is clearly visible in the sequential images.
Figure 18: Mosaic of sequential VIS images
#### 4.3.2 Lessons learned
This was the first mission ever to demonstrate a hyperspectral camera on a
nanosatellite. It was also the first space-borne demonstration of a tunable
FPI-based nanosatellite-compatible hyperspectral camera.
The main lesson learned was that this technology works in space environment
and it can be used for nanosatellite-based hyperspectral imagers. All of the
primary mission objectives were completed, so the AaSI mission can be
considered successful.
Not all functionalities of the imager could be verified though. The tumbling
platform prevented imaging of planned targets, and the limited downlink
allowed only the use of minimal spectral mode with six wavelengths. However,
the tumbling platform showcased the benefits of frame-based spectral imaging,
as the images in different wavelengths can be overlapped in post processing.
This is a great benefit in nanosatellite missions, as the imager can still be
used in the case of attitude control malfunction.
## 5 Platform in-orbit performance
The in-orbit performance of spacecraft platform which consisted of commercial
and in house developed subsystems, is briefly presented. While the key
platform subsystems were successfully commissioned, the spacecraft
accomplished its mission with partial success. The design approach of platform
subsystems which consisted of an Electrical Power Subsystem (EPS) [35], an
ADCS [36], a Global Positioning System (GPS)-based navigation system [37], a
Ultra High Frequency (UHF) [38] and S-band [39] radios for Telemetry,
Telecommand & Communication (TT&C), and a Linux-based Onboard Data handling
(OBDH) [40] is briefly presented in [19].
### 5.1 In orbit performance of EPS
In order to monitor and keep track of the health of the spacecraft, a number
of housekeeping sensors were used. The performance of EPS is presented in
terms of telemetry values of voltage, current and temperature sensors. The
flight data of these sensors confirms that the EPS is functional and provides
power to satellite subsystems since its launch. However, there are some
issues. The telemetry data reveals partially degraded performance of one of
the solar panels as evident by green plot in Fig. 20. This behaviour is likely
due to the un-controlled spin orientation of the satellite. The telemetry data
of solar panel temperatures, EPS board temperature and battery temperatures
from launch date till Aug 2020 is plotted in Fig. 19. The highest temperature
variation takes place on the satellite surface as evident from central graphs
representing panel X and Y, where solar panel temperatures change in $\pm$20∘C
range. This range remains quite stable throughout the mission representing
that the temperature is at equilibrium. The temperature inside the satellite
depends on operation of payloads and platform subsystems. The telemetry data
of board and battery temperatures, as evident from Fig. 19, represents that
the passive thermal control maintains sufficiently stable temperature
fluctuations.
Figure 19: Aalto 1 surface and inner temperatures (in ${\circ}$C ) from launch
till Aug 2020 Figure 20: Solar panel current intensities from launch till Aug
2019. The vertical axis represents the generated current (in mA) read by each
BCR
### 5.2 In-orbit performance of ADCS
The commissioning phase of the ADCS functions were met with complications, as
some of the sensor readings were erroneous. Two of the sun sensors (on the +X
and $-$X directions of the satellite) were malfunctioned and not usable for
attitude estimation. In Fig. 21, the gyroscope readings from regular
housekeeping data until October 2017 have a low angular rate resolution. This
was because of improper processing of sensor raw data and a problem with the
communication channel in the ADCS module. This was fixed with a small firmware
update. Over the course of the mission, sometimes the ADCS module is reset and
defaulted to idle mode. Such event will turn off the ADCS sensors which needs
to be manually turned on. This shows up as frozen sensor data, visible in Fig.
21 around October–November, 2017 and February–May 2018.
The first attempt to detumble the satellite was attempted on October 2017. The
attempt failed because of constant rebooting of the ADCS module when the B-dot
control was turned on. The problem was caused by the magnetorquer driver
channel which was later fixed with a major firmware and magnetorquer driver
update on June 2018 consequently solving the reboot issue. The detumbling
operation was tested again with positive results. The spin rate of the
satellite was reduced close to 0 deg/sec as confirmed by the telemetry data of
Fig. 21. The detumbling control was kept on until September 2018, after which
the satellite started to spin up.
Figure 21: Gyroscope data from July 2017 to November 2019.
The main cause of the uncontrolled spin up of the satellite remains unknown
when the B-dot is disabled. Some possible causes are environmental disturbance
torque, residual dipole moment generated by unknown magnetic materials or
current loop from the solar panels power routing.
Although detumbling with the B-dot controller was successful, many other
mission modes, including controlled spin up for tether deployment, were not
successful. The ADCS commissioning modes, including the spin up manoeuvre, has
been tried with the magnetorquers only [6, 5]. The reaction wheels showed
inconsistencies in their power reading during the early commissioning phase
and thus have not been thoroughly tested yet.
An important lesson learned was to procure the commercial modules at the early
stages of development and test all the functional modes during the
qualification phase. Moreover, designing an in house subsystem gives more
flexibility in interfacing and testing.
### 5.3 In-orbit performance of OBDH, TT&C and GPS subsystems
Figure 22: OBC reboot events during July-Nov 2017 [40]
The OBC $-$1 branch that was enabled at satellite deployment had reset itself
after around a month after launch. The cause was initially perceived as single
event upset due to radiation in South Atlantic Anomaly, but the problem was
later found to be actually caused by a software bug in the command line. The
reboots due to software bugs, radiation and EPS reset etc. plotted in relation
to proton fluxes, are given in Fig. 22 [40]. During first five months after
launch, a total of 38 boot events occurred. A boot event may have one or more
boots, with the group having a likely common cause. A further detailed
analysis on the boot events can be read in [40]. The satellite suffered from
instability in EPS, resulting in several resets in EPS and the arbiter. A
Coronal Mass Ejection (CME) occurred in early September 2017, providing an
excellent opportunity for RADMON testing [41]. The satellite was quickly
retasked to collect as much data as possible with RADMON. A precious RADMON
set of data collection was also interrupted during a CME due to OBC reboots. A
few unexplained boot events of the OBC, which were resolved without
involvement of the arbiter, occurred during the CME event. It is suspected
that these may be related to either radiation or EPS reset.
Immediately after the launch, multiple objects launched on the same rocket as
Aalto-1 had similar Two Line Element (TLE), and it was unclear which TLE set
belonged to Aalto-1. The GPS subsystem was one of the first instruments
successfully operated after contacting the satellite, and navigation solutions
provided by the receiver allowed determining the correct TLE set. The
determined identity was also communicated to the TLE data provider [42].
It has been observed that the TLE accuracy has been sufficient for most
routine operations, and the use of GPS has been less frequent than expected. A
sub-optimal GPS antenna placement (resulting from a compromise with solar
panel placement) and satellite tumbling have caused delays in obtaining the
first fix after powering the receiver.
The commissioning of the UHF transceiver was successful since the first
contact with the CubeSat was established during first pass over the ground
station. The commissioning phase was met with many challenges which have been
briefly detailed in [19]. From the telemetry logs, a radio interference in the
Northern direction, close to the horizon, was noted at around 437.22 MHz.
Similar kind of interference around the 437.0–437.4 MHz was measured by the
UWE-3 CubeSat mission though the source of interference has not been confirmed
[43]. The S-band transmitter has not been successfully commissioned despite
multiple attempts in July 2017 and July 2018.
## 6 Discussion and conclusions
Although the mission was a partial success in terms of executing the
experiments, the important lessons learned during this mission have been
applied in the design of next variants of payloads and platforms. The RADMON
instrument was successful in commissioning and measurement phases. Its
heritage has been used to design a more complex Particle Telescope (PATE)
payload for the upcoming FORESAIL-1 mission [44]. The EPB tether could not be
deployed due to a failure in tether deployment hardware. The lessons learned
have been taken into consideration in development of the plasma brake for
upcoming FORESAIL-1 and ESTCube-2 missions [45]. The AaSI was the first
nanosatellite-compatible hyper-spectral imager to be flown in space. Aalto-1
project successfully demonstrated the expertise of VTT in both visible and
hyper-spectral miniature imager designs. The technology has many potential
future applications to serve CubeSat and/or scientific industry/community.
Since Aalto-1, VTT’s hyper-spectral imagers have been developed for Reaktor
Hello World, PICASSO, Hera and Comet Interceptor missions. The platform has
provided successful in-orbit demonstration, although some subsystems lacked
the desired performance. An important lesson learned was to perform a rigorous
test campaign while integrating the commercial and in-house built subsystems.
## Acknowledgements
The RADMON team thanks P.-O. Eriksson and S. Johansson at the Accelerator
Laboratory, Åbo Akademi University, for operating the cyclotron. Computations
necessary for the presented modeling were conducted on the Pleione cluster at
the University of Turku.
Aalto University and its Multidisciplinary Institute of Digitalisation and
Energy are thanked for Aalto-1 project funding, as are Aalto University,
Nokia, SSF, the University of Turku and RUAG Space for supporting the launch
of Aalto-1.
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# Anticoncentration versus the Number of Subset Sums
Vishesh Jain Ashwin Sah Supported by NSF Graduate Research Fellowship Program
DGE-1745302 Mehtaab Sawhney Supported by NSF Graduate Research Fellowship
Program DGE-1745302
###### Abstract
Let $\vec{w}=(w_{1},\dots,w_{n})\in\mathbb{R}^{n}$. We show that for any
$n^{-2}\leq\epsilon\leq 1$, if
$\\#\\{\vec{\xi}\in\\{0,1\\}^{n}:\langle\vec{\xi},\vec{w}\rangle=\tau\\}\geq
2^{-\epsilon n}\cdot 2^{n}$
for some $\tau\in\mathbb{R}$, then
$\\#\\{\langle\vec{\xi},\vec{w}\rangle:\vec{\xi}\in\\{0,1\\}^{n}\\}\leq
2^{O(\sqrt{\epsilon}n)}.$
This exponentially improves the $\epsilon$ dependence in a recent result of
Nederlof, Pawlewicz, Swennenhuis, and Wegrzycki and leads to a similar
improvement in the parameterized (by the number of bins) runtime of bin
packing.
title = Anticoncentration versus the Number of Subset Sums, author = Vishesh
Jain, Ashwin Sah, and Mehtaab Sawhney, plaintextauthor = Vishesh Jain, Ashwin
Sah, Mehtaab Sawhney, keywords = anticoncentration, year=2021, number=6,
received=19 January 2021, published=28 June 2021, doi=10.19086/aic.24872,
[classification=text]
## 1 Introduction
For $\vec{w}:=(w_{1},\dots,w_{n})\in\mathbb{R}^{n}$ and a real random variable
$\xi$, recall that the Lévy concentration function of $\vec{w}$ with respect
to $\xi$ is defined for all $r\geq 0$ by
$\mathcal{L}_{\xi}(\vec{w},r)=\sup_{\tau\in\mathbb{R}}\mathbb{P}[|w_{1}\xi_{1}+\dots+w_{n}\xi_{n}-\tau|\leq
r],$
where $\xi_{1},\dots,\xi_{n}$ are i.i.d. copies of $\xi$. In combinatorial
settings (where $\vec{w}\in\mathbb{Z}^{n}$) a particularly natural and
interesting case is when $r=0$ and $\xi$ is a Bernoulli random variable, i.e.,
$\xi=0$ with probability $1/2$ and $\xi=1$ with probability $1/2$. For
lightness of notation, we will denote this special case by
$\rho(\vec{w})=\mathcal{L}_{\operatorname{Ber}(1/2)}(\vec{w},0)=\sup_{\tau\in\mathbb{R}}\mathbb{P}[\langle\vec{w},\vec{\xi}\rangle=\tau].$
In this note, we study the following question.
###### Question 1.1.
For a vector $\vec{w}=(w_{1},\dots,w_{n})\in\mathbb{R}^{n}$ with
$\rho(\vec{w})\geq\rho$, how large can the range
$\mathcal{R}(\vec{w})=\\{w_{1}\xi_{1}+\dots+w_{n}\xi_{n}:\xi_{i}\in\\{0,1\\}\\}$
be?
The two extremal examples here are $\vec{w}=(0,0,\dots,0)$, which corresponds
to $\rho(\vec{w})=1$, $|\mathcal{R}(\vec{w})|=1$ and
$\vec{w}=(1,10,\dots,10^{n-1})$, which corresponds to $\rho(\vec{w})=2^{-n}$,
$|\mathcal{R}(\vec{w})|=2^{n}$. Motivated by these examples, one may ask if
there is a smooth trade-off between $\rho(\vec{w})$ and
$|\mathcal{R}(\vec{w})|$. This turns out not to be the case. Indeed, for any
$\epsilon>0$, Wiman [6] gave an example of a $\vec{w}\in\mathbb{Z}^{n}$ for
which $|\mathcal{R}(\vec{w})|\geq 2^{(1-\epsilon)n}$ and $\rho(\vec{w})\geq
2^{-0.7447n}$. At the other end of the spectrum, when $\rho(\vec{w})\geq
2^{-\epsilon n}$, the so-called inverse Littlewood–Offord theory [4, 5, 3]
_heuristically_ suggests that $\vec{w}$ is essentially contained in a low-rank
generalized arithmetic progression of ‘small’ volume so that
$|\mathcal{R}(\vec{w})|$ is also ‘small’. However, the number of ‘exceptional
elements’ in the inverse Littlewood–Offord theorems [4, 5, 3] is unfortunately
too large to be able to rigorously establish such a statement.
Nevertheless, in a recent work on the parameterized complexity of the bin
packing problem (see Section 1.1), Nederlof, Pawlewics, Swennenhuis and
Wegrzycki [2] showed that for any $\epsilon>0$,
$\rho(\vec{w})\geq 2^{-\epsilon n}\implies|\mathcal{R}(\vec{w})|\leq
2^{\delta(\epsilon)n},$
where
$\delta(\epsilon)=O\left(\frac{\log\log(\epsilon^{-1})}{\sqrt{\log(\epsilon^{-1})}}\right).$
(1.1)
In particular, $\delta(\epsilon)\to 0$ as $\epsilon\to 0$. Moreover, we must
have $\delta(\epsilon)\geq(2-o(1))\epsilon$, as can be seen by considering
$\vec{w}=(C_{1},\dots,C_{1},C_{2},\dots,C_{2},\dots,C_{n/k},\dots,C_{n/k})\in\mathbb{R}^{n},$
where each $C_{i}$ is repeated $k$ times, and $C_{i}$ is sufficiently small
compared to $C_{i+1}$ for all $i$. Indeed, for such $\vec{w}$, we have
$\rho(\vec{w})=2^{-(\frac{1}{2}+o_{k}(1))\frac{n}{k}\log_{2}{k}}$ while
$|\mathcal{R}(\vec{w})|\leq 2^{(1+o_{k}(1))\frac{n}{k}\log_{2}{k}}$.
We conjecture that this example is essentially the worst possible, so that
$\delta(\epsilon)\leq 2\epsilon$. We are able to show that
$\delta(\epsilon)=O(\sqrt{\epsilon}),$ (1.2)
thereby obtaining an exponential improvement over 1.1. More precisely,
###### Theorem 1.2.
Let $\epsilon>0$. For any $n\geq\epsilon^{-1/2}$ and any
$\vec{w}\in\mathbb{R}^{n}$ satisfying $\rho(\vec{w})\geq\exp(-\epsilon n)$, we
have
$|\mathcal{R}(\vec{w})|\leq\exp(C_{\ref{thm:main}}\epsilon^{1/2}n),$
where $C_{\ref{thm:main}}$ is an absolute constant.
We prove this theorem in Section 2.
### 1.1 Application to bin packing
The bin packing problem is a classic NP-complete problem whose decision
version may be stated as follows: given $n$ items with weights
$w_{1},\dots,w_{n}\in[0,1]$ and $m$ bins, each of capacity $1$, is there a way
to assign the items to the bins without violating the capacity constraints?
Formally, is there a map $f:[n]\to[m]$ such that $\sum_{i\in
f^{-1}(j)}w_{i}\leq 1$ for all $j\in[m]$?
Björklund, Husfeldt, and Koivisto [1] provided an algorithm for solving bin
packing in time $\tilde{O}(2^{n})$ where the tilde hides polynomial factors in
$n$. It is natural to ask whether the base of the exponent may be improved at
all i.e. is there a (possibly randomized) algorithm to solve bin packing in
time $\tilde{O}(2^{(1-\epsilon)n})$ for some absolute constant $\epsilon>0$?
In recent work, Nederlof, Pawlewics, Swennenhuis and Wegrzycki [2] showed that
this is true provided that the number of bins $m$ is fixed. More precisely,
they showed that there exists a function $\sigma:\mathbb{N}\to\mathbb{R}^{>0}$
and a randomized algorithm for solving bin packing which, on instances with
$m$ bins, runs in time $\tilde{O}(2^{(1-\sigma(m))n})$, where $\tilde{O}$
hides polynomials in $n$ as well as exponential factors in $m$. Their
analysis, which crucially relies on 1.1, gives a very small value of
$\sigma(m)$ satisfying
$\sigma(m)\leq 2^{-m^{9}}.$ (1.3)
Using Theorem 1.2 instead of 1.1 in a black-box manner in the analysis of [2],
we exponentially improve the bound on $\sigma(m)$.
###### Corollary 1.3.
With notation as above, the randomized algorithm of [2] solves bin packing
instances with $m$ bins in time $\tilde{O}(2^{(1-\sigma(m))n})$ with high
probability, for $\sigma\colon\mathbb{N}\to\mathbb{R}^{>0}$ satisfying
$\sigma(m)=\tilde{\Omega}(m^{-12}),$ (1.4)
where $\tilde{\Omega}$ hides logarithmic factors in $m$.
###### Remark.
This follows by noting that the function $f_{C}(m)$ in [2, Section 3.6] is
$\tilde{\Theta}(m^{-2})$ so that $\delta$ in [2, Section 3.6] is
$\tilde{\Theta}(m^{-3})$. With Theorem 1.2, the function $\varepsilon(\delta)$
in the runtime analysis of [2, Section 3.4] satisfies
$\varepsilon(\delta)=O(\delta^{2})$. Therefore, the function $f_{B}(\delta)$
in the same section is $\tilde{O}(\delta^{4})$, which is
$\tilde{\Omega}(m^{-12})$. Note that if one were able to establish the
conjecturally optimal bound $\delta=O(\epsilon)$, this would lead to
$f_{B}(\delta)=\tilde{O}(\delta^{2})$, thereby giving the quadratically better
$\sigma(m)=\tilde{\Omega}(m^{-6})$.
### 1.2 Notation
We use big-$O$ notation to mean that an absolute multiplicative constant is
being hidden. We use $\operatorname{Ber}(1/2)$ to denote the balanced
$\\{0,1\\}$ Bernoulli distribution and $\operatorname{Bin}(k)$ to denote the
binomial distribution on $k$ trials with parameter $1/2$. Recall that
$\operatorname{Bin}(k)$ is the sum of $k$ independent
$\operatorname{Ber}(1/2)$ random variables. Given a distribution $\mu$, we let
$\mu^{\otimes n}$ denote the distribution of a random vector with $n$
independent samples from $\mu$ as its coordinates. We also use the following
standard additive combinatorics notation: $C+D=\\{c+d:c\in C,d\in D\\}$ is the
sumset (if $C,D$ are subsets of the same abelian group), and for a positive
integer $k$, we let $k\cdot C=C+\cdots+C$ ($k$ times) be the iterated sumset.
Finally, in some cases we will use the notation $\Sigma\cdot$ or $\int\cdot$
to denote that the expression in the sum or integral is the same as in the
previous line to simplify the presentation of long expressions.
### 1.3 Outline of the proof
As in [2], the starting point of our proof is the following observation: let
$A$ denote a fixed (but otherwise arbitrary) set of unique preimages for
points in $\mathcal{R}(\vec{w})$ (hence, $|A|=|\mathcal{R}(\vec{w})|$) and let
$B$ denote the the set of preimages of a value $\tau\in\mathbb{R}$ realising
$\rho(\vec{w})$. Then (Lemma 2.2) for any $k\geq 1$, the map $A\times(k\cdot
B)\to A+k\cdot B$ is a bijection. In particular, if $\vec{a}$ is sampled from
the uniform distribution on $A$ and $\vec{b}_{1},\dots,\vec{b}_{k}$ are
independently sampled from the uniform distribution on $B$, then
$\displaystyle|A|$
$\displaystyle=|A|\cdot\mathbb{P}[\vec{a}+\vec{b}_{1}+\dots+\vec{b}_{k}\in\\{0,\dots,k+1\\}^{n}]$
$\displaystyle=|A|\cdot\sum_{\vec{x}\in\\{0,\dots,k+1\\}^{n}}\mathbb{P}[\vec{a}+\vec{b}_{1}+\dots+\vec{b}_{k}=\vec{x}]$
$\displaystyle\leq|A|\cdot\sum_{\vec{x}\in\\{0,\dots,k+1\\}^{n}}\mathbb{P}[\vec{a}=\vec{a}(\vec{x})]\cdot\mathbb{P}[\vec{b}_{1}+\dots+\vec{b}_{k}=\vec{x}-\vec{a}(\vec{x})]$
$\displaystyle\leq\sum_{\vec{x}\in\\{0,\dots,k+1\\}^{n}}\mathbb{P}[\vec{b}_{1}+\dots+\vec{b}_{k}=\vec{x}-\vec{a}(\vec{x})]$
In [2], the largeness of $B$ is exploited by finding, for every $a\in A$, a
large subset of $B$ which is ‘balanced’ (in a certain sense) with respect to
$a$. Instead, we exploit the largeness of $B$ directly by using the
observation that the density of the uniform measure on $B$ with respect to the
uniform measure on $\\{0,1\\}^{n}$ is at most $2^{n}/|B|\leq 2^{\varepsilon
n}$. In particular, if we let $\mu_{k}$ denote the measure on $k\cdot B$
induced by the product measure on $B\times\dots\times B$ via the map
$(b_{1},\dots,b_{k})\mapsto b_{1}+\dots+b_{k}$ and if we let
$\operatorname{Bin}(k)^{\otimes n}$ denote the $n$-fold product of the
$\operatorname{Binomial}(k,1/2)$ distribution, then the density of $\mu_{k}$
with respect to $\operatorname{Bin}(k)^{\otimes n}$ is at most
$2^{k\varepsilon n}$. This allows us to replace the measure $\mu_{k}$
appearing in the last line of the above equation by
$\operatorname{Bin}(k)^{\otimes n}$, at the cost of a factor of
$2^{k\varepsilon n}$. Thus,
$|A|\leq 2^{k\epsilon
n}\cdot\sum_{\vec{x}\in\\{0,\dots,k+1\\}^{n}}\mathbb{P}_{\vec{x}\sim\operatorname{Bin}(k)^{\otimes
n}}[\vec{x}-\vec{a}(\vec{x})]$
The above expression is still complicated by the presence of the shift
$\vec{a}(\vec{x})$, about which we have no information except that it lies in
the set $A$. The key technical lemma in the proof is Lemma 2.1, which
essentially allows us to remove this shift after paying a factor which depends
on $|A|$. Ultimately, this gives an upper bound on the sum in terms of $|A|$
and $k$, which amounts to an upper bound on $|A|$ in terms of $k,\epsilon$,
and $|A|$. Optimizing the value of the free parameter $k$ now gives the
desired conclusion.
## 2 Proof of Theorem 1.2Theorem 1.2
We begin by recording the following key comparison bound, which will be proved
at the end of this section.
###### Lemma 2.1.
Let $n\geq k\geq C_{\ref{lem:sup-ratio}}$, where $C_{\ref{lem:sup-ratio}}$ is
a sufficiently large absolute constant and let $\delta>0$. For any
$A\subseteq\\{0,1\\}^{n}$ with $|A|\leq\exp(\delta n)$, the following holds.
Let $\vec{x},\vec{b}\sim\operatorname{Bin}(k)^{\otimes n}$ be independent
$n$-dimensional random vectors. Then,
$\mathbb{E}_{\vec{x}}\bigg{[}\sup_{\vec{a}\in
A}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\bigg{]}\leq\exp\left(C_{\ref{lem:sup-
ratio}}\left(\frac{1}{k}+\sqrt{\frac{\delta}{k}}\right)n\right).$
Let $n$, $\epsilon$, and $\vec{w}$ be as in Theorem 1.2. Let $\tau$ be such
that $\mathbb{P}[\langle\vec{w},\vec{\xi}\rangle=\tau]=\rho(\vec{w})$, where
$\vec{\xi}$ is a random vector with i.i.d. $\operatorname{Ber}(1/2)$
components. Let
$B=\\{\vec{\xi}\in\\{0,1\\}^{n}:\langle\vec{w},\vec{\xi}\rangle=\tau\\}.$
In particular, $|B|\geq\exp(-\epsilon n)\cdot 2^{n}$. Let
$|\mathcal{R}(\vec{w})|=\exp(\delta n)$. For each $r\in\mathcal{R}(\vec{w})$,
let $\vec{\xi}(r)$ be a fixed (but otherwise arbitrary) element of
$\\{0,1\\}^{n}$ such that $\langle\vec{w},\vec{\xi}(r)\rangle=r$. Let
$A=\\{\vec{\xi}(r)\in\\{0,1\\}^{n}:r\in\mathcal{R}(\vec{w})\\}.$
Note that, by definition, for any distinct $\vec{a}_{1},\vec{a}_{2}\in A$, we
have that
$\langle\vec{w},\vec{a}_{1}\rangle\neq\langle\vec{w},\vec{a}_{2}\rangle$ and
that $|A|=|\mathcal{R}(\vec{w})|=\exp(\delta n)$.
We will make use of the simple, but crucial, observation from [2] that $A$ and
$k\cdot B$ have a full sumset for all $k\geq 1$.
###### Lemma 2.2 ([2, Lemma 4.2]).
The map $(\vec{a},\vec{c})\mapsto\vec{a}+\vec{c}$ from $A\times(k\cdot B)$ to
$A+k\cdot B$ is injective.
###### Proof.
Indeed, if
$\vec{a}_{1}+(\vec{b}^{(1)}_{1}+\dots+\vec{b}^{(1)}_{k})=\vec{a}_{2}+(\vec{b}^{(2)}_{1}+\dots+\vec{b}^{(2)}_{k}$),
where $\vec{a}_{i}\in A$ and $\vec{b}^{(i)}_{j}\in B$, then taking the inner
product of both sides with $\vec{w}$ and using
$\langle\vec{w},\vec{b}\rangle=\tau$ for all $b\in B$, we see that
$\langle\vec{w},\vec{a}_{1}\rangle=\langle\vec{w},\vec{a}_{2}\rangle$, which
implies that $\vec{a}_{1}=\vec{a}_{2}$ by the definition of $A$. ∎
We are now ready to prove Theorem 1.2.
###### Proof of Theorem 1.2.
Let $k\geq 2$ be a parameter which will be chosen later depending on
$\epsilon$. We may assume $\epsilon\in(0,(2C_{\ref{lem:sup-ratio}})^{-2})$ by
adjusting $C_{\ref{thm:main}}$ appropriately at the end to make larger values
trivial. By Lemma 2.2, for each $\vec{x}\in\\{0,\ldots,k+1\\}^{n}$ for which
there exist $\vec{a}\in A$ and $\vec{c}\in k\cdot B$ with
$\vec{a}+\vec{c}=\vec{x}$, there exists a unique such choice
$\vec{a}=\vec{a}(\vec{x})\in A$. (For $\vec{x}\notin A+k\cdot B$, we let
$\vec{a}(\vec{x})$ be an arbitrary element of $A$.)
Now, let $\vec{a}$ be uniform on $A$, let $\vec{b}_{1},\ldots,\vec{b}_{k}$ be
uniform on $B$, and let $\vec{v}_{1},\ldots,\vec{v}_{k}$ be uniform on
$\\{0,1\\}^{n}$. Let $C_{i}\subseteq\\{0,\ldots,k+1\\}^{n}$ be the set of
vectors with $i$ coordinates equal to $k+1$. For
$\vec{x}\in\\{0,\ldots,k+1\\}^{n}$, we let
$\vec{x}^{\ast}\in\\{0,\dots,k\\}^{n}$ denote the vector obtained by setting
every occurrence of $k+1$ in $\vec{x}$ to $k$. We have
$\displaystyle 1$
$\displaystyle=\mathbb{P}[\vec{a}+\vec{b}_{1}+\cdots+\vec{b}_{k}\in\\{0,\ldots,k+1\\}^{n}]$
$\displaystyle=\sum_{i=0}^{n}\sum_{\vec{x}\in
C_{i}}\mathbb{P}[\vec{a}+\vec{b}_{1}+\cdots+\vec{b}_{k}=\vec{x}]$
$\displaystyle\leq\sum_{i=0}^{n}\sum_{\vec{x}\in
C_{i}}\mathbb{P}[\vec{a}=\vec{a}(\vec{x})]\mathbb{P}[\vec{b}_{1}+\cdots+\vec{b}_{k}=\vec{x}-\vec{a}(\vec{x})]$
$\displaystyle\leq\frac{1}{|A|}\sum_{i=0}^{n}\sum_{\vec{x}\in
C_{i}}\bigg{(}\frac{2^{n}}{|B|}\bigg{)}^{k}\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}-\vec{a}(\vec{x})]$
$\displaystyle\leq\frac{e^{k\epsilon n}}{|A|}\sum_{i=0}^{n}\sum_{\vec{x}\in
C_{i}}\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}^{\ast}]\sup_{\vec{a}\in
A}\frac{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}-\vec{a}]}{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}^{\ast}]}$
$\displaystyle=\frac{e^{k\epsilon
n}}{|A|}\sum_{i=0}^{n}(1/2^{k})^{i}\sum_{S\in\binom{[n]}{i}}\mathbb{E}_{\vec{x}\sim\operatorname{Bin}(k)^{\otimes([n]\setminus
S)}\times\\{k+1\\}^{S}}\bigg{[}\sup_{\vec{a}\in
A}\frac{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}-\vec{a}]}{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}^{\ast}]}\bigg{]}.$
Let $A_{S}$ be the set of elements in $A\subseteq\\{0,1\\}^{n}$ whose support
contains $S$. Let
$A^{\prime}_{S}=\\{\vec{a}^{\prime}\in\\{0,1\\}^{[n]\setminus
S}:\exists\vec{a}\in A_{S}\text{ with }\vec{a}|_{[n]\setminus
S}=\vec{a}^{\prime}\\}.$
Recall that $|A|=\exp(\delta n)$. Abusing notation so that the supremum of an
empty set is $0$, we can continue the above chain of inequalities to get that
$\displaystyle 1$ $\displaystyle\leq\frac{e^{k\epsilon
n}}{|A|}\sum_{i=0}^{n}(1/2^{k})^{i}\sum_{S\in\binom{[n]}{i}}\mathbb{E}_{\vec{x}\sim\operatorname{Bin}(k)^{\otimes([n]\setminus
S)}\times\\{k+1\\}^{S}}\bigg{[}\sup_{\vec{a}\in
A}\frac{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}-\vec{a}]}{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}^{\ast}]}\bigg{]}$
$\displaystyle=\frac{e^{k\epsilon
n}}{|A|}\sum_{i=0}^{n}(1/2^{k})^{i}\sum_{S\in\binom{[n]}{i}}\mathbb{E}_{\vec{x}\sim\operatorname{Bin}(k)^{\otimes([n]\setminus
S)}}\bigg{[}\sup_{\vec{a}\in
A_{S}^{\prime}}\frac{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=(\vec{x}-\vec{a})\times\\{k\\}^{S}]}{\mathbb{P}[\vec{v}_{1}+\cdots+\vec{v}_{k}=\vec{x}\times\\{k\\}^{S}]}\bigg{]}$
$\displaystyle=\frac{e^{k\epsilon
n}}{|A|}\sum_{i=0}^{n}(1/2^{k})^{i}\sum_{S\in\binom{[n]}{i}}\mathbb{E}_{\vec{x}\sim\operatorname{Bin}(k)^{\otimes([n]\setminus
S)}}\bigg{[}\sup_{\vec{a}\in
A_{S}^{\prime}}\frac{\mathbb{P}[(\vec{v}_{1}+\cdots+\vec{v}_{k})|_{[n]\setminus
S}=\vec{x}-\vec{a}]}{\mathbb{P}[(\vec{v}_{1}+\cdots+\vec{v}_{k})|_{[n]\setminus
S}=\vec{x}]}\bigg{]}$ $\displaystyle\leq\frac{e^{k\epsilon
n}}{|A|}\left(\sum_{i=0}^{n/2}\cdot+\sum_{i=n/2}^{n}2^{-ki}\cdot
2^{n}\cdot\left(\max_{\ell}\frac{\max\Big{\\{}\binom{k}{\ell-1},\binom{k}{\ell}\Big{\\}}}{\binom{k}{\ell}}\right)^{n-i}\right)$
$\displaystyle\leq\frac{e^{k\epsilon
n}}{|A|}\left(\sum_{i=0}^{n/2}\cdot+n\cdot 2^{-kn/2}\cdot 2^{n}\cdot
k^{n}\right)$ $\displaystyle\leq\frac{e^{k\epsilon
n}}{|A|}\bigg{(}\sum_{i=0}^{n/2}\binom{n}{i}2^{-ki}\exp(C_{\ref{lem:sup-
ratio}}(k^{-1}+(2\delta)^{1/2}k^{-1/2})(n/2))+2^{-kn/4}\bigg{)}$
$\displaystyle\leq\exp(-\delta
n)\exp\left(O(k\epsilon+k^{-1}+\delta^{1/2}k^{-1/2})n\right)$
by Lemma 2.1 applied to $A_{S}$, as long as $n/2\geq k\geq C_{\ref{lem:sup-
ratio}}\geq 20$. To deduce the last line, note that
$\binom{n}{i}2^{-ki}\leq(2^{-k}en/i)^{i}$, so for $i\geq\lceil
en/2^{k-1}\rceil$ the sum of weighted binomials is bounded by a geometric
series. Additionally, for $1\leq i\leq en/2^{k}$, if this interval is
nonempty, the sum of binomials is certainly bounded by $\exp(O(k^{-1}n))$.
Hence, the above inequality yields
$\delta\leq C(k\epsilon+k^{-1}+\delta^{1/2}k^{-1/2})$
for some absolute constant $C>0$. Now letting $k=\epsilon^{-1/2}/2$ (note that
this satisfies $2C_{\ref{lem:sup-ratio}}\leq 2k=\epsilon^{-1/2}\leq n$), we
find that
$\delta=O(\epsilon^{1/2}),$
as desired. ∎
The proof of Lemma 2.1 relies on the following preliminary estimate.
###### Lemma 2.3.
If $1\leq s\leq k/(16\pi)$, then
$\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{(}\frac{x}{k+1-x}\bigg{)}^{s}\leq\exp(10\pi
s^{2}/k)+2k^{s}(4/5)^{k}.$
###### Proof.
We let $x\sim\operatorname{Bin}(k)$ and $y=x-k/2\sim\operatorname{Bin}(k)-k/2$
throughout. We let $z\sim\mathcal{N}(0,k\pi/8)$. We have
$\displaystyle\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{[}\bigg{(}\frac{x}{k+1-x}\bigg{)}^{s}\bigg{]}$
$\displaystyle=\mathbb{E}_{y}\bigg{[}\bigg{(}1+\frac{2y-1}{k/2+1-y}\bigg{)}^{s}\bigg{]}$
$\displaystyle\leq\mathbb{E}_{y}\bigg{[}\bigg{(}1+\frac{2y}{k/2+1-y}\bigg{)}^{s}\mathbbm{1}_{|y|\leq
k/3}\bigg{]}+k^{s}\mathbb{P}[|y|\geq k/3]$
$\displaystyle\leq\mathbb{E}_{y}\bigg{[}\bigg{(}1+\frac{2y}{k/2+1-y}\bigg{)}^{s}\mathbbm{1}_{|y|\leq
k/3}\bigg{]}+2k^{s}(4/5)^{k}.$
Note that the probability estimate for $\mathbb{P}[\mathbbm{1}_{|y|\geq k/3}]$
follows from the sharp (entropy) version of the Chernoff-Hoeffding theorem.
Since for $|y|\leq k/3$,
$\frac{2y}{(k/2+1-y)}\leq\frac{2y}{k/2+1}+\frac{8y^{2}}{(k/2+1)^{2}},$
and using $(1+x)\leq\exp(x)$, we can continue the previous inequality as
$\displaystyle\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{[}\bigg{(}\frac{x}{k+1-x}\bigg{)}^{s}\bigg{]}$
$\displaystyle\leq\mathbb{E}_{y}\bigg{[}\bigg{(}1+\frac{2y}{k/2+1}+\frac{8y^{2}}{(k/2+1)^{2}}\bigg{)}^{s}\mathbbm{1}_{|y|\leq
k/3}\bigg{]}+2k^{s}(4/5)^{k}$
$\displaystyle\leq\mathbb{E}_{y}\bigg{[}\exp\bigg{(}\frac{4sy}{k+2}+\frac{32sy^{2}}{k^{2}}\bigg{)}\bigg{]}+2k^{s}(4/5)^{k}.$
Now, let $z_{1},\dots,z_{k}$ be i.i.d. $\mathcal{N}(0,1)$ random variables.
Then,
$y\sim\frac{1}{2}\left(\operatorname{sgn}{z_{1}}+\dots+\operatorname{sgn}{z_{k}}\right).$
Moreover, for any $-k\leq\ell\leq k$,
$\mathbb{E}[z_{1}+\dots+z_{k}\mid\operatorname{sgn}(z_{1})+\dots+\operatorname{sgn}(z_{k})=\ell]=\sqrt{\frac{2}{\pi}}\ell.$
In particular, under this coupling of $y,z_{1},\dots,z_{k}$, we have
$\mathbb{E}[z_{1}+\dots+z_{k}\mid y]=\sqrt{\frac{8}{\pi}}y.$
Let $z=z_{1}+\dots+z_{k}$, so that $z\sim\mathcal{N}(0,k)$. Then, by the
convexity of
$f(y)=\exp\bigg{(}\frac{4sy}{k+2}+\frac{32sy}{k^{2}}\bigg{)}$
and using Jensen’s inequality, we have
$\displaystyle\mathbb{E}_{y}f(y)$
$\displaystyle=\mathbb{E}_{y,z_{1},\dots,z_{k}}f(y)$
$\displaystyle=\mathbb{E}_{y,z_{1},\dots,z_{k}}f\left(\sqrt{\frac{\pi}{8}}\mathbb{E}[z\mid
y]\right)$ $\displaystyle\leq\mathbb{E}_{z}f(\sqrt{\pi}z/\sqrt{8})$
$\displaystyle=\mathbb{E}_{w\sim\mathcal{N}(0,1)}\exp\bigg{(}\frac{s\sqrt{2k\pi}}{k+2}w+\frac{4s\pi}{k}w^{2}\bigg{)}$
$\displaystyle=\bigg{(}1-\frac{8\pi s}{k}\bigg{)}^{-1/2}\exp\bigg{(}\frac{\pi
s^{2}k^{2}}{(k+2)^{2}(k-8\pi s)}\bigg{)}$
$\displaystyle\leq\exp\bigg{(}\frac{8\pi s}{k}+\frac{2\pi s^{2}}{k}\bigg{)}$
$\displaystyle\leq\exp(10\pi s^{2}/k).\qed$
Finally, we can prove Lemma 2.1
###### Proof of Lemma 2.1.
We may assume that $\delta\geq 2000/k$ since the statement for $\delta<2000/k$
follows from the statement for $\delta=2000/k$. Also, note that we may assume
that $\delta\leq\log 2$. For any $t\in\mathbb{R}$, we have
$\displaystyle\mathbb{P}_{\vec{x}}\bigg{[}\sup_{\vec{a}\in
A}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\geq
e^{tn}\bigg{]}$ $\displaystyle\leq|A|\sup_{\vec{a}\in
A}\mathbb{P}_{\vec{x}}\bigg{[}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\geq
e^{tn}\bigg{]}$ $\displaystyle\leq|A|\sup_{\vec{a}\in A}\inf_{s\geq
2}\exp(-stn)\mathbb{E}_{\vec{x}}\bigg{[}\bigg{(}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\bigg{)}^{s}\bigg{]}$
$\displaystyle=|A|\sup_{\vec{a}\in A}\inf_{s\geq
2}\exp(-stn)\prod_{i=1}^{n}\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{[}\bigg{(}\frac{\mathbb{P}[\operatorname{Bin}(k)=x-a_{i}]}{\mathbb{P}[\operatorname{Bin}(k)=x]}\bigg{)}^{s}\bigg{]}$
$\displaystyle\leq|A|\inf_{s\geq
2}\exp(-stn)\bigg{(}\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{(}\frac{x}{k+1-x}\bigg{)}^{s}\bigg{)}^{n}.$
In the last line, we have used that
$\displaystyle\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{[}\bigg{(}\frac{x}{k+1-x}\bigg{)}^{s}\bigg{]}$
$\displaystyle\geq\bigg{(}\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{[}\frac{x^{2}}{(k+1-x)^{2}}\bigg{]}\bigg{)}^{s/2}$
$\displaystyle=\bigg{(}\sum_{\ell=0}^{k-1}\frac{\ell+1}{k-\ell}\binom{k}{\ell}2^{-k}\bigg{)}^{s/2}$
$\displaystyle=\bigg{(}\sum_{\ell=0}^{k-1}\bigg{(}\frac{k+2}{k}+\frac{4(k+1)(\ell-k/2)}{k^{2}}+\frac{(k+1)(k-2\ell)^{2}}{k^{2}(k-\ell)}\bigg{)}\binom{k}{\ell}2^{-k}\bigg{)}^{s/2}$
$\displaystyle\geq\bigg{(}\sum_{\ell=0}^{k-1}\bigg{(}\frac{k+2}{k}+\frac{4(k+1)(\ell-k/2)}{k^{2}}\bigg{)}\binom{k}{\ell}2^{-k}\bigg{)}^{s/2}$
$\displaystyle=\bigg{(}\frac{k+2}{k}-\frac{3k+4}{k}2^{-k}\bigg{)}^{s/2}$
$\displaystyle\geq 1$
if $k\geq 3$. Therefore, by Lemma 2.3, we have
$\displaystyle\mathbb{P}_{\vec{x}}\bigg{[}\sup_{\vec{a}\in
A}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\geq
e^{tn}\bigg{]}$ $\displaystyle\leq|A|\inf_{s\geq
2}\exp(-stn)\bigg{(}\mathbb{E}_{x\sim\operatorname{Bin}(k)}\bigg{(}\frac{x}{k+1-x}\bigg{)}^{s}\bigg{)}^{n}$
$\displaystyle\leq|A|\inf_{2\leq s\leq k/(16\pi)}\exp(-stn)\bigg{(}\exp(10\pi
s^{2}/k)+2k^{s}(4/5)^{k}\bigg{)}^{n}$ $\displaystyle\leq|A|\inf_{2\leq s\leq
k/(10\log k)}\exp(-stn)\bigg{(}\exp(12\pi s^{2}/k)\bigg{)}^{n}$
$\displaystyle\leq\begin{cases}|A|\exp\left(-\frac{kt^{2}n}{48\pi}\right)&\quad\text{
if }\sqrt{\frac{96\pi\delta}{k}}\leq t\leq(\log{k})^{-1}\\\
|A|\exp\left(-\frac{kn}{48\pi(\log{k})^{2}}\right)&\quad\text{ if
}(\log{k})^{-1}\leq t\leq\log{k}.\end{cases}$
Here, the second case follows by plugging in $s=k/(24\pi\log{k})$ and
simplifying (assuming $C_{\ref{lem:sup-ratio}}$ is large enough so $s\geq 2$),
and the first case follows from plugging in $s=kt/(24\pi)$ which satisfies
$2\leq s\leq k/(10\log{k})$ by the restriction on $t$ and $\delta$. Finally,
since
$0\leq\sup_{\vec{a}\in
A}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\leq\left(\max_{\ell}\frac{\max\Big{\\{}\binom{k}{\ell-1},\binom{k}{\ell}\Big{\\}}}{\binom{k}{\ell}}\right)^{n}\leq
k^{n},$
we have
$\displaystyle\mathbb{E}_{\vec{x}}\bigg{[}\sup_{\vec{a}\in
A}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\bigg{]}$
$\displaystyle=\int_{-\infty}^{\log k}\mathbb{P}\bigg{[}\sup_{\vec{a}\in
A}\frac{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}-\vec{a}]}{\mathbb{P}_{\vec{b}}[\vec{b}=\vec{x}]}\geq
e^{tn}\bigg{]}ne^{tn}dt$
$\displaystyle\leq\int_{1/\log{k}}^{\log{k}}\cdot+\int_{\sqrt{96\pi\delta/k}}^{1/\log{k}}\cdot+\int_{-\infty}^{\sqrt{96\pi\delta/k}}ne^{tn}dt$
$\displaystyle\leq
e^{\sqrt{96\pi\delta/k}n}+\int_{\sqrt{96\pi\delta/k}}^{1/\log
k}|A|\exp\bigg{(}-\frac{kt^{2}n}{48\pi}\bigg{)}ne^{tn}dt$
$\displaystyle\quad+\int_{1/\log k}^{\log
k}|A|\exp\bigg{(}-\frac{kn}{48\pi(\log k)^{2}}\bigg{)}ne^{tn}dt$
$\displaystyle\leq\exp\left(O(\sqrt{\delta/k})n\right)+\int_{\sqrt{96\pi\delta/k}}^{1/\log{k}}ne^{-tn}dt+1$
$\displaystyle\leq\exp\left(O(\sqrt{\delta/k})n\right).\qed$
## Acknowledgments
The authors are grateful to the anonymous reviewers for several suggestions
which helped improve the presentation of the paper.
## References
* [1] Andreas Björklund, Thore Husfeldt, and Mikko Koivisto, _Set partitioning via inclusion-exclusion_ , SIAM Journal on Computing 39 (2009), no. 2, 546–563.
* [2] Jesper Nederlof, Jakub Pawlewicz, Céline M. F. Swennenhuis, and Karol Węgrzycki, _A faster exponential time algorithm for bin packing with a constant number of bins via additive combinatorics_ , pp. 1682–1701.
* [3] Hoi Nguyen and Van Vu, _Optimal inverse Littlewood-Offord theorems_ , Adv. Math. 226 (2011), no. 6, 5298–5319. MR 2775902
* [4] Mark Rudelson and Roman Vershynin, _The Littlewood–Offord problem and invertibility of random matrices_ , Advances in Mathematics 218 (2008), no. 2, 600–633.
* [5] Terence Tao and Van H. Vu, _Inverse Littlewood–Offord theorems and the condition number of random discrete matrices_ , Annals of Mathematics (2009), 595–632.
* [6] Mårten Wiman, _Improved constructions of unbalanced uniquely decodable code pairs_ , 2017, https://www.diva-portal.org/smash/get/diva2:1120593/FULLTEXT01.pdf.
[vj] Vishesh Jain
Department of Statistics, Stanford University
Stanford, California
visheshjstanfordedu
https://jainvishesh.github.io [as] Ashwin Sah
Department of Mathematics, Massachusetts Institute of Technology
Cambridge, Massachusetts
asahmitedu
http://www.mit.edu/~asah/ [ms] Mehtaab Sawhney
Department of Mathematics, Massachusetts Institute of Technology
Cambridge, Massachusetts
msawhneymitedu
http://www.mit.edu/~msawhney/
|
# On a classical solution to the Abelian Higgs model
N. Mohammedi
Institut Denis Poisson (CNRS - UMR 7013),
Université de Tours,
Faculté des Sciences et Techniques,
Parc de Grandmont, F-37200 Tours, France. e-mail: noureddine.mohammedi@univ-
tours.fr
###### Abstract
A particular solution to the equations of motion of the Abelian Higgs model is
given. The solution involves the Jacobi elliptic functions as well as the Heun
functions.
## 1 Introduction
The search for analytical solutions to the classical equations of motion of
field theories which are of relevance to certain domains of physics is of
great interest. One of these field theories is the Abelian Higgs model whose
importance embraces particle physics, condensed matter physics and cosmology
[1, 2, 3, 4, 5]. The full fledged Euler-Lagrange equations of motion of this
model have so far resisted attempts to solve them. This is because these are
highly non-linear coupled second order partial differential equations. In the
literature, the classical studies of the Abelian Higgs model are mostly
dedicated to its topological properties. In this context, vortices have been
identified and a great deal has been learned through numerical analyses (see
[6] for a sample). An account of the problem of constructing stationary
topological and non-topological classical solutions in various field theories
can be found in [7, 8, 9, 10].
In order to reduce the degree of complexity of the equations of motion of the
Abelian Higgs model some simplifying strategies have to be adopted. In this
area, the authors of ref.[11] explicitly constructed periodic sphaleron
solutions (minima and saddle points of the energy functional) of the
$(1+1)$-dimensional Abelian Higgs model on a circle. They found some
analytical solutions (for some special values of the Higgs mass) for the the
small perturbations (normal modes) around the sphaleron solutions. Their work
is directly inspired by an earlier investigation by Manton and Samols [12] who
found sphaleron solutions in the scalar theory $\phi^{4}$ defined on a circle.
In this note we have identified another situation where the equations of
motion of the four-dimensional Abelian Higgs model become relatively simple.
If the complex scalar field is parametrised as $\phi=\rho\,e^{i\theta}$ and we
define the gauge invariant quantity
$\widetilde{A}_{\mu}=A_{\mu}+\frac{1}{e}\partial_{\mu}\theta$, with $A_{\mu}$
being the gauge field, then imposing the constraint
$\widetilde{A}_{\mu}\widetilde{A}^{\mu}=0$ renders the equations of motion
tractable111Upon publication of the present work, I became aware that the
authors of refs.[13, 14, 15, 16, 17, 18, 19, 20] have used, among others, the
condition $A_{\mu}A^{\mu}=0$, to build analytic solutions in numerous field
theories. I am greatful to Fabrizio Canfora for pointing out this to me.. As a
matter of fact, the equation of motion of the complex scalar field decouples
and reduces to the usual equation of motion of a $\phi^{4}$ scalar field
theory which is known to possess kink solutions. The problem is then brought
to solving the gauge field equation in the presence of a kink background.
We have first solved the gauge field equation of motion when the scalar field
$\rho$ takes the usual kink profile
$\rho=\pm\,v\,\tanh\left(p_{\mu}x^{\mu}+w_{0}\right)$, where $p_{\mu}$ is a
space-like four vector. The gauge field $A_{\mu}$ is, up to a gauge function,
expressed in terms of associated Legendre functions and has two independent
polarisations. However, the equation of motion of a $\phi^{4}$ scalar field
theory is also solved by the twelve Jacobi elleptic functions (the kink
solution is a very special case of this). Next, we solved the gauge field
equation of motion when the scalar field $\rho$ is represented by one of the
Jacobi elleptic functions. Here we found that the gauge field is determined by
means of Heun functions.
In passing, we should mention that some analytical solutions involving gauge
fields have been explicitly constructed in different contexts. Brihaye [21],
in the case of the $SU(2)$ Yang-Mills theory coupled to a triplet Higgs field,
has found (for particular values of the ratio of the two coupling constants)
solutions involving the Jacobi elleptic functions. The authors of ref.[22],
studying a $(1+1)$-dimensional Abelian Higgs model, have obtained exact
solutions (subject to the approximation that the two wells of the Higgs
potential are deep). These were consequently used to build approximate
analytical solution, in the form of oscillons and oscillating kinks, for the
dynamics of both the gauge and Higgs fields. Similar studies can also be found
in [23, 24]. Along these lines, one finds in [25] a numerical solutions
corresponding to a kink (domain wall) in a theory consisting of two
interacting complex scalars coupled to two independant gauge fields. The
stability of this solution was later analysed in [26]. The partial relevance
of all of these solutions to the present work is worth investigating.222I am
greatful to an anonymous referee for bringing some of these works to my
attention. Finally, various analytical solutions involving a generalised
Maxwell-Higgs models (theories with non-standard kinetic terms) have been
obtained in [27, 28]. Nevertheless, we should insist on the fact that none of
the above mentioned solutions in [11, 22, 23, 24, 25, 26] is exact. The
closest study to our analyses is ref.[11] (in that it involves the Jacobi
elleptic functions). Their sphaleron solution (and the perturbation about it)
to not fit in the caterory $\widetilde{A}_{\mu}\widetilde{A}^{\mu}=0$
considered here.
The paper is organised as follows: In the next section we briefly describe the
Abelian Higgs model and lay out its simple classical solutions. In section
three, we solve the equation of motion of the gauge field in the background of
a kink scalar field. We then review, in section four, the classical solution
to the equation of motion of a $\phi^{4}$ scalar field theory in terms of the
Jacobi elleptic functions (solution involving the Weierstrass elliptic
function are presented in an appendix). These are used in section five to
build the solution to the gauge field equation of motion. Our main results are
summarised in the last section.
## 2 The Abelian Higgs model
The Abelian Higgs model is described by the Lagrangian333The space-time
coordinates are
$x^{\mu}=\left(x^{0}\,,\,x^{1}\,,\,x^{2}\,,\,x^{3}\right)=\left(x^{0}\,,\,\vec{x}\right)$
and the indices are raised and lowered with the metric
$g_{\mu\nu}={\rm{diag}}\left(1\,,\,-1\,,\,-1\,,\,-1\right)$. A four-vector is
$V^{\mu}=\left(V^{0}\,,\,V^{1}\,,\,V^{2}\,,\,V^{3}\right)=\left(V^{0}\,,\,\vec{V}\right)$.
$\displaystyle{\cal
L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\left(\partial_{\mu}\phi^{\star}-ieA_{\mu}\phi^{\star}\right)\left(\partial^{\mu}\phi+ieA^{\mu}\phi\right)-\frac{\lambda}{2}\left(\varphi^{\star}\varphi-v^{2}\right)^{2}\,\,\,.$
(2.1)
Here $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength of the gauge field $A_{\mu}$ and $\phi$ is the complex scalar field.
It is understood that the two parameters $v^{2}$ and $\lambda$ are both
positive.
It is convenient for our purpose to parametrise the complex scalar field
$\phi$ as444One must be aware that
$\phi=\rho\,e^{i\theta}=\rho\,e^{i(\theta+2\pi N)}$ with $N\in\mathbb{Z}$. In
this note we assume that
$\partial_{\mu}\partial_{\nu}\theta-\partial_{\nu}\partial_{\mu}\theta=0$.
$\phi=\rho\,e^{i\theta}\,\,\,\,.$ (2.2)
The Lagrangian (2.1) becomes then
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+e^{2}\rho^{2}\left(A_{\mu}+\frac{1}{e}\partial_{\mu}\theta\right)\left(A^{\mu}+\frac{1}{e}\partial^{\mu}\theta\right)$
(2.3) $\displaystyle+$ $\displaystyle\partial_{\mu}\rho\,\partial^{\mu}\rho\
-\frac{\lambda}{2}\left(\rho^{2}-v^{2}\right)^{2}\,\,\,\,$
and the local gauge symmetry is
$\displaystyle A_{\mu}\longrightarrow
A_{\mu}+\frac{1}{e}\partial_{\mu}\Lambda\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\theta\longrightarrow\theta-\Lambda\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\rho\longrightarrow\rho\,\,\,\,\,\,.$
(2.4)
One could use this gauge freedom to set the field $\theta$ to zero.
Nevertheless, we will keep $\theta$ through out.
The equations of motion for the Abelian Higgs model are
$\displaystyle\partial_{\mu}\widetilde{F}^{\mu\nu}+2e^{2}\,\rho^{2}\,\widetilde{A}^{\nu}$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,,$ (2.5)
$\displaystyle\partial_{\mu}\partial^{\mu}\rho+{\lambda}\,\rho\left(\rho^{2}-v^{2}\right)-e^{2}\,\rho\,\widetilde{A}_{\mu}\widetilde{A}^{\mu}$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,,$ (2.6)
$\displaystyle\partial_{\mu}\left(\rho^{2}\widetilde{A}^{\mu}\right)$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,.$ (2.7)
We have defined the gauge invariant variable
$\widetilde{A}_{\mu}=A_{\mu}+\frac{1}{e}\partial_{\mu}\theta\,\,\,\,.$ (2.8)
and
$\widetilde{F}_{\mu\nu}=F_{\mu\nu}=\partial_{\mu}\widetilde{A}_{\nu}-\partial_{\nu}\widetilde{A}_{\mu}$.
The last equation (2.7) corresponds to the field $\theta$ and is also a
consequence of (2.5). Notice that we have expressed the equations of motion in
terms of the two gauge invariant variables $\widetilde{A}_{\mu}$ and $\rho$.
The simplest known solution to the equations of motion is when the scalar
field $\rho$ is frozen at one of the two minima of the potential energy. That
is, $\rho^{2}=v^{2}$. In this case the equations of motion become
$\displaystyle\partial_{\mu}\widetilde{F}^{\mu\nu}+2e^{2}\,v^{2}\,\widetilde{A}^{\nu}$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,,$ (2.9)
$\displaystyle\widetilde{A}_{\mu}\widetilde{A}^{\mu}$ $\displaystyle=$
$\displaystyle 0\,\,\,\,\,,$ (2.10)
$\displaystyle\partial_{\mu}\widetilde{A}^{\mu}$ $\displaystyle=$
$\displaystyle 0\,\,\,\,\,.$ (2.11)
These equations have, for instance, the plane wave solution
$\displaystyle\widetilde{A}_{\mu}=\varepsilon_{\mu}\,cos\left(p_{\nu}x^{\nu}+w_{0}\right)\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\varepsilon_{\mu}\,\varepsilon^{\mu}=\varepsilon_{\mu}\,p^{\mu}=0\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,p_{\mu}\,p^{\mu}=2e^{2}\,v^{2}\,\,\,\,.$
(2.12)
The polarisation vector $\varepsilon_{\mu}$ has two independent components for
a given wave vector $p_{\mu}$. The mass-shell relation
$p_{\mu}\,p^{\mu}=2e^{2}\,v^{2}$ is that of a massive particle with a positive
mass squared.
The other known solution is the ”kink” solution for which
$\widetilde{A}_{\mu}=0$. The equations of motion come then to the single
equation
$\partial_{\mu}\partial^{\mu}\rho+{\lambda}\,\rho\left(\rho^{2}-v^{2}\right)\,=\,0\,\,\,\,\,.$
(2.13)
This is solved by
$\displaystyle\rho=\pm\,v\,\tanh\left(p_{\mu}x^{\mu}+w_{0}\right)\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,p_{\mu}p^{\mu}=-\frac{\lambda
v^{2}}{2}\,\,\,\,\,.$ (2.14)
Here we have a mass-shell condition of a relativistic particle of negative
mass squared.
In this note, we will look for a solution which satisfies the gauge invariant
condition
$\widetilde{A}_{\mu}\widetilde{A}^{\mu}=0\,\,\,\,\,.$ (2.15)
The equations of motion reduce then to
$\displaystyle\partial_{\mu}\widetilde{F}^{\mu\nu}+2e^{2}\,\rho^{2}\,\widetilde{A}^{\nu}$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,,$ (2.16)
$\displaystyle\partial_{\mu}\partial^{\mu}\rho+{\lambda}\,\rho\left(\rho^{2}-v^{2}\right)$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,,$ (2.17)
$\displaystyle\partial_{\mu}\left(\rho^{2}\widetilde{A}^{\mu}\right)$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,.$ (2.18)
We notice that the equation of motion for the scalar field $\rho$ decouples
from the rest. We will report here on a non-trivial solution obeying the
condition (2.15).
## 3 A gauge field corresponding to the “kink” scalar field
The second equation (2.17) admits the ”kink” solution as written in (2.14). We
would like here to find the gauge field corresponding to it. We assume the
following form for the gauge field $\widetilde{A}_{\mu}$ :
$\widetilde{A}_{\mu}\left(w\right)=\varepsilon_{\mu}\,h\left(w\right)\,\,\,\,\,.$
(3.1)
Here (and in the rest of the paper) we will use the notation
$w=p_{\mu}x^{\mu}+w_{0}\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,p^{2}=p_{\mu}p^{\mu}$
(3.2)
with $w_{0}$ a constant. The four-vector $p_{\mu}$ obeyes the mass-shell
relation $p^{2}=-\frac{\lambda v^{2}}{2}$. The polarisation vector
$\varepsilon_{\mu}$ is required to satisfy
$\varepsilon_{\mu}\,\varepsilon^{\mu}\,=p_{\mu}\,\varepsilon^{\mu}\,=0\,\,\,\,\,.$
(3.3)
Equations (2.15) and (2.18) are then automatically obeyed.
Substituting (3.1) into (2.16) results in the differential equation
$\frac{d^{2}h}{dw^{2}}\left(w\right)-\frac{4e^{2}}{\lambda}\,\tanh^{2}\left(w\right)\,h\left(w\right)=0\,\,\,\,\,.$
(3.4)
By the change of variables
$z=\tanh\left(w\right)$ (3.5)
one transforms the differential equation (3.4) into
$\displaystyle\left(1-z^{2}\right)\,\frac{d^{2}h}{dz^{2}}\left(z\right)-2z\,\frac{dh}{dz}\left(z\right)+\left[l\left(l+1\right)-\frac{m^{2}}{\left(1-z^{2}\right)}\right]h\left(z\right)=0\,\,\,\,\,.$
(3.6)
This differential equation is known as associated Legendre equations [29, 30,
31]. It is, in general, singular at the points $z=\pm 1\,,\,\pm\infty$. In the
case at hand, the variable $z$ is real and lies in the domain
$\left[-1\,,\,+1\right]$ and the real constants $l$ and $m$ are defined
as555Greek indices $\nu$ and $\mu$ are usually used instead of $l$ and $m$.
$\displaystyle
l=-\frac{1}{2}\pm\sqrt{\frac{1}{4}+\frac{4e^{2}}{\lambda}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m^{2}=\frac{4e^{2}}{\lambda}\,\,\,\,\,\,.$
(3.7)
The general solution to (3.6) is
$\displaystyle
h\left(z\right)=a\,P^{m}_{l}\left(z\right)+b\,Q^{m}_{l}\left(z\right)\,\,\,\,\,\,,$
(3.8)
where $P^{m}_{l}\left(z\right)$, $Q^{m}_{l}\left(z\right)$ are associated
Legendre functions [29, 30, 31] of the first and second kind, respectively.
The two constants $a$ and $b$ are arbitrary.
When $z$ is real and belonging to the interval $\left[-1\,,\,+1\right]$, the
associated Legendre function $P^{m}_{l}\left(z\right)$ is real and is
expressed as [29, 30, 31]
$\displaystyle
P^{m}_{l}\left(z\right)=\frac{1}{\Gamma\left(1-m\right)}\left(\frac{1+z}{1-z}\right)^{\frac{m}{2}}F\left(-l,l+1;1-m;\frac{1-z}{2}\right)\,\,\,\,\,\,,$
(3.9)
where $F\left(a,b;c;x\right)$ is the hypergeometric function and
$\Gamma\left(\nu\right)$ is the gamma function. Similarly, the associated
Legendre function $Q^{m}_{l}\left(z\right)$ is given by
$\displaystyle
Q^{m}_{l}\left(z\right)=\frac{\pi}{2\sin\left(m\pi\right)}\left[\cos\left(m\pi\right)\,P^{m}_{l}\left(z\right)-\frac{\Gamma\left(l+m+1\right)}{\Gamma\left(l-m+1\right)}\,P^{-m}_{l}\left(z\right)\right]\,\,\,\,\,\,.$
(3.10)
To summarise, a particular solution to the equations of motion of the Abelian
Higgs model (2.1) is given by
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\pm\,v\,\tanh\left(w\right)\,e^{i\theta}\,\,\,\,\,\,,$
$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{e}\,\partial_{\mu}\theta+\varepsilon_{\mu}\left[a\,P^{m}_{l}\left(\tanh\left(w\right)\right)+b\,Q^{m}_{l}\left(\tanh\left(w\right)\right)\right]\,\,\,\,\,\,,$
$\displaystyle p^{2}$ $\displaystyle=$ $\displaystyle-\frac{\lambda
v^{2}}{2}\,\,\,\,\,\,,\,\,\,\,\,\,\varepsilon_{\mu}\,\varepsilon^{\mu}\,=p_{\mu}\,\varepsilon^{\mu}\,=0\,\,\,\,\,\,.$
(3.11)
The parameters $l$ and $m$ are given (3.7). The field $\theta\left(x\right)$
is any arbitrary smooth function (obeying
$\partial_{\mu}\partial_{\nu}\theta-\partial_{\nu}\partial_{\mu}\theta=0$).
The gauge field $A_{\mu}$ has two independent polarisations for a given vector
$p_{\mu}$. Furthermore, the solution carries both electric and magnetic
fields.
## 4 Solutions to $\phi^{4}$ in terms of the Jacobi elliptic functions
The equation of motion for a phi-to-the-four scalar field theory is given by
$\displaystyle\partial_{\mu}\partial^{\mu}\rho+{\lambda}\,\rho\left(\rho^{2}-v^{2}\right)=0\,\,\,\,\,.$
(4.1)
We will look for solution which depend on the single variable
$w=p_{\mu}x^{\mu}+w_{0}$. That is, $\rho=\rho\left(w\right)$. The equation of
motion becomes then
$\displaystyle\frac{\text{d}^{2}\rho}{\text{d}w^{2}}+\frac{\lambda}{p^{2}}\,\rho\left(\rho^{2}-v^{2}\right)$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,\,.$ (4.2)
Notice that any solution to this equation is obviously subject to the
following remark:
$\displaystyle\text{if}\,\,\,\rho\left(w\,,\,p^{2}\right)\,\,\,\text{is a
solution then
}\,\,\,\rho\left(s\,w\,,\,\frac{p^{2}}{s^{2}}\right)\,\,\,\text{is also a
solution }\,\,\,,$ (4.3)
where $s$ is a constant. Therefore, the mass-shell relation, $p^{2}$, will be
determined up to a constant.
It is well-known that equation (4.2) is solved by the twelve Jacobi elliptic
functions [29, 30, 31]. Indeed, The scalar field $\rho(w)$ satisfies the first
order differential equation
$\displaystyle\left(\frac{\text{d}\rho}{\text{d}w}\right)^{2}=-\frac{\lambda}{2p^{2}}\,\rho^{2}\left(\rho^{2}-2v^{2}\right)+av^{2}\,\,\,\,\,$
(4.4)
with $av^{2}$ a constant of integration. By writing
$\displaystyle\rho\left(w\right)=v\,\varepsilon\,y\left(w\right)\,\,\,\,\,,$
(4.5)
where $\varepsilon$ is a constant, one obtains the first order differential
equation
$\displaystyle\left(\frac{\text{d}y}{\text{d}w}\right)^{2}=-\frac{\lambda
v^{2}\varepsilon^{2}}{2p^{2}}\,y^{4}+\frac{\lambda
v^{2}}{p^{2}}\,y^{2}+\frac{a}{\varepsilon^{2}}\,\,\,\,\,.$ (4.6)
The twelve Jacobi elliptic functions are obtained as solutions to this last
equation [29, 30, 31] for different choices of the three constants $p^{2}$,
$\varepsilon^{2}$ and $a$ (and boundary conditions). The table in Appendix B
gather these values. For instance, the solution given in terms of the “sine”
Jacobi elliptic function ${\text{sn}}{(x\,,\,m)}$ corresponds to the choice
$\displaystyle a$ $\displaystyle=$
$\displaystyle\varepsilon^{2}=\frac{2m^{2}}{1+m^{2}}\,\,\,\,,$ $\displaystyle
p^{2}$ $\displaystyle=$ $\displaystyle-\frac{\lambda v^{2}}{1+m^{2}}\,\,\,\,$
(4.7)
and the differential equation (4.6) takes the form
$\displaystyle\left(\frac{\text{d}y}{\text{d}w}\right)^{2}=\left(1-y^{2}\right)\left(1-m^{2}\,y^{2}\right)\,\,\,\,\,.$
(4.8)
The general solution [29, 30, 31] to this last equation is
$y\left(w\right)={\text{sn}}{(w+d\,,\,m)}$, where $d$ is an arbitrary
constant. Hence our scalar field $\rho$ is given by
$\displaystyle\rho\left(w\right)=\pm
v\,\sqrt{\frac{2m^{2}}{1+m^{2}}}\,\,{\text{sn}}{(w+d\,,\,m)}\,\,\,\,\,\,,\,\,\,\,\,\,p^{2}=-\frac{\lambda
v^{2}}{1+m^{2}}\,\,\,\,.$ (4.9)
The parameter $m$ must be different from zero here. The mass-shell relation
$p^{2}=-\frac{\lambda v^{2}}{1+m^{2}}$ is that of a particle with negative
mass squared. It is worth mentioning that the “kink” solution (2.14)
corresponds to $m=1$ as
${\text{sn}}{(w+d\,,\,1)}=\tanh\left(w+d\right)\,\,\,\,.$ (4.10)
## 5 The general solution to the gauge field equation
The gauge field, $A_{\mu}=\varepsilon_{\mu}\,h\left(w\right)$ with
$\varepsilon_{\mu}\varepsilon^{\mu}=p_{\mu}\varepsilon^{\mu}=0$, is determined
by the differential equation
$\frac{\text{d}^{2}h}{\text{d}w^{2}}\left(w\right)+\frac{2e^{2}}{p^{2}}\,\rho^{2}\left(w\right)\,h\left(w\right)=0\,\,\,\,\,.$
(5.1)
For the moment, the expression of $p^{2}$ is not fixed. This allows one to
treat all possible mass-shell conditions at the same time.
Let us assume that
$h\left(w\right)=h\left(Z\right)\,\,\,\,\,,\,\,\,\,\,Z=\frac{1}{\tau^{2}}\,\rho^{2}\left(w\right)\,\,\,\,\,,$
(5.2)
where $\tau$ is a constant to be properly chosen later. Then by using the fact
that the field $\rho\left(w\right)$ satisfies the two equations
$\displaystyle\frac{\text{d}^{2}\rho}{\text{d}w^{2}}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{p^{2}}\,\rho\left(\rho^{2}-v^{2}\right)\,\,\,\,\,,$
$\displaystyle\left(\frac{\text{d}\rho}{\text{d}w}\right)^{2}$
$\displaystyle=$
$\displaystyle-\frac{\lambda}{2p^{2}}\,\rho^{2}\left(\rho^{2}-2v^{2}\right)+av^{2}\,\,\,\,\,$
(5.3)
we arrive at the differential equation
$\displaystyle\frac{\text{d}^{2}h}{\text{d}Z^{2}}+\left(\frac{\gamma}{Z}-\frac{\delta}{1-Z}-\frac{\epsilon
k^{2}}{1-k^{2}Z}\right)\frac{\text{d}h}{\text{d}Z}+\frac{\left(s+\alpha\beta
k^{2}Z\right)}{Z\left(1-Z\right)\left(1-k^{2}Z\right)}\,h=0\,\,\,\,\,.$ (5.4)
The different constants are given by
$\displaystyle s$ $\displaystyle=$ $\displaystyle 0\,\,\,\,\,,$
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\gamma=\epsilon=\frac{1}{2}\,\,\,\,\,,$ $\displaystyle\alpha$
$\displaystyle=$
$\displaystyle\alpha_{\pm}=\frac{1}{4}\left(1\pm\sqrt{1+16\,\frac{e^{2}}{\lambda}}\right)\,\,\,\,\,,$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\beta_{\mp}=\frac{1}{4}\left(1\mp\sqrt{1+16\,\frac{e^{2}}{\lambda}}\right)\,\,\,\,\,,$
$\displaystyle\tau^{2}$ $\displaystyle=$
$\displaystyle\frac{2k^{2}}{\left(1+k^{2}\right)}\,v^{2}\,\,\,\,\,.$ (5.5)
They satisfy the relation
$\gamma+\delta+\epsilon=\alpha_{\pm}+\beta_{\mp}+1\,\,\,\,\,.$ (5.6)
We also have the mass-shell relation
$\displaystyle p^{2}$ $\displaystyle=$ $\displaystyle-\frac{2\lambda
v^{2}k^{2}}{a\left(1+k^{2}\right)^{2}}\,\,\,\,\,.$ (5.7)
The constant $a$ will be fixed later.
The equation (5.4) is a Fuchsian differential equation and its general
solution is given by
$\displaystyle h\left(Z\right)$ $\displaystyle=$ $\displaystyle
C_{1}\,\text{Hn}\left(k^{2},0;\alpha_{+},\beta_{-},\frac{1}{2},\frac{1}{2};Z\right)+C_{2}\,\text{Hn}\left(k^{2},0;\alpha_{-},\beta_{+},\frac{1}{2},\frac{1}{2};Z\right)\,\,\,\,\,,$
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\frac{1}{\tau^{2}}\,\rho^{2}\left(w\right)=\frac{\left(1+k^{2}\right)}{2k^{2}}\,\frac{1}{v^{2}}\,\rho^{2}\,\left(w\right)\,\,\,\,\,,$
(5.8)
where $\text{Hn}\left(k^{2},s;\alpha,\beta,\gamma,\delta;z\right)$ is the Heun
function666Sometimes the notation $a=1/k^{2}$ and $q=-s/k^{2}$ is used. A
useful note on Heun’s functions and further references can be found in [32].
and $C1$ and $C2$ are two arbitrary constants. It is important to emphasise at
this point that the scalar field $\rho\left(w\right)$ is any solution to the
equation of motion (4.2), that is, any of the twelve Jacobi elliptic
functions.
When, for instance, the scalar field $\rho\left(w\right)$ is expressed in
terms of the Jacobi elliptic function $\text{sn}\left(w+d,m\right)$ in (4.9)
then the expression of $p^{2}$ there has to match that written in (5.7). This
leads to the identifications
$\displaystyle
a=\frac{2k^{2}\left(1+m^{2}\right)}{\left(1+k^{2}\right)^{2}}=\frac{2m^{2}}{\left(1+m^{2}\right)}\,\,\,\,\,\,\,\,\Longrightarrow\,\,\,\,k^{2}=m^{2}\,\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,\,k^{2}=\frac{1}{m^{2}}\,\,\,\,\,.$
(5.9)
If we choose $k=m$, for example, then, using (4.9) and (5.8), the two fields
of the Abelian Higgs model (2.1) are found to be given by
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\pm
v\,\sqrt{\frac{2m^{2}}{1+m^{2}}}\,{\text{sn}}{(w+d\,,\,m)}\,e^{i\theta}\,\,\,\,\,\,,$
$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{e}\,\partial_{\mu}\theta+\varepsilon_{\mu}\left[C_{1}\,\text{Hn}\left(m^{2},0;\alpha_{+},\beta_{-},\frac{1}{2},\frac{1}{2};{\text{sn}}^{2}{(w+d\,,\,m)}\right)\right.$
$\displaystyle+$ $\displaystyle
C_{2}\,\text{Hn}\left(m^{2},0;\alpha_{-},\beta_{+},\frac{1}{2},\frac{1}{2};{\text{sn}}^{2}{(w+d\,,\,m)}\right)\bigg{]}\,\,\,\,\,,$
$\displaystyle p^{2}$ $\displaystyle=$ $\displaystyle-\frac{\lambda
v^{2}}{1+m^{2}}\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\varepsilon_{\mu}\,\varepsilon^{\mu}\,=p_{\mu}\,\varepsilon^{\mu}\,=0\,\,\,\,\,\,.$
(5.10)
The two constants $\alpha_{\pm}$ and $\beta_{\pm}$ depend on the Abelian Higgs
parameters and are listed in (5.5). The field $\theta\left(x\right)$ is
arbitrary.
Finally, when $m=1$ one has ${\text{sn}}{(w+d\,,\,1)}=\tanh\left(w+d\right)$,
and the particular solution given in (3.11) is recovered by converting the
Heun functions into hypergeometric functions with the help of the relation
[32]
$\displaystyle\left\\{\begin{array}[]{l}\text{Hn}\left(1,s;\alpha,\beta,\gamma,\delta;z\right)=\left(1-z\right)^{r}F\left(r+\alpha,r+\beta;\gamma;z\right)\,\,\,\,,\\\
\\\
r=\xi-\sqrt{\xi^{2}-\alpha\beta-s}\,\,\,\,\,\,,\,\,\,\,\,\,\,\xi=\frac{1}{2}\left(\gamma-\alpha-\beta\right)\,\,\,\,\,.\end{array}\right.$
(5.14)
We have represented in Figure 1 the Jacobi elliptic function
${\text{sn}}{(x,2)}$ and in Figure 2 the Heun function
$\text{Hn}\left(4,0;\frac{3}{4},-\frac{1}{4},\frac{1}{2},\frac{1}{2};{\text{sn}}^{2}{(x,2)}\right)$.
Figure 1: A sketch of the Jacobi elliptic function ${\text{sn}}{(x\,,\,2)}$.
Figure 2: A sketch of the Heun function
$\text{Hn}\left(4,0;\frac{3}{4},-\frac{1}{4},\frac{1}{2},\frac{1}{2};{\text{sn}}^{2}{(x,2)}\right)$
for $\sqrt{1+16\frac{e^{2}}{\lambda}}=2$.
## 6 Conclusion
We have presented classical solutions to the equations of motion of the
Abelian Higgs model (2.1). The complex scalar field $\phi$ and the gauge field
$A_{\mu}$ are given by
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\pm
v\,\varepsilon\,{\text{pq}}{\left(w+d\,,\,m\right)}\,e^{i\theta}\,\,\,\,\,\,,$
$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{e}\,\partial_{\mu}\theta+\varepsilon_{\mu}\bigg{[}C_{1}\,\text{Hn}\left(k^{2},0;\alpha_{+},\beta_{-},\frac{1}{2},\frac{1}{2};Z\right)$
$\displaystyle+$ $\displaystyle
C_{2}\,\text{Hn}\left(k^{2},0;\alpha_{-},\beta_{+},\frac{1}{2},\frac{1}{2};Z\right)\bigg{]}\,\,\,\,\,,$
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\varepsilon^{2}\,\frac{\left(1+k^{2}\right)}{2k^{2}}\,{\text{pq}}^{2}{\left(w+d\,,\,m\right)}\,\,\,\,\,,$
$\displaystyle w$ $\displaystyle=$ $\displaystyle
p_{\mu}x^{\mu}+w_{0}\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\varepsilon_{\mu}\,\varepsilon^{\mu}\,=p_{\mu}\,\varepsilon^{\mu}\,=0\,\,\,\,\,\,.$
(6.1)
Here ${\text{pq}}{\left(w+d\,,\,m\right)}$, with pq any pair of the letters
(c,d,n,s), is one of the twelve Jacobi elliptic functions and
$\text{Hn}\left(k^{2},s;\alpha,\beta,\gamma,\delta;z\right)$ is the Heun
function. We have listed in table 1 the values $p^{2}$ (the mass-shell
relation) and $\varepsilon$ for each function
${\text{pq}}{\left(w+d\,,\,m\right)}$. Table 1 gives also the expression of
$k^{2}$ in terms of the parameter $m$ entering the Jacobi elliptic function
${\text{pq}}{\left(w+d\,,\,m\right)}$. Finally,
$\alpha_{\pm}=\frac{1}{4}\left(1\pm\sqrt{1+16\,\frac{e^{2}}{\lambda}}\right)$
and
$\beta_{\mp}=\frac{1}{4}\left(1\mp\sqrt{1+16\,\frac{e^{2}}{\lambda}}\right)$.
Notice that the differential equation (5.1) is linear in $h(w)$. Therefore,
the gauge field $A_{\mu}$ is in fact a linear combination of all the
polarisation vectors $\varepsilon_{\mu}$ satisfying
$\varepsilon_{\mu}\,\varepsilon^{\mu}\,=p_{\mu}\,\varepsilon^{\mu}\,=0$.
Finally, the solution (6.1) could have been expressed in terms of the
Weierstrass elliptic function as shown in Appendix A.
There are two immediate questions that one might ask regarding the solutions
found in this article. The first regards their physical relevance.
Unfortunately, our solution cannot find applications in condensed matter
physics. This is because the ’trick’ used to decouple the complex scalar field
equation of motion does not apply in the non-relativistic version of the
Abelian Higgs model (Ginzburg-Landau theory, gauged non-linear Schrödinger,
London theory, $\dots$). The equivalent of
$\widetilde{A}_{\mu}\widetilde{A}^{\mu}=0$ would be $A_{0}=c\vec{A}^{2}$, $c$
being a constant. However, this is not compatible with Maxwell’s equations. It
is though plausible that our solution might be of use in a theory of gravity
coupled to the Abelian Higgs model. This is currently under investigation.
The second (hard) question concerns the stability of the solution presented
here. Our solution is not protected by any topological argument (unlike the
vortex solution) and therefore is expected to be unstable. The only way to set
one’s mind at rest is by carrying a perturbation expansion around our exact
solutions. This is done through the substitution777Our solution can be make
static by setting $p_{0}=0$ in the variable $w=p_{\mu}x^{\mu}+w_{0}$. (in the
spirit of refs.[11, 26])
$\displaystyle\rho$ $\displaystyle\longrightarrow$
$\displaystyle\rho\left(w\right)+\kappa\left(w\right)\,e^{i\omega_{1}t}$
$\displaystyle\widetilde{A}_{\mu}$ $\displaystyle\longrightarrow$
$\displaystyle\widetilde{A}_{\mu}\left(w\right)+a_{\mu}\left(w\right)\,e^{i\omega_{2}t}\,\,\,.$
(6.2)
Here $\rho\left(w\right)$ and $\widetilde{A}_{\mu}\left(w\right)$ are
solutions to the equations of motion of the Abelian Higgs model. The
perturbations $\kappa\left(w\right)$ and $a_{\mu}\left(w\right)$ are
accompanied by their normal modes represented by the two constants
$\omega_{1}$ and $\omega_{2}$, respectively. If $\omega_{1}$ and $\omega_{2}$
are real then we have an oscillatory regime and the solution is stable. The
next step is to plug the above substitution into the full equations of motion
(2.5), (2.6) and (2.7). To first order in the perturbation, one obtains then
the differential equations which, in principle, allow one to determine the
normal modes. The differential equations obtained here involve the Jacobi
elleptic functions and Heun functions and are, at the moment, difficult to
analyse.
It might be easier, though, to choose a perturbation which respects the
constraint $\widetilde{A}_{\mu}\widetilde{A}^{\mu}=0$. This can be achieved
for instance by writing $a_{\mu}\left(w\right)=v_{\mu}g\left(w\right)$ and
demanding that $v_{\mu}v^{\mu}=\varepsilon_{\mu}v^{\mu}=0$, where
$\varepsilon_{\mu}$ is the polarisation vector of
$\widetilde{A}_{\mu}\left(w\right)$. The substitution (6.2) is then carried
out in the simpler equations of motion (2.16), (2.17) and (2.18). We hope to
report on the progress in this case in the near future.
As a last remark, we mention that our solution is easily adapted to the case
of a $(1+1)$-dimensional Abelian Higgs model and the complex scalar field
$\phi$ could be compactified on a circle of length $L$. This would make close
contact with the works in [11, 12, 41] and might be of help in the
determination of the normal modes.
APPENDICES
## Appendix A Solutions to $\phi^{4}$ in terms of the Weierstrass elliptic
function
The Weierstrass elliptic function is a general solution to the first order
differential equation [29, 30, 31] (see also [33] for some lectures on the
suject)
$\displaystyle\left(\frac{\text{d}\wp}{\text{d}z}\right)^{2}$ $\displaystyle=$
$\displaystyle 4\,\wp^{3}\left(z\right)-g_{2}\,\wp\left(z\right)-g_{3}$ (A.1)
$\displaystyle=$ $\displaystyle 4\left(\wp-e_{1}\right)\left(\wp-
e_{2}\right)\left(\wp-e_{3}\right)\,\,\,\,.$
The constants $g_{2}$ and $g_{3}$ are known as the lattice invariants and
$e_{i}\,\,,i=1,2,3$ are the roots of the Weierstrass normal cubic equation
$4z^{3}-g_{2}z-g_{3}=0$.
In order to obtain a Weierstrass differential equation in the case of
$\phi^{4}$ theory, we start from equation (4.6)
$\displaystyle\left(\frac{\text{d}y}{\text{d}w}\right)^{2}=-\frac{\lambda
v^{2}\varepsilon^{2}}{2p^{2}}\,y^{4}+\frac{\lambda
v^{2}}{p^{2}}\,y^{2}+\frac{a}{\varepsilon^{2}}$ (A.2)
and multiply both sides with $4y^{2}$ to get
$\displaystyle\left(\frac{\text{d}f}{\text{d}w}\right)^{2}$ $\displaystyle=$
$\displaystyle-\frac{2\lambda
v^{2}\varepsilon^{2}}{p^{2}}\,f^{3}+\frac{4\lambda
v^{2}}{p^{2}}\,f^{2}+\frac{4a}{\varepsilon^{2}}\,f\,\,\,\,,$ $\displaystyle
f\left(w\right)$ $\displaystyle\equiv$ $\displaystyle
y^{2}\left(w\right)\,\,\,\,.$ (A.3)
The next step is to get rid of the term proportional to $f^{2}$. This is
achieved by the change of functions
$f\left(w\right)=g\left(w\right)+\frac{2}{3}\frac{1}{\varepsilon^{2}}\,\,\,.$
(A.4)
The resulting differential equation is given by
$\displaystyle\left(\frac{\text{d}g}{\text{d}w}\right)^{2}$ $\displaystyle=$
$\displaystyle-\frac{2\lambda
v^{2}\varepsilon^{2}}{p^{2}}\,g^{3}+\frac{4}{\varepsilon^{2}}\left(\frac{2}{3}\frac{\lambda
v^{2}}{p^{2}}+{a}\right)g+\frac{8}{3}\frac{1}{\varepsilon^{4}}\left(\frac{4\lambda
v^{2}}{9p^{2}}+{a}\right)\,\,\,\,.$ (A.5)
In order to obtain a Weierstrass differential equation, we choose the constant
$\varepsilon$ such that
$p^{2}=-\varepsilon^{2}\,\frac{\lambda v^{2}}{2}\,\,\,.$ (A.6)
This leads to
$\displaystyle\left(\frac{\text{d}g}{\text{d}w}\right)^{2}$ $\displaystyle=$
$\displaystyle
4\,g^{3}+\frac{4}{\varepsilon^{4}}\left({\varepsilon^{2}}\,a-\frac{4}{3}\right)\,g+\frac{8}{3}\frac{1}{\varepsilon^{6}}\left({\varepsilon^{2}}\,a-\frac{8}{9}\right)\,\,\,\,,$
$\displaystyle=$ $\displaystyle
4\left[g-\frac{1}{\varepsilon^{2}}\left(\frac{1}{3}+\sqrt{1-{\varepsilon^{2}}\,a}\right)\right]\left[g-\frac{1}{\varepsilon^{2}}\left(\frac{1}{3}-\sqrt{1-{\varepsilon^{2}}\,a}\right)\right]\left[g+\frac{2}{3}\frac{1}{\varepsilon^{2}}\right]\,\,\,\,.$
The solution to this differential equation is given by the Weierstrass
elliptic function
$\displaystyle
g\left(w\right)=\wp\left(w+d\,;\,-\frac{4}{\varepsilon^{4}}\left({\varepsilon^{2}}\,a-\frac{4}{3}\right)\,,\,-\frac{8}{3}\frac{1}{\varepsilon^{6}}\left({\varepsilon^{2}}\,a-\frac{8}{9}\right)\right)\,\,\,\,,$
(A.8)
where $d$ is a constant. Recalling that $\rho\left(w\right)=v\varepsilon
y\left(w\right)$, the square of our scalar field $\rho\left(w\right)$ is
finally given by
$\displaystyle\rho^{2}\left(w\right)$ $\displaystyle=$ $\displaystyle
v^{2}\left[\frac{2}{3}+\varepsilon^{2}\,\wp\left(w+d\,;\,-\frac{4}{\varepsilon^{4}}\left({\varepsilon^{2}}\,a-\frac{4}{3}\right)\,,\,-\frac{8}{3}\frac{1}{\varepsilon^{6}}\left({\varepsilon^{2}}\,a-\frac{8}{9}\right)\right)\right]\,\,\,\,\,\,\,\,\,$
$\displaystyle=$ $\displaystyle
v^{2}\left[\frac{2}{3}+\wp\left(\frac{1}{\varepsilon}(w+d)\,;\,-{4}\left({\varepsilon^{2}}\,a-\frac{4}{3}\right)\,,\,-\frac{8}{3}\left({\varepsilon^{2}}\,a-\frac{8}{9}\right)\right)\right]\,\,\,\,\,,\,\,\,\,$
$\displaystyle p^{2}$ $\displaystyle=$
$\displaystyle-\varepsilon^{2}\,\frac{\lambda v^{2}}{2}\,\,\,.$ (A.9)
In the last equality we have used the homogeneity relation [29, 30, 31, 33]
$\wp\left(z;g_{2},g_{3}\right)=\mu^{-2}\wp\left(\mu^{-1}z;\mu^{4}g_{2},\mu^{6}g_{3}\right)\,\,\,\,.$
(A.10)
Therefore we could have used the Weierstrass elliptic function (instead of the
Jacobi elliptic functions) to express the solution to the Abelian Higgs model
given in (6.1). In this case, the identification of the two expressions of
$p^{2}$ in (A.9) and (5.7) leads to
${\varepsilon^{2}}\,a=\frac{4k^{2}}{\left(1+k^{2}\right)^{2}}\,\,\,\,.$ (A.11)
Finally, we should mention that the Weierstrass elliptic function could be
converted into Jacobi elliptic functions [29, 30, 31, 33].
## Appendix B The twelve Jacobi elliptic functions Solutions to $\phi^{4}$
The table below summarises the solutions to the equation
$\displaystyle\left(\frac{\text{d}y}{\text{d}w}\right)^{2}=-\frac{\lambda
v^{2}\varepsilon^{2}}{2p^{2}}\,y^{4}+\frac{\lambda
v^{2}}{p^{2}}\,y^{2}+\frac{a}{\varepsilon^{2}}\,\,\,\,\,$ (B.1)
and gives the corresponding values of the parameters $p^{2}$ (the mass-shell
relation), $\varepsilon^{2}$ and $a$ (see also [34, 35] and [36] for some
particular cases).
$p^{2}$ | $\varepsilon^{2}$ | $\frac{a}{\varepsilon^{2}}$ | $y\left(w\right)$ | $\rho\left(w\right)$ | $k^{2}\,\,\text{or}\,\,\frac{1}{k^{2}}$
---|---|---|---|---|---
$-\frac{\lambda v^{2}}{1+m^{2}}$ | $\frac{2m^{2}}{1+m^{2}}$ | $1$ | $\text{sn}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{sn}\left(w+d,m\right)$ | $m^{2}$
$-\frac{\lambda v^{2}}{1-2m^{2}}$ | $-\frac{2m^{2}}{1-2m^{2}}$ | $1-m^{2}$ | $\text{cn}\left(w+d,m\right)$ | $\pm\,v\,\varepsilon\,\text{cn}\left(w+d,m\right)$ | $\frac{m^{2}-1}{m^{2}}$
$\frac{\lambda v^{2}}{2-m^{2}}$ | $\frac{2}{2-m^{2}}$ | $m^{2}-1$ | $\text{dn}\left(w+d,m\right)$ | $\pm\,v\,\varepsilon\,\text{dn}\left(w+d,m\right)$ | ${1-m^{2}}$
$-\frac{\lambda v^{2}}{1+m^{2}}$ | $\frac{2m^{2}}{1+m^{2}}$ | $1$ | $\text{cd}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{cd}\left(w+d,m\right)$ | $m^{2}$
$\frac{\lambda v^{2}}{2m^{2}-1}$ | $\frac{2m^{2}(1-m^{2})}{2m^{2}-1}$ | $1$ | $\text{sd}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{sd}\left(w+d,m\right)$ | $\frac{m^{2}-1}{m^{2}}$
$\frac{\lambda v^{2}}{2-m^{2}}$ | $\frac{2(1-m^{2})}{2-m^{2}}$ | $-1$ | $\text{nd}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{nd}\left(w+d,m\right)$ | ${1-m^{2}}$
$-\frac{\lambda v^{2}}{1+m^{2}}$ | $\frac{2}{1+m^{2}}$ | $m^{2}$ | $\text{dc}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{dc}\left(w+d,m\right)$ | $m^{2}$
$\frac{\lambda v^{2}}{2m^{2}-1}$ | $-\frac{2(1-m^{2})}{2m^{2}-1}$ | $-m^{2}$ | $\text{nc}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{nc}\left(w+d,m\right)$ | $\frac{m^{2}-1}{m^{2}}$
$\frac{\lambda v^{2}}{2-m^{2}}$ | $-\frac{2(1-m^{2})}{2-m^{2}}$ | $1$ | $\text{sc}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{sc}\left(w+d,m\right)$ | ${1-m^{2}}$
$-\frac{\lambda v^{2}}{1+m^{2}}$ | $\frac{2}{1+m^{2}}$ | $m^{2}$ | $\text{ns}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{ns}\left(w+d,m\right)$ | $m^{2}$
$\frac{\lambda v^{2}}{2m^{2}-1}$ | $\frac{-2}{2m^{2}-1}$ | $-m^{2}(1-m^{2})$ | $\text{ds}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{ds}\left(w+d,m\right)$ | $\frac{m^{2}-1}{m^{2}}$
$\frac{\lambda v^{2}}{2-m^{2}}$ | $\frac{-2}{2-m^{2}}$ | $1-m^{2}$ | $\text{cs}\left(w,m\right)$ | $\pm\,v\,\varepsilon\,\text{cs}\left(w+d,m\right)$ | ${1-m^{2}}$
Table 1: The twelve Jacobi elliptic functions solutions to (B.1) and (4.2) and
their corresponding parameters $p^{2}$ (the mass-shell relation),
$\varepsilon^{2}$ and $\frac{a}{\varepsilon^{2}}$. The table gives also the
relation between $k^{2}$ and $m^{2}$ appearing in (6.1).
It is important to notice that $\varepsilon^{2}$ has to be strictly positive
(for $\rho(w)$ to be a real field and different from zero). This,
consequently, puts restrictions on the allowed values of the parameter $m$ for
some of the solutions.
## Appendix C Solutions to $\phi^{4}$ in terms of trigonometric and
hyperbolic functions
We have seen that the equation of motion for the field $\rho(w)$ is given by
the differential equation
$\displaystyle\frac{\text{d}^{2}\rho}{\text{d}w^{2}}+\frac{\lambda}{p^{2}}\,\rho\left(\rho^{2}-v^{2}\right)$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,\,.$ (C.1)
The general solution to this equation is expressed in terms of the twelve
Jacobi elliptic functions depending on a parameter $m$. On the other hand, the
Jacobi elliptic functions reduce to ordinary trigonometric or hyperbolic
functions for the two special values $m=0$ and $m=1$ [29, 30, 31].
There are twelve different trigonometric and hyperbolic functions
corresponding to the values $m=0$ and $m=1$. However, only five888The other
seven functions, for $m=0$ and $m=1$, lead either to $\rho(w)=0$ or to a
complex $\rho(w)$. For example,
${\text{ds}}{(w+d\,,\,1)}=\frac{1}{\sinh{(w+d)}}$ but $\varepsilon^{2}=-2$ for
$m=1$, as can be seen from table 1. This leads to a complex scalar field
$\rho(w)$. of them are solutions the above differential equation. These are
$\displaystyle\rho\left(w\right)$ $\displaystyle=$ $\displaystyle\pm
v\,{\text{sn}}\left[\tau\left(w+\alpha\right)+\beta\,,1\,\right]=\pm
v\,\tanh\left[\tau\left(w+\alpha\right)+\beta\right]\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,p^{2}=-\frac{\lambda\,v^{2}}{2\tau^{2}}\,\,,$
(C.2) $\displaystyle\rho\left(w\right)$ $\displaystyle=$
$\displaystyle\pm\,v\,{\text{ns}}\left[\tau\left(w+\alpha\right)+\beta\,,1\,\right]=\frac{\pm\,v\,}{\tanh\left[\tau\left(w+\alpha\right)+\beta\right]}\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,p^{2}=-\frac{\lambda\,v^{2}}{2\tau^{2}}\,\,\,\,,$
(C.3) $\displaystyle\rho\left(w\right)$ $\displaystyle=$
$\displaystyle\pm\sqrt{2}\,v\,{\text{cn}}\left[\tau\left(w+\alpha\right)+\beta\,,1\,\right]=\frac{\pm\sqrt{2}\,v\,}{\cosh\left[\tau\left(w+\alpha\right)+\beta\right]}\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,p^{2}=\frac{\lambda\,v^{2}}{\tau^{2}}\,\,\,\,,$
(C.4) $\displaystyle\rho\left(w\right)$ $\displaystyle=$
$\displaystyle\pm\sqrt{2}\,v\,{\text{ds}}\left[\tau\left(w+\alpha\right)+\beta\,,0\,\right]=\frac{\pm\sqrt{2}\,v\,}{\sin\left[\tau\left(w+\alpha\right)+\beta\right]}\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,p^{2}=-\frac{\lambda\,v^{2}}{\tau^{2}}\,\,\,\,,$
(C.5) $\displaystyle\rho\left(w\right)$ $\displaystyle=$
$\displaystyle\pm\sqrt{2}\,v\,{\text{dc}}\left[\tau\left(w+\alpha\right)+\beta\,,0\,\right]=\frac{\pm\sqrt{2}\,v\,}{\cos\left[\tau\left(w+\alpha\right)+\beta\right]}\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,p^{2}=-\frac{\lambda\,v^{2}}{\tau^{2}}\,\,.$
(C.6)
The solutions depend on a parameter $\tau$ and a constant $\tau\alpha+\beta$.
However, sometimes these solutions are found under different writings [37, 38,
39, 40]. For instance, the first solution (C.2) can be expressed as
$\displaystyle\rho\left(w\right)$ $\displaystyle=$ $\displaystyle\pm
v\,\tanh\left[\tau\left(w+\alpha\right)+\beta\right]=\pm
v\,\frac{\mu+\tanh\left[\tau\left(w+\alpha\right)\right]}{1+\mu\tanh\left[\tau\left(w+\alpha\right)\right]}\,\,,\,\,\mu=\tanh\left(\beta\right)\,\,.$
(C.7)
Other expressions can be reached by simply expanding the arguments of the
trigonometric and hyperbolic functions.
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|
Daniel S. Shen∗ and Min Chi+ ∗William G. Enloe High School, Raleigh, NC, USA
+North Carolina State University, Raleigh, NC, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
# TC-DTW: Accelerating Multivariate Dynamic Time Warping Through Triangle
Inequality and Point Clustering
###### Abstract
Dynamic time warping (DTW) plays an important role in analytics on time
series. Despite the large body of research on speeding up univariate DTW, the
method for multivariate DTW has not been improved much in the last two
decades. The most popular algorithm used today is still the one developed
seventeen years ago. This paper presents a solution that, as far as we know,
for the first time consistently outperforms the classic multivariate DTW
algorithm across dataset sizes, series lengths, data dimensions, temporal
window sizes, and machines. The new solution, named TC-DTW, introduces
Triangle Inequality and Point Clustering into the algorithm design on lower
bound calculations for multivariate DTW. In experiments on DTW-based nearest
neighbor finding, the new solution avoids as much as 98% (60% average) DTW
distance calculations and yields as much as 25$\times$ (7.5$\times$ average)
speedups.
## 1 Introduction
Temporal data analytics is important for many domains that have temporal data.
One of its fundamental questions is calculating the similarity between two
time series, due to temporal distortions. In the medical records of two
patients, for instance, the times when their checks were done often differ;
the points on two motion tracking series often differ in the arrival times of
their samples. Consequently, it is important to find the proper alignment of
the points in two time series before their similarity (or differences) can be
computed. In such an alignment, as illustrated in Figure 1(a), one point may
match with one or multiple points in the other series.
The most influential method to find the proper alignments is Dynamic Time
Warping (DTW). It uses dynamic programming to find the best mapping, that is,
an alignment that makes the accumulated differences of the aligned points in
the two series minimum. Such an accumulated difference is called DTW distance.
Experimental comparisons of DTW to most other distance measures on many
datasets have concluded that DTW almost always outperforms other measures
Wang+:DMKD2013 . It hence has been used as one of the most preferable measures
in pattern matching tasks for time series data Chadwick+:DEST2011 ;
Adams+:ISMIR05 ; Alon+:PAMI2009 ; Laerhoven+:ICMLA2009 ; Dupasquier+:Chaos2011
; Huber+:MVA2011 ; Inglada+:sense2015 ; Rak+:TKDD2013 .
DTW remains popular even in the era of deep learning, for several reasons.
DTW-based temporal data analytics are still widely used for its advantages in
better interpretability over deep neural networks. Moreover, there is also
interest in combining DTW with deep learning by, for instance, using DTW as
part of the loss function. Studies have shown that DTW, compared to
conventional Euclidean distance loss, leads to a better performance in a
number of applications Cuturl+:arxiv2017 through softmin Chang+:arxiv2019 ;
Shah+:CODS16 . A more recent way to combine DTW and deep learning is to use
DTW as a replacement of kernels in convolutional neural networks, creating
DTWNet that shows improved inference accuracy Cai+:NIPS2019 .
(a) DTW is to find the best mapping (b) Difference matrix (c) Bounding envelope of a 1D series and the LB_Keogh lower bounds |
---|---
Figure 1: (a) DTW is about finding the mapping between the points on two
sequences that minimizes the accumulated differences between the corresponding
points. (b) The difference matrix between two sequences Q and C, and the
Sakoe-Chuba Band. (c) The bounding envelope of sequence Q is used for
computing the Keogh lower bounds (summation of the shaded area) of the DTW
distance between Q and C. U and L: two series that form an envelope of Q. (d)
The bounding envelope on a 2D series is even looser due to the stretching
effects on both dimensions.
Even with dynamic programming, DTW is still time-consuming to compute. On its
own, DTW has an $O(mn)$ time complexity ($m,n$ are the lengths of the involved
two series). Maximizing the DTW speed is essential for many uses. In real-time
health monitoring systems, for example, speed deficiency forces current
systems to do coarse-grained sampling, leaving many sensed data unprocessed,
causing risks of missing crucial symptoms early. For some diseases, it could
be life threatening. For Septic Shock, for instance, every hour of delay in
antibiotic treatment leads to an 8% increase in mortality Kumar2006 .
Many studies have been conducted to enhance the speed of DTW-based analytics
Rak+:TKDD2013 ; Tan+:SDM17 ; Yi+:VLDB2000 ; Kim+:ICDE2001 ; Lemire+:PR2009 ;
Zhou+:ICDE2011 . The most commonly used is the use of lower bounds. Consider a
common use of DTW, searching for the candidate series in a collection that is
the most similar to a query series. We can avoid computing the DTW distance
between the query and a candidate series if the lower bound of their DTW
distance exceeds the best-so-far (i.e., the smallest DTW distance between the
query and the already examined candidate series). Various lower bounding
methods have been proposed Yi+:VLDB2000 ; Kim+:ICDE2001 ; Lemire+:PR2009 ;
Zhou+:ICDE2011 , with LB_Keogh Keogh+:VLDB2006 being the most popular choice.
Most prior work on accelerating DTW has been, however, on univariate DTW;
research on optimizing multivariate DTW has been scarce. A multidimensional or
as we will refer to in the following, a multivariate time series is one in
which each temporal point of the series carries multiple values. Multivariate
time series data are ubiquitous in real-world dynamic systems such as health
care and distributed sensor networks. For example, measurements taken from a
weather station would be a multivariate time series with the dimensions being
temperature, air pressure, wind speed, and so on. There is a large demands for
fast DTW on multivariate time series Shokoohi+:DMKD17 ; Ridgely+:2009 ;
Kela+:2006 ; Ko+:2005 ; Liu+:Mob2009 ; Peti+:TGRS12 ; Wang+:interspeech2013 .
The current multivariate DTW analysis is based on an algorithm named LB_MV
Rath+:2003 . It was proposed in 2003 as a straightforward extension of the
univariate lower bound-based DTW algorithms to multivariate data. There have
been a few attempts to speed up multivariate DTW, but they are primarily
concentrated on the reduction or transformations of the high dimension problem
space Li+:physica2015 ; Li+:elsevier2017 ; Hu+:ICDM2013 ; Gong+:Springer2015 ,
rather than optimizing the designs of DTW algorithms.
The goal of this work is to come up with new algorithms that can
significantly, consistently outperform the two-decade prevailing algorithm,
whereby advancing the state of the art and better meeting the needs of
temporal data analytics.
Our approach is creatively tightening the DTW lower bounds. The classic
method, LB_MV, gets a lower bound by efficiently computing a bounding envelope
of a time series by getting the max and min on each dimension in a series of
sliding windows. The envelope is loose, as Figure 1(d) shows (further
explained in Section 3). The key challenge in tightening the lower bounds is
in keeping the time overhead sufficiently low while tightening the bounds as
much as possible. It is especially challenging for multivariate data: As the
DTW distance is affected by all dimensions, the computations of the lower
bounds would need to take them all into consideration.
Our solution is TC-DTW. TC-DTW addresses the problem by creatively introducing
geometry and quantization into DTW bounds calculations. More specifically, TC-
DTW introduces Triangle Inequality and Point Clustering into the algorithm
design of multivariate DTW. It features two novel ideas: (i) leveraging the
overlapping windows of adjacent points in a time series to construct triangles
and then using Triangle Inequality to compute tighter lower bounds with only
scalar computations; (ii) employing quantization-based point clustering to
efficiently reduce the size of bounding boxes in the calculations of lower
bounds. These two ideas each lead to an improved algorithm for DTW lower bound
calculations. Putting them together via a simple dataset-based adaptation, we
show that the resulting TC-DTW consistently outperforms the classic algorithm
LB_MV, across datasets, data sizes, dimensions, temporal window sizes, and
hardware.
Experiments on the 13 largest datasets in the online UCR collection
Bagnall+:arxiv2018 on two kinds of machines show that TC-DTW produces lower
bounds that can help avoid as much as 98% (60% average) DTW distance
calculations in DTW-based nearest neighbor findings, the most common usage of
DTW. The speedups over the default windowed DTW (without lower bounds) are as
much as 25$\times$ (8$\times$ on average), significantly higher than the
speedups from the classic algorithm. Sensitivity studies on different
dimensions, window sizes, and machines have all confirmed the consistently
better performance of TC-DTW at no loss of any result quality, which suggests
the potential of TC-DTW in serving as a more efficient drop-in replacement of
existing multivariate DTW algorithms. The code has been released on
github111https://github.com/DanielSongShen/MultDTW. To our best knowledge, TC-
DTW is the first known multivariate DTW algorithm that consistently
outperforms the classic algorithm LB_MV Rath+:2003 with no loss of the result
precision.
In the rest of this paper, we first describe the background in Section 2,
provide a formal problem statement 3, and then present the integration of
Triangle Inequality and Point Clustering into DTW respectively in Sections 4
and 5. After that, we explain the adaptive deployment of two ideas, and their
combined result TC-DTW in Section 6. We report experimental results in Section
7, discuss some other issues in Section 8 and related work in Section 9, and
finally conclude the paper in Section 10.
## 2 Background
In this section, we will review the basic DTW algorithm as well as two common
optimizations, window DTW and lower bound filtering. Then, we will describe
the prior solutions to multivariate DTW and their limitations.
Dynamic Time Warping (DTW) Let $Q=<q_{1},q_{2},...q_{m}>$ be a query time
series and $C=<c_{1},c_{2},...c_{n}>$ be a candidate time series we want to
compare it to. As Figure 1 (a) illustrates, DTW tries to find the best
alignment between $Q$ and $C$ such that the accumulated difference between the
mapping points is minimum. In the mapping, it is possible that multiple points
in one series map to one point in the other series. DTW finds such a mapping
by building a cost matrix $C$ between each pair of points in $Q$ and $C$ where
each element $d(i,j)$ of the matrix is the cost of aligning $q_{i}$ with
$c_{j}$, as illustrated by the meshed box in Figure 1(b). DTW then uses
dynamic programming, shown as the following recursive formula, to minimize the
sum of the elements on a path from the bottom left to the top right of the
cost matrix. The path is known as the warping path; at each element of the
cost matrix, the path goes right, up, or up right.
$DTW(i,j)=d(i,j)+\min\begin{cases}DTW(i-1,j)\\\ DTW(i,j-1)\\\ DTW(i-1,j-1)\\\
\end{cases}$ (2.1)
Window DTW The basic DTW has a complexity O$(mn)$, where $m$ and $n$ are the
lengths of the two sequences. A common approach to reducing the computations
is to use a window to constrain the warping path. This restrains the distance
in the time axis that a point $q_{i}$ can be mapped to in $C$. There are
several versions of window DTW, the most popular of which, also the one we
will be using in this paper, is the Sakoe-Chiba Band SakoeBand . As shown in
Figure 1 (b), the Sakoe-Chiba Band assumes the best path through the matrix
will not stray far from the middle diagonal. Let two points aligning in C and
Q be their $i$th and $j$th point respectively. The window constraint states
that the following must hold: $j-W\leq i\leq j+W$, where $W$ is called the
warping window size. Such a constraint is used in almost all applications of
DTW, in both single and multiple variate cases.
DTW-NN and Univariate Lower Bound Filtering A common use of DTW is in finding
a reference sequence that is most similar to a given query sequence—which is
called DTW-NN (or DTW Nearest Neighbor) problem. It is, for instance, a key
step in DTW-based time series clustering. As many DTW distances need to be
computed when there are many reference sequences, even with the window
constraints, the process can still take a long time. Many studies have tried
to improve the speed. Lower bound filtering is one of the most widely used
optimizations, designed for saving unnecessary computations and hence getting
large speedups in this setting. The basic idea is that if the lower bound of
the DTW distance between Q and a reference C is already greater than the best-
so-far (i.e., the smallest distance from Q to the references examined so far),
the calculations of the DTW distance between Q and C can be simply skipped as
it is impossible for C to be the nearest neighbor of Q.
For univariate DTW, researchers have proposed a number of ways to compute DTW
lower bounds (as listed in Section 9). The most commonly used is LB_Keogh
bound Keogh+:VLDB2006 . Given two univariate time series $Q$ and $C$, of
lengths $m$ and $n$ respectively, the LB_Keogh lower Bound $LB(Q,C)$ is
calculated in two steps.
1) As shown in Figure 1 (c), a bounding envelope is created around series $Q$
(red line) by constructing two time series $U$ and $L$ (for upper and lower,
represented by two dotted lines) such that $Q$ must lie in the between. $U$
and $L$ are constructed as:
$u_{i}=\max(q_{i-W}:q_{i+W})\;\;\;\;\;\;l_{i}=\min(q_{i-W}:q_{i+W})$
2) Once the bounding envelope has been created, $LB(Q,C)$ can be simply
computed from the parts of $C$ outside the envelope. Formally, it is
calculated as the squared sum of the distances from every part of the
candidate sequence $C$ outside the bounding envelope to the nearest orthogonal
edge of the bounding envelope, illustrated as the shaded area in Figure 1(c).
Multivariate DTW Multivariate DTW deals with multi-dimensional time series
(MDT), where each point in a series is a point in a $D$-dimensional ($D>1$)
space. We can represent a time series $Q$ as $q_{1}$, $\ldots$ $q_{n}$ and
each $q_{i}$ is a vector in $R^{D}$, $(q_{i,1}$, $\ldots$, $q_{i,D})$.
Multivariate DTW measures the dynamic warping distance between two MDTs. There
are two ways to define multivariate DTW distance.
One is independent DTW, calculated as the cumulative distances of all
dimensions independently measured under univariate DTW, expressed as
$DTW_{I}(Q,C)=\sum_{i=1,D}DTW(Q_{i},C_{i})$, where $Q_{i},C_{i}$ are
univariate time series equaling the $i$th dimension of $Q$ and $C$
respectively. The other is dependent DTW, calculated in a similar way to the
standard DTW for single-dimensional time series, except that the distance
between two points on $Q$ and $C$ is the Euclidean distances of two
$D$-dimensional vectors. Both kinds of DTWs have been used. A comprehensive
study Shokoohi+:DMKD17 has concluded that both kinds are valuable for
practical applications.
As independent DTW is the sum of multiple univariate DTWs, it can directly
benefit from optimizations designed for univariate DTWs. Dependent DTW is
different; the multi-dimensional distance calculations are not only more
expensive than univariate distance, but also offer a different problem setting
for optimizations. This paper focuses on optimizations of dependent DTW,
particularly the optimizations that are tailored to its multi-dimensional
nature. Without further notice, in the rest of this paper, multivariate DTW
refers to dependent DTW. (For the completeness of the discussion, we note that
there is another kind of multivariate DTW, which aligns points not on the time
dimension, but on a multi-dimensional space Cai+:NIPS2019 , such as matching
pixels across two 2-D images. It is beyond the scope of this work.)
## 3 Problem Statement and Design Considerations
This section starts with a formal definition of the optimal lower bounding
problem of multivariate DTW and the tradeoffs to consider in the solution
design. It prepares the foundation for the follow-up presentations of the set
of new algorithm designs.
The optimal lower bounding problem is to come up with a way to calculate the
lower bound of the DTW of two multivariate series such that when it is applied
to DTW-NN (DTW for Nearest Neighbor introduced in Sec 2), the overall time is
minimized. Formally, it is defined as follows.
###### Definition 1.
Optimal Lower Bounding Problem for Multivariate DTW in DTW-NN. Let
$\mathbb{Q}$ and $\mathbb{C}$ be two sets of multivariate time series. The
problem is to find
$\mathbb{LB}^{\ast}=\\{LB_{i,j}^{\ast}|i=1,\ldots,|\mathbb{Q}|$,
$j=1,\ldots,|\mathbb{C}|$} such that:
$\displaystyle\forall
Q_{i}\in\mathbb{Q},C_{j}\in\mathbb{C},\;\;\;\;\;LB_{i,j}^{\ast}\leq
DTW(Q_{i},C_{j})$ (3.2)
$\displaystyle\mathbb{LB}^{\ast}=\operatorname*{arg\,min}_{\mathbb{LB}}T_{\mathbb{Q},\mathbb{C}}(\mathbb{LB})+T_{\mathbb{LB}}(\mathbb{Q},\mathbb{C})$
(3.3)
where, $DTW(Q_{i},C_{j})$ is the DTW distance between $Q_{i}$ and $C_{j}$,
$T_{\mathbb{Q},\mathbb{C}}(\mathbb{LB})$ is the time taken to compute all
lower bounds in $\mathbb{LB}$ from $\mathbb{Q}$ and $\mathbb{C}$, and
$T_{\mathbb{LB}}(\mathbb{C},\mathbb{Q})$ is the time taken to find the nearest
neighbor of $Q_{i}$ ($i=1,\ldots,|\mathbb{Q}|$) in $\mathbb{C}$ while
leveraging $\mathbb{LB}$.
Formula 3.3 in the problem definition indicates the fundamental tradeoff in
lower bounding, that is, the tension between the time needed to compute the
lower bounds and the time the lower bounds may save DTW-NN. The tighter the
lower bound is, the more effective it can be in helping avoid unnecessary DTW
distance calculations; but if it takes too much time to compute, the benefits
could be outweighed by the overhead.
Algorithm LB_MV LB_MV is the classic algorithm for computing the lower bounds
of multivariate DTW. Since it was proposed in 2003 Rath+:2003 , it has been
the choice in almost all applications of multivariate DTW. It is a simple
extension of the univariate method for computing the LB_Keogh lower bound
(which is explained in Section 2). Like in the univariate case, LB_MV also
first builds a bounding envelope for the query series. Each point on the top
and bottom of the envelope, $u_{i}$ and $l_{i}$, are defined as follows:
$u_{i}=(u_{i,1},u_{i,2},\ldots,u_{i,D})\;\;\;\;\;l_{i}=(l_{i,1},l_{i,2},\ldots,l_{i,D})$
where, $u_{i,p}$ and $l_{i,p}$ are the max and min of the points in a window
(centered at $i$) on the query series Q on dimension $p$, that is:
$u_{i,p}=max(q_{i-W,p}:q_{i+W,p})\;\;\;\;\;l_{i,p}=min(q_{i-W,p}:q_{i+W,p})$
Figure 1(d) illustrates the bounding envelope of a 2D series in an actual time
series of Japanese vowel audio UCRArchive2018 .
With the envelope, the lower bound of the DTW distance between series Q and
series C is calculated as ($n$ for the length of $C$)
$LB\\_MV(Q,C)=\sqrt{\sum_{i=1}^{n}\sum_{p=1}^{D}\begin{cases}\begin{array}[]{ll}(c_{i,p}-u_{i,p})^{2}&ifc_{i,p}>u_{i,p}\\\
(c_{i,p}-l_{i,p})^{2}&ifc_{i,p}<l_{i,p}\\\ 0&otherwise\end{array}\end{cases}}$
This straightforward extension has been adopted widely in the applications of
multivariate DTW. It computes the bounds fast, but is limited in
effectiveness. As simple $max$ and $min$ are taken as the bounds in every
dimension, the envelope is subject to the stretching effects on all
dimensions, resulting in the loose bounding envelope as seen in Figure 1(d).
(Experiments in Section 7 confirm the limitation quantitatively.)
Algorithm LB_AD. To facilitate the following discussions, we present a simple
alternative method to LB_MV illustrated in Figure 2(a). In the graph, $Q$ and
$C$ are a candidate series and a query series respectively. For a point
$q_{i}$ on $Q$, this method computes the distances between $q_{i}$ and every
candidate point in the relevant window, that is, every $c_{j}$ on $C$
($i-W\leq j\leq i+W$), and selects the smallest distance as the point lower
bound at i. It then sums up all these point lower bounds of $C$ to get the
lower bound of $DTW(C,Q)$. The soundness of the method is easy to see if we
notice that the point that $q_{i}$ maps to in the DTW path must be one of the
$c_{j}$s on $C$ ($i-W\leq j\leq i+W$).
The lower bounds obtained by this method can be much tighter than LB_MV, but
the computation time to get the lower bounds is comparable to the time needed
to compute the actual DTW distance. We call this method LB_AD (AD for all
distances).
The algorithms that we will introduce next can be regarded as optimizations to
$LB\\_AD$. They either leverage Triangle Inequality or Point Clustering to
conservatively approximate the lower bound LB_AD while substantially reducing
the needed computations.
| |
---|---|---
(a) | (b) | (c)
Figure 2: (a) Cost matrix with window size equaling five. In LB_AD, all the
distances are computed between a candidate point and the query points within
its window, the minimum of which is taken as the lower bound at the candidate,
and the summation of these lower bounds across all candidate points is taken
as the lower bound of $DTW(Q,C)$. (b) Illustration of LB_TI at point $q_{i}$
and $c_{i-1}$. (c) Illustration of LB_TI at the top boundary of a window,
which is $c_{i+2}$ for query point $q_{i}$.
## 4 Triangle Lower Bounding
The algorithms to be presented in this section center around the basic
triangle inequality theorem in geometry.
###### Theorem 1.
Triangle Inequality: for any triangle, the sum of the lengths of any two sides
must be no smaller than the length of the remaining side, denoted as
$d(a,b)\leq d(a,c)+d(b,c)$, where d(x,y) is the distance between two points
$x$ and $y$.
Triangle inequality holds on metrics, such as Euclidean distance. Although it
does not directly apply to DTW distance as it is not a metric, an important
insight we note is that it still holds for point distances in commonly seen
time series, and leveraging it can avoid many unnecessary distance
calculations and hence help efficient derivation of DTW lower bounds. We next
explain how we turn that insight into more efficient DTW algorithms, which we
call triangle DTW.
### 4.1 Basic Triangle DTW (LB_TI)
Figure 2(b) illustrates the essential idea of the basic algorithm of triangle
DTW, named LB_TI (TI for Triangle Inequality). Instead of computing the
distance between $q_{i}$ and every $c_{j}$ ($i-W\leq j\leq i+W$) as in LB_AD,
it uses triangle inequality to quickly estimate the lower bound of the
distance $d(q_{i},c_{j})$.
Consider an example when $j=i-1$, that is, the distance between $q_{i}$ and
$c_{i-1}$ in the cost matrix in Figure 2(a). These two points and $q_{i-1}$
together form a triangle, shown in Figure 2(b). According to Triangle
Inequality, we have
$\displaystyle d(q_{i},c_{i-1})$
$\displaystyle\geq|d(q_{i-1},c_{i-1})-d(q_{i-1},q_{i})|$ (4.4)
So, $|d(q_{i-1},c_{i-1})-d(q_{i-1},q_{i})|$ could be taken as the lower bound
of $d(c_{i},q_{i-1})$. If we do not know $d(q_{i-1},c_{i-1})$, we can
substitute it with its lower bound $L(q_{i-1},c_{i-1})$ or upper bound
$U(q_{i-1},c_{i-1})$ as follows:
$\displaystyle d(q_{i},c_{i-1})$
$\displaystyle\geq|d(q_{i-1},c_{i-1})-d(q_{i-1},q_{i})|$
$\displaystyle=\max(d(q_{i-1},c_{i-1})-d(q_{i-1},q_{i}),\;d(q_{i-1},q_{i})-d(q_{i-1},c_{i-1}))$
$\displaystyle\geq\max(L(q_{i-1},c_{i-1})-d(q_{i-1},q_{i}),\;d(q_{i-1},q_{i})-U(q_{i-1},c_{i-1}))$
(4.5)
Similarly, via Triangle Inequality, we can get the upper bound of
$d(q_{i},c_{i-1})$ as follows:
$\displaystyle d(q_{i},c_{i-1})$ $\displaystyle\leq
d(q_{i-1},c_{i-1})+d(q_{i-1},q_{i})$ $\displaystyle\leq
U(q_{i-1},c_{i-1})+d(q_{i-1},q_{i})$ (4.6)
Based on Equations 4.5 and 4.6, we have the following recursive formula for
the lower bound and upper bound of $d(q_{i},c_{i-1})$
$\displaystyle L(q_{i},c_{i-1})$
$\displaystyle=\max(L(q_{i-1},c_{i-1})-d(q_{i-1},q_{i}),\;d(q_{i-1},q_{i})-U(q_{i-1},c_{i-1}))$
(4.7) $\displaystyle U(q_{i},c_{i-1})$
$\displaystyle=U(q_{i-1},c_{i-1})+d(q_{i-1},q_{i})$ (4.8)
In application of Equations 4.7 and 4.8, $d(q_{i-1},q_{i})$ will need to be
computed, but as it is needed to be done only once for a query series $Q$ and
can be reused for all candidates series, the overhead is marginal.
$L(q_{i-1},c_{i-1})$ and $U(q_{i-1},c_{i-1})$ are already computed when the
algorithm works on $q_{i-1}$. The true distances at $q_{0}$ can be computed
and used as the lower bounds and upper bounds at $q_{0}$ to initiate the
application of the formulae.
It is easy to see that the formulae hold if we replace $c_{i-1}$ with any
other candidate points. What we care about are only the candidate points
within the window of $q_{i}$, that is, $c_{j}(j=i-W,i-W+1,\ldots,i+W)$. As
points $c_{i-W},c_{i-W+1},\ldots,c_{i+W-1}$ also reside in the window of
$q_{W-1}$, the lower bound and upper bound values on the right hand side (rhs)
of Equations 4.7 and 4.8 should be available when the algorithm handles
$q_{i}$. But $c_{i+W}$ is not, and hence needs a special treatment.
Figure 2 (c) illustrates the special treatment of $c_{i+W}$ at $q_{i}$. It
constructs a triangle with $c_{i+W}$, $q_{i}$, and $c_{i+W-1}$. The bounds can
be hence computed as follows:
$\displaystyle L(q_{i},c_{i+W})$
$\displaystyle=\max(L(q_{i},c_{i+W-1})-d(c_{i+W-1},c_{i+W}),\;d(c_{i+W-1},c_{i+W})-U(q_{i},c_{i+W-1}))$
(4.9) $\displaystyle U(q_{i},c_{i+W})$
$\displaystyle=U(q_{i},c_{i+W-1})+d(c_{i+W-1},c_{i+W})$ (4.10)
where, $L(q_{i},c_{i+W-1})$ and $U(q_{i},c_{i+W-1})$ are the results at
$q_{i}$ and $c_{i+W-1}$, and $d(c_{i+W-1},c_{i+W})$ needs to be computed ahead
of time; but as it can be reused for many query series, the overhead is
marginal.
Figure 3: Algorithm of Basic Triangle DTW, $LB\\_TI$.
Full algorithm. Figure 3 outlines the entire algorithm. It first calculates
the distances between every two adjacent points on each query and candidate
series, then goes through each query-candidate series pair to compute the
lower bound of their DTW. In the process, it applies Equations 4.7, 4.8, 4.9,
and 4.10 to compute the lower bounds and upper bounds at each relevant query-
candidate point pair; the minimum of the lower bounds of a candidate point is
taken as its overall lower bound; the summation of all these overall lower
bounds across all candidate points is taken as the lower bound of the DTW of
the query-candidate series pair.
Characteristics. Compared to $LB\\_AD$, this algorithm saves computations by
replacing vector (distance) computations with scalar (bound) computations. It
replaces pair-wise distance computations in the distance matrix with lower
bound and upper bound calculations. This basic Triangle DTW algorithm favors
problems in which the time series are obtained through dense sampling such
that the distance between two adjacent points on a time series is small: small
$d(c_{i},c_{i-1})$ and $d(q_{i+r},q_{i+W-1})$ in Figure 2(b,c) lead to tighter
bounds. The recursive nature of Equations 4.7 and 4.8 entails that as the
algorithm moves through the points on a candidate series, the bounds get
looser and looser. We call this phenomenon the diminishing tightness problem.
Some variations of the algorithm may mitigate the problem. We next describe
several example variations.
### 4.2 Variations of LB_TI with Extra Distances
The variations in this section all try to tighten the bounds of the basic
LB_TI algorithm by adding some extra distance calculations.
(1) Triangle DTW with Top (LB_TI_TOP). The differences of this variation from
the basic LB_TI is that it computes the true Euclidean distance between
$q_{i}$ and the top boundary candidate point of its window, rather than
estimates it with the ($q_{i}$, $c_{i+W}$, $c_{i+W-1}$) triangle (Equations
4.9 and 4.10). Note that as a side effect, it forgoes the need for computing
the distances between adjacent candidate points.
(2) Periodical Triangle DTW (LB_TI$P$). The difference of this variation from
the basic LB_TI is that it periodically tightens the bounds by computing the
true candidate-query point distances at some candidate points, $q_{i}$
($mod(i,P)==0)$, where $P$ is a positive integer, called the TIP period
length. These distances are then used as the lower and upper bounds of $q_{i}$
in the follow-up bound calculations. The smaller $P$ is, the tighter the
bounds could be, and the more computations it incurs. When $P$ equals the
query series length $|Q|$, the algorithm is identical to the Basic Triangle
algorithm LB_TI.
(3) Periodical Triangle DTW with Top (LB_TI$P$_TOP). It is a simple
combination of LB_TI$P$ and LB_TOP, that is, besides periodically computing
the true candidate-query point Euclidean distances, it also computes the true
distances from every query point to the top boundary of its candidate window.
Among these variations, our experiments show that variation three
(LB_TI$P$_TOP) performs the best overall, striking an overall best tradeoff
between the tightness of the bounds and the runtime overhead. In the following
discussion, without further notice, we will use LB_TI to refer to this
variation.
## 5 Clustering DTW (LB_PC)
The second approach we present tightens the lower bounds by exploiting the
structure or distributions of data in time series through point clustering
(LB_PC).
| |
---|---|---
(a) Insight behind LB_PC | | (b) Algorithm in LB_PC setup stage
Figure 4: (a) The insight behind LB_PC. The five query points correspond to
$q_{j}$ (j=i-2, i-1, i, i+1, i+2), the five points in the window of $c_{i}$.
Clustering the query points allows the algorithm to better bound the query
points and hence compute a lower bound much tighter than LB_MV gets. (b)
Algorithm used in the setup stage of LB_PC for identifying the bounding boxes
of candidate series.
Figure 4(a) illustrates the basic insight behind the design of LB_PC. For the
purpose of explanation, we assume that each point in the series of interest is
a 2-D point, and the five query points, $q_{i-2}$ to $q_{i+2}$, are located in
two areas as shown in Figure 4(a). An envelope used in LB_MV is essentially a
bounding box as shown by the dash lines in Figure 4(a), and the lower bound
computed by LB_MV is essentially the distance from $c_{i}$ to the corner
closest to it which is corner $X$ in Figure 4(a).
The bounds would be a lot tighter if we instead first cluster the five query
points into two groups (the two solid-lined boxes in Figure 4(a)), apply LB_MV
to each of the groups to estimate the minimum distance between $c_{i}$ and
each of the groups, and then take the minimum of those distances ($c_{i}$ to
the two corners of the two solid-lined boxes in Figure 4(a)) as the lower
bound between $c_{i}$ and the five query points. Our illustration uses 2-D
space, but the insight obviously also applies to spaces of higher dimensions.
LB_PC is designed on this insight. The challenge is to minimize the time
overhead as clustering takes time. LB_PC addresses it by using adaptive
quantization as the clustering method, assisted by uniform window grouping.
Quantization-based clustering.
Figure 4(b) outlines the algorithm of quantization-based clustering. For some
given points to cluster, it first gets the overall value range in each
dimension. It then regards the value range as a grid with $L^{D}$ cells (for
$D$ dimensions)—each dimension’s value range is evenly divided into $L$
segments. The cell that a point belongs to can be then directly computed by
dividing its value by the length of a cell in each dimension. Each non-empty
cell is regarded as a cluster, and the boundaries of the cell define a
bounding box.
To control the number of clusters (e.g., no more than a certain number $K$),
if the number of non-empty cells exceeds $K$, the last (in increasing
lexicographic order of the cells) extra boxes are combined. In addition, we
find that it is beneficial to add a limit on the smallest cell length in the
quantization. When, for instance, all points in a window have very similar
values in one dimension, separating the tiny range into multiple even smaller
cells adds no benefits but overhead. In our experiments, we set the limit to
0.00001 of the normalized range of values.
This clustering algorithm is applied by LB_PC on the query points of each
window. In essence, when a query comes, the LB_PC algorithm works as LB_MV
does, except that it uses the bounding boxes (produced by the clustering
algorithm) rather than the simple envelope. The value of $K$ and the
quantization level $L$ determine the tradeoff between the tightness of the
lower bounds and the lower bounds computation overhead. Its appropriate value
can be empirically selected for a given dataset as other DTW parameters are.
Uniform window grouping.
To further reduce the cost, we introduce an optimization named uniform window
grouping. It computes the bounding boxes for each expanded window. An expanded
window is the result of merging $w$ consecutive windows. Take Figure 2 (a) as
an example. In the original example, each window covers 5 points. The expanded
window with $w=3$ would include 7 points {$q_{i-2}$, $q_{i-1}$, $q_{i}$,
$q_{i+1}$, $q_{i+2}$, $q_{i+3}$, $q_{i+4}$}. The shifting stride becomes $w$
accordingly. The next expanded window would include points {$q_{i}$,
$q_{i+1}$, $q_{i+2}$, $q_{i+3}$, $q_{i+4}$, $q_{i+5}$, $q_{i+6}$}, and so on.
This method reduces the number of bounding boxes to compute and save and hence
both the time and space cost by a factor of $w$. We call $w$ the window
expansion factor.
Note that expanded windows are used in only the computations of bounding
boxes; the DTW lower bound computations still use the original windows. In the
DTW lower bound computation, the bounding boxes of an expanded window are used
as the bounding boxes of the original windows that compose that expanded
window. It may cause looser bounding boxes and hence looser lower DTW bounds,
but experiments (Section 7) show that the impact is small.
It is worth mentioning that an alternative grouping method we considered is
non-uniform grouping, which groups adjacent windows if and only if their
bounding boxes are the same. Experiments show that although this method keeps
the bounding boxes sizes unchanged, the space saved is modest (about 20-30%)
and it cannot save time cost. The uniform grouping is hence preferred.
## 6 Selective Deployment, TC-DTW, and Computational Complexities
Selective Deployment Both of the two proposed methods can help tighten the
lower bounds, but also introduce extra time overhead. Inspired by the
cascading idea in univariate DTW Rak+:TKDD2013 , we use selective deployment
of these algorithms with the combination of LB_MV. For a given query-candidate
series pair, the method first applies LB_MV to get a lower bound. It applies
the advanced algorithms only if $e<LB\\_MV/d_{best}<1$, where $d_{best}$ is
the DTW distance between the query series and the most similar candidate
series found so far, and $e$ is called triggering threshold, a value between 0
and 1. The reason for that design is that if $LB\\_MV/d_{best}\geq 1$, as the
lower bound is greater than the best-so-far, this candidate series is
impossible to be the most similar candidate; there is obviously no need to
apply other methods. If the lower bound is a lot smaller than the best-so-far
(i.e., $LB\\_MV\leq e$), there is a good chance for this current candidate to
be indeed better than the best-so-far. Applying LB_TI or LB_PC may produce a
larger lower bound, which is however unlikely to exceed $D_{best}$; the DTW
between the query and the candidate series would then have to be computed. In
both of these two cases, applying LB_TI or LB_PC adds overhead but avoids no
extra DTW computations. This selective deployment strategy can help deploy DTW
only when it is likely to give benefits. Similar to other DTW parameters, the
appropriate value of triggering threshold $e$ for a dataset can be determined
during the setup stage.
TC-DTW LB_TI and LB_PC offer two different approaches to improve LB_MV. As the
next section shows, both are effective, but the one that performs the best
could vary on different datasets. To get the best of both worlds, a simple
method is to use a small samples of series, in the setup stage, to try out
both methods and settle on the better one. We use TC-DTW to reference the
resulting choice.
Table 1: Summary of the algorithms on computing lower bound of DTW. (Notations
at bottom)
Method | Lower Bounds Calculation
---|---
| Operations | Time Complexity
LB_MV | Compute envelope of Q and distances to bounding envelopes (1 bounding box/point) | $\mathcal{O}(D\ast n\ast M\ast N)$
LB_AD | Compute distances to each candidate point | $\mathcal{O}(W\ast D\ast n\ast M\ast N)$
LB_TI | Compute neighbor distances on Q, 1/P LB_AD distances, TI bounds, and window top distances | $\mathcal{O}(4+((2+W/P)\ast D)\ast n\ast M\ast N)$
LB_PC | Compute bounding boxes and distances to bounding boxes ($K$ boxes per $w$ points) | $\mathcal{O}(K\ast D\ast n\ast M\ast N/w)$
n: series length (for explanation purpose, assume —C—=—Q—); $M:|\mathbb{C}|$;
$N:|\mathbb{Q}|$; $W:$ window size; $D:$ dimension; $P:$ the TIP period
length;
$K:$ # clusters in LB_PC; $w:$ window expansion factor.
Computational Complexities Table 1 summarizes the various methods for
computing the lower bounds of DTW between two collections of time series
$\mathbb{C}$ and $\mathbb{Q}$. For simplicity, the table assumes the same
length shared by query and candidate series. LB_AD and LB_MV lie at two ends
of the spectrum of overhead and tightness, while the methods proposed in this
work offer options in the between. They are equipped with knobs (e.g., the
period length $P$ and the number of clusters $K$) which can be adjusted to
strike the best tradeoff for a given problem.
## 7 Experiments
To evaluate the efficacy of the proposed algorithms, we conduct a series of
experiments and comparisons in DTW-based nearest neighbor findings, in which,
the goal is to find the most similar (i.e., having the smallest DTW distance)
candidate series for each query series. In this most common use of DTW, lower
bounds are used to avoid computing the DTW distance between a query and a
candidate if their DTW lower bound already exceeds the so-far-seen smallest
DTW distance. The experiments are designed to answer the following major
questions:
1. 1.
How effective are the two new methods, LB_TI and LB_PC, in accelerating
multivariate DTW in nearest neighbor finding?
2. 2.
How effective are the two techniques when they are applied without prior
preparations in the streaming scenario? What effects does the runtime overhead
of the lower bound calculations impose on the end-to-end performance?
3. 3.
How do the benefits change with window sizes or data dimensions?
4. 4.
Do they perform well on different machines?
### 7.1 Methodology
Methods to compare. We focus our comparisons on the following methods:
* •
LB_MV: the most commonly used method in multivariate DTW Rath+:2003 .
* •
LB_TI: our proposed triangle DTW (period length $P$ is set to five for all
datasets).
* •
LB_PC: our proposed point clustering DTW with uniform window grouping.
* •
TC-DTW: the combination of LB_TI and LB_PC. For a given dataset, it picks the
better one of the two methods as described in Section 6.
Both LB_TI and LB_PC use the selective deployment scheme as described in
Section 6. The parameters used in the two methods are selected from the
following ranges: {0.05, 0.1, 0.2} for the triggering threshold in LB_TI,
{0.1, 0.5} for the triggering threshold in LB_PC, {2,3} for the quantization
levels in LB_PC. The selection, as part of the database configuration process,
is through empirically experimenting with the parameters on a small set (23)
of randomly chosen candidate series. Other parameters are the maximum number
of clusters $K$ and the window expansion factor $w$ in LB_PC, which are both
set to 6 for all datasets.
In all the methods, early abandoning Wang+:DMKD2013 is used: During the
calculations of either lower bounds or DTW distances, as soon as the value
exceeds the so-far-best, the calculation is abandoned. This optimization has
been shown beneficial in prior uni-variate DTW studies Wang+:DMKD2013 ; our
observations on multivariate DTW are also positive.
Datasets. Our experiments used all the 13 largest multivariate datasets in the
online UCR multivariate data collection Bagnall+:arxiv2018 , which are the
ones in the most need for accelerations. The numbers of pair-wise DTW
distances in these datasets range from 40,656 to 525 million, as Table 2
shows. These datasets cover a wide variety in domains, the length of a time
series (Length), and the numbers of dimensions of data points (Dim). Values on
the time series were normalized before being used, and NA values were replaced
with zeros. For each dataset, 30% are used as queries, and the rest are used
as candidates.
Table 2: Datasets Dataset | #DTW | #Points | #Series | Dim | Length | Description
---|---|---|---|---|---|---
Articularyword. | 69,431 | 82,800 | 575 | 9 | 144 | Motion tracking of lips and mouth while speaking 12 different words
Charactertraj. | 1,715,314 | 520,156 | 2858 | 3 | 182 | Pen force and trajectory while writing various characters on a tablet
Ethanolconcen. | 57619 | 917,524 | 524 | 3 | 1751 | Raw spectra of water-and-ethanol solutions in whisky bottles
Handwriting | 210,000 | 152,000 | 1000 | 3 | 152 | Motion of writing hand when writing 26 letters of the alphabet.
Insectwingbeat | 525,000,000 | 1,500,000 | 50000 | 200 | 22 | Reconstruction of the sound of insects passing through a sensor
Japanesevowels | 86,016 | 18,560 | 640 | 12 | 29 | Audio, recording of Japanese speakers saying ”a” and ”e”
Lsst | 5,093,681 | 177,300 | 4925 | 6 | 36 | Simulated astronomical time-series data in preparation for observations from the LSST Telescope
Pems-sf | 40,656 | 63,360 | 440 | 963 | 144 | Occupancy rate of car lanes in San Francisco bay area
Pendigits | 25,373,053 | 87,936 | 10992 | 2 | 8 | Pen xy coordinates while writing digits 0-9 on a tablet
Phonemespectra | 9,337,067 | 1,446,956 | 6668 | 11 | 217 | Segmented audio collected from Google Translate Audio files
Selfregulationscp1 | 66024 | 502,656 | 561 | 6 | 896 | cortical potentials signal
Spokenarabicdigits | 16,255,009 | 818,214 | 8798 | 13 | 93 | Audio recording of Arabic digits
Uwavegesturelib. | 4,211,021 | 1,410,570 | 4478 | 3 | 315 | XYZ coordinates from accelerometers while performing various simple gestures.
Hardware and measurement. To consider the performance sensitivity to hardware,
we have measured the performance of the methods on two kinds of machines. (1)
AMD machine with 2GHz Opteron CPUs, 32GB DDR3 DRAM, and 120GB SSD; (2) Intel
machine with 2.10GHz Skylake Silver CPUs, 96GB DDR4 2666 DRAM, and 240GB INTEL
SDSC2KB240G7 SSD. We focus the discussions on the results produced on the AMD
machine and report the results on the Intel machine later. The machines run
Redhat Linux 4.8.5-11.
In the experiments, the methods are applied on the fly by computing the
envelopes, neighboring point distances, and bounding boxes of the query series
at runtime as the query arrives. All runtime overhead is counted in the
performance report of all the methods. The reported times are the average of
10 repetitions.
Window Sizes and Data Dimensions One of the aspects in the experiment design
is to decide what window sizes to use for the DTW calculations. Our decision
is based on observations some earlier work has made on what window sizes make
DTW-based classification give the highest classification accuracy; the results
are listed as ”learned_w” on the UCR website UCRArchive2018 . One option
considered is to use certain percentages of a sequence length as the window
size as some prior work does. But observations on the UCR list show that there
is no strong correlations between sequence length and the best window size. In
fact, a percentage range often used by prior work is 3-8% of the sequence
length, which covers only 10% of the 128 datasets. On the other hand, the best
window sizes of almost all (except two) datasets fall into the range of [0,
20] regardless of the sequence length. Although that list does not contain
most of the multivariant datasets used in this study, the statistics offer
hints on the range of proper window sizes. Based on those observations, we
experiment with two window sizes, 10 and 20.
Even though some of the datasets have points with a high dimension, in
practice, practitioners typically first use some kind of dimension reduction
to lower the dimensions to a small number Hu+:ICDM2013 to avoid the so-called
curse of dimensionality. In all the literature we have checked on the
applications of multivariate DTW, the largest dimension used is less than 10.
Hence, in our examination of the impact of dimensionality on the methods, we
study three settings: $dim$=3,5,10.
|
---|---
(a) Japanesevowels | (b) Insectwingbeat
Figure 5: The lower bounds of the DTW distance between sequence 0 and other
sequences in Japanesevowels and Insectwingbeat datasets. For legibility, the
candidate series are sorted based on the LB_TI bounds. Table 3: Comparisons of
LB_TI and LB_PC over LB_MV with all runtime overhead counted.
W=20*
---
| Speedups | Percentage of skipped DTWs calc.
Dataset | LB_MV | TC-DTW | LB_TI | LB_PC | LB_MV | TC-DTW | LB_TI | LB_PC
Articularyword. | 7.60 | 10.51 | 7.32 | 10.51 | 84% | 91% | 85% | 91%
Charactertraj. | 16.98 | 25.31 | 22.11 | 25.31 | 89% | 99% | 92% | 99%
Ethanolconcen. | 6.67 | 11.08 | 11.08 | 7.06 | 79% | 87% | 87% | 80%
Handwriting | 3.41 | 3.55 | 3.55 | 3.55 | 36% | 50% | 42% | 50%
Insectwingbeat | 6.03 | 9.02 | 7.66 | 9.02 | 56% | 90% | 71% | 90%
Japanesevowels | 2.59 | 2.91 | 2.69 | 2.91 | 10% | 40% | 17% | 40%
Lsst | 1.68 | 1.64 | 1.64 | 1.63 | 7% | 15% | 7% | 15%
Pems-sf | 6.05 | 7.33 | 6.02 | 7.33 | 78% | 85% | 77% | 85%
Pendigits | 4.06 | 4.13 | 4.13 | 3.89 | 27% | 48% | 34% | 48%
Phonemespectra | 1.95 | 2.02 | 1.92 | 2.02 | 12% | 13% | 13% | 13%
Selfregulationscp1 | 1.86 | 1.92 | 1.92 | 1.92 | 24% | 29% | 25% | 29%
Spokenarabicdigits | 5.76 | 7.71 | 7.71 | 7.28 | 64% | 83% | 75% | 83%
Uwavegesturelib. | 11.85 | 17.63 | 17.63 | 13.89 | 92% | 95% | 95% | 94%
Average | 5.88 | 8.06 | 7.34 | 7.41 | 51% | 63% | 55% | 63%
W=10*
| Speedups | Percentage of skipped DTWs calc.
Dataset | LB_MV | TC-DTW | LB_TI | LB_PC | LB_MV | TC-DTW | LB_TI | LB_PC
Articularyword. | 11.24 | 11.78 | 10.68 | 11.78 | 96% | 95% | 94% | 95%
Charactertraj. | 16.77 | 18.52 | 18.52 | 17.78 | 100% | 100% | 100% | 99%
Ethanolconcen. | 13.29 | 15.87 | 15.87 | 13.05 | 97% | 100% | 100% | 94%
Handwriting | 4.16 | 4.69 | 3.99 | 4.69 | 51% | 72% | 55% | 72%
Insectwingbeat | 5.44 | 10.91 | 7.17 | 10.91 | 66% | 94% | 81% | 94%
Japanesevowels | 2.55 | 3.21 | 2.88 | 3.21 | 30% | 71% | 45% | 71%
Lsst | 1.67 | 2.06 | 1.70 | 2.06 | 21% | 39% | 21% | 39%
Pems-sf | 7.28 | 7.47 | 6.78 | 7.47 | 86% | 91% | 85% | 91%
Pendigits | 4.08 | 4.29 | 4.29 | 3.98 | 27% | 49% | 34% | 49%
Phonemespectra | 2.08 | 2.16 | 2.10 | 2.16 | 25% | 27% | 25% | 27%
Selfregulationscp1 | 2.10 | 2.38 | 2.18 | 2.38 | 41% | 52% | 41% | 52%
Spokenarabicdigits | 8.19 | 10.42 | 10.07 | 10.42 | 86% | 97% | 90% | 97%
Uwavegesturelibrary | 15.40 | 16.17 | 16.17 | 15.23 | 98% | 100% | 100% | 98%
Average | 7.25 | 8.46 | 7.88 | 8.09 | 63% | 76% | 67% | 75%
*: Actual window sizes are capped at the length of a time series.
Table 4: Speedups ($\times$) in Various Dimensions on Datasets with More than
8 Dimensions. (W=20) Dim=3
---
Dataset | LB_MV | TC-DTW | LB_TI | LB_PC
Articularyword. | 7.67 | 10.59 | 7.49 | 10.59
Insectwingbeat | 12.12 | 15.24 | 14.39 | 15.24
Japanesevowels | 3.17 | 3.53 | 3.41 | 3.53
Pems-sf | 5.99 | 7.17 | 5.98 | 7.17
Phonemespectra | 1.97 | 1.97 | 1.97 | 1.95
Spokenarabic. | 8.41 | 11.54 | 11.01 | 11.54
Average | 6.55 | 8.34 | 7.38 | 8.34
Dim=5
Articularyword. | 6.86 | 9.24 | 6.86 | 9.24
Insectwingbeat | 6.13 | 8.94 | 8.04 | 8.94
Japanesevowels | 2.63 | 2.97 | 2.66 | 2.97
Pems-sf | 7.82 | 8.58 | 7.77 | 8.58
Phonemespectra | 1.93 | 1.98 | 1.98 | 1.95
Spokenarabic. | 5.55 | 7.81 | 7.81 | 7.40
Average | 5.15 | 6.59 | 5.85 | 6.51
Dim=10
Articularyword. | 5.80 | 8.10 | 5.87 | 8.10
Insectwingbeat | 2.55 | 4.45 | 3.21 | 4.45
Japanesevowels | 2.12 | 2.28 | 2.20 | 2.28
Pems-sf | 5.27 | 5.94 | 5.17 | 5.94
Phonemespectra | 2.00 | 1.97 | 1.97 | 1.87
Spokenarabic. | 3.12 | 4.09 | 4.09 | 3.93
Average | 3.48 | 4.47 | 3.75 | 4.43
Table 5: Performance on Intel machine. (W=20) | Speedup ($\times$)
---|---
Dataset | LB_MV | TC-DTW | LB_TI | LB_PC
Articularyword. | 3.52 | 4.88 | 3.82 | 4.88
Charactertr. | 6.97 | 9.99 | 9.21 | 9.99
Ethanolconc. | 6.70 | 11.16 | 11.16 | 7.02
Handwriting | 2.83 | 2.96 | 2.96 | 2.86
Insectwingbeat | 9.09 | 13.70 | 11.65 | 13.70
Japanesevowels | 3.96 | 4.48 | 4.19 | 4.48
Lsst | 2.37 | 2.39 | 2.26 | 2.39
Pems-sf | 3.53 | 3.86 | 3.66 | 3.86
Phonemespectra | 2.44 | 2.46 | 2.46 | 2.36
Pendigits | 6.00 | 6.34 | 6.17 | 6.34
Selfregula. | 1.82 | 1.97 | 1.88 | 1.97
Spokenarabic. | 5.62 | 10.84 | 10.45 | 10.84
Uwavegesture. | 5.06 | 7.89 | 7.20 | 7.89
Average | 4.61 | 6.38 | 5.93 | 6.04
### 7.2 Experimental Results
Soundness and Sanity Checks Because the optimizations used in all the methods
keep the semantic of the original DTW distance, they do not affect the
precision of the results. It is confirmed in our experiments: The optimized
methods find the nearest neighbor for every query as the default method does,
and return the correct DTW distance. In a sanity check on Nearest Neighbor-
based classification, the classification results are identical among all the
methods. We focus our following discussions on speed.
Overall Speed Table 3 reports the results when the window size is set to 20
and 10; the actual window size for a dataset is capped at the length of a
series. The first five dimensions are used; Table 4 reports results on other
dimensions.
The speedup of a method X on a dataset is computed as $T(X)/T$(default), where
$T$(default) is the time taken by the default windowed DTW algorithm (which
uses no lower bounds) in finding the candidate series most similar to each
query series. $T(X)$ is the time taken by method X which includes all the
runtime overhead, such as the time of lower bounds calculations if method X
uses lower bounds.
In Table 3, the ”Speedups” columns report the speedups of the methods over the
default algorithm. The ”Skips” columns in Table 3 report the percentage of the
query-candidate DTW distances that those methods successfully avoid computing
via their calculated lower bounds.
From Table 3, We see that on average, LB_MV gives 5.86$\times$ speedups on the
datasets. The speedups come from avoiding the DTW calculations for 50% of
query-candidate pairs and also the early abandoning scheme. LB_TI avoids 56%
DTW calculations, and gives 7.33$\times$ average speedup. LB_PC avoids 64% DTW
calculations, achieving 7.42$\times$ average speedups, slightly higher than
LB_TI. As examples, Figure 5 shows the lowerbounds from the three methods on
two representative datasets. Similar trends appear on other datasets. The
larger lower bounds from LB_TI and LB_PC are the main reasons for the time
savings.
In most cases, LB_PC avoids more DTW computations and gets a larger speedup
than LB_TI. There are four exceptions, Ethanolconcentration, Pendigits,
Spokenarabicdigits and Uwavegesturelibrary, on which LB_TI performs better
than LB_PC does. The results are influenced by the value distributions and
changing frequencies of the datasets. Method TC_DTW gets the best of both
worlds through its dataset-adaptive selection, achieving an average
8.03$\times$ speedups. The speedups differ substantially across datasets. On
Lsst, the avoidance rate is only 15%, and the speedup is 1.67$\times$; on
Charactertrajectories, the avoidance rate is as large as 98%, and the speedup
is 25.1$\times$. In general, if the series in a dataset do not have many
differences, lower bounds are less effective in avoiding computations, and the
speedups are less pronounced. Despite the many differences among the datasets,
TC-DTW consistently outperforms the popular method LB_MV on all datasets.
The results also confirm the usefulness of early abandoning. For instance,
even though LB_MV avoids only 7% DTW distance calculations on dataset Lsst,
the speedup it gives is as much as 1.65$\times$, showing the benefits from the
early abandoning. Nevertheless, the benefits from the improved DTW algorithms
are essential. For instance, on dataset Charactertrajectories, the 25$\times$
speedups come primarily from the avoidance of 98% of the DTW distance
calculations. In fact, because early abandoning is adopted by all four methods
shown in Table 3, the differences in their produced speedups are caused only
by their algorithmic differences rather than early abandoning.
Time overhead To study the impact from the runtime overhead, Figure 6 puts the
speedups of LB_TI and LB_PC in the backdrop of their ideal speedups where
runtime overhead is excluded. (For readability, we include only some of the
datasets in that graph.) The vertical axis is in log scale to make all the
bars visible. The largest gaps appear on datasets Charactertrajectories and
Uwavegesturelibrary, with a 10-55$\times$ speedup left untapped due to the
runtime overhead, indicating that further optimization of the implementations
of the two methods could potentially yield more speedups.
Figure 6: Speedups of LB_TI and LB_PC, and their ideal results when all
runtime overhead is removed.
Window size The bottom half of Table 3 shows the results when the window size
is 10 (again, capped at the length of a time series). As window size reduces,
all methods become more effective in finding skip opportunities with the
average skips increasing to over 60% in all the methods. The three proposed
methods, LB_TI, LB_PC, and TC-DTW, still show a clear edge over LB_MV.
Dimensions As mentioned earlier, even though some of the datasets have points
with a high dimension, in practice, practitioners typically first use
dimension reduction methods to lower the dimensions to a small number
Hu+:ICDM2013 to avoid the so-called curse of dimensionality. In all the
literature on multivariate DTW applications that we have checked, the largest
dimension used is less than 10. Therefore in our examination of the impact of
dimensionality on the methods, we focus on three settings: $dim$=3,5,10. Our
experiments run on the datasets that have dimensions greater than 8 by
changing the used dimensions to the first $dim$ dimensions of the data. Table
4 reports the speedups from the methods over the default window-based DTW. All
methods see some reductions of speedups as the number of dimensions increase.
It is intuitive: As data become sparser in higher-dimension spaces, the
bounding boxes become larger and lower bounds become looser. Nevertheless, TC-
DTW still consistently outperforms LB_MV on all the datasets and dimensions.
Performance cross machines To examine whether the new methods consistently
show benefits across different machines, besides the experiments on the AMD
machine, we have measured the performance of the methods on the Intel machine
(configuration given in Section 7.1). Table 5 reports the results. The setting
is identical to that in Table 3 (W=40). Compared to the results on the AMD
machine in Table 3, the speedups of the three methods are all somewhat
smaller, reflecting the influence of the hardware differences in terms of the
relative speeds between CPU and memory accesses. But the trends are the same.
The results confirm that the better efficacy of the three new methods over
LB_MV holds across machines, with TC-DTW excelling at an average 6.29$\times$
speedups.
Overall, the experiments confirm the advantages of the new methods over the
current approach LB_MV in accelerating multivariate DTW. The advantages
consistently hold across datasets, data sizes, dimensions, window sizes, and
machines. The advantages of the new methods over LB_MV are especially
pronounced on large datasets with large windows, where, time is especially of
concern.
## 8 Discussions
The two methods presented in the paper are the crystallized results of the
numerous ideas we have explored. The other ideas are not as effective. We
briefly describe one of them for its potential for future development.
This idea is also about clustering, but instead of point-level clustering, it
considers clustering of query series. The basic insight behind it is that the
tight bounds from LB_AD meet Triangle Inequality with the Euclidean distances
between two query series. That is, $LB\\_AD(Q_{j},C)-ED(Q_{i},Q_{j})\leq
LB\\_AD(Q_{i},C)\leq LB\\_AD(Q_{j},C)+ED(Q_{i},Q_{j})$, where $ED$ stands for
Euclidean distance. We omit the proof here but note that it can be easily
proved if one notices that such an inequality holds at every corresponding
point between $Q_{i}$ and $Q_{j}$. Therefore, for two queries series similar
to each other, we can use the inequality to estimate one’s lower bounds with a
candidate series based on those of the other series. We attempted with some
quick ways to do clustering (e.g., Euclidean distances among sample points),
but did not observe significant benefits except on only a few time series. The
reason is that even though the method can sometimes give bounds tighter than
LB_MV, the degree is too small to change the filtering result. Applying the
idea to each segment of a series might give better results. We leave it for
future exploration.
It is worth mentioning another possible application scenario of the proposed
methods. In the scenario studied in our experiments, the bounding boxes and
neighboring point distances are computed on the fly when a query series
arrives. Another option is to switch the role of query and candidate series
and compute the bounding boxes and neighboring distances of the candidate
series ahead of time (e.g., when setting up the database). The pre-computed
info can then be used in processing queries. Such an option incurs extra space
cost as the bounding boxes and neighbor point distances need to be permanently
stored but if that is not a concern, it could further amplify the performance
benefits of the methods.
## 9 Related Work
The most significant advantage of DTW is its invariance against signal warping
(shifting and scaling in the time axis, or Doppler effect). Experimental
comparison of DTW to most other highly cited distance measures on many
datasets concluded that DTW almost always outperforms other measures
Wang+:DMKD2013 . Therefore, DTW has become one of the most preferable measures
in pattern matching tasks for time series data. It has been used in robotics,
medicine Chadwick+:DEST2011 , biometrics, speech processing Adams+:ISMIR05 ,
climatology, aviation, gesture recognition Alon+:PAMI2009 , user interfaces
Laerhoven+:ICMLA2009 , industrial processing, cryptanalysis
Dupasquier+:Chaos2011 , mining of historical manuscripts Huber+:MVA2011 ,
space exploration, wildlife monitoring, and so on Inglada+:sense2015 ;
Rak+:TKDD2013 .
DTW remains popular in the era of deep learning. DTW-based temporal data
analytics are still widely used for its advantages in better interpretability
over DNN. Moreover, there is strong interest in combining DTW with deep
learning. The most common integration of DTW and deep learning is to use DTW
as a loss function. Studies have shown that DTW, compared to conventional
Euclidean distance loss, leads to a better performance in a number of
applications Cuturl+:arxiv2017 through softmin Chang+:arxiv2019 ;
Shah+:CODS16 . A more recent way to combine DTW and deep learning is to use
DTW as a replacement of kernels in convolutional neural networks, creating
DTWNet that shows improved inference accuracy Cai+:NIPS2019 . There are also
some other integrations of DTW with DNN Thiollire+:AHD2015 . In addition,
there is interest in using DTW as a distance measure in learning time series
shapelets Shah+:CODS16 . Besides serving as a similarity measure, DTW could be
leveraged as a feature extracting tool. For example, predefined patterns can
be identified in the data via DTW computing. Subsequently these patterns could
be used to classify the temporal data into categories Kate:DMKD16 ;
Giraldo+:SIU18 .
Many studies have been conducted to enhance the speed of DTW-based analytics,
but mostly for single-dimensional data, including the many lower bound
functions for univariate time series, such as LB_Yi Yi+:VLDB2000 , LB_Kim
Kim+:ICDE2001 , LB_Keogh Keogh+:VLDB2006 , LB_Improved Lemire+:PR2009 and
LB_ECorner Zhou+:ICDE2011 , various methods to enable early abandoning
Rak+:TKDD2013 , the combination of hierarchy k-means and lower bounding
Tan+:SDM17 , and so on.
There are hundreds of research efforts that use DTW in a multi-dimensional
setting Ridgely+:2009 ; Kela+:2006 ; Ko+:2005 ; Liu+:Mob2009 ; Peti+:TGRS12 ;
Wang+:interspeech2013 . Research is however scarce in designing optimizations
specifically for accelerating multivariate DTW, despite the even higher cost
of multivariate DTW than univariate DTW. Almost all of the existing
applications of multivariate DTW are based on a straightforward extension of
the single-dimensional lower bound (LB_Keogh [27]) to the multi-dimensional
data Rath+:2003 , described as LB_MV in Section 2. The previously proposed
optimizations have concentrated on the reduction or transformations of the
high dimension space. Li and others Li+:physica2015 ; Li+:elsevier2017 tries
to reduce all dimensions to a single dimension (center series) and then use
univariate methods; Hu and others have studied the effects of giving more
emphasis on important dimensions Hu+:ICDM2013 ; Gong and others demonstrate
that by rotating the space, one could improve the tightness of the lower
bounds Gong+:Springer2015 . Some optimizations proposed for univariate DTW
could potentially benefit applications of multivariate DTW. For instance, in
DTW-based Nearest Neighbor, the early abandoning strategy stops the
computation of the DTW between a query sequence and a reference sequence if
evidence shows that the DTW distance will definitely exceed the best-so-far.
All of these prior studies have concentrated on factors outside the
calculations of lower bounds for more effective filtering. This work gives the
first systematic exploration on the optimizations of lower bounds calculations
specific to multivariate DTW. Being complementary to the previously proposed
optimizations, it can be used with some of the existing optimizations
together.
Triangle inequality has been leveraged in other data mining and machine
learning optimizations (e.g., Yinyang K-Means Ding+:ICML15 ). We are not aware
of a previous introduction of it in accelerating multivariate DTW.
## 10 Conclusions
This paper has introduced Triangle Inequality and Pointer Clustering into the
algorithm design of multivariate DTW. The incorporation of Triangle Inequality
allows the use of scalar computations to replace vector distance calculations
in computing tighter lower bounds; the integration of Point Clustering allows
the use of quantization-based clustering to tighten the lower bounds. It also
presents several design optimizations to the two methods, including periodical
distance calculations, adaptive quantization, uniform window grouping, and
data-adaptive selection. Experiments on 13 datasets show that the resulting
method, TC-DTW, consistently outperforms the current most commonly used method
LB_MV significantly, across datasets, data sizes, dimensions, window sizes,
and hardware, suggesting the potential of TC-DTW as a drop-in replacement for
LB_MV, a method that has prevailed in the most part of the last two decades.
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# Palindromic and Colored Superdiagonal Compositions
Jazmín Mantilla Escuela de Matemáticas, Universidad Industrial de Santander,
Bucaramanga, COLOMBIA<EMAIL_ADDRESS>, Wilson Olaya-León Escuela de
Matemáticas, Universidad Industrial de Santander, Bucaramanga, COLOMBIA
<EMAIL_ADDRESS>and José L. Ramírez Departamento de Matemáticas,
Universidad Nacional de Colombia, Bogotá, COLOMBIA<EMAIL_ADDRESS>http://sites.google.com/site/ramirezrjl
###### Abstract.
A superdiagonal composition is one in which the $i$-th part or summand is of
size greater than or equal to $i$. In this paper, we study the number of
palindromic superdiagonal compositions and colored superdiagonal compositions.
In particular, we give generating functions and explicit combinatorial
formulas involving binomial coefficients and Stirling numbers of the first
kind.
###### Key words and phrases:
Compositions, palindromic compositions, colored compositions, generating
functions, combinatorial identities.
###### 2010 Mathematics Subject Classification:
05A15, 05A19
## 1\. Introduction and Notation
A _composition_ of a positive integer $n$ is a sequence of positive integers
$\sigma=(\sigma_{1},\sigma_{2},\ldots,\sigma_{\ell})$ such that
$\sigma_{1}+\sigma_{2}+\cdots+\sigma_{\ell}=n$. The summands $\sigma_{i}$ are
called _parts_ of the composition and $n$ is referred to as the _weight_ of
$\sigma$. For example, the compositions of $4$ are
$(\texttt{4}),\quad(\texttt{3},\texttt{1}),\quad(\texttt{1},\texttt{3}),\quad(\texttt{2},\texttt{2}),\quad(\texttt{2},\texttt{1},\texttt{1}),\quad(\texttt{1},\texttt{2},\texttt{1}),\quad(\texttt{1},\texttt{1},\texttt{2}),\quad(\texttt{1},\texttt{1},\texttt{1},\texttt{1}).$
A _palindromic_ (or _self-inverse_) _composition_ is one whose sequence of
parts is the same whether it is read from left to right or right to left. For
example, the palindromic compositions of $4$ are
$(\texttt{4}),\quad(\texttt{2},\texttt{2}),\quad(\texttt{1},\texttt{2},\texttt{1}),\quad(\texttt{1},\texttt{1},\texttt{1},\texttt{1}).$
Hoggatt and Bicknell [8] studied palindromic compositions having parts in a
subset of positive integers. In particular, they showed that the total number
of palindromic compositions of $n$ is given by $2^{\lfloor n/2\rfloor}$.
Moreover, the number of palindromic compositions of $n$ with parts 1 and 2 are
the interleaved Fibonacci sequence
$1,\quad 1,\quad 2,\quad 1,\quad 3,\quad 2,\quad 5,\quad 3,\quad 8,\quad
5,\quad 13,\quad 8,\quad 21,\dots$
The literature contains several generalizations and restrictions of the
compositions. Much of them are related to the kind of parts or summands, for
example compositions with even or odd parts [8, 7], with parts in arithmetical
progressions [1, 9], compositions with colored parts [2, 4, 11], colored
palindromic compositions [10, 6, 3], superdiagonal compositions [5], among
other restrictions. For further information on compositions, we refer the
reader to the text by Heubach and Mansour [7].
In this paper, we study _palindromic superdiagonal compositions_ , that is a
palindromic composition $(\sigma_{1},\sigma_{2},\ldots,\sigma_{\ell})$ of $n$,
with the additional condition $\sigma_{i}\geq i$, for $i=1,2,\dots,\ell$. For
example, the palindromic superdiagonal compositions of $10$ are
$(\texttt{10}),\quad(\texttt{5},\texttt{5}),\quad(\texttt{4},\texttt{2},\texttt{4}),\quad(\texttt{3},\texttt{4},\texttt{3}).$
Deutsch et al. [5] proved that the total number of superdiagonal compositions
is given by the combinatorial sum:
$\sum_{k\geq 1}\binom{n-\binom{k}{2}-1}{k-1}.$
Agarwal [2] generalized the concept of a composition by allowing the parts to
come in various colors. By a _colored composition_ of a positive integer $n$
we mean a composition $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{\ell})$
such that the part of size $i$ can come in one of $i$ different colors. The
colors of the summand i are denoted by subscripts $i_{1},i_{2},\ldots,i_{i}$
for each $i\geq 1$. For example, the colored compositions of $4$ are given by
$\displaystyle(\texttt{4}_{\texttt{1}}),\quad(\texttt{4}_{\texttt{2}})\quad(\texttt{4}_{\texttt{3}}),\quad(\texttt{4}_{\texttt{4}}),\quad(\texttt{3}_{\texttt{1}},\texttt{1}_{\texttt{1}}),\quad(\texttt{3}_{\texttt{2}},\texttt{1}_{\texttt{1}}),\quad(\texttt{3}_{\texttt{3}},\texttt{1}_{\texttt{1}}),\quad(\texttt{1}_{\texttt{1}},\texttt{3}_{\texttt{1}}),\quad(\texttt{1}_{\texttt{1}},\texttt{3}_{\texttt{2}}),\quad(\texttt{1}_{\texttt{1}},\texttt{3}_{\texttt{3}}),$
$\displaystyle(\texttt{2}_{\texttt{1}},\texttt{2}_{\texttt{1}}),\quad(\texttt{2}_{\texttt{1}},\texttt{2}_{\texttt{2}}),\quad(\texttt{2}_{\texttt{2}},\texttt{2}_{\texttt{1}}),\quad(\texttt{2}_{\texttt{2}},\texttt{2}_{\texttt{2}}),\quad(\texttt{2}_{\texttt{1}},\texttt{1}_{\texttt{1}},\texttt{1}_{\texttt{1}}),\quad(\texttt{2}_{\texttt{2}},\texttt{1}_{\texttt{1}},\texttt{1}_{\texttt{1}}),\quad(\texttt{1}_{\texttt{1}},\texttt{2}_{\texttt{1}},\texttt{1}_{\texttt{1}}),$
$\displaystyle(\texttt{1}_{\texttt{1}},\texttt{2}_{\texttt{2}},\texttt{1}_{\texttt{1}}),\quad(\texttt{1}_{\texttt{1}},\texttt{1}_{\texttt{1}},\texttt{2}_{\texttt{1}}),\quad(\texttt{1}_{\texttt{1}},\texttt{1}_{\texttt{1}},\texttt{2}_{\texttt{2}}),\quad(\texttt{1}_{\texttt{1}},\texttt{1}_{\texttt{1}},\texttt{1}_{\texttt{1}},\texttt{1}_{\texttt{1}}).$
A _colored superdiagonal composition_ is a colored composition such that the
$i$-th part $\sigma_{i}$ satisfies $\sigma_{i}\geq i$, for each $i\geq 1$. The
colored superdiagonal compositions of $4$ are
(1)
$\displaystyle\begin{split}&(\texttt{4}_{\texttt{1}}),\quad(\texttt{4}_{\texttt{2}}),\quad(\texttt{4}_{\texttt{3}}),\quad(\texttt{4}_{\texttt{4}}),\quad(\texttt{1}_{\texttt{1}},\texttt{3}_{\texttt{1}}),\quad(\texttt{1}_{\texttt{1}},\texttt{3}_{\texttt{2}}),\quad(\texttt{1}_{\texttt{1}},\texttt{3}_{\texttt{3}}),\\\
&(\texttt{2}_{\texttt{1}},\texttt{2}_{\texttt{1}}),\quad(\texttt{2}_{\texttt{1}},\texttt{2}_{\texttt{2}}),\quad(\texttt{2}_{\texttt{2}},\texttt{2}_{\texttt{1}}),\quad(\texttt{2}_{\texttt{2}},\texttt{2}_{\texttt{2}}).\end{split}$
The goal of this paper is to enumerate the palindromic superdiagonal
compositions and colored superdiagonal compositions. In particular we give the
generating functions and explicit combinatorial formulas involving binomial
coefficients and Stirling numbers of the first kind.
## 2\. Enumeration of the Palindromic Superdiagonal Compositions
Let ${\mathcal{S}_{\sf{Pal}}}$ denote the set of palindromic superdiagonal
compositions. The composition $(\sigma_{1},\sigma_{2},\ldots,\sigma_{\ell})$
of $n$ can be represented as a _bargraph_ of $\ell$ columns, such that the
$i$-th column contains $\sigma_{i}$ cells for $1\leq i\leq\ell$. For example,
in Figure 1 we show the superdiagonal compositions of $n=10$ with their
bargraph representations.
Figure 1. Palindromic compositions of $n=10$.
Let $\sigma$ be a composition and let us denote the weight of $\sigma$ by
$|\sigma|$ and the number of parts of $\sigma$ by $\rho(\sigma)$. Using these
parameters, we introduce this bivariate generating function
$S(x,y):=\sum_{\sigma\in{\mathcal{S}_{\sf{Pal}}}}x^{|\sigma|}y^{\rho(\sigma)}.$
In Theorem 2.1 we give an expression for the generating function $S(x,y)$.
###### Theorem 2.1.
The bivariate generating function $S(x,y)$ is given by
$S(x,y)=\sum_{\sigma\in{\mathcal{S}_{\sf{Pal}}}}x^{|\sigma|}y^{\rho(\sigma)}=\sum_{m\geq
0}\left(\frac{x^{3m^{2}+m}}{(1-x^{2})^{m}}y^{2m}+\frac{x^{3m^{2}+4m+1}}{(1-x)(1-x^{2})^{m}}y^{2m+1}\right).$
###### Proof.
Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{2m})$ be a palindromic
superdiagonal composition of $n$ with $2m$ parts. From the definition we have
the condition $\sigma_{i}=\sigma_{2i}\geq 2i$, for $i=1,\dots,m$, see Figure 2
for a graphical representation of this case.
Figure 2. Decomposition of a palindromic superdiagonal composition.
The columns $i$-th and $2i$-th contribute to the generating function the term
$x^{2i}y^{2}+x^{2i+2}y^{2}+x^{2i+4}y^{2}+\cdots=\frac{x^{2i}y^{2}}{1-x^{2}}.$
Therefore the composition $\sigma$ contributes to the generating function the
term
$\frac{x^{2m}y^{2}}{1-x^{2}}\frac{x^{2m+2}y^{2}}{1-x^{2}}\cdots\frac{x^{4m}y^{2}}{1-x^{2}}=\frac{x^{3m^{2}+m}y^{2m}}{(1-x^{2})^{m}}.$
If the number of parts is odd, that is
$\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{2m-1})$, then from a similar
argument the contribution to the generating function is given by
$\frac{x^{3m^{2}+4m+1}}{(1-x)(1-x^{2})^{m}}y^{2m+1}.$
Summing in the above two cases over $m$ we obtain the desired result. ∎
As a series expansion, the generating function $S(x,y)$ begins with
$S(x,y)=1+xy+x^{2}y+x^{3}y+x^{4}\left(y^{2}+y\right)+x^{5}y+x^{6}\left(y^{2}+y\right)+x^{7}y+x^{8}\left(y^{3}+y^{2}+y\right)\\\
+x^{9}\left(y^{3}+y\right)+\bm{x^{10}\left(2y^{3}+y^{2}+y\right)}+x^{11}\left(2y^{3}+y\right)+x^{12}\left(3y^{3}+y^{2}+y\right)+\cdots$
Notice that Figure 1 shows the palindromic superdiagonal compositions
corresponding to the bold coefficient in the above series. Let $s(n)$ and
$s(n,k)$ denote the number of palindromic superdiagonal compositions of $n$
and the number of palindromic superdiagonal compositions of $n$ with exactly
$k$ parts. Note that $s(n)=\sum_{k\geq 1}s(n,k)$. In Table 1 we show the first
few values of the sequence $s(n,k)$.
$k\backslash n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
2 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1
3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 10
4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 | 3 | 0 | 4 | 0 | 5 | 0 | 6 | 0 | 7
5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 3 | 3 | 6 | 6
Table 1. Values of $s(n,k)$, for $1\leq n\leq 26$ and $1\leq k\leq 5$.
Setting $y=1$ in Theorem 2.1 implies that the generating function for the
total number of palindromic superdiagonal compositions is the following
$\displaystyle S(x,1)$ $\displaystyle=\sum_{n\geq 0}s(n)x^{n}=\sum_{m\geq
0}\frac{x^{3m^{2}+m}(1-x+x^{3m+1})}{(1-x)(1-x^{2})^{m}}.$
The first few values of the sequence $s(n)$ for $0\leq n\leq 28$ are
$1,\,1,\,1,\,1,\,2,\,1,\,2,\,1,\,3,\,2,\,4,\,3,\,5,\,4,\,7,\,5,\,9,\,6,\,11,\,7,\,13,\,9,\,16,\,12,\,20,\,16,\,25,\,21,\,31,\dots{}$
In Theorem 2.2 we give a combinatorial expression for the sequence $s(n,k)$.
###### Theorem 2.2.
The number of palindromic superdiagonal compositions
1. (1)
of $2n$ with $2k$ parts equals
$s(2n,2k)=\binom{n-\binom{k+1}{2}-2\binom{k}{2}-1}{k-1},$
2. (2)
of either $2n$ or $2n-1$ with $2k-1$ parts equals
$s(n,2k-1)=\binom{\lfloor\frac{n-3k^{2}}{2}\rfloor+2k-1}{k-1}.$
###### Proof.
From the proof of Theorem 2.1 we have
$\displaystyle s(2n,2k)$
$\displaystyle=[x^{2n}]\frac{x^{3k^{2}+k}}{(1-x^{2})^{k}}$
$\displaystyle=[x^{2n-3k^{2}-k}]\sum_{\ell\geq
0}\binom{k+\ell-1}{k-1}x^{2\ell}$
$\displaystyle=\binom{k+\frac{2n-3k^{2}-k}{2}-1}{k-1}$
$\displaystyle=\binom{n-\binom{k+1}{2}-2\binom{k}{2}-1}{k-1}.$
The combinatorial formula for $s(n,2k-1)$ is obtained in a similar manner. ∎
## 3\. Colored Superdiagonal Compositions
In this section we give the generating function for the total number of
colored superdiagonal compositions. Remember that a composition
$\sigma=(\sigma_{1},\dots,\sigma_{\ell})$ of $n$ is a colored superdiagonal
composition if $\sigma_{i}\geq i$ for all $1\leq i\leq\ell$, with the
additional condition that if a part is of size $i$ then it can come in one of
$i$ different colors. For example,
$(\texttt{3}_{\texttt{2}},\texttt{2}_{\texttt{1}},\texttt{5}_{\texttt{3}},\texttt{5}_{\texttt{2}},\texttt{6}_{\texttt{6}})$
is a colored superdiagonal composition of $21$.
Let $c(n)$ denote the number of colored superdiagonal compositions of $n$. In
Theorem 3.2 we give the generating for this sequence. Before we need some
definitions and one lemma.
Given integers $n,k\geq 0$, let ${n\brack k}$ denote the (unsigned) _Stirling
numbers of the first kind_ , which are defined as connection constants in the
polynomial identity
(2) $x(x+1)\cdots(x+(n-1))=\sum_{k=0}^{n}{n\brack k}x^{k}.$
This sequence counts the number of permutations on $n$ elements with $k$
cycles. It is also known that this sequence satisfies the recurrence relation
(3) $\displaystyle{n\brack k}=(n-1){n-1\brack k}+{n-1\brack k-1},$
with the initial conditions ${0\brack 0}=1$ and ${n\brack 0}={0\brack n}=0$
for $n\geq 1$.
Let $m$ be a non-negative integer. We introduce the polynomials defined by
(4) $\displaystyle Q_{m}(x):=\prod_{\ell=1}^{m}(\ell-(\ell-1)x),\quad m\geq
1,$
with the initial value $Q_{0}(x)=1$. The first few polynomials are
$\displaystyle Q_{0}(x)$ $\displaystyle=1,\quad Q_{1}(x)=1,\quad
Q_{2}(x)=2-x,\quad Q_{3}(x)=2x^{2}-7x+6,$ $\displaystyle Q_{4}(x)$
$\displaystyle=-6x^{3}+29x^{2}-46x+24,$ $\displaystyle Q_{5}(x)$
$\displaystyle=24x^{4}-146x^{3}+329x^{2}-326x+120,$ $\displaystyle Q_{6}(x)$
$\displaystyle=-120x^{5}+874x^{4}-2521x^{3}+3604x^{2}-2556x+720.$
###### Proposition 3.1.
The polynomials $Q_{m}(x)$ can be expressed as
$Q_{m}(x)=\sum_{k=0}^{m}T(m,k)x^{k},$
where
$T(m,k)=\sum_{i=0}^{m}\binom{i}{m-k}{m+1\brack m+1-i}(-1)^{m+i+k}.$
###### Proof.
Let $T(m,k)$ be the $k$-th coefficient of $Q_{m}(x)$. From (4) we have
$Q_{m}(x)=mQ_{m-1}(x)-(m-1)xQ_{m-1}(x).$
Then we obtain the recurrence relation $T(m,k)=mT(m-1,k)-(m-1)T(m-1,k-1),$
with the initial conditions $T(0,0)=T(1,0)=1$ and $T(1,1)=0$. Let
$H(m,k)=\sum_{i=0}^{m}\binom{i}{m-k}{m+1\brack m+1-i}(-1)^{m+i+k}$. The
sequences $T(m,k)$ and $H(m,k)$ satisfy the same recurrence relation and the
same initial conditions. In fact, from (3) we have
$\displaystyle H(m,k)$ $\displaystyle=\sum_{i=0}^{m}\binom{i}{m-k}{m+1\brack
m+1-i}(-1)^{m+i+k}$ $\displaystyle=\sum_{i=0}^{m}\binom{i}{m-k}\left(m{m\brack
m+1-i}+{m\brack m-i}\right)(-1)^{m+i+k}$
$\displaystyle=m\sum_{i=0}^{m-1}\binom{i+1}{m-k}{m\brack
m-i}(-1)^{m+i+k-1}+\sum_{i=0}^{m}\binom{i}{m-k}{m\brack m-i}(-1)^{m+i+k}$
$\displaystyle=m\sum_{i=0}^{m-1}\left(\binom{i}{m-k}+\binom{i}{m-k-1}\right){m\brack
m-i}(-1)^{m+i+k-1}+H(m-1,k-1)$ $\displaystyle=mH(m-1,k)-(m-1)H(m-1,k-1).$
Moreover, they satisfy the same initial conditions, i.e., $H(0,0)=H(1,0)=1$
and $H(1,1)=0$. ∎
###### Theorem 3.2.
The generating function for the number of colored superdiagonal compositions
is
$C(x)=\sum_{m\geq 0}\frac{x^{\binom{m+1}{2}}}{(1-x)^{2m}}Q_{m}(x).$
###### Proof.
Let $\sigma=(\sigma_{1},\dots,\sigma_{m})$ be a non-empty colored
superdiagonal composition of $n$ with $m$ parts. If $\sigma_{i}=\ell$ with
$\ell\geq i$, then $\sigma_{i}$ contributes to the generating function the
term $\ell x^{\ell}$, for $\ell\geq i$ and $1\leq i\leq m$. Let $C_{m}(x)$ be
the generating function of the colored superdiagonal compositions with $m$
parts. Then we have the following expression
$\displaystyle C_{m}(x)$ $\displaystyle=\left(\sum_{i\geq
1}ix^{i}\right)\left(\sum_{i\geq 2}ix^{i}\right)\cdots\left(\sum_{i\geq
m}ix^{i}\right)$
$\displaystyle=\frac{x}{(1-x)^{2}}\frac{(2-x)x^{2}}{(1-x)^{2}}\cdots\frac{(m-(m-1)x)x^{m}}{(1-x)^{2}}$
$\displaystyle=\frac{x^{\binom{m+1}{2}}}{(1-x)^{2m}}\prod_{\ell=1}^{m}(\ell-(\ell-1)x)$
$\displaystyle=\frac{x^{\binom{m+1}{2}}}{(1-x)^{2m}}Q_{m}(x).$
Finally, summing the last expression over $m\geq 0$, we get the desired
result. ∎
From Theorem 3.2 and Proposition 3.1 we obtain the following corollary.
###### Corollary 3.3.
The number of colored superdiagonal compositions of $n$ is given by
$c(n)=\sum_{m,\ell\geq
0}\binom{2m+\ell-1}{\ell}T\left(m,n-\binom{m+1}{2}-\ell\right).$
The first few values of the sequence $c(n)$ are
$1,\quad 1,\quad 2,\quad 5,\quad\textbf{11},\quad 21,\quad 42,\quad 86,\quad
171,\quad 322,\quad 596,\dots$
Notice that Equation (1) shows the colored superdiagonal compositions
corresponding to the bold term in the above sequence.
## References
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* [2] A. K. Agarwal. $n$-Colour compositions, _Indian J. Pure Appl. Math._ 31(11) (2000), 1421–1427.
* [3] J. J. Bravo, J. L. Herrera, J. L. Ramírez, and M. Shattuck. $n$-Color palindromic compositions with restricted subscripts, _Proc. Indian Acad. Sci. Math. Sci._ , Accepted.
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|
# Multi-color continuous-variable quantum entanglement in dissipative Kerr
solitons
Ming Li Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China. Yan-Lei Zhang Key Laboratory of
Quantum Information, CAS, University of Science and Technology of China, Hefei
230026, China CAS Center For Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
P. R. China. Xin-Biao Xu Key Laboratory of Quantum Information, CAS,
University of Science and Technology of China, Hefei 230026, China CAS Center
For Excellence in Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, P. R. China. Chun-Hua
Dong Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China. Hong X. Tang Department of
Electrical Engineering, Yale University, New Haven, CT 06511, USA Guang-Can
Guo Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China. Chang-Ling Zou<EMAIL_ADDRESS>Key Laboratory of Quantum Information, CAS, University of Science and
Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.
###### Abstract
In a traveling wave microresonator, the cascaded four-wave mixing between
optical modes allows the generation of frequency combs, including the
intriguing dissipative Kerr solitons (DKS). Here, we theoretically investigate
the quantum fluctuations of the comb and reveal the quantum feature of the
soliton. It is demonstrated that the fluctuations of Kerr frequency comb lines
are correlated, leading to multi-color continuous-variable entanglement. In
particular, in the DKS state, the coherent comb lines stimulate photon-pair
generation and also coherent photon conversion between all optical modes, and
exhibit all-to-all connection of quantum entanglement. The broadband multi-
color entanglement is not only universal, but also is robust against practical
imperfections, such as extra optical loss or extraordinary frequency shift of
a few modes. Our work reveals the prominent quantum nature of DKSs, which is
of fundamental interest in quantum optics and also holds potential for quantum
network and distributed quantum sensing applications.
_Introduction.-_ Over the past decades, nonlinear optics technologies become
the backbone of modern optics, enabling frequency conversion, optical
parametric oscillation, ultrafast modulation, and non-classical quantum
sources. Especially, various nonlinear optics mechanisms enabled the
generation of the optical frequency comb, which has attracted lots of research
interests due to its revolutionary applications in precision spectroscopy,
astronomy detection, optical clock, and optical communication (fortier201920,
; foltynowicz2011optical, ; picque2019frequency, ; Papp:14, ;
obrzud2019microphotonic, ; marin2017microresonator, ; menicucci2008one, ).
Recently, by harnessing the enhanced nonlinear optical effect in a
microresonator, the Kerr frequency comb, in particular, the dissipative Kerr
soliton (DKS) as phase-locked frequency comb, has been extensively studied
(kippenberg2018dissipative, ). Such DKS have been realized on photonic chips
with various $\chi^{(3)}$ materials, showing advances in compact size,
scalability, low power consumption, great spectral range and large repetition
rate (gaeta2019photonic, ; Wan:20, ; Bruch2021, ; bai2020brillouin, ).
Therefore, the DKS is appealing for photonic quantum information science. From
one aspect, the DKS produces an array of stable and locked coherent laser
sources with equally spaced frequencies, which provides a coherent laser
source for driving nonlinear light-matter interaction as well as stable
frequency reference of local oscillators for heterodyne measurements via
frequency multiplexing. By taking advantage of the coherence over a large
frequency band, the quantum key distribution utilizing DKS has been
experimentally studied (wang2020quantum, ), which promises commercialized
high-speed quantum communication. From another aspect, the Kerr nonlinear
interaction is inherently a quantum parametric process that describes the
annihilation of two photons and simultaneous generation of signal-idler photon
pairs, which imply quantum correlations among comb lines. When working below
the threshold, the DKS devices have been exploited to generate both discrete-
and continuous-variable (CV) quantum entangled states (reimer2016generation, ;
Cui2020, ; zhu2020chip, ; PhysRevResearch.2.023138, ), holding the potential
for one-way quantum computing and high-dimensional entanglement between
different colors that could be distributed over the quantum network
(kues2019quantum, ). For the case above the threshold, although previous
studies of $\chi^{(2)}$ optical combs indicate the potential applications of
comb in the multipartite cluster-state generation for quantum computing
(menicucci2008one, ; PhysRevLett.107.030505, ; PhysRevLett.108.083601, ), the
quantum nature of DKS have not been studied excepting one pioneer work by
Chembo (chembo2016quantum, ), who pointed out the mode-pairs locating
symmetrically in two sides of the pump mode exhibit significant squeezing.
In this Letter, we theoretically investigated the CV entanglement between comb
lines in a microcavity and compare the quantum features of comb in different
states, i.e. the state below the threshold, primary comb, and DKS. We showed
that the DKS could stimulate pair-generation and coherent conversion
interactions between optical modes and thus produces a complex network links
all optical modes. Evaluated by the entanglement logarithm negativity, it is
demonstrated that a large number of comb lines with different colors are
entangled together when the cavity is prepared at the soliton state. In
particular, a group of modes show all-to-all quantum entanglement, indicating
that arbitrary two modes over a large bandwidth could be entangled. The all-
to-all entanglement persists even some modes in this complex network are
eliminated. This multi-color quantum-entangled state (coelho2009three, ) holds
great potential in application in multi-user quantum communication, quantum
teleportation network (van2000multipartite, ; furusawa2007quantum, ) and
quantum-enhanced measurements (PhysRevLett.111.093603, ; simon2017quantum, ;
guo2020distributed, ; xia2020demonstration, ).
_Model and principle.-_ Figure 1(a) illustrates an optical microring cavity
sided-coupled to a waveguide for generating dissipative Kerr solitons
(kippenberg2018dissipative, ). By injecting a pump laser into a resonance of
the cavity, the intracavity optical field builds up and the photons are
converted between optical modes via cascaded four-wave mixing (FWM) due to the
Kerr nonlinearity, leading to a broad comb spectrum at the output. For
investigating the DKS in the microcavity, we consider a group of $2N+1$ modes
belongs to the same mode family, with the spatial distributions described by
$\Psi\left(\overrightarrow{r},\theta\right)=\phi\left(\overrightarrow{r}\right)e^{im\theta}$
(strekalov2016nonlinear, ). Here, $m\in\mathbf{m}$ denotes the mode index,
which corresponds to the angular momentum of the resonant modes and
$\mathbf{m}=\left\\{-N,-N-1,...,-1,0,1,...,N-1,N\right\\}$. Due to the
dispersion, the resonant frequencies could be expanded with respect to the
index as $\omega_{m}=\omega_{0}+mD_{1}+\frac{m^{2}}{2!}D_{2}+....$
(chembo2013spatiotemporal, ). The Hamiltonian describing this multimode system
reads (guo2018efficient, )
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j=-N}^{N}\delta_{i}a_{i}^{\dagger}a_{i}+g_{0}\sum_{ijkl}\delta\left(i+j-k-l\right)a_{i}^{\dagger}a_{j}^{\dagger}a_{k}a_{l}$
(1)
in the rotating frame of
$\sum_{j=-N}^{N}\left(\omega_{p}+jD_{1}\right)a_{j}^{\dagger}a_{j}$. Here,
$\delta_{j}=\omega_{0}-\omega_{p}+j^{2}D_{2}/2$ is the mode detuning by
neglecting higher order dispersion terms, $a_{i}^{\dagger}$($a_{i}$) is the
photon creation (annihilation) operator, $g_{0}$ is the vaccum coupling
strength of the Kerr nonlinearity, $\delta(\cdot)$ is the Kronecker delta
function and reflects the phase-matching condition $i+j=k+l$ according to the
angular momentum conservation (guo2018efficient, ; strekalov2016nonlinear, ).
The nonlinear interaction terms describe the simultaneous annihilation and
creation of photon-pairs in mode-pair $\left(i,j\right)$ and
$\left(k,l\right)$, with the total photon numbers are conserved. By taking all
permutations of indices $\left\\{i,j,k,l\right\\}$ in the summation, the
Hamiltonian is Hermitian and involves all FWM terms, including self-phase
modulation, cross-phase modulation, degenerate and non-degenerate FWMs.
Figure 1: (a) Schematic of dissipative Kerr soliton generation in an optical
microring cavity with a continuous-wave laser drive. (b) Four-wave mixing
processes classified by the conserved total angular momentum $\xi$, with
photon-pairs generated or annihilated simultaneously in mode-pairs
$\left(i,j\right)$ with $\xi=i+j$. The paired modes are labeled by the same
color and orientation, and the index $\xi$ can be taken from $-2N$ to $2N$.
Here we only show $\xi=-1,0,4$ as an illustration. Blue arrow: coherent photon
conversion between modes $m=-4,0$ stimulated by $m=-1,3$, with $\xi=-1$.
In Eq. (1), the number of FWM terms grows rapidly with the total mode number
as $\sim\left(2N+1\right)^{3}$. Since the total angular momentum [$\xi=i+j$ of
mode-pair $\left(i,j\right)$] conserves, we use the $\xi$ to classify
different FWM terms, as all the mode-pairs of $\xi$ could interact with each
other. As shown by Fig. 1(b), each horizontal line of $\xi$ represents the
group of signal-idler mode-pairs which are labeled by arrows with the same
color and orientation. The dashed orange arrow represents the degenerate case
$i=j$. It is anticipated that the photon-pair generation for mode-pair
$\left(i,j\right)$ would be stimulated by all the coherent light fields
belongs to the same $\xi$, which might lead to the bipartite CV entanglement
by the effective interaction
$\left(a_{i}^{\dagger}a_{j}^{\dagger}+a_{i}a_{j}\right)$ for all pairs in
$\xi$. Meanwhile, a single mode also connects to multiple $\xi$. As an
example, for mode $m=-3$ in Fig. 1(b), it could be entangled with mode $m=3$
with $\xi=0$, and $m=2$ with $\xi=-1$, as well as $m=7$ with $\xi=4$. In
addition, for the phase-matching $i+j=k+l$, there are also coherent photon
conversion processes between mode $\left(i,l\right)$ simulated by mode
$\left(j,k\right)$ and vice verse, which also distributes the quantum
correlations between different modes. Consequently, the FWM in the
microresonator results in a complex network, with optical modes serving as
nodes, and the links corresponding to photon-pair generation and coherent
photon conversion.
The complex network of FWM in a microcavity (Fig. 1(b)) implies complicated
dynamics of optical fields, which can be numerically evaluated by solving the
quantum dynamics of bosonic modes according to the Hamiltonian [Eq. (1)].
Instead of solving the intractable many-body nonlinear equations via full
quantum theory, we adopt the mean-field treatment of the strongly pumped
system, i.e. the mean and fluctuation of optical field in each mode are solved
separately, due to the weak nonlinearity $g_{0}/\kappa\ll 1$ in practice
(PhysRevApplied.13.034030, ). The operator of the cavity mode $a_{i}$ is
approximated by the sum of a classical field described by a _amplitude_
$\alpha_{i}$ and a _fluctuation_ described by a bosonic operator $\delta
a_{i}$. According to the Heisenberg equation and discarding the high-order
terms of the fluctuation operators, the dynamics of the classical fields and
their fluctuations follow
$\displaystyle\frac{d}{dt}\alpha_{i}$ $\displaystyle=$
$\displaystyle\beta_{i}\alpha_{i}-ig_{0}\sum_{jkl}\alpha_{j}^{*}\alpha_{k}\alpha_{l}+\varepsilon_{p}\delta\left(i\right),$
(2) $\displaystyle\frac{d}{dt}\delta a_{i}$ $\displaystyle=$
$\displaystyle\beta_{i}\delta a_{i}+\sqrt{2\kappa_{a}}a_{i}^{in}$ (3)
$\displaystyle-ig_{0}\sum_{jkl}\left(\alpha_{k}\alpha_{l}\delta
a_{j}^{\dagger}+\alpha_{j}^{*}\alpha_{l}\delta
a_{k}+\alpha_{j}^{*}\alpha_{k}\delta a_{l}\right),$
respectively. Here, the summation takes over all possible permutation
$\left(j,k,l\right)$ with $i=k+l-j$, $\beta_{i}=-i\delta_{i}-\kappa_{i}$,
$\kappa_{j}$ is the total amplitude decay rate of the $j$-th mode,
$\varepsilon_{p}$ is the pump strength, $a_{i}^{in}$ is the input noise on
mode $i$ and fulfills $\langle
a_{i}^{in}(t)(a_{i}^{in})^{\dagger}(t^{\prime})\rangle=\delta(t-t^{\prime})$.
From Eq. (3), the last three terms represent the classical-field-stimulated
photon-pair generation in modes $\left(i,j\right)$, and also the coherent
photon conversion between modes $\left(i,k\right)$ and $\left(i,l\right)$.
Rather than merely produce down-conversion photons by drive lasers
(reimer2016generation, ), these terms imply very rich quantum dynamics in the
complex network, with the quantum correlations directly generated through
photon-pair generation and also indirectly generated through coherently re-
distributing the generated photons among the modes.
Figure 2: Quantum entanglement in different comb states. (a)-(d) Optical
spectra of below-threshold comb state (a), primary comb (b), two-soliton state
(c), and single-soliton state (d). (e)-(h) The corresponding entanglement
measure $E_{\mathcal{N}}$ between all mode-pairs. The $E_{\mathcal{N}}>0$ is
painted with colors whereas $E_{\mathcal{N}}=0$ is painted white.
The classical fields have been extensively studied in previous works
(hansson2014numerical, ; guo2018efficient, ). With the amplitudes of classical
fields $\alpha_{i}$ obtained, the CV quantum correlation
(braunstein2005quantum, ) between the comb lines can be evaluated by solving
the dynamics of fluctuations [Eq. (3)]. It is convenient to represent the
fluctuations by the “amplitude” and “phase” field quadratures $\delta
X_{i}=\left(\delta a^{\dagger}+\delta a\right)/\sqrt{2}$, $\delta
Y_{i}=i\left(\delta a^{\dagger}-\delta a\right)/\sqrt{2}$. The dynamics of the
quadratures $\overrightarrow{Q}=\left\\{\delta X_{-N},\delta Y_{-N},\delta
X_{-N+1},\delta Y_{-N+1},...,\delta X_{N},\delta Y_{N}\right\\}^{T}$ follow
$\frac{d}{dt}\overrightarrow{Q}=\mathbf{M}\cdot\overrightarrow{Q}+\overrightarrow{n}(t),$where
$\overrightarrow{n}^{T}=\left\\{\sqrt{2\kappa_{-N}}X_{-N}^{in},\sqrt{2\kappa_{-N}}Y_{-N}^{in},...\right\\}$
is the input noise, and $\mathbf{M}$ is a $(4N+2)\times(4N+2)$ matrix derived
from Eq. (3). Then, the correlation matrix $\mathbf{V}$ for all modes can be
solved following the deviation in Ref. (vitali2007optomechanical, ) and we
obtain
$\displaystyle\mathbf{M}\mathbf{V}+\mathbf{V}\mathbf{M}^{T}$ $\displaystyle=$
$\displaystyle-\mathbf{D},$ (4)
where the element of the correlation matrix $V_{ij}=\langle
Q_{i}Q_{j}+Q_{j}Q_{i}\rangle/2$, the noise term $D_{ij}=\langle
n_{i}n_{j}+n_{j}n_{i}\rangle/2$ can be derived from $\langle
X_{i}^{in}(t)X_{j}^{in}(t^{\prime})\rangle=\delta(i-j)\delta(t-t^{\prime})$
and $\langle X_{i}^{in}(t)Y_{j}^{in}(t^{\prime})\rangle=0$. The bipartite CV
entanglement between comb lines could be evaluated by the logarithmic
negativity
$\displaystyle E_{\mathcal{N}}^{mn}$ $\displaystyle=$
$\displaystyle\mathrm{max}[0,-\ln\sqrt{2}\eta],$ (5)
where $\eta=\sqrt{\Theta-\sqrt{\Theta^{2}-4\mathrm{det}V}}$,
$\Theta=\mathrm{det}A+\mathrm{det}B-2\mathrm{det}C$, with $A$, $B$ and $C$ are
the elements of $V^{mn}=\\{\\{A,C\\},\\{C^{T},B\\}\\}$, which is a sub-matrix
of $V$ representing the bipartite correlation matrix between $\left\\{\delta
X_{m},\delta Y_{m},\delta X_{n},\delta Y_{n}\right\\}$. $E_{\mathcal{N}}$ is a
measure of the CV entanglement (PhysRevA.70.022318, ; PhysRevA.65.032314, ;
vitali2007optomechanical, ), and the mode-pair is entangled only if
$E_{\mathcal{N}}^{mn}>0$. By calculating all combinations of the modes, one
obtains the entanglement matrix $\mathbf{E}_{\mathcal{N}}$ of the cavity
field.
_Multi-color entanglement.-_ Figure 2 depicts the typical results of Kerr
combs in a microring resonator at different states (PhysRevA.89.063814, ;
guo2018efficient, ), with the upper and bottom rows show the classical
intracavity fields and the entanglement $E_{\mathcal{N}}$ between mode-pairs.
Here, we consider a monochromatic field driving on the $0$-th mode to
initially excite the mode-pairs with $\xi=0$, and the system parameters are
chosen from a typical AlN microring in the experimental work
(guo2018efficient, ).
(i) Below the threshold [Fig. 2(a) and (e)]. For weak pump power, the
parametric gain on mode-pair of $\xi=0$ can not compensate the dissipation in
these modes, thus the system stays below the optical parametric oscillation
threshold and generates thermal photon-pairs by the spontaneous parametric
down-conversion (SPDC) (reimer2016generation, ). From the comb spectrum of
classical field [Fig. 2(a)], only the $0$-th mode is sufficiently excited, and
the intracavity fields in all the other modes are negligible. Figure 2(e)
shows the matrix $\mathbf{E}_{\mathcal{N}}$, which only has positive value at
its diagonal elements, indicating that only the photon-pair generation between
mode-pair $\left(+l,-l\right)$ for $\xi=0$ are initiated. Due to the
dispersion of the cavity resonance, modes with indices away from $0$
experience poorer phase-matching condition and thus the $E_{\mathcal{N}}$
decays with $l$.
(ii) Primary comb [Figs. 2(b) and (f)]. As the pump power increases above the
OPO threshold and the cavity field can be prepared into a stable Turing
pattern (Fig. 2(b)). In this state, several equally-spaced comb lines are
efficiently excited, with the space approximately be $m\times$FSR determined
by the dissipation rate $\kappa$ and the dispersion $D_{2}$
(PhysRevA.89.063814, ). Even though only $\xi=0$ is initially excited, the
generated comb lines can stimulate the cascaded FWM and produce entanglement
for mode-pairs belongs to other $\xi$. Consequently,
$\mathbf{E}_{\mathcal{N}}$ in Fig. 2(f) shows non-zero values at elements on
several diagonal lines, with each diagonal line corresponding to $\xi=2nm$
with $n\in\mathbb{Z}$. In the primary comb, the power of the comb lines
decreases with the mode index, the degree of entanglement decreases from the
main-diagonal line ($\xi=0$) to the high-order diagonal ($\xi=2nm\neq 0$).
(iii) Soliton state [Figs. 2(c), (d), (g) and (h)]. With appropriate laser
power and frequency detuning, the intracavity field can be driven to the
soliton states, which are ultrashort pulses circulating inside the cavity
(herr2014temporal, ; kippenberg2018dissipative, ). As shown by the spectra in
Fig.2(c)-(d), when tens of modes are efficiently excited in both the two and
single soliton states, the envelopes show a profile of
$\mathrm{sech}^{2}$-function. The strong comb line with index number $l$
generated by the the pump mode can further drive the mode-pairs with $\xi=2l$,
leading to the positive $E_{\mathcal{N}}$ on the corresponding diagonals of
the entanglement matrix in Fig. 2(g) and 2(h). Due to the intensity
distribution of the spectra, the entanglement matrix
$\mathbf{E}_{\mathcal{N}}$ of the two-soliton state has positive values only
on diagonal lines corresponding to even mode index ($m/2\in\mathbb{Z})$, while
all diagonals near the center have positive values for the single-soliton
state. Since multi-soliton state has higher energy than the single-soliton
state (herr2014temporal, ), the FWMs are excited more efficiently, resulting
in a higher $E_{\mathcal{N}}$. For the case in Fig. 2, the two-soliton state
has a maximum $E_{\mathcal{N}}$ of 0.259, in comparison with 0.117 for the
single-soliton state.
_All-to-all entanglement.-_ For single-soliton state, modes are excited
efficiently so that the multi-color entanglement is distinct from other cases.
In particular, we can find a group of modes [mode index $|i|<12$ in Fig. 2(h)]
where all-pair of modes are entangled. The all-to-all entanglement indicates a
fully connected complex network. This phenomena manifests the distinct
physical mechanisms of entanglement generation in the soliton state: two-mode
and single-mode ($l$ [$\propto\left(a_{l}^{2}+a_{l}^{\dagger 2}\right)$] due
to the comb lines in $\xi=2l$) squeezing generation, and the linear conversion
between two modes $\left(i,j\right)$ stimulated by comb lines
$\left(\xi-i,\xi-j\right)$
[$\propto\left(\alpha_{\xi-i}^{*}a_{i}^{\dagger}\alpha_{\xi-j}a_{j}+h.c.\right)$]
for all $\xi$, compared to the SPDC in conventional studies [Fig. 2(a)]. Since
the modes have relatively uniform intensities around the $0$-th mode, i.e.
$\left|\alpha_{j}\right|$ decays with $j$ as indicated by the
$\mathrm{sech}^{2}$-function, the soliton state has higher degree of
entanglement at the center compared with modes away from the pump in Figs.
2(g) and (h).
According to Eq. 3, the entanglement is stimulated by classical fields and
thus the all-to-all entanglement region could be further spread by flatten the
comb spectrum. Since the spectrum bandwidth of DKS could be efficiently
controlled by the dispersion $D_{2}$ (herr2014temporal, ), the multi-color
entanglement for various $D_{2}$ is investigated. Figures 3(a)-(b) show the
the entanglement matrix $\mathbf{E}_{\mathcal{N}}$ for $D_{2}/\kappa=1.645$
and $D_{2}/\kappa=0.329$, respectively. The $D_{2}/\kappa=0.329$ case supports
an all-connected group involving more modes than that of the
$D_{2}/\kappa=1.645$ case but has less degree of entanglement, as marked by
the black square. Defining the edge length of the square as the size of the
fully connected group, the size of the quantum state decreases with the
dispersion $D_{2}$, as summarized in Fig. 3(c). All-to-all entanglement among
more than $50$ colors is predicted for $D_{2}/\kappa=0.329$. Thus, by
designing a microring with smaller dispersion and larger cavity size,
potentially hundreds of modes can be all-to-all entangled. By distributing the
photons to different users via wavelength multiplexing, this all-to-all
entangled state is promising to build a multi-party quantum teleportation
network (yonezawa2004demonstration, ).
Figure 3: All-to-all quantum entanglement via dispersion engineering. (a)-(b)
The matrices $\mathbf{E}_{\mathcal{N}}$ of single-soliton state with
$D_{2}/\kappa=1.645$ (a) and $D_{2}/\kappa=0.329$ (b), respectively. (c) The
relationship between the size of the group and the cavity dispersion. The
frequency and strength of the pump field are fixed.
_Entanglement and comb percolation.-_ For a complex network, it is always
curious about how the system performs if some nodes are removed. The network
with defects corresponds to the optical cavities in practice, where some modes
show extraordinary low quality factor or large frequency shifting, due to the
perturbation of environments or avoid-mode crossing induced by other mode
families. The absorption and shifting of the resonance can significantly
suppress the intensities in these modes and modify the soliton spectrum.
Therefore, we further investigate the multi-color entanglement of the soliton
state in a defective mode family. The defects are simulated by magnifying the
dissipation rate of certain modes to $1000\>\kappa$ to eliminate the mode
density of state and thus suppresses the corresponding comb lines. As shown by
the numerical results in Figs. 4(a)-(b), the single-soliton state can still
exist even if $5$ modes are eliminated, and the spectra remain a envelope of
$\mathrm{sech}^{2}$-function except the lines of the lossy modes being
suppressed. Due to the nature of the complex network, each mode can
participate many different frequency mixing processes with different $\xi$,
thus the network is still fully connected even in the absence of a few nodes,
and showing a percolation of the comb generation in a microcavity. From the
entanglement matrix of the soliton state, as shown in Figs. 4(c) and (d), only
the elements that are associated with the very lossy modes are affected,
whereas the entanglement between the rest of modes are almost unaffected. The
percolation of the entanglement demonstrates the robustness of the multi-color
entanglement of DKS against loss or other distortion of the mode density of
states in practical experiments. Therefore, such a robust, self-organizing,
entangled quantum source holds great potential for future applications in non-
ideal environments.
Figure 4: Multi-color entanglement in a defective FWM network. (a)-(b) The
soliton spectrum with the dissipation rate of $1$ (a) and $5$ (b) consecutive
modes are magnified by 1000 times. (c)-(d) The entanglement matrices of the
corresponding soliton states.
_Conclusion.-___ The multi-color CV entanglement in the DKS comb is
investigated. In the microresonator, a complex network of optical modes are
formed via the Kerr nonlinearity, with the optical fluctuations of different
modes are connected through-pair-generation and conversion interactions. It is
demonstrated that the modes in the complex network show all-to-all
entanglement, and also exhibits robustness against the defects of the network,
which is induced by extra optical losses or other experimental imperfections.
Combing with other nonlinear optical effects, the multi-color entanglement of
DKS could be further extended to other frequency bands, such as mid-infrared
and ultraviolet wavelengths, by $\chi^{(2)}$-nonlinearity (guo2018efficient, ;
Bruch2021, ) or Raman scattering (jung2014green, ; karpov2016raman, ;
xue2017second, ). By multiplexing the color of DKS to other degrees of freedom
of photons, such as the polarization, transverse waveguide mode, and time-bin,
high-dimension quantum cluster state could be generated
(larsen2019deterministic, ; asavanant2019generation, ). Our results reveal the
interesting dynamics of quantum entanglement generation associated with DKS,
and also indicate the great potential of DKS in quantum information science,
such as high-dimensional CV quantum sources, multiparty quantum teleportation
network (pirandola2015advances, ), and distributed quantum metrology
(guo2020distributed, ; zhang2020distributed, ).
###### Acknowledgements.
This work was funded by the National Key Research and Development Program
(Grant No. 2016YFA0301300), the National Natural Science Foundation of China
(Grant No.11874342, 11934012, 11904316, 11704370, and 11922411), Anhui
Initiative in Quantum Information Technologies (Grant No. AHY130200), and
Anhui Provincial Natural Science Foundation (Grant No. 2008085QA34) and the
China Postdoctoral Science Foundation (Grant No. 2019M662153). ML and CLZ was
also supported by the Fundamental Research Funds for the Central Universities,
and the State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, China. This work was partially
carried out at the USTC Center for Micro and Nanoscale Research and
Fabrication.
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|
# Estimating black hole masses in obscured AGN using X-rays
Mario Gliozzi1 and James K. Williams1 and Dina A. Michel1
1 Department of Physics and Astronomy, George Mason University, 4400
University Drive, Fairfax, VA 22030
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Determining the black hole masses in active galactic nuclei (AGN) is of
crucial importance to constrain the basic characteristics of their central
engines and shed light on their growth and co-evolution with their host
galaxies. While the black hole mass ($M_{\mathrm{BH}}$) can be robustly
measured with dynamical methods in bright type 1 AGN, where the variable
primary emission and the broad line region (BLR) are directly observed, a
direct measurement is considerably more challenging if not impossible for the
vast majority of heavily obscured type 2 AGN. In this work, we tested the
validity of an X-ray-based scaling method to constrain the $M_{\mathrm{BH}}$
in heavily absorbed AGN. To this end, we utilized a sample of type 2 AGN with
good-quality hard X-ray data obtained by the NuSTAR satellite and with
$M_{\mathrm{BH}}$ dynamically constrained from megamaser measurements. Our
results indicate that, when the X-ray broadband spectra are fitted with
physically motivated self-consistent models that properly account for
absorption, scattering, and emission line contributions from the putative
torus and constrain the primary X-ray emission, then the X-ray scaling method
yields $M_{\mathrm{BH}}$ values that are consistent with those determined from
megamaser measurements within their respective uncertainties. With this method
we can therefore systematically determine the $M_{\mathrm{BH}}$ in any type 2
AGN, provided that they possess good-quality X-ray data and accrete at a
moderate to high rate.
###### keywords:
Galaxies: active – Galaxies: nuclei – X-rays: galaxies
††pubyear: 2015††pagerange: Estimating black hole masses in obscured AGN using
X-rays–B
## 1 Introduction
Historically, radio-quiet active galactic nuclei (AGN) have been divided into
two main categories based on their optical spectroscopy: type 1 AGN, whose
spectra are characterized by the presence of broad permitted lines (with full
width at half maximum FWHM $>2000\,\mathrm{km~{}s^{-1}}$) along with narrow
forbidden lines, and type 2 AGN, where only narrow forbidden lines are
detected (e.g., Khachikian & Weedman, 1974; Antonucci, 1983).
According to the basic AGN unification model, type 2 AGN can be considered as
the obscured counterpart of type 1 AGN and their main differences can be
simply ascribed to different viewing angles, due to the presence of an
obscuring toroidal structure made of gas and dust surrounding the AGN (e.g.,
Osterbrock, 1978; Antonucci, 1993; Tadhunter, 2008; Urry & Padovani, 1995).
However, over the years, theoretical and observational studies have revealed
that the simplest version of the unification model, based on a smooth donut-
shaped torus, is unable to explain several observations, favoring instead a
scenario where the torus is clumpy, with a covering factor depending on
various AGN properties, and where the overall obscuration occurs on different
scales with significant contribution from the galaxy itself. See Netzer (2015)
and Ramos Almeida & Ricci (2017) for recent comprehensive reviews on the
unification model of AGN.
Regardless of the nature of the obscuration, in type 2 AGN, the central engine
– an optical/UV emitting accretion disk, coupled with an X-ray emitting
Comptonization corona – and the broad line region (BLR) are not directly
accessible to observations. This makes it more difficult to determine the
properties of obscured AGN, which represent the majority of the AGN population
and thus play a crucial role in our understanding of the AGN activity, census,
and cosmological evolution (see Hickox & Alexander 2018 for a recent review on
obscured AGN).
Table 1: Properties of the sample
Source | Distance | $M_{\mathrm{BH}}$ | $\lambda_{\mathrm{Edd}}$ | NuSTAR | Exposure
---|---|---|---|---|---
name | (Mpc) | ($10^{6}$ M☉) | ($L_{\mathrm{bol}}$/$L_{\mathrm{Edd}}$) | observation ID | (ks)
(1) | (2) | (3) | (4) | (5) | (6)
NGC 1068 | $14.4^{\textrm{a}}$ | $8.0\pm 0.3$ | $0.210\pm 0.053$ | 60002033002 | 52.1
NGC 1194 | $53.2^{\textrm{b}}$ | $65.0\pm 3.0$ | $0.007\pm 0.002$ | 60061035002 | 31.5
NGC 2273 | $25.7^{\textrm{b}}$ | $7.5\pm 0.4$ | $0.132\pm 0.034$ | 60001064002 | 23.2
NGC 3079 | $17.3^{\textrm{c}}$ | $2.4_{-1.2}^{+2.4}$ | $0.011\pm 0.009$ | 60061097002 | 21.5
NGC 3393 | $50.0^{\textrm{d}}$ | $31.0\pm 2.0$ | $0.062\pm 0.016$ | 60061205002 | 15.7
NGC 4388 | $19.0^{\textrm{b}}$ | $8.5\pm 0.2$ | $0.035\pm 0.009$ | 60061228002 | 21.4
NGC 4945 | $3.7^{\textrm{e}}$ | $1.4\pm 0.7$ | $0.135\pm 0.075$ | 60002051004 | 54.6
IC 2560 | $26.0^{\textrm{f}}$ | $3.5\pm 0.5$ | $0.175\pm 0.050$ | 50001039004 | 49.6
Circinus | $4.2^{\textrm{g}}$ | $1.7\pm 0.3$ | $0.143\pm 0.044$ | 60002039002 | 53.9
Columns: 1 = megamaser AGN name. 2 = distance used computing the
$M_{\mathrm{BH}}$ from the maser measurements. References for the distances
and black hole masses are (a) Lodato & Bertin (2003), (b) Kuo et al. (2011),
(c) Kondratko, Greenhill, & Moran (2005), (d) Kondratko, Greenhill, & Moran
(2008), (e) Greenhill et al. (1997), (f) Yamauchi et al. (2012), and (g)
Greenhill et al. (2003). 3 = black hole mass. 4 = Eddington ratio with
Brightman’s bolometric correction of $10\times$ to $L_{\mathrm{X}}$ from
Brightman et al. (2016). 5 = NuSTAR observation ID. 6 = exposure time.
In order to shed light on the properties of the AGN central engine and its
accretion state, we need to accurately determine the black hole mass
($M_{\mathrm{BH}}$). In type 1 AGN, a reliable dynamical method frequently
used is the so-called reverberation mapping method, where intrinsic changes in
the continuum emission of the central engine, measured with some time delay in
the line emission produced by the BLR, are used to constrain the
$M_{\mathrm{BH}}$, modulo a geometric factor (Blandford & McKee, 1982;
Peterson et al., 2004). On the other hand, in type 2 AGN, by definition the
BLR is not visible and hence the reverberation mapping technique cannot be
applied. Nevertheless, there is a small fraction of heavily obscured AGN for
which it is still possible to measure the $M_{\mathrm{BH}}$ in a reliable way
via a dynamical method. These are the sources that display water megamaser
emission; if this emission is located in the accretion disk and is
characterized by the Keplerian motion, then the $M_{\mathrm{BH}}$ can be
constrained with great accuracy (e.g. Kuo et al., 2011).
In this work, we use a sample of heavily obscured type 2 AGN with
$M_{\mathrm{BH}}$ constrained by megamaser measurements and with good-quality
hard X-ray spectra obtained with the Nuclear Spectroscopic Telescope Array
(NuSTAR), a focusing hard X-ray telescope launched in 2012 with large
effective area and excellent sensitivity in the energy range 3–78 keV, where
the signatures of absorption and reflection are most prominent. Our main goal
is to test whether an X-ray scaling method that yields $M_{\mathrm{BH}}$
values broadly consistent with those obtained from reverberation mapping in
type 1 AGN can be extended to type 2 AGN.
The paper is structured as follows. In Section 2, we describe the sample
properties and the X-ray data reduction. In Section 3, we report on the
spectral analysis of NuSTAR data. The application of the X-ray scaling method
and the comparison between the $M_{\mathrm{BH}}$ values derived with this
method and those obtained from megamaser measurements are described in Section
4. We discuss the main results and draw our conclusions in Section 5.
## 2 Sample Selection and Data Reduction
We chose our sample of type 2 AGN based on the following two criteria: these
objects must have 1) the $M_{\mathrm{BH}}$ dynamically determined by megamaser
disk measurements, and 2) good-quality hard X-ray data. The former criterion
is crucial to quantitatively test the validity of the X-ray scaling method
applied to heavily obscured AGN, whereas the latter criterion is necessary to
robustly constrain the properties of the primary X-ray emission by accurately
assessing the contributions of absorption and reflection caused by the
putative torus. These criteria are fulfilled by the sample described by
Brightman et al. (2016), which is largely based on the sample of megamasers
analyzed by Masini et al. (2016) and spans a range in X-ray luminosity between
$10^{42}~{}\mathrm{erg~{}s^{-1}}$ and a few units in
$10^{43}~{}\mathrm{erg~{}s^{-1}}$. The general properties of this sample,
including the distance used to determine the $M_{\mathrm{BH}}$ from maser
measurements, the $M_{\mathrm{BH}}$ itself, and the Eddington ratio
$\lambda_{\mathrm{Edd}}=L_{\mathrm{bol}}/L_{\mathrm{Edd}}$, are reported in
Table 1.
The archival NuSTAR data of these nine objects were calibrated and screened
using the NuSTAR data analysis pipeline nupipeline with standard filtering
criteria and the calibration database CALDB version 20191219. From the
calibrated and screened event files we extracted light curves and spectra,
along with the RMF and ARF files necessary for the spectral analysis, using
the nuproduct script. The extraction regions used for both focal plane
modules, FPMA and FPMB, are circular regions of radii ranging from 40″ to 100″
depending on the brightness of the source, and centered on the brightest
centroid. Background spectra and light curves were extracted by placing
circles of the same size used for the source in source-free regions of the
same detector. No flares were found in the background light curves. All
spectra were binned with a minimum of 20 counts per bin using the HEASoft task
grppha 3.0.1 for the $\chi^{2}$ statistics to be valid.
Figure 1: The top panels show the NuSTAR spectra (black data points indicate
FPMA data whereas the red ones indicate FPMB data) with the best-fit models,
whereas the bottom panels show the data-to-model ratios.
## 3 Spectral analysis
Table 2: Spectral Results
Source | $N_{{\textrm{H}}_{\textrm{Gal}}}$ | $\log(N_{{\textrm{H}}_{\textrm{bor}}})$ | $\cos{\theta}$ | CFtor | $A_{\textrm{Fe}}$ | $N_{{\textrm{H}}_{\textrm{mytz}}}$ | $\Gamma$ | $N_{\mathrm{BMC}}$ | $\log{A}$ | $f_{\textrm{s}}$ | ($\chi^{2}/$dof)
---|---|---|---|---|---|---|---|---|---|---|---
name | ($10^{20}$ cm-2) | | | (%) | | ($10^{24}$ cm-2) | | | | (%) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
NGC 1068 | $2.59$ | $23.3\pm 0.1$ | $0.1$ | 15 | 1.0 | $3.5\pm 0.1$ | $1.98_{-0.01}^{+0.01}$ | $2.2_{-0.1}^{+0.1}\times 10^{-3}$ | 0.08 | 1.6 | 754.9/713
NGC 1194 | $5.53$ | $23.9\pm 0.1$ | $0.1$ | 91 | 3.2 | $0.8\pm 0.1$ | $1.62_{-0.05}^{+0.05}$ | $6.4_{-0.8}^{+1.1}\times 10^{-5}$ | 0.57 | 4.0 | 194.3/167
NGC 2273 | $5.80$ | $25.0\pm 0.7$ | $0.1$ | 15 | 1.0 | $6.8\pm 0.4$ | $1.95_{-0.05}^{+0.05}$ | $3.1_{-0.2}^{+0.2}\times 10^{-3}$ | 2.0 | … | 54.0/57
NGC 3079 | $0.87$ | $24.5\pm 0.1$ | $0.1$ | 20 | 1.0 | $2.9\pm 0.1$ | $1.91_{-0.06}^{+0.05}$ | $1.1_{-0.2}^{+0.2}\times 10^{-3}$ | 0.8 | 0.3 | 80.1/75
NGC 3393 | $6.13$ | $25.2\pm 0.2$ | $0.1$ | 15 | 1.0 | $2.4\pm 0.1$ | $1.86_{-0.10}^{+0.10}$ | $9.6_{-1.2}^{+3.1}\times 10^{-4}$ | 0.22 | … | 54.3/65
NGC 4388 | $2.57$ | $23.6\pm 0.1$ | $0.1$ | 91 | 1.0 | $0.4\pm 0.1$ | $1.66_{-0.04}^{+0.04}$ | $3.3_{-0.4}^{+0.5}\times 10^{-4}$ | -0.55 | 17.0 | 435.2/420
NGC 4945 | $14.0$ | $24.4\pm 0.1$ | $0.1$ | 91 | 0.8 | $3.0\pm 0.7$ | $1.74_{-0.05}^{+0.05}$ | $1.6_{-0.1}^{+0.1}\times 10^{-3}$ | 2.15 | 0.5 | 1699.8/1716
IC 2560 | $6.51$ | $25.1\pm 0.1$ | $0.1$ | 15 | 2.3 | $6.9\pm 0.1$ | $2.08_{-0.08}^{+0.08}$ | $1.8_{-0.3}^{+0.4}\times 10^{-3}$ | 2.0 | … | 88.3/64
Circinus | $52.5$ | $23.6\pm 0.1$ | $0.1$ | 24 | 1.7 | $1.6\pm 0.1$ | $2.17_{-0.01}^{+0.01}$ | $9.8_{-0.2}^{+0.1}\times 10^{-3}$ | -0.43 | 3.3 | 1713.2/1714
Columns: 1 = megamaser AGN name. 2 = Galactic column density from NASA’s
HEASARC. 3 = column density calculated with the Borus model. 4 = cosine of the
inclination angle. 5 = covering factor. 6 = iron abundance relative to the
solar value. 7 = column density calculated with the MYTorus model. 8 = photon
index. 9 = normalization of the BMC model. 10 = logarithm of $A$, where
$A=(f+1)/f$ and $f$ is the fraction of seed photons that are scattered. 11 =
fraction of the primary emission scattered along the line of sight by an
extended ionized reflector. 12 = $\chi^{2}$ divided by degrees of freedom.
The X-ray spectral analysis was performed using the xspec v.12.9.0 software
package (Arnaud, 1996), and the errors quoted on the spectral parameters
represent the 1$\sigma$ confidence level.
The NuSTAR spectra of this sample have already been reasonably well fitted
with self-consistent physically motivated models such as MYTorus (Murphy &
Yaqoob, 2009) and Torus (Brightman & Nandra, 2011) to account for the
continuum scattering and absorption, as well as the fluorescent line emission
produced by the torus, whereas the primary emission was parametrized with a
phenomenological power-law model. However, in order to apply the X-ray scaling
method (whose key features are described in the following section), the
primary emission needs to be parametrized by the Bulk Motion Comptonization
model (BMC), which is a generic Comptonization model that convolves thermal
seed photons producing a power law (Titarchuk, Mastichiadis, & Kylafis, 1997).
This model, which can be used to parametrize both the bulk motion and the
thermal Comptonization, is described by four spectral parameters: the
normalization $N_{\mathrm{BMC}}$, the spectral index $\alpha$, the temperature
of the seed photons $kT$, and $\log A$, where $A$ is related to the fraction
of scattered seed photons $f$ by the relationship $A=(f+1)/f$. Unlike the
phenomenological power-law model, the BMC parameters are computed in a self-
consistent way, and the power-law component produced by the BMC does not
extend to arbitrarily low energies.
We carried out a homogeneous systematic reanalysis of the NuSTAR spectra of
these sources. We started from the best-fit models reported in the literature
but utilized the Borus model (Baloković et al., 2018), which can be considered
as an evolution of the previous torus models. Specifically, Borus has the same
geometry implemented in Torus but can also be used in a decoupled mode, where
the column density $N_{\mathrm{H}}$ responsible for the continuum scattering
and fluorescent line emission is allowed to be different from the
$N_{\mathrm{H}}$ responsible for the attenuation of the primary component.
Additionally, unlike Torus, this model correctly accounts for the absorption
experienced by the photons backscattered from the far side of the inner torus.
With respect to MYTorus, Borus contains additional emission lines, has a
larger range for $N_{\mathrm{H}}$, and directly yields the value of the
covering fraction. However, since Borus only parametrizes the scattered
continuum and the fluorescent line components associated with the torus, to
account for the absorption and scattering experienced by the primary emission,
we utilized the zeroth-order component of MYTorus (MYTZ), which properly
includes the effects of the Klein-Nishina Compton scattering cross section
that are relevant in heavily absorbed AGN at energies above 10 keV. In
summary, our procedure can be summarized in three steps: 1) we started from
the spectral best fits reported in the literature; 2) we then substituted
Borus (more specifically, we used the borus02_v170323a.fits table) for either
Torus or MYTorus to account for the scattered and line components, and used
the zeroth-order component of MYTorus for the transmitted one; 3) finally, we
substituted BMC for the power-law model used for the primary emission.
In the spectral fitting, in order to preserve the self-consistency of these
physically motivated torus models, which are created by Monte Carlo
simulations using a power law to parametrize the X-ray primary emission, one
needs to link the primary emission parameters – the photon index $\Gamma$ and
the normalization $N_{\mathrm{PL}}$ – to the input parameters of the scattered
continuum and emission-line components. In the case of the BMC model, the
power-law slope is described by the spectral index $\alpha$, which is related
to the photon index by the relationship $\Gamma=\alpha+1$. However, there is
not a known mathematical equation linking the normalizations
$N_{\mathrm{BMC}}$ and $N_{\mathrm{PL}}$. We therefore derived this
relationship empirically by using a sample of clean type 1 AGN (i.e., AGN
without cold or warm absorbers), whose details are described in Williams,
Gliozzi, & Rudzinsky (2018); Gliozzi & Williams (2020). We fitted the 2–10 keV
XMM-Newton spectra twice, first with the BMC model and then with a power law.
The results of this analysis are illustrated in Fig. 2, where
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ is plotted versus $N_{\mathrm{BMC}}$,
showing that, regardless of the value of $N_{\mathrm{BMC}}$, the normalization
ratios cluster around the average value, $30.8\pm 0.9$, represented by the
longer-dashed line, with moderate scattering of $\sigma=7.2$, represented by
the shorter-dashed lines. Fig. 2, where the data point’s size and color
provide information about the photon index, also reveals a tendency for the
AGN with steeper spectra to have larger values of
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$. This trend is formally confirmed by a
least-squares best fit of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ vs. $\Gamma$,
which yields $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =-18.9 + 26.3$\Gamma$, with a
Pearson’s correlation coefficient of 0.85.
These results are in agreement with those obtained from a series of
simulations carried out with the fakeit command in xspec. Simulating spectra
of the BMC model with the parameters varying over a broad range, and then
fitting them with a power-law model, we found that
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ shows a horizontal trend when plotted vs.
$N_{\mathrm{BMC}}$ with an average value consistent with 30 for $\Gamma=1.9$,
whereas the horizontal trend is consistent with an average value of 24 for
$\Gamma=1.6$ and 33 for $\Gamma=2.2$.
Based on these findings, in our spectral fitting of the megamaser sample we
forced $N_{\mathrm{BMC}}$ to be equal to $N_{\mathrm{PL}}$/30 by linking these
parameters to reflect this relationship. For completeness, and to take into
account the weak dependence of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ on $\Gamma$,
we have also carried out the spectral analysis assuming
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 24 (i.e., the average value minus one
standard deviation) for flat spectrum sources and
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 38 (average $+\sigma$) for steep spectrum
sources.
We note that, compared to $\Gamma$ and $N_{\mathrm{BMC}}$, the remaining BMC
parameters $kT$ and $\log A$ play a marginal role in the shape of the spectrum
and in the determination of the $M_{\mathrm{BH}}$, as explicitly assessed in
Gliozzi et al. (2011). Therefore, to limit the number of free parameters, we
fixed $kT$ to 0.1 keV, which is consistent with the values generally obtained
when the BMC model is fitted to X-ray AGN spectra (e.g., Gliozzi et al., 2011;
Williams, Gliozzi, & Rudzinsky, 2018), whereas $\log A$ was fixed to the best-
fit value obtained in the first fitting iteration.
Figure 2: $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ plotted vs. $N_{\mathrm{BMC}}$
for a sample of “clean” type 1 AGN (i.e., AGN with negligible warm or cold
absorbers). The black longer-dashed line represents the average value, whereas
the shorter-dashed lines indicate the one standard deviation levels from the
average. Both the data point’s size and color provide information on the
source’s photon index $\Gamma$: the larger the symbol and the darker the
color, the steeper the $\Gamma$.
Our baseline model for all type 2 AGN fitted in this work is expressed in the
xspec syntax as follows:
phabs * (atable(Borus) + MYTZ * BMC + const * BMC)
where the first absorption model phabs accounts for our Galaxy contribution,
the Borus table model parametrizes the continuum scattering and fluorescent
emission line components associated with the torus, and MYTZ models the
absorption and Compton scattering acting on the transmitted primary emission,
which is described by the Comptonization model BMC. The last additive
component const*BMC parametrizes the fraction of primary emission directly
scattered towards the observer by a putative optically thin ionized medium,
which is often observed below 5 keV in spectra of heavily obscured AGN (e.g.,
Yaqoob, 2012).
Depending on the source and the complexity of its X-ray spectrum, additional
components (such as the host galaxy contribution, individual lines, additional
absorption and scattering components, or models describing off-nuclear sources
contained in the NuSTAR extraction region) are included and described in the
individual notes of each source reported in the Appendix.
The spectral parameters obtained by fitting this baseline model are reported
in Table 2, and the best fits and model-to-data ratios are shown in Fig 1.
## 4 Black hole masses
### 4.1 $M_{\mathrm{BH}}$ from the X-ray scaling method
The X-ray scaling method was first introduced by Shaposhnikov & Titarchuk
(2009), who showed that the BH mass and distance $D$ of any stellar mass BH
can be obtained by scaling these properties from those of an appropriate
reference source (i.e., a BH system with $M_{\mathrm{BH}}$ dynamically
determined and distance tightly constrained). In its original form this
technique exploits the similarity of the trends displayed by different BH
systems in two plots – the photon index $\Gamma$ vs. quasi-periodic
oscillation (QPO) frequency plot and the $N_{\mathrm{BMC}}$–$\Gamma$ diagram –
to derive their $M_{\mathrm{BH}}$ and $D$.
Based on the assumption that the process leading to the ubiquitous emission of
X-rays – the Comptonization of seed photons produced by the accretion disk –
is the same in all BH systems regardless of their mass, this method can in
principle be extended to any BH including the supermassive BHs at the cores of
AGN. In the latter case, since the detection of QPOs is extremely rare but the
distance is generally well constrained by redshift or Cepheid measurements,
only the $N_{\mathrm{BMC}}$–$\Gamma$ diagram is used to determine the
$M_{\mathrm{BH}}$. Indeed, over the years, this method has been successfully
applied to stellar mass BHs (e.g., Seifina, Titarchuk, & Shaposhnikov 2014;
Titarchuk & Seifina 2016) and to ultraluminous X-ray sources (e.g., Titarchuk
& Seifina 2016; Jang et al. 2018), as well as to a handful of AGN that showed
high spectral and temporal variability during deep X-ray exposures (e.g.,
Gliozzi et al. 2010; Giacché, Gilli, & Titarchuk 2014; Seifina, Chekhtman, &
Titarchuk 2018.)
Although the vast majority of AGN do not possess long-term X-ray observations
and do not show strong intrinsic spectral variability (i.e., variability
described by substantial changes of $\Gamma$ not caused by obscuration
events), the X-ray scaling method can be extended to any type 1 AGN with one
good-quality X-ray observation. Indeed, Gliozzi et al. (2011) demonstrated
that the $M_{\mathrm{BH}}$ values determined with this method are fully
consistent with the corresponding values obtained from the reverberation
mapping technique. The reference sources, used in that study and then also in
this work, are three stellar mass BHs residing in X-ray binaries – GRO
J1655-40, GX 339-4, and XTE J1550-564 – with $M_{\mathrm{BH}}$ dynamically
determined and spectral evolution during the rising and decaying phases of
their outbursts mathematically parametrized by Shaposhnikov & Titarchuk
(2009). The physical properties of the stellar references and the mathematical
description of their spectral trends, as well as the details of the method,
are reported in Gliozzi et al. (2011).
In summary, all the reference trends yielded $M_{\mathrm{BH}}$ measurements
consistent with the reverberation mapping values within their nominal
uncertainties, with the decaying trends showing a slightly better agreement
than the rising trends, which have a tendency to underestimate
$M_{\mathrm{BH}}$ to a moderate degree. Unfortunately, the most reliable
reference source – GRO J1655-40 during the 2005 decaying phase (hereafter
GROD05) – has a fairly small range of $\Gamma$ during its spectral transition
limiting its application to sources with relatively flat photon indices. Using
the reverberation mapping values as calibration, it was determined that for
AGN with steep spectra ($\Gamma>2$) the best estimate of $M_{\mathrm{BH}}$ is
obtained using the value derived from the rising phase of the 1998 outburst of
XTE J1550-564 multiplied by a factor of 3 (hereafter 3*XTER98). Below, we
summarize the general principles at the base of this technique; a more
detailed explanation can be found in Shaposhnikov & Titarchuk (2009) and
Gliozzi et al. (2011). For completeness, in the Appendix we report the basic
information on the reference sources, including the mathematical expression of
their spectral trends, which is necessary to derive $M_{\mathrm{BH}}$ using
the equation reported below.
The scaling method assumes that all BH systems accreting at a moderate or high
rate undergo similar spectral transitions, characterized by the “softer when
brighter” trend (i.e., the X-ray spectrum softens when the accretion and hence
the luminosity increases). These spectral transitions are routinely observed
in stellar BHs (e.g., Remillard & McClintock, 2006) and often found in samples
of AGN (e.g., Shemmer et al. 2008; Risaliti, Young, & Elvis 2009; Brightman et
al. 2013, 2016), which are characterized by considerably longer dynamical
timescales, making it nearly impossible to witness a genuine state transition
in a supermassive BH system, although a few long monitoring studies have
observed this spectral trend in individual AGN (e.g., Sobolewska & Papadakis,
2009). The “softer when brighter” trend, usually illustrated by plotting the
photon index versus the Eddington ratio $\lambda_{\mathrm{Edd}}$, is seen with
some scattering in numerous type 1 AGN samples and also in the heavily
absorbed type 2 AGN, which are the focus of our work (Brightman et al., 2016).
This lends support to the hypothesis that the photon index $\Gamma$ is a
reliable indicator of the accretion state of any BH.
Indeed, this is the fundamental assumption of the X-ray scaling method:
$\Gamma$ is indicative of the accretion state of the source, and BH systems in
the same accretion state are characterized by the same accretion rate (in
Eddington units) and the same radiative efficiency $\eta$. As a consequence,
when we compare the accretion luminosity ($L\propto\eta
M_{\mathrm{BH}}\dot{m}$) in BH systems that are in the same accretion state
(i.e., with the same $\Gamma$), we are directly comparing their
$M_{\mathrm{BH}}$. This explains why the comparison of the values of the
normalization of the BMC model, $N_{\mathrm{BMC}}$ (which is defined as the
accretion luminosity in units of $10^{39}$ erg s-1 divided by the distance
squared in units of 10 kpc), computed at the same value of $\Gamma$ between
the AGN and a known stellar BH reference source, yields the $M_{\mathrm{BH}}$.
This is illustrated in Fig. 3 and mathematically described by
$M_{\mathrm{BH,AGN}}=M_{\mathrm{BH,ref}}\times\left(\frac{N_{\mathrm{BMC,AGN}}}{N_{\mathrm{BMC,ref}}}\right)\times\left(\frac{d_{\mathrm{AGN}}^{2}}{d_{\mathrm{ref}}^{2}}\right)$
where $N_{\mathrm{BMC,ref}}$ and $d_{\mathrm{ref}}$ are the BMC model
normalization and distance of the stellar mass BH system used as a reference.
Figure 3: $N_{\mathrm{BMC}}$–$\Gamma$ plot, showing the data point
corresponding to NGC 4945 and two reference patterns; the darker trend refers
to GROD05, the spectral evolution of GRO 1655-40 during the decay of an
outburst that occurred in 2005, and the lighter color trend indicates GROR05,
the spectral evolution shown by the same source during the outburst rise. The
dashed lines indicate the uncertainties in the reference spectral trends,
whereas the error bars represent the uncertainties of the AGN spectral
parameters.
Fig. 3 illustrates the X-ray scaling method and its inherent uncertainties
that are related to the statistical errors on the spectral parameters $\Gamma$
and $N_{\mathrm{BMC}}$ and on the uncertainty of the reference source spectral
trend (shown by the dashed lines), as well as on the specific reference source
trend utilized. Although similar in shape, the reference spectral trends show
some differences (e.g., in their plateau levels and slopes), leading to
slightly different $M_{\mathrm{BH}}$ values. From Fig. 3, it is clear that
these differences exist also between the rise and decay phases of the same
reference source.
It is important to note that at very low accretion rates both stellar mass and
supermassive BHs show an anti-correlation between $\Gamma$ and
$\lambda_{\mathrm{Edd}}$ (e.g., Constantin et al. 2009; Gu & Cao 2009;
Gültekin et al. 2012). Since the X-ray scaling method is based on the positive
correlation between these two quantities, it cannot be applied to determine
the $M_{\mathrm{BH}}$ of objects in the very low-accretion regime. This was
explicitly demonstrated by the work of Jang et al. (2014), who analyzed a
sample of low-luminosity low-accreting AGN.
In the following, we systematically estimate the $M_{\mathrm{BH}}$ using all
the reference sources available (depending on the AGN’s $\Gamma$, not all
reference sources can be used since their photon index ranges vary from
reference source to reference source) and then compute the $M_{\mathrm{BH}}$
average value and its uncertainty $\sigma/\sqrt{n}$ (where $\sigma$ is the
standard deviation and $n$ is the number of reference trends utilized). As
already explained above, for AGN with steep spectra, the most reliable
estimate of $M_{\mathrm{BH}}$ is obtained using the 3*XTER98 reference trend;
therefore, we also include this value in Table 3. All $M_{\mathrm{BH}}$ values
listed in this table were computed assuming $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$
= 30; however, for completeness, we also report the $M_{\mathrm{BH}}$ obtained
assuming $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 24 and 38 for flat- and steep-
spectrum sources, respectively. We note that such changes in
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ lead to $M_{\mathrm{BH}}$ values that are
consistent with the values obtained with the original assumption
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30, within the respective
$M_{\mathrm{BH}}$ uncertainties that are of the order of 10%–40%.
The $M_{\mathrm{BH}}$ values obtained with the different reference sources and
their average are illustrated in Fig. 4. As already found in Gliozzi et al.
(2011) for the reverberation mapping AGN sample, the reference trends of
decaying outbursts yield systematically larger $M_{\mathrm{BH}}$ values
compared to those obtained from the rising trends. For each obscured AGN,
several $M_{\mathrm{BH}}$ values obtained from different reference sources and
their average appear to be broadly consistent with the value obtained from
megamaser measurements (a quantitative comparison is carried out in the next
subsection). The only noticeable exception is NGC 1194, for which the X-ray
scaling method yields values significantly lower than the maser one. This
discrepancy however is not surprising, since this source has a fairly low
accretion rate and in that regime the X-ray scaling method cannot be safely
applied.
Figure 4: $M_{\rm BH}$ values obtained with the X-ray scaling method using
different reference sources, compared with $M_{\rm BH}$ obtained from
megamaser measurements, which are represented by the black symbols at the top
of each panel. Table 3: Black hole masses with the X-ray scaling method
Source | $M_{{\textrm{BH}}_{\textrm{GROD05}}}$ | $M_{{\textrm{BH}}_{\textrm{GROR05}}}$ | $M_{{\textrm{BH}}_{\textrm{GXD03}}}$ | $M_{{\textrm{BH}}_{\textrm{GXR04}}}$ | $M_{{\textrm{BH}}_{\textrm{XTER98}}}$ | $M_{{\textrm{BH}}_{\textrm{aver}}}$ | $M_{{\textrm{BH}}_{\textrm{3*XTER98}}}$
---|---|---|---|---|---|---|---
name | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
NGC 1068 | … | $1.6_{-0.2}^{+0.3}$ | $8.5_{-1.0}^{+0.8}$ | $4.1_{-0.2}^{+0.3}$ | $1.9_{-0.5}^{+0.8}$ | $4.0_{-1.6}^{+1.6}$ | $5.6_{-1.5}^{+2.4}$
NGC 1194 | $5.6_{-0.7}^{+1.0}$ | $1.7_{-0.4}^{+1.0}$ | $9.0_{-2.3}^{+5.7}$ | $2.0_{-0.1}^{+0.2}$ | $1.2_{-0.4}^{+0.6}$ | $3.9_{-1.5}^{+1.5}$ | $3.7_{-1.1}^{+1.9}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $6.9_{-1.0}^{+1.5}$ | $2.2_{-0.6}^{+1.6}$ | $12_{-3.5}^{+13}$ | $2.4_{-0.1}^{+0.4}$ | $1.5_{-0.4}^{+0.7}$ | $5.0_{-2.0}^{+2.0}$ | $4.4_{-1.3}^{+2.2}$
NGC 2273 | $22.8_{-3.4}^{+5.2}$ | $7.3_{-1.0}^{+1.3}$ | $39.6_{-4.8}^{+4.0}$ | $19.0_{-0.8}^{+1.4}$ | $8.6_{-2.3}^{+3.8}$ | $19.5_{-5.8}^{+5.8}$ | $25.8_{\ -6.9}^{+11.3}$
NGC 3079 | $4.8_{-0.6}^{+0.6}$ | $1.3_{-0.2}^{+0.3}$ | $7.1_{-0.9}^{+0.8}$ | $3.2_{-0.1}^{+0.2}$ | $1.5_{-0.4}^{+0.7}$ | $3.6_{-1.1}^{+1.1}$ | $4.5_{-1.2}^{+2.0}$
NGC 3393 | $40.1_{-4.2}^{+4.8}$ | $10.7_{-1.6}^{+2.3}$ | $57.1_{-7.5}^{+7.3}$ | $23.3_{-1.0}^{+1.8}$ | $11.6_{-3.1}^{+5.2}$ | $28.5_{-8.9}^{+8.9}$ | $34.7_{\ -9.4}^{+15.7}$
NGC 4388 | $3.3_{-0.4}^{+0.5}$ | $1.0_{-0.2}^{+0.4}$ | $5.0_{-1.1}^{+2.0}$ | $1.3_{-0.1}^{+0.1}$ | $0.8_{-0.2}^{+0.4}$ | $2.3_{-0.8}^{+0.8}$ | $2.3_{-0.7}^{+1.1}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $4.1_{-0.5}^{+0.7}$ | $1.2_{-0.3}^{+0.5}$ | $6.3_{-1.4}^{+2.5}$ | $1.6_{-0.1}^{+0.2}$ | $1.0_{-0.3}^{+0.5}$ | $2.9_{-1.0}^{+1.0}$ | $2.9_{-0.8}^{+1.4}$
NGC 4945 | $0.5_{-0.05}^{+0.06}$ | $0.14_{-0.03}^{+0.04}$ | $0.7_{-0.1}^{+0.2}$ | $0.2_{-0.01}^{+0.01}$ | $0.13_{-0.04}^{+0.06}$ | $0.3_{-0.1}^{+0.1}$ | $0.4_{-0.1}^{+0.2}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $0.6_{-0.1}^{+0.1}$ | $0.17_{-0.03}^{+0.05}$ | $0.9_{-0.1}^{+0.2}$ | $0.3_{-0.01}^{+0.02}$ | $0.16_{-0.05}^{+0.08}$ | $0.4_{-0.1}^{+0.1}$ | $0.5_{-0.1}^{+0.2}$
IC 2560 | … | $3.2_{-0.4}^{+0.5}$ | $16.1_{-3.0}^{+2.0}$ | $10.2_{-0.5}^{+1.0}$ | $4.2_{-1.1}^{+1.7}$ | $8.4_{-3.0}^{+3.0}$ | $12.5_{-3.2}^{+5.2}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $2.8_{-0.3}^{+0.4}$ | $14.6_{-2.0}^{+1.6}$ | $8.8_{-0.6}^{+0.7}$ | $3.6_{-0.9}^{+1.5}$ | $7.4_{-2.7}^{+2.7}$ | $10.8_{-2.8}^{+4.6}$
Circinus | … | $0.4_{-0.1}^{+0.1}$ | … | … | $0.5_{-0.1}^{+0.2}$ | $0.4_{-0.1}^{+0.1}$ | $1.6_{-0.4}^{+0.6}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $0.3_{-0.1}^{+0.1}$ | … | … | $0.4_{-0.1}^{+0.2}$ | $0.6_{-0.4}^{+0.4}$ | $1.2_{-0.3}^{+0.5}$
Columns: 1 = AGN name. 2–8 = black hole masses determined with the X-ray
scaling method. Subscripts denote GROD05 = reference source GRO J1655-40 in
the decreasing phase; GROR05 = reference source GRO J1655-40 in the rising
phase; GXD03 = reference source GX 339-4 in the decreasing phase; GXR03 =
reference source GX 339-4 in the rising phase; XTER98 = reference source XTE
J1550-564 in the rising phase; 3*XTER98 = reference source XTE J1550-564 in
the rising phase with a multiplicative correction of a factor 3 applied. Note,
the average value (in column 7) is obtained averaging all the
$M_{\mathrm{BH}}$ obtained from all the reference sources but excluding
3*XTER98. Note: For each source the first line reports the $M_{\mathrm{BH}}$
values obtained using $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30 in the spectral
fitting; the second line (present only for sources with relatively flat or
steep spectra) explicitly states the different value of
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ used.
Figure 5: Plots showing the difference between the BH mass determined from
megamaser measurements and the values obtained with the X-ray scaling method
for the different reference sources, divided by the uncertainty of the
difference, $\Delta M_{\rm BH}/\sigma_{\rm diff}$. The horizontal dashed lines
enclose the region where the difference between the BH masses is within
3$\sigma$.
### 4.2 Black hole mass comparison
To compare the $M_{\mathrm{BH}}$ values obtained from the X-ray scaling method
with the maser ones in a quantitative way, we computed, using all the
available reference trends, the difference $\Delta
M_{\mathrm{BH}}=M_{\mathrm{BH,maser}}-M_{\mathrm{BH,scaling}}$ and its
uncertainty $\sigma_{\mathrm{diff}}$, obtained by adding the respective errors
in quadrature. As explained before, the error on the $M_{\mathrm{BH}}$
inferred from the scaling method includes the uncertainties on the spectral
AGN parameters and on the reference trends in the $N_{\mathrm{BMC}}$–$\Gamma$
diagram. Depending on the reference trend utilized, the percentage
uncertainties range from 10%–15% for GROD05 and GXD03 to 30%–40% for XTER98,
which is also the percentage uncertainty of the average $M_{\mathrm{BH}}$.
The error on the $M_{\mathrm{BH}}$ obtained with megamaser measurements
accounts for the uncertainties associated with the source position and with
the fitting of the Keplerian rotation curve (Kuo et al., 2011). For the
uncertainties on the $M_{\mathrm{BH}}$ determined via megamaser measurements
we used the errors quoted in the literature with the exception of NGC 4388,
for which we multiplied the quoted uncertainty by a factor of 10; this yields
a percentage error of $\sim$24%, which better reflects the actual uncertainty
on the $M_{\mathrm{BH}}$ in this source, where there is no systemic maser
detected and the five maser spots detected are not sufficient to demonstrate
that the rotation is Keplerian (Kuo et al., 2011). Note that both methods
explicitly depend on the sources’ distances and hence, in principle, their
total uncertainties should account also for the distance uncertainties (indeed
some of the sources of this maser sample are fairly close and thus their
distances cannot be obtained from the redshift and Hubble’s law). However,
since our goal is to compare the two methods, we can avoid the uncertainty
associated with the distance by assuming the exact same distance used in the
maser papers.
Figure 6: $M_{\rm BH,X}$, the BH mass obtained with the scaling method plotted
versus $M_{\rm BH,maser}$ obtained from the megamaser. The top left panel
shows the X-ray scaling values derived from the GROD05 reference, the top
right panel those from GXD03, the bottom left the values from 3*XTER98, and
the bottom right panel the $M_{\mathrm{BH}}$ values obtained from the average
of all the available reference sources. The longer-dashed line represents the
perfect one-to-one correspondence between the two methods, i.e., a ratio
$M_{\rm BH,maser}/M_{\rm BH,X}=1$, whereas the shorter-dashed lines indicate
the ratios of 3 and 1/3, respectively.
We used the criterion $\Delta M_{\mathrm{BH}}/\sigma_{\mathrm{diff}}<3$ to
assess whether the $M_{\mathrm{BH}}$ values derived with these two methods are
statistically consistent. In other words, the X-ray scaling measurements of
the $M_{\mathrm{BH}}$ are considered formally consistent with the
corresponding megamaser values if their difference is less than three times
the uncertainty $\sigma_{\mathrm{diff}}$.
The results of these comparisons are summarized in Table 4 and illustrated in
Fig. 5, where the dashed lines represent the 3$\sigma$ levels. From this
figure it is evident that every source has at least one $M_{\mathrm{BH}}$
scaling value that is consistent with the maser one, with GROD05 and GXD03
being the most reliable ones, along with the average $M_{\mathrm{BH}}$ and the
value obtained with 3*XTER98. The latter ones are always within 3$\sigma$ from
the megamaser value, also by virtue of their slightly larger uncertainties.
An alternative way to compare the two methods is offered by the ratio
$M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}}$. The ratios, obtained by
dividing the megamaser $M_{\mathrm{BH}}$ by each of the available reference
sources, as well as by the $M_{\mathrm{BH}}$ average and by 3*XTER98, are
reported in Table 5 and illustrated in Fig. 6, where the $M_{\mathrm{BH}}$
values obtained with the scaling method for the most reliable references
(GROD05, GXD03, 3*XTER98) and the average values are plotted versus their
respective megamaser values. From this figure, one can see that, for GROD05
(top left panel), 3*XTER98 (bottom left panel), and the average (bottom right
panel), all values are consistent with the ratio of 1 within a factor of 3,
and a good agreement is found also with GXD03 (top right panel) with two
sources (IC 2560 and NGC 2273) that have slightly larger values.
Based on the values reported in Table 5, all ratios obtained from these
reference trends are consistent with unity at the 3$\sigma$ limit (i.e., their
ratio $\pm 3\sigma$ is consistent with 1) confirming the statistical agreement
between the two methods. Finally, we note that using
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 24 (for flat spectrum sources) and 38
(for steep spectrum sources) confirms and reinforces the conclusions derived
from the original assumption $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30.
Table 4: $\Delta{M_{\mathrm{BH}}}/\sigma_{\mathrm{diff}}$: Comparison between
$M_{\textrm{BH}}$ from maser and X-ray scaling
Source | $\Delta{M_{\mathrm{BH}}}/\sigma$
---|---
name | GROD05 | GROR05 | GXD03 | GXR04 | XTER98 | average | 3$*$XTER98
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
NGC 1068 | … | 16.7 | $-0.5$ | 10 | 8.6 | 2.5 | 1.2
NGC 1194 | $19.0$ | 20.5 | $11.3$ | $21.0$ | $21.0$ | $18.3$ | $20.4$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $17.9$ | 19.7 | $6.0$ | $20.8$ | $20.8$ | $16.7$ | $20.0$
NGC 2273 | $-3.5$ | 0.1 | $-7.3$ | $-9.7$ | $-0.4$ | $-2.0$ | $-2.0$
NGC 3079 | $-1.2$ | 0.6 | $-2.4$ | $-0.4$ | 0.5 | $-0.6$ | $-0.9$
NGC 3393 | $-1.8$ | 7.3 | $-3.4$ | 3.2 | 4.2 | 0.3 | $-0.3$
NGC 4388 | $2.5$ | 3.7 | $1.4$ | 3.6 | 3.8 | 2.9 | 2.8
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $2.1$ | 3.6 | $0.8$ | 3.4 | 3.7 | 2.5 | 2.4
NGC 4945 | $1.3$ | 1.8 | $1.0$ | 1.7 | 1.8 | 1.5 | 1.4
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $1.1$ | 1.8 | $0.6$ | 1.6 | 1.8 | 1.4 | 1.3
IC 2560 | … | 0.5 | $-4.9$ | $-7.5$ | $-0.4$ | $-1.6$ | $-2.1$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | 1.1 | $-5.7$ | $-6.3$ | $-0.1$ | $-1.4$ | $-2.0$
Circinus | … | 4.4 | … | … | 3.4 | 4.1 | 0.3
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | 4.7 | … | … | 4.0 | 2.3 | 1.0
Columns: 1 = AGN name. 2–8 = Change in black hole mass over error for each
reference source. Reference sources: GROD05 = reference source GRO J1655-40 in
the decreasing phase; GROR05 = reference source GRO J1655-40 in the rising
phase; GXD03 = reference source GX 339-4 in the decreasing phase; GXR03 =
reference source GX 339-4 in the rising phase; XTER98 = reference source XTE
J1550-564 in the rising phase; 3*XTER98 = reference source XTE J1550-564 in
the rising phase with a multiplicative correction of a factor 3 applied. Note,
the average value (in column 7) is obtained averaging all the
$M_{\mathrm{BH}}$ obtained from all the reference sources but excluding
3*XTER98. Note: For each source the first line reports the values obtained
using $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30 in the spectral fitting; the
second line (present only for sources with relatively flat or steep spectra)
explicitly states the different value of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$
used.
Table 5: Ratio between $M_{\mathrm{BH}}$ values obtained from maser
measurements and the X-ray scaling method:
$M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}}$
Source | Ratio
---|---
name | GROD05 | GROR05 | GXD03 | GXR04 | XTER98 | average | 3$*$XTER98
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
NGC 1068 | … | $5.1\pm 0.8$ | $0.9\pm 0.1$ | $1.9\pm 0.1$ | $4.3\pm 1.5$ | $2.0\pm 0.8$ | $1.4\pm 0.5$
NGC 1194 | $12\pm 2$ | $38\pm 16$ | $7\pm 3$ | $32\pm 3$ | $52\pm 21$ | $17\pm 7$ | $17\pm 7$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $10\pm 2$ | $30\pm 15$ | $6\pm 4$ | $27\pm 3$ | $44\pm 18$ | $13\pm 6$ | $15\pm 6$
NGC 2273 | $0.3\pm 0.1$ | $1.0\pm 0.2$ | $0.20\pm 0.02$ | $0.40\pm 0.03$ | $0.9\pm 0.3$ | $0.4\pm 0.1$ | $0.3\pm 0.1$
NGC 3079 | $0.5\pm 0.4$ | $1.8\pm 1.4$ | $0.3\pm 0.3$ | $0.8\pm 0.6$ | $1.6\pm 1.3$ | $0.7\pm 0.5$ | $0.5\pm 0.4$
NGC 3393 | $0.8\pm 0.1$ | $2.9\pm 0.6$ | $0.5\pm 0.1$ | $1.3\pm 0.1$ | $2.7\pm 1.0$ | $1.1\pm 0.3$ | $0.9\pm 0.3$
NGC 4388 | $2.6\pm 0.7$ | $8.8\pm 3.6$ | $1.7\pm 0.7$ | $6.5\pm 1.6$ | $10.9\pm 5.0$ | $3.7\pm 1.6$ | $3.6\pm 1.7$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $2.1\pm 0.6$ | $7.0\pm 2.8$ | $1.3\pm 0.5$ | $5.2\pm 1.3$ | $8.7\pm 4.0$ | $3.0\pm 1.3$ | $2.9\pm 1.3$
NGC 4945 | $2.8\pm 1.5$ | $10.3\pm 5.7$ | $2.0\pm 1.1$ | $6.2\pm 3.1$ | $11\pm 6.9$ | $4.1\pm 2.5$ | $3.7\pm 2.3$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $2.3\pm 1.2$ | $8.2\pm 4.6$ | $1.6\pm 1.2$ | $4.8\pm 2.4$ | $8.7\pm 5.4$ | $3.3\pm 2.0$ | $2.9\pm 1.8$
IC 2560 | … | $1.1\pm 0.2$ | $0.2\pm 0.1$ | $0.3\pm 0.1$ | $0.8\pm 0.3$ | $0.4\pm 0.2$ | $0.3\pm 0.1$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $1.3\pm 0.2$ | $0.2\pm 0.1$ | $0.4\pm 0.1$ | $1.0\pm 0.4$ | $0.5\pm 0.2$ | $0.3\pm 0.1$
Circinus | … | $4.6\pm 0.8$ | … | … | $3.3\pm 1.2$ | $3.9\pm 0.9$ | $1.1\pm 0.4$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $5.9\pm 1.3$ | … | … | $4.3\pm 1.6$ | $2.7\pm 1.6$ | $1.4\pm 0.5$
Columns: 1 = AGN name. 2–8 = Ratio of maser to X-ray scaling for each
reference source. Reference sources: GROD05 = reference source GRO J1655-40 in
the decreasing phase; GROR05 = reference source GRO J1655-40 in the rising
phase; GXD03 = reference source GX 339-4 in the decreasing phase; GXR03 =
reference source GX 339-4 in the rising phase; XTER98 = reference source XTE
J1550-564 in the rising phase; 3*XTER98 = reference source XTE J1550-564 in
the rising phase with a multiplicative correction of a factor 3 applied. Note,
the average value (in column 7) is obtained averaging all the
$M_{\mathrm{BH}}$ obtained from all the reference sources but excluding
3*XTER98. Note: For each source the first line reports the values obtained
using $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30 in the spectral fitting; the
second line (present only for sources with relatively flat or steep spectra)
explicitly states the different value of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$
used.
## 5 Discussion and Conclusions
Constraining the $M_{\mathrm{BH}}$ in AGN is of crucial importance, since it
determines the space and temporal scales of BHs, constrains their accretion
rate via the Eddington ratio, and plays an essential role in our understanding
of the BH growth and co-evolution with the host galaxy. The most reliable ways
to determine the $M_{\mathrm{BH}}$ are direct dynamical methods, which measure
the orbital parameters of “test particles”, whose motion is dominated by the
gravitational force of the supermassive BH. For example, the mass of the
supermassive BH at the center of our Galaxy has been tightly constrained by
detailed studies of the orbits of a few innermost stars observed over several
years (e.g., Ghez et al. 2008; Gillessen et al. 2009). In nearby weakly active
galaxies, the $M_{\mathrm{BH}}$ is determined by the gas dynamics within the
sphere of influence of the BH (e.g., Gebhardt et al. 2003). On the other hand,
in bright type 1 AGN, the $M_{\mathrm{BH}}$ measurement is obtained from the
dynamics of the BLR via the reverberation mapping technique (e.g., Peterson et
al. 2004). Finally, in heavily absorbed type 2 AGN, where the BLR is
completely obscured, the only possible direct measurement of the
$M_{\mathrm{BH}}$ is based on megamaser measurements (e.g., Kuo et al. 2011
and references therein).
The main problem with direct dynamical methods is that they are fairly limited
in their application. For instance, direct measurements of $M_{\mathrm{BH}}$
via gas dynamics are limited to nearby weakly active galaxies, where the
sphere of influence is not outshined by the AGN and are sufficiently close to
be resolvable at the angular resolution of ground-based observatories.
Similarly, the reverberation mapping technique, which is heavily time and
instrument consuming, is limited to type 1 AGN with small or moderate masses.
Finally, the megamaser emission in type 2 AGN is relatively rare, and only
when the megamaser originates in the accretion disk (as opposed to the jet and
outflows) can this technique be used to constrain the $M_{\mathrm{BH}}$ (e.g.,
Panessa et al. 2020 and references therein).
Fortunately, there are a few robust indirect methods that make it possible to
constrain the $M_{\mathrm{BH}}$ beyond the range of applicability of the
direct dynamical ones. For example, the tight correlation between
$M_{\mathrm{BH}}$ and the stellar velocity dispersion in the bulge
$\sigma_{\ast}$, observed in nearby nearly quiescent galaxies (e.g., Tremaine
et al. 2002), can be extrapolated to constrain the $M_{\mathrm{BH}}$ in many
distant and more active galaxies. Similarly, the empirical relationship
between the BLR radius and optical luminosity makes it possible to determine
the mass of numerous type 1 AGN with only one spectral measurement without the
need of long monitoring campaigns (e.g., Kaspi et al. 2000).
Although indirect methods have proven to be very useful to derive general
results for large samples of AGN, caution must be applied when these methods
are extrapolated well beyond the original range of applicability of the direct
methods. To check for consistency and avoid potential biases associated with
the various assumptions inherent in these indirect methods, it is important to
develop and utilize alternative techniques to constrain the $M_{\mathrm{BH}}$.
In this perspective, X-ray-based methods may offer a useful complementary way
to the more commonly used optically based ones, since X-rays that are produced
very close to the BH are less affected by absorption and by star and galaxy
contamination. Indeed, model-independent methods based on X-ray variability
yielded $M_{\mathrm{BH}}$ values broadly consistent with those obtained with
dynamical methods (e.g. Papadakis 2004; Nikołajuk et al. 2006; McHardy et al.
2006; Ponti et al. 2012). In a previous work focused on a sample of AGN with
reverberation mapping measurements and good quality XMM-Newton data, we
demonstrated that the X-ray scaling method also provides results in agreement
with reverberation mapping within the respective uncertainties (Gliozzi et
al., 2011).
It is important to bear in mind that the X-ray scaling method is not
equivalent to making some general assumptions on the accretion rate and the
bolometric correction and deriving the BH mass from the X-ray luminosity using
the formula $M_{\mathrm{BH}}=\kappa_{\mathrm{bol}}L_{\mathrm{X}}/(1.3\times
10^{38}\lambda_{\mathrm{Edd}})$, where $\kappa_{\mathrm{bol}}$ is the
bolometric correction that may range from 15 to 150 depending on the accretion
rate of the source (Vasudevan & Fabian, 2009), and $L_{\mathrm{X}}$ the X-ray
luminosity in erg/s. With this simple equation, without an a priori knowledge
of the accretion rate of the source, one could at best obtain the order of
magnitude of the $M_{\mathrm{BH}}$. Since $\lambda_{\mathrm{Edd}}$ can vary
over a broad range (for example, for this small sample of obscured AGN, the
Eddington ratio varies from 0.01 to 0.3), it is not possible to obtain a
specific value of $M_{\mathrm{BH}}$ that can be quantitatively compared with
the value obtained from the dynamical method and find a good agreement, as we
did with the scaling method.
One may then argue that the only important parameter in the scaling method is
$N_{\mathrm{BMC}}$ (because of its direct dependence on the accretion
luminosity and distance) and that it is still possible to obtain a good
agreement with the dynamically estimated $M_{\mathrm{BH}}$ with any value of
the photon index. To test this hypothesis, we have selected the two sources
with the flattest spectra of our sample (NGC 4388 and NGC 4945) and the two
sources with the steepest spectra (IC 2560 and Circinus), and recalculated
their $M_{\mathrm{BH}}$ with the scaling method assuming $\Gamma=2.17$ for the
flattest sources and $\Gamma=1.66$ for the steepest sources. This led to
changes of $M_{\mathrm{BH}}$ by a factor slightly larger than 2 (note that
considerably larger changes of $M_{\mathrm{BH}}$ would have resulted if we had
used a larger difference in the photon indices instead of the minimum and
maximum values of this small sample). If the photon index did not play any
role, then these $M_{\mathrm{BH}}$ changes should have not made a difference
in the agreement with the values obtained via the dynamical method, with some
objects showing a slightly better agreement and others a slightly worse
agreement. Instead, all four sources, which were originally consistent with
their maser respective estimates based on the mass ratio criterion described
above (see Table 5 and Figure 6), showed a clear departure from the dynamical
$M_{\mathrm{BH}}$ values with three sources (NGC 4388, NGC 4945, and IC 2560)
that were not formally consistent with the maser values anymore (their new
mass ratios were 8.0, 7.1, and 6.7, respectively) and only Circinus (ratio of
0.5) still consistent, but only by virtue of the fact that the original ratio
was basically 1. We therefore conclude that the scaling method works because
the photon index accurately characterizes the accretion state of accreting
black holes and allows the correct selection of the reference source’s
$N_{\mathrm{BMC}}$ value to be compared with the AGN’s value.
In this study, we have extended the X-ray scaling method to a sample of
heavily obscured type 2 AGN with $M_{\mathrm{BH}}$ already constrained by
megamaser measurements. This dynamical method is rightly considered one of the
most reliable; however, the accuracy of the $M_{\mathrm{BH}}$ derived with
this technique depends on the quality of the radio data, on the assumption
that the megamaser emission is produced in an edge-on disk, and that its
rotation curve is strictly Keplerian. Additionally, one should bear in mind
that this technique measures the mass enclosed within the megamaser emission.
As a consequence, the actual $M_{\mathrm{BH}}$ may be slightly smaller if the
measured enclosed mass encompasses a nuclear cluster or the inner part of a
massive disk, or alternatively slightly larger if radiation pressure (not
included in the $M_{\mathrm{BH}}$ derivation) plays an important role (Kuo et
al., 2011).
Specifically, for the sources of our sample, the rotation curve traced by the
megamaser in NGC 1068 is non-Keplerian; the $M_{\mathrm{BH}}$ was derived
assuming a self-gravitating accretion disk model (Lodato & Bertin, 2003). NGC
1194 displays one of the largest maser disks (with inner and outer radii of
0.54 and 1.33 parsecs) which appears to be slightly bent and is consistent
with Keplerian rotation (Kuo et al., 2011). NGC 2273 also shows indications of
a warped but much smaller disk (with inner and outer disk radii of 0.028 and
0.084 pc) with Keplerian rotation (Kuo et al., 2011). In NGC 3079 the disk
appears to be thick and flared (Kondratko, Greenhill, & Moran, 2005), whereas
in NGC 3393 the maser seems to describe a flat disk perpendicular to the kpc
radio jet, and the positions of the maser points have substantial
uncertainties (Kondratko, Greenhill, & Moran, 2008). NGC 4388, located in the
Virgo cluster, has only five megamaser spots, which make it impossible to
demonstrate that they lie on a disk or that the rotation is Keplerian (Kuo et
al., 2011). For this reason, to reflect the actual uncertainty on the
$M_{\mathrm{BH}}$ derived by megamaser measurements, we have increased the
statistical error by a factor of 10, leading to an uncertainty of $\sim$24%.
The megamaser in NGC 4945 has been modeled as an edge-on thin disk, although
this is not the only possible interpretation of the data; the non-Keplerian
rotation of the blue-shifted emission and the substantial position errors lead
to a relatively large uncertainty in the $M_{\mathrm{BH}}$ of $\sim$50%
(Greenhill et al., 1997). In IC 2560 the megamaser emission has been
attributed to an edge-on thin disk with Keplerian rotation with some
additional contribution from a jet (Yamauchi et al., 2012). Finally, the
megamaser emission in Circinus appears to be associated with a warped
accretion disk and a wide-angle outflow (Greenhill et al., 2003). In summary,
because of the presence of outflows, jets, disk warps, or non-Keplerian
rotation curves, we should consider the $M_{\mathrm{BH}}$ values determined
from megamaser measurements as robust estimates but not as extremely accurate
values, and the errors reported in Table 1 are likely lower limits on their
actual uncertainties.
With respect to type 1 AGN, the main difficulty of applying the X-ray scaling
method to heavily obscured AGN is the need to properly constrain the
parameters of the primary emission in sources whose X-ray spectra are
dominated by absorption and reflection. However, the NuSTAR spectra of these
specific sources, often complemented with Chandra and XMM-Newton data, were
the object of very detailed analyses, which led to the disentanglement and a
careful characterization of the different contributions of the AGN direct and
reprocessed emission, of the host galaxy, and of the off-nuclear sources
located in the spectral extraction region (e.g., Yaqoob 2012; Puccetti et al.
2014; Arévalo et al. 2014; Bauer et al. 2015). Guided by these findings, we
were able to parametrize the torus contribution using the physically motivated
self-consistent model Borus (Baloković et al., 2018) instead of the MYTorus or
Torus models used in the previous analyses. To characterize the primary
emission, instead of the phenomenological power-law model, we utilized the BMC
Comptonization model, since the scaling method directly scales the
normalization of this model $N_{\mathrm{BMC}}$ between AGN and an appropriate
stellar reference to determine $M_{\mathrm{BH}}$.
With our baseline spectral model, where we assumed
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30, as described in detail in Section 3
(see also the Appendix for details on the spectral fittings of individual
sources), and applying the scaling technique summarized in Section 4.1, we
obtained the following results:
* •
Many of the $M_{\mathrm{BH}}$ values, obtained with different reference
trends, are broadly in agreement with the corresponding megamaser ones. In
particular, the estimates derived using GROD05, 3*XTER98, and the ones
obtained by averaging the values inferred from all the available reference
sources, are consistent at the 3$\sigma$ level, based on measurements of
$\Delta
M_{\mathrm{BH}}/\sigma_{\mathrm{diff}}=(M_{\mathrm{BH,maser}}-M_{\mathrm{BH,scaling}})/\sigma_{\mathrm{diff}}$,
which are reported in Table 4 and shown in Fig. 5.
* •
The agreement between the two methods is confirmed by the
$M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}}$ ratio: for all type 2 AGN of
our sample $(M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}})\pm 3\sigma\leq 1$,
when using the best reference sources or the average $M_{\mathrm{BH}}$, as
summarized in Table 5. Fig. 6 illustrates the good agreement between the two
methods, showing that GROD05, GXD04 (partially), 3*XTER98, and the average
obtained from all reference patterns are all consistent with the one-to-one
ratio within a factor of three.
* •
The only object of our sample for which the $M_{\mathrm{BH}}$ inferred from
the X-ray scaling method is statistically inconsistent with the megamaser
value is NGC 1194, which is the AGN with the lowest accretion rate
($\lambda_{\mathrm{{Edd}}}\simeq 7\times 10^{-3}$). However, this discrepancy
is expected, since the X-ray scaling method cannot be applied in this regime,
where $\Gamma$ generally shows an anti-correlation with
$\lambda_{\mathrm{Edd}}$.
In conclusion, our work demonstrates that the same X-ray scaling method works
equally well for type 1 AGN (given the formal agreement with the reverberation
mapping sample) and type 2 AGN (based on the agreement with the megamaser
sample). We thus conclude that this method can be safely applied to any type
of AGN regardless of their level of obscuration, provided that these sources
accrete above a minimum threshold and that their primary X-ray emission can be
robustly characterized via spectral analysis. This also proves that this
method is robust and can be used to complement the various indirect methods,
especially when they are applied well beyond the range of validity of the
direct methods, from which they were calibrated. Finally, the X-ray scaling
method offers the possibility to investigate in a systematic and homogeneous
way the existence of any intrinsic difference in the fundamental properties of
the central engines in type 1 and type 2 AGN. We plan to carry out this type
of investigation in our future work.
## Acknowledgements
We thank the anonymous referee for constructive comments and suggestions that
improved the clarity of the paper and helped strengthen our conclusions. This
research has made use of data, software, and/or web tools obtained from the
High Energy Astrophysics Science Archive Research Center (HEASARC), a service
of the Astrophysics Science Division at NASA/GSFC and of the Smithsonian
Astrophysical Observatory’s High Energy Astrophysics Division, and of the
NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science
Data Center (ASDC, Italy) and the California Institute of Technology (Caltech,
USA).
## Data Availability
The data underlying this article are available in the High Energy Astrophysics
Science Archive Research Center (HEASARC) Archive at
https://heasarc.gsfc.nasa.gov/docs/archive.html.
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## Appendix A Additional Spectral Results
NGC 1068: A detailed analysis of the NuSTAR, XMM-Newton, and Chandra spectra
of this source was carried out by Bauer et al. (2015). Thanks to the excellent
sensitivities of XMM-Newton and NuSTAR over broad complementary energy ranges,
and to the sub-arcsecond spatial resolution of Chandra, the authors were able
to disentangle the contributions of the host galaxy and off-nuclear sources
from the AGN emission within the NuSTAR extraction region. The overall best-
fit model is fairly complex and comprises several Fe and Ni emission lines, a
Bremsstrahlung component to account for the radiative recombination continuum
and lines, a cutoff power-law model to account for the off-nuclear X-ray
sources, in addition to the AGN-related emission, which is parametrized by two
different MYTorus scattered and line components, in addition to the
transmitted one described by the zeroth-order component of that model. In our
fitting, in addition to our baseline model we added the Bremsstrahlung and
cutoff power law with all parameters fixed at the values provided by Bauer et
al. (2015), and a Gaussian line to roughly model the excess around 6.5 keV. To
account for the multiple absorption components, we also added a second Borus
model, whose best-fit parameters are
$\log(N_{{\textrm{H}}_{\textrm{bor}}})=24.9\pm 0.1$, CFtor = 83%, and
$A_{\textrm{Fe}}=1$. Our best-fit parameters are broadly consistent with the
results presented by Bauer et al. (2015). The observed flux in the 2–10 keV
energy band is $5.4\times 10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the
intrinsic one (i.e., corrected for absorption) $1.3\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
NGC 1194: The starting model for the spectral fit of this source is provided
by the work of Masini et al. (2016), who fitted the NuSTAR spectrum with the
MYTorus model in the decoupled mode, with the addition of a Gaussian line at
6.8 keV, and a scattering fraction of the primary continuum of
$f_{\mathrm{s}}\sim 3\%$. In our fitting, we used our baseline model and found
the main parameters ($\Gamma$, $N_{\mathrm{H}}$, and $f_{\mathrm{s}}$) to be
fully consistent with their best-fit results. The 2–10 keV observed flux is
$1.2\times 10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one
$1.0\times 10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
NGC 2273: The starting spectral model for this source is again provided by the
work of Masini et al. (2016), who fitted the NuSTAR spectrum with the Torus
model that favored a heavily absorbed scenario with $N_{\mathrm{H}}>7\times
10^{24}\,{\mathrm{cm^{-2}}}$. In our fitting, we used our baseline model,
which yielded a best fit broadly consistent with their results. The 2–10 keV
observed flux is $9.2\times 10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the
intrinsic one $3.6\times 10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
NGC 3079: The starting spectral model for this source is again provided by the
work of Masini et al. (2016), who fitted the NuSTAR spectrum with the MYTorus
model in a coupled mode. The results obtained with our baseline model are
consistent within the respective uncertainties with their results. The 2–10
keV observed flux is $6.4\times 10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and
the intrinsic one $1.2\times 10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
NGC 3393: The starting spectral model for this source is provided by the work
of Koss et al. (2015) and Masini et al. (2016), who fitted the NuSTAR spectrum
with both MYTorus and Torus models. The results obtained with our baseline
model are broadly consistent with the results presented by these authors with
a slightly larger value of $N_{\mathrm{H}}$ ($10^{25}$ vs. $2.2\times
10^{24}\,{\mathrm{cm^{-2}}}$). The 2–10 keV observed flux is $4.4\times
10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $8.3\times
10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
NGC 4388: The starting spectral model for this source is once more provided by
the work of Masini et al. (2016), who fitted the NuSTAR spectrum with the
MYTorus and Torus models, which favor a Compton-thin scenario with a
substantial scattered primary emission that dominates below 5 keV. The results
from our baseline model are fully consistent with their results. The 2–10 keV
observed flux is $7.9\times 10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the
intrinsic one $1.4\times 10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
NGC 4945: A detailed analysis of the NuSTAR, Suzaku, and Chandra spectra of
this source was carried out by Puccetti et al. (2014), who in turn, were
guided by the results obtained by Yaqoob (2012) based on a comprehensive
analysis of all the hard X-ray spectra available at that time. The wealth of
high-quality broad-band spectra obtained with several observatories made it
possible to parametrize separately the different contributions of the host
galaxy, the AGN, and contaminating sources within the NuSTAR extraction
region. The best-fit model is fairly complex and comprises several emission
lines, the galaxy optically thin thermal continuum, which is described by the
APEC model, the contamination from off-nuclear sources parametrized by a power
law, and the AGN emission seen through a torus described by the MYTorus model
in the decoupled mode. In our fitting procedure, in addition to our baseline
model we included the APEC and power-law models with all parameters fixed at
the values provided by Yaqoob. Our results are broadly consistent with those
obtained by both Yaqoob and Puccetti. The 2–10 keV observed flux is $3.7\times
10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $2.7\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
IC 2560: The starting spectral model for this source is again provided by the
work of Masini et al. (2016), who fitted the NuSTAR spectrum with the Torus
model, which favors a heavily absorbed primary emission characterized by a
steep photon index. The results from our baseline model are broadly consistent
with their results. The 2–10 keV observed flux is $3.7\times
10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $1.7\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
Circinus: A detailed analysis of the NuSTAR, XMM-Newton, and Chandra spectra
of this source was carried out by Arévalo et al. (2014). Combining the
complementary properties of these observatories (i.e., the high sensitivities
of XMM-Newton and NuSTAR over broad energy ranges and the sub-arcsecond
spatial resolution of Chandra), the authors were able to disentangle the
contributions of different contamination sources (diffuse emission from the
host galaxy, supernova remnant contribution, and off-nuclear X-ray binary
sources) from the AGN emission within the NuSTAR extraction region. The
overall best-fit model is complex and comprises several emission lines, an
APEC model for the diffuse emission, three Mekal models to parametrize the
supernova remnant, and a power-law model to account for the off-nuclear point-
like sources, in addition to two different MYTorus models used in the
decoupled mode. In our fitting, in addition to our baseline model we added all
the contamination models with all the parameters fixed at the values provided
by Arévalo et al. (2014) and three Gaussian lines to roughly model the line
excess in the 5.5–7.5 keV range. To account for the multiple absorption
components, we also added a second Borus model, whose best-fit parameters are
$\log(N_{{\textrm{H}}_{\textrm{bor}}})=24.6\pm 0.1$, CFtor = 10%, and
$A_{\textrm{Fe}}=1$. Our best-fit parameters are broadly consistent with their
results. The 2–10 keV observed flux is $2.0\times
10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $2.1\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.
## Appendix B The X-ray scaling method
The X-ray scaling method for determining the mass of a black hole
($M_{\mathrm{BH}}$) was first described by Shaposhnikov & Titarchuk (2009) and
first applied to AGN by Gliozzi et al. (2011), where the method is described
in detail. Here, we only report the essential information on the stellar
reference sources – their $M_{\mathrm{BH}}$ values and distances (Table 6) and
the mathematical expression of the spectral trend with the best fit parameters
for the different sources (Table 7) – that is needed to reproduce the
$M_{\mathrm{BH}}$ values. The two steps below accomplish the scaling described
in Section 4.1.
Step 1. Use the following equation to solve for $N_{\textrm{BMC,r}}$, the BMC
normalization the reference source would have at the same photon index as the
target AGN. The reference source is a Galactic, stellar-mass black hole with
known mass and distance.
$N_{\textrm{BMC,r}}(\Gamma)=N_{\textrm{tr}}\times\left\\{1-\ln\left[\exp\left(\frac{A-\Gamma}{B}\right)-1\right]\right\\}^{(1/\beta)}$
(1)
where $\Gamma$ is the photon index of the target AGN as determined by the
spectral fit, and $A$, $B$, $N_{\mathrm{tr}}$, and $\beta$ are given in Table
7. Note: this equation was first presented by Jang et al. (2018) with an
error: there should be a minus sign before the logarithm.
Step 2. Use the equation presented in Section 4.1 to solve for
$M_{\textrm{BH,t}}$.
$M_{\textrm{BH,t}}=M_{\textrm{BH,r}}\times\left(\frac{N_{\textrm{BMC,t}}}{N_{\textrm{BMC,r}}}\right)\times\left(\frac{d_{\textrm{t}}}{d_{\textrm{r}}}\right)^{2}$
(2)
where $M_{\mathrm{BH}}$ is the black hole mass, $N_{\mathrm{BMC}}$ is the BMC
normalization, and $d$ is the distance. The $t$ subscript denotes the target
AGN and the $r$ subscript denotes the reference source.
Table 6: Characteristics of reference sources
Name | $M_{\mathrm{BH}}$ | $d$
---|---|---
| (M☉) | (kpc)
GRO J1655-40 | $6.3\pm 0.3$ | $3.2\pm 0.2$
GX 339-4 | $12.3\pm 1.4$ | $5.7\pm 0.8$
XTE J1550-564 | $10.7\pm 1.5$ | $3.3\pm 0.5$
Source: Gliozzi et al. (2011)
Table 7: Parametrization of $\Gamma$–$N_{\mathrm{BMC}}$ reference patterns
Transition | $A$ | $B$ | $N_{\mathrm{tr}}$ | $\beta$
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
GRO J1655-40 D05 | $1.96\pm 0.02$ | $0.42\pm 0.02$ | $0.023\pm 0.001$ | $1.8\pm 0.2$
GRO J1655-40 R05 | $2.35\pm 0.04$ | $0.74\pm 0.04$ | $0.131\pm 0.001$ | $1.0\pm 0.1$
GX 339-4 D03 | $2.13\pm 0.03$ | $0.50\pm 0.04$ | $0.0130\pm 0.0002$ | $1.5\pm 0.3$
GX 339-4 R04 | $2.10\pm 0.03$ | $0.46\pm 0.01$ | $0.037\pm 0.001$ | $8.0\pm 1.5$
XTE J1550-564 R98 | $2.96\pm 0.02$ | $2.8\pm 0.2$ | $0.055\pm 0.010$ | $0.4\pm 0.1$
Columns: 1 = reference source spectral transition. 2 = parameter that
determines the rigid translation of the spectral pattern along the y-axis. 3 =
parameter characterizing the lower saturation level of the pattern. 4 =
parameter that determines the rigid translation of the spectral pattern along
the x-axis. 5 = slope of the spectral pattern. Source: Gliozzi et al. (2011)
|
# On the cohomology of $\operatorname{\mathsf{GL}}(N)$ and adjoint Selmer
groups
J. Tilouine and E. Urban
(Date: August 3, 2021)
###### Abstract.
We prove -under certain conditions (local-global compatibility and vanishing
of integral cohomology), a generalization of a theorem of Galatius and
Venkatesh. We consider the case of $\operatorname{\mathsf{GL}}(N)$ over a CM
field and we relate the localization of penultimate non vanishing cuspidal
cohomology group for a locally symmetric space to the Selmer group of the Tate
dual of the adjoint representation. More precisely we construct a Hecke-
equivariant injection from the divisible group associated to the first
fundamental group of a derived deformation ring to the Selmer group of the
twisted dual adjoint motive with divisible coefficients and we identify its
cokernel as its first Tate-Shafarevich group. Actually, we also construct
similar maps for higher homotopy groups with values in exterior powers of
Selmer groups, although with less precise control on their kernel and
cokernel. We generalize this to Hida families as well.
The first author is partially supported by the grants PerCoLaTor ANR-14-CE25,
Coloss AAPG2019, and NSF grant DMS 1464106. The second author is partially
funded by the grant DMS 1407239 from the National Science Foundation.
## 1\. Introduction
Let $G$ be a connected reductive group over ${\mathbb{Q}}$. Let $Z$ be its
center and $A\subset Z$ be the maximal split torus of $Z$. For any
${\mathbb{Q}}$-group $H$, we decompose its group of adelic points as
$H(\operatorname{A})=H_{f}\times H_{\infty}$ into its finite and archimedean
parts. Let $K_{\infty}$ be a maximal connected compact subgroup of
$G_{\infty}$. Let $X_{G}=G_{\infty}/C_{\infty}$ where
$C_{\infty}=A_{\infty}K_{\infty}$ be the riemannian space associated to $G$.
Let $d=\dim\,X_{G}=\dim\,G_{\infty}-\dim\,C_{\infty}$; let
$\ell_{0}=\dim\,T_{G_{\infty}}-\dim\,T_{C_{\infty}}$ where $T_{G_{\infty}}$,
resp. $T_{C_{\infty}}$ denote a maximal torus of $G_{\infty}$ resp.
$C_{\infty}$. If $G$ admits a Shimura variety, $\ell_{0}=0$ and $d$ is even.
Let $q_{0}=(d-\ell_{0})/2$. For instance, for $F$ a number field and
$G=\operatorname{Res}_{F/{\mathbb{Q}}}\operatorname{\mathsf{GL}}(N)$ with
$[F\colon{\mathbb{Q}}]=r_{1}+2r_{2}$, we have
$\ell_{0}=(N-[{N\over 2}])r_{1}+Nr_{2}-1,\quad q_{0}=[{N^{2}\over
4}]r_{1}+{N(N-1)\over 2}r_{2},\quad d=2q_{0}+\ell_{0}={N(N+1)\over
2}r_{1}+N^{2}r_{2}-1$
Let $U$ be a compact open subgroup of $G_{f}$ and
$Y_{G}(U)=G({\mathbb{Q}})\backslash(X_{G}\times G_{f}/U)$
be the locally symmetric space of level $U$ associated to $G$. Let $\lambda$
be a dominant weight for $G$ and $V_{\lambda}$ the irreducible algebraic
representation of $G$ of highest weight $\lambda$. Let
$\mathrm{H}^{\bullet}(Y_{G}(U),V_{\lambda}({\mathbb{C}}))$ be the graded space
of cohomology of the corresponding local system over $Y_{G}(U)$. Let
$H^{\bullet}_{temp}(Y_{G}(U),V_{\lambda}({\mathbb{C}}))$ be its tempered part
(corresponding to tempered automorphic algebraic representations). Let
$\mathrm{H}^{temp}_{\bullet}=\mathrm{H}^{d-\bullet}_{temp}(Y_{G}(U),V_{\lambda}({\mathbb{C}}))$.
Let $h_{\mathbb{Z}}$ be the spherical ${\mathbb{Z}}$-Hecke algebra outside
$Ram(U)$ acting faithfully on $\mathrm{H}^{temp}_{\bullet}$.
Recall that Prasanna-Venkatesh [PV16, Sect.3] defined a rank $\ell_{0}$
abelian Lie subalgebra ${\mathfrak{a}}_{G}\subset Lie(G_{\mathbb{C}})$ and an
action of the graded ring $\bigwedge^{\bullet}{\mathfrak{a}}_{G}$ on
$\mathrm{H}^{temp}_{\bullet}$; they conjectured that there is a Hecke
equivariant isomorphism
$\mathrm{H}^{temp}_{\bullet}\cong\mathrm{H}_{q_{0}}^{temp}\otimes_{{\mathbb{C}}}\bigwedge^{\bullet}{\mathfrak{a}}_{G}$
or
$\mathrm{H}^{temp}_{\bullet}\cong\mathrm{H}_{q_{0}}^{temp}\otimes_{h_{\mathbb{Z}}\otimes{\mathbb{C}}}\bigwedge_{h_{\mathbb{Z}}\otimes{\mathbb{C}}}^{\bullet}h_{\mathbb{Z}}\otimes{\mathfrak{a}}_{G}$
Let $p$ be a prime. We fix embeddings
$\overline{{\mathbb{Q}}}\hookrightarrow\overline{{\mathbb{Q}}}_{p}$ and
$\overline{{\mathbb{Q}}}\hookrightarrow{\mathbb{C}}$. Let $K$ be a
sufficiently big $p$-adic field, $\mathcal{O}$ its valuation ring, $\varpi$ a
uniformizing parameter, and $k=\mathcal{O}/(\varpi)$ its residue field.
Let $\mathrm{H}^{\bullet}(A)=\mathrm{H}^{\bullet}(Y_{G}(U),V_{\lambda}(A))$
for $A=\mathcal{O}$ or $k=\mathcal{O}/(\varpi)$. Note that
$\mathrm{H}^{\bullet}(\mathcal{O})$ may have torsion. Let $h$ be the
$\mathcal{O}$-Hecke algebra outside $Ram(U)$ acting faithfully on this module.
Let $S_{p}$ be the set of places of $F$ above $p$. Let $\pi$ be a cuspidal
representation occuring in $H^{q_{0}}(\mathcal{O})$, let $\lambda_{\pi}\colon
h\to\mathcal{O}$ be the corresponding Hecke eigensystem and
$\overline{\lambda}\colon h\to k$ its reduction modulo $\varpi$. We put
${\mathfrak{m}}=\operatorname{Ker}(\overline{\lambda})$ the corresponding
maximal ideal.
In what follos, we will assume that $\pi$ has an associated Galois
representation
$\rho_{\pi}\colon\operatorname{Gal}(\overline{F}/F)\to{}^{L}G(\mathcal{O})$
(unramified outside $Ram(U)\cup S_{p}$) satisfying:
$(RLI)$ The residual representation $\rho_{\pi}$ has residual large image.
Recall that this means that its image contains $\phi(H(k^{\prime}))$ where
$k^{\prime}\subset k$ is a subfield of $k$ and $\phi\colon
H\to\widehat{G}^{\prime}$ is the universal covering of the derived subgroup
$\widehat{G}^{\prime}$ of $\widehat{G}$). When this condition is satisfied, we
also say that ${\mathfrak{m}}$ is strongly non Eisenstein.
Let $H^{\bullet}_{\mathfrak{m}}$ (resp. $H^{\bullet}_{\mathfrak{m}}(k)$) be
the localization of $\mathrm{H}^{\bullet}(\mathcal{O})$ (resp.
$\mathrm{H}^{\bullet}(k)$) at ${\mathfrak{m}}$. Recall that Caraiani and
Scholze [CaSch15] have proven that if $Y_{G}$ is a unitary Shimura variety,
then $\mathrm{H}^{i}_{\mathfrak{m}}=0$ for $i\neq q_{0}={d\over 2}$ which
implies that $\mathrm{H}^{q_{0}}_{\mathfrak{m}}$ is torsion-free. In general,
we put $q_{m}=q_{0}$ and $q_{s}=q_{0}+\ell_{0}$ and consider the assumption
$(Van_{\mathfrak{m}})\quad\mathrm{H}^{i}_{\mathfrak{m}}(k)=0\ \mbox{\rm
for}\,i\notin[q_{m},q_{s}]$
When $(RLI)$ holds, it is generally believed that that $(Van_{\mathfrak{m}})$
is satisfied, but it seems very hard to prove in general. Nevertheless, it is
known if $q_{0}=1$. If $q_{0}\geq 2$, $\mathrm{H}^{1}_{\mathfrak{m}}=0$ can be
proven for many groups [Cai21, Section 1.2.4]. In particular, if $q_{0}\leq
2$, $(Van_{\mathfrak{m}})$ holds for many groups $G$. This is the case for:
* 1)
$\operatorname{\mathsf{GL}}(2)_{/F}$ for $F$ imaginary quadratic (called in
the sequel the Bianchi case, for which $q_{0}=\ell_{0}=1,d=3$) or $F$ CM of
degree $4$ (called in the sequel the higher Bianchi case, where $q_{0}=2$,
$\ell_{0}=3$, $d=7$),
* 2)
$\operatorname{\mathsf{GL}}(3)_{/{\mathbb{Q}}}$, ($q_{m}=2$,
$\ell_{0}=1,d=5$),
* 3)
$O(p,q)$, $p+q\leq 5$ (see [Cai21, Section 7])
From now on, we furthermore assume that
$G=\operatorname{Res}_{F/{\mathbb{Q}}}\operatorname{\mathsf{GL}}_{N}$ and $F$
is CM with $[F\colon{\mathbb{Q}}]=2d_{0}$. Therefore, $\ell_{0}=Nd_{0}-1$ and
$q_{0}={N(N-1)\over 2}d_{0}$. Let $\operatorname{T}=h_{\mathfrak{m}}$ be the
localized Hecke algebra acting faithfully on
$\mathrm{H}^{\bullet}_{\mathfrak{m}}$. These modules are finite $\mathcal{O}$
but may contain torsion. We assume
$(\operatorname{Gal}_{\mathfrak{m}})$ There exists a continuous Galois
representation
$\rho_{\mathfrak{m}}\colon\operatorname{Gal}(\overline{F}/F)\to\operatorname{\mathsf{GL}}_{N}(\operatorname{T})$
unramified outside $Ram(U)\cup S_{p}$ and such that for any $v\notin
Ram(U)\cup S_{p}$,
$\operatorname{char}(\rho_{\mathfrak{m}}(\operatorname{Frob}_{v}))=Hecke_{v}(X)$
where the Hecke polynomial at $v$ is given by
$Hecke_{v}(X)=X^{N}-T_{v,1}X^{N-1}+\ldots+(-1)^{i}T_{v,i}\mathbb{N}v^{{i(i-1)\over
2}}X^{N-i}+\ldots+(-1)^{N}T_{v,N}\mathbb{N}v^{{N(N-1)\over 2}}.$
This conjecture is proven in [Sch15] if one replaces $\operatorname{T}$ by
$\operatorname{T}/I$ where $I$ is a nilpotent ideal of exponent bounded in
terms of $N$ and $[F\colon{\mathbb{Q}}]$. The exponent of $I$ is bounded by
$4$ in [NT16]. By [CGH+19], the ideal can be taken to be $0$ if $p$ splits
totally in $F$.
Let ${\mathfrak{n}}\subset\mathcal{O}_{F}$ be a squarefree ideal coprime to
$p$. We denote by $S$ the set of places of $F$ dividing ${\mathfrak{n}}$. Let
$U_{0}({\mathfrak{n}})\subset GL(N,\widehat{\mathcal{O}}_{F})$ be the Iwahori
subgroup of level ${\mathfrak{n}}$, and
$Y=Y_{0}({\mathfrak{n}})=\operatorname{\mathsf{GL}}_{N}(F)\backslash
GL_{N}(\operatorname{A}_{F})/U_{0}({\mathfrak{n}})C_{\infty}$
be the Shimura space of level $\Gamma_{0}({\mathfrak{n}})$ (here
$C_{\infty}={\mathbb{R}}^{\times}U_{N}(F^{+}_{\infty})$). We assume
$(MIN)$ $\rho_{\pi}$ is ${\mathfrak{n}}$-minimal.
Recall that this means that for any place $v|{\mathfrak{n}}$, the image
$\overline{\rho}_{\pi}(I_{v})$ of the inertia subgroup $I_{v}$ contains a
regular unipotent element. More precisely, let $J_{N}$ be the standard Jordan
block of size $N$ and $t_{v}\colon I_{v}\to{\mathbb{Z}}_{p}(1)$ be the
$p$-adic tame homomorphism of $F_{v}$ given by
$\tau(p_{v}^{1/p^{n}})=\zeta_{p^{n}}^{t_{v}(\tau)}\cdot p_{v}^{1/p^{n}}$ for
all $n$’s. Then the condition of ${\mathfrak{n}}$-minimality at $v$ is that
there exists $g_{v}\in\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$ such that
for any $\tau\in I_{v}$,
$g_{v}\cdot\rho_{\pi}(\tau)\cdot g_{v}^{-1}=\exp(t_{v}(\tau)J_{N}).$
Let $\lambda=(\lambda_{\tau,i})$ where $\tau\colon
F\to\overline{{\mathbb{Q}}}$ and $i=1,\ldots,N$. Let
$\pi=\pi_{f}\otimes\pi_{\infty}$ be a cuspidal automorphic representation such
that $dim\,\pi_{f}^{U_{0}({\mathfrak{n}})}=1$ and of cohomological weight
$\lambda$. In other words, there are Hecke equivariant embeddings
$\pi_{f}^{U_{0}({\mathfrak{n}})}\subset\mathrm{H}^{i}(Y,V_{\lambda}({\mathbb{C}})),\quad
i=1,2$
Let $S_{p}=S_{F,p}$ be the set of places of $F$ above $p$. By [HLLT16], we
know that the Galois representation
$\rho_{\pi}\colon\operatorname{Gal}(\overline{F}/F)\to\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$
associated to $\pi$ exists and by [Ca14], $\rho_{\pi}|_{G_{F_{w}}}$ is
Fontaine-Laffaille at $w\in S_{p}$. We will assume either of the two following
local conditions at $p$ :
$(FL)$ $p$ is unramified in $F$ and $\lambda_{\tau,1}-\lambda_{\tau,N}<p-N$
for any $\tau\in I_{F}$.
$(ORD_{\pi})$ $F$ contains an imaginary quadratic field in which $p$ splits,
$\pi$ is unramified and ordinary at all places $v\in S_{p}$.
To recall what the latter means, we follow the notations of [ACC+18, Section
5.1, before Th.5.5.1] (see also [Ge19, Lemma 2.7.6]). We assume $S\cap
S_{F,p}=\emptyset$. Then the ordinarity of $\pi$ at $v\in S_{p}$ means that
$u_{\lambda,\varpi_{v}}^{(i)}\in\mathcal{O}^{\times}$ for $i=1,\ldots,N$ where
the $u_{\lambda,\varpi_{v}}^{(i)}$’s are defined by
$Hecke_{v}(X)=\prod_{i=1}^{N}(X-\mathbb{N}v^{i-1}\frac{u_{\lambda,\varpi_{v}}^{(i)}}{u_{\lambda,\varpi_{v}}^{(i-1)}}\cdot\prod_{\tau\colon
F_{v}\to\overline{{\mathbb{Q}}}_{p}}\tau(\varpi_{v})^{\lambda_{\tau,N-i+1}})$
with $u_{\lambda,\varpi_{v}}^{(0)}=1$ by convention.
For $i=1,\ldots,N$, let $\chi_{\pi,v,i}\colon
G_{F_{v}}\to\mathcal{O}^{\times}$ be the character given by
$\chi_{\pi,v,i}\circ\operatorname{Art}_{F_{v}}(u)=\epsilon^{-(i-1)}(\operatorname{Art}_{F_{v}}(u))\prod_{\tau}\tau(u)^{\lambda_{\tau,N+1-i}}$
for all $u\in\mathcal{O}_{F_{v}}^{\times}$ , and by
$\chi_{\pi,v,i}\circ\operatorname{Art}_{F_{v}}(\varpi_{v})=\mathbb{N}v^{i-1}\frac{u_{\lambda,\varpi_{v}}^{(i)}}{u_{\lambda,\varpi_{v}}^{(i-1)}}\cdot\prod_{\tau\colon
F_{v}\to\overline{{\mathbb{Q}}}_{p}}\tau(\varpi_{v})^{\lambda_{\tau,N-i+1}}.$
We put
$\underline{\chi}_{\pi,v}=\operatorname{diag}(\chi_{\pi,v,i})_{i=1,\ldots,N}$
Then the translation of the condition $(ORD_{\pi})$ on the Galois side is as
follows. There exists $g_{v}\in\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$
such that $g_{v}\cdot\rho_{\pi}|_{G_{F_{v}}}\cdot g_{v}^{-1}$ is upper
triangular. Let $B=TN$ be the Levi decomposition of the Borel $B$ of upper
triangular matrices with $T$ the subgroup of diagonal matrices. And if we
denote by $\underline{\chi}_{v}\colon G_{F_{v}}\to T(\mathcal{O})$ the
homomorphism $g_{v}\cdot\rho_{\pi}|_{G_{F_{v}}}\cdot g_{v}^{-1}$ modulo
$N(\mathcal{O})$. Then
$\underline{\chi}_{v}=\underline{\chi}_{\pi,v}$
In that situation, we consider the so-called $p$-distinguished condition
$(DIST)$ We assume that $\overline{\rho}_{\pi}$ is distinguished at each $v\in
S_{p}$ (or in brief, $p$-distinguished).
With the notations above, it means that for each $v\in S_{p}$, the residual
characters $\overline{\chi}_{v,i}$ modulo $\varpi$ for $i=1,\ldots,N$ are
mutually distinct. In other words, the assumption of $p$-distinguishability
expresses the fact that for all $v\in S_{p}$, the homomorphism
$\overline{\underline{\chi}}_{v}=(\overline{\chi}_{v,i})_{i=1,\ldots,N}$
separates the roots of $(\operatorname{\mathsf{GL}}_{N},B,T)$. We will also
need sometimes the assumption of strong $p$-distinguishability:
(STDIST) For all $i<j$’s, the quotients
$\overline{\chi}_{\widetilde{v},i}\cdot\overline{\chi}_{\widetilde{v},j}^{-1}$
are neither trivial nor equal to $\overline{\epsilon}^{-1}$.
We now recall briefly the deformation theory of Galois representations that
plays a central role in this paper. Let
$\mathcal{F}_{\overline{\rho}}\colon{}_{\mathcal{O}}\operatorname{Art}_{k}\to
Sets$ resp. $\mathcal{F}_{v}$, be the problem of deformations of
$\overline{\rho}=\overline{\rho}_{\pi}$, resp. of
$\overline{\rho}_{v}=\overline{\rho}_{\pi}|_{G_{F_{v}}}$. For $v\in S_{p}$, we
consider the subfunctors $\mathcal{F}^{?}_{v}\subset\mathcal{F}_{v}$ where
$?=\operatorname{ord},FL$; assuming $(ORD_{\pi})$,
$\mathcal{F}^{\operatorname{ord}}_{v}$ is the functor of
$\underline{\chi}_{\pi,v}$-ordinary deformations $\rho$’s, that is, those for
which $\underline{\chi}_{\rho}=\underline{\chi}_{\pi,v}$ (see Section 2.1.2);
similarly, assuming $(FL)$, $\mathcal{F}^{FL}_{v}$ is the functor of.
Fontaine-Laffaille local deformations (see Section 2.1.3). Similarly, for
$v\in S$, we consider the subfunctor
$\mathcal{F}_{v}^{min}\subset\mathcal{F}_{v}$ of minimal deformations (see
2.1.1). Let
$\mathcal{F}_{loc}=\prod_{v\in S\cup S_{p}}\mathcal{F}_{v}\quad\mbox{\rm and
}\mathcal{F}_{loc}^{min,?}=\prod_{v\in
S}\mathcal{F}_{v}^{min}\times\prod_{v\in S_{p}}\mathcal{F}_{v}^{?}$
and consider the global deformation problem defined as the fiber product
$\mathcal{D}^{?}=\mathcal{F}_{\overline{\rho}}^{min,?}=\mathcal{F}_{\overline{\rho}}\times_{\mathcal{F}_{loc}}\mathcal{F}_{loc}^{min,?}.$
It is pro-prepresentable by a pair $(R^{?},\rho^{?})$ where
$?=\operatorname{ord},FL$ (see for instance [CHT08, Prop.2.2.9 and Lemma
2.4.1] for $?=FL$ and [CHT08, Section 2.4.4] or [Ti96, Chapt.6] for
$?=\operatorname{ord}$). As in [CaGe18], we introduce allowable Taylor-Wiles
sets $Q$ disjoint of the set of places dividing ${\mathfrak{n}}p$ and consider
the functors $\mathcal{D}^{?}_{Q}$ similar to $\mathcal{D}^{?}$, where we
allow arbitrary ramification at places $v\in Q$. Similarly, let $R^{?}_{Q}$ be
the universal deformation ring of $\mathcal{D}^{?}_{Q}$ and
$(\operatorname{T}_{Q},{\mathfrak{m}}_{Q})$ be the analogue of
$(\operatorname{T},{\mathfrak{m}})$ with Taylor-Wiles auxiliary level at $Q$
(see [CaGe18, Section 9.1]). The rings $R_{Q}^{?}$ and $\operatorname{T}_{Q}$
both admit a natural structure of $\mathcal{O}[\Delta_{Q}]$-algebra where
$\Delta_{Q}$ is the $p$-Sylow of $\prod_{v\in Q}T(k_{v})$. By
$(\operatorname{Gal}_{\mathfrak{m}})$, there is a Galois representation
$\rho_{{\mathfrak{m}}_{Q}}\colon\operatorname{Gal}(\overline{F}/F)\to\operatorname{\mathsf{GL}}_{N}(\operatorname{T}_{Q})$
associated to $(\operatorname{T}_{Q},{\mathfrak{m}}_{Q})$. We will assume the
following
###### Conjecture 1.
(LLC) For any Taylor-Wiles set $Q$, the conjugacy class
$[\rho_{{\mathfrak{m}}_{Q}}]$ is in
$\mathcal{D}^{?}_{Q}(\operatorname{T}_{Q})$.
The local-global compatibility conjecture (LLC) for
$\rho_{{\mathfrak{m}}_{Q}}$ has been proven (at least if $F$ is not imaginary
quadratic but contains an imaginary quadratic field in which $p$ splits)
modulo a nilpotent ideal (see [ACC+18]) of exponent bounded by
$[F\colon{\mathbb{Q}}]$ and $N$. Assuming (LLC), for each Taylor-Wiles set
$Q$, there is a canonical surjective $\mathcal{O}[\Delta_{Q}]$-homomorphism
$\phi_{Q}\colon R^{?}_{Q}\to\operatorname{T}_{Q}$. Let
$\phi_{=}\phi_{\emptyset}$. By Taylor-Wiles-Kisin patching, the
$\mathcal{O}[\Delta_{Q}]$-homomorphism $\phi_{Q}$ give rise to morphism
$S_{\infty}\to
R_{\infty}^{?}\to\operatorname{T}_{\infty}\subset\operatorname{\mathsf{End}}_{S_{\infty}}(C^{\bullet}_{\infty})$
(see [CaGe18]). The graded module
$\operatorname{Tor}_{\bullet}^{S_{\infty}}(R_{\infty},\mathcal{O})$ has a
natural structure of graded algebra over
$\operatorname{Tor}_{0}^{S_{\infty}}(R_{\infty},\mathcal{O})=R_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R^{?}$
. Let $\mathrm{H}_{\bullet}^{\mathfrak{m}}=H^{d-\bullet}_{\mathfrak{m}}$, then
we have:
###### Theorem 1.
([CaGe18]) Assume $\zeta_{p}\notin F$, $p>N$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(LLC)$ $(RLI)$, $(MIN)$, $(FL)$ or
$(ORD_{\pi})+(DIST)$. Then, the canonical map
$R_{\infty}^{?}\to\operatorname{T}_{\infty}$ is an isomorphism, and in
particular
$R^{?}_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R^{?}\cong\operatorname{T}$.
Moreover, the module $\mathrm{H}^{\mathfrak{m}}_{q_{m}}$ is finite and free
over $\operatorname{T}$, and there is an isomorphism of graded modules:
$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}}\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$
in other words $\mathrm{H}_{\bullet}^{\mathfrak{m}}$ is a free graded
$\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$-module.
On the other hand, in [GV18], the authors introduced derived notions of
deformation problems for $\overline{\rho}_{{\mathfrak{m}}}$ denoted
$\mathcal{D}^{s}$, $\mathcal{D}^{s,?}$; they are pro-represented by simplicial
pro-artinian rings $\mathcal{R}$, resp. $\mathcal{R}^{s,?}$. Generalizing
their main result Th.12.1 and Th.14.1, Y. Cai [Cai21, Theorem 0.6] proved
###### Theorem 2.
([GV18, Cai21]) Assume that $p>N$, $\zeta_{p}\notin F$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(LLC)$ and $(Van_{\mathfrak{m}})$
hold. Under the assumptions $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$,
we have an isomorphism of graded commutative rings
$\pi_{\bullet}\mathcal{R}\cong\operatorname{Tor}_{\bullet}^{S_{\infty}}(R_{\infty},\mathcal{O})$,
in particular we have an isomorphism of graded objects
$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}}\pi_{\bullet}(\mathcal{R})$
Our first main result concerns the relation between $\pi_{1}(\mathcal{R})$ and
the minimal and ordinary (resp. Fontaine-Laffaille) Selmer group of
$\operatorname{Ad}(\rho_{\pi})^{*}(1)$. Recall $\pi_{0}(\mathcal{R})=R$ is the
classical $(min,?)$-deformation ring. Let
$\mathcal{O}_{n}=\mathcal{O}/(\varpi^{n})$. We consider the simplicial ring
homomorphism
$\phi_{n}\colon\mathcal{R}\to R\to\mathcal{O}_{n}$
given by the universal property for the deformation
$\rho_{n}=\rho_{\pi}\pmod{(\varpi^{n})}$. Let
$M_{n}=\varpi^{-n}\mathcal{O}/\mathcal{O}$; it is a finite
$\mathcal{O}_{n}$-module. Consider the simplicial ring
$\Theta_{n}=\mathcal{O}_{n}\oplus M_{n}[1]$ concentrated in degrees $0$ and
$1$ up to homotopy. It is endowed with a simplicial ring homomorphism
$\operatorname{pr}_{n}\colon\Theta_{n}\to\mathcal{O}_{n}$ given by the first
projection. Let $L_{n}(\mathcal{R})$ be the set of homotopy equivalence
classes of simplicial ring homomorphisms $\Phi\colon\mathcal{R}\to\Theta_{n}$
such that $\operatorname{pr}_{n}\circ\Phi=\phi_{n}$. Following [GV18, Lemma
15.1], we define (see Section 4) compatible homomorphisms
$\pi(n,\mathcal{R})\colon
L_{n}(\mathcal{R})\to\operatorname{\mathsf{Hom}}_{R}(\pi_{1}(\mathcal{R}),M_{n})$
Let $\mathop{\rm
Sel}(F,\operatorname{Ad}\,\rho_{n}(1)\otimes_{\operatorname{T}}M_{n}^{\vee})$
be the minimal and $?=ord,FL$ Selmer group of
$\operatorname{Ad}\,\rho_{n}(1)\otimes_{\operatorname{T}}M_{n}^{\vee}$. Formal
arguments show that it is the Pontryagin dual of $L_{n}(\mathcal{R})$.
Therefore, by passing to Pontryagin dual, we obtain a homomorphism
$GV_{n}=\pi(n,\mathcal{R})^{\vee}$:
$GV_{n}\colon\operatorname{\mathsf{Hom}}_{\operatorname{T}}(\pi_{1}(\mathcal{R}),M_{n})^{\vee}\hookrightarrow
Sel(F,(\operatorname{Ad}\,\rho_{n})^{\ast}(1)\otimes M_{n}^{\vee})$
By taking the direct limit we obtain a homomorphism
$GV\colon\pi_{1}(\mathcal{R})\otimes_{R,\lambda}\mathcal{O}\otimes_{\mathcal{O}}K/\mathcal{O}\hookrightarrow\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes K/\mathcal{O})$
Then our first result (Theorem 13 and Proposition 7 in the text) is
###### Theorem 3.
Assume $p>N$, $\zeta_{p}\notin F$, $(\operatorname{Gal}_{\mathfrak{m}})$,
$(LLC)$, $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$. The natural
homomorphism
$GV\colon\pi_{1}(\mathcal{R})\otimes_{\operatorname{T}}K/\mathcal{O}\hookrightarrow\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes K/\mathcal{O})$
is injective. The left-hand side module is $\varpi$-divisible of
$\mathcal{O}$-corank $\ell_{0}$. If we are in the ordinary case, let us
furthermore assume $(STDIST)$, then the cokernel of $GV$ is the Tate-
Shafarevitch group ${}^{1}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})$ and is finite. Moreover, its $\mathcal{O}$-Fitting ideal is
the same as the one of $\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})\otimes K/\mathcal{O})$.
The key step to prove this theorem is Proposition 5. A similar result, under
stronger assumptions, is proven in the course of the proof of [GV18, Theorem
15.2]. However, under our assumptions, the proof given there doesn’t seem to
work. Our approach is different and involves only commutative algebra. The
last part of the statement is proved in Proposition 7.
Actually, under the same assumptions , we can also define a graded version
$GV^{\bullet}$ of the Galatius-Venkatesh homomorphism $GV$. Let us introduce
some notations before stating our result. For a graded algebra $G$, we
introduce the truncation
$\tau_{\leq\ell_{0}}G=\bigoplus_{j\leq\ell_{0}}G_{j}$
with a partial algebra structure defined only for $g_{j_{k}}\in G_{j_{k}}$
such that $j=j_{1}+j_{2}\leq\ell_{0}$. For a graded algebra $G$, we also
introduce its largest graded-commutative quotient $\widetilde{G}$, that is,
its largest (graded) quotient in which
$<a,b>_{j_{1},j_{2}}=(-1)^{j_{1}j_{2}}<b,a>_{j_{2},j_{1}}$ for
$a\in\widetilde{G}_{j_{1}}$, $b\in\widetilde{G}_{j_{2}}$. We endow
$\pi_{\bullet}(\mathcal{R}^{\otimes\bullet})$ with a structure of graded
algebra and we prove (see Theorem 14):
###### Theorem 4.
For every cuspidal representation $\pi^{\prime}$ occuring in
$H^{\bullet}_{\mathfrak{m}}$, for any $m\geq 1$, there is an homomorphism of
$A_{m}$-modules
$(HGV_{j})\quad GV_{m}^{j}\colon\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A_{m}}\to\bigwedge^{j}_{A_{m}}\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})^{\oplus j}$
It induces a morphism of truncated graded $A_{m}$-algebras
$(HGV_{\bullet})\quad
GV_{m}^{\bullet}\colon\tau_{\leq\ell_{0}}\widetilde{\pi}_{\bullet}(\mathcal{R}^{\otimes\bullet})\otimes
A_{m}\to\tau_{\leq\ell_{0}}\bigwedge^{\bullet}_{A_{m}}\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})^{\oplus\bullet}$
Moreover, for any $j$, the composition $a_{j}\circ GV_{m}^{j}\circ\mu_{j}$ of
$GV_{m}^{j}$ with the tensor shuffle multiplication map
$\mu_{j}\colon\bigwedge^{j}_{A}\pi_{1}(\mathcal{R})_{A}\to\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A}$ and the homomorphism $a_{j}$ induced by the addition
$\mathop{\rm Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
A_{m})^{\oplus j}\to\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})$
coincides with $\bigwedge^{j}GV_{m}^{1}$. For $j=\ell_{0}$, the cokernel of
$GV_{m}^{\ell_{0}}\circ\mu_{\ell_{0}}$ is annihilated by
$\operatorname{Fitt}_{\mathcal{O}}(\mathop{\rm
Sel}(F,Ad(\rho_{\pi^{\prime}}))$.
This theorem implies that for any $1\leq j\leq\ell_{0}$, the shuffle
multiplication $\mu_{j}$ is injective111However it would be more interesting
to study the injectivity of the homomorphisms $GV_{m}^{j}$..
We have similar results for Hida families in Section 7. Let $R_{h}$ be the
universal minimal $\Lambda$-ordinary deformation ring prorepresenting the
minimal $\Lambda$-ordinary deformation problem $\mathcal{D}_{h}$ (see Section
6.2). Let $\mathcal{R}_{h}$ the simplical proartinian ring prorepresenting the
simplicial minimal $\Lambda$-ordinary deformation problem
$\mathcal{D}_{h}^{s}$ (see Section 6.3). The rings $\mathcal{R}_{h}$ and
$R_{h}$ are naturally $\Lambda$-algebra, where $\Lambda$ is the Hida-Iwasawa
algebra (see Section 6.1). We have $\pi_{0}(\mathcal{R}_{h})=R_{h}$ ( Section
6.3), hence we have a surjective $\Lambda$-algebra homomorphism
$\mathcal{R}_{h}\to R_{h}$. Note that $\pi_{\bullet}(\mathcal{R}_{h})$ is a
graded algebra over $\pi_{0}(\mathcal{R}_{h})=R_{h}$. Let ${\mathfrak{m}}_{h}$
be the maximal ideal of $\Lambda$-ordinary Hida Hecke algebra acting
faithfully on
$\mathbb{H}_{h}^{\bullet}=e\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$
associated to ${\mathfrak{m}}$ and let $\operatorname{T}_{h}$ be its
corresponding ${\mathfrak{m}}_{h}$-localization. Assuming
$(\operatorname{Gal}_{\mathfrak{m}})$ and (LLC), there is $\Lambda$-algebra
homomorphism $R_{h}\to\operatorname{T}_{h}$. We prove following the Calegari-
Geraghty method (Th. 15 and 16 in the text) :
###### Theorem 5.
Assuming $(\operatorname{Gal}_{\mathfrak{m}})$ and (LLC), $(MIN)$,
$(ORD_{\pi})+(DIST)$. Assume $(Van_{\mathfrak{m}})$ holds for a classical
specialization $\pi$ of $\operatorname{T}_{h}$ of level prime to $p$. Then
1) $R_{h}\to\operatorname{T}_{h}$ is an isomorphism,
2) there is a isomorphism of graded $\operatorname{T}_{h}$-modules
$\mathbb{H}_{h,{\mathfrak{m}}_{h}}^{\bullet}=\mathbb{H}^{q_{s}}_{h,{\mathfrak{m}}_{h}}\otimes_{\operatorname{T}_{h}}\pi_{d-\bullet}(\mathcal{R}_{h})$
Given a Hida family $\operatorname{T}_{h}\to\mathbf{I}$ of the
${\mathfrak{m}}_{h}$-localization of the ordinary Hida Hecke algebra, let
$\Lambda_{\mathbf{I}}$ be the image of $\Lambda$ in $\mathbf{I}$ and
$\mathcal{R}_{\mathbf{I}}=\mathcal{R}_{h}\underline{\otimes}_{\Lambda}\Lambda_{\mathbf{I}}$.
We prove (Theorem 17 in the text):
###### Theorem 6.
There are natural $\mathbf{I}$-linear injective homomorphisms
$GV_{h}\colon\pi_{1}(\mathcal{R}_{h})\otimes_{R_{h}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}).$
and
$GV_{\mathbf{I}}\colon\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}).$
and we have $GV_{\mathbf{I}}\circ\pi_{\mathbf{I}}=GV_{h}$ for
$\pi_{\mathbf{I}}$ the homomorphism induced by the map
$\mathcal{R}_{h}\to\mathcal{R}_{\mathbf{I}}$.
The new feature in this Hida-theoretic context is that the Hida-theoretic
cohomology module
$\mathbb{H}_{h}^{\bullet}=e\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$
is conjectured to be concentrated in the maximal degree $q_{s}$. This is one
form of the non-abelian Leopoldt conjecture due to Hida and one of the
authors. They are discussed in Section 6.4 in the text. Recall that a
simplicial ring is called discrete if it is weakly equivalent to a constant
ring. Then, this non-abelian Leopoldt conjecture is equivalent to the
following:
###### Conjecture 2.
The simplicial ring $\mathcal{R}_{h}$ is discrete and for any arithmetic
weight $\lambda^{\prime}$, there is an isomorphism of commutative graded rings
$\operatorname{Tor}_{\bullet}^{\Lambda}(R_{h},\Lambda/P_{\lambda^{\prime}})=\pi_{\bullet}(\mathcal{R}_{\lambda^{\prime}}).$
Because of this conjecture, the first statement of the theorem above is
probably empty. However, the second appears to be interesting and yields the
inconditional Theorem below (Theorem 18 in the text). Let $\mathcal{L}$ be the
minimal $\Lambda$-ordinary local deformation data. Let ${\mathop{\rm
Sel}}_{\mathbf{I}}=\mathrm{H}^{1}_{\mathcal{L}}(F,\operatorname{Ad}\rho_{\mathbf{I}}\otimes\widehat{\mathbf{I}})$
and $M_{\mathbf{I}}$ its Pontryagin dual. Let $\Phi_{\mathbf{I}}$ be the
Pontryagin dual of
$\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}$.
Let finally
$N_{\mathbf{I}}=\operatorname{Ker}(M_{\mathbf{I}}\to\Phi_{\mathbf{I}})$ be the
Pontryagin dual of $\operatorname{Coker}\,GV_{\mathbf{I}}$.
###### Theorem 7.
Assume moreover (STDIST). Then
(1) The $\mathbf{I}$-modules $M_{\mathbf{I}}$ and $\Phi_{\mathbf{I}}$ have
rank $\ell_{0}$ and $\Phi_{\mathbf{I}}$ is free.
(2) The Poitou-Tate-Pontryagin duality induces an isomorphism
$N_{\mathbf{I}}\cong\varprojlim_{n}{\mathop{\rm
Sel}}_{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}]$
Let $\widetilde{\mathbf{I}}$ be the integral closure of $\mathbf{I}$ and let
$X=\operatorname{Supp}(\widetilde{\mathbf{I}}/\mathbf{I})$. For $P\notin X$,
we have $\mathbf{I}/P=\widetilde{\mathbf{I}}/P\widetilde{\mathbf{I}}$. For
such $P$, the reduction modulo $P$ of the exact sequence
$0\to N_{\mathbf{I}}\to M_{\mathbf{I}}\to\Phi_{\mathbf{I}}\to 0$
is exactly the Pontryagin dual of the exact sequence given by
$0\to\pi_{1}(\mathcal{R}_{P})\stackrel{{\scriptstyle
GV_{P}}}{{\rightarrow}}{\mathop{\rm Sel}}_{P}\to\operatorname{Coker}GV_{P}\to
0$
where the natural notations are explained before Corollary 4.
In learning the material used here, we benefited from a seminar on Galatius-
Venkatesh paper held in 2019 at Paris 13, and of discussions with Y. Harpaz
and A. Vezzani. Special thanks are due to Yichang Cai whose thesis provided
much of the framework for our work. He also taught us several technical
aspects of Model Categories and Derived Deformation theory, although
misunderstandings and mistakes left in the text are ours.
## 2\. Calegari-Geraghty Theory for $\operatorname{\mathsf{GL}}_{N}$
### 2.1. Local deformation conditions
For each finite place $v$ of $F$, let $\overline{F}_{v}$ be an algebraic
closure of $F_{v}$ and $G_{F_{v}}$ be the local Galois group of
$\overline{F}_{v}$ over $F_{v}$, identified to a decomposition group at $v$ in
$G_{F}$.
By definition, for an artinian local $\mathcal{O}$-algebra $A$ with residue
field $k$, let $\mathcal{F}_{\overline{\rho}}(A)$ (resp. $\mathcal{F}_{v}(A)$
) be the set of conjugacy classes of liftings $\rho\colon
G_{F}\to\operatorname{\mathsf{GL}}_{N}(A)$ of $\overline{\rho}_{\pi}$ (resp.
$\rho_{v}\colon G_{F_{v}}\to\operatorname{\mathsf{GL}}_{N}(A)$ of
$\overline{\rho}_{v}$). Let $\mathcal{F}_{v}^{\square}$ be the functor of
liftings of $\overline{\rho}_{v}$. Recall that it carries an action of the
formal group scheme $\widehat{\operatorname{PGL}_{N}}$ by conjugation and that
$\mathcal{F}_{v}^{\square}/\widehat{\operatorname{PGL}_{N}}\cong\mathcal{F}_{v}$.
We define below subfunctors $\mathcal{F}_{v}^{?,\square}$, resp.
$\mathcal{F}_{v}^{?}$ of local liftings, resp. deformations of
$\bar{\rho}_{v}$, for $?=min,\operatorname{ord},FL$.
###### Definition 1.
A functor
$\mathcal{F}\colon{}_{\mathcal{O}}\operatorname{Art}_{k}\to\operatorname{\mathsf{SETS}}$
is called liftable if for any surjection $\alpha\colon A_{1}\to A_{0}$, the
morphism $\mathcal{F}(A_{1})\to\mathcal{F}(A_{0})$ is surjective.
Note that if $\mathcal{F}_{v}^{?,\square}$ is formally smooth,
$\mathcal{F}_{v}^{?}$ is liftable. We prove this below for
$?=min,\operatorname{ord},FL$.
#### 2.1.1. minimal case
For $v\in S$, let $t_{v}\colon I_{v}\to{\mathbb{Z}}_{p}^{\times}$ be the
$p$-adic tame character and let $q=\mathcal{O}_{F_{v}}/(\varpi_{v})$; for
$\tau\in I_{v}^{tame}$, and for $\phi_{v}$ an arithmetic Frobenius, we have
$\phi_{v}\circ\tau\circ\phi_{v}^{-1}=\tau^{q}$. We consider the problem
$\mathcal{F}_{v}^{min,\square}\subset\mathcal{F}_{v}^{\square}$, resp.
$\mathcal{F}_{v}^{min}\subset\mathcal{F}_{v}$ of $v$-minimal liftings, resp.
deformations:
$\mathcal{F}_{v}^{min,\square}(A)=\\{\rho_{v}\in\mathcal{F}_{v}^{\square}(A);\mbox{\rm
for}\,\tau\in I_{v},\mbox{\rm there
exists}\,g_{v}\in\operatorname{\mathsf{GL}}_{N}(A)\,\mbox{\rm such that for
any}\,\tau\in I_{v},$ $g_{v}\cdot\rho(\tau)\cdot
g_{v}^{-1}=\exp(t_{v}(\tau)J_{N})\\}$
and
$\mathcal{F}_{v}^{min}(A)=\mathcal{F}_{v}^{min,\square}(A)/\widehat{\operatorname{PGL}_{N}}(A).$
Let $\Phi_{v}=\operatorname{diag}(q^{N-1},\ldots,1)$.
###### Lemma 1.
The functor $\mathcal{F}_{v}^{min,\square}$ is prorepresented by a formally
smooth $\mathcal{O}$-algebra $R_{v}^{min,\square}$, isomorphic to a power
series ring in $N^{2}$ variables. The functor $\mathcal{F}_{v}^{min}$ is
liftable.
Comments:
1) Note that the proof uses the fact that the matrix $J_{N}$ is regular
nilpotent.
2) This result is also proven in [CHT08, Lemma 2.4.19 and Cor.2.4.20] and
[Bo19, Introduction].
3) The functor $\mathcal{F}_{v}^{min}$ is pro-representable if and only if $p$
does not divide $q_{v}^{i}-1$ ($i=1,\ldots,N-1$).
###### Proof.
Let $C_{0}$ be the centralizer of $J_{N}$ in the standard Borel $B\subset G$.
The formal scheme $\mathcal{C}$ of centralizers of liftings of
$\overline{J}_{N}$ is isomorphic to the principal homogeneous formal scheme
$\widehat{\operatorname{\mathsf{GL}}_{N}/C_{0}}$, hence is formally smooth of
dimension $N^{2}-N$. There is a fibration of functors
$\mathcal{F}_{v}^{min,\square}\to\mathcal{C}$ given by $\rho_{v}\to
g_{v}^{-1}C_{0}g_{v}$. Therefore, to study the formal smoothness of
$\mathcal{F}_{v}^{min,\square}$, one can fix $C\subset B$ and study its fiber
$\mathcal{F}_{v,[C]}^{min,\square}$ in $\mathcal{F}_{v}^{min,\square}$. In the
notations below, we take $C=C_{0}$. We therefore consider $\rho_{v}$ such that
for any $\tau\in I_{v}$, $\rho_{v}(\tau)=\exp(t_{v}(\tau)J_{N})$. Let
$\Phi_{v}=\operatorname{diag}(q_{v}^{N-1},\ldots,1)$. We have
$\rho_{v}(\phi_{v})=\Psi_{v}$ with $\Psi_{v}=\Phi_{v}g$ with $g\in C_{0}$. Let
$\widehat{C}_{0}$ be the formal group scheme associated to $C_{0}$. By the
considerations above, we see easily that the map $g\mapsto([\rho_{g}]$ where
$\rho_{g}(\phi_{v})=\Phi_{v}\cdot g$ and
$\rho_{g}(\tau)=\exp(t_{v}(\tau)J_{N})$ for $\tau\in I_{v}$, is an isomorphism
of functors
$\widehat{C}\cong\mathcal{F}_{v,[C]}^{min,\square}.$
This isomorphism is compatible to conjugation by $\widehat{G}$. It implies
that $\mathcal{F}_{v}^{min,\square}$ is a torsor under the smooth formal group
scheme $\widehat{G}$, hence is formally smooth of dimension $N^{2}$. This
proves the formal smoothness of
$R_{v}^{min,\square}\cong\mathcal{O}[[X_{1},\ldots,X_{N^{2}}]]$.
Any deformation $[\rho_{v}]\in\mathcal{F}_{v}^{min}(A)$ has a representative
$\rho_{v}$ such that for any $\tau\in I_{v}$,
$\rho_{v}(\tau)=\exp(t_{v}(\tau)J_{N})$. For such a representative,
$\rho_{v}(\phi_{v})=\Psi_{v}$ can be written $\Psi_{v}=\Phi_{v}g$ with $g\in
C_{0}$. If $\rho_{v}^{\prime}$ is another such representative, we see that
there exists $h\in\widehat{C}(A)$ such that
$\rho_{v}^{\prime}(\phi_{v})=h\Psi_{v}h^{-1}=\Phi_{v}g^{\prime}$
Therefore, $g^{\prime}=\Phi_{v}^{-1}h\Phi_{v}h^{-1}\cdot.g$ for an
$h\in\widehat{C}(A)$ and conversely. Conjugation by $\Phi_{v}$ is an
automorphism of $\widehat{C}(A)$. Thus we find that
$\mathcal{F}_{v}^{min}\cong\widehat{C}/(\Phi_{v}-1)\widehat{C}$
which is isomorphic to the functor
$A\mapsto A/(q-1)A\times A/(q^{2}-1)A\times\ldots A/(q^{N-1}-1)A$
This functor is not pro-representable unless $p$ does not divide the $q^{i}-1$
(for all $i=1,\ldots,N-1$). However, it is liftable. ∎
#### 2.1.2. The ordinary case
For $v\in S_{p}$, we consider the problem
$\mathcal{F}_{v}^{\operatorname{ord}}\subset\mathcal{F}_{v}$ of ordinary
deformations of $\overline{\rho}_{v}$ without fixing the Hodge-Tate weights.
This means that $[\rho_{v}]\in\mathcal{F}_{v}(A)$ if and only if for a
representative $\rho_{v}$, there exists
$g_{v}\in\operatorname{\mathsf{GL}}_{N}(A)$ such that
$g_{v}\cdot\rho{|_{G_{F_{v}}}}\cdot g_{v}^{-1}$ takes values in $B(A)$ and
that its reduction $\underline{\chi}_{\rho,v}\colon G_{F_{v}}\to T(A)$ modulo
$N(A)$ is a lifting of $\overline{\underline{\chi}}_{v}\colon G_{F_{v}}\to
T(k)$.
Let $\mu=(\mu_{v})_{v\in S_{p}}$ where
$\mu_{v}\colon\mathcal{O}_{F_{v}}^{\times}\to T(\mathcal{O})$ is a continuous
character lifting the character
$\bar{\underline{\chi}}_{v}\colon\mathcal{O}_{F_{v}}^{\times}\to T(k)$ given
by $\bar{\rho}|_{I_{v}}$. We define the subfunctor
$\mathcal{F}_{v}^{\operatorname{ord},\mu}\subset\mathcal{F}_{v}^{\operatorname{ord}}$
of ordinary deformations of weight $\mu$ to be given by $[\rho_{v}]$’s for
which
$\underline{\chi}_{\rho,v}\circ\operatorname{Art}_{F_{v}}|_{\mathcal{O}_{F_{v}}^{\times}}=\mu_{v}$
where $\mu_{v}$ is considered as taking values in $T(A)$ via the canonical
morphism $T(\mathcal{O})\to T(A)$.
In this paper, besides the sections dealing with Hida theory, we only consider
the case
$\mu_{v}=\underline{\chi}_{\pi,v}\circ\operatorname{Art}_{F_{v}}|_{\mathcal{O}_{F_{v}}^{\times}}.$
The homomorphism $\mu$ is then algebraic, given by the Hodge-Tate weights of
$\rho_{\pi_{v}}$.
In the Lemma below, we also consider the subfunctor
$\mathcal{F}_{v}^{\det,\operatorname{ord}}\subset\mathcal{F}_{v}^{\operatorname{ord}}$
where the determinant of the deformations is fixed equal to $\det\,\rho_{v}$.
###### Lemma 2.
For $v\in S_{p}$, under the condition (STDIST) (and $p>N$) the functors
$\mathcal{F}_{v}^{\det,\operatorname{ord}}$,
$\mathcal{F}_{v}^{\operatorname{ord}}$,
$\mathcal{F}_{v}^{\operatorname{ord},\mu}$ are pro-representable by a complete
noetherian local ring $R_{v}^{\det,\operatorname{ord}}$,
resp.$R_{v}^{\operatorname{ord}}$, $R_{v}^{\operatorname{ord},\mu}$. These
problems are unobstructed, hence formally smooth. In particular, one has
$R_{v}^{\operatorname{ord}}=R_{v}^{\det,\operatorname{ord}}[[T_{1},\ldots,T_{f}]]$
(where $f=[F_{v}\colon{\mathbb{Q}}_{p}]$ is the degree of $F_{v}$).
###### Proof.
Let ${\mathfrak{b}}$ be the Lie algebra of the standard Borel $B$ and
${\mathfrak{b}}^{\prime}$ the subalgebra of trace zero elements. To prove that
$\mathcal{F}_{v}^{\det,\operatorname{ord}}$, resp.
$\mathcal{F}_{v}^{\operatorname{ord},\mu}$, is proprepresentable and
unobstructed, it is enough to show that
$H^{i}(\Gamma_{v},{\mathfrak{b}}^{\prime})=0$ for $i\neq 1$. By Tate duality,
this follows from the strong distinguishability condition. Since $p>N$, any
$[\rho]\in\mathcal{F}_{v}^{\operatorname{ord}}(A)$ has a twist $rho_{\chi}$ by
a $p$-power order character $\chi$ whose determinant is equal to
$\det\,\rho_{\pi,v}$. This character $\chi$ factors through the universal
character $G_{F_{v}}\to\mathcal{O}[[T_{1},\ldots,T_{f}]]^{\times}$. Hence
$\mathcal{F}_{v}^{\operatorname{ord}}$ is proprepresentable by
$R_{v}^{\operatorname{ord}}=R_{v}^{\det,\operatorname{ord}}[[T_{1},\ldots,T_{f}]]$.
∎
Comment: The dimension of $R_{v}^{\operatorname{ord}}$ (given in [CHT08, Lemma
2.4.8]) will be recalculated later when applying the Poitou-Tate formula.
###### Corollary 1.
The functor $\mathcal{F}_{v}^{\operatorname{ord},\square}$ is pro-
representable by a formally smooth ring $R_{v}^{\operatorname{ord},\square}$.
###### Proof.
The morphism
$\mathcal{F}_{v}^{\operatorname{ord},\square}\to\mathcal{F}_{v}^{\operatorname{ord}}$
is a $\widehat{G^{ad}}$-torsor, hence is smooth and the base is smooth. ∎
In all sections below (except those dealing with Hida theory), we fix
$\mu_{v}=\underline{\chi}_{\pi,v}\circ\operatorname{Art}_{F_{v}}|_{\mathcal{O}_{F_{v}}^{\times}}$
for each $v\in Sp$. We call the deformation problem
$\mathcal{F}_{v}^{\operatorname{ord},\mu}$ the
$\underline{\chi}_{\pi}$-ordinary deformation problem, or for brevity the
ordinary deformation problem (with Hodge-Tate weights fixed by $\pi$). Hence
we write only $\mathcal{F}_{v}^{\operatorname{ord}}$, although $\mu$ is fixed
by $\mu=\underline{\chi}_{\pi}$.
#### 2.1.3. The Fontaine-Laffaille case
Finally, for $v\in S_{p}$, we consider the problem
$\mathcal{F}_{v}^{FL,\square}$, resp. $\mathcal{F}_{v}^{FL}$, of Fontaine-
Laffaille liftings $\rho_{v}$, resp. deformations $[\rho_{v}]$ of
$\overline{\rho}_{v}$. This means that there exists a $\phi$-filtered
$A$-module $M$ free of rank $N$ over $A$, such that $\rho_{v}$ is isomorphic
to $V_{crys}(M)$.
###### Lemma 3.
The functor $\mathcal{F}_{v}^{FL,\square}$ is pro-representable by a formally
smooth ring $R_{v}^{FL,\square}$ isomorphic to a power series
$\mathcal{O}$-algebra in $N^{2}+[F_{v}\colon F]{N(N-1)\over 2}$ variables.
###### Proof.
This is [CHT08, Coroll.2.4.3]. ∎
### 2.2. The Theorem of Calegari-Geraghty
For $?=\operatorname{ord},FL$, let
$\mathcal{F}_{loc}=\prod_{v\in S\cup S_{p}}\mathcal{F}_{v}\quad\mbox{\rm and
}\mathcal{F}_{loc}^{min,?}=\prod_{v\in
S}\mathcal{F}_{v}^{min}\times\prod_{v\in S_{p}}\mathcal{F}_{v}^{?}$
We define the global deformation problem as the fiber product
$\mathcal{D}^{?}=\mathcal{F}_{\overline{\rho}}^{min,?}=\mathcal{F}_{\overline{\rho}}\times_{\mathcal{F}_{loc}}\mathcal{F}_{loc}^{min,?}.$
By Schlessinger’s criterion, assuming $p$-distinguishedness, resp.
$p$-smallness, the functor $\mathcal{D}^{\operatorname{ord}}$, resp.
$\mathcal{D}^{FL}$, is proprepresentable by a pair $(R^{?},\rho^{?})$ where
$?=\operatorname{ord},FL$ (see for instance [CHT08, Prop.2.2.9 and Lemma
2.4.1] for $?=FL$ and [CHT08, Section 2.4.4] or [Ti96, Chapt.6] for
$?=\operatorname{ord}$). For any proartinian $\mathcal{O}$-algebra $A$ and any
object $[\rho]\in\mathcal{D}^{?}(A)$ there is a unique local
$\mathcal{O}$-algebra homomorphism $\phi_{\rho}\colon R\to A$ such that
$\phi_{\rho}\circ\rho^{?}\equiv\rho$. Assuming $(LLC)$, we have
$[\rho_{\pi}]\in\mathcal{D}^{?}(\mathcal{O})$ and
$[\rho_{\mathfrak{m}}]\in\mathcal{D}^{?}(T)$ and we therfore have the
following commutative diagram of local surjective $\mathcal{O}$-algebra
homomorphisms.
$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\rho_{\mathfrak{m}}}}$$\scriptstyle{\phi_{\rho_{\pi}}}$$\textstyle{\operatorname{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\pi}}$$\textstyle{\mathcal{O}}$
Recall that the local-global compatihility condition $(LLC)$ for
$\rho_{\mathfrak{m}}$ has been proven (see [ACC+18]), at least if $F$ is not
imaginary quadratic.
Let now $S_{\infty}$ be the Taylor-Wiles power series ring and let
$R_{\infty}\to\operatorname{T}_{\infty}$ be the surjective homomorphism of
$S_{\infty}$-algebras introduced by Calegari-Geraghty using Taylor-Wiles
systems (see [CaGe18]), and recalled in the introduction.
###### Theorem 8.
([CaGe18]) Assume $\zeta_{p}\notin F$, $p>N$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(LLC)$ $(RLI)$, $(MIN)$, $(FL)$ or
$(ORD_{\pi})+(DIST)$. Then, the canonical map
$R_{\infty}^{?}\to\operatorname{T}_{\infty}$ is an isomorphism, and in
particular
$R^{?}_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R^{?}\cong\operatorname{T}$.
Moreover, the module $\mathrm{H}^{\mathfrak{m}}_{q_{m}}$ is finite and free
over $\operatorname{T}$, and there is an isomorphism of graded modules:
$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}}\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$
in other words $\mathrm{H}_{\bullet}^{\mathfrak{m}}$ is a free graded
$\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$-module.
Remarks:
1) Poincaré duality shows that $\mathrm{H}^{\mathfrak{m}}_{q_{s}}$ is
isomorphic to $\operatorname{\mathsf{Hom}}(T,\mathcal{O})^{m}$ as $T$-module
(this reflects also that it is torsion free as $\mathcal{O}$-module).
2) for $G=\operatorname{\mathsf{GL}}(2)$ over a number field $F$, the rank $m$
is $2^{r_{1}}$ (this follows from calculations of
$({\mathfrak{g}},C_{\infty})$-cohomology, see [BW00] or [H94a]); hence
$m=2^{[F\colon{\mathbb{Q}}]}$ if $F$ is totally real, or $1$ if $F$ is CM.
In the sequel we will focus on the case
$G=\operatorname{\mathsf{GL}}(N)_{/F}$, with $F$ CM, quadratic over the
totally real field $F^{+}$, assuming $(Van_{\mathfrak{m}})$ and $(LLC)$. More
general cases are treated in [Cai21].
## 3\. Simplicial Deformation problems and tangent complexes
We follow [GV18, Sections 5 and 7] and [Cai21]. Given a category
$\mathcal{C}$, we denote by $\mathcal{C}^{s}$ the category of simplicial
objects of $\mathcal{C}$. Note any object $C$ of $\mathcal{C}$ defines a
simplicial object $\mathcal{F}_{C}\in\mathcal{C}^{s}$ by putting
$\mathcal{F}_{C}([n])=C$ and $\mathcal{F}_{C}(\phi)=\operatorname{Id}_{C}$ for
any $\phi\in\operatorname{\mathsf{Hom}}_{\Delta}([n],[m])$. Such an object is
called discrete. We will be mostly concerned with
$\mathcal{C}=\operatorname{Mod}_{A}$, $\operatorname{\mathsf{SETS}}$ and the
category ${}_{A}\\!\operatorname{CR}_{B}$ of pairs $(X,\pi)$ where $X$ is an
$A$-algebra and $\pi\colon X\to B$ is an $A$-algebra homomorphism. The
resulting simplicial categories are model categories (see [Cai21, 3.1.2]).
Note it is not the case for the subcategory
${}_{A}\\!\operatorname{Art}_{B}\subset{}_{A}\\!\operatorname{CR}_{B}$ of
simplicial artinian rings. In the sequel, we mostly consider $A=\mathcal{O}$
and $B=\mathcal{O}/(\varpi)=k$.
A functor
$\mathcal{F}\colon{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
is called homotopy invariant if it sends weakly equivalent objects to weakly
equivalent objects. Let $\ast\in\operatorname{\mathsf{SETS}}^{s}$. A
simplicial deformation problem of $\ast$ is a homotopy invariant functor
$\mathcal{F}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
such that $\mathcal{F}(k)=\\{\ast\\}$. A diagram
$\begin{array}[]{ccc}A_{1}\times_{A_{0}}^{h}A_{2}&\stackrel{{\scriptstyle\pi_{2}}}{{\to}}&A_{2}\\\
\pi_{1}\downarrow&&f_{2}\downarrow\\\ A_{1}&\stackrel{{\scriptstyle
f_{1}}}{{\to}}&A_{0}\end{array}$
in ${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$ is a homotopy pullback if
$f_{1}\circ\pi_{1}$ is homotopic to $f_{2}\circ\pi_{2}$. It is unique up to
weak equivalence. A simplicial deformation problem $\mathcal{F}$ of an object
$\ast$ is formally cohesive [GV18, Definition 3.8] if it is homotopy invariant
and if it preserves homotopy pullback. Let us give an example of such a
functor.
###### Definition 2.
Given two simplicial rings $\mathcal{R},\mathcal{R}^{\prime}$, the simplicial
set
$\operatorname{\mathsf{sHom}}_{\operatorname{CR}^{s}}(\mathcal{R},\mathcal{R}^{\prime})$
is defined as
$[p]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(\mathcal{R},(\mathcal{R}^{\prime})^{\Delta[p]})$.
Here, for each $p$, $(\mathcal{R}^{\prime})^{\Delta[p]}$ denotes the
simplicial ring given by
$[q]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{\mathsf{SETS}}^{s}}(\Delta[p]\times\Delta[q],\mathcal{R}^{\prime})$;
as a simplicial ring, it is weakly equivalent to $\mathcal{R}^{\prime}$ (see
cite[Lemma 2.13]GV18). The structure of simplicial set on
$[p]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(\mathcal{R},(\mathcal{R}^{\prime})^{\Delta[p]})$
is given by the natural face and degeneracy maps coming from those between the
$\Delta[p]$’s.
Recall that $\mathcal{R}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$ is
cofibrant if for any $n\geq 0$, $\mathcal{R}_{n}$ is a free
$\mathcal{O}$-algebra. Note this is only a sufficient condition. For precise
definition and properties see [Cai21, Section 2.1]. One can show that for any
simplicial ring $\mathcal{R}$, there exists a fibration which is a weak
equivalence $c(\mathcal{R})\to\mathcal{R}$ where $c(\mathcal{R})$ is a
cofibrant simplicial ring. One can either assume that the
$c(\mathcal{R})_{n}$’s are noetherian for all $n$’s, or that
$\mathcal{R}\mapsto c(\mathcal{R})$ is a functorial construction, but in
general not both. Any such $c(\mathcal{R})$ is called a cofibrant replacement
of $\mathcal{R}$.
Let $\mathcal{R}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$ be cofibrant;
then the functor
${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
given by
$A\mapsto\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}}(\mathcal{R},A):{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
is formally cohesive. Any pro-representable (see [GV18, 2.14 and 2.19] and
Section 3.2 below) functor $\mathcal{F}$ should be formally cohesive. J. Lurie
gave a criterion for (pro)representability of a simplicial deformation functor
$\mathcal{F}$ of, say, a point $\ast\in\operatorname{\mathsf{SETS}}^{s}$ (see
[GV18, 4.6]). The first (necessary) condition is formal cohesiveness. The
second involves its tangent complex ${\mathfrak{t}}\mathcal{F}$ which we
recall below.
### 3.1. Tangent complex
Let $A$ be a commutative ring and $Z$ be a commutative $A$-algebra. Then for
any $Z$-module $M$ and $X\in{}_{A}\\!\operatorname{CR}_{Z}$, we have natural
isomorphisms
$\operatorname{\mathsf{Hom}}_{{}_{A}\\!\operatorname{CR}_{Z}}(X,Z\oplus
M)\cong\operatorname{Der}_{A}(X,M)\cong\operatorname{\mathsf{Hom}}_{Z}(\Omega_{X/A}\otimes_{X}Z,M).$
So the functor $X\mapsto\Omega_{X/A}\otimes_{X}Z$ is left ajoint to the
functor $M\mapsto Z\oplus M$. The functors $M\mapsto Z\oplus M$,
$X\to\Omega_{X/A}\otimes_{X}Z$ both have extensions to functors between
simplicial categories, and we have natural isomorphism
$\operatorname{\mathsf{Hom}}_{{}_{A}\\!\operatorname{CR}^{s}_{Z}}(X,Z\oplus
M)\cong\operatorname{\mathsf{Hom}}_{\operatorname{Mod}^{s}_{Z}}(\Omega_{X/A}\otimes_{X}Z,M).$
The functor $M\mapsto Z\oplus M$ preserves fibrations and weak equivalences.
So the total left derived functor of its left adjoint functor
$X\mapsto\Omega_{X/A}\otimes_{X}Z$ exists [GJ10, II.7.3]. It is denoted
$X\mapsto L_{X/Z}$. In order to give a concrete description of $L_{X/Z}$ and
its fundamental property, one often uses the Dold-Kan equivalence. Let
$\operatorname{Ch}(Z)$, resp. $\operatorname{Ch}_{\geq 0}(Z)$, be the category
of chain complexes of $Z$-modules, resp. the subcategory of effective
complexes $(\ldots\to V_{n}\to\ldots\to V_{1}\to V_{0})$. Recall that the
Dold-Kan equivalence is a functor [GV18, Section 4.3.1] and [Cai21, 3.1.4]
$\\\ DK\colon\operatorname{Ch}_{\geq 0}(Z)\to\operatorname{Mod}^{s}_{Z}$
inducing an equivalence of categories that sends weak equivalences to quasi-
isomorphisms. It therefore provides an identification222When no confusion is
possible, we will doing so without specifying it. of the categories of
simplicial $Z$-modules $\operatorname{Mod}^{s}_{Z}$ with the category
$\operatorname{Ch}_{\geq 0}(Z)$ of non-negative chain complexes $(\ldots
V_{n}\to\ldots\ldots V_{1}\to V_{0})$ of $Z$-modules. Now, let
$X\in{}_{A}\\!\operatorname{CR}_{Z}$; to construct $L_{X/Z}$, we choose a
cofibrant replacement $c(X)\to X$ and form the simplicial module
$\Omega_{c(X)/A}\otimes_{A}Z$. By the Dold-Kan equivalence, it comes from a
non-negative chain complex, well defined up to quasi-isomorphism, denoted
$L_{X/Z}$.
Moreover, given $M\in\operatorname{Ch}_{\geq 0}(Z)$, we can form the
simplicial ring $Z\oplus\operatorname{DK}(M)$ where the ideal
$\operatorname{DK}(M)$ is of square zero. We have a weak equivalence of
functors of $M\in\operatorname{Ch}_{\geq 0}(Z)$:
$\operatorname{\mathsf{sHom}}_{{}_{A}\\!\operatorname{CR}^{s}_{Z}}(X,Z\oplus\\\
DK(M))\cong\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{Z}}(L_{X/Z},M).$
For $A=\mathcal{O}$ and $Z=k$, we will use the following definition.
###### Definition 3.
The $k$-cotangent complex functor $\mathcal{R}\mapsto L_{\mathcal{R}/k}$ is
the total left derived functor of the functor
$\mathcal{R}\mapsto\Omega_{\mathcal{R}/\mathcal{O}}\otimes_{\mathcal{R}}k$.
More precisely, for
$\mathcal{R}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$, we have a
simplicial $k$-module
$L_{\mathcal{R}/k}=\Omega_{c(\mathcal{R})/\mathcal{O}}\otimes_{c(\mathcal{R})}k$,
where $c(\mathcal{R})$ is a cofibrant replacement of $\mathcal{R}$ in the
model category ${}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$, and
$L_{\mathcal{R}/k}$ is well-defined in the homotopy category.
The simplicial $k$-module
$\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(L_{\mathcal{R}/k},k)$
may be viewed as an element333Given a non-positive finite dimensional chain
complex $V\in\operatorname{Ch}_{\leq 0}(k)$, its $k$-dual $V^{\vee}$ is a
finite dimensional non-negative chain complex
$V^{\vee}\in\operatorname{Ch}_{\geq 0}(Z)$. of $\operatorname{Ch}_{\geq
0}(k)$, we denote it by ${\mathfrak{t}}\mathcal{R}$ and call it the tangent
complex of $\mathcal{R}$ over $k$. It is well defined in the derived category
of $k$-vector spaces. We label its terms either as
${\mathfrak{t}}^{(i)}\mathcal{R}$ ($i\in{\mathbb{Z}}_{\geq 0}$), or as
${\mathfrak{t}}_{(-i)}\mathcal{R}={\mathfrak{t}}^{(i)}\mathcal{R}$. With these
notations, we have, via Dold-Kan identification of simplicial modules and
chain complexes:
$\pi_{-i}{\mathfrak{t}}\mathcal{R}=H_{-i}({\mathfrak{t}}\mathcal{R})=H^{i}({\mathfrak{t}}\mathcal{R})$.
More generally, if $\mathcal{R}_{1}\to\mathcal{R}_{2}$ is a morphism of
objects
$\mathcal{R}_{1},\mathcal{R}_{2}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$,
we fix compatible cofibrant replacements $c(\mathcal{R}_{1})\to
c(\mathcal{R}_{2})$ in the model category
${}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$. The relative $k$-cotangent
complex
$L_{\mathcal{R}_{2}/\mathcal{R}_{1}}=\Omega_{c(\mathcal{R}_{2})/c(\mathcal{R}_{1})}\otimes_{c(\mathcal{R}_{2})}k$
is independent of the choice of the cofibrant replacements, hence is a well-
defined object in the homotopy category. The simplicial $k$-module
$\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(L_{\mathcal{R}_{2}/\mathcal{R}_{1}},k)$
may be viewed as an element of $\operatorname{Ch}_{\leq 0}(k)$, we denote it
by ${\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1})$ and call it the relative
tangent complex of $\mathcal{R}_{2}$ with respect to $\mathcal{R}_{1}$ over
$k$ (we won’t specify this below). It is well defined in the derived category
of $k$-vector spaces.
###### Lemma 4.
If $\mathcal{R}_{1}\to\mathcal{R}_{2}$ is a morphism of objects
$\mathcal{R}_{1},\mathcal{R}_{2}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$,
we have a distinguished triangle
${\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1})\to{\mathfrak{t}}\mathcal{R}_{2}\to{\mathfrak{t}}{\mathcal{R}_{1}}\stackrel{{\scriptstyle+1}}{{\rightarrow}}$
hence a long exact sequence
$\ldots H^{i}({\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1}))\to
H^{i}({\mathfrak{t}}\mathcal{R}_{2})\to
H^{i}({\mathfrak{t}}\mathcal{R}_{1})\to
H^{i+1}({\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1}))\to\ldots$
###### Proof.
Apply the exact functor
$\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(-,k)$ to the
distinguished triangle
$L_{\mathcal{R}_{1}/k}\underline{\otimes}_{\mathcal{R}_{1}}k\to
L_{\mathcal{R}_{2}/k}\to
L_{\mathcal{R}_{2}/\mathcal{R}_{1}}{\rightarrow}(L_{\mathcal{R}_{1}/k}\underline{\otimes}_{\mathcal{R}_{1}}k)[1]$
similar to [COT, Section 90.7]. To obtain this distinguished triangle, first,
take a cofibrant replacement $c(\mathcal{R}_{1})\to c(\mathcal{R}_{2})$ of
$\mathcal{R}_{1}\to\mathcal{R}_{2}$, then form the exact sequence of
simplicial $c(\mathcal{R}_{2})$-modules
$0\to\Omega_{c(\mathcal{R}_{1})/\mathcal{O}}\underline{\otimes}_{c(\mathcal{R}_{1})}c(\mathcal{R}_{2})\to\Omega_{c(\mathcal{R}_{2})/\mathcal{O}}\to\Omega_{c(\mathcal{R}_{2})/c(\mathcal{R}_{1})}{\rightarrow}0$
where the first tensor product is taken for simplicial
$c(\mathcal{R}_{1})$-modules; then, apply $-\otimes_{c(\mathcal{R}_{2})}k$. ∎
Let $k[n]$ be the object of $\operatorname{Ch}_{\geq 0}(k)$ defined by $k$ in
degree $n$ and $0$ elsewhere. Then, we have
$\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}}(\mathcal{R},k\oplus
DK(k[n]))\cong\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(L_{\mathcal{R}/k},DK(k[n]))$
for $\mathcal{R}$ cofibrant, and hence for $i\geq 0$, we have
$\pi_{i}\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}}(\mathcal{R},k\oplus
DK(k[n]))\cong H^{i+n}({\mathfrak{t}}\mathcal{R}).$
###### Proposition 1.
Let
$\mathcal{F}:{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
be a formally cohesive functor. Then there exists an object
${\mathfrak{t}}\mathcal{F}\in\operatorname{Ch}(k)$, unique up to weak
equivalence, such that for any non-positive finite dimensional chain complex
$V\in\operatorname{Ch}_{\geq 0}(k)$, we have
$\operatorname{\mathsf{sHom}}_{\operatorname{Ch}(k)}(V,{\mathfrak{t}}\mathcal{F})\cong\mathcal{F}(k\oplus\\\
DK(V^{\vee})).$
###### Proof.
See [Cai21, Corollary 3.43]. ∎
The chain complex ${\mathfrak{t}}\mathcal{F}$ is called the tangent complex of
$\mathcal{F}$. We denote the terms of this complex as
${\mathfrak{t}}_{i}\mathcal{F}$ (with degree $-1$ differential) for
$i\in{\mathbb{Z}}$, or
${\mathfrak{t}}^{(i)}\mathcal{F}={\mathfrak{t}}_{(-i)}\mathcal{F}$ (with
degree $+1$ differential). We also write
${\mathfrak{t}}^{i}\mathcal{F}=\mathrm{H}^{i}({\mathfrak{t}}\mathcal{F})=\mathrm{H}_{-i}({\mathfrak{t}}\mathcal{F})$.
If we let $V=k[-n]$ and take the $i$-th homotopy group, then we get
$\mathrm{H}^{i+n}{\mathfrak{t}}\mathcal{F}=\pi_{-i}\mathcal{F}(k\oplus
DK(k[n]))$ for any $i,n\geq 0$. We write ${\mathfrak{t}}^{i}\mathcal{F}$ to
abbreviate
$\mathrm{H}^{i}({\mathfrak{t}}\mathcal{F})=\pi_{i}{\mathfrak{t}}\mathcal{F}$
(the last equality is via Dold-Kan, viewing ${\mathfrak{t}}\mathcal{F}$ as a
simplicial $k$-vector space). To end this section, we recall the important
conservativity property of the tangent complex functor.
###### Lemma 5.
Let $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ be formally cohesive functors from
${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$ to
$\operatorname{\mathsf{sSETS}}$. Then a natural transformation
$T:\mathcal{F}_{1}\to\mathcal{F}_{2}$ is a weak equivalence if and only if $T$
induces isomorphisms
${\mathfrak{t}}^{i}\mathcal{F}_{1}\cong{\mathfrak{t}}^{i}\mathcal{F}_{2}$ for
all $i$.
###### Proof.
See [Cai21, Corollary 3.46]. ∎
### 3.2. Representability
Given a projective system $\mathcal{R}=(\mathcal{R}_{\alpha})_{\alpha}$ with
$\mathcal{R}_{\alpha}\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$, we put
for any $A\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$
$\operatorname{\mathsf{sHom}}(\mathcal{R},A)=\operatorname{holim}_{\alpha}\operatorname{\mathsf{sHom}}(\mathcal{R}_{\alpha},A)$
###### Definition 4.
A functor
$\mathcal{F}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{sSETS}}$
is proprepresentable if there exists a projective system of cofibrant
simplical artinian rings $\mathcal{R}=(\mathcal{R}_{\alpha})_{\alpha}$ with
$\mathcal{R}_{\alpha}\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$ and a
functorial weak equivalence:
$\operatorname{\mathsf{sHom}}(\mathcal{R},A)\to\mathcal{F}(A)$
for all $A\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$.
Note that ${\mathfrak{t}}^{i}\mathcal{R}=H_{-i}({\mathfrak{t}}\mathcal{R})$
vanishes for $i<0$. So if a formally cohesive functor
$\mathcal{F}:{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
is pro-representable, then ${\mathfrak{t}}^{i}\mathcal{F}$ should vanishes for
$i<0$. The converse statement is proved by Lurie.
###### Theorem 9 (Lurie’s derived Schlessinger criterion).
Let $\mathcal{F}$ be a formally cohesive functor. Then $\mathcal{F}$ is pro-
representable if and only if ${\mathfrak{t}}^{i}\mathcal{F}$ vanishes for each
$i<0$. In addition, if all the $k$-vector spaces
${\mathfrak{t}}^{i}\mathcal{F}$ are finite, the pro-artinian ring
$\mathcal{R}$ has a countable indexing category.
### 3.3. Simplicial Galois deformation functors
In this section we consider a smooth group scheme $G$ defined over
$\mathcal{O}$. For our application, $G=\operatorname{\mathsf{GL}}_{N}$. Given
an $\mathcal{O}$-algebra $A$, let ${\bf B}G(A)$ be the simplicial set
associated to the group $G(A)$ by the bar construction. By definition, it is
given by $[p]\mapsto N_{p}G(A)=G(A)^{p}$, with standard face and degeneracy
maps [Wei94, Example 8.1.7]. Note that for any fixed $p$,
$N_{p}G(A)=\operatorname{\mathsf{Hom}}_{rings}(\mathcal{O}_{G}^{\otimes
p},A)$. We can therefore say that the functor $N_{p}G\colon A\mapsto
N_{p}G(A)$ is represented by the ring
$\mathcal{O}_{N_{p}G}=\mathcal{O}_{G}^{\otimes p}$.
If now $A$ is a simplicial $\mathcal{O}$-algebra, we can define a bisimplicial
set $\operatorname{Bi}^{naive}_{G}(A)$, i.e. a contravariant functor
$\operatorname{Bi}^{naive}_{G}(A)\colon\Delta\times\Delta\to\operatorname{\mathsf{SETS}},\quad([p],[q])\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}}(\mathcal{O}_{N_{p}G},A_{q})=G(A_{q})^{p}$
where the $p$-degeneracy and $p$-face maps are given by the codegeneracy and
coface maps of $\mathcal{O}_{N_{\bullet}G}$ and the $q$-degeneracy and
$q$-face maps are given by the degeneracy and face maps of $A$. Recall that
given two simplicial rings $R,R^{\prime}$, we defined in Definition 2 the
simplicial set $\operatorname{\mathsf{sHom}}(R,R^{\prime})$ as
$[p]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(R,(R^{\prime})^{\Delta[p]})$
with face and degeneracy maps coming from those between the $\Delta[p]$’s. We
can consider for each $p$ the bisimplicial set $\operatorname{Bi}_{G}(A)$
given by
$([p],[q])\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(c(\mathcal{O}_{N_{p}G}),A^{\Delta[q]})$
where $c(\mathcal{O}_{N_{p}G})$ denotes a cofibrant replacement of
$\mathcal{O}_{N_{p}G}$, with degeneracy and face maps as for the bisimplicial
set $\operatorname{Bi}^{naive}_{G}(A)$. Note that the natural morphism of
simplicial rings $A_{q}\to A^{\Delta[q]}$ induces an injective morphism of
bisimplicial sets
$\operatorname{Bi}^{naive}_{G}(A)\hookrightarrow\operatorname{Bi}_{G}(A).$
Consider the simplicial set ${\bf
B}^{\prime}G(A)=\operatorname{hocolim}_{q}\operatorname{Bi}_{G}(A)_{q}$. It
may not be fibrant. So, we fix a fibrant replacement
${\bf B}G(A)=F({\bf B}^{\prime}G(A)).$
For $A$ homotopically discrete, this obviously generalizes the usual
definition of ${\bf B}G(\pi_{0}(A))$.
Remark: Recall that in [GV18, Definition 5.1], the authors defined a
simplicial set ${\bf B}^{GV}G(A)$ as the fibrant replacement $Ex^{\infty}$ of
the simplicial set
$[p]\mapsto([p],[p])\mapsto\operatorname{Bi}_{G}(A)([p],[p])=\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(c(\mathcal{O}_{N_{p}G}),A^{\Delta[p]}).$
The two definitions are weakly equivalent but there is a slight difference: in
our definition, we have a canonical fibration ${\bf B}G(A)\to{\bf B}G(k)$,
while for ${\bf B}G^{GV}(A)$, one has a canonical fibration ${\bf B}G(A)\to
Ex^{\infty}{\bf B}G(k)$.
The key properties of this construction are that:
* •
The functor ${\bf B}G$ is invariant by homotopy : if $A\to B$ is a weak
equivalence, the resulting morphism of simplicial sets ${\bf B}G(A)\to{\bf
B}G(B)$ also is, by smoothness of $\mathcal{O}_{G}$ (see proof of [GV18,
Corollary 5.3]).
* •
It preserves homotopy pullbacks.
* •
The ${\bf B}G$ construction is functorial in $G$.
Let us write the $S\cup S_{p}$-ramified Galois group $\Gamma=G_{F,S\cup
S_{p}}$ as inverse limit of finite Galois groups
$\Gamma_{\alpha}=\operatorname{Gal}(F_{\alpha}/F)$, where $F_{\alpha}/F$ is a
finite $S\cup S_{p}$-ramified extension. Let ${\bf B}\Gamma$ be the
prosimplicial set given by the projective system $({\bf
B}\Gamma_{\alpha})_{\alpha}$ of simplicial sets ${\bf B}\Gamma_{\alpha}$. Note
that the residual Galois representation
$\overline{\rho}\colon\Gamma\to G(k)$
gives rise to an element $[\overline{\rho}]$ of
$\operatorname{\mathsf{sHom}}({\bf B}\Gamma_{\alpha},{\bf B}G(k))$ for some
$\alpha$. A natural idea to define the analogue of the classical deformation
functor
$\mathcal{F}_{\overline{\rho}}\colon{}_{\mathcal{O}}\\!\operatorname{Art}_{k}\to\operatorname{\mathsf{SETS}},A\mapsto\operatorname{\mathsf{Hom}}_{\overline{\rho}}(\Gamma,G(A))/adG(A)$
is to define for each $A\in
Ob({}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k})$, and any $\alpha$, the
simplicial set
$\mathcal{F}^{s}_{\alpha,\overline{\rho}}(A)=\operatorname{\mathsf{sHom}}_{[\overline{\rho}]}({\bf
B}\Gamma_{\alpha},{\bf B}G(A))$
which is the homotopy fiber in
$\mathcal{F}^{s}_{\alpha}(A)=\operatorname{\mathsf{sHom}}({\bf
B}\Gamma_{\alpha},{\bf B}G(A))$
of $[\overline{\rho}]\in\operatorname{\mathsf{sHom}}({\bf
B}\Gamma_{\alpha},{\bf B}G(k))$. Then, one would pass to the inductive limit
over $\alpha$ to obtain $\mathcal{F}^{s}_{\overline{\rho}}$ as the homotopy
fiber at $[\overline{\rho}]$ of
$\mathcal{F}^{s}=\varinjlim_{\alpha}\mathcal{F}^{s}_{\alpha}.$
It is homotopy invariant and preserves homotopy pullbacks. However, as
explained in [GV18, Section 5.4], when the center $Z$ of $G$ is non trivial,
the functor $\mathcal{F}^{s}_{\overline{\rho}}$ cannot be representable
because $Z$ gives rise to non trivial automorphisms. It can be modified in two
different ways to be made pro-representable: one is to fix the determinant
${\bf B}\det\colon{\bf B}G(A)\to{\bf B}{{\mathbb{G}}}_{m}(A)$
of the simplicial deformations, the other is to "quotient the action of $G$ by
the center $Z$" (with or without fixing the determinant) as in [GV18, Section
5.4]. The two modifications will be denoted
$\mathcal{F}^{s}_{\det,\overline{\rho}}$ and
$\mathcal{F}^{s}_{Z,\overline{\rho}}$. Let us recall their constructions.
### 3.4. Modification by the center
We first define $\mathcal{F}^{s}_{\det,\overline{\rho}}$.
Let $\rho_{\pi}\colon\Gamma\to G(\mathcal{O})$ be the Galois representation
that we fixed. For any $A\in
Ob({}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k})$, we have a morphism ${\bf
B}G(\mathcal{O})\to{\bf B}G(A)$. On the other hand, the determinant morphism
induces morphisms (both denoted by $\det$) of simplicial sets
${\bf B}G(A)\to{\bf B}{{\mathbb{G}}}_{m}(A)\quad\mbox{\rm
and}\,\mathcal{F}^{s}_{G}(A)=\operatorname{\mathsf{sHom}}({\bf B}\Gamma,{\bf
B}G(A))\to\mathcal{F}^{s}_{{{\mathbb{G}}}_{m}}(A)=\operatorname{\mathsf{sHom}}({\bf
B}\Gamma,{\bf B}{{\mathbb{G}}}_{m}(A))$
We recall now the definition of $\mathcal{F}^{s}_{Z,\overline{\rho}}$ ([GV18,
Section 5.4]). Consider the short exact sequence
$1\to Z(A)\to G(A)\to PG(A)\to 1$
It gives rise to a fibration sequence (see the section 3.1.3 of [Cai21]).
${\bf B}G(A)\to{\bf B}PG(A)\to{\bf B}^{2}Z(A).$
Let $\ast=\pi_{0}{\bf B}\Gamma$. Then the functor $\mathcal{F}^{s}_{Z}$ is
defined by the homotopy pullback diagram
$\begin{array}[]{ccc}\mathcal{F}^{s}_{Z}(A)&\to&\operatorname{\mathsf{Hom}}(\ast,{\bf
B}^{2}Z(A)\\\ \downarrow&&\downarrow\\\
\mathcal{F}^{s}(A)&\to&\operatorname{\mathsf{Hom}}({\bf B}\Gamma,{\bf
B}^{2}Z(A))\end{array}$
Its crucial property is that there is a functorial fibration sequence:
$\operatorname{\mathsf{sHom}}(\ast,{\bf
B}Z(A))\to\operatorname{\mathsf{sHom}}({\bf B}\Gamma,{\bf
B}G(A))\to\mathcal{F}^{s}_{Z}(A)$
We then define $\mathcal{F}^{s}_{Z,\overline{\rho}}$ as the homotopy fiber of
$[\overline{\rho}]$. It follows from the remarks above that
###### Lemma 6.
The functor $\mathcal{F}^{s}_{Z,\overline{\rho}}$ is homotopy invariant and
preserves homotopy pullbacks, hence is formally cohesive.
Let $\mathcal{F}^{s,S\cup
S_{p}}_{Z,\overline{\rho}}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{sSETS}}$
be the functor given by modifying as above the functor
$\mathcal{F}^{s}_{\overline{\rho}}(A)={\mathfrak{s}}_{[\overline{\rho}]}({\bf
B}\Gamma,{\bf
B}G(A))=\operatorname{holim}_{\alpha}\operatorname{\mathsf{sHom}}_{[\overline{\rho}]}({\bf
B}\Gamma_{\alpha},{\bf B}G(A))$
The functor $\mathcal{F}^{s,S\cup S_{p}}_{Z,\overline{\rho}}$ is called the
functor of simplicial deformations of $\overline{\rho}$ unramified outside
$S\cup S_{p}$. To study the representability of this deformation functor, we
now need to determine its tangent complex. The tangent complex of
$\mathcal{F}^{s}_{\alpha}$ can be computed as a Cech complex $C^{\bullet}({\bf
B}\Gamma_{\alpha},{\mathfrak{t}}{\bf B}G)$ ([GV18, Section 4.7] and [GV18,
Example 4.38]). Let ${\mathfrak{g}}_{k}=\operatorname{Lie}_{k}G$ be the Lie
algebra of $G$. The tangent complex ${\mathfrak{t}}{\bf B}G$ is concentrated
in degree $1$ and one has ${\mathfrak{t}}{\bf B}G={\mathfrak{g}}_{k}[1]$
([GV18, Lemma 5.5]). Therefore
${\mathfrak{t}}\mathcal{F}^{s}_{\alpha}=C^{\bullet+1}({\bf
B}\Gamma_{\alpha},{\mathfrak{g}}_{k})$
Let ${\mathfrak{z}}_{k}$ be the center of ${\mathfrak{g}}_{k}$. Then, the
functorial fibration 3.4 provides a distinguished triangle
$C^{\bullet+1}(\ast,{\mathfrak{z}}_{k})\to C^{\ast+1}({\bf
B}\Gamma,{\mathfrak{g}}_{k})\to{\mathfrak{t}}\mathcal{F}^{s}_{Z,\bar{\rho}}.$
From these facts it follows that the Cech complex $C^{\bullet}({\bf
B}\Gamma,{\mathfrak{t}}{\bf B}G)$ can be computed as the Galois cohomology
standard complex (see [Cai21]). Then by [GV18, Lemma 4.30, (iv)] (Mayer-
Vietoris long exact sequence) we have the following key proposition:
###### Proposition 2.
Suppose that $\mathrm{H}^{0}(\Gamma,{\mathfrak{g}}_{k})={\mathfrak{z}}_{k}$.
Then, the homology groups of the tangent complex
${\mathfrak{t}}\mathcal{F}^{s}_{Z,\bar{\rho}}$ satisfy
${\mathfrak{t}}^{i}\mathcal{F}^{s}_{Z,\bar{\rho}}=0$ when $i\leq 0$, and
${\mathfrak{t}}^{i}\mathcal{F}_{Z,\bar{\rho}}=\mathrm{H}^{i+1}(F_{S}/F,\operatorname{Ad}\bar{\rho})$
when $i\geq 0$.
and its corollary:
###### Corollary 2.
Suppose that the adjoint action of $\bar{\rho}$ fixes precisely the center of
the Lie algebra of $G(k)$. The functor $\mathcal{F}^{s}_{Z,\bar{\rho}}$ is
pro-representable by a simplicial pro-artinian ring $\mathcal{R}$. This means
that there exists a functorial weak equivalence
$\operatorname{\mathsf{sHom}}(\mathcal{R},A)\to\mathcal{F}^{s}_{Z,\bar{\rho}}(A)$
Note that, given a profinite group $\Gamma$, the question of
prorepresentability of the functor
$A\mapsto\mathcal{F}_{\bar{\rho}}^{s}(A)=\operatorname{\mathsf{sHom}}_{[\overline{\rho}]}({\bf
B}\Gamma,{\bf B}G(A))$
by a proartinian simplicial ring $\mathcal{R}$ can only be posed in terms of
the existence of a functorial weak equivalence
$\operatorname{\mathsf{sHom}}_{pro\operatorname{Art}^{s}_{k}}(R,A)\to\mathcal{F}_{\bar{\rho}}^{s}(A)$
and not of a functorial isomorphism, because even is $A$ is discrete, there is
a bijection between the set of conjugacy classes of homomorphisms $\Gamma\to
G(A)$ and the set of homotopy classes of the simplicial set of morphisms of
prosimplicial sets ${\bf B}\Gamma\to{\bf B}G(A)$, not with the simplicial set
of morphisms of prosimplicial sets itself.
###### Lemma 7.
The functor
$\pi_{0}\mathcal{F}^{s}_{Z,\overline{\rho}}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}\quad
A\mapsto\pi_{0}\mathcal{F}^{s}_{Z,\overline{\rho}}(A)$
coincides with Mazur’s functor of $S\cup S_{p}$-ramified deformations of
$\overline{\rho}$.
###### Proof.
This is [GV18, Lemma 7.1]. It follows from the fact that for a discrete
artinian ring $A$, ${\bf B}G(A)$ is weakly equivalent to the classical
classifying space of $G(\pi_{0}(A))$ defined in [Wei94, Example 8.1.7].
Therefore $\pi_{0}(\operatorname{\mathsf{sHom}}({\bf B}\Gamma,{\bf B}G(A)))$
is $\operatorname{\mathsf{Hom}}({\bf B}\Gamma,{\bf B}G(\pi_{0}(A)))$, which is
naturally $\operatorname{\mathsf{Hom}}_{cont}(\Gamma,G(A))/G(A)$. By the
definition of $\mathcal{F}^{s}_{Z,\overline{\rho}}$, this implies that
$\pi_{0}\mathcal{F}^{s}_{Z,\overline{\rho}}(A)$ is naturally
$\operatorname{\mathsf{Hom}}_{cont}(\Gamma,G(A))/PG(A)=\operatorname{\mathsf{Hom}}_{cont}(\Gamma,G(A))/G(A)$,
as desired. ∎
### 3.5. Simplicial Galois deformation functors with local conditions
Let us define simplicial local deformation functors $\mathcal{F}_{v}^{s,?}$.
For $v\in S\cup S_{p}$, let us write $\Gamma_{v}=G_{F_{v}}$ as inverse limit
of finite Galois groups $\operatorname{Gal}(F^{\prime}_{v}/F_{v})$. Recall
that for $?=min,\operatorname{ord},\operatorname{ord}-\mu,FL$, under suitable
assumptions we have seen (Lemma 1, Lemma 2, Lemma 3, that the ring
$R_{v}^{?,\square}$ representing $\mathcal{F}_{v}^{?,\square}$ is formally
smooth. The functor $\mathcal{F}_{v}^{s}$ of arbitrary simplicial deformations
of $\overline{\rho}|_{\Gamma_{v}}$ is given by
$\mathcal{F}^{s}_{v}(A)=\operatorname{\mathsf{sHom}}_{\overline{\rho}_{v}}({\bf
B}\Gamma_{v},{\bf
B}G(A))=\operatorname{holim}_{F^{\prime}_{v}}\operatorname{\mathsf{sHom}}_{\overline{\rho}_{v}}({\bf
B}\operatorname{Gal}(F^{\prime}_{v}/F_{v}),{\bf B}G(A))$
We denote by $\mathcal{F}^{s}_{v,Z}$ its modification by the center as in
[GV18, Section 5.4] (or Section 2.4 above). Note that by [GV18, Lemma 5.2], we
have
${\mathfrak{t}}^{-1}\mathcal{F}_{v,Z}^{s}=\operatorname{Coker}(\mathrm{H}^{0}(\\{1\\},{\mathfrak{z}})\to\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}))$
which doesn’t vanish (except in certain FL cases). Therefore, by the converse
of Lurie’s criterion (Theorem 9 above), $\mathcal{F}_{v,Z}^{s}$ is not pro-
representable. However, by [GV18, Lemma 5.2] we have
###### Lemma 8.
For any $i\geq 0$,
${\mathfrak{t}}^{i}\mathcal{F}_{v,Z}^{s}={\mathfrak{t}}^{i}\mathcal{F}_{v}^{s}\cong\mathrm{H}^{i+1}(\Gamma_{v},{\mathfrak{g}}_{k}))$
For $?=min,\operatorname{ord},FL$, following [GV18, Section 9.2], we shall
define local deformation functors $\mathcal{F}_{v}^{s,?}$ with morphisms
$\mathcal{F}_{v}^{s,?}\to\mathcal{F}_{v}^{s}$
which induce embeddings of the tangent complexes. They are defined as "derived
quotients" of functors $\mathcal{F}_{v}^{s,?,\square}$ by the adjoint action
of $G$. Recall that for a simplicial set $X$ with action of a group $G$, the
simplicial quotient $G\backslash X$ is defined as the simplicial set
$N(\ast,G,X)$ given by the diagonal of the bisimplicial set
$([p],[q])\mapsto\ast\times G^{p}\times X_{q}$ where for $q$ fixed, the
$p$-faces and degeneracy maps are given by the bar construction $[p]\mapsto
N_{p}(\ast,G,X_{q})=\ast\times G^{p}\times X_{q}$ while the $q$-ones are given
by the simplicial set $X$. We now form, so to speak, a formally cohesive
replacement of the simplicial analogue of the quotient functor
$A\mapsto\widehat{G}(A)\backslash\mathcal{F}_{v,\ast}^{\square,?}(A).$
Namely, for each simplicial $\mathcal{O}$-algebra $A$ in
${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$, we define the simplicial set
$\mathcal{F}_{v}^{?}(A)$ as
$[p]\mapsto\operatorname{\mathsf{Hom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(R_{v}^{?,\square}\otimes\mathcal{O}_{N_{p}G}),A^{\Delta[p]})$
By the use of the cofibrant replacement of
$R_{v}^{?,\square}\otimes\mathcal{O}_{N_{p}G}$, we know that
$\mathcal{F}_{v}^{?}$ is formally cohesive (by [Cai21, Example 2.2.9]). The
existence of the natural transformation of functors
$\mathcal{F}_{v}^{s,?}\to\mathcal{F}_{v}^{s}$
is established in great generality (which includes $?=min,ord$) in [Cai21,
Proposition 5.15].
###### Lemma 9.
We have a fibration
$\mathcal{F}_{v}^{s,\square,?}\to\mathcal{F}_{v}^{s,?}\to\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(-)\to BG(k)).$
###### Proof.
By definition (see [GV18, Definition 5.4 (i)]),
$\mathcal{F}^{s,?,\square}_{v}(A)$ is the homotopy fiber of the composed
morphism
$\mathcal{F}^{s,?,\square}_{v}(A)\to\operatorname{\mathsf{sHom}}_{[\overline{\rho}_{v}]}(B\Gamma_{v},{\bf
B}G(A))\to\operatorname{\mathsf{hofib}}({\bf B}G(A)\to{\bf B}G(k))$
where the second map is the evaluation on the base point $e=(e_{n})_{n}\in
B\Gamma_{v}$ (with $e_{n}=(e,\ldots,e)\in\Gamma_{v}^{n}$). By definition,
$\mathcal{F}^{s,?}_{v}(A)$ is the diagonal of the bisimplicial set
$[p]\mapsto\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{N_{p}G}\otimes
R^{?,\square}_{v}),A)\simeq\prod_{i=1}^{p}\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{G}),A)\times\mathcal{F}_{v}^{s,\square,?}(A).$
Similarly, $\operatorname{\mathsf{hofib}}_{\ast}({\bf B}G(A)\to BG(k))$
(homotopy fiber of the standard marked point in $BG(k)$) is the diagonal of
the bisimplicial set
$[p]\mapsto\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{N_{p}G}),A)\simeq\prod_{i=1}^{p}\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{G}),A).$
Therefore, the morphism
$s\mathcal{F}^{s,\square,?}_{v}(A)\to\operatorname{\mathsf{sHom}}_{[\overline{\rho}_{v}]}(B\Gamma_{v},{\bf
B}G(A))$ factors through
$s\mathcal{F}^{s,\square,?}_{v}(A)\to\mathcal{F}^{s,?}_{v}(A)$. This gives the
desired fibration. ∎
###### Remark 1.
Note that in the simplicial context, the classical fibration
$\widehat{G}(A)\to\mathcal{F}_{v}^{\square,?}(A)\to\mathcal{F}_{v}^{?}(A)$ is
replaced by
$\mathcal{F}_{v}^{s,\square,?}\to\mathcal{F}_{v}^{s,?}\to\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(-)\to BG(k))$
where, for a classical ring $A$, $\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(A)\to BG(k))$ is to be thought of as $B\widehat{G}(A)[1]$.
Let us put
$\mathrm{H}^{1}_{min}(\Gamma_{v},{\mathfrak{g}}_{k}):=\mathrm{H}^{1}_{unr}(\Gamma_{v},{\mathfrak{g}}_{k}),\quad\mathrm{H}^{1}_{\operatorname{ord}}(\Gamma_{v},{\mathfrak{g}}_{k}):=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})),$
where
$L^{\prime}_{v}=\operatorname{Ker}(\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{b}})\to\mathrm{H}^{1}(I_{v},{\mathfrak{b}}/{\mathfrak{n}}))$,
and
$\mathrm{H}^{1}_{FL}(\Gamma_{v},{\mathfrak{g}}_{k})=\mathrm{H}^{1}_{f}(\Gamma_{v},{\mathfrak{g}}_{k}).$
By the long exact sequence above, we conclude:
###### Lemma 10.
For $?=min,\operatorname{ord},\operatorname{ord}-\mu,FL$, we have
1. (1)
${\mathfrak{t}}^{-1}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,
2. (2)
${\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})$,
3. (3)
${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s,?}=0$ for $i\geq 1$.
###### Proof.
Recall that for any deformation functor $\mathcal{F}$,
${\mathfrak{t}}^{n-i}\mathcal{F}\cong\pi_{i}\mathcal{F}(k\oplus k[n])$. By
Lemma 4.30 (4) of [GV18], or [GJ10, Lemma 7.3], the fibration of Lemma 9 gives
rise to a long exact sequence
$\displaystyle
0\to{\mathfrak{t}}^{-1}s\mathcal{F}_{v}^{s,?}\to{\mathfrak{t}}^{-1}\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(-)\to BG(k))$ $\displaystyle\to$
$\displaystyle{\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,\square,?}\to{\mathfrak{t}}^{0}s\mathcal{F}_{v}^{s,?}\to
0\to
t^{1}\mathcal{F}^{s,\square,?}\to{\mathfrak{t}}^{1}s\mathcal{F}_{v}^{s,?}\to
0$
(we have used that $\operatorname{\mathsf{hofib}}_{\ast}(BG(k[\epsilon])\to
BG(k))$ is a $K(\pi,1)$). Note that
${\mathfrak{t}}^{-1}\operatorname{\mathsf{hofib}}_{\ast}({\bf B}G(-)\to
BG(k))\cong\pi_{1}\operatorname{\mathsf{hofib}}_{\ast}(BG(k[\epsilon])\to
BG(k))$, and $\pi_{1}\operatorname{\mathsf{hofib}}_{\ast}(BG(k[\epsilon])\to
BG(k))$ is $\operatorname{Ker}(G(k[\epsilon])\to G(k))={\mathfrak{g}}_{k}$.
Therefore, the tangent spaces ${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s,?}$ can be
calculated as follows. Since $\mathcal{F}_{v}^{s,\square,?}$ is prorepresented
by $c(R_{v}^{\square,?})$ which is weakly equivalent to the discrete ring
$R_{v}^{\square,?}$, its tangent complex is given by
${\mathfrak{t}}\mathcal{F}_{v}^{s,\square,?}={\mathfrak{t}}R_{v}^{\square,?}$
Moreover, since $R_{v}^{\square,?}$ is formally smooth, we know that
${\mathfrak{t}}R_{v}^{\square,?}$ is quasi-isomorphic to the complex
${\mathfrak{t}}^{0}R_{v}^{\square,?}[0]$ concentrated in degree $0$. It is
well-known that
${\mathfrak{t}}^{0}R_{v}^{\square,?}=Z^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})$.
Hence we can rewrite the long exact sequence above as
$\displaystyle
0\to{\mathfrak{t}}^{-1}\mathcal{F}_{v}^{s,?}\to{\mathfrak{g}}_{k}\to
Z^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})\to{\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,?}\to
0\to 0\to{\mathfrak{t}}^{1}\mathcal{F}_{v}^{s,?}\to 0$
This yields the Lemma.
∎
As in the global case, we define also the deformation functors
$\mathcal{F}_{v,Z}^{s}$ and $\mathcal{F}_{v,Z}^{s,?}$ by modifying the center
$Z$. They are inserted in a fibration
$\operatorname{\mathsf{sHom}}(\ast,{\bf
B}Z)\to\mathcal{F}_{v}^{s,?}\to\mathcal{F}_{v,Z}^{s,?}$
similar to (3.4).
For $v\in S_{p}$ and $?=\operatorname{ord}$, we can give a more concrete
alternative definition of the local nearly ordinary deformation functor. We
define $\widetilde{\mathcal{F}}^{s,\operatorname{ord}}_{v}(A)$ as the set of
simplicial deformations $\phi\in\operatorname{\mathsf{sHom}}({\bf
B}G_{F_{v}},{\bf B}G(A))$ of $\overline{\rho}|_{G_{F_{v}}}$, which factor
through the morphism of simplicial sets ${\bf B}B(A)\to{\bf B}G(A)$ associated
to the inclusion of the standard Borel $B\subset G$. for
$?=\operatorname{ord}-\mu$, let
$\operatorname{Def}^{s}_{v,T}=\operatorname{\mathsf{sHom}}_{\operatorname{\mathsf{sSETS}}{}_{/BT(k)}}(BI_{v},{\bf
B}T(-))$ be the derived deformation functor of $\bar{\chi}_{v}|_{I_{v}}\colon
I_{v}\to T(k)$. Then there is a natural transformation
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}\to\operatorname{Def}^{s}_{v,T}$.
Note that $\mu_{v}$ defines a natural transformation
$\ast\to\operatorname{Def}^{s}_{v,T}$. We define
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}=\operatorname{\mathsf{hofib}}_{\mu_{v}}(\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}\to\operatorname{Def}^{s}_{v,T}).$
In addition, the natural transformation
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}\to\mathcal{F}^{s}_{v}$
induces
$\widetilde{\mathcal{F}}_{v,Z}^{s,\operatorname{ord},\mu}\to\mathcal{F}^{s}_{v,Z}$.
###### Lemma 11.
The functors $\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}$ and
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$ are formally
cohesive. Assuming $(STDIST)$, the tangent complex
${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}\cong{\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$,
is quasi-isomorphic to
${\mathfrak{t}}\mathcal{F}^{s,\operatorname{ord}}_{v}\cong{\mathfrak{t}}\mathcal{F}_{v}^{s,\operatorname{ord},\mu}$
:
* •
${\mathfrak{t}}^{-1}\widetilde{\mathcal{F}}_{v,\ast}^{s,\operatorname{ord}}\cong\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,
* •
${\mathfrak{t}}^{0}\widetilde{\mathcal{F}}_{v,\ast}^{s,\operatorname{ord}}\cong\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k}^{\prime})$,
* •
${\mathfrak{t}}^{i}\widetilde{\mathcal{F}}_{v,\ast}^{s,\operatorname{ord}}=0$
for $i\geq 1$.
Moreover, we have
$\pi_{0}\widetilde{\mathcal{F}}^{s,\operatorname{ord}}_{v}=\mathcal{F}_{v}^{\operatorname{ord}}.$
###### Proof.
Note that $\operatorname{Def}^{s}_{v,T}$ is formally cohesive. We see that
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}$ is formally cohesive: the
functor
$A\mapsto\operatorname{\mathsf{sHom}}_{[\overline{\rho}_{v}]}({\bf
B}G_{F_{v}},{\bf B}B(A))$
is homotopy invariant and preserves homotopy pullbacks. From this, it follows
by definition that $\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$ is
also formally cohesive.
The tangent complex
${\mathfrak{t}}^{i}\widetilde{\mathcal{F}}_{v}^{\operatorname{ord}}$ is
calculated in [GV18, Lemma 5.6] (replacing $G$ by $B$). More precisely, let
${\mathfrak{b}}=Lie(B)$. Then we have a quasi-isomorphism of complexes
${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{\operatorname{ord}}\sim
C^{\ast+1}({B}\Gamma_{v},{\mathfrak{b}})$
The assumption $(reg^{\ast})$ implies
$\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}/{\mathfrak{b}}_{k})=0$. Thus,
${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}$ is quasi-
isomorphic to ${\mathfrak{t}}\mathcal{F}^{s,\operatorname{ord}}_{v}$. The same
holds for
${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$. If
$A\in Ob({}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k})$ is a discrete
artinian ring, $\mathcal{F}^{s,\operatorname{ord}}_{v}(A)$ is the set of
$B$-conjugacy classes of liftings of $\overline{\rho}_{v}$; because of
$(reg)$, the set of $B$-conjugacy classes coincide with the set of
$G$-conjugacy classes of representations $\rho_{v}\colon G_{F_{v}}\to G(A)$,
such that after conjugation by some $g_{v}\in G(A)$, $\rho_{v}$ takes values
in $B(A)$ and such that the composition of ${}^{g_{v}}\\!\rho_{v}$ and the
reduction $B(A)\to T(A)=B(A)/U_{B}(A)$ is a lifting of
$\underline{\overline{\chi}_{v}}\colon G_{F_{v}}\to T(k)$. (see [Ti96, Claim
in Proof of Proposition 6.2]). Hence we conclude
$\widetilde{\mathcal{F}}^{s,\operatorname{ord}}_{v}(A)=\mathcal{F}^{\operatorname{ord}}_{v}(A)$.
Similarly for
$\widetilde{\mathcal{F}}^{s,\operatorname{ord},\mu}_{v}(A)=\mathcal{F}^{\operatorname{ord},\mu}_{v}(A)$.
∎
We define $\mathcal{F}_{loc}^{s}$, resp. $\mathcal{F}_{loc}^{s,min,?}$ (and
their center modified counterparts) as
$\mathcal{F}_{loc}^{s}=\prod_{v\in S\cup
S_{p}}\mathcal{F}^{s}_{v},\quad\mathcal{F}_{loc,Z}^{s}=\prod_{v\in S\cup
S_{p}}\mathcal{F}^{s}_{v,Z}$
resp.
$\mathcal{F}_{loc}^{s,min,?}=\prod_{v\in
S}\mathcal{F}^{s,min}_{v,Z}\times\prod_{v\in
S_{p}}\mathcal{F}^{s,?}_{v},\quad\mathcal{F}_{loc,Z}^{s,min,?}=\prod_{v\in
S}\mathcal{F}^{s,min}_{v,Z}\times\prod_{v\in S_{p}}\mathcal{F}^{s,?}_{v,Z}$
as in [GV18, Definition 9.1]. By definition, the morphisms
$\mathcal{F}_{\overline{\rho}}^{s}\to\mathcal{F}^{s}_{v}$
induce morphisms
$\mathcal{F}^{s}_{\overline{\rho},Z}\to\mathcal{F}^{s}_{v,Z}$
hence a morphism
$\mathcal{F}^{s}_{Z,\overline{\rho}}\to\mathcal{F}^{s}_{loc,Z}$
We finally put
$\mathcal{F}^{s,?}=\mathcal{F}^{s,gl,?}_{\overline{\rho}}=\mathcal{F}^{s}_{\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{loc}}\mathcal{F}_{loc}^{s,min,?},\quad\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{s,gl,?}_{Z,\overline{\rho}}=\mathcal{F}^{s}_{Z,\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{loc,Z}}\mathcal{F}_{loc}^{s,min,?}$
By Lemma 3 and 4 we have $\pi_{0}\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{?}_{Z}$.
For $?=\operatorname{ord},FL$, we put
$\mathrm{H}^{1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho})=\operatorname{Ker}(\mathrm{H}^{1}(\Gamma,\operatorname{Ad}\bar{\rho})\to\bigoplus_{v\in
S}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over\mathrm{H}^{1}_{unr}(\Gamma_{v},\operatorname{Ad}\bar{\rho})}\oplus\bigoplus_{v\in
S_{p}}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over L^{?}_{v}}$
where $\operatorname{Ad}\bar{\rho}={\mathfrak{g}}_{k}$ with the adjoint Galois
action and if $?=\operatorname{ord}$,
$L^{?}_{v}=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k}))$
for
$L_{v}^{\prime}=\operatorname{Ker}(\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{b}}_{k}))\to\mathrm{H}^{1}(I_{v},{\mathfrak{b}}_{k}/{\mathfrak{n}}_{k}))$,
and if $?=FL$, $L_{v}^{?}=\mathrm{H}_{f}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$
(that is, the image of
$\mathrm{H}_{f}^{1}(\Gamma_{v},{\mathfrak{g}}_{\mathcal{O}})\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$
where $\mathrm{H}_{f}^{1}(\Gamma_{v},{\mathfrak{g}}_{\mathcal{O}})$ is defined
by Bloch and Kato, [BK90, Definition (3.7.3)].
For $v\in S$, let $\widetilde{L}_{v}^{min}$ be the inverse image of
$\mathrm{H}^{1}_{unr}(\Gamma_{v},{\mathfrak{g}}_{k})$ by
$Z^{1}(\Gamma_{v},{\mathfrak{g}}_{k})\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$,
and for $v\in S_{p}$, let $\widetilde{L}_{v}^{?}$ be the inverse image of
$L_{v}^{?}$ by
$Z^{1}(\Gamma_{v},{\mathfrak{g}}_{k})\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$.
###### Definition 5.
Given a morphism of cochain complexes $f\colon A^{\bullet}\to B^{\bullet}$,
its mapping cone $MC(f)^{\bullet}$ is defined by $MC(f)^{n}=B^{n}\oplus
A^{n+1}$ with $d(b,a)=(db+fa,-da)$. This give rise to distinguished triangles
$A^{\bullet}\to B^{\bullet}\to MC(f)^{\bullet}\to A^{\bullet}[1]$
and
$MC(f)^{\bullet}[-1]\to A^{\bullet}\to B^{\bullet}\to MC(f)^{\bullet}$
We define the complex
$C^{\bullet}_{min,?}(\Gamma,{\mathfrak{g}}_{k})=MC(f)[-1]$
for the morphism
$f\colon C^{\bullet}(\Gamma,{\mathfrak{g}}_{k})\to\bigoplus_{v\in
S}C^{\bullet}(\Gamma_{v},{\mathfrak{g}}_{k}))/\widetilde{L}_{v}^{\bullet,min}\oplus\bigoplus_{v\in
S_{p}}C^{\bullet}_{?}(\Gamma_{v},{\mathfrak{g}}_{k}))/\widetilde{L}_{v}^{\bullet,?}$
where for $?=min,\operatorname{ord},FL$,
$\widetilde{L}_{v}^{0,?}=C^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,
$\widetilde{L}_{v}^{1,?}=\widetilde{L}_{v}^{?}$ and
$\widetilde{L}_{v}^{2,min}=0$.
We write its cohomology as
$\mathrm{H}^{\ast}_{min,?}(\Gamma,{\mathfrak{g}}_{k})=\mathrm{H}^{\ast}(C^{\bullet}_{min,?}(\Gamma,{\mathfrak{g}}_{k}))$.
In accordance with Wiles’ notations, we also write sometimes this cohomology
as $\mathrm{H}^{\ast}_{\mathcal{L}}(\Gamma,{\mathfrak{g}}_{k})$ where
$\mathcal{L}=(L_{v})_{v\in S\cup S_{p}}$. We have the following exact
sequence:
$\begin{array}[]{l}{0\to\mathrm{H}^{1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho})\to\mathrm{H}^{1}(\Gamma,\operatorname{Ad}\bar{\rho})\to\bigoplus_{v\in
S}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over\mathrm{H}^{1}_{unr}(\Gamma_{v},\operatorname{Ad}\bar{\rho})}\oplus\bigoplus_{v\in
S_{p}}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over
L^{?}_{v}}}\\\
{\to\mathrm{H}^{2}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho})\to\mathrm{H}^{2}(\Gamma,\operatorname{Ad}\bar{\rho})\to\bigoplus_{v\in
S\cup S_{p}}\mathrm{H}^{2}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\dots}\\\
\end{array}$
Recall that we are interested in the deformation functor
$\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{s,gl,?}_{Z,\overline{\rho}}$.
###### Theorem 10.
We have ${\mathfrak{t}}^{-1}\mathcal{F}_{Z}^{s,?}=0$ and for any $i\geq 0$,
the $i$-th cohomology of the tangent complex
${\mathfrak{t}}\mathcal{F}_{Z}^{s,?}$ is naturally identified with the
cohomology with local conditions:
${\mathfrak{t}}^{i}\mathcal{F}^{s,?}\cong\mathrm{H}^{i+1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho}).$
###### Proof.
Since
$\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{s}_{Z,\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{loc,Z}}\mathcal{F}_{loc,Z}^{min,?,s}$,
we can apply [GV18, Lemma 4.30 (iv)] and we get a long exact sequence of
finite dimensional $k$-vector spaces
$0\to{\mathfrak{t}}^{-1}\mathcal{F}_{Z}^{s,?}\to{\mathfrak{t}}^{-1}\mathcal{F}^{s}_{Z,\overline{\rho}}\oplus{\mathfrak{t}}^{-1}\mathcal{F}_{loc,Z}^{min,?,s}\to{\mathfrak{t}}^{-1}\mathcal{F}^{s}_{loc,Z}\to$
$\to{\mathfrak{t}}^{0}\mathcal{F}_{Z}^{s,?}\to{\mathfrak{t}}^{0}\mathcal{F}^{s}_{Z,\overline{\rho}}\oplus{\mathfrak{t}}^{0}\mathcal{F}_{loc,Z}^{min,?,s}\to{\mathfrak{t}}^{0}\mathcal{F}^{s}_{loc,Z}\to$
$\to{\mathfrak{t}}^{1}\mathcal{F}_{Z}^{s,?}\to{\mathfrak{t}}^{1}\mathcal{F}^{s}_{Z,\overline{\rho}}\oplus{\mathfrak{t}}^{1}\mathcal{F}_{loc,Z}^{min,?,s}\to{\mathfrak{t}}^{1}\mathcal{F}^{s}_{loc,Z}\to{\mathfrak{t}}^{1}\mathcal{F}^{s,?}\ldots$
We know by Proposition 2 that
${\mathfrak{t}}^{-1}\mathcal{F}^{s}_{Z,\overline{\rho}}=0$ and
${\mathfrak{t}}^{i}\mathcal{F}^{s}_{Z,\overline{\rho}}\cong\mathrm{H}^{i+1}(\Gamma,{\mathfrak{g}}_{k})$
for $i\geq 0$. By Lemma 8, we have
${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s}\cong\mathrm{H}^{i+1}(\Gamma_{v},{\mathfrak{g}}_{k})$
for all $i\geq-1$ and
${\mathfrak{t}}^{-1}\mathcal{F}_{v,Z}^{s}\cong\operatorname{Coker}({\mathfrak{z}}_{k}\to\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}))$.
By Lemma 10, we have for $v\in S\cup S_{p}$,
${\mathfrak{t}}^{-1}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,
${\mathfrak{t}}^{-1}\mathcal{F}_{v,Z}^{s,?}\cong\operatorname{Coker}({\mathfrak{z}}\to\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}))$,
${\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})$,
${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s,?}=0$ for $i\geq 1$. We have therefore
${\mathfrak{t}}^{-1}\mathcal{F}^{s,?}=0$ and
$0\to{\mathfrak{t}}^{0}\mathcal{F}_{Z}^{s,?}\to\mathrm{H}^{1}(\Gamma,{\mathfrak{g}}_{k})\oplus\bigoplus_{v\in
S\cup
S_{p}}\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})\to\bigoplus_{v\in
S\cup S_{p}}\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})\to$
$\to{\mathfrak{t}}^{1}\mathcal{F}_{Z}^{s,?}\to\mathrm{H}^{2}(\Gamma,{\mathfrak{g}}_{k})\to\bigoplus_{v\in
S\cup S_{p}}\mathrm{H}^{2}(\Gamma_{v},{\mathfrak{g}}_{k}).$
By comparing to the long exact sequence of cohomology, we conclude that for
any $i\geq 0$,
${\mathfrak{t}}^{i}\mathcal{F}_{Z}^{s,?}\cong\mathrm{H}^{i+1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho}).$
∎
###### Proposition 3.
Under the assumption $(STDIST)$, the global deformation problem
$\mathcal{F}_{Z}^{s,\operatorname{ord}}$ is pro-representable. Under the
assumption $(FL)$, the deformation problem $\mathcal{F}_{Z}^{s,FL}$ is pro-
representable.
Note that $\mathcal{F}^{s}_{Z,\bar{\rho}}$ is pro-representable, but not
$\mathcal{F}_{loc,Z}^{min,?,s}$.
###### Proof.
We apply Lurie’s criterion Theorem 9. We know that
$\mathcal{F}^{s}_{Z,\bar{\rho}}$, $\mathcal{F}_{loc,Z}^{s}$ and
$\mathcal{F}_{loc,Z}^{min,?,s}$ are formally cohesive, hence so is their
homotopy fiber product $\mathcal{F}_{Z}^{s,?}$. By Theorem 10,
${\mathfrak{t}}^{-1}\mathcal{F}_{Z}^{s,?}=0$ and for any $i\geq 0$,
${\mathfrak{t}}^{i}\mathcal{F}_{Z}^{s,?}$ is finite dimensional over $k$ (and
vanishes for $i>1$). ∎
### 3.6. Theorems of Galatius-Venkatesh and Cai
In this section we state generalizations of the main theorems of Galatius and
Venkatesh [GV18] by Y. Cai [Cai21]. We recall the notations and assumptions.
Let $\pi$ be a cohomological cuspidal representation on
$\operatorname{\mathsf{GL}}_{N}(F)$ ($F$ a CM field), with squarefree
conductor ${\mathfrak{n}}$, and level group $U=U_{0}({\mathfrak{n}})$. Let $S$
be the set of places dividing ${\mathfrak{n}}$. It occurs in
$\mathrm{H}^{\bullet}(X_{U},V_{\lambda}({\mathbb{C}}))$ for a weight
$\lambda=(\lambda_{\tau,i})_{\tau\in I_{F},1\leq i\leq
N}\in(X^{\ast}(\operatorname{\mathbf{T}}_{K})^{+})^{\operatorname{\mathsf{Hom}}(F,{\mathbb{C}})}$,
i.e. $\lambda_{\tau,1}\geq\ldots\geq\lambda_{\tau,N}$ for any $\tau\in I_{F}$.
Let $p>2$ be a rational prime relatively prime to ${\mathfrak{n}}$, unramified
in $F$; let $S_{p}$ be the set of places of $F$ above $p$. We assume either
$(FL)\quad\lambda_{\tau,1}-\lambda_{\tau,N}<p-N,\forall\tau\in I_{F},$
or
$(\operatorname{ord})\quad\pi_{v}\,\mbox{\rm is ordinary for all}\,v\in
S_{p}.$
Let $\Gamma=\operatorname{Gal}(F_{S\cup S_{p}}/F)$. Let $K_{0}$ be a
sufficiently big number field containing the Hecke eigenvalues of $\pi$. Fix a
$p$-adic place $v_{0}$ and let $K$ be its $p$-adic completion, $\mathcal{O}$
its valuation ring, $\varpi$ a uniformizing parameter; $k$ its residue field.
Let $\rho_{\pi}\colon\Gamma\to\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$ be
the Galois representation associated to $\pi$ and
$\bar{\rho}\colon\Gamma\to\operatorname{\mathsf{GL}}_{N}(k)$ its reduction
modulo $\varpi$. Assume
$(MIN)$ for any places $v\in S$, the image $\overline{\rho}_{\pi}(I_{v})$ of
the inertia subgroup $I_{v}$ contains a regular unipotent element.
Recall that for any place $v\in S_{p}$, it is known that $(FL)$, resp.
$(\operatorname{ord})$, implies that $\rho_{\pi}|_{\Gamma_{v}}$ and
$\bar{\rho}|_{\Gamma_{v}}$ are both Fontaine-Laffaille, resp. ordinary. Assume
that the image of $\bar{\rho}$ is enormous:
$(RLI)$ $\bar{\rho}(\operatorname{Gal}_{F(\zeta_{p})})\supset
k^{\times}\operatorname{\mathsf{SL}}_{N}(k^{\prime})$ for some subfield
$k^{\prime}$ of the residue field $k$.
Recall that a Taylor-Wiles prime $v$ is a place $v\notin S\cup S_{p}$ of
residual characteristic $q_{v}$ such that $q_{v}\equiv 1\pmod{p}$ and
$\bar{\rho}(\operatorname{Frob}_{v})$ is conjugated to a strongly regular
element of $T(k)$ (i.e. an element $t\in T(k)$ whose centralizer in $G(k)$
coincides with $T(k)$). In particular, for a Taylor-Wiles prime $v$, we may
choose a representation $\bar{\rho}_{v}:\Gamma_{v}\to T(k)$ such that the
composition with $T\hookrightarrow G$ is conjugate to
$\bar{\rho}|_{\Gamma_{v}}$. Note that by definition, $\bar{\rho}_{v}$ factors
through the Galois group $\Gamma_{v}^{\operatorname{unr}}$ of the maximal
unramified extension $F_{v}^{\operatorname{unr}}/F_{v}$. Let
$\mathcal{F}_{v}^{s}$, resp. $\mathcal{F}_{v}^{\operatorname{unr},s}$ be the
simplicial deformation functor of $\bar{\rho}|_{\Gamma_{v}}$. As in [GV18,
Section 8.1], in order to study them, one needs to compare them to the
(simpler) simplicial deformation functors of $\bar{\rho}_{v}$ with target in
$T$. Let $\mathcal{F}_{v}^{s,T}$, resp.
$\mathcal{F}_{v}^{s,T,\operatorname{unr}}$, be the $T$-valued derived
deformation functor, resp. its unramified analogue, of $\bar{\rho}_{v}$. Since
$T$ is commutative, its center is $T$ itself, so their center-modified
analogues make use of $T$; they are denoted by $\mathcal{F}_{v,T}^{s,T}$,
resp. $\mathcal{F}_{v,T}^{s,T,\operatorname{unr}}$. They are pro-
representable. Let $Q$ be a finite set of Taylor-Wiles primes; we consider the
functors
$\mathcal{F}_{Q,loc}^{s}=\prod_{v\in S\cup S_{p}\cup Q}\mathcal{F}^{s}_{v},$
$\mathcal{F}_{Q,loc}^{s,min,?}=\prod_{v\in
S}\mathcal{F}^{s,min}_{v}\times\prod_{v\in
S_{p}}\mathcal{F}^{s,?}_{v}\times\prod_{v\in Q}\mathcal{F}_{v}^{s},$
let $\Gamma_{Q}=\operatorname{Gal}(F_{S\cup S_{p}\cup Q}/F)$ and
$\mathcal{F}^{s}_{Q,\overline{\rho}}$ be the global derived deformation
problem $A\mapsto\operatorname{\mathsf{Hom}}_{[\bar{\rho}]}(B\Gamma_{Q},{\bf
B}G(A))$; we then define
$\mathcal{F}^{s,min,?}_{Q}=\mathcal{F}^{s,gl/loc,?}_{Q,\overline{\rho}}=\mathcal{F}^{s}_{Q,\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{Q,loc}}\mathcal{F}_{Q,loc}^{s,min,?}$
and their center-modified analogues $\mathcal{F}_{Z,Q}^{s,min,?}$ and
$\mathcal{F}_{Z,Q}^{s,min,?}$, involving products of center-modified global
and local factors. We also introduce $\mathcal{F}_{Q,T}^{s,T}=\prod_{v\in
Q}\mathcal{F}_{v,T}^{s,T}$ and
$\mathcal{F}_{Q,T}^{s,T,\operatorname{unr}}=\prod_{v\in
Q}\mathcal{F}_{v,T}^{s,T,\operatorname{unr}}$.
###### Definition 6.
An allowable Taylor-Wiles datum of level $n$ is a set of Taylor-Wiles primes
$Q=\\{v_{1},\dots,v_{r}\\}$, together with a strongly regular element
$t_{v_{i}}\in T(k)$ conjugate to $\bar{\rho}(\operatorname{Frob}_{v_{i}})$ for
each $i\in\\{1,\dots,r\\}$, such that:
1. (1)
Each $q_{v_{i}}\equiv 1\pmod{p^{n}}$, $i\in\\{1,\dots,r\\}$.
2. (2)
We have
$\mathrm{H}^{2}_{\mathcal{L}_{Q}}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})=0$,
where $\mathcal{L}_{Q}=(L_{Q,v})_{v}$ with
$L_{Q,v}=\left\\{\begin{array}[]{l}{L_{v},\,v\in S\cup S_{p};}\\\
{\mathrm{H}^{1}(F_{v},\operatorname{Ad}\bar{\rho}),\,v\in
Q.}\end{array}\right.$
As noted in [GV18, Remark after Definition 6.2], we have
###### Remark 2.
Under the assumption that
$k^{\times}\cdot\operatorname{\mathsf{SL}}_{N}(k^{\prime})\subset\operatorname{Im}\bar{\rho}$,
for any level $n\geq 1$, there exist infinitely many allowable Taylor-Wiles
data $(Q,(t_{v})_{v\in Q})$ of degree $n$.
Let $\Gamma_{Q}=\operatorname{Gal}(F_{S\cup S_{p}\cup Q})/F)$ and for each
$v\in Q$, For an allowable Taylor-Wiles datum $Q$, we have the long exact
sequence:
$\begin{array}[]{l}{0\to\mathrm{H}^{1}_{\mathcal{L}_{Q}}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})\to\mathrm{H}^{1}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})\stackrel{{\scriptstyle
A}}{{\to}}\bigoplus\limits_{l\in
S}\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})/L_{v})}\\\ {\to
0\to\mathrm{H}^{2}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})\stackrel{{\scriptstyle
B}}{{\to}}\bigoplus\limits_{v\in S\cup
Q}\mathrm{H}^{2}(\Gamma_{v},\operatorname{Ad}\bar{\rho})}.\end{array}$
Moreover, the cokernel of $B$ is dual to
$\mathrm{H}^{0}(\mathbb{Z}[\frac{1}{S(Q)}],(\operatorname{Ad}\bar{\rho})^{\ast})$
by global duality. We want to approximate $\mathcal{R}$ by larger rings
allowing ramification at Taylor-Wiles primes. Let $(Q_{n},(t_{v})_{v\in
Q_{n}})$ be an allowable Taylor-Wiles datum of level $n$. Let
$\mathcal{F}^{s}_{n}=\mathcal{F}_{Q_{n},Z}^{s,?}$ and let $\mathcal{R}_{n}$ be
a representing ring for $\mathcal{F}_{n}^{s}$. Let
$\mathcal{F}_{n}^{s,\mathrm{loc}}:=\mathcal{F}_{Q_{n},T}^{s,T}$ with
representing ring $\mathcal{S}_{n}$. Let
$\mathcal{F}_{n}^{s,\mathrm{loc,ur}}:=\mathcal{F}_{Q_{n},T}^{s,T,\operatorname{unr}}$
with representing ring $\mathcal{S}_{n}^{\mathrm{ur}}$. As noticed in [Cai21],
$\mathcal{F}_{n}^{s,\mathrm{loc}}$ is objectwise weakly equivalent to the
derived framed deformation functor $\prod_{v\in
Q_{m}}\mathcal{F}_{v}^{s,T,\square}$, and similarly for
$\mathcal{F}_{Q_{n},T}^{s,T,\operatorname{unr}}$ and $\prod_{v\in
Q_{n}}\mathcal{F}_{v}^{s,T,\square}$. Therefore, the simplicial deformation
rings $\mathcal{S}_{n}$ and $\mathcal{S}_{n}^{\operatorname{unr}}$ can be
described explicitely as in [GV18, Remark 8.7].
###### Lemma 12.
We have a homotopy pullback square
$\textstyle{\mathcal{F}_{Z}^{s,?}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{n}^{s,\mathrm{loc,ur}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{n}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{n}^{s,\mathrm{loc}}.}$
We have an objectwise weak equivalence
$\mathcal{F}_{n}^{s,?}\times^{h}_{\mathcal{F}_{n}^{s,\mathrm{loc}}}\mathcal{F}_{n}^{s,\mathrm{loc,ur}}\sim\mathcal{F}_{Z}^{s,?}.$
###### Proof.
This is [Cai21, Lemma 6.6 and Corollary 6.7]. ∎
Now we pass to the level of rings. Recall the derived tensor product
$\underline{\otimes}$ of [GV18, Definition 3.3], we remark that
$\mathcal{R}_{1}\underline{\otimes}_{\mathcal{R}_{3}}\mathcal{R}_{2}$
represents the homotopy pullback of represented functors after cofibrant
replacements, and since all pro-rings for us are indexed by natural numbers,
the derived tensor product can be explicitly constructed levelwise (see [GV18,
Definition 3.3] for details). The above corollary gives a weak equivalence
$\mathcal{R}\sim\mathcal{R}_{n}\underline{\otimes}_{\mathcal{S}_{n}}\mathcal{S}_{n}^{\operatorname{unr}}.$
We say a pro-object $\mathcal{R}$ of $\operatorname{Art}^{s}$ is homotopy
discrete if the map $\mathcal{R}\to\pi_{0}\mathcal{R}$ induces an equivalence
on represented functors after applying level-wise cofibrant replacement (see
[GV18, Definition 7.4]). Let $F_{Q_{n}}^{\times}=\prod_{v\in
Q_{n}}F_{v}^{\times}$, $\mathcal{O}_{Q_{n}}^{\times}=\prod_{v\in
Q_{n}}\mathcal{O}_{v}^{\times}$. By choosing uniformizing parameters at $v$’s
in $Q_{n}$, we have an isomorphism
$F_{Q_{n}}^{(p)}/\mathcal{O}_{Q_{n}}^{(p)}\cong{\mathbb{Z}}^{Q_{n}}.$
hence a decomposition
$F_{Q_{n}}^{\times}=\mathcal{O}_{Q_{n}}^{\times}\times{\mathbb{Z}}^{Q_{n}}.$
Let $\Delta^{\prime}_{Q_{n}}=\mathcal{O}_{Q_{n}}^{(p)}$ be the pro-$p$
completion of $\mathcal{O}_{Q_{n}}^{\times}$. It is a finite group isomorphic
to the $p$-Sylow of $\prod_{v\in Q}k_{v}^{\times}$. Let
$\Delta_{Q_{n}}=\Delta^{\prime}_{Q_{n}}/(p^{n})$ and
$F_{Q_{n}}^{(p)}=\Delta_{Q_{n}}\times{\mathbb{Z}}^{Q_{n}}.$
By definition of the level, this group is free of rank $r$ over
${\mathbb{Z}}/p^{n}{\mathbb{Z}}$. Let $S_{n}=\mathcal{O}[\Delta_{Q_{n}}]$; it
is a complete intersection ring. Actually let
$S_{\infty}=\mathcal{O}[[Y_{1},\ldots,Y_{Nr}]]$. The ideal
$J_{n}=((1+Y_{1})^{n}-1,\ldots,(1+Y_{Nr})^{n}-1)$, and an isomorphism
$i_{n}\colon S_{\infty}/J_{n}\cong S_{n}$
Let
$\Sigma_{n}=\mathcal{O}[[X^{\ast}(T)\otimes
F_{Q_{n}}^{(p)}]]=S_{n}[[F_{Q_{n}}^{(p)}/\mathcal{O}_{Q_{n}}^{(p)}]]\stackrel{{\scriptstyle
i}}{{\cong}}S_{n}[[X_{1},\ldots,X_{Nr}]],$
$\Sigma_{\infty}=S_{\infty}[[X_{1},\ldots,X_{Nr}]]$
The isomorphism $i_{n}$ induces an isomorphism
$\Sigma_{\infty}/J_{n}\Sigma_{\infty}\cong\Sigma_{n}.$ Let also
$\Sigma_{n}^{\operatorname{unr}}=\mathcal{O}[[X^{\ast}(T)\otimes
F_{Q_{n}}^{(p)}/\mathcal{O}_{Q_{n}}^{(p)}]]\stackrel{{\scriptstyle
i_{\operatorname{unr}}}}{{\cong}}\mathcal{O}[[X_{1},\ldots,X_{Nr}]].$
Note that the isomorphisms $i$ and $i_{\operatorname{unr}}$ are compatible to
the augmentation homomorphism
$S_{n}=\mathcal{O}[\Delta_{Q_{n}}]\to\mathcal{O}$.
###### Lemma 13.
The simplicial rings $\mathcal{S}_{n}$ and
$\mathcal{S}_{n}^{\operatorname{unr}}$ are homotopy discrete. Actually
$\mathcal{S}_{n}$, resp. $\mathcal{S}_{n}^{\operatorname{unr}}$, is a
cofibrant replacement of the complete intersection classical ring
$\Sigma_{n}$, resp. the formally smooth classical ring
$\Sigma_{n}^{\operatorname{unr}}$.
###### Proof.
See [GV18] and [Cai21]. ∎
Since we can choose $\mathcal{S}_{n}$ and
$\mathcal{S}_{n}^{\operatorname{unr}}$ up to weak equivalence, we suppose in
the following that $\mathcal{S}_{n}=c(\Sigma_{n})$ and
$\mathcal{S}_{n}^{\operatorname{unr}}=c(\Sigma_{n}^{\operatorname{unr}})$. Let
$\bar{S}_{m}=S_{\infty}/(p^{m},J_{m})$ and
$\bar{\Sigma}_{m}=\Sigma_{\infty}/(p^{m},J_{m}\Sigma_{\infty})$. Note that
$\bar{S}_{m}$ is finite and if $M_{m}$ is a $\Sigma_{m}$-module,
$\bar{M}_{m}=M_{m}\otimes_{\Sigma_{m}}\bar{\Sigma}_{m}=M_{m}\otimes_{S_{m}}\bar{S}_{m}$
and
$\bar{M}_{m}\otimes_{\bar{\Sigma}_{m}}\Sigma_{m}^{\operatorname{unr}}=\bar{M}_{m}\otimes_{\bar{S}_{m}}\mathcal{O}/(p^{m})$.
This can be rewritten in terms of simplicial rings as follows. By quotienting
by the ideal $(p^{m},J_{m})$ the morphism $S_{\infty}\to\mathcal{S}_{m}$, we
get the commutative diagram
$\textstyle{\bar{S}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{m}/(p^{m},J_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}/p^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}}$
which induces a homotopy pullback square of represented functors. In
consequence, for any $\mathcal{S}_{m}/(p^{m},J_{m})$-module $\mathcal{M}_{m}$,
we have a weak equivalence
$\mathcal{M}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}\sim\mathcal{M}_{m}\underline{\otimes}_{\mathcal{S}_{m}/(p^{m},J_{m})}\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}$.
Let $R_{m}$, resp. $\bar{R}_{m}$, be the quotient of the classical universal
deformation ring which represents the $(min,?)$-deformations with
$Q_{m}$-ramification, resp. the $(min,?)$-deformations modulo $p^{m}$ whose
$Q_{m}$-ramification has of inertial level $\leq m$ (i.e. such that the
restriction to inertia factors modulo $p^{m}$ for all $v\in Q_{m}$). By
definition, the ring $\bar{R}_{m}$ is a natural quotient of
$R_{m}/(p^{m},J_{m})$ and is an $\mathcal{S}_{m}/(p^{m},J_{m})$-algebra. The
map from
$\mathcal{R}_{m}\leftarrow\mathcal{S}_{m}\to\mathcal{S}_{m}^{\operatorname{unr}}$
to
$\bar{R}_{m}\leftarrow\mathcal{S}_{m}/(p^{m},J_{m})\to\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}$
induces
(1)
$\mathcal{R}\sim\mathcal{R}_{m}\underline{\otimes}_{\mathcal{S}_{m}}\mathcal{S}_{m}^{\operatorname{unr}}\to\bar{R}_{m}\underline{\otimes}_{\mathcal{S}_{m}/(p^{m},J_{m})}\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}\sim\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}.$
A Taylor-Wiles patching argument provides projective systems out of the
$\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}$’s. Let
$R_{\infty}$ be the projective limit for one of these projective systems (it
depends on the choice of such a projective system). The main result of [GV18]
(Theorem 14.1) compares $\pi_{\ast}\mathcal{R}$ to
$\operatorname{Tor}_{\ast}^{S_{\infty}}(R_{\infty},\mathcal{O})$ under certain
conditions.
For $n\geq m$, there is a natural
$\bar{R}_{n}\underline{\otimes}_{\bar{S}_{n}}\mathcal{O}/p^{n}\to\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}$.
Let $f_{n,m}$ be the composition
$\mathcal{R}\to\bar{R}_{n}\underline{\otimes}_{\bar{S}_{n}}\mathcal{O}/p^{n}\to\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}.$
In order to apply [GV18, Theorem 12.1], recall the key result [Cai21,
Proposition 6.11] generalizing [GV18, Theorem 11.1]:
###### Proposition 4.
Under the assumptions $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})$, let $n\geq
m$, the map $f_{n,m}$ induces an isomorphism on ${\mathfrak{t}}^{0}$ and a
surjection on ${\mathfrak{t}}^{1}$.
Based on the fact that the homology of
$\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}$ is
$\operatorname{Tor}_{\bullet}^{\bar{S}_{m}}(\bar{R}_{m},\mathcal{O}/p^{m})$,
this Proposition yields the following
###### Theorem 11.
We have an isomorphism of graded commutative rings
$\pi_{\bullet}\mathcal{R}\cong\operatorname{Tor}_{\bullet}^{S_{\infty}}(R_{\infty},\mathcal{O})$.
###### Proof.
Recall that $\pi_{\bullet}(\mathcal{R})$ is a $\pi_{0}(\mathcal{R})$-algebra
(see [Cai21, Lemma 3.45]) and that $\pi_{0}(\mathcal{R})=R_{0}$. The theorem
is proven in [Cai21, Proposition 6.13] by applying Proposition 4 to [GV18,
Theorem 12.1]. Note that Theorem 12.1 requires a numerical coincidence (12.1),
which is satisfied when the (framed) local deformation rings are formally
smooth (here it is the case by Section 2.1). ∎
This Theorem implies the Main Theorem ([GV18, Th.14.1], generalized in [Cai21,
Theorem 7.3]) below. We need to add few more notations and assumptions. (this
implies Taylor-Wiles data exist, see [ACC+18, 6.2.28]). Let
$C^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))$ be the integral cochain complex;
its cohomology is $\mathrm{H}^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))$.
There is a natural homomorphism
$\mathcal{H}^{S,\operatorname{univ}}\to\operatorname{\mathsf{End}}_{D(\mathcal{O})}(C^{\bullet}(X_{U},V_{\lambda}(\mathcal{O})))$
from the abstract spherical Hecke algebra outside $S$ to the endomorphisms of
$C^{\bullet}$ in the derived category. Let
$\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))$ be the image of this map.
Then $\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))$ is finite over
$\mathcal{O}$. see [NT16]. Let ${\mathfrak{m}}$ be a non-Eisenstein maximal
ideal of $\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))$ associated to
$\pi$ modulo $v_{0}$. For simplicity, we write
$C_{\lambda,{\mathfrak{m}}}^{\bullet}$ for
$C^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))_{{\mathfrak{m}}}$ and write
$\operatorname{T}_{\lambda,{\mathfrak{m}}}$ for
$\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))_{{\mathfrak{m}}}$. We assume
$(\operatorname{Gal}_{\mathfrak{m}})$ For $?=FL,\operatorname{ord}$, there
exists a lifting $\rho_{\mathfrak{m}}\colon G_{F,S\cup
S_{p}}\to\operatorname{\mathsf{GL}}_{N}(\operatorname{T}_{\lambda,{\mathfrak{m}}})$
of $\bar{\rho}$ such that
$[\rho_{\mathfrak{m}}]\in\mathcal{F}^{?}(\operatorname{T}_{\mathfrak{m}})$. In
particular, there is a natural map
$\mathcal{R}\to\operatorname{T}_{\lambda,{\mathfrak{m}}}\hookrightarrow\operatorname{\mathsf{End}}_{D(\mathcal{O})}(C_{\lambda,{\mathfrak{m}}}^{\bullet}).$
Let
$\mathrm{H}^{\bullet}_{\mathfrak{m}}=\mathrm{H}^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))_{\mathfrak{m}}$.
Let $\mathrm{H}_{i}^{\mathfrak{m}}=\mathrm{H}^{d-i}_{\mathfrak{m}}$ for all
$i\geq 0$ and let $\mathrm{H}_{\bullet}^{\mathfrak{m}}=\bigoplus_{i\geq
0}\mathrm{H}_{i}^{\mathfrak{m}}$ the corresponding graded module. We assume
$(Van_{\mathfrak{m}})$ $\mathrm{H}^{i}_{\mathfrak{m}}(k)=0$ if
$i\notin[q_{0},q_{0}+\ell_{0}]$.
###### Theorem 12.
Assume $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(Van_{\mathfrak{m}})$ as above. Then,
(i) we have an isomorphism of graded commutative rings
$\pi_{\ast}\mathcal{R}\cong\operatorname{Tor}_{\ast}^{S_{\infty}}(R_{\infty},\mathcal{O})$,
(ii)
$\pi_{0}(\mathcal{R})=R_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R\cong\operatorname{T}_{\lambda,{\mathfrak{m}}}$,
$\mathrm{H}_{q_{0}}^{\mathfrak{m}}=\mathrm{H}^{q_{0}+\ell_{0}}_{\mathfrak{m}}$
is free of rank $1$ over $\operatorname{T}_{\lambda,{\mathfrak{m}}}$,
(iii) There is an isomorphism of graded $\pi_{\bullet}\mathcal{R}$-modules
$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}_{\lambda,{\mathfrak{m}}}}\pi_{\bullet}(\mathcal{R}).$
###### Remark 3.
As in [GV18, Theorem 14.1], this statement relies on [CaGe18, Theorem 12.1].
See examples where these assumptions are satisfied in [Cai21, Section 8].
### 3.7. Divisible part of the Dual Adjoint Selmer group
Let $P$ be a finite free $\mathcal{O}$-module with action of
$\Gamma=G_{F,S\cup S_{p}}$. For each $v\in S_{p}$, let $Fil_{v}^{+}P\subset
Fil_{v}P\subset P$ be direct factors submodules stable by $\Gamma_{v}$. We
define the minimal $(Fil_{v}^{+}P)_{v}$-ordinary Selmer group as
${\mathop{\rm
Sel}}_{\operatorname{ord}}(P)=\mathrm{H}^{1}_{min,\operatorname{ord}}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Ker}(\mathrm{H}^{1}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})\to\bigoplus_{v}\frac{\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}{\mathrm{H}^{1}_{\operatorname{ord}}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}))$
where
$\mathrm{H}^{1}_{\operatorname{ord}}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}}))$
$L^{\prime}_{v}=\operatorname{Ker}(\mathrm{H}^{1}(F_{v},Fil_{v}P\otimes{\mathbb{Q}}/{\mathbb{Z}})\to\mathrm{H}^{1}(F_{v},Fil_{v}P\otimes{\mathbb{Q}}/{\mathbb{Z}}))/\mathrm{H}^{1}(I_{v},Fil_{v}^{+}P\otimes{\mathbb{Q}}/{\mathbb{Z}}))$
Similarly,
${\mathop{\rm
Sel}}_{f}(P)=\mathrm{H}^{1}_{f}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Ker}(\mathrm{H}^{1}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})\to\bigoplus_{v}\frac{\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}{\mathrm{H}^{1}_{f}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}))$
where
$\mathrm{H}^{1}_{f}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Im}(\mathrm{H}^{1}_{f}(F_{v},P\otimes{\mathbb{Q}})\to\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})).$
In the sequel we shall take
$P=\operatorname{Ad}(\rho_{\pi})={\mathfrak{g}}_{\mathcal{O}}$,
$Fil_{v}P={\mathfrak{b}}_{\mathcal{O}}$,
$Fil_{v}^{+}P={\mathfrak{n}}_{\mathcal{O}}$, or
$P=\operatorname{Ad}(\rho_{\pi})^{\vee}(1)={\mathfrak{g}}_{\mathcal{O}}^{\vee}(1)$,
$Fil_{v}={\mathfrak{n}}_{\mathcal{O}}^{\perp}(1)$,
$Fil_{v}^{+}P={\mathfrak{b}}_{\mathcal{O}}^{\perp}(1)$. For
$?=\operatorname{ord},f$, Theorem (12) implies that $\mathop{\rm
Sel}_{?}(\operatorname{Ad}(\rho_{\pi}))$ is finite. Let
${}_{?}(P)=\mathrm{H}^{1}_{?}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})/\mathrm{H}^{1}_{?}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})_{\varpi-
div}$
It is the torsion quotient of
$\mathrm{H}^{1}_{?}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})$. For
$P={\mathfrak{g}}_{\mathcal{O}}^{\vee}(1)$, We will recall below, using
Poitou-Tate duality, that it is Pontryagin dual to $\mathop{\rm
Sel}_{?}(\operatorname{Ad}(\rho_{\pi}))$. Recall that Beilinson conjecture
predicts that
$(B)\quad cork_{\mathcal{O}}\mathop{\rm
Sel}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))=\ell_{0}.$
In the next section, we’ll see that this conjecture follows from Theorem 12.
In the next section, assuming the assumption of Theorem 12, we will provide an
automorphic description of a $\varpi$-divisible subgroup of a corank
$\ell_{0}$ of $\mathop{\rm Sel}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))$.
By Proposition 7 below, $\mathop{\rm
Sel}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))$, assuming Theorem 12,
Conjecture $(B)$ holds. It implies that the cokernel of this embedding is
therefore ${}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))$, which is
Pontryagin dual to $\mathop{\rm Sel}_{?}({\mathfrak{g}}_{\mathcal{O}})$.
## 4\. The Galatius-Venkatesh homomorphism
We redefine the map of Lemma 15.1 of [GV18] as follows. Let
$\mathcal{O}_{n}=\mathcal{O}/(\varpi^{n})$. We consider the simplicial ring
homomorphism
$\phi_{n}\colon\mathcal{R}\to R\to\mathcal{O}_{n}$
given by the universal property for the deformation
$\rho_{n}=\rho_{\pi}\pmod{(\varpi^{n})}$. Let $M_{n}$ be a finite
$\mathcal{O}_{n}$-module. Consider the simplicial ring
$\Theta_{n}=\mathcal{O}_{n}\oplus{\operatorname{DK}}(M_{n}[1])$ where
$M_{n}[1]$ is the chain complex concentrated in degree $1$ up to homotopy. For
an explicit description of the simplicial module ${\\\ DK}(M_{n}[1])$, see
[Cai21, Section 3.4]. The simplicial ring $\Theta_{n}$ is endowed with a
simplicial ring homomorphism
$\operatorname{pr}_{n}\colon\Theta_{n}\to\mathcal{O}_{n}$ given by the first
projection. Let $L_{n}(\mathcal{R})$ be the set of homotopy equivalence
classes of simplicial ring homomorphisms $\Phi\colon\mathcal{R}\to\Theta_{n}$
such that $\operatorname{pr}_{n}\circ\Phi=\phi_{n}$. There is a canonical
bijection
$L_{n}(\mathcal{R})\cong\mathrm{H}^{2}_{f}(F,\operatorname{Ad}(\rho_{n})\otimes
M_{n})$
Moreover, as in [GV18, Lemma 15.1], there is a map
$\pi(n,\mathcal{R})\colon
L_{n}(\mathcal{R})\to\operatorname{\mathsf{Hom}}_{R}(\pi_{1}(\mathcal{R}),M_{n})$
sending (the homotopy class of) $\Phi$ to the homomorphism
$\pi(n,\mathcal{R})(\Phi)$ which sends the homotopy class $[\gamma]$ of a loop
$\gamma$ to
$\Phi\circ\gamma\in\operatorname{\mathsf{Hom}}_{\operatorname{\mathsf{sSETS}}}(\Delta[1],M_{n}[1])=M_{n}$.
Recall a loop $\gamma$ is a morphism of $\operatorname{\mathsf{sSETS}}$
$\gamma\colon\Delta[1]\to\Theta_{n}$
from the simplicial interval $\Delta[1]$ to the simplicial set $\Theta_{n}$
which sends the boundary $\partial\Delta[1]$ to $0$. Note that
$\pi(n,\mathcal{R})(\Phi)$ is $R$-linear by definition of the structure of
$\pi_{0}(\mathcal{R})$-module on $\pi_{1}(\mathcal{R})$.
###### Proposition 5.
For any $n\geq 1$, the map $\pi(n,\mathcal{R})$ is surjective.
We’ll give two proofs of this Proposition. Note first that by the isomorphisms
(14.4), we have a diagram
$\begin{array}[]{ccccc}\pi(n,\mathcal{R})&\colon&L_{n}(\mathcal{R})&\rightarrow&\operatorname{\mathsf{Hom}}_{\pi_{0}(\mathcal{R})}(\pi_{1}(\mathcal{R}),M_{n})\\\
&&\uparrow&&\uparrow\\\ \pi(R_{n}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{n}}\mathcal{O}_{n})&\colon&L_{n}(R_{n}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{n}}\mathcal{O}_{n})&\rightarrow&\operatorname{\mathsf{Hom}}_{\pi_{0}(\mathcal{R})}(\pi_{1}(R_{n}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{n}}\mathcal{O}_{n}),M_{n})\\\ &&\downarrow&&\downarrow\\\
\pi(R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O})&\colon&L_{n}(R_{\infty}.\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O})&\rightarrow&\operatorname{\mathsf{Hom}}_{\pi_{0}(\mathcal{R})}(\pi_{1}(R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O}),M_{n})\end{array}$
Note that $\pi_{1}\mathcal{R}=\pi_{1}(R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O})$ by [GV18, Theorem 14.1]. So it is
enough to prove that the map on the last line is surjective. In the sequel,
since $n$ is fixed, we drop the indices $n$ and write simply $A=A_{n}$,
$\Theta=\Theta_{n}$, $M=M_{n}$.
###### Proof.
First Proof: Recall that by finiteness of the Calegari-Geraghty complex of
finite free $S_{\infty}$-modules constructed in [CaGe18], $R_{\infty}$ is a
finite $S_{\infty}$ algebra. Let be a cofibrant replacement $P_{\bullet}\to
R_{\infty}$ by noetherian polynomial rings $\mathbf{P}_{n}$ over $S_{\infty}$.
Then, one can take as $R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O}$ the complex associated to the
simplicial ring
${\bf{R}}_{\bullet}=\mathbf{P}_{\bullet}\otimes_{S_{\infty}}\mathcal{O}$,
which consists in (noetherian) polynomial $\mathcal{O}$-algebras and is a
cofibrant fibration of $R_{\infty}\otimes_{S_{\infty}}\mathcal{O}$ (but is not
a weak equivalence). By the adjunction formula of Section 3.1, we have
$\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{A}}({\bf{R}}_{\bullet},A\oplus{\operatorname{DK}}(M[1]))\cong\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{A}}(L_{{\bf{R}}_{\bullet}/\mathcal{O}}\otimes_{{\bf{R}}_{\bullet}}A,M[1]).$
Note that by construction, since ${\bf{R}}_{\bullet}$ is cofibrant, we have
$L_{{\bf{R}}_{\bullet}/\mathcal{O}}=\Omega_{{\bf{R}}_{\bullet}/\mathcal{O}}.$
Therefore,the datum of a homomorphism of chain complexes $\Phi\colon
L_{{\bf{R}}_{\bullet}/\mathcal{O}}\otimes_{{\bf{R}}_{\bullet}}A\to M[1]$ is
equivalent to that of a homomorphism of modules
$\psi_{1}\colon\Omega^{1}_{{\bf{R}}_{1}/\mathcal{O}}\otimes_{{\bf{R}}_{1}}A\to
M$ placed in degree $1$ which is itself equivalent to the datum of a classical
$\mathcal{O}$-algebra homomorphism $\Phi_{1}\colon{\bf{R}}_{1}\to A\oplus M$.
Thus we need to lift a given $\varphi_{1}\colon\pi_{1}({\bf{R}}_{\bullet})\to
M$ to a homomorphism $\Phi_{1}$.
Following [COT, Section 09D4 Example 5.9], we give an explicit description of
$\mathbf{P}_{i}$ ($i=0,1,2$) and their face and degeneracy maps. We denote
$\mathbf{P}_{0}=S_{\infty}[u_{1},\ldots,u_{k}]$ which we abbreviate as
$\mathbf{P}_{0}=S_{\infty}[u_{j}]$, and similarly for the other rings:
$\mathbf{P}_{1}=S_{\infty}[u_{j},x_{t}]$,
$\mathbf{P}_{2}=S_{\infty}[u_{j},x_{t},v_{r},y_{t},z_{t},w_{t,t^{\prime}}]$,
where $u_{i},x_{t},v_{r},y_{t},z_{t},w_{t,t^{\prime}}$ are independent
variables. One fixes $\mathbf{P}_{0}\to R_{\infty}$ a surjective
$S_{\infty}$-algebra homomorphism, and a system $(f_{t})$ of generators of
$\operatorname{Ker}(\mathbf{P}_{0}\to R_{\infty})$. Here $r=(r_{t})$ runs over
the (finitely generated) module of systems of relations between the $f_{t}$’s
in $\mathbf{P}_{0}$ : $\sum_{t}r_{t}\cdot f_{t}=0$.
Let $d_{0},d_{1}\colon\mathbf{P}_{1}\to\mathbf{P}_{0}$ be the two face maps
given by $u_{i}\mapsto u_{i}$ and $x_{t}\mapsto 0$, resp. $x_{t}\mapsto
f_{t}$. Let
$\delta_{0},\delta_{1},\delta_{2}\colon\mathbf{P}_{2}\to\mathbf{P}_{1}$ the
three face maps. See their definitions in [COT, Section 09D4 Example 5.9]. We
simply recall that $\delta_{0}(v_{r})=\delta_{1}(v_{r})=0$ and
$\delta_{2}(v_{r})=\sum_{t}r_{t}x_{t}$.
By tensoring by $\mathcal{O}$ over $\mathbf{S}_{\bullet}$, these maps yield
the face and degeneracy maps of $\bf{R}_{i}$, $i=0,1,2$. We still denote by
$d_{0},d_{1}\colon\bf{R}_{1}\to\bf{R}_{0}$ and
$\delta_{0},\delta_{1},\delta_{2}\colon\bf{R}_{2}\to\bf{R}_{1}$ the resulting
face maps. We have
$\pi_{1}({\bf{R}}_{\bullet})=\operatorname{Ker}(d_{0}-d_{1})/\operatorname{Im}(\delta_{0}-\delta_{1}+\delta_{2})$.
For any homomorphism $\varphi_{1}\colon\operatorname{Ker}(d_{0}-d_{1})\to M$,
such that $\varphi_{1}\circ(\delta_{0}-\delta_{1}+\delta_{2})=0$, we consider
$\Phi_{0}\colon\mathbf{P}_{0}\otimes_{\mathbf{S}_{0}}\mathcal{O}\to A$ as the
composition of $\mathbf{P}_{0}\otimes_{S_{\infty}}\mathcal{O}\to
R_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R$ and $R\to A$. We need to define
$\Phi_{1}\colon\mathbf{P}_{1}\otimes_{\mathbf{S}_{1}}\mathcal{O}\to A\oplus
M$, compatible to $\Phi_{0}$ for the $d_{i}$ ($i=0,1$) and $s_{0}$, in terms
of $\varphi_{1}$.
One can rewrite this more explicitely,
${\bf{R}}_{0}=\mathbf{P}_{0}\otimes_{S_{\infty}}\mathcal{O}=\mathcal{O}[[u_{j}]]$
and
${\bf{R}}_{1}=\mathbf{P}_{1}\otimes_{S_{\infty}}\mathcal{O}=\mathcal{O}[[u_{j},x_{t}]]$.
Then $\operatorname{Ker}(d_{0}-d_{1})$ is the $\mathcal{O}[[u_{j}]]$-module of
$P(u_{j},x_{t})$ such that
$P(u_{j},0)=P(u_{j},f_{t}).$
Moreover, $R_{\infty}\otimes_{S\infty}\mathcal{O}$ is a quotient of
$\bf{R}_{0}$. We can extend $\varphi_{1}$ to $\mathcal{O}[[u_{j},x_{t}]]$ as
follows. We first write $P=P_{1}+K_{1}$ where
$P_{1}=P(u_{j},0)+\sum_{t}Q_{t}(u_{j})\cdot x_{t}$ and
$K_{1}\in\operatorname{Ker}(d_{0}-d_{1})$; indeed, if we write
$P(u_{j},f_{t})-P(u_{j},0)=\sum_{t}Q_{t}(u_{j})\cdot f_{t}$, then if we put
$P_{1}(u_{j},x_{t})=P(u_{j},0)+\sum_{t}Q_{t}(u_{j})\cdot x_{t}$, we see that
$K_{1}=P-P_{1}\in\operatorname{Ker}(d_{0}-d_{1})$. Then we put
$\Phi_{1}(P)=\Phi_{0}(d_{0}(P_{1}))+\varphi_{1}(K_{1})$
Note that $\Phi_{0}(d_{0}(P_{1}))=\Phi_{0}(d_{0}(P))$. Let us check that
$\Phi_{1}$ is well defined. If we write
$P(u_{j},f_{t})-P(u_{j},0)=\sum_{k}Q_{t}(u_{j})\cdot
f_{t}=\sum_{k}Q^{\prime}_{t}(u_{j})\cdot f_{t}$
for another system $(Q^{\prime}_{t})$ of elements of $\mathcal{O}[[u_{j}]]$;
let $K_{1}^{\prime}=P-P^{\prime}_{1}$ where
$P_{1}^{\prime}=P(u_{j},0)+\sum_{t}Q_{t}^{\prime}x_{t}$. We need to show that
$\varphi_{1}(K_{1})=\varphi_{1}(K^{\prime}_{1})$. The system $r=(r_{t})$
defined by $r_{t}=Q_{t}-Q_{t}^{\prime}$ is a relation between the $f_{t}$’s:
$\sum_{t}r_{t}f_{t}=0$. By [COT, 09D4], we see that
$\varphi_{1}(K_{1})-\varphi_{1}(K^{\prime}_{1})=\varphi_{1}(\sum_{t}r_{t}x_{t})=\varphi_{1}(\delta_{0}-\delta_{1}+\delta_{2})(v_{r}))=0$
hence $\Phi_{1}$ is well defined. This yields the desired lifting
$\Phi\colon R_{\bullet}\to A\oplus{\operatorname{DK}}(M[1]))$
of $\varphi_{1}\colon\pi_{1}(R_{\bullet})\to M$.
To conclude the proof, we have compatible morphisms $\mathcal{R}\to
R_{\bullet}/{\mathfrak{a}}_{n}$ by [GV18, (14.4)] which induce an isomorphism
$\pi_{1}(\mathcal{R})\cong\pi_{1}(R_{\bullet})$. Therefore, the surjectivity
of $\pi(n,\mathcal{R})$ follows from that of $\pi(n,R_{\bullet})$. ∎
###### Proof.
Second Proof: For a $B$-algebra $C$, we denote by $B[C]^{(j)}$ the ring
defined by induction as follows : $B[C]^{(0)}$ is the polynomial $B$-algebra
with variables the elements of $C$, and for $j\geq 1$, $B[C]^{(j)}$ is the
polynomial $B$-algebra with variables the elements of $B[C]^{(j-1)}$. As
explained in [SIMP, Section 14.34.5], these sets form a simplicial set. In
particular the $d_{k}$’s, $k=0,\ldots,j$ from $B[C]^{(j)}$ to $B[C]^{(j-1)}$
are given as follows. Let
$P_{j}=P_{j}([P_{j-1}(\ldots([P_{0}([\underline{c})])]\ldots])\in B[C]^{(j)}.$
Then, $d_{k}P_{j}$ is obtained by removing the $k$-th bracket (the zeroth
being the innermost one).
Let $B=S_{\infty}$ and $C=R_{\infty}$ We choose the functorial cofibrant
replacement $P_{\bullet}$ of $R_{\infty}$ by rings
$\mathbf{P}_{j}=S_{\infty}[R_{\infty}]^{(j)}$ Then, one can take as
$R_{\infty}\stackrel{{\scriptstyle L}}{{\otimes}}_{S_{\infty}}\mathcal{O}$ the
complex associated to the simplicial ring
${\bf{R}}_{\bullet}=\mathbf{P}_{\bullet}\otimes_{S_{\infty}}\mathcal{O}$; tt
follows from [Gil13, Corollary 7.10.5] that ${\bf{R}}_{\bullet}$ is a
cofibrant fibration of $R_{\infty}\otimes_{S_{\infty}}\mathcal{O}$ (but is not
a weak equivalence). It can also be proven by verifying first that the
morphism of simplicial rings $\mathcal{O}\to{\bf{R}}_{\bullet}$ is free in the
sense of [Gil13, Definition 7.6.2] (that is, by verifying Conditions (1)-(3)
which follow this definition), then, by quoting [Gil13, Theorem 7.6.13]
stating that any free simplicial morphism is a cofibration.
Let $P_{1}=P_{1}([P_{0}([\underline{c}])])\in{\bf{R}}_{1}$ where $P_{1}$,
resp. $P_{0}$, has coefficients in $\mathcal{O}$, resp. in $B$. Let
$P^{(1,0)}_{0},\ldots,P^{(m,0)}_{0}$ be the finite list of polynomials $P_{0}$
which actually occur as variables in $P_{1}$. We write
$P_{1}=P_{1}([P_{0}(\underline{c})])+\sum_{s=1}^{m}([P^{(s,0)}_{0}]-[P^{(s,0)}(\underline{c})])E_{s}$,
where $E_{s}\in{\bf{R}}_{1}$. Let $P^{(1,1)}_{0},\ldots,P^{(n,1)}_{0}$ be the
list of polynomials $P_{0}$ which occur as variables in $E_{s}$. Then we
redecompose $E_{s}$ as
$E_{s}=E_{s}(P_{0}([\underline{c}]))+\sum_{t=1}^{n}([P^{(t,1)}_{0}]-P^{(t,1)}([\underline{c}]))F_{s,t}$.
If we put
$Q_{1}=P_{1}([P_{0}(\underline{c})])+\sum_{s=1}^{m}([P^{(s,0)}_{0}]-[P^{(s,0)}(\underline{c})])\cdot
E_{s}(P_{0}([\underline{c}]))$
and
$K_{1}=\sum_{s,t}([P^{(s,0)}_{0}]-[P^{(s,0)}(\underline{c})])\cdot([P^{(t,1)}_{0}]-[P^{(t,1)}(\underline{c})])\cdot
F_{s,t}$
we see that $P_{1}=Q_{1}+K_{1}$, with $Q_{1}\in{\bf{R}}_{0}$ and $K_{1}\in
ZN_{1}({\bf{R}}_{\bullet})$ since we have clearly $d_{k}P_{1}=d_{k}Q_{1}$ for
$k=0,1$. Moreover, if $P_{1}=Q_{1}+K_{1}=Q^{\prime}_{1}+K^{\prime}_{1}$, we
have $Q_{1}=d_{0}\widetilde{Q}_{1}$ and
$Q_{1}^{\prime}=d_{0}\widetilde{Q^{\prime}}_{1}$ (where $\widetilde{Q}_{1}$,
resp. $\widetilde{Q^{\prime}}_{1}$, is defined by inserting an inner bracket:
if $Q_{1}=Q_{1}([c])$, put $\widetilde{Q}_{1}=Q_{1}([[[c]])])$ and similarly
for $\widetilde{Q^{\prime}}_{1}$), hence $Q_{1}-Q^{\prime}_{1}\in
BN_{1}({{\bf{R}}}_{\bullet})$. ∎
Let $\lambda\colon R\to\mathcal{O}$ be a Hecke eigensystem associated to an
automorphic representation $\pi^{\prime}$occuring in $T$. Let us consider
$M_{n}=\varpi^{-n}\mathcal{O}/\mathcal{O})$ as a $R$-module through $\lambda$;
we take the Pontryagin dual $\pi(n,\mathcal{R})^{\vee}$ and apply Poitou-Tate
duality. We obtain a $T$-linear injection :
$GV_{n}\colon\operatorname{\mathsf{Hom}}_{R}(\pi_{1}(\mathcal{R}),M_{n})^{\vee}\hookrightarrow\mathrm{H}^{1}_{f}(F,\operatorname{Ad}\,\rho_{n}(1)\otimes
M_{n}^{\vee})$
We have
$\operatorname{\mathsf{Hom}}_{R}(\pi_{1}(\mathcal{R}),M_{n})=\operatorname{\mathsf{Hom}}_{\mathcal{O}}(\pi_{1}(\mathcal{R})\otimes_{R,\lambda}\mathcal{O}/\varpi^{n},\varpi^{-n}\mathcal{O}/\mathcal{O}_{n})$.
The Pontryagin dual of this module is equal to
$\pi_{1}(\mathcal{R})\otimes_{R,\lambda}\mathcal{O}/\varpi^{n}$.
The right hand side is
$\mathrm{H}^{1}_{f}(F,\operatorname{Ad}\,\rho_{\lambda}(1)\otimes\mathcal{O}/\varpi^{n})=\mathrm{H}^{1}_{f}(F,\operatorname{Ad}\,\rho_{\lambda}(1)/(\varpi^{n}))=\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\lambda})(1)\otimes\varpi^{-n}\mathcal{O}/\mathcal{O})$.
Taking inductive limit on both sides we obtain a linear injection
$GV\colon\pi_{1}(\mathcal{R})\otimes_{R,\lambda}\mathcal{O}\otimes_{\mathcal{O}}K/\mathcal{O}\hookrightarrow\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\lambda})(1)\otimes K/\mathcal{O})$
###### Theorem 13.
Assume $p>N$, $\zeta_{p}\notin F$, $(\operatorname{Gal}_{\mathfrak{m}})$,
$(LLC)$, $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$.The natural
homomorphism
$GV\colon\pi_{1}(\mathcal{R})\otimes_{R,\lambda}\mathcal{O}\otimes_{\mathcal{O}}K/\mathcal{O}\hookrightarrow\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\lambda})(1)\otimes K/\mathcal{O})$
is injective. The left hand side module is $\varpi$-divisible of
$\mathcal{O}$-corank $\ell_{0}$.
###### Proof.
The left-hand side module is the tensor product of a finitely generated
$\mathcal{O}$-module by $K/\mathcal{O}$, hence it is $\varpi$-divisible.
Moreover, we have $R=T$. Let us show that
$\pi_{1}(\mathcal{R})\otimes_{R,\lambda}\mathcal{O}\otimes_{\mathcal{O}}K/\mathcal{O}$
has corank $\ell_{0}$: Indeed, by Borel-Wallach, we have
$\dim\,\mathrm{H}^{q_{s}-1}_{temp}=\ell_{0}\cdot\dim\,\mathrm{H}^{q_{s}}_{temp}$
and and, for $F$ a CM field,
$\dim\,\mathrm{H}^{q_{s}}_{cusp}=\dim\,A_{cusp}(\lambda)$
where $A_{cusp}(\lambda)$ denotes the space of cuspidal automorphic forms of
level $K$ and cohomological weight $\lambda$. This implies that
$H^{q_{s}}_{\mathfrak{m}}$ is free of rank one over $T$, hence
$\mathrm{H}^{q_{s}-1}_{\mathfrak{m}}$ is free of rank $\ell_{0}$ over $T$.
This concludes the proof. ∎
As in [TU19, Lemma 10], one shows by using Poitou-Tate duality that the right-
hand side $\mathcal{O}$-module is of cofinite rank $\ell_{0}$. In other words,
the left-hand side is the maximal $\varpi$-divisible submodule of the right-
hand side. Therefore by [BK90, Definition 5.13], we see that
$Coker\,GV=\Sha(\operatorname{Ad}(\rho_{\mathfrak{m}})(1))$. By Poitou-Tate
duality, this group is finite and is isomorphic to the Pontryagin dual of
$\mathop{\rm Sel}(\operatorname{Ad}\,\rho_{\lambda}\otimes K/\mathcal{O})$. By
deformation theory, this Pontryagin dual is isomorphic to
$C_{1}(R,\lambda)=\Omega_{R/\mathcal{O}}\otimes_{T,\lambda}\mathcal{O}$. By
the $R=T$ theorem of Calegari-Geraghty, this group is isomorphic to
$C_{1}(R,\lambda)=\Omega_{T/\mathcal{O}}\otimes_{T,\lambda}\mathcal{O}$, which
is finite.
## 5\. A graded version of the Galatius-Venkatesh homomorphism
Under the assumptions $p>N$, $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$,
we can also define a graded version $GV^{\bullet}$ of the Galatius-Venkatesh
homomorphism $GV$. For every cuspidal representation $\pi^{\prime}$ occuring
in $H^{\bullet}_{\mathfrak{m}}$, let
$\phi=\lambda^{\operatorname{gal}}_{\pi^{\prime}}\colon R^{?}\to\mathcal{O}$
be the algebra homomorphism associated to $\rho_{\pi^{\prime}}$. Recall that
the derived universal ring $\mathcal{R}$ is given by a projective system of
cofibrant simplicial artinian rings $(\mathcal{R}_{\alpha})$. For any $j\geq
1$, we denote by $\mathcal{R}^{\otimes j}$ the pro-artinian simplical ring
defined by the projective system of the strict tensor product simplicial
artinian $\mathcal{O}$-algebras $(\mathcal{R}_{\alpha}^{\otimes j})$. Recall
that the strict tensor product of two simplicial $\mathcal{O}$-modules
$\operatorname{A}=(\mathcal{A}_{n})$ and $\mathcal{B}=(\mathcal{B}_{n})$ is
$\mathcal{A}\otimes\mathcal{B}$ such that for any $n\geq 0$,
$(\mathcal{A}\otimes\mathcal{B})_{n}=\mathcal{A}_{n}\otimes_{\mathcal{O}}\mathcal{B}_{n}$.
If $\mathcal{A}$ and $\mathcal{B}$ are simplicial $\mathcal{O}$-algebras, the
result is a simplicial $\mathcal{O}$-algebra.
Let $m\geq 1$ and $A=\mathcal{O}/(\varpi^{m})$ and
$\pi_{k}(\mathcal{R})_{A}=\pi_{k}(\mathcal{R})\otimes_{\pi_{0}(\mathcal{R}),\phi}A$.
Note that for any $j\geq 0$, we have a natural ring homomorphism
$\pi_{0}(\mathcal{R})^{\otimes j}\to\pi_{0}(\mathcal{R}^{\otimes j})$, hence a
natural structure of $\pi_{0}(\mathcal{R})^{\otimes j}$-module on
$\pi_{j}(\mathcal{R}^{\otimes j})$. Let us consider $\phi^{\otimes
j}\colon\pi_{0}(\mathcal{R})^{\otimes j}\to A$, via the identification
$A^{\otimes j}\cong A$. We can form $\pi_{j}(\mathcal{R}^{\otimes
j})_{A}=\pi_{j}(\mathcal{R}^{\otimes j})\otimes_{\pi_{0}(\mathcal{R})^{\otimes
j},\phi^{\otimes j}}A$.
In this subsection, we assume $p>\ell_{0}=Nd_{0}-1$.
Let $\mathcal{T}$ be a simplicial $\mathcal{O}$-algebra. For $j_{1},j_{2}\geq
0$ and $j=j_{1}+j_{2}$, consider the subset
$P_{j_{1},j_{2}}\subset{{\mathbb{S}}}_{j}$ of permutations $(\sigma,\tau)$
where $\sigma(i)=\sigma_{i}$, $i=1,\ldots,j_{1}$ with
$\sigma_{1}<\ldots\sigma_{j_{1}}$, and $\tau(i+j_{1})=\tau_{i}$,
$i=1,\ldots,j_{2}$, with $\tau_{1}<\ldots<\tau_{j_{2}}$. Recall that the
graded $\mathcal{O}$-module $\pi_{\bullet}(\mathcal{T})$ is endowed with a
structure of graded $\mathcal{O}$-algebra by the shuffle product
$[-,-]_{j_{1},j_{2}}\colon\pi_{j_{1}}(\mathcal{T})\times\pi_{j_{2}}(\mathcal{T})\to\pi_{j}(\mathcal{T})$
induced by the Eilenberg-Zilber map: for $t_{i}\in\mathcal{T}_{j_{i}}$ by
$\sum_{(\sigma,\tau)\in P_{j_{1},j_{2}}}sgn(\sigma,\tau)\cdot
A(\sigma)(t_{1})\cdot A(\tau)(t_{2})$
see [Gil13, Section 8.3] or [EZ].
Consider the graded $\mathcal{O}$-algebra
$\pi_{\bullet}(\mathcal{T}^{\otimes\bullet})=\bigoplus_{j}\pi_{j}(\mathcal{T}^{\otimes
j})$
for the multiplication given, for $j_{1},j_{2}\geq 0$ and $j_{1}+j_{2}=j$, by
$<-,->_{j_{1},j_{2}}\colon\pi_{j_{1}}(\mathcal{T}^{\otimes
j_{1}})\times\pi_{j_{2}}(\mathcal{T}^{\otimes
j_{2}})\to\pi_{j}(\mathcal{T}^{\otimes j})$
defined as the composition of the normalized shuffle product for the simplical
$\mathcal{O}$-algebra $\mathcal{T}^{\otimes j}$:
$[-,-]_{j_{1},j_{2}}\colon\pi_{j_{1}}(\mathcal{T}^{\otimes
j})\times\pi_{j_{2}}(\mathcal{T}^{\otimes j})\to\pi_{j}(\mathcal{T}^{\otimes
j})$
with the morphism
$\pi_{j_{1}}(\mathcal{R}^{\otimes
j_{1}})\times\pi_{j_{2}}(\mathcal{T}^{\otimes
j_{2}})\to\pi_{j_{1}}(\mathcal{T}^{\otimes
j})\times\pi_{j_{2}}(\mathcal{T}^{\otimes j})$
induced by $\iota_{1}\otimes\iota_{2}$ where
$\iota_{1}\colon\mathcal{T}^{\otimes j_{1}}\to\mathcal{T}^{\otimes
j_{1}+j_{2}},x\mapsto x\otimes 1$ and $\iota_{2}\colon\mathcal{R}^{\otimes
j_{2}}\to\mathcal{R}^{\otimes j_{1}+j_{2}},x\mapsto 1\otimes x$. Note that
$\pi_{\bullet}(\mathcal{T}^{\otimes\bullet})$ endowed with $<-,->$ is a graded
associative algebra but is not graded-commutative in general. It comes with a
graded-algebra homomorphism
$c_{\bullet}\colon\pi_{\bullet}(\mathcal{T}^{\otimes\bullet})\to\pi_{\bullet}(\mathcal{T})$
given degreewise by the multiplication $m_{j}\colon\mathcal{T}^{\otimes
j}\to\mathcal{T}$. It is compatible with the (tensor) shuffle product: for
$j=j_{1}+j_{2}$, we have
$[-,-]_{j_{1},j_{2}}\circ(c_{j_{1}}\otimes
c_{j_{2}})=c_{j}\circ<-,->_{j_{1},j_{2}}.$
Let $\widetilde{\pi}_{\bullet}(\mathcal{T}^{\otimes\bullet})$ be the largest
graded-commutative quotient of $\pi_{\bullet}(\mathcal{T}^{\otimes\bullet})$.
It is the graded $\mathcal{O}$-algebra sum of the quotients
$\widetilde{\pi}_{j}(\mathcal{T}^{\otimes j})$ of
$\widetilde{\pi}_{j}(\mathcal{T}^{\otimes j})$, defined by induction as the
sub-$\mathcal{O}$-module generated by the products
$<t_{1},t_{2}>-(-1)^{j_{1}j_{2}}<t_{2},t_{1}>_{j_{2},j_{1}}$,
$t_{i}\in\widetilde{\pi}_{j_{i}}(\mathcal{T}^{\otimes j_{i}})$. The
homomorphism $c_{\bullet}$ factors through
$\widetilde{\pi}_{\bullet}(\mathcal{T}^{\otimes\bullet})$ as
$\widetilde{c}_{\bullet}\colon\widetilde{\pi}_{\bullet}(\mathcal{T}^{\otimes\bullet})\to\pi_{\bullet}(\mathcal{T})$
Note that the iterated tensor shuffle product
$\pi_{1}(\mathcal{T})^{\otimes\bullet}\to{\pi}_{\bullet}(\mathcal{T}^{\otimes\bullet})$
induces a homomorphism of graded-commutative algebras
$s_{\bullet}\colon\bigwedge^{\bullet}\pi_{1}(\mathcal{T})\to\widetilde{\pi}_{\bullet}(\mathcal{T}^{\otimes\bullet})$
The composition $c_{\bullet}\circ s_{\bullet}$ coincides with the iterated
classical shuffle product
$\bigwedge^{\bullet}\pi_{1}(\mathcal{T})\to\pi_{\bullet}(\mathcal{T}).$
For certain simplicial rings $\mathcal{T}$, as seen below,
$\widetilde{c}_{\bullet}$ is an isomorphism. However, this may not be always
the case.
###### Remark 4.
1) Given a simplicial $\mathcal{O}$-algebra $\mathcal{T}$, the relation
between $\pi_{\bullet}(\mathcal{T})$ and
$\pi_{\bullet}(\mathcal{T}^{\otimes\bullet})$ is similar to the relation, for
a finite free $\mathcal{O}$-module $T$, between $\bigwedge^{\bullet}T$ and
$\bigwedge^{\bullet}(T^{\oplus\bullet})$, using inclusions $\iota_{k}\colon
T^{\oplus j_{k}}\to T^{\oplus j}$ for $j=j_{1}+j_{2}$ and wedge products
$\bigwedge^{j_{1}}T^{\oplus j}\times\bigwedge^{j_{2}}T^{\oplus
j}\to\bigwedge^{j}T^{\oplus j}.$
The graded algebra homomorphism
$c_{\bullet}\bigwedge^{\bullet}T^{\oplus\bullet}\to\bigwedge^{\bullet}T$
analogue to
$c_{\bullet}\colon\pi_{\bullet}(\mathcal{T}^{\otimes\bullet})\to\pi_{\bullet}(\mathcal{T})$
is induced by the addition morphisms $a_{j}\colon T^{\oplus j}\to T$ (instead
of the multiplications $m_{j}$).
2) For $\mathcal{T}=\mathcal{R}$ the universal derived deformation ring, we
know that the graded-commutative $\mathcal{O}$-algebra
$\pi_{\bullet}(\mathcal{R})$ is finite. Here, the (non-commutative) graded
algebra $\pi_{\bullet}(\mathcal{R}^{\otimes\bullet})$ is not finite in
general. Consider the graded-commutative algebra homomorphism
$\widetilde{c}_{\bullet}\colon\widetilde{\pi}_{\bullet}(\mathcal{R}^{\otimes\bullet})\to\pi_{\bullet}(\mathcal{R})$
One can ask whether it is an isomorphism.
3) As graded $\mathcal{O}$-modules
$\pi_{\bullet}(\mathcal{R}^{\otimes\bullet})$ and $\pi_{\bullet}(\mathcal{R})$
can be related by induction, using the Tor-spectral sequences ([Qu67, Theorem
6,Sect.6,II]):
$E^{2}=\operatorname{Tor}^{\mathcal{O}}_{\bullet}(\pi_{\bullet}(\mathcal{R}^{\otimes
j-1}),\pi_{\bullet}(\mathcal{R}))\Rightarrow\pi_{\bullet}(\mathcal{R}^{\otimes
j})$
4) The graded $\mathcal{O}$-algebra
$\pi_{\bullet}(\mathcal{R}^{\otimes\bullet})$ is also a graded algebra over
the graded $\mathcal{O}$-algebra $\pi_{0}(\mathcal{R})^{\otimes\bullet}$. Let
$A=\mathcal{O}/(\varpi^{m})$. The system of homomorphisms $\phi^{\otimes
j}\colon\pi_{0}(\mathcal{R})^{\otimes j}\to A$, $j=1,\ldots$, defined above
give rise to an $\mathcal{O}$-algebra homomorphism
$\phi^{\otimes\bullet}\colon\pi_{0}(\mathcal{R})^{\otimes\bullet}\to A,$
and the tensor products $\pi_{j}(\mathcal{R}^{\otimes
j})_{A}=\pi_{j}(\mathcal{R}^{\otimes j})\otimes_{\pi_{0}(\mathcal{R})^{\otimes
j},\phi^{\otimes j}}A$. give rise to a tensor product over the graded tensor
algebra $\pi_{0}(\mathcal{R})^{\otimes\bullet}$
$\pi_{(}\mathcal{R}^{\otimes\bullet})_{A}=\pi_{\bullet}(\mathcal{R}^{\otimes\bullet})\otimes_{\pi_{0}(\mathcal{R})^{\otimes\bullet},\phi^{\otimes\bullet}}A.$
Let $V$ be a finite free $A$-module; consider the simplicial $A$-algebra
$\mathcal{S}_{V}=A\oplus\operatorname{DK}(V[1])$ and the graded algebra
$\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\bullet})$ endowed with $<-,->$.
###### Lemma 14.
The graded algebra $\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\bullet})$ is
canonically isomorphic to the graded $A$-algebra $\bigotimes^{\bullet}V$; its
largest graded-commutative quotient
$\widetilde{\pi}_{\bullet}(\mathcal{S}_{V}^{\otimes\bullet})$ is isomorphic to
$\bigwedge^{\bullet}V$. Moreover, $\widetilde{c}_{\bullet}$ is an isomorphism,
as well as $s_{\bullet}$; they yield canonical identifications
$\widetilde{\pi}_{\bullet}(\mathcal{S}_{V}^{\otimes\bullet})=\pi_{\bullet}(\mathcal{S}_{V})=\bigwedge^{\bullet}{\pi}_{1}(\mathcal{S}_{V}).$
###### Remark 5.
For instance, for $V=A$, $\pi_{\bullet}(\mathcal{S}_{A}^{\otimes\bullet})$ is
isomorphic to the (strictly) commutative graded algebra $A[X]$ of one variable
polynomials, hence $\pi_{\bullet}(\mathcal{S}_{A})\cong A[X]/(X^{2})$ as
graded-commutative algebras, in such a way that the algebra homomorphism
$c_{\bullet}\colon\pi_{\bullet}(\mathcal{S}_{A}^{\otimes\bullet})\to\pi_{\bullet}(\mathcal{S}_{A})$
identifies to the quotient homomorphism $A[X]\to A[X]/(X^{2})$.
###### Proof.
We first note that given two simplicial $\mathcal{O}$-modules $\mathcal{A}$
and $\mathcal{B}$, and $\mathcal{A}\otimes\mathcal{B}$ their strict tensor
product (degreewise tensor product), the Eilenberg-Zilber map
$C(\mathcal{A})\otimes C(\mathcal{B})\to C(\mathcal{A}\otimes\mathcal{B})$
induces an isomorphism of complexes
$N(\mathcal{A})\otimes N(\mathcal{B})\to N(\mathcal{A}\otimes\mathcal{B})$
by the Eilenberg-MacLane theorem (its inverse is the Alexander-Whitney map,
see for instance [Gil13, Section 5.8, Page 76] or [GJ10, Theorem 4.2.4, Page
205]). Therefore, by applying the Dold-Kan functor to this isomorphism we see
that for two chain complexes $C,D\in Ch_{+}(\mathcal{O})$, there is a natural
weak equivalence
$\operatorname{DK}(C\otimes D)\cong\operatorname{DK}(C)\otimes DK(D).$
In particular, if $V[1]^{\otimes k}$ denotes the total tensor $k$th-power
complex of $V[1]$, we have a weak equivalence
$\operatorname{DK}(V[1])^{\otimes k}\cong\operatorname{DK}(V[1]^{\otimes k}).$
Moreover, for any $k\geq 0$, the tensor product induces an isomorphism of
complexes
$m_{k}\colon V[1]^{\otimes k}\cong V^{\otimes k}[k]$
By composing these maps, we obtain a weak equivalence of simplicial
$A$-modules
$\mathcal{S}_{V}^{\otimes j}\cong\bigoplus_{k=0}^{j}{j\choose
k}\cdot\operatorname{DK}(V^{\otimes k}[k])$
which we can rewrite as a quasi-isomorphism
$N(\mathcal{S}_{V}^{\otimes j})\cong\bigoplus_{k=0}^{j}{j\choose k}\cdot
V^{\otimes k}[k].$
Now $\pi_{j}(\operatorname{DK}(W[k]))=\mathrm{H}_{j}(W[k])=0$ for any $k<j$
and any finite free $A$-module $W$, hence $m_{j}\circ pr_{2}^{\otimes j}$
induces an isomorphism $\pi_{j}(\mathcal{S}_{V}^{\otimes
j})=\pi_{j}(\operatorname{DK}(V^{\otimes j}[j]))$ which yields
$\pi_{j}(\mathcal{S}_{V}^{\otimes j})\cong\mathrm{H}_{j}(V^{\otimes
j}[j])=V^{\otimes j}$. Hence we have an isomorphism of graded $A$-modules
$\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\bullet})\cong\bigotimes^{\bullet}V$
It remains to show it is multiplicative. For this, we use again the Eilenberg-
MacLane theorem: Let us consider the Eilenberg-Zilber map
$EZ_{j_{1},j_{2}}\colon\colon C(\mathcal{S}_{V}^{\otimes j_{1}})\otimes
C(\mathcal{S}_{V}^{\otimes j_{2}})\to C(\mathcal{S}_{V}^{\otimes
j_{1}}\otimes\mathcal{S}_{V}^{\otimes j_{2}})$
followed by the canonical identification
$C(\mathcal{S}_{V}^{\otimes j_{1}}\otimes\mathcal{S}_{V}^{\otimes j_{2}})\cong
C(\mathcal{S}_{V}^{\otimes j}).$
The homomorphism $<-,->_{j_{1},j_{2}}$ is given by the restriction to the
$(j_{1},j_{2})$-component on the left-hand side of $EZ_{j_{1},j_{2}}$. Using
the identifications $V^{\otimes k}=\mathrm{H}_{0}(V^{\otimes
k}[0])=\mathrm{H}_{k}(V^{\otimes
k}[k])=\mathrm{H}_{k}(C(\mathcal{S}_{V})^{\otimes k})$, we see that the
isomorphism $V^{\otimes j_{k}}\cong\pi_{j_{k}}(\mathcal{S}_{V}^{\otimes
j_{k}})$, $k=1,2$, is given by the $j_{k}$-th homology
$\mathrm{H}_{j_{k}}(EZ_{j_{k}})$, of the morphism of chain complexes
$EZ_{j_{k}}\colon C(\mathcal{S}_{V})^{\otimes j_{k}}\to
C(\mathcal{S}_{V}^{\otimes j_{k}}).$
Let $m\colon V^{\otimes j_{1}}\otimes_{A}V^{\otimes j_{2}}\to V^{\otimes j}$
be the multiplication isomorphism. By the identification
$\mathrm{H}_{0}(W[0])=W$, we have $EZ_{0,0}=m$, hence
$\mathrm{H}_{j_{1}}(V^{\otimes
j_{1}}[j_{1}])\otimes\mathrm{H}_{j_{2}}(V^{\otimes
j_{2}}[j_{2}])\to\mathrm{H}_{j}(V^{\otimes j}[j])$
is given by $m$ via $\mathrm{H}_{k}(V^{\otimes
k}[k])=\mathrm{H}_{0}(V^{\otimes k}[0])=V^{\otimes k}$. On the other hand, we
have by associativity of the shuffles
$EZ_{j_{1},j_{2}}\circ(EZ_{j_{1}}\otimes EZ_{j_{2}})=EZ_{j}\circ EZ_{0,0}.$
This shows that, via the identifications $EZ_{j_{k}}\colon V^{\otimes
j_{k}}\cong\pi_{j_{k}}(\mathcal{S}_{V}^{\otimes j_{k}})$ and $EZ_{j}\colon
V^{\otimes j}\cong\pi_{j}(\mathcal{S}_{V}^{\otimes j})$, the multiplication
$<-,->_{j_{1},j_{2}}\colon\pi_{j_{1}}(\mathcal{S}_{V}^{\otimes
j_{1}})\times\pi_{j_{2}}(\mathcal{S}_{V}^{\otimes
j_{2}})\to\pi_{j}(\mathcal{S}_{V}^{\otimes j})$
becomes the multiplication $V^{\otimes j_{1}}\times V^{\otimes j_{2}}\to
V^{\otimes j}$, as desired.
∎
Recall that if $p>\ell_{0}$, for any $j\leq\ell_{0}$, for any $A$-module $W$,
the largest submodule $\bigwedge^{j}W$ and the largest quotient
$\bigwedge^{j}W$ of $W^{\otimes j}$ are naturally isomorphic. For this reason,
for a graded algebra $G$, we introduce the truncation
$\tau_{\leq\ell_{0}}G=\bigoplus_{j\leq\ell_{0}}G_{j}$
with a partial algebra structure defined only for $g_{j_{k}}\in G_{j_{k}}$
such that $j=j_{1}+j_{2}\leq\ell_{0}$.
###### Theorem 14.
For every cuspidal representation $\pi^{\prime}$ occuring in
$H^{\bullet}_{\mathfrak{m}}$, for any $m\geq 1$, there is an homomorphism of
$A_{m}$-modules
$(HGV_{j})\quad GV_{m}^{j}\colon\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A_{m}}\to\bigwedge^{j}_{A_{m}}\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})^{\oplus j}$
It induces a morphism of truncated graded $A_{m}$-algebras
$(HGV_{\bullet})\quad
GV_{m}^{\bullet}\colon\tau_{\leq\ell_{0}}\widetilde{\pi}_{\bullet}(\mathcal{R}^{\otimes\bullet})\otimes
A_{m}\to\tau_{\leq\ell_{0}}\bigwedge^{\bullet}_{A_{m}}\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})^{\oplus\bullet}$
Moreover, for any $j$, the composition $a_{j}\circ GV_{m}^{j}\circ\mu_{j}$ of
$GV_{m}^{j}$ with the tensor shuffle multiplication map
$\mu_{j}\colon\bigwedge^{j}_{A}\pi_{1}(\mathcal{R})_{A}\to\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A}$ and the homomorphism $a_{j}$ induced by the addition
$\mathop{\rm Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
A_{m})^{\oplus j}\to\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})$
coincides with $\bigwedge^{j}GV_{m}^{1}$.
For $j=\ell_{0}$, the cokernel of $GV_{m}^{\ell_{0}}\circ\mu_{\ell_{0}}$ is
annihilated $\operatorname{Fitt}(\mathop{\rm Sel}(Ad(\rho_{\pi^{\prime}}))$.
###### Proof.
Let $m\geq 1$. We define first for any $j\geq 1$ a homomorphism
$GV_{m}^{j}\colon\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})\otimes_{\pi_{0}(\mathcal{R}),\lambda^{\operatorname{gal}}_{\pi^{\prime}}}\mathcal{O}/\varpi^{m}\mathcal{O}\rightarrow\bigwedge^{j}_{\mathcal{O}/(\varpi^{m})}\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes\mathcal{O}/\varpi^{m}\mathcal{O})^{\oplus
j}.$
Let $A=\mathcal{O}/(\varpi^{m})$. Let
$\rho=\rho_{\pi^{\prime}}\pmod{\varpi^{m}}$. We will proceed by Pontryagin
duality, and define an $A$-linear homomorphism
$\Theta_{m,V}^{j}\colon\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V})^{\otimes_{A}j}\to\operatorname{\mathsf{Hom}}(\pi_{j}(\mathcal{R}^{\otimes
j}),V^{\otimes j})$. Let
$\Phi=\phi_{1}\otimes\ldots\otimes\phi_{j}\in\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V})^{\otimes_{A}j}$.
It defines a homomorphism
$\Phi\colon\mathcal{R}^{\otimes_{A}j}\to\mathcal{S}_{V}^{\otimes_{A}j}$
and, by passing to the homotopy groups:
$\pi_{j}(\Phi)\colon\pi_{j}(\mathcal{R}^{\otimes
j})\to\pi_{j}(\mathcal{S}_{V}^{\otimes_{A}j})$. By Lemma 14, we have
canonically $\pi_{j}(\mathcal{S}_{V}^{\otimes j})=V^{\otimes j}$. Thus, we can
view $\pi_{j}(\Phi)$ as a homomorphism
$\pi_{j}(\Phi)\colon\pi_{j}(\mathcal{R}^{\otimes j})\to V^{\otimes j}.$
and put $\Theta_{m,V}^{j}(\Phi)=\pi_{j}(\Phi)$. By Lemma 14 again, the direct
sum of these homomorphisms provides a graded algebra homomorphism
$\Theta_{m,V}^{\bullet}\colon\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V})^{\otimes\bullet}\to\operatorname{\mathsf{Hom}}_{gralg}(\pi_{\bullet}(\mathcal{R}^{\otimes\bullet}),V^{\otimes\bullet})$
By composing with the quotient morphism $q_{j}\colon
V^{\otimes\bullet}\to\bigwedge^{\bullet}V$, any graded algebra homomorphism
$\pi_{\bullet}(\mathcal{R}^{\otimes\bullet})\to V^{\otimes\bullet}$ induces a
homomorphism
$\widetilde{\pi}_{\bullet}(\mathcal{R}^{\otimes\bullet})\to\bigwedge{}^{\bullet}V.$
For any $A$-module $W$, let $\widetilde{\bigwedge}^{\bullet}W$ be the largest
graded-commutative submodule of $W^{\otimes\bullet}$. By restricting
$\Phi\mapsto q_{j}\circ\pi_{j}(\Phi)$ to
$\widetilde{\bigwedge}^{\bullet}\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V})$,
we obtain a homomorphism
$\widetilde{\bigwedge}^{\bullet}\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V})\to\operatorname{\mathsf{Hom}}_{gralg}(\widetilde{\pi}_{\bullet}(\mathcal{R}^{\otimes\bullet}),\bigwedge{}^{\bullet}V).$
which we still denote by $\Theta_{m,V}^{\bullet}$. In particular, for every
$j\geq 1$, we have a homomorphism
$\Theta_{m,V}^{j}\colon\widetilde{\bigwedge}{}^{j}\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V})\to\operatorname{\mathsf{Hom}}_{A}(\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A},\bigwedge{}^{j}V).$
For each fixed $j$, let us choose $V=V_{j}:=A^{j}$ so that
$\bigwedge{}^{j}V_{j}=A$. We obtain
$\Theta_{m,V_{j}}^{j}\colon\widetilde{\bigwedge}{}^{j}\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V_{j}})\to\operatorname{\mathsf{Hom}}_{A}(\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A},A).$
Since $p>\ell_{0}$, we know that for any $A$-module $W$ and for any
$j\leq\ell_{0}$, $\widetilde{\bigwedge}^{j}W$ is a direct factor of
$W^{\otimes j}$ and is canonically isomorphic to the largest exterior product
quotient ${\bigwedge}^{j}W$. Now we remark that
$\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V_{j}})\cong\mathrm{H}^{2}_{f}(F,\operatorname{Ad}(\rho_{m})\otimes
V_{j}=\mathrm{H}^{2}_{f}(F,\operatorname{Ad}(\rho_{m})^{\oplus j},$
hence, its Pontryagin dual is canonically isomorphic to $\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))^{\oplus j}$. Therefore, by passing
to the $A$-dual, we obtain $A$-linear homomorphisms
$GV^{j}_{m}\colon\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A}\to\left(\bigwedge^{j}(\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))^{\oplus j})^{\vee}\right)^{\vee}$
For finite $A=\mathcal{O}/(\varpi^{m})$-modules, we see by the structure
theorem of these modules that the Kronecker homomorphism
$\bigwedge^{j}\mathop{\rm Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))^{\oplus
j}\to\left(\bigwedge^{j}(\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))^{\oplus j})^{\vee}\right)^{\vee}$
is an isomorphism. Hence we can rewrite $GV_{m}^{j}$ as
$GV^{j}_{m}\colon\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A}\to\bigwedge^{j}\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))^{\oplus j}.$
We now need to show that $GV^{\bullet}_{m}=\bigoplus_{j}GV^{j}_{m}$ is a
graded-commutative algebra homomorphism. On one hand, let us compute
$GV_{m}^{j}(<a_{1},a_{2}>_{j_{1},j_{2}})$; let
$\Phi_{k}\in\widetilde{\bigwedge}{}^{j}_{k}\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V_{j_{k}}})$,
$k=1,2$. By composing with the inclusions $\iota_{k}\colon V_{j_{k}}\to
V_{j}$, we can view $\Phi_{k}$ as taking values in $\mathcal{S}_{V_{j}}$ and
form $\Phi=\Phi_{1}\wedge\Phi_{2}$; then
$GV_{m}^{j}(<a_{1},a_{2}>_{j_{1},j_{2}})(\Phi)$ is given by
$q_{j_{1}}\circ\Phi_{1}(a_{1})\wedge
q_{j_{2}}\circ\Phi_{2}(a_{2})\in\bigwedge^{j}V_{j}=A$. On the other hand,
$GV^{j_{k}}_{m}(a_{k})$ sends $\Phi_{k}$ to
$q_{j_{k}}\circ\Phi_{k}(a_{k})\in\bigwedge^{j_{k}}V_{j_{k}}=A$. By Lemma 14,
we conclude that
$GV_{m}^{j}(<a_{1},a_{2}>_{j_{1},j_{2}})(\Phi)=GV^{j_{k}}_{m}(a_{k})(\Phi_{1})\cdot
GV^{j_{k}}_{m}(a_{k})(\Phi_{2}).$
Finally, for $j\leq\ell_{0}$, let
$\mu_{j}\colon\bigwedge^{j}\pi_{1}(\mathcal{R})\to\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})$ and
$a_{j}\colon\bigwedge^{j}\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))^{\oplus
j}\to\bigwedge^{j}\mathop{\rm Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))$
be the projection induced by the addition $a_{j}\colon\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))^{\oplus j}\to\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{m})^{\ast}(1))$. Let us show that the resulting
map $a_{j}\circ GV^{j}_{m,V_{j}}\circ\mu_{j}$ coincides with
$\bigwedge^{j}GV_{m}^{1}$. For $a_{k}\in\pi_{1}(\mathcal{R})$ and
$\phi_{k}\in\pi_{0}\operatorname{\mathsf{sHom}}_{\phi_{\rho}}(\mathcal{R},\mathcal{S}_{V_{1}^{(k)}})$
(where $V_{1}^{(k)}=A$) for $k=1\ldots,j$, we need to compute
$GV^{j}_{m}(a_{1}\cdot\ldots\cdot a_{j})(\phi_{1}\wedge\ldots\wedge\phi_{j})$.
We view $A^{j}=V_{j}$ as the sum $V_{1}^{(1)}\oplus\ldots\oplus V_{1}^{(j)}$.
Therefore $\bigwedge^{j}V_{j}=V_{1}^{(1)}\otimes\ldots\otimes V_{1}^{(j)}$,
and by Lemma 14 again, we find
$GV^{j}_{m}(a_{1}\cdot\ldots\cdot
a_{j})(\phi_{1}\wedge\ldots\wedge\phi_{j})=GV^{1}_{m}(a_{1})(\phi_{1})\cdot\ldots\cdot
GV^{1}_{m}(a_{j})(\phi_{j}).$
This implies that the multiplication homomorphism
$\mu_{j}\colon\bigwedge^{j}\pi_{1}(\mathcal{R})\to\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})$
is injective since its composition with $a_{j}\circ GV_{m}^{j}$ is equal to
$\bigwedge^{j}_{A}GV_{m}^{1}$ which is injective (since $GV_{m}^{1}$ is
injective and that a tensor product of $A$-linear injections is injective).
The formula $GV^{1}_{m}=GV[\varpi^{m}]$ is obvious.
We now prove the last statement. Recall that by our assumptions
$(\operatorname{Gal}_{\mathfrak{m}})$ and (LLC), we have
$\lambda^{\operatorname{gal}}_{\pi^{\prime}}=\lambda_{\pi^{\prime}}$ and that
$\operatorname{Im}\mu_{j}$ has bounded index in $\pi_{j}(\mathcal{R})_{A}$
when $m$ grows and that the restriction of $GV_{m}^{j}$ to this submodule is
an injection into $\bigwedge^{j}_{A}\mathop{\rm
Sel}(\operatorname{Ad}\rho_{\pi^{\prime}}^{\ast}(1)_{A})$. By definition of
the Fitting ideal, we therefore see that
$\operatorname{Coker}GV_{m}^{\ell_{0}}\circ\mu_{\ell_{0}}$ is annihilated by
$\operatorname{Fitt}(\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi^{\prime}}))$.
∎
###### Remark 6.
1) As proved above, for any $1\leq j\leq\ell_{0}$, the shuffle multiplication
$\mu_{j}$ is injective. However it would be interesting to study the
injectivity of the homomorphisms $GV_{m}^{j}$.
2) By Remark 4, 1) above, one can apply the homomorphisms $c_{j}$ to both
sides of $(HGV_{j})$ however it is not clear if $GV_{m}^{\bullet}$ factors
through $c_{\bullet}$ into a homomorphism
$\pi_{\bullet}(\mathcal{R})\otimes
A_{m}\to\bigwedge^{\bullet}_{A_{m}}\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m}).$
We finish this section with an observation. Let $K=Frac(\mathcal{O})$ and
$V=K$ be a one dimensional $K$-vector space. Consider the simplicial
$K$-algebras $\mathcal{S}_{V}^{\otimes\ell_{0}}$ and
$\mathcal{R}_{K}=\mathcal{R}\otimes_{\phi_{\rho_{\pi^{\prime}}}}K$ for a
cuspidal representation $\pi^{\prime}$ occuring in
$\mathrm{H}^{\bullet}_{\mathfrak{m}}$. Then we have:
###### Proposition 6.
There are canonical isomorphisms of graded $K$-algebras
$\pi_{\bullet}(\mathcal{R}_{K})\cong\bigwedge^{\bullet}_{K}\pi_{1}(\mathcal{R})_{K}$
and
$\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\ell_{0}})\cong\bigwedge^{\bullet}_{K}\pi_{1}(\mathcal{S}_{V}^{\otimes\ell_{0}}).$
By fixing identifications
$\pi_{1}(\mathcal{R})_{K}=K^{\ell_{0}}=\pi_{1}(\mathcal{S}_{V}^{\otimes\ell_{0}})$,
we have an identification of graded $K$-algebras
$\pi_{\bullet}(\mathcal{R}_{K})=\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\ell_{0}}).$
There exists a weak equivalence of simplicial modules
$\mathcal{R}_{K}\sim\mathcal{S}_{V}^{\otimes\ell_{0}}$
If $\ell_{0}=1$, there is a (non canonical) weak equivalence of simplicial
$K$-algebras
$\phi\colon\mathcal{R}_{K}\to\mathcal{S}_{V}^{\otimes\ell_{0}}.$
###### Remark 7.
There exists simplicial $K$-algebra homomorphisms
$\mathcal{R}_{K}^{\otimes\ell_{0}}\to\mathcal{R}_{K}$ (given by the
multiplication $\mu_{\ell_{0}}$) and
$\mathcal{R}_{K}^{\otimes\ell_{0}}\to\mathcal{S}_{V}^{\otimes\ell_{0}}$. The
second is given by the choice of a basis of the $\ell_{0}$-dimensional
$K$-vector space $\mathop{\rm
Sel}(\operatorname{Ad}(\rho^{\ast}_{\pi_{\prime}})(1))^{\ast}_{K}$. Indeed,
this choice gives rise to homomorphisms
$\phi_{1},\ldots,\phi_{\ell_{0}}\colon\mathcal{R}_{K}\to\mathcal{S}_{V}$
above $\phi_{\rho_{\pi^{\prime}}}$. By tensor product, we have a simplicial
ring homomorphism
$\mathcal{R}_{K}^{\otimes\ell_{0}}\to\mathcal{S}_{V}^{\otimes\ell_{0}}$
It is not clear if it can be adjusted to factor through the multiplication
homomorphism
$\mu_{\ell_{0}}\colon\mathcal{R}_{K}^{\otimes\ell_{0}}\to\mathcal{R}_{K}$. In
other words, it is not clear that this second homomorphism factors through the
first into a homomorphism of simplicial $K$-algebras
$\mathcal{R}_{K}\to\mathcal{S}_{V}^{\otimes\ell_{0}}$. We can ask whether
there exists a zig-zag of weak equivalence maps of simplicial $K$-algebras
$\mathcal{R}_{K}\leftarrow\ldots\rightarrow\mathcal{S}_{V}^{\otimes\ell_{0}}.$
In other words, this poses the question of whether $\mathcal{R}_{K}$ is weakly
equivalent to $\mathcal{S}_{K}^{\otimes\ell_{0}}$ as simplicial $K$-algebra.
The answer is yes if $\ell_{0}=1$.
###### Proof.
We first note that
$\mathcal{S}_{V}^{\otimes\ell_{0}}=\bigoplus_{j=0}^{\ell_{0}}{\ell_{0}\choose
j}\cdot\operatorname{DK}(V^{\otimes j}[j])$
hence
$\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\ell_{0}})=\bigoplus_{j=0}^{\ell_{0}}{\ell_{0}\choose
j}\cdot\pi_{j}(\operatorname{DK}(V^{\otimes
j}[j]))=\bigoplus_{j=0}^{\ell_{0}}K^{\oplus{\ell_{0}\choose
j}}=\bigwedge^{\bullet}K^{\oplus\ell_{0}}$
as graded $K$-module. We also know by [GV18] that
$\pi_{\bullet}(\mathcal{R}_{K})=\bigwedge^{\bullet}K^{\oplus\ell_{0}}$
as graded $K$-module, hence the simplicial $K$-modules $\mathcal{R}_{K}$ and
$\mathcal{S}_{V}^{\otimes\ell_{0}}$ have the same homotopy graded module. By
Section 2.5 of [Ke04], it implies that there exists a zig-zag of quasi-
isomorphisms between them.
Actually, by [Cai21, Theorem 5.31], if we put
$\operatorname{Tor}^{S_{\infty}}_{\bullet}(R_{\infty},\mathcal{O})_{K}=\operatorname{Tor}^{S_{\infty}}_{\bullet}(R_{\infty},\mathcal{O})\otimes_{\phi_{\rho_{\pi^{\prime}}}}K$,
we have an isomorphism of graded $K$-algebras
$\pi_{\bullet}(\mathcal{R}_{K})=\operatorname{Tor}^{S_{\infty}}_{\bullet}(R_{\infty},\mathcal{O})_{K}$
and by [GV18, Formula (15.11)], we have an isomorphism of graded $K$-algebras
$\operatorname{Tor}^{S_{\infty}}_{\bullet}(R_{\infty},\mathcal{O})_{K}=\bigwedge^{\bullet}\pi_{1}(\mathcal{R})_{K}$
hence we have an isomorphism of graded $K$-algebras
$\pi_{\bullet}(\mathcal{R}_{K})=\bigwedge^{\bullet}K^{\oplus\ell_{0}}$
Similarly, by definition of the Eilenberg-Zilber map, the shuffle product
$\bigwedge^{j}\pi_{1}(\mathcal{S}_{V}^{\otimes\ell_{0}})\to\pi_{j}(cS_{V}^{\otimes\ell_{0}})$
is induced by the canonical isomorphism of complexes
$\bigotimes^{j}(K[1]^{\oplus\ell_{0}})\to\bigoplus_{k_{1},\ldots,k_{j}}K[1]^{\otimes
j}$
(recall $V=K$); this induces an isomorphism
$\bigwedge^{j}(K[1]^{\oplus\ell_{0}})\to\bigoplus_{k_{1}<\ldots<k_{j}}K[j].$
this shows that we have a graded algebra isomorphism
$\bigwedge^{\bullet}\pi_{1}(\mathcal{S}_{V}^{\otimes\ell_{0}})\to\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\ell_{0}})$
Hence we conclude that we have graded algebra isomorphisms
$\pi_{\bullet}(\mathcal{S}_{V}^{\otimes\ell_{0}})=\bigwedge^{\bullet}(K^{\oplus\ell_{0}})=\pi_{\bullet}(\mathcal{R}_{K}).$
For $\ell_{0}=1$, recall that by Proposition 5, we have a morphism
$\phi\colon\mathcal{R}_{K}\to\mathcal{S}_{V}$ of simplicial $K$-algebras above
$\phi_{\rho}$ which induces an isomorphism $\pi_{1}(\mathcal{R}_{K})\cong K$
(this isomorphism is defined up to a scalar). Since
$\pi_{k}(\mathcal{R}_{K})=\pi_{k}(\mathcal{S}_{V})=0$ for $k>1$, we see that
$\phi$ is a weak equivalence. ∎
### 5.1. The Bloch-Kato conjecture
Let $K_{\pi}$ be a number field over which the cohomological cuspidal
representation $\pi$ of $\operatorname{\mathsf{GL}}_{N}(F)$ is defined. Assume
there exists a rank $N$ motive $M_{\pi}$ defined over $F$ with coefficients in
$K_{\pi}$ associated to $\pi$. Let
$\operatorname{Ad}\,M_{\pi}=\operatorname{\mathsf{End}}(M_{\pi})$; let
$(\operatorname{Ad}\,M_{\pi})^{\ast}(1)$ be the twisted dual of
$\operatorname{Ad}\,M_{\pi}$.
A conjecture of Beilinson predicts that the motivic cohomology group
$\mathrm{H}^{1}_{mot}(\operatorname{Ad}\,M_{\pi}^{\ast},K_{\pi}(1))=Ext^{1}_{\mathcal{MM}}(\operatorname{Ad}(M_{\pi}),K_{\pi}(1))$
is of rank $\ell_{0}$ over $K_{\pi}$. Here $\mathcal{MM}$ is the conjectural
category of mixed motives. Part of the conjectural framework is the existence
of archimedean and $p$-adic regulator maps:
$\textstyle{\mathrm{H}^{1}_{mot}((\operatorname{Ad}\,M_{\pi})^{\ast},K_{\pi}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{p}}$$\scriptstyle{r_{\infty}}$$\textstyle{\mathrm{H}^{1}_{f}(F,(\operatorname{Ad}\,\rho_{\pi})^{\ast}(1))}$$\textstyle{\mathrm{H}^{1}_{\mathcal{D}}(\operatorname{Ad}(M_{\pi})^{\ast}_{\mathbb{R}},{\mathbb{R}}(1))}$
where $\mathrm{H}^{1}_{f}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1))$ is the
Bloch-Kato Selmer group attached to the Galois representation
$\operatorname{Ad}(\rho_{\pi})^{\ast}(1)$ and $\mathrm{H}^{1}_{\mathcal{D}}$
stands for Deligne cohomology and, for any motive $M$, $M_{\mathbb{R}}$
denotes the real Hodge structure of its Betti realization $\mathrm{H}_{B}(M)$.
By a standard computation the Deligne cohomology is equal to the cokernel of
Deligne’s period map
$\displaystyle
c_{\infty}^{+}\colon\mathrm{H}_{B}(\operatorname{Ad}(M_{\pi})^{\ast}(1))^{+}_{\mathbb{R}}\longrightarrow\mathrm{H}_{dR}(\operatorname{Ad}(M_{\pi})^{\ast}(1))_{\mathbb{R}}/F^{0}\mathrm{H}_{dR}(\operatorname{Ad}(M_{\pi})^{\ast}(1))_{\mathbb{R}}$
which is injective because the weight of
$\operatorname{Ad}(M_{\pi})^{\ast}(1)$ is negative. Note that
$F^{+}=F^{-}\mathrm{H}_{dR}(\operatorname{Ad}(M_{\pi})^{\ast}(1))_{\mathbb{R}}=F^{0}\mathrm{H}_{dR}(\operatorname{Ad}(M_{\pi})^{\ast}(1))_{\mathbb{R}}.$
Our motive is not critical. From this description, one sees immediately that
the Deligne cohomology, namely $\operatorname{Coker}\ c_{\infty}^{+}$, has a
rational structure and even a $p$-integral structure if $p$ prime to $N$ and
$p>max(HT(\rho_{\pi}))$ so that the Hodge-Tate weights of
$\operatorname{Ad}(\rho_{\pi})$ are in the Fontaine-Lafaille range.
A first conjecture by Beilinson and Bloch-Kato is that:
* (Bei)
The regulator map $r_{p}$ (resp. $r_{\infty}$) induces an isomorphism after
the extension of scalar to $K$ (resp, to ${\mathbb{R}}$).
Assuming (Bei), we can define a regulator
$R_{\pi}\in{\mathbb{R}}^{\times}/O_{{\pi,(p)}}^{\times}$ as follows. Let
$L_{\pi}\subset\rho_{\pi}$ be a stable $\mathcal{O}$-lattice. Because the
Hodge-Tate weights are in the Fontaine-Lafaille range, this lattice induces a
lattice of $D_{dR}(\rho_{\pi})$ and therefore define a integral
$\mathcal{O}_{(\pi,(p))}$\- lattice of
$\mathrm{H}_{dR}((\operatorname{Ad}\,\rho_{\pi})^{\ast}(1))$ and of
$\mathrm{H}_{B}((\operatorname{Ad}\,\rho_{\pi})^{\ast}(1))$ via the comparison
isomorphisms $c_{\infty},c_{p}$ and $c_{dR}$. This yields an
$\mathcal{O}_{(\pi,(p))}$ -integral structure on
$coker(\beta)=\mathrm{H}^{1}_{\mathcal{D}}((\operatorname{Ad}(M_{\pi})^{\ast},{\mathbb{R}}(1))$.
Let $A(F)$ be the inverse image of
$\mathrm{H}^{1}_{f}(F,\operatorname{Ad}(L_{\pi})^{\ast}(1))$ via $r_{p}$. The
archimedean regulator of $(\operatorname{Ad}(M_{\pi})^{\ast}(1)$ is then
defined as the co-volume of the image of $A(F)$ by $r_{\infty}$ with respect
to the Haar measure induced by the $\mathcal{O}_{(\pi,(p))}$ -integral
structure defined above. It gives an element
$R_{\pi}\in{\mathbb{R}}^{\times}/\mathcal{O}_{(\pi,(p))}^{\times}$
We now review the definition of the Tate-Shafarevitch group
${}^{1}_{\ast}(F,(\operatorname{Ad}(M_{\pi})^{\ast}(1))$ following Bloch-Kato
($\ast=\operatorname{ord},f$). It depends on the choice of the stable lattice
$L_{\pi}$. It is defined as:
$\operatorname{Ker}\left({\mathrm{H}^{1}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})\over\Phi_{\mathcal{O}}\otimes
K/\mathcal{O}}\to\bigoplus_{v}{\mathrm{H}^{1}(F_{v},\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})\over\mathrm{H}^{1}_{\ast}(F_{v},\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})}\right)$
where
$\Phi_{\mathcal{O}}=\mathrm{H}^{1}_{\ast}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1))\cap\mathrm{H}^{1}(F,\operatorname{Ad}(L_{\pi})^{\ast}(1))$.
$A(F\otimes{\mathbb{R}})/A(F)$ with
$A(F\otimes{\mathbb{R}}):=D_{\mathbb{R}}/(F^{0}D_{\mathbb{R}}+W^{+})$
with $D=\mathrm{H}_{dR}((\operatorname{Ad}(M_{\pi})^{\ast}(1))$,
$W=c_{p}^{-1}(\operatorname{Ad}(L_{\pi})(1))\cap\mathrm{H}_{B}((\operatorname{Ad}(M_{\pi})^{\ast}(1))$
and where we have identified $A(F)$ by its image under the regulator map
$r_{\infty}$ We have the exact sequence
$0\rightarrow W^{+}_{\mathbb{R}}/W^{+}\rightarrow
A(F\otimes{\mathbb{R}})/A(F)\rightarrow\mathrm{H}^{1}_{\mathcal{D}}(\operatorname{Ad}(M_{\pi})^{\ast},{\mathbb{R}}(1))/A(F)\rightarrow
0$
###### Proposition 7.
Let $p>N$ and $\zeta_{p}\notin F$. In the Fontaine-Laffaille case, assume
$(FL)$. In the ordinary case, make the hypothesis of strong distinguishability
$(STDIST)$ : $\alpha\circ\underline{\chi}_{v}\neq 1,\omega^{-1}$ for any
positive root $\alpha$. Let $\ast=o,f$ (ordinary or crystalline Fontaine-
Laffaille). Then, the Selmer group
$\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})$ has $\mathcal{O}$-corang $\ell_{0}$. Moreover
${}^{1}_{\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})$ and
$\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})\otimes
K/\mathcal{O}))$ have same $\mathcal{O}$-Fitting ideal.
###### Proof.
Since the $\mathcal{O}$-algebra $R=\operatorname{T}$ is finite, we know that
$\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\operatorname{T}})\otimes
K/\mathcal{O}))$ and
$\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})\otimes
K/\mathcal{O}))$ are finite. Let us write
$W_{n}:=\operatorname{Ad}(\rho_{\pi})\otimes\varpi^{-n}\mathcal{O}/\mathcal{O}={\mathfrak{g}}_{n}$
and $q=\\#\mathcal{O}/(\varpi)$. Since
$\mathrm{H}^{0}(F,W_{n})=\mathcal{O}/(\varpi^{n})$ and
$\mathrm{H}^{0}(F,W_{n}(1))=\\{0\\}$ because $\zeta_{p}\notin F$, we have by
the Euler-Poincaré characteristic formula due to Greenberg-Wiles
$\displaystyle{\\#\mathrm{H}^{1}_{m\ast}(F,W_{n})\over\\#\mathrm{H}^{1}_{m\ast}(F,W_{n}^{\ast}(1))\\#\mathrm{H}^{0}(F,W_{n})}={1\over\prod_{v|\infty}\\#\mathrm{H}^{0}(F_{v},W_{n})}\cdot\prod_{v|N}{\\#\mathrm{H}^{1}_{unr}(F_{v},W_{n})\over\\#\mathrm{H}^{0}(F_{v},W_{n})}\cdot\prod_{v|p}{\\#\mathrm{H}^{1}_{\ast}(F_{v},W_{n})\over\\#\mathrm{H}^{0}(F_{v},W_{n})}$
Here $m\ast$ means minimal and $\ast$. We note that we have
$\\#\mathrm{H}^{1}_{unr}(F_{v},W_{n})=\\#\mathrm{H}^{0}(F_{v},W_{n})$ for all
$v|N$. Moreover, for each $v|\infty$,
$\mathrm{H}^{0}(F_{v},W_{n})=(\mathcal{O}/\varpi^{n}\mathcal{O})^{N^{2}}$.
(1) The case $\ast=\operatorname{ord}$.
For any $v|p$, let us compute
${\\#\mathrm{H}^{1}_{\operatorname{ord}}(F_{v},W_{n})\over\\#\mathrm{H}^{0}(F_{v},W_{n})}$.
Let ${\mathfrak{b}}=\operatorname{Lie}(B;\mathcal{O})$,
${\mathfrak{n}}=\operatorname{Lie}(U_{B};\mathcal{O})$,
${\mathfrak{g}}=\operatorname{Lie}(G;\mathcal{O})$ and
${\mathfrak{b}}_{n}={\mathfrak{b}}\otimes\mathcal{O}/\varpi^{n}\mathcal{O}$,
${\mathfrak{n}}_{n}={\mathfrak{n}}\otimes\mathcal{O}/\varpi^{n}\mathcal{O}$;
consider the fiber product
$L^{\prime}_{v}=\mathrm{H}^{1}(F_{v},{\mathfrak{b}}_{n})\times_{\mathrm{H}^{1}(F_{v},{\mathfrak{b}}_{n}/{\mathfrak{n}}_{n})}\mathrm{H}^{1}_{unr}(F_{v},{\mathfrak{b}}_{n}/{\mathfrak{n}}_{n})$
Then, we have
$\mathrm{H}^{1}_{\operatorname{ord}}(F_{v},W_{n})=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(F_{v},W_{n}))$.
We can insert $L_{v}^{\prime}$ in the long exact sequence
$0\to\mathrm{H}^{0}(F_{v},{\mathfrak{n}}_{n})\to\mathrm{H}^{0}(F_{v},{\mathfrak{b}}_{n})\to\mathrm{H}^{0}(F_{v},{\mathfrak{b}}_{n}/{\mathfrak{n}}_{n})\to\mathrm{H}^{1}(F_{v},{\mathfrak{n}}_{n})\to
L_{v}^{\prime}\to\mathrm{H}^{1}(G_{F_{v}}/I_{v},{\mathfrak{b}}^{\prime}_{n}/{\mathfrak{n}}_{n})\to
0$
Hence, taking the orders of the terms of the sequence, we have :
$\\#L_{v}^{\prime}\cdot(\\#\mathrm{H}^{0}(F_{v},{\mathfrak{b}}_{n}))^{-1}=\\#\mathrm{H}^{1}(G_{F_{v}}/I_{v},{\mathfrak{b}}_{n}/{\mathfrak{n}}_{n})\cdot(\\#\mathrm{H}^{0}(F_{v}),{\mathfrak{b}}_{n}/{\mathfrak{n}}_{n}))^{-1}\cdot\\#\mathrm{H}^{1}(F_{v},{\mathfrak{n}}_{n})\cdot(\\#\mathrm{H}^{0}(F_{v},{\mathfrak{n}}_{n}))^{-1}$
We have
$\\#\mathrm{H}^{1}(G_{F_{v}}/I_{v},{\mathfrak{b}}_{n}/{\mathfrak{n}}_{n})\cdot(\\#\mathrm{H}^{0}(F_{v},{\mathfrak{b}}_{n}/{\mathfrak{n}}_{n}))^{-1}=1$
$\\#\mathrm{H}^{1}(G_{v},{\mathfrak{n}}_{n})\cdot(\\#\mathrm{H}^{0}(G_{v},{\mathfrak{n}}_{n}))^{-1}=\\#{\mathfrak{n}}_{n}^{[F_{v}\colon{\mathbb{Q}}_{p}]}\cdot\\#\mathrm{H}^{0}(G_{v},{\mathfrak{n}}_{n}^{\ast}(1))$
and $\\#\mathrm{H}^{0}(G_{v},{\mathfrak{n}}_{n}^{\ast}(1))=1$ by strong
distinguishability. Note also for the same reason that
$\mathrm{H}^{0}(F_{v},W_{n})=\mathrm{H}^{0}(F_{v},{\mathfrak{b}}_{n})$
(because $\mathrm{H}^{0}(F_{v},W_{n}/{\mathfrak{b}}_{n})=0$). Since
$L_{v}^{\prime}\to L_{v}$ is injective by strong distinguishability, we
conclude that for every $v|p$,
$\\#\mathrm{H}^{1}_{\operatorname{ord}}(F_{v},W_{n})\cdot(\\#\mathrm{H}^{0}(F_{v},W_{n}))^{-1}=q^{({N(N-1)\over
2})\cdot n[F_{v}\colon{\mathbb{Q}}_{p}]}$.
We have $\sum_{v|p}[F_{v}\colon{\mathbb{Q}}_{p}]=2d_{0}$. Note that
$d_{0}N^{2}-1-2d_{0}({N(N-1)\over 2})=Nd_{0}-1$. Recall that
$\ell_{0}=Nd_{0}-1$.
Therefore, we get
(2)
$\displaystyle{\\#\mathrm{H}^{1}_{mo}(\Gamma,W_{n}^{\ast}(1))=q^{n\ell_{0}}\cdot\\#\mathrm{H}^{1}_{mo}(\Gamma,W_{n})}$
where the sequence $(\\#\mathrm{H}^{1}_{mo}(\Gamma,W_{n}))_{n}$ is bounded.
(2) The Fontaine-Laffaille case.
For any $v|p$, let us compute
${\\#\mathrm{H}^{1}_{f}(F_{v},W_{n})\over\\#\mathrm{H}^{0}(F_{v},W_{n})}$. Let
$\mathbb{G}_{v}$ be the Fontaine-Laffaille functor defined in [CHT08, Section
2.4.1)]. Let $V_{n}=\rho_{\pi}\pmod{\varpi^{n}}$ and
$M_{n}=\mathbb{G}(V_{n})$.
We have
$\\#\mathrm{H}^{1}_{f}(F_{v},W_{n})\cdot(\\#\mathrm{H}^{0}(F_{v},W_{n}))^{-1}=\\#\operatorname{Ext}_{MF}^{1}(M_{n},M_{n})\cdot(\\#\operatorname{\mathsf{Hom}}_{MF}(M_{n},M_{n}))^{-1}.$
We apply [CHT08, Lemma 2.4.2 and Corollary 2.4.3] for $M=N=M_{n}$ and we
obtain
$=q^{({N(N-1)\over 2})\cdot n[F_{v}\colon{\mathbb{Q}}_{p}]}.$
Note that $d(N^{2}-1)-2d({N(N-1)\over 2})=(N-1)d=\ell_{0}$. Therefore,
(3)
$\displaystyle{\\#\mathrm{H}^{1}_{mf}(\Gamma,W_{n}^{\ast}(1))=q^{n\ell_{0}}\cdot\\#\mathrm{H}^{1}_{mf}(\Gamma,W_{n})}$
In case (1) and (2), we conclude by passing to the inductive limit on $n$. Let
$\varinjlim_{n}W_{n}=W_{\infty}=\operatorname{Ad}\,\rho_{\pi}\otimes
K/\mathcal{O}$ and $\varinjlim_{n}W_{n}^{\ast}(1)=W_{\infty}^{\ast}(1)$. Then
by (2) and (3), there exists $c\geq 0$ such that for all $n$ sufficiently
large, the order of
$\mathrm{H}^{1}_{m\ast}(\Gamma,W_{\infty}^{\ast}(1))[\varpi^{n}]=\mathrm{H}^{1}_{m\ast}(\Gamma,W_{n}^{\ast}(1))$
is $q^{n\ell_{0}+c}$. Passing to the Pontryagin dual and using Nakayama’s
lemma one sees easily that this implies there exists an exact sequence
$0\rightarrow\Phi_{\mathcal{O}}\otimes\varpi^{-n}\mathcal{O}/\mathcal{O}\rightarrow\mathrm{H}^{1}_{m\ast}({\mathbb{Q}},W_{n}(1))\rightarrow{}^{1}_{\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})[\varpi^{n}]\rightarrow 0$
where $\Phi_{\mathcal{O}}$ is a finitely generated $\mathcal{O}$-module of
rank $\ell_{0}$. Note that $\Phi_{\mathcal{O}}$ can be supposed to be free and
that for $n$ sufficiently large,
${}^{1}_{\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})[\varpi^{n}]={}^{1}_{\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O}).$
From (2) and (3), we indeed deduce that
$\\#{}^{1}_{\ast}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})=\\#\mathrm{H}^{1}_{m\ast}(\Gamma,W_{n})$ for all $n>0$
sufficiently large. ∎
For any finite $S$-ramified extension $F^{\prime}/F$, let $S^{\prime}$ the set
of places of $F^{\prime}$ above $S$ and $U_{F^{\prime}}^{S}$ be the image of
$\prod_{v^{\prime}\notin
S^{\prime}}\mathcal{O}^{\times}_{F^{\prime}_{v^{\prime}}}$ in the group
$C_{F}$ of idèle classes of $F$. Let
$C_{S}(F^{\prime})=C_{F^{\prime}}/U_{F^{\prime}}^{S}$
$C_{S}=\varinjlim_{F^{\prime}\subset F_{S}}C_{S}(F^{\prime})$. It is known
[NSW99, Sect.X.9] that $C_{S}$ is divisible (assuming $S_{p}\subset S$). Let
$C_{S}(p)$ be the $p$-divisible part of $C_{S}$. Moreover, there is an
isomorphism [NSW99, Proposition 8.1.21 and Proposition 8.3.8]:
$tr\colon\mathrm{H}^{2}(\Gamma,C_{S}(p))\cong{\mathbb{Q}}_{p}/{\mathbb{Z}}_{p}$
Let us write
$W_{n}:=\operatorname{Ad}(\rho_{\pi})\otimes\varpi^{-n}\mathcal{O}/\mathcal{O}$
as in the previous Lemma. By [NSW99, Theorem 3.4.6], we have a perfect pairing
induced by the cup-product and trace map:
$\mathrm{H}^{1}(\Gamma,W_{n}^{\ast}(1))\times\mathrm{H}^{1}(\Gamma,W_{n})\to\mathrm{H}^{2}(\Gamma,C_{S}(p)\otimes
K/\mathcal{O})\cong K/\mathcal{O}$
by using Tate local duality, we obtain by restriction a pairing
$<\bullet,\bullet>_{n}\colon\mathrm{H}^{1}_{m\ast}(\Gamma,W_{n}^{\ast}(1))\times\mathrm{H}^{1}_{m\ast}(\Gamma,W_{n})\to
K/\mathcal{O}$
which is non-degenerate on the right: if $<x,y>_{n}=0$ for any $x$, then
$y=0$. We have seen in the proof of Lemma 7 that
$\mathrm{H}^{1}_{m\ast}(\Gamma,W_{\infty}^{\ast}(1))[\varpi^{n}]=\mathrm{H}^{1}_{m\ast}(\Gamma,W_{n}^{\ast}(1))$
Therefore, by passing to the direct limit over $n$, we obtain a pairing
$<\bullet,\bullet>\colon\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})\times\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})\otimes
K/\mathcal{O})\to K/\mathcal{O}$
which is non-degenerate on the right.
###### Corollary 3.
With the notations of Lemma 7, for $\ast=FL,\operatorname{ord}$, the pairing
$<\bullet,\bullet>$ induces a perfect pairing
${}^{1}_{\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})\times\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})\otimes
K/\mathcal{O})\to K/\mathcal{O}$
between two finite $\mathcal{O}$-modules.
###### Proof.
If
$x\in\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})$ is divisible, we see that
$<x,y>=<\varpi^{n}x_{n},y>=<x_{n},\varpi^{m}y>=0$ for $n$ sufficiently large
(because $\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})\otimes
K/\mathcal{O})$ is finite). Hence, the pairing factors as
${}^{1}_{\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})\times\mathrm{H}^{1}_{m\ast}(\Gamma,\operatorname{Ad}(\rho_{\pi})\otimes
K/\mathcal{O})\to K/\mathcal{O}.$
By Lemma 7, the two groups have the same order and the pairing is non-
degenerate on the right. Therefore it is perfect. ∎
## 6\. Galatius and Venkatesh Theory for Hida families
### 6.1. Nearly ordinary Hida-Hecke algebra
Let $E$ be an imaginary quadratic field in which $p$ splits and $F^{+}$ be a
totally real field; let $d_{0}=[F^{+}\colon F]$. We consider the CM field
$F=F^{+}E$. It has a natural CM type (embeddings fixing $E$). It is for this
type of CM fields that the local properties of the Scholze Galois
representation are established (at least modulo a nilpotent ideal of
explicitely bounded nilpotency exponent). Since $p$ splits in $E$, say
$p\mathcal{O}_{E}={\mathfrak{p}}{\mathfrak{p}}^{c}$. Thus, $F$ has also a
natural $p$-adic CM type, namely the set of primes above ${\mathfrak{p}}$.
Let $B=TU_{B}$ be the upper triangular Borel subgroup of the $F$-algebraic
group $G=\operatorname{\mathsf{GL}}(N)$. Let $S_{p}^{+}$ be the set of primes
obove $p$in $F^{+}$. We assume that all the places $v\in S_{p}^{+}$ split in
$F$. Let $U=U_{0}({\mathfrak{n}})$ and for each $m\geq 0$
$U_{1}(p^{m})=\\{g\in U,g\pmod{p^{m}}\in U_{B}(\mathcal{O}_{F,p}/(p^{m}))\\}$
Let $Y_{1}(p^{m})=G(F)\backslash X_{G}\times G(F_{f})/U_{1}(p^{m})$; we write
$Y=Y_{1}(1)$. Let $\pi_{m}\colon Y_{1}(p^{m})\to Y_{1}(p^{m-1})$ be the
transition maps. They are finite. These locally symmetric spaces form a
projective system $Y_{1}(p^{\infty})\to Y$ of Galois group
$G(\mathcal{O}_{F,p})$. Let
$\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$ be the projective limit
of the $\mathcal{O}$-modules $\mathrm{H}^{\bullet}(Y_{1}(p^{m}),\mathcal{O})$
for the transition maps
$\operatorname{Tr}_{\pi_{m}}\colon\mathrm{H}^{\bullet}(Y_{1}(p^{m}),\mathcal{O})\to\mathrm{H}^{\bullet}(Y_{1}(p^{m-1}),\mathcal{O})$.
Let $h(Y_{1}(p^{\infty}),\mathcal{O})$ be the closed $\mathcal{O}$-algebra
generated by the Hecke operators outside ${\mathfrak{n}}$ acting faithfully on
$\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$. Let $U_{p}$ be the
Hecke operator associated to $\operatorname{diag}(p^{N-1},p^{N-2},\ldots,1)$
and $e$ be the idempotent of the Hecke algebra
$h(Y_{1}(p^{\infty}),\mathcal{O})$ associated to $U_{p}$. In particular, the
torus $T(\mathcal{O}_{F,p})$ acts on
$\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$ by right translation.
This action factors through the quotient by the closure of central global
units: $T(\mathcal{O}_{F,p})/\overline{Z(\mathcal{O}_{F})}$. Actually, the
class group
$Cl_{U,p^{\infty}}=F_{\operatorname{A}}^{\times}/{\overline{F}^{\times}\cdot(U^{p}\cap
Z(F_{f}))\cdot F_{\infty}^{\times}}$ (where $Z\subset T\subset G$ denotes the
center) also acts by right translation on $Y_{1}(p^{\infty})$. This gives an
action of the Hida-Iwasawa group ${\bf\mathrm{H}}$ defined as the amalgamated
product
${\bf\mathrm{H}}=T(\mathcal{O}_{F,p})\times_{Z(\mathcal{O}_{F,p})}Cl_{U,p^{\infty}}$.
We can decompose ${\bf\mathrm{H}}={\bf\mathrm{H}}_{t}\times\bf W$, where
${\bf\mathrm{H}}_{t}$ is finite and $\bf W$ is a free
${\mathbb{Z}}_{p}$-module of rank $2Nd-(d-1-\delta)=2(N-1)d+d+1+\delta$ where
$\delta$ is the Leopoldt defect for $(F,p)$.
###### Definition 7.
The completed group algebra $\Lambda=\mathcal{O}[[\bf W]]$ is called the Hida-
Iwasawa algebra.
Let
$\mathbb{H}^{\bullet}=e\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$.
Let $I\subset G(\mathcal{O}_{F,p})$ be the Iwahori subgroup and
$N=U_{B}(\mathcal{O}_{F,p})$; let $\mathcal{C}(I/N,\mathcal{O})$ be the Banach
$\mathcal{O}$-module of continuous functions on $I/N$. Let $\mathbb{D}$ be the
$\mathcal{O}$-linear dual of $\mathcal{C}(I/N,\mathcal{O})$. It is a ring for
the convolution multiplication: $<f,d\ast
d^{\prime}>=<<f(gh),d_{g}>,d^{\prime}_{h}>$. It is called the ring of integral
distributions (or measures) on $I/N$.
For any $p$-adic weight $\lambda\in
Hom_{\mathrm{c}ont}(T(\mathcal{O}_{F,p}),\mathcal{O})$, we denote by
$\mathbb{D}_{\lambda}(\mathcal{O})$ the continuous $\mathcal{O}$-dual of
$\mathcal{C}_{\lambda}(I/N,\mathcal{O})$ which is the sub-module of continuous
functions $f\in\mathcal{C}(I/N,\mathcal{O})$ such that $f(tg)=\lambda(t)f(g)$
for all $t\in T(\mathcal{O}_{F,p})$ and $g\in I$.
By applying Poincaré duality to the result [H98, (6.14)], we have that
$e\mathrm{H}^{q_{s}}(Y_{1}(p^{\infty}),\mathcal{O})=e\mathrm{H}^{q_{s}}(Y_{0}(p),\mathbb{D}).$
Let $\lambda$ be a dominant weight of $T$ and $V_{\lambda}$ the irreducible
representation of highest weight $\lambda$. Let $\lambda^{\vee}$ be the
highest weight of the dual of $V_{\lambda}$.
###### Definition 8.
We say that $\lambda\colon T(\mathcal{O}_{F,p})\to\mathcal{O}^{\times}$ is
arithmetic if it is dominant algebraic and it is trivial on (the $p$-adic
closure of) $Z(\mathcal{O}_{F})$. In other words, it defines an
$\mathcal{O}$-algebra homomorphism
$\phi_{\lambda}\colon\Lambda\to\mathcal{O}$. The prime ideal
$P_{\lambda}=\operatorname{Ker}\phi_{\lambda}$ of $\Lambda$ is also called
arithmetic.
For an arithmetic character $\lambda$, the inclusion
$V_{\lambda^{\vee}}(\mathcal{O})\hookrightarrow\mathcal{C}_{\lambda}(I/N,\mathcal{O})\subset\mathcal{C}(I/N,\mathcal{O})$
provides by duality a $\mathcal{O}[U]$-linear homomorphism
$\mathbb{D}\to\mathbb{D}_{\lambda}(\mathcal{O})\to V_{\lambda}(\mathcal{O})$
hence a $\Lambda$-linear homomorphism
$e\mathrm{H}^{q_{s}}(Y_{0}(p),\mathbb{D})\to
e\mathrm{H}^{q_{s}}(Y_{0}(p),V_{\lambda}(\mathcal{O})).$
By [H95, Theorem 5.2] or [H98, Theorem 6.2], we have an isogeny
$\mathbb{H}^{q_{s}}/P\mathbb{H}^{q_{s}}\to
e\mathrm{H}^{q_{s}}(Y_{0}(p),V_{\lambda}(\mathcal{O})).$
This implies that $\mathbb{H}^{q_{s}}$ is a finitely generated
$\Lambda$-module. But for $F$ CM, it is a torsion $\Lambda$-module (see [H98,
Section 6]).
Let $\pi$ be a $p$-ordinary cuspidal cohomological automorphic representation
occuring in
$e\mathrm{H}^{q_{s}}(Y_{0}(p),V_{\lambda}(\mathcal{O}))\otimes{\mathbb{C}})$.
Assume its Hecke values belong to $\mathcal{O}$. Its Hecke eigensystem
$\theta_{\pi}\colon h_{\lambda}(Y_{0}(p),\mathcal{O})\to\mathcal{O}$ gives
rise by composition with $h(Y_{1}(p^{\infty}),\mathcal{O})\to
h_{\lambda}(Y_{0}(p),\mathcal{O})$ to a homomorphism $\theta\colon
eh(Y_{1}(p^{\infty}),\mathcal{O})\to\mathcal{O}$. Let ${\mathfrak{m}}$ be the
maximal ideal of the nearly ordinary Hecke algebra
$eh(Y_{1}(p^{\infty}),\mathcal{O})$ given by $\theta$ modulo $\varpi$. We
assume it is non Eisenstein. Let
$\operatorname{T}_{h}=eh(Y_{1}(p^{\infty}),\mathcal{O})_{\mathfrak{m}}$ and
$\operatorname{T}_{\lambda}=eh_{\lambda}(Y_{0}(p),\mathcal{O})_{\mathfrak{m}}$.
We have an algebra homomorphism $\Lambda\to\operatorname{T}_{h}$.
###### Definition 9.
A prime ideal ${\mathfrak{P}}$ of $\operatorname{T}_{h}$ is called of weight
$\lambda$ if its inverse image in $\Lambda$ is arithmetic $P_{\lambda}$ and we
write ${\mathfrak{P}}|P_{\lambda}$. It is said arithmetic if $\lambda$ is an
arithmetic weight. The prime ${\mathfrak{P}}$ is called Hida-automorphic if
${\mathfrak{P}}\in\operatorname{Supp}_{\operatorname{T}_{h}}(\mathbb{H}^{\bullet}(Y_{0}(p),\mathbb{D}_{\lambda}(\mathcal{O}))$.
We denote by $\widetilde{\Sigma}_{h}$ the set of Hida-automorphic primes and
by $\widetilde{\Sigma}_{h}^{a}\subset\widetilde{\Sigma}_{h}$ its subset of
arithmetic Hida-automorphic primes. We also introduce subset $\Sigma_{h}$ and
$\Sigma_{h}^{a}$ of $Spec(\Lambda)$ defined as the respective images of
$\widetilde{\Sigma}_{h}$ and $\widetilde{\Sigma}_{h}^{a}$ by the map
$Spec(\operatorname{T}_{h})\to Spec(\Lambda)$.
We can interprete these sets in terms the support over $\Lambda$ or
$\operatorname{\mathbf{T}}_{h}$ of the cohomology with coefficient in
$\mathbb{D}$.
###### Lemma 15.
We have
$\tilde{\Sigma}_{h}=\operatorname{Supp}_{\operatorname{T}_{h}}H^{\bullet}(Y_{0}(p),\mathbb{D}))\cap
Spec(\operatorname{T}_{h})(\overline{\mathbb{Q}}_{p})$
and
$\Sigma_{h}=\operatorname{Supp}_{\Lambda}(\mathbb{H}^{\bullet}(Y_{0}(p),\mathbb{D}))\cap
Spec(\Lambda)(\overline{\mathbb{Q}}_{p})$
###### Proof.
It is sufficient to prove the second statement. Let $P_{\lambda}\in
Spec(\Lambda)(\mathcal{O})$. We have a Spectral squence
$Tor_{j}^{\Lambda}((H^{i}(Y_{0}(p),\mathbb{D})_{P_{\lambda}},\Lambda/P_{\lambda})\Rightarrow
H^{i-j}(Y_{0}(p),\mathbb{D}_{\lambda}(\mathcal{O})))\otimes Frac(\mathcal{O})$
If $\lambda\in\Sigma_{h}$, then clearly one of the terms of the spectral
sequence does not vanish. It therefore implies that
$H^{i}(Y_{0}(p),\mathbb{D}))_{P_{\lambda}}\neq 0$ for some $i$ and thus
$\Sigma_{h}\subset\operatorname{Supp}_{\Lambda}(\mathbb{H}^{\bullet}(Y_{0}(p),\mathbb{D}))\cap
Spec(\Lambda)(\overline{\mathbb{Q}}_{p})$
Conversely if
$P_{\lambda}\in\operatorname{Supp}_{\Lambda}(\mathbb{H}^{\bullet}(Y_{0}(p),\mathbb{D}))\cap
Spec(\Lambda)(\overline{\mathbb{Q}}_{p})$, let $q_{\lambda}$ the largest
integer $i$ such that $H^{i}(Y_{0}(p),\mathbb{D}))_{P_{\lambda}}\neq 0$. Then
from the above spectral, we have
$H^{q_{\lambda}}Y_{0}(p),\mathbb{D}))_{P_{\lambda}}\otimes\Lambda/P_{\lambda}\cong
H^{q_{\lambda}}(Y_{0}(p),\mathbb{D}_{\lambda})\otimes Frac(\mathcal{O})\neq
0.$
So $P_{\lambda}\in\Sigma_{h}$.
∎
### 6.2. Minimal ordinary deformation rings over $\Lambda$
We consider the problem $\mathcal{D}_{h}$ of minimal ($\Lambda$-)ordinary
deformations of $\overline{\rho}$.
A deformation $\rho\colon\Gamma\to\widehat{G}(A)$ is called minimal at $v\in
S$, if there exists $g_{v}\in\widehat{G}(A)$ such that for any $\sigma\in
I_{v}$, $g_{v}\cdot\rho(\sigma)\cdot g_{v}^{-1}=\exp(t_{v}(\sigma)J)$ where
$J$ is the standard regular nilpotent Jordan matrix and $t_{v}\colon
I_{v}\to{\mathbb{Z}}_{p}(1)$ is the $p$-adic tame inertia homomorphism.
It is called ($\Lambda$-)ordinary at $v\in S_{p}$ if there exists
$g_{v}\in\widehat{G}(A)$ such that $g_{v}\cdot\rho\cdot g_{v}^{-1}\colon
G_{F_{v}}\to\widehat{G}(A)$ takes values in the standard Borel $\widehat{B}$
and its reduction modulo its unipotent radical $\widehat{N}$:
$\chi_{\rho}\colon G_{F_{v}}\to\widehat{T}(A)$, is a lifting of the regular
homomorphism $\overline{\chi}_{\overline{\rho}}\colon
G_{F_{v}}\to\widehat{T}(k)$.
Note that in the definition of $\Lambda$-ordinary deformations, contrary to
$\underline{\chi}_{\pi}$-ordinary deformations, the Hodge-Tate weights are
left variable.
By [CHT08, Section 2.4.4] or [Ti96, Chapt.6], $\mathcal{D}_{h}$ is pro-
representable. Let $R_{h}$ be the universal deformation ring for minimal
ordinary deformations. We also have a group homomorphism $\mathbb{H}\to
R_{h}^{\times}$ defined by duality: Let $\rho$ be the universal deformation
$\Gamma\to\widehat{G}(R_{h})$. For each place $v\in S_{p}$, we have a
homomorphism
$\chi_{\rho,v}\colon
G_{v}=\operatorname{Gal}(\overline{F}_{v}/F_{v})\to\widehat{T}(R_{h})=\widehat{B}(R_{h})/U_{\widehat{B}}(R_{h})$
obtained by reduction mod. $U_{\widehat{B}}(R_{h})$ of a conjugate of
$\rho|_{G_{\widetilde{v}}}$. By class field theory, it gives rise to
$\mathcal{O}_{F_{v}}^{\times}\to\widehat{T}(R_{h})$ which can be interpreted
as a homomorphism $T(\mathcal{O}_{F_{v}})\to R_{h}^{\times}$. Putting all
places together we have $T(\mathcal{O}_{F,p})\to R_{h}^{\times}$. Using the
determinant $\det\,\rho$, we also have a homomorphism $Cl_{U,p^{\infty}}\to
R_{h}^{\times}$; both coincide on $Z(\mathcal{O}_{F,p})$. On the other hand,
by [Sch15] and [CGH+19] and [ACC+18], for each dominant weight $\lambda$,
there exists a continuous Galois representation
$\rho_{\lambda}\colon\Gamma\to\widehat{G}(\operatorname{T}_{\lambda})$
which is minimal and ordinary. Since the residual representation
$\overline{\rho}$ is absolutely irreducible, these representation glue into a
deformation of $\overline{\rho}$
$\rho_{\operatorname{T}}\colon\Gamma\to\widehat{G}(\operatorname{T}_{h})$
which is minimal and $p$-ordinary. This is due to Carayol for
$\widehat{G}=\operatorname{\mathsf{GL}}_{N}$; for the general case, see
[BHKT19, Theorem 4.10]. It gives rise to a surjective ring homomorphism
$\phi\colon R_{h}\to\operatorname{T}_{h}$.
For Galois representations coming from automorphic representations on a
unitary group $U(N)$, it is well-known that it is $\Lambda$-linear (see [Ge19,
Cor.2.5], [HT17, Proposition 2.8]) For Galois representations coming from the
cohomology of $\operatorname{\mathsf{GL}}_{N}$ by [Sch15] and [CGH+19], the
$\Lambda$-linearity follows from [ACC+18, Theorem 5.5.1] (at least modulo a
nilpotent ideal of nilpotency order explicitely bounded in terms of $d$ and
$N$).
In a manner similar to [CaGe18, Section 5], we introduce a Taylor-Wiles system
$\\{Q_{m}\\}$ of finite sets of finite places of $F$ in order to define rings
$S_{\infty}^{\Lambda}$, $R_{h,\infty}$ and a ring homomorphism $\alpha\colon
S_{\infty}^{\Lambda}\to R_{h,\infty}$. Let
$r=\dim\,\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,(\operatorname{Ad}\,\overline{\rho})^{\ast}(1))$.
An element $t\in\widehat{T}(k)$ is called strongly regular if for any root
$\alpha^{\vee}$ of $\widehat{T}$, $\alpha^{\vee}(t)\neq 1$. Let
$\mathcal{L}=(L_{v})_{v\,finite}$ where
-for $v\in S_{p}$, $L_{v}=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(F_{v},{\mathfrak{g}}))$ where $L^{\prime}_{v}=\operatorname{Ker}(\mathrm{H}^{1}(G_{F_{v}},{\mathfrak{b}})\to\mathrm{H}^{1}(I_{v}),{\mathfrak{b}}/{\mathfrak{n}}))$
-for $v\notin S_{p}$, $L_{v}=\mathrm{H}^{1}_{unr}(G_{F_{v}},\operatorname{Ad}\,\overline{\rho})$, and
For a finite set $Q$ disjoint of $S\cup S_{p}$, we write
$\mathcal{L}_{Q}=(L^{Q}_{v})_{v\,finite}$ where $L^{Q}_{v}$ is as above for
$v\in S_{p}$ or $v\notin S_{p}\cup Q$, and
$L^{Q}_{v}=\mathrm{H}^{1}(G_{F_{v}},{\mathfrak{g}})$ for $v\in Q$.
###### Definition 10.
A finite set $Q$ of finite places of $F$ is called a Taylor-Wiles set if ‘
* •
$Q\cap(S\cup S_{p})=\emptyset$, $\sharp\,Q=r$,
* •
for any $v\in Q$, $Nv\equiv 1\pmod{p}$
* •
for any $v\in Q$, $\overline{\rho}(\operatorname{Frob}_{v})$ is conjugate to a
strongly regular element of $\widehat{T}(k)$
* •
we have $\mathrm{H}^{1}_{\mathcal{L}_{Q}^{\perp}}(G_{F,S\cup S_{p}\cup
S_{Q}},Ad(\overline{\rho})^{\ast}(1))=0$.
As in [CHT08, 2.5], the existence of such sets follows from the assumption of
big image of $\overline{\rho}$.
Let $Q$ be such a set; for $v\in Q$, let $\Delta_{v}$, resp. $T(k(v))^{p}$, be
the $p$-Sylow, resp. the prime to $p$ part, of $T(k(v))$, where
$k(v)\mathcal{O}_{F_{v}}/(\varpi_{v})$ is the residue field of $\O_{F}$ at
$v$. We have canonically $T(k(v))=\Delta_{v}\times T(k(v))^{p}$.
Let $\Delta_{Q}=\prod_{v\in Q}\Delta_{v}$.
The $p$-rank of $\Delta_{Q}$ is $Nr$. Let $R_{h,Q}$ be the universal
deformation ring of $\overline{\rho}$ representing the covariant functur
$\mathcal{D}_{h,Q}\colon{}_{\mathcal{O}}\operatorname{Art}_{k}\to\operatorname{\mathsf{SETS}}$
of deformations $\rho$ which are minimal ordinary and unramified outside
$S\cup Q$. As above, it comes with a natural $\Lambda$-algebra structure
$\Lambda\to R_{h,Q}$. For each $v\in Q$, let us fix an ordering
$\underline{\alpha}_{v}=(\alpha_{v,1},\ldots,\alpha_{v,N})$ of the distinct
roots of $\overline{\rho}(\operatorname{Frob}_{v})$ in $k$. Let
$\rho_{Q}\colon G_{F,S\cup S_{p}\cup S_{Q}}\to\widehat{G}(R_{h,Q})$ be the
universal Galois representation. Then for each $v\in Q$ one can diagonalize
$\rho_{Q}(\operatorname{Frob}_{v})$ in a basis adapted to the ordering
$\underline{\alpha}_{v}$. An immediate generalization of the induction in
[TW95, Lemma 7] (mentioned in [ACC+18, Lemma 6.2.19]) shows that for any fixed
$v\in Q$, the restriction $\rho_{h,Q}|_{G_{F_{v}}}$ to a decomposition
subgroup at $v\in Q$ takes values after conjugation, in
$\widehat{T}(R_{h,Q})$. Moreover, for any $v\in Q$, the restriction of the
homomorphism $G_{F_{v}}\to\widehat{T}(R_{h,Q})$ to the inertia subgroup
$I_{v}$ of $G_{F_{v}}$ can be viewed as a homomorphism $T(k(v))\to
R_{h,Q}^{\times}$ of $p$-power order. We thus obtain a group homomorphism
$\Delta_{Q}\to R_{h,Q}^{\times}$, hence an $\mathcal{O}$-algebra homomorphism
$\alpha_{Q}\colon\Lambda[\Delta_{Q}]\to R_{h,Q}.$
Let $I_{Q}$ be the augmentation ideal of $\Lambda[\Delta_{Q}]$. the natural
homomorphism $R_{h,Q}\to R_{h,\emptyset}=R_{h}$ induces an isomorphism
$R_{h,Q}/I_{Q}R_{h,Q}\cong R_{h}$.
Let $I_{v}$, resp. $I_{v}^{+}$, be the Iwahori, resp. the pro-$p$ Iwahori
subgroup of $\operatorname{\mathsf{GL}}_{N}(\mathcal{O}_{F_{v}})$.
Let $U_{0}(Q)=U^{Q}\times\prod_{v\in Q}I_{v},\quad
U_{1}(Q)=U^{Q}\times\prod_{v\in Q}I_{v}^{+}$ For any $v\in Q$, we have a
canonical isomorphism $i_{v}\colon I_{v}/I_{v}^{+}\cong T(k(v))$. Let $U_{Q}$
be the level subgroup $U_{1}(Q)\subset U_{Q}\subset U_{0}(Q)$ such that via
the isomorphism $i_{Q}=\prod_{v\in Q}i_{v}$ we have
$U_{Q}/U_{1}(Q)\cong\prod_{v\in Q}T(k(v))^{p}.$
Let $Y_{0,1}(Q,p^{\infty})$, resp. $Y^{Q}_{1}(p^{\infty})$, be the provariety
associated to the level group $U_{0}(Q)\cap U_{1}(p^{\infty})$, resp.
$U_{Q}\cap U_{1}(p^{\infty})$. We denote by $<\bullet>_{Q}$ the isomorphism
$<\bullet>_{Q}\colon\Delta_{Q}\cong\operatorname{Gal}(Y^{Q}_{1}(p^{\infty})/Y_{0,1}(Q,p^{\infty}))$
Let $h^{-}(Y^{Q}_{1}(p^{\infty}),\mathcal{O})$ be the Hecke algebra outside
$S\cup Q$ acting faithfully on
$e\mathrm{H}^{q_{s}}(Y^{Q}_{1}(p^{\infty}),\mathcal{O})$. There is a natural
surjective algebra homomorphism
$h^{-}(Y^{Q}_{1}(p^{\infty}),\mathcal{O})\to h(Y_{1}(p^{\infty}),\mathcal{O})$
We still denote by ${\mathfrak{m}}$ the inverse image of the maximal ideal
${\mathfrak{m}}$ by this homomorphism.
For $v\in Q$, let $\alpha_{v,i}\in k_{v}^{\times}$ ($i=1,\ldots,N$) be the
(simple) roots of
$\operatorname{char}(\overline{\rho}(\operatorname{Frob}_{v}))$. Let
$\widetilde{\alpha_{v,i}}\in\operatorname{T}_{h}$ the simple root of
$\operatorname{char}(\rho_{h}(\operatorname{Frob}_{v}))$ lifting
$\alpha_{v,i}$. Let $U_{v,i}=[U_{Q}t_{v,i}U_{Q}]$ where
$t_{v,i}=\operatorname{diag}(\varpi_{v}\cdot 1_{i},1_{n-i})$. Let
$h(Y^{Q}_{1}(p^{\infty}),\mathcal{O})=h^{-}(Y^{Q}_{1}(p^{\infty}),\mathcal{O})[U_{v,i}\,(v\in
Q,i=1,\ldots,N-1),<x>_{Q},x\in\Delta_{Q}]$
We define the maximal ideal ${\mathfrak{m}}_{Q}$ of
$h(Y^{Q}_{1}(p^{\infty}),\mathcal{O})$ as
${\mathfrak{m}}_{Q}={\mathfrak{m}}+(U_{v,i}-\prod_{j\leq
i}N(v)^{-(j-1)}\widetilde{\alpha_{v,j}})_{v\in Q}$
(see [Ge19, Corollary 2.7.8]). Let
$\operatorname{T}_{h,Q}=eh(Y^{Q}_{1}(p^{\infty}),\mathcal{O})_{{\mathfrak{m}}_{Q}}$
It acts faithfully on
$e\mathrm{H}^{q_{s}}(Y^{Q}_{1}(p^{\infty},\mathcal{O})_{{\mathfrak{m}}_{Q}}$.
By Scholze and [CGH+19], there exists a Galois representation
$\rho_{\operatorname{T}_{h,Q}}\colon G_{F,S\cup S_{p}\cup
Q}\to\operatorname{\mathsf{GL}}_{N}(\operatorname{T}_{h,Q})$
associated to $\operatorname{T}_{h,Q}$. Modulo a nilpotent ideal (or assuming
Conjecture A), it is ordinary at places above $p$ and minimal at places
dividing ${\mathfrak{n}}$. Moreover, for any $v\in Q$, let $I_{v}$ be the
inertia subgroup at $v$; again by (a trivial generalization of) [TW95, Lemma
7], we may assume after conjugation that it takes values in
$\widehat{T}(\operatorname{T}_{h,Q})$. Let us determine the character
$\Delta_{v}\to\operatorname{T}_{h,Q}^{\times}$.
-On the maximal conjugate self-adjoint quotient of $\operatorname{T}_{h,Q}$, it follows from [CHT08, Proposition 3.4.4 (8)] that it is given by $a\mapsto<a>_{Q}$ for $a\in\Delta_{v}$. Note that these authors define a Hecke algebra $eh(Y^{Q},\mathcal{O})$ using a level group $U_{Q}$ associated to maximal parahoric subgroups instead of Iwahori subgroups at $v\in Q$ , hence their diamond operator is defined on $k(v)^{\times}$ instead of $T(k(v))$, but the proof is the same.
-For $\operatorname{T}_{h,Q}$ itself, it follows from [ACC+18, Proposition 6.5.11] that this is also the case, modulo a nilpotent ideal with nilpotence index bounded in terms of $N$ and $[F:{\mathbb{Q}}]$. Conjecture A of [CaGe18], generalized to $\operatorname{\mathsf{GL}}_{N}$ of a CM field asserts in particular that this nilpotent ideal can be chosen to be $0$.
This implies that, assuming Conjecture A, the Galois representation
$\rho_{\operatorname{T}_{h,Q}}$ satisfies all the local conditions of the
problem $\mathcal{D}_{h,Q}$. Therefore, there exists a canonical
$\Lambda$-algebra homomorphism
$\phi_{Q}\colon R_{h,Q}\to\operatorname{T}_{h,Q}$
associated to $\rho_{\operatorname{T}_{h,Q}}$.
###### Definition 11.
A Taylor-Wiles system is a collection $\\{Q_{m}\\}$ of mutually disjoint
Taylor-Wiles sets such that for any $v\in Q_{m}$, $Nv\equiv 1\pmod{p^{m}}$.
Let $Y^{Q_{m}}_{1}(p^{\infty})$ be the provariety associated to the level
group $U_{Q_{m}}\cap U_{1}(p^{\infty})$. Let
$\operatorname{T}_{h,Q_{m}}=h(Y^{Q_{m}}(p^{\infty}),\mathcal{O})_{{\mathfrak{m}}_{Q_{m}}}$.
We thus have a collection of surjective ring homomorphisms $\phi_{Q_{m}}\colon
R_{h,Q_{m}}\to\operatorname{T}_{h,Q_{m}}$ which reduce to $\phi\colon
R_{h}\to\operatorname{T}_{h}$ modulo $I_{Q_{m}}$.
Recall that for
$G=\operatorname{Res}_{F/{\mathbb{Q}}}\operatorname{\mathsf{GL}}_{N}$,
$\ell_{0}=Nd_{0}-1$. Recall we assume all primes above $p$ in $F^{+}$ split in
$F$.
###### Proposition 8.
Assume $(ST)$. Then, for any $m\geq 1$, $R_{h,Q_{m}}$ can be generated by
$s=rN-\ell_{0}$ elements.
###### Proof.
By [DDT95, Theorem 2.18], we know that $R_{h,Q_{m}}$ can be generated by $s$
elements with
$s=h^{0}(F,{\mathfrak{g}})-\sum_{v|\infty}\dim_{k}{\mathfrak{g}}+\sum_{v\in
S_{p}}(\ell_{v}-h^{0}(G_{v},{\mathfrak{g}}))+\sum_{v\in
Q}(\dim_{k}(\ell_{v}-h^{0}(G_{v},{\mathfrak{g}})).$
where $\ell_{v}=\dim_{k}L_{v}$. We have $h^{0}(F,{\mathfrak{g}})=1$ and
$-\sum_{v|\infty}\dim_{k}{\mathfrak{g}}=-N^{2}d_{0}$
for $v\in S_{p}$, let us compute $\ell_{v}-h^{0}(G_{v},{\mathfrak{g}})$; we
recall that $L_{v}$ is the image in $\mathrm{H}^{1}(G_{v},{\mathfrak{g}})$ of
the fiber product
$L^{\prime}_{v}=\mathrm{H}^{1}(G_{v},{\mathfrak{b}})\times_{\mathrm{H}^{1}(G_{v},{\mathfrak{b}}/{\mathfrak{n}})}\mathrm{H}^{1}_{unr}(G_{v},{\mathfrak{b}}/{\mathfrak{n}})$
We can insert $L_{v}^{\prime}$ in the long exact sequence
$0\to\mathrm{H}^{0}(G_{v},{\mathfrak{n}})\to\mathrm{H}^{0}(G_{v},{\mathfrak{b}})\to\mathrm{H}^{0}(G_{v},{\mathfrak{b}}/{\mathfrak{n}})\to\mathrm{H}^{1}(G_{v},{\mathfrak{n}})\to
L_{v}^{\prime}\to H^{1}(G_{v}/I_{v},{\mathfrak{b}}/{\mathfrak{n}})\to 0$
Hence, we have :
$\dim_{k}L_{v}^{\prime}-h^{0}(G_{v},{\mathfrak{b}})=h^{1}(G_{v}/I_{v},{\mathfrak{b}}/{\mathfrak{n}})-h^{0}(G_{v}/I_{v},{\mathfrak{b}}/{\mathfrak{n}})+h^{1}(G_{v},{\mathfrak{n}})-h^{0}(G_{v},{\mathfrak{n}})$
We have
$h^{1}(G_{v}/I_{v},{\mathfrak{b}}/{\mathfrak{n}})=h^{0}(G_{v}/I_{v},{\mathfrak{b}}),$
$h^{1}(G_{v},{\mathfrak{n}})-h^{0}(G_{v},{\mathfrak{n}})=[F_{v}\colon{\mathbb{Q}}_{p}]\dim_{k}{\mathfrak{n}}+h^{0}(G_{v},{\mathfrak{n}}^{\ast}(1))$
and $h^{0}(G_{v},{\mathfrak{n}}^{\ast}(1))=0$ by strong distinguishability.
Since $L_{v}^{\prime}\to L_{v}$ is injective by strong distinguishability, we
conclude
$\ell_{v}-h^{0}(G_{v},{\mathfrak{g}})=[F_{v}\colon{\mathbb{Q}}_{p}]\cdot
N(N-1)/2$. So
$\sum_{v\in S_{p}}(\ell_{v}-h^{0}(G_{v},{\mathfrak{g}}))=N(N-1)d_{0}$
Finally, vor $v\in Q$, $L_{v}=\mathrm{H}^{1}(G_{v},{\mathfrak{g}})$. By
inflation restriction,
$0\to\mathrm{H}^{1}_{unr}(G_{v},{\mathfrak{g}})\to\mathrm{H}^{1}(G_{v},{\mathfrak{g}})\to\operatorname{\mathsf{Hom}}_{G_{v}/I_{v}}({\mathbb{Z}}_{p}(1),{\mathfrak{g}})\to
0.$
The kernel and cokernel are $N$-dimensional. Moreover
$\mathrm{H}^{0}(G_{v},{\mathfrak{g}})$ is diagonal, hence $N$-dimensional.
Therefore $\ell_{v}-h^{0}(G_{v},{\mathfrak{g}})=2N-N=N$.
We conclude that
$s=1-N^{2}d_{0}+N(N-1)d_{0}+rN=Nr-Nd_{0}+1=rN-\ell_{0}$
as desired. ∎
Let $S_{\infty}^{\Lambda}=\Lambda[[S_{1},\ldots,S_{Nr}]]$ and
$R_{\infty}^{\Lambda}=\Lambda[[X_{1},\ldots,X_{s}]]$. For any TW set $Q_{m}$
as above, there exists (several) surjective $\Lambda$-algebra homomorphisms
$r_{Q_{m}}\colon R_{\infty}^{\Lambda}\to R_{h,Q_{m}}$.
###### Lemma 16.
There exists a $\Lambda$-algebra homomorphism $\alpha_{\infty}\colon
S_{\infty}^{\Lambda}\to R_{\infty}^{\Lambda}$, a sequence of Taylor-Wiles sets
$Q_{m_{j}}$, and $\Lambda$-algebra homomorphisms $\Lambda$-algebra
homomorphisms $r_{Q_{m_{j}}}\colon R_{\infty}^{\Lambda}\to R_{h,Q_{m_{j}}}$
such that for any $j$, the diagram
$\begin{array}[]{ccc}S_{\infty}^{\Lambda}&\stackrel{{\scriptstyle\alpha_{\infty}}}{{\longrightarrow}}&R_{\infty}^{\Lambda}\\\
\downarrow&&\downarrow r_{Q_{m_{j}}}\\\
\Lambda[\Delta_{Q_{m_{j}}}]&\stackrel{{\scriptstyle\alpha_{Q_{m_{j}}}}}{{\longrightarrow}}&R_{h,Q_{m_{j}}}\end{array}$
commutes.
The argument of [CaGe18, Part II] and [GV18, Section 13 (Theorem 13.1 and its
proof)] goes through for $S_{\infty}^{\Lambda}\to R_{\infty}^{\Lambda}$ to
prove
###### Theorem 15.
1) $\phi\colon R_{h}\to\operatorname{T}_{h}$ is an isomorphism of
$\Lambda$-algebras (which may not locally complete intersections over
$\mathcal{O}$).
2) As graded module over the commutative graded ring
$\operatorname{Tor}_{\bullet}^{S_{\infty}^{\Lambda}}(R_{\infty}^{\Lambda},\Lambda)$,
we have
$\mathbb{H}^{q_{s}-\bullet}_{\mathfrak{m}}\cong\mathbb{H}^{q_{s}}_{\mathfrak{m}}\otimes\operatorname{Tor}_{\bullet}^{S_{\infty}^{\Lambda}}(R_{\infty}^{\Lambda},\Lambda).$
### 6.3. Simplicial deformation ring
Let
$\mathcal{D}^{s}_{h}\colon{}_{\mathcal{O}_{k}}\to\operatorname{\mathsf{sSETS}}$
be the problem of minimal ordinary simplicial deformations (without specifying
the Hodge-Tate weights). By the same argument proving the prorepresentability
of $\mathcal{D}^{s}_{\lambda}$, one sees that $\mathcal{D}^{s}_{h}$ is pro-
representable by a simplicial deformation ring $\mathcal{R}_{h}$ which
satisfies $\pi_{0}(\mathcal{R}_{h})=R_{h}$. Exactly as in Theorem 11, we prove
under the same assumptions as Theorem 15:
###### Theorem 16.
There is a natural isomorphism of graded $T_{h}$-algebras
$\pi_{\bullet}(\mathcal{R}_{h})\cong\operatorname{Tor}_{\bullet}^{S_{\infty}^{\Lambda}}(R_{\infty}^{\Lambda},\Lambda)$
Let $\mathcal{R}_{\lambda^{\prime}}$ be the simplicial deformation ring at
bottom level $U_{0}(p)$ and weight $\lambda^{\prime}$, as in Section 3.6.
Recall that if $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are two simplicial
$\mathcal{O}$-algebras, one can form a tensor product simplicial
$\mathcal{O}$-algebra $\mathcal{R}_{1}\underline{\otimes}\mathcal{R}_{2}$.
Note that for the prime ideal $P_{\lambda^{\prime}}$ of $\Lambda$ associated
to the arithmetic weight $\lambda^{\prime}$, we have a weak equivalence of
simplicial rings
$\mathcal{R}_{h}\underline{\otimes}_{\Lambda}\Lambda/P_{\lambda^{\prime}}\mathcal{R}_{h}\cong\mathcal{R}.$
We first formulate the conjecture of Concentration in Supremum Degree (CinS):
###### Conjecture 3.
The following two equivalent statements are true
1) $\mathrm{H}^{\bullet}_{\mathfrak{m}}$ is concentrated in degree $q_{s}$,
2) $\mathcal{R}_{h}$ is discrete and for any arithmetic weight
$\lambda^{\prime}$, there is an isomorphism of commutative graded rings
$\operatorname{Tor}_{\bullet}^{\Lambda}(R_{h},\Lambda/P_{\lambda^{\prime}})=\pi_{\bullet}(\mathcal{R}_{\lambda^{\prime}}).$
Comments:
1) The equivalence between the two statements follows from Theorem 15.
2) By Theorem 15, this conjecture also implies that
(a) $\mathrm{H}^{\bullet}_{\mathfrak{m}}$ is free of rank one over
$R_{h}=\operatorname{T}_{h}$,
(b) for any arithmetic weight $\lambda^{\prime}$, for any
$i=0,\ldots,\ell_{0}$,
$\operatorname{Tor}^{\Lambda}_{i}(\mathrm{H}^{q_{s}}_{\mathfrak{m}},\Lambda/P_{\lambda^{\prime}})=\mathrm{H}^{q_{s}-i}(Y_{0}(p),V_{\lambda^{\prime}}(\mathcal{O}))_{\mathfrak{m}}.$
3) All the statements are easy to prove for $N=2$ and $F$ quadratic because
$\mathrm{H}^{1}_{\mathfrak{m}}=0$ since it is torsion free over
$\Lambda=\mathcal{O}[[T_{1},T_{2},X_{1},X_{2}]]$ (where $X_{i}$’s are the
twist variables) but is annihilated by $T_{1}-T_{2}$.
This conjecture is motivated by non abelian Leopoldt conjectures due to Hida
and one of the authors (E. Urban).
### 6.4. Non-abelian Leopoldt Conjectures
Recall a first version of the non abelian Leopoldt conjecture which we call
the deformation theoretic non abelian Leopoldt conjecture (DNAL), cf. [Ti96,
Section 9, Example 1]:
###### Conjecture 4.
$\dim R_{h}=\dim\Lambda-\ell_{0}$
###### Proposition 9.
Assume that Conjecture (DNAL) holds; then Conjecture (CinS) holds.
###### Proof.
Let $q\in[q_{m},q_{s}]$ be the smallest integer such that
$\mathrm{H}^{q}_{h}\neq 0$. Then, the complex $C^{q_{m}}\to\ldots\to
C^{q}_{h}$ is a resolution of $M=C^{q}_{h}/B^{q}_{h}$ where
$B^{q}_{h}=\operatorname{Im}(C^{q-1}_{h}\to C^{q}_{h})$. We have
$H^{q}_{h}\subset M$ and $\operatorname{\mathsf{projdim}}_{\Lambda}M\leq
q-q_{h}$ hence by Auslander-Buchsbaum formula, we have
$\operatorname{depth}_{\Lambda}M\geq\dim_{\Lambda}M-(q-q_{h})$. Recall
Ischebeck’s Lemma ([Mat, (15.E) Lemma 2]: if $A,B$ are two finitely generated
$\Lambda$-modules, for any $i<depth_{\Lambda}B-\dim_{\Lambda}A$,
$\operatorname{Ext}^{i}(A,B)=0.$
Here, we consider $\operatorname{Ext}^{0}(\mathrm{H}^{q}_{h},M))\neq 0$ hence
$0\geq\operatorname{depth}_{\Lambda}M-\dim_{\Lambda}\mathrm{H}^{q}_{h}$, so
$\dim_{\Lambda}\mathrm{H}^{q}_{h}\geq\operatorname{depth}_{\Lambda}M\geq\dim_{\Lambda}-\ell_{0}.$
∎
Recall
$\Sigma_{coh}\subset\operatorname{Supp}_{\Lambda}(\mathrm{H}^{\bullet}_{\mathfrak{m}})$.
Note that it is Zariski-dense in
$\operatorname{Supp}_{\Lambda}(\mathrm{H}^{\bullet}_{\mathfrak{m}})$. We know
that for $P_{\lambda}\in\Sigma_{coh}$,
$\mathrm{H}^{i}(Y_{0}(p),V_{\lambda}(K))_{\mathfrak{m}}\neq 0$ if and only f
$i\in[q_{m},q_{s}]$ (where $q_{m}=q_{0}$ and $q_{s}=q_{0}+\ell_{0}$). Let us
consider the integral $\operatorname{Tor}$-spectral sequence
$E_{2}^{i,j}(\lambda)=\operatorname{Tor}_{i}^{\Lambda}(\mathrm{H}^{j}_{\mathfrak{m}},\Lambda/P_{\lambda})\Rightarrow\mathrm{H}^{j-i}(Y_{0}(p),V_{\lambda}(\mathcal{O}))_{\mathfrak{m}}$
($i\leq 0$, $j\geq 0$). Let $\lambda$ be an arietic weight. Consider the
condition
$(INTDEG)_{\lambda}$ the $\operatorname{Tor}$-spectral sequence degenerates at
$E_{2}$ and we have
$\mathrm{H}^{q}(Y_{0}(p),V_{\lambda}(\mathcal{O}))_{\mathfrak{m}}=\bigoplus_{j-i=q}\operatorname{Tor}_{i}^{\Lambda}(\mathrm{H}^{j}_{\mathfrak{m}},\Lambda/P_{\lambda}).$
Note that Conjecture (CinS) implies $(INTDEG)_{\lambda}$ for all arithmetic
weights $\lambda$. Recall that for a finitely generated module $M$ over a
noetherian ring $A$, one defines
$\operatorname{codim}_{A}M=min\\{ht{{\mathfrak{p}}};{\mathfrak{p}}\in\operatorname{Supp}(M)\\}$.
If $A$ is Cohen-Macaulay, one has $\dim A=\dim_{A}M+\operatorname{codim}_{A}M$
and $\dim_{A}M=\operatorname{depth}_{A}M$. If $A$ is regular, it follows from
the Auslander-Buchsbaum formula that
$\operatorname{codim}_{A}M=\operatorname{\mathsf{projdim}}_{A}M$ (which is
finite). Then, assuming Conjecture (CinS), we see that for any
$i\in[0,\ell_{0}]$,
$\operatorname{Tor}_{i}^{\Lambda}(\mathrm{H}^{q_{s}}_{\mathfrak{m}},\Lambda/P)\neq
0$
for any $P\in\Sigma_{coh}$ Since the subset $\Sigma_{coh}$ is Zariski-dense in
$\operatorname{Supp}_{\Lambda}(\mathrm{H}^{\bullet}_{\mathfrak{m}})$, we
conclude that
$\operatorname{\mathsf{projdim}}_{\Lambda}\mathrm{H}^{q_{s}}_{\mathfrak{m}}=\ell_{0}$
This implies that $\operatorname{Supp}(\mathrm{H}^{q_{s}}_{\mathfrak{m}})$ has
$\Lambda$-codimension $\leq\ell_{0}$. Let us recall the cohomological non
abelian Leopoldt Conjecture due to Hida and E. Urban.
###### Conjecture 5.
(CNLA) $\operatorname{codim}_{\Lambda}\Sigma_{h}=\ell_{0}$
###### Remark 8.
Hida conjectured that
$\operatorname{codim}_{\Lambda}\mathrm{H}^{\bullet}_{\mathfrak{m}}\otimes\mathbb{Q}_{p}=\ell_{0}$
and Urban conjectured that
$\operatorname{codim}_{\Lambda}\Sigma_{h}=\ell_{0}$.But by Lemma 15, we have
$\Sigma_{h}=\operatorname{Supp}_{\Lambda}(\mathbb{H}^{\bullet}_{\mathfrak{m}})\cap
Spec(\Lambda)(\overline{\mathbb{Q}}_{p})$, therefore these two statements are
equivalent.
###### Lemma 17.
Assume that Condition $(INTDEG)_{\lambda}$ holds for some arithmetic weight
$\lambda$ and that Conjecture $(CNAL_{h})$ holds. Then Conjecture (CinS)
holds.
###### Proof.
Let $q$ with $q_{m}\leq q<q_{s}$ be the minimal integer such that
$\mathrm{H}^{q}_{\mathfrak{m}}\neq 0$. By (INTDEG) , we see that for all i’s
$i>\ell_{0}-(q_{s}-q)$, we have
$\operatorname{Tor}^{\Lambda}_{i}(\mathrm{H}^{q_{s}},\Lambda/P)\neq 0$ for all
$P\in\Sigma_{coh}(\subset\operatorname{Supp}_{\Lambda}(\mathrm{H}^{q_{s}}_{\mathfrak{m}}))$.
This implies
$\operatorname{\mathsf{projdim}}\mathrm{H}^{q}_{\mathfrak{m}}\leq\ell_{0}-(q_{s}-q)<\ell_{0}$.
Contradiction.
∎
## 7\. The Galatius-Venkatesh homomorphism for Hida families
Let $\operatorname{Spec}\mathbf{I}$ be an irreducible component of
$\operatorname{Spec}\operatorname{T}_{h}$; let ${\mathfrak{m}}_{\mathbf{I}}$
be its maximal ideal. Note that $\mathbf{I}$ is not necessary flat over
$\Lambda$. We have a $\Lambda$-linear algebra homomomorphism
$\theta\colon\operatorname{T}_{h}\to\mathbf{I}$. Let $\Lambda_{\mathbf{I}}$ be
the image of $\Lambda$ in $\mathbf{I}$. Let
$\mathcal{R}_{\mathbf{I}}=\mathcal{R}_{h}\underline{\otimes}_{\Lambda}\Lambda_{\mathbf{I}}$
and $R_{\mathbf{I}}=R_{h}\otimes_{\Lambda}\Lambda_{\mathbf{I}}$. We have
$\pi_{0}(\mathcal{R}_{\mathbf{I}})=R_{\mathbf{I}}$ and there is a natural
$R_{h}$-algebra homomorphism
$\pi_{\mathbf{I}}\colon\pi_{1}(\mathcal{R}_{h})\to\pi_{1}(\mathcal{R}_{\mathbf{I}})$.
However, we conjecture that $\pi_{1}(\mathcal{R}_{h})=0$ (hence
$\pi_{\mathbf{I}}=0$) while
$\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}$
will be shown to interpolate the classical
$\pi_{1}(\mathcal{R}_{\lambda^{\prime}})\otimes_{R_{\lambda^{\prime}},\theta_{\pi^{\prime}}}K/\mathcal{O}$
for all arithmetic weights $\lambda^{\prime}$ congruent to $\lambda$ modulo
$p$ and all classical Hecke eigensystems $\theta_{\pi^{\prime}}$ of weight
$\lambda^{\prime}$.
###### Theorem 17.
There are natural $\mathbf{I}$-linear injective homomorphisms
$GV_{h}\colon\pi_{1}(\mathcal{R}_{h})\otimes_{R_{h}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}).$
and
$GV_{\mathbf{I}}\colon\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}).$
and we have $GV_{\mathbf{I}}\circ\pi_{\mathbf{I}}=GV_{h}$.
###### Proof.
Let $n\geq 1$, let
$\mathbf{I}_{n}=\mathbf{I}/{\mathfrak{m}}_{\mathbf{I}}^{n}$. For any finite
$\mathbf{I}_{n}$-module $M$ (which can be viewed as a
$\operatorname{T}_{h}$-module), we consider the simplicial ring
$\Theta_{n}=\mathbf{I}_{n}\oplus M[1]$. Let $L_{n}(\mathcal{R}_{h})$ be the
set of homotopy equivalence classes of simplicial ring homomorphisms
$\Phi\colon\mathcal{R}_{h}\to\Theta_{n}$ such that
$\operatorname{pr}_{n}\circ\Phi=\phi_{n}$. Note that any such $\Phi$ factors
through $\mathcal{R}_{h}\to\mathcal{R}_{\mathbf{I}}$ into a homomorphism
$\Phi_{\mathbf{I}}\colon\mathcal{R}_{\mathbf{I}}\to\Theta_{n}$. So,
$L_{n}(\mathcal{R}_{h})$ can be reinterpreted as the set
$L_{n}(\mathcal{R}_{\mathbf{I}})$ of homotopy equivalence classes of
$\operatorname{\mathsf{Hom}}_{\operatorname{pr}_{n}}(\mathcal{R}_{\mathbf{I}},\Theta_{n}).$
There is a canonical bijection
$L_{n}(\mathcal{R}_{h})\cong\mathrm{H}^{2}_{\mathcal{L}}(F,\operatorname{Ad}(\rho_{\theta})\otimes_{\mathbf{I}_{n}}M)$
where $\mathcal{L}$ is the minimal ordinary condition. Moreover, as in [GV18,
Lemma 15.1] and in Formula (4), for each $n\geq 1$, we have a
$\operatorname{T}_{h}$-linear homomorphism
$\pi(n,\mathcal{R}_{\ast})\colon
L_{n}(\mathcal{R}_{\ast})\to\operatorname{\mathsf{Hom}}_{\operatorname{T}_{h}}(\pi_{1}(\mathcal{R}_{\ast}),M)$
where $\ast=h,\mathbf{I}$. As in Proposition 5, this homomorphism is
surjective. We take $M=\mathbf{I}_{n}$. Its Pontryagin dual is
$M^{\vee}=\operatorname{\mathsf{Hom}}_{\mathcal{O}}(\mathbf{I},K/\mathcal{O})[{\mathfrak{m}}_{\mathbf{I}}^{n}]$.
Let
$\widehat{\mathbf{I}}=\operatorname{\mathsf{Hom}}_{\mathcal{O}}(\mathbf{I},K/\mathcal{O})$.
Let
$\pi_{1}\mathcal{R}_{\ast,\theta}=\pi_{1}\mathcal{R}_{\ast}\otimes_{\operatorname{T}_{h},\theta}\mathbf{I}$.
We see that we can rewrite the Pontryagin dual of the right hand side of (7)
as
$\operatorname{\mathsf{Hom}}_{\mathcal{O}}(\operatorname{\mathsf{Hom}}_{\operatorname{T}_{h}}(\pi_{1}(\mathcal{R}_{\ast}),M),K/\mathcal{O})=\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\pi_{1}(\mathcal{R}_{\ast})_{\theta},\mathbf{I}/{\mathfrak{m}}_{\mathbf{I}}^{n}),\widehat{\mathbf{I}})$
Since
$\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\widehat{\mathbf{I}},\widehat{\mathbf{I}})=\mathbf{I}$,
we have
$\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\widehat{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}],\widehat{\mathbf{I}})=\mathbf{I}/{\mathfrak{m}}_{\mathbf{I}}^{n}$
hence
$\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\pi_{1}(\mathcal{R}_{\ast})_{\theta},\mathbf{I}/{\mathfrak{m}}_{\mathbf{I}}^{n})=\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\pi_{1}(\mathcal{R}_{\ast})_{\theta},\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\widehat{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}],\widehat{\mathbf{I}})))=$
$\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\pi_{1}(\mathcal{R}_{\ast})_{\theta}\otimes_{\mathbf{I}}\widehat{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}],\widehat{\mathbf{I}})),$
but for
$S=\pi_{1}(\mathcal{R}_{\ast})_{\theta}\otimes_{\mathbf{I}}\widehat{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}]$,
we have
$\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\operatorname{\mathsf{Hom}}_{\mathbf{I}}(S,\widehat{\mathbf{I}}),\widehat{\mathbf{I}})=S$,
so we conclude
$\operatorname{\mathsf{Hom}}_{\mathcal{O}}(\operatorname{\mathsf{Hom}}_{\operatorname{T}_{h}}(\pi_{1}(\mathcal{R}_{\ast}),M),K/\mathcal{O})=\pi_{1}\mathcal{R}_{\ast,\theta}\otimes_{\mathbf{I}}\widehat{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}].$
Similarly, the Pontryagin dual of the left hand side of (7) is
$\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}\rho_{\theta}^{\ast}(1)\otimes_{\mathcal{O}}M^{\vee})$
which can be rewritten as
$\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}\rho_{\theta}^{\ast}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}])$
Let $GV_{\ast,n}$ be the Pontryagin dual of $\pi(n,\mathcal{R}_{\ast})$. Let
$GV_{\ast}=\varinjlim_{n}GV_{\ast,n}$, it provides the desired injection
$GV_{\ast}\colon\pi_{1}(\mathcal{R}_{\ast})\otimes_{\operatorname{T}_{h}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}).$
∎
### 7.1. Interpolation properties
Let $P\in\operatorname{Spec}\mathbf{I}$ be a codimension one prime; assume
that $\mathcal{O}_{P}=\mathbf{I}/P$ is a dvr. Let
$K_{P}=\operatorname{Frac}(\mathcal{O}_{P})$. We have
$\operatorname{\mathsf{Hom}}(\mathcal{O}_{P},K/\mathcal{O})=K_{P}/\mathcal{O}_{P})$.
The following exact control theorem holds for the minimal ordinary Selmer
group. We put
${\mathop{\rm
Sel}}^{\ast}_{\mathbf{I}}=\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}),\quad{\mathop{\rm
Sel}}^{\ast}_{P}=\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{P}(1)\otimes_{\mathcal{O}_{P}}K_{P}/\mathcal{O}_{P})$
###### Proposition 10.
For any prime ideal $P\subset\mathbf{I}$,
${\mathop{\rm Sel}}^{\ast}_{\mathbf{I}}[P]={\mathop{\rm Sel}}^{\ast}_{P}.$
###### Proof.
We apply [SU14, Proposition 3.2.8]: Let $T$ be a $\mathbf{I}[\Gamma]$-module,
which is finite free over $\mathbf{I}$. Then, for any ideal
${\mathfrak{a}}\subset\mathbf{I}$, $\mathop{\rm
Sel}(T\otimes_{\mathbf{I}}\widehat{\mathbf{I}})[{\mathfrak{a}}]=\mathop{\rm
Sel}(T\otimes_{\mathbf{I}}\widehat{\mathbf{I}/{\mathfrak{a}}})$ under the
assumption that $T\otimes_{\mathbf{I}}\widehat{\mathbf{I}}$ has no non trivial
subquotient with trivial Galois action.
Let us check this condition for
$T=\operatorname{Ad}\rho_{\mathbf{I}}^{\ast}(1)$. Let $M$ be a
$\mathbf{I}[\Gamma]$-submodule of $T\otimes_{\mathbf{I}}\widehat{\mathbf{I}}$
and $N$ a $\mathbf{I}[\Gamma]$-quotient of $M$. By applying
$\operatorname{\mathsf{Hom}}_{\mathbf{I}}(-,\widehat{\mathbf{I}})$, one sees
that $\widehat{M}$ is a $\mathbf{I}[\Gamma]$-quotient of
$\operatorname{Ad}{\rho}_{\mathbf{I}}$. The reduction of this homomorphism
modulo ${\mathfrak{m}}_{\mathbf{I}}$ is null by absolute irreducibility of
$\operatorname{Ad}\overline{\rho}$. By Nakayama’s lemma, this implies
$\widehat{M}=0$. The same holds for the $\mathbf{I}[\Gamma]$-submodule
$\widehat{N}\subset\widehat{M}$. ∎
Let $M_{\mathbf{I}}$ be the Pontryagin dual of ${\mathop{\rm
Sel}}^{\ast}_{\mathbf{I}}$. Let $\Phi_{\mathbf{I}}$ be the Pontryagin dual of
$\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}$.
We have
$\Phi_{\mathbf{I}}=\operatorname{\mathsf{Hom}}_{R_{\mathbf{I}}}(\pi_{1}(\mathcal{R}_{\mathbf{I}}),\mathbf{I})=\operatorname{\mathsf{Hom}}_{{\mathbf{I}}}(\pi_{1}(\mathcal{R}_{\mathbf{I}})_{\theta},\mathbf{I}).$
It is a finitely generated torsion free $\mathbf{I}$-module. Let
$N_{\mathbf{I}}=\operatorname{Ker}(M_{\mathbf{I}}\to\Phi_{\mathbf{I}})$ be the
Pontryagin dual of $\operatorname{Coker}\,GV_{\mathbf{I}}$. We have a short
exact sequence
$0\to N_{\mathbf{I}}\to M_{\mathbf{I}}\to\Phi_{\mathbf{I}}\to 0$
Let ${\mathop{\rm
Sel}}_{\mathbf{I}}=\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}\rho_{\mathbf{I}}\otimes\widehat{\mathbf{I}})$.
###### Theorem 18.
(1) The $\mathbf{I}$-modules $M_{\mathbf{I}}$ and $\Phi_{\mathbf{I}}$ have
rank $\ell_{0}$ and $\Phi_{\mathbf{I}}$ is free.
(2) Let $T_{{\mathfrak{m}}_{\mathbf{I}}}{\mathop{\rm
Sel}}_{\mathbf{I}}=\varprojlim_{n}{\mathop{\rm
Sel}}_{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}]$. The Poitou-Tate-
Pontryagin duality induces an isomorphism
$N_{\mathbf{I}}\cong T_{{\mathfrak{m}}_{\mathbf{I}}}{\mathop{\rm
Sel}}_{\mathbf{I}}$
###### Proof.
(1) Let $P$ be an arithmetic prime of weight $\lambda^{\prime}$ congruent to
$\lambda$ modulo $p$; let $\mathcal{O}_{P}=\mathbf{I}/P$. Generically, it is a
dvr. We compare
$GV_{\mathbf{I}}\colon\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}})$
to the "classical" map
$GV_{P}\colon\pi_{1}(\mathcal{R}_{\lambda^{\prime}})\otimes_{R_{\lambda^{\prime}}}K_{P}/\mathcal{O}_{P}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{P}(1)\otimes_{\mathcal{O}_{P}}K_{P}/\mathcal{O}_{P})$
We have a commutative diagram
$\begin{array}[]{ccccc}GV_{\mathbf{I}}&\colon&\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}&\hookrightarrow&\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}})\\\
&&\uparrow&&\uparrow\\\
GV_{P}&\colon&\pi_{1}(\mathcal{R}_{\lambda^{\prime}})\otimes_{R_{\lambda^{\prime}}}K_{P}/\mathcal{O}_{P}&\hookrightarrow&\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{P}(1)\otimes_{\mathcal{O}_{P}}K_{P}/\mathcal{O}_{P})\end{array}$
Let $M_{P}$ be the Pontryagin dual of ${\mathop{\rm Sel}}^{\ast}_{P}$.
Similarly, let $\Phi_{P}$ denote the Pontryagin dual of
$\pi_{1}(\mathcal{R}_{\lambda^{\prime}})\otimes_{R_{\lambda^{\prime}}}K_{P}/\mathcal{O}_{P}$.
By Pontryagin duality we obtain a diagram
$\begin{array}[]{ccc}M_{\mathbf{I}}&\to&\Phi_{\mathbf{I}}\\\
\downarrow&&\downarrow\\\ M_{P}&\to&\Phi_{P}\end{array}$
where the two horizontal arrow are surjective and where the left downarrow
induces an isomorphism by exact control of the dual Selmer groups:
$M_{\mathbf{I}}/PM_{\mathbf{I}}\cong M_{P}.$
It implies that the right downarrow
$\Phi_{\mathbf{I}}/P\Phi_{\mathbf{I}}\rightarrow\Phi_{P}$
is surjective. Let us prove it is an isomorphism. Note that $\Phi_{P}$ is free
of rank $\ell_{0}$ by Theorem 13. We have
$\Phi_{\mathbf{I}}/P\Phi_{\mathbf{I}}=\operatorname{\mathsf{Hom}}_{\mathbf{I}}(\pi_{1}(\mathcal{R}_{\mathbf{I}})_{\theta},\mathbf{I}/P)=\operatorname{\mathsf{Hom}}_{\mathbf{I}/P}(\pi_{1}(\mathcal{R}_{\mathbf{I}})_{\theta}\otimes_{\mathbf{I}}\mathbf{I}/P,\mathbf{I}/P)$
Let $\widetilde{\mathbf{I}}$ be the integral closure of $\mathbf{I}$ and let
$X=\operatorname{Supp}(\widetilde{\mathbf{I}}/\mathbf{I})$. For $P\notin X$,
we have $\mathbf{I}/P=\widetilde{\mathbf{I}}/P\widetilde{\mathbf{I}}$.
Therefore, we can assume $\mathbf{I}$ is integrally closed provided we
consider only arithmetic primes $P\notin X$. We know that $M_{\mathbf{I}}$ and
$\pi_{1}(\mathcal{R}_{\mathbf{I}})_{\theta}$ (hence $\Phi_{{}_{b}I}$) are
finitely generated $\mathbf{I}$-modules. By the structure theorem of modules
over a Krull ring, $M_{\mathbf{I}}$, resp. $\Phi_{\mathbf{I}}$, is pseudo-
isomorphic to $\mathbf{I}^{r_{1}}\oplus\Theta_{1}$, resp.
$\mathbf{I}^{r_{2}}\oplus\Theta_{2}$ where $\Theta_{i}$ ($i=1,2$) is a torsion
$\mathbf{I}$-module. By reduction modulo $P\notin X$, we know by Lemma 7 that
$r_{1}=\ell_{0}$ and $r_{2}\leq\ell_{0}$. Note that
$\Phi_{\mathbf{I}}/P\Phi_{\mathbf{I}}=\operatorname{\mathsf{Hom}}_{\mathcal{O}_{P}}(\pi_{1}(\mathcal{R}_{\mathbf{I}})_{\theta}\otimes_{\mathbf{I}}\mathcal{O}_{P},\mathcal{O}_{P})$
the right member is free of rank $r_{2}$ over $\mathcal{O}_{P}$. Hence,
$\Phi_{\mathbf{I}}/P\Phi_{\mathbf{I}}\cong\mathcal{O}_{P}^{r_{2}}$; since this
$\mathcal{O}_{P}$-module maps surjectively to $\mathcal{O}_{P}^{\ell_{0}}$, we
conclude that $r_{2}=\ell_{0}$ and that
$\Phi_{\mathbf{I}}/P\Phi_{\mathbf{I}}\rightarrow\Phi_{P}$
is an isomorphism. Thus, by Nakayama’s lemma, there exists a surjective
$\mathbf{I}$-linear homomorphism
$j\colon\mathbf{I}^{\ell_{0}}\to\Phi_{\mathbf{I}}$
Let $\mathcal{K}=\mathop{\rm Ker}j$. For $P\notin X$, we have
$\mathcal{K}\subset P\cdot\mathbf{I}^{\ell_{0}}$. Note that $\bigcap_{P\notin
X}P=0$ in $\mathbf{I}$, hence $\mathcal{K}=0$ and $j$ is an isomorphism as
desired.
(2) We use notations from the proof of Theorem 17. By Lemma 10,
$N_{\mathbf{I}}/{\mathfrak{m}}_{\mathbf{I}}^{n}N_{\mathbf{I}}$ is Pontryagin
dual of $\operatorname{Coker}GV_{\mathbf{I},n}$. By Poitou-Tate duality (Lemma
3), the group $\operatorname{Coker}GV_{\mathbf{I},n}$ is Pontryagin dual to
${\mathop{\rm
Sel}}_{\mathbf{I}_{n}}=\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}\rho_{n}\otimes_{\mathbf{I}_{n}}\widehat{\mathbf{I}}_{n})$
By taking projective limits over $n$, we find that $N_{\mathbf{I}}$ is
Pontryagin dual to $T_{{\mathfrak{m}}_{\mathbf{I}}}{\mathop{\rm
Sel}}_{\mathbf{I}}$.
∎
Let $\widetilde{\mathbf{I}}$ be the integral closure of $\mathbf{I}$ and let
$X=\operatorname{Supp}(\widetilde{\mathbf{I}}/\mathbf{I})$. For $P\notin X$,
we have $\mathbf{I}/P=\widetilde{\mathbf{I}}/P\widetilde{\mathbf{I}}$. Let
$L^{alg}_{\mathbf{I}}$ be the Fitting ideal of the torsion $\mathbf{I}$-module
$N_{\mathbf{I}}$. It plays the role of the algebraic $p$-adic $L$ function for
the family of motives $\operatorname{Ad}(\rho_{\mathbf{I}})$. For any
arithmetic prime $P$ of $\mathbf{I}$, let $N_{P}$ be the Pontryagin dual of
$\operatorname{Coker}GV_{P}$. Let $L^{alg}_{\mathbf{I}}(P)$ be the image of
$L^{alg}_{\mathbf{I}}$ in $\mathbf{I}/P$.
###### Corollary 4.
For any $P\notin X$,
(1) the map
$\operatorname{Coker}GV_{P}\to\operatorname{Coker}GV_{\mathbf{I}}[P]$ is an
isomorphism
(2) the specialization $L^{alg}_{\mathbf{I}}(P)$ of $L^{alg}_{\mathbf{I}}$ at
$P$ is a generator of
$\operatorname{Fitt}_{\mathcal{O}_{P}}N_{P}=\operatorname{Fitt}_{\mathcal{O}_{P}}\operatorname{Coker}GV_{P}$.
###### Proof.
By Pontryagin duality, both statements are equivalent to
$N_{\mathbf{I}}/PN_{\mathbf{I}}=N_{P}$. This statement follows from the
formulas $M_{\mathbf{I}}/PM_{\mathbf{I}}=M_{P}$,
$\Phi_{\mathbf{I}}/P\Phi_{\mathbf{I}}=\Phi_{P}$ and
$\operatorname{Tor}_{1}^{\mathbf{I}}(\Phi_{\mathbf{I}},\mathbf{I}/P)=0$ proven
in Theorem 18. ∎
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E. Urban, Department of Mathematics, Columbia University.
|
# Characterizing and Measuring the Similarity of Neural Networks with
Persistent Homology
David Pérez-Fernández
SEGITTUR
<EMAIL_ADDRESS>
&Asier Gutiérrez-Fandiño††footnotemark:
Barcelona Supercomputing Center
<EMAIL_ADDRESS>
&Jordi Armengol-Estapé
Barcelona Supercomputing Center
<EMAIL_ADDRESS>
&Marta Villegas
Barcelona Supercomputing Center
<EMAIL_ADDRESS>
Contributed equally.
###### Abstract
Characterizing the structural properties of neural networks is crucial yet
poorly understood, and there are no well-established similarity measures
between networks. In this work, we observe that neural networks can be
represented as abstract simplicial complex and analyzed using their
topological ’fingerprints’ via Persistent Homology (PH). We then describe a
PH-based representation proposed for characterizing and measuring similarity
of neural networks. We empirically show the effectiveness of this
representation as a descriptor of different architectures in several datasets.
This approach based on Topological Data Analysis is a step towards better
understanding neural networks and serves as a useful similarity measure.
## 1 Introduction
Machine learning practitioners can train different neural networks for the
same task. Even for the same neural architecture, there are many
hyperparameters, such as the number of neurons per layer or the number of
layers. Moreover, the final weights for the same architecture and
hyperparameters can vary depending on the initialization and the optimization
process itself, which is stochastic. Thus, there is no direct way of comparing
neural networks accounting for the fact that neural networks solving the same
task should be measured as being similar, regardless of the specific weights.
This also prevents one from finding and comparing modules inside neural
networks (e.g., determining if a given sub-network does the same function as
other sub-network in another model). Moreover, there are no well-known methods
for effectively characterizing neural networks.
This work aims to characterize neural networks such that they can be measured
to be similar once trained for the same task, with independence of the
particular architecture, initialization, or optimization process. We focus on
Multi-Layer Perceptrons (MLPs) for the sake of simplicity. We start by
observing that we can represent a neural network as a directed weighted graph
to which we can associate certain topological concepts.111See Jonsson [21] for
a complete reference on graph topology. Considering it as a simplicial
complex, we obtain its associated Persistent Diagram. Then, we can compute
distances between Persistent Diagrams of different neural networks.
The proposed experiments aim to show that the selected structural feature,
Persistent Homology, serves to relate neural networks trained for similar
problems and that such a comparison can be performed by means of a predefined
measure between the associated Persistent Homology diagrams. To test the
hypothesis, we study different classical problems (MNIST, Fashion MNIST,
CIFAR-10, and language identification and text classification datasets),
different architectures (number and size of layers) as well as a control
experiment (input order).
In summary, the main contributions of this work are the following:
* •
We propose an effective graph characterization strategy of neural networks
based on Persistent Homology.
* •
Based on this characterization, we suggest a similarity measure of neural
networks.
* •
We provide empirical evidence that this Persistent Homology framework captures
valuable information from neural networks and that the proposed similarity
measure is meaningful.
The remainder of this paper is organized as follows. In Section 2, we go
through the related work. Then, in Section 3 we describe our proposal and the
experimental framework to validate it. Finally, in sections 4 and 5 we report
and discuss the results and arrive to conclusions, respectively.
## 2 Related Work
One of the fundamental papers of Topological Data Analysis (TDA) is presented
in Carlsson [8] and suggests the use of Algebraic Topology to obtain
qualitative information and deal with metrics for large amounts of data. For
an extensive overview of simplicial topology on graphs, see Giblin [18],
Jonsson [21]. Aktas et al. [2] provide a thorough analysis of PH methods.
More recently, a number of publications have dealt with the study of the
capacity of neural networks using PH. Guss and Salakhutdinov [19] characterize
learnability of different neural architectures by computable measures of data
complexity. Rieck et al. [30] introduce the neural persistence metric, a
complexity measure based on TDA on weighted stratified graphs. This work
suggests a representation of the neural network as a multipartite graph and
the filtering of the Persistent Homology diagrams are performed for each layer
independently. As the filtration contains at most 1-simplices (edges), they
only capture zero-dimensional topological information, i.e. connectivity
information. Donier [14] propose the concept of spatial capacity allocation
analysis. Konuk and Smith [22] propose an empirical study of how NNs handle
changes in topological complexity of the input data.
In terms of pure neural network analysis, there are relevant works, like Hofer
et al. [20], that study topological regularization. Clough et al. [11]
introduce a method for training neural networks for image segmentation with
prior topology knowledge, specifically via Betti numbers. Corneanu et al. [13]
try to estimate (with limited success) the performance gap between training
and testing via neuron activations and linear regression of the Betti numbers.
On the other hand, topological analysis of decision boundaries has been a very
prolific area. Ramamurthy et al. [28] propose a labeled Vietoris-Rips complex
to perform PH inference of decision boundaries for quantification of the
complexity of neural networks.
Naitzat et al. [27] experiment on the PH of a wide range of point cloud input
datasets for a binary classification problems to see that NNs transform a
topologically rich dataset (in terms of Betti numbers) into a topologically
simpler one as it passes through the layers. They also verify that the
reduction in Betti numbers is significantly faster for ReLU activations than
hyperbolic tangent activations.
Liu [25] obtain certain geometrical and topological properties of decision
regions for neural models, and provide some principled guidance to designing
and regularizing them. Additionally, they use curvatures of decision
boundaries in terms of network weights, and the rotation index theorem
together with the Gauss-Bonnet-Chern theorem.
Regarding neural network representations, one of the most related works to
ours, Gebhart et al. [16], focuses on topological representations of neural
networks. They introduce a method for computing PH over the graphical
activation structure of neural networks, which provides access to the task-
relevant substructures activated throughout the network for a given input.
Interestingly, in Watanabe and Yamana [35], authors work on neural network
representations through simplicial complexes based on deep Taylor
decomposition and they calculate the PH of neural networks in this
representation. In Chowdhury et al. [10], they use directed homology to
represent MLPs. They show that the path homology of these networks is non-
trivial in higher dimensions and depends on the number and size of the network
layers. They investigate homological differences between distinct neural
network architectures.
As far as neural network similarity measures are concerned, the literature is
not especially prolific. In Kornblith et al. [23], authors examine similarity
measures for representations (meaning, outputs of different layers) of neural
networks based on canonical correlation analysis. However, note that this
method compares neural network representations (intermediate outputs), not the
neural networks themselves. Remarkably, in Ashmore and Gashler [3], authors do
deal with the intrinsic similarity of neural networks themselves based on
Forward Bipartite Alignment. Specifically, they propose an algorithm for
aligning the topological structures of two neural networks. Their algorithm
finds optimal bipartite matches between the nodes of the two MLPs by solving
the well-known graph cutting problem. The alignment enables applications such
as visualizations or improving ensembles. However, the methods only works
under very restrictive assumptions,222For example, the two neural networks
”must have the same number of units in each of their corresponding layers”,
and the match is done layer by layer. and this line of work does not appear to
have been followed up.
Finally, we note that there has been a considerable growth of interest in
applied topology in the recent years. This popularity increase and the
development of new software
libraries,333https://www.math.colostate.edu/~adams/advising/appliedTopologySoftware/
along with the growth of computational capabilities, have empowered new works.
Some of the most remarkable libraries are Ripser [32, 5], and Flagser [26].
They are focused on the efficient computation of PH. For GPU-Accelerated
computation of Vietoris-Rips PH, Ripser++ [37] offers an important speedup.
The Python library we are using, Giotto-TDA [31], makes use of both above
libraries underneath.
We have seen that there is a trend towards the use of algebraic topology
methods for having a better understanding of phenomena of neural networks and
having more principled deep learning algorithms. Nevertheless, little to no
works have proposed neural network characterizations or similarity measures
based on intrinsic properties of the networks, which is what we intend to do.
## 3 Methodology
In this section, we propose our method, which is heavily based on concepts
from algebraic topology. We refer the reader to the Supplementary Material for
the mathematical definitions. In this section, we also describe the conducted
experiments.
Intrinsically characterizing and comparing neural networks is a difficult,
unsolved problem. First, the network should be represented in an object that
captures as much information as possible and then it should be compared with a
measure depending on the latent structure. Due to the stochasticity of both
the initialization and training procedure, networks are parameterized
differently. For the same task, different functions that effectively solve it
can be obtained. Being able to compare the trained networks can be helpful to
detect similar neural structures.
We want to obtain topological characterizations associated to neural networks
trained on a given task. For doing so, we use the Persistence Homology (from
now on, PH) of the graph associated to a neural network. We compute the PH for
various neural networks learned on different tasks. We then compare all the
diagrams for each one of the task.
More specifically, for each of the studied tasks (image classification on
MNIST, Fashion MNIST and CIFAR-10; language identification, and text
classification on the Reuters dataset),444For more details, see Section 3.2.
we proceed as follows:
* •
We train several neural network models on the particular problem.
* •
We create a directed graph from the weights of the trained neural networks
(after changing the direction of the negative edges and normalising the
weights of the edges).
* •
We consider the directed graph as a simplicial complex and calculate its PH,
using the weight of the edges as the filtering parameter, which range from 0
to 1. This way we obtain the so-called Persistence Diagram.
* •
We compute the distances between the Persistence Diagrams (prior
discretization of the Persistence Diagram so that it can be computed) of the
different networks.
* •
Finally, we analyze the similarity between different neural networks trained
for the same task, for a similar task, and for a completely different task,
independently of the concrete architecture, to see whether there is
topological similarity.
As baselines, we set two standard matrix comparison methods that are the
1-Norm and the Frobenius norm. Having adjacency matrix $A$ and $B$, we compute
the difference as $norm(A-B)$. However, these methods only work for matrices
of similar size and thus, they are not general enough. We could also have used
the Fast Approximate Quadratic assignment algorithm suggested in Vogelstein et
al. [34], but for large networks this method becomes unfeasible to compute.
### 3.1 Proposal
Our method is as follows. We start by associating to a neural network a
weighted directed graph that is analyzed as an abstract simplicial complex
consisting on the union of points, edges, triangles, tetrahedrons and larger
dimension polytopes (those are the elements referred as simplices). Abstract
simplicial complexes are used in opposition to geometric simplicial complexes,
generated by a point cloud embedded in the Euclidean space $\mathbb{R}^{n}$.
Given a trained neural network, we take the collection of neural network
parameters as directed and weighted edges that join neurons, represented by
graph nodes. Biases are considered as new vertices that join target neurons
with an edge having a given weight. Note that, in this representation, we lose
the information about the activation functions, for simplicity and to avoid
representing the network as a multiplex network. Bias information could also
have been ignored because we want large PH groups that characterize the
network, while these connections will not change the homology group dimension
of any order.
For negative edge weights, we reverse edge directions and maintain the
absolute value of the weights. We discard the use of weight absolute value
since neural networks are not invariant under weight sign transformations.
This representation is consistent with the fact that every neuron can be
replaced by a neuron from which two edges with opposite weights emerge and
converge again on another neuron with opposite weights. From the point of view
of homology, this would be represented as a closed cycle.
We then normalize the weights of all the edges as expressed in Equation 1
where $w$ is the weight to normalize, $W$ are all the weights and $\zeta$ is
an smoothing parameter that we set to 0.000001. This smoothing parameter is
necessary as we want to avoid normalized weights of edges to be 0. This is
because 0 implies a lack of connection.
$max(1-\frac{|w|}{max(|max(W)|,|min(W)|)},\zeta)$ (1)
Given this weighted directed graph, we then define a directed flag complex
associated to it. Topology of this directed flag complex can be studied using
homology groups $H_{n}$. In this work we calculate homology groups up to
degree 3 ($H_{0}$-$H_{3}$) due to computational complexity and our neural
network representation method’s layer connectivity limit.
The dimensions of these homology groups are known as Betti numbers. The $i$-th
Betti number is the number of $i$-dimensional voids in the simplicial complex
($\beta_{0}$ gives the number of connected components of the simplicial
complex, $\beta_{1}$ gives the number of non reducible loops and so on). For a
deeper introduction to algebraic topology and computational topology, we refer
to Edelsbrunner and Harer [15], Ghrist [17].
We work with a family of simplicial complexes, $K_{\varepsilon}$, for a range
of values of $\varepsilon\in\mathbb{R}$ so that the complex at step
$\varepsilon_{t}$ is embedded in the complex at $\varepsilon_{t+1}$ for
$\varepsilon_{t}\leq\varepsilon_{t+1}$, i.e. $K_{\varepsilon}\subseteq
K_{\varepsilon_{t+1}}$. In our case, $\varepsilon$ is the minimum weight of
included edges of our graph representation of neural networks.
The nested family of simplicial complexes is called a filtration. We calculate
a sequence of homology groups by varying the $\varepsilon$ parameter,
obtaining a persistence homology diagram. PH calculations are performed on
$\mathbb{Z}_{2}$.
This filtration gives a collection of contained directed weighted graph or
simplicial complex $K_{\varepsilon_{min}}\subseteq\ldots\subseteq
K_{\varepsilon_{t}}\subseteq K_{\varepsilon_{t+1}}\subseteq\ldots\subseteq
K_{\varepsilon_{max}}$, where $t\in[0,1]$ and $\varepsilon_{min}=0$,
$\varepsilon_{max}=1$ (recall that edge weights are normalized).
Given a filtration, one can look at the birth, when a homology class appears,
and death, the time when the homology class disappears. The PH treats the
birth and the death of these homological features in $K_{\varepsilon}$ for
different $\varepsilon$ values. Lifespan of each homological feature can be
represented as an interval $(birth,death)$, of the homological feature. Given
a filtration, one can record all these intervals by a Persistence Barcode (PB)
[8], or in a Persistence Diagram (PD), as a collection of multiset of
intervals.
As mentioned previously, our interest in this work is to compare PDs from two
different simplicial complexes. There are two distances traditionally used to
compare PDs, Wasserstein distance and Bottleneck distance. Their stability
with respect to perturbations on PDs has been object of different studies [9,
12].
In order to make computations feasible and to obviate noisy intervals, we
filter the PDs by limiting the minimum PD interval size. We do so by setting a
minimum threshold $\eta=0.01$. Intervals with a lifespan under this value are
not considered. Additionally, for computing distances, we need to remove
infinity values. As we are only interested in the deaths until the maximum
weight value, we replace all the infinity values by $1.0$.
Wasserstein distance calculations are computationally hard for large PDs (each
PD of our NN models has a million persistence intervals per diagram).
Therefore we use a vectorized version of PDs instead, also called PD
discretization. This vectorized version summaries have been proposed and used
on recent literature [1, 6, 7, 24, 29].
For the persistence diagram distance calculation, we use the Giotto-TDA
library [31] and compute the following supported vectorized persistence
summaries: 1. Persistence landscape. 2. Weighted silhouette. 3. Heat
vectorizations.
### 3.2 Experimental Framework
#### Datasets
To determine the topological structural properties of trained NNs, we select
different kinds of datasets. We opt for four well-known benchmarks in the
machine learning community and one regarding language identification: (1) the
MNIST555http://yann.lecun.com/exdb/mnist/ dataset for classifying handwritten
digit images, (2) the Fashion MNIST [36] dataset for classifying clothing
images into 10 categories, (3) the
CIFAR-10666https://www.cs.toronto.edu/~kriz/cifar.html (CIFAR) dataset for
classifying 10 different objects, (4) the Reuters dataset for classifying
news into 46 topics, and (5) the Language Identification Wikipedia
dataset777https://www.floydhub.com/floydhub/datasets/language-
identification/1/data for identifying 7 different languages.
We selected these datasets because, apart from being well-known benchmarks,
the performances without transfer learning are good enough and they have
different data types and sizes. For CIFAR-10 and Fashion MNIST datasets we
train a Convolutional Neural Network (CNN) first, and the convolutional layers
are shared between all the models of the same dataset as a feature extractor.
Recall that in this work we are focusing on MLPs, so we do not consider that
convolutional weights. For the MNIST, Reuters and Language Identification
datasets, we use an MLP. For Reuters and Language identification datasets, we
vectorize the sentences with character frequency.
#### Experiments Pipeline
We study the following variables (hyperparameters): 1. Layer width, 2.
Number of layers, 3. Input order888Order of the input features, the control
experiment. This one should definitely not affect the performance in the
neural networks, so if our method is correct, it should be uniform as per the
proposed topological distances.), 4. Number of labels (number of considered
classes).
We define the base architecture as the one with a layer width of 512, 2
layers, the original features order, and considering all the classes (10 in
the case of MNIST, Fashion MNIST and CIFAR, 46 in the case of Reuters and 7 in
the case of the language identification task). Then, doing one change at a
time, keeping the rest of the base architecture hyperparameters, we experiment
with architectures with the following configurations:
* •
Layer width: 128, 256, 512 (base) and 1024.
* •
Number of layers: 2 (base), 4, 6, 8 and 10.
* •
Input order: 5 different randomizations (with base structure), the control
experiment.
* •
Number of labels (MNIST, Fashion MNIST, CIFAR-10): 2, 4, 6, 8 and 10 (base).
* •
Number of labels (Reuters): 2, 6, 12, 23 and 46 (base).
* •
Number of labels (Language Identification): 2, 3, 4, 6 and 7 (base).
Note that this is not a grid search over all the combinations. We always
modify one hyperparameter at a time, and keep the rest of them as in the base
architecture. In other words, we experiment with all the combinations such
that only one of the hyperparameters is set to a non-base value at a time.
For each dataset, we train 5 times (each with a different random weight
initialization) each of these neural network configurations. Then, we compute
the topological distances (persistence landscape, weighted silhouette, heat)
among the different architectures. In total, we obtain $5\times 5\times 3$
distance matrices (5 datasets, 5 random initializations, 3 distance measures).
Finally, we average the 5 random initializations, such that we get $5\times 3$
matrices, one for each distance on each dataset. All the matrices have
dimensions $19\times 19$, since 19 is the number of experiments for each
dataset (corresponding to the total the number of architectural configurations
mentioned above). Note that the base architecture appears 8 times (1, on the
number of neurons per layer, 1 on the number of layers, 1 on the number of
labels and the 5 randomizations of weight initializations).
All experiments were executed in a machine with 2 NVIDIA V100 of 32GB, 2
Intel(R) Xeon(R) Platinum 8176 CPU @ 2.10GHz, and of 1.5TB RAM, for a total of
around 3 days.
The code and results are fully open source999https://github.com/asier-
gutierrez/nn-similarity under MIT license.
## 4 Results & Discussion
Number | Experiment | Index
---|---|---
1 | Layer size | 1-4
2 | Number of layers | 5-9
3 | Input order | 10-14
4 | Number of labels | 15-19
Table 1: Indices of the experiments of the distance matrices.
(a) Reuters
(b) Language Identification
Figure 1: Distance matrices using Silhouette discretization.
Results from control experiments can be seen in the third group on Figures 1
and 4. In these figures, groups are separated visually using white dashed
lines. Experiments groups are specified in Table 1. Control experiments in all
the images appear very dimmed, which means that they are very similar, as
expected. Recall that the control experiments consist of $5$ (randomizations)
$\times$ $5$ (executions) and that 25 different neural networks have been
trained; each one of the network has more than 690,000 parameters that have
been randomly initialized. After the training, results show that these
networks have very close topological distance, as expected.
(a) 1-norm
(b) Frobenius norm
Figure 2: Control experiments using norms. Norm | Minimum | Maximum | Mean | Standard deviation
---|---|---|---|---
1-Norm | 0.6683 | 4.9159 | 1.9733 | 1.5693
Frobenius | 0.0670 | 0.9886 | 0.4514 | 0.3074
Table 2: Normalized difference comparison of self-norm against the maximum
mean distance of the experiment.
For Figure 2 we computed both 1-norm and Frobenius norm (the baselines) for
graphs’ adjacency matrices of control experiments. Note that as we ran the
experiment five times, we make the mean for each value of the matrix. In order
to show whether the resulting values are positive or negative, we subtract to
the maximum difference of each dataset the norm of each cell separately, we
take the absolute value and we divide by the maximum difference of each
dataset. Therefore, we obtain five values per dataset. Table 2 shows the
statistics reflecting that the distance among the experiments are large and,
thus, they are not characterizing any similarity but rather an important
dissimilarity.
In contrast, Figure 3, with our method (Silhouette), shows perfect diagonal of
similarity blocks. In the corresponding numeric results, we obtained show
small distances, as shown in Table 3. We can appreciate that each dataset has
its own hub. This confirms the validity of our proposed similarity measure.
Figure 3: Control experiment comparison matrix using Silhouette
discretization.
The method we present also seems to capture some parts of hyperparameter
setup. For instance, in Figure 4 we can observe gradual increase of distances
in the first group regarding layer size meaning that, as layer size increases,
the topological distance increases too. Similarly, for the number of layers
(second group) and number of labels (fourth group) the same situation holds.
Note that in Fashion MNIST and CIFAR-10, the distances are dimmer because we
are not dealing with the weights of the CNNs. Recall that the CNN acts as a
frozen extractor and are pretrained for all runs (with the same weights), such
that the MLP layers themselves are the only potential source of dissimilarity
between runs.
(a) MNIST
(b) Fashion MNIST
(c) CIFAR-10
Figure 4: Distance matrices using Heat discretization. | Heat distance | Silhouette distance
---|---|---
Dataset | Mean | Deviation | Mean | Deviation
MNIST | 0.0291 | 0.0100 | 0.1115 | 0.0364
F. MNIST | 0.0308 | 0.0132 | 0.0824 | 0.0353
CIFAR-10 | 0.0243 | 0.0068 | 0.0769 | 0.0204
Language I. | 0.0159 | 0.0040 | 0.0699 | 0.0159
Reuters | 0.0166 | 0.0051 | 0.0387 | 0.0112
Table 3: PH distances across input order (control) experiments, normalized by
dataset.
Thus, our characterization is sensitive to the architecture (e.g., if we
increase the capacity, distances vary), but at the same time, as we saw
before, it is not dataset-agnostic, meaning that it also captures whether two
neural networks are learning the same problem or not.
In Figure 4, Fashion MNIST (Figure 4(b)) and CIFAR (Figure 4(c)) dataset
results are interestingly different from those of MNIST (Figure 4(a)) dataset.
This is, presumably, because both Fashion MNIST and CIFAR use a pretrained CNN
for the problem. Thus, we must analyze the results taking into account this
perspective. The first fully connected layer size is important as it can avoid
a bottleneck from the previous CNN output. Some works in the literature show
that adding multiple fully connected layers does not necessarily enhance the
prediction capability of CNNs [4], which is congruent with our results when
adding fully connected layers (experiments 5 to 9) that result in dimmer
matrices than the one from. Concerning the experiments on input order, there
is slightly more homogeneity than in MNIST, again showing that the order of
sample has negligible influence. Moreover, there could have been even more
homogeneity taking into account that the fully connected network reduced its
variance thanks to the frozen weights of the CNN. This also supports the fact
that the CNN is the main feature extractor of the network. As in MNIST
results, CIFAR results show that the topological properties are, indeed, a
mapping of the practical properties of neural networks.
Figure 5: Language Identification dataset PH Landscape distance matrix.
We refer to the Supplementary Material for all distance matrices for all
datasets and all distances, as well as for the standard deviations matrices
and experiment group statistics.
## 5 Conclusions & Future Work
Results from different experiments, in five different datasets from computer
vision and natural language, lead to similar topological properties and are
trivially interpretable, which yields to general applicability.
The bests discretizations chosen for this work are the Heat and Silhouette.
They show better separation of experiment groups, and are effectively
reflecting changes in a sensitive way. We also explored the Landscape
discretization but it offers a very low interpretability and clearance. In
other words, it is not helpful for comparing PH diagrams associated to neural
networks.
The most remarkable conclusion comes from the control experiments. The
corresponding neural networks, with different input order but the same
architecture, are very close to each other. The PH framework does, indeed,
abstract away the specific weight values, and captures latent information from
the networks, allowing comparisons to be based on the function they
approximate. The selected neural network representation is reliable and
complete, and yields coherent and meaningful results. Instead, the baseline
measures, the 1-Norm and the Frobenius norm, implied an important
dissimilarity between the experiments in the control experiments, meaning that
they did not capture the fact that these neural networks were very similar in
terms of the solved problem.
We conclude that our proposed characterization, does, indeed, capture
meaningful information from neural network, and the computed distances can
serve as an effective similarity measure between networks. To the best of our
knowledge, this similarity measure between neural networks is the first of its
kind.
As future work, we suggest adapting the method to different deep learning
libraries and make it support popular neural architectures such as CNNs,
Recurrent Neural Networks, and Transformers [33]. Finally, we suggest
performing more analysis regarding the learning of a neural network, and
trying to topologically answer the question of how a neural network learns.
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|
Motion Estimation of Connected and Automated Vehicles under Communication
Delay and Packet Loss of V2X Communications Ziran Wang, Kyungtae Han, and
Prashant TiwariToyota Motor North America R&D, InfoTech Labs, Mountain View,
CA 2021-01-0202 2021
## 1 Abstract
The emergence of the connected and automated vehicle (CAV) technology enables
numerous advanced applications in our transportation system, benefiting our
daily travels in terms of safety, mobility, and sustainability. However,
vehicular communication technologies such as Dedicated Short-Range
Communications (DSRC) or Cellular-Based Vehicle-to-Everything (C-V2X)
communications unavoidably introduce issues like communication delay and
packet loss, which will downgrade the performances of any CAV applications. In
this study, we propose a consensus-based motion estimation methodology to
estimate the vehicle motion when the vehicular communication environment is
not ideal. This methodology is developed based on the consensus-based
feedforward/feedback motion control algorithm, estimating the position and
speed of a CAV in the presence of communication delay and packet loss. The
simulation study is conducted in a traffic scenario of unsignalized
intersections, where CAVs coordinate with each other through V2X
communications and cross intersections without any full stop. Game engine-
based human-in-the-loop simulation results shows the proposed motion
estimation methodology can cap the position estimation error to $\pm 0.5$ m
during periodic packet loss and time-variant communication delay.
## 2 Introduction
### 2.1 Background
During the past decades, the emergence of connected and automated vehicle
(CAV) technology brings new possibilities to our transportation systems.
Specifically, the level of connectivity within our vehicles has greatly
increased, allowing these “equipped” vehicles to behave in a cooperative
fashion. The cooperation is not only among vehicles themselves through
vehicle-to-vehicle (V2V) communications, but also among vehicles and other
transportation entities through vehicle-to-infrastructure (V2I)
communications, vehicle-to-network/cloud (V2N/V2C) communications, vehicle-to-
pedestrian (V2P) communications, and etc. In summary, all these communication
technologies are categorized as vehicle-to-everything (V2X) communications.
In the system architecture of a CAV, the communication module plays an equally
important role as other modules, such as the localization module, perception
module, planning module, and control module. It facilitates real-time and
reliable wireless V2X communications among CAVs and other entities, either
through Dedicated Short-Range Communications (DSRC) or Cellular-Based Vehicle-
to-Everything (C-V2X) communication technologies. The communication module of
a CAV system is capable of providing additional information that cannot be
readily detected by the perception module, and can generally provide
information more quickly than through sensor detection and processing. These
scenarios may include the following:
* •
Information from other CAVs that are beyond perception ranges of the ego CAV,
or that are occluded from view by intermediate vehicles or by
horizontal/vertical road curvatures.
* •
Information from other CAVs that cannot be directly sensed by the perception
module, such as vehicle status information (e.g., wheel speeds, fault status,
performance capabilities, etc.).
* •
Immediate notification of speed changes or steering commands as soon as they
have been issued to other CAV’s actuators, even before their motions have
begun to change.
* •
Negotiations between CAVs regarding desired maneuvers (e.g., merging, lane
changing), so that these can be done more safely and efficiently.
Given the strength of information sharing among CAVs through their
communication modules, cooperative automated driving can be enabled in
multiple traffic scenarios, such as Cooperative Adaptive Cruise Control (CACC)
[1, 2], cooperative ramp merging [3, 4], speed harmonization on highways [5,
6], cooperative eco-driving at signalized intersections [7, 8], and automated
coordination at unsignalized intersections [9, 10]. In these applications of
cooperative automated driving, CAVs are coordinated to maintain safe inter-
vehicle distances and accomplish tasks together based on V2X communications.
However, in a wireless communication network, due to the uncertain reliability
of wireless communication links, communication time delays and packet losses
are always unavoidable in the information sharing process among CAVs. Along
with such time-delayed and partially dropped measurements of vehicles’
motions, the control module of a CAV system needs to be carefully designed to
reduce the adverse impact of vulnerable feedback channels on the system
performance.
### 2.2 Literature Review
To address this time delay issue of V2X communications, two typical methods
for delayed system (Razumikhin-based method and Krasovskii-based method) were
studied in the literature. For example, Gao et al. considered uniform and
constant time delays with a Razumikhin-based method applied to synthesize an
H-infinity controller [11]. Besides uniform and constant time delays, Petrillo
et al. considered multiple time-varying delays and used adaptive feedback
gains to compensate for the errors arising from outdated information [12]. Di
Bernardo et al. designed a consensus-based CACC controller by using the
concept of aggregate delay, and a Razumikhin-based method was applied to the
stability analysis [13], which was further extended to the case of
heterogeneous time delays in [14]. To compare the Razumikhin-based and
Krasovskii-based methods, Chehardoli et al. further designed a consensus-based
controller and demonstrated the less conservatism of the latter method in the
sense of a greater upper bound of time delays [15].
As for packet losses studied in the literature, one feasible solution is to
achieve smooth transition to remove the reliance on V2X communications. For
example, Ploeg et al. designed an acceleration estimation algorithm using on-
board sensors in the case of communication failures so as to achieve seamless
transition from CACC to Adaptive Cruise Control (ACC) [16]. Harfouch et al.
designed an adaptive switched controller for the transition from CACC to ACC
to address the network switching due to communication failures [17].
Quite a few works have been done regarding motion estimation of vehicles with
sensor-fusion method, such as Extended Kalman filter (EKF) [18], unscented
Kalman Filter (UKF) [19], and particle filter [20]. These filters are the
modified versions of Kalman filter, and are based on the first-order Markov
chain, namely the state at current time step is conditioned only on the state
at the previous time step. Although they are useful for real-time
implementation, a crucial limitation is that sensor data prior to the previous
time step cannot be directly used. According to that, moving horizon
estimation methodologies have been proposed, which utilize all data acquired
in previous time steps, and can handle irregular sampling rates and delayed
data without any additional modifications [21, 22]. However, all
aforementioned works have been primarily focused on the motion estimation of
the ego vehicle, where data acquired from various sensors are fused to provide
the estimation result. To the best of the authors’ knowledge, very few
literature has dealt with motion estimation of a target connected vehicle
without the help of additional sensors.
### 2.3 Contributions of the Paper
Compared to the existing literature that studied the communication delay and
packet loss aspects of V2X communications, we make the following contributions
in this paper:
* •
Different from most existing literature, we consider both communication delay
and packet loss aspects in our motion estimation algorithm to design a
reliable and robust CAV system.
* •
We develop a consensus-based motion estimator and controller that allow the
cooperative automated driving system to keep functioning, without necessarily
degrading it to a non-cooperative system that is much less advanced like [16,
17].
* •
We conduct a case study of automated coordination at unsignalized
intersections, where CAVs cooperate with each other coming from different legs
of an intersection and cross without any full stops.
* •
Instead of only conducting numerical simulations to prove the effectiveness of
the system, we measure V2X communication parameters by real connected
vehicles, and then develop a high-fidelity simulation platform based on Unity
game engine.
In the rest of this paper, we firstly formulate the problem to solve in this
paper, covering system assumptions, system preliminaries, and problem
statement. Then, we propose the methodology of the paper, which is the
consensus-based motion estimation methodology for CAVs under communication
delay and packet loss of V2X communications. Next, the game engine-based
simulation is conducted, which validates the effectiveness of the proposed
methodology. Finally, we draw the conclusion of this paper, and point out some
future steps that can further improve this study.
## 3 Problem Formulation
### 3.1 System Assumptions
As described earlier, CAVs rely on on-board perception sensors, such as
camera, radar, and LIDAR, to measure neighboring vehicles’ states. With the
introduction of V2X communications, CAVs can obtain the states of those beyond
their direct measurement ranges and to obtain information that cannot be
detected by remote sensors (such as the issuance of internal control
commands). This helps enhance the sensing range of CAVs, but also brings about
various communication issues to CAV systems. Fortunately, because the
communicated information is supplementary to the information obtained from on-
board perception sensors, it is possible for cooperative automated driving
systems to be degraded to non-cooperative automated driving systems when
communication issues occur. However, this system degradation unavoidably
decreases the effectiveness of the original CAV systems, since information
flows among different vehicles will disappear, and CAV systems will become
automated vehicle (AV) systems.
In this paper, we develop a motion estimation methodology that still enables
cooperative automated driving even when communication issues occur.
Information flows among CAVs still exist “virtually”, which are based on the
estimated motions of CAVs, instead of the ground-truth motions measured by the
motion sensors of CAVs. Whenever the V2X communication recovers from the
undesired status (e.g., communication delay decreases to a normal range, or
packet loss disappears), those information flows among CAVs will return to the
ground-truth motions. With the proposed motion estimation methodology,
information flows are working all the time in the CAV systems, avoiding any
system degradation from cooperative automated driving to non-cooperative
automated driving.
It shall be noted that, since this paper mainly focuses on designing the
motion estimation methodology, some reasonable specifications and assumptions
are made while modeling the system to enable the theoretical analysis:
* •
Only the longitudinal motion of vehicles is considered in this paper, where
the motion control algorithm and motion estimator are both designed for the
longitudinal vehicle dynamics.
* •
The focus of this paper is estimating the vehicle motion under
observed/measured V2X communication issues. The reasons that cause such V2X
communication issues are not studied in this paper.
* •
A specific V2X communication technology (either DSRC or C-V2X) is not required
for our proposed method. The motion estimation methodology is supposed to
handle issues caused by any V2X communication technologies.
* •
Although the perception module of a CAV (i.e., camera, radar, and/or LIDAR)
can provide auxiliary information when the performance of V2X communications
is impaired, we do not consider its measurements in our motion estimation
methodology due to the scope of this paper.
### 3.2 System Preliminaries
First, given the longitudinal dynamics of a vehicle $i$ as the following
equations:
$\displaystyle\dot{r}_{i}(t)$ $\displaystyle=v_{i}(t)$ (1)
$\displaystyle\dot{v}_{i}(t)$ $\displaystyle=a_{i}(t)$ $\displaystyle
a_{i}(t)$
$\displaystyle=\frac{1}{m}\left[F_{net_{i}}(t)-R_{i}T_{br_{i}}(t)-c_{vi}v_{i}(t)^{2}-c_{fi}v_{i}(t)-d_{mi}(t)\right]$
where ${r}_{i}(t)$, $v_{i}(t)$, and $a_{i}(t)$ denote the longitudinal
position, longitudinal speed and longitudinal acceleration of vehicle $i$ at
time $t$, respectively, $m_{i}$ denotes the mass of vehicle $i$, $F_{net_{i}}$
denotes the net engine force of vehicle $i$ at time $t$, which mainly depends
on the vehicle speed and the throttle angle, $R_{i}$ denotes the effective
gear ratio from the engine to the wheel of vehicle $i$, $T_{br_{i}}(t)$
denotes the brake torque of vehicle $i$ at time $t$, $c_{vi}$ denotes the
coefficient of aerodynamic drag of vehicle $i$, $c_{fi}$ denotes the
coefficient of friction force of vehicle $i$, $d_{mi}(t)$ denotes the
mechanical drag of vehicle $i$ at time $t$.
We can then derive the following equations from the principle of vehicle
dynamics when the braking maneuver is deactivated, i.e., vehicle $i$ is
accelerating by the net engine force:
$F_{net_{i}}(t)=\ddot{x}_{i}(t)m_{i}+c_{vi}\dot{x}_{i}(t)^{2}+c_{pi}\dot{x}_{i}(t)+d_{mi}(t)$
(2)
and we have the following equation when the braking maneuver is activated
(vehicle $i$ decelerates by the brake torque):
$T_{br_{i}}(t)=\frac{\dot{x}_{i}(t)m_{i}+c_{vi}\dot{x}_{i}(t)^{2}+c_{pi}\dot{x}_{i}(t)+d_{mi}(t)}{R_{i}}$
(3)
Note that the net engine force is a function of the vehicle speed and the
throttle angle, which is typically based on the steady-state characteristics
of engine and transmission systems. The associated mathematical derivation can
be referred to [23].
Based on the existing literature [24], the motion control module of a CAV is
based on a hierarchical strategy, where the high-level controller generates a
target acceleration (the first two equations in (1)), while the low-level
controller commands the vehicle actuators to track the target acceleration
(the last equations in (1)). In this paper, we focus on the high-level vehicle
controller, where we propose the motion estimator based on the dynamics
consensus of CAVs.
### 3.3 Problem Statement
Given a two-CAV scenario, where the ego vehicle $i$ gets information from its
target vehicle $j$ through V2X (e.g., V2V, V2I, or V2N) communications, the
dynamics consensus can be generalized as a longitudinal control problem shown
as Fig. 1.
Figure 1: The illustration of vehicle parameters in a V2X communication
environment with an ego vehicle $i$ and its target vehicle $j$.
In this figure, $r$, $v$, $a$, and $l$ denote the vehicle longitudinal
position, longitudinal speed, longitudinal acceleration, and length,
respectively. Simply speaking, the dynamics consensus of CAVs can be
demonstrated as follows:
$\begin{array}[]{l}r_{i}(t)\rightarrow r_{j}(t)-r_{\text{headway}}\\\
v_{i}(t)\rightarrow v_{j}(t)\\\ a_{i}(t)\rightarrow a_{j}(t)\end{array}$ (4)
where $r_{\text{headway}}$ denotes the desired distance headway between these
two vehicles. Although this dynamics consensus does not apply to all traffic
scenarios (where some of them might require the ego vehicle to have a
higher/lower speed/acceleration than the target vehicle), it can be applied to
most of the cooperative longitudinal motion control applications of CAVs, such
as the ones mentioned in the first section of this paper (e.g., CACC, speed
harmonization, etc.).
However, the cooperative longitudinal motion control of CAVs heavily relies on
the performance of V2X communications, which enable the information
transmission between these two vehicles shown in Fig. 1. Time delay [11, 12]
and packet loss [16, 17] are two major issues that impair the performance of
V2X communications in CAV applications. Different from most of the existing
literature that tackle these two issues separately, we propose a motion
estimation methodology in this paper to address them at the same time.
## 4 Methodology
### 4.1 Consensus-Based Motion Control
In order to achieve the dynamics consensus of CAVs in equation (4), a double-
integrator consensus-baesd longitudinal motion control algorithm can be
proposed as
$\begin{array}[]{l}\dot{r}_{i}(t)=v_{i}(t)\\\
\dot{v}_{i}(t)=-\alpha_{ij}k_{ij}\cdot\left[\left(r_{i}(t)-r_{j}(t)+l_{j}+v_{i}(t)\cdot
t_{ij}^{g}(t)\right)+\gamma_{i}\right.\cdot\\\
\left.\left(v_{i}(t)-v_{j}(t)\right)\right]\end{array}$ (5)
where $\alpha_{ij}$ is the adjacency matrix of the directed graph (i.e., V2X
communication topology between vehicle $i$ and $j$), $t_{ij}^{g}(t)$ is the
time-variant desired time gap between two vehicles, which can be adjusted by
many factors like road grade, vehicle mass, braking ability, etc. The term
$[l_{j}+v_{i}(t)\cdot t_{ij}^{g}(t)]$ is another form of the term
$r_{\text{headway}}$ in equation (4). The control gains $k_{ij}$ and
$\gamma_{i}$ in this algorithm can be either defined as constants, or further
tuned by a feedforward control algorithm to guarantee the safety, efficiency,
and comfort of this slot-following process. A lookup-table approach is adopted
to dynamically calculate these control gains, based on the initial speeds of
two vehicles, as well as their initial headway. In short, it can be summarized
as
$\\{k_{ij},\gamma_{i}\\}=f\big{(}v_{i}(0),v_{j}(0),r_{i}(0)-r_{j}(0)\big{)}$
(6)
where the details can be referred to our previous work [25]. With this
longitudinal motion control algorithm equation (5), vehicle $i$ in Fig. 1 is
able to converge its longitudinal speed $v_{i}(t)$ to vehicle $j$’s
longitudinal speed $v_{j}(t)$, and converge its longitudinal position
${r}_{i}(t)$ to vehicle $j$’s longitudinal position ${r}_{j}(t)$ minus a
desired headway between them.
However, the aforementioned algorithm does not consider the communication
delay issue. It is without a doubt that, whatever information vehicle $i$
receives is not the exactly current information of vehicle $j$, due to the
unavoidable transmission time $\tau_{ij}(t)$. Therefore, equation (5) can be
further written into the following form while considering the time-variant
communication delay:
$\begin{array}[]{l}\dot{r}_{i}(t)=v_{i}(t)\\\
\dot{v}_{i}(t)=-\alpha_{ij}k_{ij}\cdot\left[\left(r_{i}(t)-r_{j}(t-\tau_{ij}(t))+l_{j}+v_{i}(t)\cdot
t_{ij}^{g}(t))\right)+\gamma_{i}\right.\cdot\\\
\left.\left(v_{i}(t)-v_{j}(t-\tau_{ij}(t))\right)\right]\end{array}$ (7)
### 4.2 Motion Estimation for Communication Delay and Packet Loss
As stated in the problem statement of this paper, time delay and packet loss
are two major issues that impair the performance of V2X communications in CAV
applications. In this subsection, we develop a motion estimation methodology
that overcomes these two issues at the same time.
As stated below, Algorithm 1 is the main function of the proposed motion
estimation methodology, where Algorithm 2, Algorithm 3, and Algorithm 4 are
called in this main function.
1
Result: Vehicle $j$’s estimated longitudinal motion
$\mathbf{\tilde{V}_{j}(t)}$ and $\mathbf{\tilde{R}_{j}(t)}$ in the future
horizon $[t+1,t+N]$
2
3Vehicle $i$ associates with its target vehicle $j$, where $j=i-1$;
4
5while _communication between vehicle $i$ and $j$ is currently on_ do
6
7 if _$n=j==0$ , namely vehicle $j$ is the leader of a communication topology
and does not have any target vehicle_ then
8 Vehicle $j$ estimate its future longitudinal speed trajectory
$\mathbf{\tilde{V}_{j}(t)}$ based on Algorithm 2;
9 Vehicle $j$ cumulatively estimates its future longitudinal position
trajectory $\mathbf{\tilde{R}_{j}(t)}$ based on Algorithm 3;
10
11 else
12 for _$n==1\rightarrow j$_ do
13
14 if _Vehicle $n$ is connected to its target vehicle $n-1$, namely no packet
loss is in presence at this time step $t$_ then
15 Vehicle $n$ estimates its future speed longitudinal trajectory
$\mathbf{\tilde{V}_{n}(t)}$ based on $\mathbf{\tilde{V}_{n-1}(t)}$ and
$\mathbf{\tilde{R}_{n-1}(t)}$ with Algorithm 4;
16 else
17 Vehicle $n$’s future speed longitudinal trajectory estimate stays the same
since no information update
$\mathbf{\tilde{V}_{n}(t)}=\mathbf{\tilde{V}_{n}(t-1)}$;
18 end if
19
20 Vehicle $n$ cumulatively estimates its future longitudinal position
trajectory $\mathbf{\tilde{R}_{n}(t)}$ based on $\mathbf{\tilde{V}_{n}(t)}$
with Algorithm 3;
21 end for
22 Vehicle $j$’s estimated motion $\mathbf{\tilde{V}_{j}(t)}$ and
$\mathbf{\tilde{R}_{j}(t)}$ can be derived when $n==j$;
23 end if
24
25 Vehicle $j$ sends $\mathbf{\tilde{V}_{j}(t)}$ and
$\mathbf{\tilde{R}_{j}(t)}$ to its following vehicle $i$;
26 end while
27
28while _communication between vehicle $i$ and $j$ is currently off due to
packet loss_ do
29 At time step $(t+k)\subseteq[t+1,t+N]$, vehicle $i$ extras
$\tilde{v}_{j}(t+k)$ from $\mathbf{\tilde{V}_{j}(t)}$, and
$\tilde{r}_{j}(t+k)$ from $\mathbf{\tilde{R}_{j}(t)}$, and uses them as the
inputs of the motion control algorithm;
30 end while
31
32Vehicle $i$ disassociates with its target vehicle $j$;
Algorithm 1 Main function of the motion estimation methodology, with three
algorithms being called inside
Algorithm 2: This algorithm estimates vehicle $j$’s future longitudinal speed
trajectory when it does not have any target vehicle to follow. In this case,
vehicle $j$ will converge to its target speed $v_{j}(t\rightarrow\infty)$,
which is a known and preset value. Specifically, this algorithm is proposed
based upon the Intelligent Driver Model (IDM) [26] as follows:
$\tilde{v}_{j}(t+k)=\tilde{v}_{j}(t+k-1)+a_{j}^{\max}\cdot\left[1-\left(\frac{\tilde{v}_{j}(t+k-1)}{v_{j}(t\rightarrow\infty)}\right)^{\sigma}\right]\cdot\delta
t$ (8)
where $k\subseteq[1,N]$, and $\tilde{v}_{j}(t)=v_{j}(t)$; $a_{j}^{max}$ is a
preset constant denoting vehicle $j$’s maximum changing rate of longitudinal
speed, which is set as 0.73 m/s2 in this study; $\sigma$ is the free
acceleration exponent defined by IDM, which characterizes how the acceleration
of the vehicle decreases with speed, which is set as 4 in this study; $\delta
t$ is the duration of a prediction time step. Based on this algorithm, the
future longitudinal speed trajectory of vehicle $j$ can be given as
$\mathbf{\tilde{V}_{j}(t)}=\Big{(}\tilde{v}_{j}(t+1),\tilde{v}_{j}(t+2),...,\tilde{v}_{j}(t+k),...,\tilde{v}_{j}(t+N)\Big{)}$.
Algorithm 3: This algorithm estimates vehicle $n$’s future longitudinal
position trajectory $\mathbf{\tilde{R}_{n}(t)}$ based on its estimated speed
trajectory $\mathbf{\tilde{V}_{n}(t)}$, which is given as:
$\tilde{r}_{n}(t+k)=\tilde{r}_{n}(t+k-1)+\tilde{v}_{n}(t+k-1)\cdot\delta t$
(9)
where $k\subseteq[1,N]$, and $\tilde{r}_{n}(t)=r_{n}(t)$. Based on this
algorithm, the future longitudinal position trajectory of vehicle $n$ can be
given as
$\mathbf{\tilde{R}_{n}(t)}=(\tilde{r}_{n}(t+1),\tilde{r}_{n}(t+2),...,\tilde{r}_{n}(t+k),...,\tilde{r}_{n}(t+N))$.
Algorithm 4: This algorithm estimates vehicle $n$’s future longitudinal speed
trajectory when it has a target vehicle $(n-1)$ to follow, meanwhile also
considers the presence of communication delay $\tau_{n(n-1)}(t+k)$. When
$\tau_{n(n-1)}(t+k)<\delta t$, namely the communication delay is less than the
duration of a prediction time step, then target vehicle $(n-1)$’s longitudinal
speed is assumed unchanged during this delayed period:
$\tilde{v}_{n-1}(t+k)=\tilde{v}_{n-1}\Big{(}t+k-\tau_{n(n-1)}(t+k)\Big{)}$
(10)
When $\tau_{n(n-1)}(t+k)>=\delta t$, namely the communication delay is equal
to or longer than the duration of a prediction time step, then
$\begin{split}\tilde{v}_{n-1}(t+k)&=\tilde{v}_{n-1}\Big{(}t+k-\tau_{n(n-1)}(t+k)\Big{)}\\\
&+\dfrac{\tau_{n(n-1)}(t+k)}{\delta
t}\cdot\dot{\tilde{v}}_{n-1}\Big{(}t+k-\tau_{n(n-1)}(t+k)\Big{)}\end{split}$
(11)
In either case, target vehicle $(n-1)$’s longitudinal position can be adjusted
by:
$\tilde{r}_{n-1}(t+k)=\tilde{r}_{n-1}(t+k-1)+\tilde{v}_{n-1}(t+k)\cdot\tau_{n(n-1)}(t+k)$
(12)
Then, vehicle $(n-1)$’s future longitudinal motion at each time step is used
to estimate vehicle $n$’s future longitudinal speed as:
$\begin{split}\tilde{v}_{n}(t+k)&=\tilde{v}_{n}(t+k-1)-\alpha_{n(n-1)}k_{n(n-1)}\\\
&\cdot\Bigg{[}\bigg{(}\tilde{r}_{n}(t+k)-\tilde{r}_{n-1}\Big{(}t+k\Big{)}\\\
&+l_{n-1}+\tilde{v}_{n}(t+k)\cdot t_{n(n-1)}^{g}(t+k)\Big{)}\bigg{)}\\\
&+\gamma_{n}\cdot\bigg{(}\tilde{v}_{n}(t+k)-\tilde{v}_{n-1}(t+k\Big{)}\bigg{)}\Bigg{]}\end{split}$
(13)
Parameters of this algorithm are set according to the consensus-based motion
control algorithm equation (5), where $i=n$ and $j=n-1$. Based on this
algorithm, the future longitudinal speed trajectory of vehicle $n$ can be
given as
$\mathbf{\tilde{V}_{n}(t)}=\Big{(}\tilde{v}_{n}(t+1),\tilde{v}_{n}(t+2),...,\tilde{v}_{n}(t+k),...,\tilde{v}_{n}(t+N)\Big{)}$.
## 5 Case Study and Simulation Results
### 5.1 Automated Coordination at Unsignalized Intersections
Traffic signals has been playing a crucial role in achieving safer performance
at intersections. Researchers and practitioners have shown in their works
that, the appropriate installation and operation of traffic signals can reduce
the severity of crashes [27]. However, the addition of unnecessary or
inappropriately-designed signals have adverse effects on traffic safety and
mobility. In addition, the dual objectives of safety and mobility conflict in
many cases. The design and operation of traffic signals at intersections
requires choosing elements that may lead to trade-offs in safety and mobility.
In this paper, we conduct a case study in the automated coordination scenario
at unsignalized intersections, where different CAVs cooperate with each other
based on V2X communications and cross the intersection without any full stop
[9, 10]. As shown in Fig. 2, we adopt the “virtual vehicle” idea that projects
CAVs coming from four different legs to a virtual lane based on their
longitudinal positions (more precisely, their longitudinal distances to the
intersection crossing point). The consensus-based motion control algorithm (7)
can then be applied to CAVs to adjust their longitudinal positions and speeds
with respect to their target vehicles (either on the same leg or another leg),
and therefore form a vehicle string on the virtual lane while approaching to
the intersection. In this manner, all CAVs can collaboratively cross the
unsignalized intersection without any full stop.
Figure 2: The illustration of vehicle parameters in a vehicle-to-vehicle (V2V)
communication environment.
However, V2X communications among vehicles have limited performances, with
time-variant communication delay and packet loss. The proposed motion
estimation methodology will estimate target CAV’s motion when its information
is received with delay, or not received by the ego vehicle at all.
### 5.2 Game Engine Simulation Environment
During the past decade, the rapid development of game engines makes them a
popular option to model and simulate advanced vehicular technology [28]. Game
engines typically consist of three modules: 1) a rendering engine for 2-D or
3-D graphics, 2) a physics engine for collision detection and response, 3) and
a scene graph for the management of multiple elements (e.g., models, sound,
scripting, threading, etc.). They have been increasingly used by researchers
to simulate autonomous driving [29, 30, 31], prototype connected vehicle
systems [4, 32], study driver behaviors [33], and etc.
In this paper, we use Unity game engine to conduct the case study of automated
coordination at unsignalized intersections to evaluate our motion estimation
methodology. As shown in Fig. 3, a map is built based on the South of Market
(SoMa) district in San Francisco with 1:1 ratio [30]. Shown as yellow lines on
the road surface, centimeter-level routes along the 2nd Street, Harrison
Street, Folsom Street, Howard Street, and Mission Street are further modeled
in this case study, so CAVs can be located with relatively high precision. The
consensus-based motion control algorithm and the proposed motion estimation
methodology are applied to CAVs in this environment through Unity’s C# API.
Figure 3: The map with a four-intersection (each with four legs) corridor
built in Unity game engine based on the SoMa district in San Francisco
In this simulation, we set the time-variant communication delay based on the
results of our previous field test regarding 4G LTE-based V2C (not standard
C-V2X) communications [34]. Note that this communication technology normally
has higher communication delays than DSRC or C-V2X, so we adopt it in the
simulation as a stress test of our motion estimation methodology.
Specifically, we set the communication delay as a normal distribution with a
mean value of 40 ms and a standard deviation of 0.0259.
Additionally, we also model the packet loss in the simulation with a hybrid
model, which includes both random packet loss over the whole simulation
period, and certain packet loss during some specific time periods. The random
packet loss modeling part is to simulate the relatively periodic communication
issues, while the certain modeling part is to simulate the non-line-of-sight
(NLOS) scenario when the communication is obstructed by physical objects
(e.g., tall buildings or bridges). The packet loss is modelled and simulated
in a more frequent manner than it is supposed to be in the real world, so we
can conduct stress test of our motion estimation methodology, similar to
communication delay.
### 5.3 Simulation Results and Evaluation
The simulation is conducted on a Windows desktop (processor Intel Core i7-9750
@2.60 GHz, 32.0 GB memory, NVIDIA Quadro RTX 5000 Max-Q graphics card) and
Unity version 2019.2.11f1. A snapshot of the simulation is shown as Fig. 4,
where CAVs are randomly generated from three legs of this 2nd-Harrison
intersection (since Harrison St is a one-way street), and they automatically
coordinate with each other to cross this unsignalized intersection without any
collision or full stop. Besides this intersection, CAVs are also randomly
generated from different legs of other three intersections along 2nd St (as
illustrated in Fig. 3), so the ego vehicle on 2nd St can travel through all
four unsignalized intersections in a row by automated coordination with other
CAVs.
Figure 4: Unity simulation of automated coordination at an unsignalized
intersection, where CAVs are implemented with the proposed motion estimation
methodology to deal with communication delay and packet loss
Multiple simulation trips are conducted under various prediction time steps of
the proposed motion estimation methodology, investigating the impacts on the
estimation error and computational load. All CAVs in this simulation are
implemented with a first-come-first-served motion planning algorithm to
generate the crossing sequence, a consensus-based motion control algorithm
shown earlier in this paper to generate the reference speed, and the proposed
motion estimation methodology to deal with communication delay and packet
loss.
Figure 5: Longitudinal position trajectory of vehicles crossing the first
intersection (2nd-Harrison intersection), where the ego vehicle (dark red
dashed curve) has communication delay and packet loss. Figure 6: Error of the
ego vehicle’s position estimation with different prediction time step values
compared to the ground truth. Figure 7: Maximum error of the ego vehicle’s
position estimation with different prediction time step values. Figure 8:
Minimum update frequency with different prediction time step values.
As can be seen from the results, Fig. 5 shows the ground-truth longitudinal
position trajectory of CAVs crossing 2nd-Harrison intersection, which is the
first of four consecutive intersections in the simulation network. The ego
vehicle is applied with the time-variant communication delay and the hybrid
packet loss introduced in the previous subsection. Specifically, two major
periods of packet loss happen during 4-6 second and 6-8 second, respectively,
where NPC vehicle 4 (which considers the ego vehicle as its target vehicle)
gets a little bit close to the ego vehicle due to the estimation error (shown
in Fig. 6). However, this does not cause any rear-end collisions in the
simulation, since these two vehicles do not travel on the same leg of this
intersection.
Besides aforementioned two major periods of packet loss during 4-8 second,
estimation error can also be observed during the whole period when the ego
vehicle crosses the first intersection, as shown in Fig. 6. These errors are
generated by the combination of periodic packet loss and time-variant
communication delay, and are shown to be capped by $\pm 0.5$ m with the help
of the proposed motion estimation methodology.
Additionally, the impacts of different values of the prediction time step
$\delta t$ are also investigated. Shown in Fig. 6 and Fig. 7, when the
prediction happens more frequently, the maximum position estimation error is
decreased. Particularly, when the prediction time step is 0.01 s (i.e.,
prediction frequency is 100 Hz), the estimation error is always less than 0.2
m based on the simulation setting of communication delay and packet loss.
However, when the prediction time step is 1 s (i.e., prediction frequency is 1
Hz), the estimation error can reach 5.8 m during the major packet loss. This
is caused by the difference of information granularity, where a more frequent
prediction also means a more frequent information update, which could happen
immediately after the packet resumes to be received.
However, a more frequent prediction also leads to a higher computational load.
Since this simulation is conducted in Unity game engine, the computational
load is measured by the minimum update frame rate of the simulation. As shown
in Fig. 8, When the prediction time step is 0.01 s, the frame rate drops to a
minimum of 6 FPS, indicating the simulation can be run only six frames per
second at that time instant (with the simulation hardware setting). However,
when the prediction time step is 1 s, the frame rate is always higher than 19
FPS. This is caused by the difference of iterations that the proposed motion
estimation methodology needs to be called. However, since this simulation is
conducted on a single computer for multiple vehicles as a centralized fashion,
the concern of computational load can be further relieved if distributed
computing can be adopted by each individual vehicle in the future.
## 6 Conclusion
In this paper, we have investigated the issues of communication delay and
packet loss in V2X communications among CAVs. A motion estimation methodology
has been developed based on the consensus-based feedforward/feedback motion
control algorithm, which handles aforementioned two communication issues at
the same time. A case study of automated coordination at unsignalized
intersection has been conducted, where CAVs coordinate with each other through
V2X communications to cross the intersection without any full stop. The
simulation study in Unity game engine has shown that, the proposed motion
estimation methodology (with 0.01 s prediction time step) can cap the position
estimation error at $\pm 0.5$ m with the presence of periodic packet loss and
time-variant communication delay. Additionally, different impacts of the
prediction time steps on the estimation error and computational load have also
been studied in the simulation.
One major future step of this work is to implement the proposed motion
estimation methodology on real CAVs, and conduct real-world field
implementations. Real vehicle dynamics, instead of the simplified version
adopted in this paper, needs to be considered to improve the fidelity of the
algorithm. Additionally, data measured by CAVs’ on-board perception sensors
can also be leveraged to provide additionally sources of motion estimation,
which could significantly decreases the motion estimation error of the
proposed methodology.
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## 7 Contact Information
Ziran Wang, Ph.D.
Research Scientist
Toyota Motor North America R&D, InfoTech Labs
Work Address: 465 N Bernardo Ave, Mountain View, CA 94043
Work Phone: (650) 439-9524
Work Email<EMAIL_ADDRESS>
## 8 Acknowledgments
This work is conducted under the “Digital Twin” project at Toyota Motor North
America R&D. The authors would like to thank the feedback and guidance from
Dr. Nejib Ammar and Dr. Bin Cheng to improve this work.
The contents of this paper only reflect the views of the authors, who are
responsible for the facts and the accuracy of the data presented herein. The
contents do not necessarily reflect the official views of Toyota Motor North
America R&D, InfoTech Labs.
## 9 Definitions, Acronyms, Abbreviations
CAV | Connected and Automated Vehicle
---|---
DSRC | Dedicated Short Range Communications
C-V2X | Cellular Vehicle-to-Everything
V2V | Vehicle-to-Vehicle
V2I | Vehicle-to-Infrastructure
V2N | Vehicle-to-Network
V2C | Vehicle-to-Cloud
V2P | Vehicle-to-Pedestrian
ACC | Adaptive Cruise Control
CACC | Cooperative Adaptive Cruise Control
NLOS | Non-Line-of-Sight
FPS | Frames Per Second
|
# Cosmic Ray Transport in Mixed Magnetic Fields and their role on the Observed
Anisotropies
Margot Fitz Axen,1,3 Julia Speicher,2 Aimee Hungerford3 and Chris L. Fryer3
1Department of Astronomy, The University of Texas at Austin, 2515 Speedway,
Stop C1400, Austin, Texas 78712-1205, USA
2American Astronomical Society, 2000 Florida Ave., NW, Suite 300, Washington,
DC 20009-1231, USA
3Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los
Alamos, NM 87544, USA E-mail<EMAIL_ADDRESS>
(Accepted 2020 November 5. Received 2020 October 29; in original form 2020 May
13)
###### Abstract
There is a growing set of observational data demonstrating that cosmic rays
exhibit small-scale anisotropies (5-30∘) with amplitude deviations lying
between 0.01-0.1% that of the average cosmic ray flux. A broad range of models
have been proposed to explain these anisotropies ranging from finite-scale
magnetic field structures to dark matter annihilation. The standard diffusion
transport methods used in cosmic ray propagation do not capture the transport
physics in a medium with finite-scale or coherent magnetic field structures.
Here we present a Monte Carlo transport method, applying it to a series of
finite-scale magnetic field structures to determine the requirements of such
fields in explaining the observed cosmic-ray, small-scale anisotropies.
###### keywords:
astroparticle physics — ISM: magnetic fields — ISM: cosmic rays
††pubyear: 2020††pagerange: Cosmic Ray Transport in Mixed Magnetic Fields and
their role on the Observed Anisotropies–Cosmic Ray Transport in Mixed Magnetic
Fields and their role on the Observed Anisotropies
## 1 Introduction
Observations of cosmic rays in the TeV-PeV energy range have demonstrated both
small- and large-scale anisotropies. The large ($>60^{\circ}$) and small-scale
anisotropies have been observed by a large number of instruments (Amenomori et
al., 2005, 2006; Guillian et al., 2007; Abdo et al., 2008; Aglietta et al.,
2009; Abdo et al., 2009; Amenomori et al., 2010; Munakata et al., 2010; Abbasi
et al., 2010; Cui, 2011; Abbasi et al., 2011; Aartsen et al., 2013). The
large-scale dipole anisotropy is reasonably well fit by the asymmetries in
nearby sources smoothed by the subsequent diffusion of these cosmic rays
(Erlykin & Wolfendale, 2006; Blasi & Amato, 2012; Pohl & Eichler, 2013;
Sveshnikova et al., 2013). It is possible that magnetic fields in the
heliosphere could affect the anisotropy (Desiati & Lazarian, 2013; Schwadron
et al., 2014).
The small-scale anisotropies are much more difficult to explain. These small-
scale anisotropies typically have size scales between $5-30^{\circ}$ and
amplitudes between 0.01-0.1% of the background cosmic ray flux (Blasi & Amato,
2012). A number of models have been proposed to explain such anisotropies. For
example, anisotropies could arise from a nearby, as yet undetected, supernova
remnant (Salvati & Sacco, 2008), perhaps mediated by a local, coherent
magnetic field or asymmetry in the propagation (Drury & Aharonian, 2008;
Malkov et al., 2010; Biermann et al., 2013). Another set of proposals argue
that properties of the heliosphere can drive the observed anisotropies (Drury
& Aharonian, 2008; Lazarian & Desiati, 2010; Desiati & Lazarian, 2013). More
exotic models have also been proposed invoking strangelet production or dark
matter annihilation (Harding, 2013; Kotera et al., 2013). More recently, a
series of models have proposed that turbulent magnetic fields are sufficient
to explain the anisotropies (Giacinti & Sigl, 2012; Ahlers, 2014).
Magnetic fields generated in turbulence are believed to exist on all scales,
from the smallest scales in the Kolmogorov spectrum to the largest eddy
scales(Kulsrud & Zweibel, 2008; Saveliev et al., 2012; Krasheninnikov, 2013;
Kritsuk et al., 2017; Kim et al., 2020). In the Milky Way, this leads to
magnetic field structures ranging from well below the parsec scale up to a
kiloparsec(Kulsrud & Zweibel, 2008; Saveliev et al., 2012; Krasheninnikov,
2013). If the scale of the magnetic fields were limited to the smallest
turbulence scales, particle transport within these fields could be treated in
the diffusion limit where the magnetic fields are treated as a scattering term
with a net energy loss as is done in codes like GalProp (Strong et al., 2007).
This treatment does not accurately model any possible larger scale, coherent
structures in the magnetic field.
However, transport methods that can model both small and large scale fields
face numerical challenges. For instance, the method described in Fryer et al.
(2007) and Harding et al. (2016) allows for transport in one of two extreme
solutions for each spatial zone: either dominated by small scales (isotropic
scattering limit) or dominated by coherent fields. In this paper, we
generalize the Monte Carlo method from Fryer et al. (2007) and Harding et al.
(2016) to allow for more general magnetic field profiles that include both
small- and large-scale features. As in Harding et al. (2016), we focus on a
magnetic field configuration that studies the interaction of a single point
source for cosmic ray production with a coherent magnetic field of varying
strength relative to the small-scale field (sufficiently small with respect to
the spatial scale that the interaction can be treated as isotropic). With this
study, we hope to determine the magnetic field properties needed to explain
the observed anisotropies in the cosmic ray flux more realistically.
In section 2, we describe the properties of our grid, particles and magnetic
field configurations. In section 3, we describe the methods we use for
particle propagation and their differences from previous work. In section 4 we
describe the verification tests we used on the code. In section 5, we study
the propagation of cosmic rays in a box to apply this code to a cosmic-ray
transport application. In section 6, we study the observational implications
of these transport calculations. Finally, in section 7, we conclude with a
discussion of the properties needed to produce cosmic ray anisotropies.
## 2 Code Description and Initial Conditions
### 2.1 Grid Geometry and Particle Properties
The setup of our grid is similar to that used by Harding et al. (2016). We
define a three dimensional grid of cubic zones, with physical properties that
are constant within each zone. The grid we use is a 150x150x150 pc grid,
divided into 50 zones of 3x3x3 pc. Though the zone size is arbitrary, it is
set to be much larger than the cosmic ray scattering length and Larmor radius
of particle motion. We define the tally plane as being the upper x face of the
grid, at x=150 pc. The particles are propagated from a chosen starting
location through this grid until they hit one of the grid faces or until they
are absorbed into the ISM.
The current code physics is currently limited to the propagation of protons,
which are expected to be the dominant form of cosmic rays that create the
observed anisotropies in the TeV-PeV energy range (Harding et al., 2016). We
assume a point source for cosmic rays placed in the center of the simulation
grid, emitting particles in random directions according to an isotropic
distribution. All particles are assumed to stay traveling at relativistic
velocities (v $\approx$ c). We tested primarily particles with energy of 10
TeV, which is at the lower end of the energies observed for cosmic ray
anisotropies. Additionally, we did one study in which we varied the cosmic ray
energy to higher energy values, up to 100 PeV. Note that the proton mean free
path is what sets the fundamental scale of the computed solutions and there is
a straightforward relationship between proton energy and proton mean free path
shown in equation (6). Variation of the assumed proton energy has been used as
our primary means to study the impact of scaling the proton mean free path. In
principle, these particles lose energy due to Coulomb scattering, ionization,
and other processes, as was studied in Harding et al. (2016). However, for the
size-scale and conditions in their, and our simulation grid, they found that
energy losses of protons were minimal (~1-10 MeV over the course of their
entire propagation) and, for the calculations in this paper, we do not include
energy losses.
### 2.2 Magnetic Fields
The code is designed so that the properties of the magnetic field can be
varied in each zone. This includes the small-scale magnetic field amplitude
and the coherent magnetic field amplitude and direction. For the calculations
in this paper, we focus on a bimodal distribution of field structures: small-
scale fields with size-scales well below the zone size, and coherent magnetic
fields that are equal to or greater than the zone size. In the extreme case of
no coherent magnetic field, we can transport using a statistically sampled
isotropic magnetic field direction equivalent to a symmetric diffusion
coefficient. Focusing on a single coherent field allows us to test the ability
of the code to model small- and large-scale magnetic field structures combined
in a single zone and to study in detail the effects of these global structures
on the transport. This setup is a considerable improvement over the grid
design used by Harding et al. (2016), whose setup only allowed for either a
turbulent field or a coherent field in each zone of the grid.
For our simulations, we assume that the entire grid is filled with a small-
scale magnetic field component of constant amplitude $B_{\text{t}}=3\mu$G.
This field is assumed to be produced in the smallest turbulence scales and is
much smaller than our simulation grid-scale. With our grid size-scales, a
typical cosmic ray undergoes many pitch-angle scatterings as it transports
through the grid. The time for energy evolution is much longer than the
scattering timescale for our high-energy protons (Schlickeiser & Miller,
1998). As discussed above, this long energy-evolution timescale and the
relatively small simulation grid-size means that, although the proton
undergoes many "scatterings", its energy does not vary considerably as the
particle transports through the grid (Harding et al., 2016). Under these
conditions, we can mimic the cosmic ray interaction with the small-scale
fields as isotropic and assume the energy is constant during this period.
Therefore, we can model the turbulent magnetic field through a random vector
sampled at every particle step through the simulation:
$\text{cos}(\theta)=2\chi_{1}-1,$ (1) $\phi=2\pi\chi_{2},$ (2)
where $\chi_{i}$ are standard deviates between 0 and 1. Converting these
expressions to Cartesian coordinates gives a three-component, random vector.
In addition to this turbulent field, we define six grid zone boundaries that
bound a region that includes a coherent magnetic field component. This single
coherent field is only speculative but is meant to approximate some of the
features of the broad range of magnetic field scales that may exist in the
ISM. Inside this region, the coherent field component has a constant magnitude
and direction given by $B_{\text{g}}(x,y,z)=(B_{\text{g}},0,0)$. Clearly,
modeling the coherent magnetic field in this way is an oversimplification of
the net effects large scale magnetic field structures in the ISM would have,
but it allows us to test the importance of these possible large-scale coherent
fields. We study a variety of different configurations for the box region
containing the coherent component to the magnetic field. We have chosen to
keep the y and z coordinates of the box zone numbers constant at 108 pc $<$
y,z $<$ 120 pc. The geometry of the coherent field region is varied using the
x zone value of the box, for which we vary the offset from the tally plane and
the extent of the box. We tested box extents of 6 and 24 pc and varied the box
offset from the tally plane from 0 to 18 pc. This is in contrast to Harding et
al. (2016), whose coherent magnetic field region spanned the entire x length
of the simulation space. A comparison of this is shown in Figure 1.
The amplitude of the coherent magnetic field is varied as a fraction of the
maximum amplitude of the small scale field. The exact values for the three
parameters studied are shown in Table 1.
Figure 1: Comparison of a few of our simulation setups to that used by Harding et al. (2016). The source of cosmic rays is placed in the center of our simulation grid, shown by the red star. Along a rectangular path, shown by the blue box, we place a coherent magnetic field on top of the small-scale field. The amplitude of this coherent magnetic field is varied as a fraction of the amplitude of the small-scale field. We vary the length of the coherent magnetic field and its offset from the tally plane, shown by the green region. The top left plot shows Harding’s setup, running the length of the simulation space, while the other three show some variations of the box offset and extent which we use. Table 1: Shows the full set of test configurations we chose. Altogether there were 42 different coherent magnetic field configurations. Variable | Tested Values
---|---
Extent [pc] | 6, 24
Offset [pc] | 0, 3, 6, 9, 12, 15, 18
$B_{\text{g}}$/$B_{\text{t}}$ | 0.1, 0.5, 1.0
## 3 Particle Propagation
In this section, we describe the propagation of particles through our code.
The methods we use are similar to those used in Harding et al. (2016). At the
beginning of the particle’s lifetime, its starting location is determined by
the input source location and its initial direction is sampled randomly from
an isotropic distribution. It is then propagated through the grid using one of
two methods, depending on whether it is in the coherent magnetic field region
or not.
In Harding et al. (2016), the methods used included a direct transport Monte
Carlo method if the particle was in the coherent magnetic field region and, if
not, a Discrete Diffusion Monte Carlo (DDMC) method (Evans et al., 2003).
Given the simplicity of the magnetic field configuration, such an approach is
reasonable. This DDMC approach reproduces the results of simple diffusion
transport schemes such as GalProp Strong et al. (2007). While we were able to
use a DDMC algorithm very similar to that used by Harding et al. (2016) for
the majority of our grid, the direct transport method did not allow for the
magnetic field configurations that include the isotropic, turbulent field
component in addition to the coherent field component inside the box.
To study these hybrid regions, we modified the transport method used by
Harding et al. (2016) to account for particle propagation inside the coherent
field region. We tested two different methods for this hybrid
transport/diffusion Monte Carlo algorithm. Outside the coherent field region,
the particles were propagated using a DDMC method similar to that used by
Harding et al. (2016). In 3.1 we first discuss the direct transport Monte
Carlo scheme which was used in Harding et al. (2016) and which is the basis
for our hybrid transport method. In 3.2 we discuss the Monte Carlo diffusion
approximation which we employ for most of the grid. Finally in 3.3 we discuss
the two methods we use for our coherent field region in order to account for
both components to the magnetic field.
### 3.1 Particle Transport Monte Carlo Methods
A particle transport Monte Carlo code tracks the particle motion as the
particle goes through a zone and passes into the next. Particles within a
magnetic field of strength $B$ propagate according to the equation of motion:
$\frac{dv}{dt}=\frac{c}{r_{\text{L}}}v\times\hat{B},$ (3)
where $\hat{B}$ is the direction of the net magnetic field, $v$ is the
direction of the particles velocity, and $r_{\text{L}}$ is the Larmor radius
of the particle motion. This causes particles to move in a circular path of
radius $r_{\text{L}}$ around the magnetic field line, perpendicular to it. The
component of the particle’s initial velocity parallel to the magnetic field
line does not change. Particles within a magnetic field of strength $B$,
energy $E$, and charge $Ze$ have a Larmor radius of (Schlickeiser, 2002):
$r_{\text{L}}=1.1\times
10^{-3}Z^{-1}\left(\frac{E}{\text{TeV}}\right)\left(\frac{B}{\mu\text{G}}\right)^{-1}\text{pc}.$
(4)
Solving equation (3) is computationally difficult due to the number of
directional changes the particles make spiraling around the magnetic field
lines. Therefore, the approach that Harding et al. (2016) and many others take
is to approximate the particle motion as simply following the field line it
experiences. Using this approximation, one step through the simulation is a
step over which the amplitude and direction of the total magnetic field is
constant. Effectively, the particle’s velocity vector is either parallel or
antiparallel to the magnetic field line, depending on its initial direction in
approaching it. The particle’s position can then be reset as:
$x(t)=x_{0}+\frac{(v_{0}\cdot B)}{B}\hat{B}t,$ (5)
where $x_{0}$ is the particle’s previous position, $v_{0}$ is the initial
direction of the particle’s velocity when approaching the field line,
$\hat{B}$ is the magnetic field direction, and $t$ is the time the particle
took for the step. The full path length traversed by the particle is $ct$.
Equation (5) is assumed to be a valid approximation if the Larmor radius is on
par with or smaller than the path length of the particle, under the assumption
that solving equation (3) directly would be unlikely to move the particle into
a region with a different magnetic field. We note that it is possible that
solving equation (3) explicitly can alter the particle motion, and should be
studied in future work.
The time $t$ that a particle is expected to take for one step is dependant on
the mean free path of the particle motion. By describing the turbulent
magnetic field interaction as a scattering term, we can alse describe the
distance to scattering interaction. The scattering mean free path describes
how far the particle is expected to travel before encountering a change in the
turbulent component of the magnetic field, and is determined by the particle
energy and the strength of the magnetic field. It is given by (Schlickeiser &
Miller, 1998; Fryer et al., 2007)
$\lambda_{\text{sc}}=2\times
10^{7}\left(\frac{\lambda_{\text{max}}}{1\text{cm}}\right)^{1/2}\left(\frac{B}{0.1\text{mG}}\right)^{-1/2}\left(\frac{E}{10\text{TeV}}\right)^{1/2}\text{cm},$
(6)
where $\lambda_{\text{max}}$ is the scale length of the turbulent magnetic
field, B is the amplitude of the total magnetic field (turbulent and coherent)
and E is the proton energy. We use $\lambda_{\text{max}}=10^{15}$ cm, as was
done in Harding et al. (2016). For the cosmic ray energies we consider,
magnetic field scattering is the only significant particle interaction, and
other interactions such as proton-proton scattering are negligible. Therefore,
we take the total mean free path to be equal to the scattering mean free path:
$\lambda=\lambda_{\text{sc}}$. Particles with a mean free path $\lambda$
follow an exponential probability distribution for their motion,
$P(x)=e^{-x/\lambda}$, where x is the distance traveled in one step. Solving
this distribution for its cumulative distribution function (CDF) gives the
expression $\chi=1-e^{-x/\lambda}$, where $\chi$ is a random variable sampled
between 0 and 1. This can be rearranged to obtain a distance traveled for that
step, along with the time it took for the particle to go that distance:
$t=x/c=-\text{ln}(\chi)\left(\frac{\lambda}{c}\right).$ (7)
### 3.2 Discrete Diffusion Monte Carlo
In regions with only a turbulent magnetic field, we can treat the transport in
the diffusion approximation and we use a DDMC method which is very similar to
that used by Harding et al. (2016). In these regions, particle motion changes
direction too quickly for it to be computationally feasible to track the
particle motion directly. However, particles in an isotropic magnetic field
exhibit “random walk" motion, with their displacement proportional to the
square root of their travel time. DDMC methods combine a number of smaller
random walk steps that the particle would take into a larger step based on
this principle. Rather than picking a distance to collision using equation
(7), we instead pick a travel time and then sample a particle distance
traveled in that time.
For a three dimensional random walk, the probability of moving distance $R$
after $N$ steps is (Rycroft & Bazant, 2005; Harding et al., 2016)
$P(R)=4\pi R^{2}\left(\frac{3}{2\pi
Na^{2}}\right)^{3/2}\text{exp}\left(\frac{-3R^{2}}{2Na^{2}}\right),$ (8)
where $a$ is the expectation value for displacement in a single step. To
approximate the transport motion, we relate the number of steps, $N$, to the
total distance traveled by the particle and the average step size:
$N=ct/\lambda$. The expected distance traveled in one step $a=\sqrt{\langle
x^{2}\rangle-\langle x\rangle^{2}}=\sqrt{\langle x^{2}\rangle}$ is calculated
from the transport equation probability function to be $a=\sqrt{2}\lambda$.
Putting in these values gives the expression:
$P(R)=4\pi R^{2}\left(\frac{3}{4\pi
ct\lambda}\right)^{3/2}\text{exp}\left(\frac{-3R^{2}}{4ct\lambda}\right).$ (9)
This expression does not lend itself to sampling directly via an inversion
technique, so we instead cast it in a hybrid form using both inversion
sampling and rejection sampling techniques. Specifically, we break the full
probability distribution $P(R)$ into two components, $g(R)$ and $h(R)$.
Defining the constant $C=3/(4ct\lambda)$, it can be seen that:
$P(R)\propto Re^{\frac{-2C}{3}R^{2}}Re^{\frac{-C}{3}R^{2}}=g(R)h(R).$ (10)
The function g(R) is the 2D random walk probability distribution, while h(R)
has the correction for the full 3D solution (Rycroft & Bazant, 2005).
To sample the complete function, we first sample a value $R_{\text{samp}}$ via
inversion of the CDF for $g(R)$, and then choose to accept $R_{\text{samp}}$
based on a rejection sample of the function $h(R)$. The efficiency of the
rejection step is 80$\%$, and does not dramatically increase the computational
cost of each run. However, for algorithm simplicity, most of our runs employ
an approximate sampling technique where the rejection step is removed. To
justify this, we compared select test cases using the complete function and
discovered that, within our error range, there was no noticeable difference in
results when the rejection step was removed. This is largely because $g(R)$ is
a close approximation to $P(R)$ in terms of mean diffusion distance
properties. The primary effect of removing the rejection check was to slightly
decrease the diffusive component’s mean free path. This results in a slight
enhancement of the influence of the diffusive field component relative to the
coherent field component. As such, using the 2D approximation to the full 3D
solution provides a conservative estimate of the influence of coherent
magnetic fields on the cosmic ray flux. These results can be seen intuitively
from Figure 2, which shows the three functions $P(R)$, $g(R)$, and $h(R)$ for
C=1. Plotted over $P(R)$ and $g(R)$ in the red points is the average value of
$R$ sampled for the two distributions. The function $g(R)$ has a lower peak
and is skewed farther towards lower distances than $P(R)$, leading to a
slightly lower average value of $R$ sampled.
Figure 2: Shows the normalized probability density functions using C=1 for the
full 3D random walk distribution $P(R)$ (Equation 9), the 2D approximation we
use $g(R)$ (Equation 11), and the 3D correction $h(R)$. The red points show
the average value of R sampled for $P(R)$ and $g(R)$.
With this approximation, the employed probability distribution function is
given by:
$P(R)=Re^{\frac{-2C}{3}R^{2}},$ (11)
which yields a sampled distance given by:
$R=\sqrt{-\text{ln}(\chi)2ct\lambda}.$ (12)
The values we use for $t$ are $t=1000$ years and $t=2$ years, depending on the
particle’s situation. We discuss this more in Section 4.
If the particle’s travel takes it to the edge of a zone wall before the end of
its step, it stops and checks whether it is going into the region with a
coherent magnetic field. If it is, it goes to the edge of the zone and exits
the diffusion loop, with a travel time of $x/c$. Otherwise, it continues its
journey along the path until it either hits another zone wall or finishes its
timestep. Therefore all particles will keep going until they either reach the
end of the sampled distance or hit the edge of a zone in which the transport
mechanism is necessary. For each substep along its sampled distance that it
hits a zone wall, the particle’s remaining time along its current path is
recalculated as the difference between its originally sampled time and its
substep time along that path. Once the particle reaches the end of its
timestep, its direction is resampled.
Note that, although equations (1) and (2) are used in both a transport method
and DDMC method to approximate the turbulent magnetic field, they have a
different meaning for the two. Because the transport algorithm approximates
the particle motion directly, the randomly sampled direction represents the
direction of the turbulent magnetic field influence for that particle step. In
the diffusion algorithm, this vector represents a direction of particle travel
over many steps, through many different magnetic field lines.
### 3.3 Modified Transport Monte Carlo: Two Methods
For the region inside the coherent magnetic field, there is no clear analytic
solution to describe the particle motion, arising from the fact that we can
neither assume the diffusion approximation or assume the particle flows along
field lines. Instead, the solution lies somewhere between these two extremes.
For the particle transport Monte Carlo method, the magnetic field direction is
sampled, and the next direction of particle motion is based on the particle’s
initial velocity direction with respect to this vector. In contrast, for the
diffusion approximation, the particles direction change after many steps is
sampled; hence the particle’s previous direction of travel doesn’t matter.
Therefore, for the region containing the coherent magnetic field, we explored
two methods for modeling the particle motion which attempt to “combine" the
transport and diffusive methods in different ways. Both of these methods give
solutions for particle transport that are somewhere between the isotropic
magnetic field solution and the transport solution.
The first method we explored (“Method 1") forces particles to follow either
transport motion or diffusive motion inside the coherent magnetic field box.
For every step, we sample a random number $r$ between 0 and 1 and compare that
to the coherent field amplitude ratio
($g=B_{\text{g}}/(B_{\text{g}}+B_{\text{t}})$). If $r<g$, then the particle
uses transport motion and follows the coherent magnetic field with one of two
directions, based on the velocity of its previous step. If $r>g$, then the
travel direction is sampled randomly. Thus, for example, if the amplitudes of
the coherent and turbulent field are the same
($B_{\text{g}}/B_{\text{t}}=1.0$), then $g=0.5$, and a particle that travels
inside the box will follow transport motion about half of the time and random
walk motion the other half.
The second method we explored (“Method 2") uses a direction of travel which is
the vector sum of the coherent magnetic field vector and the turbulent
magnetic field vector. The turbulent magnetic field vector is randomly
sampled. The coherent field vector, however, in this case, represents the
direction the field line would give the particle rather than the field line
direction itself. It is one of two directions; either parallel to the field or
antiparallel to it, based on the particle’s previous direction of travel.
A consequence of using a timestep $t=1000$ years outside the coherent magnetic
field box is that it was inadequate to simply use the transport solution
inside the coherent field box and the diffusion approximation outside. The
boundary conditions of the transport solution require that when particles
leave the transport box they do not resample their direction; which means they
are biased to move away from the box. The distance which particles have to
travel to recover random walk motion can be relatively large in our case,
since with a timestep $t=1000$ years the average distance a particle travels
in one diffusion step is $R(t)=0.78$ pc. This causes a deficit of particles
inside the coherent magnetic field region.
In order to rectify this situation, we made two modifications to stop the
particles that entered the coherent field region from quickly propagating away
from it. First, we put a “sheath" of one grid zone around the coherent
magnetic field in which the particles followed the transport solution, if they
had just exited the field region. Second, once the particles had exited the
sheath region into the diffusion regime, they used a timestep of $t=2$ years
rather than $t=1000$ years. These modifications allowed us to get the correct
precision for particles that entered the field region while running the
particles that never entered the field region quickly.
## 4 Code Verification
### 4.1 Isotropic Magnetic Field Travel Time
One testable property of an isotropic magnetic field is that the particle
motion follows a certain distance distribution. Because the mean free path of
the particle motion is the same everywhere, the particles should always follow
a root-mean square (rms) distance distribution for their travel time following
equation (12) of $R(t)=\sqrt{2ct\lambda}$. To confirm our code reproduces this
property, we extended the simulation box to calculate a broad particle
distribution.
In order to make this simulation run faster with a larger transport box, we
set the mean free path in every grid zone to $\lambda=0.01\text{pc}$, which is
approximately the mean free path of a 100 TeV- 1 PeV proton in a 3$\mu$G
magnetic field (our run simulations, with 10 TeV protons, have a mean free
path of approximately $\lambda=0.001\text{pc}$. We specified a “box region"
given by 42 pc $<$ y,z $<$ 102 pc and 90 pc $<$ x $<$ 150 pc, in which the
packets must follow the transport solution; just as would be done if there was
a coherent magnetic field in this region. With this setup, about 65 percent of
the total number of particles run entered this region at some point during
their lifetime, and they spent on average 12 percent of their lifetime in the
box region.
In total, we did 100 runs of the code, each with 10000 particles. Each
particle’s position was recorded every 20000 years. Figure 3 shows the
difference between the rms distance traversed for the 10000 particles of a run
at each time and the analytic solution at that time. Each point on the plot
represents a separate run. The line represents the average of these 100 points
at each time. This test was done for both just DDMC and the IMC/DDMC hybrid.
As can be seen, the points for the IMC/DDMC hybrid method does not reproduce
the exact solution (averaging above 0). However, the average is below 1%, and
does not show a strongly increasing trend with time. The distribution of the
points never exceeds 4 percent. For this paper, this error lies below the
numerical uncertainty in our Monte Carlo statistics (Section 4.2) and the
errors from our methods are always less than these statistical errors.
Figure 3: Shows, for the isotropic magnetic field verification, the rms
distance errors from the analytic solution calculated at each time of the 100
runs done. The plot on the left uses just the DDMC algorithm, while the plot
on the right uses the hybrid DDMC/transport algorithm. The line shows the
average of these points at each time. The average error never reaches 1
percent and tends to level out around 80000 years. The distribution of points
at every time never spans more than 4 percent.
### 4.2 Statistical Uncertainties
The goal of this paper is to study the effect of a coherent magnetic field on
the particle flux at the tally plane. Although it alters the flux across this
entire boundary, the primary affect is at the position of the coherent
magnetic field box (in the space 108 pc $<$ y,z $<$ 120 pc). With a finite set
of particles, statistical uncertainties often dominate the errors in a Monte
Carlo approach. To test this, we first ran the particles through a purely
isotropic magnetic field, using the diffusion algorithm alone. We then defined
the particle flux $F$ for a magnetic field model $i$ as being the ratio
between the number of particles that hit this region $N_{{\text{hit, i}}}$ to
the number that hit this region in the isotropic run $N_{{\text{hit, back}}}$,
relative to the total numbers of particles $N_{\text{tot, i}}$ and
$N_{\text{tot, back}}$ run for both:
$F=\frac{\frac{N_{{\text{hit, i}}}}{N_{\text{tot, i}}}}{\frac{N_{{\text{hit,
back}}}}{N_{\text{tot, back}}}}=\frac{N_{\text{tot, back}}}{N_{\text{tot,
i}}}\frac{N_{{\text{hit, i}}}}{N_{{\text{hit, back}}}},$ (13)
Statistical errors in Monte Carlo methods cause different runs to give
slightly different numbers for this quantity. However, a set of runs with the
same magnetic field configuration should form roughly a Gaussian distribution
for the computed value of $F$, with 95$\%$ of the values within $2\sigma$ of
the mean. Because for every box configuration we only wanted to do one run, we
had to relate the $1\sigma$ value $\epsilon_{\text{F}}$ of this distribution
to an uncertainty measure in a single run based on the number of particle
counts.
For counting statistics, the uncertainty in a counted number of particles is
equal to the square root of the number of particles for the statistic. We can
therefore compute upper and lower uncertainty bounds for the computed flux
above the background using counts for the number of particles that exit the
box region and counts that exit the box region for the background run:
$F_{\text{up}}=\frac{N_{\text{tot, back}}}{N_{\text{tot,
i}}}\frac{N_{{\text{hit, i}}}+\sqrt{N_{{\text{hit, i}}}}}{N_{{\text{hit,
back}}}-\sqrt{N_{{\text{hit, back}}}}},$ (14)
$F_{\text{low}}=\frac{N_{\text{tot, back}}}{N_{\text{tot,
i}}}\frac{N_{{\text{hit, i}}}-\sqrt{N_{{\text{hit, i}}}}}{N_{{\text{hit,
back}}}+\sqrt{N_{{\text{hit, back}}}}}.$ (15)
The difference between these two quantities $\Delta F$ gives another measure
for the $1\sigma$ uncertainty interval around the computed value for the flux
above the background: $F\pm 1\sigma=F\pm\Delta
F/2=F\pm(F_{\text{up}}-F_{\text{low}})/2$. In order to see whether $\Delta
F/2$ computed for one run was approximately equal to what the value of
$\epsilon_{\text{F}}$ would be for the distribution, we set up a test
configuration used in the full set of runs; a box extent of 24 pc, offset of 0
pc, and $B_{\text{g}}/B_{\text{t}}=1.0$; and did 100 separate particle runs
for this same configuration. We did this for both Method 1 and Method 2.
For both configurations, we computed the mean flux $\mu$ and standard
deviation $\sigma$ in the flux for the set of runs. We found that using these
quantities, the data accurately fit a Gaussian distribution. We also confirmed
that the average computed value of $\Delta F/2$ for the set very closely
matched the $1\sigma$ value $\epsilon_{\text{F}}$ of the Gaussian
distribution, and consequently the average computed value of $\Delta F/2F$ for
the set was close to the value of $\sigma/\mu$ from the Gaussian distribution;
in fact, for the tests, they were only about 10$\%$ apart from each other. For
our production runs we wanted to be within 10$\%$ of the "correct" value of F,
with 95$\%$ certainty; so, for every box configuration, we ran enough
particles such that the $2\sigma$ value $2\epsilon_{\text{F}}/F=\Delta F/F$
was less than 0.1.
Figure 4 shows histograms of the computed flux F above the background for the
configuration tested for Method 1 and Method 2 along with a Gaussian fitted to
the mean and standard deviations for each. Table 2 shows the results of the
data from the Gaussian distribution.
Table 2: Describes the data found by the uncertainty test described in the text. The second column shows the mean value of $\Delta F/2F$ for the set of 100 runs for each configuration (computed for each run individually). The next column shows the ratio between the mean and standard deviation for the Gaussian distribution of all the flux values computed. The fourth column shows the percentage difference between these quantities, and the final column shows percentages within range of the mean for the Gaussian distribution of the flux values. M | $\langle\Delta F/2F\rangle$ | $\sigma/\mu$ | $\frac{\langle\Delta F/2F\rangle-(\sigma/\mu)}{(\sigma/\mu)}$ | $1,2,3\sigma$
---|---|---|---|---
1 | 5.60e-2 | 5.08e-2 | 10.22 | 62,96,100
2 | 5.60e-3 | 5.05e-3 | 10.91 | 70,95,99
Figure 4: Distributions for the uncertainty test described in the text,
showing histogram of flux uncertainties obtained for 100 runs of the two
methods. The vertical red line in each plot marks the average computed mean of
the data and the other tick markers show the 1, 2, and 3 $\sigma$ intervals
around the mean. Note: The obtained fluxes are different for the cases due to
the different methods used, which is discussed more in Section 5.
## 5 Results
With this code, we can calculate the range of effects varying the coherent
magnetic field properties outlined in Section 2.1. Figure 5 shows two contour
plots of the particle flux a coherent magnetic field produces above the
background at the tally plane. As can be seen, the coherent magnetic field
produces a noticeable flux above the background, shown by the yellow and green
patches in the upper right corners of the plots. These patches span 4x4 grid
zones in our case, and the flux $F$ previously discussed contains the relative
sum of all of the extra particles in this region.
In these plots, the likelihood of the particles to follow the direction of the
coherent field can clearly be observed. One noticeable feature of the plots is
that the effect on the tally plane is not always simply a higher flux (darker
patch) on the contour plot. Rather, the magnetic field configurations with a
stronger effect tend to form a “ring" of increased flux due to the likelihood
of the particles to follow the coherent field when they encounter it in the
outer zones of the coherent field region. Another feature of the stronger
magnetic field configurations is the decreased flux surrounding the box and
extending up into the right corner of the plot. This shadowing effect was also
observed in Harding et al. (2016), and is due to a lack of particles
scattering past the coherent field without entering it and getting “trapped".
Because our coherent field box contains a turbulent component, there is some
expectation that the shadowing effect would be muted, but it is still present
in many cases.
Figure 5: Shows two examples of the particle flux above the background $F$ for
our simulation runs. Both were done using the same coherent magnetic field
configuration of $B_{\text{g}}/B_{\text{t}}=1.0$, box extent=24 pc, and box
offset=0 pc, but the left plot was computed using Method 1 while the right
plot was computed using Method 2. The coherent magnetic field region can be
seen by the yellow and green patches in the upper right corner of the plots.
The computed flux results of our code are shown in Figure 6. The plot on the
left shows the results using “Method 1" while the plot on the right shows the
results using “Method 2". The figure shows the computed flux above the
background F for every box extent, tally plane offset, and coherent magnetic
field ratio tested. Each color shows a different coherent magnetic field box
extent, with each color having three different lines representing the three
different coherent magnetic field ratios. The dotted lines are put in for
comparison and were computed using the methods of Harding et al. (2016); i.e.,
no turbulent magnetic field in the coherent field box. In general, the topmost
line of one color is the highest coherent magnetic field and the coherent
field decreases moving down in height; this holds up except for the lower
points close to the background of $F=1$, where statistical uncertainty causes
more variation.
These results demonstrate the large effect of a coherent magnetic field
component on the particle flux above the background. Indeed, one can see from
Figure 6 that for all runs done in the study of coherent magnetic field
amplitude, coherent field box length, and coherent field box offset from the
tally plane, there was a noticeable flux observed above the background level;
and above the observed anisotropy level. Only a few of the runs with the
lowest flux values did not have a $2\sigma$ range above $F=1$, which signifies
a 95$\%$ chance of a noticeable observed flux. These points are shown by
crosses instead of dots in Figure 6.
Figure 6: Shows the results of our code, with the particle flux above the
background defined in Section 4.2 as a function of the coherent field offset
from the tally plane. Each color represents a different box extent, which was
tested for three different values of the magnetic field. The dotted lines
shows values computed using the methods of Harding et al. (2016). Every run is
done to 10$\%$ uncertainty. The points show the computed values, with the dots
being points whose uncertainty range is above the background while the stars
have an uncertainty range below the background.
### 5.1 Method Differences
Both of the hybrid transport methods, Method 1 and Method 2, produce the same,
generally predictable trends for the variation of the tested box configuration
variables. Increasing the coherent magnetic field ratio, increasing the box
length, and decreasing its distance from the tally plane all increase the
observed flux at the plane. However, the two methods do produce
quantitatively-different results from each other, depending on the coherent
magnetic field ratio used.
In addition to our main configuration parameter space, we did one study where
we picked a single box offset of 0 pc and extent value of 6 pc and varied the
value of the coherent magnetic field ratio for 10 different values, ranging
from 0.1 to 1.0. This is shown in Figure 7. It can be seen that, for higher
magnetic field ratios ($B_{\text{g}}/B_{\text{t}}\gtrsim 0.6$), Method 2
predicts a higher particle flux above the background than Method 1 while, for
lower magnetic field ratios ($B_{\text{g}}/B_{\text{t}}\lesssim 0.6$), Method
1 predicts a higher particle flux above the background than Method 2. This
means that over the span of magnetic field ratio values we tested for the main
study, Method 2 has a much broader range in predicted flux values (because
higher coherent magnetic field ratios predict a higher particle flux above the
background in general).
These differences can be seen in the dependence of the results on the spatial
distribution of the coherent magnetic field. Figure 8 shows the particle flux
at tally plane as a function of the distance away from the center of the box
region (108 pc $<$ y,z $<$ 120 pc). The top two plots were computed using
Method 1, and the bottom two plots were computed using Method 2. The left
column uses a coherent magnetic field ratio of $B_{\text{g}}/B_{\text{t}}=0.5$
while the right column uses a coherent magnetic field ratio of
$B_{\text{g}}/B_{\text{t}}=1.0$. As can be seen, for the higher magnetic field
ratio, Method 2 produces a higher particle flux around the box region, and a
greater deficit outside. However, for the lower field value, Method 1 still
has an observable flux around the box while it is much harder to see for
Method 2.
The differences between these two methods provide an indication of the
uncertainty in the predicted particle anisotropy at the tally plane. For our
tests, Method 1 predicts an anisotropy for all box geometries with
$B_{\text{g}}/B_{\text{t}}=0.5$ and $B_{\text{g}}/B_{\text{t}}=1.0$, and for
most of the geometries with $B_{\text{g}}/B_{\text{t}}=0.1$. Method 2, on the
other hand, only predicts an anisotropy for all box configurations with
$B_{\text{g}}/B_{\text{t}}=1.0$; all of the configurations with
$B_{\text{g}}/B_{\text{t}}=0.1$ and many with $B_{\text{g}}/B_{\text{t}}=0.5$
are not predicted to be high enough to be seen within our 10% simulation
errors.
Figure 7: Shows the results of varying the magnetic field ratio for one box
geometry configuration (extent=6 pc, offset=0 pc). Figure 8: Shows the number
of particles at the Earth sky as a function of box offset and radial distance
from the center of the box region. The top plots were computed using Method 1
and the bottom plots were computed using Method 2. The left column uses a
magnetic field ratio of $B_{\text{g}}/B_{\text{t}}=0.5$ while the right column
uses a magnetic field ratio of $B_{\text{g}}/B_{\text{t}}=1.0$. Both have a
box extent of 24 pc.
### 5.2 Variation of Cosmic Ray Energy
Finally, we tested one magnetic field configuration using higher energy
particles, for both Method 1 and Method 2. We tested 100 TeV, 1 PeV, 10 PeV,
and 100 PeV particles. The magnetic field configuration we chose to test was a
box extent of 24 pc, offset of 0 pc, and coherent magnetic field ratio of
$B_{\text{g}}/B_{\text{t}}=1.0$. We found that the trend is that higher energy
particles produce a smaller flux above the background in the box. This is
shown in the Figure 9. The reason for this likely has to do with the higher
energy particles having a longer path length, which means they are more likely
to "escape" from the coherent magnetic field.
Figure 9: Shows the results of varying the energy for one box geometry and
magnetic field ratio (extent=6 pc, offset=0 pc, $B_{g}/B_{t}=1.0$)
## 6 Observational Implications
We now attempt to approximate the effect that a coherent magnetic field might
have on the observed cosmic ray flux at Earth. Though we have demonstrated
theoretically that a coherent magnetic field can create anisotropies in the
cosmic ray flux across a tally plane, this doesn’t correlate exactly to the
anisotropies observed in the cosmic ray arrival direction at Earth. The
observed flux at a point requires tallying the angular distribution of the
flux as well. For statistical results, section 5 tallied the entire flux. In
this section, we also study the angular distribution.
To understand the angular distribution of the material on our tally plan, we
ran one configuration where we also tallied the velocity $\vec{u}$ of the
particles as they hit the plane. We ran particles using the configuration
tested for Method 1 with a box extent=24 pc, offset=3 pc, and
$B_{\text{g}}/B_{\text{t}}=1.0$ (though any of the runs with an offset greater
than 0 pc could have been used). For every particle that hit the plane, we
calculated $\text{cos}(\theta_{u})=\hat{u}\cdot\hat{n}$, where
$\hat{n}=[1,0,0]$ and $\theta_{u}$ is the angle between its velocity and the
plane surface normal. We then tallied the number of particles between
$\text{cos}(\theta_{u})$ and
$\text{cos}(\theta_{u})+\Delta\text{cos}(\theta_{u})$ for values of
$\text{cos}(\theta_{u})$ between 0 and 1. We used 15 bins in
$\text{cos}(\theta_{u})$, so $\Delta\text{cos}(\theta_{u})=1/15$.
We found that the velocity distribution of the particles being emitted from
the plane follows the relationship 111This is a Lambertian distribution in
angle, which is expected for particles following isotropic scattering motion.
$F(\hat{u})\propto\hat{u}\cdot\hat{n}=\text{cos}(\theta_{u}).$ (16)
The results of this test are shown in Figure 10. Each point represents a total
number of particles emitted in one bin. In addition to showing the calculation
done for the entire tally plane, we also show the same calculation for two
regions on the plane; one that is the box region used to compute the flux, and
another region that is in the opposite corner of the plane. All lines are
normalized to the number of particles that were considered for that region.
These details are to emphasize the fact that although these regions have
different total numbers of particles and particle density, they all follow the
same distribution in momentum. The main difference between the different
regions is the offset of the points from the best fit line; the opposite
corner region has a higher offset than the box region or entire plane because
there are less particles being considered, leading to higher statistical
error.
Figure 10: Shows the number of particles emitted in certain different
directions as a function of cos($\theta$), run for one box configuration. We
show the results for the entire x=150 pc plane (red), as well as the box
region alone (blue) and one other region at a different location in the grid
(green). Each line is normalized to the number of particles being considered
in that region. The black lines are a linear fit to the red points.
We can use the angular distribution of the particles at the tally surface
inferred from this study to calculate the observed flux in the observer frame
(as a function of viewing angle). To do so, we determine an observer location
("position of the Earth") at coordinates $[x_{e},y_{e},z_{e}]$. We calculate
each particle’s position in spherical coordinates [$\phi,\theta$] from its
Cartesian coordinates $[x_{p},y_{p},z_{p}]$ on the tally plane using $\rm
tan(\phi)=(y_{p}-y_{e})/(x_{p}-x_{e})$ and $\rm
tan(\theta)=(z_{p}-z_{e})/(x_{p}-x_{e})$. Each particle is then in a spherical
grid zone $\Omega_{z}$, where each zone $\Omega_{z}$ bounds a two dimensional
region in [$\phi$, $\theta$] space. By summing only these angle-dependent
contributions from the tally surface, we calculate the observed sky
distribution for part of the sky that can be mapped from the tally plane. For
each spherical grid zone $\Omega_{z}$, we tallied the proportional flux of
particles that would arrive at the Earth point from that zone as
$R(\Omega_{z})\propto\sum_{i=1}^{N_{\Omega_{z}}}\rm cos(\theta_{e}),$ (17)
where the sum is over all of the $N_{\Omega_{z}}$ particles that hit the plane
in the zone $\Omega_{z}$ and $\theta_{e}$ is the angle between the surface
normal and the vector connecting the particle hit position to the Earth point.
For the plots we present, the Earth point is at position
$[x_{e},y_{e},z_{e}]=[168,114,114]$ pc, and the angular resolution in both
dimensions are $\Delta\theta=\pi/50$ and $\Delta\phi=\pi/50$ (50 angular bins
each), with $\theta\in[0,\pi]$ and $\phi\in[0,\pi]$. To demonstrate the cosmic
ray flux in a way that is commonly done observationally, and reduce the effect
of the higher noise floor due to the geometric effect of the angular bins, for
each solid angle bin we chose to consider the difference between the flux and
the background run flux rather than the ratio:
$F(\Omega_{z})=R(\Omega_{z})-R_{\text{back}}(\Omega_{z})\propto\sum_{i=1}^{N_{\Omega_{z}}}\text{cos}(\theta_{e})-\sum_{i=1}^{N_{\text{back,}\Omega_{z}}}\text{cos}(\theta_{e}),$
(18)
where here all calculations were done using the same number of particles
originally started for both the main and the background runs ($N_{\text{tot,
i}}=N_{\text{tot, back}}$), since the number of particles will not normalize
out in these plots.
Figure 11 illustrates this process for one box configuration using Method 2.
The top two panels show $R(\Omega_{z})$ and $R_{\text{back}}(\Omega_{z})$
computed using Equation 17. The higher particle counts in the bottom right
corners of these panels appear because of the geometric location of the earth
sky point compared to the tally plane and the location of the cosmic ray
source. The placement of the earth sky point near the top left corner of the
tally plane causes more zones on the plane to be included in the lower right
angular bins, and fewer in the top left angular bins. Additionally, there are
more particle counts in the tally plane zones near the cosmic ray source. In
the bottom panel, computed using Equation 18, these effects are removed and
the location of the coherent magnetic field can clearly be seen by the higher
particle flux.
Figure 11: Plots of $R(\Omega_{z})$ (top left), $R_{\text{back}}(\Omega_{z})$
(top right), and $F(\Omega_{z})$ (bottom) using Method 1 and
$B_{\text{g}}/B_{\text{t}}$=1.0, box extent=24 pc and box offset = 3 pc.
Finally, we calculate for each plot a generic noise level $\sigma_{\Omega}$,
where $\sigma_{\Omega}=|\text{min}(F(\Omega_{z}))|$. This simply means that
the noise level for all of the angular bins for one plot is the magnitude of
the most negative value computed, or the flux in the angular bin where the
isotropic run had the highest flux above the main run. Though the uncertainty
is technically a function of angular zone, this method is a simple way to
calculate the uncertainty, and is an overestimate for most of the grid.
Plotting $F(\Omega_{z})/\sigma_{\Omega}$ allows us to neglect the
proportionality constant in these expressions, as it should be the same for
$F(\Omega_{z})$ and $\sigma_{\Omega}$. The results are presented in Figures 12
and 13, which show contour plots of $F(\Omega_{z})/\sigma_{\Omega}$ at the
earth point for Methods 1 and 2.
Each row in the figure shows the effect of varying one of the magnetic field
configuration parameters; the box extent, coherent magnetic field ratio, and
earth sky offset. In these plots, the colored region in the upper left corner
shows the location of the coherent magnetic field box. The effect of varying
the three magnetic field configuration parameters produced what might be the
expected changes on the background flux at the Earth sky. Increasing the box
extent, decreasing its distance from the sky, and increasing the coherent
magnetic field ratio produce a higher particle flux observed on the sky. This
is in contrast to Harding et al. (2016), where all coherent magnetic field
values produce the same result as there is no turbulent field in the box to
drive the particles out.We have varied the observer position and the magnitude
of the anisotropy varies, but not significantly.
The anisotropies presented here are much higher than the observed anisotropy
in the cosmic ray flux, implying that the strength of the coherent magnetic
fields can be much lower than what we assumed. Even so, we are assuming the
magnetic field energy in the coherent fields is much less than that of the
small-scale structures. The strength of our coherent magnetic fields was not
chosen to match the observed anisotropies, but to demonstrate the role such
magnetic fields can play on the flux observed at the earth. Statistical
limitations of our Monte Carlo method required higher coherent magnetic field
strengths than those needed to explain the observations. Finally, we note that
limitations with our grid setup, such as particles not being allowed to
scatter back across the tally plane, may also have a minor effect on our
results.
Figure 12: Shows plots of $F(\Omega_{z})/\sigma_{\Omega}$ for different box
configurations using Method 1. The different rows show the effects on the
Earth sky image of varying the box extent, coherent magnetic field ratio, and
offset. The top row uses box extent values of 6 pc and 24 pc while keeping
$B_{\text{g}}/B_{\text{t}}=1.0$ and offset=3 pc. The middle row uses values of
$B_{\text{g}}/B_{\text{t}}=0.1$, 0.5, and 1.0 while keeping the box extent=24
pc and box offset=3 pc. The bottom row uses box offset values of 15 pc, 9 pc,
and 3 pc while keeping the box extent=24 pc and
$B_{\text{g}}/B_{\text{t}}$=1.0. Figure 13: Shows plots of
$F(\Omega_{z})/\sigma_{\Omega}$ for different box configurations using Method
2. The different rows show the effects on the Earth sky image of varying the
box extent, coherent magnetic field ratio, and offset. The top row uses box
extent values of 6 pc and 24 pc while keeping $B_{\text{g}}/B_{\text{t}}=1.0$
and offset=0 pc. The middle row uses values of
$B_{\text{g}}/B_{\text{t}}=0.1$, 0.5, and 1.0 while keeping the box extent=24
pc and box offset=0 pc. The bottom row uses box offset values of 2 pc, 1 pc,
and 0 pc while keeping the box extent=24 pc and
$B_{\text{g}}/B_{\text{t}}=1.0$.
## 7 Discussion and Conclusions
In this study , we have extended the work of Harding et al. (2016) to include
more realistic magnetic field structures which included both an isotropic and
an anisotropic component. Additionally, we varied our magnetic field
structures in a different way than was previously studied, varying the length
of the field structure and moving it away from the tally plane. To accomplish
this hybrid solution, we propose two methods which "combine" the standard
transport and diffusion algorithms usually used. One of these methods involved
the particles switching off between the two methods, picking the method of
travel based on a probabilistic approach. The other method involved the
addition of two vectors intended to represent the two components of the
magnetic field separately.
We have shown that the anisotropies in the particle flux at the tally plane
are much easier to create than might be expected, no matter which method is
used. In particular, with a strong enough coherent magnetic field component,
there are noticeable anisotropies above the background for all coherent field
length scales and offset values tested. Because the length scale of the
coherent field is so much larger than the mean free path of particle motion,
even a weak coherent magnetic field component can give the particle a strong
enough directional push over time to move it into the box region at the tally
plane. Additionally, even magnetic field structures removed from the plane can
cause a noticeable particle flux above the background. This is because the
coherent magnetic field alters the particle transport, effectively creating
what appears to be a new cosmic ray source.
Finally, we took the results at the tally plane and used them to make a
construction of what an observer at Earth might see. Though this method had
many limitations, we showed that the stronger, closer magnetic field
configurations produce a noticeable anisotropy. Many other works have found a
similar effect to the results presented here and in Harding et al. (2016). In
several studies (Giacinti & Sigl (2012),Ahlers & Mertsch (2015), Ahlers
(2014), López-Barquero et al. (2016)), it was shown that anisotropies in the
flux can arise from local turbulent magnetic field structures. Our simulations
were only sensitive to anisotropies that are much larger than those observed.
The fact that modest coherent magnetic fields can produce strong anisotropies
show that the features needed to explain the observed fields can be quite
small and the small amplitudes of the observed cosmic ray anisotropies place
limits on the nature of these coherent magnetic fields.
We stress that although we have made improvements over previous studies
modeling cosmic ray transport, we have not done the “ultimate" treatment of
the fields. This involves resolving the turbulent magnetic field structure and
using this in combination with the coherent field, directly solving the
Lorentz equation of motion for every particle step. Although such full
treatments are beyond computational power for large grids, it is possible to
use small-scale calculations to improve on the recipes (Methods 1 and 2) used
here. We defer this more detailed study for a later paper.
Another way to reduce the uncertainties in this study would be to obtain
better measurements of the magnetic field structures in the solar
neighborhood. With these structures, the cosmic ray anisotropies may be better
understood.
## Data Availability
The data underlying this article are available in the article.
## Acknowledgements
This work was supported by the US Department of Energy through the Los Alamos
National Laboratory. Los Alamos National Laboratory is operated by Triad
National Security, LLC, for the National Nuclear Security Administration of
U.S. Department of Energy (Contract No. 89233218CNA000001) This research was
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958 and by NASA ATP grant 80NSSC20K0507.
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The Kolkata Paise Restaurant Problem is a challenging game, in which $n$ agents must decide where to have lunch during their lunch break. The game is very interesting because there are exactly $n$ restaurants and each restaurant can accommodate only one agent. If two or more agents happen to choose the same restaurant, only one gets served and the others have to return back to work hungry. In this paper we tackle this problem from an entirely new angle. We abolish certain implicit assumptions, which allows us to propose a novel strategy that results in greater utilization for the restaurants. We emphasize the spatially distributed nature of our approach, which, for the first time, perceives the locations of the restaurants as uniformly distributed in the entire city area. This critical change in perspective has profound ramifications in the topological layout of the restaurants, which now makes it completely realistic to assume that every agent has a second chance. Every agent now may visit, in case of failure, more than one restaurants, within the predefined time constraints. From the point of view of each agent, the situation now resembles more that of the iconic travelling salesman, who must compute an optimal route through $n$ cities. Following this shift in paradigm, we advocate the use of metaheuristics. This is because exact solutions of the TSP are prohibitively expensive, whereas metaheuristics produce near-optimal solutions in a short amount of time. Thus, via metaheuristics each agent can compute her own personalized solution, incorporating her preferences, and providing alternative destinations in case of successive failures. We analyze rigorously the resulting situation, proving probabilistic formulas that confirm the advantages of this policy and the increase in utilization. The detailed mathematical analysis of our scheme demonstrates that it can achieve utilization ranging from $0.85$ to $0.95$ from the first day, while rapidly attaining steady state utilization $1.0$. Finally, we note that the equations we derive generalize formulas that were previously presented in the literature, which can be seen as special cases of our results.
Keywords:: Game theory, Kolkata Paise Restaurant Problem, TSP, metaheuristics, optimization, probabilistic analysis.
§ INTRODUCTION
§.§ The Kolkata Paise Restaurant Problem
The El Farol Bar problem is a well-established problem in Game Theory. It was William Brian Arthur who introduced El Farol Bar problem in Inductive Reasoning and Bounded Rationality [1]. It cab be described as follows: $N$ people, the players, need to decide simultaneously but independently whether they will visit tonight a bar that offers live music. In order to have an enjoyable night the bar must not be too crowded. Each potential visitor does not know the number of attendances each night in advance, so the visitor must predict and decide whether she wants to go to the bar or stay home. Although the players decide using previous knowledge, their choice is not affected by previous visits and they cannot communicate with each other [2]. In the El Farol Bar problem, the number of choices $n$ is equal to $2$, so the players have to choose between staying home or going out.
The Kolkata Paise Restaurant Problem, as well as the Minority Game, are variants of the El Farol Bar problem. The Minority Game was first introduced in 1997 by Damien Challet and Yi-Cheng Zhang [3]. They developed the mathematical formulation of the El Farol Bar which they named Minority Game. This game has an odd number $N$ of agents and at each stage of the game they decide whether they will go to the bar or stay home. The minority wins and the majority loses. Agents have to decide whether they want to go to the bar or not, regardless of the predictions for the attendance size. The Minority Game is a binary symmetric version of the El Farol Bar problem, with the symmetry relying on the fact that the bar can contain half of the players.
The Kolkata Paise Restaurant Problem (KPRP for short) is a repeated game that was named after the city Kolkata in India. In KPRP there are $n$ cheap restaurants (Paise Restaurants) and $N$ laborers who choose among these places for their quick lunch break. If the restaurant they go to is crowded, they have to return to work hungry, since they do not have time to visit another restaurant, or lack the resources needed to travel to another area. This generalization of the El Farol Bar is described as follows: each of a large number $N$ of laborers has to choose between a large number $n$ of restaurants, where usually $N = n$. In order for a player to win, that is to eat lunch, only one player should go to each restaurant. If more than one players attend the same restaurant at the same time, an agent is chosen randomly and only this agent is served. The player who gets to eat has a payoff equal to $1$, whereas all others who also chose this restaurant have a payoff equal to $0$. Each agent prefers to go to an unoccupied restaurant, than visit a restaurant where there are other agents as well. This realization in turn implies that the pure strategy Nash equilibria of the stage game are Pareto efficient. Consequently, there are exactly $n!$ pure strategy Nash equilibria for the stage game. This, combined with the rationality of the players, leads to the conclusion that it is possible to sustain a pure strategy Nash equilibrium of the stage game as a sub-game perfect equilibrium of the KPRP.
In [4] each agent has a rational preference over the restaurants and, despite the fact that the first restaurant is the most preferred, all agents prefer to be served even at their least preferable restaurant than not to be served at all. The prices are considered to be identical and each restaurant is allowed to serve only one agent. If more than one laborers attend the same restaurant, one laborer is chosen randomly, while the others remain starved for that day. The Kolkata Paise Restaurant problem is symmetric, given the preferences of the agents over the set of restaurants. The game is non-trivial because there is a hierarchy among the restaurants, with the first being the most preferable. Another approach stipulates that if multiple agents choose the same restaurant they have to share the same meal and as a result, none of them is happy. The choice of each player is secret and they have to choose simultaneously. The players choose their strategy based on the payoffs. It is assumed that the restaurants charge their meals with the same price. There is even a version where some restaurants offer much tastier meals than others. This game is a repeated game with a period of one day, and the choices of each player are known to the other players at the end of the day. The agents have their personal strategy as to where they intend to have lunch. In order to attain the optimal solution, the agents have to communicate and coordinate their actions, something which is forbidden. As a result, some agents may end up hungry and, at the same time, some restaurants may waste their food.
Some authors study the case where the number of restaurants $n$ is small and the agents take coordinated actions. Then, they analyze the game as a sub-game of KPRP and estimate the possibility to preserve the cyclically fair norm. As a result, punishment schemes need to be designed in this case. Every evening the agent makes up her mind based on her past experiences and the available information about each restaurant, which is supposed to be known to every agent. Each agent decides on her own, with no interaction with the other players. If more than one customers arrive at the same restaurant, an agent is randomly chosen to eat and the rest have to starve. There is a ranking system among the restaurants shared by the customers. The $n!$ Pareto efficient states can be achieved when all customers get served. The probability of this event is very low, due to the absence of cooperation and disclosure among the agents.
§.§ The Travelling Salesman Problem
In discrete optimization problems, the variables take discrete values and, usually, the objective is to find a graph or another similar visualization,
from an infinite or finite set [5].
The Travelling Salesman Problem (TSP) is a famous optimization problem described as follows: a salesman has to visit all the nearby cities starting from a specific city to which the salesman must return [6]. The only constraint is that the salesman must start and finish at the same specific city and visit each city only once. The visiting order is to be determined by the salesman each time the problem arises. The cities are connected through railway or roads and the cost of each travel is modeled by the difficulty in traversing the edges of the graph. The salesman has just one purpose and that is to visit all the cities with the minimum possible travel cost. In this problem, the optimum solution is the fastest, shortest and cheapest solution. TSP is easily expressed as a mathematical problem that typically assumes the form of a graph, where each of its nodes are the cities that the salesman has to visit. TSP was formulated during the 1800s by Sir William Rowan Hamilton and Thomas Kirkman and it was first studied by Karl Menger during the 1930s at Harvard and Vienna [6]. The purpose of TSP is for the salesman to determine the route with the lowest possible cost. Some of the typical applications of TSP are network optimization and hardware identification problems. It has kept researchers busy for decades and many solutions have emerged. TSP is an NP-hard problem and the results of the practical, heuristic solutions are not always optimal, but approximate [6]. The simplest “naive” solution to this problem is, of course, to try all possibilities and explore all paths, but the cost in time and complexity is so huge that is practically impossible. In order to overcome that, when solving a TSP the pragmatic focus is a near-optimal route, instead of always the best. For the graph depicted in Figure <ref> the optimal tour is $1 \rightarrow 3 \rightarrow 4 \rightarrow 2 \rightarrow 1$ with cost $7 + 12 + 19 + 11 = 49$.
state/.style =
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minimum size = 7 mm,
every loop/.style = min distance = 12 mm,
every edge/.style =
[state] (s1) at (2.0, 0.0) 1;
[state] (s2) at (0.0, 2.0) 2;
[state] (s3) at (0.0, 0.0) 3;
[state] (s4) at (-2.0, 0.0) 4;
(s1) edge node [right] $11$ (s2)
(s1) edge node [above] $7$ (s3)
(s1) edge [bend left = 50] node [below] $42$ (s4)
(s2) edge node [right] $35$ (s3)
(s2) edge node [left] $19$ (s4)
(s3) edge node [above] $12$ (s4);
A small scale instance of a TSP.
§.§ Related Work
§.§.§ Games and the Kolkata Paise Restaurant Problem
The Kolkata Paise Restaurant Problem (KPRP) was initially introduced in an earlier form in 2007 [7]. Its current formulation appeared in 2009 in [8] and [9]. Subsequently, many creative ideas and different lines of thought have been published and even a quantum version of the game has arisen. In [8], the importance of diversity is emphasized while herd behavior is penalized. Furthermore, the differences between the KPRP and the Minority Game are highlighted. One major difference is that in the KPRP the emphasis is placed on the simultaneous move many choice problem, in contrast to the Minority Game, which studies a simultaneous move two choice problem. Another important difference is the existence of a ranking system in the KPRP, but not in the Minority Game. Some of the strategies developed for the KPRP are discussed in [10] which also discusses problems where these strategies can be successfully applied. Ghosh et al. in [11] present a dictator's, or a social planner's as they call it, solution. In this solution the agents form a queue and the planner assigns each of them to a ranked restaurant depending on the queue of the first evening. The following evening the agents go to the next ranked restaurant and the last in the queue goes to the first ranked restaurant. This solution is called the fair social norm. In real life, each agent decides in parallel or democratically every evening, so this solution may be considered somewhat unrealistic. However, the parallel decision or democratic decision strategy is not as efficient as the dictated one, with the last leading to one of the best solutions to this problem. Banerjee et al. in [4] offer a generalization of the problem in such a way that the cyclically fair norm is sustained. Each strategy is viewed as a sub-game of perfect equilibrium of the KPRP. In 2013, Ghosh et al. published an article about stochastic optimization strategies in the Minority Game and the KPRP [12]. There, they point out that a stochastic crowd avoiding strategy results in a efficient utilization in the KPRP. Reinforcement learning was first introduced in the KPRP by [13], together with six revision protocols aiming at efficient resource utilization. These protocols combine local information with reinforcement learning, Each revision protocol has two variants depending on whether or not customers who were once served by a restaurant remain loyal to that restaurant in all subsequent periods. Some of these protocols were experimentally tested and shown to improve the utilization rate. Another generalization was introduced by Yang et al. in [14] aiming at dynamic markets this time. They studied what happens when agents can either divert to another district or stay in the current one. Each agent may replace another agent with no prior knowledge of the game, following a Poisson distribution. Agarwal et al. in [15] showed that the KPRP can be reduced to a Majority Game. In the latter, capacity is not restricted and agents aim at choosing with the herd. If more than one agents choose the same option, the utility decreases (see also [16] and [17]). Abergel et al. in [18] applied the KPRP in hospitals and beds. The local patients choose among the local hospitals those with the best ranking and compete with the other patients. If the patients are not treated in time it is a clear case of social waste of service for the rest of the hospitals. A brief presentation of the KPRP was given by Sharma et al. in [19], which included the origin and an overview of the game, strategies that may arise, several extensions and its applications in a variety of phenomena. The authors also presented an experimental analysis. Park et al. in [20] introduced the KPRP in the Internet of Things (IoT) and IoT devices. They used a KPRP approach to develop a scheme for these devices, because it allowed them to model situations where multiple resources are shared among multiple users, each with individual preferences. In [21], Sinha et al. propose a phase transition behavior, where if two or more agents visit the same restaurant, one is randomly picked to eat. The agents evolve their strategy based on the publicly available information about past choices in order for each of them to reach the best minority choice. In the same paper, they also develop two strategies for crowd-avoiding.
A significant trend, which has been quite evident in the last two decades, is to enhance classical games using unconventional means. The most prominent direction is to cast a classical game in a quantum setting. Since the pioneering works of Meyer [22] and Eisert et al. [23], quantum versions for a plethora of well-known classical games have been studied in the literature. Starting from the most famous of all games, the Prisoners' Dilemma [23], [24], [25], [26], many researchers have sought to achieve better solutions by employing quantumness (see the recent [27] and [26] and references therein), or other tools, such as automata ([28]). It not surprising that unconventional approaches to classical games are undertaken because they promise clear advantages over the classical ones. Another line of research is to turn to biological systems for inspiration. The Prisoners' Dilemma features prominently in this setting also (see [29] for a brief survey), but in reality most game situations can easily find analogues in biological and bio-inspired processes [30] and [31]. A quantum version of the KPRP was proposed in [32], where the quantum Minority Game was expanded to a multiple choice version. The agents cannot communicate with each other and have to choose among $m$ choices, but an agent wins if she makes a unique choice. Higher payoffs than the classical version were observed due to shared entanglement and quantum operations. In Sharif's [33] review, quantum protocols for quantum games were introduced, including a protocol for a three-player quantum version of the KPRP. In [34] the authors study the effect of quantum decoherence in a three-player quantum KPRP using tripartite entangled qutrit states. They observe that in the case of maximum decoherence the influence of the amplitude damping channel dominates over depolarizing and flipping channels. Furthermore, the Nash equilibrium of the problem does not change under decoherence.
§.§.§ The Travelling Salesman Problem
The Travelling Salesman Problem is a well-known combinatorial optimization problem. In this problem a salesman must compute a route that begins from a particular node (the starting location), passes through all other nodes only once before returning to the starting location, and has the minimum cost. The first appearance of the term “Travelling Salesman Problem” probably occurred between 1931 and 1932. The core of the TSP problem, however, was first mentioned over a century before, in a 1832's German book [35]. The mathematical formulation was introduced by Hamilton and Kirkman [35] and is typically expressed as follows. A cycle in a graph is a path that begins and ends at the same node and passes through all other nodes once. A Hamiltonian cycle contains all the vertices of the graph. The Travelling Salesman Problem amounts to figuring the cheapest way to visit every city and return back. Research efforts on TSP and closely related problems include Ascheuer et al. [36] that addressed the asymmetric TSP-TW using more than three alternative integer programming formulations and more than ten neighborhood structures. Gutin and Punnen [37] studied the effect of sorting-based initialization procedures. The authors claimed that understanding the algorithmic behavior is the best way to find solutions, since this would help in determining the best solution out of those available. Jones and Adamatzky [38] showed experimentally that using a sorting function within their algorithm was not functional and failed to return a feasible solution in some cases.
The difficulty in tackling the TSP motivated researchers to explore other avenues. One such notable and particularly promising approach is based on metaheuristics. A metaheuristic is a high-level heuristic that is designed to recognize, build, or select a lower-level heuristic (such as a local search algorithm) that can provide a fairly good solution, particularly with missing or incomplete information or with limited computing capacity [39]. The term “metaheuristics” was coined by Glover. Metaheuristics can be used for a wide range of problems. Of course, it must be noted that metaheuristic procedures, in contrast to exact methods, do not guarantee a global optimal solution [40]. Papalitsas et al. [41] designed a metaheuristic based on VNS for the TSP with emphasis on Time Windows. Another quantum-inspired method, based on the original General Variable Neighborhood Search (GVNS), was proposed in order to solve the standard TSP [42]. This quantum-inspired procedure was also applied successfully to the solution of real-life problems that can be modeled as TSP instances [43]. A quantum-inspired procedure for solving the TSP with Time Windows was also presented in [44]. More recently, [45] applied a quantum-inspired metaheuristic for tackling the practical problem of garbage collection with time windows that produced particularly promising experimental results, as further comparative analysis demonstrated in [46]. A thorough statistical and computational analysis on asymmetric, symmetric, and national TSP benchmarks from the well known TSPLIB benchmark library, was conducted in [47]. Very recently, Papalitsas et al. parameterized the TSPTW into the QUBO (Quadratic Unconstrained Binary Optimization) model [48]. The QUBO formulation enables TSPTW to run on a Quantum Annealer and is a critical step towards the ultimate goal of running the TSPTW with pure quantum optimization methods. Stochastic optimization can be implemented through several metaheuristic processes. The solution generated depends on the set of created random variables [39]. Metaheuristic processes may find successful solutions with less computational effort than accurate algorithms, iterative methods or basic heuristic procedures by looking for a wide variety of feasible solutions [40]. Hence, metaheuristic procedures are extremely useful and practical approaches for optimization because they can guarantee good solutions in a small amount of time. For example, a problem instance with thousands of nodes can be run for $30-40$ seconds and produce a solution with $3-5\%$ deviation from the optimal. This deviation depends on the implemented local search procedures inside the main part of the algorithm. An efficient design and choice of those improvement heuristics will define the deviation from the optimal solution. In view of the small amount of time they require and of the good quality of the solution they produce, we advocate their functional use in the Distributed Kolkata Paise Restaurant game.
§.§ Contribution
Let us now briefly summarize the contributions of this paper.
* We study the Kolkata Paise Restaurant Problem from an entirely new perspective. We identify and state explicitly certain implicit assumption that are inherent in the standard formulation of the game. We then take the unconventional step to abolish them entirely. This provides the opportunity for an entirely new setting and the adoption of a novel approach that leads to a new and more efficient strategy and, ultimately, to greater utilization for the restaurants.
* For the first time, to the best of our knowledge, we focus on the spatial setting of the game and we propose a more realistic and plausible topological layout for the restaurants. We perceive the restaurants to be uniformly distributed in the entire city area. This, rather pragmatic and more probable in reality situation, has profound ramifications on the topological layout of the game: the restaurants now get closer and, as their number $n$ increases, a standard assumption in the literature, the distances between nearby restaurants decrease. Due to the distribution of the restaurants, the resulting version of the game is aptly named the Distributed Kolkata Paise Restaurant Game.
* Thus, now it is realistic to assume that every agent has a second, a third, maybe even a fourth, chance. Every agent may visit, within the predefined time constraints, more than one restaurants. The agent is no longer a single destination and back traveller. The agent now resembles the iconic travelling salesman, who must pass through a network of cities, visiting every city once, coming back to the starting point, and all the time following the optimal route. This leads to the completely novel idea that each agent faces her own personalized TSP. We emphasize that the situation is specific for each agent, since the resulting network will vary. This is because each agent may have a different starting position and a different preference ranking of the restaurants. Of course, it is practically impossible to compute exact solutions for the TSP, as TSP is a famous NP-hard problem. However, this is a very small setback, as we may use metaheuristics. Metaheuristics can produce near-optimal solutions in a very short amount of time and this makes them indispensable tools of great practical value.
* This entirely new setting is formalized and then rigorously analyzed via probabilistic tools. We derive general formulas that mathematically confirm the advantages of this policy and the increase in utilization. Detailed examples of typical instances of the game are given in a series of Tables and the derived equations are graphically depicted in order to demonstrate their qualitative and quantitative characteristics. Our scheme demonstrably achieves utilization ranging from $0.85$ and going to $0.95$ and even beyond from the first day. The steady state utilization, to which the game rapidly converges, is, as expected, $1.0$.
* Finally, let us point out that the equations we derive generalize formulas that were previously presented in the literature, showing that the latter are actually special cases of our results.
§.§ Organization of the paper
The structure of this paper is as follows. In section <ref> we provide a comprehensive description of the KPRP and the TSP. In subsection <ref> we mention some important works that deal with the KPRP and the TSP. The rigorous formulations of the KPRP and the TSP are presented in section <ref>. In section <ref> we give a thorough explanation and presentation of the distributed version of the game, which we call Distributed Kolkata Paise Restaurant Game.
We analyze mathematically the topological situation regarding the restaurants in section <ref>, where the profound ramifications of the hypothesis that they follow the uniform probability distribution are developed. We formally prove the main results of the paper, which showcase the advantages of the distributed framework in a definitive manner in section <ref> . Finally, in Section <ref> we summarize our results and discuss future extensions of this work.
§ BACKGROUND
§.§ Formulation of the standard Kolkata Paise Restaurant Problem
In its most usual formulation, the Kolkata Paise Restaurant Problem is a repeated game with infinite rounds. There is a set of players, typically called agents or customers, that is denoted by $A = \{ a_1, \ldots, a_n \}$, a set of restaurants that is denoted by $R = \{ r_1, \ldots, r_n \}$, and a utility vector $u = ( u_1, \ldots, u_n ) \in \mathbb{R}^n$, which is associated with the restaurants and is common to every agent. On any given day, all agents decide to go to one of the $n$ restaurants for lunch. If it happens that just one agent arrives at a specific restaurant, then she will have lunch and she will be happy. If, however, two or more agents choose the same restaurant for lunch, then, it is generally assumed that just one of them will eat. The one to eat is chosen randomly. So, in such a case all but one will not be happy. Each agent has a utility and if they have lunch their utility is one, otherwise it is zero. In Chakrabarti et al. [8] the KPRP is modeled as a general one-shot restaurant game, where the set of agents is considered to be finite and the utilities are ranked as follows: $ 0 < u_n \leq \ldots \leq u_2 \leq u_1 $. The set of agents $A$ and the ranking of the utilities can be used to define the game. The latter can be represented as $G(u) = (A , S, \prod)$, where $A$ is the set of agents, $S$ is the set of strategies available to all agents, and $\prod = (\prod_1, \ldots, \prod_n )$ stands for the payoff vector. If the $i^{th}$ agent $a_i$ decides to go to the $j^{th}$ restaurant $r_j$, then the corresponding strategy is $s_i = j$. Every day each agent decides to which of the $n$ restaurants will go to eat. If $s_i = j$, this means that agent $a_i$ has decided to go to restaurant $r_j$. Given any strategy combination $s = (s_1, \ldots, s_n ) \in S^n$, the associated payoff vector is defined as $\prod ( s ) = ( \prod_1 ( s ), \ldots, \prod_n ( s ) )$, where the payoff $\prod_i ( s )$ of player $a_i$ is $\frac { u_{s_i} } { N_i (s) }$ and $N_i (s)$ is the total number of players that have made the same choice, i.e., restaurant $r_j$, as player $a_i$, including $a_i$. The strategy combination is in fact the restaurants the agents chose to eat to, and their payoff depends on their decision and the number of other agents that have made the same choice. In the literature, a game like KPRP, where there are potentially infinite rounds and in each round the same stage game is played, is referred to a supergame [49]. A supergame is a situation where the same game is repeatedly played as a one-shot game and the agents count the payoff in the long run of the game. This makes the payoff function more complex due to the repetitions.
§.§ Formulation of the TSP
The problem of finding the shortest Hamiltonian cycle is closely related to the TSP. The Hamiltonian graph problem, i.e., determining if a graph has a Hamiltonian cycle, is reducible to the traveling salesman problem. The trick is to assign zero length to the graph edges and, at the same time, create a new edge of length one for each missing edge. If the TSP solution for the resulting graph is zero, then there is a Hamiltonian cycle in the original graph; if the TSP solution is a positive number, then there is no Hamiltonian cycle in the original graph (see [50]). In different fields, such as operational research and theoretical computer science, TSP, which is NP-hard, is of great significance. Usually TSP is represented by a graph. The fact that TSP is NP-hard implies that there is no known polynomial-time algorithm for finding an optimal solution regardless of the size of the problem instance [51]. There are two types of models for the TSP, symmetric and asymmetric. The former is represented by a complete undirected graph $G = (V, E)$ and the latter by a complete directed graph $G = (V, A)$. Assuming that $n$ denotes the number of cities (nodes), $V = \{ 1, 2, 3, \ldots, n \}$ is the set of vertices, $E = \{ (i, j) : i, j \in V, \text{ where } i < j \}$ is the set of edges, and $A = \{ (i, j) : i, j \in V, \text{ where } i \neq j \}$ is the set of arcs. A cost matrix $C = [ c_{i, j} ]$, which satisfies the triangle inequality $c_{i, j} \leq c_{i, k} + c_{k, j}$ for every $i, j, k$, is defined for each edge or arc. If $c_{i, j}$ is equal to $c_{j, i}$, the TSP is symmetric (sTSP), otherwise it is called asymmetric (aTSP). In particular, this is the case for problems where the vertices are points $P_i = (X_i, Y_i)$ of the Euclidean plane, and $c_{i, j} = \sqrt{ (X_j - X_i)^2 + (Y_j - Y_i)^2 }$ is the Euclidean distance. The triangle inequality holds if the quantity $c_{i, j}$ represents the length of the shortest path from $i$ to $j$ in the graph $G$ [52]. In the case of the symmetrical TSP, the number of all possible routes covering all cities and corresponding to all feasible solutions is given by $\frac {(n-1)!} {2}$ (recall that the number of cities is $n$). The cost of the route is the sum of the costs of the edges followed.
§ FORMULATION OF THE DKPRG
The Kolkata Paise Restaurant problem (KPRP) is considered an extension of the minority game, as it involves multiple players ($n$) each having multiple choices ($N$). In its most general form it is possible that $n \neq N$. In this paper we follow the pretty much standard approach that the number of agents is equal to the number of restaurants, i.e., $n = N$. The novelty of our work lies on the fact that we advocate a spatially distributed and, in our view, more realistic version of the KPRP by taking into account the topology of the restaurants and by allowing the agents to begin their routes from different starting points. We call our version the Distributed Kolkata Paise Restaurant Game, or DKPRG for from now on.
In the original formulation of the KPRP one may readily point out the following important underlying assumptions.
(A1) All agents start from the same location.
(A2) All restaurants are near enough to the point of origin of every customer, so that each customer can, in principle, go to any restaurant, eat there and return back to work in time, that is within the time window of the lunch break.
(A3) Every restaurant is sufficiently far away from every other restaurant, so as to make prohibitive in terms of time constraints the possibility of any customer trying a second restaurant, in case her first choice proved fruitless.
In the two dimensional setting of Kolkata, or, as a matter of fact, of any city, the above assumptions taken together imply something very close to the situation depicted in Figure <ref>. There, one can see that the agents are concentrated within a very narrow region, which can be viewed as the center of a conceptual “circle.” The restaurants are located on this “circle” and since no two of them are allowed to be close they form something that resembles a “regular polygon.”
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[rectangle, fill = WordRed] (Restaurants) at (0.0, 3.5) Restaurants ;
[rectangle, fill = WordRed] (r1) at (3.0, 0.0) $r_1$ ;
[rectangle, fill = WordRed] (r2) at ( 3*cos(45) , 3*sin(45) ) $r_2$ ;
[rectangle, fill = WordRed] (r3) at (0.0, 3.0) $r_3$ ;
[rectangle, fill = WordRed] () at ( 3*cos(145) , 3*sin(145) ) ;
[rectangle, fill = WordRed] () at ( 3*cos(180) , 3*sin(180) ) ;
[rectangle, fill = WordRed] () at ( 3*cos(225) , 3*sin(225) ) ;
[rectangle, fill = WordRed] (rn-1) at (0.0, -3.0) $r_{n-1}$ ;
[rectangle, fill = WordRed] (rn) at ( 3*cos(-45) , 3*sin(-45) ) $r_n$ ;
[WordAquaLighter60, line width = 1.5pt, rounded corners = 30pt](-1.5, -1.5) rectangle (1.5, 1.5);
[rectangle, fill = WordBlueDarker50] (Agents) at (0.0, 0.0) Agents ;
[circle, fill = WordBlueDarker50] (a1) at ( 1.3*cos(45) , 1.3*sin(45) ) ${\tiny a_1}$ ;
[circle, fill = WordBlueDarker50] () at ( 1.3*cos(135) , 1.3*sin(135) ) ;
[circle, fill = WordBlueDarker50] () at ( 1.3*cos(180) , 1.3*sin(180) ) ;
[circle, fill = WordBlueDarker50] () at ( 1.3*cos(225) , 1.3*sin(225) ) ;
[circle, fill = WordBlueDarker50] (an) at ( 1.3*cos(315) , 1.3*sin(315) ) ${\tiny a_n}$ ;
The assumed topology in the standard KPRP. The restaurants are located on a “circle,” forming a “regular polygon.” The agents are concentrated in a very narrow region around center of the “circle.”
This last remark is significant because it disallows a situation as the one shown in Figure <ref>. The spatial layout depicted in this Figure is strictly forbidden. The proximity of two, three or more restaurants would contradict the impossibility of a second chance. In the standard KPRP no agent is allowed a second chance. We write “circle” and “regular polygon” inside quotation marks because we are not obviously dealing with a perfect geometric circle or a perfect regular polygon, but two dimensional approximations resembling the aforementioned symmetric shapes. Clearly, this a very special topological layout, one that is highly unlikely to be observed in practice. There is no compelling reason for the restaurants to exhibit this regularity or the agents to be confined to approximately the same location. On the contrary, it would seem far more reasonable to assume that at least the restaurants and perhaps even the agents are uniformly distributed within a given area. Finally, the usual assumption that the preference ranking of the restaurants is common to all customers seems a bit too special and probably too restrictive.
[scale = 1.25]
[line width = 1.5pt, MyBlue] (0, 0) circle [radius = 3cm];
[rectangle, fill = WordRed] (Restaurants) at (0.0, 3.5) Restaurants ;
[rectangle, fill = WordRed] (r1) at ( 3*cos(0) , 3*sin(0) ) ;
[rectangle, fill = WordRed] (r1) at ( 3*cos(35) , 3*sin(35) ) ;
[rectangle, fill = WordRed] (rj-1) at ( 3*cos(70) , 3*sin(70) ) $r_{j-1}$ ;
[rectangle, fill = WordRed] (rj) at ( 3*cos(90) , 3*sin(90) ) $r_j$ ;
[rectangle, fill = WordRed] (rj+1) at ( 3*cos(110) , 3*sin(110) ) $r_{j+1}$ ;
[rectangle, fill = WordRed] () at ( 3*cos(145) , 3*sin(145) ) ;
[rectangle, fill = WordRed] () at ( 3*cos(180) , 3*sin(180) ) ;
[rectangle, fill = WordRed] () at ( 3*cos(225) , 3*sin(225) ) ;
[rectangle, fill = WordRed] (rn-1) at ( 3*cos(260) , 3*sin(260) ) $r_p$ ;
[rectangle, fill = WordRed] (rn-1) at ( 3*cos(280) , 3*sin(280) ) $r_q$ ;
[rectangle, fill = WordRed] (rn) at ( 3*cos(-45) , 3*sin(-45) ) ;
[WordAquaLighter60, line width = 1.5pt, rounded corners = 30pt](-1.5, -1.5) rectangle (1.5, 1.5);
[rectangle, fill = WordBlueDarker50] (Agents) at (0.0, 0.0) Agents ;
[circle, fill = WordBlueDarker50] (a1) at ( 1.3*cos(45) , 1.3*sin(45) ) ${\tiny a_1}$ ;
[circle, fill = WordBlueDarker50] () at ( 1.3*cos(135) , 1.3*sin(135) ) ;
[circle, fill = WordBlueDarker50] () at ( 1.3*cos(180) , 1.3*sin(180) ) ;
[circle, fill = WordBlueDarker50] () at ( 1.3*cos(225) , 1.3*sin(225) ) ;
[circle, fill = WordBlueDarker50] (an) at ( 1.3*cos(315) , 1.3*sin(315) ) ${\tiny a_n}$ ;
The spatial layout depicted in this Figure is strictly forbidden. The proximity of two, three or more restaurants would contradict the impossibility of a second chance. In the standard KPRP no agent is allowed a second chance.
With that motivation in mind, in this work we propose to abolish all these assumptions. The resulting game is spatially distributed in terms of restaurants and as such is called the Distributed Kolkata Paise Restaurant Game (DKPRG). In our setting, each customer may have her own staring point, which is, in general, different from the starting locations of the other customers. The staring locations can either be concentrated in a small region of the Kolkata city area, precisely like the standard KPRP, or they may be assumed to follow a random distribution. The fundamental difference with prior approaches is that now the restaurants are viewed as being uniformly distributed over the city of Kolkata. This uniform randomness in the placement of the restaurants implies that there must be clusters of restaurants sufficiently near each other. This conclusion becomes inescapable, particularly in the case where the number of restaurants is large ($n \to \infty$). As will be shown in the following sections, the expected distance between “adjacent” restaurants will be relatively short and will only decrease as the number $n$ of restaurants increases.
The assumption of the random placement of restaurants leads to a personalized situation for each individual agent: each agent is in effect faced with a personalized Travelling Salesman Problem. To every agent corresponds an individual graph, which is assumed to be complete. The completeness assumption is not absolutely essential for the TSP, but, in any case, seems reasonable in the sense that one can go from any given restaurant to any other. This graph has $n + 1$ nodes, which are the locations of the $n$ restaurants plus the location of the starting point of the customer. The costs assigned to the edges of the graph are also personalized; each agent combines an objective factor, the spatial distances between the restaurants, with a subjective factor, her personal preferences. Recall that in the DKPRG we forego the common preference restriction and we let every customer have a distinct preference, i.e., she may prefer a particular restaurant and dislike another. Let us clarify however, that getting served, even at the least preferable restaurant, is more desirable than not getting served at all! This in turn will lead to a possibly unique ordering of the restaurants from the most preferable to the least for each agent. For instance, if two restaurants $r$ and $r'$ are equidistant from the staring point $s$ of a certain customer, something that is obviously an objective fact, but the customer in question has a clear preference for $r$ over $r'$, then she adjusts the costs $c_{s, r}$ and $c_{s, r'}$ corresponding to the edges $(s, r)$ and $(s, r')$, respectively, so that $c_{s, r} < c_{s, r'}$.
Hence, every agent is faced with a distinctive network topology, which is the combined result of the inherent randomness of the spatial locations and the subjectiveness of her preferences. The topology of the restaurants has a further implication of the utmost importance: a customer whose first choice is a particular restaurant, will now have with very high probability the opportunity to visit a second, a third, or even a fourth restaurant in the same area if need be. For each agent the time cost is dominated by the time taken to visit the first restaurant; the trip to other nearby restaurants in the same region incurs a relatively negligible time cost due to their spatial proximity. The customer has a second (even a third) chance to be served within the time window of the lunch break. Thus, an efficient, if not optimal, method for every customer to make well-informed decisions regarding her first, second, third, etc. choice is to solve the Travelling Salesman Problem for her personalized graph. Obviously, the TSP being an NP-hard problem, precludes the possibility of exact solutions. Nonetheless, near-optimal solutions of great practical value can easily be achieved in very short time by employing metaheuristics, as we have pointed out in subsection <ref>.
From this perspective, we proceed now to propose an effective distributed strategy that, if adopted by every agent, will lead to an efficient global solution. All of them will use a common high-level strategy that is tailored and fine-tuned according to their individual preferences. To enhance clarity we explicitly state below the hypotheses and that define and characterize the DKPRG variant.
(H1) DKPRG is an infinitely repeated game.
(H2) There are two main protagonists in the game. First, the $n$ agents (also referred to as customers) with different, in general, starting locations. The set of agents is denoted by $A = \{ a_1, \ldots, a_n \}$. Second, the $n$ restaurants that are uniformly distributed within the same area. The set of restaurants is denoted by $R = \{ r_1, \ldots, r_n \}$. All agents know the locations of the restaurants, but each one of them need not know the starting locations of the other agents.
(H3) To each agent $a \in A$ corresponds a distinct personal preference ordering $P_a = (r_{j_1}, r_{j_2}, \ldots, r_{j_n})$ such that restaurant $r_{j_1}$ is her first preference, $r_{j_2}$ is her second preference, and so on, with $r_{j_n}$ being the least preferable restaurant for $a$.
(H4) We adopt the standard convention that each restaurant can accommodate only one customer at a time. The immediate ramification of this convention is that if two or more customers arrive at a restaurant, only one can be served. The one to be served is chosen randomly.
(H5) The aforementioned hypotheses immediately bring to the front the novelty and contribution of our approach. The positions of the restaurants relative to the starting point of each customer create for every customer a distinct topology, a distinct network of restaurants. Specifically, each agent $a \in A$ perceives a personalized graph $G_a = (V_a, E_a)$. $G_a$ is a complete undirected graph having $n + 1$ vertices $v_0, v_1, \ldots, v_n$, where $v_0$ is the starting location of $a$ and $v_j$ is the location of restaurant $j, 1 \leq j \leq n$. For each pair of distinct vertices $u, v \in V_a$ there exists an undirected edge $(u, v) \in E_a$. The graph $G_a$ is complemented with the (symmetric) cost matrix $C_a$, that assigns to each edge $(u, v)$ a cost $c_{u, v}$. We may surmise that the costs are computed by a function $f_a$ that incorporates geographical data, i.e., the distances between the restaurants, and the preference ordering $P_a$. The topological layout of the restaurants is an objective and global reality that is common to all customers and is undeniably crucial to a rational computation of the travel costs. On the other hand, it would be illogical if an agent did not take into account her preferences. The weight assigned to the spatial distances need not be equal to the weight assigned to the preferences. A conservative approach could assign a far greater weight to the distances compared to the preferences. A more idiosyncratic approach would deal with both on an equal footing by assigning equal weights to distances and preferences. It is plausible that for customer $a$ the personal preferences may play a more prominent role than for customer $a'$, in which case we may allow for the possibility that, in the process of computing the costs, each customer assigns completely different weights. In any event, we regard each cost matrix $C_a$ as distinct, which, along with the uniqueness of each $V_a$, explains why the resulting networks $G_a$ are all considered different, that is every customer is confronted with her own unique and personalized TSP.
(H6) Each customer $a \in A$ solves the corresponding TSP using an efficient metaheuristic that outputs a near-optimal solution. In that manner, $a$ computes a (near-optimal) tour $T_a = ( l_0, l_1, \dots, l_n, l_{n+1} )$. The tour is represented by the ordered list $( l_0, l_1, \dots, l_n, l_{n+1} )$, where $l_0 = l_{n+1}$ is the starting point of $a$ and $l_k, 1 \leq k \leq n$, is the index of the restaurant in the $k^{th}$ position of the tour. Endowed with their individual route $T_a$, which is an integral part of their strategy, all customers follow a simple common strategy. From their starting location $l_0$ they first travel to the restaurant $r_{l_1}$. Once there, those that get served conclude their route successfully. Those that do not get served, proceed to the restaurant $r_{l_2}$. If their attempt at getting lunch also fails at $r_{l_2}$, then they proceed to $r_{l_3}$, and so on. Obviously, the time constraints, that is the fact that the agent must have returned to her staring point by the time the lunch break is over, means that the agent will not have the opportunity to exhaust the entire tour. The customer must interrupt the tour at some point in order to return. This may happen after travelling unsuccessfully to two, three, or more restaurants, depending on the topology of the network.
(H7) The Revision Strategy. We adopt the standard assumption that the agents operate independently and no communication takes place between any two of them. Therefore, each customer is completely unaware of the routes of the other customers. They revise their strategy every evening taking into account only what happened during the present day. This means that they decide using only information from the last day and no prior information or history need to be kept. We assume that all agents follow the same policy. If they got served at a specific restaurant this day, then tomorrow they go straight to the same restaurant. This applies even if this restaurant is not in the first place of their tour. For example, those agents that failed to get lunch at their first choice, but managed to do so at their second, or third choice, tomorrow go straight the restaurant that served them despite the fact that this particular restaurant is not their most preferable. Those that failed to get lunch, only know which restaurants were left vacant, i.e., not visited by any agent today. Further or more elaborate information, such as the choices of other players or if they got served and at which restaurant, seems unnecessary. The unserved agents construct and solve their new personalized TSP, this time using as vertices only the vacant restaurants (plus of course their starting location).
Having explained the details of the DKPRG, we shall proceed to analyze the mathematical characteristics and evaluate the resulting utilization of this policy in the following sections.
§ TOPOLOGICAL CONSIDERATIONS
We begin this section by fixing the notation and giving some definitions to clarify the most important concepts of our exposition.
* The one-shot DKPRG takes place every day. We use the parameter $t = 1, 2, \ldots,$ to designate the day under consideration.
* To each agent $a \in A$ corresponds the personalized network $G_a = (V_a, E_a)$ together with the personalized cost matrix $C_a$, which are constructed in the way we outlined in the previous section. Agent $a$ follows the tour $T_a = ( l_0, l_1, \dots, l_n, l_{n+1} )$, which is the solution to her personalized TSP. As we have emphasized, by using metaheuristics it is possible to obtain near-optimal solutions in a very short amount of time.
* The quality and efficiency of the strategy is measured by the utilization ratio $f$. This is of course the fraction of agents being served in a day, or, equivalently, the fraction of restaurants serving customers in a day. The equivalence is obvious because there are $n$ customers and $n$ restaurants.
In section <ref> we shall revisit the concept of utilization and we shall be more precise by asserting the expected utilization per day as a function of the game parameters.
An agent $a$ who has opted to follow tour $T_{a} = ( l_0, l_1, \dots, l_n, l_{n+1} )$ will initially try to get lunch at restaurant $r_{l_1}$. If she succeeds, she will eat and then return to her starting point. If she fails, she will visit the next restaurant in the tour, i.e., $r_{l_2}$. If she gets lunch there, she will subsequently go back to work. This process will go on until either she gets served or runs out of time, in which case she must interrupt the tour and return to work. If the time constraints allow her to pass through the first $m$ restaurants in her tour, in the worst-case scenario of $m - 1$ consecutive failures, then we say that $T_{a}$ is an $m$-stop tour. To facilitate our mathematical analysis we take for granted that all customers follow $m$-stop tours. We have already explained why, in our view, $m$ must be $\geq 2$. The case where $m = 1$ reduces to the standard treatment of the KPRP, which has already been analyzed extensively in the literature. In the rest of this work we study the case where $m \geq 2$. All these considerations motivate the next definition.
* The tour $T_{a} = ( l_0, l_1, \dots, l_n, l_{n+1} )$ associated with agent $a$ is an $m$-stop tour, $m \geq 2$, if, in the worst case, agent $a$ can visit restaurants $l_1, l_2, \ldots, l_m$ in this order without violating her time constraints. In such a tour, $l_1$ is the first stop, $l_2$ is the second stop, and so on, with $l_m$ being the final $m^{th}$ stop.
* If $\forall a \in A$, $T_{a}$ is an $m$-stop tour, then the resulting game is the $m$-stop DKPRG.
Let us now explore the spatial ramifications of our assumption that the restaurants are uniformly distributed within the overall city area. We now give the formal definition of uniform distribution.
Given a region $B$ on the plane, a random variable $L$ has uniform distribution on $B$, if given any subregion[If one wants to be overly technical, one should assume that both $B$ and $C$ are measurable sets. In the current setting, we believe that it is unnecessary to go to such a technical depth.] $C$ the following holds:
\begin{align} \label{eq:Uniform Distribution Definition}
P(L \in C) = \frac{ \text{area} ( C ) } { \text{area} ( B ) }, \quad C \subset B \ .
\end{align}
We assume of course that $L$ takes values in $B$.
The above definition is adapted from [53]. For a more general and sophisticated definition in terms of measures we refer the interested reader to [54].
Assuming that the $n$ restaurants are uniformly distributed on the whole city area, then if the city area is partitioned into $n$ regions of equal area, the expected number $\overline{N_p}$ of restaurants in each region is exactly $1$.
\begin{align} \label{eq:Expected Value Np}
\overline{N_p} = 1 \ , \ 1 \leq p \leq n \ .
\end{align}
Let $B$ stand for the whole city area and let $B_1, \ldots, B_n$ be the $n$ regions. The hypotheses assert that:
* $B_1 \cup \ldots \cup B_n = B$,
* $B_p \cap B_q = \emptyset$, if $1 \leq p \neq q \leq n$, and
* $\text{area} ( B_1 ) = \text{area} ( B_2 ) = \ldots = \text{area} ( B_n ) = \frac {\text{area} ( B ) } { n }$.
Invoking the fact that the $n$ restaurants are uniformly distributed on the whole city, we deduce from (<ref>) that for every restaurant $r_j, 1 \leq j \leq n$, and for every region $B_p, 1 \leq p \leq n$,
\begin{align} \label{eq:Probability rj In Bp}
P(r_j \in B_p) = \frac{ \text{area} ( B_p ) } { \text{area} ( B ) } = \frac { 1 } { n }
\ , \quad 1 \leq j, p \leq n \ .
\tag{ \ref{thr: Expected Number of Restaurants}.i }
\end{align}
We may now define the following collection of auxiliary random variables $N_{pj}$, where $1 \leq p, j \leq n$.
\begin{align} \label{eq:Random Variables Npj}
N_{p j} =
\left\{
\begin{matrix*}[l]
1 & \text{if restaurant } r_j \text{ is located in region } B_p \\
0 & \text{otherwise}
\end{matrix*}
\right.
\ .
\tag{ \ref{thr: Expected Number of Restaurants}.ii }
\end{align}
By combining the result from (<ref>) with definition (<ref>), we may conclude that
\begin{align} \label{eq:Probability of Random Variables Npj}
N_{p j} =
\left\{
\begin{matrix*}[l]
1 & \text{with probability } \frac{ 1 } { n } \\
0 & \text{with probability } \frac{ n - 1 } { n }
\end{matrix*}
\right.
\ , \quad 1 \leq p, j \leq n \ .
\tag{ \ref{thr: Expected Number of Restaurants}.iii }
\end{align}
Then, the random variable
\begin{align} \label{eq:Random Variables Np}
N_p = \sum_{ j = 1 }^{ n } N_{p j} \quad ( 1 \leq p \leq n ) \tag{ \ref{thr: Expected Number of Restaurants}.iv }
\end{align}
gives the number of restaurants in region $B_p$, $1 \leq p \leq n$. We are not interested in the actual value of the random variable $N_p$ per se, but in its expected value $E \left[ N_p \right]$. The latter can be easily computed if we use the above results and the linearity of the expected value operator.
\begin{align} \label{eq:Expected Value of Random Variables Np}
\overline{N_p} = E \left[ N_{ p } \right]
\overset{ (\ref{eq:Random Variables Np}) } { = }
E \left[ \sum_{ j = 1 }^{ n } N_{p j} \right] =
\sum_{ j = 1 }^{ n } E \left[ N_{p j} \right]
\overset{ (\ref{eq:Probability of Random Variables Npj}) } { = }
\sum_{ j = 1 }^{ n } \left( 1 \cdot \frac{ 1 }{ n } \right) =
1 \tag{ \ref{thr: Expected Number of Restaurants}.v }
\end{align}
This establishes that the expected number of restaurants in each region is precisely $1$ and proves formula (<ref>).
Partitioning a city area into $n$ disjoint regions of equal area might not be an easy task. The point is that for large values of $n$, as is the standard assumption in the literature, it is certainly doable. We stress the fact the shape of the regions need not be the same. Indeed, the validity of Proposition <ref> holds irrespective of whether the regions have the same shape or any particular shape for that matter.
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[rectangle, fill = WordRed] (Restaurants) at (0.0, 4.5) Uniformly Distributed Restaurants ;
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[rectangle, fill = WordRed] (r3) at (1.45, 2.3) $r_3$ ;
[rectangle, fill = WordRed] (rn-1) at (-1.0, -2.35) $r_{n-2}$ ;
[rectangle, fill = WordRed] (rn-1) at (0.8, -3.65) $r_{n-1}$ ;
[rectangle, fill = WordRed] (rn) at (2.5, -2.5) $r_n$ ;
Kolkata can be conceptually partitioned into $n$ regions $B_1, \ldots, B_n$ of equal area. If the $n$ restaurants are uniformly distributed in the overall Kolkata area, then the expected number of restaurants in each region $B_j, 1 \leq j \leq n$, is $1$.
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[rectangle, minimum size = 2.0 cm] () at (3.5, -3.25) $B_n$ ;
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[rectangle, fill = WordRed] (Restaurants) at (0.0, 4.5) Uniformly Distributed Restaurants ;
[rectangle, fill = WordRed] (r1) at (-3.5, 3.25) $r_1$ ;
[rectangle, fill = WordRed] (r2) at (-2.5, 3.75) $r_2$ ;
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[rectangle, fill = WordRed] (rn-1) at (1.56, -3.22) $r_{n-2}$ ;
[rectangle, fill = WordRed] (rn-1) at (2.5, -3.75) $r_{n-1}$ ;
[rectangle, fill = WordRed] (rn) at (3.65, -3.75) $r_n$ ;
As $n \to \infty$, the expected number of restaurants in each region remains $1$, but the expected distance between restaurants in neighbouring regions decreases.
This topological layout of the restaurants is shown in Figures <ref> and <ref>. In these Figures, the regions are drawn are squares, but this is just for convenience and to facilitate their graphic depiction. As we have explained, the regions are not required to have the same shape and nor does their shape need to resemble a regular two dimensional figure. For very large values of $n$, partitioning a city into very small identical squares is a good approximation, as we know from the field of image representation.
It is useful to contrast the two Figures. The latter depicts the situation where the number of restaurants is much larger compared to the number of restaurants in the former Figure. This demonstrates clearly what happens when $n$ increases significantly, i.e., when $n \to \infty$. Irrespective of the size of magnitude of $n$, the expected number of restaurants in each of the $n$ regions (recall that they are pairwise disjoint and of equal area) remains $1$. What does change however is the area of each region, which decreases with $n$ and, as a consequence, the expected distance between restaurants located in adjacent regions.
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[rectangle, minimum size = 2.0 cm] () at (1.0, 3.5) $B_3$ ;
[rectangle, minimum size = 2.0 cm] () at (-0.45, -2.9) $B_{n \! - \! 2}$ ;
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[rectangle, minimum size = 2.0 cm] () at (3.0, -2.5) $B_n$ ;
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[rectangle, fill = WordRed] (rn-1) at (1.1, -3.75) $r_{n \! - \! 1}$ ;
[rectangle, fill = WordRed] (rn) at (2.35, -2.5) $r_n$ ;
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This figure shows the expected distances between restaurants located in adjacent regions.
Let us make the rather obvious observation that there is a meaningful notion of distance defined between any two points, or locations if you prefer, in the entire city area. In reality, this can be the geographical distance between any two locations, expressed in meters or kilometers or in some other unit of length. For instance, let us consider two points $x$ and $y$ with spatial coordinates $(x_1, x_2)$ and $(y_1, y_2)$, respectively. A typical manifestation of the notion of distance is the Euclidean distance: $\sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2}$ between $x$ and $y$. In any event, we take for granted the existence of such a distance function defined on every pair of points $(x, y)$ of the city, which is denoted by $d (x, y)$.
* The distance between two regions $B_p$ and $B_q$ is defined as
\begin{align} \label{eq:Region Distance}
d ( B_p, B_q ) = \inf \{ d (x, y) : x \in B_p \text{ and } y \in B_q \} \ .
\end{align}
* Two regions $B_p$ and $B_q$ are adjacent if
\begin{align} \label{eq:Region Adjacency}
d ( B_p, B_q ) = 0 \ .
\end{align}
* We define the concept of diameter (see [55] for details) for the regions $B_p, 1 \leq p \leq n$. In particular, we define
\begin{align} \label{eq:Diameter}
diam B_p = \sup \{ d (x, y) : x, y \in B_p \} \ .
\end{align}
Let the $n$ restaurants be uniformly distributed on the city area and assume that the whole area is partitioned into $n$ regions of equal area. If $r_p$ and $r_q$ are the restaurants located at adjacent regions $B_p$ and $B_q$ respectively, where $1 \leq p \neq q \leq n$, then the distance $d ( r_p, r_q )$ between them is bounded above by $diam B_p + diam B_q$:
\begin{align} \label{eq:Upper Bound on Restaurant Distance}
d ( r_p, r_q ) \leq diam B_p + diam B_q \ , \quad 1 \leq p \neq q \leq n \ .
\end{align}
Consider two adjacent regions $B_p$ and $B_q$. By (<ref>), this means that $d ( B_p, B_q ) = 0$, which in turn implies that $\forall \varepsilon \ \exists x \in B_p \ \exists y \in B_q \text{ such that } d ( x, y ) \leq \varepsilon \quad (\star)$. In view of Proposition <ref>, one expects to find exactly one restaurant in $B_p$ and exactly one restaurant in $B_q$. So, let $r_p$ and $r_q$ be the restaurants located at regions $B_p$ and $B_q$, respectively, and consider the distance $d ( r_p, r_q )$ between them. By the triangle inequality, which is a fundamental property of every distance function, we may write that $d ( r_p, r_q ) \leq d ( r_p, x ) + d ( x, y ) + d ( y, r_q ), \forall x \in B_p \ \forall y \in B_q \quad (\star\star)$. From $(\star)$ and $(\star\star)$ we conclude that $\forall \varepsilon \ \exists x \in B_p \ \exists y \in B_q \text{ such that } d ( r_p, r_q ) \leq d ( r_p, x ) + d ( y, r_q ) + \varepsilon \quad (\star\star\star)$. Now, according to (<ref>), $d ( r_p, x ) \leq diam B_p$ and $d ( y, r_q ) \leq diam B_q$. These last two relations combined with $(\star\star\star)$, give that $d ( r_p, r_q ) \leq diam B_p + diam B_q$, as desired.
The above upper bound can be simplified if we further assume that all regions have the same geometric shape. This regularity does not impose any serious restriction on the overall setting of the game and allows us to assert that $diam B_1 = \ldots = diam B_n = D$, in which case inequality (<ref>) becomes:
\begin{align} \label{eq:Regular Upper Bound on Restaurant Distance}
d ( r_p, r_q ) \leq 2 D \ , \quad 1 \leq p \neq q \leq n \ .
\end{align}
In the special case where the regions are squares, as depicted in Figures <ref> and <ref>, one can easily see that the diameter $D$ is proportional to $\sqrt {\frac { 2 } { n } }$:
\begin{align} \label{eq:Square Diameter}
D \propto \sqrt {\frac { 2 } { n } } \ .
\end{align}
A comparison between Figures <ref> and <ref> demonstrates that the expected distance between restaurants which lie in adjacent regions is quite short, as it is bounded above by the sum of the diameters of the corresponding regions. The diameter of the regions decreases as $n$ increases, and in the special case shown in these two Figures, the diameter decreases in proportion to $\frac { 1 } { \sqrt { n } }$. In layman terms, this means that adjacent restaurants get very close to each other as $n \to \infty$. Once the agent arrives at a restaurant, then, with high probability, visiting an adjacent restaurant will only incur a negligible extra cost that will not violate her time constraints.
[scale = 1.2]
[WordIceBlue] (-4.0, -4.0) rectangle (4.0, 4.0);
[step = 1.0 cm, MyBlue, line width = 0.1pt] (-4.0, -4.0) grid (4.0, 4.0);
[rectangle, minimum size = 2.0 cm] () at (-3.8, 3.85) $B_1$ ;
[rectangle, minimum size = 2.0 cm] () at (-2.8, 3.15) $B_2$ ;
[rectangle, minimum size = 2.0 cm] () at (-1.8, 3.85) $B_3$ ;
[rectangle, minimum size = 2.0 cm] () at (1.3, -3.85) $B_{n \! - \! 2}$ ;
[rectangle, minimum size = 2.0 cm] () at (2.3, -3.15) $B_{n \! - \! 1}$ ;
[rectangle, minimum size = 2.0 cm] () at (3.8, -3.85) $B_n$ ;
[rectangle] at (-0.5, 3.5) $\cdot$ ;
[rectangle] at (1.5, 3.5) $\cdot$ ;
[rectangle] at (3.5, 3.5) $\cdot$ ;
[rectangle] at (-3.0, 2.0) $\cdot$ ;
[rectangle] at (-3.0, 0.0) $\cdot$ ;
[rectangle] at (-3.0, -2.0) $\cdot$ ;
[rectangle] at (0.0, 2.0) $\cdot$ ;
[rectangle] at (0.0, 0.0) $\cdot$ ;
[rectangle] at (0.0, -2.0) $\cdot$ ;
[rectangle] at (3.0, 2.0) $\cdot$ ;
[rectangle] at (3.0, 0.0) $\cdot$ ;
[rectangle] at (3.0, -2.0) $\cdot$ ;
[rectangle] at (-3.5, -3.5) $\cdot$ ;
[rectangle] at (-1.5, -3.5) $\cdot$ ;
[rectangle] at (0.5, -3.5) $\cdot$ ;
[rectangle, fill = MyLightRed] (r1) at (-3.3, 3.25) $r_1$ ;
[rectangle, fill = MyLightRed] (r2) at (-2.275, 3.78) $r_2$ ;
[rectangle, fill = MyLightRed] (r3) at (-1.28, 3.22) $r_3$ ;
[rectangle, fill = MyLightRed] (rn-1) at (1.41, -3.22) $r_{n \! - \! 2}$ ;
[rectangle, fill = MyLightRed] (rn-1) at (2.4, -3.75) $r_{n \! - \! 1}$ ;
[rectangle, fill = MyLightRed] (rn) at (3.3, -3.2) $r_n$ ;
[GreenTeal, dashed, line width = 0.5pt] (-4.0, 3.0) – node [above, rotate = 45] $diam B_1$ (-3.0, 4.0);
[GreenTeal, dashed, line width = 0.5pt] (3.0, -4.0) – node [below, rotate = 45] $ diam B_n$ (4.0, -3.0);
[MyDarkBlue, line width = 1.0pt] (-3.07, 3.42) – node [above, rotate = 20] $\mathbf{d(r_1, r_2)}$ (-2.51, 3.61);
[MyDarkBlue, line width = 1.0pt] (-2.03, 3.59) – node [below, rotate = -15] $\mathbf{d(r_2, r_3)}$ (-1.53, 3.4);
[MyDarkBlue, line width = 1.0pt] (1.79, -3.39) – node [below, rotate = -35] $\mathbf{d(r_{n \! - \! 2}, r_{n \! - \! 1})}$ (2.02, -3.55);
[MyDarkBlue, line width = 1.0pt] (2.75, -3.55) – node [above, rotate = 35] $\mathbf{d(r_{n \! - \! 1}, r_n)}$ (3.04, -3.36);
When $n$ increases, the area and the diameter of the regions $B_1, \ldots, B_n$ decrease. As a result the expected distances between restaurants located in adjacent regions decrease. In other words, as $n \to \infty$, the restaurants in adjacent regions get closer and closer.
We clarify that we are not making any assumption about the probabilistic distribution of the agents. One possibility is that the agents might be concentrated in the “center,” or in another specific location of the city area, as is tacitly assumed by the original KPRP. Another possibility is that the agents follow a random distribution over the area, for instance they might also follow the uniform distribution. The former case is depicted in Figure <ref> and the latter in Figure <ref>. The crucial observation is that in both cases any of the $n$ agent can, potentially, have lunch in any of the $n$ restaurants and return back in time. This fact implies that, assuming each agent follows the (near-optimal) tour produced as a solution to her individual TSP, she may visit a second, or even a third, restaurant if her previous choices proved fruitless. To see why this is indeed so, one may consider for instance agent $a_1$ in both Figures <ref> and <ref> and the restaurant that is furthest apart. Without loss of generality let us say that in both cases this is restaurant $r_n$. Being able to visit $r_n$ while adhering to her time constraints, implies being also able to pass through adjacent restaurants within the same time window.
[scale = 1.0]
[WordIceBlue] (-4.0, -4.0) rectangle (4.0, 4.0);
[step = 2.0 cm, MyBlue, line width = 0.25pt] (-4.0, -4.0) grid (4.0, 4.0);
[rectangle, minimum size = 2.0 cm] () at (-3.0, 3.0) $B_1$ ;
[rectangle, minimum size = 2.0 cm] () at (-1.0, 3.0) $B_2$ ;
[rectangle, minimum size = 2.0 cm] () at (1.0, 3.0) $B_3$ ;
[rectangle, minimum size = 2.0 cm] () at (-1.0, -3.0) $B_{n-2}$ ;
[rectangle, minimum size = 2.0 cm] () at (1.0, -3.0) $B_{n-1}$ ;
[rectangle, minimum size = 2.0 cm] () at (3.0, -3.0) $B_n$ ;
[rectangle] at (2.5, 3.0) $\cdot$ ;
[rectangle] at (3.0, 3.0) $\cdot$ ;
[rectangle] at (3.5, 3.0) $\cdot$ ;
[rectangle] at (-3.0, 1.0) $\cdot$ ;
[rectangle] at (-3.0, 0.0) $\cdot$ ;
[rectangle] at (-3.0, -1.0) $\cdot$ ;
[rectangle] at (0.0, 1.0) $\cdot$ ;
[rectangle] at (0.0, 0.0) $\cdot$ ;
[rectangle] at (0.0, -1.0) $\cdot$ ;
[rectangle] at (3.0, 1.0) $\cdot$ ;
[rectangle] at (3.0, 0.0) $\cdot$ ;
[rectangle] at (3.0, -1.0) $\cdot$ ;
[rectangle] at (-3.5, -3.0) $\cdot$ ;
[rectangle] at (-3.0, -3.0) $\cdot$ ;
[rectangle] at (-2.5, -3.0) $\cdot$ ;
[rectangle, fill = WordRed] (r1) at (-3.5, 3.5) $r_1$ ;
[rectangle, fill = WordRed] (r2) at (-1.0, 2.5) $r_2$ ;
[rectangle, fill = WordRed] (r3) at (1.45, 2.3) $r_3$ ;
[rectangle, fill = WordRed] (rn-1) at (-1.0, -2.35) $r_{n-2}$ ;
[rectangle, fill = WordRed] (rn-1) at (0.8, -3.65) $r_{n-1}$ ;
[rectangle, fill = WordRed] (rn) at (2.5, -2.5) $r_n$ ;
[WordAquaLighter60, line width = 1.5pt, rounded corners = 30pt](-1.15, -1.15) rectangle (1.15, 1.15);
[rectangle, fill = WordBlueDarker50] (Agents) at (0.0, 4.5) Agents Concentrated in a Small Region;
[circle, fill = WordBlueDarker50] (a1) at ( 0.8*cos(45) , 0.8*sin(45) ) $a_1$ ;
[circle, fill = WordBlueDarker50] () at ( 0.8*cos(135) , 0.8*sin(135) ) ;
[circle, fill = WordBlueDarker50] () at ( 0.8*cos(180) , 0.8*sin(180) ) ;
[circle, fill = WordBlueDarker50] () at ( 0.8*cos(225) , 0.8*sin(225) ) ;
[circle, fill = WordBlueDarker50] (an) at ( 0.8*cos(315) , 0.8*sin(315) ) $a_n$ ;
The above figure depicts the situation where all $n$ agents are concentrated within a small region of the Kolkata city, while the $n$ restaurants are uniformly distributed in the overall Kolkata area.
[scale = 1.0]
[WordIceBlue] (-4.0, -4.0) rectangle (4.0, 4.0);
[step = 2.0 cm, MyBlue, line width = 0.25pt] (-4.0, -4.0) grid (4.0, 4.0);
[rectangle, minimum size = 2.0 cm] () at (-3.0, 3.0) $B_1$ ;
[rectangle, minimum size = 2.0 cm] () at (-1.0, 3.0) $B_2$ ;
[rectangle, minimum size = 2.0 cm] () at (1.0, 3.0) $B_3$ ;
[rectangle, minimum size = 2.0 cm] () at (-0.75, -3.0) $B_{n-2}$ ;
[rectangle, minimum size = 2.0 cm] () at (0.75, -3.0) $B_{n-1}$ ;
[rectangle, minimum size = 2.0 cm] () at (3.0, -3.0) $B_n$ ;
[rectangle] at (2.5, 3.0) $\cdot$ ;
[rectangle] at (3.0, 3.0) $\cdot$ ;
[rectangle] at (3.5, 3.0) $\cdot$ ;
[rectangle] at (-3.0, 1.0) $\cdot$ ;
[rectangle] at (-3.0, 0.0) $\cdot$ ;
[rectangle] at (-3.0, -1.0) $\cdot$ ;
[rectangle] at (0.0, 1.0) $\cdot$ ;
[rectangle] at (0.0, 0.0) $\cdot$ ;
[rectangle] at (0.0, -1.0) $\cdot$ ;
[rectangle] at (3.0, 1.0) $\cdot$ ;
[rectangle] at (3.0, 0.0) $\cdot$ ;
[rectangle] at (3.0, -1.0) $\cdot$ ;
[rectangle] at (-3.5, -3.0) $\cdot$ ;
[rectangle] at (-3.0, -3.0) $\cdot$ ;
[rectangle] at (-2.5, -3.0) $\cdot$ ;
[rectangle, fill = WordRed] (r1) at (-3.5, 3.5) $r_1$ ;
[rectangle, fill = WordRed] (r2) at (-1.0, 2.5) $r_2$ ;
[rectangle, fill = WordRed] (r3) at (1.45, 2.3) $r_3$ ;
[rectangle, fill = WordRed] (rn-1) at (-1.0, -2.35) $r_{n-2}$ ;
[rectangle, fill = WordRed] (rn-1) at (0.8, -3.65) $r_{n-1}$ ;
[rectangle, fill = WordRed] (rn) at (2.5, -2.5) $r_n$ ;
[rectangle, fill = WordBlueDarker50] (Agents) at (0.0, 4.5) Agents Uniformly Distributed;
[circle, fill = WordBlueDarker50] (a1) at (-2.5, 2.5) $a_1$ ;
[circle, fill = WordBlueDarker50] (a1) at (-0.5, 3.5) $a_2$ ;
[circle, fill = WordBlueDarker50] (a1) at (0.5, 2.5) $a_3$ ;
[circle, fill = WordBlueDarker50] (an) at (-1.5, -3.5) $a_{n-2}$ ;
[circle, fill = WordBlueDarker50] (an) at (1.5, -2.5) $a_{n-1}$ ;
[circle, fill = WordBlueDarker50] (an) at (3.5, -3.5) $a_n$ ;
This figure reflects the situation where both the $n$ restaurants and the $n$ agents are uniformly distributed in the overall Kolkata area.
§ MATHEMATICAL ANALYSIS OF THE UTILIZATION
The current section is devoted to the analytic estimation of the evolution of the game parameters and the daily utilization of the proposed strategy scheme. Let us briefly summarize the policy that regulates the $m$-DKPRG.
* At the beginning of day $1$ all $n$ agents are in the same position, in that they have not got lunch yet, and they in a precarious state not knowing if they will manage to eat eventually. So, at this point in time they are all unsatisfied. The situation with the restaurants is symmetrical. All $n$ restaurants face uncertainty in that it is yet unknown whether they will be chosen by at least one customer. Therefore, at this point they are all vacant.
* The situation is quite different at the end of day $1$. A significant percentage of the $n$ agents, as will be shown in this section, managed to get lunch. An equal percentage of the $n$ restaurants was utilized. The common strategy followed by all agents ensures that the same agents will get lunch next day, the day after the next, etc. These agents are satisfied, since they have effectively “won” the game. Symmetrically, the same restaurants will be utilized every day from now on. They will be permanently reserved.
* At the beginning of day $2$, only those agents that failed to eat yesterday will essentially play the game. These will the active players of day $2$. The active players will strive to get lunch exclusively to the restaurants that did not serve any customer yesterday. The rest of the agents are already satisfied and will certainly have lunch today, each one at the specific restaurant that (eventually) served her yesterday.
* By the end of day $2$, a significant percentage of the active agents will have succeeded in getting lunch. Thus, the total number of satisfied agents will increase by the amount of today's gains. Of course, an equal percentage of yesterday's vacant restaurants will also be utilized for the first time today.
* This process will continue ad infinitum.
The next concepts will prove useful in our analysis.
* The expected number of agents that managed to eat lunch during day $1$ is denoted by $\overline{A_{1}^{s}}$ and the expected number of agents that failed to eat lunch during day $1$ is denoted by $\overline{A_{1}^{u}}$.
* The expected number of agents that got lunch for the first time during day $t, t = 2, 3, \ldots$, is denoted by $\overline{A_{t}^{s}}$. The expected number of agents that failed to get lunch during day $t, t = 2, 3, \ldots$, is denoted by $\overline{A_{t}^{u}}$.
* Symmetrically, the expected number of restaurants that served lunch during day $1$ is denoted by $\overline{R_{1}^{r}}$ and the expected number of restaurants that did not serve lunch during day $1$ is denoted by $\overline{R_{1}^{v}}$.
* The expected number of restaurants that served a customer for the first time during day $t, t = 2, 3, \ldots$, is denoted by $\overline{R_{t}^{r}}$. The expected number of agents that failed to serve lunch during day $t, t = 2, 3, \ldots$, is denoted by $\overline{R_{t}^{v}}$.
* The vacancy probability of day $1$ is the probability that a restaurant did not accommodate any customer during day $1$ and is designated by $VP_{1}$.
* The vacancy probability of day $t, t = 2, 3, \ldots$, designated by $VP_{t}$, is the probability that a restaurant that has not served any customer before day $t$ did not serve a customer during day $t$ either.
* In the $m$-stop DKPRG, only the customers that have yet to get lunch participate actively in today's game. The agents that actually play the game at the beginning of day $t$, seeking a restaurant to get lunch, are called active players and their expected number is denoted by $n_t$.
* The expected utilization of day $t, t = 1, 2, \ldots$, denoted by $\overline{f_t}$, is the fraction of the expected number of agents that were served during day $t$. The steady state utilization is defined as $f_\infty = \sup \{ f_t : t \in \mathbb{N} \}$.
Equiprobability of tours. The following analysis is based on the premise that all $n!$ tours are equiprobable. In the rest of this paper we shall refer to this assumption as the equiprobability of tours assumption (EPT for short). In view of the discussion in the previous sections, this premise is well justified.
An immediate consequence of the EPT assumption is the equiprobability of each restaurant appearing in any position. In particular, let us recall that in the tour $T_{a} = ( l_0, l_1, \dots, l_n, l_{n+1} )$, corresponding to agent $a$, $l_0 = l_{n+1}$ is the starting point of $a$ and $l_k, 1 \leq k \leq n$, is the index of the restaurant in the $k^{th}$ position of the tour. We may easily calculate the probability that a restaurant is in a specific position of the tour, as well as the probability of the complementary event. For easy reference, these facts are collected in the next Proposition whose proof is trivial and thus omitted.
Assuming the equiprobability of tours, the following hold.
\begin{align} \label{eq:Probability of rj in position k of Ta}
\forall a \in A \ \ \forall r \in R \ \ \forall k, 1 \leq k \leq n, \quad P(r \text{ is in position } k \text{ of } T_{a} ) = \frac{1}{n}
\end{align}
\begin{align} \label{eq:Probability of rj not in position k of Ta}
\forall a \in A \ \ \forall r \in R \ \ \forall k, 1 \leq k \leq n, \quad P(r \ \mathrm{not} \text{ in position } k \text{ of } T_{a} ) = \frac{n - 1}{n}
\end{align}
The above can be generalized to handle the case of a restaurant $r$ appearing in one of $w$ distinct positions $k_1, k_2, \ldots, k_w$, where $1 < w \leq n$.
\begin{align} \label{eq:Probability of rj in positions kw of Ta}
\forall a \in A \ \ \forall r \in R \quad P( r \text{ is in } \mathrm{one} \text{ of positions } k_1, \ldots, k_w \text{ of } T_{a} ) = \frac{w}{n}
\end{align}
\begin{align} \label{eq:Probability of rj not in positions kw of Ta}
\forall a \in A \ \ \forall r \in R \quad P( r \ \mathrm{not} \text{ in } \mathrm{any} \text{ of positions } k_1, \ldots, k_w \text{ of } T_{a} ) = \frac{n - w}{n}
\end{align}
We only mention that the above hold for every restaurant, every position, and, of course, for every tour. Since the probability that restaurant $r \in R$ is in the $k^{th}$ position of the tour of agent $a$ is $\frac{1}{n}$, the probability of the complementary event, i.e., that restaurant $r$ is not in the $k^{th}$ position of $T_{a}$ is $\frac{n - 1}{n}$. If we deem as “success” the case where $r$ is indeed in the $k^{th}$ position of $T_{a}$ and as “failure” the case where $r$ is not, then this situation is a typical example of a Bernoulli trial, having probability of success $\frac{1}{n}$ (also referred to as parameter, see [56]) and probability of failure $\frac{n - 1}{n}$. In view of (<ref>) we denote this as
\begin{align} \label{eq:Bernoulli Trial Parameter}
P(r \text{ is in position } k \text{ of } T_{a}) \sim Ber(\frac{1}{n}) \ , \quad \forall a \in A \ \ \forall r \in R \ \ \forall k, 1 \leq k \leq n \ .
\end{align}
Analogously, the probability that restaurant $r \in R$ appears in one of $w, 1 < w \leq n$, distinct positions of the tour of agent $a$ is $\frac{w}{n}$. The probability of the complementary event, i.e., that restaurant $r$ is not in any one of these $w$ positions of $T_{a}$ is $\frac{n - w}{n}$. This time, one may view as “success” the case where $r$ is indeed in one of the designated $w$ positions of $T_{a}$ and as “failure” the case where $r$ is not. So, once again we are facing with a Bernoulli trial, this time with parameter $\frac{w}{n}$.
\begin{align} \label{eq:General Bernoulli Trial Parameter}
P( r \text{ is in } \mathrm{one} \text{ of positions } k_1, \ldots, k_w \text{ of } T_{a} ) \sim Ber(\frac{w}{n}) \ , \quad \forall a \in A \ \ \forall r \in R \ .
\end{align}
The fact that the $n$ agents calculate their tours independently, implies that $n$ independent Bernoulli trials take place simultaneously, all with the same success and failure probabilities. This situation is described by the binomial distribution with parameters $(n, p)$[We refer the reader to [56] and [53] for a more detailed analysis.], denoted by $Bin(n, p)$, where $p = \frac{1}{n}$ in the simple case of one position and $p = \frac{w}{n}$ in the general case of $w$ positions. By employing well-known formulas from probability textbooks we may assert the following Proposition, whose proof is also trivial.
Given a restaurant $r$, if its appearance in a specified position $k$ in one tour counts as one success, whereas its failure to appear in the specified position $k$ in one tour counts as one failure, then the probability of exactly $l$ appearances in position $k$ in total is given by
\begin{align} \label{eq:Probability of l successes in n trials}
\forall r \in R \ \ \forall k, 1 \leq k \leq n, \quad P( r \text{ appears } l \text{ times in position } k \text{ in } n \text{ tours} ) = \binom { n } { l } \left( \frac{1}{n} \right)^l \left( \frac{n - 1}{n} \right)^{n - l} \ .
\end{align}
In the special case, where $r$ never appears, that is it appears $0$ times, in the specified position $k$, the above formula becomes:
\begin{align} \label{eq:Probability of 0 successes in n trials}
\forall r \in R \ \ \forall k, 1 \leq k \leq n, \quad &P( r \text{ \emph{never} appears in position } k \text{ in } n \text{ tours} ) \nonumber \\
&= \binom { n } { 0 } \left( \frac{1}{n} \right)^0 \left( \frac{n - 1}{n} \right)^n = \left( \frac{n - 1}{n} \right)^n \ .
\end{align}
More generally, the probability that restaurant $r$ appears exactly $l$ times in total in one of the $w$ distinct positions $k_1, \ldots, k_w, 1 < w \leq n$ is given by
\begin{align} \label{eq:Probability of l successes in w positions in n trials}
\forall r \in R \quad P( r \text{ appears } l \text{ times in } \mathrm{one} \text{ of positions } k_1, \ldots, k_w \text{ in } n \text{ tours} ) = \binom { n } { l } \left( \frac{w}{n} \right)^l \left( \frac{n - w}{n} \right)^{n - l} \ .
\end{align}
If $r$ never appears, that is it appears $0$ times, in anyone of the $w$ designated positions $k_1, \ldots, k_w, 1 < w \leq n$, the previous formula reduces to:
\begin{align} \label{eq:Probability of 0 successes in w positions in n trials}
\forall r \in R \quad &P( r \ \mathrm{never} \text{ appears in } \mathrm{any} \text{ of positions } k_1, \ldots, k_w \text{ in } n \text{ tours} ) \nonumber \\
&= \binom { n } { 0 } \left( \frac{w}{n} \right)^0 \left( \frac{n - w}{n} \right)^n = \left( \frac{n - w}{n} \right)^n \ .
\end{align}
We must emphasize that the above hold for every restaurant $r \in R$, for every position $k, 1 \leq k \leq n$, and for every set of positions $\{ k_1, \ldots, k_w \}, 1 < w \leq n$ . In other words, for every restaurant, the probability that it does not appear in one specific position in any of the $n$ tours is $\left( \frac{n - 1}{n} \right)^n$, and the probability that it does not appear in any of $w$ distinct positions in any of the $n$ tours is $\left( \frac{n - w}{n} \right)^n$.
According to the strategy scheme employed in the $m$-stop DKPRG, at the start of the second (third, etc.) day, the satisfied customers always go straight to the restaurant that eventually served them the previous day. We stress the word eventually because an agent may have failed to get lunch during stop $1$ of the previous day, but she may have succeeded during the second, third, or $m^{th}$ stop. This strategy is followed by all agents, something that guarantees that those customers that were satisfied on the previous day will remain satisfied today. Effectively, this strategy implies that the satisfied agents have “won” the game and from now on they do not need to solve their personalized TSP. The game will be played competitively by the unsatisfied agents of the previous day. We assume that they are aware of the unoccupied restaurants and, therefore, each one of them will once again solve her personalized TSP to compute her near-optimal tour. Of course, today the network of restaurants will consist of only the unoccupied restaurants, i.e., it will be significantly smaller that yesterday. The one-shot $m$-stop DKPRG of today will be different from the one-shot game of the previous day in a critical factor: the number of “actively competing” players will be significantly smaller. By the nature of the game, the number of active players at the beginning of stop $1$ of the present day is equal to the number of unsatisfied customers at the end of the previous day. The way the expected number of active players varies with each passing day is captured by the following Theorem <ref>.
The daily progression of the $m$-DKPRG is described by the following formulas, where $t$ stands for the day in question.
\begin{align}
VP_{t} &= \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \ , \ n_t \geq m, \ t = 1, 2 , \ldots \ , \label{eq:Vacancy Probability VPt} \\
\overline{R_{t}^{v}} &= n_{t} \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \ , \ n_t \geq m, \ t = 1, 2 , \ldots \ , \label{eq:Expected Number of Vacant Restaurants at End of Day t} \\
\overline{R_{t}^{r}} &= n_{t} \left( 1 - \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \right) \ , \ n_t \geq m, \ t = 1, 2 , \ldots \ , \label{eq:Expected Number of Reserved Restaurants at End of Day t} \\
\overline{A_{t}^{u}} &= n_{t} \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \ , \ n_t \geq m, \ t = 1, 2 , \ldots \ , \label{eq:Expected Number of Unsatisfied Customers at End of Day t} \\
\overline{A_{t}^{s}} &= n_{t} \left( 1 - \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \right) \ , \ n_t \geq m, \ t = 1, 2 , \ldots \ , \label{eq:Expected Number of Satisfied Customers at End of Day t} \\
n_{1} &= n \ , \label{eq:Expected Number of Active Agents on Day 1} \\
n_{t + 1} &= n_{t} \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \ , \ n_t \geq m, \ t = 2, 3, \ldots \ , \label{eq:Expected Number of Active Agents on Day t+1} \\
\overline{f_{t}} &=
\frac { \sum_{d = 1}^{t} n_{d} \left( 1 - \left( \frac{ n_{d} - m }{ n_{d} } \right)^{ n_{d} } \right) } { n }
= \frac { n - n_t \left( \frac{n_t - m}{n_t} \right)^{n_t} } { n }
\ , \ n_t \geq m, \ t = 1, 2 , \ldots \ . \label{eq:Expected Utilization on Day t}
\end{align}
The proof of the above formulas goes as follows.
* We first prove the auxiliary result that the vacancy probability at the beginning of stop $z, 1 \leq z \leq m$, of day $t$ is
\begin{align} \label{eq:Vacancy Probability VPtz}
VP_{t, z} = \left( \frac{ n_{t} + 1 - z }{ n_{t} } \right)^{ n_{t} } \ , \ 1 \leq z \leq m \ .
\end{align}
* Indeed, at the beginning of stop $1$ of day $t$, the expected number of restaurants that have not served any customer yet is equal to the expected number $n_{t}$ of active agents. On day $t$, the game is all about the active agents and the restaurants that have never been utilized up to now. At this moment in time all these restaurants are still unutilized, so vacancy is a certainty. Thus, indeed $VP_{t, 1} = 1$, which is in agreement with (<ref>) when $z = 1$.
* We recall that, according to our strategy, at the beginning of day $t$ the expected number of restaurants that have not served any customer yet is equal to the expected number $n_t$ of active players. At the beginning of stop $2$ of day $t$, the probability that one of these restaurants $r$ has not served lunch yet is precisely the probability that $r$ never appears in position $1$ in any tour of the active players. This last probability is given from (<ref>), where of course $n$ must now be replaced by $n_{t}$. Hence,
\begin{align} \label{eq:Vacancy Probability VPt2}
VP_{t, 2} = \left( \frac{n_{t} - 1}{n_{t}} \right)^{n_{t}} \ , \tag{ \ref{thr: DKPRG Quantitative Characteristics}.i }
\end{align}
which is also in agreement with (<ref>) when $z = 2$.
* Let us now carefully examine what happens during stop $2$ of day $t$ of the game. According to our scheme, those customers who have failed to get lunch at their first destination will immediately proceed to their second destination. For example, if customer $a$, who follows tour $T_{a} = ( l_0, l_1, \dots, l_n, l_{n+1} )$, was not served at restaurant $r_{l_1}$, she will try restaurant $r_{l_2}$. However, an added complication arises now. It may well be the case that $r_{l_2}$ is already occupied from stop $1$. In such a case $r_{l_2}$ is completely unavailable, i.e., it is now serving another active agent. In view of this fact, we may conclude that the restaurants that are vacant at the beginning of stop $3$ must satisfy two properties:
(P1) they must be vacant at the beginning of stop $2$, which means that must never appear in position $1$ in any tour of the active players, and
(P2) they must never appear in position $2$ in any tour of the active players.
The above are summarized more succinctly in the following rule.
(C) The restaurants that have not served any customer up to day $t$ and are still vacant at the beginning of stop $3$ of day $t$, never appear in position $1$ or position $2$ in any tour of the active players.
\begin{align} \label{eq:Vacancy Probability VPt3}
VP_{t, 3} = P( r \ \emph{never} \text{ appears in positions } 1 \text{ or } 2 \text{ in } n_t \text{ tours} )
\overset{ (\ref{eq:Probability of 0 successes in w positions in n trials}) } { = } \left( \frac{n_t - 2}{n_t} \right)^{n_t} \ , \tag{ \ref{thr: DKPRG Quantitative Characteristics}.ii }
\end{align}
which is again in agreement with (<ref>) when $z = 3$.
* The same reasoning can be employed to show that the vacancy probability $VP_{t, z}$ at the beginning of stop $z$ of day $t$ is
\begin{align}
VP_{t, z} = P( r \text{ \emph{never} appears in positions } 1, \ldots, z - 1 ) \overset{ (\ref{eq:Probability of 0 successes in w positions in n trials}) } { = } \left( \frac{n_t + 1 - z}{n_t} \right)^{n_t} \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.iii }
\end{align}
Hence, we have proved the validity of (<ref>).
* Finally, to calculate the probability that one of the restaurants $r$ that have not served any customer up to day $t$ is still vacant at the end of stop $m$ of day $t$, which in effect means at the end of day $t$, we must determine the probability that $r$ never appears in positions $1$, or, $2$, or $\ldots$, or $m$ in any tour of the active players. Thus,
\begin{align}
VP_{t} = P( r \text{ \emph{never} appears in positions } 1, \ldots, m ) \overset{ (\ref{eq:Probability of 0 successes in w positions in n trials}) } { = } \left( \frac{n_t - m}{n_t} \right)^{n_t} \ , \tag{ \ref{thr: DKPRG Quantitative Characteristics}.iv }
\end{align}
which verifies (<ref>), as desired. However, there is one final detail that we must emphasize here. Probabilities are real numbers taking values in the real line interval $[0, 1]$. For the equation (<ref>) to be valid, it must hold that $n_t \geq m$, otherwise it cannot be regarded as a probability. The physical meaning of this restriction is that (<ref>) is meaningful and correct as long as there are at least as many active players as stops $m$. If on some day $t$ we have that $n_t \leq m$, then the strategy we adhere to will make sure that all $n_t$ active players will manage to get lunch during day $t$.
* Let us clarify that our sample space consists precisely of the restaurants that have not served any customer up to day $t$. The expected number of restaurants in our sample space that remained vacant at the end of day $t$ is given by $\overline{R_{t}^{v}}$. First, we express probabilistically those restaurants of our sample space that remain vacant after all agents visit their first $m$ destinations. We define the family of random variables $R_{t j}^{v}, \ 1 \leq j \leq n_t$. The random variable $R_{t j}^{v}$ indicates whether restaurant $r_j$ is vacant or not at the end of day $t$. Specifically, if $R_{t j}^{v}$ has the value $1$, then restaurant $r_j$ is vacant at the end of day $t$, whereas if $R_{t j}^{v}$ is $0$, then $r_j$ is occupied.
\begin{align} \label{eq:Random Variables Rtjv}
R_{t j}^{v} =
\left\{
\begin{matrix*}[l]
1 & \text{if restaurant } r_j \text{ is \emph{vacant} at the end of day } t \\
0 & \text{otherwise}
\end{matrix*}
\right.
\ , \quad 1 \leq j \leq n_t \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.v }
\end{align}
By combining definition (<ref>) and equation (<ref>), we deduce that
\begin{align} \label{eq:Probability of Random Variables Rtjv}
R_{t j}^{v} =
\left\{
\begin{matrix*}[l]
1 & \text{with probability } \left( \frac{n_t - m}{n_t} \right)^{n_t} \\
0 & \text{with probability } 1 - \left( \frac{n_t - m}{n_t} \right)^{n_t}
\end{matrix*}
\right.
\ , \quad 1 \leq j \leq n_t \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.vi }
\end{align}
Having done that, we define the random variable $R_{t}^{v}$, which counts the the number of restaurants that are vacant at the end of day $t$.
\begin{align} \label{eq:Random Variable Rtv}
R_{t}^{v} = \sum_{ j = 1 }^{ n_t } R_{t j}^{v} \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.vii }
\end{align}
As always, in this probabilistic setting, we are interested not in the actual value of the random variable $R_{t}^{v}$, but in its expected value $E [ R_{t}^{v} ]$. In view of definition (<ref>) and the linearity of the expected value operator, we derive that
\begin{align} \label{eq:Expected Value of Random Variable R_{t}^{v}}
\overline{R_{t}^{v}} &= E \left[ R_{t}^{v} \right]
\overset{ (\ref{eq:Random Variable Rtv}) } { = }
E \left[ \sum_{ j = 1 }^{ n_t } R_{t j}^{v} \right] =
\sum_{ j = 1 }^{ n_t } E \left[ R_{t j}^{v} \right] \nonumber \\
&\overset{ (\ref{eq:Probability of Random Variables Rtjv}) } { = }
\sum_{ j = 1 }^{ n_t } \left( 1 \cdot \left( \frac{n_t - m}{n_t} \right)^{n_t} \right) = n_t \left( \frac{n_t - m}{n_t} \right)^{n_t}
\ , \tag{ \ref{thr: DKPRG Quantitative Characteristics}.viii }
\end{align}
which verifies (<ref>).
* Recall that our sample space contains exactly those restaurants that have not served any customer up to day $t$. $\overline{R_{t}^{r}}$ denotes the expected number of the restaurants of the sample space that were visited by an agent by the end of day $t$. Now, we define the family of random variables $R_{t j}^{r}, \ 1 \leq j \leq n_t$, which indicate whether restaurant $r_j$ is occupied or not at the end of day $t$. Specifically, if $R_{t j}^{r}$ has the value $1$, then restaurant $r_j$ is occupied at the end of day $t$, whereas if $R_{t j}^{r}$ is $0$, then $r_j$ is vacant.
\begin{align} \label{eq:Random Variables Rtjr}
R_{t j}^{r} =
\left\{
\begin{matrix*}[l]
1 & \text{if restaurant } r_j \text{ is \emph{occupied} at the end of day } t \\
0 & \text{otherwise}
\end{matrix*}
\right.
\ , \quad 1 \leq j \leq n_t \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.ix }
\end{align}
By combining definition (<ref>) and equation (<ref>), we deduce that
\begin{align} \label{eq:Probability of Random Variables Rtjr}
R_{t j}^{r} =
\left\{
\begin{matrix*}[l]
1 & \text{with probability } 1 - \left( \frac{n_t - m}{n_t} \right)^{n_t} \\
0 & \text{with probability } \left( \frac{n_t - m}{n_t} \right)^{n_t}
\end{matrix*}
\right.
\ , \quad 1 \leq j \leq n_t \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.x }
\end{align}
Having done that, we define the random variable $R_{t}^{r}$, which counts the the number of restaurants that are occupied at the end of day $t$.
\begin{align} \label{eq:Random Variable Rtr}
R_{t}^{r} = \sum_{ j = 1 }^{ n_t } R_{t j}^{r} \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.xi }
\end{align}
We are not interested in the actual value of the random variable $R_{t}^{r}$, but in its expected value $E [ R_{t}^{r} ]$. In view of definition (<ref>) and the linearity of the expected value operator, we derive that
\begin{align} \label{eq:Expected Value of Random Variable R_{t}^{r}}
\overline{R_{t}^{r}} &= E \left[ R_{t}^{r} \right]
\overset{ (\ref{eq:Random Variable Rtr}) } { = }
E \left[ \sum_{ j = 1 }^{ n_t } R_{t j}^{r} \right] =
\sum_{ j = 1 }^{ n_t } E \left[ R_{t j}^{r} \right] \nonumber \\
&\overset{ (\ref{eq:Probability of Random Variables Rtjr}) } { = }
\sum_{ j = 1 }^{ n_t } 1 \cdot \left( 1 - \left( \frac{n_t - m}{n_t} \right)^{n_t} \right) = n_t \left( 1 - \left( \frac{n_t - m}{n_t} \right)^{n_t} \right)
\ , \tag{ \ref{thr: DKPRG Quantitative Characteristics}.xii }
\end{align}
which verifies (<ref>).
* The rules of the game stipulate that the number of customers that have not managed to eat lunch at the end of day $t$ is equal to the number of restaurants that have not served any customer at the end of day $t$. Hence, their expected values are also equal, which means that $\overline{A_{t}^{u}} = \overline{R_{t}^{v}}$ and (<ref>) is proved.
* Likewise, the adopted strategy ensures that the number of the active players that succeeded in getting lunch at the end of day $t$ is equal to the number of restaurants that, although they had not served any agent up to day $t$, they managed to accommodate a customer by the end of day $t$. Hence, their expected values are also equal, which means that $\overline{A_{t}^{s}} = \overline{R_{t}^{r}}$ and (<ref>) is proved.
* We are now in a position that enables us to assert the expected number of active players.
* At the beginning of the first day, the numbers of active players is exactly $n$. This trivial observation confirms the initial condition (<ref>).
* As we have previously explained, the adopted strategy in the $m$-DKPRG ensures that the number of agents that have not got lunch at the end of day $t$ is always equal to the number of active players on day $t + 1$. Thus, the expected number of active agents on day $t + 1$ is equal to the expected number of unsatisfied agents at the end of day $t$:
\begin{align} \label{eq:Expected Number of Active Agents Equal to Unsatisfied Agents}
n_{t + 1} = \overline{A_{t}^{u}} \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.xiii }
\end{align}
By combining the previous result (<ref>) with (<ref>), we derive
\begin{align} \label{Expected Number of Active Agents}
n_{t + 1} = n_{t} \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \ , \tag{ \ref{thr: DKPRG Quantitative Characteristics}.xiv }
\end{align}
which establishes the validity of (<ref>), as desired.
* The expected utilization $\overline{f_t}$ for day $t = 1, 2 , \ldots$, is the ratio of the expected number of agents that were served during day $t$. This last numbers is equal to the expected number of customers that got lunch on day $1$, plus the expected number of the additional customers that got lunch on day $2$, and so on. The additional agents of day $t$ are precisely those agents that had failed to get lunch prior to day $t$, but succeeded in eating on day $t$. Their expected number is $\overline{A_{t}^{s}}$, which is given by equation (<ref>). Hence, the total number of agents that have eaten lunch up to and including day $t$ is given by
\begin{align} \label{eq:Total Expected Number of Satisfied Agents on Day t}
\sum_{d = 1}^{t} A_{d}^{s}
\overset{ (\ref{eq:Expected Number of Satisfied Customers at End of Day t}) } { = }
\sum_{d = 1}^{t} n_{d} \left( 1 - \left( \frac{ n_{d} - m }{ n_{d} } \right)^{ n_{d} } \right)
\ , \ t = 1, 2 , \ldots \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.xv }
\end{align}
An equivalent way to compute this exact number is by subtracting from the total number of agents $n$ the expected number of agents that failed to get lunch on day $t$, which is $\overline{A_{t}^{u}}$, which is given by equation (<ref>). Thus,
\begin{align} \label{eq:Expected Number of Remaing Unsatisfied Agents on Day t}
n - \overline{A_{t}^{u}}
\overset{ (\ref{eq:Expected Number of Unsatisfied Customers at End of Day t}) } { = }
n - n_t \left( \frac{n_t - m}{n_t} \right)^{n_t}
\ , \ t = 1, 2 , \ldots \ . \tag{ \ref{thr: DKPRG Quantitative Characteristics}.xvi }
\end{align}
Together the two formulas (<ref>) and (<ref>), allow us to conclude that
\begin{align} \label{Expected Utilization}
\overline{f_{t}}
\overset{ (\ref{eq:Total Expected Number of Satisfied Agents on Day t}) } { = }
\frac { \sum_{d = 1}^{t} n_{d} \left( 1 - \left( \frac{ n_{d} - m }{ n_{d} } \right)^{ n_{d} } \right) } { n }
\overset{ (\ref{eq:Expected Number of Remaing Unsatisfied Agents on Day t}) } { = }
\frac { n - n_t \left( \frac{n_t - m}{n_t} \right)^{n_t} } { n }
\ , \ t = 1, 2 , \ldots \ , \tag{ \ref{thr: DKPRG Quantitative Characteristics}.xvii }
\end{align}
which establishes the validity of (<ref>), as desired.
Let us now make an important observation: formula (<ref>) that we derived above, and which gives the expected number of vacant restaurants at the end of day $t$, is completely general and subsumes more special formulas found in the literature. Take for example the special case where $t = 1$ and $m = 1$. For these values, (<ref>) computes the expected number of vacant restaurants at the end of day $1$ for the standard one-stop KPRP. By subtracting this quantity from $n$, the number of initially available restaurants, and then dividing by $n$, we derive the expected utilization ratio for day $1$. Indeed
\begin{align} \label{eq:General Form Day 1 Utilization}
\overline{f_1} = \frac { n - n \left( \frac{n - 1}{n} \right)^n } { n } = 1 - \left( \frac{n - 1}{n} \right)^n \ .
\end{align}
One assumption that is taken for granted in the literature is that the number of agents $n$ tends to infinity. It is straightforward to see how the above formula simplifies when $n \to \infty$. We recall a very useful fact from calculus (see for instance [57]), namely that
\begin{align} \label{eq:The function Expx as a Limit}
\forall x \in \mathbb{R}, \quad \lim_{n \to \infty} \left( 1 + \frac{x}{n}\right)^{n} \to e^{x} \ .
\end{align}
Under this premise, we see that $\lim_{n \to \infty} f \to 1 - e^{-1}$, which is in complete agreement with a well-known result of the literature.
If we assume that $n \to \infty$, then the following approximations hold, where $t$ is the day in question.
\begin{align}
n_{t + 1} &\approx n e^{-t m} \ , \ t = 1, 2 , \ldots \ , \label{eq:Approximation of Expected Number of Active Agents on Day t+1} \\
\overline{A_{t}^{u}} &\approx n e^{-t m} \ , \ t = 1, 2 , \ldots \ , \label{eq:Approximation of Expected Number of Unsatisfied Customers on Day t} \\
\overline{A_{t}^{s}} &\approx n \left( 1 - e^{ - m } \right) e^{ - ( t - 1 ) m }
= n e^{ - ( t - 1 ) m } - n e^{ -t m } \ , \ t = 1, 2 , \ldots \ , \label{eq:Approximation of Expected Number of Satisfied Agents on Day t} \\
\overline{R_{t}^{v}} &\approx n e^{-t m} \ , \ t = 1, 2 , \ldots \ , \label{eq:Approximation of Expected Number of Vacant Restaurants on Day t} \\
\overline{R_{t}^{r}} &\approx n e^{ -(t - 1) m} - n e^{ -t m} \ , \ t = 1, 2 , \ldots \ , \label{eq:Approximation of Expected Number of Reserved Restaurants on Day t} \\
VP_{t} &\approx e^{-t m} \ , \ t = 1, 2 , \ldots \ , \label{eq:Approximation of Vacancy Probability on Day t} \\
\overline{f_{t}} &\approx 1 - e^{ -t m } \ , \ t = 1, 2 , \ldots \ . \label{eq:Approximate Expected Utilization on Day t} \\
\overline{f_{\infty}} &\approx 1 \ . \label{eq:Approximate Steady State Utilization}
\end{align}
The above approximations are easily proved as shown below.
* If we invoke property (<ref>), we deduce that $\left( \frac{n - m}{n} \right)^n \xrightarrow [n \to \infty] {} e^{-m}$ and $\left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \xrightarrow [n_t \to \infty] {} e^{-m}$. When $n$ and $n_t$ do not take very large values, these limits can only serve as good approximations. Thus, it is more accurate to write
\begin{align} \label{eq:Exponential Approximation}
\left( \frac{n - m}{n} \right)^n \approx e^{-m}
\quad \text{ and } \quad
\left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \approx e^{-m} \ . \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.i }
\end{align}
Formulas (<ref>) and (<ref>) imply that
\begin{align} \label{eq:Approximate Number of Active Agents on Day 2}
n_2 = n \left( \frac{n - m}{n} \right)^n
\overset{ (\ref{eq:Exponential Approximation}) } { \approx }
n e^{-m} \ . \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.ii }
\end{align}
In an identical manner, we see that
\begin{align} \label{eq:Approximate Number of Active Agents on Day 3}
n_3 = n_2 \left( \frac{n_2 - m}{n_2} \right)^{n_2}
\overset{ (\ref{eq:Exponential Approximation}) } { \approx } n_2 e^{-m}
\overset{ (\ref{eq:Approximate Number of Active Agents on Day 2}) } { \approx } n e^{-m} e^{-m} = n e^{-2 m} \ . \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.iii }
\end{align}
Following this line of thought, it is now routine to see that (<ref>) holds.
* The proof is trivial because the number of customers that have not eaten lunch by the end of day $t$ is equal to the number of active agents at the beginning of day $t + 1$.
Considering the fact that $\left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \xrightarrow [n_t \to \infty] {} e^{-m}$, we may deduce that $1 - \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \xrightarrow [n_t \to \infty] {} 1 - e^{-m}$. In this way we have derived the next approximation.
\begin{align} \label{eq:Approximate Rate of success}
1 - \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \approx 1 - e^{-m} \ . \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.iv }
\end{align}
It follows from (<ref>) that
\begin{align} \label{eq:Approximation of Expected Number of Active Agents on Day t}
n_{t} \approx n e^{ -(t - 1) m} \ . \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.v }
\end{align}
Together (<ref>) and (<ref>) imply that
\begin{align} \label{eq:Approximate Expected Number of Ssatisfied Customers}
\overline{A_{t}^{s}} &= n_{t} \left( 1 - \left( \frac{ n_{t} - m }{ n_{t} } \right)^{ n_{t} } \right)
\overset{ (\ref{eq:Approximation of Expected Number of Active Agents on Day t}), (\ref{eq:Approximate Rate of success}) } { \approx }
n \left( 1 - e^{ - m } \right) e^{ - ( t - 1 ) m } \nonumber \\
&= n e^{ - ( t - 1 ) m } - n e^{ - ( t - 1 ) m } e^{ - m }
= n e^{ - ( t - 1 ) m } - n e^{ -t m }
\ , \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.vi }
\end{align}
which proves equation (<ref>), as desired.
* Again, this result is obvious because the number of vacant restaurants at the end of day $t$ is equal to the number of unsatisfied customers at the end of day $t$.
* The number of restaurants that served a customer for the first time during day $t$ is equal to the number of agents that managed to get lunch for the first time during day $t$. Hence, the result is immediate.
* This can be easily shown by substituting in formula (<ref>) the approximation (<ref>).
* A good approximation for the total number of restaurants that were utilized during day $t$ can be found by subtracting from $n$ the approximate expected number of customers that did not have lunch on day $t$. In view of (<ref>), this implies that
\begin{align} \label{eq:Approximate Expected Utilization I}
\overline{f_{t}}
\overset{ (\ref{eq:Approximation of Expected Number of Vacant Restaurants on Day t}) } { \approx }
\frac { n - n e^{ -t m } } { n }
= 1 - e^{ -t m }
\ , \ t = 1, 2 , \ldots \ . \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.vii }
\end{align}
Alternatively, one may use the expected number of agents that ate lunch on day $t$, which can be computed as the sum $\sum_{d = 1}^{t} \overline{A_{d}^{s}}$. Using the relation (<ref>), this sum can be written as
\begin{align} \label{eq:Approximate Sum of Satisfied Customers}
\sum_{d = 1}^{t} n \left( 1 - e^{ - m } \right) e^{ - ( d - 1 ) m }
&= n \left( 1 - e^{ - m } \right) \sum_{d = 1}^{t}e^{ - ( d - 1 ) m } \nonumber \\
&= n \left( 1 - e^{ - m } \right) \frac { 1 - e^{ - t m } } { 1 - e^{ m } }
= n \left( 1 - e^{ - t m } \right)
\ , \ t = 1, 2 , \ldots \ . \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.viii }
\end{align}
In the above derivation, we used a well-know fact (see for instance [58]), namely that the sum of the first $t$ terms of a geometric sequence with first term $g_1$ and ration $\rho$ is given by the formula $g_1 + g_1 \rho + g_1 \rho^2 + \dots + g_1 \rho^{t - 1} = g_1 \frac { 1 - \rho^t } { 1 - \rho }$. This second way to estimate the expected utilization confirms, as expected, that
\begin{align} \label{eq:Approximate Expected Utilization II}
\overline{f_{t}} = \sum_{d = 1}^{t} \overline{A_{d}^{s}}
\overset{ (\ref{eq:Approximate Sum of Satisfied Customers}) } { \approx }
\frac { n \left( 1 - e^{ - t m } \right) } { n }
= 1 - e^{ -t m }
\ , \ t = 1, 2 , \ldots \ , \tag{ \ref{thr: Approximation of DKPRG Quantitative Characteristics}.ix }
\end{align}
which proves equation (<ref>), as desired.
* A trivial consequence of (<ref>), as $t \to \infty$.
To demonstrate how the exact formulas (<ref>) - (<ref>) reflect the daily evolution of the $m$-DKPRG we study five typical instances of the game. The first four are instances of $2$-DKPRG games with substantially different number of players. In the first four games, the number of steps $m$ is $2$, meaning that each agent may visit two restaurants if the need arises. In the first example the number of agents $n$ is $100$, a relatively small number, and its detailed progression is shown in Table <ref>. The steady state utilization is, as expected, $1$ and it is achieved by the end of day $3$.
2c||@ Number of stops $m$ 5c:=2
2c||@ Number of agents $n$ 5c:=100
2c||@ Day ($t$)
@ $n_t$
@ $VP_{t}$
@ $\overline{A_{t}^{s}} = \overline{R_{t}^{r}}$
@ $\overline{A_{t}^{u}} = \overline{R_{t}^{v}}$
@ $\overline{f_{t}}$
@ Start of day 1 c2 1 0 c2 0
@ End of day 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
This Table demonstrates the progression of the $m$-DKPRG for $m = 2$ and $n = 100$. It can be seen that all restaurants are utilized by the end of day $3$.
In the second example the number of agents $n$ is $1000$ and its progression is shown in Table <ref>. The steady state utilization is, as expected, $1$ and it is achieved by the end of day $5$, i.e., $2$ days later compared to the previous example.
2c||@ Number of stops $m$ 5c:=2
2c||@ Number of agents $n$ 5c:=1000
2c||@ Day ($t$)
@ $n_t$
@ $VP_{t}$
@ $\overline{A_{t}^{s}} = \overline{R_{t}^{r}}$
@ $\overline{A_{t}^{u}} = \overline{R_{t}^{v}}$
@ $\overline{f_{t}}$
@ Start of day 1 c2 1 0 c2 0
@ End of day 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
This Table demonstrates the progression of the $m$-DKPRG for $m = 2$ and $n = 1000$. One may ascertain that all customers eat lunch by the end of day $5$.
The third example is more meaningful and interesting because in this case the number $n$ of agents is $10^6$, which may be thought of as representing the average case. Table <ref> contains the analytical evolution of this instance. Even when confronted with a significant number of agents, the steady state utilization $1$ is rapidly achieved by the end of day $8$. Although it takes longer to reach that stage, utilization upwards of $0.98$ is established from day $2$.
2c||@ Number of stops $m$ 5c:=2
2c||@ Number of agents $n$ 5c:=1000000
2c||@ Day ($t$)
@ $n_t$
@ $VP_{t}$
@ $\overline{A_{t}^{s}} = \overline{R_{t}^{r}}$
@ $\overline{A_{t}^{u}} = \overline{R_{t}^{v}}$
@ $\overline{f_{t}}$
@ Start of day 1 c2 1 0 c2 0
@ End of day 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
This Table demonstrates the progression of the $m$-DKPRG for $m = 2$ and $n = 1000000$. Although in this case it takes longer to reach the steady state, utilization upwards of $0.98$ is established from day $2$.
The fourth example is instructive about the behavior of our strategy when a large number of agents is involved. In this case the number $n$ of agents is $10^9$ and, unsurprisingly, it takes $11$ days to reach the steady state utilization $1$. All the details of the progression of this game are given in Table <ref>. Careful observation of the data confirms a major characteristic of our distributed game: for $m = 2$ steps the first day utilization is at least $0.86$ and it goes over $0.98$ from day $2$.
2c||@ Number of stops $m$ 5c:=2
2c||@ Number of agents $n$ 5c:=1000000000
2c||@ Day ($t$)
@ $n_t$
@ $VP_{t}$
@ $\overline{A_{t}^{s}} = \overline{R_{t}^{r}}$
@ $\overline{A_{t}^{u}} = \overline{R_{t}^{v}}$
@ $\overline{f_{t}}$
@ Start of day 1 c2 1 0 c2 0
@ End of day 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
This Table demonstrates the progression of the $m$-DKPRG for $m = 2$ and $n = 1000000000$. One can see that the first day utilization is at least $0.86$ and it goes over $0.98$ from day $2$.
It is quite straightforward to convince ourselves that playing a $3$-stop game is better than playing a a $2$-stop game. A precise quantitative analysis of the resulting advantages can be performed by considering the exact formulas (<ref>) - (<ref>). Nonetheless, we believe it is expedient to showcase the difference with the following example. The present example resembles the previous one in that the number of agents is the same, namely $10^9$. However, this time each agent may visit up to three restaurants if need be. Such an instance, with a large number of agents, can serve as the best demonstration of the dramatic improvement that can be obtained by an increase in the number of steps. Indeed, the data in the Table <ref> corroborate this expectation, as one can now see that all restaurants are utilized by the end of day $8$, compared to day $11$ before, the utilization at the end of first day is already up to an impressive $0.95$ and becomes $1$, for all practical purposes, at the end of day $6$. This last example can be considered as a compelling argument that advocates the importance of topological analysis for the network of restaurants.
2c||@ Number of stops $m$ 5c:=3
2c||@ Number of agents $n$ 5c:=1000000000
2c||@ Day ($t$)
@ $n_t$
@ $VP_{t}$
@ $\overline{A_{t}^{s}} = \overline{R_{t}^{r}}$
@ $\overline{A_{t}^{u}} = \overline{R_{t}^{v}}$
@ $\overline{f_{t}}$
@ Start of day 1 c2 1 0 c2 0
@ End of day 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
@ Start of day [0,-2] + 1
@ End of day [0,-2] + 1
ifgt([0,-1], c1, [0,-1] * ( ( [0,-1] - c1 ) / [0,-1] )^[0,-1], 0)
ifgt([-1,-1], c1, ( ( [-1,-1] - c1 ) / [-1,-1] )^[-1,-1], 0)
ifgt([-2,-1], c1, [-2,-1] - ( [-2,-1] * [-1,0] ), [-2,-1])
ifgt([-3,-1], c1, [-3,-1] * [-2,0], 0)
ifgt([-4,-1], c1, ( c2 - [-1,0] ) / c2, 1)
This Table demonstrates the progression of the $m$-DKPRG for $m = 3$ and $n = 1000000000$.
The above examples were studied using the exact formulas (<ref>) - (<ref>). Tables <ref> - <ref> reflect the daily evolution of the above five instances of the $m$-DKPRG according to rigorous mathematical description provided by formulas (<ref>) - (<ref>). The next Figure <ref> is a graphical representation of the exact utilization $\overline{f_{t}}$ from all the previous examples, as shown in the Tables <ref> - <ref>. In this Figure, the generally excellent behavior of this scheme can be easily verified. We point out the rapid convergence to the steady state in a matter of few days and, especially, the superiority of the three stop policy. The latter achieves $0.95$ utilization from the first day and above $0.99$ from the second day.
[scale = 0.75]
scientific axes = clean,
x axis = attribute = Days, length = 12cm, ticks = tick unit = day, step = 1, label = [node style = fill = WordVeryLightTeal ] Days $t = 1, 2, \ldots$ ,
y axis = attribute = Utilization, length = 12cm, ticks = step = 0.01, label = [node style = fill = WordVeryLightTeal ] Expected utilization $\overline{f_{t}}$ ,
all axes = grid,
style sheet = strong colors,
visualize as line/.list = a, b, c, d, e,
a = label in legend = text = $10^2 \ Agents$ ,
b = label in legend = text = $10^3 \ Agents$ ,
c = label in legend = text = $10^6 \ Agents$ ,
d = label in legend = text = $10^9 \ Agents$ ,
e = label in legend = text = $10^9 \ Agents$
data [set = a]
Days, Utilization
1, 0.8673804
2, 0.9848263
3, 1.0
4, 1.0
5, 1.0
6, 1.0
7, 1.0
8, 1.0
9, 1.0
10, 1.0
11, 1.0
data [set = b]
Days, Utilization
1, 0.8649355
2, 0.9819923
3, 0.9978386
4, 0.9999921
5, 1.0
6, 1.0
7, 1.0
8, 1.0
9, 1.0
10, 1.0
11, 1.0
data [set = c]
Days, Utilization
1, 0.864665
2, 0.9816847
3, 0.9975216
4, 0.9996649
5, 0.9999549
6, 0.9999942
7, 0.9999995
8, 1.0
9, 1.0
10, 1.0
11, 1.0
data [set = d]
Days, Utilization
1, 0.8646647
2, 0.9816844
3, 0.9975212
4, 0.9996645
5, 0.9999546
6, 0.9999939
7, 0.9999992
8, 0.9999999
9, 1.0
10, 1.0
11, 1.0
data [set = e]
Days, Utilization
1, 0.9502129
2, 0.9975212
3, 0.9998766
4, 0.9999939
5, 0.9999997
6, 1.0
7, 1.0
8, 1.0
9, 1.0
10, 1.0
11, 1.0
This figure depicts the exact expected utilization $\overline{f_{t}}$, as given by equation (<ref>), for all five instances of the $m$-DKPRG studied in Tables <ref> - <ref>.
The above remarks must not diminish the value of the approximate formulas (<ref>) - (<ref>). Their value lies on the fact that they can provide easy to compute and particularly good approximations for large $n$. A simple comparison of Figure <ref> to the approximations shown in Figure <ref>, which corresponds to the case $m = 2$, and in Figure <ref>, which depicts the case where $m = 3$, ascertains their accuracy.
[scale = 0.75]
scientific axes = clean,
x axis = label = [node style = fill = WordVeryLightTeal ] Days $t = 1, 2, \ldots$ , length = 12cm, ticks = tick unit = day, step = 1 ,
y axis = label = [node style = fill = WordVeryLightTeal ] Number of stops $m = 2 \quad \overline{f_{t}} \approx 1 - e^{ - 2 t }$ , length = 9cm, ticks = step = 0.02 ,
all axes = grid,
visualize as smooth line,
data/format = function]
var x : interval [0.4 : 10] samples 100;
func y = 1 - exp( - 2 * x ) ;
This figure depicts the approximate expected utilization $\overline{f_{t}} \approx 1 - e^{ - 2 t }$ as given by equation (<ref>) for $m = 2$.
[scale = 0.75]
scientific axes = clean,
x axis = label = [node style = fill = WordVeryLightTeal ] Days $t = 1, 2, \ldots$ , length = 12cm, ticks = tick unit = day, step = 1 ,
y axis = label = [node style = fill = WordVeryLightTeal ] Number of stops $m = 3 \quad \overline{f_{t}} \approx 1 - e^{ - 3 t }$ , length = 9cm, ticks = step = 0.02 ,
all axes = grid,
visualize as smooth line,
data/format = function]
var x : interval [0.4 : 10] samples 100;
func y = 1 - exp( - 3 * x ) ;
This figure depicts the approximate expected utilization $\overline{f_{t}} \approx 1 - e^{ - 3 t }$ as given by equation (<ref>) for $m = 3$.
§ CONCLUSION
This work explored a completely new angle of the Kolkata Paise Restaurant Problem. The topological layout of the restaurants takes center stage in this new paradigm. Initially, we explicitly stated certain assumptions that are implicitly present in the standard formulation of the game. Having done that, we undertook the radical step to go past them and create an entirely new setting. The critical examination of the topological setting of the game unavoidably enhanced our perception regarding the locations of the restaurants and suggested a more realistic topological layout. We argued that their uniform distribution in the entire city area is the most logical, fair, and probable situation. As a result, we defined a new version of the game that is spatially distributed and, for this, is is aptly named the Distributed Kolkate Paise Restaurant Game (DKPRG).
The uniform probabilistic distribution of the restaurants enabled us to rigorously prove that, as their number $n$ increases, the restaurants get closer and the distance between adjacent restaurants decreases. In such a network, every customer has the opportunity to pass through more than one restaurants within the allowed time window. The agents now become travelling salesmen and this led us to suggest the innovative idea that TSP can be used to increase the chances of success in this game. We propose that each agent should use metaheuristics to solve her personalized TSP because metaheuristics produce near-optimal solutions very fast and as such can be easily used in practice. This culminated in the development of a new and more efficient strategy that achieves greater utilization.
After rigorously formulating DKPRG, we proved completely general formulas that assert the increase in utilization of our scheme. We established that utilization ranging from $0.85$ to $0.95$ is achievable. This was shown in great detail in Tables <ref> - <ref>, which depict the daily progress of characteristic instances of the DKPRG according to the rigorous mathematical description provided by the exact formulas (<ref>) - (<ref>). Apart from the exact formulas, we also derived the approximate formulas (<ref>) - (<ref>). They can be quite useful because they are considerably easier to compute and are exceedingly good approximations for large $n$. This fact is easily corroborated by comparing Figure <ref> to the approximations shown in Figures <ref> and <ref>. Let us remark that the derived equations generalize previously presented formulas in the literature.
It is worth mentioning that the fact that our strategy exhibits very rapid convergence to the steady state of utilization $1.0$ can be potentially used to address the following situation. An issue that remains and is common to almost all works in the literature is the simple matter that a near optimal utilization may not, in general, be optimal for every agent individually. A socially efficient outcome where every agent eats lunch and every restaurant gets a customer to serve, is not necessarily optimal for the individual customer, in the sense that an agent may get served in a restaurant of low preference. A possible solution to this might be to reset the game periodically. We expect that adopting a reset period, i.e., setting a specific period of days, after which the system is reset and the game starts from scratch, may alleviate this drawback. In any event, this idea for a future work will require further study and experimental evaluation of its usefulness. Finally, another possible direction for future work could include extensive experimental tests and further investigation of other versions of TSP. For instance, that there exists a more restrictive version of the TSP, the Travelling Salesman Problem with Time Windows (TSP-TW). TSP-TW is a constrained version of TSP in which the salesman must visit the cities within a specific time window. This version is even more complicated and difficult to solve. However, the inherent time constraints built-in the TSP-TW may provide for an even more realistic modeling of the DKPRG, so it is a research avenue that we believe is worth pursuing.
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|
11institutetext: University of Potsdam, Institute of Computer Science,
Potsdam, Germany
11email<EMAIL_ADDRESS>orcid:0000-0001-8888-7475
# The Coq Proof Script Visualiser (coq-psv)††thanks: Special thanks to
Sebastian Böhne who gave the idea and concept to the tool. 111This work was
presented during a talk at the Coq Workshop 2020
Mario Frank
The Coq proof assistant [2] is used in various scientific, industrial, and
teaching contexts and enables scientists, teachers, and students to exchange
formalisations in an easy way. But this exchange is limited to the vernacular
files and the respective LaTeX representation generated by coqdoc [3], for
example. Also, those representations merely contain the definitions and proof
scripts but not the information about the specific proof steps, i.e. the
current goals and hypotheses. Thus, in order to fully understand the specific
proofs, including the way how hypotheses change, the recipient needs to
acquire a compatible version of Coq. This can be cumbersome as the
installation of the specific Coq version and necessary additional tools is not
always straight-forward due to operating system and Opam version differences.
To our knowledge, there currently is no tool that is able to represent Coq
proofs on paper including all used tactics, goals and hypotheses in a readable
and printable manner as tree representations usually do not scale well for
long proofs.
## Applications.
The Coq Proof Script Visualiser aims at supporting the development, teaching,
and review process of proofs by exporting the proof scripts together with the
current proof state for each proof step. As all information about the proof is
then at hand, readers merely need to know the semantics of the used tactics to
understand the proof. Proof developers can benefit from additional warnings
given by coq-psv when superfluous structuring is used. Reviewers can retrace
the proof more easily without the need to install Coq. In teaching, the
generated output can be easily modified to be used as a cloze where students
have to fill in the matching tactic or introduced hypothesis, for example. And
most notably, the exported proof representations can be used for archiving
purposes and enriching the output of coqdoc.
## Approach.
Given a Coq file, coq-psv processes it using the Coq parsing functionality and
data structures. Although the output of coq-psv is linear, it internally
generates a labelled proof tree for each proof in the file as using a tree
structure simplifies the analysis and augmenting of the proof information. The
label itself contains the name and type of the proven statement (e.g. theorem
or lemma) together with the statement itself and the information, how the
proof was completed (Qed or Admitted). The tree consists of nodes that contain
the used tactic or command, and the resulting hypotheses and goals. While the
extraction of the label content is easy, building the tree nodes is much more
complex. This is due to the numerous ways in Coq to branch in a proof
situation, which includes bullets, brackets and the (deprecated) Focus
command. After extracting the labelled trees, they are translated to LaTeX
tables according to the given command line options. We will give insight into
the way we handle different branching options and why we do so in the talk.
Figure 1: Excerpt of a proof visualisation
## Versatile Use.
The coq-psv supports many command line options that define the output
behaviour. It is able to process both single Coq files and complete Coq
projects. Furthermore, it is possible to generate either complete LaTeX
documents or only the LaTeX tables. Also, all proofs from a vernacular file
can be exported either to one single file or to separate files per proof. The
latter simplifies the integration into other LaTeX documents. There are two
different table layouts: The first one mimics the goal and hypothesis
visualisation of CoqIDE while the second one represents the goals and
hypotheses in sequent style. Also, it is possible to hide goals and hypotheses
in the step in which they are created if all of them are handled by bullets.
This can reduce the size of the output significantly. Hiding hypotheses that
remain unchanged until the end of the (sub-)proof and marking them as
invariant on introduction is also supported, which again condenses the size of
the tables (see Figure 1). Finally, the layout of the tables can be modified
after the generation. This can be done by overriding the default LaTeX
commands used in the generated files before importing them. For example the
size of the table itself and of specific table columns can be set to a desired
value without having to modify the table itself.
## Related work.
There are already some approaches to representing proof scripts in a more
readable and independent way (see [5, 7, 8, 9, 11, 10], for example). But
except for [5] and [11], all of them represent the proofs only as trees or
diagrams. While trees and diagrams can give a good overview about the
structure of a proof, printing them in a readable way can be problematic for
long proofs. Although the approach described in [5] supports both a tree and a
Fitch style representation, the latter only shows the current step and it does
not seem to be possible to export the complete proof as LaTeX file, for
example. Also, only ProofWeb [5], Proviola [11], ProofTree [10] and Traf [7]
are targeting Coq and the two latter only work in a live session with Coq and,
for example, ProofGeneral [1]. Furthermore, Traf uses ProofTree and both seem
to have been discontinued for more than two years. according to the project
pages [10, 12]. Proviola is able to generate an animated version of a proof
script giving good insight into the proving process, but it does not focus on
printing the output on paper. Anther project makes Coq proofs available in the
browser (see [6]) which makes the local installation of Coq unnecessary and
improves usability as documents can be shared. But it does not seem to be
possible to generate a printable version with all necessary information as
coq-psv does.
## Current state and future work.
The tool should be currently seen as proof of concept and supports Coq 8.10.
It is implemented in OCaml and can be installed from our Opam repository as
described on the coq-psv project page [4]. There are currently some minor
issues in the vertical alignment of the used tactic. But even non-trivial
proofs can be visualised decently as can be seen in some examples on the
project page. It is possible to generate LaTeX and PDF files. Obviously, there
are many options for future work. For example, one could hide tactics like
clear, rename and move and give them a special treatment instead: for
instance, marking cleared hypotheses, annotating identifier changes with the
substitution pattern and just change the position of the displayed hypotheses.
Then, a support for modules in files should be incorporated. Furthermore,
supporting especially HTML output is a main aspect of future work. We aim at
integrating such an improved coq-psv into coqdoc. Additionally, coq-psv could
be extended to improve the proof scripts themselves. For instance, structure
improvements can be suggested. Especially, unused or irrelevant hypotheses
might be detected and automatically deleted, if desired. This is not an easy
task since it may change the names of the hypotheses appearing later on.
Finally, one could give the option to conduct an automatic reordering of the
lines in a proof script.
## References
* [1] David Aspinall. Proof General: A Generic Tool for Proof Development. In Susanne Graf and Michael Schwartzbach, editors, Tools and Algorithms for the Construction and Analysis of Systems, pages 38–43. Springer, 2000.
* [2] The Coq proof assistant. http://coq.inria.fr. (Last visited 18.06.2020).
* [3] The coqdoc wiki page. https://github.com/coq/coq/wiki/Coqdoc. (Last visited 18.06.2020).
* [4] Coq Proof Script Visualiser project page. https://www.cs.uni-potsdam.de/coq-psv. (Last visited 18.06.2020).
* [5] Maxim Hendriks, Cezary Kaliszyk, Femke Van Raamsdonk, and Freek Wiedijk. Teaching Logic Using a State-of-the-Art Proof Assistant. Acta Didactica Napocensia, 3(2):35–48, 2010.
* [6] The JSCoq project page. https://jscoq.github.io/. (Last visited 18.06.2020).
* [7] Hideyuki Kawabata, Yuta Tanaka, Mai Kimura, and Tetsuo Hironaka. Traf: A Graphical Proof Tree Viewer Cooperating with Coq Through Proof General. In Sukyoung Ryu, editor, Programming Languages and Systems, pages 157–165, Cham, 2018. Springer International Publishing.
* [8] Tomer Libal, Martin Riener, and Mikheil Rukhaia. Advanced Proof Viewing in ProofTool. In Christoph Benzmüller and Bruno Woltzenlogel Paleo, editors, Proceedings Eleventh Workshop on User Interfaces for Theorem Provers (UITP 2014), volume 167 of EPTCS, pages 35–47, 2014.
* [9] Jorge Pais and Álvaro Tasistro. Proof Assistant Based on Didactic Considerations. Journal of Universal Computer Science, 19(11):1570–1596, 2013.
* [10] The ProofTree project page. https://askra.de/software/prooftree/. (Last visited 18.06.2020).
* [11] Carst Tankink, Herman Geuvers, James McKinna, and Freek Wiedijk. Proviola: A tool for proof re-animation. CoRR, abs/1005.2672, 2010.
* [12] The Traf project page. https://github.com/hide-kawabata/traf. (Last visited 18.06.2020).
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# Reconfigurable magnonic mode-hybridisation and spectral control in a
bicomponent artificial spin ice
Jack C. Gartside Blackett Laboratory, Imperial College London, London SW7
2AZ, United Kingdom These authors contributed equally Corresponding author
e-mail<EMAIL_ADDRESS>Alex Vanstone Blackett
Laboratory, Imperial College London, London SW7 2AZ, United Kingdom These
authors contributed equally Troy Dion Blackett Laboratory, Imperial College
London, London SW7 2AZ, United Kingdom London Centre for Nanotechnology,
University College London, London WC1H 0AH, United Kingdom Kilian D. Stenning
Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
Daan M. Arroo London Centre for Nanotechnology, University College London,
London WC1H 0AH, United Kingdom Hide Kurebayashi London Centre for
Nanotechnology, University College London, London WC1H 0AH, United Kingdom
Will R. Branford Blackett Laboratory, Imperial College London, London SW7
2AZ, United Kingdom
###### Abstract
Strongly-interacting nanomagnetic arrays are finding increasing use as model
host systems for reconfigurable magnonics. The strong inter-element coupling
allows for stark spectral differences across a broad microstate space due to
shifts in the dipolar field landscape. While these systems have yielded
impressive initial results, developing rapid, scaleable means to access a
broad range of spectrally-distinct microstates is an open research problem.
We present a scheme whereby square artificial spin ice is modified by widening
a ‘staircase’ subset of bars relative to the rest of the array, allowing
preparation of any ordered vertex state via simple global-field protocols.
Available microstates range from the system ground-state to high-energy
‘monopole’ states, with rich and distinct microstate-specific magnon spectra
observed. Microstate-dependent mode-hybridisation and anticrossings are
observed at both remanence and in-field with dynamic coupling strength tunable
via microstate-selection. Experimental coupling strengths are found up to
$g/2\pi=0.15\leavevmode\nobreak\ \text{GHz}$. Microstate control allows fine
mode-frequency shifting, gap creation and closing, and active mode number
selection.
## Introduction
The field of magnonics aims to employ spin-waves to propagate and process
information[1, 2]. Spin-waves offer a host of attractive benefits as data
carriers including low heat generation, power consumption[3] and coherent
coupling to photons[4], phonons[5] and other magnons[6, 7, 8, 9]. Functional
magnonics has proliferated in recent years, with wide-ranging applications
from transistors[10] to multiplexers[11] and logic gates[12]. As the
complexity of magnonic designs increases, so does the demand for versatile,
reconfigurable host systems.
Recently, a family of metamaterials termed reconfigurable magnonic crystals
(RMC)[13, 14, 15, 16, 17] has made strong progress in answering this need.
Typically comprising discrete nanopatterned magnetic elements closely-packed
in arrays to promote strong dipolar coupling, RMC support multiple microstates
and exhibit distinct microstate-dependent magnonic dynamics and spectra with
diverse functional benefits. A subset of RMC has emerged based on artificial
spin ice (ASI) arrays[18, 19, 20, 21, 17, 22] where geometrical frustration
gives rise to a vastly degenerate microstate space that features a long-range
ordered ground state [23, 24] and high-energy ‘magnetic monopole’-like excited
states[25, 26]. The potential to leverage these states for their magnonic
properties is great and studies into fundamentals of state-spectra
correspondence[27, 28, 29, 30, 31, 32, 33] have set the scene for a new
generation of ASI-based RMC designs. An open problem in the field is
developing reliable, versatile and rapid means for microstate access. While
ASI possesses a huge range of states, they remain largely unavailable for
magnonic exploitation due to state preparation techniques being overly
simplistic (for example global-field protocols which may only prepare
saturated or randomly demagnetised, unrepeatable states), overly slow
(surface-probe microscope nanomagnetic writing techniques[34, 24, 35, 36]
which may prepare any state but on timescales unsuitable for technological
integration) or difficult to realise with current nanofabrication techniques
(for example, recently proposed multi-level stripline technique[37]). In the
absence of such techniques, ASI systems have been modified to allow access to
an enhanced microstate range using global fields, ‘magnetic charge ice’ which
rotates a subset of bars in square ASI to allow global-field preparation of
three microstates[35, 38, 39] (types 1-3 as seen in fig. 1) or bar subset
modification via either material[40], shape-anisotropy[30] or exchange-
bias[41]. The magnetic charge ice case is elegant, but the way in which bars
are rotated leads to greater separations between neighbouring elements so that
greater density is required to achieve an appreciable interelement coupling
required for collective excitations. Moreover, the rotation means different
sublattices will in general experience different effective global excitation
and bias fields.
Here, we present a square ASI with a ‘staircase’-pattern subset of width-
modified bars. Shown in figure 1, this enables preparation of four distinct
type 1-4 microstates (fig. 1 b-e), g-j) including the typically elusive
ground-state (type 1) and ‘monopole’ states (types 3 and 4). The four states
display rich and distinct magnonic spectra with fine control over mode
frequency shifting, gap opening and tuning, and number of active modes.
Microstate-dependent mode-hybridisation and anticrossings are observed with
coupling-strength and gap width tunable via state selection. Selective mode-
hybridisation offers reconfigurable mode profile and index control in-field
and crucially at remanence.
Figure 1: Schematic of width-modified square and high-density square
reconfigurable magnonic crystals and their type 1-4 microstates and hysteresis
loops. $y$ and $x$ array axes are referred to as ‘ground-state’ and ‘monopole’
orientations throughout this work.
a) Scanning electron micrograph of the square sample. Bars are 830 nm long,
230 nm (wide-bar) and 145 nm (thin-bar) wide, 20 nm thick with 120 nm vertex
gap (bar-end to vertex-centre).
b) Scanning electron micrograph of the high-density square sample. Bars are
600 nm long, 200 nm (wide-bar) and 125 nm (thin-bar) wide, 20 nm thick with
100 nm vertex gap.
c-f) MOKE hysteresis loops of S sample in ‘ground state’ (c) and ‘monopole’
(d) orientations, HDS sample in ‘ground state’ (e) and ‘monopole’ (f)
orientations. Blue points show fully-saturating hysteresis loop, orange points
show minor loops with maximum positive field value chosen to prepare sample in
type 1 (c,e), type 4 (d) and type 3 (f) states before sweeping back to
negative saturation.
g-j) Magnetic force microscope images of type 1-4 microstates. Type 1 and 4
states have inset SEM images showing the relative orientation of
$\mathbf{H}_{ext}$ to the width-modified subsets required for state
preparation. Type 2 and 3 states may be prepared in either $\pm 90^{\circ}$
field orientation. Type 1 and 4 states are often termed ‘ground state’ and
‘monopole’ state in artificial spin ice.
k-n) Magnetic charge dumbbell schematic of type 1-4 microstates.
## Results and Discussion
### Microstate access via width-modification
Samples were designed by taking square ASI and increasing the width of one
nanobar subset. Two samples were fabricated, square (S, fig. 1 a) and high-
density square (HDS, fig. 1 b), using an electron beam lithography liftoff
process and thermal deposition of Ni81Fe19. Sample dimensions were selected
such that the wider bar subset may be magnetically reversed via a global-field
without thin-bars also reversing from the combination of global-field
$\mathbf{H}_{ext}$ and local dipolar field $\mathbf{H}_{loc}$, satisfying
$\mathbf{H}_{c2}>\mathbf{H}_{ext}+\mathbf{H}_{loc}>\mathbf{H}_{c1}$ with
$\mathbf{H}_{c1}$ and $\mathbf{H}_{c2}$ the wide and thin-bar coercive fields
respectively ($\mathbf{H}_{c1}$ and $\mathbf{H}_{c2}$ visible in fig. 1 MOKE
loops). This enables preparation of the entire range of ordered, ‘pure’ (i.e.
single vertex type across the array) microstates. Arrays comprising identical
bars may only access a single pure microstate by saturating with global-field
(type 2). The width-modification employed here allows global-field access to
four pure states with distinct local-field profiles and corresponding magnonic
spectral dynamics. Microstates are shown via magnetic force microscope (MFM)
(fig. 1 g-j) and magnetic charge schematics (fig. 1 k-n). The S sample
comprises bars of 830 nm $\times$ 230 nm (wide-bar), 145 nm (thin-bar)
$\times$ 20 nm, 120 nm vertex gap (defined bar end to vertex centre). The HDS
sample comprises bars of 600 nm $\times$ 200 nm (wide-bar), 125 nm (thin-bar)
$\times$ 20 nm, 100 nm gap.
Bars are widened in alternating $y$-axis columns (axes defined in fig. 1 a)
such that wide-bars may be reversed from a saturated background state (type 2,
fig. 1 h,l) without reversing thin-bars. If the global-field
$\mathbf{H}_{ext}$ is oriented along the $y$-axis, reversing only wide-bars
from a $\hat{y}$-saturated state leaves the system in the antiferromagnetic
type 1 state (fig. 1 g,k), which forms the ASI ground state with and without
width-modification[23, 42, 43]. Here the microstate allows the maximum amount
of inter-bar dipolar flux-closure and lowest system energy. If
$\mathbf{H}_{ext}$ is oriented along the $x$-axis, reversing wide-bars results
in the type 4 state (fig. 1 j,n), termed a ‘monopole’ or ‘all-in, all-out’
state[44, 43, 45, 46] with four like-polarity magnetic charges at each vertex,
highly-repulsive inter-bar dipolar field interactions and maximum system
energy. If $\mathbf{H}_{ext}$ has any angular misalignment from the width-
modified columns, one of the $\pm 45^{\circ}$ wide-bars will experience a
higher field along its easy-axis, resulting in that bar reversing at lower
$\mathbf{H}_{ext}$. The resulting state with just one of the $\pm 45^{\circ}$
wide-bars reversed is the type 3 state (fig. 1 i,m) with three like-polarity
and one opposite polarity magnetic charge per vertex. In experiment there is
always some angular misalignment and the array will transition between states
2 and 4 via state 3. The field window in which state 3 exists may be broadened
by deliberately increasing angular misalignment. The S array may access type
1-4 states, the HDS array may access states 1-3 but not 4 as the increased
$\mathbf{H}_{loc}$ magnitudes arising from smaller inter-bar separation leads
to spontaneous reversal of thin-bars from a thin-bar majority charge type 3 to
a wide-bar majority type 3 when attempting state 4 access, i.e.
$\mathbf{H}_{ext}+\mathbf{H}_{loc}>\mathbf{H}_{c-thin}$.
We analyse mode frequencies following the Kittel equation[47]
$f=\frac{\mu_{0}\gamma}{2\pi}\sqrt{\mathbf{H}(\mathbf{H}+\mathbf{M})}$ in the
k=0 limit applicable to this work, with $\gamma$ the gyromagnetic ratio and
$\mathbf{H}=\mathbf{H}_{ext}+\mathbf{H}_{loc}$. The local dipolar field
landscape varies greatly between microstates, with resulting distinct
microstate-dependent magnon spectra.
### Microstate-dependent magnonic spectra
Figure 2: Differential ferromagnetic resonance spectra of square (S) and high-
density square (HDS) samples with corresponding micromagnetic simulations.
Peak amplitude occurs at boundary between white and black bands.
Samples were saturated in 1000 mT negative field then swept in positive field
direction, with relative field orientation indicated in inset scanning
electron micrographs. Measurements were performed at room temperature.
Fields were swept from $\pm 300\leavevmode\nobreak\ $ mT, with full sweeps
(a-d), 0-40 mT sweeps around the coercive fields (e-h) and micromagnetic
MuMax3 simulations of the coercive field region (i-l) presented. Sample
geometry and $\mathbf{H}_{ext}$ orientation are shown inset. Switching fields
are labelled by vertical dashed lines, a and b subscripts refer to separate
$\pm 45$ and $\mp 45$ subset reversal where applicable, type 3- and 3+ refer
to thin-bar majority and wide-bar majority type 3 states respectively. From
left to right, vertical columns of spectra relate to samples: S (‘ground-
state’ orientation), S (‘monopole’ orientation), HDS (‘ground-state’
orientation), HDS (‘monopole’ orientation).
Broadband FMR spectra were measured using a flip-chip method with samples
excited by a coplanar waveguide. Frequency resolution is 20 MHz. For S and HDS
samples, spectra were taken with $\mathbf{H}_{ext}$ in $\hat{y}$ (‘ground-
state’ orientation, as in fig. 1 a) and $\hat{x}$ (‘monopole’ orientation).
Samples were saturated at $\mathbf{H}_{ext}=-1000\leavevmode\nobreak\ $ mT
then swept from -300 mT to 300 mT, above the saturation fields observed in
MOKE data to examine spectral effects when shape anisotropy is overcome.
Accompanying micromagnetic simulations were performed using MuMax3.
Figure 2 shows differential FMR spectra for each sample and orientation at
$\pm 300\leavevmode\nobreak\ $ mT field range (fig. 2 a-e, relative
$\mathbf{H}_{ext}$ orientation inset) and $0-40\leavevmode\nobreak\ $ mT range
(fig. 2 f-j) with corresponding simulated spectra (fig. 2 k-o).
Spectra exhibit two main Kittel-like modes, the lower and higher frequency
modes corresponding to bar-centre localised modes in the wide $W_{1}$ and
thin-bars $T_{1}$ respectively. This correspondence is evidenced by frequency
jumps and $\frac{\partial f}{\partial\mathbf{H}}$ sign inversions indicating
bar reversal[48] in the low- and high-$f$ modes at $\mathbf{H}_{c1}$ and
$\mathbf{H}_{c2}$ respectively, matching switching fields observed via MOKE
(fig. 1 c-f). Higher relative amplitude of the low-$f$ mode matches the larger
sample volume share of the wide-bar, simulated spatial mode-power maps support
the mode-bar correspondence. The wide-bar exhibits two higher-index modes W2
and W3, occurring at lower frequency relative to W1 due to the backward-volume
wave nature of the modes. Simulated spatial mode profiles are shown in
supplementary figure 1. Higher-order modes are expected in the thin-bar and
observed in simulation, but fall below the amplitude threshold for
experimental detection. In addition to offering two well-defined frequency
channels, the different width bar subsets allow clear identification of which
subset has reversed or undergone microstate-dependent frequency shifting.
For $\mathbf{H}_{ext}>$ 200 mT, thin and wide-bar modes tend to the same
frequency as shape-anisotropy is overcome and bar magnetisation rotates from
an Ising-like state to lie parallel to $\mathbf{H}_{ext}$. At these
$\mathbf{H}_{ext}$, the bar demagnetising fields become negligble and the
Kittel equation is dominated by $\mathbf{H}_{ext}$.
At lower fields around $\mathbf{H}_{c1}$ and $\mathbf{H}_{c2}$, rich and
distinct spectra are observed between samples and orientations. Figure 2 e)
shows S sample spectra in ‘ground-state’ orientation. At
$\mathbf{H}_{ext}=0\leavevmode\nobreak\ $ the system is in a negatively-
saturated type 2 microstate (fig. 1 h,l), the $M_{x}$ component of all bars
oriented against positive $\mathbf{H}_{ext}$ and both modes exhibiting
negative $\frac{\partial f}{\partial\mathbf{H}}$. At
$\mathbf{H}_{c1}=16\leavevmode\nobreak\ $ mT the wide-bars reverse, its mode
jumping 6.2-7.8 GHz and displaying positive $\frac{\partial
f}{\partial\mathbf{H}}$. The thin-bar mode is blueshifted 0.1 GHz at
$\mathbf{H}_{c1}$ due to the change in local dipolar field landscape as the
system enters a type 1 microstate (fig. 1 g,k). For
$\mathbf{H}_{c1}<\mathbf{H}_{ext}<\mathbf{H}_{c2}$ the system is in a type 1
state, the two modes exhibiting opposite $\frac{\partial
f}{\partial\mathbf{H}}$ and crossing at $\mathbf{H}_{ext}=23mT$. Opposing
frequency gradients and presence of a mode crossing in this field range afford
sensitive mode and gap tunability via $\mathbf{H}_{ext}$. At
$\mathbf{H}_{c2}=29\leavevmode\nobreak\ $ mT the thin-bars reverse, preparing
a type 2 state aligned with $\mathbf{H}_{ext}$ and redshifting the wide-bar
mode 0.1 GHz via the shift in dipolar field landscape. Above $\mathbf{H}_{c2}$
both modes exhibit positive $\frac{\partial f}{\partial\mathbf{H}}$.
Rotating $\mathbf{H}_{ext}$ $90^{\circ}$ accesses the ‘monopole’ orientation.
Fig. 2 b) shows that at high-field saturated states, ‘monopole’ and ‘ground-
state’ orientations behave similarly. Around the coercive fields, fig. 2 f)
shows stark spectral differences between the orientations. While the ‘ground-
state’ orientation transitions directly between a type 2 and type 1 state via
simultaneous reversal of both wide-bars, the highly-unfavourable type 4
dipolar field landscape separates the wide-bar switching into two distinct
reversal events occurring at different fields. At
$\mathbf{H}_{ext}=13\leavevmode\nobreak\ $ mT the $\pm 45^{\circ}$ subset of
wide-bars better aligned to $\mathbf{H}_{ext}$ reverses, placing the system in
a ‘thin-bar majority’ type 3 state (fig. 1 i,m) where both thin-bars and a
single wide-bar share like-polarity charges. This splits the low frequency
mode as half the wide-bars ($\pm 45^{\circ}$) reverse while the rest ($\mp
45^{\circ}$) remain aligned against $\mathbf{H}_{ext}$. The reversed wide-bar
mode jumps from 6.0-7.7 GHz and exhibits positive $\frac{\partial
f}{\partial\mathbf{H}}$. The unswitched wide-bar mode is blueshifted 0.3 GHz
and the thin-bar mode redshifted 0.1 GHz by the type 3 dipolar field
landscape. This reduces the gap between unswitched wide and thin-bar modes by
0.4 GHz without modifying the magnetisation state of either bar, demonstrating
the degree of spectral control available via microstate engineering. Remaining
wide-bars reverse at 20 mT, placing the system in a type 4 microstate (1 j,n)
and unifying the wide-bars in a single 7.5 GHz mode, redshifted 0.1 GHz
relative to the already-reversed wide-bar mode. Thin and wide-bar modes now
occupy the same frequency, obscuring the expected thin mode blueshift in the
experimental data. The mode gap width may be modified by varying the relative
bar widths at the fabrication stage, and the overlap here is a consequence of
the specific dimensions employed. At $\mathbf{H}_{c2}=\leavevmode\nobreak\ $
23 mT the thin-bar reverses, placing the system in a type 2 state aligned with
$\mathbf{H}_{ext}$ and restoring the mode gap.
Figure 3: Differential ferromagnetic resonance spectra taken while negatively
sweeping $\mathbf{H}_{ext}$ after microstate preparation in positive field.
Microstates were prepared by -1000 mT saturation then applying the positive
field required to reverse the desired bars, hence differing positive field
limits for different microstate spectra.
a-d) Experimental spectra for S sample microstates taken in ‘ground-state’
(a,b) and ‘monopole’ (c,d) orientations.
e-h) Simulated S sample microstate spectra for ‘ground-state’ (e,f) and
‘monopole’ (g,h) orientations.
i) Mode peak-extractions for all S sample microstate-spectra.
j-l) Experimental spectra for HDS sample microstates taken in ‘ground-state’
(j,k) and ‘monopole’ (l) orientations. ‘Monopole’ orientation signal-to-noise
is lower due to array-waveguide alignment issues. Modes are still well-
resolved and correspond well with simulation.
m-o) Simulated HDS sample microstate spectra for ‘ground-state’ (m,n) and
‘monopole’ (o) orientations.
p) Mode peak-extractions for all HDS sample microstate-spectra.
To demonstrate the spectral control available via array design choices, the
HDS sample was fabricated. By reducing bar-length and inter-bar separation,
stronger dipolar interactions between bars and greater variation in the
dipolar field landscape is achieved, resulting in larger spectral shifts when
transitioning state. Fig. 2 g) shows the HDS ‘ground-state’ orientation,
qualitatively matching that of the S sample but with frequency shifts of 0.2
GHz (twice that observed in the S sample) and a broadened type 1 field window
due to the increased stability. Fig. 2 h) shows the HDS ‘monopole’
orientation, again qualitatively matching that of the S sample (with enhanced
0.3 GHz frequency shifts) up to 23 mT where the system transitions from a
thin-bar majority type 3 state to a wide-bar majority type 3 rather than type
4. A wide-bar majority type 3 persists between 23-30 mT. At 30 mT the
remaining $\pm 45^{\circ}$ thin-bar reverses, causing a 0.3 GHz redshift in
the thin-bar mode and transitioning to a type 2 state.
### Negative field evolution of microstate-dependent magnonic spectra
So far spectra have been measured while positively sweeping $\mathbf{H}_{ext}$
after negative saturation. This allows study of the system as it evolves
through a range of microstates, but each microstate is stable in a limited
field window. Alternatively, states may be prepared via negatively saturating
then applying a microstate-specific positive field (i.e. 22 mT for the S
sample type 1 state, fig. 2 e), then recording spectra while negatively
sweeping $\mathbf{H}_{ext}$ until saturation. This allows mode dynamics to be
studied for each microstate over its entire stable field range, revealing
additional spectral details not accessed in fig. 2 e-h) 0-40 mT sweeps. Field
protocols correspond to the orange traces on the figure 1 c,d,e,f) MOKE loops.
Figure 3 shows negatively-swept spectra for the S (FMR panels a-d),
simulations e-h) and HDS samples (FMR j-m), simulations n-q) for all
microstates and orientations alongside mode extractions (S sample i), HDS r)
allowing state comparison. Figure 3 a) shows the S sample ‘ground-state’
orientation prepared in a type 1 state at $\mathbf{H}_{ext}=22$ mT. At 22 mT
thin and wide-bar modes occupy a single frequency at 8 GHz. As field is
negatively swept, modes exhibit opposite gradient due to opposing wide and
thin bar magnetisation, reaching a maximum mode-frequency gap of 3 GHz at -16
mT, after which the wide-bars reverse. This prepares a type 2 state, with a
wide-bar frequency jump and thin-bar redshift as observed in the 0-40 mT
positive sweeps. The broadly tunable 0-3 GHz gap and wide field-stability
window of type 1 state are desirable for functional magnonic systems where
mode-frequency gap control is crucial.
Figure 3 b) shows the ‘ground-state’ orientation S sample prepared in a type 2
state at 10 mT. Sweeping $\mathbf{H}_{ext}$ negatively, both modes exhibit a
constant gradient and frequency gap of 2 GHz. Figure 3 c) shows a monopole-
orientation type 3 spectra. A 21 mT preparation field reverses half the wide-
bars, preparing a thin-bar majority type 3 state. The preparation here is
distinct from earlier discussion of type 3 states in the HDS sample, where a
well-defined $\pm 45^{\circ}$ subset reverses due to closer alignment to
$\mathbf{H}_{ext}$. Here, the Gaussian spread[25] of $\mathbf{H}_{c1}$
throughout the system due to nanofabrication imperfections (termed quenched
disorder[49, 25, 50]) is leveraged for state-preparation. By selecting a 21 mT
field at the centre of the $\mathbf{H}_{c1}$ distribution, half the wide-bars
are reversed and on average the system placed in a type 3 state, with a random
distribution of $\pm 45^{\circ}$ wide-bars reversed. While sweeping field back
from 21 mT the thin-bar exhibits negative $\frac{\partial
f}{\partial\mathbf{H}}$. The wide-bar mode is split into reversed and
unreversed modes exhibiting opposite $\frac{\partial f}{\partial\mathbf{H}}$
sign. The two modes should cross at 5 mT if no deviation from Kittel-like
behaviour is observed. However, the modes are bent away from each other around
5 mT with an anticrossing frequency gap remaining between them. The gap is
observed in both experimental and simulated (fig. 3 g) spectra with 0.27 GHz
width, and a corresponding 0.30 GHz gap in the HDS type 3 spectra
(experimental and simulated in fig. 3 l and p respectively). Whereas
previously discussed mode-frequency shifting occurs due to magnetostatic
inter-bar interactions, i.e. the microstate-dependent dipolar field landscapes
giving different $\mathbf{H}_{loc}$ values for the Kittel equation, mode
anticrossings are an effect of dynamic mode-hybridisation[6, 51, 9, 8, 14,
31]. High-resolution anticrossing spectra are shown in figure 4 with
accompanying discussion below. In addition to the anticrossing the type 3
state offers a high-degree of spectral control, with 3 active modes and
tunable mode-gaps.
Figure 3 d) shows the type 4 state, prepared at 22 mT. Qualitatively the
spectra resembles that of the type 1 state but modes exhibit enhanced
separation due to different local dipolar field landscape and are redshifted
relative to type 1. This is best visualised through peak extractions shown in
fig. 3 i). 0.4 and 0.2 GHz mode gaps at 22 mT are observed for type 4 and type
1 states respectively, along with a 0.35 GHz blueshift of the type 1 wide mode
relative to type 4 demonstrating the fine control available. At -12 mT one
$\pm 45^{\circ}$ subset of wide-bars reverses, preparing a type 3 state with
accompanying mode shifts and wide-bar mode-splitting. The remaining wide
subset reverses at -16 mT, preparing a type 2 state.
The HDS sample exhibits qualitatively similar behaviour to the S sample with
increased magnitude microstate-dependent frequency shifts due to stronger
inter-element interaction. Figure 3 j) shows the ‘ground-state’ orientation
HDS sample prepared in a type 1 state at 30 mT. Here a 0-3.6 GHz mode gap is
observed over a -12 - 30 mT field range. Figure 3 k) shows a type 2 state
prepared at 16 mT, exhibiting a constant 2 GHz gap across the field sweep.
Figure 3 l) shows the system in a thin-bar majority type 3 state at 21 mT. The
reversed and unreversed wide-bar modes exhibit opposite gradient with a 7.3
GHz crossing at 6 mT.
Figure 4: Mode-hybridisation and anticrossings.
a-d) Negative-swept field dependent experimental FMR spectra of type 3 states
for S and HDS samples in ground-state and ‘monopole’ orientations. Red scatter
points are peak extractions of the upper mode branch, blue points extractions
of the lower branch. Mode frequency gaps or anticrossings and mode bending in
the field range around the crossing point are observed in all samples and
orientations. Monopole-orientation crossing points (panels b and d) are offset
in positive $\mathbf{H}_{ext}$ due to the net $\mathbf{H}_{loc}$ at the
vertex, ground state orientation crossings (a and c) are offset at lower
magnitude, negative $\mathbf{H}_{ext}$ due to the different $\mathbf{H}_{loc}$
profile along this axis.
e) Simulated spectra of monopole-orientation HDS sample. Anticrossing gap of
0.32 GHz is observed at 4.3 mT. Error bars correspond to frequency range
integrated over to produce spatial mode plots shown in f).
f) Simulated spatial mode-power maps for monopole-orientation HDS sample. Maps
I-VI relate to corresponding points labelled on spectra in panel e). High-
frequency, single-node mode branch is seen in I-III. Low-frequency, multi-node
branch is IV-VI. g) Simulated type 1 spectra of ground-state orientation HDS
sample. Crossing occurs at 24 mT with no observable gap. Error bars correspond
to frequency range integrated over to produce spatial mode plots shown in h).
f) Simulated spatial mode-power maps for ground-state orientation type 1 HDS
sample. Maps I-V relate to corresponding points labelled on spectra in panel
e). Mode-hybridisation is not observed, with matching profile pairs I and V,
IV and III.
### Microstate-dependent mode hybridisation and anticrossings
In the type 3 spectra (fig. 3 c),g),l) and p), where reversed and unreversed
wide-bar modes approach a single frequency they do not overlap. The modes
instead bend away from a Kittel-like form as they approach, leaving an
anticrossing gap[52, 9, 53, 51, 31] which has been predicted to occur in ASI
due to a microstate-dependent band structure[31]. Figure 4 shows the
anticrossing region in type 3 microstates. Samples were prepared in type 3
states in ground-state and monopole orientations, spectra measured while
negatively-sweeping field. Mode-bending and anticrossings are exhibited in all
experimental spectra (fig. 4 a-d) and corresponding simulations (fig. 4 f-h).
Experimental spectra show anticrossing gaps of
$\Delta_{S,GS}=0.27\leavevmode\nobreak\ $ GHz,
$\Delta_{S,monopole}=0.27\leavevmode\nobreak\ $GHz,
$\Delta_{HDS,GS}=0.22\leavevmode\nobreak\ $ GHz,
$\Delta_{HDS,monopole}=0.3\leavevmode\nobreak\ $ GHz.
Simulated spatial mode-power maps (fig. 4 f) show this effect occurs due to
mode-hybridisation between reversed and unreversed wide-bars, causing the two
modes to act like distinct upper and lower frequency v-shaped branches (red
and blue peak extraction points respectively in experimental spectra) rather
than diagonally intersecting Kittel-like modes. The upper-branch mode profile
shows a single-node bulk mode, localised in the reversed bar at fields below
the crossing point and the unreversed bar at fields above the crossing. The
lower-branch profile shows a double-node bulk mode, localised in the
unreversed bar at fields below crossing and vice-versa. At the crossing point,
the single-node bulk mode appears in both wide-bars at the upper-branch
frequency and the multi-node mode appears in both wide-bars at the lower-
branch frequency. The field at which anticrossings occur may be modified by
rotating the sample between ground-state and monopole-orientations. The two
orientations have different net $\mathbf{H}_{loc}$ values along the
$\mathbf{H}_{ext}$-axis due to broken microstate symmetry and resultingly
crossings occur at different fields, ground-state at -1 mT (-1 mT), ‘monopole’
at 3 mT (5 mT) in S (HDS) samples. The antiferromagnetic macrospin ordering in
type 3 states is crucial for mode-hybridisation. The difficulty in preparing
such antiferromagnetic states is a key barrier to observing dynamic coupling
effects such as anticrossings, and a key strength of the microstate-access
protocol presented here.
The microstate control demonstrated allows tailoring of spectra such that
modes may also cross with no resolvable anticrossing. In type 1 states (fig. 1
g,k), crossings are observed between thin and wide-bar modes with no
observable gap or deviation from Kittel-like behaviour. Simulated spectra of
the crossing point (fig. 4 g) show no anticrossing gap and spatial power maps
(fig. 4 h) show a single-node bulk mode throughout the type 1 field range.
While the type 1 state exhibits antiferromagnetic order between the thin and
wide bars, it occurs at weaker effective interaction than type 3 states as the
wide-thin bar vertex separation and dipolar-coupling are reduced relative to
the type 3 wide-wide bar case. The lack of a resolvable type 1 anticrossing is
testament to the sensitivity of dynamic coupling phenomena to interaction
strength. Supplementary figure 2 shows simulated 0-10 GHz spectra of the HDS
sample (‘monopole’ orientation) in a type 3 state with 0-3 GHz edge modes
present and spatial magnetisation profiles of the nanoisland edges.
Realignments of the static edge magnetisation occur at -1.5 mT and -6.5 mT for
the wide and thin bars respectively, while the anticrossing occurs at 5 mT. As
such, static magnetisation realignments are unlikely to be involved in the
observed mode-hybridisation.
We have demonstrated that introducing a width-modified sublattice to ASI
permits rapid, scalable and reconfigurable control over rich and diverse
spectral features. This approach offers an attractive addition to the host of
spectral and microstate control methodologies, requiring only widely-available
global-field and nanofabrication protocols. The state-dependent spectra
observed suggest microwave-assisted state preparation[54, 55, 56, 57] as a
promising direction for integrated read-write functionality.
The magnitude and diversity of microstate-dependent mode and gap control
exhibited invite a host of functional applications including tunable microwave
filters and enable further study of how spin-wave characteristics and band
structure of nanomagnetic systems may be employed in magnonic logic[58] and
neuromorphic devices[59]. In particular, the observation of previously elusive
microstate-dependent mode-hybridisation and anticrossings suggests a magnonic
device which may reconfigurably transmit or reflect spin-waves depending on
its state. In this regard we emphasise that anticrossing behaviour depends
only on the microstate and does not require width-modification except as a
means for microstate access. As state-preparation techniques develop, we
expect mode-hybridisation to become observable and exploitable in other
artificial spin systems.
## Methods
Simulations were performed using MuMax3. To maintain field sweep history,
ground state files are generated in a separate script and used as inputs for
dynamic simulations. S sample dimensions are; wide: 800 by 230 by 20 nm,
narrow: 800 by 130 by 20 nm and lattice parameter: 1120 nm (gap = 160 nm). HDS
sample dimensions are; wide: 600 by 200 by 20 nm and narrow: 600 by 130 by 20
nm and lattice parameter: 800 nm (gap = 100 mn). Perturbation of dimensions
from SEM images were introduced to more accurately reproduce both static and
dynamic magnetisation behaviour. Material parameters for NiFe used are;
saturation magnetisation, Msat = 750 kA/m, exchange stiffness. Aex = 13 pJ and
damping, $\alpha$ = 0.001 All simulations are discretized with lateral
dimensions, cx,y = 5 nm and normal direction, cz = 10 nm and periodic boundary
conditions applied to generate lattice from unit cell. A broadband field
excitation sinc pulse function is applied along z-direction with cutoff
frequency = 20 GHz, amplitude = 0.5 mT. Simulation is run for 25 ns saving
magnetisation every 25 ps. Static relaxed magnetisation at t = 0 is subtracted
from all subsequent files to retain only dynamic components, which are then
subject to a FFT along the time axis to generate a frequency spectra. Power
spectra across the field range are collated and plotted as a colour contour
plot with resolution; $\Delta f$ = 40 MHz and $\Delta\mu_{0}H$ = 1 mT. Spatial
power maps are generated by integrating over a range determined by the full
width half maximum of peak fits and plotting each cell as a pixel whose colour
corresponds to its power. Each colour plot is normalised to the cell with
highest power. High-resolution simulations performed for figure 4 have lower
damping, $\alpha=0.0001$, and are run for 100 ns saving every 50 ps. This
produces colour plots with resolution; $\Delta f$ = 10 MHz and
$\Delta\mu_{0}H$ = 0.2 mT. $\mathbf{H}_{ext}$ is offset from the array
$\hat{x},\hat{y}$-axes by $1^{\circ}$ to better match experiment. Lowering
alpha reduces mode linewidth and allows for better resolution of mode
behaviour particularly when multiple modes are present in close frequency
proximity, as in the anticrossing case.
Samples were fabricated via electron-beam lithography liftoff method on a
Raith eLine system with PMMA resist. Ni81Fe19 (permalloy) was thermally
evaporated and capped with Al2O3. A ‘staircase’ subset of bars was increased
in width to reduce its coercive field relative to the thin subset, allowing
independent subset reversal via global field.
Ferromagnetic resonance spectra were measured using a NanOsc Instruments
cryoFMR in a Quantum Design Physical Properties Measurement System. Broadband
FMR measurements were carried out on large area samples $(\sim 2\times
2\leavevmode\nobreak\ \text{ mm}^{2})$ mounted flip-chip style on a coplanar
waveguide. The waveguide was connected to a microwave generator, coupling RF
magnetic fields to the sample. The output from waveguide was rectified using
an RF-diode detector. Measurements were done in fixed in-plane field while the
RF frequency was swept in 20 MHz steps. The DC field was then modulated at 490
Hz with a 0.48 mT RMS field and the diode voltage response measured via lock-
in. The experimental spectra show the derivative output of the microwave
signal as a function of field and frequency. The normalised differential
spectra are displayed as false-colour images with symmetric log colour scale.
Magnetic force micrographs were produced on a Dimension 3100 using
commercially available normal-moment MFM tips.
MOKE measurements were performed on a Durham Magneto-Optics NanoMOKE system.
The laser spot is approximately 20 $\mu$m diameter. The longitudinal Kerr
signal was normalised and the linear background subtracted from the saturated
magnetisation. The applied field is a quasistatic sinusoidal field cycling at
11 Hz and the measured Kerr signal is averaged over 300 field loops to improve
signal to noise.
### Author contributions
JCG, AV, TD and WRB conceived the work.
JCG, KDS and AV fabricated the samples. AV performed CAD design of the
structures.
AV performed experimental MOKE measurements and the majority of FMR
measurements.
JCG performed FMR measurements on the HDS sample, MS orientation.
AV performed analysis of FMR measurements and generation of spectral heatmaps.
AV, JCG and KDS performed MFM measurements.
TD wrote code for simulation of the magnon spectra and performed micromagnetic
simulations. Noticed avoided crossings in simulation prompting further
investigation into experimental data.
DMA wrote code for simulation of the magnon spectra.
JCG drafted the manuscript, with contributions from all authors in editing and
revision stages.
### Acknowledgements
This work was supported by the Leverhulme Trust (RPG-2017-257) to WRB.
TD and AV were supported by the EPSRC Centre for Doctoral Training in Advanced
Characterisation of Materials (Grant No. EP/L015277/1).
Simulations were performed on the Imperial College London Research Computing
Service[60].
The authors would like to thank Professor Lesley F. Cohen of Imperial College
London for enlightening discussion and comments, and David Mack for excellent
laboratory management.
### Competing interests
The authors declare no competing interests.
### Data availability statement
The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.
### Code availability statement
The code used in this study is available from the corresponding author on
reasonable request.
## Supplementary Information
Supplementary figure 1 Simulated spatial mode profiles of the S sample
(‘monopole’ orientation) taken at zero field along with corresponding 0-40 mT
spectra.
### Supplementary note 1 - Simulated spatial mode profiles
Supplementary figure 2 Simulated spectra of HDS sample (‘monopole’
orientation) prepared in type 3 microstate. 0-10 GHz frequency range shows
edge mode behaviour (0-3 GHz range) of wide and thin bars and anticrossing of
wide bar bulk modes at 5 mT, 7.2 GHz.
Micromagnetic simulations of spatial mode profiles taken at zero field and
corresponding 0-40 mT spectra for the S sample (‘monopole’ orientation) are
shown in supplementary figure 1. Mode labelling numbers W1, W2 and W3 are
sequential and do not denote mode index number.
### Supplementary note 2 - Extended frequency and field range spectra of type
3 spectra
Supplementary figure 2 shows simulated spectra for the HDS sample (‘monopole’
orientation) prepared in the type 3 state. Corresponding to an wider frequency
range version of the spectra shown in figure 3 e), the edge modes of the thin
and wide bars are observed in the 0-3 GHz frequency range. The wide bar edge
mode reverses gradient at -1.5 mT, the thin bar edge mode reverses gradient at
-6.5 mT. These gradient reversals are caused by changes in the static
magnetisation curl states of the nanoisland edge regions, which are shown at
-8, -4, 0, 4 and 10 mT below the spectra. These state changes occur far from
the anticrossing point at 5 mT and as such, rearrangements of the static edge
magnetisation states are unlikely to be linked to the mode hybridisation
behaviour.
### Supplementary note 3 - Positions of the MOKE magnetization plateaux
Bar reversal occurs by a domain wall nucleation process at a field determined
by the aspect ratios of the bars. The wide bar subset can be characterized by
a mean coercive field $H_{c1}$, and a magnetization $M_{wide}$ and the thin
bar subset by a mean coercive field $H_{c2}$ and a magnetization $M_{thin}$.
Because the other bar dimensions are identical the ratio of the volumes and
magnetisations is the same as the ratio of the bar widths. For a sample with
all bars identical, the type 1 and type 4 state should have zero net
magnetization. For the width modified sample the different volumes of reversed
and unreversed bars would be expected (for perfect Ising spins and nominal bar
widths) to give $M/M_{type2}$ =
$\frac{M_{wide}-M_{thin}}{M_{wide}+M_{thin}}=\frac{t_{wide}-t_{thin}}{t_{wide}+t_{thin}}\leavevmode\nobreak\
$ = 0.226 for the S sample and 0.231 for the HDS sample (for both type 1 and
type 4). The relative magnetization of the ground-state minor loop to the
saturated major loop at zero field is approx. 0.3 in the S sample and 0.2 in
the HDS sample. For the type 4 state it is approx. 0.5 and 0.6 for S and HDS
respectively. In a type 3 that is formed by both wide bars switching and
triggering one thin bar to also reverse, then $M/M_{S}$ would be
$\frac{t_{wide}}{t_{wide}+t_{thin}}$ = 0.613 for S and 0.615 for HDS.
Imperfections in the nanofabrication (quenched disorder) give the sublattice
switching fields a Gaussian distribution about the mean, with a standard
deviations $\sigma_{wide}$ around $H_{c1}$ and $\sigma_{thin}$ around
$H_{c2}$. If bars were all sufficiently spaced to be not interacting then
desired states could be accessed by applying $H_{ext}=H_{c1}+H_{c2}/2$ as long
as $H_{c2}-H_{c1}>>\sigma_{wide}+\sigma_{thin}$. However we are in the
strongly-interacting regime and so each bar experiences an effective field
$H_{eff}=H_{app}+H_{loc}$. The reversal of wide bars will change $H_{loc}$
experienced by the thin bars. If $\Delta H_{loc}$ increases $H_{eff}$ then
this makes it more difficult to realise the ordered state, as do the cases
where we are preparing a type 4 state. Where we are writing the type 1 state
from the saturated type 2 state, $\Delta H_{loc}$ decreases $H_{eff}$ and so
the interactions increase the operating window where the ordered state may be
prepared. The difference can be seen in the MOKE hysteresis loops in fig. 1
c,d,e,f). In fig. 1 c,e) we have very clear plateaux in the major (blue)
hysteresis loops and can very easily and reproducibly send minor loops to the
type 1 microstate and back to saturated. Note that the data is the average of
thousands of individual loops and so the sharp switching and flat plateuax
show there is no significant stochasticity in this major hysteresis loop and
we go through the same microstates at the same fields in each measurement.
Similarly in the minor (orange) loops we repeatedly go the the same expected
plateau magnetization. For the same sample, with the same extrinsic disorder
and sigmas, in the monopole geometry, switching the wide bars causes the
dipolar field of all neighbouring bars to help the reversal of the thin bars
and so the two Gaussian distributions start to overlap. We know from MFM we
can access large areas of pure type 4 with the correct protocol, but the
hysteresis loop shows very broad reversal with no clear plateaux. It is not
clear from our data whether the broadening we see is from averaging similar
broad loops or different loops with sharper individual features. The disorder
could be spatial within the measurement spot, temporal with loop cycle number
or both. Certainly there is a significant stochastic contribution in the
measurement.
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|
# Dynamic State Estimation for Radial Microgrid Protection
Arthur K. Barnes 1* — and Adam Mate 1* — Manuscript submitted: December 14,
2020. 1 The authors are with the Advanced Network Science Initiative at Los
Alamos National Laboratory, Los Alamos, NM 87544 USA. Email:{abarnes,
<EMAIL_ADDRESS>versions of one or more of the figures in this paper are
available online at https://ieeexplore.ieee.org.LANL ANSI LA-UR-20-30126.
###### Abstract
Microgrids are localized electrical grids with control capability that are
able to disconnect from the traditional grid to operate autonomously. They
strengthen grid resilience, help mitigate grid disturbances, and support a
flexible grid by enabling the integration of distributed energy resources.
Given the likely presence of critical loads, the proper protection of
microgrids is of vital importance; however, this is complicated in the case of
inverter-interfaced microgrids where low fault currents preclude the use of
conventional time-overcurrent protection. This paper introduces and
investigates the application of dynamic state estimation, a generalization of
differential protection, for the protection of radial portions of microgrids
(or distribution networks); both phasor-based and dynamic approaches are
investigated for protection. It is demonstrated through experiments on three
case-study systems that dynamic state estimation is capable of correctly
identifying model parameters for both normal and faulted operation.
###### Index Terms:
power system operation, dynamic state estimation, microgrid, distribution
network, protection.
## I Introduction
Dynamic state estimation (abbr. DSE) is a generalization of differential
protection, which offers a reduced likelihood of misoperation, particularly in
the case of assets with nonlinear characteristics (e.g., transformers that are
being energized) [1]. It is also useful in cases where distance protection
performs poorly (e.g., transmission lines with series compensation [2] or
mutually coupled transmission lines [3]). DSE has previously been introduced
to microgrid branch protection [4, 5, 6].
This paper investigates the application of DSE for the protection of radial
portions of a microgrid (or a distribution network). This can be a challenge
in electrical grids with distributed generation on account of lack of fault
current from inverter-interfaced generation [7], varying fault current between
grid-connected and islanded modes [7], and the potential for normally-meshed
operation [8] and unbalanced operation due to single-phase loads [8].
Admittance relaying has been investigated as a solution for the protection of
microgrids [9], however, it has been observed to present issues with grounded-
wye connected loads [10]; consequently, additional relaying is necessary to
prevent misoperation [8].
This paper treats the radial portions of a microgrid as load busses. It is
assumed that these portions contain no loops or downstream generation; they
are modeled as constant-impedance networks with unknown impedances but known
connectivity. To ensure that the number of measured variables is greater than
approximately 1.6 times the number of free parameters (where 1.6 is a commonly
selected number for redundancy to ensure sufficient measurements for system
identification [11, 12]), most models presented here make the assumption that
the loads are balanced.
Every load and fault configuration requires a separate model. For a given load
configuration, a model for each fault configuration is fit to measured values;
the model with the lowest error, in terms of fitting the observed variables,
is assumed to be the correct one. On a grounded-wye-connected load, the
following models are necessary to distinguish between normal operation, line-
ground faults and line-line faults:
1. 1.
Normal operation: each branch of the load has the same impedance, which is
modeled as a series resistive-inductive (abbr. RL) network.
2. 2.
Phase A-ground fault: the faulted branch A is modeled as a resistance, while
the unfaulted branches B and C are modeled as series RL networks with equal
parameters.
3. 3.
Phase B-ground fault: the faulted branch B is modeled as a resistance, while
the unfaulted branches C and A are modeled as series RL networks with equal
parameters.
4. 4.
Phase C-ground fault: the faulted branch C is modeled as a resistance, while
the unfaulted branches A and B are modeled as series RL networks with equal
parameters.
5. 5.
Phase A-B fault: the fault impedance is modeled as a resistance across the
load terminals A and B, while each branch of the load is modeled as a series
RL network.
6. 6.
Phase B-C fault: the fault impedance is modeled as a resistance across the
load terminals B and C, while each branch of the load is modeled as a series
RL network.
7. 7.
Phase C-A fault: the fault impedance is modeled as a resistance across the
load terminals C and A, while each branch of the load is modeled as a series
RL network.
On a delta-connected system, the following models are necessary to distinguish
between normal operation, line-ground and line-line faults:
1. 1.
Normal operation: each branch of the load has the same impedance, which is
modeled as series RL network.
2. 2.
Phase A-ground fault: the fault impedance is modeled as a resistance between
load terminal A and ground, while the load branches are modeled as series RL
networks.
3. 3.
Phase B-ground fault: the fault impedance is modeled as a resistance between
load terminal B and ground, while the load branches are modeled as series RL
networks.
4. 4.
Phase C-ground fault: the fault impedance is modeled as a resistance between
load terminal C and ground, while the load branches are modeled as series RL
networks.
5. 5.
Phase A-B fault: the fault impedance is modeled as a resistance across the
load terminals A and B, while the branches across load terminals B-C and C-A
are modeled as series RL networks.
6. 6.
Phase B-C fault: the fault impedance is modeled as a resistance across the
load terminals B and C, while the branches across load terminals C-A and A-B
are modeled as series RL networks.
7. 7.
Phase C-A fault: the fault impedance is modeled as a resistance across the
load terminals C and A, while the branches across load terminals A-B and B-C
are modeled as series RL networks.
Both phasor-based and dynamic approaches are investigated for radial microgrid
protection. Section II describes the implementation of phasor-based state
estimation: it is conceptually similar to DSE but more straightforward to
derive and implement as it only requires a single time period. Section III
describes the implementation of DSE. Section IV describes how two different
transient models of loads are developed as test cases and run to test both
phasor and dynamic state estimation; next, Section V presents the performance
of state estimation on the test cases. Finally, Section VI summarizes the
conclusions of this paper.
## II Phasor Implementation
The phasor implementation of state estimation-based protection is simpler:
only a single time period is used, which limits the number of measurements and
therefore the number of parameters that can be estimated. In this section, it
is applied to single-phase, grounded-wye and delta-connected load
configurations.
### II-A Single-Phase Impedance
The output of the system (illustrated in Fig. 1a):
$\mathbf{y}=\begin{bmatrix}V\\\ I\end{bmatrix}$
where $V$ and $I$ are phasor quantities.
The state of the system:
$\mathbf{x}=\begin{bmatrix}Z\\\ I_{z}\end{bmatrix}$
The output-state mapping for the system is the following vector-valued
function:
$\mathbf{y}=\mathbf{h}(\mathbf{x})$
where
$\displaystyle h_{1}(\mathbf{x})$ $\displaystyle=V_{z}=ZI_{z}$ $\displaystyle
h_{2}(\mathbf{x})$ $\displaystyle=I_{z}$
The Jacobian of $\mathbf{h}(\mathbf{x})$ is determined as follows:
$\displaystyle\frac{\partial V_{z}}{\partial Z}$
$\displaystyle=\frac{\partial}{\partial Z}ZI_{z}=I_{z}$
$\displaystyle\frac{\partial V_{z}}{\partial I_{z}}$
$\displaystyle=\frac{\partial}{\partial I_{z}}ZI_{z}=Z$
$\displaystyle\frac{\partial I_{z}}{\partial Z}$
$\displaystyle=\frac{\partial}{\partial Z}I_{z}=0$
$\displaystyle\frac{\partial I_{z}}{\partial I_{z}}$
$\displaystyle=\frac{\partial}{\partial I_{z}}I_{z}=1$
The mapping between variables and the state vector:
$\displaystyle Z$ $\displaystyle=x_{1}$ $\displaystyle I_{z}$
$\displaystyle=x_{2}$
Given the variable and state mapping, the Jacobian can be built as follows:
$H(n,m)=0$, unless specified below.
$H=\begin{bmatrix}[1.5]\frac{\partial V_{z}}{\partial Z}&\frac{\partial
V_{z}}{\partial I_{z}}\\\ \frac{\partial I_{z}}{\partial Z}&\frac{\partial
I_{z}}{\partial I_{z}}\end{bmatrix}$
Given the Jacobian, the state of the system can be solved for iteratively:
$\displaystyle{\bm{\epsilon}}_{i}$
$\displaystyle=\mathbf{y}-\mathbf{h}(\mathbf{x}_{i})\hskip 36.135pt$
$\displaystyle J_{i}$ $\displaystyle=||{\bm{\epsilon}}_{i}||^{2}$
$\displaystyle\mathbf{x}_{i+1}$
$\displaystyle=\mathbf{x}_{i}+(H^{\prime}_{i}H_{i})^{-1}H^{\prime}_{i}{\bm{\epsilon}}_{i}$
### II-B Grounded-Wye with Line-Ground Fault
The output of the system (illustrated in Fig. 1b):
$\mathbf{y}=\begin{bmatrix}I_{a}&I_{b}&I_{c}&V_{a}&V_{b}&V_{c}\end{bmatrix}^{T}$
The easiest way to model this is as an unbalanced load where the fault
impedance is not treated specially. The state of the system therefore:
$\mathbf{x}=\begin{bmatrix}Y_{a}&Y_{b}&Y_{c}&V_{za}&V_{zb}&V_{zc}\end{bmatrix}^{T}$
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle\mathbf{h}_{1}(\mathbf{x})$ $\displaystyle=I_{a}=y_{a}V_{a}$
$\displaystyle\mathbf{h}_{2}(\mathbf{x})$ $\displaystyle=I_{b}=y_{b}V_{b}$
$\displaystyle\mathbf{h}_{3}(\mathbf{x})$ $\displaystyle=I_{c}=y_{c}V_{c}$
$\displaystyle\mathbf{h}_{4}(\mathbf{x})$ $\displaystyle=V_{a}=V_{za}$
$\displaystyle\mathbf{h}_{5}(\mathbf{x})$ $\displaystyle=V_{b}=V_{zb}$
$\displaystyle\mathbf{h}_{6}(\mathbf{x})$ $\displaystyle=V_{c}=V_{zc}$
### II-C Grounded-Wye with Line-Line Fault
The output of the system (illustrated in Fig. 1c):
$\mathbf{y}=\begin{bmatrix}I_{a}&I_{b}&I_{c}&V_{a}&V_{b}&V_{c}\end{bmatrix}^{T}$
The state of the system:
$\mathbf{x}=\begin{bmatrix}Y_{l}&Y_{f}&V_{za}&V_{zb}&V_{zc}\end{bmatrix}^{T}$
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{1}(\mathbf{x})$
$\displaystyle=I_{a}=(y_{l}+y_{f})V_{za}-y_{f}V_{zb}$ $\displaystyle
h_{2}(\mathbf{x})$ $\displaystyle=I_{b}=-y_{f}V_{za}+y_{l}V_{zb}$
$\displaystyle h_{3}(\mathbf{x})$ $\displaystyle=I_{c}=y_{l}V_{zc}$
$\displaystyle h_{4}(\mathbf{x})$ $\displaystyle=V_{a}=V_{za}$ $\displaystyle
h_{5}(\mathbf{x})$ $\displaystyle=V_{b}=V_{zb}$ $\displaystyle
h_{6}(\mathbf{x})$ $\displaystyle=V_{c}=V_{zc}$
### II-D Delta-Connected Load with Line-Line Fault
The output of the system (illustrated in Fig. 1d):
$\mathbf{y}=\begin{bmatrix}I_{a}&I_{b}&I_{c}&V_{a}&V_{b}&V_{c}\end{bmatrix}^{T}$
The state of the system:
$\mathbf{x}=\begin{bmatrix}Y_{f}&Y_{ll}&V_{za}&V_{zb}&V_{zc}\end{bmatrix}^{T}$
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{1}(\mathbf{x})$
$\displaystyle=I_{a}=y_{aa}V_{za}-y_{ab}V_{zb}-y_{ca}V_{zc}$
$\displaystyle=(y_{ab}+y_{ca})V_{za}-y_{ab}V_{zb}-y_{ca}V_{zc}$
$\displaystyle=(y_{f}+y_{ll})V_{za}-y_{f}V_{sb}-y_{ll}V_{zc}$ $\displaystyle
h_{2}(\mathbf{x})$
$\displaystyle=I_{b}=-y_{ab}V_{za}+y_{bb}V_{zb}-y_{bc}V_{zc}$
$\displaystyle=-y_{ab}V_{za}+(y_{ab}+y_{bc})V_{zb}-y_{bc}V_{zc}$
$\displaystyle=-y_{f}V_{za}+(y_{f}+y_{ll})V_{zb}-y_{ll}V_{zc}$ $\displaystyle
h_{3}(\mathbf{x})$
$\displaystyle=I_{c}=-y_{ca}V_{za}-y_{bc}V_{zb}+y_{cc}V_{zc}$
$\displaystyle=-y_{ca}V_{za}-y_{bc}V_{zb}+(y_{ac}+y_{bc})V_{zc}$
$\displaystyle=-y_{ll}V_{za}-y_{ll}V_{zb}+2y_{ll}V_{zc}$ $\displaystyle
h_{4}(\mathbf{x})$ $\displaystyle=V_{a}=V_{za}$ $\displaystyle
h_{5}(\mathbf{x})$ $\displaystyle=V_{b}=V_{zb}$ $\displaystyle
h_{6}(\mathbf{x})$ $\displaystyle=V_{c}=V_{zc}$
### II-E Delta-Connected Load with a Line-Ground Fault
The output of the system (illustrated in Fig. 1e):
$\mathbf{y}=\begin{bmatrix}I_{a}&I_{b}&I_{c}&V_{a}&V_{b}&V_{c}\end{bmatrix}^{T}$
The state of the system:
$\mathbf{x}=\begin{bmatrix}Y_{ll}&Y_{f}&V_{za}&V_{zb}&V_{zc}\end{bmatrix}^{T}$
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{1}(\mathbf{x})$
$\displaystyle=I_{a}=y_{aa}V_{za}-y_{ab}V_{zb}-y_{ca}V_{zc}$
$\displaystyle=(y_{ab}+y_{ca}+y_{ag})V_{za}-y_{ab}V_{zb}-y_{ca}V_{zc}$
$\displaystyle=(y_{f}+2y_{ll})V_{za}-y_{ll}V_{sb}-y_{ll}V_{zc}$ $\displaystyle
h_{2}(\mathbf{x})$
$\displaystyle=I_{b}=-y_{ab}V_{za}+y_{bb}V_{zb}-y_{bc}V_{zc}$
$\displaystyle=-y_{ab}V_{za}+(y_{ab}+y_{bc})V_{zb}-y_{bc}V_{zc}$
$\displaystyle=-y_{ll}V_{za}+2y_{ll}V_{zb}-y_{ll}V_{zc}$ $\displaystyle
h_{3}(\mathbf{x})$
$\displaystyle=I_{c}=-y_{ca}V_{za}-y_{bc}V_{zb}+y_{cc}V_{zc}$
$\displaystyle=-y_{ca}V_{za}-y_{bc}V_{zb}+(y_{ac}+y_{bc})V_{zc}$
$\displaystyle=-y_{ll}V_{za}-y_{ll}V_{zb}+2y_{ll}V_{zc}$ $\displaystyle
h_{4}(\mathbf{x})$ $\displaystyle=V_{a}=V_{za}$ $\displaystyle
h_{5}(\mathbf{x})$ $\displaystyle=V_{b}=V_{zb}$ $\displaystyle
h_{6}(\mathbf{x})$ $\displaystyle=V_{c}=V_{zc}$
(a) Single-phase load
(b) Grounded-wye load with line-ground fault fault
(c) Grounded-wye load with line-line fault
(d) Delta-connected load with a line-line fault
(e) Delta-connected load with a line-ground fault
Figure 1: Phasor-based implementation of state estimation-based protection,
applied to single-phase, grounded-wye and delta-connected load configurations.
## III Dynamic Implementation
While the phasor implementation uses a single time period for state
estimation, with the dynamic implementation several periods are used; in this
paper, 12 cycles are sampled at a 2 [kHz] sample rate. As in Section II, here
the DSE-based protection is applied to single-phase, grounded-wye and delta-
connected load configurations.
### III-A Single-Phase Series RL Load
The output of the system (illustrated in Fig. 2a):
$y(t)=\begin{bmatrix}v(t)\\\ i(t)z(t)\end{bmatrix}$
For the purposes of state estimation, this is sampled at points
$n\in\\{1,...,N\\}$ giving the following vector-value equation:
$\mathbf{y}=\begin{bmatrix}\mathbf{v}\\\ \mathbf{i}\end{bmatrix}$
where
$\displaystyle\mathbf{v}$
$\displaystyle=\begin{bmatrix}v(1)&v(2)&\cdots&v(N)\end{bmatrix}^{T}$
$\displaystyle\mathbf{i}$
$\displaystyle=\begin{bmatrix}i(1)&i(2)&\cdots&i(N)\end{bmatrix}^{T}$
$\displaystyle\mathbf{z}$
$\displaystyle=\begin{bmatrix}z(1)&z(2)&\cdots&z(N)\end{bmatrix}^{T}$
The state of the system:
$\mathbf{x}=\begin{bmatrix}R&L&\mathbf{v}_{r}&\mathbf{v}_{l}\end{bmatrix}^{T}$
The output-state mapping for the system is the following vector-valued
function:
$\mathbf{y}=\mathbf{h}(\mathbf{x})$
where
$\displaystyle h_{n}(\mathbf{x})=v_{r}(n)+v_{l}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{N+n}(\mathbf{x})=Gv_{r}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{2N+n}(\mathbf{x})=Gv_{r}(n)-Gv_{r}(n-2)+\frac{2\Lambda\Delta
t}{6}(v_{l}(n)+$ $\displaystyle+4v_{l}(n-1)+v_{l}(n-2)),\quad\forall
n\in\\{3,4,\ldots,N\\}$
In the above, $v_{R}(n)=Ri_{L}(n)$ follows from discretizing
$v_{R}(t)=Ri_{L}(t)$, and
$v_{l}(n)=\frac{2\Lambda\Delta t}{6}(v_{l}(n)+4v_{l}(n-1)+v_{l}(n-2))$
follows from discretizing
$i_{l}(t)=\frac{1}{L}\int_{t-\Delta t}^{t}v_{l}(\tau)d\tau$
via Simpson’s 1/3 rule [13].
Given the variable and state vector mapping, the Jacobian can be built as
follows: $H(n,m)=0$, unless specified otherwise below.
$\displaystyle H(n,2+n)=\frac{\partial v(n)}{\partial v_{r}(n)}=1\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle H(n,2+N+n)=\frac{\partial
v(n)}{\partial v_{l}(n)}=1\quad\forall n\in\\{1,2,\ldots,N\\}$ $\displaystyle
H(N+n,1)=\frac{\partial i(n)}{\partial G}=v_{r}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle H(N+n,2+n)=\frac{\partial
i(n)}{\partial v_{r}(n)}=G\quad\forall n\in\\{1,2,\ldots,N\\}$ $\displaystyle
H(2N+n-2,1)=\frac{\partial z(n-2)}{\partial R}=v_{r}(n)-v_{r}(n-2)$
$\displaystyle\quad\forall n\in\\{3,4,\ldots,N\\}$ $\displaystyle
H(2N+n-2,n)=\frac{\partial z(n-2)}{\partial v_{r}(n)}=G\quad\forall
n\in\\{3,4,\ldots,N\\}$ $\displaystyle H(2N+n-2,2)=\frac{\partial
z(n-2)}{\partial\Lambda}=\frac{2\Delta t}{6}(v_{l}(n)+$
$\displaystyle+4v_{l}(n-1)+v_{l}(n-2))\quad\forall n\in\\{3,4,\ldots,N\\}$
$\displaystyle H(2N+n-2,2+N+n)=\frac{\partial z(n-2)}{\partial
v_{l}(n)}=\frac{\Delta t\Lambda}{3}$ $\displaystyle\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle H(2N+n-2,1+N+n)=\frac{\partial
z(n-2)}{\partial v_{l}(n-1)}=\frac{4\Delta t\Lambda}{3}$
$\displaystyle\quad\forall n\in\\{1,2,\ldots,N\\}$ $\displaystyle
H(2N+n-2,N+n)=\frac{\partial z(n-2)}{\partial v_{l}(n-2)}=\frac{\Delta
t\Lambda}{3}$ $\displaystyle\quad\forall n\in\\{1,2,\ldots,N\\}$
Given the Jacobian, the state of the system can be solved for iteratively, by
applying the same equations as in Section II-A.
### III-B Grounded-Wye Load without Fault
The sampled output of the system (illustrated in Fig. 2b):
$\mathbf{y}=\begin{bmatrix}\mathbf{v}_{a}&\mathbf{v}_{b}&\mathbf{v}_{c}&\mathbf{i}_{a}&\mathbf{i}_{b}&\mathbf{i}_{c}&\mathbf{z}_{a}&\mathbf{z}_{b}&\mathbf{z}_{c}\end{bmatrix}^{T}$
where
$\displaystyle\mathbf{v}_{\phi}$
$\displaystyle=\begin{bmatrix}v_{\phi}(1)&v_{\phi}(2)&\cdots&v_{\phi}(N)\end{bmatrix}^{T}$
$\displaystyle\mathbf{i}_{\phi}$
$\displaystyle=\begin{bmatrix}i_{\phi}(1)&i_{\phi}(2)&\cdots&i_{\phi}(N)\end{bmatrix}^{T}$
$\displaystyle\mathbf{z}_{\phi}$
$\displaystyle=\begin{bmatrix}z_{\phi}(1)&z_{\phi}(2)&\cdots&z_{\phi}(N-2)\end{bmatrix}^{T}for\hskip
3.61371pt\phi\in\\{a,b,c\\}$
The state of the system:
$x(t)=\begin{bmatrix}G&\Lambda&\mathbf{v}_{ra}&\mathbf{v}_{rb}&\mathbf{v}_{rc}&\mathbf{v}_{la}&\mathbf{v}_{lb}&\mathbf{v}_{lc}\end{bmatrix}^{T}$
where $G=R^{-1}$ is the conductance, $\Lambda=L^{-1}$ is the reciprocal of the
inductance, $\mathbf{v}_{r\phi}$ is the voltage across the resistance on phase
$\phi$ at each time period $1,\ldots,N$ and $\mathbf{v}_{l\phi}$ is the
voltage across the inductance on phase $\phi$ at each time period
$1,\ldots,N$.
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{n}(\mathbf{x})=v_{ra}(n)+v_{la}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+N}(\mathbf{x})=v_{rb}(n)+v_{lb}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+2N}(\mathbf{x})=v_{rb}(n)+v_{lb}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+3N}(\mathbf{x})=Gv_{ra}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+4N}(\mathbf{x})=Gv_{rb}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+5N}(\mathbf{x})=Gv_{rc}(n))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+6N}(\mathbf{x})=G(v_{ra}(n)-v_{ra}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{la}(n)+$ $\displaystyle+4v_{la}(n-1)+v_{la}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+7N}(\mathbf{x})=G(v_{rb}(n)-v_{rb}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lb}(n)+$ $\displaystyle+4v_{lc}(n-1)+v_{lb}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+8N}(\mathbf{x})=G(v_{rc}(n)-v_{rc}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lc}(n)+$ $\displaystyle+4v_{lc}(n-1)+v_{lc}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}.$
### III-C Grounded-Wye Load with Line-Ground Fault
The sampled output of the system (illustrated in Fig. 2c):
$\mathbf{y}=\begin{bmatrix}\mathbf{v}_{a}&\mathbf{v}_{b}&\mathbf{v}_{c}&\mathbf{i}_{a}&\mathbf{i}_{b}&\mathbf{i}_{c}&\mathbf{z}_{b}&\mathbf{z}_{c}\end{bmatrix}^{T}$
Note that there are no $z_{a}(n)$ output variables as the reactive impedance
on phase A is large compared to the parallel fault conductance $G_{f}$.
The state of the system:
$x(t)=\begin{bmatrix}G&\Lambda&G_{f}&\mathbf{v}_{ra}&\mathbf{v}_{rb}&\mathbf{v}_{rc}&\mathbf{v}_{lb}&\mathbf{v}_{lc}\end{bmatrix}^{T}$
where $G_{f}=R^{-1}$ is the conductance and the remaining states are the same
as those in that of the grounded-wye no-fault state.
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{n}(\mathbf{x})=v_{ra}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+N}(\mathbf{x})=v_{rb}(n)+v_{lb}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+2N}(\mathbf{x})=v_{rb}(n)+v_{lb}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+3N}(\mathbf{x})=Gv_{ra}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+4N}(\mathbf{x})=Gv_{rb}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+5N}(\mathbf{x})=Gv_{rc}(n))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+6N}(\mathbf{x})=G(v_{rb}(n)-v_{rb}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lb}(n)+$ $\displaystyle+4v_{lc}(n-1)+v_{lb}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+7N}(\mathbf{x})=G(v_{rc}(n)-v_{rc}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lc}(n)+$ $\displaystyle+4v_{lc}(n-1)+v_{lc}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}$
### III-D Grounded-Wye Load with Line-Line Fault
The sampled output of the system (illustrated in Fig. 2d):
$\mathbf{y}=\begin{bmatrix}\mathbf{v}_{a}&\mathbf{v}_{b}&\mathbf{v}_{c}&\mathbf{i}_{a}&\mathbf{i}_{b}&\mathbf{i}_{c}&\mathbf{z}_{a}&\mathbf{z}_{b}&\mathbf{z}_{c}\end{bmatrix}^{T}$
The state of the system:
$x(t)=\begin{bmatrix}G&\Lambda&\mathbf{v}_{ra}&\mathbf{v}_{rb}&\mathbf{v}_{rc}&\mathbf{v}_{la}&\mathbf{v}_{lb}&\mathbf{v}_{lc}\end{bmatrix}^{T}$
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{n}(\mathbf{x})=v_{ra}(n)+v_{la}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+N}(\mathbf{x})=v_{rb}(n)+v_{lb}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+2N}(\mathbf{x})=v_{rb}(n)+v_{lb}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+3N}(\mathbf{x})=Gv_{ra}(n)+G_{f}(v_{ra}(n)+v_{la}(n)-v_{rb}(n)-$
$\displaystyle-v_{lb}(n))\quad\forall n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+4N}(\mathbf{x})=Gv_{rb}(n)-G_{f}(v_{ra}(n)+v_{la}(n)-v_{rb}(n)-$
$\displaystyle-v_{lb}(n))\quad\forall n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+5N}(\mathbf{x})=Gv_{rc}(n))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+6N}(\mathbf{x})=G(v_{ra}(n)-v_{ra}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{la}(n)+$ $\displaystyle+4v_{la}(n-1)+v_{la}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+7N}(\mathbf{x})=G(v_{rb}(n)-v_{rb}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lb}(n)+$ $\displaystyle+4v_{lc}(n-1)+v_{lb}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+8N}(\mathbf{x})=G(v_{rc}(n)-v_{rc}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lc}(n)+$ $\displaystyle+4v_{lc}(n-1)+v_{lc}(n-2))\quad\forall
n\in\\{1,2,\ldots,N\\}$
### III-E Delta Load without Fault
The sampled output of the system (illustrated in Fig. 2f):
$\mathbf{y}=\begin{bmatrix}\mathbf{v}_{ab}&\mathbf{v}_{bc}&\mathbf{v}_{ca}&\mathbf{i}_{a}&\mathbf{i}_{b}&\mathbf{i}_{c},&\mathbf{z}_{ab}&\mathbf{z}_{bc}&\mathbf{z}_{ca}\end{bmatrix}^{T}$
The state of the system:
$x(t)=\begin{bmatrix}G&\Lambda\mathbf{v}_{rab}&\mathbf{v}_{rbc}&\mathbf{v}_{rca}&\mathbf{v}_{lab}&\mathbf{v}_{lbc}&\mathbf{v}_{lca}\end{bmatrix}^{T}$
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{n}(\mathbf{x})=v_{rab}(n)+v_{lab}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+N}(\mathbf{x})=v_{rbc}(n)+v_{lbc}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+2N}(\mathbf{x})=v_{rca}(n)+v_{lca}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+3N}(\mathbf{x})=G(v_{rab}(n)-v_{rca}(n))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+4N}(\mathbf{x})=G(v_{rbc}(n)-v_{rab}(n))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+5N}(\mathbf{x})=G(v_{rca}(n)-v_{rbc}(n))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+6N}(\mathbf{x})=G(v_{rab}(n)-v_{rab}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lab}(n)+$
$\displaystyle+4v_{lab}(n-1)+v_{lab}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+7N}(\mathbf{x})=G(v_{rbc}(n)-v_{rbc}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lbc}(n)+$
$\displaystyle+4v_{lbc}(n-1)+v_{lbc}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+8N}(\mathbf{x})=G(v_{rca}(n)-v_{rca}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lca}(n)+$
$\displaystyle+4v_{lca}(n-1)+v_{lca}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
### III-F Delta Load with Line-Line Fault
The sampled output of the system (illustrated in Fig. 2d):
$\mathbf{y}=\begin{bmatrix}\mathbf{v}_{ab}&\mathbf{v}_{bc}&\mathbf{v}_{ca}&\mathbf{i}_{a}&\mathbf{i}_{b}&\mathbf{i}_{c},&\mathbf{z}_{bc}&\mathbf{z}_{ca}\end{bmatrix}^{T}$
Note that there are no $z_{ab}(n)$ output variables as the reactive impedance
on phase A is large compared to the parallel fault conductance $G_{f}$.
The state of the system:
$x(t)=\begin{bmatrix}G&\Lambda&G_{f}&\mathbf{v}_{rab}&\mathbf{v}_{rbc}&\mathbf{v}_{rca}&\mathbf{v}_{lbc}&\mathbf{v}_{lca}\end{bmatrix}^{T}$
where $G_{f}=R^{-1}$ is the conductance and the remaining states are the same
as those in that of the delta no-fault state.
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{n}(\mathbf{x})=v_{rab}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+N}(\mathbf{x})=v_{rbc}(n)+v_{lbc}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+2N}(\mathbf{x})=v_{rca}(n)+v_{lca}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+3N}(\mathbf{x})=Gv_{ra}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+4N}(\mathbf{x})=Gv_{rb}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+5N}(\mathbf{x})=Gv_{rc}(n))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+6N}(\mathbf{x})=G(v_{rbc}(n)-v_{rbc}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lbc}(n)+$
$\displaystyle+4v_{lbc}(n-1)+v_{lbc}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+7N}(\mathbf{x})=G(v_{rca}(n)-v_{rca}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lca}(n)+$
$\displaystyle+4v_{lca}(n-1)+v_{lca}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
### III-G Delta Load with Line-Ground Fault
The sampled output of the system (illustrated in Fig. 2g):
$\mathbf{y}=\begin{bmatrix}\mathbf{v}_{ab}&\mathbf{v}_{bc}&\mathbf{v}_{ca}&\mathbf{v}_{a}&\mathbf{i}_{a}&\mathbf{i}_{b}&\mathbf{i}_{c},&\mathbf{z}_{ab}&\mathbf{z}_{bc}&\mathbf{z}_{ca}\end{bmatrix}^{T}$
The state of the system:
$\displaystyle
x(t)=\begin{bmatrix}G&\Lambda&G_{f}&\mathbf{v}_{rab}&\mathbf{v}_{rbc}&\mathbf{v}_{rca}&...\end{bmatrix}^{T}$
$\displaystyle\begin{bmatrix}...&\mathbf{v}_{lab}&\mathbf{v}_{lbc}&\mathbf{v}_{lca}&\mathbf{v}_{f}\end{bmatrix}^{T}$
where $G_{f}=R^{-1}$ is the fault conductance, $\mathbf{v}_{f}$ is the voltage
across the fault and the remaining states are the same as those in that of the
delta no-fault state.
The output function $\mathbf{h}(\mathbf{x})$ can be written as:
$\displaystyle h_{n}(\mathbf{x})=v_{rab}(n)+v_{lab}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+N}(\mathbf{x})=v_{rbc}(n)+v_{lbc}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+2N}(\mathbf{x})=v_{rca}(n)+v_{lca}(n)\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+3N}(\mathbf{x})=v_{f}(n)\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+4N}(\mathbf{x})=G(v_{rab}(n)-v_{rca}(n))+G_{f}v_{f}(n)$
$\displaystyle\quad\forall n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+5N}(\mathbf{x})=G(v_{rbc}(n)-v_{rab}(n))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+6N}(\mathbf{x})=G(v_{rca}(n)-v_{rbc}(n))\quad\forall
n\in\\{1,2,\ldots,N\\}$ $\displaystyle
h_{n+7N}(\mathbf{x})=G(v_{rab}(n)-v_{rab}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lab}(n)+$
$\displaystyle+4v_{lab}(n-1)+v_{lab}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+8N}(\mathbf{x})=G(v_{rbc}(n)-v_{rbc}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lbc}(n)+$
$\displaystyle+4v_{lbc}(n-1)+v_{lbc}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
$\displaystyle h_{n+9N}(\mathbf{x})=G(v_{rca}(n)-v_{rca}(n-2))-\frac{2\Delta
t\Lambda}{6}(v_{lca}(n)+$
$\displaystyle+4v_{lca}(n-1)+v_{lca}(n-2))\quad\forall n\in\\{1,2,\ldots,N\\}$
(a) Single-phase RL series load
(b) Grounded-wye-connected RL load
(c) Grounded-wye-connected RL load with a line-ground fault
(d) Grounded-wye-connected RL load with a line-line fault
(e) Delta-connected RL load
(f) Delta-connected RL load with a line-line fault
(g) Delta-connected RL load with a line-ground fault
Figure 2: Dynamic implementation of state estimation-based protection, applied
to single-phase, grounded-wye and delta-connected load configurations.
## IV Experiments
Three different case-study systems are considered: 1) a single-phase load, 2)
a grounded-wye constant-impedance load, and 3) a delta-connected constant
impedance load. These load configurations are studied for both phasor-based
state estimation and DSE. To verify the noise immunity of the methods, random
noise with an amplitude of approximately 10% of the signal peak is added to
the measurements in both cases. Experiments are performed in Julia v1.5 [14]
and 64-bit MATLAB R2019b [15].
### IV-A Phasor Implementation
For the single-phase phasor model, it is assumed that the source voltage is
240 [V] and the load impedance $R+jX$ is such that it draws a current of
$10-j5$ [A]. For the three-phase phasor models, both line-ground and line-line
fault configurations are considered. These assume that the voltage source is
480 [V] rms line-line and the load impedance $R+jX$ is such that it draws
$30-i15$ [A] per phase. The fault resistance $R_{f}$ is selected such that
$R_{f}=R/10$.
Measured data is obtained by assuming a balanced input voltage and calculating
the current by multiplying the input voltage phasor vector with the admittance
matrix of the load-fault network. This is also the case for the single-phase
dynamic load, however, in that case the measured phasor voltage is converted
to instantaneous voltage to obtain the input for DSE.
### IV-B Dynamic Implementation
The first case-study system is solved ad-hoc, assuming an ideal source with
the parameters listed in Table I.
Variable | Symbol | Value | Units
---|---|---|---
Total load real power | $P$ | 10 | kW
Total load reactive power | $Q$ | 5 | kVAR
Line-line RMS source voltage | $V_{ll}$ | 480 | V
Simulation time | T | 10 | ms
Sample rate | $T_{s}$ | 100 | $\mu$s
TABLE I: Parameters for Single-Phase Dynamic Load
The latter two case-study systems are modeled in the MATLAB/Simulink® SimScape
multi-physics simulation environment, using the Specialized Power Systems
library with the parameters listed in Table II and Table III.
Variable | Symbol | Value | Units
---|---|---|---
Total load real power | $P$ | 10 | kW
Total load reactive power | $Q$ | 5 | kVAR
Line-line RMS source voltage | $V_{ll}$ | 240 | V
Source resistance | $R_{s}$ | 19.2 | $\Omega$
Source inductance | $L_{s}$ | 25.465 | mH
Fault resistance | $R_{f}$ | 1 | m$\Omega$
Ground resistance | $R_{g}$ | 10 | m$\Omega$
Cable positive-sequence resistance | $R_{c}$ | 183.7 | m$\Omega$
Cable positive-sequence reactance | $L_{c}$ | 26.6 | m$\Omega$
Simulation time | T | 200 | ms
Fault start time | $T_{f}$ | 50 | ms
TABLE II: Common Parameters for Three-Phase Dynamic Models Variable | Grounded-Wye | Delta
---|---|---
Load resistance R ($\Omega$) | 18.432 | 55.296
Load inductance L (mH) | 24.457 | 73.3
TABLE III: Varying Parameters for Three-Phase Dynamic Models
In the systems, the load is connected to a 480 [V] rms line-line source
through 1000 [ft] of 1/0 AWG quadruplex overhead service drop cable. Three
different cases are considered: 1) no-fault, 2) line-ground fault, and 3)
line-line fault.
### IV-C Grounded-Wye Load
For the grounded-wye case, the system used is depicted in Fig. 3. The load
consists of three balanced series RL branches wired in a grounded-wye
configuration. This system has the parameters listed in Tables II and III.
Note that the total fault resistance for the line-ground fault is
$R_{f}+R_{g}=110$ [m$\Omega$], while the total fault resistance for the line-
line fault is $2R_{f}=200$ [m$\Omega$].
Figure 3: MATLAB Simulink model for a grounded-wye load with faults.
### IV-D Delta Load
For the delta-connected case, the system used is depicted in Fig. 4. The load
consists of three balanced series RL branches wired in a delta configuration.
This system has the parameters listed in Tables II and III. Note that the
total fault resistance for the line-ground fault is $R_{f}+R_{g}=110$
[m$\Omega$], while the total fault resistance for the line-line fault is
$2R_{f}=200$ [m$\Omega$].
(a) Main model (b) Delta load
Figure 4: MATLAB Simulink model for a delta load with faults.
## V Results
Results for the phasor models are presented in Table IV, while results for the
dynamic models are presented in Table V. The phasor models provide an
excellent estimate of the system parameters. DSE has difficulty estimating the
fault resistance for the dynamic model of grounded-wye network with a line-
line fault; a potential solution is to reduce the model order by neglecting
the load impedance on the faulted phases. Some moderate error is observed for
the case of the delta-connected load with a line-ground fault; again, it may
be possible to improve performance by neglecting load impedance on the faulted
phases.
Case | $R$ | $\hat{R}$ | $X$ | $\hat{X}$ | $R_{f}$ | $\hat{R}_{f}$
---|---|---|---|---|---|---
Single-Phase RL Load | 19.200 | 19.200 | 9.600 | 9.600 | – | –
Grounded-Wye Line-Ground Fault | 7.387 | 7.387 | 3.693 | 3.693 | 0.923 | 0.923
Grounded-Wye Line-Line Fault | 7.387 | 5.184 | 3.693 | 4.787 | 0.923 | 0.935
Delta Line-Line Fault | 22.160 | 22.160 | 11.080 | 11.080 | 2.770 | 2.770
Delta Line-Ground Fault | 22.160 | 22.160 | 11.080 | 11.080 | 2.770 | 2.770
TABLE IV: Results for Phasor State Estimation Case | $R$ ($\Omega$) | $\hat{R}$ ($\Omega$) | $L$ (mH) | $\hat{L}$ (mH) | $R_{f}$ ($m\Omega$) | $\hat{R}_{f}$ ($m\Omega$)
---|---|---|---|---|---|---
Single-Phase RL Load | 19.200 | 19.265 | 25.465 | 25.988 | – | –
Grounded-Wye No Fault | 18.432 | 18.404 | 24.446 | 24.485 | – | –
Grounded-Wye Line-Ground Fault | 18.432 | 18.432 | 24.446 | 24.446 | 11.000 | 10.997
Grounded-Wye Line-Line Fault | 18.432 | 18.432 | 24.446 | 24.446 | 11.000 | 3.165
Delta No Fault | 55.296 | 55.412 | 73.339 | 73.495 | – | –
Delta Line-Line Fault | 55.296 | 55.895 | 73.339 | 73.666 | 2.000 | 2.001
Delta Line-Ground Fault | 55.296 | 55.479 | 73.339 | 73.495 | 11.000 | 11.405
TABLE V: Results for Dynamic State Estimation
## VI Conclusions
The results of performed experiments prove that DSE is capable of correctly
identifying model parameters of three different load configurations for both
normal and faulted operations. These load configurations model a lumped
equivalent of a radial electrical grid supplying multiple loads.
Several models showed sensitivity to inital conditions, particularly the
delta-connected load, so it is important that consideration be given to
providing the presented methods with good initial conditions. One issue is
making sure that there is a sufficient number of measurements to estimate
model states; for example, it is not possible to infer impedances for an
unbalanced delta-connected load given a single time snapshot. To reduce the
number of states, the models presented here assume that loads are balanced;
this assumption can be an issue for systems with a high degree of load
imbalance.
Existing work has demonstrated that DSE can operate with nonlinear elements
[16]. One option for future work is to expand the here presented methods with
other load models. These could include nonlinear voltage-dependent models
where power is a polynomial function of voltage (ZIP loads) or where power is
a polynomial function of both voltage and frequency (e.g. the WSCC load model
[17]). Alternately, these could include dynamic load models (e.g. an induction
motor model (MOTORW) or a composite load model (CMPLDW) [18]). Last, there is
the possibility of protecting line sections that include loads with
coordinated breakers at both ends; this could correspond to a distributed
parameter line or a Pi/Tee lumped equivalent model [19].
## References
* [1] A. P. S. Meliopoulos, G. J. Cokkinides, P. Myrda, Y. Liu, R. Fan, L. Sun, R. Huang, and Z. Tan. Dynamic State Estimation-Based Protection: Status and Promise. IEEE Transactions on Power Delivery, 32(1):320–330, Feb. 2017\.
* [2] Y. Liu, A. P. S. Meliopoulos, R. Fan, L. Sun, and Z. Tan. Dynamic State Estimation Based Protection on Series Compensated Transmission Lines. IEEE Transactions on Power Delivery, 32(5):2199–2209, 2017.
* [3] Y. Liu, A. P. S. Meliopoulos, L. Sun, and R. Fan. Dynamic State Estimation Based Protection of Mutually Coupled Transmission Lines. CSEE Journal of Power and Energy Systems, 2(4):6–14, Dec. 2016\.
* [4] Y. Liu, A. P. S. Meliopoulos, R. Fan, and L. Sun. Dynamic State Estimation Based Protection of Microgrid Circuits. In 2015 IEEE Power Energy Society General Meeting, pages 1–5, Jul. 2015.
* [5] O. Vasios, S. Kampezidou, and A. P. S. Meliopoulos. A Dynamic State Estimation Based Centralized Scheme for Microgrid Protection. In Proceedings of the 2018 North American Power Symposium, pages 1–6, Sep. 2018.
* [6] S. Choi and A. P. S. Meliopoulos. Effective Real-Time Operation and Protection Scheme of Microgrids Using Distributed Dynamic State Estimation. IEEE Transactions on Power Delivery, 32(1):504–514, Feb. 2017\.
* [7] R. M. Tumilty, M. Brucoli, G. M. Burt, and T. C. Green. Approaches to Network Protection for Inverter Dominated Electrical Distribution Systems. In Proceedings of the 3rd IET International Conference on Power Electronics, Machines and Drives, 2006, pages 622–626, Apr. 2006.
* [8] M. Dewadasa, A. Ghosh, and G. Ledwich. Line Protection in Inverter Supplied Networks. In Proceedings of the 2008 Australasian Universities Power Engineering Conference, pages 1–6, Dec. 2008.
* [9] A. K. Barnes and A. Mate. Implementing Admittance Relaying for Microgrid Protection. In Proceedings of the 2021 IEEE/IAS 57th Industrial and Commercial Power Systems Technical Conference, pages 1–9, Apr. 2021.
* [10] J. M. Dewadasa, A. Ghosh, and G. Ledwich. Distance Protection Solution for a Converter Controlled Microgrid. In Proceedings of the 15th National Power Systems Conference, 2008\.
* [11] D. P. Kothari and I. J. Nagrath. Modern Power System Analysis. Tata McGraw-Hill Education, 1989.
* [12] A. Monticelli. Electric Power System State Estimation. Proceedings of the IEEE, 88(2):262–282, Feb. 2000.
* [13] S. Chapra and R. Canale. Numerical Methods for Engineers. McGraw-Hill Education, 2009.
* [14] J. Bezanson, S. Karpinski, V. B. Shah, and A. Edelman. Julia: A Fast Dynamic Language for Technical Computing. arXiv:1209.5145, 2012.
* [15] MathWorks. MATLAB R2019b Documentation. MATLAB – https://www.mathworks.com, 2020.
* [16] H. F. Albinali. State Estimation-Based Centralized Substation Protection Scheme. PhD thesis, Georgia Institute of Technology, Aug. 2017.
* [17] L. Pereira, D. Kosterev, P. Mackin, D. Davies, J. Undrill, and Z. Wenchun. An Interim Dynamic Induction Motor Model for Stability Studies in the WSCC. IEEE Transactions on Power Systems, 17(4):1108–1115, Nov. 2002\.
* [18] North American Electric Reliability Corporation. Technical Reference Document: Dynamic Load Modeling. Technical report, NERC, 2016.
* [19] D. T. Rizy and R. H. Staunton. Evaluation of Distribution Analysis Software for DER Applications. Technical report, Oak Ridge National Laboratory, 2002.
|
The Grothendieck Construction in Categorical Network Theory
Joseph Patrick Moeller
Doctor of Philosophy
Dr. John C. Baez
Dr. Wee Liang Gan
Dr. Carl Mautner
First of all, I owe all of my achievements to my wife, Paola. I couldn't have gotten here without my parents: Daniel, Andrea, Tonie, Maria, and Luis, or my siblings: Danielle, Anthony, Samantha, David, and Luis.
I would like to thank my advisor, John Baez, for his support, dedication, and his unique and brilliant style of advising. I could not have become the researcher I am under another's instruction. I would also like to thank Christina Vasilakopoulou, whose kindness, energy, and expertise cultivated a deeper appreciation of category theory in me. My experience was also greatly enriched by my academic siblings: Daniel Cicala, Kenny Courser, Brandon Coya, Jason Erbele, Jade Master, Franciscus Rebro, and Christian Williams, and by my cohort: Justin Davis, Ethan Kowalenko, Derek Lowenberg, Michel Manrique, and Michael Pierce.
I would like to thank the UCR math department. Professors from whom I learned a ton of algebra, topology, and category theory include Julie Bergner, Vyjayanthi Chari, Wee-Liang Gan, José Gonzalez, Jacob Greenstein, Carl Mautner, Reinhard Schultz, and Steffano Vidussi. Special thanks goes to the department chair Yat-Sun Poon, as well as Margarita Roman, Randy Morgan, and James Marberry, and many others who keep the whole thing together.
The material in <ref> consists of work from both Network models joint with John Baez, John Foley, and Blake Pollard <cit.>. <ref> consists of work done in my paper Noncommutative network models <cit.>. <ref> arose from Network models from Petri nets with catalysts joint with Baez and Foley <cit.>. <ref> consists of joint work with Christina Vasilakopoulou appearing in our paper Monoidal Grothendieck construction <cit.>. Part of this work was performed with funding from a subcontract with Metron Scientific Solutions working on DARPA’s Complex Adaptive System Composition and Design Environment (CASCADE) project.
To Teresa Danielle Moeller.
In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a network model, a monoidal variant of Joyal's combinatorial species. The construction of the operad required that we develop a monoidal lift of the Grothendieck construction. We then demonstrate how concepts like priority and dependency can be represented in this framework. For the former, we generalize Green's graph products of groups to the context of universal algebra. For the latter, we examine the emergence of monoidal fibrations from the presence of catalysts in Petri nets.
CHAPTER: INTRODUCTION
§ SEARCH AND RESCUE
Imagine that you have a network of boats, planes, and drones tasked with rescuing sailors who have fallen overboard in a hurricane. You want to be able to task these agents to search certain areas for survivors in an intelligent way. You do not want to waste time and resources by double searching some areas while other areas get neglected. Also, if one of the searchers gets taken out by the storm, you must update the tasking so that other agents can cover the areas which the downed agent has yet to search, as well as recording that there is a new known person in need of rescue.
In 2015, DARPA launched a program called Complex Adaptive System Composition and Design Environment, or CASCADE. The goal of this program was to write software that would be able to handle this sort of tasking of agents in a network in a flexible and responsive way. The bulk of this thesis was developed while I was working on this project with Metron Scientific Solutions Inc., developing a mathematically principled foundation around which this software could be designed. John Baez, John Foley, Blake Pollard, and I developed the theory of network models to address this challenge <cit.>.
§ NETWORK OPERADS
Large complex networks can be viewed as being built up from small simple pieces. This sort of many-to-one composition is perfectly suited to being modeled using operads. While a category can be described as a system of composition for a collection of arrows which have a specified input type and a specified output type, an operad is a system of composition for a collection of trees which have a specified family of input types and a single specified output type.
\[
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We use the word “operations” instead of “trees”, hence the name operad. Like categories, operads were originally developed in algebraic topology <cit.>. Also like categories, operads have since found applications elsewhere, including physics and computer science <cit.>. We include a review of the basics of operads needed for this thesis in <ref>.
In a network operad, the operations describe ways of sticking together a collection of networks to form a new larger network. To get a network operad, we treat a network as one of these operations and define the composition as overlaying a bunch of small networks on top of a large base network. For example, in the following picture, we are considering simple graphs as a sort of network. On the left, we are starting with a base network consisting of nine nodes and four edges, and we are attempting to attach more edges by overlaying three smaller graphs. The result of the operadic composition is on the right.
\[
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\node [style=species] (8) at (3, -1) {$8$};
\node [style=species] (9) at (5, -1) {$9$};
\node [rectangle, dashed, fill = none, draw = black, minimum width = 220, minimum height = 150] () at (4, 0.75) {};
\node [style=species] (11) at (-1, 6.5) {$1$};
\node [style=species] (12) at (1, 6.5) {$2$};
\node [style=species] (13) at (0, 5) {$3$};
\node [rectangle, dashed, fill = none, draw = black, minimum width = 90, minimum height = 80] () at (0, 5.75) {};
\node [style=species] (14) at (3, 6.5) {$1$};
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\node [style=species] (16) at (3, 5) {$3$};
\node [style=species] (17) at (5, 5) {$4$};
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\node [style = none] (s) at (4,3.37) {};
\node [style = none] (t) at (6,3.37) {};
\node [style = none] (u) at (4,-1.85) {};
\node [style = none] (v) at (4,-3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=simple] (1) to (2);
\draw [style=simple] (3) to (6);
\draw [style=simple] (4) to (6);
\draw [style=simple] (6) to (8);
\draw [style=simple] (11) to (12);
\draw [style=simple] (12) to (13);
\draw [style=simple] (14) to (15);
\draw [style=simple] (15) to (16);
\draw [style=simple] (16) to (17);
\draw [style=simple] (18) to (19);
\draw [style=simple] (a) to (h);
\draw [style=simple] (b) to (i);
\draw [style=simple] (c) to (j);
\draw [style=simple] (d) to (k);
\draw [style=simple] (e) to (l);
\draw [style=simple] (f) to (m);
\draw [style=simple] (g) to (n);
\draw [style=simple] (o) to (r);
\draw [style=simple] (p) to (s);
\draw [style=simple] (q) to (t);
\draw [style=simple] (u) to (v);
\end{pgfonlayer}
\end{tikzpicture}}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style = none] () at (0,0) {=};
\node [style = none] () at (1.5,-3) {};
\node [style = none] () at (-1,0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\end{pgfonlayer}
\end{tikzpicture}
\scalebox{0.6}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (1) at (1, 2.5) {$1$};
\node [style=species] (2) at (3, 2.5) {$2$};
\node [style=species] (3) at (2, 1) {$3$};
\node [style=species] (4) at (5, 2.5) {$4$};
\node [style=species] (5) at (7, 2.5) {$5$};
\node [style=species] (6) at (5, 1) {$6$};
\node [style=species] (7) at (7, 1) {$7$};
\node [style=species] (8) at (3, -1) {$8$};
\node [style=species] (9) at (5, -1) {$9$};
\node [rectangle, dashed, fill = none, draw = black, minimum width = 220, minimum height = 150] () at (4, 0.75) {};
\node [style = none] (a) at (1,4.36) {};
\node [style = none] (b) at (2,4.36) {};
\node [style = none] (c) at (3,4.36) {};
\node [style = none] (d) at (4,4.36) {};
\node [style = none] (e) at (5,4.36) {};
\node [style = none] (f) at (6,4.36) {};
\node [style = none] (g) at (7,4.36) {};
\node [style = none] (h) at (1,3.37) {};
\node [style = none] (i) at (2,3.37) {};
\node [style = none] (j) at (3,3.37) {};
\node [style = none] (k) at (4,3.37) {};
\node [style = none] (l) at (5,3.37) {};
\node [style = none] (m) at (6,3.37) {};
\node [style = none] (n) at (7,3.37) {};
\node [style = none] (u) at (4,-1.85) {};
\node [style = none] (v) at (4,-3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=simple] (1) to (2);
\draw [style=simple] (2) to (3);
\draw [style=simple] (3) to (6);
\draw [style=simple] (4) to (5);
\draw [style=simple] (4) to (6);
\draw [style=simple] (5) to (6);
\draw [style=simple] (6) to (7);
\draw [style=simple] (6) to (8);
\draw [style=simple] (8) to (9);
\draw [style=simple] (a) to (h);
\draw [style=simple] (b) to (i);
\draw [style=simple] (c) to (j);
\draw [style=simple] (d) to (k);
\draw [style=simple] (e) to (l);
\draw [style=simple] (f) to (m);
\draw [style=simple] (g) to (n);
\draw [style=simple] (u) to (v);
\end{pgfonlayer}
\end{tikzpicture}}
\]
This example is fairly elementary, and it is probably not too difficult for someone comfortable with the notions to define this operad. However, it is not just simple graphs that one needs when talking about managing and tasking complex networks of various sorts of agents with various forms of communication and capabilities. One could continue replicating the procedure for constructing network operads for each type of network whenever needed, but this is an inefficient strategy. Instead, we devised a general recipe for constructing such an operad for a given network type, and a general method for specifying a network type in an efficient way, using what we call a network model. All of this is done in the language of category theory, so we also have a theory of how morphisms between network models give morphisms between their operads.
§ CONSTRUCTING NETWORK OPERADS
There is a well-known trick for extracting an operad from any symmetric monoidal category. An operation in the operad is defined to be a morphism from a tensor product of a finite family of objects to a single object. This is called the underlying operad of the symmetric monoidal category. So now we have shifted the problem of defining an operad where the operations are networks to defining a symmetric monoidal category where the morphisms are networks. To achieve this, we can use the famous Grothendieck construction—though we need to enhance it to suit our purposes.
§ MONOIDAL GROTHENDIECK CONSTRUCTION
The Grothendieck construction is a well-known trick for turning a family of categories indexed by the objects of some other category into a single category in an intelligent way <cit.>. What we really would like is that morphisms in the indexing category translate into morphisms in our total category between objects from the corresponding indices. The classic example is the family of categories $\Mod_R$ of $R$-modules, indexed by the objects of $\Ring$, the category of rings. Sometimes, one would like to talk about a single category of modules over all possible rings to study the interactions between such modules. The naive thing to do would be to just take the coproduct, defining
\[\Mod = \coprod_{R \in \Ring} \Mod_R.\]
However, in this category an $R$-module and an $S$-module would have no morphisms between them. This runs counter to the goal of having a single category for reasoning about the interactions of modules over potentially different rings. If $f \maps R \to S$ is a ring homomorphism, there is a way of turning $S$-modules into $R$-modules using $f$, called pullback. If $M$ is an $S$-module, $m \in M$, and $r \in R$, pulling back $M$ along $f$ defines an $R$-module structure on the underlying abelian group of $M$. We define the action of $r \in R$ on $m \in M$ by the following formula.
\[
r \cdot m = f(r) \cdot m
\]
This construction turns out to give a functor
\[f^\ast \maps \Mod_S \to \Mod_R.\]
We should hope also that the data of these functors is included in the total category we construct. Indeed, the Grothendieck construction accomplishes precisely this.
However, it is not simply a category that we need, but a symmetric monoidal category. So we built an enhanced version of the Grothendieck construction, which takes family of categories indexed by a symmetric monoidal category and constructs a symmetric monoidal category <cit.>. Christina Vasilakopoulou and I extended this modification to solve the monoidality problem in the Grothendieck correspondence <cit.>.
These two steps constitute the construction of the desired network operad: we start with a monoidal indexed category, use the monoidal Grothendieck construction to produce a symmetric monoidal category, and then take its underlying operad. This leads to another question though: what monoidal indexed categories should we feed into this construction in order to produce network operads?
§ NETWORK MODELS
The answer is that we should take a monoidal version of Joyal's combinatorial species <cit.>. A combinatorial species is a functor $F \maps \FinBij \to \Set$. One way of looking at this is as a family of symmetric group actions, one for each natural number. Another way of looking at it is as a particular type of indexed category, where there is a family of discrete categories (sets) indexed by the natural numbers, and functors (functions) between them corresponding to the morphisms in $\FinBij$. So this is something to which we can apply the Grothendieck construction. The resulting total category is a groupoid which has all the elements in all the sets as the objects, and an isomorphism between these elements if they are in the same orbit under the symmetric group action.
Recall our example of a network operad where an operation is a simple graph. To build this, we can start with the species of simple graphs $\SG \maps \FinBij \to \Set$. We give this the structure of a lax monoidal functor $(\FinBij, +) \to (\Set, \times)$ by equipping it with a natural map $\SG(m) \times \SG(n) \to \SG(m+n)$ given by disjoint union. We include the data of the overlaying of graphs as a monoid structure on the set $\SG(n)$ of simple graphs on $n$ nodes. The product of two graphs on $n$ nodes is another graph on $n$ nodes given by identifying corresponding nodes, and including an edge wherever either of the original graphs had one. So now we have a lax symmetric monoidal functor $(\SG, \sqcup) \maps (\FinBij, +) \to (\Mon, \times)$. We call such a map a network model. When we take the Grothendieck construction of this, we treat the monoids as one-object categories. By doing this, the resulting category has objects given by finite sets, a morphism $n \to n$ is given by a simple graph on $n$ nodes, composition overlays the graphs, and tensor sets them side by side.
§ CONSTRUCTING NETWORK MODELS
Network operads are constructed from network models. How do we get our hands on some network models? We know about a few examples of network models: simple graphs, directed graphs, multigraphs, colored vertices, etc. Ideally, we would have a (functorial) way to generate network models from some simple description of what we want a network to look like.
We can begin by examining the basic example: simple graphs. It consists of a family of monoids $\SG(n)$ where the elements are simple graphs on $n$ nodes, with symmetric group actions which permute the nodes, and a “disjoint union” operation $\sqcup \maps \SG(m) \times \SG(n) \to \SG(m+n)$. The level-$0$ and level-$1$ monoids are both trivial. The first interesting one is level-$2$, where the monoid is isomorphic to the Boolean truth values with the “or” operation. The rest of the monoids in this network model can be seen as built from $\SG(2)$. A simple graph with $n$ nodes has $n\choose2$ places where it can either have or not have an edge. We can define the monoid $\SG(n)$ to be the product of $n\choose2$ copies of $\SG(1)$, indexed by distinct pairs of nodes. This leads to the general construction: given a monoid $M$, let $\overline M(n)$ be the monoid given by the product $n \choose2$ copies of $M$. Then the collection of these monoids $\overline M$ is a network model, where a network has an element of $M$ between every pair of nodes, and overlaying two networks simply requires performing the monoid operation at every pair of nodes. This construction covers the example of simple graphs by design, but also includes multigraphs, directed graphs, graphs with colored edges, and many other examples.
§ NONCOMMUTATIVE NETWORK MODELS
Another property we wanted to be able to represent within the network operads framework was forms of communication which had a built-in limitation on the number of connections. This is a natural issue in the search and rescue domain problem <cit.>.
There is no natural way to decide which edges not to include when the limit of connections is reached. This means that the network must have some extra data built into it. In particular, it must remember the order in which the connections were added to each node. For this, we need the edge components of the constituent monoids to not commute with each other. Due to a variant of the Eckmann-Hilton argument, edge components of a network model's constituent monoid actually must commute with each other if they do not share any of their nodes. This means the most we can ask for is that edge components of the network model do not commute with each other when edges have a node in common.
We cannot simply take iterated products of the monoid as we did before because the edge components of the resulting monoids always commute with each other. We also cannot simply take coproducts because the edge components do not commute with each other in way that are necessary for a network model. Therefore, we must have a mix of products and coproducts depending on which edges share a node and which do not. Specifically, if two edges share a node, then elements of the corresponding edge components of the monoid must not commute with each other, and if they do not share a node, they must commute with each other. Such a monoid can be constructed using graph products of monoids, introduced for groups in Elisabeth Green's thesis <cit.>. The idea is to produce a new monoid from a finite set of monoids by assigning them to the nodes in a graph, taking the coproduct of them all, then imposing commutativity relations between elements coming from monoids which had an edge between them in the graph.
What indexing graph should we use though? We want a copy of the monoid for every possible edge. So our indexing graph should have $n\choose2$ nodes, one for every subset of cardinality $2$. We want to impose commutativity between two edge components whenever the corresponding edges do not share a node, so we add an edge for each pair of cardinality $2$ subsets which have empty intersection. This is precisely the definition of what are called the Kneser graphs! The first few non-empty ones are depicted below.
\[
\vcenter{\hbox{\begin{tikzpicture} % KG_3,2
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (a) at (0, 0.44) {};
\node [style=reddot] (e) at (-0.5, -.44) {};
\node [style=reddot] (f) at (0.5, -.44) {};
\node [style=none] () at (2, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\end{pgfonlayer}
\end{tikzpicture}}}
\vcenter{\hbox{\begin{tikzpicture} % KG_4,2
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (a) at (1, 0) {};
\node [style=reddot] (b) at (0.5, .87) {};
\node [style=reddot] (c) at (-0.5, .87) {};
\node [style=reddot] (d) at (-1, 0) {};
\node [style=reddot] (e) at (-0.5, -.87) {};
\node [style=reddot] (f) at (0.5, -.87) {};
\node [style=none] () at (2, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (a) to (b);
\draw (c) to (d);
\draw (e) to (f);
\end{pgfonlayer}
\end{tikzpicture}}}
\vcenter{\hbox{\begin{tikzpicture} % Petersen graph
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (a) at (0, -1) {};
\node [style=reddot] (b) at (0.95, -0.31) {};
\node [style=reddot] (c) at (0.59, 0.81) {};
\node [style=reddot] (d) at (-0.59, 0.81) {};
\node [style=reddot] (e) at (-0.95, -0.31) {};
\node [style=reddot] (A) at (0, -2) {};
\node [style=reddot] (B) at (1.9, -0.62) {};
\node [style=reddot] (C) at (1.18, 1.62) {};
\node [style=reddot] (D) at (-1.18, 1.62) {};
\node [style=reddot] (E) at (-1.9, -0.62) {};
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\draw (a) to (A);
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\draw (d) to (D);
\draw (e) to (E);
\draw (A) to (B);
\draw (B) to (C);
\draw (C) to (D);
\draw (D) to (E);
\draw (E) to (A);
\draw (a) to (c);
\draw (b) to (d);
\draw (c) to (e);
\draw (d) to (a);
\draw (e) to (b);
\end{pgfonlayer}
\end{tikzpicture}}}
\]
For a given monoid $M$, we thus define the corresponding network model to be the graph product of $M$ with itself indexed by the corresponding Kneser graph. In fact, this construction gives the free network model on $M$, forming a left adjoint to the functor which evaluates a network model at $2$. This provides a solution to the problem of representing degree limited networks in the language of network operads. This construction gives a network operad where the networks are graphs such that every vertex has degree $\leq N$, and the network does not take an edge if this limit would be exceeded.
§ PETRI NETS WITH CATALYSTS
Network models are also able to describe scenarios where there is an agent or agents that can manipulate and transport resources within the network <cit.>. Baez, Foley, and I use a simple structure called a Petri net to represent resources and processes that transform them <cit.>. A Petri net can be drawn as a directed graph with vertices of two kinds: places or species, which we draw as yellow circles below, and transitions, which we draw as blue squares:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\quad\;$};
\node [style=species] (B) at (-4, -0.5) {$\quad\;$};
\node [style=species] (A) at (-4, 0.5) {$\quad\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
Petri nets are intended to model resources in a network of processes. Sometimes, we represent the resources by a finite number of tokens in each place:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\quad\;$};
\node [style=species] (B) at (-4, -0.5) {$\;\bullet\;$};
\node [style=species] (A) at (-4, 0.5) {$\bullet\bullet$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
This is called a marking. We can then “run” the Petri net by repeatedly changing the marking using the transitions. For example, the above marking can change to this:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\;\bullet\;$};
\node [style=species] (B) at (-4, -0.5) {$\quad\;$};
\node [style=species] (A) at (-4, 0.5) {$\;\bullet\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
and then this:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\quad\;$};
\node [style=species] (B) at (-4, -0.5) {$\bullet\bullet$};
\node [style=species] (A) at (-4, 0.5) {$\;\bullet\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
Thus, the places represent different types of resource, and the transitions describe ways that one collection of resources of specified types can turn into another such collection.
An agent might pick up a box and carry it over to a truck, and then drive the truck over to a new warehouse, and then unload the box. In this scenario, the gasoline in the truck might be a resource that considered to be consumed by this process, but the agent is not. This qualitative difference between the agent as a resource and the gasoline as a resource leads to a quantitative difference. Specifically, the number of agents in this network is never changing, but the number of gallons of gasoline is. What this means for the Petri net model of this network is that there is no combination of transition firing that change the number of agents. This gives us a fibration of the commutative monoidal category of executions for the Petri net. However, unlike the monoidal fibrations described earlier, the fibres here are only premonoidal in general, not quite monoidal. This gives an example of a generalized network model, one where the monoids in the original definition are replaced with categories.
§ OUTLINE OF THE THESIS
I begin by laying out the theory of network models and network operads in <ref>. <ref> and <ref> contain basic definitions and examples. The construction of a network operad from a network model and several examples of algebras of network operads are given in <ref>.
In <ref>, more constructions of network models are given. The construction of free network models from a given monoid is detailed in <ref>. This depends on a generalization of Green's graph products given in <ref>. In section 3.4, an example of an algebra for a noncommutative network model arising from limitations on communication networks is given.
<ref> discusses the construction of network models from Petri nets with catalysts. In <ref>, the basic notions for the categorical treatment of Petri nets are recalled. <ref> explains what it means for a Petri net to have catalysts. <ref> describes how catalysts induce a premonoidal fibration on the category of executions, and explain how this gives an example of a generalized network model.
I finish with a self-contained treatment of the monoidal Grothendieck construction in <ref>. As the theoretical underpinning of the theory of network models, it is the most technically dense, and thus saved for the most enduring of readers. <ref> and <ref> describe monoidal fibrations and indexed categories. <ref> details the corresponding Grothendieck constructions for each monoidal variant. <ref> discusses the special case of when the base category is co/cartesian. In <ref>, we give a nuts-and-bolts description of the monoidal structures constructed in various scenarios. <ref> demonstrates the potential usefulness of the construction with examples from categorical algebra and dynamical systems.
I wanted to include my own explanations and several references for preliminary materials, but did not want this to clutter the primary narrative of the thesis. I have included much of this in several appendices. I discuss some of the basic theory of monoidal categories in <ref>; monoidal 2-categories and Gray monoids in <ref>; fibrations, indexed categories, and the Grothendieck construction in <ref>; and combinatorial species and operads in <ref>.
CHAPTER: NETWORK MODELS
§ INTRODUCTION
In this chapter, we study operads suited for designing networks. These could be networks where the vertices represent fixed or moving agents and the edges represent communication channels. More generally, they could be networks where the vertices represent entities of various types, and the edges represent relationships between these entities, e.g. that one agent is committed to take some action involving the other. The work done is this chapter arose from an example where the vertices represent planes, boats and drones involved in a search and rescue mission in the Caribbean <cit.>. However, even for this one example, we want a flexible formalism that can handle networks of many kinds, described at a level of detail that the user is free to adjust.
To achieve this flexibility, we introduce a general concept of network model. Simply put, a network model is a kind of network. Any network model gives an operad whose operations are ways to build larger networks of this kind by gluing smaller ones. This operad has a canonical algebra where the operations act to assemble networks of the given kind. But it also has other algebras, where it acts to assemble networks of this kind equipped with extra structure and properties. This flexibility is important in applications.
What exactly is a kind of network? At the crudest level, we can model networks as simple graphs. If the vertices are agents of some sort and the edges represent communication channels, this means we allow at most one channel between any pair of agents. However, simple graphs are too restrictive for many applications. If we allow multiple communication channels between a pair of agents, we should replace simple graphs with multigraphs. Alternatively, we may wish to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. This requires replacing simple graphs with directed graphs. To combine these features we could use directed multigraphs. It is also important to consider graphs with colored vertices, to specify different types of agents, and colored edges, to specify different types of channels. This leads us to colored directed multigraphs. All these are examples of what we mean by a kind of network. Even more complicated kinds, such as hypergraphs or Petri nets, are likely to become important as we proceed. Thus, instead of separately studying all these kinds of networks, we introduce a unified notion that subsumes all these variants: a network model. Namely, given a set $C$ of vertex colors, a network model $F$ is a lax symmetric monoidal functor $F \maps \S(C) \to \Cat$, where $\S(C)$ is the free strict symmetric monoidal category on $C$ and $\Cat$ is the category of small categories, considered with its cartesian monoidal structure. Unpacking this definition takes a little work. It simplifies in the special case where $F$ takes values in $\Mon$, the category of monoids. It simplifies further when $C$ is a singleton, since then $\S(C)$ is the groupoid $\S$, where objects are natural numbers and morphisms from $m$ to $n$ are bijections $\sigma \maps \{1,\dots,m\} \to \{1,\dots,n\}$. If we impose both these simplifying assumptions, we have what we call a one-colored network model: a lax symmetric monoidal functor $F \maps \S \to \Mon$. As we shall see, the network model of simple graphs is a one-colored network model, and so are many other motivating examples.
Joyal began an extensive study of functors $F \maps \S \to \Set$, which are now commonly called species <cit.>. Any type of extra structure that can be placed on finite sets and transported along bijections defines a species if we take $F(n)$ to be the set of structures that can be placed on the set $\{1, \dots, n\}$. From this perspective, a one-colored network model is a species with some extra operations.
This perspective is helpful for understanding what a one-colored network model $F \maps \S \to \Mon$ is actually like. If we call elements of $F(n)$ networks with $n$ vertices, then:
* Since $F(n)$ is a monoid, we can overlay two networks with the same number of vertices and get a new one. We denote this operation by
\[
\cup \colon F(n) \times F(n) \to F(n).
\]
For example:
\[\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
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\draw [style=simple] (11) to (12);
\end{pgfonlayer}
\end{tikzpicture}}\]
* Since $F$ is a functor, the group $S_n$ acts on the monoid $F(n)$. Thus, for each $\sigma \in S_n$, we have a monoid automorphism that we call
\[\sigma \maps F(n) \to F(n) . \]
For example, if $\sigma = (2\,3) \in S_3$, then
\[\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
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\end{pgfonlayer}
\end{tikzpicture}}\]
* Since $F$ is lax monoidal, we have an operation
\[
\sqcup \colon F(m) \times F(n) \to F(m+n)
\]
We call this operation the disjoint union of networks. For example:
\[\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none, scale = 1.2] () at (5,2) {$\sqcup$};
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\end{pgfonlayer}
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The first two operations are present whenever we have a functor $F \maps \S \to \Mon$. The last two are present whenever we have a lax symmetric monoidal functor $F \maps \S \to \Set$. When $F$ is a one-colored network model we have all three—and unpacking the definition further, we see that they obey some equations, which we list in <ref>. For example, the interchange law
\[(g \cup g') \sqcup (h \cup h') = (g \sqcup h) \cup (g' \sqcup h') \]
holds whenever $g,g' \in F(m)$ and $h, h' \in F(n)$.
In <ref> we study one-colored network models more formally, and give many examples. In <ref> we describe a systematic procedure for getting one-colored network models from monoids. In <ref> we study general network models and give examples of these. In <ref> we describe a category $\NetMod$ of network models, and show that the procedure for getting network models from monoids is functorial. We also make $\NetMod$ into a symmetric monoidal category, and give examples of how to build new networks models by tensoring old ones.
Our main result is that any network model gives a typed operad, also known as a colored operad or symmetric multicategory <cit.>. A typed operad describes ways of sticking together things of various types to get new things of various types. An algebra of the operad gives a particular specification of these things and the results of sticking them together. We review the definitions of operads and their algebras in <ref>. A bit more precisely, a typed operad $O$ has:
* a set $T$ of types,
* sets of operations $O(t_1,...,t_n ; t)$ where $t_i, t \in T$,
* ways to compose any operation
\[f \in O(t_1,\dots,t_n ;t) \]
with operations
\[g_i \in O(t_{i1},\dots,t_{i k_i}; t_i) \qquad (1 \le i \le n) \]
to obtain an operation
\[f \circ (g_1,\dots,g_n) \in O(t_{1i}, \dots, t_{1k_1}, \dots,
t_{n1}, \dots t_{n k_n}; t), \]
* and ways to permute the arguments of operations,
which obey some rules <cit.>. An algebra $A$ of $O$ specifies a set $A(t)$ for each type $t \in T$ such that the operations of $O$ act on these sets. Thus, it has:
* for each type $t \in T$, a set $A(t)$ of things of type $t$,
* ways to apply any operation
\[f \in O(t_1, \dots, t_n ; t) \]
to things
\[a_i \in A(t_i) \qquad (1 \le i \le n) \]
to obtain a thing
\[\alpha(f)(a_1, \dots, a_n) \in A(t). \]
Again, we demand that some rules hold <cit.>.
In <ref> we describe the typed operad $\O_F$ arising from a one-colored network model $F$. The set of types is $\N$, since we can think of `network with $n$ vertices' as a type. The sets of operations are given as follows:
\[
\O_F(n_1, \dots, n_k; n) = \left\{
\begin{array}{cl} S_n \times F(n) & \textrm{if }
n_1 + \cdots + n_k = n \\
\emptyset & \textrm{otherwise.}
\end{array} \right.
\]
The key idea here is that we can overlay a network in $F(n)$ on the disjoint union of networks with $n_1, \dots, n_k$ vertices and get a new network with $n$ vertices as long as $n_1 + \cdots n_k = n$. We can also permute the vertices; this accounts for the group $S_n$. But the most important fact is that networks serve as operations to assemble networks, thanks to our ability to overlay them.
Using this fact, we show in <ref> that the operad $\O_F$ has a canonical algebra $A_F$ whose elements are simply networks of the kind described by $F$:
\[A_F(n) = F(n) .\]
In this algebra any operation
\[
(\sigma,g) \in \O_F(n_1, \dots , n_k; n) = S_n \times F(n)
\]
acts on a $k$-tuple of networks
\[
h_i \in A_F(n_i) = F(n_i) \qquad (1 \le i \le k)
\]
to give the network
\[
\alpha(\sigma,g)(h_1, \dots, h_k) = g \cup \sigma(h_1 \sqcup \cdots \sqcup h_k) \in A_F(n).
\]
In other words, we first take the disjoint union of the networks $h_i$, then permute their vertices with $\sigma$, and then overlay the network $g$.
An example is in order, since the generality of the formalism may hide the simplicity of the idea. The easiest example of our theory is the network model for simple graphs. In <ref> we describe a one-colored network model $\SG \maps \S \to \Mon$ such that $\SG(n)$ is the collection of simple graphs with vertex set $\n = \{1,\dots,n\}$. Such a simple graph is really a collection of 2-element subsets of $\n$, called edges. Thus, we may overlay simple graphs $g,g' \in \SG(n)$ by taking their union $g \cup g'$. This operation makes $\SG(n)$ into a monoid.
Now consider an operation $f \in \O_\SG(3,4,2;9)$. This is an element of $S_9 \times \SG(9)$: a permutation of the set $\{1,\dots, 9\}$ together with a simple graph having this set of vertices. If we take the permutation to be the identity for simplicity, this operation is just a simple graph $g \in \SG(9)$. We can draw an example as follows:
\[\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (0) at (2, 1) {$3$};
\node [style=species] (1) at (4.75, -3.25) {$9$};
\node [style=species] (2) at (7.5, 2.5) {$5$};
\node [style=species] (3) at (1, 2.5) {$1$};
\node [style=none] (4) at (0, 3) {};
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\node [style=species] (14) at (7.5, 1) {$7$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=simple] (0) to (11);
\draw [style=simple] (3) to (13);
\end{pgfonlayer}
\end{tikzpicture}
The dashed circles indicate that we are thinking of this simple graph as an element of $\O(3,4,2;9)$: an operation that can be used to assemble simple graphs with 3, 4, and 2 vertices, respectively, to produce one with 9 vertices.
Next let us see how this operation acts on the canonical algebra $A_\SG$, whose elements are simple graphs. Suppose we have elements $a_1 \in A_\SG(3)$, $a_2 \in A_\SG(4)$ and $a_3 \in A_\SG(2)$:
\[
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\node [style=species] (3) at (1, 2.5) {$1$};
\node [style=none] (4) at (0, 3) {};
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\end{pgfonlayer}
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\end{pgfonlayer}
\end{tikzpicture}
\]
We can act on these by the operation $f$ to obtain $\alpha(f)(a_1,a_2,a_3) \in A_\SG(9)$. It looks like this:
\[\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (0) at (2, 1) {$3$};
\node [style=species] (1) at (4.75, -2.5) {$9$};
\node [style=species] (2) at (7.5, 2.5) {$5$};
\node [style=species] (3) at (1, 2.5) {$1$};
\node [style=none] (4) at (0, 3) {};
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\node [style=species] (11) at (6.25, 1) {$6$};
\node [style=species] (13) at (3, 2.5) {$2$};
\node [style=species] (14) at (7.5, 1) {$7$};
\node [style=triplebounding] (15) at (4.25, .55) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=simple] (3) to (13);
\draw [style=simple] (13) to (0);
\draw [style=simple] (10) to (2);
\draw [style=simple] (0) to (11);
\draw [style=simple] (2) to (11);
\draw [style=simple] (11) to (14);
\draw [style=simple] (6) to (1);
\end{pgfonlayer}
\end{tikzpicture}}\]
We have simply taken the disjoint union of $a_1$, $a_2$, and $a_3$ and then overlaid $g$, obtaining a simple graph with 9 vertices.
The canonical algebra is one of the simplest algebras of the operad $O_\SG$. We can define many more interesting algebras for this operad. For example, we might wish to use this operad to describe communication networks where the communicating entities have locations and the communication channels have limits on their range. To include location data, we can choose $A(n)$ for $n \in \N$ to be the set of all graphs with $n$ vertices where each vertex is an actual point in the plane $\R^2$. To handle range-limited communications, we could instead choose $A(n)$ to be the set of all graphs with $n$ vertices in $\R^2$ where an edge is permitted between two vertices only if their Euclidean distance is less than some specified value. This still gives a well-defined algebra: when we apply an operation, we simply omit those edges from the resulting graph that would violate this restriction.
Besides the plethora of interesting algebras for the operad $O_\SG$, and useful homomorphisms between these, one can also modify the operad by choosing another network model. This provides additional flexibility in the formalism. Different network models give different operads, and the construction of operads from network models is functorial, so morphisms of network models give morphisms of operads.
In <ref> we apply the machinery provided by <ref> to build operads from network models. We also describe some algebras of these operads, and in <ref> we discuss an algebra whose elements are networks of range-limited communication channels.
§ ONE-COLORED NETWORK MODELS
We begin with a special class of network models: those where the vertices of the network have just one color. To define these, we use $\S$ to stand for a skeleton of the groupoid of finite sets and bijections:
Let $\S$, the symmetric groupoid, be the groupoid for which:
* objects are natural numbers $n \in \N$,
* a morphism from $m$ to $n$ is a bijection $\sigma \colon \{1,\dots,m\} \to
\{1,\dots,n\}$
and bijections are composed in the usual way.
There are no morphisms in $\S$ from $m$ to $n$ unless $m = n$. For each $n \in \N$, the endomorphisms of $n$ form the symmetric group $S_n$. It is convenient to write $\n$ for the set $\{1,\dots,n\}$, so that a morphism $\sigma \maps n \to n$ in $\S$ is the same as a bijection $\sigma \maps \n \to \n$.
There is a functor $+ \maps \S \times \S \to \S$ defined as follows. Given $m, n \in \N$ we let $m + n$ be the usual sum, and given $\sigma \in S_m$ and $\tau \in S_n$, let $\sigma+\tau \in S_{m+n}$ be as follows:
\begin{equation}
\label{eq:plus}
(\sigma + \tau)(j)=
\begin{cases}
\sigma(j)&\text{if } j\leq m
\\\tau(j-m)+m&\text{otherwise.}
\end{cases}
\end{equation}
For objects $m, n \in \S$, let $B_{m,n}$ be the block permutation of $m+n$ which swaps the first $m$ with the last $n$. For example $B_{4,3} \maps 7 \to 7$ is the permutation $(1473625)$:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty] (1) at (1, 1.5) {};
\node [style=empty] (2) at (1.5, 1.5) {};
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\node [style=empty] (5) at (3, 1.5) {};
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\node [style=empty] (7) at (4, 1.5) {};
\node [style=empty] (1a) at (1, 0) {};
\node [style=empty] (2a) at (1.5, 0) {};
\node [style=empty] (3a) at (2, 0) {};
\node [style=empty] (4a) at (2.5, 0) {};
\node [style=empty] (5a) at (3, 0) {};
\node [style=empty] (6a) at (3.5, 0) {};
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\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=simple] (1.center) to (4a.center);
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\draw [style=simple] (3.center) to (6a.center);
\draw [style=simple] (4.center) to (7a.center);
\draw [style=simple] (5.center) to (1a.center);
\draw [style=simple] (6.center) to (2a.center);
\draw [style=simple] (7.center) to (3a.center);
\end{pgfonlayer}
\end{tikzpicture} \]
The tensor product $+$ and braiding $B$ give $\S$ the structure of a strict symmetric monoidal category. This follows as a special case of <ref>.
A one-colored network model is a lax symmetric monoidal functor
\[F \maps \S \to \Mon .\]
Here $\Mon$ is the category with monoids as objects and monoid homomorphisms as morphisms, considered with its cartesian monoidal structure.
Algebraically, a network model is a family of monoids $\{M_n\}_{n \in \N}$ each with a group action of the corresponding symmetric group $S_n$, such that the product of any two embed into the one indexed by the sum of their indices equivariantly, i.e. in a way which respects the group action: $M_m \times M_n \hookrightarrow M_{m+n}$.
Many examples of network models are given below. A pedestrian way to verify that these examples really are network models is to use the following result:
A one-colored network model $F \maps \S \to \Mon$ is the same as:
* a family of sets $\{F(n)\}_{n\in \N}$
* distinguished identity elements $e_n \in F(n)$
* a family of overlay functions $\cup \maps F(n) \times F(n) \to F(n)$
* a bijection $\sigma \maps F(n) \to F(n)$ for each $\sigma \in S_n$
* a family of disjoint union functions $\sqcup \maps F(m) \times F(n) \to F(m+n)$
satisfying the following equations:
* $e_n \cup g = g \cup e_n = g$
* $g_1 \cup (g_2 \cup g_3) = (g_1 \cup g_2) \cup g_3$
* $\sigma(g_1 \cup g_2) = \sigma g_1 \cup \sigma g_2$
* $\sigma e_n = e_n$
* $(\sigma_2 \sigma_1) g = \sigma_2 (\sigma_1 g)$
* $(g_1 \cup g_2) \sqcup (h_1 \cup h_2) = (g_1 \sqcup h_1) \cup (g_2 \sqcup h_2)$
* $1 (g) = g$
* $e_m \sqcup e_n = e_{m+n}$
* $\sigma g \sqcup \tau h = (\sigma + \tau) (g \sqcup h)$
* $g_1 \sqcup (g_2 \sqcup g_3) = (g_1 \sqcup g_2) \sqcup g_3$
* $e_0 \sqcup g = g \sqcup e_0 = g$
* $B_{m,n} (h \sqcup g) = g \sqcup h$
for $g, g_i \in F(n)$, $h, h_i \in F(m)$, $\sigma, \sigma_i \in S_n$, $\tau \in S_m$, and $1$ the identity of $S_n$.
Having a functor $F \maps \S \to \Mon$ is equivalent to having the first four items satisfying Equations 1–6. The binary operation $\cup$ gives the set $F(n)$ the structure of a monoid, with $e_n$ acting as the identity. Equation 1 tells us $e_n$ acts as an identity, and Equation 2 gives the associativity of $\cup$. Equations 3 and 4 tell us that $\sigma$ is a monoid homomorphism. Equations 5 and 6 say that the map $(\sigma,g) \mapsto \sigma g$ defines an action of $S_n$ on $F(n)$ for each $n$. All of these actions together give us the functor $F \maps \S \to \Mon$.
That the functor is lax monoidal is equivalent to having item 5 satisfying Equations 7–11. Equations 7 and 8 tell us that $\sqcup$ is a family of monoid homomorphisms. Equation 9 tells us that it is a natural transformation. Equation 10 tells us that the associativity hexagon diagram for lax monoidal functors commutes for $F$. Equation 11 implies the commutativity of the left and right unitor square diagrams. That the lax monoidal functor is symmetric is equivalent to Equation 12. It tells us that the square diagram for symmetric monoidal functors commutes for $F$.
This is one of the simplest examples of a network model:
[Simple graphs]
Let a simple graph on a set $V$ be a set of 2-element subsets of $V$, called edges. There is a one-colored network model $\SG \maps \S \to \Mon$ such that $\SG(n)$ is the set of simple graphs on $\n$.
To construct this network model, we make $\SG(n)$ into a monoid where the product of simple graphs $g_1, g_2 \in \SG(n)$ is their union $g_1 \cup g_2$. Intuitively speaking, to form their union, we `overlay' these graphs by taking the union of their sets of edges. The simple graph on $\n$ with no edges acts as the unit for this operation. The groups $S_n$ acts on the monoids $\SG(n)$ by permuting vertices, and these actions define a functor $\SG \maps \S \to \Mon$.
Given simple graphs $g \in \SG(m)$ and $h \in \SG(n)$ we define $g \sqcup h \in \SG(m + n)$ to be their disjoint union. This gives a monoid homomorphism $\sqcup \maps \SG(m) \times \SG(n) \to \SG(m + n)$ because
\[(g_1 \cup g_2) \sqcup (h_1 \cup h_2) = (g_1 \sqcup h_1) \cup (g_2 \sqcup h_2). \]
This in turn gives a natural transformation with components
\[\sqcup_{m, n} \maps \SG(m) \times \SG(n) \to \SG(m + n), \]
which makes $\SG$ into lax symmetric monoidal functor.
One can prove this construction really gives a network model using either <ref>, which requires verifying a list of equations, or <ref>, which gives a general procedure for getting a network model from a monoid $M$ by letting elements of $\Gamma_M(n)$ be maps from the complete graph on $\n$ to $M$. If we take $M = \Boole = \{F,T\}$ with `or' as the monoid operation, this procedure gives the network model $\SG = \Gamma_\Boole$. We explain this in <ref>.
There are many other kinds of graph, and many of them give network models:
[Directed graphs]
Let a directed graph on a set $V$ be a collection of ordered pairs $(i,j) \in V^2$ such that $i \ne j$. These pairs are called directed edges. There is a network model $\DG \maps \S \to \Mon$ such that $\DG(n)$ is the set of directed graphs on $\n$. As in <ref>, the monoid operation on $\DG(n)$ is union.
Let a multigraph on a set $V$ be a multiset of 2-element subsets of $V$. If we define $\MG(n)$ to be the set of multigraphs on $\n$, then there are at least two natural choices for the monoid operation on $\MG(n)$.
The most direct generalization of $\SG$ of <ref> is the network model $\MG \maps \S \to \Mon$ with
values $(\MG(n), \cup)$ where $\cup$ is now union of edge multisets.
That is, the multiplicity of $\{ i, j \}$ in $g \cup h$ is maximum of the multiplicity of $\{ i, j \}$ in $g$ and the multiplicity of $\{ i, j \}$ in $h$.
Alternatively, there is another network model $\MGplus \maps \S \to \Mon$ with
values $(\MG(n), +)$ where $+$ is multiset sum. That is, $g + h$ obtained by adding multiplicities of corresponding edges.
[Directed multigraphs]
Let a directed multigraph on a set $V$ be a multiset of ordered pairs $(i,j) \in V^2$ such that $i \ne j$. There is a network model $\DMG \maps \S \to \Mon$ such that $\DMG(n)$ is the set of directed multigraphs on $\n$ with monoid operation the union of multisets. Alternatively, there is a network model with values $(\DMG(n), +)$ where $+$ is multiset sum.
Let a hypergraph on a set $V$ be a set of nonempty subsets of $V$, called hyperedges. There is a network model $\HG \maps \S \to \Mon$ such that $\HG(n)$ is the set of hypergraphs on $\n$. The monoid operation $\HG(n)$ is union.
[Graphs with colored edges]
Fix a set $B$ of edge colors and let $\SG \maps \S \to \Mon$ be the network model of simple graphs as in <ref>. Then there is a network model $H \maps \S \to \Mon$ with
\[
H(n) = \SG(n)^B
\]
making the product of $B$ copies of the monoid $\SG(n)$ into a monoid in the usual way. In this model, a network is a $B$-tuple of simple graphs, which we may view as a graph with at most one edge of each color between any pair of distinct vertices. We describe this construction in more detail in <ref>.
There are also examples of network models not involving graphs:
A poset is a lattice if every finite subset has both an infimum and a supremum. If $L$ is a lattice, then $(L, \vee)$ and $(L, \wedge)$ are both monoids, where $x \vee y$ is the supremum of $\{x,y\} \subseteq L$ and $x \wedge y$ is the infimum.
Let $P(n)$ be the set of partitions of the set $\n$. This is a lattice where $\pi \le \pi'$ if the partition $\pi$ is finer than $\pi'$. Thus, $P(n)$ can be made a monoid in either of the two ways mentioned above. Denote these monoids as $P^{\vee}(n)$ and $P^{\wedge}(n)$. These monoids extend to give two network models $P^{\vee}, P^{\wedge} \maps \S \to \Mon$.
§.§ One-colored network models from monoids
There is a systematic procedure that gives many of the network models we have seen so far. To do this, we take networks to be ways of labelling the edges of a complete graph by elements of some monoid $M$. The operation of overlaying two of these networks is then described using the monoid operation.
For example, consider the Boolean monoid $\Boole$: that is, the set $\{F,T\}$ with `inclusive or' as its monoid operation. A complete graph with edges labelled by elements of $\Boole$ can be seen as a simple graph if we let $T$ indicate the presence of an edge between two vertices and $F$ the absence of an edge. To overlay two simple graphs $g_1, g_2$ with the same set of vertices we simply take the `or' of their edge labels. This gives our first example of a network model, <ref>.
To formalize this we need some definitions. Given $n \in \N$, let $\E(n)$ be the set of 2-element subsets of $\n = \{1, \dots, n\}$. We call the members of $\E(n)$ edges, since they correspond to edges of the complete graph on the set $\n$. We call the elements of an edge $e \in \E(n)$ its vertices.
Let $M$ be a monoid. For $n \in \N$, let $\Gamma_M(n)$ be the set of functions $g \maps \E(n) \to M$. Define the operation $\cup \colon \Gamma_M(n) \times \Gamma_M(n) \to \Gamma_M(n)$ by $(g_1 \cup g_2)(e) = g_1(e) g_2(e)$ for $e \in \E(n)$. Define the map $\sqcup \colon \Gamma_M(m) \times \Gamma_M(n) \to \Gamma_M(m+n)$ by
\[
(g_1 \sqcup g_2)(e) =
\left\{\begin{array}{cl}
g_1(e) & {\rm if \; both \; vertices \; of \;} e {\rm \; are \;}\leq m
\\
g_2(e) & {\rm if \; both \; vertices \; of \;} e {\rm \; are \;} > m
\\
{\rm the \; identity \; of \; } M & {\rm otherwise}
\\
\end{array}
\right.
\]
The symmetric group $S_n$ acts on $\Gamma_M(n)$ by $\sigma(g)(e) = g(\sigma^{-1}(e))$.
For each monoid $M$ the data above gives a one-colored network model $\Gamma_M \maps \S \to \Mon$.
We can define $\Gamma_M$ as the composite of two functors, $\E \maps \S \to \Inj$ and $M^{-} \maps \Inj \to \Mon$, where $\Inj$ is the category of sets and injections.
The functor $\E \maps \S \to \Inj$ sends each object $n \in \S$ to $\E(n)$, and it sends each morphism $\sigma \maps n \to n$ to the permutation of $\E(n)$ that maps any edge $e = \{x,y\} \in \E(n)$ to $\sigma(e) = \{\sigma(x), \sigma(y)\}$. The category $\Inj$ does not have coproducts, but it is closed under coproducts in $\Set$. It thus becomes symmetric monoidal with $+$ as its tensor product and the empty set as the unit object. For any $m, n \in \S$ there is an injection
\[\mu_{m,n} \maps \E(m) + \E(n) \to \E(m+n) \]
expressing the fact that a 2-element subset of either $\m$ or $\n$ gives a 2-element subset of $\m+\n$. The functor $\E \maps \S \to \Inj$ becomes lax symmetric monoidal with these maps $\mu_{m,n}$ giving the lax preservation of the tensor product.
The functor $M^- \maps \Inj \to \Mon$ sends each set $X$ to the set $M^X$ made into a monoid with pointwise operations, and it sends each function $f \maps X \to Y$ to the monoid homomorphism $M^f \maps M^X \to M^Y$ given by
\[(M^f g)(y) = \left\{ \begin{array}{ccl}
g(f^{-1}(y)) & \textrm{if } y \in \mathrm{im}(f) \\
1 & \textrm{otherwise}
\end{array} \right.\]
for any $g \in M^X$. Using the natural isomorphisms $M^{X + Y} \cong M^X \times M^Y$ and $M^{\emptyset} \cong 1$ this functor can be made symmetric monoidal.
As the composite of the lax symmetric monoidal functor $\E \maps \S \to \Inj$ and the symmetric monoidal functor $M^- \maps \Inj \to \Mon$, the functor $\Gamma_M \maps \S \to \Mon$ is lax symmetric monoidal, and thus a network model. With the help of <ref>, it is easy to check that this description of $\Gamma_M$ is equivalent to that in the theorem statement.
[Simple graphs, revisited]
Let $\Boole = \{F,T\}$ be the Boolean monoid. If we interpret $T$ and $F$ as `edge' and `no edge' respectively, then $\Gamma_{\Boole}$ is just $\SG$, the network model of simple graphs discussed in <ref>.
Recall from <ref> that a multigraph on the set $\n$ is a multisubset of $\E(n)$, or in other words, a function $g \maps \E(n) \to \N$. There are many ways to create a network model $F \maps \S \to \Mon$ for which $F(n)$ is the set of multigraphs on the set $\n$, since $\N$ has many monoid structures. Two of the most important are these:
[Multigraphs with addition for overlaying]
Let $(\N, +)$ be $\N$ made into a monoid with the usual notion of addition as $+$. In this network model, overlaying two multigraphs $g_1, g_2 \maps \E(n) \to \N$ gives a multigraph $g \maps \E(n) \to \N$ with $g(e) = g_1(e) + g_2(e)$. In fact, this notion of overlay corresponds to forming the multiset sum of edge multisets and $\Gamma_{(\N,+)}$ is the network model of multigraphs called $\MGplus$ in <ref>.
[Multigraphs with maximum for overlaying]
Let $(\N, \max)$ be $\N$ made into a monoid with $\max$ as the monoid operation. Then $\Gamma_{(\N,\max)}$ is a network model where overlaying two multigraphs $g_1, g_2 \maps \E(n) \to \N$ gives a multigraph $g \maps \E(n) \to \N$ with $g(e) = g_1(e) \max g_2(e)$.
For this monoid structure overlaying two copies of the same multigraph gives the same multigraph. In other words, every element in each monoid $\Gamma_{(\N,\max)}(n)$ is idempotent and $\Gamma_{(\N,\max)}$ is the network model of multigraphs called $\MG$ in <ref>.
[Multigraphs with at most $k$ edges between vertices]
For any $k \in \N$, let $\Boole_k$ be the set $\{0,\dots,k\}$ made into a monoid with the monoid operation $\oplus$ given by
\[x \oplus y = (x + y) \min k \]
and $0$ as its unit element. For example, $\Boole_0$ is the trivial monoid and $\Boole_1$ is isomorphic to the Boolean monoid. There is a network model $\Gamma_{\Boole_k}$ such that $\Gamma_{\Boole_k}(n)$ is the set of multigraphs on $\n$ with at most $k$ edges between any two distinct vertices.
§ GENERAL NETWORK MODELS
The network models described so far allow us to handle graphs with colored edges, but not with colored vertices. Colored vertices are extremely important for applications in which we have a network of agents of different types. Thus, network models will involve a set $C$ of vertex colors in general. This requires that we replace $\S$ by the free strict symmetric monoidal category generated by the color set $C$. Thus, we begin by recalling this category.
For any set $C$, there is a category $\SC$ for which:
* Objects are formal expressions of the form
\[
c_1 \otimes \cdots \otimes c_n
\]
for $n \in \N$ and $c_1, \dots, c_n \in C$.
We denote the unique object with $n = 0$ as $I$.
* There exist morphisms from $c_1 \otimes \cdots \otimes c_m$ to $c'_1 \otimes \cdots \otimes c'_n$ only if $m = n$, and in that case a morphism is a permutation $\sigma \in S_n$ such that $c'_{\sigma(i)} = c_i$ for all $i$.
* Composition is the usual composition of permutations.
Note that elements of $C$ can be identified with certain objects of $\S(C)$, namely the one-fold tensor products. We do this in what follows.
$\S(C)$ can be given the structure of a strict symmetric monoidal category making it into the free strict symmetric monoidal category on the set $C$. Thus, if $\A$ is any strict symmetric monoidal category and $f \maps C \to \Ob(\A)$ is any function from $C$ to objects of the $\A$, there exists a unique strict symmetric monoidal functor $F \maps \S(C) \to \A$ with $F(c) = f(c)$ for all $c \in C$.
This is well-known; see for example Sassone <cit.> or Gambino and Joyal <cit.>. The tensor product of objects is $\otimes$, the unit for the tensor product is $I$, and the braiding
\[(c_1 \otimes \cdots \otimes c_m) \otimes (c'_1 \otimes \cdots \otimes c'_n) \to (c'_1 \otimes \cdots \otimes c'_n) \otimes (c_1 \otimes \cdots \otimes c_m) \]
is the block permutation $B_{m, n}$. Given $f \maps C \to \Ob(\A)$, we define $F\maps \S(C) \to \A$ on objects by
\[F(c_1 \otimes \cdots \otimes c_n) = f(c_1) \otimes \cdots \otimes f(c_n) , \]
and it is easy to check that $F$ is strict symmmetric monoidal, and the unique functor with the required properties.
Let $C$ be a set, called the set of vertex colors. A
$C$-colored network model is a lax symmetric monoidal functor
\[F \maps \SC \to \Cat. \]
A network model is a $C$-colored network model for some set $C$.
If $C$ has just one element, $\S(C) \cong \S$ and a $C$-colored network model is a one-colored network model in the sense of <ref>. Here are some more interesting examples:
[Simple graphs with colored vertices]
There is a network model of simple graphs with $C$-colored vertices. To construct this, we start with the network model of simple graphs $\SG \maps \S \to \Mon$ given in <ref>. There is a unique function from $C$ to the one-element set. By <ref>, this function extends uniquely to a strict symmetric monoidal functor
\[F \maps \S(C) \to \S . \]
An object in $\S(C)$ is formal tensor product of $n$ colors in $C$; applying $F$ to this object we forget the colors and obtain the object $n \in \S$. Composing $F$ and $\SG$, we obtain a lax symmetric monoidal functor
\[
\S(C) \stackrel{F}{\longrightarrow} \S \stackrel{\SG}{\longrightarrow} \Mon
\]
which is the desired network model. We can use the same idea to `color' any of the network models in <ref>.
Alternatively, suppose we want a network model of simple graphs with $C$-colored vertices where an edge can only connect two vertices of the same color. For this we take a cartesian product of $C$ copies of the functor $\SG$, obtaining a lax symmetric monoidal functor
\[{\SG}^C \maps \S^C \to \Mon^C. \]
There is a function $h \maps C \to \Ob(\S^C)$ sending each $c \in C$ to the object of $S^\C$ that equals $1 \in \S$ in the $c$th place and $0 \in \S$ elsewhere. Thus, by <ref>, $h$ extends uniquely to a strict symmetric monoidal functor
\[H_C \maps \S(C) \to \S^C .\]
Furthermore, the product in $\Mon$ gives a symmetric monoidal functor
\[\Pi \maps \Mon^C \to \Mon .\]
Composing all these, we obtain a lax symmetric monoidal functor
\[
\SC \stackrel{H_C}{\longrightarrow} \S^C \stackrel{\SG^C}{\longrightarrow} \Mon^C \stackrel{\Pi}{\longrightarrow} \Mon
\]
which is the desired network model.
More generally, if we have a network model $F_c \maps \S \to \Mon$ for each color $c \in C$, we can use the same idea to create a network model:
\[
\begin{tikzcd}
\S(C)
\arrow[r, "H_C"]
\S^C
\arrow[r, "\prod_{c \in C} F_c"]
\Mon^C
\arrow[r, "\prod"]
\Mon
\end{tikzcd}\]
in which the vertices of color $c \in C$ partake in a network of type $F_c$.
[Petri nets]
Petri nets are a kind of network widely used in computer science, chemistry and other disciplines <cit.>. A Petri net $(S, T, i, o)$ is a pair of finite sets and a pair of functions $i, o \maps S \times T \to \N$. Let $P(m, n)$ be the set of Petri nets $(\m, \n, i, o)$. This becomes a monoid with product
\[
(\m, \n, i, o) \cup (\m, \n, i', o')
= (\m, \n, i+i', o+o')
\]
The groups $S_m\times S_n$ naturally act on these monoids, so we have a functor
\[P \maps \S^2 \to \Mon . \]
There are also `disjoint union' operations
\[\sqcup \maps P(m, n) \times P(m', n') \to P(m+m', n+n') \]
making $P$ into a lax symmetric monoidal functor. In <ref> we described a strict symmetric monoidal functor $H_C \maps \S(C) \to \S^C$ for any set $C$. In the case of the 2-element set this gives
\[H_2 \maps \S(2) \to \S^2 .\]
We define the network model of Petri nets to be the composite
\[\S(2) \stackrel{H_2}{\longrightarrow} \S^2 \stackrel{P}{\longrightarrow} \Mon .\]
§.§ Categories of network models
For each choice of the set $C$ of vertex colors, we can define a category $\NetMod_C$ of $C$-colored network models. However, it is useful to create a larger category $\NetMod$ containing all these as subcategories, since there are important maps between network models that involve changing the vertex colors.
For any set $C$, let $\NetMod_C$ be the category for which:
* an object is a $C$-colored network model, that is, a lax symmetric monoidal functor $F \maps \SC\to \Cat$,
* a morphism is a monoidal natural transformation between such functors:
\[\begin{tikzcd}
\SC
\arrow[r, "F", bend left=40]
\arrow[bend left=40]{r}[name=LUU, below]{}
\arrow[r, "F'", bend right=40, swap, pos=0.45]
\arrow[bend right=40, pos = 0.53]{r}[name=LDD]{}
\arrow[Rightarrow, to path=(LUU) -- (LDD)\tikztonodes]{r}{\gn}
\Cat
\end{tikzcd}\]
and composition is the usual composition of monoidal natural transformations.
In particular, $\NetMod_1$ is the category of one-colored network models. For an example involving this category, consider the network models built from monoids in <ref>. Any monoid $M$ gives a one-colored network model $\Gamma_M$ for which an element of $\Gamma_M(n)$ is a way of labelling the edges of the complete graph on $\n$ by elements of $M$. Thus, we should expect any homomorphism of monoids $f \maps M \to M'$ to give a morphism of network models $\Gamma_f \maps \Gamma_M \to \Gamma_{M'}$ for which
\[\Gamma_f(n) \maps \Gamma_M(n) \to \Gamma_{M'}(n) \]
applies $f$ to each edge label.
Indeed, this is the case. As explained in the proof of <ref>, the network model $\Gamma_M$ is the composite
\[\S \stackrel{\E}{\longrightarrow} \Inj
\stackrel{M^{-}}{\longrightarrow} \Mon .\]
The homomorphism $f$ gives a natural transformation
\[f^{-} \maps M^{-} \To M'^{-} \]
that assigns to any finite set $X$ the monoid homomorphism
\[\begin{array}{rccl}
f^X \maps & M^X & \to & M'^X \\
& g & \mapsto & f \circ g .
\end{array}
\]
It is easy to check that this natural transformation is monoidal. Thus, we can whisker it with the lax symmetric monoidal functor $\E$ to get a morphism of network models:
\[\begin{tikzcd}
\S \arrow[r, "\E"] &
\Inj
\arrow[r, "M^-", bend left=40]
\arrow[bend left=40]{r}[name=LUU, below]{}
\arrow[r, "M'^-", bend right=40, swap, pos=0.45]
\arrow[bend right=40]{r}[name=LDD]{}
\arrow[Rightarrow, to path=(LUU) -- (LDD)\tikztonodes]{r}{f^-}
\Mon
\end{tikzcd}\]
and we call this $\Gamma_f \maps \Gamma_M \to \Gamma_{M'}$.
There is a functor
\[\Gamma \maps \Mon \to \NetMod_1 \]
sending any monoid $M$ to the network model $\Gamma_M$ and any homomorphism of monoids $f \maps M \to M'$ to the morphism of network models $\Gamma_f \maps \Gamma_M \to \Gamma_{M'}$.
To check that $\Gamma$ preserves composition, note that
\[\begin{tikzcd}[column sep=huge]
\S
\arrow[r, "\E"]
\Inj
\arrow[r, "M^-", bend left=80]
\arrow[r, ""{name=TOP}, bend left=80, swap, pos=0.455]
\arrow[r, "M'^-"{name=Ml}] % Ml = middle with label
\arrow[Rightarrow, from=TOP, to=Ml, "f^-", pos=0.3]
\arrow[r, ""{name=M}, swap]
\arrow[r, "M''^-", bend right=80, swap]
\arrow[r, ""{name=BOT}, bend right=80, pos=0.45]
\arrow[Rightarrow, from=M, to=BOT, "f'^-", pos=0.5]
\Mon
\end{tikzcd}\]
\[\begin{tikzcd}[column sep=huge]
\S \arrow[r, "\E"] &
\Inj
\arrow[r, "M^-", bend left=80]
\arrow[bend left=80, pos=0.47]{r}[name=LUU, below]{}
\arrow[r, "M''^-", bend right=80, swap, pos=0.5]
\arrow[bend right=80, pos=0.44]{r}[name=LDD]{}
\arrow[Rightarrow, from=LUU, to=LDD, "(f'f)^-"]
\Mon
\end{tikzcd} \]
since $f'^- f^- = (f'f)^-$. Similarly $\Gamma$ preserves identities.
It has been said that category theory is the subject in which even the examples need examples. So, we give an example of the above result:
[Imposing a cutoff on the number of edges]
In <ref> we described the network model of multigraphs $\MGplus$ as $\Gamma_{(\N, +)}$. In <ref> we described a network model $\Gamma_{\Boole_k}$ of multigraphs with at most $k$ edges between any two distinct vertices. There is a homomorphism of monoids
\begin{align*}
f \maps (\N, +) &\to \Boole_k\\
n &\mapsto n \min k
\end{align*}
and this induces a morphism of network models
\[\Gamma_f \maps \Gamma_{(\N, +)} \to \Gamma_{\Boole_k} .\]
This morphism imposes a cutoff on the number of edges between any two distinct vertices: if there are more than $k$, this morphism keeps only $k$ of them. In particular, if $k = 1$, $\Boole_k$ is the Boolean monoid, and
\[\Gamma_f \maps \MGplus \to \SG \]
sends any multigraph to the corresponding simple graph.
One useful way to combine $C$-colored networks is by `tensoring' them. This makes $\NetMod_C$ into a symmetric monoidal category:
For any set $C$, the category $\NetMod_C$ can be made into a symmetric monoidal category with the tensor product defined pointwise, so that for objects $F, F' \in \NetMod_C$ we have
\[(F \otimes F')(x) = F(x) \times F'(x) \]
for any object or morphism $x$ in $\S(C)$, and for morphisms $\phi, \phi'$ in $\NetMod_C$ we have
\[(\phi \otimes \phi')_x = \phi_x \times \phi'_x \]
for any object $x \in \S(C)$.
More generally, for any symmetric monoidal categories $\A$ and $\B$, there is a symmetric monoidal category $\Sym\Mon\Cat(\A, \B)$ whose objects are lax symmetric monoidal functors from $\A$ to $\B$ and whose morphisms are monoidal natural transformations, with the tensor product defined pointwise. The proof in the `weak' case was given by Hyland and Power <cit.>, and the lax case works the same way.
If $F, F' \maps \S(C) \to \Mon$ then their tensor product again takes values in $\Mon$. There are many interesting examples of this kind:
[Graphs with colored edges, revisited]
In <ref> we described network models of simple graphs with colored edges. The above result lets us build these network models starting from more basic data. To do this we start with the network model for simple graphs, $\SG \maps \S \to \Mon$, discussed in <ref>. Fixing a set $B$ of `edge colors', we then take a tensor product of copies of $\SG$, one for each $b \in B$. The result is a network model $\SG^{\otimes B} \maps \S \to \Mon$ with
\[\SG^{\otimes B}(n) = \SG(n)^B \]
for each $n \in \N$.
[Combined networks]
We can also combine networks of different kinds. For example, if $\DG \maps \S \to \Mon$ is the network model of directed graphs given in <ref> and $\MG \maps \S \to \Mon$ is the network model of multigraphs given in <ref>, then
\[\DG \otimes \MG \maps \S \to \Mon \]
is another network model, and we can think of an element of $(\DG \otimes \MG)(n)$ as a directed graph with red edges together with a multigraph with blue edges on the set $\n$.
Next we describe a category $\NetMod$ of network models with arbitrary color sets, which includes all the categories $\NetMod_C$ as subcategories. To do this, first we introduce `color-changing' functors. Recall that elements of $C$ can be seen as certain objects of $\S(C)$, namely the 1-fold tensor products. If $f \maps C \to C'$ is a function, there exists a unique strict symmetric monoidal functor $f_* \maps \S(C) \to \S(C')$ that equals $f$ on objects of the form $c \in C$. This follows from <ref>.
Next, we define an indexed category $\NetMod_{-} \maps \Set\op \to \CAT$ that sends any set $C$ to $\NetMod_C$ and any function $f \maps C \to D$ to the functor that sends any $D$-colored network model $F \maps \S(D) \to \Cat$ to the $C$-colored network model given by the composite
\[\S(C) \xrightarrow{f_*} \S(D) \xrightarrow{F} \Cat .\]
Applying the Grothendieck construction (see <ref>) to this indexed category, we define the category of network models to be
\[\NetMod = \int \NetMod_-. \]
In elementary terms, $\NetMod$ has:
* pairs $(C, F)$ for objects, where $C$ is a set and $F \maps \S(C) \to \Cat$ is a $C$-colored network model.
* pairs $(f, g) \maps (C, F) \to (D, G)$ for morphisms, where $f \maps C \to D$ is a function and $g \maps F \Rightarrow G \circ f_*$ is a morphism of network models.
[Simple graphs with colored vertices, revisited]
In <ref> we constructed the network model of simple graphs with colored vertices. We started with the network model for simple graphs, which is a one-colored network model $\SG \maps \S \to \Mon$. The unique function $! \maps C \to 1$ gives a strict symmetric monoidal functor $!_* \maps \S(C) \to \S(1) \cong \S$. The network model of simple graphs with $C$-colored vertices is the composite
\[
\S(C) \xrightarrow{!_*} \S \xrightarrow{\SG} \Mon
\]
and there is a morphism from this to the network model of simple graphs, which has the effect of forgetting the vertex colors.
In fact, $\NetMod$ can be understood as a subcategory of the following category:
Let $\Sym\Mon\ICat$ be the category where:
* objects are pairs $(\C, F)$ where $\C$ is a small symmetric monoidal category and $F \maps \C \to \Cat$ is a lax symmetric monoidal functor, where $\Cat$ is considered with its cartesian monoidal structure.
* morphisms from $(\C, F)$ to $(\C', F')$ are pairs $(G, \gn)$ where $G \maps \C \to \C'$ is a lax symmetric monoidal functor and $\gn \maps F \To F' \circ G $ is a symmetric monoidal natural transformation:
\[\begin{tikzcd}
\C
\arrow[dr, "F"]
\arrow[dr, ""{name=F}, swap]
\arrow[dd, "G", swap]
\\&
\Cat
\\
\C'
\arrow[ur, "F'", swap]
\arrow[ur, ""{name=F'}, pos=0.43]
\arrow[Rightarrow, from = F, to = F', "\gn", swap]
\end{tikzcd}\]
We shall use this way of thinking in the next two sections to build operads from network models. It must be said that $\Sym\Mon\ICat$ is naturally a 2-category where a 2-morphism $\xi \maps (G, \gn) \To (G', \gn')$ is a natural transformation $\xi \maps G \to G'$ such that
\[\begin{tikzcd}
\C
\arrow[dd, bend right=90, "G", pos=0.495, swap]
\arrow[dd, bend right=90, ""{name=L, right}, swap, phantom, pos = 0.49]
\arrow[dd, "G'"{name=R, left}, swap]
\arrow[dr, "F", pos=0.4]
\arrow[dr, ""{name=U, below}, pos=0.4, phantom]
& & &
\C'
\arrow[dd, "G"{name=R2, left}, pos=0.52, swap]
\arrow[dr, "F", pos=0.4]
\arrow[dr, ""{name=U2, below}, pos=0.44, phantom]
\\
& \Cat & = & & \Cat.
\\
\C
\arrow[ur, "F'", swap, pos=0.4]
\arrow[ur, ""{name=D}, pos=0.35, phantom]
\arrow[Rightarrow, from=U, to=D, "\gn'", swap]
\arrow[Rightarrow, from=L, to=R, "\xi"{above}, swap]
& & &
\C'
\arrow[ur, "F'", swap, pos=0.4]
\arrow[ur, ""{name=D2}, phantom, pos = 0.4]
\arrow[Rightarrow, from=U2, to=D2, "\gn", swap, pos=0.55]
\end{tikzcd}
\]
and here we are considering its 1-dimensional truncation. The 2-dimensional structure is detailed in <ref>, and utilized in <ref>. This lets us define 2-morphisms between network models, extending $\NetMod$ to a 2-category. We do not seem to need these 2-morphisms in our applications, so we suppress 2-categorical considerations in most of what follows.
§ OPERADS FROM NETWORK MODELS
Next we describe the operad associated to a network model. This construction is given in two steps. For the first step, we can use the strict symmetric Grothendieck construction of <ref> to define a strict symmetric monoidal category $\Int F$ from a given network model $F \maps \S(C) \to \Cat$. For the second step, we then use the underlying operad construction (recalled in <ref>) to build an operad $\O_F$.
Given a network model $F \maps \S(C) \to \Cat$, define the network operad $\O_F$ to be $\Op(\Int F)$.
For the sake of the unfamiliar reader, we give a brief description of these constructions in the specific context of network models, which does not assume prior knowledge. We recall the ordinary Grothendieck construction in <ref>, and <ref> is entirely dedicated to studying the (braided/symmetric) monoidal variants of it. We give a nuts-and-bolts description of the symmetric monoidal category $(\int F, \otimes_\phi)$ built from a network model $(F, \phi) \maps (\S(C), \otimes ) \to (\Mon, \times)$.
The objects of $\Int F$ correspond to objects of $\S(C)$, which are formal expressions of the form $c_1 \otimes \cdots \otimes c_n$ with $n \in \N$ and $c_i \in C$. The morphisms of $\int F$ are pairs $(\sigma, g)$ where $\sigma \maps c_1 \otimes \dots \otimes c_n \to c_{\sigma1} \otimes \dots \otimes c_{\sigma n}$ is a morphism in $\S(C)$, and $g$ is an element of the monoid $F(c_{\sigma1} \otimes \dots \otimes c_{\sigma n})$. Composition is given by $(\sigma, g) \circ (\tau, h) = (\sigma \tau, g \cdot F \sigma(h))$. The tensor product of two objects is given by concatenation. The tensor product of two morphisms is given by $(\sigma, g) \otimes (\tau, h) = (\sigma \otimes \tau, \phi(g,h))$. The unit object is $(I, \phi_0)$, where $I$ is the monoidal unit for $\S(C)$ and $\phi_0$ is the unit laxator for $F$.
For a one-object network model $F$, a more compact description of the category $\int F$ can be given by the following formula, where monoids and groups are being considered as one-object categories by default.
\[\int F \cong \coprod_{n \in \N} F(n) \rtimes S_n\]
The network operad $\O_F$ is a typed operad where the types are ordered $k$-tuples of elements of $C$. For objects $x_i, x$ of $\int F$, the operations in $\O_F$ are given by $\O_F(x_1, \dots, x_n; x) = \Int F(x_1 \otimes \cdots \otimes x_n, x)$.
Now suppose that $F$ is a one-colored network model, so that $F \maps \S \to \Mon$. Then the objects of $\S$ are simply natural numbers, so $\O_F$ is an $\N$-typed operad. Given $n_1, \dots, n_k, n \in \N$, we have
\[
\O_F(n_1, \dots, n_k; n) = \hom_{\Int \! F}(n_1 + \cdots + n_k, n).
\]
By definition, a morphism in this homset is a pair consisting of a bijection $\sigma \maps n_1 + \cdots + n_k \to n$ and an element of the monoid $F(n)$. So, we have
\begin{equation}
\label{eq:operations_in_CN}
\O_F(n_1, \dots, n_k; n) = \left\{
\begin{array}{cl} S_n \times F(n) & \textrm{if } n_1 + \cdots n_k = n \\
\emptyset & \textrm{otherwise.} \\
\end{array} \right.
\end{equation}
Here is the basic example:
[Simple network operad]
If $\SG \maps \S \to \Mon$ is the network model of simple graphs in <ref>, we call $\O_\SG$ the simple network operad. By <ref>, an operation in $\O_\SG(n_1, \dots, n_k; k)$ is an element of $S_n$ together with a simple graph having $\mathbf{n} = \{1, \dots, n\}$ as its set of vertices.
The operads coming from other one-colored network models work similarly. For example, if $\DG \maps \S \to \Mon$ is the network model of directed graphs from <ref>, then an operation in $\O_\SG(n_1, \dots, n_k; n)$ is an element of $S_n$ together with a directed graph having $\n$ as its set of vertices.
In <ref> we gave a pedestrian description of one-colored network models. We can describe the corresponding network operads in the same style:
Suppose $F$ is a one-colored network model. Then the network operad $\O_F$ is the $\N$-typed operad for which the following hold:
* The sets of operations are given by
\[\O_F(n_1, \dots, n_k; n) = \left\{
\begin{array}{cl} S_n \times F(n) & \textrm{if }
n_1 + \cdots n_k = n \\
\emptyset & \textrm{otherwise.}
\end{array} \right. \]
* Composition of operations is given as follows. Suppose that
\[(\sigma, g) \in S_n \times F(n) = \O_F(n_1, \dots, n_k; n) \]
and for $1 \le i \le k$ we have
\[(\tau_i, h_i) \in S_{n_i} \times F(n_i) =
\O_F(n_{i1}, \dots, n_{ij_i}; n_i). \]
\[(\sigma, g) \circ ((\tau_1, h_1), \dots, (\tau_k, h_k)) = (\sigma (\tau_1 + \cdots + \tau_k), g \cup \sigma(h_1 \sqcup \cdots \sqcup h_k)) \]
where $+$ is defined in <ref>, while $\cup$ and $\sqcup$ are defined in <ref>.
* The identity operation in $\O_F(n;n)$ is
$(1, e_n)$, where $1$ is the identity in $S_n$ and $e_n$ is the identity in the monoid $F(n)$.
* The right action of the symmetric group $S_k$ on $\O_F(n_1, \dots, n_k;n)$ is given as follows. Given $(\sigma, g) \in \O_F(n_1, \dots, n_k;n)$ and $\tau \in S_k$, we have
\[(\sigma, g) \tau = (\sigma\tau, g) . \]
This is a straightforward combination of the underlying operad of a symmetric monoidal category and the symmetric monoidal structure on $\int F$.
The construction of operads from symmetric monoidal categories described in <ref> is functorial, so the construction of operads from network models is as well.
The assignment of a network model $F \maps \S(C) \to \Cat$ to the operad $\O_F = \Op(\Int G)$ and a morphism of network models $(G, \gn) \maps (C, F) \to (C', F' G')$ to the operad morphism $\O_G = \Op(\widehat\Gamma)$ is a functor
\[
\O \maps \NetMod \to \Opd.
\]
There is a functor
\[
\textstyle{\Int} \maps \NetMod \to \Sym\Mon\Cat
\]
given by restricting the strict symmetric monoidal Grothendieck construction of <ref> to $\NetMod$. Composing this with the functor
\[
\Op \maps \Sym\Mon\Cat \to \Opd
\]
constructed in <ref> we obtain a functor $\O \maps \NetMod \to \Opd$ with the properties stated in the theorem. Since these properties specify how $\O$ acts on objects and morphisms, it is unique.
§.§ Algebras of network operads
Our interest in network operads comes from their use in designing and tasking networks of mobile agents. The operations in a network operad are ways of assembling larger networks of a given kind from smaller ones. To describe how these operations act in a concrete situation we need to specify an algebra of the operad. The flexibility of this approach to system design takes advantage of the fact that a single operad can have many different algebras, related by homomorphisms.
An algebra $A$ of a typed operad $O$ specifies a set $A(t)$ for each type $t \in T$ such that the operations of $O$ can be applied to act on these sets. That is, each algebra $A$ specifies:
* for each type $t \in T$, a set $A(t)$, and
* for any types $t_1, \dots, t_n, t \in T$, a function
\[
\alpha \maps O(t_1, \dots, t_n;t) \to \hom(A(t_1) \times \cdots \times A(t_n), A(t))
\]
obeying some rules that generalize those for the action of a monoid on a set <cit.>. All the examples in this section are algebras of network operads constructed from one-colored network models $F \maps \S \to \Mon$. This allows us to use <ref>, which describes $\O_F$ explicitly.
The most basic algebra of such a network operad $\O_F$ is its `canonical algebra', where it acts on the kind of network described by the network model $F$:
[The canonical algebra]
Let $F \maps \S \to \Mon$ be a one-colored network model. Then the operad $\O_F$ has a canonical algebra $A_F$ with
\[
A_F(n) = F(n)
\]
for each $n \in N$, the type set of $\O_F$.
In this algebra any operation
\[
(\sigma, g) \in \O_F(n_1, \dots , n_k; n) = S_n \times F(n)
\]
acts on a $k$-tuple of elements
\[
h_i \in A_F(n_i) = F(n_i) \qquad (1 \le i \le k)
\]
to give
\[
\alpha(\sigma, g)(h_1, \dots, h_k) = g \cup \sigma(h_1 \sqcup \cdots \sqcup h_k) \in A(n) .
\]
Here we use <ref>, which gives us the ability to overlay networks using the monoid structure $\cup \maps F(n) \times F(n) \to F(n)$, take their `disjoint union' using maps $\sqcup \maps F(m) \times F(m') \to F(m + m')$, and act on $F(n)$ by elements of $S_n$. Using the equations listed in this theorem one can check that $\alpha$ obeys the axioms of an operad algebra.
When we want to work with networks that have more properties than those captured by a given network model, we can equip elements of the canonical algebra with extra attributes. Three typical kinds of network attributes are vertex attributes, edge attributes, and `global network' attributes. For our present purposes, we focus on vertex attributes. Vertex attributes can capture internal properties (or states) of agents in a network such as their locations, capabilities, performance characteristics, etc.
[Independent vertex attributes]
For any one-colored network model $F\maps \S \to \Mon$ and any set $X$, we can form an algebra $A_X$ of the operad $\O_F$ that consists of networks whose vertices have attributes taking values in $X$. To do this, we define
\[
A_X(n) = F(n) \times X^n .
\]
In this algebra, any operation
\[
(\sigma, g) \in \O_F(n_1, \dots , n_k; n) = S_n \times F(n)
\]
acts on a $k$-tuple of elements
\[
(h_i, x_i) \in F(n_i) \times X^{n_i} \qquad (1 \le i \le k)
\]
to give
\[
\alpha_X(\sigma, g) = (g \cup \sigma(h_1 \sqcup \cdots \sqcup h_k), \sigma(x_1, \dots, x_k)).
\]
Here $(x_1, \dots, x_k) \in X^n$
is defined using the canonical bijection
\[
X^n \cong \prod_{i=1}^k X^{n_i}
\]
when $n_1 + \cdots + n_k = n$, and $\sigma \in S_n$ acts on $X^n$ by permutation of coordinates. In other words, $\alpha_X$ acts via $\alpha$ on the $F(n_i)$ factors while permuting the vertex attributes $X^n$ in the same way that the vertices of the network $h_1 \sqcup \cdots \sqcup h_k$ are permuted.
One can easily check that the projections
$F(n) \times X^n \to F(n)$
define a homomorphism of $\O_F$-algebras, which we call
\[
\pi_X \maps A_X \to A .
\]
This homomorphism `forgets the vertex attributes' taking values in the set $X$.
[Simple networks with a rule obeyed by edges]
Let $\O_\SG$ be the simple network operad as explained in <ref>. We can form an algebra of the operad $\O_\SG$ that consists of simple graphs whose vertices have attributes taking values in some set $X$, but where an edge is permitted between two vertices only if their attributes obey some condition. We specify this condition using a symmetric function
\[
p \maps X\times X \to \Boole
\]
where $\Boole = \{F, T\}$. An edge is not permitted between vertices with attributes $(x_1, x_2) \in X \times X$ if this function evaluates to $F$.
To define this algebra, which we call $A_p$, we let $A_p(n) \subseteq \SG(n) \times X^n$ be the set of pairs $(g, x)$ such that for all edges $\{i, j\} \in g$ the attributes of the vertices $i$ and $j$ make $p$ true:
\[
p(x(i), x(j)) = T .
\]
There is a function
\[
\tau_p \maps A_X(n) \to A_p(n)
\]
that discards edges $\{i, j\}$ for which $p(x(i), x(j)) = F$. Recall that $A_X(n) = \SG(n) \times X^n$, and recall from <ref> that we can regard $\SG(n)$ as the set of functions $g \maps \E(n) \to \Boole$. Then we define $\tau_p$ by
\[
\tau_p(g, x) = (\overline{g}, x)
\]
\[
\overline{g}\{i, j\} =
\left\{ \begin{array}{ccl} g\{i, j\} & \textrm{if} & p(x(i), x(j)) = T \\
F & \textrm{if} & p(x(i), x(j)) = F.
\end{array} \right.
\]
We can define an action $\alpha_p$ of $\O_\SG$ on the sets $A_p(n)$ with the help of this function. Namely, we take $\alpha_p$ to be the composite
\[
\begin{tikzcd}
\O_\SG(n_1, \dots, n_k ; n) \times A_p(n_1) \times \cdots \times A_p(n_k) \arrow[d, hookrightarrow]
\\
\O_\SG(n_1, \dots, n_k ; n) \times A_X(n_1) \times \cdots \times A_X(n_k) \arrow[d, "\alpha_X"]
\\
A_X(n)\arrow[d, "\tau_p"]
\\
\end{tikzcd}\]
where the action $\alpha_X$ was defined in <ref>. One can check that $\alpha_p$ makes the sets $A_p(n)$ into an algebra of $\O_\SG$, which we call $A_p$. One can further check that the maps $\tau$ define a homomorphism of $\O_\SG$-algebras, which we call
\[
\tau_p \maps A_X \to A_p .
\]
[Range-limited networks]
We can use the previous examples to model range-limited communications between entities in a plane. First, let $X = \R^2$ and form the algebra $A_X$ of the simple network operad $\O_\SG$. Elements of $A_X(n)$ are simple graphs with vertices in the plane.
Then, choose a real number $L \ge 0$ and let $d$ be the usual Euclidean distance function on the plane. Define $p \maps X \times X \to \Boole$ by setting $p(x, y)=T$ if $d(x, y) \le L$ and $p(x, y) = F$ otherwise. Elements of $A_p(n)$ are simple graphs with vertices in the plane such that no edge has length greater than $L$.
[Networks with edge count limits]
Recall the network model for multigraphs $\MGplus$, defined in <ref> and clarified in <ref>. An element of $\MGplus(n)$ is a multigraph on the set $\n$, namely a function $g \maps \E(n) \to \N$ where $\E(n)$ is the set of 2-element subsets of $\n$.
If we fix a set $X$ we obtain an algebra $A_X$ of $\O_{\MGplus}$ as in <ref>. The set $A_X(n)$ consists of multigraphs on $\n$ where the vertices have attributes taking values in $X$.
Starting from $A_X$ we can form another algebra where there is an upper bound on how many edges are allowed between two vertices, depending on their attributes. We specify this upper bound using a symmetric function
\[b \maps X \times X \to \N. \]
To define this algebra, which we call $A_b$, let
$A_b(n) \subseteq \MGplus(n) \times X^n$ be the set of pairs $(g, x)$ such that for all $\{i, j\} \in \E(n)$ we have
\[g(i, j) \le b(x(i), x(j)) .\]
Much as in <ref> there is function
\[\pi \maps A_X(n) \to A_b(n) \]
that enforces this upper bound: for each $g \in A_X(n)$ its image $\pi(g)$ is obtained by reducing the number of edges between vertices $i$ and $j$ to the minimum of $g(i, j)$ and $\beta(i, j)$:
\[\pi(g)(i, j) = g(i, j) \min \beta(i, j) .\]
We can define an action $\alpha_b$ of $\O_\MG$ on the sets $A_b(n)$ as follows:
\[\begin{tikzcd}
\O_\MG(n_1, \dots, n_k ; n) \times A_p(n_1) \times \cdots \times A_p(n_k) \arrow[d, hookrightarrow]
\\
\O_\MG(n_1, \dots, n_k ; n) \times A_X(n_1) \times \cdots \times A_X(n_k) \arrow[d, "\alpha_X"]
\\
A_X(n)\arrow[d, "\pi"]
\\
\end{tikzcd}\]
One can check that $\alpha_b$ indeed makes the sets $A_b(n)$ into an algebra of $\O_\MGplus$, which we call $A_b$, and that the maps $\pi_p$ define a homomorphism of $\O_\MGplus$-algebras, which we call
\[\pi_p \maps A_X \to A_b .\]
[Range-limited networks, revisited]
We can use <ref> to model entities in the plane that have two types of communication channel, one of which has range $L_1$ and another of which has a lesser range $L_2 < L_1$. To do this, take $X = \R^2$ and define $b \maps X \times X \to \N$ by
\[b(x, y)= \left\{ \begin{array}{cl}
0 & \textrm{if } d(x, y) > L_1 \\
1 & \textrm{if } L_2 < d(x, y) \le L_1 \\
2 & \textrm{if } d(x, y) \le L_2
\end{array} \right.
\]
Elements of $A_b(n)$ are multigraphs with vertices in the plane having no edges between vertices whose distance is $> L_1$, at most one edge between vertices whose distance is $\le L_1$ but $> L_2$, and at most two edges between vertices whose distance is $\le L_2$.
Moreover, the attentive reader may notice that the action $\alpha_b$ of $\O_\MGplus$ for this specific choice of $b$ factors through an action of $\O_{\Gamma_{\Boole_2}}$, where $\Gamma_{\Boole_2}$ is the network model defined in <ref>. That is, operations $\O_{\Gamma_{\Boole_2}}(n_1, \dots , n_k; n) = S_n \times \Gamma_{\Boole_2}(n)$ where
$\Gamma_{\Boole_2}(n)$ is the set of multigraphs on $\n$ with at most $2$ edges between vertices are sufficient to compose these range-limited networks. This is due to the fact that the values of this $b \maps X \times X \to \N$ are at most 2.
These examples indicate that vertex attributes and constraints can be systematically added to the canonical algebra to build more interesting algebras, which are related by homomorphisms. <ref> illustrates how adding extra attributes to the networks in some algebra $A$ can give networks that are elements of an algebra $A'$ equipped with a homomorphism $\pi \maps A' \to A$ that forgets these extra attributes. <ref> illustrates how imposing extra constraints on the networks in some algebra $A$ can give an algebra $A'$ equipped with a homomorphism $\tau \maps A \to A'$ that imposes these constraints: this works only if there is a well-behaved systematic procedure, defined by $\tau$, for imposing the constraints on any element of $A$ to get an element of $A'$.
The examples given so far scarcely begin to illustrate the rich possibilities of network operads and their algebras. In particular, it is worth noting that all the specific examples of network models described here involve commutative monoids. However, noncommutative monoids are also important. Suppose, for example, that we wish to model entities with a limited number of point-to-point communication interfaces—e.g. devices with a finite number $p$ of USB ports. More formally, we wish to act on sets of degree-limited networks $A_{\rm deg} (n)\subset \SG(n) \times \N^n$ made up of pairs $(g, p)$ such that the degree of each vertex $i$, ${\rm deg}(i), $ is at most the degree-limiting attribute of $i$: ${\rm deg}(i) \le p(i)$. Naïvely, we might attempt to construct a map $\tau_{\rm deg} \maps A_\N \to A_{\rm deg}$ as in <ref> to obtain an action of the simple network operad $\O_\SG$. However, this is turns out to be impossible. For example, if attempt to build a network from devices with a single USB port, and we attempt to connect multiple USB cables to one of these devices, the relevant network operad must include a rule saying which attempts, if any, are successful. Since we cannot prioritize links from some vertices over others—which would break the symmetry built into any network model—the order in which these attempts are made must be relevant. Since the monoids $\SG(n)$ are commutative, they cannot capture this feature of the situation.
The solution is to use a class of noncommutative monoids dubbed `graphic monoids' by Lawvere <cit.>: namely, those that obey the identity $aba = ab$. These allow us to construct a one-colored network model $\Gamma \maps \S \to \Mon$ whose network operad $\O_\Gamma$ acts on $A_{\rm deg}$. For our USB device example, the relation $aba = ab$ means that first attempting to connect some USB cables between some devices ($a$), second attempting to connect some further USB cables ($b$), and third attempting to connect some USB cables precisely as attempted in the first step ($a$, again) has the same result as only performing the first two steps ($ab$). We explore more applications of noncommutativity in network models in <ref>.
CHAPTER: NONCOMMUTATIVE NETWORK MODELS
§ INTRODUCTION
In <ref>, we gave a functorial construction of a network model from a monoid, which we call the ordinary network model for weighted graphs. In this chapter, we provide a different construction in order to realize a larger class of networks as algebras of network operads, which we call the free varietal network model for weighted graphs. In Section <ref>, we give an example of a family of networks which cannot form an algebra for any ordinary network model for weighted graphs, but does for a varietal one. In this chapter, we give a construction for the free network model on a given monoid. This describes networks which look like the given monoid when you restrict to looking at the combinatorial behavior at a single pair of nodes. In <ref>, we give a concrete construction of a left adjoint to the functor which evaluates a network model at its second level. This requires a categorical treatment and generalization of Green's theory of products of groups indexed by a graph, (i.e. graph products of groups) <cit.>, which we give in <ref>.
This construction is designed to model networks which carry information on the edges. For example, with $\N$ a monoid under addition, $\Gamma_\N$ is a network model for loopless undirected multigraphs where overlaying is given by adding the number of edges. A similar example is $\Gamma_\Boole = \SG$. There is a monoid homomorphism $\N \to \Boole$ which sends all but $0$ to $T$. This induces a map of network models $\Gamma_\N \to \Gamma_\Boole$. Essentially this map reduces the information of a graph from the number of connections between each pair of vertices to just the existence of any connection.
[Algebra for range-limited communication]
Consider a communication network where each node represents a boat and an edge between two nodes represents a working communication channel between the corresponding boats. Some forms of communication are restricted by the distance between those communicating. Assume that there is a known maximal distance over which our boats can communicate. Networks of this sort form an algebra of the simple graphs operad in the following way.
Let $(X,d)$ be a metric space, and $0 \leq L \in \R$. Our boats will be located at points in this space. The operad $\O_\SG$ has an algebra $(A_{d,L}, \alpha)$ defined as follows. The set $A_{d,L}(\n)$ is the set of pairs $(h,f)$ where $h \in \SG(\n)$ is a simple graph and $f \maps \n \to X$ is a function such that if $\{v_1,v_2\}$ is an edge in $g$ then $d(f(v_1),f(v_2)) \leq L$. The number $L$ represents the maximal distance over which the boat's communication channels operate. Notice that this condition does not demand that all connections within range must be made. An operation $(\sigma, g)\in \O_\SG(\n_1, \dots, \n_k; \n)$ acts on a $k$-tuple $(h_i, f_i) \in A_{d,L}(\n_i)$ by
\[
\alpha(\sigma, g)((h_1, f_1), \dots, (h_k, f_k)) = (g \cup \sigma(h_1 \sqcup \dots \sqcup h_k), f_1 \sqcup \dots \sqcup f_k).
\]
Elements of this algebra are simple graphs in the space $X$ with an upper limit on edge lengths. When an operation acts on one of these, it tries to put new edges into the graph, but fails to when the range limit is exceeded <cit.>.
A characteristic of the construction given in Theorem <ref> is that elements of the resulting monoids that correspond to different edges automatically commute with each other.
For example, for a monoid $M$, the fourth constituent monoid of the ordinary $M$ network model is $\Gamma_M(4) = M^6$.
Then the element $(m_1, 0, 0, 0, 0, 0)$ represents a graph with one edge with weight $m_1 \in M$, the element $(0, m_2, 0, 0, 0, 0)$ represents a graph with a different edge with weight $m_2 \in M$, and
\begin{align*}
(m_1, 0, 0, 0, 0, 0) \cup (0, m_2, 0, 0, 0, 0)
&= (m_1, m_2, 0, 0, 0, 0)
\\&= (0, m_2, 0, 0, 0, 0) \cup (m_1, 0, 0, 0, 0, 0).
\end{align*}
This commutativity between edges means that networks given by ordinary network models cannot record information about the order in which edges were added to it.
The ability to record such information about a network is desirable, for example, if one wishes to model networks which have a limit on the number of connections each agent can make to other agents.
The degree of a vertex in a simple graph is the number of edges which include that vertex. The degree of a graph is the maximum degree of its vertices. A graph is said to have degree bounded by $k$, or simply bounded degree, if the degree of each vertex is less than or equal to $k$. Let $B_k(\n)$ denote the set of networks with $\n$ vertices and degree bound $k$. One might guess that the family of such networks could form an algebra for the simple graphs operad.
Does the collection of networks of bounded degree form an algebra of a network operad? If so, is there such an algebra which is useful in applications?
Specifically, can networks of bounded degree form an algebra of $\O_\SG$, the simple graph operad? Setting two graphs next to each other will not change the degree of any of the vertices. Overlaying them almost definitely will, which makes defining an action of $\SG(\n)$ on $B_k(\n)$ less obvious.
Ordinary network models are not sufficient to model this type of network because the graph monoids it produced could not remember the order that edges were added into a network. Even if $M$ is a noncommutative monoid, since $\Gamma_M$ is a product of several copies of $M$, one for each pair of vertices, it cannot distinguish the order that two different edges touching $v_1$ were added to a network if their other endpoints are different.
Instead of taking the product of $\binom{n}{2}$ copies of $M$, we consider taking the coproduct, so as not to impose any commutativity relations between the edges. Since the lax structure map $\sqcup \maps F(\m) \times F(\n) \to F(\m+\n)$ associated to a network model $F \maps \S \to \Mon$ must be a monoid homomorphism, then
\[(a \sqcup b) \cup (c \sqcup d) = (a \cup c) \sqcup (b \cup d).\]
In particular, if we let $\emptyset$ denote the the identity of $F(\n)$ for any $\n$, then
\begin{align*}
(a \sqcup \emptyset) \cup (\emptyset \sqcup b)
&= (a \cup \emptyset) \sqcup (\emptyset \cup b)
\\&= (\emptyset \cup a) \sqcup (b \cup \emptyset)
\\&= (\emptyset \sqcup b) \cup (a \sqcup \emptyset).
\end{align*}
This is reminiscent of the Eckmann–Hilton argument (see <ref>), but notice that the domains of the operations $\cup$ and $\sqcup$ are not the same. This equation says that elements which correspond to disjoint edges must commute with each other. Simply taking the coproduct of $\binom{n}{2}$ copies of $M$ cannot give the constituent monoids of a network model.
For a collection of monoids $\{M_i\}_{i \in I}$, elements of the product monoid which come from different components always commute with each other. In the coproduct, they never do. A graph product (in the sense of Green <cit.>) of such a collection allows one to impose commutativity between certain components and not others by indicating such relations via a simple graph. The calculation above shows that the constituent monoids of a network model must satisfy certain partial commutativity relations. We use graph products to construct a family of monoids with the right amount of commutativity to both answer the question above and satisfy the conditions of being a network model. The following theorems are proven in Section <ref>.
The functor $\NetMod \to \Mon$ defined by $F \mapsto F(\2)$ has a left adjoint
\[\Gamma_{-,\Mon} \maps \Mon \to \NetMod.\]
The fact that this construction is a left adjoint tells us that the network models constructed are ones in which the only relations that hold are those that follow from the defining axioms of network models.
A variety of monoids is the class of all monoids satisfying a given set of identities. For example, $\Mon$ has subcategories $\CMon$ of commutative monoids and $\GMon$ of graphic monoids which are varieties of monoids satisfying the equations
\[ab=ba \quad\text{and}\quad aba=ab\]
respectively. Given a variety of monoids $\V$, let $\NetMod_\V$ be the subcategory of $\NetMod$ consisting of $\V$-valued network models. We recreate graph products in varieties of monoids to obtain a more general result.
The functor $\NetMod_\V \to \V$ defined by $F \mapsto F(\2)$ has a left adjoint $\Gamma_{-,\V} \maps \V \to \NetMod_\V$.
In particular, if $\V = \CMon$, since products and coproducts are the same in $\CMon$, the ordinary $M$ network model and the $\CMon$ varietal $M$ network model are also the same. Note that this does not indicate that $\Gamma_{-,\V}$ is a complete generalization of $\Gamma_-$ from Theorem <ref>, since $\Gamma_M$ is not an example of $\Gamma_{-,\V}$ when $M$ is not commutative.
The ordinary construction for a network model given a monoid $M$ has constituent monoids given by finite cartesian powers of $M$. To include the networks described in the question above into the theory of network models, we must construct a network model from a given monoid which does not impose as much commutativity as the ordinary construction does, specifically among elements corresponding to different edges. The first attempt at a solution is to use coproducts instead of products. However, in this section we saw that we cannot create the constituent monoids of a network model simply by taking them to be coproducts of $M$ instead of products. There must be some commutativity between different edges, specifically between edges which do not share a vertex.
Given a monoid $M$, we want to create a family of monoids indexed by $\N$, the $n$th of which looks like a copy of $M$ for each edge in the complete graph on $\n$, has minimal commutativity relations between these edge components, but does have commutativity relations between disjoint edges. Partial commutativity like this can be described with Green's graph products, which we describe in Section <ref>. The type of graph which describes disjointness of edges in a graph as we need is called a Kneser graph, which we describe in Section <ref>. Besides concerning ourselves with relations between edge components, sometimes we also want the constituent monoids in a network model to obey certain relations which $M$ obeys. In Section <ref> we describe varieties of monoids and a construction which produces monoids in a chosen variety. In Section <ref> we prove this construction is functorial, and in Section <ref> we use this construction to give a positive answer to the question.
§ GRAPH PRODUCTS
This section is dedicated to constructing the constituent monoids for the network models we want. In this section there are two different ways that graphs are being used. It is important that the reader does not get these confused. One way is the graphs which are elements of the constituent monoids of the network models we are constructing. The other way we use graphs is to index the Green product (which we define in <ref>) to describe commutativity relations in the constituent monoids of the network models we are constructing.
A network model is essentially a family of monoids with properties similar to the simple graphs example, so we think of the elements of these monoids as graphs, and we think of the operation as overlaying the graphs.
These monoids have partial commutativity relations they must satisfy, as we see in <ref>.
The graphs we use in the Green product, the Kneser graphs, are there to describe the partial commutativity in the constituent monoids.
§.§ Green Products
Given a family of monoids $\{M_v\}_{v\in V}$ indexed by a set $V$, there are two obvious ways to combine them to get a new monoid, the product and the coproduct.
From an algebraic perspective, a significant difference between these two is whether or not elements that came from different components commute with each other. In the product they do. In the coproduct they do not.
Green products, or commonly graph products, of groups were introduced in 1990 by Green <cit.>, and later generalized to monoids by Veloso da Costa <cit.>.
The idea provides something of a sliding scale of relative commutativity between components. We follow <cit.> in the following definitions.
By a simple graph $G=(V, E)$, we mean a set $V$ which we call the set of vertices, and a set $E \subseteq \binom{V}{2}$, which we call the set of edges. A map of simple graphs $f \maps (V, E) \to (V', E')$ is
a function $f \maps V \to V'$ such that if $\{u, v\} \in E$ then $\{f(u), f(v)\} \in E'$.
Let $\sGrph$ denote the category of simple graphs and maps of simple graphs.
For a set $V$, a family of monoids $\{M_v\}_{v\in V}$, and a simple graph $G = (V, E)$, the $G$ Green product (or simply Green product when unambiguous) of $\{M_v\}_{v\in V}$, denoted $G(M_v)$, is
\[
G(M_v) = \left( \coprod_{v\in V} M_v \right) /R_G
\]
where $R_G$ is the congruence generated by the relation
\[
\{ (m n, n m) |\, m \in M_v, n\in M_u, u, v \text{ are adjacent in }G \}
\]
where the operation in the free product is denoted by concatenation.
If $G$ is the complete graph on $n$ vertices, then $G(M_v) \cong \prod M_v$. If $G$ is the $n$-vertex graph with no edges, then $G(M_v) \cong \coprod M_v$.
We call each $M_v$ a component of the Green product. Elements of $G(M_v)$ are written as expressions as in the free product, $m^{v_1}_1\dots m^{v_k}_k \in G(M_v)$ where the superscript indicates that $m_i \in M_{v_i}$. We often consider Green products of several copies of the same monoid, so this notation allows one to distguish elements coming from different components of the product, even if they happen to come from the same monoid. The intention and result of the imposed relations is that for an expression $m^{v_1}_1\dots m^{v_k}_k$ of an element, if there is an $i$ such that $\{v_i, v_{i+1}\} \in E$, then we can rewrite the expression by replacing $m^{v_i}_i m^{v_{i+1}}_{i+1}$ with $m^{v_{i+1}}_{i+1} m^{v_i}_i$. This move is called a shuffle, and two expressions are called shuffle equivalent if one can be obtained from the other by a sequence of shuffles. An expression $m^{v_1}_1\dots m^{v_k}_k$ is reduced if whenever $i<j$ and $v_i = v_j$, there exists $l$ with $i<l<j$ and $\{v_i, v_l\} \notin E$. If two reduced expressions are shuffle equivalent, they are clearly expressions of the same element. The converse is also true.
Every element of $M$ is represented by a reduced expression. Two reduced expressions represent the same element of $M$ if and only if they are shuffle equivalent.
In this section, we use a categorical description of Green products to define a similar construction in a more general context. The relevant property of $\Mon$ that we need for this generalization is that $\Mon$ is a pointed category.
Let $\C$ be a category. An object of $\C$ which is both initial and terminal is called a zero object. If $\C$ has such an object, $\C$ is called a pointed category <cit.>. For any two objects $A, B$ of a pointed category, there is a unique map $0 \maps A \to B$ which is the composite of the unique map from $A$ to the zero object, and the unique map from the zero object to $B$.
If $\C$ is a pointed category with finite products, then for two objects $A, B$ of $\C$, the objects admit canonical maps $A \to A \times B$.
\[
\begin{tikzcd}
\arrow[ddl, "1", bend right, swap]
\arrow[d, "\exists!i_A"]
\arrow[ddr, "0", bend left]
\\&
A \times B
\arrow[dl, "\pi_A"]
\arrow[dr, "\pi_B", swap]
\\
\end{tikzcd}\]
So we have the following maps
\[
\begin{tikzcd}
\arrow[dr, "i_A"]
\arrow[dl, "i_B", swap]
\\&
A \times B
\arrow[dr, "\pi_B", swap]
\arrow[dl, "\pi_A"]
\\
\end{tikzcd}\]
satisfying the following properties.
\begin{align*}
\pi_A i_A &= 1_A&
\pi_B i_B &= 1_B\\
\pi_B i_A &= 0&
\pi_A i_B &= 0
\end{align*}
This is suggestive of a biproduct, but in a general pointed category $A \times B$ is not necessarily isomorphic to $A + B$.
In <ref>, we use a generalized Green product to construct network models.
A generalized Green product is a colimit of a diagram whose shape is derived from a given graph. We describe the shapes of the diagrams here with directed multi-graphs. We refer to them here as quivers to help distinguish them from other variants of graphs and the role they play in this chapter. A quiver is a pair of sets $E$, $V$, respectively called the set of edges and set of vertices, and a pair of functions $s, t \maps E \to V$ assigning to each edge its starting vertex and its terminating vertex respectively. A map of quivers is a pair of functions
[d, shift right = 1, "s_1", swap]
[d, shift left = 1, "t_1"]
[r, "f_E"]
[d, shift right = 1, "s_2", swap]
[d, shift left = 1, "t_2"]
[r, "f_V", swap]
such that the $s$-square and the $t$-square both commute.
We will use the word cospan to refer to the quiver with the following shape.
\[\bullet \to \bullet \leftarrow \bullet\]
Define a functor $IC \maps \sGrph \to \Quiv$ which replaces every edge with a cospan ($IC$ stands for `insert cospan'). Specifically, given a simple graph $(V, E)$ where $E \subseteq \binom{V}{2}$, define the quiver $Q_1 \rightrightarrows Q_0$ where $Q_0 = V \sqcup E$ and $Q_1 = \{(v, e) \in V \times E |\, v \in e\}$, then define the source map $s \maps Q_1 \to Q_0$ by projection onto the first component, and the target map $t \maps Q_1 \to Q_0$ by projection onto the second component. For example, the simple graph
\[
\begin{tikzpicture}
\node[style=species] (1) {$1$};
\node[style=species] (2) [right = 1.5cm of 1] {$2$};
\node[style=species] (3) [below = 1.5cm and 1.5cm of 2] {$3$};
\node[style=species] (4) [below = 1.5cm of 1] {$4$};
\path[draw, thick]
(1) edge node {} (2)
(2) edge node {} (3)
(3) edge node {} (1)
(4) edge node {} (1);
\end{tikzpicture}\]
gives the quiver
\[
\begin{tikzcd}
\arrow[r]
\arrow[dr]
\arrow[d]
\{1, 2\}
\arrow[l]
\arrow[d]
\\
\{1, 4\}
\{1, 3\}
\{2, 3\}
\\
\arrow[u]
\arrow[u]
\arrow[ul]
\end{tikzcd}\]
Let $G=(V, E)$ and $G' = (V', E')$ be simple graphs, and $f \maps G \to G'$ a map of simple graphs. Define a map of quivers $ICf \maps IC(G) \to IC(G')$ by $ICf_0 = f_V \sqcup f_E$ and $ICf_1(v, e) = (f_V(v), f_E(e))$.
\[
\begin{tikzcd}[row sep = large]
\arrow[d, "s_G", bend right, swap]
\arrow[d, "t_G", bend left]
\arrow[r, "ICf_1"]
\arrow[d, "s_{G'}", bend right, swap]
\arrow[d, "t_{G'}", bend left]
\\
\arrow[r, "IC f_0", swap]
\end{tikzcd}\]
This construction gives a coproduct preserving functor $IC \maps \sGrph \to \Quiv$.
Let $F \maps \Quiv \to \Cat$ denote the free category (or path category) functor <cit.>. Since $F$ is a left adjoint, it preserves colimits. Notice that any quiver of the form $IC(G)$ would never have a path of length greater than 1. Thus the free path category on $IC(G)$ simply has identity morphisms adjoined.
The objects in the category $F(IC(G))$ come from two places. There is an object for each vertex of $G$, and there is an object at the apex of the cospan for each edge in $G$. We call these two subsets of objects vertex objects and edge objects. We abuse notation and refer to the object given by the vertex $u$ by the same name, and similar for edge objects.
If $\{M_v\}_{v\in V}$ is a family of monoids indexed by the set $V$, that means that there is a functor $M \maps V \to \Mon$ from the set $V$ thought of as a discrete category. Notice that if $G$ is a simple graph with vertex set $V$, then the discrete category $V$ is a subcategory of $F(IC(G))$.
We can then extend the functor $M$ to
\[D \maps F(IC(G)) \to \Mon\]
in the following way. Obviously we let $D(u) = M_u$ for a vertex object $u$. If $\{u, v\}$ is an edge in $G$, then $D(\{u, v\}) = M_u \times M_v$. The morphism $(u, \{u, v\})$ is sent to the canonical map $M_u \to M_u \times M_v$. For example, for a family of monoids $\{M_1, \dots M_4\}$, we have the following diagram.
\[
\begin{tikzcd}
\arrow[r]
\arrow[dr]
\arrow[d]
M_1 \times M_2
\arrow[l]
\arrow[d]
\\
M_1 \times M_4
M_1\times M_3
M_2\times M_3
\\
\arrow[u]
\arrow[u]
\arrow[ul]
\end{tikzcd}\]
Since there are no non-trivial pairs of composable morphisms in categories of the form $F(IC(G))$, nothing further needs to be checked to confirm $D$ is a functor.
Despite the way we are denoting these products, we are not considering them to be ordered products. Alternatively, we could have used a more cumbersome notation that does not suggest any order on the factors.
Let $V$ be a set, $\{M_v\}_{v \in V}$ be a family of monoids indexed by $V$, and $G = (V, E)$ be a simple graph with vertex set $V$. The $G$ Green product of $M_v$ is the colimit of the diagram $D \maps F(IC(G)) \to \Mon$ defined as above.
\[
G(M_v) \cong \colim D.
\]
We show that $G(M_v)$ satisfies the necessary universal property. The vertex objects in the diagram have inclusion maps into the edge objects $i_{u, v} \maps M_u \to M_u \times M_v$, and all the objects have inclusion maps into $G(M_v)$, $j_u \maps M_u \to G(M_v)$ and $j_{u, v} \maps M_u \times M_v \to G(M_v)$ such that $j_{u, v} \circ i_{u, v} = j_u$. Note that due to the fact that we have unordered products for objects, there is some redundancy in our notation, namely $j_{u, v} = j_{v, u}$.
If we have a monoid $Q$ and maps $f_u \maps M_u \to Q$ and $f_{u, v} \maps M_u \times M_v \to Q$ such that
\begin{align*}
f_{u, v} &= f_{v, u}\\
f_{u, v} \circ i_{u, v} &= f_u,
\end{align*}
then we define a map $\phi \maps G(M_v) \to Q$ by $\phi(m^{v_1}_1 \dots m^{v_k}_k) = f_{v_1}(m_1) \dots f_{v_k}(m_k)$. Since this map is defined via expressions of elements, <ref> tells us that to check this map is well-defined, we need only check that the values of two expressions that differ by a shuffle are the same. Let $m^{v_1}_1 \dots m^{v_k}_k$ be an expression, and $i$ such that $\{v_i, v_{i+1}\} \in E$.
\begin{align*}
\phi (m^{v_i}_i m^{v_{i+1}}_{i+1})
&= f_{v_i}(m_i) f_{v_{i+1}}(m_{i+1})
\\&= f_{v_i, v_{i+1}} (m_i, m_{i+1})
\\&= f_{v_{i+1}}(m_{i+1}) f_{v_i}(m_i)
\\&= \phi (m^{v_{i+1}}_{i+1} m^{v_i}_i)
\end{align*}
It is clear that
\[
\phi (m^{v_1}_1 \dots m^{v_k}_k) = \phi (m^{v_1}_1 \dots m^{v_{i-1}}_{i-1}) \phi( m^{v_i}_im^{v_{i+1}}_{i+1}) \phi(m^{v_{i+2}}_{i+2} \dots m^{v_k}_k),
\]
so two shuffle equivalent expressions have the same value under $\phi$, and $\phi$ is well-defined. It is clearly a monoid homomorphism, and has the property $\phi \circ j_u = f_u$ and $\phi \circ j_{u, v} = f_{u, v}$. To show this map is unique, assume there is another such map $\psi \maps G(M_v) \to Q$. Since $\psi \circ j_u = f_u$, then $\psi(m_u) = f(u)$, and
\begin{align*}
\psi(m^{v_1}_1 \dots m^{v_k}_k)
&= \psi(m^{v_1}_1) \dots \psi(m^{v_k}_k)
\\&= f_{v_1}(m_1) \dots f_{v_k}(m_k)
\\&= \phi(m^{v_1}_1 \dots m^{v_k}_k).\qedhere
\end{align*}
This result makes it reasonable to generalize Green products in the following way.
Let $\C$ be a pointed category with finite products and finite colimits, $V$ a set, $\{A_v\}_{v \in V}$ a family of objects of $\C$ indexed by $V$, and $G$ a simple graph with vertex set $V$. Let $D \maps F(IC(G)) \to \C$ be the diagram defined by $v \mapsto A_v$, $\{u, v\} \mapsto A_u \times A_v$, and the morphism $(u, \{u, v\})$ is mapped to the inclusion $A_u \to A_u \times A_v$ as above. The $G$ Green product of $\{A_v\}_{v \in V}$ is the colimit of $D$ in $\C$,
\[
G^\C(A_v) = \colim D.
\]
If $\C=\Mon$, we denote the Green product simply as $G(A_v)$.
In <ref>, we use this general notion of graph products in varieties of monoids to construct network models whose constituent monoids are in those varieties.
Note that since $F \circ IC$ is a functor, the group $\Aut(G)$ of graph automorphisms of $G$ naturally acts on $G^\C(A_v)$.
§.§ Kneser Graphs
We focus here on a special family of simple graphs known as the Kneser graphs <cit.>. The Kneser graph $KG_{n,m}$ has vertex set $\binom{n}{m}$, the set of $m$-element subsets of an $n$-element set, and an edge between two vertices if they are disjoint subsets. Since a simple graph is defined as a collection of two-element subsets of an $n$-element set, the Kneser graph $KG_{n,2}$ has a vertex for each edge in the complete graph on $n$, and has an edge between every pair of vertices which correspond to disjoint edges. So the Kneser graph $\KG_{n,2}$ can be thought of as describing the disjointness of edges in the complete graph on $n$. For instance, the complete graph on $5$ is
\[
\begin{tikzpicture} % Petersen graph
\begin{pgfonlayer}{nodelayer}
\node [style=species] (a) at (0, -1) {};
\node [style=species] (b) at (0.95, -0.31) {};
\node [style=species] (c) at (0.59, 0.81) {};
\node [style=species] (d) at (-0.59, 0.81) {};
\node [style=species] (e) at (-0.95, -0.31) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (a) to (b);
\draw (b) to (c);
\draw (c) to (d);
\draw (d) to (e);
\draw (e) to (a);
\draw (a) to (c);
\draw (b) to (d);
\draw (c) to (e);
\draw (d) to (a);
\draw (e) to (b);
\end{pgfonlayer}
\end{tikzpicture}\]
and the corresponding Kneser graph $\KG_{5,2}$ is the Petersen graph:
\[
\begin{tikzpicture} % Petersen graph
\begin{pgfonlayer}{nodelayer}
\node [style=species] (a) at (0, -1) {};
\node [style=species] (b) at (0.95, -0.31) {};
\node [style=species] (c) at (0.59, 0.81) {};
\node [style=species] (d) at (-0.59, 0.81) {};
\node [style=species] (e) at (-0.95, -0.31) {};
\node [style=species] (A) at (0, -2) {};
\node [style=species] (B) at (1.9, -0.62) {};
\node [style=species] (C) at (1.18, 1.62) {};
\node [style=species] (D) at (-1.18, 1.62) {};
\node [style=species] (E) at (-1.9, -0.62) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (a) to (A);
\draw (b) to (B);
\draw (c) to (C);
\draw (d) to (D);
\draw (e) to (E);
\draw (A) to (B);
\draw (B) to (C);
\draw (C) to (D);
\draw (D) to (E);
\draw (E) to (A);
\draw (a) to (c);
\draw (b) to (d);
\draw (c) to (e);
\draw (d) to (a);
\draw (e) to (b);
\end{pgfonlayer}
\end{tikzpicture}\]
For sets $X,Y$ and a function $f \maps X \to Y$, let $f[U] = \{f(x) |\, x \in U\}$ for $U \subseteq X$. Let $\FinInj$ denote the category of finite sets and injective functions.
For $k \in \N$, there is a functor $\bink{-} \maps \FinInj \to \FinInj$ which sends $X$ to $\bink{X}$ the set of $k$-element subsets of $X$, and injections $f \maps X \to Y$ to the functions $\bink{f} \maps \bink{X} \to \bink{Y}$ defined by $\bink{f}(U) = f[U]$.
Note that this result holds for $\Inj$ the category of sets and injective functions, but we only require $\FinInj$ for our purposes.
If $f \maps X \to Y$ is an injection, then $|f[U]| = |U|$ for $U \subseteq X$. It then makes sense to restrict the induced map on power sets to subsets of a fixed cardinality. The map $\bink{f} \maps \bink{m} \to \bink{n}$ defined by $\bink{f}(U) = f[U]$ is then well defined. If $f[U]=f[V]$ and $x \in U$, then $f(x) \in f[U] = f[V]$, which implies there is a $y \in V$ such that $f(y) = f(x)$. Since $f$ is injective, then $x = y \in V$. Thus $U = V$ by symmetry.
Let $i_X$ and $i_Y$ denote the following inclusion maps.
\[
\begin{tikzcd}
\arrow[dr,"i_X"]
\arrow[dl,"i_Y",swap]
\\&
\end{tikzcd}\]
Since these maps are injective, they induce maps $\bink{i_X}, \bink{i_Y}$, and we get a map $\Phi_{X,Y} \maps \bink{X}+\bink{Y} \to \bink{X+Y}$ by the universal property in the following way.
\[
\begin{tikzcd}
\bink{X}
\arrow[dr,"j_X"]
\arrow[ddr,"\bink{i_X}",bend right,swap]
\bink{Y}
\arrow[dl,"j_Y",swap]
\arrow[ddl,"\bink{i_Y}",bend left]
\\&
\bink{X}+\bink{Y}
\arrow[d,"\exists!\Phi_{X,Y}",dashed]
\\&
\bink{X+Y}
\end{tikzcd}\]
The functor $\bink{-}$ is made lax symmetric monoidal
\[(\bink{-},\Phi, \phi) \maps (\FinInj,+, \emptyset) \to (\FinInj, +, \emptyset)\]
where the components of $\Phi$ are defined as above.
The family of maps $\{\Phi_{X,Y}\}$ is clearly a natural transformation.
There is no choice for the map $\phi \maps \emptyset \to \bink{\emptyset}$.
The left and right unitor laws hold trivially. Checking the coherence conditions for the associator and the symmetry are straightforward computations.
For $n,k \in \N$, the simple graph $KG_{n,k}$ has vertex set $V = \bink{n}$ and edge set $\{\{u,v\} \subseteq \binom{V}{2} |\, u \cap v = \emptyset\}$.
If $f \maps m \to n$ is injective, then we get a map $\bink{f}$ between the vertex sets of $KG_{m,k}$ and $KG_{n,k}$. Let $\{u,v\} \in \binom{V}{2}$ be an edge in $KG_{m,k}$. Then $f[u] \cap f[v] = \emptyset$ by injectivity, so $\{f[u],f[v]\}$ is an edge of $KG_{n,k}$. An injection $f$ then induces a map of graphs, denoted $KG_{f,k} \maps KG_{m,k} \to KG_{n,k}$.
Since $\bink{f}$ is injective, $KG_{f,k}$ is an embedding.
Nothing about this construction requires finiteness of the sets involved, but our applications only call for finite graphs.
For $k \in \N$, there is functor $KG_{-,k} \maps \FinInj \to \sGrph$ which sends $n$ to $KG_{n,k}$ and $f\maps m \to n$ to $KG_{f,k}$.
Not only does $KG_{m,k}$ embed into $KG_{n,k}$ when $m<n$, but $KG_{m,k} + KG_{n,k}$ embeds into $KG_{m+n,k}$.
We construct the embedding $KG_{m,k} + KG_{n,k} \to KG_{m+n,k}$ by using the lax structure map from <ref> for the vertex map, $\Phi_{m, n} \maps \bink{m} + \bink{n} \to \bink{m+n}$. Restricting this map to either $\bink{m}$ (resp. $\bink{n}$) gives the map $\bink{i_m}$ (resp. $\bink{i_n}$) which we already know induces a map of graphs. Thus $\Phi_{m, n}$ induces a map of graphs, which we call $\Psi_{m, n}$.
The functor $KG_{-,k}$ is made lax (symmetric) monoidal
\[(KG_{-,k},\Psi) \maps (\Inj,+) \to (\sGrph, +)\]
where the components of $\Psi$ are defined as above.
All the necessary properties for $\Psi$ are inherited immediately from $\Phi$.
Let $(L, \Lambda) \maps (\Inj,+) \to (\Cat,+)$ be the composite $L = F \circ IC \circ KG_{-,2}$ with the obvious laxator. Let $M$ be a monoid. Then from the construction given in the previous subsection, for each $n$ we get a diagram $D_n \maps L(n) \to \Mon$ which sends all vertex objects to $M$, all edge objects to $M \times M$, and all nontrivial morphisms to inclusions $M \to M \times M$.
Taking the colimit of $D_n$ then gives the Green product $\KG_{n,2}(M)$. Note that we identify constituent monoids with the corresponding submonoid of the graph product when this can be done without confusion.
Let $M_{p,q}$ be a $\binom{m+n}{2}$ family of monoids, and $G_1$ and $G_2$ be graphs with $m$ and $n$ vertices respectively. Let $a_1 \in M_{p_1,q_1}$ with $p_1,q_1 \leq m$ and $a_2 \in M_{p_2,q_2}$ with $p_2,q_2>m$, and let $\overline a_1, \overline a_2$ be their values under the canonical inclusions $M_{p,q} \hookrightarrow (G_1 \sqcup G_2)(M_{p,q})$. Then $\overline a_1 \overline a_2 = \overline a_2 \overline a_1$ in $(G_1 \sqcup G_2)(M_{p,q})$.
By definition, there is an edge in the Kneser graph $KG_{m+n,2}$ between the vertices ${p_1,q_1}$ and ${p_2,q_2}$. This imposes the desired commutativity relation.
§.§ Varieties of Monoids
A finitary algebraic theory or Lawvere theory is a category $T$ with finite products in which every object is isomorphic to a finite cartesian power $x^n = \prod^n x$ of a distinguished object $x$ <cit.>. An algebra of a theory $T$, or $T$-algebra, is a product preserving functor $T \to \Set$. Let $T\Alg$ denote the category of $T$-algebras with natural transformations for morphisms. We are primarily concerned with monoids in this chapter. The theory of monoids $T_\Mon$ has morphisms $m \maps x \times x \to x$ and $e \maps x^0 \to x$, which makes the following diagrams commute.
\[
\begin{tikzcd}
\arrow[d, "m \times 1_x", swap]
\arrow[r, "1_x \times m"]
\arrow[d, "m"]
x \times x^0
\arrow[r, "1_x \times e"]
\arrow[dr, "\simeq", swap]
\arrow[d, "m"]
x^0 \times x
\arrow[l, "e \times 1_x", swap]
\arrow[dl, "\simeq"]
\\
\arrow[r, "m", swap]
\end{tikzcd}\]
A variety of $T$-algebras is a full subcategory of $T\Alg$ which is closed under products, subobjects, and homomorphic images. Birkhoff's theorem implies that this is equivalent to the category $T'\Alg$ of algebras of another theory $T'$ which has the same morphisms, but satisfies more commutative diagrams <cit.>. For example, commutative monoids are given by algebras of the theory of commutative monoids $T_\CMon$, which has morphisms $m,e$ as in $T_\Mon$, satisfies the same commutative diagrams as $T_\Mon$, but also satisfies the following commutative diagram
\[
\begin{tikzcd}
\arrow[dr, "m", swap]
\arrow[r, "b"]
\arrow[d, "m"]
\\&
\end{tikzcd}\]
where $b: x^2 \to x^2$ is the braid isomorphism. We only use varieties of monoids in this chapter, so we give these “extra” conditions by equations, e.g. commutative monoids are those which satisfy the equation $ab=ba$ for all elements $a,b$. We call the extra equations the defining equations of the variety.
A graphic monoid is a monoid which satisfies the graphic identity: $aba=ab$ for all elements $a,b$. Graphic monoids are algebras of a theory $T_\GMon$. A semigroup obeying this relation is known as a left regular band <cit.>. The term graphic monoid was introduced by Lawvere <cit.>. Let $M$ be a graphic monoid. If we let $b$ be the unit of $M$, then the graphic relation says that $a^2=a$. Every element of $M$ is idempotent. If $a,c \in M$, then $ca=c$ if $c$ already has $a$ as a factor.
Graphic monoids are present when talking about types of information where a piece of information cannot contain the same piece of information twice. A simple example can be seen in the powerset of a given set $X$, given the structure of a monoid by union. Of course, this example is overly simple because the operation is commutative idempotent, which is stronger than graphic. A more interesting example can be seen by considering the following simple graph.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (1) at (-2, 0) {a};
\node [style=construct] (2) at (0, 0) {b};
\node [style=construct] (3) at (2, 0) {c};
\node [style=none] (4) at (-1, 0.25) {x};
\node [style=none] (5) at (1, 0.25) {y};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (2);
\draw (2) to (3);
\end{pgfonlayer}
\end{tikzpicture}\]
We will define a monoid structure on the set $M = \{1, a, b, c, x, y\}$ in the following way. First, $1$ is a freely adjoined identity element. For $p,q \in M \setminus \{1\}$, define $pq$ as follows. Pick a generic point $f$ in $p$ and a generic point $g$ in $q$. Then move a small distance along a straight line path from $f$ to $g$. We define the product $pq$ to be the component of the graph you land in. Here are some example computations:
\begin{align*}
&ab = x
&aa = a
\\
&bc = y
&xb = x
\\
&ac = x
&ca = y
\end{align*}
The last two demonstrate that this monoid is not commutative. More complicated examples can be constructed by using the same idea for the operation, but applying it to different spaces.
The following fact is critical in <ref>. It follows immediately from the definitions.
Every variety of monoids is a pointed category and has finite colimits.
§.§ Varietal Network Models
Our motivation for using graphic monoids is that we use the graphic relation to model “commitment” in the following way. Let $M$ be a graphic monoid, where we think of an element of $M$ as a task or list of tasks. If we first commit to doing task $x$, and then commit to doing task $y$, then we have the element $xy$ as our task list, indicating that we committed to $x$ before $y$. If we then try to commit to to doing $x$, the graphic relation saves us from recording this information twice. The relation also preserves the order in which we committed to $x$ and $y$: if $x$ is a task list of the form $x = ab$, and we have committed to $xy$, and then try to commit to $bc$, we get $(xy)(bc) = (aby)(bc) = a (byb) c = a (by) c = abyc = xyc$.
We want to construct a network model from a monoid in a variety $\V$ which has constituent monoids that are also in $\V$. If $M$ is a monoid in a variety $\V$, then each constituent monoid $\Gamma_M(n)$ is a product of several copies of $M$, and so is also in $\V$ by definition. Thus the ordinary network model (given in <ref>) restricted to a variety gives a functor $\V \to \NetMod_\V$, where $\NetMod_\V$ denotes the category of $\V$-valued network models.
The free product of two monoids is a monoid, $M+N$ an element of which is given by a list with entries in the set $M \sqcup N$ such that if two consecutive entries of a list are either both elements of $M$ or both elements of $N$, then the list is identified with the list that is the same everywhere except that those two entries are reduced to one entry occupied by their product. Note that the empty list is identified with both the singleton list consisting of the identity element of $M$, and the singleton list consisting of the identity element of $N$. Free products of monoids gives the coproduct in the category of monoids $\Mon$. Free products of monoids are very similar to free products of groups, which can be found in most books introducing group theory <cit.>.
If two monoids $M$ and $N$ are in a variety $\V$, taking their free product will not necessarily produce a monoid in $\V$, i.e. varieties are not necessarily closed under the coproduct of $\Mon$. It is easy to find an example demonstrating this.
Consider $\IMon$, the variety of idempotent monoids, i.e. monoids satisfying the equation $x^2 = x$ for all elements $x$. The boolean monoid $\B$ is an object in $\IMon$. The free product of $\B$ with itself $\B+\B$ can be generated by elements $a$ and $b$ which correspond to the element $1$ in each copy of $\B$. The element $ab \in \B+\B$ is not idempotent, as $abab \neq ab$. However, every variety $\V$ does have coproducts. The coproduct in a variety of monoids is the quotient of the free product by the congruence relation generated by the variety's defining equations. In <ref> we give a construction $\V \to \NetMod_\V$ which uses colimits in order to impose minimal relations.
<ref> tells us that it makes sense to talk about Green products in a variety, which we call varietal Green products. In the next section, we use varietal Green products with Kneser graphs to construct network models.
§ FREE NETWORK MODELS
In this section, we state and prove the main result of this chapter.
It says that given a monoid $M$ in a variety $\V$, we can construct a network model whose constituent monoids are also in $\V$, while avoiding to impose commutativity relations when possible. In the following section, we see how this construction resolves the dilemma presented in the question.
Let $M$ be a monoid in a variety $\V$. Define $\GMV(n)$ to be the $KG_{n,2}$ Green product of $\binom{n}{2}$ copies of $M$.
For $\V$ a variety of monoids, $\Gamma_{-,\V} \maps \V \to \NetMod_\V$ is a functor, as given above. The network model $\GMV$ is called the $\V$-varietal network model for $M$-weighted graphs, or just the varietal $M$ network model.
In order to prove this, we must first show that a monoid $M$ gives a network model, i.e. a lax symmetric monoidal functor. The laxator for $\GMV$ is canonically defined, but perhaps it is not as immediate as the one for the ordinary $M$ network model. We treat this first before returning to the proof of the main theorem.
Let $A$ and $B$ be objects in a pointed category with finite products and coproducts. Let $p_A \maps A \times B \to A$ and $p_B \maps A \times B \to B$ denote the canonical projections, and $i_A \maps A \to A+B$ and $i_B \maps B \to A + B$ the canonical inclusions.
The category $\CMon$ of commutative monoids is such a category. Recall that the operation of a monoid is a monoid homomorphism if and only if the monoid is commutative.
We have
\[
\begin{tikzcd}
A \times B
\arrow[dl, "p_A", swap]
\arrow[dr, "p_B"]
\arrow[d, dashed]
\\
\arrow[d, "i_A", swap]
(A + B) \times (A + B)
\arrow[d,"\ast"]
\arrow[dl]
\arrow[dr]
\arrow[d, "i_B"]
\\
A + B
A + B
A + B
\end{tikzcd}
\]
where $\ast$ denotes the operation in the commutative monoid $A+B$, and the dashed arrow is $<i_A p_A, i_B p_B>$ given by universal property.
The composite of the two maps going down the middle is the inverse to the canonical map $A + B \to A \times B$.
The operation in a noncommutative monoid is not a monoid homomorphism, but all the above maps still exist as functions.
Recall that we let $\cup$ denote the operation in the monoids $\GMV(n)$. There is always a homomorphism $\phi_{m, n} \maps \GMV(m) + \GMV(n) \to \GMV(m+n)$ by universal property of coproducts.
\[\gamma \maps (\GMV(m) + \GMV(n)) \times (\GMV(m) + \GMV(n)) \to \GMV(m) + \GMV(n)\]
denote the monoid operation of the coproduct.
\[\adjustbox{scale = 0.75}{
\begin{math}\begin{tikzcd}[row sep = 50]
\GMV(m) \times \GMV(n)
\arrow[dl, "p_1", swap]
\arrow[dr, "p_2"]
\arrow[d, dashed]
\\
\GMV(m)
\arrow[d, "i_1", swap]
\arrow[d,"\gamma"]
\arrow[dl]
\arrow[dr]
\GMV(n)
\arrow[d, "i_2"]
\\
\GMV(m)+\GMV(n)
\GMV(m)+\GMV(n)
\arrow[d, "\phi"]
\GMV(m)+\GMV(n)
\\&
\GMV(m+n)
\end{tikzcd}\end{math}
The monoids $\GMV(n)$ are constructed specifically so that $\phi \circ \gamma \circ <i_1 \circ p_1, i_2 \circ p_2>$ is a monoid homomorphism despite the fact that $\gamma$ is not.
In the proof of the following theorem, we utilize a string diagrammatic calculus suited for reasoning in a symmetric monoidal category. We refer the reader to Selinger's thorough exposition of such string diagramatic languages and their use in category theory <cit.>.
The function $\GMV(m) \times \GMV(n) \to \GMV(m + n)$ given by $\phi \circ (i_1 \circ p_1 \cup i_2 \circ p_2)$ is a monoid homomorphism. Moreover, the family of maps of this form gives a natural transformation, denoted $\sqcup$.
We have the following actors in play:
* the monoid operations $\cup_k \maps \GMV(\mathbf k)$ for $k = m, n, m+n$ (we leave off the subscripts below)
* the monoid operation of the coproduct
\[\gamma \maps (\GMV(m) + \GMV(n)) \times (\GMV(m) + \GMV(n)) \to \GMV(m) + \GMV(n)\]
* the canonical inclusion maps $i_1 \maps \GMV(m) \to \GMV(m) \to \GMV(n)$ and $i_2 \maps \GMV(n) \to \GMV(m) \to \GMV(n)$
* the canonical map $\phi \maps \GMV(m) + \GMV(n) \to \GMV(m+n)$
We represent these string diagramatically (read from top to bottom) as follows. Note that these are digrams in $\Set$ with its cartesian monoidal structure, because the monoid operations $\cup_k$ and $\gamma$ are not necessarily monoid homomorphisms.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (U) at (0, 0) {$\cup$};
\node [style=none] (1) at (-0.5, 1) {};
\node [style=none] (2) at (-0.5, 0.5) {};
\node [style=none] (3) at (0.5, 0.5) {};
\node [style=none] (4) at (0, -1) {};
\node [style=none] (5) at (0.5, 1) {};
\node [style=none] () at (0.75, 0) {,};
\node [style=none] () at (1, 0) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (U);
\draw [bend left] (3.center) to (U);
\draw [] (2.center) to (1);
\draw [] (3.center) to (5);
\draw (4.center) to (U);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (g) at (0, 0) {$\gamma$};
\node [style=none] (1) at (-0.5, 1) {};
\node [style=none] (2) at (-0.5, 0.5) {};
\node [style=none] (3) at (0.5, 0.5) {};
\node [style=none] (4) at (0, -1) {};
\node [style=none] (5) at (0.5, 1) {};
\node [style=none] () at (0.75, 0) {,};
\node [style=none] () at (1, 0) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (g);
\draw [bend left] (3.center) to (g);
\draw [] (2.center) to (1);
\draw [] (3.center) to (5);
\draw (4.center) to (g);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (1) at (0, 0) {$i_1$};
\node [style=none] (2) at (0, 1) {};
\node [style=none] (3) at (0, -1) {};
\node [style=none] (5) at (0.75, 0) {,};
\node [style=none] (5) at (1, 0) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (1);
\draw (3.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (1) at (0, 0) {$i_2$};
\node [style=none] (2) at (0, 1) {};
\node [style=none] (3) at (0, -1) {};
\node [style=none] (5) at (0.75, 0) {,};
\node [style=none] (5) at (1, 0) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (1);
\draw (3.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (1) at (0, 0) {$\phi$};
\node [style=none] (2) at (0, 1) {};
\node [style=none] (3) at (0, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (1);
\draw (3.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\]
We define $\sqcup \maps \GMV(m) \times \GMV(n) \to \GMV(m + n)$ as follows.
\begin{equation}\label{def:disjoint}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (U) at (0, 0) {$\sqcup$};
\node [style=none] (1) at (-0.5, 1) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (-0.5, 0.5) {};
\node [style=none] (4) at (0.5, 0.5) {};
\node [style=none] (5) at (0, -1) {};
\node [style=none] () at (1.2, 0) {=};
\node [style=none] () at (1.8, 0) {}; %spacing
\node [style=none] () at (0, -1.6) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (3.center) to (U);
\draw [bend left] (4.center) to (U);
\draw [] (3.center) to (1);
\draw [] (4.center) to (2);
\draw (5.center) to (U);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (4) at (0, -1) {};
\node [style=construct] (6) at (-0.5, 1) {$\phi$};
\node [style=construct] (7) at (0.5, 1) {$\phi$};
\node [style=construct] (15) at (0, 0) {$\cup$};
\node [style=none] (16) at (-0.5, 3) {};
\node [style=none] (17) at (0.5, 3) {};
\node [style=construct] (18) at (-0.5, 2) {$i_1$};
\node [style=construct] (19) at (0.5, 2) {$i_2$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (15) to (4.center);
\draw [bend right] (6) to (15);
\draw [bend left] (7) to (15);
\draw (18) to (6);
\draw (19) to (7);
\draw (16.center) to (18);
\draw (17.center) to (19);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}
<ref> gives the following equation.
\begin{equation}\label{prop13}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (4) at (0, -1) {};
\node [style=construct] (6) at (-0.5, 1) {$\phi$};
\node [style=construct] (7) at (0.5, 1) {$\phi$};
\node [style=construct] (15) at (0, 0) {$\cup$};
\node [style=none] (16) at (-0.5, 3) {};
\node [style=none] (17) at (0.5, 3) {};
\node [style=construct] (18) at (-0.5, 2) {$i_2$};
\node [style=construct] (19) at (0.5, 2) {$i_1$};
\node [style=none] (1) at (1.8, 1) {=};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (15) to (4.center);
\draw [bend right] (6) to (15);
\draw [bend left] (7) to (15);
\draw (18) to (6);
\draw (19) to (7);
\draw (16.center) to (18);
\draw (17.center) to (19);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=none] () at (-1.8, 1) {}; %spacer
\node [style=construct] (f1) at (-0.5, 1) {$\phi$};
\node [style=construct] (f2) at (0.5, 1) {$\phi$};
\node [style=construct] (U) at (0, 0) {$\cup$};
\node [style=construct] (i1) at (-0.5, 2) {$i_1$};
\node [style=construct] (i2) at (0.5, 2) {$i_2$};
\node [style=none] (1) at (0, -1) {};
\node [style=none] (2) at (-0.5, 2.5) {};
\node [style=none] (3) at (0.5, 2.5) {};
\node [style=none] (4) at (0, 3) {};
\node [style=none] (5) at (-0.5, 3.5) {};
\node [style=none] (6) at (0.5, 3.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (5) to (4.center);
\draw [bend left] (6) to (4.center);
\draw [bend left] (4.center) to (3.center);
\draw [bend right] (4.center) to (2.center);
\draw (2.center) to (i1);
\draw (3.center) to (i2);
\draw (i1) to (f1);
\draw (i2) to (f2);
\draw [bend right] (f1) to (U);
\draw [bend left] (f2) to (U);
\draw (U) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}
Since $\phi$ is a homomorphism, we get the following equation.
\begin{equation}\label{phiishom}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (1) at (0, 0) {$\cup$};
\node [style=construct] (2) at (-0.5, 1) {$\phi$};
\node [style=construct] (3) at (0.5, 1) {$\phi$};
\node [style=none] (4) at (-0.5, 2) {};
\node [style=none] (5) at (0.5, 2) {};
\node [style=none] (6) at (0, -1) {};
\node [style=none] (7) at (1.8, 0.5) {=};
\node [style=none] (8) at (2.8, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4) to (2);
\draw (5) to (3);
\draw [bend right] (2) to (1);
\draw [bend left] (3) to (1);
\draw (1) to (6);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (g) at (0, 1) {$\gamma$};
\node [style=construct] (f) at (0, 0) {$\phi$};
\node [style=none] (1) at (-0.5, 2) {};
\node [style=none] (2) at (0.5, 2) {};
\node [style=none] (3) at (-0.5, 1.5) {};
\node [style=none] (4) at (0.5, 1.5) {};
\node [style=none] (5) at (0, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (3.center);
\draw (2) to (4.center);
\draw [bend right] (3.center) to (g);
\draw [bend left] (4.center) to (g);
\draw (g) to (f);
\draw (f) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}
Since $i_1$ and $i_2$ are homomorphisms, we get the following equations.
\begin{equation}\label{isarehoms}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (1) at (0, 0) {$\gamma$};
\node [style=construct] (2) at (-0.5, 1) {$i_j$};
\node [style=construct] (3) at (0.5, 1) {$i_j$};
\node [style=none] (4) at (-0.5, 2) {};
\node [style=none] (5) at (0.5, 2) {};
\node [style=none] (6) at (0, -1) {};
\node [style=none] (7) at (1.8, 0.5) {=};
\node [style=none] (8) at (2.8, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4) to (2);
\draw (5) to (3);
\draw [bend right] (2) to (1);
\draw [bend left] (3) to (1);
\draw (1) to (6);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (g) at (0, 1) {$\gamma$};
\node [style=construct] (f) at (0, 0) {$i_j$};
\node [style=none] (1) at (-0.5, 2) {};
\node [style=none] (2) at (0.5, 2) {};
\node [style=none] (3) at (-0.5, 1.5) {};
\node [style=none] (4) at (0.5, 1.5) {};
\node [style=none] (5) at (0, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (3.center);
\draw (2) to (4.center);
\draw [bend right] (3.center) to (g);
\draw [bend left] (4.center) to (g);
\draw (g) to (f);
\draw (f) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}
We want to show that $(g \sqcup h) \cup (g' \sqcup h') = (g \cup g') \sqcup (h \cup h')$. We compute:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (s1) at (-0.75, 1) {$\sqcup$};
\node [style=construct] (s2) at (0.75, 1) {$\sqcup$};
\node [style=construct] (U) at (0, 0) {$\cup$};
\node [style=none] (1) at (-1.25, 1.5) {};
\node [style=none] (2) at (0.25, 1.5) {};
\node [style=none] (3) at (-0.25, 1.5) {};
\node [style=none] (4) at (1.25, 1.5) {};
\node [style=none] (5) at (0, -1) {};
\node [style=none] (6) at (-1.25, 2) {};
\node [style=none] (7) at (0.25, 2) {};
\node [style=none] (8) at (-0.25, 2) {};
\node [style=none] (9) at (1.25, 2) {};
\node [style=none] () at (2.2, 0.5) {=};
\node [style=none] () at (2.2, 0.9) {(\ref{def:disjoint})};
\node [style=none] () at (2.8, 1) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6) to (1.center);
\draw (8) to (3.center);
\draw (7) to (2.center);
\draw (9) to (4.center);
\draw [bend right] (1.center) to (s1);
\draw [bend left] (3.center) to (s1);
\draw [bend right] (2.center) to (s2);
\draw [bend left] (4.center) to (s2);
\draw [bend right] (s1) to (U);
\draw [bend left] (s2) to (U);
\draw (U) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (U1) at (-1, -1) {$\cup$};
\node [style=construct] (U2) at (1, -1) {$\cup$};
\node [style=construct] (U3) at (0, -2) {$\cup$};
\node [style=construct] (i1) at (-1.5, 1) {$i_1$};
\node [style=construct] (i2) at (-0.5, 1) {$i_2$};
\node [style=construct] (i1') at (0.5, 1) {$i_1$};
\node [style=construct] (i2') at (1.5, 1) {$i_2$};
\node [style=construct] (f1) at (-1.5, 0) {$\phi$};
\node [style=construct] (f2) at (-0.5, 0) {$\phi$};
\node [style=construct] (f3) at (0.5, 0) {$\phi$};
\node [style=construct] (f4) at (1.5, 0) {$\phi$};
\node [style=none] (1) at (-1.5, 2) {};
\node [style=none] (2) at (0.5, 2) {};
\node [style=none] (3) at (-0.5, 2) {};
\node [style=none] (4) at (1.5, 2) {};
\node [style=none] (5) at (0, -3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (i1);
\draw (3) to (i2);
\draw (2) to (i1');
\draw (4) to (i2');
\draw (i1) to (f1);
\draw (i2) to (f2);
\draw (i1') to (f3);
\draw (i2') to (f4);
\draw [bend right] (f1) to (U1);
\draw [bend left] (f2) to (U1);
\draw [bend right] (f3) to (U2);
\draw [bend left] (f4) to (U2);
\draw [bend right = 40] (U1) to (U3);
\draw [bend left = 40] (U2) to (U3);
\draw (U3) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (i1) at (-1.5, 2) {$i_1$};
\node [style=construct] (i2) at (-0.5, 2) {$i_2$};
\node [style=construct] (i1') at (0.5, 2) {$i_1$};
\node [style=construct] (i2') at (1.5, 2) {$i_2$};
\node [style=construct] (f1) at (-1.5, 1) {$\phi$};
\node [style=construct] (f2) at (-0.5, 1) {$\phi$};
\node [style=construct] (f3) at (0.5, 1) {$\phi$};
\node [style=construct] (f4) at (1.5, 1) {$\phi$};
\node [style=construct] (U1) at (0, 0) {$\cup$};
\node [style=construct] (U2) at (-0.75, -0.75) {$\cup$};
\node [style=construct] (U3) at (0.375, -1.75) {$\cup$};
\node [style=none] (1) at (-1.5, 3) {};
\node [style=none] (2) at (-0.5, 3) {};
\node [style=none] (3) at (0.5, 3) {};
\node [style=none] (4) at (1.5, 3) {};
\node [style=none] (5) at (0.375, -2.75) {};
\node [style=none] (6) at (-1.5, -0.25) {};
\node [style=none] (7) at (0, -0.25) {};
\node [style=none] (8) at (1.5, -1) {};
\node [style=none] (9) at (-0.75, -1) {};
\node [style=none] () at (2.5, 0) {=};
\node [style=none] () at (2.5, 0.4) {(\ref{prop13})};
\node [style=none] () at (-2.5, 0) {=};
\node [style=none] () at (3.3, 0) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (i1);
\draw (2) to (i2);
\draw (3) to (i1');
\draw (4) to (i2');
\draw (i1) to (f1);
\draw (i2) to (f2);
\draw (i1') to (f3);
\draw (i2') to (f4);
\draw (f1) to (6.center);
\draw [bend right] (f2) to (U1);
\draw [bend left] (f3) to (U1);
\draw [bend right] (6.center) to (U2);
\draw (U1) to (7.center);
\draw [bend left] (7.center) to (U2);
\draw (U2) to (9.center);
\draw [bend right = 40] (9.center) to (U3);
\draw (f4) to (8.center);
\draw [bend left = 40] (8.center) to (U3);
\draw (U3) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (3) at (-1.5, 0.75) {$i_1$};
\node [style=construct] (4) at (0.5, 0.75) {$i_2$};
\node [style=construct] (5) at (-0.5, 0.75) {$i_1$};
\node [style=construct] (6) at (1.5, 0.75) {$i_2$};
\node [style=construct] (17) at (-1.5, -0.15) {$\phi$};
\node [style=construct] (18) at (-0.5, -0.15) {$\phi$};
\node [style=construct] (19) at (0.5, -0.15) {$\phi$};
\node [style=construct] (20) at (1.5, -0.15) {$\phi$};
\node [style=construct] (21) at (0, -1) {$\cup$};
\node [style=construct] (22) at (-0.75, -1.75) {$\cup$};
\node [style=construct] (23) at (0.375, -2.5) {$\cup$};
\node [style=none] (8) at (-1.5, 2) {};
\node [style=none] (9) at (0.25, 1.25) {};
\node [style=none] (11) at (-0.25, 1.25) {};
\node [style=none] (12) at (1.5, 2) {};
\node [style=none] (13) at (-0.25, 1.5) {};
\node [style=none] (14) at (0.25, 1.5) {};
\node [style=none] (15) at (-0.5, 2) {};
\node [style=none] (16) at (0.5, 2) {};
\node [style=none] (24) at (0.375, -3.5) {};
\node [style=none] (25) at (1.5, -1.75) {};
\node [style=none] (26) at (-1.5, -1) {};
\node [style=none] () at (2.5, -0.7) {=};
\node [style=none] () at (3.3, -0.7) {}; %spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (8.center) to (3.center);
\draw [bend left, looseness=1] (9.center) to (4.center);
\draw [bend right, looseness=1] (11.center) to (5.center);
\draw (12.center) to (6.center);
\draw (11.center) to (14.center);
\draw [bend right, looseness=1] (14.center) to (16.center);
\draw [bend right, looseness=1] (15.center) to (13.center);
\draw (13.center) to (9.center);
\draw (3.center) to (17.center);
\draw (5.center) to (18.center);
\draw (4.center) to (19.center);
\draw (6.center) to (20.center);
\draw (17.center) to (26.center);
\draw [bend right=45] (26.center) to (22.center);
\draw [bend right=45] (18.center) to (21.center);
\draw [bend left=45] (19.center) to (21.center);
\draw (20.center) to (25.center);
\draw [bend left=45] (21.center) to (22.center);
\draw [bend left=45] (25.center) to (23.center);
\draw [bend right=45] (22.center) to (23.center);
\draw (23.center) to (24.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (3) at (-1.5, 0.75) {$i_1$};
\node [style=construct] (4) at (0.5, 0.75) {$i_2$};
\node [style=construct] (5) at (-0.5, 0.75) {$i_1$};
\node [style=construct] (6) at (1.5, 0.75) {$i_2$};
\node [style=construct] (17) at (-1.5, -0.15) {$\phi$};
\node [style=construct] (18) at (-0.5, -0.15) {$\phi$};
\node [style=construct] (19) at (0.5, -0.15) {$\phi$};
\node [style=construct] (20) at (1.5, -0.15) {$\phi$};
\node [style=construct] (21) at (-1, -1) {$\cup$};
\node [style=construct] (22) at (1, -1) {$\cup$};
\node [style=construct] (23) at (0, -2) {$\cup$};
\node [style=none] (8) at (-1.5, 2) {};
\node [style=none] (9) at (0.25, 1.25) {};
\node [style=none] (11) at (-0.25, 1.25) {};
\node [style=none] (12) at (1.5, 2) {};
\node [style=none] (13) at (-0.25, 1.5) {};
\node [style=none] (14) at (0.25, 1.5) {};
\node [style=none] (15) at (-0.5, 2) {};
\node [style=none] (16) at (0.5, 2) {};
\node [style=none] (24) at (0, -3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (8.center) to (3.center);
\draw [bend left, looseness=1] (9.center) to (4.center);
\draw [bend right, looseness=1] (11.center) to (5.center);
\draw (12.center) to (6.center);
\draw (11.center) to (14.center);
\draw [bend right, looseness=1] (14.center) to (16.center);
\draw [bend right, looseness=1] (15.center) to (13.center);
\draw (13.center) to (9.center);
\draw (3.center) to (17.center);
\draw (5.center) to (18.center);
\draw (4.center) to (19.center);
\draw (6.center) to (20.center);
\draw [bend right] (17.center) to (21.center);
\draw [bend left] (18.center) to (21.center);
\draw [bend right] (19.center) to (22.center);
\draw [bend left] (20.center) to (22.center);
\draw [bend right] (21.center) to (23.center);
\draw [bend left] (22.center) to (23.center);
\draw (23.center) to (24.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (3) at (-1.5, 0.75) {$i_1$};
\node [style=construct] (4) at (0.5, 0.75) {$i_2$};
\node [style=construct] (5) at (-0.5, 0.75) {$i_1$};
\node [style=construct] (6) at (1.5, 0.75) {$i_2$};
\node [style=construct] (17) at (-1, 0) {$\gamma$};
\node [style=construct] (18) at (1, 0) {$\gamma$};
\node [style=construct] (19) at (-1, -1) {$\phi$};
\node [style=construct] (20) at (1, -1) {$\phi$};
\node [style=construct] (21) at (0, -2) {$\cup$};
\node [style=none] (22) at (0, -3) {};
\node [style=none] (8) at (-1.5, 2) {};
\node [style=none] (9) at (0.25, 1.25) {};
\node [style=none] (11) at (-0.25, 1.25) {};
\node [style=none] (12) at (1.5, 2) {};
\node [style=none] (13) at (-0.25, 1.5) {};
\node [style=none] (14) at (0.25, 1.5) {};
\node [style=none] (15) at (-0.5, 2) {};
\node [style=none] (16) at (0.5, 2) {};
\node [style=none] () at (-2.5, -1) {=};
\node [style=none] () at (-2.5, -0.6) {(\ref{phiishom})};
\node [style=none] () at (2.5, -1) {=};
\node [style=none] () at (2.5, -0.6) {(\ref{isarehoms})};
\node [style=none] () at (3.3, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (8.center) to (3.center);
\draw [bend left, looseness=1] (9.center) to (4.center);
\draw [bend right, looseness=1] (11.center) to (5.center);
\draw (12.center) to (6.center);
\draw (11.center) to (14.center);
\draw [bend right, looseness=1] (14.center) to (16.center);
\draw [bend right, looseness=1] (15.center) to (13.center);
\draw (13.center) to (9.center);
\draw [bend right] (3.center) to (17.center);
\draw [bend left] (5.center) to (17.center);
\draw [bend right] (4.center) to (18.center);
\draw [bend left] (6.center) to (18.center);
\draw (18.center) to (20.center);
\draw [bend left](20.center) to (21.center);
\draw (17.center) to (19.center);
\draw [bend right](19.center) to (21.center);
\draw (21.center) to (22.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (0) at (0, -2.7) {$\sqcup$};
\node [style=construct] (1) at (-1, 0) {$\cup$};
\node [style=construct] (2) at (1, 0) {$\cup$};
\node [style=construct] (17) at (-1, -0.9) {$i_1$};
\node [style=construct] (18) at (-1, -1.8) {$\phi$};
\node [style=construct] (19) at (1, -0.9) {$i_2$};
\node [style=construct] (20) at (1, -1.8) {$\phi$};
\node [style=none] (3) at (-1.5, 0.25) {};
\node [style=none] (4) at (0.5, 0.25) {};
\node [style=none] (5) at (-0.5, 0.25) {};
\node [style=none] (6) at (1.5, 0.25) {};
\node [style=none] (7) at (0, -3.6) {};
\node [style=none] (8) at (-1.5, 1.5) {};
\node [style=none] (9) at (0.25, 0.75) {};
\node [style=none] (11) at (-0.25, 0.75) {};
\node [style=none] (12) at (1.5, 1.5) {};
\node [style=none] (13) at (-0.25, 1) {};
\node [style=none] (14) at (0.25, 1) {};
\node [style=none] (15) at (-0.5, 1.5) {};
\node [style=none] (16) at (0.5, 1.5) {};
\node [style=none] () at (2.5, -1.6) {=};
\node [style=none] () at (2.5, -1.2) {(\ref{def:disjoint})};
\node [style=none] () at (3.3, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=55, looseness=1.25] (3.center) to (1.center);
\draw [bend right=55, looseness=1.25] (4.center) to (2.center);
\draw [bend left=55, looseness=1.25] (5.center) to (1.center);
\draw [bend left=55, looseness=1.25] (6.center) to (2.center);
\draw (0.center) to (7.center);
\draw (8.center) to (3.center);
\draw [bend left, looseness=1] (9.center) to (4.center);
\draw [bend right, looseness=1] (11.center) to (5.center);
\draw (12.center) to (6.center);
\draw (11.center) to (14.center);
\draw [bend right, looseness=1] (14.center) to (16.center);
\draw [bend right, looseness=1] (15.center) to (13.center);
\draw (13.center) to (9.center);
\draw (1.center) to (17.center);
\draw (2.center) to (19.center);
\draw (17.center) to (18.center);
\draw (19.center) to (20.center);
\draw [bend right] (18.center) to (0.center);
\draw [bend left] (20.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=construct] (0) at (0, -1) {$\sqcup$};
\node [style=construct] (1) at (-1, 0) {$\cup$};
\node [style=construct] (2) at (1, 0) {$\cup$};
\node [style=none] (3) at (-1.5, 0.25) {};
\node [style=none] (4) at (0.5, 0.25) {};
\node [style=none] (5) at (-0.5, 0.25) {};
\node [style=none] (6) at (1.5, 0.25) {};
\node [style=none] (7) at (0, -2) {};
\node [style=none] (8) at (-1.5, 1.5) {};
\node [style=none] (9) at (0.25, 0.75) {};
\node [style=none] (11) at (-0.25, 0.75) {};
\node [style=none] (12) at (1.5, 1.5) {};
\node [style=none] (13) at (-0.25, 1) {};
\node [style=none] (14) at (0.25, 1) {};
\node [style=none] (15) at (-0.5, 1.5) {};
\node [style=none] (16) at (0.5, 1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=55, looseness=1.25] (3.center) to (1.center);
\draw [bend right=55, looseness=1.25] (4.center) to (2.center);
\draw [bend left=55, looseness=1.25] (5.center) to (1.center);
\draw [bend left=55, looseness=1.25] (6.center) to (2.center);
\draw [bend right = 55] (1.center) to (0.center);
\draw [bend left = 55] (2.center) to (0.center);
\draw (0.center) to (7.center);
\draw (8.center) to (3.center);
\draw [bend left, looseness=1] (9.center) to (4.center);
\draw [bend right, looseness=1] (11.center) to (5.center);
\draw (12.center) to (6.center);
\draw (11.center) to (14.center);
\draw [bend right, looseness=1] (14.center) to (16.center);
\draw [bend right, looseness=1] (15.center) to (13.center);
\draw (13.center) to (9.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Let $\sigma \in S_m$ and $\tau \in S_n$. Then
\begin{align*}
\GMV(\sigma + \tau)(g \sqcup h)
&= \GMV(\sigma + \tau)\phi(i_1(g) \cup i_2(h))
\\&= \GMV(\sigma)\phi(i_1(g)) \cup \GMV(\tau)\phi(i_2(h))
\\&= \GMV(\sigma(g)) \sqcup \GMV(\tau(h)),
\end{align*}
so the following diagram commutes.
\[
\begin{tikzcd}
\GMV(m) \times \GMV(n)
\arrow[r, "\sqcup"]
\arrow[d, "\GMV(\sigma) \times \GMV(\tau)", swap]
\GMV(m+n)
\arrow[d, "\GMV(\sigma + \tau)"]
\\
\GMV(m) \times \GMV(n)
\arrow[r, "\sqcup", swap]
\GMV(m+n)
\end{tikzcd}
\]
Thus $\sqcup$ is a natural transformation.
Checking the coherence conditions for $\sqcup$ to be a laxator is a straightforward computation.
Let $f \maps M \to N$. Then define the natural transformation $f_\V \maps \GMV \to \Gamma_{N,\V}$ with components $(f_\V)_n \maps \GMV(n) \to \Gamma_{N,\V}(n)$ given by the universal property. Composition is clearly preserved.
The functor $\Gamma_{-,\V}$ is left adjoint to $E \maps \NetMod_\V \to \V$ where $E(F) = F(\mathbf 2)$ for $(F,\Phi) \maps (\S, +) \to (\V, \times)$ a $\V$-network model.
Because of this, we call $\Gamma_{M,\V}$ the free $\V$-valued network model on the monoid $M$ or the free $\V$ network model on $M$.
By construction, $\Gamma_{M,\V}(\2) = M$, so let the unit $\eta = 1_{1_\V} \maps 1_\V \to \Gamma_{-,\V}(\2)$.
We use the universal property of $\GMV$ to construct the counit. We define a map $F(\2) \to F(n)$ for each vertex in $\KG_{n,2}$, and a map $F(\2) \times F(\2) \to F(n)$ for each edge in $\KG_{n,2}$.
If $i,j \leq n$, then $F((1\; i)(2\; j)) \maps F(n) \to F(n)$. If $e$ is the unit of the monoid $F (\mathbf{n - 2} )$, and $m \in F(\2)$, then $\Phi_{\2, n - \2} (m, e) \in F(n)$. Define maps $c_{i,j} \maps F(\2) \to F(n)$ by
\[
c_{i, j} = F((1\; i)(2\; j)) (\Phi_{\2, n - \2}(m, e)).
\]
The intuition here is that $m$ is a value on one edge of the graph, and $e$ is a graph with $n-2$ vertices and no edges. Then $\Phi(m,e)$ is the graph with $n$ vertices, and just one $m$-valued edge between vertices $1$ and $2$. Then the permutation $(1\; i)(2\; j)$ permutes this one-edge graph to put $m$ between vertex $i$ and vertex $j$. So the map $c_{i,j}$ places the one-edge monoid $M$ at the $i,j$-position in the $n$-vertex monoid.
Define maps $c_{i,j,p,q} \maps F(\2) \times F(\2) \to F(n)$ by $c_{i,j,p,q} (m, m') = c_{i,j} (m) c_{p,q} (m')$. The second gives a monoid homomorphism precisely because $(F,\Phi)$ is a network model.
Then we get a map $(\epsilon_F)_n \maps \Gamma_{F(\2),\V}(n) \to F(n)$ by universal property, which gives a monoidal natural transformation automatically. That these maps form the components of a natural transformation can be seen by a routine computation.
Notice that
\begin{align*}
(\epsilon \Gamma_{-,\V})_M = \epsilon_{\Gamma_{M,\V}} = 1_{\Gamma_{M,\V}},\\
(\Gamma_{-,\V} \eta)_M = \Gamma_{1_M,\V} = 1_{\Gamma_{M,\V}},\\
(E \epsilon)_F = E(\epsilon_F) = (\epsilon_F)_2 = 1_{F(2)},\\
(\eta E)_F = \eta_{F(2)} = 1_{F(2)}.
\end{align*}
Thus, checking that the snake equations hold is routine.
In $\CMon$, products and coproducts are isomorphic. In particular, for a commutative monoid $M$, $\Gamma_{M,\CMon} \cong \Gamma_M$.
Note that this does not indicate that varietal network models completely encompass ordinary network models. If $M$ is a noncommutative monoid,then $\Gamma_{M, \CMon}$ is not defined, but $\Gamma_M$ is.
§ COMMITMENT NETWORKS
The motivating example of network models in general is $\SG$, the network model of simple graphs. By Example <ref>, this network model is an example of the main construction of this chapter, $\SG = \Gamma_{\B,\CMon}$. The boolean monoid is not only an object in $\CMon$, it is also an object in $\GMon$, the variety of graphic monoids. Then we can consider the network models $\Gamma_{\B,\Mon}$ and $\Gamma_{\B,\GMon}$.
Elements of the monoid $\Gamma_{\B,\Mon}(n)$ are words $e_{p_1,q_1} \dots e_{p_k,q_k}$. These words are interpreted as graphs with edges that look like they were built with popsicle sticks, and if two edges lie directly on top of each other, they are identified.
Besides that relation, you can stack edges as high as you want by placing them between different pairs of vertices, but sharing one vertex.
There are networks one could imagine building with this popsicle stick intuition which are not allowed by this formalism. For instance, consider a network with three nodes and an edge for each pair of nodes, each overlapping exactly one of its neighbors, forming an Escher-esque ever-ascending staircase. This sort of network is not allowed by the formalism, since networks are actually equivalence classes of words, where letters have a definite position relative to each other. This is an important feature for this network model as it is necessary to guarantee that the procedure in the following example is well-defined, giving an algebra of the related network operad. What this means in terms of popsicle stick intuition is that allowed networks are built by placing popsicle sticks one at a time.
Elements of the $\Gamma_{\B,\GMon}(n)$ are similar to those in the previous example, except that they must obey the graphic identity, $xyx = xy$ for all $x,y \in \Gamma_{\B,\GMon}(n)$. What this means in the graphical interpretation is that all edges can be identified with the lowest occuring instance of an edge on the same vertex pair. This means that these networks in reduced form have at most as many edges as the complete simple graph with the same number of edges. Essentially these networks are simple graphs with a partial order on the edges which respects disjointness of edges.
The networks in the previous example have exactly what we need in a network model to realize networks of bounded degree as an algebra of a network operad.
[Networks of bounded degree, revisited]
The degree of a vertex in a simple graph is the number of edges in the graph which contain that vertex. For $k \in \N$, we say that a simple graph is $k$-bounded if all vertices have degree less than or equal to $k$. Then we can consider the set $B_k(n)$ of $k$-bounded simple graphs. We can define an action of $\Gamma_{\B, \GMon}(n)$ on $B_k(n)$ in the following way. Let $g = e_1 \dots e_l \in \Gamma_{\B, \GMon}(n)$ and $h \in B_k(n)$. Choose a graph $h' \in \Gamma_{\B, \GMon}(n)$ which has the same edges as $h$. Define $h_0 = h'$, then define $h_i = h_{i-1}e_i$ if that is $k$-bounded, else $h_i= h_{i-1}$. Let $hg$ denote $h_l$, which is a $k$-bounded element of $\Gamma_{\B, \GMon}(n)$. Let $\Gamma^k_{\B, \GMon}(n)$ denote the set of $k$-bounded elements of $\Gamma_{\B, \GMon}(n)$. There is a function $s \maps \Gamma^k_{\B, \GMon}(n) \to B_k(n)$. So we define $h g$ to be $s(h_l)$. This is independent of the choice of $h'$ and defines an action of $\Gamma_{\B,\GMon}$ on $B_k(n)$.
The networks in the question in <ref> can be represented by simple graphs with vertex degrees bounded by $k$. Then $B_k(n)$ gives an algebra of the operad $\O_{\B,\GMon}$. This resolves the conflict encountered in the question in <ref>. Ordinary network models could not record the order in which edges were added to a network, which was necessary to define a systematic way of attempting to add new connections to a network which has degree limitations on each vertex.
CHAPTER: PETRI NETS
§ INTRODUCTION
Petri nets are a widely studied formalism for describing collections of entities of different types, and how they turn into other entities <cit.>. In this chapter, we combine Petri nets with network models. This is worthwhile because while both formalisms involve networks, they serve different functions, and are in some sense complementary.
A Petri net can be drawn as a bipartite directed graph with vertices of two kinds: places, drawn as circles below, and transitions drawn as squares:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\quad\;$};
\node [style=species] (B) at (-4, -0.5) {$\quad\;$};
\node [style=species] (A) at (-4, 0.5) {$\quad\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
In applications to chemistry, places are also called species. When we run a Petri net, we start by placing a finite number of tokens in each place:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\quad\;$};
\node [style=species] (B) at (-4, -0.5) {$\;\bullet\;$};
\node [style=species] (A) at (-4, 0.5) {$\bullet\bullet$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
This is called a marking. Then we repeatedly change the marking using the transitions. For example, the above marking can change to this:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\;\bullet\;$};
\node [style=species] (B) at (-4, -0.5) {$\quad\;$};
\node [style=species] (A) at (-4, 0.5) {$\;\bullet\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
and then this:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\quad\;$};
\node [style=species] (B) at (-4, -0.5) {$\bullet\bullet$};
\node [style=species] (A) at (-4, 0.5) {$\;\bullet\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
Thus, the places represent different types of entity, and the transitions describe ways that one collection of entities of specified types can turn into another such collection.
Network models serve a different function than Petri nets: they are a general tool for working with networks of many kinds. A network model is a lax symmetric monoidal functor $G \maps \S(C) \to \Cat$, where $\S(C)$ is the free strict symmetric monoidal category on a set $C$. Elements of $C$ represent different kinds of “agents”. Unlike in a Petri net, we do not usually consider processes where these agents turn into other agents. Instead, we wish to study everything that can be done with a fixed collection of agents. Any object $x \in \S(C)$ is of the form $c_1 \otimes \cdots \otimes c_n$ for some $c_i \in C$; thus, it describes a collection of agents of various kinds. The functor $G$ maps this object to a category $G(x)$ that describes everything that can be done with this collection of agents.
In many examples considered so far, $G(x)$ is a category whose morphisms are graphs whose nodes are agents of types $c_1, \dots, c_n$. Composing these morphisms corresponds to overlaying graphs. Network models of this sort let us design networks where the nodes are agents and the edges are communication channels or shared commitments. In <ref>, the operation of overlaying graphs was always commutative. In <ref> we introduced more general noncommutative overlay operations. This lets us design networks where each agent has a limit on how many communication channels or commitments it can handle; the noncommutativity allows us to take a first come, first served approach to resolving conflicting commitments.
Here we take a different tack: we instead take $G(x)$ to be a category whose morphisms are processes that the given collection of agents, $x$, can carry out. Composition of morphisms corresponds to carrying out first one process and then another.
This idea meshes well with Petri net theory, because any Petri net $P$ determines a symmetric monoidal category $FP$ whose morphisms are processes that can be carried out using this Petri net. More precisely, the objects in $FP$ are markings of $P$, and the morphisms are sequences of ways to change these markings using transitions, e.g.:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\;\bullet\;$};
\node [style=species] (B) at (-4, -0.5) {$\;\bullet\;$};
\node [style=species] (A) at (-4, 0.5) {$\;\bullet\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw[->, line width=1.00] (-0.3, 0) to (0.2, 0);
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\bullet\bullet$};
\node [style=species] (B) at (-4, -0.5) {$\quad\;$};
\node [style=species] (A) at (-4, 0.5) {$\quad\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\node [style=empty] at (0, 0) {{$\to$}};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw[->, line width=1.00] (-0.3, 0) to (0.2, 0);
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\;\bullet\;$};
\node [style=species] (B) at (-4, -0.5) {$\bullet\bullet$};
\node [style=species] (A) at (-4, 0.5) {$\quad\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
Given a Petri net, then, how do we construct a network model $G \maps \S(C) \to \Cat$, and in particular, what is the set $C$? In a network model the elements of $C$ represent different kinds of agents. In the simplest scenario, these agents persist in time. Thus, it is natural to take $C$ to be some set of “catalysts”. In chemistry, a reaction may require a catalyst to proceed, but it neither increases nor decrease the amount of this catalyst present. For a Petri net, catalysts are species that are neither increased nor decreased in number by any transition. For example, species $a$ is a catalyst in the following Petri net, so we outline it in red:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, -0.1) {$\;\;c\;\;$};
\node [style=species] (B) at (-4, -0.1) {$\;\;b\;\;$};
\node [style=catalyst] (A) at (-2.5, 2) {$\;\;a\;\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\;\phantom{\Big{|}}\tau_1\;$};
\node [style=transition] (tau2) at (-2.5, -0.8){$\;\phantom{\Big{|}}\tau_2\;$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow, bend right=70, looseness=1.00, red] (A) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (B) to (tau1);
\draw [style=inarrow, bend right=70, looseness=1.00, red] (tau1) to (A);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
but neither $b$ nor $c$ is a catalyst. The transition $\tau_1$ requires one token of type $a$ as input to proceed, but it also outputs one token of this type, so the total number of such tokens is unchanged. Similarly, the transition $\tau_2$ requires no tokens of type $a$ as input to proceed, and it also outputs no tokens of this type, so the total number of such tokens is unchanged.
In Theorem <ref> we prove that given any Petri net $P$, and any subset $C$ of the catalysts of $P$, there is a network model $G \maps \S(C) \to \Cat$. An object $x \in \S(C)$ says how many tokens of each catalyst are present; $G(x)$ is then the subcategory of $FP$ where the objects are markings that have this specified amount of each catalyst, and morphisms are processes going between these.
From the functor $G \maps \S(C) \to \Cat$ we can construct a category $\int G$ by the Grothendieck construction. Because $G$ is symmetric monoidal we can make $\int G$ into a symmetric monoidal category by the monoidal Grothendieck construction of <ref>. The tensor product in $\int G$ describes doing processes in parallel. The category $\int G$ is similar to $FP$, but it is better suited to applications where agents each have their own individuality, because $FP$ is actually a commutative monoidal category, where permuting agents has no effect at all, while $\int G$ is not so degenerate. In Theorem <ref> we make this precise by more concretely describing $\int G$ as a symmetric monoidal category, and clarifying its relation to $FP$.
There are no morphisms between an object of $G(x)$ and an object of $G(x')$ unless $x \cong x'$, since no transitions can change the amount of catalysts present. The category $FP$ is thus a disjoint union, or more precisely a coproduct, of subcategories $FP_i$ where $i$, an element of free commutative monoid on $C$, specifies the amount of each catalyst present. The tensor product on $FP$ has the property that tensoring an object in $FP_i$ with one in $FP_j$ gives an object in $FP_{i+j}$, and similarly for morphisms.
However, in Prop. <ref> we show that each subcategory $FP_i$ also has its own tensor product, which describes doing one process and then another while reusing catalyst tokens. This tensor product makes $FP_i$ into a premonoidal category—an interesting generalization of a monoidal category which we recall. Finally, in Theorem <ref> we show that these monoidal structures define a lift of the functor $G \maps \S(C) \to \Cat$ to a functor $\hat{G} \maps \S(C) \to \PreMonCat$, where $\PreMonCat$ is the category of strict premonoidal categories.
§ PETRI NETS
A Petri net generates a symmetric monoidal category whose objects are tensor products of species and whose morphisms are built from the transitions by repeatedly taking composites and tensor products. There is a long line of work on this topic starting with the papers of Meseguer–Montanari <cit.> and Engberg–Winskel <cit.>, both dating to roughly 1990. It continues to this day, because the issues involved are surprisingly subtle <cit.>. In particular, there are various kinds of symmetric monoidal categories to choose from. Following the work of Master and Baez <cit.> we use `commutative' monoidal categories. These are just commutative monoid objects in $\Cat$, so their associator:
\[
\alpha_{a, b, c} \colon (a \otimes b) \otimes c \stackrel{\sim}{\longrightarrow} a \otimes (b \otimes c),
\]
their left and right unitor:
\[
\lambda_a \maps I \otimes a \stackrel{\sim}{\longrightarrow} a , \qquad
\rho_a \maps a \otimes I \stackrel{\sim}{\longrightarrow} a ,
\]
and even their braiding:
\[
\sigma_{a, b} \maps a \otimes b \stackrel{\sim}{\longrightarrow} b \otimes a
\]
are all identity morphisms. While every symmetric monoidal category is equivalent to one with trivial associator and unitors, this ceases to be true if we also require the braiding to be trivial. However, it seems that Petri nets most naturally serve to present symmetric monoidal categories of this very strict sort. Thus, we shall describe a functor from the category of Petri nets to the category of commutative monoidal categories, which we call $\CMon\Cat$:
\[
F \colon \Petri \to \CMon\Cat .
\]
To begin, let $\CMon$ be the category of commutative monoids and monoid homomorphisms. There is a forgetful functor from $\CMon$ to $\Set$ that sends commutative monoids to their underlying sets and monoid homomorphisms to their underlying functions. It has a left adjoint $\N \maps \Set \to \CMon$ sending any set $X$ to the free commutative monoid on $X$. An element $a \in \N[X]$ is formal linear combination of elements of $X$:
\[ a = \sum_{x \in X} a_x \, x ,\]
where the coefficients $a_x$ are natural numbers and all but finitely many are zero. The set $X$ naturally includes in $\N[X]$, and for any function $f \maps X \to Y$, $\N[f] \maps \N[X] \to \N[Y]$ is the unique monoid homomorphism that extends $f$. We often abuse language and use $\N[X]$ to mean the underlying set of the free commutative monoid on $X$.
A Petri net is a pair of functions of the following form:
\[\begin{tikzcd}
\arrow[r, shift left = 1, "s"]
\arrow[r, shift right = 1, "t", swap]
\N[S].
\end{tikzcd}\]
We call $T$ the set of transitions, $S$ the set of places or species, $s$ the source function, and $t$ the target function. We call an element of $\N[S]$ a marking of the Petri net.
For example, in this Petri net:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-1, 0) {$\quad\;$};
\node [style=species] (B) at (-4, -0.5) {$\quad\;$};
\node [style=species] (A) at (-4, 0.5) {$\quad\;$};
\node [style=transition] (tau1) at (-2.5, 0.6) {$\big.\;\;\;\, $};
\node [style=transition] (tau2) at (-2.5, -0.7) {$\big.\;\;\;\, $};
\node [style = none] () at (-2.5, 0.6) {$\tau_1$};
\node [style = none] () at (-2.5, -0.69) {$\tau_2$};
\node [style = none] () at (-4, 0.5) {$a$};
\node [style = none] () at (-4, -0.5) {$b$};
\node [style = none] () at (-1, 0) {$c$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (A) to (tau1);
\draw [style=inarrow] (B) to (tau1);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau1) to (C);
\draw [style=inarrow, bend left=15, looseness=1.00] (C) to (tau2);
\draw [style=inarrow, bend left=15, looseness=1.00] (tau2) to (B);
\draw [style=inarrow, bend right =15, looseness=1.00] (tau2) to (B);
\end{pgfonlayer}
\end{tikzpicture}
\]
we have $S = \{a,b,c\}$, $T = \{\tau_1, \tau_2\}$, and
\[
\begin{array}{ll} s(\tau_1) = a+b & t(\tau_1) = c \\
s(\tau_2) = c & t(\tau_2) = 2b.
\end{array}
\]
The term `species' is used in applications of Petri nets to chemistry. Since the concept of `catalyst' also arose in chemistry, we henceforth use the term `species' rather than `places'.
A Petri net morphism from the Petri net $P$ to the Petri net $P'$ is a pair of functions ($f \maps T \to T'$, $g \maps S \to S'$) such that the following diagrams commute:
\[
\begin{tikzcd}
\arrow[r, "s"]
\arrow[d, "f", swap]
\N[S]
\arrow[d, "\N\lbrack g \rbrack"]
\\
\arrow[r, swap, "s'"]
\N[S']
\end{tikzcd}
\quad
\begin{tikzcd}
\arrow[r, "t"]
\arrow[d, "f", swap]
\N[S]
\arrow[d, "\N\lbrack g \rbrack"]
\\
\arrow[r, swap, "t'"]
\N[S']
\end{tikzcd}
\]
Let $\Petri$ denote the category of Petri nets and Petri net morphisms with composition defined by
\[(f, g) \circ (f', g') = (f \circ f', g \circ g').\]
A commutative monoidal category is a commutative monoid object in $(\Cat, \times)$. Let $\CMon\Cat$ denote the category of commutative monoid objects in $(\Cat,\times)$.
More concretely, a commutative monoidal category is a strict monoidal category for which $a \otimes b = b \otimes a$ for all pairs of objects and all pairs of morphisms, and the braid isomorphism $a \otimes b \to b \otimes a$ is the identity map.
Every Petri net $P = \left( s, t \maps T \to \N[S] \right)$ gives rise to a commutative monoidal category $FP$ as follows. We take the commutative monoid of objects $\Ob(FP)$ to be the free commutative monoid on $S$. We construct the commutative monoid of morphisms $\Mor(FP)$ as follows. First we generate morphisms recursively:
* for every transition $\tau \in T$ we include a morphism $\tau \maps s(\tau) \to t(\tau)$;
* for any object $a$ we include a morphism $1_a \maps a \to a$;
* for any morphisms $f \maps a \to b$ and $g \maps a' \to b'$ we include a morphism denoted $f+g \maps a +a' \to b +b'$ to serve as their tensor product;
* for any morphisms $f \maps a \to b$ and $g \maps b \to c$ we include a morphism $g\circ f \maps a \to c$ to serve as their composite.
Then we quotient by an equivalence relation on morphisms that imposes the laws of a commutative monoidal category, obtaining the commutative monoid $\Mor(FP)$.
Similarly, morphisms between Petri nets give morphisms between their commutative monoidal categories. Given a Petri net morphism
\[
\begin{tikzcd}
\arrow[r, shift left = 1]
\arrow[r, shift right = 1]
\arrow[d, "f", swap]
\N[S]
\arrow[d, "\N\lbrack g\rbrack"]
\\
\arrow[r, shift left = 1]
\arrow[r, shift right = 1]
\N[S']
\end{tikzcd}
\]
we define the functor $F(f, g) \maps FP \to FP'$ to be $\N[g]$ on objects, and on morphisms to be the unique map extending $f$ that preserves identities, composition, and the tensor product. This functor is strict symmetric monoidal.
There is a functor $F \maps \Petri \to \CMon\Cat$ defined as above.
This is straightforward; the proof that $F$ is a left adjoint is harder <cit.>, but we do not need this here.
§ CATALYSTS
One thinks of a transition $\tau$ of a Petri net as a process that consumes the source species $s(\tau)$ and produces the target species $t(\tau)$. An example of something that can be represented by a Petri net is a chemical reaction network <cit.>. Indeed, this is why Carl Petri originally invented them. A `catalyst' in a chemical reaction is a species that is necessary for the reaction to occur, or helps lower the activation energy for reaction, but is neither increased nor depleted by the reaction. We use a modest generalization of this notion, defining a catalyst in a Petri net to be a species that is neither increased nor depleted by any transition in the Petri net.
Given a Petri net $s, t \maps T \to \N[S]$, recall that for any marking $a \in \N[S]$ we have
\[ a = \sum_{x \in S} a_x x \]
for certain coefficients $a_x \in \N$. Thus, for any transition $\tau$ of a Petri net, $s(\tau)_x$ is the coefficient of the place $x$ in the source of $\tau$, while $t(\tau)_x$ is its coefficient in the target of $\tau$.
A species $x \in S$ in a Petri net $P = (s, t \maps T \to \N[S])$ is called a catalyst if $s(\tau)_x = t(\tau)_x$ for every transition $\tau \in T$. Let $S_{\mathrm{cat}} \subseteq S$ denote the set of catalysts in $P$.
A Petri net with catalysts is a Petri net $P = (s, t \maps T \to \N[S])$ with a chosen subset $C \subseteq S_{\mathrm{cat}}$. We denote a Petri net $P$ with catalysts $C$ as $(P, C)$.
Suppose we have a Petri net with catalysts $(P, C)$. Recall that the set of objects of $FP$ is the free commutative monoid $\N[C]$. We have a natural isomorphism
\[
\N[S] \cong \N[C] \times \N[S \setminus C].
\]
We write
\[
\pi_C \maps \N[S] \to \N[C]
\]
for the projection. Given any object $a \in FP$, $\pi_C(a)$ says how many catalysts of each species in $C$ occur in $a$.
Given a Petri net with catalysts $(P, C)$ and any $i \in \N[C]$, let $FP_i$ be the full subcategory of $FP$ whose objects are objects $a \in FP$ with $\pi_C (a) = i$.
Morphisms in $FP_i$ describe processes that the Petri net can carry out with a specific fixed amount of every catalyst. Since no transition in $P$ creates or destroys any catalyst, if $f \maps a \to b$ is a morphism in $FP$ then
\[
\pi_C(a) = \pi_C(b) .
\]
Thus, $FP$ is the coproduct of all the
subcategories $FP_i$:
\[
FP \cong \coprod_{i \in \N[C]} FP_i
\]
as categories. The subcategories $FP_i$ are not generally monoidal subcategories because if $a, b \in FP$ and $a+b$ is their tensor product then
\[
\pi_C(a+b) = \pi_C(a) + \pi_C(b)
\]
so for any $i, j \in \N[C]$ we have
\[
a \in FP_i, \; b \in FP_j \Rightarrow a + b \in FP_{i + j}
\]
and similarly for morphisms.
Thus, we can think of $FP$ as a commutative monoidal category `graded' by $\N[C]$. But note we are free to reinterpret any process as using a greater amount of various catalysts, by tensoring it with identity morphism on this additional amount of
catalysts. That is, given any morphism in $FP_i$, we can always tensor it with the identity on $j$ to get a morphism in $FP_{i+j}$.
Since $\N[C]$ is a commutative monoid we can think of it as a commutative monoidal category with only identity morphisms, and we freely do this in what follows. Network models rely on a similar but less trivial way of constructing a symmetric monoidal category from a set $C$. Namely, for any set $C$ there is a category $\S(C)$ for which:
* Objects are formal expressions of the form
\[
c_1 \otimes \cdots \otimes c_n
\]
for $n \in \N$ and $c_1, \dots, c_n \in C$. When $n = 0$ we write this expression as $I$.
* There exist morphisms
\[
f \maps c_1 \otimes \cdots \otimes c_m \to c'_1 \otimes \cdots \otimes c'_n
\]
only if $m = n$, and in that case a morphism is a permutation $\sigma \in S_n$ such that $c'_{\sigma(i)} = c_i$ for all $i=1, \dots , n$.
* Composition is the usual composition of permutations.
In short, an object of $\S(C)$ is a list of catalysts, possibly empty, and allowing repetitions. A morphism is a permutation that maps one list to another list.
As shown in <ref>, $\S(C)$ is the free strict symmetric monoidal category on the set $C$. There is thus a strict symmetric monoidal functor
\[
p \maps \S(C) \to \N[C]
\]
sending each object $c_1 \otimes \cdots \otimes c_n$ to the object $c_1 + \cdots + c_n$, and sending every morphism to an identity morphism. This can also be seen directly. In what follows, we use this functor $p$ to construct a lax symmetric monoidal functor $G \maps \S(C) \to \Cat$, where $\Cat$ is made symmetric monoidal using its cartesian product.
Given a Petri net with catalysts $(P, C)$, there exists a unique functor $G \maps \S(C) \to \Cat$ sending each object $x \in \S(C)$ to the category $FP_{p(x)}$ and each morphism in $\S(C)$ to an identity functor.
The uniqueness is clear. For existence, note that since $\N[C]$ has only identity morphisms there is a functor $H \maps \N[C] \to \Cat$ sending each object $x \in \N[C]$ to the category $FP_{p(x)}$. If we compose $H$ with the functor $p \maps \S(C) \to \N[C]$ described above we obtain the functor $G$.
The functor $G \maps \S(C) \to \Cat$ becomes lax symmetric monoidal with the lax structure map
\[
\Phi_{x, y} \maps FP_{p(x)} \times FP_{p(y)} \to FP_{p(x \otimes y)}
\]
given by the tensor product in $FP$, and the map
\[
\phi \maps 1 \to FP_0
\]
sending the unique object of the terminal category $1 \in \Cat$ to the unit for the tensor product in $FP$, which is the object $0 \in FP_0$.
Recall that $G$ is the composite of $p \maps \S(C) \to \N[C]$ and $H \maps \N[C] \to \Cat$. The functor $p$ is strict symmetric monoidal. The functor $p$ is strict symmetric monoidal. One can check that the functor $H$ becomes lax symmetric monoidal if we equip it with the lax structure map
\[
FP_i \times FP_j \to FP_{i+j}
\]
given by the tensor product in $FP$, and the map
\[
1 \to FP_0
\]
sending the unique object of $1 \in \Cat$ to the unit for the tensor product in $FP$, namely $0 \in \N[S] = \Ob(FP)$. Composing the lax symmetric monoidal functor $H$ and with the strict symmetric monoidal functor $p$, we obtain the lax symmetric monoidal functor $G$ described in the theorem statement.
We defined $C$-colored network model in <ref> to be a lax symmetric monoidal functor from $\S(C)$ to $\Cat$.
We call the $C$-colored network model $G \maps \S(C) \to \Cat$ of Theorem <ref> the Petri network model associated to the Petri net with catalysts $(P, C)$.
The following Petri net $P$ has species $S = \{a, b, c, d, e\}$ and transitions $T = \{\tau_1, \tau_2\}$:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=catalyst] (A) at (-2, 2) {$\;\;a\;\;$};
\node [style=catalyst] (B) at (2, 2) {$\;\;b\;\;$};
\node [style=species] (C) at (-4, 0.6) {$\;\;c\;\;$};
\node [style=species] (D) at (0, 0.6) {$\;\;d\;\;$};
\node [style=species] (E) at (4, 0.6) {$\;\;e\;\;$};
\node [style=transition] (tau1) at (-2, 0.6) {$\;\phantom{\Big{|}}\tau_1\;$};
\node [style=transition] (tau2) at (2, 0.6) {$\;\phantom{\Big{|}}\tau_2\;$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow, bend right=70, looseness=1.00, red] (A) to (tau1);
\draw [style=inarrow, bend right=70, looseness=1.00, red] (tau1) to (A);
\draw [style=inarrow, bend right=70, looseness=1.00, red] (B) to (tau2);
\draw [style=inarrow, bend right=70, looseness=1.00, red] (tau2) to (B);
\draw [style=inarrow, bend left=12, looseness=1] (C) to (tau1);
\draw [style=inarrow, bend right=12, looseness=1] (C) to (tau1);
\draw [style=inarrow, bend left=12, looseness=1] (tau1) to (D);
\draw [style=inarrow, bend right=12, looseness=1] (tau1) to (D);
\draw [style=inarrow] (D) to (tau2);
\draw [style=inarrow] (tau2) to (E);
\end{pgfonlayer}
\end{tikzpicture}
\]
Species $a$ and $b$ are catalysts, and the rest are not. We thus can take $C = \{a, b\}$ and obtain a Petri net with catalysts $(P, C)$, which in turn gives a Petri network model $G \maps \S(C) \to \Cat$. We outline catalyst species in red, and also draw the edges connecting them to transitions in red.
Here is one possible interpretation of this Petri net. Tokens in $\, c\, $ represent people at a base on land, tokens in $\, d\, $ are people at the shore, and tokens in $\, e\, $ are people on a nearby island. Tokens in $\, a\, $ represent jeeps, each of which can carry two people at a time from the base to the shore and then return to the base. Tokens in $\, b\, $ represent boats that carry one person at a time from the shore to the island and then return.
Let us examine the effect of the functor $G \maps \S(C) \to \Cat$ on various objects of $\S(C)$. The object $a \in \S(C)$ describes a situation where there is one jeep present but no boats. The category $G(a)$ is isomorphic to $FX$, where $X$ is this Petri net:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-4, 0.6) {$\;\;c\;\;$};
\node [style=species] (D) at (0, 0.6) {$\;\;d\;\;$};
\node [style=species] (E) at (4, 0.6) {$\;\;e\;\;$};
\node [style=transition] (tau1) at (-2, 0.6) {$\;\phantom{\Big{|}}\tau_1\;$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow, bend left=12, looseness=1] (C) to (tau1);
\draw [style=inarrow, bend right=12, looseness=1] (C) to (tau1);
\draw [style=inarrow, bend left=12, looseness=1] (tau1) to (D);
\draw [style=inarrow, bend right=12, looseness=1] (tau1) to (D);
\end{pgfonlayer}
\end{tikzpicture}
\]
That is, people can go from the base to the shore in pairs, but they cannot go to the island. Similarly, the object $b$ describes a situation with one boat present but no jeeps, and the category $G(b)$ is isomorphic to $FY$, where $Y$ is this Petri net:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-4, 0.6) {$\;\;c\;\;$};
\node [style=species] (D) at (0, 0.6) {$\;\;d\;\;$};
\node [style=species] (E) at (4, 0.6) {$\;\;e\;\;$};
\node [style=transition] (tau2) at (2, 0.6) {$\;\phantom{\Big{|}}\tau_2\;$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow] (D) to (tau2);
\draw [style=inarrow] (tau2) to (E);
\end{pgfonlayer}
\end{tikzpicture}
\]
Now people can only go from the shore to the island, one at a time.
The object $a \otimes b \in \S(C)$ describes a situation with one jeep and one boat. The category $G(a \otimes b)$ is isomorphic to $FZ$ for this Petri net $Z$:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=species] (C) at (-4, 0.6) {$\;\;c\;\;$};
\node [style=species] (D) at (0, 0.6) {$\;\;d\;\;$};
\node [style=species] (E) at (4, 0.6) {$\;\;e\;\;$};
\node [style=transition] (tau1) at (-2, 0.6) {$\;\phantom{\Big{|}}\tau_1\;$};
\node [style=transition] (tau2) at (2, 0.6) {$\;\phantom{\Big{|}}\tau_2\;$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=inarrow, bend left=12, looseness=1] (C) to (tau1);
\draw [style=inarrow, bend right=12, looseness=1] (C) to (tau1);
\draw [style=inarrow, bend left=12, looseness=1] (tau1) to (D);
\draw [style=inarrow, bend right=12, looseness=1] (tau1) to (D);
\draw [style=inarrow] (D) to (tau2);
\draw [style=inarrow] (tau2) to (E);
\end{pgfonlayer}
\end{tikzpicture}
\]
Now people can go from the base to the shore in pairs and also go from the shore to the island one at a time.
Surprisingly, an object $x \in \S(C)$ with additional jeeps and/or boats always produces a category $G(x)$ that is isomorphic to one of the three just shown: $G(a), G(b)$ and $G(a \otimes b)$. For example, consider the object $b \otimes b \in \S(C)$, where there are two boats present but no jeeps. There is an isomorphism of categories
\[
- + b \maps G(b) \to G(b \otimes b)
\]
defined as follows. Recall that $G(b) = FP_b$ and $G(b \otimes b) = FP_{b+b}$, where $FP_b$ and $FP_{b+b}$ are subcategories of $FP$. The functor
\[
- + b \maps FP_b \to FP_{b+b}
\]
sends each object $x \in FP_b$ to the object $ x + b$, and sends each morphism $f \maps x \to y$ in $FP_b$ to the morphism $1_b + f \maps b + x \to b + y$. That this defines a functor is clear; the surprising part is that it is an isomorphism. One might have thought that the presence of a second boat would enable one to carry out a given task in more different ways.
Indeed, while this is true in real life, the category $FP$ is commutative monoidal, so tokens of the same species have no `individuality': permuting them has no effect. There is thus, for example, no difference between the following two morphisms in $FP_{b+b}$:
* using one boat to transport one person from the base to shore and another boat to transport another person, and
* using one boat to transport first one person and then another.
It is useful to draw morphisms in $FP$ as string diagrams, since such diagrams serve as a general notation for morphisms in monoidal categories <cit.>. For expository treatments, see <cit.>. The rough idea is that objects of a monoidal category are drawn as labelled wires, and a morphism $f \maps x_1 \otimes \cdots \otimes x_m \to y_1 \otimes \cdots \otimes y_n$ is drawn as a box with $m$ wires coming in on top and $n$ wires coming out at the bottom. Composites of morphisms are drawn by attaching output wires of one morphism to input wires of another, while tensor products of morphisms are drawn by setting pictures side by side. In symmetric monoidal categories, the braiding is drawn as a crossing of wires. The rules governing string diagrams let us manipulate them while not changing the morphisms they denote. In the case of symmetric monoidal categories, these rules are well known <cit.>. For commutative monoidal categories there is one additional rule:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty] (a) at (0, 1) {$x$};
\node [style=empty] (b) at (1, 1) {$y$};
\node [style=empty] (c) at (0, -1) {$y$};
\node [style=empty] (d) at (1, -1) {$x$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt] (a) to (d);
\draw [line width=1.5 pt] (b) to (c);
\end{pgfonlayer}
\end{tikzpicture}
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty] (a) at (0, 1) {$x$};
\node [style=empty] (b) at (1, 1) {$y$};
\node [style=empty] (c) at (1, -1) {$y$};
\node [style=empty] (d) at (0, -1) {$x$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt] (a) to (d);
\draw [line width=1.5 pt] (b) to (c);
\end{pgfonlayer}
\end{tikzpicture}
\]
This says both that $x \otimes y = y \otimes x$ and that the braiding $\sigma_{x,y} \maps x \otimes y \to y \otimes x$ is the identity.
Here is the string diagram notation for the equation we mentioned between two morphisms in $FP$:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (b) at (0, 4) {$b$};
\node [style=empty, red] (b') at (1.5, 4) {$b$};
\node [style=empty] (d) at (3, 4) {$d$};
\node [style=empty] (d') at (4.5, 4) {$d$};
\node [style=morphism] (tau1) at (2.25, 2.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|}\;$};
\node [style=empty] (equals) at (5.5, 1) {$=$};
\node [style=morphism] (tau2) at (2.25, -0.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|\;}$};
\node [style=empty, red] (b'') at (0, -2) {$b$};
\node [style=empty, red] (b''') at (1.5, -2) {$b$};
\node [style=empty] (e) at (3, -2) {$e$};
\node [style=empty] (e') at (4.5, -2) {$e$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, red] (b) to (tau2);
\draw [line width=1.5 pt, red] (b') to (tau1);
\draw [line width=1.5 pt] (d) to (tau1);
\draw [line width=1.5 pt] (d') to (tau2);
\draw [line width=1.5 pt, red] (tau1) to (b'');
\draw [line width=1.5 pt] (tau1) to (e');
\draw [line width=1.5 pt, red] (tau2) to (b''');
\draw [line width=1.5 pt] (tau2) to (e);
\end{pgfonlayer}
\end{tikzpicture}
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (b) at (0, 4) {$b$};
\node [style=empty, red] (b') at (1.5, 4) {$b$};
\node [style=empty] (d) at (3, 4) {$d$};
\node [style=empty] (d') at (4.5, 4) {$d$};
\node [style=morphism] (tau1) at (2.25, 2.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|}\;$};
\node [style=morphism] (tau2) at (2.25, -0.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|\;}$};
\node [style=empty, red] (b'') at (0, -2) {$b$};
\node [style=empty, red] (b''') at (1.5, -2) {$b$};
\node [style=empty] (e) at (3, -2) {$e$};
\node [style=empty] (e') at (4.5, -2) {$e$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, bend left=20, looseness=2, red] (b) to (b'');
\draw [line width=1.5 pt, red] (b') to (tau1);
\draw [line width=1.5 pt] (d) to (tau1);
\draw [line width=1.5 pt] (d') to (tau2);
\draw [line width=1.5 pt, bend right =30, looseness=1.5, red] (tau1) to (tau2);
\draw [line width=1.5 pt] (tau1) to (e');
\draw [line width=1.5 pt, red] (tau2) to (b''');
\draw [line width=1.5 pt] (tau2) to (e);
\end{pgfonlayer}
\end{tikzpicture}
\]
We draw the object $b$ (standing for a boat) in red to emphasize that it serves as a catalyst. At left we are first using one boat to transport one person from the base to shore, and then using another boat to transport another person. At right we are using the same boat to transport first one person and then another, while another boat stands by and does nothing. These morphisms are equal because they differ only by the presence of the braiding $\sigma_{b,b} \maps b + b \to b + b$ in the left hand side, and this is an identity morphism.
The above example illustrates an important point: in the commutative monoidal category $FP$, permuting catalyst tokens has no effect. Next we construct a symmetric monoidal category $\int G$ in which permuting such tokens has a nontrivial effect. One reason for wanting this is that in applications, the catalyst tokens may represent agents with their own individuality. For example, when directing a boat to transport a person from base to shore, we need to say which boat should do this. For this we need a symmetric monoidal category that gives the catalyst tokens a nontrivial braiding.
To create this category, we use the symmetric monoidal Grothendieck construction of <ref>. Given any symmetric monoidal category $X$ and any lax symmetric monoidal functor $F \maps X \to \Cat$, this construction gives a symmetric monoidal category $\int F$ equipped with a functor (indeed an opfibration) $\int F \to X$. In <ref>, we used this construction to build an operad from any network model, whose operations are ways to assemble larger networks from smaller ones. Now this construction has a new significance.
Starting from a Petri network model $G \maps \S(C) \to \Cat$, the symmetric monoidal Grothendieck construction gives a symmetric monoidal category $\int G$ in which:
* an object is a pair $(x, a)$ where $x \in \S(C)$ and $a \in FP_{p(x)}$.
* a morphism from $(x, a)$ to $(x', a')$ is a pair $(\sigma, f)$ where $\sigma \maps x \to x'$ is a morphism in $\S(C)$ and $f \maps a \to a'$ is a morphism in $FP$.
* morphisms are composed componentwise.
* the tensor product is computed componentwise: in particular, the tensor product of objects $(x, a)$ and $(x', a')$ is $(x \otimes x', a + a')$.
* the associators, unitors and braiding are also computed componentwise (and hence are trivial in the second component, since $FP$ is a commutative monoidal category).
The functor $\int G \to \S(C)$ simply sends each pair to its first component.
This is simpler than one typically expects from the Grothendieck construction. There are two main reasons: first, $G$ maps every morphism in $\S(C)$ to an identity morphism in $\Cat$, and second, the lax structure map for $G$ is simply the tensor product in $FP$. However, this construction still has an important effect: it makes the process of switching two tokens of the same catalyst species into a nontrivial morphism in $\int G$.
More formally, we have:
If $G \maps \S(C) \to \Cat$ is the Petri network model associated to the Petri net with catalysts $(P, C)$, then $\int G$ is equivalent, as a symmetric monoidal category, to the full subcategory of $\S(C) \times FP$ whose objects are those of the form $(x, a)$ with $x \in \S(C)$ and $a \in FP_{p(x)}$.
One can read this off from the description of $\int G$ given above.
The difference between $\int G$ and $FP$ is that the former category keeps track of processes where catalyst tokens are permuted, while the latter category treats them as identity morphisms. In the terminology of Glabbeek and Plotkin, $\int G$ implements the `individual token philosophy' on catalysts, in which permuting tokens of the same catalyst is regarded as having a nontrivial effect <cit.>. By contrast, $FP$ implements the `collective token philosophy', where all that matters is the number of tokens of each catalyst, and permuting them has no effect.
There is a map from $\int G$ to $FP$ that forgets the individuality of the catalyst tokens. A morphism in $\int G$ is a pair $(\sigma, f)$ where $\sigma \maps x \to x'$ is a morphism in $\S(C)$ and $f \maps a \to a'$ is a morphism in $FP$ with $a \in G(x), a' \in G(x')$. There is a symmetric monoidal functor
\[
\textstyle{\int} G \to FP
\]
that discards this extra information, mapping $(\sigma, f)$ to $f$. The symmetric monoidal Grothendieck construction also gives a symmetric monoidal opfibration
\[ \textstyle{\int} G \to \S(C) \]
which maps $(\sigma, f)$ to $\sigma$, by <ref>.
Let $(P, C)$ be the Petri net with catalysts in Ex. <ref>, and $G \maps \S(C) \to \Cat$ the resulting Petri network model. In $\int G$ the following two morphisms are not equal:
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (b) at (0, 4) {$b$};
\node [style=empty, red] (b') at (1.5, 4) {$b$};
\node [style=empty] (d) at (3, 4) {$d$};
\node [style=empty] (d') at (4.5, 4) {$d$};
\node [style=morphism] (tau1) at (2.25, 2.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|}\;$};
\node [style=morphism] (tau2) at (2.25, -0.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|\;}$};
\node [style=empty, red] (b'') at (0, -2) {$b$};
\node [style=empty, red] (b''') at (1.5, -2) {$b$};
\node [style=empty] (e) at (3, -2) {$e$};
\node [style=empty] (e') at (4.5, -2) {$e$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, red] (b) to (tau2);
\draw [line width=1.5 pt, red] (b') to (tau1);
\draw [line width=1.5 pt] (d) to (tau1);
\draw [line width=1.5 pt] (d') to (tau2);
\draw [line width=1.5 pt, red] (tau1) to (b'');
\draw [line width=1.5 pt] (tau1) to (e');
\draw [line width=1.5 pt, red] (tau2) to (b''');
\draw [line width=1.5 pt] (tau2) to (e);
\end{pgfonlayer}
\end{tikzpicture}
\ne
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (b) at (0, 4) {$b$};
\node [style=empty, red] (b') at (1.5, 4) {$b$};
\node [style=empty] (d) at (3, 4) {$d$};
\node [style=empty] (d') at (4.5, 4) {$d$};
\node [style=morphism] (tau1) at (2.25, 2.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|}\;$};
\node [style=morphism] (tau2) at (2.25, -0.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|\;}$};
\node [style=empty, red] (b'') at (0, -2) {$b$};
\node [style=empty, red] (b''') at (1.5, -2) {$b$};
\node [style=empty] (e) at (3, -2) {$e$};
\node [style=empty] (e') at (4.5, -2) {$e$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, bend left=20, looseness=2, red] (b) to (b'');
\draw [line width=1.5 pt, red] (b') to (tau1);
\draw [line width=1.5 pt] (d) to (tau1);
\draw [line width=1.5 pt] (d') to (tau2);
\draw [line width=1.5 pt, bend right =30, looseness=1.5, red] (tau1) to (tau2);
\draw [line width=1.5 pt] (tau1) to (e');
\draw [line width=1.5 pt, red] (tau2) to (b''');
\draw [line width=1.5 pt] (tau2) to (e); (a) to (f);
\end{pgfonlayer}
\end{tikzpicture}
\]
because the braiding of catalyst species in $\int G$ is nontrivial. This says that in $\int G$ we consider these two processes as different:
* using one boat to transport one person from the base to shore and another boat to transport another person, and
* using one boat to transport first one person and then another.
On the other hand, in $\int G$ we have
\[
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (b) at (0, 4) {$b$};
\node [style=empty, red] (b') at (1.5, 4) {$b$};
\node [style=empty] (d) at (3, 4) {$d$};
\node [style=empty] (d') at (4.5, 4) {$d$};
\node [style=morphism] (tau1) at (2.25, 2.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|}\;$};
\node [style=morphism] (tau2) at (2.25, -0.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|\;}$};
\node [style=empty, red] (b'') at (0, -2) {$b$};
\node [style=empty, red] (b''') at (1.5, -2) {$b$};
\node [style=empty] (e) at (3, -2) {$e$};
\node [style=empty] (e') at (4.5, -2) {$e$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, red] (b) to (tau2);
\draw [line width=1.5 pt, red] (b') to (tau1);
\draw [line width=1.5 pt] (d) to (tau1);
\draw [line width=1.5 pt] (d') to (tau2);
\draw [line width=1.5 pt, red] (tau1) to (b'');
\draw [line width=1.5 pt] (tau1) to (e');
\draw [line width=1.5 pt, red] (tau2) to (b''');
\draw [line width=1.5 pt] (tau2) to (e);
\end{pgfonlayer}
\end{tikzpicture}
\vcenter{\hbox{\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (b) at (0, 4) {$b$};
\node [style=empty, red] (b') at (1.5, 4) {$b$};
\node [style=empty] (d) at (3, 4) {$d$};
\node [style=empty] (d') at (4.5, 4) {$d$};
\node [style=morphism] (tau1) at (2.5, 2.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|}\;$};
\node [style=morphism] (tau2) at (2.5, -0.2) {$\;\phantom{\Big|}\tau_2\phantom{\Big|\;}$};
\node [style=empty, red] (b'') at (0, -2) {$b$};
\node [style=empty, red] (b''') at (1.5, -2) {$b$};
\node [style=empty] (e) at (3.5, -2) {$e$};
\node [style=empty] (e') at (4.75, -2) {$e$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, red] (b) to (tau2);
\draw [line width=1.5 pt, red] (b') to (tau1);
\draw [line width=1.5 pt, bend left=45, looseness=1] (d) to (tau2);
\draw [line width=1.5 pt] (d') to (tau1);
\draw [line width=1.5 pt, red] (tau1) to (b'');
\draw [line width=1.5 pt] (tau1) to (e');
\draw [line width=1.5 pt, red] (tau2) to (b''');
\draw [line width=1.5 pt] (tau2) to (e);
\end{pgfonlayer}
\end{tikzpicture}
\]
because these morphisms differ only by two people on the shore switching place before they board the boats, and the braiding of non-catalyst species is the identity. In short, the $\int G$ construction implements the individual token philosophy only for catalyst tokens; tokens of other species are governed by the collective token philosophy.
§ PREMONOIDAL CATEGORIES
We have seen that for a Petri net $P$, a choice of catalysts $C$ lets us write the category $FP$ as a coproduct of subcategories $FP_i$, one for each possible amount $i \in \N[C]$ of the catalysts. The subcategory $FP_i$ is only a monoidal subcategory when $i = 0$. Indeed, only $FP_0$ contains the monoidal unit of $FP$. However, we shall see that each subcategory $FP_i$ can be given the structure of a premonoidal category, as defined by Power and Robinson <cit.>. We motivate our use of this structure by describing two failed attempts to make $FP_i$ into a monoidal category.
Given two morphisms in $FP_i$ we typically cannot carry out these two processes simultaneously, because of the limited availability of catalysts. But we can do first one and then the other. For example, imagine that two people are trying to walk through a doorway, but the door is only wide enough for one person to walk through. The door is a resource that is not depleted by its use, and thus a catalyst. Both people can use the door, but not at the same time: they must make an arbitrary choice of who goes first.
We can attempt to define a tensor product on $FP_i$ using this idea. Fix some amount of catalysts $i \in \N[C]$. Objects of $FP_i$ are of the form $i + a$ with $a \in \N[S - C]$. On objects we define
\[(i + a) \otimes_i (i + a' ) = i + a + a'.\]
The unit object for $\otimes_i$ is therefore $i + 0$, or simply $i$. For morphisms
\begin{align*}
f &\maps i + a \to i + b \\
f' &\maps i + a' \to i + b'
\end{align*}
we define
\[
f \otimes_i f' = (f + 1_{b'}) \circ (1_{a} + f').
\]
The tensor product
$f \otimes_i f' = (f + 1_{b'}) \circ (1_{a} + f') $
of morphisms in $FP_i$ involves an arbitrary choice: namely, the choice to do $f'$ first. This is perhaps clearer if we draw this morphism as a string diagram in $FP$.
\[
\vcenter{\hbox{\scalebox{1}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (i) at (0, -3) {$i$};
\node [style=empty] (a) at (1.5, 1.5) {$a$};
\node [style=empty] (a') at (3.5, 1.5) {$a'$};
\node [style=empty] (m) at (2.5, -0.75) {};
\node [style=morphism] (f) at (1.5, -1.5) {$\;\phantom{\Big|}f\phantom{\Big|}\;$};
\node [style=morphism] (f') at (3.5, 0) {$\;\phantom{\Big|}f'\phantom{\Big|\;}$};
\node [style=empty] (b) at (1.5, -3) {$b$};
\node [style=empty] (b') at (3.5, -3) {$b'$};
\node [style=empty, red] (i') at (5, 1.5) {$i$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, bend left = 40, looseness=1, red] (i') to (f');
\draw [line width=1.5 pt] (a') to (f');
\draw [line width=1.5 pt] (a) to (f);
\draw [line width=1.5 pt, bend right = 25, red] (f) to (m.center);
\draw [line width=1.5 pt, bend left = 25, red] (m.center) to (f');
\draw [line width=1.5 pt] (f') to (b');
\draw [line width=1.5 pt, bend right = 40, looseness=1, red] (f) to (i);
\draw [line width=1.5 pt] (f) to (b);
\end{pgfonlayer}
\end{tikzpicture}
\]
If instead we choose to do $f$ first, we can define a tensor product ${}_i \otimes$ which is the same on objects but given on morphisms by
\[
f \, {}_i \!\otimes f' = (1_b + f') \circ (f + 1_{a'}) .
\]
It looks like this:
\[
\vcenter{\hbox{\scalebox{1}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty, red] (i) at (0, 1.5) {$i$};
\node [style=empty] (a) at (1.5, 1.5) {$a$};
\node [style=empty] (a') at (3.5, 1.5) {$a'$};\node [style=empty] (m) at (2.5, -0.75) {};
\node [style=morphism] (f) at (1.5, 0) {$\;\phantom{\Big|}f\phantom{\Big|}\;$};
\node [style=morphism] (f') at (3.5, -1.5) {$\;\phantom{\Big|}f'\phantom{\Big|\;}$};
\node [style=empty] (b) at (1.5, -3) {$b$};
\node [style=empty] (b') at (3.5, -3) {$b'$};
\node [style=empty, red] (i') at (5, -3) {$i$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt, bend right = 40, looseness=1, red] (i) to (f);
\draw [line width=1.5 pt] (a') to (f');
\draw [line width=1.5 pt] (a) to (f);
\draw [line width=1.5 pt, bend left = 25, red] (f) to (m.center);
\draw [line width=1.5 pt, bend right = 25, red] (m.center) to (f');
\draw [line width=1.5 pt] (f') to (b');
\draw [line width=1.5 pt, bend left = 40, looseness=1, red] (f') to (i');
\draw [line width=1.5 pt] (f) to (b);
\end{pgfonlayer}
\end{tikzpicture}
\]
Unfortunately, neither of these tensor products makes $FP_i$ into a monoidal category! Each makes the set of objects $\Ob(FP_i)$ and the set of morphisms $\Mor(FP_i)$ into a monoid in such a way that the source and target maps $s,t \maps \Mor(FP_i) \to \Ob(FP_i)$, as well as the identity-assigning map $i \maps \Ob(FP_i) \to \Mor(FP_i)$, are monoid homomorphisms. The problem is that neither obeys the interchange law, so neither of these tensor products defines a functor from $FP_i \times FP_i$ to $FP_i$. For example,
\[
(1 \otimes_i f')\circ (f \otimes_i 1) \ne (f \otimes_i 1) \circ (1 \otimes_i f') .
\]
The other tensor product suffers from the same problem.
What is going on here? It turns out that $FP_i$ is a `strict premonoidal category'. While these structures first arose in computer science <cit.>, they are also mathematically natural, for the following reason. There are only two symmetric monoidal closed structures on $\Cat$, up to isomorphism <cit.>. One is the the cartesian product. The other is the `funny tensor product' <cit.>. A monoid in $\Cat$ with its cartesian product is a strict monoidal category, but a monoid in $\Cat$ with its funny tensor product is a strict premonoidal category. The funny tensor product $\C \square \D$ of categories $\C$ and $\D$ is defined as the following pushout in $\Cat$:
\[\begin{tikzcd}
\C_0 \times \D_0
\arrow[d, "i \times 1", swap]
\arrow[r, "1 \times j"]
\C_0 \times \D
\arrow[d]
\\
\C \times \D_0
\arrow[r]
\C \square \D
\end{tikzcd}\]
Here $\C_0$ is the subcategory of $\C$ consisting of all the objects and only identity
morphisms, $i \maps \C_0 \to \C$ is the inclusion, and similarly for $j \maps \D_0 \to \D$.
Thus, given morphisms $f \maps x \to y$ in $\C$ and $f' \maps x' \to y'$ in $\C$, the category $C \square D$ in contains a square of the form
\[ \begin{tikzcd}
x \square x'
\arrow[d, swap, "f \square 1"] \arrow[r, "1 \square f' "]
x \square y'
\arrow[d, "f \square 1"]
\\
x' \square y
\arrow[r, swap, "1 \square f' "]
x' \square y',
\end{tikzcd}\]
but in general this square does not commute, unlike the corresponding square in
$\C \times \D$.
A strict premonoidal category is a category $\C$ equipped with a functor
$\boxtimes \maps \C \square \C \to \C$ that obeys the associative law and an object $I \in \C$ that serves as a left and right unit for $\boxtimes$.
Given two morphisms $f \maps x \to y$, $f' \maps x' \to y'$ in a strict premonoidal category $\C$ we obtain a square
\[\begin{tikzcd}
x \boxtimes x'
\arrow[d, swap, "f \boxtimes 1"] \arrow[r, "1 \boxtimes f' "]
x \boxtimes y'
\arrow[d, "f \boxtimes 1"]
\\
x' \boxtimes y
\arrow[r, swap, "1 \boxtimes f' "]
x' \boxtimes y',
\end{tikzcd}\]
but this square may not commute. There are thus two candidates for a morphism from
$x \boxtimes x'$ to $y \boxtimes y'$. When these always agree, we can make $\C$
monoidal by setting $f \boxtimes f'$ equal to either (and thus both) of these candidates. We shall give $FP_i$ a strict premonoidal structure where these two candidates do not agree: one is $f \otimes_i f'$ while the other is $f \, {}_i \! \otimes f'$. This explains the meaning of these two failed attempts to give $FP_i$ a monoidal structure.
Thanks to the description of $\C \square \C$ as a pushout, to know the tensor product $\boxtimes$ in a strict premonoidal category $\C$ it suffices to know $x \boxtimes y$, $x \boxtimes f$ and $f \boxtimes y$ for all objects $x,y$ and morphisms $f$ of $\C$. (Here we find it useful to write
$x \boxtimes f$ for $1_x \boxtimes f$ and $f \boxtimes y$ for $f \boxtimes 1_y$.) In the case at hand, we define
\[ \boxtimes_i \maps FP_i \square FP_i \to FP_i \]
on objects by setting
\[(i + a) \boxtimes_i (i + a' ) = i + a + a'\]
for all $a,a' \in \N[S-C]$, while for morphisms
\begin{align*}
f &\maps i + a \to i + b \\
f' &\maps i + a' \to i + b'
\end{align*}
we set
\[
a \boxtimes f' = f' + 1_a, \qquad
f \boxtimes a' = f + 1_{a'}.
\]
The tensor product $\boxtimes_i$ makes $FP_i$ into a strict premonoidal category.
This can be checked directly, but this is also a special case of a construction in Power and Robinson's paper on premonoidal categories <cit.>. They describe a construction, sometimes called `linear state passing' <cit.>, that takes any object $i$ in any symmetric monoidal category $C$ and yields a premonoidal category $C_i$ where objects are of the form $i \otimes c$ for $c \in C$ and morphisms are morphisms in $C$ of the form $f \maps i \otimes c \to i \otimes c'$. We are considering the special case where $C = FP$, and because $FP$ is commutative monoidal the resulting premonoidal category is strict: all the coherence isomorphisms are identities.
Using the description of $FP_i \square FP_i$ as a pushout, to show $\boxtimes_i
\maps FP_i \square FP_i \to FP_i$ is a well-defined functor it suffices to check these equations:
\begin{align*}
s(f \boxtimes_i 1_{i+a'}) &= s(f) \boxtimes_i a' \\
t(f \boxtimes_i 1_{i+a'}) &= t(f) \boxtimes_i a' \\
(f \circ g) \boxtimes_i 1_{i+a'} &= (f \boxtimes_i 1_{i+a'}) \circ (g \boxtimes_i 1_{i+a'}) \\
1_{(i+a) \boxtimes_i (i+a')} &= 1_{i+a} \boxtimes 1_{i+a'}
\end{align*}
and the analogous equations with the factors switched. To check associativity of $\boxtimes_i$, given morphisms
\begin{align*}
f &\maps i + a \to i + b
\\
f' &\maps i + a' \to i + b'
\\
f'' &\maps i + a'' \to i + b''
\end{align*}
it suffices to check that
\begin{align*}
(a \boxtimes_i a') \boxtimes_i a'' &= a \boxtimes_i (a' \boxtimes_i a'') \\
(f \boxtimes_i a') \boxtimes_i a'' &= f \boxtimes_i (a' \boxtimes_i a'') \\
(a \boxtimes_i f') \boxtimes_i a'' &= a \boxtimes_i (f' \boxtimes_i a'') \\
(a \boxtimes_i a') \boxtimes_i f'' &= a \boxtimes_i (a' \boxtimes_i f'') .
\end{align*}
To complete the proof it suffices to show that the object $i \in FP_i$ serves as a left
and right unit for $\boxtimes_i$, which amounts to checking these equations:
\[ i \boxtimes_i (i+a) = i+a = (i+a) \boxtimes_i i \]
\[ i \boxtimes f = f = f \boxtimes_i i .\]
All these calculations are straightforward.
A curious feature of this monoidal category $FP_i$ is that there is in general no way to extend it to a symmetric, or even braided, monoidal category. To see this, note that the only isomorphisms in $FP$ are identities: this follows from how the morphisms in $FP$ are recursively generated, with the only equations imposed being the laws of a commutative monoidal category. Thus, the only option for a braiding on $FP_i$, say
\[
B_{i+a, i+b} \maps (i + a) \otimes_i (i + b) \to (i + b) \otimes_i (i + a),
\]
is the identity. Since the tensor product of objects in $FP_i$ is commutative, this is a plausible candidate. However, given morphisms
\begin{align*}
f &\maps i + a \to i + b \\
f'&\maps i + a'\to i + b'
\end{align*}
we in general do not have $f \otimes f' = f' \otimes f$, as would be required by naturality of the braiding if the braiding were the identity. This can be checked in examples with the help of string diagrams:
\[
\scalebox{0.8}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty] (i) at (0, 4) {$i$};
\node [style=empty] (a) at (1.5, 4) {$a$};
\node [style=empty] (a') at (3, 4) {$a'$};
\node [style=morphism] (f') at (1.2, 2.2) {$\;\phantom{\Big|}f'\phantom{\Big|}\;$};
\node [style=empty] (i2) at (0.9, 1) {$i$};
\node [style=morphism] (f) at (1.2, -0.2) {$\;\phantom{\Big|}f\phantom{\Big|\;}$};
\node [style=empty] (i3) at (0, -2) {$i$};
\node [style=empty] (b) at (1.5, -2) {$b$};
\node [style=empty] (b') at (3, -2) {$b'$};
\node [style = empty] at (4, 1){{$\ne$}};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt] (i) to (f');
\draw [line width=1.5 pt] (a') to (f');
\draw [line width=1.5 pt] (f') to (f);
\draw [line width=1.5 pt] (f') to (b');
\draw [line width=1.5 pt, bend left=45, looseness=1] (a) to (f);
\draw [line width=1.5 pt] (f) to (i3);
\draw [line width=1.5 pt] (f) to (b);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=empty] (i) at (0, 4) {$i$};
\node [style=empty] (a) at (1.5, 4) {$a$};
\node [style=empty] (a') at (3, 4) {$a'$};
\node [style=morphism] (f') at (1.2, 2.2) {$\;\phantom{\Big|}f\phantom{\Big|}\;$};
\node [style=empty] (i2) at (0.9, 1) {$i$};
\node [style=morphism] (f) at (1.2, -0.2) {$\;\phantom{\Big|}f'\phantom{\Big|\;}$};
\node [style=empty] (i3) at (0, -2) {$i$};
\node [style=empty] (b) at (1.5, -2) {$b$};
\node [style=empty] (b') at (3, -2) {$b'$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [line width=1.5 pt] (i) to (f');
\draw [line width=1.5 pt] (a') to (f);
\draw [line width=1.5 pt] (f') to (f);
\draw [line width=1.5 pt, bend left=45, looseness=1] (f') to (b);
\draw [line width=1.5 pt] (a) to (f');
\draw [line width=1.5 pt] (f) to (i3);
\draw [line width=1.5 pt] (f) to (b');
\end{pgfonlayer}
\end{tikzpicture}
\]
Finally, we show that the tensor products $\boxtimes_i$ on the categories $FP_i$ let us lift our network model $G$ from $\Cat$ to the category of strict premonoidal categories.
Let $\PreMonCat$ be the category of strict premonoidal categories and strict premonoidal functors, meaning functors between strict premonoidal categories that strictly preserve the tensor product. Let $U \maps \PreMonCat \to \Cat$ denote the forgetful functor which sends a strict premonoidal category to its underlying category.
The network model $G \maps \S(C) \to \Cat$ lifts to a functor $\hat G \maps \S(C) \to \PreMonCat$:
\[\begin{tikzcd}
\PreMonCat
\arrow[d, "U"]
\\
\S(C)
\arrow[r, "G", swap]
\arrow[ur, "\hat G"]
\Cat
\end{tikzcd}\]
where $\hat G(x) = FP_{p(x)}$ with the strict premonoidal structure described in Prop. <ref>.
Since $G$ sends each morphism in $\S(C)$ to an identity functor, so must $\hat G$.
CHAPTER: MONOIDAL GROTHENDIECK CONSTRUCTION
§ INTRODUCTION
The Grothendieck construction <cit.> exhibits one of the most fundamental relations in category theory, namely the equivalence between contravariant pseudofunctors into $\Cat$ and fibrations. In previous chapters, we have described how the to construct a total category, denoted $\int F$, from a functor of the form $F \maps \X\op \to \Cat$. Actually, we really could have been using pseudofunctors, since $\Cat$ is more naturally thought of as a 2-category. We refer to pseudofunctors of the form $F \maps \X\op \to \Cat$ as indexed categories. The construction of $\int F$ from a given indexed category essentially forgets the distinction between the categories $Fx$ for $x \in \X$, and incorporates the functors $Ff \maps Fy \to Fx$ as maps between the objects of $Fy$ and $Fx$. The distinction between these categories could be remembered via a fibration, a special sort of functor $P \maps \int F \to \X$, which tells you how to take preimage categories of the objects, $P\inv(x)$, and turn certain maps in $\int F$ into functors between the preimage categories. For a general fibration $P \maps \A \to \X$, the category $\X$ is called the base category and the category $\A$ is called the total category. For an object $x \in \X$, the preimage category $P\inv(x)$ is called the fibre of $P$ over $x$. A fibration is precisely what is needed to reconstruct all the data in the indexed category from its total category. Indeed, the Grothendieck construction gives an equivalence between the 2-categories $\ICat$ of indexed categories, and $\Fib$ of fibrations. This equivalence allows us to move between the worlds of indexed categories and fibred categories, providing access to tools and results from both. We recall the basic theory of fibrations, indexed categories, and the Grothendieck construction in <ref>.
Due to the importance of the Grothendieck construction, it is only natural that one would be interested in extra structure these objects may have, and how the correspondence extends. In particular, a version which handles monoidal structures on the various categories in play could potentially be very useful, as monoidal categories are of central interest in both pure and applied category theory. There are several categories to consider as equipped with monoidal structures in this scenario: fibers $P\inv(x)$ of a fibration $P \maps \A \to \X$, its base category $\X$, its total category $\A$, the indexing category $\X$ of an indexed category $F \maps \X\op \to \Cat$, and the categories $Fx$ indexed by $F$. Of course these options are not really all distinct. The base of the fibrations correspond to the indexing category under the equivalence, and the fibres correspond to the individual categories selected by the indexed category. This boils our options down to two monoidal variants: fibre-wise, and global.
In the first variant—the fibre-wise approach—the fibres are equipped with a monoidal structure, and the reindexing functors are equipped with a strict monoidal structure. The additional structure this gives to the corresponding indexed categories turns them into pseudofunctors into $\Mon\Cat$, which were called indexed monoidal categories by Hofstra and de Marchi <cit.>. In the second variant—the global approach—the total category and the base category of the fibration are each equipped with the structure of a monoidal category, and the fibration is equipped with a strict monoidal structure. The corresponding structure equipped to the related indexed category is a little less obvious. The indexing category is equipped with a monoidal structure as in the fibration side of the picture, and the pseudofunctor is now equipped with the structure of a lax monoidal structure into $\Cat$ with its cartesian structure. We call these monoidally indexed categories or just monoidal indexed categories.
Both of these variants can be seen as special cases of a much more general phenomenon. Pseudomonoids are a categorification of monoid objects internal to a monoidal category. It would be reasonable to call it “monoidal category internal to a monoidal 2-category”. We can see both of the monoidal variants of both fibrations and indexed categories described above as examples of pseudomonoids in certain 2-categories of fibrations or indexed categories.
The 2-category $\ICat(\X)$ of indexed categories over a fixed base category has finite products, and thus a cartesian monoidal structure. Pseudomonoids taken with respect to this monoidal structure are precisely pseudofunctors $\X\op \to \Mon\Cat$, i.e. the fibre-wise monoidal indexed categories described above. Similarly, the 2-category $\Fib(\X)$ of fibrations over a fixed base category has a cartesian monoidal structure, for which pseudomonoids are precisely the fibre-wise monoidal fibrations described above.
The 2-category $\ICat$ of indexed categories over different base categories has finite products, and thus a cartesian monoidal structure. Pseudomonoids taken with respect to this monoidal structure are precisely lax monoidal pseudofunctors $(\X\op, \otimes) \to (\Cat, \times)$, i.e. the global monoidal indexed categories described above. Similarly, the 2-category $\Fib$ of fibrations over different base categories has a cartesian monoidal structure, for which pseudomonoids are precisely the global monoidal fibrations described above.
An immediate consequence of this perspective on these objects is that the Grothendieck construction lifts naturally into both settings. The 2-category of fibre-wise monoidal fibrations is equivalent to the 2-category of fibre-wise monoidal indexed categories, c.f. <ref>. Similarly, the 2-category of global monoidal fibrations is equivalent to the 2-category of global monoidal indexed categories, c.f. <ref>.
When $\X$ is cartesian monoidal, a global monoidal structure can be constructed from fibre-wise monoidal data, and vice versa, c.f. <ref>. We use our high-level perspective to give a new proof of the result of Shulman giving an equivalence between fibre-wise monoidal indexed categories and global monoidal fibrations over cartesian base categories <cit.>.
The fact that the monoidal Grothendieck construction naturally arises in diverse settings is what motivated the theoretical clarification presented here. We gather a few examples in the last section of the chapter to exhibit the various constructions concretely, and we are convinced that many more exist and would benefit from such a viewpoint. The examples include standard (op)fibrations such as the (co)domain (op)fibration and families classified in their monoidal contexts, as well as certain special algebraic cases of interest such as monoid-(co)algebras as objects in monoidal Grothendieck categories. Moreover, global categories of (co)modules for (co)monoids in any monoidal category, as well as (co)modules for (co)monads in monoidal double categories also naturally fit in this context. Finally, certain categorical approaches to systems theory employ algebras for monoidal categories, namely monoidal indexed categories, as their basic compositional tool for nesting of systems; clearly these also fall into place, giving rise to total monoidal categories of systems with new potential to be explored.
In <ref> and <ref>, we give the fibre-wise and global monoidal versions of fibrations and indexed categories. In the first version, the fibers are equipped with a monoidal structure. In the second, the base and total categories are monoidal, and the fibration is (strict) monoidal. In <ref>, we lift the Grothendieck construction into these monoidal settings as well. In <ref>, we give a detailed description of the monoidal structures given by the correspondences.
§ MONOIDAL FIBRES AND MONOIDAL FIBRATIONS
We begin by describing the two monoidal variants of fibrations. This requires familiarity with notions such as monoidal 2-categories, pseudomonoids, and the 2-categories $\Fib$ and $\Fib(\X)$. The 2-categories $\Fib$ and $\OpFib$ of (op)fibrations over arbitrary bases, explained in <ref>, have a cartesian monoidal structure inherited from $\Cat^\2$. For two fibrations $P$ and $Q$, their product in $\Cat^\2$
\begin{equation}\label{Fib_cart}
P \times Q \maps \A \times \B \to \X \times \Y
\end{equation}
is also a fibration, where a cartesian lifting is a pair consisting of a $P$-lifting and a $Q$-lifting; similarly for opfibrations. The monoidal unit is the trivial (op)fibration $1_\1 \maps \1 \to \1$. Since the monoidal structure is cartesian, they are both symmetric monoidal 2-categories. We refer to a pseudomonoid in $(\Fib, \times, 1_\1)$ as a monoidal fibration. By the following result, this aligns with the common notion of monoidal fibration <cit.>.
A monoidal fibration $P \maps \A \to \X$ is a fibration for which both the total $\A$ and base category $\X$ are monoidal, $P$ is a strict monoidal functor and the tensor product $\otimes_\A$ of $\A$ preserves cartesian liftings.
The multiplication and unit are fibred 1-cells $\mlt = (\otimes_\A, \otimes_\X) \maps P \times P \to P$ and $\uni = (I_\A, I_\X) \maps \1 \to P$
displayed as follows.
\begin{equation} \label{multunitmonoidalfibr}
\begin{tikzcd}
\A \times \A
\arrow[r, "\otimes_\A"]
\arrow[d, swap, "P \times P"]
\A
\arrow[d, "P"]
\1
\arrow[r, "I_\A"]
\arrow[d, swap, "1_\1"]
\A
\arrow[d, "P"]
\\
\X \times \X
\arrow[r, swap, "\otimes_\X"]
\X
\1
\arrow[r, swap, "I_\X"]
\X
\end{tikzcd}
\end{equation}
A morphism $(\phi_1, \phi_2)$ in $\A \times \A$ is $P\times P$-cartesian if and only if $\phi_1$ and $\phi_2$ are both $P$-cartesian. The condition of $(\otimes_\A, \otimes_\X)$ forming a fibred 1-cell tells us precisely that $\phi_1 \otimes_\A \phi_2$ is $P$-cartesian. The pieces of associativity and unitality 2-cells corresponding to $\A$ and to $\X$ give precisely the associativity and unitality structures for each to be given the structure of a monoidal category. The functor $P$ is strict with respect to these monoidal structures on $\A$ and $\X$ due to the fact that the diagrams above commute.
A monoidal fibred 1-cell between two monoidal fibrations $P \maps \A \to \X$ and $Q \maps \B \to \Y$ is a (strong) morphism of pseudomonoids between them, as defined in <ref>.
A monoidal fibred 1-cell between two monoidal fibrations $P$ and $Q$ is a fibred 1-cell $(H, F)$ where both functors are monoidal, $(H, \phi, \phi_0)$ and $(F, \psi, \psi_0)$, such that $Q (\phi_{a, b})=\psi_{ P a, P b}$ and $Q \phi_0=\psi_0$.
A monoidal fibred 1-cell amounts to a fibred 1-cell, i.e. a commutative square
\begin{equation}\label{eq:HF}
\begin{tikzcd}
\A
\arrow[r, "H"]
\arrow[d, "P"']
\B
\arrow[d, "Q"]
\\
\X
\arrow[r, "F"']
\Y
\end{tikzcd}
\end{equation}
where $H$ preserves cartesian liftings,
along with invertible 2-cells <ref> in $\Fib$ satisfying <ref>. By <ref>, these are fibred 2-cells
×[dr, bend left, "⊗_"]
[ddd, "Q"]
[ddl, Rightarrow, swap, "ϕ"]
×[ddd, swap, "P ×P"]
[urr, bend left, "H ×H"]
[dr, bend right, "⊗_"]
[ddd, "Q"]
[urr, bend right, -, white, line width = 5]
[urr, swap, bend right, "H", pos = 0.6]
×[dr, bend left, swap, "⊗_"]
[ddl, Rightarrow, "ψ"]
×[urr, bend left, "F ×F"]
[dr, bend right, swap, "⊗_"]
[urr, bend right, swap, "F"]
[uuu, -, white, line width = 5]
[uuu, leftarrow, "P"]
[dd, swap, equal]
[rrr, bend left, "I_"]
[rrr, bend left, swap, phantom, ""name = IB, pos = 0.7]
[dr, bend right, "I_"]
[dd, "Q"]
[urr, bend right, -, white, line width = 5]
[urr, swap, bend right, "H", pos = 0.6]
[to = IB, Leftarrow, "ϕ_0"]
[rrr, bend left, swap, "I_", pos = 0.6]
[rrr, bend left, phantom, swap, ""name = IY, pos = 0.7]
[dr, bend right, swap, "I_"]
[urr, bend right, swap, "F"]
[uu, -, white, line width = 5]
[uu, leftarrow, "P"]
[to = IY, Leftarrow, "ψ_0", swap]
where $\phi$ and $\psi$ are natural isomorphisms with components
ϕ_a, b Ha ⊗Hb H(a ⊗b), ψ_x, y Fx ⊗Fy F(x ⊗y)
such that $\phi$ is above $\psi$, i.e. the following diagram commutes:
[column sep=.6in]
Q (Ha ⊗Hb)
[r, " Q ϕ_a, b"]
[d, equal, "<ref>"']
Q H(a ⊗b)
[d, equal, "<ref>"]
Q H a ⊗Q H b
[d, equal, "<ref>"' ]
F P(a ⊗b)
[d, equal, "<ref>"]
F P a ⊗F P b
[r, "ψ_ P a, P b"']
F( P a ⊗P b)
Similarly, $\phi_0$ and $\psi_0$ have single components $\phi_0 \maps I_\B \xrightarrow{\sim} H(I_\A)$ and $ \psi_0 \maps I_\Y \xrightarrow{\sim} F(I_\X)$ such that $Q (\phi_0) = \psi_0$.
These two conditions in fact say that the identity transformation, a.k.a. commutative square <ref> is a monoidal one, as expressed in <cit.>.
The relevant axioms dictate that $(\phi, \phi_0)$ and $(\psi, \psi_0)$ give $H$ and $F$ the structure of strong monoidal functors.
For lax or oplax morphisms of pseudomonoids in $\Fib$, we obtain appropriate notions of monoidal fibred 1-cells, where the top and bottom functors of <ref> are lax or oplax monoidal respectively.
Finally, a monoidal fibred 2-cell is a 2-cell between morphisms $(H, F)$ and $(K, G)$ of pseudomonoids $P$, $Q$ in $\Fib$.
A monoidal fibred 2-cell between two monoidal fibred 1-cells is an ordinary fibred 2-cell $(\alpha, \beta)$ where both natural transformations are monoidal.
Unpacking the definition, we see that a monoidal fibred 2-cell is a fibred 2-cell as described in <ref>
[row sep=.45in, column sep=.7in]
[d, " P"']
[r, bend left = 20, "H"]
[r, bend right = 20, "K"']
[r, phantom, "⇓β" description]
[d, " Q "]
[r, bend left = 20, "F"]
[r, bend right = 20, "G"']
[r, phantom, "⇓α" description]
satisfying the axioms <ref>. These amount to the fact that both $\beta$ and $\alpha$ are monoidal natural transformations between the respective lax monoidal functors.
We denote by $\PsMon(\Fib)= \MonFib$ the 2-category of monoidal fibrations, monoidal fibred 1-cells and monoidal fibred 2-cells.
By changing the notion of morphisms between pseudomonoids to lax or oplax, we obtain 2-categories $\MonFib_\lax$ and $\MonFib_\opl$.
There are also 2-categories $\BrMonFib$ and $\SymMonFib$ of braided (resp. symmetric) monoidal fibrations, braided (resp. symmetric) monoidal fibred 1-cells and monoidal fibred 2-cells, defined to be $\BrPsMon(\Fib)$ and $\SymPsMon(\Fib)$ respectively; see <ref>.
Dually, we have appropriate 2-categories of monoidal opfibrations, monoidal opfibred 1-cells and monoidal opfibred 2-cells and their braided and symmetric variations, $\MonOpFib$, $\BrMonOpFib$ and $\SymMonOpFib$. All the structures are constructed dually, where a monoidal opfibration, namely
a pseudomonoid in the cartesian monoidal $(\OpFib, \times, 1_\1)$,
is a strict monoidal functor such that the tensor product of the total category preserves cocartesian liftings.
All the above 2-categories have sub-2-categories of monoidal (op)fibrations over a fixed monoidal base $(\X, \otimes, I)$, e.g. $\MonFib(\X)$ and $\MonOpFib(\X)$. The morphisms are monoidal (op)fibred functors, i.e. fibred 1-cells of the form $(H, 1_\X)$ with $H$ monoidal, and the 2-cells are monoidal (op)fibred natural transformations, i.e. fibred 2-cells of the form $(\beta, 1_{1_\X})$ with $\beta$ monoidal. These 2-categories correspond to the `global' monoidal part of the story.
Moreover, the above constructions can be adjusted accordingly to the context of split fibrations. Explicitly, the 2-category $\PsMon(\Fib_\spl)=\MonFib_\spl$ has as objects monoidal split fibrations, namely split fibrations $P \maps \A\to\X$ between monoidal categories which are strict monoidal functors and $\otimes_\A$ strictly preserves cartesian liftings (compare to <ref>). Furthermore, the hom-categories $\MonFib_\spl(P, Q)$ between monoidal split fibrations are full subcategories of $\MonFib(P, Q)$ spanned by the monoidal fibred 1-cells which are split as fibred 1-cells, namely $(H, F)$ as in <ref> where $H$ strictly preserves cartesian liftings.
We end this section by considering a different monoidal object in the context of (op)fibrations, starting over from the usual 2-categories of (op)fibrations over a fixed base $\X$, (op)fibred functor and (op)fibred natural transformations $\Fib(\X)$ and $\OpFib(\X)$. Notice that contrary to the earlier devopment, there is no monoidal structure on $\X$. Both these 2-categories are also cartesian monoidal, but in a different manner than $\Fib$ and $\OpFib$, due to the cartesian monoidal structure of $\Cat/\X$; see for example <cit.>. Explicitly, for fibrations $P \maps \A\to\X$ and $Q \maps \B \to \X$, their tensor product $P \boxtimes Q$ is given by any of the two equal functors to $\X$ from the following pullback
\begin{equation}\label{Fib_X_cart}
\begin{tikzcd}[column sep=.5in, row sep=.5in] \A \times_\X \B
\arrow[r]
\arrow[d]
\arrow[dr, phantom, very near start, "\lrcorner"]\arrow[dr, dashed, bend left, "P\boxtimes Q"description]
\A
\arrow[d, "P"]
\\
\B
\arrow[r, "Q"']
\X
\end{tikzcd}
\end{equation}
since fibrations are closed under pullbacks and of course composition. The monoidal unit is $1_\X \maps \X \to \X$.
A pseudomonoid in $(\Fib(\X), \boxtimes, 1_\X)$ is an ordinary fibration $P \maps \A\to\X$ equipped with two fibred functors $(\mlt, 1_\X) \maps P\boxtimes P\to P$ and $(\uni, 1_\X) \maps 1_\X\to P$ displayed as
\begin{equation}\label{eq:fibrewisetensor}
\begin{tikzcd}
\A \times_\X \A
\arrow[dr, "P\boxtimes P"']
\arrow[rr, "\mlt"]
\A
\arrow[dl, "P"]
\\
\X
\end{tikzcd}\qquad
\begin{tikzcd}
\X
\arrow[rr, "\uni"]
\arrow[dr, "1_X"']
\A
\arrow[dl, "P"]
\\
\X
\end{tikzcd}
\end{equation}
along with invertible fibred 2-cells satisfying the usual axioms.
In more detail, the pullback $\A \times_\X \A$ consists of pairs of objects of $\A$ which are in the same fibre of $P$, and $P\boxtimes P$ sends such a pair to their underlying object defining their fibre.
The functor $\mlt$ maps any $(a, b)\in\A_x$ to some $m(a, b):=a\otimes_x b\in\A_x$ and the map $\uni$ sends an object $x \in \X$ to a chosen one, $I_x$, in its fibre. The invertible 2-cells and the axioms guarantee that these maps define a monoidal structure on each fibre $\A_x$, providing the associativity, left and right unitors. The fact that $\mlt$ and $\uni$ preserve cartesian liftings translate into a strong monoidal structure on the reindexing functors: for any $f \maps x\to y$ and $a, b\in\A_y$, $f^*a\otimes_x f^*b\cong f^*(a\otimes_y b)$ and $I_y\cong f^*(I_x)$.
A (lax) morphism between two such fibrations is a fibred functor <ref> such that the induced functors $H_x \maps \A_x \to \B_x$ between the fibres as in <ref> are (lax) monoidal, whereas a 2-cell between them is a fibred natural transformation $\beta \maps H\Rightarrow K$ <ref> which is monoidal when restricted to the fibers, $\beta_x|_{\A_x} \maps H_x\Rightarrow K_x$. In this way, we obtain the 2-category $\PsMon(\Fib(\X))$ and dually $\PsMon(\OpFib(\X))$. These 2-categories correspond to the `fibrewise' monoidal part of the story.
Finally, taking pseudomonoids in the 2-category of split fibrations over a fixed base, we obtain the 2-category $\PsMon(\Fib_\spl(\X))$ with objects split fibrations equipped with a fibrewise tensor product and unit as above, but now the reindexing functors strictly preserve that monoidal structure since the top functors of <ref> strictly preserve cartesian liftings: $f^*a \otimes_x f^*b = f^* (a \otimes_y b)$ and $I_y = f^* (I_x)$. Moreover, $\PsMon (\Fib_s(\X)) (P, Q)$ is the full subcategory of $\PsMon(\Fib(\X))(P, Q)$ spanned by split fibred functors, namely $H \maps \A\to\B$ which strictly preserve cartesian liftings but still $H_x$ are monoidal functors between the monoidal fibres as before.
As is evident from the above descriptions, the 2-categories $\MonFib(\X)$ and $\PsMon(\Fib(\X))$ are different in general. A monoidal fibration over $\X$ is a strict monoidal functor, whereas a pseudomonoid in fixed-base fibrations is a fibration with monoidal fibres in a coherent way: none of the base or the total category need to be monoidal.
§ INDEXED CATEGORIES AND MONOIDAL STRUCTURES
The 2-categories of indexed and opindexed categories $\ICat$ and $\OpICat$, explained in <ref>, are both monoidal. Explicitly, given two indexed categories $\M \maps \X\op \to \Cat$ and $\psN \maps \Y\op \to \Cat$, their tensor product $\M \otimes \psN \maps (\X \times \Y)\op \to \Cat$ is the composite
\begin{equation}\label{ICat_cart}
(\X \times \Y)\op \cong \X\op \times \Y\op \xrightarrow{\M \times \psN} \Cat \times \Cat \xrightarrow{\times} \Cat
\end{equation}
i.e. $(\M \otimes \psN)(x, y) = \M(x) \times \psN(y)$ using the cartesian monoidal structure of $\Cat$. The monoidal unit is the indexed category $\Delta \1 \maps \1\op \to \Cat$ that picks out the terminal category $\1$ in $\Cat$, and similarly for opindexed categories. Notice that this monoidal 2-structure, formed pointwise in $\Cat$, is also cartesian.
We call a pseudomonoid in $(\ICat, \otimes, \Delta \1)$ a monoidal indexed category.
A monoidal indexed category is a lax monoidal pseudofunctor
\[(\M, \mu, \mu_0) \maps (\X\op, \otimes\op, I) \to (\Cat, \times, \1),\]
where $(\X, \otimes, I)$ is an (ordinary) monoidal category.
Unpacking the definition, we see that a monoidal indexed category is an indexed category $\M \maps \X\op \to \Cat$ equipped with multiplication and unit indexed 1-cells $(\otimes_\X, \mu) \maps \M \otimes \M \to \M$, $(\eta, \mu_0) \maps \Delta \mathbf 1 \to \M$ which by <ref> are as follows.
\[
\begin{tikzcd}[column sep=.7in, row sep=.2in]
\X\op \times \X\op
\arrow[dr, "\M \otimes \M"]
\arrow[dd, "\otimes\op"']
\\
\arrow[r, phantom, "\Downarrow{\scriptstyle\mu}"description]
\Cat
\\
\X\op
\arrow[ur, "\M"']
\end{tikzcd}\qquad
\begin{tikzcd}[column sep=.7in, row sep=.2in]
\1\op
\arrow[dr, "\Delta\1"]
\arrow[dd, "I\op"']
\\
\arrow[r, phantom, "\Downarrow{\scriptstyle\mu_0}"description]
\Cat
\\
\X\op
\arrow[ur, "\M"']
\end{tikzcd}
\]
These come equipped with invertible indexed 2-cells
as in <ref>; the axioms this data is required to satisfy, on the one hand, render $\X$ a monoidal category with $\otimes \maps \X \times \X \to \X$ its tensor product functor and $I \maps \1 \to \X$ its unit. On the other hand,
the resulting axioms for the components
\begin{equation}\label{eq:laxatorunitor}
\mu_{x, y} \maps \M x \times \M y \to \M (x \otimes y), \qquad
\mu_0 \maps \1 \to \M (I)
\end{equation}
of the above pseudonatural transformations precisely give $\M$ the structure of a lax monoidal pseudofunctor, recalled in <ref>.
We then define a monoidal indexed 1-cell to be a (strong) morphism between pseudomonoids in $(\ICat, \otimes, \Delta\1)$.
A monoidal indexed 1-cell between two monoidal indexed categories $\M$ and $\psN$ is an indexed 1-cell $(F, \tau)$, where the functor $F$ is (strong) monoidal and the pseudonatural transformation $\tau$ is monoidal.
Unpacking the definition, we see that a monoidal indexed 1-cell is an indexed 1-cell $(F, \tau) \maps \M \to \psN$
[column sep=.7in, row sep=.2in]
[dr, ""]
[dd, "F"']
[r, phantom, "⇓τ"description]
[ur, ""']
between two monoidal indexed categories
$(\M, \mu, \mu_0)$ and $(\psN, \nu, \nu_0)$ equipped with two invertible indexed 2-cells $(\psi, m)$ and $(\psi_0, m_0)$ as in <ref>, which explicitly consist of natural isomorphisms $\psi$, $\psi_0$ and invertible modifications
\[
\begin{tikzcd}[row sep=.3in, column sep=.25in]
\X\op \times \X\op
\arrow[d, "\otimes\op"']
\arrow[dr, "F\op \times F\op" description]
\arrow[drrr, "\M \otimes \M", bend left=20]
\arrow[drrr, phantom, bend left=5, "\Downarrow{\scriptstyle \tau \times \tau}"]
\X\op \times \X\op
\arrow[drrr, "\M \otimes \M", bend left]
\arrow[d, "\otimes\op"']\arrow[drrr, phantom, "\Downarrow{\scriptstyle\mu}"description]
\\
\X\op
\arrow[d, "F\op"']
\arrow[r, phantom, "\Downarrow{\scriptstyle\psi}"description, bend right]
\Y\op \times \Y\op
\arrow[rr, "\psN \otimes \psN"description]
\arrow[dr, "\otimes\op" description]
\arrow[drr, phantom, "\Downarrow{\scriptstyle\nu}"description]
\Cat
\arrow[r, phantom, "\stackrel{m}{\Rrightarrow}"]
\X\op
\arrow[rrr, "\M"description]\arrow[d, "F\op"']\arrow[drrr, phantom, "\Downarrow{\scriptstyle\tau}"description]
\Cat
\\
\Y\op\arrow[rr, "\mathrm{id}"']
\Y\op\arrow[ur, "\psN"', bend right]
& \phantom{A} &
\Y\op\arrow[urrr, "\psN"', bend right]
&&& \phantom{A}
\end{tikzcd}
\]
\[
\begin{tikzcd}[row sep=.25in, column sep=.3in]
\1\op\arrow[d, "I\op"']\arrow[drrr, "\Delta\1", bend left=20]
\arrow[drrr, phantom, "\Downarrow{\scriptstyle\nu_0}"]\arrow[ddrr, "I\op"description]
\1\op\arrow[drrr, "\Delta\1", bend left]\arrow[d, "I\op"']\arrow[drrr, phantom, "\Downarrow{\scriptstyle\mu_0}"description]
\\
\X\op\arrow[d, "F\op"']\arrow[r, phantom, "\Downarrow{\scriptstyle\psi_0}"description, bend right]
& \phantom{A} &&
\Cat\arrow[rr, phantom, "\stackrel{m_0}{\Rrightarrow}"]
\X\op\arrow[rrr, "\M"description]\arrow[d, "F\op"']\arrow[drrr, phantom, "\Downarrow{\scriptstyle\tau}"description]
\Cat
\\
\Y\op\arrow[rr, "\mathrm{id}"']
\Y\op\arrow[ur, "\psN"', bend right]
& \phantom{A} &&
\Y\op\arrow[urrr, "\psN"', bend right]
&&& \phantom{A}
\end{tikzcd}
\]
as dictated by the general form <ref> of indexed 2-cells.
The natural isomorphisms $\psi$ and $\psi_0$ have components
ψ_x, z Fx⊗Fy F(x⊗y), ψ_0 I F(I) in $\Y\op$
whereas the modifications $m$ and $m_0$ are given by families of invertible natural transformations
[column sep=.3in, row sep=.2in]
Fx ×Fy
[r, "ν_Fx, Fy"]
[dr, dashed]
(Fx ⊗Fy)
[d, "ψ_x, y"]
[ur, "τ_x×τ_y"]
[dr, "μ_x, y"']
[rr, phantom, "⇓m_x, y"description]
F (x⊗y)
[ur, "τ_x⊗y"']
[column sep=.2in, row sep=.2in]
[dr, "ψ_0"]
[ur, "ν_0"]
[dr, "μ_0"']
[rr, phantom, "⇓m_0"description]
(I)[ur, "τ_I"']
The appropriate coherence axioms ensure that the functor $F \maps \X \to \Y$ has a strong monoidal structure $(F, \psi, \psi_0)$, and that the pseudonatural transformation $\tau \maps \M \Rightarrow \psN \circ F\op$ is monoidal with $m_{x, y}$, $m_0$ as in <ref>. Notice that $F\op$ being monoidal makes $F$ monoidal with inverse structure isomorphisms.
Finally, a monoidal indexed 2-cell is a 2-cell between morphisms of pseudomonoids in $(\ICat, \otimes, \Delta\1)$.
A monoidal indexed 2-cell between two monoidal indexed 1-cells $(F, \tau)$ and $(G, \sigma)$ is an indexed 2-cell $(\alpha, m)$ such that $\alpha$ is an ordinary monoidal natural transformation and $m$ is a monoidal modification.
Following the definition of <ref>, an indexed 2-cell $(a, m) \maps (F, \tau)\Rightarrow(G, \sigma) \maps \M\to\psN$ as in <ref>, which consists of a natural transformation $\alpha \maps F\Rightarrow G$ and a modification $m$ with components
[column sep=.5in, row sep=.15in]
x[dr, bend right=10, "σ_x"'][rr, bend left, "τ_x"] [rr, phantom, "⇓m_x"description] Fx
Gx[ur, bend right=10, "α_x"']
is monoidal, exactly when $\alpha \maps F\Rightarrow G$ is compatible with the strong monoidal structures of $F$ and $G$, and the modification $m \maps \tau \Rrightarrow \psN \alpha\op \circ \sigma$ satisfies <ref> for the induced monoidal structures on its domain and target pseudonatural transformations.
We write $\PsMon(\ICat) = \Mon\ICat$, the 2-category of monoidal indexed categories, monoidal indexed 1-cells and monoidal indexed 2-cells. Moreover, their braided and symmetric counterparts form $\Br\Mon\ICat$ and $\Sym\Mon\ICat$ respectively, as the 2-categories of braided and symmetric pseudomonoids in $(\ICat, \otimes, \Delta\1)$ formally discussed in <ref>.
Similarly, we have 2-categories of (braided or symmetric) monoidal opindexed categories, 1-cells and 2-cells $\Mon\OpICat$, $\Br\Mon\OpICat$ and $\Sym\Mon\OpICat$.
All these 2-categories have sub-2-categories of monoidal (op)indexed categories with a fixed monoidal domain $(\X, \otimes, I)$, and specifically
\begin{gather}\label{eq:monicatX}
\MonICat(\X)=\MonTCat_\pse(\X\op, \Cat) \\ \Mon\OpICat(\X)=\MonTCat_\pse(\X, \Cat)\nonumber
\end{gather}
the functor 2-categories of lax monoidal pseudofunctors, monoidal pseudonatural transformations and monoidal modifications.
Moreover, we can consider pseudomonoids in the strict context. Explicitly, the 2-category $\PsMon(\ICat_\spl)=\MonICat_\spl$ has as objects monoidal strict indexed categories namely (2-)functors $\M \maps \X\op\to\Cat$ from an ordinary monoidal category $\X$ which are lax monoidal as before, but the laxator and unitor <ref> are strictly natural rather than pseudonatural transformations. The hom-categories $\PsMon(\ICat_\spl)(\M, \psN)$ between monoidal strict indexed categories are full subcategories of $\MonICat(\M, \psN)$ spanned by strict natural transformations—which are however still lax monoidal, i.e. equipped with isomorphisms <ref>.
Similarly to the previous <ref> on fibrations, we end this section with the study of pseudomonoids in a different but related monoidal 2-category, namely $\ICat(\X)=\TCat_\pse(\X\op, \Cat)$ of indexed categories with a fixed domain $\X$. Working in this 2-category, or in $\OpICat(\X)$, there is no assumed monoidal structure on $\X$. Their monoidal structure is again cartesian: for two $\X$-indexed categories $\M, \psN \maps \X\op \to \Cat$, their product is
\begin{equation}\label{eq:icatxprod}
\M\boxtimes\psN \maps \X\op \xrightarrow{\Delta} \X\op \times \X\op \xrightarrow{\M \times \psN} \Cat \times \Cat \xrightarrow{\times} \Cat
\end{equation}
with pointwise components $(\M \boxtimes \psN) (x) = \M (x) \times \psN (x)$ in $\Cat$. The monoidal unit is just $\X\op \xrightarrow{!} \1 \xrightarrow{\Delta \1} \Cat$, which we will also call $\Delta \1$.
A pseudomonoid in $(\ICat(\X), \boxtimes, \Delta\1)$ is a pseudofunctor $\M \maps \X\op \to \Cat$ equipped with indexed functors <ref> $\mlt \maps \M\boxtimes\M\to\M$ and $\uni \maps \Delta\1\to\M$ namely
[row sep=.1in]
×[r, "×"]
×[dr, "×"]
[dr, "Δ"]
[ur, "Δ"]
[rrr, phantom, "⇓"description]
[rrr, bend right=20, ""']
[ur, "!"]
[rr, bend right, ""']
[rr, phantom, "⇓"description]
with components $\mlt_x \maps \M x \times \M x \to \M x$ and $\uni_x \maps \1 \to \M x$ which are pseudonatural via
\begin{equation}\label{eq:pseudonaturalmult}
\begin{tikzcd}[column sep=.6in]
\M x
\times \M x
\arrow[d, "\mlt_x"']
\arrow[r, "\M f\times\M f"]
\arrow[dr, phantom, "\cong"description]
\M y \times \M y
\arrow[d, "\mlt_y"]
\\
\M x
\arrow[r, "\M f"']
\M y
\end{tikzcd}\quad
\begin{tikzcd}
\1
\arrow[r, "="]
\arrow[d, "\uni_x"']
\arrow[dr, phantom, "\cong"description]
\1
\arrow[d, "\uni_y"]
\\
\M x
\arrow[r, "\M f"']
\M y
\end{tikzcd}
\end{equation}
If we denote $\mlt_x=\otimes_x$ and $\uni_x=I_x$, the pseudomonoid invertible 2-cells <ref> and the axioms these data satisfy make each $\M x$ into a monoidal category $(\M x, \otimes_x, I_x)$, and each $\M f$ into a strong monoidal functor: the above isomorphisms have components $\M f(a)\otimes_y\M f(b)\cong\M f(a\otimes_x b)$ and $I_y\cong\M f(I_x)$ for any $a, b\in\M x$.
Such a structure, namely a pseudofunctor $\M \maps \X\op \to \Mon\Cat$ into the 2-category of monoidal categories, strong monoidal functors and monoidal natural transformations, was directly defined as an indexed strong monoidal category in <cit.>, and as indexed monoidal category in <cit.>. We will avoid this notation in order to not create confusion with the term monoidal indexed categories.
A strong morphism of pseudomonoids <ref> in $(\ICat(\X), \boxtimes, \Delta\1)$ ends up being a pseudonatural trasformation $\tau \maps \M\Rightarrow\psN \maps \X\op\to\Cat$ (indexed functor) whose components $\tau_x \maps \M x\to\psN x$ are strong monoidal functors, whereas a 2-cell between strong morphisms of pseudomonoids is an ordinary modification
[rr, bend left=40, "", ""'name = F]
[rr, bend right=40, ""', ""name = G]
[rr, phantom, "m ⇛"description]
[from = F, to = G, Rightarrow, "τ"', bend right=50]
[from = F, to = G, Rightarrow, "σ", bend left=50]
whose components $m_x \maps \tau_x\Rightarrow\sigma_x$ are monoidal natural transformations.
We thus obtain the 2-categories $\PsMon (\ICat(\X))$ as well as $\PsMon(\OpICat(\X))$; from the above descriptions, it is clear that
\begin{gather}\label{eq:imoncats}
\PsMon(\ICat(\X))=\TCat_\pse(\X\op, \Mon\Cat) \\
\PsMon(\OpICat(\X))=\TCat_\pse(\X, \Mon\Cat)\nonumber
\end{gather}
which will also be rediscovered by <ref>.
Finally, taking pseudomonoids in strict $\X$-indexed categories $\ICat_\spl(\X)=[\X\op, \Cat]$ produces the 2-category $\PsMon(\ICat_\spl(\X))$ with objects functors $\M \maps \X\op\to\Mon\Cat_\mathrm{st}$ into monoidal categories with strict monoidal functors: the isomorphisms <ref> are now equalities due to strict naturality of the multiplication and unit. Then the hom-categories $\PsMon(\ICat_\spl(\X))(\M, \psN)$ are full subcategories of $\PsMon(\ICat(\X))(\M, \psN)$ spanned by strictly natural transformations $\tau \maps \M\Rightarrow\psN$, still with strong monoidal components $\tau_x$. For example, it would not be correct to write $\PsMon(\ICat_\spl(\X))=[\X\op, \Mon\Cat_{(\mathrm{st})}]$.
It is evident that $\MonICat(\X)$ and $\PsMon(\ICat(\X))$ are in principle different. A monoidal indexed category with base $\X$ is a lax monoidal pseudofunctor into $\Cat$ (and $\X$ is required to be monoidal already), whereas a pseudomonoid in $\X$-indexed categories is a pseudofunctor from an ordinary category $\X$ into $\Mon\Cat$.
§ TWO MONOIDAL GROTHENDIECK CONSTRUCTIONS
In <ref>, we recalled
the standard equivalence between fibrations and indexed categories via the Grothendieck construction. We will now lift this correspondence to their monoidal versions studied in Sections <ref> and <ref>, using general results about pseudomonoids in arbitrary monoidal 2-categories described in <ref>.
Since both $\Fib$ and $\ICat$ are cartesian monoidal 2-categories, via <ref> and <ref> respectively, our first task is to ensure that they are monoidally equivalent.
The 2-equivalence $\Fib\simeq\ICat$ between the cartesian monoidal 2-categories of fibrations and indexed categories is (symmetric) monoidal.
Since they form an equivalence, both 2-functors from <ref> preserve limits, therefore are monoidal 2-functors. Moreover, it can be verified that the natural isomorphisms with components $\F\cong\F_{P_\F}$ and $P\cong P_{\F_P}$ are monoidal with respect to the cartesian structure, due to universal properties of products.
There are 2-equivalences
\begin{gather*}
\MonFib\simeq \MonICat \\
\BrMonFib\simeq\Br\Mon\ICat \\
\SymMonFib\simeq\Sym\Mon\ICat
\end{gather*}
between the 2-categories of monoidal fibrations and monoidal indexed categories, as well as their braided and symmetric versions. Dually, there is a 2-equivalence $\MonOpFib\simeq\MonOpICat$ between the 2-categories of monoidal opfibrations and monoidal opindexed categories, as well as their braided and symmetric versions.
Since $\MonFib=\PsMon(\Fib)$ and $\MonICat=\PsMon(\ICat)$, we obtain the equivalence as a special case of <ref>; similar for $\OpFib\simeq\OpICat$.
The above 2-equivalences restrict to the sub-2-categories of fixed bases or domains, which by <ref> are
\begin{gather*}
\MonFib(\X) \simeq\MonTCat_\pse(\X\op,\Cat) \\
\MonOpFib(\X) \simeq\MonTCat_\pse(\X\op,\Cat)
\end{gather*}
These results correspond to the global monoidal structure of fibrations and indexed categories. Even though they were directly derived via abstract reasoning, for exposition purposes we briefly describe this equivalence on the level of objects; some relevant details can also be found in <cit.>. Independently and much earlier, in his thesis <cit.> Shulman explores such a fixed-base equivalence on the level of double categories (of monoidal fibrations and monoidal pseudofunctors over the same base).
Suppose that $(\M, \mu, \mu_0) \maps (\X\op, \otimes, I) \to (\Cat, \times, \1)$ is a monoidal indexed category, i.e. a lax monoidal pseudofunctor with structure maps <ref>. The induced monoidal product $\otimes_\mu \maps \inta \M \times \inta \M \to \inta \M$ on the Grothendieck category is defined on objects by
\begin{equation}
\label{eq:globalmonstr}
(x,a) \otimes_\mu (y,b) = (x \otimes y, \mu_{x,y}(a,b))
\end{equation}
and $I_\mu=(I, \mu_0(*))$ is the unit object. Clearly, the induced fibration $\inta \M \to \X$ which maps each pair to the underlying $\X$-object strictly preserves the monoidal structure. Moreover, pseudonaturality of $\mu$ implies that $\otimes_\mu$ preserves cartesian liftings, so all clauses of <ref> are satisfied. For a more detailed exposition of the structure, as well as the braided and symmetric version, we refer the reader to the <ref>.
We can also restrict to the context of split fibrations and strict indexed categories.
There are 2-equivalences
\begin{gather*}
\MonFib_\spl \simeq \MonICat_\spl \\
\MonOpFib_\spl \simeq \MonOpICat_\spl
\end{gather*}
between monoidal split (op)fibrations and monoidal strict (op)indexed categories, as well as for the fixed-base case.
Again by applying $\PsMon(\textrm{-})$ to the 2-equivalence $\ICat_\spl\simeq\Fib_\spl$, we obtain equivalences between the respective structures discussed in <ref>, as the strict counterparts of <ref> and <ref>. Recall that a monoidal strict indexed category is a lax monoidal 2-functor $\X\op\to\Cat$ whose structure maps $(\phi,\phi_0)$ are strictly natural transformations, and corresponds to a split fibration which is monoidal like before, only the tensor product of the total category strictly preserves cartesian liftings.
We close this section in a similar manner to Sections <ref> and <ref>, namely by working in the cartesian monoidal 2-categories $(\Fib(\X), \boxtimes, 1_\X)$ and $(\ICat(\X), \boxtimes, \Delta\1)$ of fibrations and indexed categories with a fixed base category.
There are 2-equivalences between (op)fibrations with monoidal fibres and strong monoidal reindexing functors, and pseudofunctors into $\Mon\Cat$
\begin{align*}
\PsMon(\Fib(\X)) &\simeq \TCat_\pse(\X\op,\Mon\Cat)\quad
\\
\PsMon(\OpFib(\X)) &\simeq \TCat_\pse(\X\op,\Mon\Cat)
\end{align*}
Moreover, these restrict to 2-equivalences between split (op)fibrations with monoidal fibres and strict monoidal reindexing functors, and ordinary functors into $\Mon\Cat_\mathrm{st}$.
Since $\Fib(\X) \simeq \ICat(\X)$ is also a monoidal 2-equivalence, <ref> applies once more – recall <ref>.
These equivalences correspond to the fibrewise monoidal structure on fibrations and indexed categories. In more detail, a pseudofunctor $\M \maps \X\op \to \Mon\Cat$ maps every object $x$ to a monoidal category $\M x$ and every morphism $f \maps x \to y$ to a strong monoidal functor $\M f \maps \M y \to \M x$; under the usual Grothendieck construction, these are precisely the fibre categories and the reindexing functors between them for the induced fibration, as described at the end of <ref>. Notice how, in particular, $\X$ is not a monoidal category, as was the case in <ref>.
A very similar, relaxed version of the fibrewise monoidal correspondence seems to connect the concepts of an indexed monoidal category, defined in <cit.> as a pseudofunctor $\M\maps\X\op\to\Mon\Cat_\lax$, and that of of a lax monoidal fibration, defined in <cit.>. Notice that these terms are misleading with respect to ours: an indexed monoidal category is not a monoidal indexed category, and also a lax monoidal fibration is not a functor with a lax monoidal stucture.
Briefly, there is a full sub-2-category $\Fib_\opl(\X)\subseteq\Cat/\X$ of fibrations, namely fibred 1-cells <ref> which are not required to have a cartesian functor on top. As discussed in <cit.>, this is 2-equivalent to $\TCat_{ps,opl}(\X\op,\Cat)$, the 2-category of pseudofunctors, oplax natural transformations and modifications. Describing pseudomonoids therein appears to give rise to a fibration with monoidal fibres and lax monoidal reindexing functors between them, or equivalently a pseudofunctor into $\Mon\Cat_\lax$. We omit the details so as to not digress from our main development.
§ SUMMARY OF STRUCTURES
The bulk of this chapter is dedicated to proving various monoidal variations of the equivalence between fibrations and indexed categories, using general results in monoidal 2-category theory. In this section, we detail the descriptions of the (braided/symmetric) monoidal structures on the total category of the Grothendieck construction, assuming the appropriate data is present. We also provide a hands-on correspondence that underlies the proof of <ref> regarding the transfer of monoidal structure from a functor to its target and vice versa. We hope this section can serve as a quick and clear reference on some fundamental constructions of this chapter.
§.§ Monoidal Structures
As sketched under <ref>, let $(\X, \otimes, I)$ be a monoidal category, and
\[(\M, \mu, \mu_0) \maps (\X\op, \otimes\op, I) \to (\Cat, \times, \1)\]
a monoidal indexed category, a.k.a. lax monoidal pseudofunctor. Recall that $\mu$ is pseudonatural transformation consisting of functors $\mu_{x,y} \maps \M x \times \M y \to \M(x \otimes y)$ for any objects $x$ and $y$ of $\X$, and natural isomorphisms
\[
\begin{tikzcd}[column sep = 70]
\M z \times \M w
\arrow[r, "\M f \times \M g"]
\arrow[d, "\mu_{z,w}", swap]
\M x \times \M y
\arrow[d, "\mu_{x,y}"]
\arrow[dl, phantom, "{\scriptstyle\stackrel{\mu_{f,g}}{\cong}}"]
\\
\M(z \otimes w)
\arrow[r, "\M(f \otimes g)", swap]
\M(x \otimes y)
\end{tikzcd}\]
for any arrows $f \maps x \to z$ and $g \maps y \to w$ in $\X$. Also the unique component of $\mu_0$ is the functor $\mu_0\maps\1\to\M(I)$.
The induced tensor product functor on the total category, denoted as $\otimes_\mu \maps \inta \M \times \inta \M \to \inta \M$, is given on objects by
(x,a) ⊗_μ(y,b) = (x ⊗y, μ_x,y(a,b))
On morphisms $\left(f\maps x\to z, k\maps a\to(\M f)c\right)$ and $\left(g\maps y\to w,\ell\maps b\to(\M g)d\right)$, we get
\[
(f, k) \otimes_\mu (g, \ell) = (x\otimes y\xrightarrow{f \otimes g} z\otimes w, \mu_{f,g}(\mu_{x, y}(k, \ell)))
\]
where the latter is the composite morphism
μ_x,y(a,b)μ_x,y((f)(c),(g)(d))(f⊗g)(μ_z.w(c,d)) in (x⊗y).
The monoidal unit is $I_\mu=(I, \mu_0)$.
If $a_{x,y,z} \maps (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the associator in $\X$, the associator for $(\inta \M, \otimes_\mu, I_\mu)$ is given by
α_(x,b), (y,c), (z,d) = (α_x,y,z, ω_x,y,z (b,c,d))
where $\omega$ is the invertible modification <ref>.
If $l_x \maps I \otimes x \to x$ and $r_x \maps x \otimes I \to x$ are the left and right unitors in $\X$, the unitors in $\inta \M$ are defined as
\begin{gather*}
\lambda_x = (l_x, \xi_x^{\text{-}1}(a))\maps (I,\mu_0)\otimes_\mu(x,a)\to(x,a) \\
\rho_x = (r_x, \zeta_x(a))\maps(x,a)\otimes_\mu(I,\mu_0)\to(x,a)
\end{gather*}
where $\zeta$ and $\xi$ are invertible modifications as in <ref>.
We now turn to the correspondence between 1-cells of <ref>:
given a monoidal indexed 1-cell
\[
\begin{tikzcd}
(\X, \otimes, I)\op
\arrow[dr, "{(\M, \mu, \mu_0)}"]
\arrow[dd, "{(F, \psi, \psi_0)\op}", swap]
\\
\arrow[r, phantom, "\Downarrow{\scriptstyle\tau}"]
(\Cat, \times, \1)
\\
(\Y, \otimes, I)\op
\arrow[ur, "{(\psN, \nu, \nu_0)}", swap]
\end{tikzcd}\]
where $\M$ and $\psN$ are lax monoidal pseudofunctors and $F$ is a monoidal functor, as in <ref>, we first of all obtain an ordinary fibred 1-cell $(P_\tau,F)\maps P_\M\to P_\psN$ as explained above <ref>
with $P_\tau (x, a) = (F x, \tau_x (a))$. The functor $F$ is already monoidal, and $P_\tau$ obtains a monoidal structure too: for example, there are isomorphisms
P_τ(x,a) ⊗_νP_τ(y, b) P_τ((x, a) ⊗_μ(y, b)) in
between the objects
\begin{align*}
P_\tau (x, a) \otimes_\nu P_\tau (y, b)
&= (Fx, \tau_x (a)) \otimes_\nu (Fy, \tau_y (b)
= (Fx \otimes Fy, \nu_{Fx, Fy} (\tau_x (a), \tau_y (b))
\\
&= P_\tau(x\otimes y,\mu_{x,y}(a,b))
= (F(x\otimes y),\tau_{x\otimes y}(\mu_{x,y}(a,b)))
\end{align*}
given by $\psi_{x,y} \maps Fx \otimes Fy \xrightarrow{\sim} F (x \otimes y)$ and by
\[
\nu_{Fx, Fy} (\tau_x (a), \tau_y (b)) \cong \psN (\psi_{x, y}) (\tau_{x \otimes y} (\mu_{x, y} (a, b)))
\]
essentially given by the monoidal pseudonatural isomorphism <ref> for $\tau \maps \M \Rightarrow \psN F\op$. As a result, $(P_\tau,F)$ is indeed a monoidal fibred 1-cell as in <ref>.
Finally, it can be verified that starting with a monoidal indexed 2-cell as in <ref>, the induced fibred 2-cell <ref> is monoidal, i.e. $P_m$ satisfies the conditions of a monoidal natural transformation.
Regarding the induced braided and symmetric monoidal structures, suppose that $(\X,\otimes,I)$ is a braided monoidal category, with braiding $b$ with components
\[
\braid_{x,y} \maps x \otimes y \xrightarrow{\sim} y \otimes x;
\]
then $\X\op$ is braided monoidal with the inverse braiding, namely $(\X\op,\otimes\op,I,\braid^{-1})$.
Now if $(\M,\mu,\mu_0)\maps\X\op\to\Cat$ is a braided lax monoidal pseudofunctor, i.e. a braided monoidal indexed category,
by <ref> we have an induced braided monoidal structure on $(\inta\M, \otimes_\mu, I_\mu)$, namely
\[
B_{(x, a), (y, b)} \maps (x, a) \otimes_\mu (y,b)=(x\otimes y,\mu_{x,y}(a,b)) \to (y,b) \otimes_\mu (x, a)=(y\otimes x,\mu_{y,x}(b,a))
\]
are given by $\braid_{x,y} \maps x \otimes y \cong y \otimes x$ in $\X$ and $(v_{x,y})_{(a,b)} \maps \mu_{x,y} (a,b) \cong \M (\braid^{-1}_{x,y}) (\mu_{y,x} (b,a))$, where $v$ is as in <ref>.
If $\M$ is a symmetric lax monoidal pseudofunctor, it can be verified that
\[
B_{(y, b),(x, a)} \circ B_{(x, a),(y, b)} = 1_{(x,a) \otimes_\mu (y,b)}
\]
therefore $\inta \M$ is also symmetric monoidal, as is the monoidal fibration $P_\M \maps \inta \M \to \X$.
§.§ Monoidal Indexed Categories as Ordinary Pseudofunctors
Here we detail the correspondence between monoidal opindexed categories and a pseudofunctors into $\Mon\Cat$ when the domain is a cocartesian monoidal category, as established by <ref>; the one for indexed categories is of course similar. We denote by $\nabla_x \maps x+x \to x$ the induced natural components due to the universal property of coproduct, and $\iota_x \maps x \to x+y$ the inclusion into a coproduct.
Start with a lax monoidal pseudofunctor $\M \maps (\X, +, 0) \to (\Cat, \times, \1)$ equipped with $\mu_{x,y}\maps \M(x) \times \M(y) \to \M(x + y)$ and $\mu_0 \maps \1 \to \M(0)$, which gives the global monoidal structure <ref> of the corresponding opfibration. There exists an induced monoidal structure on each fibre $\M(x)$ as follows:
\begin{gather}
\label{eq:explicitstructure1}
\otimes_x \maps \M(x) \times \M(x) \xrightarrow{\mu_{x,x}} \M(x + x) \xrightarrow{\M(\nabla)} \M(x)
\\
I_x \maps \1 \xrightarrow{\mu_0} \M(0) \xrightarrow{\M(!)} \M(x) \nonumber
\end{gather}
Moreover, each $\M f \maps \M x \to \M y$ is a strong monoidal functor, with $\phi_{a,b} \maps (\M f)(a) \otimes_y (\M f)(b) \xrightarrow{\sim} \M f(a \otimes_xb)$ and $\phi_0 \maps I_y \xrightarrow{\sim} (\M f) I_x$ essentially given by the following isomorphisms
\begin{equation}\label{eq:strongmonreindex}
\begin{tikzcd}[row sep=.3in,column sep=.8in]
\M x \times \M x
\arrow[r,"\M f\times\M f"]
\arrow[d,"\mu_{x,x}"']
\arrow[dr,phantom,"{\scriptstyle\stackrel{\mu^{f,f}}{\cong}}"description]
\M y \times \M y
\arrow[d,"\mu_{y,y}"]
\\
\M(x+x)
\arrow[d,"\M(\nabla_x)"']
\arrow[r,"\M(f+f)"description]
\arrow[dr,phantom,"{\scriptstyle\cong}"description]
& \M(y+y)\arrow[d,"\M(\nabla_y)"]
\\
\M x \arrow[r,"\M f"']
\M y
\end{tikzcd}
\qquad
\begin{tikzcd}
\1
\arrow[r,"\mu_0"]
\arrow[d,"\mu_0"']
\arrow[ddr,phantom,"{\scriptstyle\cong}"description]
\M(0)
\arrow[dd,"\M(!)"]
\\
\M(0)
\arrow[d,"\M(!)"']
\M x
\arrow[r,"\M f"']
\M y
\end{tikzcd}
\end{equation}
since $\nabla$ and $!$ are natural and $\M$ is a pseudofunctor.
In the opposite direction, take an ordinary pseudofunctor $\M\maps\X\to\Mon\Cat$ into the 2-category of monoidal categories, strong monoidal functors and monoidal natural transformations, with $\otimes_x\maps\M(x)\times\M(x)\to\M(x)$ and $I_x$ the fibrewise monoidal structures in every $\M x$. We can use those to endow $\M$ with a lax monoidal structure via
\begin{gather*}
\mu_{x,y} \maps \M(x) \times \M(y) \xrightarrow{\M(\iota_x) \times \M(\iota_y)} \M(x+y) \times \M(x+y) \xrightarrow{\otimes_{x+y}} \M(x+y)
\\
\mu_0 \maps \1 \xrightarrow{I_0} \M(0)
\end{gather*}
The fact that all $\M f$ are strong monoidal imply that the above components form pseudonatural transformations, and all appropriate conditions are satisfied.
In the strict context, a lax monoidal 2-functor $\M\maps(\X,+,0)\to(\Cat,\times,\1)$ with natural laxator and unitor bijectively corresponds to a functor $\X\to\Mon\Cat_\textrm{st}$ since <ref> are in fact strictly commutative, by naturality of $\mu,\mu_0$ and functoriality of $\M$.
In the even more special case of an ordinary lax monoidal functor $\M\maps(\X,+,0)\to(\Cat,\times,\1)$, the fibres $\M(x)$ turn out to be strict monoidal. For example, strict associativity of the tensor is established by
[column sep=.15in]
x ×x ×x
[rr, "1 ×⊗_x"]
[dr, "1 ×μ_x, x"']
[ddrr, bend left, "⊗_x"]
[dr, "μ_x, x"description]
[dd, phantom, "(*)"]
[ur, "1×(∇)"description]
[dr, "μ_x, x+x"description]
[dr, "∇"description]
[ur, "(1+∇)"description]
[dr, "(∇+1)"description]
[ur, "μ_x+x, x"description]
[dr, "(∇)×1"description]
[ur, "∇"description]
[ur, "μ_x, x×1"]
[rr, "⊗_x×1"']
[ur, "μ_x"description]
[uurr, bend right, "⊗_x"']
where the three diamond-shaped diagrams on the right commute due to naturality of $\mu$ as well as associativity of $\nabla$ and functoriality of $\M$ already in the monoidal strict opindexed case, whereas $(*)$ is in general $\omega$ from <ref> which in this case is an identity, and the four triangular diagrams commute due to <ref>.
§.§ Comparison with Higher-Dimensional Grothendieck Constructions
Monoidal categories are precisely bicategories with one object. As recalled in <ref>, there is a theory of fibred bicategories and indexed bicategories, and a corresponding Grothendieck construction. It is natural to consider the possibility that the monoidal Grothendieck constructions presented here are special cases of this bicategorical version. However, it is easy to see that this cannot be the case. When one restricts their view to just the objects, the bicategorical Grothendieck construction is just taking the disjoint union of the object sets of the fibres. If you consider an indexed monoidal category as a special case of an indexed bicategory, where each fibre has one object, then generally you would not expect the total bicategory to have one object. It would have as many objects as the base category. Thus, the result would not be a monoidal category. The construction given here always produces a monoidal category.
§ THE (CO)CARTESIAN CASE
In the previous section, we obtain two different equivalences between fixed-base fibrations and fixed-domain indexed categories of monoidal flavor: <ref> where both total and base categories are monoidal, and <ref> where only the fibres are monoidal. Clearly, neither of these two cases implies the other in general. The global monoidal structure as defined in <ref> sends two objects in arbitrary fibres to a new object lying in the fibre of the tensor of their underlying objects in the base, whereas having a fibre-wise tensor products does not give a way of multiplying objects in different fibres of the total category.
In <cit.>, Shulman introduces monoidal fibrations (<ref>) as a building block for fibrant double categories. Due to the nature of the examples, the results restrict to the case where the base of the monoidal fibration $P \maps \A \to \X$ is equipped with specifically a cartesian or cocartesian monoidal structure; the main idea is that these fibrations form a “parameterized family of monoidal categories”. Formally, a central result therein lifts the Grothendieck construction to the monoidal setting, by showing an equivalence between monoidal fibrations over a fixed (co)cartesian base and ordinary pseudofunctors into $\Mon\Cat$.
If $\X$ is cartesian monoidal,
\begin{equation}\label{eq:Shulmanequiv}
\MonFib(\X) \simeq \TCat_\pse(\X\op, \Mon\Cat)
\end{equation}
Dually, if $\X$ is cocartesian monoidal, $\MonOpFib(\X) \simeq \TCat_\pse(\X, \Mon\Cat)$.
Bringing all these structures together, we obtain the following.
If $\X$ is a cartesian monoidal category,
[ampersand replacement=&, sep=.25in]
[r, "≃"]
[d, "≃"'anchor=south, rotate=90, inner sep=.5mm]
_(, )
[d, "≃"anchor=south, rotate=270, inner sep=.5mm]
[r, "≃"]
_(, )
Dually, if $\X$ is a cocartesian monoidal category,
[ampersand replacement=&, sep=.25in]
[r, "≃"]
[d, "≃"'anchor=south, rotate=90, inner sep=.5mm]
_(, )
[d, "≃"anchor=south, rotate=270, inner sep=.5mm]
[r, "≃"]
_(, )
In the strict context, the restricted equivalences give a correspondence between monoidal split fibrations over $\X$ and functors $\X^{\op} \to \Mon\Cat_\mathrm{st}$, and between monoidal split opfibrations over $\X$ and functors $\X \to \Mon\Cat_\mathrm{st}$.
The original proof of <ref> is an explicit, piece-by-piece construction of an equivalence, and employs the reindexing functors $\Delta^*$ and $\pi^*$ induced by the diagonal and projections in order to move between the appropriate fibres and build the required structures. The global monoidal structure is therein called external and the fibre-wise internal.
Here we present a different argument that does not focus on the fibrations side. The equivalence between lax monoidal pseudofunctors $\X\op \to \Cat$ and ordinary pseudofunctors $\X\op \to \Mon\Cat$, which essentially provides a way of transferring the monoidal structure from the target category to the functor itself and vice versa, brings a new perspective on the behavior of such objects.
For any two monoidal 2-categories $\K$ and $\L$, the following are true.
* For an arbitrary 2-category $\A$,
\begin{equation}\label{eq:equiv1}
\TCat_\pse(\A, \MonTCat_\pse(\K, \L))\simeq\MonTCat_\pse(\K, \TCat_\pse(\A, \L))
\end{equation}
* For a cocartesian 2-category $\A$,
\begin{equation}\label{eq:equiv2}
\TCat_\pse(\A, \MonTCat_\pse(\K, \L))\simeq\MonTCat_\pse(\A\times\K, \L)
\end{equation}
First of all, recall <cit.> that there are equivalences
_(, _(, Ł))≃_(×, Ł)≃_(, _(, Ł))
which underlie <ref> and <ref> for the respective pseudofunctors; so the only part needed is the correspondence between the respective monoidal structures. Notice that $\A \times \K$ is a monoidal 2-category since both $\A$ and $\K$ are, and also $\TCat_\pse (\A, \L)$ is monoidal since $\L$ is: define $\otimes_{[]}$ and $I_{[]}$ by $(\F \otimes_{[]} \G)(a) = \F a \otimes_\L \G a$ (similarly to <ref>) and $I_{[]} \maps \A \xrightarrow{!} \1 \xrightarrow{I_\L} \L$.
First, we prove 1. Take a pseudofunctor $\F \maps \A \to \MonTCat_\pse (\K, \L)$. For every $a \in \A$, its image pseudofunctor $\F a$ is lax monoidal, i.e. comes equipped with maps in $\L$:
\begin{equation}
\label{eq:Faweakmon}
\phi_{x, y}^a \maps (\F a) (x) \otimes_\L (\F a) (y) \to (\F a) (x \otimes_\K y), \quad \phi_0^a \maps I_\L \to (\F a) I_\K
\end{equation}
for every $x, y \in \K$, satisfying coherence axioms.
Now define the pseudofunctor $\overline{\F} \maps \K \to \TCat_\pse(\A, \L)$, with $(\overline{\F} x) (a) := (\F a) (x)$. It has a lax monoidal structure, given by pseudonatural transformations
x ⊗_[] y ⇒ (x ⊗_y), I_[] ⇒(I_)
whose components evaluated on some $a \in \A$ are defined to be <ref>. Pseudonaturality and lax monoidal axioms follow, and in a similar way we can establish the opposite direction and verify the equivalence.
Now, we prove 2. If $\A$ is a cocartesian monoidal 2-category, a lax monoidal pseudofunctor $\F \maps \A \to \MonTCat_\pse (\K, \L)$ induces a pseudofunctor $\tilde{\F} \maps \A \times \K \to \L$ by $\tilde{\F} (a, x) := (\F a) (x)$. Its lax monoidal structure is given by the composite
[row sep=.1in, column sep=.2in]
(a, x) ⊗_Ł(b, y)
[d, equal]
[rr, dashed, "ψ_(a, x), (b, y)"]
(a + b, x ⊗_y)
[d, equal]
(a)(x) ⊗_Ł(b)(y)
[rdd, "(ι_a)_x ⊗(ι_b)_y"']
((a+b))(x ⊗_y)
((a+b))(x) ⊗_Ł((a+b))(y)
[uur, "ϕ^a+b_x, y"']
where $a \xrightarrow{\iota_a} a + b \xleftarrow{\iota_b} b$ are the inclusions, and $\psi_0 \maps I_\L \xrightarrow{\phi_0^0} \tilde{\F} (0, I_\K)$; the respective axioms follow.
In the opposite direction, starting with some pseudofunctor $\G \maps \A \times \K \to \L$ equipped with a lax monoidal structure $\psi_{(a, x), (b, y)}$ and $\psi_0$, we can build $\hat{\G} \maps \A \to \MonTCat_\pse(\K, \L)$ for which every $\hat{\G}a$ is a lax monoidal pseudofunctor, via
\begin{gather*}
\begin{tikzcd}[row sep=.1in, column sep=.5in, ampersand replacement=\&]
(\hat{\G}a)(x) \otimes_\L (\hat{\G}b)(y)
\arrow[d, equal]
\arrow[rr, dashed, "\phi^a_{(x, y)}"]
\&\&
(\hat{\G}a)(x\otimes_\K y)
\arrow[d, equal]
\\
\G(a, x) \otimes_\L \G(a, y)
\arrow[r, "{\psi_{(a, x), (a, y)}}"']
\&
\G(a + a, x \otimes_\K y)
\arrow[r, "{G(\nabla, 1)}"']
\&
\G(a, x\otimes_\K y)
\end{tikzcd}\\
\phi_0^a \maps I_\L \xrightarrow{\psi_0} G(0, I_\K) \xrightarrow{G(!, 1)} G(a, I_\K)
\end{gather*}
The equivalence follows, using the universal properties of coproducts and initial object.
The top and bottom right 2-categories of the first square are equivalent as follows, where $\X\op$ is cocartesian.
\begin{align*}
\TCat_\pse(\X\op, \Mon\Cat)
& \simeq \TCat_\pse (\X\op, \PsMon(\Cat)) & \text{\cref{eq:PsMon}}\\
& \simeq \TCat_\pse (\X\op, \MonTCat_\pse(\1, \Cat)) & \text{\cref{eq:equiv2}} \\
& \simeq \MonTCat_\pse (\X\op \times \1, \Cat) \\
& \simeq \MonTCat_\pse(\X\op, \Cat)
\end{align*}
The strict context equivalence can be explicitly verified as a special case of the above, where the corresponding 1-cells and 2-cells are as described in <ref> and <ref>.
The decisive step in the above proof is
the much broader <ref>; for a grounded explanation of the correspondence of the relevant structures, see <ref>.
In simpler words, a lax monoidal structure of a pseudofunctor $F\maps(\A, +, 0)\to(\Cat, \times, \1)$ gives a pseudofunctor $F\maps\A\to\Mon\Cat$ and vice versa: in a sense, `monoidality' can move between the functor and its target.
As another corollary of <ref>, we can formally deduce that pseudomonoids in $(\ICat(\X), \boxtimes, \Delta\1)$ are functors into $\Mon\Cat$, as described at the end of <ref>.
For any $\X$, $\PsMon(\ICat(\X))\simeq\TCat_\pse(\X\op, \Mon\Cat)$.
There are equivalences
\begin{align*}
\PsMon(\ICat(\X))
& = \PsMon(\TCat_\pse(\X\op, \Cat))
\\& \simeq \MonTCat_\pse (\1, \TCat_\pse(\X\op, \Cat)) & \text{\cref{eq:equiv1}}
\\& \simeq \TCat_\pse(\X\op, \MonTCat_\pse(\1, \Cat)) & \text{\cref{eq:PsMon}}
\\& \simeq \TCat_\pse(\X\op, \PsMon(\Cat))
\\& \simeq \TCat_\pse(\X\op, \Mon\Cat)
\end{align*}
as desired.
As a first and meaningful example of <ref>, recall that the categories $\Fib$ and $\ICat$ are themselves fibred over $\Cat$, with fibres $\Fib(\X)$ and $\ICat(\X)$ respectively. The base category in both cases is the cartesian monoidal category $(\Cat, \times, 1)$, therefore <ref> applies. The following proposition shows that the monoidal structures of $\Fib$, $\ICat$ and $\Fib(\X)$, $\ICat(\X)$, instrumental for the study of global and fibre-wise monoidal structures, follow the very same abstract pattern.
The fibrations $\Fib\to\Cat$ and $\ICat\to\Cat$ are monoidal, and moreover their fibres $\Fib(\X)$ and $\ICat(\X)$ are monoidal and the reindexing functors are strong monoidal.
The pseudofunctors inducing $\Fib\to\Cat$ and $\ICat\to\Cat$ are
[row sep=.05in]
[mapsto, rr]
[dd, "F"']
[mapsto, rr]
[dd, "F"']
[mapsto, rr]
[uu, "F^*"']
[mapsto, rr]
[uu, "-∘F"']
where $\mathsf{CAT}$ is the 2-category of possibly large categories, $F^*$ takes pullbacks along $F$ and $-\circ F\op$ precomposes with the opposite of $F$. These are both lax monoidal, with the respective structures essentially being <ref> and <ref> giving the global monoidal structure on the fibrations.
Since the base of both monoidal fibrations is cartesian, the global monoidal structure is equivalent to a fibre-wise monoidal structure, as per the theme of this whole section. The induced monoidal structure on each $\Fib(\X)$ is given by <ref> and on each $\ICat(\X)$ by <ref>, and $F^*$, $-\circ F\op$ are strong monoidal functors accordingly.
The above essentially lifts the global and fibre-wise monoidal structure development one level up, exhibiting fibrations and indexed categories as examples of the monoidal Grothendieck construction themselves.
Concluding this investigation on monoidal structures of fibrations and indexed categories, we consider the (co)cartesian monoidal (op)fibration case; for example, a monoidal fibration $P\maps(\A, \times, 1)\to(\X, \times, 1)$ as in <ref> where $P$ preserves products (or coproducts for opfibrations) on the nose. As remarked in <cit.>, the equivalence <ref> restricts to one between pseudofunctors which land to cartesian monoidal categories, and monoidal fibrations where the total category is cartesian monoidal. With the appropriate 1-cells and 2-cells that preserve the structure, we can write the respective equivalences as
\begin{gather}
\TCat_\pse(\X\op, \Cart) \simeq \cMonFib(\X) \textrm{ for cartesian $\X$}\label{eq:cocartspecialcase} \\
\TCat_\pse(\X, \Cocart) \simeq \cocMonOpFib(\X) \textrm{ for cocartesian $\X$}\nonumber
\end{gather}
where the prefixes $\mathsf{c}$ and $\mathsf{coc}$ correspond to the respective (co)cartesian structures. Explicitly, in order for the total category to specifically be endowed with (co)cartesian monoidal structure, it is required not only that the base category is but also the fibres are and the reindexing functors preserve finite (co)products.
This special case of the monoidal Grothendieck construction that connects the existence of (co)products and initial/terminal object in the fibres and in the total category, is reminiscent (and also an example of) the general theory of fibred limits originated from <cit.>. Explicitly, <cit.> deduces that if the base of a fibration $P \maps \A \to \X$ has $\J$-limits for any small category $\J$, then the fibres have and the reindexing functors preserve $\J$-limits if and only if $\A$ has $\J$-limits and $P$ strictly preserves them, and dually for opfibrations and colimits. Hence for finite (co)products in (op)fibrations, <ref> re-discovers that result using the monoidal Grothendieck correspondence.
Moreover, since the squares of <ref> reduce to their (co)cartesian variants, we would like to identify the conditions that the corresponding lax monoidal pseudofunctor into $\Cat$ needs to satisfy in order to give rise to a (co)cartesian monoidal (op)fibration. We employ <ref> to tackle the opfibration case: if, in a symmetric monoidal category $\X$, there exist monoidal natural transformations with components
∇_x x ⊗x →x, u_x I →x
satisfying the commutativity of
\begin{equation}\label{eq:nabla}
\begin{tikzcd}
I \otimes x
\arrow[r, "u_x\otimes1"]
\arrow[dr, "\sim"{rotate=-30}, "\ell_x"']
x\otimes x
\arrow[d, "\nabla_x"]
x\otimes I
\arrow[dr, "\sim"{rotate=-30}, "r_x"']
\arrow[r, "1\otimes u_x"]
x\otimes x
\arrow[d, "\nabla_x"]
\\&
\end{tikzcd}
\end{equation}
then $\X$ is cocartesian monoidal. In fact, it is the case that a symmetric monoidal category is cocartesian if and only if $\Mon(\X)\cong\X$.
Suppose $(\M, \mu, \mu_0) \maps \X\to\Cat$ is a (symmetric) lax monoidal pseudofunctor, such that the corresponding Grothendieck category $(\inta \M, \otimes_\mu, I_\mu)$ described in <ref> is cocartesian monoidal. This means there are monoidal natural transformations with components
\begin{equation*}
\nabla_{(x, a)}
\maps (x, a) \otimes_{\mu} (x, a)
\to (x, a)
\quad\textrm{and}\quad u_{(x, a)}
\maps (I, \mu_0(*))
\to (x, a)
\end{equation*}
making the diagrams <ref> commute.
Explicitly, by <ref>, $\nabla_{(x, a)}$ consists of morphisms
$f_x \maps x \otimes x \to x$ in $\X$ and
$\kappa_a\maps (\M f_x)(\mu_{x, x}(a, a)) \to a$ in $\M x$,
whereas $u_{(x, a)}$ consists of
$i_x \maps I \to x$ in $\X$ and
$\lambda_a \maps (\M i_x)\mu_0 \to a$
in $\M x$.
The conditions <ref> say that the composites
\[
(I, \mu_0) \otimes_\mu (x, a) \xrightarrow{u_{(x, a)} \otimes_\mu 1_{(x, a)}} (x, a) \otimes_\mu (x, a) \xrightarrow{\nabla_{(x, a)}} (x, a)
\]
\[
(x, a) \otimes_\mu (I, \mu_0) \xrightarrow{1_{(x, a)} \otimes_\mu u_{(x, a)}} (x, a) \otimes_\mu (x, a) \xrightarrow{\nabla_{(x, a)}} (x, a)
\]
are equal to the left and right unitor on $x$, where all respective structures are detailed in <ref>. Using the composition inside $\inta \M$ analogously to <ref>, these conditions translate, on the one hand, to the base being cocartesian monoidal $(\X, +, 0)$ with $f_x=\nabla_x$ and $i_x=u_x$. On the other hand, $\kappa_a$ and $\lambda_a$ form natural transformations
\begin{equation}\label{kappalambda}
\begin{tikzcd}[row sep=.1in, column sep=.2in]
\M x \times \M x
\arrow[r, "\mu_{x, x}"]
\M (x+x)
\arrow[dr, "\M(\nabla_x)"]
\\
\M x
\arrow[ur, "\Delta"]
\arrow[rrr, bend right=20, "1"']
\arrow[rrr, phantom, "\Downarrow {\scriptstyle \kappa^x}"]
\M x
\end{tikzcd}\quad
\begin{tikzcd}[row sep=.1in, column sep=.2in]
& \1
\arrow[r, "\mu_0"]
\M(0)
\arrow[dr, "\M(u_x)"]
\\
\M x
\arrow[ur, "!"]
\arrow[rrr, bend right=20, "1"']
\arrow[rrr, phantom, "\Downarrow{\scriptstyle \lambda^x}"]
\M x
\end{tikzcd}
\end{equation}
satisfying the commutativity of
[column sep=.2in, row sep=.2in]
(∇_x ∘(u_x + 1)) (μ_0, x (μ_0(*), a))
[ddd, "id"']
[rr, "∼"', "δ"]
((∇_x) ∘(u_x + 1)) ((μ_0, x(μ_0(*), a))
[d, "∼"anchor=south, rotate=90, inner sep=.5mm, "(∇_x)(μ_u_x, 1)"]
(∇_x) (μ_x, x ((u_x) (μ_0(*), a)))
[d, "(∇_x) (μ_x, x (λ^x_a, γ))"]
(∇_x) (μ_x, x (a, a))
[d, "κ^x_a"]
(ℓ_x) (μ_0, x (μ_0(*), a))
[rr, "ξ", "∼"']
and a similar one with $\mu_0$ on second arguments. The above greatly simplifies if $\M$ is just a lax monoidal functor:
the first condition becomes $1_a \cong \kappa^x_a \circ \M(\nabla_x) (\mu_{x, x} (\lambda_a^x, 1))$, and the second one $1_a \cong \kappa^x_a \circ \M(\nabla_x) (\mu_{x, x} (1_a, \lambda_a^x))$.
A lax monoidal pseudofunctor $\M\maps(\X, +, 0)\to(\Cat, \times, \1)$ equipped with natural transformations $\kappa$ and $\lambda$ as in <ref> corresponds to an ordinary pseudofunctor $\M\maps\X\to\Cocart$, or equivalently <ref> to a cocartesian monoidal opfibration.
§ EXAMPLES
In this section, we explore certain settings where the equivalence between monoidal fibrations and monoidal indexed categories naturally arises. Instead of going into details that would result in a much longer text, we mostly sketch the appropriate example cases up to the point of exhibition of the monoidal Grothendieck correspondence, providing indications of further work and references for the interested reader.
§.§ Fundamental Bifibration
For any category $\X$, the codomain or fundamental opfibration is the usual functor from its arrow category
\[\cod \maps \X^2 \longrightarrow \X\]
mapping every morphism to its codomain and every commutative square to its right-hand side leg. It uniquely corresponds to the strict opindexed category, i.e. mere functor
\begin{equation}\label{eq:fundamentalindexedcat}
\begin{tikzcd}[row sep=.05in]
\X
\arrow[r]
\Cat
\\
\arrow[r, mapsto]
\arrow[dd, "f"']
\X/x
\arrow[dd, "f_!"]
\\\\
\arrow[mapsto, r]
\X/y
\end{tikzcd}
\end{equation}
that maps an object to the slice category over it and a morphism to the post-composition functor $f_!=f\circ-$ induced by it.
If the category has a monoidal structure $(\X, \otimes, I)$, this (2-)functor naturally becomes lax monoidal with structure maps
\begin{equation}\label{eq:slicelaxator}
\X/x\times\X/y\xrightarrow{\otimes}\X/(x\otimes y), \quad \1\xrightarrow{1_I}\X/I.
\end{equation}
These components form strictly natural transformations, and for example the invertible modification $\omega$ <ref> has components the evident isomorphisms, for $(f, g, h)\in\X/x\times\X/y\times\X/z$, between
\begin{align}
a \otimes (b \otimes c)& \xrightarrow{f \otimes (g \otimes h)} x \otimes (y \otimes z) \cong (x \otimes y) \otimes z \label{eq:pseudoassociativity}\\
(a \otimes b) \otimes c & \xrightarrow{(f \otimes g) \otimes h} (x \otimes y) \otimes z\nonumber
\end{align}
By <ref>, this monoidal strict opindexed category correspondes to a monoidal split fibration, i.e. $(\X^\2, \otimes, 1_I)$ is monoidal and $\cod$ strict monoidal, where $\otimes_{\X^\2}$ strictly preserves cartesian liftings via $f_!k \otimes g_! \ell = (f \otimes g)_! (k \otimes \ell)$ – which can of course be independently verified. However in general, the slice categories $\X/x$ do not inherit the monoidal structure: there is no way to restrict the global monoidal structure to a fibrewise one.
According to <ref>, there is an induced monoidal structure on the categories $\X/x$ and a strict monoidal structure on all $f_!$ only when the monoidal structure on $\X$ is given by binary coproducts and an initial object (i.e. cocartesian). In that case, for each $k\maps a\to x$ and $\ell\maps b\to x$ in the same fibre $\X/x$, their tensor product in $\X/x$ is given by
as a simple example of <ref>.
In fact, this is precisely the coproduct of two objects in $\X/x$, and $0\xrightarrow{!}x$ the initial object, due to the way colimits in the slice categories are constructed. Therefore this falls under the cocartesian-fibres special case <ref>, bijectively corresponding to the cocartesian structure on $\X^\2$ inherited from $\X$.
Now suppose an ordinary category $\X$ has pullbacks. This endows the codomain functor also with a fibration structure, corresponding to the indexed category
[row sep=.05in]
[r, mapsto]
[dd, "f"']
[mapsto, r]
[uu, "f^*"']
with the same mapping on objects as <ref> but by taking pullbacks rather than post-composing along morphisms, a pseudofunctorial assignment. This gives $\cod\maps \X^2\to\X$ a bifibration structure, also by that classic fact that $f_!\dashv f^*$.
In this case, if $\X$ has a general monoidal structure, there is no naturally induced lax monoidal structure of that pseudofunctor as before: there is no reason for the pullback of a tensor to be isomorphic to the tensor of two pullbacks.
However, if $\X$ is cartesian monoidal (hence has all finite limits), the components
/x×/y/(x×y), /1
are pseudonatural since pullbacks commute with products. Moreover, this bijectively corresponds to monoidal fibres and strong monoidal reindexing functors, in fact also cartesian ones: for morphisms $k \maps a\to x$ and $\ell \maps b\to x$ in $\X/x$, their induced product is given by
[d, "δ^*(k×ℓ)"']
[dr, phantom, very near start, "⌟"]
a ×b
[d, "k×ℓ"]
[r, "δ"]
x ×x
and $1_x \maps x\to x$ is the unit of each slice $\X/x$, this indexed monoidal category also described in <cit.>.
The monoidal fibration structure on $\cod \maps (\X^2, \times, 1_1) \to (X, \times, 1)$ is the evident one, so it again falls in the special case <ref> now for cartesian fibres, by construction of products in slice categories.
As a final remark, analogous constructions hold for the domain functor which is again a bifibration: its fibration structure comes from pre-composing along morphisms, whereas its opfibration structure comes from taking pushouts along morphisms.
§.§ Family Fibration: Zunino and Turaev Categories
Recall that for any category $\C$, the standard family fibration is induced by the (strict) functor
\begin{equation}\label{eq:functV}
[-, \C]\maps \Set\op\to\Cat
\end{equation}
which maps every discrete category $X$ to the functor category $[X, \C]$ and every function $f \maps X \to Y$ to the functor $f^* = [f, 1]$, i.e. pre-composition with $f$.
The total category of the induced fibration $\Fam(\C)\to\C$ has as objects pairs $(X, M\maps X\to\C)$ essentially given by a family of $X$-indexed objects in $\C$, written $\{M_x\}_{x\in X}$, whereas the morphisms are
[column sep=.7in, row sep=.2in]
[dr, "M"]
[dd, "f"']
[r, phantom, "⇓α"description]
[ur, "N"']
namely a function $f\maps X\to Y$ together with families of morphisms $\alpha_x\maps M_x\to N_{fx}$ in $\C$. Notice the similarity of this description with <ref>, which for the strict indexed categories case looks like a non-discrete version of the family fibration, for $\C=\Cat$. Moreover, it is a folklore fact that $\Fam(\C)$ is the free coproduct cocompletion on the category $\C$.
On the other hand, we could consider the opfibration induced by the very same functor <ref>, denoted by $\Maf(\C)\to\Set\op$. The objects of $\Maf(\C)$ are the same as $\Fam(\C)$, but morphisms $\{M_x\}_{x\in X}\to\{N_y\}_{y\in Y}$ between them are functions $g\maps Y\to X$ (i.e. $X\to Y$ in $\Set\op$) together with families of arrows $\beta_y\maps M_{gy}\to N_y$ in $\C$. Notice that these are now indexed over the set $Y$ rather than $X$ like before, and in fact $\Maf (\X) = \Fam (\X\op)\op$.
In the case that the category is monoidal $(\C, \otimes, I)$, the (2-)functor $[-, \C]$ has a canonical lax monoidal structure. Explicitly, by taking its domain $\Set\op$ to be cocartesian by the usual cartesian monoidal structure $(\Set, \times, 1)$, the structure maps are
ϕ_X, Y [X, ] ×[Y, ] →[X ×Y, ], ϕ_0 [, ] ≅
where $\phi_{X, Y}$ corresponds, under the tensor-hom adjunction in $\Cat$, to
[X, ] ×[Y, ] ×X ×Y [X, ] ×X ×[Y, ] ×Y × .
These are again natural components, and for example <ref> has components the natural isomorphisms between the assignments $Mx\otimes (Ny\otimes Uz)$ and $(Mx\otimes Ny)\otimes Uz$.
By <ref>, this monoidal strict indexed category endows the corresponding split fibration $\Fam(\X) \to \Set$ with a monoidal structure via $\{M_x\} \otimes \{N_y\} := \{M_x \otimes N_y \}_{X \times Y}$. On the other hand, we could use the dual part of the same theorem, and instead consider the induced monoidal split opfibration $\Maf(\X) \to \Set\op$ corresponding to the same $([-, \C], \phi, \phi_0)$.
Moreover, since $\Set$ is cartesian, <ref> also applies in both cases, giving a monoidal structure to the fibres as well: for $M\maps X\to\C$ and $N\maps X\to\C$, their fibrewise tensor product and unit are given by
XX×X×, X1
which are precisely constructed as in <ref>. Once again, notice the direct similary with <ref>, the fibrewise monoidal structure on $\ICat(\X)$.
As an interesting example, consider $\C=\Mod_R$ for a commutative ring $R$, with its usual tensor product $\otimes_R$. In <cit.>, the authors introduce a category $\mathcal{T}$ of Turaev $R$-modules, as well as a category $\mathcal{Z}$ of Zunino $R$-modules, which serve as symmetric monoidal categories where group-(co)algebras and Hopf group-(co)algebras, <cit.>, live as (co)monoids and Hopf monoids respectively.
In more detail, the objects of both $\mathcal{T}$ and $\mathcal{Z}$ are defined to be pairs $(X, M)$ where $X$ is a set and $\{M_x\}_{x\in X}$ is an $X$-indexed family of $R$-modules, and their morphisms are respectively
s M_g(y) →N_y in _R
g Y →X in
tM_x→N_f(x) in _R
fX→Y in
There is a symmetric pointwise monoidal structure, $\{M_x\otimes_R N_y\}_{X\times Y}$, and there are strict monoidal forgetful functors $\mathcal{T}\to\Set\op$, $\mathcal{Z}\to\Set$.
It is therein shown that comonoids in $\mathcal{T}$ are monoid-coalgebras and monoids in $\mathcal{Z}$ are monoid-algebras, i.e. families of $R$-modules indexed over a monoid, together with respective families of linear maps
\begin{gather*}
(\mathcal{T})\quad C_{g*h}\to C_g\otimes C_h \qquad (\mathcal{Z})\quad
A_g\otimes A_h\to A_{g*h} \\
C_e\to R \qquad \phantom{ZZZZZZZ}R\to A_e
\end{gather*}
satisfying appropriate axioms. Based on the above, it is clear that $\mathcal{T}=\Maf(\Mod_R)$ and $\mathcal{Z}=\Fam(\Mod_R)$, which clarifies the origin of these categories and can be directly used to further generalize the notions of Hopf group-(co)monoids in arbitrary monoidal categories.
§.§ Global Categories of Modules and Comodules
For any monoidal category $\mathcal{V}$,
there exist global categories of modules and comodules, denoted by $\Mod$ and $\Comod$ <cit.>.
Their objects are all (co)modules over (co)monoids in $\V$, whereas a morphism between an $A$-module $M$ and a $B$-module $N$ is given by a monoid map $f\maps A\to B$ together with a morphism $k\maps M\to N$ in $\V$ satisfying the commutativity of
A⊗M[rr, "μ"][d, "1⊗k"'] M[d, "k"]
A⊗N[r, "f⊗1"'] B⊗N[r, "μ"'] N
where $\mu$ denotes the respective action, and dually for comodules.
Both these categories arise as the total categories induced by the Grothendieck construction on the functors
\begin{equation}\label{eq:functors}
\begin{tikzcd}[row sep = tiny]
\Mon(\V)\op
\arrow[r]
\Cat
\Comon(\V)
\arrow[r]
\Cat
\\
\arrow[r, dashed, mapsto]
\arrow[dd, swap, "f"]
\Mod_\V(A)
\arrow[r, dashed, mapsto]
\arrow[dd, swap, "g"]
\Comod_\V(C)
\arrow[dd, "g_!"]
\\{}\\
\arrow[r, dashed, mapsto]
\Mod_\V(B)
\arrow[uu, swap, "f^*"]
\arrow[r, dashed, mapsto]
\Comod_\V(D)
\end{tikzcd}
\end{equation}
where $f^*$ and $g_!$ are (co)restriction of scalars: if $M$ is a $B$-module, $f^*(M)$ is an $A$-module via the action
A ⊗M B ⊗M M.
The induced split fibration and opfibration, $\Mod \to \Mon(\V)$ and $\Comod \to \Comon(\V)$, map a (co)module to its respective (co)monoid.
Recall that when $(\V, \otimes, I, \sigma)$ is braided monoidal, its categories of monoids and comonoids inherit the monoidal structure: if $A$ and $B$ are monoids, then $A\otimes B$ has also a monoid structure via
A ⊗B ⊗A ⊗B A ⊗A ⊗B ⊗B A ⊗B, I ≅I ⊗I A ⊗B
where $m$ and $j$ give the respective monoid structures.
In that case, the induced split fibration and opfibration are both monoidal. This can be deduced by directly checking the conditions of <ref>, as was the case in the relevant references, or in our setting by using <ref> since both (2-)functors <ref> are lax monoidal. For example, for any $A, B \in \Mon(\V)$ there are natural maps
ϕ_A, B _(A) ×_(B) →_(A ⊗B) ϕ_0 →_(I)
with $\phi_{A, B} (M, N) = M \otimes N$, with the $A \otimes B$-module structure being
A ⊗B ⊗M ⊗N A ⊗M ⊗B ⊗N M ⊗N
and $\phi_0(*)=I$, which are pseudoassociative and pseudounital in the sense that e.g. for any $M, N, P\in\Mod_\mathcal{V}(A)\times\Mod_\mathcal{V}(B)\times\Mod_\mathcal{V}(C)$, $M\otimes(N\otimes P)$ is only isomorphic to $(M\otimes N)\otimes P$ as $(A\otimes B)\otimes C$-modules.
Notice that in general, the monoidal bases $\Mon(\V)$ and $\Comon(\V)$ are not (co)cartesian, since they have the same tensor as $(\V, \otimes, I, \sigma)$. Therefore this case does not fall under <ref>, hence the fibre categories are not monoidal. For example in $(\mathsf{Vect}_k, \otimes_k, k)$, the $k$-tensor product of two $A$-modules for a $k$-algebra $A$ is not an $A$-module as well.
We remark that the induced monoidal opfibration $\Comod \to \Comon(\V)$ in fact serves as the monoidal base of an enriched fibration structure on $\Mod \to \Mon (\V)$ as explained in <cit.>, built upon an enrichment between the monoidal bases $\Mon(\V)$ in $\Comon(\V)$ established in <cit.>. Moreover, analogous monoidal structures are induced on the (op)fibrations of monads and comonads in any fibrant monoidal double category, see <cit.>.
§.§ Systems as Monoidal Indexed Categories
In <cit.> as well as in earlier works e.g. <cit.>, the authors investigate a categorical framework for modeling systems of systems using algebras for a monoidal category. In more detail, systems in a broad sense are perceived as lax monoidal pseudofunctors
where $\mathcal{W}_\C$ is the monoidal category of $\C$-labeled boxes and wiring diagrams with types in a finite product category $\mathcal{C}$. Briefly, the objects in $\mathcal{W}_\mathcal{C}$ are pairs $X=(X^\mathrm{in}, X^\mathrm{out})$ of finite sets equipped with functions to $\ob\mathcal{C}$, thought of as boxes
[oriented WD, bbx=.1cm, bby =.1cm, bb port sep=.15cm]
[bb=33] (X) $X$;
node[left=.1 of X_in1] $a_1$
node[left=.1 of X_in2] $\dotso$
node[left=.1 of X_in3] $a_m$
node[right=.1 of X_out1] $b_1$
node[right=.1 of X_out2] $\dotso$
node[right=.1 of X_out3] $b_n$;
where $X^\mathrm{in}=\{a_1, \ldots, a_m\}$ are the input ports, $X^\mathrm{out}=\{b_1, \ldots, b_n\}$
the output ones and all wires are associated to a $\mathcal{C}$-object expressing the type of information that can go through them. A morphism $\phi\maps X\to Y$ in this category consists
of a pair of functions
that respect the $\mathcal{C}$-types,
which roughly express which port is `fed information' by which.
Graphically, we can picture it as
\begin{equation}\label{eq:wiringdiagpic}
\begin{tikzpicture}[oriented WD, baseline=(Y.center), bbx=2em, bby=1.2ex, bb port sep=1.2]
\node[bb={6}{6}] (X) {};
\node[bb={2}{3}, fit={($(X.north east)+(0.7, 1.7)$) ($(X.south west)-(.7, .7)$)}] (Y) {};
\node [circle, minimum size=4pt, inner sep=0, fill] (dot1) at ($(Y_in1')+(.5, 0)$) {};
\node [circle, minimum size=4pt, inner sep=0, fill] (dot2) at ($(X_out4)+(.5, 0)$) {};
\draw[ar] (Y_in1') to (dot1);
\draw[ar] (X_out4) to (dot2);
\draw[ar] (Y_in2') to (X_in5);
\draw[ar] (Y_in2') to (X_in4);
\draw[ar] (X_out5) to (Y_out3');
\draw[ar] (X_out2) to (Y_out1');
\draw[ar] (X_out2) to (Y_out2');
\draw[ar] let \p1=(X.north west), \p2=(X.north east), \n1={\y1+\bby}, \n2=\bbportlen in
(X_out1) to[in=0] (\x2+\n2, \n1) -- (\x1-\n2, \n1) to[out=180] (X_in1);
\draw[ar] let \p1=(X.north west), \p2=(X.north east), \n1={\y1+2*\bby}, \n2=\bbportlen in
(X_out1) to[in=0] (\x2+\n2, \n1) -- (\x1-\n2, \n1) to[out=180] (X_in2);
\draw[ar] let \p1=(X.south west), \p2=(X.south east), \n1={\y1-\bby}, \n2=\bbportlen in
(X_out6) to[in=0] (\x2+\n2, \n1) -- (\x1-\n2, \n1) to[out=180] (X_in6);
\draw [label]
node at ($(Y.north east)-(.5cm, .3cm)$) {$Y$}
node at ($(X.north east)-(.4cm, .3cm)$) {$X$}
node[left=.1 of X_in3] {$\dotso$}
node[right=.1 of X_out3] {$\dotso$}
node[above=of Y.north] {$\phi\maps X\to Y$}
\end{tikzpicture}
\end{equation}
Composition of morphisms can be thought of a zoomed-in picture of three boxes, and the monoidal structure amounts to parallel placement of boxes as in
[oriented WD, baseline=(Y.center), bbx=1.3em, bby=1ex, bb port sep=.06cm]
[bb=33] (X1) ;
[bb=33, below =.5 of X1] (X2) ;
[fit=(X1)(X2), draw] ;
node at ($(X1.west)+(1, 0)$) $X_1$
node at ($(X2.west)+(1, 0)$) $X_2$
node[left=.1 of X1_in2] $\dotso$
node[right=.1 of X1_out2] $\dotso$
node[left=.1 of X2_in2] $\dotso$
node[right=.1 of X2_out2] $\dotso$;
(X1_in1) – (-2.5, 1.7);
(X1_out1) – (2.5, 1.7);
(X1_in3) – (-2.5, -1.7);
(X1_out3) – (2.5, -1.7);
(X2_in1) – (-2.5, -5.6);
(X2_out1) – (2.5, -5.6);
(X2_in3) – (-2.5, -9.1);
(X2_out3) – (2.5, -9.1);
There is a close connection between the definition of $\mathcal{W}_\mathcal{C}$ and that of Dialectica categories as well as lenses; such considerations are the topic of work in progress <cit.>.
The systems-as-algebras formalism uses lax monoidal pseudofunctors from this category
$\mathcal{W}_\mathcal{C}$ to $\Cat$ that essentially receive a general picture such as
[oriented WD, bb min width =.5cm, bbx=.5cm, bb port sep =1, bb port length=.08cm, bby=.15cm]
[bb=22, bb name = $X_1$] (X11) ;
[bb=33, below right=of X11, bb name = $X_2$] (X12) ;
[bb=21, above right=of X12, bb name = $X_3$] (X13) ;
(X11_out1) to (X13_in1);
(X11_out2) to (X12_in1);
(X12_out1) to (X13_in2);
[bb=22, below right = -1 and 1.5 of X12, bb name = $X_4$] (X21) ;
[bb=12, above right=-1 and 1 of X21, bb name = $X_5$] (X22) ;
(X21_out1) to (X22_in1);
let 1=(X22.north east), 2=(X21.north west), 1=1+, 2=in
(X22_out1) to[in=0] (1+2, 1) – (2-2, 1) to[out=180] (X21_in1);
[bb=22, fit = ($(X11.north east)+(-1, 3)$) (X12) (X13) ($(X21.south)$) ($(X22.east)+(.5, 0)$), bb name =$Y$] (Z) ;
(Z_in1') to (X11_in2);
(Z_in2') to (X12_in2);
(X12_out2) to (X21_in2);
let 1=(X22.south east), 1=1-, 2=in
(X21_out2) to (1+2, 1) to (Z_out2');
let 1=(X12.south east), 2=(X12.south west), 1=1-, 2=in
(X12_out3) to[in=0] (1+2, 1) – (2-2, 1) to[out=180] (X12_in3);
let 1=(X22.north east), 2=(X11.north west), 1=2+, 2=in
(X22_out2) to[in=0] (1+2, 1) – (2-2, 1) to[out=180] (X11_in1);
let 1=(X13_out1), 2=(X22.north east), 2=in
(X13_out1) to (1+2, 1) – (2+2, 1) to (Z_out1');
(which really takes place in the underlying operad of $\mathcal{W}_\mathcal{C}$) and assign systems of a certain kind to all inner boxes; the lax monoidal and pseudofunctorial structure of this assignment formally produce a system of the same kind for the outer box.
Examples of such systems are discrete dynamical systems (Moore machines in the finite case), continuous dynamical systems but also more general systems with deterministic or total conditions; details can be found in the provided references. Since all these systems are lax monoidal pseudofunctors from the non-cocartesian monoidal category of wiring diagrams to $\Cat$, i.e. monoidal indexed categories, the monoidal Grothendieck construction <ref> induces a corresponding monoidal fibration in each system case, and this global structure does not reduce to a fibrewise one.
For example, the algebra for discrete dynamical systems
\begin{equation}\label{eq:DDS}
\mathrm{DDS}\maps \mathcal{W}_\Set\to\Cat
\end{equation}
assigns to each box $X=(X^\mathrm{in}, X^\mathrm{out})$
the category of all discrete dynamical systems with fixed input and output sets being $\prod_{x\in X^\mathrm{in}}x$ and $\prod_{y\in X^\mathrm{out}}y$ respectively. There exist morphisms between systems of the same input and output set, but not between those with different ones. To each morphism, i.e. wiring diagram as in <ref>, $\mathrm{DDS}$
produces a functor that maps an inner discrete dynamical system to a new outer one, with changed input and output sets accordingly.
(Pseudo)functoriality of this assignment
allows the coherent zoom-in and zoom-out on dynamical systems built out of smaller dynamical systems, and monoidality allows the creation of new dynamical systems on parallel boxes.
Being a monoidal indexed category, <ref> gives rise to a monoidal opfibration over $\mathcal{W}_\Set$. Its total category $\inta \mathrm{DDS}$ has objects all dynamical systems with arbitrary input and output sets, morphisms that can now go between systems of different inputs/outputs, and also a natural tensor product inherited from that in $\mathcal{W}_\Set$ and the laxator of $\inta\mathrm{DDS}$. In a sense, this category has all the required flexibility for the direct communication (via morphisms in the total category) between any discrete dynamical system, or any composite of systems or parallel placement of them, whereas the wiring diagram algebra <ref> focuses on the machinery of building new discrete dynamical systems systems from old.
This classic change of point of view also transfers over to maps of algebras, i.e. indexed monoidal 1-cells. As an example, see <cit.>, discrete dynamical systems can naturally be viewed as general total and deterministic machines denoted by $\mathrm{Mch}^\mathrm{td}$,
via a monoidal pseudonatural transformation
\[
\begin{tikzcd}
\mathcal{W}_\Set
\arrow[dr, "\mathrm{DDS}"]
\arrow[dd]
\\
\arrow[r, phantom, "\Downarrow"]
\Cat
\\
\mathcal{W}_{\widetilde{\mathrm{Int}_N}}
\arrow[ur, "\mathrm{Mch}^\mathrm{td}", swap]
\end{tikzcd}\]
which also changes the type of input and output wires from sets to discrete interval sheaves $\widetilde{\mathrm{Int}_N}$. This gives rise to a monoidal opfibred 1-cell
which provides a direct functorial translation between the one sort of system to the other in a way compatible with the monoidal structure.
As a final note, this method of modeling certain objects as algebras for a monoidal category (a.k.a. strict or general monoidal indexed categories) carries over to further contexts than systems and the wiring diagram category. Examples include hypergraph categories as algebras on cospans <cit.> and traced monoidal categories as algebras on cobordisms <cit.>. In all these cases, the monoidal Grothendieck construction gives a potentially fruitful change of perspective that should be further investigated.
§.§ Graphs
As we show in <ref>, the category of (directed, multi) graphs, is bifibred over set, where the bifibration $\mathsf V \maps \Grph \to \Set$ is given by sending a graph to its vertex set.
Since $\mathsf V \maps \Grph \to \Set$ preserves products, then it can be given the structure of a strict monoidal monoidal functor with respect to the cartesian monoidal structures on $\Grph$ and $\Set$. Since the cartesian morphisms are those that form pullback squares, and products in $\Grph$ are given pointwise, then the monoidal structure in $\Grph$ preserves cartesian morphisms. We can then apply <ref> to obtain a symmetric lax monoidal structure for the pseudofunctor $\Grph^* \maps \Set\op \to \Cat$. The lax structure map $\gamma_{X,Y} \maps \Grph_X \times \Grph_Y \to \Grph_{X \times Y}$ is given by taking the product of the two graphs within $\Grph$. Notice the product has vertex set given by $X \times Y$. Since the base category is cartesian monoidal, we can apply <ref>, granting a symmetric monoidal structure to the fibres $\Grph_X$. The monoidal product is given by the following composite.
\[
\Grph_X \times \Grph_X \xrightarrow{\gamma_{X,X}} \Grph_{X\times X} \xrightarrow{\Delta^*} \Grph_X
\]
Simply put, this operation is given by taking the product of the two graphs on $X$, and then restricting to the vertices on the diagonal. Indeed, this is the cartesian monoidal structure on $\Grph_X$.
Since the category $\Set$ also has all finite colimits, we obtain a symmetric lax monoidal structure for the pseudofunctor $\Grph_* \maps \Set\op \to \Cat$. The lax structure map $\phi_{X,Y} \maps \Grph_X \times \Grph_Y \to \Grph_{X + Y}$ is given by taking the disjoint union of the two graphs. Notice the disjoint union has vertex set given by $X + Y$. Since the base category is cocartesian monoidal, we can apply <ref>, granting a symmetric monoidal structure to the fibres $\Grph_X$. The monoidal product is given by the following composite.
\[
\Grph_X \times \Grph_X \xrightarrow{\phi_{X,X}} \Grph_{X + X} \xrightarrow{\Delta_*} \Grph_X
\]
Simply put, this operation is given by taking the disjoint union of the edges. Indeed, this is the cocartesian monoidal structure on $\Grph_X$. This is also the overlay operation for the network model of directed multi graphs.
CHAPTER: MONOIDAL CATEGORIES
Monoidal categories lie at the center of applied category theory. This section is included mainly to establish notation and terminology used throughout this thesis. Some standard references are <cit.> and <cit.>.
§ DEFINITIONS
§.§ Monoidal, Braided, and Symmetric Categories
A monoidal category $(\C, \otimes, I, \assoc, \lambda, \rho)$ consists of
* a category $\C$
* a functor $\otimes \maps \C \times \C \to \C$ called the tensor
* a functor $I \maps 1 \to \C$ called the unit
* a natural transformation $\assoc$ with components of the form $\assoc_{x,y,z} \maps (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ called the associator
* a natural transformation $\lambda$ with components of the form $\lambda_x \maps I \otimes x \to x$ called the left unitor
* a natural transformation $\rho$ with components of the form $\rho_x \maps x \otimes I \to x$ called the right unitor
such that the following diagrams commute.
Pentagon identity:
\begin{equation}
\label{monoidalpentagon}
\begin{tikzcd}[column sep = large]
(w \otimes x) \otimes (y \otimes z)
\arrow[ddr, "\alpha_{w,x,y \otimes z}"]
\\
((w \otimes x) \otimes y) \otimes z
\arrow[ur, "\alpha_{w \otimes x,y,z}"]
\arrow[dd, swap, "\alpha_{w,x,y} \otimes 1_z"]
\\&&
w \otimes (x \otimes (y \otimes z))
\\
(w \otimes (x \otimes y)) \otimes z
\arrow[dr, swap, "\alpha_{w,x \otimes y,z}"]
\\&
w \otimes ((x \otimes y) \otimes z)
\arrow[uur, swap, "1_w \otimes \alpha_{x,y,z}"]
\end{tikzcd}
\end{equation}
Triangle identity:
\begin{equation}
\label{leftorrightor}
\begin{tikzcd}
(x \otimes I) \otimes y
\arrow[rr, "\assoc_{x, I, y}"]
\arrow[dr, swap, "\rho_x \otimes 1_y"]
x \otimes (I \otimes y)
\arrow[dl, "1_x \otimes \lambda_y"]
\\&
x \otimes y
\end{tikzcd}
\end{equation}
A strict monoidal category is one where the associator, left unitor, and right unitor are all identity.
A braided monoidal category <cit.> is a monoidal category equipped with a natural transformation $\braid$ called the braiding with components $\braid_{x,y} \maps x \otimes y \to y \otimes x$, such that the following diagrams commute.
\begin{equation}
\begin{tikzcd}
(x \otimes y) \otimes z
\arrow[r, "\assoc_{x,y,z}"]
\arrow[d, swap, "\braid_{x,y} \otimes 1_z"]
x \otimes (y \otimes z)
\arrow[d, "\braid_{x, y \otimes z}"]
\\
(y \otimes x) \otimes z
\arrow[d, swap, "\assoc_{y,x,z}"]
(y \otimes z) \otimes x
\arrow[d, "\assoc_{y,z,x}"]
\\
y \otimes (x \otimes z)
\arrow[r, swap, "1_y \otimes \braid_{x,z}"]
y \otimes (z \otimes x)
\end{tikzcd}
\qquad
\begin{tikzcd}
x \otimes (y \otimes z)
\arrow[r, "\assoc_{x,y,z}\inv"]
\arrow[d, swap, "1_x \otimes \braid_{y,z}"]
(x \otimes y) \otimes z
\arrow[d, "\braid_{x \otimes y, z}"]
\\
x \otimes (z \otimes y)
\arrow[d, swap, "\assoc_{x,z,y}\inv"]
z \otimes (x \otimes y)
\arrow[d, "\assoc_{z,x,y}\inv"]
\\
(x \otimes z) \otimes y
\arrow[r, swap, "\braid_{x,z} \otimes 1_y"]
(z \otimes x) \otimes y
\end{tikzcd}
\end{equation}
A symmetric monoidal category is a braided monoidal category where the braiding satisfies the equation $\braid_{y,x} \circ \braid_{x,y} = 1_{x \otimes y}$ for all objects $x,y \in \C$. A commutative monoidal category is a symmetric monoidal category where the braiding is identity.
For general (braided/symmetric) monoidal categories, we write $\C$, $\D$, or $\E$.
§.§ Monoidal, Braided, and Symmetric Functors
Let $(\C, \otimes_\C, I_\C, \assoc^\C, \leftor^\C, \rightor^\C)$ and $(\D, \otimes_\D, I_\D, \assoc^\D, \leftor^\D, \rightor^\D)$ be monoidal categories. A lax monoidal functor from $\C$ to $\D$ consists of
* a functor $F \maps \C \to \D$
* a natural transformation with components $\phi_{x,y} \maps Fx \otimes_\D Fy \to F(x \otimes_\C y)$ called the laxator
* a natural transformation with unique component $\phi_0 \maps I_\D \to F I_\C$ called the unit laxator
such that the following diagrams commute.
\begin{equation}
\begin{tikzcd}[column sep = large]
(Fx \otimes_\D Fy) \otimes_\D Fz
\arrow[r, "\assoc^\D_{Fx,Fy,Fz}"]
\arrow[d, swap, "\phi_{x,y} \otimes_\D 1_{Fz}"]
Fx \otimes_\D (Fy \otimes_\D Fz)
\arrow[d, "1_{Fx} \otimes_\D \phi_{y,z}"]
\\
F(x \otimes_\C y) \otimes_\D Fz
\arrow[d, swap, "\phi_{x \otimes_\C y, z}"]
Fx \otimes_\D F(y \otimes_\C z)
\arrow[d, "\phi_{x, y \otimes_\C z}"]
\\
F((x \otimes_\C y) \otimes_\C z)
\arrow[r, swap, "F(\assoc^\C_{x,y,z})"]
F(x \otimes_\C (y \otimes_\C z))
\end{tikzcd}
\end{equation}
\begin{equation}
\begin{tikzcd}
I_\D \otimes_\D Fx
\arrow[r, "\leftor^\D_{Fx}"]
\arrow[d, swap, "\phi_0 \otimes_\D 1_{Fx}"]
\\
FI_\C \otimes_\D Fx
\arrow[r, swap, "\phi_{I_\C, x}"]
F(I_\C \otimes_\C x)
\arrow[u, swap, "F(\leftor^\C_x)"]
\end{tikzcd}
\qquad
\begin{tikzcd}
Fx \otimes_\D I_\D
\arrow[r, "\rightor^\D_x"]
\arrow[d, swap, "1_{Fx} \otimes_\D \phi_0"]
\\
Fx \otimes_\D FI_\C
\arrow[r, swap, "\phi_{x,I_\C}"]
F(x \otimes_\C I_\C)
\arrow[u, swap, "F(\rightor^\C_x)"]
\end{tikzcd}
\end{equation}
We say that $F$ is simply a monoidal functor when $\phi$ and $\phi_0$ are natural isomorphisms. It is worth noting that there exists a notion of “oplax” monoidal functors, where the structure map is reversed: $\phi_{x,y} \maps F(x \otimes y) \to Fx \otimes Fy$. However, oplax monoidal functors do not appear in this thesis, so we spend no further time on them.
A lax braided monoidal functor is a lax monoidal functor $(F, \phi, \phi_0) \maps (\C, \otimes_\C, I_\C) \to (\D, \otimes_\D, I_\D)$ where $\C$ and $\D$ are braided monoidal categories, with $\braid^\C$ and $\braid^\D$ being the respective braidings, such that the following diagram commutes.
\begin{equation}
\begin{tikzcd}
Fx \otimes_\D Fy
\arrow[r, "\braid^\D_{Fx,Fy}"]
\arrow[d, swap, "\phi_{x,y}"]
Fy \otimes_\D Fx
\arrow[d, "\phi_{y,x}"]
\\
F(x \otimes_\C y)
\arrow[r, swap, "F\braid^\C_{x,y}"]
F(y \otimes_\C x)
\end{tikzcd}
\end{equation}
A (lax) braided monoidal functor between symmetric monoidal categories is called a (lax) symmetric monoidal functor with no further requirements.
Composition of lax monoidal functors is strictly associative.
We get categories $\Mon\Cat_\ell$, $\Mon\Cat$, $\Br\Mon\Cat_\ell$, $\Br\Mon\Cat$, $\Sym\Mon\Cat_\ell$, and $\Sym\Mon\Cat$ where the objects are monoidal categories, the functors are monoidal categories, the prefix $\Br$ (resp. $\Sym$) indicates the objects and morphisms are braided (resp. symmetric), and the subscript $\ell$ indicated the morphisms are lax monoidal.
§.§ Monoidal Natural Transformations
Let $(F, \phi, \phi_0)$ and $(G, \gamma, \gamma_0)$ be lax monoidal functors. A monoidal natural transformation is a natural transformation $\theta \maps F \To G$ such that the following diagrams commute.
\begin{equation}
\begin{tikzcd}
Fx \otimes_\D Fy
\arrow[r, "\theta_x \otimes_\D \theta_y"]
\arrow[d, swap, "\phi_{x,y}"]
Gx \otimes_\D Gy
\arrow[d, "\gamma_{x,y}"]
\\
F(x \otimes_\C y)
\arrow[r, swap, "\theta_{x \otimes_C y}"]
G(x \otimes_\C y)
\end{tikzcd}
\qquad
\begin{tikzcd}
\arrow[dr, "\phi_0"]
\arrow[dl, swap, "\gamma_0"]
\\
\arrow[rr, swap, "\theta_{I_\C}"]
\end{tikzcd}
\end{equation}
There are no new laws which can be imposed on a monoidal natural transformation between braided or symmetric monoidal functors. So we do not specialize this concept any further.
§ EXAMPLES
Let $(M, \cdot, e)$ be a monoid. If we can consider $M$ as a discrete category, then it can be given a strict monoidal structure where the tensor is given by $\cdot$ and the unit is $e$. The functor $\Mon \hookrightarrow \Mon\Cat$ which realizes a monoid as a discrete monoidal category is full and faithful. If we think of this as “forgetting discreteness”, then discreteness is a property.
Given a monoidal category $(\C, \otimes, I)$, we can define $ \otimes\rev \maps \C \times \C \to \C$ by
\[
\begin{tikzcd}
\C \times \C
\arrow[rr, "\otimes\rev"]
\arrow[dr, swap, "\braid^\Cat_{\C,\C}"]
\C
\\&
\C \times \C
\arrow[ur, swap, "\otimes"]
\end{tikzcd}\]
This defines an idempotent automorphism on $\Mon\Cat$.
Given a monoidal category $(\C, \otimes, I)$, the category $\C$ can be equipped with a monoidal structure given by $\otimes\op \maps \C\op \times \C\op \to \C\op$ and the same unit object. This defines an idempotent automorphism on $\Mon\Cat$.
Any category $C$ with finite products can be equipped with a symmetric monoidal structure as follows. For every pair of objects $c, d$, choose some object satisfying the universal property of the product of $c$ and $d$, call it $c \times d$. Given a pair of morphisms $f \maps a \to b$ and $g \maps c \to d$, the universal property gives a morphism $f \times g \maps a \times b \to c \times d$ as follows.
\[
\begin{tikzcd}[row sep = tiny]
a \times b
\arrow[dl, "\pi_a", swap]
\arrow[dr, "\pi_b"]
\arrow[dd, dashed, "\exists!"]
\\
\arrow[dd, "f", swap]
\arrow[dd, "g"]
\\&
c \times d
\arrow[dl, "\pi_c"]
\arrow[dr, "\pi_d", swap]
\\
\end{tikzcd}\]
We claim that this defines a functor $\times \maps C \times C \to C$. Consider a pair of morphisms $(f_1, f_2) \maps (a_1, a_2) \to (b_1, b_2)$ and $(g_1, g_2) \maps (b_1, b_2) \to (c_1, c_2)$.
Since $(g_1 \circ f_1) \times (g_2 \circ g_2)$ and $(g_1 \times g_2) \circ (f_1 \times f_2)$ both make the following diagram commute, they must be equal.
\[
\begin{tikzcd}[row sep = small]
a_1 \times a_2
\arrow[dl]
\arrow[dr]
\arrow[dd, dashed, ""]
\\
\arrow[d, "f_1", swap]
\arrow[d, "f_2"]
\\
\arrow[d, "g_1", swap]
c_1 \times c_2
\arrow[dl]
\arrow[dr]
\arrow[d, "g_2"]
\\
\end{tikzcd}\]
Identity maps are preserved because the identity map on $a \times b$ makes the diagram below commute.
\[
\begin{tikzcd}[row sep = tiny]
a \times b
\arrow[dl, "\pi_a", swap]
\arrow[dr, "\pi_b"]
\arrow[dd, dashed]
\\
\arrow[dd, "1_a", swap]
\arrow[dd, "1_b"]
\\&
a \times b
\arrow[dl, "\pi_a"]
\arrow[dr, "\pi_b", swap]
\\
\end{tikzcd}\]
We define the unit object to be some chosen terminal object, call it $1$. The associator, unitors, pentagon, hexagon, braiding, hexagon law, and symmetric law can all be derived from the universal property of products. This gives $\C$ the structure of a symmetric monoidal category. This is called the cartesian monoidal structure, and $(\C, \times, 1)$ is called a cartesian monoidal category.
Any category with finite coproducts can be equipped with a symmetric monoidal structure by <ref> and <ref>.
If $\C$ is monoidal and $\D$ is a category, the functor category $\C^\D$ can be given a pointwise monoidal structure as follows. Define $\otimes_{pt} \maps \C^\D \times \C^\D \to \C^\D$ by $\otimes_{pt} = \otimes (F \times G) \circ \Delta$. The unit object $1 \to \C^\D$ is given by currying the composite $D \xrightarrow{!} 1 \xrightarrow{I} \C$. The rest of the structures and the necessary properties all carry over from their counterparts in $\C$. Similarly, if $\C$ is braided or symmetric, then $\C^\D$ can be given a pointwise braided or symmetric monoidal structure respectively.
Let $C$ be a small monoidal category. Then the Day convolution tensor product <cit.>
\[\otimes_\Day \maps \Set^{\C\op} \times \Set^{\C\op} \to \Set^{\C\op}\]
is the following left Kan extension.
\[\begin{tikzcd}
C\op \times C\op
\arrow[r, "{(X,Y)}"]
\arrow[d, "\otimes", swap]
\Set
\\
\arrow[ur, "X \otimes_\Day Y", swap, dashed]
\end{tikzcd}\]
This can be given by the following coend formula <cit.>.
\[X \otimes_\Day Y \maps c \mapsto \int^{c_1,c_2 \in C} C(c_1 \otimes c_2, c) \times X(c_1) \times Y(c_2)\]
Similarly, we can define the unit via left Kan extension.
\[\begin{tikzcd}
\arrow[r, "\Delta1"]
\arrow[d, "I", swap]
\Set
\\
\arrow[ur, "I_\Day", swap, dashed]
\end{tikzcd}\]
Day convolution gives the functor category $\Set^{\C\op}$ a monoidal structure. Many nice properties of this structure can be found in the literature, e.g. <cit.>. However, these properties are not heavily used in this thesis, so we choose to leave them out.
§ MONOID OBJECTS
A monoidal structure is exactly what a category needs to have if we want to define monoid objects in this category.
Let $(\C, \otimes, I)$ be a monoidal category. A monoid object internal to $\C$ consist of
* an object $x \in \C$
* a morphism $\mu \maps x \otimes x \to x$
* a morphism $\varepsilon \maps I \to x$
such that the following diagrams commute.
\begin{equation}
\begin{tikzcd}
(x \otimes x) \otimes x
\arrow[rr, "\assoc_{x,x,x}"]
\arrow[d, swap, "\mu \otimes 1_x"]
x \otimes (x \otimes x)
\arrow[d, "1_x \otimes \mu"]
\\
x \otimes x
\arrow[dr, swap, "\mu"]
x \otimes x
\arrow[dl, "\mu"]
\\&
\end{tikzcd}
\end{equation}
\begin{equation}
\begin{tikzcd}
I \otimes x
\arrow[r, "\varepsilon \otimes x"]
\arrow[dr, swap, "\leftor_x"]
x \otimes x
\arrow[d, "\mu"]
x \otimes I
\arrow[l, swap, "x \otimes \varepsilon"]
\arrow[dl, "\rightor_x"]
\\&
\end{tikzcd}
\end{equation}
Alternatively, we can express these structures with string diagrams as follows.
* multiplication
[style=blackdot] (mu) at (0, 0) ;
[style=none] (2) at (-0.5, 0.5) ;
[style=none] (3) at (0.5, 0.5) ;
[style=none] (4) at (0, -0.5) ;
[style=none] () at (1, 0) ;
[bend right] (2.center) to (mu);
[bend left] (3.center) to (mu);
(4.center) to (mu);
* neutral element
[style=blackdot] (mu) at (0, 0) ;
[style=none] (4) at (0, -0.5) ;
[style=none] () at (1, 0) ;
(4.center) to (mu);
such that
\begin{equation}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (1, 1) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=none] (5) at (0.625, -0.5) {};
\node [style=blackdot] (b1) at (0.25, 0.5) {};
\node [style=blackdot] (b2) at (0.625, 0) {};
\node [style=none] () at (1.5, 0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (b1.center);
\draw [bend left] (2.center) to (b1.center);
\draw (3.center) to (4.center);
\draw [bend left] (4.center) to (b2.center);
\draw [bend right] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (1, 1) {};
\node [style=none] (4) at (0, 0.5) {};
\node [style=none] (5) at (0.375, -0.5) {};
\node [style=blackdot] (b1) at (0.75, 0.5) {};
\node [style=blackdot] (b2) at (0.375, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw [bend right] (2.center) to (b1.center);
\draw [bend left] (3.center) to (b1.center);
\draw [bend right] (4.center) to (b2.center);
\draw [bend left] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}
\begin{equation}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (mu) at (0, 0) {};
\node [style=blackdot] (ep) at (0.5, 1) {};
\node [style=none] (1) at (-0.5, 1.5) {};
\node [style=none] (2) at (0.5, 0.5) {};
\node [style=none] (4) at (-0.5, 0.5) {};
\node [style=none] (5) at (0, -0.5) {};
\node [style=none] () at (1.25, 0.5) {=};
\node [style=none] () at (1.75, 1) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw (ep.center) to (2.center);
\draw [bend left] (2.center) to (mu.center);
\draw [bend right] (4.center) to (mu.center);
\draw (mu.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1.5) {};
\node [style=none] (2) at (0, -0.5) {};
\node [style=none] () at (0.5, 0.5) {=};
\node [style=none] () at (1, 1) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (mu) at (0.5, 0) {};
\node [style=blackdot] (ep) at (0, 1) {};
\node [style=none] (1) at (1, 1.5) {};
\node [style=none] (2) at (0, 0.5) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=none] (5) at (0.5, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw (ep.center) to (2.center);
\draw [bend right] (2.center) to (mu.center);
\draw [bend left] (4.center) to (mu.center);
\draw (mu.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}
Let $(x, \mu, \varepsilon)$ and $(y, \nu, \delta)$ be monoids in $\C$. A morphism $f \maps x \to y$ is called a monoid homomorphism if the following diagrams commute.
\[
\begin{tikzcd}
x \otimes x
\arrow[r, "f \otimes f"]
\arrow[d, swap, "\mu"]
y \otimes y
\arrow[d, "\nu"]
\\
\arrow[r, swap, "f"]
\end{tikzcd}
\qquad
\begin{tikzcd}
\arrow[dl, swap, "\varepsilon"]
\arrow[dr, "\delta"]
\\
\arrow[rr, swap, "f"]
\end{tikzcd}
\]
In strings, these equations are depicted as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (f1) at (-0.5, 0) {$f$};
\node [style=construct] (f2) at (0.5, 0) {$f$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.3, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f1.center);
\draw (y.center) to (f2.center);
\draw [bend right = 50] (f1.center) to (m.center);
\draw [bend left = 50](f2.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=blackdot] (m) at (0, 0) {};
\node [style=construct] (f) at (0, -0.75) {$f$};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right = 50] (x.center) to (m.center);
\draw [bend left = 50] (y.center) to (m.center);
\draw (m.center) to (f.center);
\draw (f.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\qquad\qquad\qquad
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (m) at (0, 0) {};
\node [style=construct] (f) at (0, -0.75) {$f$};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (0.8, -0.5) {=};
\node [style=none] () at (1.3, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (m.center) to (f.center);
\draw (f.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (m) at (0, 0) {};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Let $\Mon(\C, \otimes)$ denote the category of monoid objects in $\C$ and their homomorphisms. If $\C$ is $\Set$ with its cartesian monoidal structure, we simply denote the category of monoids by $\Mon$.
Let $(\C, \otimes, I)$ be a braided monoidal category. A commutative monoid in $\C$ is a monoid object in $\C$ where the following equation holds.
\begin{equation}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, -0.5) {};
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.25, 0.5) {};
\node [style=none] (2) at (0.25, 0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (r.center);
\draw [bend left] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -0.5) {};
\node [style=blackdot] (m) at (0, 0) {};
\node [style=none] (2) at (0, 0.55) {};
\node [style=none] (3) at (-0.25, 1) {};
\node [style=none] (4) at (0.25, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (m.center);
\draw [bend right = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend left = 45] (2.center) to (3.center);
\draw [bend left = 90, white, line width = 3, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45, white, line width = 3] (2.center) to (4.center);
\draw [bend left = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45] (2.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}
Let $\CMon(\C, \otimes)$ denote the category of commutative monoid objects in $\C$ and their homomorphisms. If $\C$ is $\Set$ with its cartesian monoidal structure, we simply denote the category of commutative monoids by $\CMon$.
§ THE ECKMANN–HILTON ARGUMENT
Let $C$ be a braided monoidal category, and let $x$ be an object equipped with two distinct monoid structures $(x, \mu, \varepsilon)$ and $(x, \nu, \eta)$ such that $\mu$ and $\nu$ are related by the following equation.
\begin{equation}
\label{interchange}
\mu \circ (\nu \otimes \nu) = \nu(\mu \otimes \mu) \circ (1 \otimes \braid \otimes 1)
\end{equation}
Then $\varepsilon = \eta$, $\mu = \nu$, and $(x, \mu, \varepsilon)$ is commutative.
It is important to note that if $\C$ is $\Set$ or some other concrete category, and the operations are instead denoted by $\circ$ and $\star$, <ref> becomes $(a \circ b) \star (c \circ d) = (a \star c) \circ (b \star d)$. Due to this formulation, this relation as it appears in many contexts is called the middle-four interchange law.
We prove it using string diagrams, just for fun. Let the following string diagram components represent $\varepsilon$, $\mu$, $\eta$, and $\nu$, respectively.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (ru) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=none] () at (1, 0.5) {,};
\node [style=none] () at (1.5, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (ru.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (rm) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=none] (2) at (-0.5, 1) {};
\node [style=none] (3) at (0.5, 1) {};
\node [style=none] () at (1, 0.5) {,};
\node [style=none] () at (1.5, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (3.center) to (rm.center);
\draw [bend right] (2.center) to (rm.center);
\draw (rm.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=bluedot] (bu) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=none] () at (1, 0.5) {,};
\node [style=none] () at (1.5, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=bluedot] (bm) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=none] (2) at (-0.5, 1) {};
\node [style=none] (3) at (0.5, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (3.center) to (bm.center);
\draw [bend right] (2.center) to (bm.center);
\draw (bm.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Then we can draw <ref> as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (r1) at (-0.75, 1) {};
\node [style=none] (b1) at (-0.25, 1) {};
\node [style=none] (b2) at (0.25, 1) {};
\node [style=none] (r2) at (0.75, 1) {};
\node [style=bluedot] (b3) at (-0.5, 0.5) {};
\node [style=bluedot] (b4) at (0.5, 0.5) {};
\node [style=reddot] (r3) at (0, 0) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] () at (1.1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (r1.center) to (b3.center);
\draw [bend left] (b1.center) to (b3.center);
\draw [bend right] (b2.center) to (b4.center);
\draw [bend left] (r2.center) to (b4.center);
\draw [bend right] (b3.center) to (r3.center);
\draw [bend left] (b4.center) to (r3.center);
\draw (r3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (r1) at (-0.75, 1) {};
\node [style=none] (b1) at (-0.25, 1) {};
\node [style=none] (b2) at (0.25, 1) {};
\node [style=none] (r2) at (0.75, 1) {};
\node [style=reddot] (r3) at (-0.5, 0.5) {};
\node [style=reddot] (r4) at (0.5, 0.5) {};
\node [style=bluedot] (b3) at (0, 0) {};
\node [style=none] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (r1.center) to (r3.center);
\draw [bend left] (b2.center) to (r3.center);
\draw [bend right, line width = 3, white] (b1.center) to (r4.center);
\draw [bend right] (b1.center) to (r4.center);
\draw [bend left] (r2.center) to (r4.center);
\draw [bend right] (r3.center) to (b3.center);
\draw [bend left] (r4.center) to (b3.center);
\draw (b3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
First, we show that the units coincide.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (ru) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=none] () at (1, 0.5) {=};
\node [style=none] () at (1.5, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (ru.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (r1) at (-0.5, 0.5) {};
\node [style=reddot] (r2) at (0.5, 0.5) {};
\node [style=reddot] (r3) at (0, 0) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (r1.center) to (r3.center);
\draw [bend left] (r2.center) to (r3.center);
\draw (r3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (r1) at (-0.75, 1) {};
\node [style=bluedot] (b1) at (-0.25, 1) {};
\node [style=bluedot] (b2) at (0.25, 1) {};
\node [style=reddot] (r2) at (0.75, 1) {};
\node [style=bluedot] (b3) at (-0.5, 0.5) {};
\node [style=bluedot] (b4) at (0.5, 0.5) {};
\node [style=reddot] (r3) at (0, 0) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (r1.center) to (b3.center);
\draw [bend left] (b1.center) to (b3.center);
\draw [bend right] (b2.center) to (b4.center);
\draw [bend left] (r2.center) to (b4.center);
\draw [bend right] (b3.center) to (r3.center);
\draw [bend left] (b4.center) to (r3.center);
\draw (r3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (r1) at (-0.75, 1) {};
\node [style=bluedot] (b1) at (-0.25, 1) {};
\node [style=bluedot] (b2) at (0.25, 1) {};
\node [style=reddot] (r2) at (0.75, 1) {};
\node [style=reddot] (r3) at (-0.5, 0.5) {};
\node [style=reddot] (r4) at (0.5, 0.5) {};
\node [style=bluedot] (b3) at (0, 0) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (r1.center) to (r3.center);
\draw [bend left] (b2.center) to (r3.center);
\draw [bend right, line width = 3, white] (b1.center) to (r4.center);
\draw [bend right] (b1.center) to (r4.center);
\draw [bend left] (r2.center) to (r4.center);
\draw [bend right] (r3.center) to (b3.center);
\draw [bend left] (r4.center) to (b3.center);
\draw (b3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=bluedot] (b1) at (-0.5, 0.5) {};
\node [style=bluedot] (b2) at (0.5, 0.5) {};
\node [style=bluedot] (b3) at (0, 0) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (b1.center) to (b3.center);
\draw [bend left] (b2.center) to (b3.center);
\draw (b3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=bluedot] (bu) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Since they are equal, we denote the unit with a black circle in the remainder of the proof. Next, we show in one calculation that the two operations are equal and commutative.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=reddot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.5, 0.5) {};
\node [style=none] (2) at (0.5, 0.5) {};
\node [style=none] (3) at (0, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (r.center);
\draw [bend left] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (b1) at (-0.25, 1) {};
\node [style=blackdot] (b2) at (0.25, 1) {};
\node [style=bluedot] (bl1) at (-0.5, 0.5) {};
\node [style=bluedot] (bl2) at (0.5, 0.5) {};
\node [style=reddot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.75, 1) {};
\node [style=none] (2) at (0.75, 1) {};
\node [style=none] (3) at (0, -0.5) {};
\node [style=none] () at (1.15, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (bl1.center);
\draw [bend left] (b1.center) to (bl1.center);
\draw [bend right] (b2.center) to (bl2.center);
\draw [bend left] (2.center) to (bl2.center);
\draw [bend right] (bl1.center) to (r.center);
\draw [bend left] (bl2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (b1) at (-0.25, 1) {};
\node [style=blackdot] (b2) at (0.25, 1) {};
\node [style=reddot] (r1) at (-0.5, 0.5) {};
\node [style=reddot] (r2) at (0.5, 0.5) {};
\node [style=bluedot] (b) at (0, 0) {};
\node [style=none] (1) at (-0.75, 1) {};
\node [style=none] (2) at (0.75, 1) {};
\node [style=none] (3) at (0, -0.5) {};
\node [style=none] () at (1.15, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (r1.center);
\draw [bend left] (b2.center) to (r1.center);
\draw [bend right, white, line width = 3] (b1.center) to (r2.center);
\draw [bend right] (b1.center) to (r2.center);
\draw [bend left] (2.center) to (r2.center);
\draw [bend right] (r1.center) to (b.center);
\draw [bend left] (r2.center) to (b.center);
\draw (b.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=bluedot] (b) at (0, 0) {};
\node [style=none] (1) at (-0.5, 0.5) {};
\node [style=none] (2) at (0.5, 0.5) {};
\node [style=none] (3) at (0, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (b.center);
\draw [bend left] (2.center) to (b.center);
\draw (b.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (b1) at (-0.75, 1) {};
\node [style=blackdot] (b2) at (0.75, 1) {};
\node [style=reddot] (r1) at (-0.5, 0.5) {};
\node [style=reddot] (r2) at (0.5, 0.5) {};
\node [style=bluedot] (b) at (0, 0) {};
\node [style=none] (1) at (-0.25, 1) {};
\node [style=none] (2) at (0.25, 1) {};
\node [style=none] (3) at (0, -0.5) {};
\node [style=none] () at (1.15, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (b1.center) to (bl1.center);
\draw [bend left] (1.center) to (bl1.center);
\draw [bend right] (2.center) to (bl2.center);
\draw [bend left] (b2.center) to (bl2.center);
\draw [bend right] (bl1.center) to (r.center);
\draw [bend left] (bl2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (b1) at (-0.25, 1) {};
\node [style=none] (b2) at (0.25, 1) {};
\node [style=blackdot] (1) at (-0.75, 1) {};
\node [style=blackdot] (2) at (0.75, 1) {};
\node [style=bluedot] (bl1) at (-0.5, 0.5) {};
\node [style=bluedot] (bl2) at (0.5, 0.5) {};
\node [style=reddot] (r) at (0, 0) {};
\node [style=none] (3) at (0, -0.5) {};
\node [style=none] () at (1.15, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (bl1.center);
\draw [bend left] (b2.center) to (bl1.center);
\draw [bend right, white, line width = 3] (b1.center) to (bl2.center);
\draw [bend right] (b1.center) to (bl2.center);
\draw [bend left] (2.center) to (bl2.center);
\draw [bend right] (bl1.center) to (r.center);
\draw [bend left] (bl2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -0.5) {};
\node [style=reddot] (m) at (0, 0) {};
\node [style=none] (2) at (0, 0.55) {};
\node [style=none] (3) at (-0.25, 1) {};
\node [style=none] (4) at (0.25, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (m.center);
\draw [bend right = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend left = 45] (2.center) to (3.center);
\draw [bend left = 90, white, line width = 3, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45, white, line width = 3] (2.center) to (4.center);
\draw [bend left = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45] (2.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\qedhere
\]
We have the following equivalences of categories.
\[\Mon(\Mon, \times) \cong \Mon(\CMon, \times) \cong \CMon(\Mon, \times) \cong \CMon(\CMon, \times) \cong \CMon\]
§ CHARACTERIZING (CO)CARTESIAN MONOIDAL CATEGORIES
In the previous section, we saw that a category with finite products can be equipped with a canonical symmetric monoidal structure, and dually so can a category with finite coproducts. In this section, we give conditions under which a symmetric monoidal category is monoidally equivalent to one given by a (co)cartesian structure <cit.>.
Let $\C$ be a category with finite coproducts. By <ref>, $\C$ can be equipped with a cocartesian monoidal structure, with tensor denoted by $+$, and the unit (which is an initial object) denoted by $0$. Let $x$ be any object in $\C$. Universal property of coproducts gives a map $\nabla \maps x+x \to x$
\[
\begin{tikzcd}
\arrow[dr, "i_1"]
\arrow[ddr, swap, bend right, "1_x"]
\arrow[dl, swap, "i_2"]
\arrow[ddl, bend left, "1_x"]
\\&
\arrow[d, dashed, "\nabla_x"]
\\&
\end{tikzcd}\]
We draw string diagrams with respect to the cocartesian monoidal structure on $\C$. Then the map $\nabla_x$ is depicted as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.5, 0.5) {};
\node [style=none] (2) at (0.5, 0.5) {};
\node [style=none] (3) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (r.center);
\draw [bend left] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Also, the universal property of an initial object gives a map $!_x \maps 0 \to x$, depicted as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (bu) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
We show that this gives $x$ the structure of a commutative monoid. We begin by finding a formula for the left unitor of the cocartesian monoidal structure on $\C$. Notice that the left unitor makes the following diagram commute (by definition)
\[
\begin{tikzcd}
\arrow[dr, "i_x"]
\arrow[ddr, swap, "1", bend right]
\arrow[dl, swap, "!"]
\arrow[ddl, "!", bend left]
\\&
\arrow[d, dashed, "\exists !"]
\\&
\end{tikzcd}
\]
and thus does so uniquely. Compare this to the diagram
\[
\begin{tikzcd}
\arrow[dr, "i_x"]
\arrow[d, swap, "1"]
\arrow[dl, swap, "!"]
\arrow[d, "!"]
\\
\arrow[dr]
\arrow[ddr, bend right, swap, "1"]
\arrow[d, "1+!"]
\arrow[dl]
\arrow[ddl, bend left, "1"]
\\&
\arrow[d, "\nabla_x"]
\\&
\end{tikzcd}
\]
whose frame is equal to that of the previous. Thus we get $\nabla_x \circ (1_x + !_x) \circ i_x= 1_x$ and similarly $\nabla_x \circ (!_x + 1_x) \circ i'_x= 1_x$, which we draw as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0.25, 0.75) {};
\node [style=blackdot] (2) at (1, 0.5) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\node [style=none] () at (4, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0.25, 0.5) {};
\node [style=none] (2) at (1, 0.75) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}\]
To show $\nabla_x$ is associative, we want to show that the following equation holds.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (1, 1) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=none] (5) at (0.625, -0.5) {};
\node [style=blackdot] (b1) at (0.25, 0.5) {};
\node [style=blackdot] (b2) at (0.625, 0) {};
\node [style=none] () at (1.5, 0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (b1.center);
\draw [bend left] (2.center) to (b1.center);
\draw (3.center) to (4.center);
\draw [bend left] (4.center) to (b2.center);
\draw [bend right] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (1, 1) {};
\node [style=none] (4) at (0, 0.5) {};
\node [style=none] (5) at (0.375, -0.5) {};
\node [style=blackdot] (b1) at (0.75, 0.5) {};
\node [style=blackdot] (b2) at (0.375, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw [bend right] (2.center) to (b1.center);
\draw [bend left] (3.center) to (b1.center);
\draw [bend right] (4.center) to (b2.center);
\draw [bend left] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
We have three inclusion maps $i_0, i_1, i_2 \maps x \to x+x+x$, which are given in strings below.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (bu) at (0, 1) {};
\node [style=none] (1) at (0, 0) {};
\node [style=blackdot] (2) at (0.5, 0.5) {};
\node [style=none] (3) at (0.5, 0) {};
\node [style=blackdot] (4) at (1, 0.5) {};
\node [style=none] (5) at (1, 0) {};
\node [style=none] () at (1.5, 0.5) {,};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\draw (2.center) to (3.center);
\draw (4.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (bu) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (0.5, 0) {};
\node [style=blackdot] (4) at (1, 0.5) {};
\node [style=none] (5) at (1, 0) {};
\node [style=none] () at (1.5, 0.5) {,};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\draw (2.center) to (3.center);
\draw (4.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (bu) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=blackdot] (2) at (0.5, 0.5) {};
\node [style=none] (3) at (0.5, 0) {};
\node [style=none] (4) at (1, 1) {};
\node [style=none] (5) at (1, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\draw (2.center) to (3.center);
\draw (4.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
The universal property of coproducts says that if the composites of the morphisms on the left and right side of the associativity equation above with any of the three inclusions is always the identity morphism on $x$, then those two morphisms must be equal. So we compute:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1.25) {};
\node [style=blackdot] (2) at (0.5, 1) {};
\node [style=blackdot] (3) at (1, 1) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=none] (5) at (0.625, -0.5) {};
\node [style=blackdot] (b1) at (0.25, 0.5) {};
\node [style=blackdot] (b2) at (0.625, 0) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (b1.center);
\draw [bend left] (2.center) to (b1.center);
\draw (3.center) to (4.center);
\draw [bend left] (4.center) to (b2.center);
\draw [bend right] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0.25, 0.75) {};
\node [style=blackdot] (2) at (1, 0.5) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\node [style=none] () at (4, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0, 1) {};
\node [style=none] (2) at (0.5, 1.25) {};
\node [style=blackdot] (3) at (1, 1) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=none] (5) at (0.625, -0.5) {};
\node [style=blackdot] (b1) at (0.25, 0.5) {};
\node [style=blackdot] (b2) at (0.625, 0) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (b1.center);
\draw [bend left] (2.center) to (b1.center);
\draw (3.center) to (4.center);
\draw [bend left] (4.center) to (b2.center);
\draw [bend right] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0.25, 0.75) {};
\node [style=blackdot] (2) at (1, 0.5) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0, 1) {};
\node [style=blackdot] (2) at (0.5, 1) {};
\node [style=none] (3) at (1, 1.25) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=none] (5) at (0.625, -0.5) {};
\node [style=blackdot] (b1) at (0.25, 0.5) {};
\node [style=blackdot] (b2) at (0.625, 0) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (b1.center);
\draw [bend left] (2.center) to (b1.center);
\draw (3.center) to (4.center);
\draw [bend left] (4.center) to (b2.center);
\draw [bend right] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0.25, 0.5) {};
\node [style=none] (2) at (1, 0.75) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1.25) {};
\node [style=blackdot] (2) at (0.5, 1) {};
\node [style=blackdot] (3) at (1, 1) {};
\node [style=none] (4) at (0, 0.5) {};
\node [style=none] (5) at (0.375, -0.5) {};
\node [style=blackdot] (b1) at (0.75, 0.5) {};
\node [style=blackdot] (b2) at (0.375, 0) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw [bend right] (2.center) to (b1.center);
\draw [bend left] (3.center) to (b1.center);
\draw [bend right] (4.center) to (b2.center);
\draw [bend left] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0.25, 0.75) {};
\node [style=blackdot] (2) at (1, 0.5) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\node [style=none] () at (4, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0, 1) {};
\node [style=none] (2) at (0.5, 1.25) {};
\node [style=blackdot] (3) at (1, 1) {};
\node [style=none] (4) at (0, 0.5) {};
\node [style=none] (5) at (0.375, -0.5) {};
\node [style=blackdot] (b1) at (0.75, 0.5) {};
\node [style=blackdot] (b2) at (0.375, 0) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw [bend right] (2.center) to (b1.center);
\draw [bend left] (3.center) to (b1.center);
\draw [bend right] (4.center) to (b2.center);
\draw [bend left] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0.25, 0.5) {};
\node [style=none] (2) at (1, 0.75) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0, 1) {};
\node [style=blackdot] (2) at (0.5, 1) {};
\node [style=none] (3) at (1, 1.25) {};
\node [style=none] (4) at (0, 0.5) {};
\node [style=none] (5) at (0.375, -0.5) {};
\node [style=blackdot] (b1) at (0.75, 0.5) {};
\node [style=blackdot] (b2) at (0.375, 0) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw [bend right] (2.center) to (b1.center);
\draw [bend left] (3.center) to (b1.center);
\draw [bend right] (4.center) to (b2.center);
\draw [bend left] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0.25, 0.5) {};
\node [style=none] (2) at (1, 0.75) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, -0.5) {};
\node [style=none] () at (1.5, 0) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (2.center) to (3.center);
\draw [bend right] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Thus we have that $\nabla_x$ is associative.
Recall that the braiding in $\C$ is derived from the universal property in the following way.
\[
\begin{tikzcd}
\arrow[dr, "i_1"]
\arrow[ddr, swap, "i_2", bend right]
\arrow[dl, swap, "i_2"]
\arrow[ddl, "i_1", bend left]
\\&
\arrow[d, dashed, "\sigma"]
\\&
\end{tikzcd}\]
Then the commutative diagram
\[
\begin{tikzcd}
\arrow[dr, "i_1"]
\arrow[d, swap, "1"]
\arrow[dl, swap, "i_2"]
\arrow[d, "1"]
\\
\arrow[dr, swap, "i_1"]
\arrow[ddr, bend right, swap, "1"]
\arrow[d, "\sigma"]
\arrow[dl, "i_1"]
\arrow[ddl, bend left, "1"]
\\&
\arrow[d, "\nabla"]
\\&
\end{tikzcd}
\]
has precisely the frame for the universal construction of $\nabla_x$. Thus $\nabla_x \circ \sigma = \nabla_x$, displayed as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, -0.5) {};
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.25, 0.5) {};
\node [style=none] (2) at (0.25, 0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (r.center);
\draw [bend left] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -0.5) {};
\node [style=blackdot] (m) at (0, 0) {};
\node [style=none] (2) at (0, 0.55) {};
\node [style=none] (3) at (-0.25, 1) {};
\node [style=none] (4) at (0.25, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (m.center);
\draw [bend right = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend left = 45] (2.center) to (3.center);
\draw [bend left = 90, white, line width = 3, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45, white, line width = 3] (2.center) to (4.center);
\draw [bend left = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45] (2.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
A symmetric monoidal category is cocartesian if and only if each object has a natural commutative monoid structure.
Given an object $x$ in a cocartesian monoidal category $\C$, we constructed a commutative monoid structure on $x$ in <ref>.
We have to show that the multiplication maps
\[\nabla_x \maps x+x \to x \]
form the components of a natural transformation
\[\nabla \maps + \circ \Delta \To 1_\C.\]
For a given morphism $f \maps x \to y$, the naturality square is
\[
\begin{tikzcd}
x + x
\arrow[r, "f+f"]
\arrow[d, swap, "\nabla_x"]
y + y
\arrow[d, "\nabla_y"]
\\
\arrow[r, swap, "f"]
\end{tikzcd}
\]
Recall that the map $f+f$ is derived from the universal property of coproducts by the following diagram.
\[
\begin{tikzcd}
\arrow[d, swap, "f"]
\arrow[dr, "i_1"]
\arrow[dl, swap, "i_2"]
\arrow[d, "f"]
\\
\arrow[dr, swap, "i_1'"]
\arrow[d, dashed, "f+f"]&
\arrow[dl, "i_2'"]
\\&
y + y
\end{tikzcd}
\]
We want to show that $\nabla_y \circ (f+f) = f \circ \nabla_x$.
\[
\begin{tikzcd}
\arrow[d, swap, "f"]
\arrow[dr, "i_1"]
\arrow[dl, swap, "i_2"]
\arrow[d, "f"]
\\
\arrow[dr, swap, "i_1'"]
\arrow[ddr, swap, "1_y", bend right]
\arrow[d, "f+f"]&
\arrow[dl, "i_2'"]
\arrow[ddl, "1_y'", bend left]
\\&
y + y
\arrow[d, "\nabla_y"]
\\&y
\end{tikzcd}
\qquad\qquad
\begin{tikzcd}
\arrow[d, swap, "1_x"]
\arrow[dr, "i_1"]
\arrow[dl, swap, "i_2"]
\arrow[d, "1_x"]
\\
\arrow[dr, swap, "1_x"]
\arrow[ddr, swap, "f", bend right]
x + x
\arrow[d, "\nabla_x"]&
\arrow[dl, "1_x"]
\arrow[ddl, "f", bend left]
\\&
\arrow[d, "f"]
\\&y
\end{tikzcd}
\]
The frames of the above diagrams are identical, and they are equal to the frame which produces $\langle f , f \rangle$, the copairing of $f$ with itself. So by universal property, they are equal.
The naturality square of the units collapses into the triangle below.
\[
\begin{tikzcd}
\arrow[dl, swap, "!_x"]
\arrow[dr, "!_y"]
\\
\arrow[rr, swap, "f"]
\end{tikzcd}
\]
which commutes by initiality of $0$.
We have shown one direction: that if $\C$ is cocartesian monoidal, then each object has a natural commutative monoid structure. Now we must show the converse. Assume that $(\C, \otimes, I)$ is a symmetric monoidal category such that each object has a natural commutative monoid structure $m \maps \otimes \circ \Delta \To 1_\C$ and $\varepsilon \maps \Delta I \To 1_\C$. We represent the components of these structures with string diagrams as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.5, 0.5) {};
\node [style=none] (2) at (0.5, 0.5) {};
\node [style=none] (3) at (0, -0.5) {};
\node [style=none] () at (0.9, 0) {,};
\node [style=none] () at (1.5, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (r.center);
\draw [bend left] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (bu) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
The unit object is initial by naturality of $\varepsilon$.
\[
\begin{tikzcd}
\arrow[d, swap, "\varepsilon_x"]
\arrow[r, "1_I"]
\arrow[d, "\varepsilon_y"]
\\
\arrow[r, swap, "f"]
\end{tikzcd}
\]
Now we must show that the monoidal structure on $\C$ is cocartesian, i.e. that the unit object is initial and tensor is coproduct. To show that $x \otimes y$ is actually the coproduct of $x$ and $y$, we first must provide inclusions, and then show that this cone satisfies the appropriate universal property. We propose that the inclusion maps $i_x \maps x \to x\otimes y$ and $i_y \maps y \to x \otimes y$ are given in string diagrams as follows.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (bu) at (0, 1) {};
\node [style=none] (1) at (0, 0) {};
\node [style=blackdot] (2) at (0.5, 0.5) {};
\node [style=none] (3) at (0.5, 0) {};
\node [style=none] () at (1, 0.5) {,};
\node [style=none] () at (1.5, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\draw (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (bu) at (0, 0.5) {};
\node [style=none] (1) at (0, 0) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (0.5, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (bu.center) to (1.center);
\draw (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Let $q$ be an object of $\C$, and $f \maps x \to q$ and $g \maps y \to q$ be maps in $\C$. Define the map $h \maps x \otimes y \to q$ to be the following composite.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50](g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Then we show the diagram
\[
\begin{tikzcd}
\arrow[dr, "i_x"]
\arrow[ddr, swap, "f", bend right]
\arrow[dl, swap, "i_y"]
\arrow[ddl, "g", bend left]
\\&
x \otimes y
\arrow[d, "h"]
\\&q
\end{tikzcd}
\]
commutes by the following calculations.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=blackdot] (y) at (0.5, 0.5) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.2, -0.25) {=};
\node [style=none] () at (1.7, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50] (g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=blackdot] (y) at (0.5, 0) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.2, -0.25) {=};
\node [style=none] () at (1.7, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 45] (y.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (0, 0.75) {};
\node [style=construct] (f) at (0, -0.25) {$f$};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (f.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (x) at (-0.5, 0.5) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.2, -0.25) {=};
\node [style=none] () at (1.7, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50] (g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (x) at (-0.5, 0) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.3, -0.25) {=};
\node [style=none] () at (1.8, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right = 45] (x.center) to (m.center);
\draw (y.center) to (g.center);
\draw [bend left = 50] (g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (0, 0.75) {};
\node [style=construct] (g) at (0, -0.25) {$g$};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (g.center);
\draw (g.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Let $k \maps x \otimes y \to q$ be a map which makes that diagram commute. Then
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (k) at (0, -0.125) {$h$};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.1, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (x.center) to (k.center);
\draw [bend left] (y.center) to (k.center);
\draw (k.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.25, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50](g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (-0.875, 0.75) {};
\node [style=blackdot] (2) at (-0.14, 0.5) {};
\node [style=blackdot] (3) at (0.14, 0.5) {};
\node [style=none] (4) at (0.9, 0.75) {};
\node [style=construct] (k1) at (-0.5, 0) {$k$};
\node [style=construct] (k2) at (0.5, 0) {$k$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.3, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right = 20] (1.center) to (k1);
\draw [bend left = 20] (2.center) to (k1);
\draw [bend right = 30] (3.center) to (k2.center);
\draw [bend left = 27] (4.center) to (k2.center);
\draw [bend right = 50] (k1.center) to (m.center);
\draw [bend left = 50] (k2.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (-0.875, 0.75) {};
\node [style=blackdot] (2) at (-0.14, 0.5) {};
\node [style=blackdot] (3) at (0.14, 0.5) {};
\node [style=none] (4) at (0.9, 0.75) {};
\node [style=blackdot] (k1) at (-0.5, 0) {};
\node [style=blackdot] (k2) at (0.5, 0) {};
\node [style=construct] (m) at (0, -0.65) {$k$};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.3, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right = 20] (1.center) to (k1);
\draw [bend left = 20] (2.center) to (k1);
\draw [bend right = 30] (3.center) to (k2.center);
\draw [bend left = 27] (4.center) to (k2.center);
\draw [bend right = 35] (k1.center) to (m.center);
\draw [bend left = 35] (k2.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (k) at (0, -0.125) {$k$};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (x.center) to (k.center);
\draw [bend left] (y.center) to (k.center);
\draw (k.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Thus $h$ is the unique such map. This demonstrates $x \otimes y$ as the coproduct of $x$ and $y$.
There is a dual statement which characterizes cartesian monoidal categories, but in order to state it, we must first define comonoid.
A comonoid object in a monoidal category $\C$ is monoid in $\C\op$. Equivalently, a comonoid is an object $x \in \C$ equipped with a comultiplication map $\mu \maps x \to x \otimes x$ and a counit map $\varepsilon \maps x \to I$, which we express as
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, 0.5) {};
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.5, -0.5) {};
\node [style=none] (2) at (0.5, -0.5) {};
\node [style=none] () at (1, 0) {,};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (1.center) to (r.center);
\draw [bend right] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.5, -0.5) {};
\node [style=none] (2) at (0.5, -0.5) {};
\node [style=none] (3) at (0, 0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
satisfying the following equations.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -1) {};
\node [style=none] (2) at (0.5, -1) {};
\node [style=none] (3) at (1, -1) {};
\node [style=none] (4) at (1, -0.5) {};
\node [style=none] (5) at (0.625, 0.5) {};
\node [style=blackdot] (b1) at (0.25, -0.5) {};
\node [style=blackdot] (b2) at (0.625, 0) {};
\node [style=none] () at (1.5, -0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (1.center) to (b1.center);
\draw [bend right] (2.center) to (b1.center);
\draw (3.center) to (4.center);
\draw [bend right] (4.center) to (b2.center);
\draw [bend left] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -1) {};
\node [style=none] (2) at (0.5, -1) {};
\node [style=none] (3) at (1, -1) {};
\node [style=none] (4) at (0, -0.5) {};
\node [style=none] (5) at (0.375, 0.5) {};
\node [style=blackdot] (b1) at (0.75, -0.5) {};
\node [style=blackdot] (b2) at (0.375, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw [bend left] (2.center) to (b1.center);
\draw [bend right] (3.center) to (b1.center);
\draw [bend left] (4.center) to (b2.center);
\draw [bend right] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0.25, -0.75) {};
\node [style=blackdot] (2) at (1, -0.5) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, 0.5) {};
\node [style=none] () at (1.5, -0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (3.center);
\draw [bend left] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\node [style=none] () at (4, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0.25, -0.5) {};
\node [style=none] (2) at (1, -0.75) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, 0.5) {};
\node [style=none] () at (1.5, -0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (3.center);
\draw [bend left] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}\]
Let $\Comon(\C, \otimes)$ denote the category of comonoid objects in $\C$ and their homomorphisms.
A cocommutative comonoid is a comonoid for which the following equation holds.
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, 0.5) {};
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.25, -0.5) {};
\node [style=none] (2) at (0.25, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (1.center) to (r.center);
\draw [bend right] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 0.5) {};
\node [style=blackdot] (m) at (0, 0) {};
\node [style=none] (2) at (0, -0.55) {};
\node [style=none] (3) at (-0.25, -1) {};
\node [style=none] (4) at (0.25, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (m.center);
\draw [bend left = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45] (2.center) to (3.center);
\draw [bend right = 90, white, line width = 3, looseness = 1.5] (m.center) to (2.center);
\draw [bend left = 45, white, line width = 3] (2.center) to (4.center);
\draw [bend right = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend left = 45] (2.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Let $\CoComon(\C, \otimes)$ denote the category of cocommutative comonoids in $\C$ and their homomorphisms.
A symmetric monoidal category is cartesian if and only if each object has a natural cocommutative comonoid structure.
This is dual to <ref>.
Let $(\C, \otimes, I)$ be a symmetric monoidal category. Then $\CMon(\C, \otimes)$ has a cocartesian monoidal structure given by $\otimes$, and $\CoComon(\C, \otimes)$ has a cartesian monoidal structure given by $\otimes$.
CHAPTER: MONOIDAL 2-CATEGORIES AND PSEUDOMONOIDS
There are many sources for the basic theory of 2-categories and bicategories <cit.>. Below we sketch some basic definitions and constructions regarding monoidal 2-categories, necessary for what follows; relevant references where explicit axioms can be found are <cit.>.
§ MONOIDAL 2-CATEGORIES
A monoidal 2-category $\K$ is a 2-category equipped with a pseudofunctor $\otimes \maps \K \times \K \to \K$ and a unit object $I \maps 1 \to \K$ which are associative and unital up to coherent equivalence. A lax monoidal pseudofunctor $\F \maps \K \to \L$ between monoidal 2-categories is a pseudofunctor equipped with pseudonatural transformations
\begin{equation}\label{eq:weakmonpseudo}
\begin{tikzcd}
\K \times \K
\arrow[r, "\F \times \F"]
\arrow[d, "\otimes_\K"']
& |[alias=doma]|
\L \times \L
\arrow[d, "\otimes_\L"]
\\
\K
\arrow[r, "\F"']
& \L
\arrow[Rightarrow, from=doma, to=coda, "\mu"', shorten >=.25in, shorten <=.25in]
\end{tikzcd}
\quad\quad
\begin{tikzcd}
\1\arrow[d, "I_{\K}"']
\arrow[dr, bend left, "I_{\L}"{name=doma}]
\\
\K
\arrow[r, "\F"']
& \L
\arrow[Rightarrow, from=doma, to=coda, "\mu_0"', shorten >=.1in, shorten <=.1in]
\end{tikzcd}
\end{equation}
with components $\mu_{a, b} \maps \F a \otimes \F b \to \F (a \otimes b)$, $\mu_0 \maps I \to \F I$, and invertible modifications
\begin{equation}\label{eq:omega}
\begin{tikzcd}
\L^3
\arrow[d, phantom, "\Downarrow{\scriptstyle\mu \times 1}"]
\arrow[r, "\otimes _\L \times 1"]
\L^2
\arrow[dd, phantom, "\Downarrow{\scriptstyle\mu}"]
\arrow[dr, "\otimes _\L"]
\L^3
\arrow[dd, phantom, "\Downarrow{\scriptstyle 1 \times \mu}"]
\arrow[r, "\otimes _\L \times 1"]
\arrow[dr, "1 \times \otimes _\L"description]
\L^2
\arrow[d, phantom, "{\scriptstyle\cong}"]
\arrow[dr, "\otimes _\L"]
\\
\K^3
\arrow[ur, "\F \times \F \times \F"]
\arrow[r, "\otimes _\K \times 1"]
\arrow[dr, "1 \times \otimes _\K", swap]
\K^2
\arrow[ur, "\F \times \F"description]
\arrow[dr, "\otimes _\K"description]
\L
\arrow[r, phantom, "\stackrel{\omega}{\Rrightarrow}"]
\K^3
\arrow[ur, "\F \times \F \times \F"]
\arrow[dr, "1 \times \otimes _\K", swap]
\L^2
\arrow[d, phantom, "\Downarrow{\scriptstyle\mu}"]
\arrow[r, "\otimes _\L"]
\L
\\&
\K^2
\arrow[u, phantom, "{\scriptstyle\cong}"]
\arrow[r, "\otimes _\K", swap]
\K
\arrow[ur, "\F", swap]
\K^2
\arrow[ur, "\F \times \F"description]
\arrow[r, "\otimes _\K", swap]
\K
\arrow[ur, "\F", swap]
\end{tikzcd}
\end{equation}
[column sep=.25in]
[dr, "1×I"']
[rr, "×I"name=doma]
[rrd, bend right=80, "1"', "≅"]
[rrr, bend left=30, ""]
[rrr, phantom, bend left=15, "≅"description]
Ł×Ł[r, "⊗_Ł"']
Ł[Rightarrow, from=doma, to=coda, "1×μ_0 "', shorten <=.5em, shorten >=.5em]
×[r, "⊗_"']
[ur, "×"description]
[urr, phantom, "⇓μ"description]
[ur, ""']
[column sep=.25in]
[rr, bend left, ""][rr, bend right, ""'][rr, phantom, "⇓1"description] Ł
[column sep=.25in]
[dr, "I×1"']
[rr, "I×"name=doma]
[rrd, bend right=80, "1"', "≅"]
[rrr, bend left=30, ""]
[rrr, phantom, bend left=15, "≅"description]
Ł×Ł[r, "⊗_Ł"']
Ł[Rightarrow, from=doma, to=coda, "μ_0×1 "', shorten <=.5em, shorten >=.5em]
×[r, "⊗_"']
[ur, "×"description]
[urr, phantom, "⇓μ"description]
[ur, ""']
subject to coherence conditions which can be found in Definition 2 in <cit.>. A monoidal pseudonatural transformation $\tau \maps \F \Rightarrow \G$ between two lax monoidal pseudofunctors $(\F, \mu, \mu_0)$ and $(\G, \nu, \nu_0)$ is a pseudonatural transformation equipped with two invertible modifications
\begin{equation}\label{eq:monpseudonat}
\begin{tikzcd}[column sep = .2in, row sep = .1in]
\K \times \K
\arrow[rr, bend left, "\F \times \F"]
\arrow[rr, bend right, "\G \times \G"']
\arrow[dd, "\otimes "']
\arrow[ddrr, phantom, bend right = 10, "\Downarrow{\scriptstyle \nu}"']
\arrow[rr, phantom, description, "\Downarrow{\scriptstyle \tau \times \tau}"]
\L \times \L
\arrow[dd, "\otimes "]
\K \times \K
\arrow[rr, "\F \times \F"]
\arrow[dd, "\otimes "']
\arrow[ddrr, phantom, bend left=10, "\Downarrow{\scriptstyle \mu}"]
\L \times \L
\arrow[dd, "\otimes "]
\\&&&
\stackrel{u}{\Rrightarrow}
\K
\arrow[rr, "\G"']
\L
\K
\arrow[rr, bend left, "\F"]
\arrow[rr, bend right, "\G"']
\arrow[rr, phantom, description, "\Downarrow{\scriptstyle\tau}"]
\L
\end{tikzcd}
\end{equation}
[row sep = .1in, column sep = .3in]
[ddrr, phantom, "⇓ν_0" description]
[rr, "I_Ł"]
[dd, "1_"']
[ddrr, phantom, near start, "⇓μ_0" description]
[rr, "I_Ł"]
[dd, "I_"']
[uurr, bend right = 40,
[uurr, "" description]
[uurr, bend right=60, ""']
[uurr, phantom, bend right, "⇓τ" description]
that consist of natural isomorphisms with components
\begin{equation}\label{eq:monpseudocomponents}
u_{a, b}\maps \nu_{a, b} \circ (\tau_a \otimes \tau_b) \xrightarrow{\sim} \tau_{a \otimes b} \circ \mu_{a, b},
\quad
u_0\maps \nu_0 \xrightarrow{\sim}\tau_I \circ \mu_0
\end{equation}
satisfying coherence conditions which can be found in <cit.>.
The above notions of course generalize those of an ordinary monoidal category, lax monoidal functor and monoidal natural transformation. However, in our higher dimensional setting, there is now room for a structure not present for monoidal 1-categories.
A monoidal modification
between two monoidal pseudonatural transformations $(\tau, u, u_0)$ and $(\sigma, v, v_0)$ is a modification
[rr, bend left=40, "", ""'name = F]
[rr, bend right=40, ""', ""name = G]
[rr, phantom, "m⇛"description]
Ł[from = F, to = G, Rightarrow, "τ"', bend right=50]
[from = F, to = G, Rightarrow, "σ", bend left=50]
which consists of pseudonatural transformations $m_a\maps \tau_a\Rightarrow\sigma_a$ compatible with the monoidal structures, in the sense that
\begin{equation}\label{eq:monoidalmodaxioms}
\begin{tikzcd}[column sep=.27in]
\G a \otimes \G b
\arrow[dr, bend left, "\nu_{a, b}"]
\G a \otimes \G b
\arrow[dr, bend left, "\nu_{a, b}"]
\\
\F a\otimes \F b
\arrow[ur, bend left, "\sigma_a\otimes \sigma_b"]
\arrow[dr, bend right, "\mu_{a, b}"']
\arrow[rr, phantom, near start, "\Downarrow{\scriptstyle v_{a, b}}"description]
\G (x\otimes y)
\arrow[r, phantom, "="description]
\F a\otimes \F b
\arrow[dr, bend right, "\mu_{a, b}"']
\arrow[ur, bend right, "\sigma_a\otimes \sigma_b"']
\arrow[rr, phantom, near end, "\Downarrow{\scriptstyle u_{a, b}}"description]
\arrow[ur, bend left, "\tau_a\otimes\tau_b"]
\arrow[ur, phantom, "\Downarrow{\scriptstyle m_x\otimes m_y}"description]
\G (a\otimes b)
\\
\F(a\otimes b)
\arrow[ur, bend left, "\tau_{a\otimes b}"]
\arrow[ur, bend right, "\sigma_{a\otimes b}"']
\arrow[ur, phantom, "\Downarrow{\scriptstyle m_{a\otimes b}}"description]
\F(a\otimes b)
\arrow[ur, bend right, "\tau_{a\otimes b}"']
\end{tikzcd}
\end{equation}
[row sep=.2in]
[dr, bend right, "μ_0"']
[rr, bend left, "ν_0"]
[rr, phantom, near start, "⇓v_0"description]
[r, phantom, "="description]
[rr, phantom, "⇓u_0"description]
[dr, bend right, "μ_0"']
[rr, bend left, "ν_0"]
[ur, bend left, "τ_I"]
[ur, bend right, "σ_I"']
[ur, phantom, "⇓m_I"description]
[ur, bend right, "τ_I"']
For any monoidal 2-categories $\K, \L$ there are 2-categories $\MonTCat_\pse(\K, \L)$ denoted by $\namedcat{WMonHom}(\K, \L)$ in <cit.> for bicategories. If we take lax monoidal 2-functors and monoidal 2-transformations, the corresponding sub-2-category is denoted by $\MonTCat(\K, \L)$.
§ PSEUDOMONOIDS
A pseudomonoid in a monoidal 2-category $(\K, \otimes , I)$ is an object $a$ equipped with multiplication $m \maps a\otimes a\to a$, unit $j \maps I \to a$, and invertible 2-cells
\begin{equation}
\label{alphalambdarho}
\begin{tikzcd}
a \otimes a \otimes a
\arrow[r, "1 \otimes m"]
\arrow[d, "m \otimes 1"']
\arrow[dr, phantom, "\scriptstyle \stackrel{\assoc}{\cong}"description]
a \otimes a \arrow[d, "m"] \arrow[d, "m"]
& a \otimes I \arrow[r, "1 \otimes j"] \arrow[dr, "\sim"']
& a \otimes a \arrow[d, "m", "\stackrel{\lambda}{\cong}\quad"'near start, "\quad\;\stackrel{\rho}{\cong}"near start]
& I \otimes a \arrow[l, "j \otimes 1"']
\arrow[dl, "\sim"] \\
a \otimes a\arrow[r, "m"']
& a && a &
\end{tikzcd}
\end{equation}
expressing associativity and unitality up to isomorphism, that satisfy appropriate coherence conditions. A lax morphism between pseudomonoids $a, b$ is a 1-cell $f\maps a\to b$ equipped with 2-cells
\begin{equation}\label{eq:laxmorphism}
\begin{tikzcd}[row sep=.5in, column sep=.5in]
a \otimes a\arrow[d, "m"']\arrow[r, "f\otimes f"] & |[alias=doma]| b\otimes b\arrow[d, "m"] \\
|[alias=coda]| a\arrow[r, "f"'] & b
\arrow[Rightarrow, from=doma, to=coda, "\phi"', shorten >=.35in, shorten <=.35in]
\end{tikzcd}
\qquad
\begin{tikzcd}[row sep=.5in, column sep=.5in]
I\arrow[d, "j"']\arrow[dr, bend left, "j"{name=doma}] \\
|[alias=coda]| a\arrow[r, "f"'] & b
\arrow[Rightarrow, from=doma, to=coda, "\phi_0"', shorten >=.15in, shorten <=.15in]
\end{tikzcd}
\end{equation}
such that the following conditions hold:
\begin{equation}\label{eq:axiomslaxmorphism}
\adjustbox{scale=.9, center}{
\begin{tikzcd}[column sep=.3in]
& b\otimes b\otimes b\arrow[r, "m\otimes1"]\arrow[d, phantom, "\Downarrow{\scriptstyle\phi\otimes 1_f}"description] & b\otimes b\otimes a\arrow[dr, "m"] & \\
a\otimes a\otimes a\arrow[r, "m\otimes 1"]\arrow[dr, "1\otimes m"']\arrow[ur, "f\otimes f\otimes f"] &
a\otimes a\arrow[dr, "m"]\arrow[ur, "f\otimes f"']\arrow[rr, phantom, "\Downarrow{\scriptstyle\phi}"description]
\arrow[d, phantom, "\stackrel{\assoc}{\cong}"description] && b \\
& a\otimes a\arrow[r, "m"'] & a\arrow[ur, "f"']
\end{tikzcd}
\begin{tikzcd}[column sep=.3in]
& b\otimes b\otimes b\arrow[r, "m\otimes1"]\arrow[dr, "1\otimes m"']\arrow[dd, phantom, "\Downarrow{\scriptstyle1_f\otimes \phi}"description] &
b\otimes b\arrow[dr, "m"]\arrow[d, phantom, "\stackrel{\alpha}{\cong}"description] & \\
a\otimes a\otimes a\arrow[ur, "f\otimes f\otimes f"]\arrow[dr, "1\otimes m"'] & & b\otimes b\arrow[r, "m"]
\arrow[d, phantom, "\Downarrow{\scriptstyle\phi}"description] & b \\
& a\otimes a\arrow[ur, "f\otimes f"]\arrow[r, "m"'] & b\otimes b\arrow[ur, "f"']
\end{tikzcd}
\end{equation}
scale=.9, center
[column sep=.25in]
a≅a⊗I[dr, "1⊗j"'][rr, "f⊗j"name=doma][rrd, bend right=60, "1_a"', "λ≅"]
[rrr, bend left=30, "f"][rrr, phantom, bend left=15, "1_f⊗λ≅"description] b⊗b[r, "m"'] b
[Rightarrow, from=doma, to=coda, "1_f⊗ϕ_0 "', shorten <=.5em, shorten >=.5em]
|[alias=coda]|a⊗a[r, "m"'][ur, "f⊗f"description][urr, phantom, "⇓ϕ"description]
a[ur, "f"']
[column sep=.25in]
a[rr, bend left, "f"][rr, bend right, "f"'][rr, phantom, "⇓1_f"description] b
[column sep=.25in]
a≅I⊗a[dr, "j⊗1"'][rr, "j⊗1"name=doma][rrd, bend right=60, "1_a"', "ρ≅"]
[rrr, bend left=30, "f"][rrr, phantom, bend left=15, "ρ⊗1_f≅"description] b⊗b[r, "m"'] b
[Rightarrow, from=doma, to=coda, "ϕ_0⊗1_f "', shorten <=.5em, shorten >=.5em]
|[alias=coda]|a⊗a[r, "m"'][ur, "f⊗f"description][urr, phantom, "⇓ϕ"description]
a[ur, "f"']
If $(f, \phi, \phi_0)$ and $(g, \psi, \psi_0)$ are two lax morphisms between pseudomonoids $a$ and $b$, a 2-cell between them $\sigma \maps f \Rightarrow g$ in $\K$ which is compatible with multiplications and units, in the sense that
\begin{equation}\label{monoidal2cell}
\begin{tikzcd}[row sep=.15in]
& b\otimes b\arrow[dr, bend left=20, "m"] & \\
a\otimes a\arrow[ur, bend left, "f\otimes f"]\arrow[ur, bend right, "g\otimes g"']\arrow[dr, bend right=20, "m"']
\arrow[ur, phantom, "\Downarrow{\scriptstyle\sigma\otimes\sigma}"description]
\arrow[rr, phantom, "{\scriptstyle\psi}\Downarrow"{description, near end}] && b \\
& a\arrow[ur, bend right=20, "g"'] &
\end{tikzcd}
\quad=\quad
\begin{tikzcd}[row sep=.15in]
& b\otimes b\arrow[rd, bend left=20, "m"] & \\
a\otimes a\arrow[ur, bend left=20, "f\otimes f"]\arrow[dr, bend right=20, "m"']
\arrow[rr, phantom, "{\scriptstyle\phi}\Downarrow"{description, near start}] && b \\
& a \arrow[ur, phantom, "\Downarrow{\scriptstyle\sigma}"description]
\arrow[ur, bend left, "f"]\arrow[ur, bend right, "g"'] \\
\end{tikzcd}
\end{equation}
[row sep=.15in, column sep=.6in]
I[rr, bend left, "j"][dr, bend right=20, "j"'][rr, phantom, "ϕ_0⇓"description, near start] b
a[ur, bend left, "f"][ur, bend right, "g"'][ur, phantom, "⇓σ"description]
= [row sep=.15in, column sep=.6in]
I[dr, bend right=20, "j"'][rr, bend left, "j"][rr, phantom, "⇓ψ_0"description] b
a[ur, bend right=20, "g"']
We obtain a 2-category $\PsMon_\lax(\K)$ for any monoidal 2-category $\K$, which is sometimes
denoted by $\Mon(\K)$ <cit.>. By changing the direction of the 2-cells in <ref> and the rest of the axioms appropriately, or asking for them to be invertible, we have 2-categories $\PsMon_\opl(\K)$ and $\PsMon(\K)$ of pseudomonoids with oplax or (strong) morphisms between them.
The prototypical example is that of the monoidal 2-category $\K = (\Cat, \times, \1)$ of categories, functors, and natural transformations with the cartesian product of categories and the unit category with a unique object and arrow. A pseudomonoid in $(\Cat, \times, \1)$ is a monoidal category, a lax (resp. oplax, strong) morphism between two of these is precisely a lax (resp. oplax, strong) monoidal functor, and a 2-cell is a monoidal natural transformation. Therefore we obtain the well-known 2-categories $\Mon\Cat_\lax$, $\Mon\Cat_\opl$ and $\Mon\Cat$.
There is an evident similarity between the structures defined above, e.g. <ref> and <ref>, or <ref> and <ref>. This is due to the fact that monoidal 2-categories, lax monoidal pseudofunctors and monoidal pseudonatural transformations are themselves appropriate pseudomonoid-related notions in a higher level; we do not get into such details, as they are not pertinent to the present work.
For our purposes, we are interested in a different observation: any pseudomonoid $a$ in a monoidal 2-category $\K$ can in fact be expressed as a lax monoidal normal pseudofunctor $A \maps \1 \to \K$ with $A(*) = a$, namely one where $A(1_*)$ is equal to $1_a$. Moreover, a monoidal pseudonatural transformation $\tau\maps A\Rightarrow B\maps \1\to\K$ bijectively corresponds to a strong morphism between the pseudomonoids $a$ and $b$, and similarly for monoidal modifications and 2-cells. Since every pseudofunctor is equivalent to a normal one, the 2-category of pseudomonoids $\PsMon(\K)$ can be equivalently viewed as $\MonTCat_{\pse}(\1, \K)$, the 2-category of lax monoidal pseudofunctors $\1\to\K$, monoidal pseudonatural transformations and monoidal modifications.
As was already shown in <cit.>, any lax monoidal 2-functor $\F\maps \K\to\L$ takes pseudomonoids to pseudomonoids, and in fact
<cit.> there is a functor $\PsMon(\F)$ that commutes with the respective forgetful functors
[column sep=.6in]
()[r, "()"][d] (Ł)[d]
[r, ""'] Ł.
Based on the above, and since every pseudofunctor from $\1$ into a 2-category trivially preserves composition on the nose and every pseudonatural transformation is really 2-natural, we can define a hom-2-functor that clarifies these assignments.
There is a 2-functor
\begin{equation}
\label{eq:PsMon}
\PsMon(-) \simeq \MonTCat_{\pse}(\1, -) \maps \MonTCat \to \TCat
\end{equation}
which maps a monoidal 2-category to its 2-category of pseudomonoids, strong morphisms and 2-cells between them.
The theory in <cit.> extends the above definitions to the case of braided and symmetric pseudomonoids in braided and symmetric monoidal 2-categories. Briefly recall that a braiding for $(\K, \otimes, I)$ is a pseudonatural equivalence with components $\braid_{a, b} \maps a \otimes b \to b \otimes a$ and invertible modifications, whereas a syllepsis is an invertible modification
\[
a \otimes b \xrightarrow{1} a \otimes b \Rrightarrow a \otimes b \xrightarrow{\braid_{a, b}} b \otimes a \xrightarrow{\braid_{b, a}} a \otimes b
\]
which is called symmetry if it satisfies extra axioms. With the appropriate notions of braided and symmetric lax monoidal pseudofunctors and monoidal pseudonatural transformations (and usual monoidal modifications), we have 3-categories $\BrMonTCat_{\pse}$ and $\SymMonTCat_{\pse}$. Indicatively, a lax monoidal pseudofunctor comes equipped an invertible modification with components
\begin{equation}\label{eq:brweakmonpseudo}
\begin{tikzcd}
\F a \otimes \F b
\arrow[r, "\mu_{a, b}"]
\arrow[d, "\braid_{\F a, \F b}"']
\arrow[dr, phantom, "\Downarrow{\scriptstyle v_{a, b}}" description]
\F b \otimes \F a
\arrow[d, "\F(\braid_{a, b})"]
\\
\F b \times \F a
\arrow[r, "\mu_{b, a}"']
\F(b\otimes a)
\end{tikzcd}
\end{equation}
As earlier, there exist 2-categories of braided and symmetric pseudomonoids with strong morphisms between them, expressed as
\[
\BrPsMon(\K)=\BrMonTCat_{(\pse)}(\1, \K)
\]
\[
\SymPsMon(\K)=\SymMonTCat_{(\pse)}(\1, \K).
\]
There are 2-functors
→, →
which map a braided or symmetric monoidal 2-category to its 2-category of braided or symmetric pseudomonoids.
Finally, recall the notion of a monoidal 2-equivalence arising as the equivalence internal to the 2-category $\MonTCat$.
A monoidal 2-equivalence is a 2-equivalence $\F\maps \K \simeq \L \maps \G$ where both 2-functors
are lax monoidal, and the 2-natural isomorphisms $1_\K \cong \F \G$, $\G \F \cong 1_\L$ are monoidal. Similarly for braided and symmetric monoidal 2-equivalences.
As is the case for any 2-functor between 2-categories, $\PsMon$ as well as $\BrPsMon$ and $\SymPsMon$ map equivalences
to equivalences.
Any monoidal 2-equivalence $\K\simeq\L$ induces a 2-equivalence between the respective 2-categories
of pseudomonoids $\PsMon(\K)\simeq\PsMon(\L)$. Similarly any braided or symmetric monoidal 2-equivalence induces $\BrPsMon(\K)\simeq\BrPsMon(\L)$ or $\SymPsMon(\K)\simeq\SymPsMon(\L)$.
CHAPTER: FIBRATIONS AND INDEXED CATEGORIES
We recall some basic facts and constructions from the theory of fibrations and indexed categories, as well as the equivalence between them via the Grothendieck construction. Several indicative references for the general theory are <cit.>.
§ FIBRATIONS
Consider a functor $P \maps \A \to \X$. A morphism $\phi \maps a \to b$ in $\A$ over a morphism $f = P(\phi) \maps x \to y$ in $\X$ is called cartesian if and only if, for all $g \maps x' \to x$ in $\X$ and $\theta \maps a' \to b$ in $\A$ with $P \theta = f \circ g$, there exists a unique arrow $\psi \maps a' \to a$ such that $P \psi = g$ and $\theta = \phi \circ \psi$:
\begin{equation}
\begin{tikzcd}[column sep = huge]
\arrow[drr, "\theta"]
\arrow[dr, dashed, swap, "\exists!\psi"]
\arrow[dd, dotted, bend right]
\\&
\arrow[r, swap, "\phi"]
\arrow[dd, dotted, bend right]
\arrow[dd, dotted, bend right]
\text{in }\A
\\
\arrow[drr, "f \circ g = P \theta"]
\arrow[dr, swap, "g"]
\\&
\arrow[r, swap, "f = P \phi"]
\text{in }\X
\end{tikzcd}
\end{equation}
For $x \in \Ob\X$, the fibre of $P$ over $x$ written $\A_x$, is the subcategory of $\A$ which consists of objects $a$ such that $P(a) = x$ and morphisms $\phi$ with $P(\phi) = 1_x$, called vertical morphisms. The functor $P \maps \A \to \X$ is called a fibration if and only if, for all $f \maps x \to y$ in $\X$ and $b\in\A_Y$, there is a cartesian morphism $\phi$ with codomain $b$ above $f$; it is called a cartesian lifting of $f$ to $b$. The category $\X$ is then called the base of the fibration, and $\A $ its total category.
Dually, the functor $U \maps \C \to \X$ is an opfibration if $U^\mathrm{op}$ is a fibration, i.e. for every $c \in \C _x$ and $h \maps x \to y$ in $\X$, there is a cocartesian morphism with domain $c$ above $h$, the cocartesian lifting of $h$ to $c$ with the dual universal property:
[column sep = huge]
[dd, dotted, bend right]
[urr, "γ"]
[r, swap, "β"]
[dd, dotted, bend right]
[ur, dashed, swap, "∃! δ"]
[dd, dotted, bend right]
[urr, "k ∘h = U γ"]
[r, swap, "h = U β"]
[ur, swap, "k"]
A bifibration is a functor which is both a fibration and opfibration.
If $P\maps \A \to \X$ is a fibration, assuming the axiom of choice we may select a cartesian arrow over each $f\maps x \to y$ in $\X$ and $b \in \A_y$, denoted by $\Cart(f, b) \maps f^*(b) \to b$. Such a choice of cartesian liftings is called a cleavage for $P$, which is then called a cloven fibration; any fibration is henceforth assumed to be cloven. Dually, if $U$ is an opfibration, for any $c \in \C_x$ and $h \maps x \to y$ in $\X$ we can choose a cocartesian lifting of $h$ to $c$, $\Cocart(h,c)\maps c \to h_!(c)$. The choice of (co)cartesian liftings in an (op)fibration induces a so-called reindexing functor between the fibre categories
\begin{equation}\label{reindexing}
f^*\maps \A _y \to \A _x\quad\textrm{ and }\quad h_! \maps \C _x \to \C _y
\end{equation}
respectively, for each morphism $f\maps x \to y$ and $h \maps x \to y$ in the base category.
It can be verified by the (co)cartesian universal property that $1_{\A _x}\cong(1_x)^*$
and that for composable morphism in the base category, $g^*\circ f^*\cong(g\circ f)^* $, as well as $(1_x)_!\cong1_{\C_x}$ and $(k\circ h)_!\cong k_!\circ h_!$. If these isomorphisms are equalities, we have the notion of a split (op)fibration.
A fibred 1-cell $(H,F) \maps P \to Q$ between fibrations $P \maps \A \to \X$ and $Q \maps \B \to \Y$ is given by a commutative square of functors and categories
\begin{equation}
\label{commutativefibredcell}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, "H"]
\arrow[d, swap, "P"]
\B
\arrow[d, "Q"]
\\
\X
\arrow[rr , "F", swap]
\Y
\end{tikzcd}
\end{equation}
where the top $H$ preserves cartesian liftings, meaning that if $\phi$ is $P$-cartesian, then $H\phi$ is $Q$-cartesian. In particular, when $P$ and $Q$ are fibrations over the same base category, we may consider fibred 1-cells of the form $(H,1_{\X})$ displayed as
\begin{equation}
\label{eq:fibredfunctor}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, "H"]
\arrow[dr, swap, "P"]
\B
\arrow[dl, "Q"]
\\&
\X
\end{tikzcd}
\end{equation}
and $H$ is then called a fibred functor. Dually, we have the notion of an opfibred 1-cell and opfibred functor. Notice that any such (op)fibred 1-cell induces functors between the fibres, by commutativity of <ref>:
\begin{equation}\label{eq:functorbetweenfibres}
H_{x} \maps \A_x\longrightarrow\B_{Fx}
\end{equation}
A fibred 2-cell between fibred 1-cells $(H,F)$ and $(K,G)$ is a pair of natural transformations ($\braid\maps H\Rightarrow K,\alpha\maps F\Rightarrow G$) with $\braid$ above $\alpha$, i.e. $Q(\braid_a)=\alpha_{Pa}$ for all $a\in\A $, displayed as
\begin{equation}
\label{eq:fibred2cell}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, bend left, "H"]
\arrow[rr, phantom, "\Downarrow \beta"]
\arrow[rr, bend right, swap, "K"]
\arrow[d, swap, "P"]
\B
\arrow[d, "Q"]
\\
\X
\arrow[rr, bend left, "F"]
\arrow[rr, phantom, "\Downarrow \alpha"]
\arrow[rr, bend right, swap, "G"]
\Y
\end{tikzcd}
\end{equation}
A fibred natural transformation is of the form $(\braid,1_{1_{\X}})\maps(H,1_{\X})\Rightarrow(K,1_\X)$
\begin{equation}\label{eq:fibrednaturaltrans}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, bend left, "H"]
\arrow[rr, phantom, "\Downarrow \beta"]
\arrow[rr, bend right, swap, "K"]
\arrow[dr, swap, "P"]
\B
\arrow[dl, "Q"]
\\&
\X
\end{tikzcd}
\end{equation}
Dually, we have the notion of an opfibred 2-cell and opfibred natural transformation between opfibred 1-cells and functors respectively.
We thus obtain a 2-category $\Fib$ of fibrations over arbitrary base categories, fibred 1-cells and fibred 2-cells. There is also a 2-category $\Fib(\X)$ of fibrations over a fixed base category $\X$, fibred functors and fibred natural transformations. Dually, we have the 2-categories $\OpFib$ and $\OpFib(\X)$. Moreover, we also have 2-categories $\Fib_\mathrm{sp}$ and $\OpFib_\mathrm{sp}$ of split (op)fibrations, and (op)fibred 1-cells that preserve the cartesian liftings `on the nose'.
Notice that $\Fib$ and $\OpFib$ are both sub-2-categories of $\Cat^\2 = [\2, \Cat]$, the arrow 2-category of $\Cat$. Similarly, $\Fib(\X)$ and $\OpFib(\X)$ are sub-2-categories of $\Cat/\X$, the slice 2-category of functors into $\X$. Due to that, both these (1-)categories form fibrations themselves. Explicitly, the functor $\cod \maps \Fib \to \Cat$ which maps a fibration to its base is a fibration, with fibres $\Fib(\X)$ and cartesian liftings pullbacks along fibrations. In fact, it is a 2-fibration <cit.>.
§ INDEXED CATEGORIES
We now turn to the world of indexed categories. Given an ordinary category $\X$, an $\X$-indexed category is a pseudofunctor
\[\M \maps \X\op \to \Cat\]
where $\X$ is viewed as a 2-category with trivial 2-cells; it comes with natural isomorphisms $\delta_{g,f} \maps (\M g) \circ (\M f) \xrightarrow\sim \M(g \circ f)$ and $\gamma_x \maps 1_{\M x} \xrightarrow\sim \M (1_x)$ for every $x\in\X$ and composable morphisms $f$ and $g$, satisfying coherence axioms.
Dually, an $\X$-opindexed category is an $\X\op$-indexed category, i.e. a pseudofunctor $\X \to \Cat$.
If an (op)indexed category strictly preserves composition, i.e. is a (2-)functor, then it is called strict.
An indexed $1$-cell $(F, \tau) \maps \M \to \psN$ between indexed categories $\M \maps \X\op \to \Cat$ and $\psN \maps \Y\op \to \Cat$ consists of an ordinary functor $F \maps \X \to \Y$ along with a pseudonatural transformation $\tau \maps \M \Rightarrow \psN \circ F\op$
\begin{equation}\label{eq:indexed1cell}
\begin{tikzcd}[column sep=.7in,row sep=.2in]
\X\op
\arrow[dr, "\M"]
\arrow[dd, "F\op"']
\\
\arrow[r, phantom, "\Downarrow{\scriptstyle\tau}"description]
\Cat
\\
\Y\op \arrow[ur, "\psN"']
\end{tikzcd}
\end{equation}
with components functors $\tau_x \maps \M x \to \psN Fx$, equipped with coherent natural isomorphisms $\tau_f \maps (\psN Ff) \circ \tau_x \xrightarrow{\sim} \tau_y \circ (\M f)$ for any $f \maps x \to y$ in $\X$.
For indexed categories with the same base, we may consider indexed 1-cells of the form $(1_\X, \tau)$
\begin{equation}\label{eq:ifun}
\begin{tikzcd}[column sep=.7in]
\X\op
\arrow[r, bend left, "\M"]
\arrow[r, bend right, swap, "\psN"]
\arrow[r, phantom, "\Downarrow \scriptstyle \tau"]
\Cat
\end{tikzcd}
\end{equation}
which are called indexed functors. Dually, we have the notion of an opindexed 1-cell and opindexed functor.
An indexed 2-cell $(\alpha,m)$ between indexed 1-cells $(F, \tau)$ and $(G, \sigma)$, pictured as
[column sep=.6in,row sep=.4in]
[drr, ""]
[drr, ""name = M, swap, pos = 0.5]
[dd,bend right=40,"F" description]
[dd, "G"description, bend left=40]
[urr,""name = H, pos = 0.4]
[from = M, to = H, Rightarrow, "σ", pos = 0.4, bend left=45]
[from = M, to = H, Rightarrow, "τ", pos = 0.56, bend right=45,swap]
[from = M, to = H, phantom, "m⇛"]
consists of an ordinary natural transformation $\alpha \maps F \Rightarrow G$ and a modification $m$
\begin{equation}\label{eq:indexed2cell}
\begin{tikzcd}[column sep=.5in,row sep=.2in]
\X\op
\arrow[rr,bend left,"\M"]
\arrow[dr,bend right=10,"F\op"']
\arrow[rr,phantom,"\Downarrow{\scriptstyle \tau}"description]
\Cat \arrow[r,phantom,"\stackrel{m}{\Rrightarrow}"description]
\X\op
\arrow[rr,bend left=30,"\M"]
\arrow[dr,bend left=35,"G\op"]
\arrow[dr,bend right=35,"F\op"']
\arrow[rr, phantom, near end, "\Downarrow{\scriptstyle \sigma}"description]
\arrow[dr,phantom, "\Downarrow{\scriptstyle \alpha\op}"description]
\Cat
\\
\Y\op
\arrow[ur,bend right=10,"\psN"']
\Y\op
\arrow[ur,bend right,"\psN"'] &
\end{tikzcd}
\end{equation}
given by a family of natural transformations $m_x \maps \tau_x \Rightarrow\psN \alpha_x \circ\sigma_x$. Notice that taking opposites is a 2-functor $(-)\op \maps \Cat \to \Cat^{co}$, on which the above diagrams rely.
An indexed natural transformation between two indexed functors is an indexed 2-cell of the form $(1_{1_\X},m)$.
Dually, we have the notion of an opindexed 2-cell and opindexed natural transformation between opindexed 1-cells and functors respectively.
Notice that an indexed 2-cell $(\alpha,m)$ is invertible if and only if both $\alpha$ is a natural isomorphism and the modification $m$ is invertible, due to the way vertical composition is formed.
We obtain a 2-category $\ICat$ of indexed categories over arbitrary bases, indexed 1-cells and indexed 2-cells. In particular, there is a 2-category $\ICat(\X)$ of indexed categories with fixed domain $\X$, indexed functors and indexed natural transformations, which coincides with the functor 2-category $\TCat_\pse(\X\op,\Cat)$.
Dually, we have the 2-categories $\OpICat$ and $\OpICat(\X)=\TCat_\pse(\X,\Cat)$. Notice that due to the absence of opposites in the world of opindexed categories, opindexed 2-cells have a different form than <ref>, namely
[column sep=.5in, row sep=.15in]
[rr, bend left=30, ""]
[dr, bend left=30, "F"]
[dr, bend right=30, "G"']
[rr, phantom, near end, "⇓τ"description]
[dr, phantom, "⇓α"description]
[r, phantom, "m⇛"description]
[rr, bend left, ""]
[dr, bend right=10, "G"']
[rr, phantom, "⇓σ"description]
[ur, bend right, ""']
[ur, bend right=10, ""']
Moreover, we have 2-categories of strict (op)indexed categories and (op)indexed 1-cells that consist of strict natural transformations $\tau$ <ref>, i.e.$\ICat(\X)=[\X\op,\Cat]$ and $\OpICat_\mathrm{sp}(\X)=[\X,\Cat]$ the usual functor 2-categories.
Notice that these categories also form fibrations over $\Cat$, this time essentially using the family fibration also seen in <ref>. The functor $\ICat \to \Cat$ sends an indexed category to its domain and an indexed 1-cell to its first component. It is a split fibration, with fibres $\ICat(\X)$ and cartesian liftings pre-composition with functors. In fact, it is also a 2-fibration as explained in <cit.>.
§ THE GROTHENDIECK CONSTRUCTION
In the first volume of the Séminaire de Géométrie Algébrique du Bois Marie <cit.>, Grothendieck introduced a construction for a fibration $P_\M \maps \inta \M \to \X$ from a given indexed category $\M \maps \X\op \to \Cat$ as follows. If $\delta$ and $\gamma$ are the structure pseudonatural transformations of the pseudofunctor $\M$,
the total category $\inta \M$ has
* objects $(x,a)$ with $x \in \X$ and $a \in \M x$;
* morphisms $(f,k) \maps (x,a) \to (y,b)$ with $f \maps x \to y$ a morphism in $\X$, and $k \maps a \to (\M f)(b)$ a morphism in $\M x$;
* composition $(g, \ell) \circ (f, k) \maps (x,a) \to (y,b) \to (z,c)$ is given by $g \circ f \maps a \to b \to c$ in $\X$ and
\begin{equation}\label{eq:comp_intM}
a\xrightarrow{k}(\M f)(b)\xrightarrow{(\M g)(\ell)}(\M g\circ\M f)(c)\xrightarrow{(\delta_{f,g})_c}\M(g\circ f)(c)\quad\textrm{in }\M x;
\end{equation}
* unit $1_{(x,a)} \maps (x,a) \to (x,a)$ is given by $1_x \maps x \to x$ in $\X$ and
\[a=1_{\M x}a\xrightarrow{(\gamma_x)_a}(\M 1_x)(a)\quad\textrm{in }\M x.\]
The fibration $P_\M \maps \inta \M \to \X$ is given by $(x,a) \mapsto x$ on objects and $(f,k) \mapsto f$ on morphisms, and the cartesian lifting of
any $(y,b)$ in $\inta \M$ along $f \maps x \to y$ in $\X$ is precisely $(f,1_{(\M f)b})$. Its fibres are precisely $\M x$ and the reindexing functors between them are $\M f$.
In the other direction, given a (cloven) fibration $P \maps \A \to \X$, we can define an indexed category $\M_P \maps \X\op \to \Cat$ that sends each object $x$ of $\X$ to its fibre category $\A_x$, and each morphism $f \maps x \to y$ to the corresponding reindexing functor $f^* \maps \A_y \to \A_x$ as in <ref>. The isomorphisms of cartesian liftings $f^* \circ g^* \cong (g \circ f)^*$ and $1_{\A_x} \cong 1_x^*$ render this assignment pseudofunctorial.
Details of the above, as well as the correspondence between 1-cells and 2-cells, can be found in the provided references. Briefly, given a pseudonatural transformation $\tau \maps \M \to \psN\circ F\op$ <ref> with components $\tau_x \maps \M x \to \psN Fx$, define a functor $P_\tau \maps \inta\M \to \inta\psN$ mapping $(x\in\X,a\in\M x)$ to the pair $(Fx\in\Y,\tau_x(a)\in\psN Fx)$ and accordingly for arrows. This makes the square
\begin{equation}
\label{eq:inducedfibred1cell}
\begin{tikzcd}
\inta\M
\arrow[r,"P_\tau"]
\arrow[d,"P_\M"']
\inta\psN
\arrow[d,"P_\psN"]
\\
\X\arrow[r,"F"']
\Y
\end{tikzcd}
\end{equation}
commute, and moreover $P_\tau$ preserves cartesian liftings due to pseudonaturality of $\tau$. Moreover, given an indexed 2-cell $(\alpha,m) \maps (F,\tau)\Rightarrow(G,\sigma)$ as in <ref>, we can form a fibred 2-cell
\begin{equation}\label{eq:inducedfibred2cell}
\begin{tikzcd}[column sep=.8in,row sep=.6in]
\inta\M
\arrow[r,bend left,"P_\tau"]
\arrow[r,bend right,"P_\sigma"']
\arrow[r,phantom,"\Downarrow{\scriptstyle P_m}"description]
\arrow[d,"P_\M"']
\inta\psN
\arrow[d,"P_\psN"]
\\
\X
\arrow[r,bend left,"F"]
\arrow[r,bend right,"G"']
\arrow[r,phantom,"\Downarrow{\scriptstyle\alpha}"description]
\Y
\end{tikzcd}
\end{equation}
where $\alpha \maps F\Rightarrow G$ is piece of the given structure, whereas $P_m$ is given by components
\[
(P_m)_{(x,a)} \maps P_\tau(x,a)=(Fx,\tau_xa) \to P_\sigma(x,a)=(Gx,\sigma_xa)\quad\textrm{in } \inta \psN
\]
explicitly formed by $\alpha_x \maps Fx \to Gx$ in $\Y$ and $(m_x)_a \maps \tau_xa \to (\psN\alpha_x)\sigma_xa$ in $\psN Fx$.
The following theorem summarizes these standard results.
* Every fibration $P \maps \A \to \X$ gives rise to a pseudofunctor $\M_P \maps \X\op { \to }\Cat$.
* Every indexed category $\M \maps \X\op \to \Cat$ gives rise to a fibration $P_\M \maps \inta\M \to \X$.
* The above correspondences yield an equivalence of 2-categories
() ≃()
so that $\M_{P_\M} \simeq \M$ and $P_{\M_P} \simeq P$.
* The above 2-equivalence extends to one between 2-categories of arbitrary-base fibrations and arbitrary-domain indexed categories
\begin{equation}\label{eq:Gr_equiv}
\ICat\simeq\Fib
\end{equation}
If we combine the above with the fact that the 2-categories $\Fib$ and $\ICat$ are fibred over $\Cat$ with fibres $\Fib(\X)$ and $\ICat(\X)$ respectively, we obtain the following $\Cat$-fibred equivalence
\begin{equation}\label{FibICat}
\begin{tikzcd}
\ICat
\arrow[rr, "\simeq"]
\arrow[dr]
\Fib
\arrow[dl]
\\&
\Cat
\end{tikzcd}
\end{equation}
There is an analogous story for opindexed categories and opfibrations that results into a 2-equivalences $\OpICat(\X)\simeq\OpFib(\X)$ and $\OpICat\simeq\OpFib$, as well as for the split versions of (op)indexed and (op)fibred categories.
§ EXAMPLES
§.§ Fundamental Fibration
Let $2$ denote the category with two objects, and one non-identity morphism $\star \to \bullet$. For a category $\X$, the functor category $\X^2$ then consists of the arrows of $\X$ as objects, and commuting squares between them as the morphisms.
For any category $\X$, the codomain or fundamental opfibration is the usual functor from its arrow category
\[\cod \maps \X^2 \longrightarrow \X\]
mapping every morphism to its codomain and every commutative square to its right-hand side leg. It uniquely corresponds to the strict opindexed category, i.e. functor
\begin{equation}
\begin{tikzcd}[row sep=.05in]
\X\arrow[r] & \Cat \\
x\arrow[r,mapsto]\arrow[dd,"f"'] & \X/x\arrow[dd,"f_!"] \\
\\
y\arrow[mapsto, r] & \X/y
\end{tikzcd}
\end{equation}
that maps an object to the slice category over it and a morphism to the post-composition functor $f_!=f\circ-$ induced by it.
§.§ Graphs
Consider (directed, multi) graphs, i.e. presheaves on the category $G = \begin{tikzcd} V \arrow[r, shift right, swap, "t"] \arrow[r, shift left, "s"] & E\end{tikzcd}$. For a presheaf $g \maps G\op \to \Set$, the set $g_V$ is the set of vertices of the graph, the set $g_E$ is the set of edges of the graph, and the maps $g_s, g_t \maps g_E \to g_V$ assign to an edge its starting and terminating vertex respectively. Let $\Grph$ denoted the category of graphs, $\Set^{G\op}$. Sometimes it is helpful to think of a graph as a single map of the form $(g_s, g_T) \maps g_E \to g_V \times g_V$. When convenient, we will abuse notation by simply referring to this map as $g \maps g_E \to g_V^2$.
Consider the inclusion of a terminal category $1$ into $G$ which selects the object $V$. This induces a functor $\mathsf V \maps \Grph \to \Set$ by precomposing, which sends a graph $g$ to its vertex set $g_V$. As we show below, this functor is in fact a bifibration. The idea here is that if you have a function $f \maps x \to y$, you can pull a graph on $y$ back to a graph on $x$, and you can also push a graph on $x$ forward to a graph on $y$.
A morphism $\phi\maps g \to h$ in $\Grph$ is $\mathsf V$-cartesian if and only if the square
\[
\begin{tikzcd}
\arrow[d, swap, "{(g_s, g_t)}"]
\arrow[r, "\phi_E"]
\arrow[d, "{(h_s, h_t)}"]
\\
\arrow[r, swap, "\phi_V^2"]
\end{tikzcd}
\]
is a pullback in $\Set$.
A simple manipulation shows that the universal property of $\phi$ forming a pullback square is the same as the universal property for it to be $\mathsf V$-cartesian.
The functor $\mathsf V \maps \Grph \to \Set$ is a fibration.
Let $f \maps x \to y$ be a function, and $g \in \Grph$ with $g_V=y$. Then we can take a pullback of the following diagram.
\[x^2 \xrightarrow{f^2} y^2 = h_V^2 \xleftarrow{(h_s,h_t)} h_E\]
By <ref>, this map is a cartesian lift of $f$.
By the Grothendieck correspondence, there is a indexed category $\Set\op \to \Cat$.This pseudofunctor assigns to a set $X$ the category $\Grph_X$ of graphs which have vertex set $X$, and graph morphisms which fix the vertices. Given a function $f \maps X \to Y$, this pseudofunctor gives a functor $f^* \maps \Grph_Y \to \Grph_X$ which sends a graph $g$ over $Y$ to the pullback, as in the proof of <ref>. Since there is also an opindexed category with the same fibres, we denote this by $\Grph^*$, referring to the action on morphisms.
To show that $\mathsf V$ is also an opfibration, it is actually easier to construct an explicit splitting. We can derive a characterization of the cocartesian maps from there.
The functor $\mathsf V \maps \Grph \to \Set$ is an opfibration.
Let $g \in \Grph$, $y \in \Set$, and $f \maps g_V \to y$ a function. Then we can obtain a graph with vertex set $y$ by taking the following composite.
\[g_E \xrightarrow{g} g_V^2 = g_V^2 \xrightarrow{f^2} y^2\]
We claim the induced map of graphs displayed below is in fact a cocartesian lift of $f$.
\[
\begin{tikzcd}
\arrow[r, "1"]
\arrow[d, swap, "g"]
\arrow[d, "f^2 \circ g"]
\\
\arrow[r, swap, "f^2"]
\end{tikzcd}
\]
Let $h$ be a graph, $\phi \maps g \to h$ a map of graphs, and $\phi \maps y \to h_V$.
\[
\begin{tikzcd}[column sep = huge]
\arrow[d, "h"]
\\&&
\arrow[ddll, bend right = 15, leftarrow, swap, "\phi_V^2"]
\\
\arrow[uurr, bend left = 15, "\phi_E"]
\arrow[r, -, white, line width = 5]
\arrow[r, "1", pos = 0.4]
\arrow[d, swap, "g"]
\arrow[d, swap, "f^2 \circ g"]
\arrow[uur, -, white, line width = 5]
\arrow[uur, dashed, ""]
\\
\arrow[r, swap, "f^2"]
\arrow[uur, swap, "1"]
\end{tikzcd}
\]
The only map which may take the place of the dashed arrow is $\phi_E$.
A morphism $\phi\maps g \to h$ in $\Grph$ is $\mathsf V$-cocartesian if and only if it is bijective on edges.
By the Grothendieck correspondence, there is a corresponding opindexed category $\Grph_* \maps \Set \to \Cat$, again referring to the action on morphisms. This must have the same fibres $\Grph_X$ as the indexed category $\Grph^*$ above. Given a function $f \maps X \to Y$, this pseudofunctor gives a functor $f_* \maps \Grph_X \to \Grph_Y$ which sends a graph $g$ over $X$ to the composite, as in the proof of <ref>.
The functor $\mathsf V \maps \Grph \to \Set$ is a bifibration.
§.§ Ring Modules
For a ring $R$, denote by $\Mod_R$ the category of $R$-modules and their homomorphisms. Given a ring homomorphism $f \maps R \to S$, and an $S$-module $N$, we can give the underlying abelian group of $N$ the structure of an $R$-module, denoted $f^\ast N$, by the formula
\[r.x := f(r).x\]
where $r \in R$ and $x \in N$.
This pullback construction is functorial:
\[f^\ast \maps \Mod_S \to \Mod_R\]
and preserves ring homomorphism composition.
\[(f\circ g)^\ast = g^\ast f^\ast \]
Indeed, the above defines a functor $\Mod_- \maps \Ring\op \to \Cat$, a (strict) indexed category. Note, one could choose to be persnickety about size here, but we do not. We can then apply the Grothendieck construction, resulting in a category $\Mod := \int \Mod_-$ where
* an object is a pair $(R \in \Ring, M \in \Mod_R)$
* a morphism is a pair $(f, \phi)$ where $f \maps R \to S$ is a ring homomorphism, and $\phi \maps M \to f^\ast N$ is an $S$-module homomorphism.
The category $\Mod$ admits a fibration $\Mod \to \Ring$ which forgets the module in a ring-module pair.
CHAPTER: SPECIES AND OPERADS
§ COMBINATORIAL SPECIES
Combinatorial species were introduced by Joyal <cit.>. A standard reference for the combinatorial perspective is <cit.>. In the previous section, we noted that the category $\Set$ can be given both a cartesian and cocartesian monoidal structure. Moreover, these satisfy a distributive law reminiscent of rings.
\[A \times (B + C) \cong A \times B + A \times C\]
Consider the subcategory $\FinBij$ consisting of finite sets and bijections. This subcategory is closed under both finite sums and products, and thus inherits both monoidal structures. However, $\FinBij$ lacks the maps that would be the projections and inclusions necessary for these structures to be cartesian or cocartesian themselves. By abuse of notation, we will continue to denote them by $+$ and $\times$.
A combinatorial species or simply species is a functor $F \maps \FinBij \to \Set$. The category of species is the functor category $\Set^\FinBij$.
There are several operations which have been defined on species. These operations make up the building blocks of a calculus for counting families of combinatorial gadgets.
Being a presheaf category, $\Set^\FinBij$ has colimits given pointwise, thus giving it a cocartesian structure. We refer to this operation simply as addition. On objects, this operation is given by $(F+G)(U) = F(U) + G(U)$.
If we apply Day convolution as in <ref> to the $+$ monoidal structure on $\FinBij$, we get the operation which we refer to as multiplication of species.
On objects, this operation is given by the following formula.
\[(F \cdot G)(U) = \sum_{V \subseteq U} F(V) \times G(U \setminus V)\]
Being a presheaf category, $\Set^\FinBij$ has products which are given pointwise, thus giving a cartesian monoidal structure. This is called the Hadamard product. On objects, it is given by $(F \times G) (U) = F(U) \times G(U)$. This tends to be less useful than multiplication, but it certainly has its purposes.
We define the Dirichlet product on species to be the Day convolution (as in <ref>) of the $\times$ monoidal structure on $\FinBij$.
To define differentiation of species, we will need to make use of the shift operator, denoted $+1$, on $\FinBij$, which is defined as the composite
\[
\begin{tikzcd}[column sep = huge]
\FinBij
\arrow[r, dashed, "+1"]
\arrow[d, swap, "\sim"]
\FinBij
\\
\FinBij \times 1
\arrow[r, swap, "1_{\FinBij} \times \Delta1"]
\FinBij \times \FinBij
\arrow[u, swap, "+"]
\end{tikzcd}
\]
where the map $\Delta1 \maps 1 \to \FinBij$ is the monoidal unit with respect to $\times$.
The differentiation operator on species is given by $D = +1^\ast \maps \Set^\FinBij \to \Set^\FinBij$. In other words, for a given species $F$, the derivative of $F$ is given by $F' = F \circ +1$, or $F'(U) = F(U+1)$ on objects. The motivation for calling this operation differentiation is that it actually corresponds to taking the formal derivative of its generation series.
The composition product or substitution product is given by the following formula.
\[(F \circ G)(U) = \sum_{\pi \text{ partition of } U} \left( F(\pi) \times \prod_{p \in \pi} G(p) \right)\]
The species which acts as unit for this product is the singleton indicator functor, i.e. $I_\circ (U)$ is a singleton is $U$ is, and is empty otherwise. This monoidal structure is not symmetric.
§ OPERADS
An operad is a generalization of category which incorporates the notion of an arrow having multiple inputs. A category is an arrow-like compositional system, consisting of a collection of directed arrows, a collection of labels for the endpoints called objects, and a rule for turning a path of such arrows into a single arrow which is associative and unital. An operad is also a compositional system, but now tree-like. An operad consists of a collection of directed “short” trees
\begin{equation*}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, 0.7) {};
\node [style=none] () at (0.5, 0.7) {\dots};
\node [style=none] (1) at (1, 0.7) {};
\node [style=none] (2) at (0, 0.5) {};
\node [style=none] (6) at (0.33, 0.5) {};
\node [style=none] (7) at (0.66, 0.5) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=downtri] (mu) at (0.5, 0) {};
\node [style=none] (5) at (0.5, -0.7) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw (3.center) to (2.center);
\draw [bend right] (2.center) to (mu.center);
\draw [bend left] (4.center) to (mu.center);
\draw (6.center) to (mu.center);
\draw (7.center) to (mu.center);
\draw (mu.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}
a collection of labels for the endpoints called objects, and a rule for turning a big tree of short trees
\begin{equation*}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=downtri] (a) at (0.5, 2.5) {};
\node [style=downtri] (b) at (1, 1.5) {};
\node [style=downtri] (c) at (0, 0.5) {};
\node [style=none] (1) at (0.25, 3) {};
\node [style=none] (2) at (0.5, 3) {};
\node [style=none] (3) at (0.75, 3) {};
\node [style=none] (5) at (1.25, 2) {};
\node [style=none] (6) at (-0.5, 1) {};
\node [style=none] (7) at (-0.125, 1) {};
\node [style=none] (8) at (0.125, 1) {};
\node [style=none] (9) at (0, -0.3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (a.center);
\draw (2.center) to (a.center);
\draw [bend left] (3.center) to (a.center);
\draw [bend right] (a.center) to (b.center);
\draw [bend left] (5.center) to (b.center);
\draw [bend left] (b.center) to (c.center);
\draw [bend right] (6.center) to (c.center);
\draw (7.center) to (c.center);
\draw (8.center) to (c.center);
\draw (c.center) to (9.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}
into a single short tree
\begin{equation*}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=downtri] (a) at (0, 0.5) {};
\node [style=none] (1) at (-0.375, 1) {};
\node [style=none] (2) at (-0.25, 1) {};
\node [style=none] (3) at (-0.125, 1) {};
\node [style=none] (4) at (0, 1) {};
\node [style=none] (5) at (0.125, 1) {};
\node [style=none] (6) at (0.25, 1) {};
\node [style=none] (7) at (0.375, 1) {};
\node [style=none] (8) at (0, -0.3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (a.center);
\draw [bend right = 15] (2.center) to (a.center);
\draw (3.center) to (a.center);
\draw (4.center) to (a.center);
\draw (5.center) to (a.center);
\draw [bend left = 15] (6.center) to (a.center);
\draw [bend left] (7.center) to (a.center);
\draw (8.center) to (a.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}
which is associative and unital. There are several good references for the theory of operads <cit.>. Here, we follow the treatment given by Yau in <cit.>.
§.§ Definition of Operad
Let $C$ be a non-empty set, whose elements we call colors. Recall that we denote the free symmetric monoidal category on $C$ by $\S(C)$. Below, we define operads to be monoids in the presheaf category $\Set^{\S(C) \times C}$ with respect to a certain monoidal structure. First we must define this monoidal structure, which is somewhat involved.
For this section, we denote objects of $\S(C)$ by either $\underline c$ or $(c_1, \dots, c_n)$ depending on context. We denote the monoidal structure on $\S(C)$ by $+$. For an object $X \in \Set^{\S(C)\op \times C}$, we denote the set assigned to $(\underline c, d) \in \S(C)\op \times C$ by $X(\underline c; d)$.
Let $X, Y \in \Set^{\S(C) \times C}$. For each $\underline c \in \S(C)$, define $Y^{\underline c}$ by the following coend formula.
\[Y^{\underline c} (\underline b) = \int^{\{\underline a_j\} \in \prod_{j=1}^m \S(C)\op} \S(C)\op (\underline a_1 + \dots + \underline a_m, \underline b) \times \left[\prod_{j=1}^m Y(\underline a_j; c_j) \right]\]
The $C$-colored circle product of $X$ and $Y$ is given by
\[X \circ Y(\underline b; d) = \int^{\underline c \in \S(C)} X(\underline c; d) \otimes Y^{\underline c}(\underline b)\]
Define the unit object $I$ as follows.
\[I(\underline c; d) = \begin{cases}
1 & \text{if } \underline c = d\\
\emptyset & \text{otherwise}
\end{cases}\]
This reduces to the composition monoidal product of species in the case where $C \cong 1$.
$(\Set^{\S(C) \times C}, \circ, I)$ is a monoidal category.
Let $C$ be a set. Define the category of operads by
\[\Opd_C = \Mon(\Set^{\S(C) \times C}, \circ, I).\]
We refer to an object of $\Opd_C$ as a $C$-colored operad, and a morphism as a color-fixing $C$-operad functor. Let $f \maps C \to D$ be a function. Let $f(\underline c)$ denote $(f(c_1), \dots, f(c_n))$ for $\underline c \in \S(C)$.
For a $D$-operad $P$, we can pullback along $f$ to get a $C$-operad given by $f^*P(\underline c; d) = P(f(\underline c); f(d))$. For a color-fixing $D$-operad functor $\phi \maps P \to Q$, we get a color-fixing $C$-operad functor $f^*\phi \maps f^*P \to f^*Q$ which sends an operation $\theta \in f^*P(\underline c; d) = P(f(\underline c); d)$ to $\phi\theta \in Q(f(\underline c); d)$. These assignments give a functor $f^* \maps \Opd_D \to \Opd_D$, and we get an indexed category $\Opd_- \maps \Set\op \to \Cat$. Define $\Opd = \int \Opd_-$. We refer to an object of $\Opd$ as an operad, and to a morphism as an operad functor.
§.§ Operads from symmetric monoidal categories
There is a standard method of constructing an single-colored operad from an object $x$ in a strict symmetric monoidal category $\C$. Namely, we define the set of $n$-ary operations to be $\hom_{\C}(x^{\otimes n}, x)$, and compose these operations using composition in $\C$. This gives the so-called endomorphism operad of $x$. Here we give the generalization of this idea to the multi-color case, using all the objects of $\C$ as the objects of the operad. In what follows we let $\Ob(\C)$ be the set of objects of a small category $\C$.
If $\C$ is a small strict symmetric monoidal category then there is an $\Ob(\C)$-colored operad $\Op(\C)$ for which:
* the set of operations $\Op(\C)(c_1, \dots, c_k; c)$ is defined to be $\hom_\C(c_1 \otimes \dots \otimes c_k, c)$,
* given operations
\[f\in \hom_\C(c_1 \otimes \cdots \otimes c_k; c) \]
\[g_i \in \hom_\C(c_{ij_1} \otimes \cdots \otimes c_{ij_i},c_i)
\]
for $1 \le i \le k$, their composite is defined by
\begin{equation}
\label{eq:composition_of_operations}
f \circ (g_1, \dots, g_k) = f \circ (g_1 \otimes \cdots \otimes g_k) .
\end{equation}
* identity operations are identity morphisms in $\C$, and
* the action of $S_k$ on $k$-ary operations is defined using the braiding in $\C$.
The various axioms of a colored operad can be checked for $\Op(\C)$ using the corresponding laws in the definition of a strict symmetric monoidal category. The associativity axiom for $\Op(\C)$ follows from associativity of composition and the functoriality of the tensor product in $\C$. The left and right unit axioms for $\Op(\C)$ follow from the unit laws for composition and the functoriality of the tensor product in $\C$. The two equivariance axioms for $\Op(\C)$ follow from the laws governing the braiding in $\C$.
The assignment $\Op \maps \Sym\Mon\Cat_\spl \to \Opd$ defined on objects as in <ref> and sending any strict symmetric monoidal functor $F \maps \C \to \C'$ to the operad morphism $\Op(F) \maps \Op(\C) \to \Op(\C')$ that acts by $F$ on types and also on operations:
\[
\Op(F) = F \maps \hom_C(c_1 \otimes \cdots \otimes c_n, c) \to \hom_{\C'}(F(c_1) \otimes \cdots \otimes F(c_n), F(c))
\]
is a functor.
This is a straightforward verification.
§.§ Operad Algebras
As a sort of monoid, operads exist to act. The elements of $\O(\underline c;d)$ for some operad $\O$ are meant to be thought of as “abstract operations” with $\underline c$ as the input types, and $d$ as the output type. When $\O$ acts on something, it is meant to be thought of as realizing these abstract operations as real operations on some family of sets indexed by the elements of $C$.
Let $C$ be a set. A $C$-colored set is a functor $C \to \Set$, where $C$ is thought of as a discrete category. This is of course the same as a function $C \to \ob\Set$. For $\underline c = (c_1, \dots, c_n) \in \S(C)$, let $X_{\underline c}$ denote the set $\prod_{j=1}^n X_{c_j}$. A map of $C$-colored sets $f \maps X \to Y$ is a natural transformation $f \maps X \To Y$. This is the same as a family of functions $\{f_c \maps X_c \to Y_c\}_{c \in C}$ with no further conditions. For $\underline c = (c_1, \dots, c_n) \in \S(C)$, let $f_{\underline c} \maps X_{\underline c} \to Y_{\underline c}$ denote the function $\prod_{j=1}^n f_{c_j} \maps \prod_{j=1}^n X_{c_j} \to \prod_{j=1}^n Y_{c_j}$.
Let $\O$ be a $C$-colored operad, with operad composition denoted by $\gamma$, and unit operation denoted by $I$. An $\O$-algebra consists of
* a $C$-colored set $X \maps C \to \Set$, also denoted $\{X_c\}_{c \in C}$
* for $\underline c \in \S(C)$ and $d \in C$, a map $\theta \maps \O(\underline c;d) \times X_{\underline c} \to X_d$
which makes the following diagrams commute for $c,d, c_j \in C$, $\underline c, \underline b_j \in \S(C)$, $\underline b = \sum^n \underline b_j$, and $\sigma \in S_n$.
* associativity:
\[
\begin{tikzcd}
\O(\underline c, d) \times \prod_{j=1}^n \O(\underline b_j, c_j) \times X_{\underline b}
\arrow[d, "\cong", "\text{permute}"']
\arrow[r, "\gamma \times 1"]
\O(\underline b; d) \times X_{\underline b}
\arrow[dd, "\theta"]
\\
\O(\underline c; d) \times \prod_{j=1}^n [\O(\underline b_j, c_j) \times X_{\underline b_j}]
\arrow[d, swap, "1 \times \prod_j \theta"]
\\
\O(\underline c; d) \times X_{\underline c}
\arrow[r, swap, "\theta"]
\end{tikzcd}\]
* unity:
\[
\begin{tikzcd}
1 \times X_c
\arrow[dl, swap, "I \times 1"]
\arrow[dr, "\cong"]
\\
\O(c;c) \times X_c
\arrow[rr, swap, "\theta"]
\end{tikzcd}\]
* equivariance:
\[
\begin{tikzcd}
\O(\underline c; d) \times X_{\underline c}
\arrow[dr, swap, "\theta"]
\arrow[rr, "\sigma \times \sigma\inv"]
\O(\underline c \sigma; d) \times X_{\underline c \sigma}
\arrow[dl, "\theta"]
\\&
\end{tikzcd}\]
A map of $\O$-algebras $(X, \theta) \to (Y, \xi)$ consists of a map of $C$-colored sets $\alpha \maps X \to Y$ such that the following diagram commutes.
\[
\begin{tikzcd}
\O(\underline c; d) \times X_{\underline c}
\arrow[r, "1 \times f_{\underline c}"]
\arrow[d, swap, "\theta"]
\O(\underline c; d) \times Y_{\underline c}
\arrow[d, "\xi"]
\\
\arrow[r, swap, "f"]
\end{tikzcd}\]
Let $\Alg(\O)$ denote the category of $\O$-algebras and maps of $\O$-algebras.
|
# The geometry and DSZ quantization of four-dimensional supergravity
C. I. Lazaroiu Center for Geometry and Physics, Institute for Basic Science,
Pohang, Republic of Korea<EMAIL_ADDRESS>and C. S. Shahbazi Department of
Mathematics, University of Hamburg, Germany<EMAIL_ADDRESS>
###### Abstract.
We develop the Dirac-Schwinger-Zwanziger (DSZ) quantization of four-
dimensional bosonic ungauged supergravity on an oriented four-manifold $M$ of
arbitrary topology and use it to obtain its manifestly duality-covariant
gauge-theoretic geometric formulation. Classical bosonic supergravity is
completely determined by a submersion $\pi$ over $M$ equipped with a complete
Ehresmann connection, a vertical euclidean metric and a vertically-polarized
flat symplectic vector bundle $\Xi$. We implement the Dirac-Schwinger-
Zwanziger quantization condition in the aforementioned classical supergravity
through the choice of an element in the degree-two sheaf cohomology group with
coefficients in a locally constant sheaf $\mathcal{L}\subset\Xi$ valued in the
groupoid of integral symplectic spaces. We show that this data determines a
Siegel principal bundle $P_{\mathfrak{t}}$ of fixed type
$\mathfrak{t}\in\mathbb{Z}$ whose connections provide the global geometric
description of the electromagnetic guage potentials of the theory. The Maxwell
gauge equations of the theory reduce to the polarized self-duality condition
determined by $\Xi$ on the connections of $P_{\mathfrak{t}}$. We investigate
the continuous and discrete U-duality groups of the theory, characterizing
them through short exact sequences and realizing the latter through the gauge
group of $P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the
geometric origin of U-duality, which we explore in several examples,
illustrating its dependence on the topology of the fiber bundles $\pi$ and
$P_{\mathfrak{t}}$ as well as on the isomorphism type of $\mathcal{L}$.
###### Key words and phrases:
Mathematical supergravity, abelian gauge theory, electromagnetic duality,
symplectic vector bundles
2010 MSC. Primary: 53C80. Secondary: 83E50.
## 1\. Introduction
The goal of this article is to construct the geometric, gauge-theoretic and
duality-covariant global formulation of the universal bosonic sector of four-
dimensional (ungauged) supergravity and study its U-duality group. In order to
do so, we develop the Dirac-Schwinger-Zwanziger (DSZ) quantization [15, 32,
35] of the gauge sector of the theory and interpret the result geometrically.
The local formulation of classical four-dimensional supergravity theories has
been studied intensively in the physics literature, see for instance the
seminal references [2, 3, 7, 8, 10, 11, 13, 14] and the reviews and books [4,
18, 20, 29, 31]. Such theories share a universal bosonic sector, which is
subject to increasingly stringent constraints according to the number of
supersymmetry generators of the theory. The global classical geometric
formulation of the universal bosonic sector of supergravity was obtained in
[24, 25], see also [28] for a mathematically rigorous approach to supergravity
based on supergeometry. In [24, 25] it was found that the local structure of
supergravity does not suffice to determine the theory on spacetimes which are
not simply-connected. As show in loc. cit., the global formulation of the
classical universal bosonic sector of ungauged supergravity on an oriented
four-manifold $M$ is a Generalized Einstein-Section-Maxwell theory, which is
determined by the following data:
* •
A _scalar bundle_ $(\pi,\mathcal{H},\mathcal{G})$, where $\pi\colon X\to M$ is
a submersion equipped with a complete flat Ehresmann connection $\mathcal{H}$
and a Euclidean vertical metric $\mathcal{G}$, i.e. a Euclidean metric on the
vertical bundle of $\pi$ which is invariant under the parallel transport of
$\mathcal{H}$.
* •
A _duality bundle_ $\Delta=(\mathcal{S},\omega,\mathcal{D})$ i.e. a flat
symplectic vector bundle over the total space $X$ of $\pi$.
* •
A vertical polarization $\mathcal{J}$ on $\Delta$, i.e. a taming of
$(\mathcal{S},\omega)$ which is invariant under the extended parallel
transport induced by $\mathcal{H}$ and $\mathcal{D}$.
The configuration space of the bosonic supergravity theory determined by a
tuple $(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$ as introduced above
is the set of triples $(g,s,\mathcal{F})$, where $g$ is a Lorentzian metric,
$s$ is a global section of $\pi$ and
$\mathcal{F}\in\Omega^{2}(M,\mathcal{S}^{s})$ is a field strength, that is, a
two-form on $M$ taking values in the pull-back $\mathcal{S}^{s}$ of
$\mathcal{S}$ by $s$ which is flat with respect to the pullback connection
$\mathcal{D}^{s}$. The classical equations of motion of the theory were given
in terms of global geometric structures in [25]. In particular, the Maxwell
equations (i.e. the equations of motion for $\mathcal{F}$) correspond to the
polarized self-duality condition determined by the taming $\mathcal{J}$.
The description of the gauge sector in terms of field strengths $\mathcal{F}$
is unsatisfactory when coupling the theory to quantized charged particles.
Indeed, the Aharonov-Bohm effect [1] implies that this sector should admit a
global description in terms of gauge potentials, which are expected to be
modeled by connections $\mathcal{A}$ on an appropriate principal bundle $P$
defined on $M$. To determine this bundle, we develop the geometric formulation
of the Dirac-Schwinger-Zwanziger (DSZ) quantization condition of the gauge
sector. We then show that this condition implies that $P$ is a Siegel bundle
in the sense of [26], i.e. a principal bundle whose structure group is the
automorphism group of an integral symplectic affine torus. The latter is
isomorphic to a certain semidirect product of an even-dimensional torus group
with a modified Siegel modular group. This process parallels the DSZ
quantization of Abelian gauge theories with manifest electromagnetic duality,
developed in [26], which depends on a Siegel system $Z$ on $X$. The latter was
defined in [26] as a local system of finitely-generated free Abelian groups
whose structure group reduces to a modified Siegel modular group and which is
isomorphic to $\Delta$ upon tensorization with $\mathbb{R}$ over $\mathbb{Z}$.
We define a classical configuration $(g,s,\mathcal{F})$ to be integral if the
cohomology class of $\mathcal{F}$ with respect to the de Rham differential
twisted by $\mathcal{D}^{s}$ belongs to the second cohomology group of $M$
with coefficients in $Z^{s}$, the pull-back of $Z$ by $s$. By the results of
[26], any element of $H^{2}(M,Z)$ is the twisted Chern class of a Siegel
bundle defined on the total space of $\pi$. Using this fact, we show that the
DSZ quantization of classical bosonic supergravity is determined by the
following data:
* •
A _scalar bundle_ $(\pi,\mathcal{H},\mathcal{G})$.
* •
A Siegel bundle $P_{\mathfrak{t}}$ (of type $\mathfrak{t}$) defined on the
total space $X$ of $\pi$.
* •
A vertical polarization $\mathcal{J}$ on the adjoint bundle
$\mathrm{ad}(P_{\mathfrak{t}})$ of $P_{\mathfrak{t}}$.
The configuration space of the DSZ quantization of bosonic supergravity
determined by $(\pi,\mathcal{H},\mathcal{G})$ and
$(P_{\mathfrak{t}},\mathcal{J})$ is given by triples $(g,s,\mathcal{A})$,
where $g$ and $s$ are as defined above and $\mathcal{A}$ is a connection on
$P^{s}_{\mathfrak{t}}$, which describes both the electric and magnetic
potentials of the theory. The Maxwell equations become a first-order condition
on the connections of $P_{\mathfrak{t}}$ which depends on the Hodge operator
of $g$ and the polarization $\mathcal{J}$. In particular, we obtain the global
and duality-covariant equations of motion of the universal bosonic sector of
four-dimensional ungauged supergravity on $(P_{\mathfrak{t}},\mathcal{J})$ in
terms of the variables $(g,s,\mathcal{A})$.
Using this geometric and gauge-theoretic formulation of bosonic supergravity,
we study its group of continuous and discrete U-duality transformations, which
we characterize through short exact sequences involving the group of
automorphisms of $P_{\mathfrak{t}}$ and its adjoint bundle. In general, these
groups can differ markedly from their local counterparts considered in the
physics literature [22]. In this regard, we emphasize the dependence of
U-duality groups on the _type_ 111This is a classical notion in the theory of
symplectic lattices see for instance [12] or [26, Appendix B] for more
details. of the corresponding Siegel modular group and of the isomorphism
class of the Siegel system $Z$, a point which does not appear to have been
noticed in the supergravity literature. In particular, the explicit
computation of the discrete U-duality groups becomes a hard arithmetic problem
in the theory of automorphisms of local systems. As an application of the
framework that we develop, we show that group of discrete electromagnetic
duality transformations is the _discrete remnant_ of the unbased group of
automorphisms of $P_{\mathfrak{t}}$. This clarifies the geometric origin of
U-duality in supergravity.
The geometric formulation described in this paper provides the basis of the
mathematical framework necessary to investigate the differential-geometric
problems arising in four-dimensional supergravity. In particular, it allows
for a mathematically rigorous formulation of the geometric constraints imposed
by supersymmetry through the corresponding Killing spinor equations, in the
spirit of [9], thus opening the way for developing the mathematical theory of
supergravity supersymmetric solutions and moduli spaces of such.
###### Acknowledgements.
We thank Vicente Cortés and Tomás Ortín for useful comments and discussions.
The work of C. I. L. was supported by grant IBS-R003-S1. The work of C.S.S. is
supported by the Germany Excellence Strategy _Quantum Universe_ \- 390833306.
## 2\. Classical bosonic supergravity
In this section we recall the construction of generalized Einstein-Section-
Maxwell theories on an oriented four-manifold $M$ given in [24, 25]. These
give the global geometric formulation of the universal bosonic sector of four-
dimensional supergravity (also called _classical geometric bosonic
supergravity_ or classical bosonic supergravity for short, see [9]).
### 2.1. Preparations
Let $M$ be an oriented and connected four-manifold. We start by introducing
the geometric data needed to formulate classical bosonic supergravity on $M$,
a detailed account of which was given in [25].
###### Definition 2.1.
A _scalar bundle_ of rank $n_{s}$ on $M$ is a triple
$(\pi,\mathcal{H},\mathcal{G})$ consisting of:
* •
A smooth submersion $\pi\colon X\to M$, where $X$ is a connected and oriented
differentiable manifold of dimension $n_{s}+4$.
* •
A complete Ehresmann connection $\mathcal{H}\subset TX$ on $\pi$.
* •
A vertical Euclidean metric, i.e. a Euclidean metric $\mathcal{G}$ defined on
the vertical bundle $\mathcal{V}\subset TX$ of $\pi$ which is preserved by the
parallel transport of $\mathcal{H}$. Recall that the vertical bundle
$\mathcal{V}\subset TX$ is defined as the kernel of the differential map
$\mathrm{d}\pi\colon TX\to TM$.
We say that a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ is _flat_ if
$\mathcal{H}$ is Frobenius integrable.
In the following we let $\mathcal{O}_{m}$ denote the restriction to the fiber
$X_{m}$ of $\pi$ at $m\in M$ of any geometric structure $\mathcal{O}$ defined
on $X$.
###### Remark 2.2.
Since the Ehresmann connection $\mathcal{H}$ of a scalar bundle is complete
and its parallel transport preserves the vertical metric $\mathcal{G}$, the
fibers $(X_{m},\mathcal{G}_{m})$ of $\pi$ are isomorphic to each other as
Riemannian manifolds. By the results of [17], it follows that $\pi$ is a fiber
bundle associated to a principal bundle with structure group given by the
isometry group of $(X_{m},\mathcal{G}_{m})$. Notice that $X_{m}$ is typically
non-compact in physics applications.
###### Definition 2.3.
A scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ is:
* •
_topologically trivial_ if $\pi$ is topologically trivial as fiber bundle,
i.e. $X$ is diffeomorphic with $M\times\mathcal{M}$ for some manifold
$\mathcal{M}$ and $\pi$ identifies with the projection
$\mathrm{pr}:M\times\mathcal{M}\rightarrow M$ on the first factor.
* •
_holonomy-trivial_ if the holonomy of $\mathcal{H}$ is trivial.
###### Remark 2.4.
Every holonomy-trivial scalar bundle is topologically trivial. Moreover, its
Ehresmann connection identifies with the pull-back of $TM$ through the
projection $\mathrm{pr}:M\times\mathcal{M}\rightarrow M$.
Let $\mathcal{P}(M)$ and $\mathcal{P}(X)$ be the sets of piece-wise smooth
paths in $M$ and $X$ defined on the unit interval. Given a scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$, let $T$ be the parallel transport defined by
$\mathcal{H}$, which associates to a path $\gamma\in\mathcal{P}(M)$ the
diffeomorphism:
$T_{\gamma}\colon X_{\gamma(0)}\xrightarrow{\sim}X_{\gamma(1)}\,,$
obtained by parallel transport along $\mathcal{H}$.
###### Definition 2.5.
A _duality bundle_ $\Delta=(\mathcal{S},\omega,\mathcal{D})$ over $\pi$ is a
triple $(\mathcal{S},\omega,\mathcal{D})$ where $\mathcal{S}$ is a vector
bundle on $X$, $\omega$ is a symplectic structure on $\mathcal{S}$ and
$\mathcal{D}$ is a flat connection on $\mathcal{S}$ preserving $\omega$. We
denote the rank of $\mathcal{S}$ by $2n_{v}$.
###### Remark 2.6.
As shown in [25], a duality bundle of rank $2n_{v}$ corresponds locally to a
supergravity theory coupled to $n_{v}$ _vector multiplets_.
Let $(\pi,\mathcal{H},\mathcal{G})$ be a scalar bundle over $M$ and
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ be a duality bundle on $\pi$, which
we shall also call a duality bundle over $(\pi,\mathcal{H},\mathcal{G})$. For
any $m\in M$, let $(\mathcal{S}_{m},\omega_{m},\mathcal{D}_{m})$ be the
restriction of $(\mathcal{S},\omega,\mathcal{D})$ to the fiber
$X_{m}=\pi^{-1}(m)$. This is a flat symplectic vector bundle on $X_{m}$ and
hence a duality structure on the latter as defined in [24]. For any path
$\Gamma\in\mathcal{P}(X)$ in the total space $X$ of $\pi$, we denote by
$\mathfrak{U}_{\Gamma}:\mathcal{S}_{\Gamma(0)}\rightarrow\mathcal{S}_{\Gamma(1)}$
the parallel transport of $\mathcal{D}$ along $\Gamma$. Since $\mathcal{D}$ is
a symplectic connection, $\mathfrak{U}_{\Gamma}$ is a symplectomorphism
between the symplectic vector spaces
$(\mathcal{S}_{\Gamma(0)},\omega_{\Gamma(0)})$ and
$(\mathcal{S}_{\Gamma(1)},\omega_{\Gamma(1)})$. For any
$\gamma\in\mathcal{P}(M)$, let $\bar{\gamma}_{x}\in\mathcal{P}(X)$ be the
horizontal lift of $\gamma$ starting at the point $x\in X_{\gamma(0)}$. By the
definition of $T$, we have $\bar{\gamma}_{x}(1)=T_{\gamma}(x)$.
###### Definition 2.7.
The extended horizontal transport along a path $\gamma\in\mathcal{P}(M)$ is
the unbased isomorphism of flat symplectic vector bundles
$\mathcal{T}_{\gamma}:\mathcal{S}_{\gamma(0)}\rightarrow\mathcal{S}_{\gamma(1)}$
defined by:
$\mathcal{T}_{\gamma}(x)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathfrak{U}_{\bar{\gamma}_{x}}\colon\mathcal{S}_{x}\rightarrow\mathcal{S}_{T_{\gamma}(x)}\,,\quad\forall\,x\in
X_{\gamma(0)}\,,$
which linearizes the Ehresmann transport $T_{\gamma}:X_{\gamma(0)}\rightarrow
X_{\gamma(1)}$ along $\gamma$.
Given a duality bundle $\Delta=(\mathcal{S},\omega,\mathcal{D})$, a
(compatible) _taming_ $\mathcal{J}\in\mathrm{Aut}_{b}(\mathcal{S})$ on
$\Delta$ is a complex structure on $\mathcal{S}$ which tames the symplectic
pairing $\omega$, i.e. it satisfies the compatibility condition:
$\omega(\mathcal{J}\xi_{1},\mathcal{J}\xi_{2})=\omega(\xi_{1},\xi_{2})\,,\qquad\forall\,\,(\xi_{1},\xi_{2})\in\mathcal{S}\times_{X}\mathcal{S},,$
and the positivity condition:
$\omega(\mathcal{J}\xi,\xi)>0\,,\qquad\forall\,\,\xi\in\dot{\mathcal{S}}~{}~{},$
where $\dot{\mathcal{S}}$ is the complement of the image of the zero section
in $\mathcal{S}$.
###### Definition 2.8.
A taming on the duality bundle $\Delta$ over $(\pi,\mathcal{H},\mathcal{G})$
is called _vertical_ if it is preserved by the extended horizontal transport
$\mathcal{T}$, i.e. if
$\mathcal{T}_{\gamma}\colon(\mathcal{S}_{\gamma(0)},\omega_{\gamma(0)},\mathcal{J}_{\gamma(0)})\rightarrow(\mathcal{S}_{\gamma(1)},\omega_{\gamma(1)},\mathcal{J}_{\gamma(1)})$
an isomorphism of tamed symplectic vector bundles for all
$\gamma\in\mathcal{P}(M)$.
###### Remark 2.9.
As explained in [24], a vertical taming on $\Delta$ is equivalent to a
_positive_ Lagrangian sub-bundle of the complexification of
$(\mathcal{S},\omega)$ which is preserved by the complexified extended
horizontal transport.
As in [24, 25], we refer to the pair $\Xi=(\Delta,\mathcal{J})$ consisting of
a duality bundle $\Delta$ defined on $(\pi,\mathcal{H},\mathcal{G})$ and a
vertical taming $\mathcal{J}$ as an _electromagnetic bundle_ on
$(\pi,\mathcal{H},\mathcal{G})$. We will refer to a choice scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$ together with a choice of electromagnetic
bundle $\Xi$ as a scalar-electromagnetic bundle $\Phi$, that is:
$\Phi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H},\mathcal{G},\Xi)\,.$
As shown in [24, 25], the universal bosonic sector of supergravity defined on
$M$ is determined by the choice of a scalar-electromagnetic bundle. Morphisms
of duality and electromagnetic bundles are defined in the natural way (see
[25]). Note that standard bundle theory implies that isomorphism classes of
duality bundles over a fixed submersion $\pi:X\rightarrow M$ are in one to one
correspondence with the character variety:
$\mathfrak{M}_{d}(X)\stackrel{{\scriptstyle{\rm def.}}}{{=}}{\rm
Hom}(\pi_{1}(X),\mathrm{Sp}(2n_{v},\mathbb{R}))/\mathrm{Sp}(2n_{v},\mathbb{R})\,.$
###### Remark 2.10.
In general, the character variety above has positive dimension, giving a
moduli space of inequivalent duality bundles. This implies [25] that one can
construct an uncountable infinity of globally inequivalent bosonic geometric
supergravities which are however all locally equivalent.
###### Remark 2.11.
A duality bundle $\Delta=(\mathcal{S},\omega,\mathcal{D})$ is called
topologically trivial if the vector bundle $\mathcal{S}$ is trivial, i.e. if
it admits a global frame. It is called _symplectically trivial_ if the
$(\mathcal{S},\omega)\in\Delta$ is symplectically trivial, i.e. if
$\mathcal{S}$ admits a global symplectic frame. Finally, we say that $\Delta$
is _holonomy trivial_ if the holonomy of $\mathcal{D}$ is the trivial group.
Holonomy-triviality implies symplectic triviality, which in turn implies
topological triviality. If $X$ is simply connected then every duality bundle
is holonomy trivial.
Smooth sections of the submersion $\pi:X\rightarrow M$ are called _scalar
sections_. For every scalar section $s\colon M\to X$ we use a superscript $s$
to denote the bundle pull-back by $s$ and the subscript $s$ to denote push-
forward by $s$ in the appropriate category. For instance,
$\Delta^{s}=(\mathcal{S}^{s},\omega^{s},\mathcal{D}^{s})$ denotes the bundle
pull-back of $\Delta=(\mathcal{S},\omega,\mathcal{S})$ by $s$, which is a flat
symplectic vector bundle over $M$. Similarly,
$\Xi^{s}=(\Delta^{s},\mathcal{J}^{s})$ denotes the pull-back of
$\Xi=(\Delta,\mathcal{J})$ by $s$, which is an electromagnetic structure on
$M$ in the sense of [24]. Let $\Phi$ be a scalar-electromagnetic bundle on
$M$. For every Lorentzian metric $g$ on $M$ and every scalar section
$s\in\Gamma(\pi)$, consider the isomorphism of vector bundles:
$\star_{g,\mathcal{J}^{s}}\colon\wedge
T^{\ast}M\otimes\mathcal{S}^{s}\to\wedge T^{\ast}M\otimes\mathcal{S}^{s}\,,$
defined through $\star_{g,\mathcal{J}^{s}}=\ast_{g}\otimes\mathcal{J}^{s}$.
Since both $\ast_{g}$ and $\mathcal{J}^{s}$ square to minus the identity, this
restricts to an involutive automorphism:
$\star_{g,\mathcal{J}^{s}}\colon\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}\to\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}$
which gives a direct sum decomposition into eigenbundles corresponding to the
eigenvalues $+1$ and $-1$:
$\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}=(\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s})_{+}\oplus(\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s})_{-}\,.$
Here the subscript denotes the sign of the corresponding eigenvalue of
$\star_{g,\mathcal{J}^{s}}$. The spaces of smooth global sections of these
sub-bundles are denoted by $\Omega^{2}_{\pm}(M,\mathcal{S}^{s})$ and their
elements are called polarized (anti)-selfdual $\mathcal{S}^{s}$-valued 2-forms
with respect to $\mathcal{J}^{s}$. We have:
$\Omega^{2}(M,\mathcal{S}^{s})=\Omega^{2}_{+}(M,\mathcal{S}^{s})\oplus\Omega^{2}_{-}(M,\mathcal{S}^{s})\,,$
The flat symplectic connection $\mathcal{D}^{s}$ of $\Delta^{s}$ defines an
exterior covariant derivative acting on $\mathcal{S}^{s}$-valued forms defined
on $M$, which we denote by:
$\mathrm{d}_{\mathcal{D}^{s}}\colon\Omega(M,\mathcal{S}^{s})\to\Omega(M,\mathcal{S}^{s})\,.$
This operator squares to zero since $\mathcal{D}^{s}$ is flat. We denote its
cohomology groups by $H^{k}(M,\Delta^{s})$ and the corresponding total
cohomology by $H(M,\Delta^{s})$. For every scalar section $s\in\Gamma(\pi)$,
we denote by $\mathfrak{G}^{s}_{\Delta}$ the sheaf of flat sections of
$\Delta^{s}$, defined as follows:
$\mathfrak{G}^{s}_{\Delta}(U)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\xi\in\Gamma(U,\mathcal{S}^{s})\,\,|\,\,\mathcal{D}^{s}\xi=0\right\\}~{}~{},$
for any open set $U\subset M$. This is a locally-constant sheaf of symplectic
vector spaces of rank $2n_{v}$, whose stalk is isomorphic to the typical fiber
of $\Delta$. Since the sheaf of smooth $\mathcal{S}^{s}$-valued forms is
acyclic, there exists a natural isomorphism of graded vector spaces:
$H(M,\Delta^{s})\simeq H(M,\mathfrak{G}^{s}_{\Delta})\,,$
where $H(M,\mathfrak{G}^{s}_{\Delta})$ is the sheaf cohomology of
$\mathfrak{G}^{s}_{\Delta}$. Note that the definition of an electromagnetic
bundle $\Xi=(\Delta,\mathcal{J})$ does not require $\mathcal{D}\in\Delta$ to
be compatible with $\mathcal{J}$, a fact which is crucial for recovering the
correct local description of bosonic geometric supergravity. The failure of
$\mathcal{D}$ to be compatible with $\mathcal{J}$ is measured by the
_fundamental form_ of an electromagnetic bundle.
###### Definition 2.12.
Let $\Phi$ be an scalar-electromagnetic bundle. The fundamental form $\Psi$ of
$\Xi$ is the following $End(\mathcal{S})$-valued one-form defined on $X$:
$\Psi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{D}\mathcal{J}\in\Omega^{1}(X,End(\mathcal{S}))\,.$
###### Remark 2.13.
For every $v\in\Gamma(TX)$, the endomorphism
$\Psi(v)\in\mathrm{End}(\mathcal{S})=\Gamma(End(\mathcal{S}))$ is an
$\mathcal{J}$-antilinear $Q$-symmetric, where the scalar product $Q$ is the
Euclidean metric induced by $\omega$ and $\mathcal{J}$ on $\mathcal{S}$ as
follows:
$Q(\xi_{1},\xi_{2})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\omega(\xi_{1},\mathcal{J}\xi_{2})~{}~{}\forall(\xi_{1},\xi_{2})\in\mathcal{S}\times_{X}\mathcal{S}\,,$
See [24, 25] for more details.
###### Definition 2.14.
An electromagnetic bundle $\Xi$ is called unitary if $\Psi=0$.
To describe the universal bosonic sector of 4d supergravity, we introduce
three natural operations which are determined by a choice of a scalar-
electromagnetic bundle $\Phi$, a Lorentzian metric $g$ on $M$ and a scalar
section $s\in\Gamma(\pi)$.
###### Definition 2.15.
The twisted exterior pairing $(\cdot,\cdot)_{g,Q^{s}}$ is the unique pseudo-
Euclidean scalar product on $\wedge T^{\ast}M\otimes\mathcal{S}^{s}$ which
satisfies:
$(\rho_{1}\otimes\xi^{s}_{1},\rho_{2}\otimes\xi^{s}_{2})_{g,Q^{s}}=(\rho_{1},\rho_{2})_{g}Q^{s}(\xi^{s}_{1},\xi^{s}_{2})=(\rho_{1},\rho_{2})_{g}Q^{s}(\xi_{1},\xi_{2})$
for all $\rho_{1},\rho_{2}\in\Omega(M)$ and all
$\xi_{1},\xi_{2}\in\Gamma(\mathcal{S}^{s})$.
Given any vector bundle $W$ on $M$, we extend this trivially to a $W$-valued
pairing (which for simplicity we denote by the same symbol) between the
bundles $W\otimes\wedge T^{\ast}M\otimes\mathcal{S}^{s}$ and $\wedge
T^{\ast}M\otimes\mathcal{S}^{s}$. Thus:
$(w\otimes\eta_{1},\eta_{2})_{g,Q^{s}}=w\otimes(\eta_{1},\eta_{2})_{g,Q^{s}}\,,\quad\forall\,\,w\in\Gamma(W)\,,\quad\forall\,\,\eta_{1},\eta_{2}\in\Omega(M,\mathcal{S}^{s})~{}~{}.$
###### Definition 2.16.
The inner $g$-contraction of (2,0)-tensors is the bundle morphism
$\oslash_{g}:(\otimes^{2}T^{\ast}M)^{\otimes
2}\rightarrow\otimes^{2}T^{\ast}M$ uniquely determined by the condition:
$(\alpha_{1}\otimes\alpha_{2})\oslash_{g}(\alpha_{3}\otimes\alpha_{4})=(\alpha_{2},\alpha_{4})_{g}\alpha_{1}\otimes\alpha_{3}\,,\quad\forall\,\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in
T^{\ast}M\,.$
We define the _inner $g$-contraction of two-forms_ to be the restriction of
$\oslash_{g}$ to
$\wedge^{2}T^{\ast}M\otimes\wedge^{2}T^{\ast}M\subset(\otimes^{2}T^{\ast}M)^{\otimes
2}$.
###### Definition 2.17.
The twisted inner contraction of $\mathcal{S}^{s}$-valued two-forms is the
unique morphism of vector bundles:
$\oslash_{Q^{s}}\colon\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}\times_{M}\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}\rightarrow\otimes^{2}(T^{\ast}M)$
which satisfies:
$(\rho_{1}\otimes s_{1})\oslash_{Q^{s}}(\rho_{2}\otimes
s_{2})=Q^{s}(s_{1},s_{2})\rho_{1}\oslash_{g}\rho_{2}\,,$
for all $\rho_{1},\rho_{2}\in\Omega^{2}(M)$ and all
$s_{1},s_{2}\in\Gamma(\mathcal{S}^{s})$.
### 2.2. The configuration space and equations of motion
We are ready to give the geometric formulation of the universal bosonic sector
of 4d classical supergravity, whose global solutions can be interpreted as
locally geometric classical supergravity U-folds [25].
###### Definition 2.18.
Let $\Phi=(\pi,\mathcal{H},\mathcal{G},\mathcal{S},\omega,\mathcal{D})$ be a
scalar-electromagnetic bundle on an oriented four-manifold $M$. The
configuration space of the universal bosonic sector determined by $(M,\Phi)$
is the set:
$\mathrm{Conf}(\Phi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(g,s,\mathcal{F})\,\,|\,\,g\in\mathrm{Lor}(M)\,,\,\,s\in\Gamma(\pi)\,,\,\,\mathcal{F}\in\Omega^{2}_{\mathrm{d}_{\mathcal{D}^{s}}\\!\mbox{-}\mathrm{cl}}(M,\mathcal{S}^{s})\right\\}\,,$
where $\mathrm{Lor}(M)$ denotes the set of Lorentzian metrics on $M$ and
$\Omega^{2}_{\mathrm{d}_{\mathcal{D}^{s}}\\!\mbox{-}\mathrm{cl}}(M,\mathcal{S}^{s})$
denotes the set of $\mathcal{D}^{s}$-closed 2-forms on $M$ valued in
$\mathcal{S}^{s}$.
###### Remark 2.19.
In general, the isomorphism class of $\mathcal{S}^{s}$ depends on the scalar
section $s\in\Gamma(\pi)$.
Given a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$, the complete Ehresmann
connection $\mathcal{H}$ can be described equivalently by a one-form
$\mathcal{C}\in\Omega^{1}(X,\mathcal{V})={\rm Hom}(TX,\mathcal{V})$ which
satisfies the condition $\mathcal{C}\circ\mathcal{C}=\mathcal{C}$. Thus
$\mathcal{C}\colon TX\to\mathcal{V}$ is a projection of the tangent bundle of
$X$ onto $\mathcal{V}$. The horizontal distribution $\mathcal{H}$ is recovered
as the kernel of $\mathcal{C}$. Given
$(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$, we define the vertical first
fundamental form $(s^{\ast}_{\mathcal{C}}\mathcal{G})\in\Gamma(T^{\ast}M\odot
T^{\ast}M)$ through [33, 34]:
$(s^{\ast}_{\mathcal{C}}\mathcal{G})(v_{1},v_{2})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{G}(\mathcal{C}\circ\mathrm{d}s(v_{1}),\mathcal{C}\circ\mathrm{d}s(v_{2}))\,,\quad(v_{1},v_{2})\in
TM\times_{M}TM\,;$
which depends explicitly on $\mathcal{C}$ or, equivalently, $\mathcal{H}$. The
trace of this tensor with respect to $g$ is called the _vertical tension_ of
the section $s\in\Gamma(\pi)$. For ease of notation, we define
$\mathrm{d}^{\mathcal{C}}s(v)=\mathcal{C}\circ\mathrm{d}s(v)\in\mathcal{V}$,
where $v\in TM$. Notice that
$\mathrm{d}^{\mathcal{C}}s\in\Omega^{1}(M,\mathcal{V}^{s})$. Denote by
$\mathrm{Lor}(X)$ the set of Lorentzian metrics on $X$. Every Lorentzian
metric $g$ on $M$ can be lifted to $\mathcal{H}$ using the isomorphism of
vector bundles
$(\mathrm{d}\pi|_{\mathcal{H}})\colon\mathcal{H}\xrightarrow{\sim}TM$ given by
the restriction of $\mathrm{d}\pi$ to $\mathcal{H}$. Thus there exists a
natural map (see [25]):
$h\colon\mathrm{Conf}(\Phi)\to\mathrm{Lor}(X)\,,\quad(g,s,\mathcal{F})\mapsto
h(g)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\pi^{\ast}g+\mathcal{G}\,,$
where $h(g)$ is written using the direct sum decomposition
$TX=\mathcal{H}\oplus\mathcal{V}$. The Lorentzian metric $h(g)$ enters the
equations of motion of bosonic supergravity, as explained below. Note that,
equipped with the lifted metric $h_{g}$, $\pi\colon(X,h_{g})\to(M,g)$ becomes
a Lorentzian submersion. Given $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$, we
denote by $\nabla^{h(g)}$ the Levi-Civita connection defined by $h(g)$ on $X$.
For every scalar section $s\colon M\to X$, we denote by $\nabla^{\Phi(g,s)}$
the connection on $TM\otimes\mathcal{V}^{s}$ given by the tensor product of
the Levi-Civita connection $\nabla^{g}$ of $g$ with the the vertical
projection of the pull-back by $s$ of the connection $\nabla^{h(g)}$. We then
have:
$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s\in\Gamma(T^{\ast}M\otimes
T^{\ast}M\otimes\mathcal{V}^{s})\,,$
as explained in [33, 34].
###### Definition 2.20.
Let $\Phi$ be a scalar-electromagnetic bundle on $M$. The universal bosonic
sector defined by $\Phi$ on $M$ is described by following system of partial
differential equations for triples $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$:
* •
The Einstein equations:
$\mathrm{Ric}^{g}-\frac{g}{2}\mathrm{R}^{g}=\frac{1}{2}\mathrm{Tr}_{g}(s^{\ast}_{\mathcal{C}}\mathcal{G})\,g-s^{\ast}_{\mathcal{C}}\mathcal{G}+2\mathcal{F}\oslash_{Q^{s}}\mathcal{F}\,,$
(1)
where $\mathrm{Ric}^{g}$ and $\mathrm{R}^{g}$ are respectively the Ricci
tensor and Ricci scalar of $g$, while $\mathrm{Tr}_{g}$ denotes trace with
respect to $g$.
* •
The scalar equations:
$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s=\frac{1}{2}(\ast\mathcal{F},\Psi^{s}\mathcal{F})_{g,Q^{s}}\,.$
(2)
* •
The Maxwell equations:
$\star_{g,\mathcal{J}^{s}}\mathcal{F}=\mathcal{F}\,.$ (3)
We denote by $\mathrm{Sol}(\Phi)\subset\mathrm{Conf}(\Phi)$ the set of
solutions to these equations.
###### Remark 2.21.
The configuration space $\mathrm{Conf}(\Phi)$ is formulated using the _field
strength_ two-forms instead of the appropriate notion of gauge potential, as
required by the Aharonov-Bohm effect [1]. The latter suggests that the gauge
potentials of the theory should be described by connections on an appropriate
principal bundle. To identify this bundle, we must impose an appropriate DSZ
quantization condition on the field strength $\mathcal{F}$. We consider this
condition and its geometric interpretation in Section 3.
###### Remark 2.22.
The fact that the formulation given above reduces locally to the usual
formulas of local bosonic supergravity found in the physics literature was
proved in detail in references [24, 25], to which we refer the reader for
further details. It is not known if this theory can be supersymmetrized when
$\mathcal{H}$ is not flat, although the Killing spinor equations can be
formulated exactly as in the case when $\mathcal{H}$ is flat.
### 2.3. The classical U-duality group
In this section we characterize the _global_ U-duality group of the bosonic
supergravity associated to a fixed scalar electromagnetic bundle
$\Phi=(\pi,\mathcal{H},\mathcal{G},\Xi)$. Given a duality bundle
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ let $\mathrm{Aut}(\mathcal{S})$
denote the group of all _unbased_ automorphisms of the vector bundle
$\mathcal{S}$. Let $f_{u}\in\mathrm{Diff}(X)$ be the diffeomorphism covered by
$u\in\mathrm{Aut}(\mathcal{S})$. Moreover, let $\mathrm{Aut}(\Delta)$ be the
group of those unbased automorphisms of $\mathcal{S}$ which preserve both
$\omega$ and $\mathcal{D}$:
$\mathrm{Aut}(\Delta)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(\mathcal{S})\,\,|\,\,\omega^{u}=\omega\,,\,\,\mathcal{D}^{u}=\mathcal{D}\right\\}\,.$
Let $\mathrm{Aut}_{\pi}(\Delta)$ be the subgroup consisting of all elements of
$\mathrm{Aut}(\Delta)$ which cover based automorphisms of the fiber bundle
$\pi$:
$\mathrm{Aut}_{\pi}(\Delta)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(\Delta)\,\,|\,\,f_{u}\in\mathrm{Aut}_{b}(\pi)\right\\}=\left\\{u\in\mathrm{Aut}(\Delta)\,\,|\,\,\pi\circ
f_{u}=\pi\right\\}\,.$
We have a short exact sequence of groups:
$1\to\mathrm{Aut}_{b}(\Delta)\to\mathrm{Aut}_{\pi}(\Delta)\to\mathrm{Aut}_{b}^{0}(\pi)\to
1\,,$ (4)
where $\mathrm{Aut}_{b}^{0}(\pi)\subset\mathrm{Aut}_{b}(\pi)$ is the subgroup
of those automorphisms of $\pi$ which are covered by elements of
$\mathrm{Aut}(\Delta)$.
Given a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ and an element
$u\in\mathrm{Aut}_{\pi}(\Delta)$, the fiber bundle automorphism
$f_{u}\in\mathrm{Aut}_{b}(\pi)$ covered by $u$ acts as a gauge transformation
on $\mathcal{H}$ through push-forward
$\mathcal{H}_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(f_{u})_{\ast}\mathcal{H}$. Similarly, since $f_{u}$ is an
automorphism of $\pi$ covering the identity, the push-forward of $\mathcal{G}$
by $f_{u}$ defines a new vertical Riemannian metric
$\mathcal{G}_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(f_{u})_{\ast}\mathcal{G}$ on $\pi$ such that
$(X_{m},\mathcal{G}_{m})$ is isometric to $(X_{m},(\mathcal{G}_{u})_{m})$ for
all $m\in M$. Given an electromagnetic bundle $\Xi=(\Delta,\mathcal{J})$,
push-forward by $f_{u}$ produces another electromagnetic bundle which we
denote by $(\Delta_{u},\mathcal{J}_{u})$. Given a scalar-electromagnetic
bundle $\Phi=(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$, the system:
$\Phi_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H}_{u},\mathcal{G}_{u},\Delta_{u},\mathcal{J}_{u})$
is a scalar-electromagnetic bundle with the same underlying submersion
$\pi\colon X\to M$. If $\mathcal{C}\in\Omega^{1}(X,\mathcal{V})$ is the
connection one-form associated to $\mathcal{H}$ then the natural push-forward
$f_{u\ast}\mathcal{C}\in\Omega^{1}(X,\mathcal{V})$ is the connection one-form
associated to $\mathcal{H}_{u}$.
###### Remark 2.23.
Since elements of $\mathrm{Aut}(\mathcal{S})$ may cover non-trivial
diffeomorphisms of $X$, the pull-back or push-forward operations must be dealt
with care (see [24]). Explicitly, define the following action of
$\mathrm{Aut}(\mathcal{S})$ on sections of $\mathcal{S}$:
$u\cdot\xi=u\circ\xi\circ f_{u}^{-1}\colon M\to\mathcal{S}\,,\qquad
u\in\mathrm{Aut}(\mathcal{S})\,,\qquad\xi\in\Gamma(\mathcal{S})\,.$
This gives an isomorphism of real vector spaces
$u\colon\Gamma(\mathcal{S})\to\Gamma(\mathcal{S})$ for every element
$u\in\mathrm{Aut}(\mathcal{S})$. We have $\omega^{u}=\omega$ if and only if:
$(\omega^{u})(\xi_{1},\xi_{2})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\omega(u\cdot\xi_{1},u\cdot\xi_{2})\circ
f_{u}=\omega(\xi_{1},\xi_{2})\,,\qquad\forall\,\,\xi_{1},\xi_{2}\in\Gamma(\mathcal{S})\,.$
Likewise, we have $\mathcal{D}^{u}=\mathcal{D}$ if and only if:
$\mathcal{D}^{u}_{v}(\xi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}u^{-1}\cdot\mathcal{D}_{f_{u\ast}\cdot
v}(u\cdot\xi)=\mathcal{D}_{v}(\xi)\,,\quad\forall\,\,\xi\in\Gamma(\mathcal{S})\,,\quad\forall\,\,v\in\Gamma(TX)\,,$
where $f_{u\ast}\cdot v=\mathrm{d}f_{u}(v)\circ f^{-1}_{u}$ and
$\mathrm{d}f_{u}\colon TX\to TX$ is the ordinary differential of
$f_{u}\in\mathrm{Diff}(X)$. Recall that if $v\in\Gamma(TX)$ then
$\mathrm{d}f_{u}(v)$ is not a vector field on $X$ but a section of $TX$ along
$f_{u}$, whereas $f_{u\ast}\cdot v\in\Gamma(TX)$ is again a vector field on
$X$. To illustrate the inner workings of the pull-backed connection $D^{u}$ we
verify that it satisfies the Leibniz identity:
$\displaystyle\mathcal{D}^{u}_{v}(\kappa\,\xi)=u^{-1}\cdot\mathcal{D}_{f_{u\ast}\cdot
v}(u\cdot(\kappa\,\xi))=u^{-1}\cdot\mathcal{D}_{f_{u\ast}\cdot v}((\kappa\circ
f_{u}^{-1})\,u\cdot\xi)=u^{-1}\cdot(\mathrm{d}(\kappa\circ
f_{u}^{-1})(f_{u\ast}\cdot v)\,u\cdot\xi)$
$\displaystyle+u^{-1}\cdot(\kappa\circ f_{u}^{-1}\,\mathcal{D}_{f_{u\ast}\cdot
v}(u\cdot\xi))=u^{-1}\cdot(\mathrm{d}\kappa(v\circ
f_{u}^{-1})\,u\cdot\xi)+u^{-1}\cdot(\kappa\circ
f_{u}^{-1}\,\mathcal{D}_{v}(u\cdot\xi))=\mathrm{d}\kappa(v)\,\xi+\kappa\,\mathcal{D}^{u}_{v}(\xi)\,,$
where $\kappa\in C^{\infty}(X)$ is a function on $X$. On the other hand, the
push-forward of $\mathcal{J}$ by $u\in\mathrm{Aut}(\mathcal{S})$ is given by:
$\mathcal{J}_{u}(\xi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(u\cdot\mathcal{J}(u^{-1}\cdot\xi))=u\circ\mathcal{J}(u^{-1}\circ\xi)\,,$
for every $\xi\in\Gamma(\mathcal{S})$.
Given a duality bundle $\Delta$ over the scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$, every element $u\in\mathrm{Aut}(\Delta)$ maps
a triplet of the form:
$(g,s,\mathcal{F})\in\mathrm{Lor}(M)\times\Gamma(\pi)\times\Omega^{2}(M,\mathcal{S}^{s})\,,$
to a triplet of the form:
$\mathbb{A}_{u}(g,s,\mathcal{F})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(g,f_{u}\circ
s,u\cdot\mathcal{F})\in\mathrm{Lor}(M)\times\Gamma(\pi)\times\Omega^{2}(M,\mathcal{S}^{f_{u}(s)})\,,$
where _dot_ denotes the natural action of $\mathrm{Aut}(\mathcal{S})$ on
$\mathcal{S}^{s}$-valued forms.
###### Remark 2.24.
Recall that
$\mathcal{F}_{m}\in\wedge^{2}T^{\ast}_{m}M\otimes\mathcal{S}^{s}_{m}$ or,
equivalently:
$\mathcal{F}_{m}\in\wedge^{2}T^{\ast}_{m}M\otimes\mathcal{S}_{s(m)}\,.$
The push-forward $u\cdot\mathcal{F}\in\Omega^{2}(M,\mathcal{S}^{f_{u}(s)})$ of
$\mathcal{F}\in\Omega^{2}(M,\mathcal{S}^{s})$ by $u$ produces a
$\mathcal{S}^{f_{u}(s)}$-valued two-form on $M$ whose value at $m\in M$ is
given by:
$(u\cdot\mathcal{F})_{m}=u_{s(m)}(\mathcal{F}_{m})\in\wedge^{2}T^{\ast}_{m}M\otimes\mathcal{S}_{f_{u}(s(m))}\,,$
where $u_{s(m)}\colon\mathcal{S}_{s(m)}\to\mathcal{S}_{f_{u}(s(m))}$ acts
trivially on the two-form components of $\mathcal{F}$.
Given $u\in\mathrm{Aut}(\Delta)$, the map $\mathbb{A}_{u}$ defined above need
not preserve the configuration space $\mathrm{Conf}(\Phi)$ defined by a fixed
scalar-electromagnetic bundle $\Phi=(\pi,\mathcal{H},\mathcal{G},\Xi)$.
Instead we have the following result.
###### Theorem 2.25.
Let $\pi\colon X\to M$ be a smooth submersion. For every connection
$\mathcal{H}$, vertical metric $\mathcal{G}$ and electromagnetic bundle $\Xi$
on $\pi$, an element $u\in\mathrm{Aut}_{\pi}(\Delta)$ defines a bijection:
$\mathbb{A}_{u}\colon\mathrm{Conf}(\Phi)\xrightarrow{\sim}\mathrm{Conf}(\Phi_{u})\,,\quad(g,s,\mathcal{F})\mapsto(g,f_{u}\circ
s,u\cdot\mathcal{F})\,,$
which restricts to a bijection:
$\mathbb{A}_{u}\colon\mathrm{Sol}(\Phi)\xrightarrow{\sim}\mathrm{Sol}(\Phi_{u})\,,$
between the solution spaces of the bosonic supergravities associated to $\Phi$
and $\Phi_{u}$ on $(\pi,\mathcal{H},\mathcal{G})$.
###### Proof.
Assume that $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$, where
$\Phi=(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$ and
$u\in\mathrm{Aut}_{\pi}(\Delta)$ covers $f_{u}\in\mathrm{Aut}_{b}(X)$.
Clearly, $f_{u}\circ s\colon M\to X$ is again a section of $\pi$ since
$f_{u}\colon X\to X$ is covers the identity over $M$. On the other hand,
$u\cdot\mathcal{F}$ is by construction a two-form on $M$ taking values in
$\mathcal{S}^{f_{u}(s)}$ whence $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi_{u})$.
The fact that this map takes solutions to solutions follows by a computation
that involves several different pull-backs through unbased automorphisms of
fiber bundles. The reader is referred to [24, Appendix D] for a detailed
account of the operations involved. Assume that
$(g,s,\mathcal{F})\in\mathrm{Sol}(\Phi)$. For the Einstein equation (1), we
compute:
$s^{\ast}_{\mathcal{C}}\mathcal{G}=(f_{u}^{-1}\circ f_{u}\circ
s)^{\ast}_{\mathcal{C}}\mathcal{G}=(f_{u}\circ
s)^{\ast}((f_{u}^{-1})^{\ast}_{\mathcal{C}}\mathcal{G})=(f_{u}\circ
s)^{\ast}_{f_{u\ast}\mathcal{C}}\mathcal{G}\,.$ (5)
On the other hand, we have:
$\mathcal{F}\oslash_{Q^{s}}\mathcal{F}=(u^{-1}\cdot
u\cdot\mathcal{F})\oslash_{Q^{s}}(u^{-1}\cdot
u\cdot\mathcal{F})=(u\cdot\mathcal{F})\oslash_{Q^{f_{u}(s)}_{u}}(u\cdot\mathcal{F})\,,$
(6)
where $Q^{f_{u}(s)}_{u}$ denotes the bilinear form on $\mathcal{S}^{s}$
defined as follows:
$Q^{f_{u}(s)}_{u}(\xi^{f_{u}(s)},\xi^{f_{u}(s)})=\omega(\xi(f_{u}(s)),\mathcal{J}(f_{u}(s))\xi(f_{u}(s)))\,,$
and where $\xi^{f_{u}(s)}\in\mathcal{S}^{f_{u}(s)}$ for every
$\xi\in\mathcal{S}$. Combining equations (5) and (6) together with the fact
that the left hand side of Equation (1) is invariant under $u$ we obtain that
$(g,f_{u}\circ s,u\cdot\mathcal{F})$ satisfies the Einstein equations with
respect to the scalar-electromagnetic structure
$(\pi,\mathcal{G},\mathcal{H}_{u},\Delta_{u},\mathcal{J}_{u})$. For the scalar
equation (2), we compute:
$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s=\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}(f_{u}^{-1}\circ
f_{u}\circ
s)=\nabla^{\Phi_{u}(g_{u},f_{u}(s))}\mathrm{d}^{f_{u\ast}\mathcal{C}}(f_{u}\circ
s)\,.$
Similarly, using the fact that $u\in\mathrm{Aut}(\Delta)$ preserves the flat
connection $\mathcal{D}$ determined by $\Delta$ together with equation (6), we
obtain:
$(\ast\mathcal{F},\Psi^{s}\mathcal{F})_{g,Q^{s}}=(\ast(u\cdot\mathcal{F}),\Psi^{f_{u}(s)}_{u}(u\cdot\mathcal{F}))_{g,Q^{f_{u}(s)}}\,,$
where we have defined:
$\Psi^{f_{u}(s)}_{u}=(\mathcal{D}\mathcal{J}_{u})^{f_{u}(s)}\,.$
Hence $(g,f_{u}\circ s,u\cdot\mathcal{F})$ satisfies the scalar equations
associated to the scalar-electromagnetic structure
$\Phi_{u}=(\pi,\mathcal{G},\mathcal{H}_{u},\Delta_{u},\mathcal{J}_{u})$. Since
the flatness condition for $\mathcal{F}$ is linear in $\mathcal{F}$ it is
enough to verify it on an homogenous element of the form
$\mathcal{F}=\alpha\otimes\xi^{s}$, where $\alpha\in\Omega^{2}(M)$ and
$\xi^{s}$ is the pull-back by $s$ of a section $\xi\in\Gamma(\mathcal{S})$. We
compute:
$\displaystyle\mathrm{d}_{\mathcal{D}^{f_{u}(s)}}(u\cdot\mathcal{F})=\mathrm{d}_{\mathcal{D}^{f_{u}(s)}}(\alpha\otimes
u\cdot\xi^{s})=\mathrm{d}_{\mathcal{D}^{f_{u}(s)}}(\alpha\otimes(u\circ\xi\circ
f^{-1}_{u}\circ f_{u}(s)))=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})$
$\displaystyle+\alpha\otimes\mathcal{D}^{f_{u}(s)}(u\circ\xi\circ
f^{-1}_{u})^{f_{u}(s)}=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})+\alpha\otimes(\mathcal{D}(u\circ\xi\circ
f^{-1}_{u}))^{f_{u}(s)}=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})$
$\displaystyle+\alpha\otimes(u\circ\mathcal{D}\xi\circ
f^{-1}_{u})^{f_{u}(s)}=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})+\alpha\otimes
u\cdot\mathcal{D}^{s}\xi^{s}=u\cdot\mathrm{d}_{\mathcal{D}^{s}}\mathcal{F}=0\,,$
where we have usted that $u\in\mathrm{Aut}_{\pi}(\Delta)$ preserves the
symplectic connection $\mathcal{D}$. Whence $u\cdot\mathcal{F}$ is flat with
respect to $\mathcal{D}^{f_{u}(s)}$. On the other hand, the Maxwell equation
is also linear in $\mathcal{F}$ hence it is enough to verify it on an
homogenous element $\mathcal{F}=\alpha\otimes\xi^{s}$. We obtain:
$\displaystyle\star_{g,\mathcal{J}^{f_{u}(s)}_{u}}(u\cdot\mathcal{F})=\ast_{g}\alpha\otimes\mathcal{J}^{f_{u}(s)}_{u}(u\cdot\xi^{s})=\ast_{g}\alpha\otimes\mathcal{J}^{f_{u}(s)}_{u}(u\circ\xi\circ
f_{u}^{-1}\circ f_{u}(s))$
$\displaystyle=\ast_{g}\alpha\otimes(\mathcal{J}_{u}(u\circ\xi\circ
f_{u}^{-1}))^{f_{u}(s)}=\ast_{g}\alpha\otimes(u\circ\mathcal{J}(\xi)\circ
f_{u}^{-1}))^{f_{u}(s)}=\ast_{g}\alpha\otimes
u\cdot\mathcal{J}^{s}(\xi^{s})=u\cdot\star_{g,\mathcal{J}^{s}}\mathcal{F}=u\cdot\mathcal{F}\,,$
whence $(g,f_{u}\circ s,u\cdot\mathcal{F})$ also satisfies the Maxwell
equations associated to
$(\pi,\mathcal{G},\mathcal{H}_{u},\Delta_{u},\mathcal{J}_{u})$. Thus
$(g,f_{u}\circ
s,u\cdot\mathcal{F})\in\mathrm{Sol}(\pi,\mathcal{H}_{u},\mathcal{G}_{u},\Delta_{u},\mathcal{J}_{u})$
and reversing the previous relations it is easy to see that the map
$(g,s,\mathcal{F})\mapsto(g,f_{u}\circ s,u\cdot\mathcal{F})$ is a bijection. ∎
###### Remark 2.26.
The group $\mathrm{Aut}_{\pi}(\Delta)$ is the global counterpart of the so-
called _pseudo-duality group_ introduced in [23] as the direct product of the
symplectic group and the diffeomorphism group of the simply connected open set
on which the theory is considered. When both $\Delta$ and $\pi$ are non-
trivial, the group $\mathrm{Aut}_{\pi}(\Delta)$ can differ markedly from the
local pseudo-duality group of loc. cit.
The following statement follows from [16, Lemma 4.2.8].
###### Lemma 2.27.
Let $\Delta$ be a duality bundle over the submersion $\pi:X\rightarrow M$ and
consider a point $x\in X$. Then there exists a canonical isomorphism:
$\mathrm{Aut}_{b}(\Delta)=\mathrm{C}(\mathrm{Hol}_{x}(\mathcal{D}),\mathrm{Aut}(S_{x},\omega_{x}))\,,$
where $\mathrm{Hol}_{x}(\mathcal{D})$ is the holonomy group of $\mathcal{D}$
at $x$, $\mathrm{Aut}(S_{x},\omega_{x})\simeq\mathrm{Sp}(2n_{v},\mathbb{R})$
is the automorphism group of the fiber $(S_{x},\omega_{x})=(S,\omega)|_{x}$
and
$\mathrm{C}(\mathrm{Hol}_{x}(\mathcal{D}),\mathrm{Aut}(S_{x},\omega_{x})))$
denotes the centralizer of $\mathrm{Hol}_{x}(\mathcal{D})$ in
$\mathrm{Aut}(S_{x},\omega_{x})$.
Fixing $x\in X$, this shows that (4) is isomorphic with the exact sequence:
$1\to\mathrm{C}(\mathrm{Hol}_{x}(\mathcal{D}),\mathrm{Aut}(S_{x},\omega_{x})))\to\mathrm{Aut}_{\pi}(\Delta)\to\mathrm{Aut}^{0}_{b}(\pi)\to
1\,.$
Given a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ and an electromagnetic
bundle $\Xi=(\Delta,\mathcal{J})$ we next introduce a subgroup of
$\mathrm{Aut}_{\pi}(\Delta)$ which preserves the Ehresmann connection
$\mathcal{H}$, the vertical metric $\mathcal{G}$ and the vertical taming
$\mathcal{J}$. This subgroup gives the global counterpart of the group of
_continuous_ U-dualities studied traditionally in the supergravity literature.
###### Definition 2.28.
Let $\Phi=(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$ be a scalar-
electromagnetic bundle on $M$. The classical U-duality group of $\Phi$ is the
subgroup $\mathrm{U}(\Phi)$ of $\mathrm{Aut}_{\pi}(\Delta)$ consisting of
those elements which preserve the Ehresmann connection $\mathcal{H}$, the
vertical metric $\mathcal{G}$ and the vertical taming $\mathcal{J}$:
$\mathrm{U}(\Phi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}_{\pi}(\Delta)\,\,|\,\,\mathcal{H}_{u}=\mathcal{H}\,,\,\,\mathcal{G}_{u}=\mathcal{G}\,,\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,.$
Let $\mathrm{Aut}_{b}(\Xi)\subset\mathrm{Aut}_{b}(\Delta)$ be the group based
automorphisms of $\Xi$, which consists of those vector bundle automorphisms of
$\mathcal{S}$ which cover the identity and preserve $\omega$, $\mathcal{D}$
and $\mathcal{J}$. The U-duality group fits into a short exact sequence:
$1\to\mathrm{Aut}_{b}(\Xi)\to\mathrm{U}(\Phi)\to\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G})\to
1\,,$
where
$\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G}))\subset\mathrm{Aut}_{b}(\pi)$
denotes the subgroup of those based automorphisms of $\pi$ that can be covered
by elements of $\mathrm{U}(\Phi)$ and preserve both the Ehresmann connection
$\mathcal{H}$ and the vertical metric $\mathcal{G}$. If the scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$ is flat then the group
$\mathrm{Aut}_{b}(\pi(\mathcal{H},\mathcal{G}))$ is finite-dimensional by
Lemma 2.27, which in turn implies that $\mathrm{U}(\Phi)$ is a finite-
dimensional Lie group. In general, this group is markedly different from the
U-duality group traditionally considered in the local formulation of the
theory. The main feature of the latter is that maps solutions to solutions and
hence it can be used as a solution generating mechanism. This key property
also holds for $\mathrm{U}(\Phi)$ as a consequence of Theorem 2.25.
###### Corollary 2.29.
The action $\mathbb{A}$ of the U-duality group $\mathrm{U}(\Phi)$ preserves
both $\mathrm{Conf}(\Phi)$ and $\mathrm{Sol}(\Phi)$. i.e. it maps
configurations to configurations and solutions to solutions. Moreover, if the
scalar bundle $(\pi,\mathcal{H},\mathcal{G})\in\Phi$ is flat then
$\mathrm{U}(\Phi)$ is a finite-dimensional Lie group.
For further reference we introduce the following definition.
###### Definition 2.30.
The _classical U-duality transformation_ defined by an element
$u\in\mathrm{U}(\Phi)$ is the bijection
$\mathbb{A}_{u}\colon\mathrm{Sol}(\Phi)\to\mathrm{Sol}(\Phi)$.
## 3\. The Dirac-Schwinger-Zwanziger integrality condition
This section discusses the geometric model obtained by imposing the DSZ
quantization condition on the universal bosonic sector of four-dimensional
supergravity defined by a fixed scalar-electromagnetic bundle. This condition
depends on the choice of a Dirac system for the underlying duality bundle
$\Delta$ and of the choice of an _integral_ cohomology class in
$H^{2}(M,\Delta)$, where integrality is defined relative to that Dirac system.
### 3.1. The vector space of integral field strengths
The DSZ quantization condition of local supergravity is implemented using a
full symplectic lattice. Similarly, we implement the DSZ quantization of the
universal bosonic sector defined by a scalar-electromagnetic bundle $\Phi$ in
terms of a smoothly varying fiber-wise choice of full symplectic lattices for
the underlying duality bundle $\Delta$, as proposed in [24]. Recall that a
full lattice $\Lambda$ in a $2n$-dimensional symplectic vector space
$(V,\omega)$ is called symplectic if the restriction of the symplectic pairing
$\omega$ to $\Lambda$ takes integer values. Such lattices are characterized up
to symplectomorphism by their type $\mathfrak{t}\in\mathrm{Div}^{n}$ (see [12,
Proposition 1.1]), where:
$\mathrm{Div}^{n}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{\mathfrak{t}=(t_{1},\ldots,t_{n})\in\mathbb{Z}_{>0}^{n}~{}|~{}t_{1}|t_{2}|\ldots|t_{n}\\}~{}~{}.$
Any full symplectic lattice of type $\mathfrak{t}\in\mathrm{Div}^{n}$ in
$(V,\omega)$ admits a basis
$\lambda_{1},\ldots,\lambda_{n},\mu_{1},\ldots,\mu_{n}$ such that:
$\omega(\lambda_{i},\mu_{j})=t_{j}\delta_{ij}~{}~{},~{}~{}\omega(\lambda_{i},\lambda_{j})=\omega(\mu_{i},\mu_{j})=0~{}~{}\forall
i,j=1,\ldots,n~{}~{}$
and any element $\mathfrak{t}\in\mathrm{Div}^{n}$ is realized as the type of
some full symplectic lattice. The symplectic lattice $\Lambda$ is called
principal if $\mathfrak{t}=\delta_{n}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(1,\ldots,1)$. The modified Siegel modular group of type
$\mathfrak{t}\in\mathrm{Div}^{n}$ is the subgroup
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\subset\mathrm{Sp}(2n,\mathbb{R})\simeq\mathrm{Aut}(V,\omega)$
consisting of those symplectic transformations which preserve a symplectic
lattice of type $\mathfrak{t}$. We have
$\mathrm{Sp}_{\delta_{n}}(2n,\mathbb{Z})=\mathrm{Sp}(2n,\mathbb{Z})$ and
$\mathrm{Sp}(2n,\mathbb{Z})\subset\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
for all $\mathfrak{t}\in\mathrm{Div}^{n}$.
###### Definition 3.1.
Let $\Delta=(\mathcal{S},\omega,\mathcal{D})$ be a duality bundle on the
scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ with submersion
$\pi:X\rightarrow M$. A Dirac system on $\Delta$ is a smooth fiber sub-bundle
$j\colon\mathcal{L}\hookrightarrow\mathcal{S}$ of full symplectic lattices in
$(\mathcal{S},\omega)$ which is preserved by the parallel transport $T$ of the
flat connection $\mathcal{D}$ in the sense that we have:
$T_{\gamma}(\mathcal{L}|_{\gamma(0)})=\mathcal{L}|_{\gamma(1)}$
for any piece-wise smooth path $\gamma\in\mathcal{P}(X)$. The common type of
these fiberwise symplectic lattices is called the type of $\mathcal{L}$. A
pair:
${\bm{\Delta}}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(\Delta,\mathcal{L})\,,$
consisting of a duality bundle $\Delta$ and a choice of Dirac system
$\mathcal{L}$ for $\Delta$ is called an integral duality bundle.
For every $x\in X$, the fiber $(\mathcal{S}_{x},\omega_{x},\mathcal{L}_{x})$
of an integral duality bundle ${\bm{\Delta}}=(\Delta,\mathcal{L})$ with
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ is an _integral symplectic space_ as
defined in [26, Appendix B]. All fibers of ${\bm{\Delta}}$ are isomorphic as
integral symplectic spaces, hence their type does not depend on $x\in X$ since
we assume that $X$ is connected.
###### Remark 3.2.
The existence of a Dirac system is obstructed. A duality bundle
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ of rank $2n_{v}$ admits a Dirac
system of type $\mathfrak{t}\in\mathrm{Div}^{n}$ if and only if the structure
group of $\mathcal{S}$ can be reduced from $\mathrm{Sp}(2n,\mathbb{R})$ to
$\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$. We say that $\Delta$ is
_semiclassical_ if it admits a Dirac system.
###### Definition 3.3.
Let ${\bm{\Delta}}_{1}=(\Delta_{1},\mathcal{L}_{1})$ and
${\bm{\Delta}}_{2}=(\Delta_{2},\mathcal{L}_{2})$ be two integral duality
bundles on $M$. A morphism of integral duality bundles from
${\bm{\Delta}}_{1}$ to ${\bm{\Delta}}_{2}$ is a morphism of duality bundles
$f\colon\Delta_{1}\to\Delta_{2}$ such that
$f(\mathcal{L}_{1})=\mathcal{L}_{2}$.
###### Remark 3.4.
Given a Dirac system $\mathcal{L}$ for a duality bundle
$\Delta=(\mathcal{S},\omega,\mathcal{D})$, let
$\mathfrak{S}_{{\bm{\Delta}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{C}(\mathcal{L})$ be the locally-constant sheaf of
continuous sections of the discrete fiber bundle $\mathcal{L}$. This is a
subsheaf of the sheaf $\mathfrak{S}_{{\bm{\Delta}}}$ of flat sections of
$(\mathcal{S},\mathcal{D})$ whose stalk at $x\in X$ identifies with the
symplectic lattice $\mathcal{L}_{x}\subset\mathcal{S}_{x}$.
For every scalar section $s\in\Gamma(\pi)$, let
$\mathfrak{S}^{s}_{{\bm{\Delta}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}s^{\ast}(\mathfrak{S}_{{\bm{\Delta}}})$ be the locally constant
sheaf on $M$ obtained as the pullback of $\mathfrak{S}_{\bm{\Delta}}$ through
$s$. The sheaf cohomology groups $H^{k}(M,\mathfrak{S}^{s}_{{\bm{\Delta}}})$
are naturally isomorphic with the cohomology groups $H^{k}(M,\mathcal{L}^{s})$
of $M$ with coefficients in the local system
$\mathcal{L}^{s}=s^{\ast}(\mathcal{L})$ and play a crucial role in what
follows.
###### Definition 3.5.
An _integral electromagnetic bundle_ is a pair:
${\bm{\Xi}}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(\Xi,\mathcal{L})\,,$
where $\Xi$ is an electromagnetic bundle on $M$ and $\mathcal{L}$ is Dirac
system for the duality bundle of $\Xi$. An _integral scalar-electromagnetic
bundle_ on $M$ is a pair:
$\bm{\Phi}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(\Phi,\mathcal{L})\,,$
where $\Phi$ is a scalar-electromagnetic bundle on $M$ and $\mathcal{L}$ is a
Dirac system for the duality bundle of $\Phi$.
Given an integral electromagnetic bundle ${\bm{\Xi}}=(\Xi,\mathcal{L})$ with
integral duality structure
${\bm{\Delta}}=(\mathcal{S},\omega,\mathcal{D},\mathcal{L})$ over a submersion
$\pi:X\rightarrow M$, the quotient:
$\mathcal{X}_{{\bm{\Delta}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{S}/\mathcal{L}$
is a flat fibration over $X$ by symplectic torus groups. The taming
$\mathcal{J}$ of $\Delta$ makes this into a fibration by polarized Abelian
varieties which however need not be flat since $\mathcal{J}$ is not flat
unless the underlying electromagnetic bundle $\Xi$ is unitary. The sheaf
$\mathfrak{S}_{\mathcal{X}_{\bm{\Delta}}}$ of smooth flat sections of
$\mathcal{X}_{{\bm{\Delta}}}$ fits into a short exact sequence of sheaves of
Abelian groups defined on $X$:
$0\to\mathfrak{S}_{{\bm{\Delta}}}\xrightarrow{j}\mathfrak{S}_{\Delta}\to\mathfrak{S}_{\mathcal{X}_{{\bm{\Delta}}}}\to
0\,.$
which pulls-back to a short exact sequence of sheaves of Abelian groups
defined on $M$:
$0\to\mathfrak{S}^{s}_{{\bm{\Delta}}}\xrightarrow{j^{s}}\mathfrak{S}^{s}_{\Delta}\to\mathfrak{S}^{s}_{\mathcal{X}_{\bm{\Delta}}}\to
0\,.$
The latter induces a long exact sequence in sheaf cohomology, of which we are
interested in the following portion:
$\ldots\rightarrow H^{1}(M,\mathfrak{S}^{s}_{\mathcal{X}_{{\bm{\Delta}}}})\to
H^{2}(M,\mathfrak{S}^{s}_{{\bm{\Delta}}})\xrightarrow{j^{s}_{\ast}}H^{2}(M,\mathfrak{S}^{s}_{\Delta})\to
H^{2}(M,\mathfrak{S}^{s}_{\mathcal{X}_{{\bm{\Delta}}}})\rightarrow\ldots\,.$
###### Definition 3.6.
The charge lattice of the integral scalar-electromagnetic structure
${\bm{\Xi}}=(\Xi,\mathcal{L})$ relative to the scalar section
$s\in\Gamma(\pi)$ is the lattice:
$L_{{\bm{\Xi}}}^{s}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}j^{s}_{\ast}(H^{2}(M,\mathfrak{S}^{s}_{{\bm{\Delta}}}))\subset
H^{2}(M,\mathfrak{S}^{s}_{\Delta})\,,$
Elements of this lattice are called _integral cohomology classes_.
It can be shown that $L_{{\bm{\Xi}}}^{s}$ is a full lattice in
$H^{2}(M,\mathfrak{S}^{s}_{\Delta})$ (see [26, Proposition 2.24]). Given an
integral scalar-electromagnetic bundle $\bm{\Phi}$, we implement DSZ
quantization by restricting the configuration space $\mathrm{Conf}(\Phi)$ to a
subset $\mathrm{Conf}(\bm{\Phi})\subset\mathrm{Conf}(\Phi)$ obtained by
imposing an _integrality condition_ on the elements of $\mathrm{Conf}(\Phi)$.
This is the appropriate implementation of the DSZ quantization condition in
our situation.
###### Definition 3.7.
Let $\bm{\Phi}$ be an integral scalar-electromagnetic bundle. The integral
configuration space $\mathrm{Conf}(\bm{\Phi})$ of defined by $\bm{\Phi}$ is
the set:
$\mathrm{Conf}(\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(g,s,\mathcal{F})\in\mathrm{Conf}(M,\Phi)\,\,|\,\,[\mathcal{F}]\in
2\pi L^{s}_{\Xi}\right\\}\,.$
The integral solution space
$\mathrm{Sol}(\bm{\Phi})\subset\mathrm{Conf}(\Phi)$ defined by $\bm{\Phi}$ is
the the set:
$\mathrm{Sol}(\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Sol}(\Phi)\cap\mathrm{Conf}(\bm{\Phi})~{}~{}.$
For further reference we introduce a refinement of the previous definition.
###### Definition 3.8.
Let $\bm{\Phi}$ be an integral scalar-electromagnetic bundle and let
$\mathfrak{V}\in H^{2}(X,\mathfrak{S}_{{\bm{\Delta}}})$. The framed integral
configuration space $\mathrm{Conf}(\mathfrak{V},\bm{\Phi})$ with _framing_
$\mathfrak{V}$ of the classical geometric supergravity theory associated to
$\bm{\Phi}$ is defined as the following subset of $\mathrm{Conf}(\bm{\Phi})$:
$\mathrm{Conf}(\mathfrak{V},\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)\,\,|\,\,[\mathcal{F}]=2\pi
j^{s}_{\ast}(\mathfrak{V}^{s})\right\\}\,,$
The framed integral solution space
$\mathrm{Sol}(\mathfrak{V},\bm{\Phi})\subset\mathrm{Conf}(\mathfrak{V},\Phi)$
is the set:
$\mathrm{Sol}(\mathfrak{V},\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Sol}(\Phi)\cap\mathrm{Conf}(\mathfrak{V},\bm{\Phi})~{}~{}.$
###### Definition 3.9.
The _arithmetic U-duality group_ of an integral scalar-electromagnetic
structure $\bm{\Phi}=(\Phi,\mathcal{L})$ is the subgroup of $\mathrm{U}(\Phi)$
defined through:
$\mathrm{U}(\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{U}(\Phi)\,\,|\,\,u(\mathcal{L})=\mathcal{L}\right\\}\,.$
The arithmetic U-duality group $\mathrm{U}(\bm{\Phi})$ is the global
counterpart of the arithmetic U-duality group of local supergravity normally
considered in the physics literature [22, 30]. We remark that the supergravity
literature seems to have considered thus far only holonomy trivial Dirac
systems $\mathcal{L}$ of principal type, though there is a priori no physical
or mathematical reason to make that assumption. We will consider some simple
examples of arithmetic U-duality groups in Section 5. More elaborated examples
will be considered in a separate publication.
## 4\. The DSZ quantization of 4d bosonic supergravity
In this section we describe the geometric and gauge-theoretic formulation of
the universal bosonic sector of 4d supergravity implied by the DSZ
quantization condition. This formulation can be constructed through a step-by-
step process as done in [26] for Abelian gauge theory. Instead of going
through the details of that process, which are similar to those in [26], we
give the description of the theory in its final form, verifying then that it
satisfies the appropriate DSZ quantization (see Theorem 4.5). The key
ingredient occurring in the construction is a _Siegel bundle_ , a special kind
of principal bundle which was introduced and discussed in detail in [26,
Section 3] and forms a particular case of the more general notion of principal
bundle with weakly-Abelian structure group studied in [27], to which we refer
the reader for background and further details. Given
$\mathfrak{t}\in\mathrm{Div}^{n_{v}}$, we define the following disconnected
Lie group:
$\mathrm{Aff}_{\mathfrak{t}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{U}(1)^{2n_{v}}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})~{}~{},$
where $\mathrm{U}(1)^{2n_{v}}\simeq\mathbb{R}^{2n_{v}}/\mathbb{Z}^{2n_{v}}$ is
an affine torus group of dimension $2n_{v}$. The group
$\mathrm{Aff}_{\mathfrak{t}}$ identifies with the set
$\mathrm{U}(1)^{2n_{v}}\times\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$
equipped with the multiplication rule:
$(a_{1},\gamma_{1})\,(a_{2},\gamma_{2})=(a_{1}+\gamma_{1}a_{2},\gamma_{1}\gamma_{2})\,,\quad\forall\,\,a_{1},a_{2}\in\mathrm{U}(1)^{2n_{v}}\,,\quad\forall\,\,\gamma_{1},\gamma_{2}\in\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})\,.$
The modified Siegel modular group
$\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$ coincides with the
automorphism group of the standard integral symplectic space
$(\mathbb{R}^{2n_{v}},\omega_{n_{v}},\wedge_{\mathfrak{t}})$ of type
$\mathfrak{t}$, where $\omega_{n_{v}}$ is the standard symplectic form on
$\mathbb{R}^{2n_{v}}$ and:
$\Lambda_{\mathfrak{t}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathbb{Z}^{n_{v}}\oplus\oplus_{i=1}^{n_{v}}{t_{i}\mathbb{Z}}\subset\mathbb{R}^{2n_{v}}$
is the standard symplectic lattice of type $\mathfrak{t}$ (see [26, Appendix
B]). Moreover, $\mathrm{Aff}_{\mathfrak{t}}$ coincides with the group of
affine symplectomorphisms of the $2n_{v}$-dimensional symplectic torus
$(\mathbb{R}^{2n_{v}}/\Lambda_{\mathfrak{t}},\Omega_{\mathfrak{t}})$, whose
symplectic form $\Omega_{\mathfrak{t}}$ is induced by $\omega_{n_{v}}$. The
connected component of the identity in $\mathrm{Aff}_{\mathfrak{t}}$ is the
torus group $\mathrm{U}(1)^{2n_{v}}$, while the group of components of
$\mathrm{Aff}_{\mathfrak{t}}$ is the discrete group
$\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$, which is infinite and non-
Abelian when $n_{v}>0$.
###### Definition 4.1.
A Siegel bundle $P_{\mathfrak{t}}$ of rank $n_{v}$ and type
$\mathfrak{t}\in\mathrm{Div}^{n_{v}}$ on $M$ is a principal bundle defined on
$M$ with structure group $\mathrm{Aff}_{\mathfrak{t}}$. A based isomorphism of
Siegel bundles is a based isomorphism of principal bundles.
Let $(\pi,\mathcal{H},\mathcal{G})$ be a scalar bundle with submersion
$\pi:X\rightarrow M$ and consider a Siegel bundle $P_{\mathfrak{t}}$ of rank
$n_{v}$ and type $\mathfrak{t}\in\mathrm{Div}^{n_{v}}$ over $X$. As shown in
[26], the adjoint bundle of $P_{\mathfrak{t}}$ admits a natural structure of
integral duality bundle of type $\mathfrak{t}$ which we denote by
${\bm{\Delta}}(P_{\mathfrak{t}})$. By definition, a vertical taming
$\mathcal{J}$ of $P_{\mathfrak{t}}$ is a vertical taming of
${\bm{\Delta}}(P_{\mathfrak{t}})$. Given such a taming, the pair
$(P_{\mathfrak{t}},\mathcal{J})$ is called a (positively) polarized Siegel
bundle (cf. [26, 27]). The integral electromagnetic bundle
${\bm{\Xi}}(P_{\mathfrak{t}},\mathcal{J})$ determined by
$(P_{\mathfrak{t}},\mathcal{J})$ is defined through:
${\bm{\Xi}}(P_{\mathfrak{t}},\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}({\bm{\Delta}}(P_{\mathfrak{t}}),\mathcal{J})\,.$
Given a scalar section $s\in\Gamma(\pi)$, we denote by $P^{s}_{\mathfrak{t}}$
the pullback of $P_{\mathfrak{t}}$ by $s$, which becomes a Siegel bundle over
$M$. Similarly, we denote by ${\bm{\Delta}}(P^{s}_{\mathfrak{t}})$ and
${\bm{\Xi}}(P^{s}_{\mathfrak{t}},\mathcal{J}^{s})$ the integral duality and
integral electromagnetic bundles defined by $P^{s}$ and $\mathcal{J}^{s}$,
which coincide with the $s$-pullbacks of the corresponding bundles defined by
$P$ and $\mathcal{J}$ on $X$. When necessary, we will write:
${\bm{\Delta}}(P^{s}_{\mathfrak{t}})=(\mathcal{S}^{s},\omega^{s},\mathcal{D}^{s})\,.$
Let $\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ be the affine space of connections
on $P^{s}_{\mathfrak{t}}$. Elements of this space are invariant one-forms on
$P_{\mathfrak{t}}$ mapping the fundamental vector fields of $P_{\mathfrak{t}}$
to their generators in $\mathrm{aff}_{\mathfrak{t}}$, where
$\mathrm{aff}_{\mathfrak{t}}\simeq\mathbb{R}^{2n_{v}}$ is the Lie algebra of
$\mathrm{Aff}_{\mathfrak{t}}$, which has trivial Lie bracket. The adjoint
curvature of a connection $\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$
will be denoted by
$\mathcal{F}_{\mathcal{A}}\in\Omega^{2}(M,\mathcal{S}^{s})$. This bundle-
valued 2-form is $\mathrm{d}_{\mathcal{D}^{s}}$-closed by the Bianchi identity
since (by the results of [26, 27]) all connections on $P^{s}$ induce the same
connection on $\mathcal{S}^{s}$, which coincides with the connection induced
by $\mathcal{D}^{s}$ on the adjoint bundle of $P^{s}_{\mathfrak{t}}$. Thus:
$\mathrm{d}_{\mathcal{D}}^{s}\mathcal{F}_{\mathcal{A}}=0\,.$
###### Definition 4.2.
A scalar-Siegel bundle of rank $n_{v}$ and type
$\mathfrak{t}\in\mathrm{Div}^{n_{v}}$ over $M$ is a system
$\zeta\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}})$, where
$(\pi,\mathcal{H},\mathcal{G})$ is a scalar bundle over $M$ with submersion
$\pi:X\rightarrow M$ and $P_{\mathfrak{t}}$ is a Siegel bundle of rank $n_{v}$
and type $\mathfrak{t}\in\mathrm{Div}^{n_{v}}$ defined on $X$. Given a
vertical taming $\mathcal{J}$ of $\Delta(P_{\mathfrak{t}})$, the system
$\bm{\upzeta}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(\Psi,\mathcal{J})$ is
called a polarized scalar-Siegel bundle of rank $n_{v}$ and type
$\mathfrak{t}$ over $M$.
###### Definition 4.3.
Let $\bm{\upzeta}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$ be a
polarized scalar-Siegel bundle over $M$. The configuration space of the
bosonic supergravity defined by $\bm{\upzeta}$ is the set:
$\mathfrak{Conf}(\bm{\upzeta})=\left\\{(g,s,\mathcal{A})\,\,|\,\,g\in\mathrm{Lor}(M)\,,\,\,s\in\Gamma(\pi)\,,\,\,\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})\right\\}\,.$
The universal bosonic sector of four-dimensional supergravity determined on
$M$ by $\bm{\upzeta}$ is defined through the following system of partial
differential equations for triples
$(g,s,\mathcal{A})\in\mathfrak{Conf}(\bm{\upzeta})$:
* •
The Einstein equations:
$\mathrm{Ric}^{g}-\frac{g}{2}\mathrm{R}^{g}=\frac{1}{2}\mathrm{Tr}_{g}(s^{\ast}_{\mathcal{C}}\mathcal{G})\,g-s^{\ast}_{\mathcal{C}}\mathcal{G}+2\mathcal{F}_{\mathcal{A}}\oslash_{Q^{s}}\mathcal{F}_{\mathcal{A}}\,.$
(7)
* •
The scalar equations:
$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s=\frac{1}{2}(\ast\mathcal{F}_{\mathcal{A}},\Psi^{s}\mathcal{F}_{\mathcal{A}})_{g,Q^{s}}\,.$
(8)
* •
The Maxwell equations:
$\star_{g,\mathcal{J}^{s}}\mathcal{F}_{\mathcal{A}}=\mathcal{F}_{\mathcal{A}}\,,$
(9)
whose set of solutions we denote by
$\mathfrak{Sol}(\bm{\upzeta})\subset\mathfrak{Conf}(\bm{\upzeta})$.
Connections $\mathcal{A}$ satisfying equation (9) will be called polarized
self-dual, following the terminology introduced in [26] in the context of
Abelian gauge theory.
###### Remark 4.4.
The Maxwell equations of the bosonic gauge sector of local supergravity are
given by a system of second-order partial differential equations for a number
$n_{v}$ of _electromagnetic_ local gauge potentials whose curvatures satisfy a
generalization of the Maxwell equations. This is locally equivalent with the
description given by the first order global equation (9), which reduces
locally to a system of first-order partial differential equations for $2n_{v}$
local gauge fields, both _electric_ and _magnetic_ (considered up to gauge
transformations of the principal bundle $P^{s}_{\mathfrak{t}}$).
The Bianchi identity and polarized self-duality condition imply that the gauge
potential of any solution $(g,s,\mathcal{A})\in\mathrm{Sol}(\bm{\upzeta})$
automatically satisfies the following second order differential equation of
Yang-Mills type:
$\mathrm{d}_{\mathcal{D}^{s}}\star_{g,\mathcal{J}}\mathcal{F}_{\mathcal{A}}=0\,.$
These differ from the usual Yang-Mills equations since
$\mathcal{F}_{\mathcal{A}}$ involves both electric and magnetic degrees of
freedom while the equations themselves involve the pulled-back taming
$\mathcal{J}^{s}$.
###### Theorem 4.5.
Let $\bm{\Phi}=(\pi,\mathcal{H},\mathcal{G},{\bm{\Delta}},\mathcal{J})$ be an
integral scalar-electromagnetic bundle of type $\mathfrak{t}$. For every
framed integral configuration space $\mathrm{Conf}(\mathfrak{V},\Phi)$ there
exists a vertically polarized Siegel bundle $(P_{\mathfrak{t}},\mathcal{J})$
on $(\pi,\mathcal{H},\mathcal{G})$ such that
${\bm{\Delta}}={\bm{\Delta}}(P_{\mathfrak{t}})$ and the twisted Chern
class222See [26, 27] for its precise definition. $c(P_{\mathfrak{t}})$ of
$P_{\mathfrak{t}}$ satisfies $c(P_{\mathfrak{t}})=\mathfrak{V}$. Moreover, the
map:
$\mathfrak{Sol}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})\to\mathrm{Sol}(\mathfrak{V},\bm{\Phi})\,,\qquad(g,s,\mathcal{A})\mapsto(g,s,\mathcal{F}_{\mathcal{A}})\,,$
is surjective.
###### Proof.
Given ${\bm{\Delta}}$ and $\mathfrak{V}$, it follows from the results of [5,
6] (see also [27]) that there exists a Siegel bundle $P_{\mathfrak{t}}$ of
type $\mathfrak{t}$ (unique up to isomorphism) whose twisted Chern class
$c(P_{\mathfrak{t}})$ equals $\mathfrak{V}$ and whose adjoint bundle is
isomorphic to ${\bm{\Delta}}$ as an integral duality bundle. The vertical
taming $\mathcal{J}$ in $\bm{\Phi}$ makes $P_{\mathfrak{t}}$ into a polarized
Siegel bundle. On the other hand, the curvature of any connection
$\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ defines a
$\mathrm{d}_{\mathcal{D}^{s}}$-cohomology class
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}$ in
$H^{2}(M,\mathfrak{S}_{\Delta^{s}})$ since, as remarked earlier:
$\mathrm{d}_{\mathcal{D}^{s}}\mathcal{F}_{\mathcal{A}}=\mathrm{d}_{\mathcal{A}}\mathcal{F}_{\mathcal{A}}=0\,.$
Given any other connection $\mathcal{A}^{\prime}$ on $P_{\mathfrak{t}}$ we
have:
$\mathcal{A}^{\prime}=\mathcal{A}+\bar{\tau}\,,$
for a unique horizontal and invariant one-form
$\bar{\tau}\in\Omega^{1}(P_{\mathfrak{t}},\mathfrak{aff}_{\mathfrak{t}})$.
Therefore, the curvatures of $\mathcal{F}_{\mathcal{A}}$ and
$\mathcal{F}_{\mathcal{A}^{\prime}}$ are related as follows:
$\mathcal{F}_{\mathcal{A}^{\prime}}=\mathcal{F}_{\mathcal{A}}+\mathrm{d}_{\mathcal{D}^{s}}\tau\in\Omega^{2}(M,\mathcal{S}^{s})\,,$
where $\tau\in\Omega^{1}(M,\mathcal{S}^{s})$ is uniquely determined by
$\bar{\tau}\in\Omega^{1}(P_{\mathfrak{t}},\mathfrak{aff}_{\mathfrak{t}})$.
This implies that the cohomology class
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}\in
H^{2}(M,\mathfrak{S}_{\Delta}^{s})$ does not depend on the connection
$\mathcal{A}$. A similar argument shows that the cohomology class
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}$ is invariant under
automorphisms of $P_{\mathfrak{t}}$ and therefore only depends on the
isomorphism class of the latter. This is further elaborated in [27] to show
that $[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}$ is equal to the _real_
twisted Chern class of $P_{\mathfrak{t}}$ as follows:
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}=2\pi
j^{s}_{\ast}(c(P^{s}_{\mathfrak{t}}))\in L^{s}_{\bm{\Delta}}\,.$
Since $c(P_{\mathfrak{t}})=\mathfrak{V}$ by construction, we immediately
conclude that:
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}=2\pi
j^{s}_{\ast}(\mathfrak{V}^{s})\in L^{s}_{\bm{\Delta}}\,.$
Hence, $(g,s,\mathcal{F}_{\mathcal{A}})$ belongs to
$\mathrm{Sol}(\mathfrak{V},\bm{\Phi})$ for all
$(g,s,\mathcal{A})\in\mathrm{Conf}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$.
Furthermore, every element in $\mathrm{Sol}(\mathfrak{V},\bm{\Phi})$ is of the
form $(g,s,\mathcal{F}_{\mathcal{A}})$ for some
$(g,s,\mathcal{A})\in\mathfrak{Sol}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$.
An explicit way to prove this is to use a good open cover
$M\subset\left\\{U_{a}\right\\}_{a\in I}$ of $M$. Then, given
$(g,s,\mathcal{F})\in\mathrm{Sol}(\mathfrak{V},\bm{\Phi})$, the restriction:
$\mathcal{F}_{a}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{F}|_{U_{a}}=\mathrm{d}\mathcal{A}_{a}\,,\qquad\mathcal{A}_{a}\in\Omega^{1}(U_{a},\mathbb{R}^{2n_{v}})\qquad
a\in I\,,$
is $\mathrm{d}_{\mathcal{D}^{s}}$-exact and hence $\mathrm{d}$-exact, since we
can trivialize $\Delta$ over $U_{a}$ as the latter is simply connected. The
family of one-forms $\left\\{\mathcal{A}_{a}\right\\}$ taking values in
$\mathbb{R}^{2n_{v}}$ can be shown to define a connection on
$P^{s}_{\mathfrak{t}}$ whose curvature is precisely $\mathcal{F}$ and hence we
conclude. ∎
The previous theorem shows that Definition 4.3 realizes geometrically the DSZ
quantization of the universal bosonic supergravity sector defined by $\Phi$
since it shows that, given a Dirac system for the duality structure of $\Phi$,
every element in the solution space $\mathrm{Sol}(\Phi,\mathcal{L})$ can be
realized through a Lorentz metric on $M$, a section of $\pi$ and a gauge
potential $\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ for some Siegel
bundle $P^{s}_{\mathfrak{t}}$ on $X$. The latter is the novel geometric object
attached to the DSZ quantization condition.
## 5\. The electromagnetic U-duality group
In this section we investigate the gauge U-duality group of the DSZ
quantization of bosonic supergravity, which yields a natural extension of its
arithmetic U-duality group and provides the geometric interpretation of
U-duality transformations as gauge transformations.
Fix a vertically polarized Siegel bundle $(P_{\mathfrak{t}},\mathcal{J})$ on
the total space $X$ of a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ and let
$\mathrm{Aut}(P_{\mathfrak{t}})$ be the automorphism group of
$P_{\mathfrak{t}}$. For every $u\in\mathrm{Aut}(P_{\mathfrak{t}})$, denote by
$\mathfrak{ad}_{u}\colon{\bm{\Delta}}(P_{\mathfrak{t}})\to{\bm{\Delta}}(P_{\mathfrak{t}})$
the automorphism of the integral duality structure
${\bm{\Delta}}(P_{\mathfrak{t}})$ defined by $u$. Let
$\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\subset\mathrm{Aut}(P_{\mathfrak{t}})$ be
the subgroup formed by all elements of $\mathrm{Aut}(P_{\mathfrak{t}})$ which
cover based automorphisms of the fiber bundle $\pi$, that is:
$\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(P_{\mathfrak{t}})\,\,|\,\,f_{u}\in\mathrm{Aut}_{b}(\pi)\right\\}=\left\\{u\in\mathrm{Aut}(P_{\mathfrak{t}})\,\,|\,\,\pi\circ
f_{u}=\pi\right\\}\,.$
We have the a short exact sequence of groups:
$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}})\to\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\to\mathrm{Aut}^{0}_{b}(\pi)\to
1\,,$
where $\mathrm{Aut}^{0}_{b}(\pi)$ is the subgroup of $\mathrm{Aut}_{b}(\pi)$
formed by those based automorphisms of $\pi$ that can be covered by elements
of $\mathrm{Aut}(P_{\mathfrak{t}})$.
###### Definition 5.1.
Let $(P_{\mathfrak{t}},\mathcal{J})$ be a vertically polarized Siegel bundle
over the scalar-bundle $(\pi,\mathcal{H},\mathcal{G})$. The gauge U-duality
group $\mathrm{U}(\bm{\upzeta})$ of the polarized scalar-Siegel bundle
$\bm{\upzeta}=(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$ is
the subgroup of $\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})$ consisting of those
elements which preserve the Ehresmann connection $\mathcal{H}$, the metric
$\mathcal{G}$ and the vertical taming $\mathcal{J}$:
$\mathrm{U}(\bm{\upzeta})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\,\,|\,\,\mathcal{H}_{u}=\mathcal{H}\,,\,\,\mathcal{G}_{u}=\mathcal{G}\,,\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,.$
Let $\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})$ be the subgroup of
$\mathrm{Aut}_{b}(P_{\mathfrak{t}})$ consisting of those based automorphisms
of $P_{\mathfrak{t}}$ which preserve $\mathcal{J}$. The gauge U-duality group
fits into a short exact sequence:
$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})\to\mathrm{U}(\bm{\upzeta})\to\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G})\to
1\,,$
where
$\mathrm{Aut}^{0}_{b}(\pi,\mathcal{H},\mathcal{G})\subset\mathrm{Aut}^{0}_{b}(\pi)$
is the subgroup consisting of those based automorphisms of $\pi$ that can be
covered by elements of $\mathrm{U}(\bm{\upzeta})$ and preserve the Ehresmann
connection $\mathcal{H}$ and the metric $\mathcal{G}$.
The main feature of the local U-duality group of a local supergravity theory
is that maps solutions to solutions and thus can be used as a solution
generating mechanism. This key property also holds for
$\mathrm{U}(\bm{\upzeta})$ as we show below. Let:
$(g,s,\mathcal{A})\in\mathfrak{Sol}(\bm{\upzeta})~{}~{}.$
Recall that $\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ is a
connection on the pull-back of $P_{\mathfrak{t}}$ through $s\in\Gamma(\pi)$.
An element $u\in\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})$ which covers
$f_{u}\in\mathrm{Aut}_{b}(\pi)$ acts on $(g,s,\mathcal{A})$ through:
$u\cdot(g,s,\mathcal{A})=(g,f_{u}(s),\mathcal{A}_{u})\,,$
where $f_{u}(s)=f_{u}\circ s\in\Gamma(\pi)$ and $\mathcal{A}_{u}$ is the push-
forward of $\mathcal{A}$ by the based isomorphism of $P^{s}_{\mathfrak{t}}$
naturally associated to $u$ as follows:
$u_{m}\colon(P^{s}_{\mathfrak{t}})_{m}=(P_{\mathfrak{t}})_{s(m)}\to(P^{f_{u}(s)}_{\mathfrak{t}})_{m}=(P_{\mathfrak{t}})_{f_{u}(s(m))}\,,\quad
p\mapsto u_{s(m)}(p)\,$
for all $m\in M$. Notice that $\mathcal{A}_{u}$ is a connection on the bundle
$P^{f_{u}(s)}_{\mathfrak{t}}$, where the latter denotes the pull-back of
$P_{\mathfrak{t}}$ by the section $f_{u}(s)\in\Gamma(s)$. Denote by:
$\bm{\upzeta}_{u}=(\pi,\mathcal{H}_{u},\mathcal{G}_{u},P_{\mathfrak{t}},\mathcal{J}_{u})$
the push-forward of $\bm{\upzeta}_{u}$ of $\bm{\upzeta}$ by
$u\in\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})$.
###### Corollary 5.2.
Let $\bm{\upzeta}=(\pi,\mathcal{H},\mathcal{G},P,\mathcal{J})$ be a polarized
scalar-Siegel bundle with submersion $\pi\colon X\to M$. Every element
$u\in\mathrm{Aut}(P)$ defines a bijection of sets:
$\mathbb{A}_{u}\colon\mathfrak{Conf}(\bm{\upzeta})\to\mathfrak{Conf}(\bm{\upzeta}_{u})\,,\quad(g,s,\mathcal{A})\mapsto(g,f_{u}(s),\mathcal{A}_{u})\,,$
which restricts to a bijection:
$\mathbb{A}_{u}\colon\mathfrak{Sol}(\bm{\upzeta})\to\mathfrak{Sol}(\bm{\upzeta}_{u})~{}~{}.$
In particular, if $u\in\mathrm{U}(\bm{\upzeta})$ then
$\mathbb{A}_{u}\colon\mathfrak{Sol}(\bm{\upzeta})\to\mathfrak{Sol}(\bm{\upzeta})$
preserves the solution space of the given supergravity theory.
###### Proof.
The result follows directly from Theorem 2.25 upon noticing that:
$\mathcal{F}_{\mathcal{A}_{u}}=u\cdot\mathcal{F}_{\mathcal{A}}\,~{}~{},$
which shows that $\mathcal{F}_{A}$ transforms as the field strength
$\mathcal{F}$ considered in Section 2 in the classical formulation of the
theory. ∎
For further reference we introduce the following definition.
###### Definition 5.3.
The gauge _U-duality transformation_ induced by $u\in\mathrm{U}(\bm{\upzeta})$
is the bijection
$\mathbb{A}_{u}\colon\mathrm{Sol}(\bm{\upzeta})\to\mathrm{Sol}(\bm{\upzeta})$.
We have a canonical morphism of groups:
$\mathfrak{ad}\colon\mathrm{U}(\bm{\upzeta})\to\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\,,\quad
u\mapsto\mathfrak{ad}_{u}\,,$
where $\bm{\Phi}(\bm{\upzeta})$ is the integral scalar-electromagnetic bundle
determined by the polarized scalar-Siegel bundle $\bm{\upzeta}$. This morphism
associates to $u$ the automorphism of the adjoint bundle of $P_{\mathfrak{t}}$
defined canonically by the latter.
###### Definition 5.4.
The _continuous subgroup_ of the gauge U-duality group
$\mathrm{U}(\bm{\upzeta})$ is:
$\mathrm{C}(\bm{\upzeta})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\ker(\mathfrak{ad})\subset\mathrm{U}(\bm{\upzeta})\,.$
The classical U-duality group was shown to be a finite-dimensional Lie group
in Section 2.3 when the scalar bundle is flat. This is no longer true for the
gauge U-duality group. Instead, if the rank of
$P_{\mathfrak{t}}\in\mathrm{U}(\bm{\upzeta})$ is positive
$\mathrm{U}(\bm{\upzeta})$ is an extension of the arithmetic duality group
$\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$ by an _infinite-dimensional_ abelian
group, a fact that allows us to pinpoint the geometric origin of U-duality.
###### Proposition 5.5.
The gauge U-duality group $\mathrm{U}(\bm{\upzeta})$ fits into a short exact
sequence:
$1\to\mathrm{C}(\bm{\upzeta})\hookrightarrow\mathrm{U}(\bm{\upzeta})\xrightarrow{\mathfrak{ad}}\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\to
1\,,$ (10)
where $\bm{\Phi}(\bm{\upzeta})$ is the polarized integral scalar-
electromagnetic bundle determined by $\bm{\upzeta}$.
###### Proof.
Since it is clear that the natural inclusion
$\mathrm{C}(\bm{\upzeta})\hookrightarrow\mathrm{U}(\bm{\upzeta})$ is injective
and the map $\mathfrak{ad}$ is a homomorphism, it suffices to prove that
$\mathfrak{ad}$ is surjective. Write the Dirac system $\mathcal{L}$ in
$\bm{\upzeta}$ as an associated bundle
$\mathcal{L}=P_{\mathfrak{t}}\times_{\ell}\mathbb{Z}^{2n}$ to
$P_{\mathfrak{t}}$ through the natural representation $\ell$ of
$\mathrm{Aff}_{\mathfrak{t}}$ on $\mathbb{Z}^{2n}$. Let
$\phi\in\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$ be an automorphism of the scalar-
electromagnetic bundle $\bm{\Phi}(\bm{\upzeta})$ associated to $\bm{\upzeta}$
and covering the diffeomorphism $f_{\phi}\in\mathrm{Diff}(M)$. Since the
latter is a diffeomorphism, the principal bundles $P_{\mathfrak{t}}$ and
$P_{\mathfrak{t}}^{f_{\phi}}$, where the latter denotes the pull-back of
$P_{\mathfrak{t}}$ by $f_{\phi}$, are isomorphic. Fix such an isomorphism,
which is equivalent to fixing an automorphism $u^{\prime}_{\phi}\colon
P_{\mathfrak{t}}\to P_{\mathfrak{t}}$ covering $f_{\phi}$. Then, for every
$[p,v]\in\mathcal{L}$ there exists a unique map $\Scr
B_{p}\in\mathbb{Z}^{2n}\to\mathbb{Z}^{2n}$ such that:
$\phi([p,v])=[u^{\prime}_{\phi}(p),\Scr B_{p}(v)]\,.$
Since $\phi$ is a linear automorphism of the integral duality structure
determined by $\bm{\upzeta}$ it follows that the map $\Scr
B_{p}\in\mathbb{Z}^{2n}\to\mathbb{Z}^{2n}$ is a linear automorphism of the
standard symplectic lattice of type $\mathfrak{t}\in\mathbb{Z}$ and therefore
belongs to $\mathrm{Aff}_{\mathfrak{t}}$. Furthermore, independence of the
representative in $[p,v]\in\mathcal{L}$ in the definition of $\Scr B_{p}$
implies:
$\Scr B_{px}=x^{-1}\circ\Scr B_{p}\circ x\,,$
for every $x\in\mathrm{Aff}_{\mathfrak{t}}$. Hence, the assignment
$p\mapsto\Scr B_{p}$ defines a smooth map $\Scr B\colon
P_{\mathfrak{t}}\to\mathrm{Aff}_{\mathfrak{t}}$ which is equivariant with
respect to the adjoint action. Therefore, we have:
$\phi([p,v])=[u^{\prime}_{\phi}(p)\Scr B_{p},(v)]\,,$
and the automorphism $u_{\phi}\colon P_{\mathfrak{t}}\to P_{\mathfrak{t}}$
defined as follows:
$u_{\phi}(p)\stackrel{{\scriptstyle{\rm def.}}}{{=}}u^{\prime}_{\phi}(p)\Scr
B_{p}\,,\quad p\in P_{\mathfrak{t}}\,,$
covers $f_{\phi}\in\mathrm{Diff}(M)$ and satisfies
$\mathfrak{ad}(u_{\phi})=\phi$ by construction. Hence $\mathfrak{ad}$ is
surjective and thus we conclude. ∎
It is clear that $\mathfrak{ad}_{u}$ is trivial when
$u\in\mathrm{C}(\bm{\upzeta})$. Intuitively speaking, elements in
$\mathrm{C}_{\pi}(P_{\mathfrak{t}},\mathcal{J})$ behave as gauge
transformations on a principal torus bundle and therefore act trivially on the
curvature of any connection. In fact, the arithmetic U-duality group
$\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$ identifies with the _discrete remnant_
(in the sense of [27]) of the gauge group $\mathrm{Aut}(P_{\mathfrak{t}})$,
which shows that U-dualities in supergravity are but gauge transformations of
the underlying Siegel bundle, a fact that elucidates their geometric origin.
We discuss next a few examples. An in-depth study of the gauge U-duality group
will be presented in a separate publication.
### 5.1. Rank-zero Siegel bundle
Let $(\pi,\mathcal{H},\mathcal{G})$ be a scalar bundle over $M$ with
submersion $\pi:X\rightarrow M$ and consider the rank zero Siegel bundle
$P_{0}=(\mathrm{id}_{X}:X\rightarrow X)$ on $X$ (which is necessarily
trivial). In this case $\mathrm{Aff}_{\mathfrak{t}}$ is the trivial group and
we have:
$\mathrm{Aut}(P_{0})=\mathrm{Diff}(X)\,,\qquad\mathrm{Aut}_{\pi}(P_{0})=\mathrm{Aut}_{b}(\pi)\,,\qquad\mathrm{Aut}_{b}(P_{0})=\\{\mathrm{id}_{X}\\}~{}~{},$
as well as:
${\bm{\Delta}}(P_{0})=X\times\left\\{0\right\\}\,.$
Let $\bm{\upzeta}_{0}=(\pi,\mathcal{H},\mathcal{G},P_{0},\mathcal{J}_{0})$,
where $\mathcal{J}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{id}_{{\bm{\Delta}}(P_{0})}$. Then:
$\mathrm{U}(\bm{\upzeta}_{0})=\mathrm{Aut}^{0}_{b}(\pi,\mathcal{H},\mathcal{G})=\left\\{u\in\mathrm{Aut}_{b}(\pi)\,\,|\,\,\mathcal{H}_{u}=\mathcal{H}\,,\,\,\mathcal{G}_{u}=\mathcal{G}\right\\}\,.$
Lemma 2.27 shows that $\mathrm{U}(\bm{\upzeta}_{0})$ is isomorphic to the
commmutant of the holonomy group of $\mathcal{H}$ inside the isometry group of
the typical fiber of $(\mathcal{M},\mathcal{G})$ of $(\pi,\mathcal{G})$. When
the holonomy of $\mathcal{H}$ is trivial, $\mathrm{U}(\bm{\upzeta}_{0})$
reduces to the orientation-preserving isometry group ${\rm
Iso}(\mathcal{M},\mathcal{G})$ of the scalar manifold but is in general
different. In particular, when the holonomy of $\mathcal{H}$ is full, that is,
equal to ${\rm Iso}(\mathcal{M},\mathcal{G})$, then
$\mathrm{U}(\bm{\upzeta}_{0})$ is isomorphic to the center of ${\rm
Iso}(\mathcal{M},\mathcal{G})$ and hence possibly trivial. This gives a simple
and explicit example illustrating how the supergravity duality group may
differ from its local counterpart considered in the literature, which in this
case would correspond always with ${\rm Iso}(\mathcal{M},\mathcal{G})$.
### 5.2. Rank zero scalar bundle
Let $(\pi,\mathcal{H},\mathcal{G})$ be the rank zero scalar bundle, that is,
$X=M$, $\pi\colon M\to M$ is the identity map, $\mathcal{H}=TM$ is canonically
identified with the tangent bundle of $M$ and $\mathcal{G}$ is the trivial
metric on the rank zero vector bundle over $M$. Then, the isometry group of
the typical fiber of $(\pi,\mathcal{H},\mathcal{G})$ is the trivial group
whence the short exact sequence:
$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})\to\mathrm{U}(\bm{\upzeta})\to\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G})\to
1\,,$
reduces to an isomorphism of groups
$\mathrm{U}(\bm{\upzeta})=\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})$
where $(P_{\mathfrak{t}},\mathcal{J})$ is a polarized Siegel bundle over $M$.
Therefore, the gauge U-duality group reduces to the gauge group of the Siegel
bundle underlying the given bosonic supergravity. This corresponds, in fact,
with the electromagnetic gauge duality group of the abelian gauge theory
determined by $(P_{\mathfrak{t}},\mathcal{J})$ as explained in detail in [26].
### 5.3. Holonomy-trivial scalar bundle
Consider a holonomy-trivial scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ in
the presentation:
$X=M\times\mathcal{M}\,,\qquad\mathcal{H}=TM^{\mathrm{pr_{1}}}\,,$
where $\mathcal{M}$ is an oriented $n_{s}$-dimensional manifold and
$\mathrm{pr}_{1}\colon M\times\mathcal{M}\to M$ is the canonical projection
onto the first factor. In this situation
$\mathcal{V}=T\mathcal{M}^{\mathrm{pr_{2}}}$, where $\mathrm{pr}_{2}\colon
M\times\mathcal{M}\to\mathcal{M}$ is the canonical projection onto the second
factor, and the vertical metric $\mathcal{G}$ descends to a Riemannian metric
on $\mathcal{M}$ which we denote by the same symbol for ease of notation.
Furthermore, consider the vertically polarized Siegel bundle
$(P_{\mathfrak{t}},\mathcal{J})$ obtained by pull-back through
$\mathrm{pr}_{2}\colon M\times\mathcal{M}\to\mathcal{M}$ of a vertically
polarized Siegel bundle on $(\mathcal{M},\mathcal{G})$, which we denote again
by $(P_{\mathfrak{t}},\mathcal{J})$ for ease of notation. Then:
$\mathrm{Aut}_{b}(\pi)=\mathrm{Diff}(\mathcal{M})\,,$
where $\mathrm{Diff}(\mathcal{M})$ the group of oriented diffeomorphisms of
$\mathcal{M}$. In particular, we obtain the following short exact sequence:
$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}})\to\mathrm{Aut}(P_{\mathfrak{t}})\to\mathrm{Diff}_{0}(\mathcal{M})\to
1\,,$
where $\mathrm{Diff}_{0}(\mathcal{M})$ denotes the subgroup of
$\mathrm{Diff}(\mathcal{M})$ that can be covered by elements in
$\mathrm{Aut}(P_{\mathfrak{t}})$. Here $P_{\mathfrak{t}}$ is considered as a
Siegel bundle over $\mathcal{M}$. In this case, the gauge U-duality group is:
$\mathrm{U}(\bm{\upzeta})=\left\\{u\in\mathrm{Aut}(P_{\mathfrak{t}})\,\,|\,\,\mathcal{G}_{u}=\mathcal{G}\,,\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,,$
and fits into the short exact sequence:
$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})\to\mathrm{U}(\bm{\upzeta})\to{\rm
Iso}_{0}(\mathcal{M},\mathcal{G})\to 1\,,$
where ${\rm Iso}_{0}(\mathcal{M},\mathcal{G})$ is group of those isometries of
$(\mathcal{M},\mathcal{G})$ which can be covered by elements in
$\mathrm{U}(\bm{\upzeta})$. Assume in addition that $P_{\mathfrak{t}}$ is
topologically trivial and write:
$P_{\mathfrak{t}}=\mathcal{M}\times\mathrm{Aff}_{\mathfrak{t}}=\mathcal{M}\times\left[\mathrm{U}(1)^{2n_{v}}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})\right]\,.$
in a fixed trivialization. We have:
$\displaystyle\mathrm{Aut}_{b}(P_{\mathfrak{t}})=\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z}))\,,$
$\displaystyle\mathrm{Aut}(P_{\mathfrak{t}})=\mathrm{Diff}(\mathcal{M})\ltimes\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z}))\,.$
Since $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ is discrete, we find:
$\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z}))=\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$
as well as:
$\mathrm{U}(\bm{\upzeta})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(f,u_{T},\mathfrak{U})\in{\rm
Iso}(\mathcal{M},\mathcal{G})\ltimes(\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z}))\,\,|\,\,\mathfrak{U}\mathcal{J}\mathfrak{U}^{-1}=\mathcal{J}\circ
f\right\\}\,.$
In particular, the morphism
$\mathfrak{ad}\colon\mathrm{U}(\bm{\upzeta})\to\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$
is given explicitly by:
$\mathfrak{ad}((f,u_{T},\mathfrak{U}))=(f,\mathfrak{U})\in{\rm
Iso}(\mathcal{M},\mathcal{G})\ltimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})\,,$
The short exact sequence:
$1\to\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})\hookrightarrow\mathrm{U}(\bm{\upzeta})\xrightarrow{\mathfrak{ad}}\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\to
1\,,$
shows how
$\mathrm{C}(\bm{\upzeta})=\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})$
captures the _non-discrete_ gauge transformations in
$\mathrm{U}(\bm{\upzeta})$, which act trivially on the adjoint bundle of $P$.
Consequently, we have:
$\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(f,\mathfrak{U})\in{\rm
Iso}(\mathcal{M},\mathcal{G})\times\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})\,\,|\,\,\mathfrak{U}\mathcal{J}\mathfrak{U}^{-1}=\mathcal{J}\circ
f\right\\}\,,$
which illustrates the explicit dependence of the arithmetic U-duality group on
the type $\mathfrak{t}\in\mathrm{Div}^{n}$ of its underlying Siegel bundle
$P_{\mathfrak{t}}$.
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# A nonabelian Brunn–Minkowski inequality
Yifan Jing Department of Mathematics, University of Illinois at Urbana-
Champaign, Urbana IL, USA<EMAIL_ADDRESS>, Chieu-Minh Tran
Department of Mathematics, University of Notre Dame, Notre Dame IN, USA
<EMAIL_ADDRESS>and Ruixiang Zhang Department of Mathematics, University of
Wisconsin Madison
AND School of Mathematics, Institute for Advanced Study
USA<EMAIL_ADDRESS>
###### Abstract.
Henstock and Macbeath asked in 1953 whether the Brunn–Minkowski inequality can
be generalized to nonabelian locally compact groups; questions in the same
line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an
inequality and prove that it is sharp for helix-free locally compact groups,
which includes real linear algebraic groups, Nash groups, semisimple Lie
groups with finite center, solvable Lie groups, etc. The proof follows an
induction on dimension strategy; new ingredients include an understanding of
the role played by maximal compact subgroups of Lie groups, a necessary
modified form of the inequality which is also applicable to nonunimodular
locally compact groups, and a proportionated averaging trick.
###### 2010 Mathematics Subject Classification:
Primary 22D05; Secondary 43A05, 49Q20, 60B15, 05D99
YJ was supported by Arnold O. Beckman Research Award (Campus Research Board
RB21011), by the Department Fellowship, and by the Trjitzinsky Fellowship from
UIUC
RZ was supported by the NSF grant DMS-1856541, the Ky Fan and Yu-Fen Fan
Endowment Fund at the Institute for Advanced Study and the NSF grant
DMS-1926686
###### Contents
1. 1 Introduction
1. 1.1 Background
2. 1.2 Statement of main results
3. 1.3 Overview of the proof
2. 2 Noncompact Lie dimension and helix dimension
3. 3 Proof of Theorem 1.3
4. 4 Reduction to outer terms of certain short exact sequences
5. 5 Reduction to unimodular subgroups
6. 6 Reduction to cocompact and codiscrete subgroups
7. 7 Proof of Theorems 1.1, 1.2, and 1.5
1. 7.1 A dichotomy lemma
2. 7.2 Proofs of the main theorems
8. A Some results about topological groups
9. B Measures and the modular function
10. C Almost-Lie groups and the Gleason–Yamabe Theorem
11. D Some results about Lie groups
12. E Solvable and Semisimple Lie groups
## 1\. Introduction
### 1.1. Background
Let $\mu$ be the usual Lebesgue measure on $\mathbb{R}^{d}$, let $X$ and $Y$
be nonempty and compact subsets of $\mathbb{R}^{d}$, and set $X+Y:=\\{x+y:x\in
X,y\in Y\\}$. The Brunn–Minkowski inequality says that
(1) $\mu(X+Y)^{1/d}\geq\mu(X)^{1/d}+\mu(Y)^{1/d}.$
For fixed $\mu(X)$ and $\mu(Y)$, the inequality provides us with the minimum
value of $\mu(X+Y)$ which is obtained, for example, when $X$, $Y$, and $X+Y$
are $d$-dimensional hypercubes with side length $\mu(X)^{1/d}$,
$\mu(Y)^{1/d}$, and $\mu(X)^{1/d}+\mu(Y)^{1/d}$, respectively.
Under the further assumption that $X$ and $Y$ are convex, the inequality in an
equivalent form was proven by Brunn [5] in 1887. In the celebrated Geometrie
der Zahlen (Geometry of Numbers) [27] published in 1896, Minkowski introduced
the current form of the inequality and established that the equality happens
if and only if $X$ and $Y$ are homothetic convex sets. Removing the convexity
assumption was done by Lyusternik [23] in 1935. However, his proof that the
same condition for equality still holds was seen to contain some flaws, a
situation eventually corrected by Henstock and Macbeath [13] in 1953. The
Brunn–Minkowski inequality is widely considered a cornerstone of convex
geometry. See [10] for an excellent survey on its numerous generalizations and
applications.
In this paper, we consider the problem of generalizing the Brunn–Minkowski
inequality to a locally compact group $G$. Here, up to a multiplication by
positive constants, we have a unique left Haar measure $\mu$ generalizing the
Lebesgue measure in $\mathbb{R}^{d}$; see Appendix B for the precise
definitions.
We temporarily further assume that $\mu$ is also invariant under right
translations. Such $G$ is called unimodular. This assumption holds when
$G=\mathbb{R}^{d}$ and in many other situations (e.g, when $G$ is compact,
discrete, a nilpotent Lie group, a semisimple Lie group, etc). Set
$XY=\\{xy:x\in X,y\in Y\\}$ for nonempty compact $X,Y\subseteq G$. The
translation invariance property of $\mu$ implies that
$\mu(XY)\geq\max\\{\mu(X),\mu(Y)\\}$
and should intuitively be even larger, hinting at a meaningful generalization
of the Brunn–Minkowski inequality to this setting. This will be shown to be
the case.
For an arbitrary locally compact group $G$, $\mu$ might no longer be right
invariant. Hence, we still have $\mu(XY)\geq\mu(Y)$, but we might have
$\mu(XY)<\mu(X)$. By a result by Macbeath [24] in 1960, the trivial inequality
$\mu(XY)\geq\mu(Y)$ for nonunimodular $G$ is already sharp in the sense that
for any $\alpha,\beta,\varepsilon>0$, there are nonempty compact $X,Y\subseteq
G$ with
$\mu(X)=\alpha,\mu(Y)=\beta,\ \text{and}\ \mu(XY)<\mu(Y)+\varepsilon.$
We will later see in this paper that there is still a meaningful
generalization of the Brunn–Minkowski inequality involving both $\mu$ and a
right Haar measure $\nu$. Surprisingly, it turns out that if one only cares
about unimodular cases, the nonunimodular cases are still needed for our
proof. We will keep the settings and notations of this paragraph throughout
the rest of the paper.
The problem of generalizing the Brunn–Minkowski inequality was proposed in
1953 by Henstock and Macbeath [13]; different variations of this problem were
also later suggested by Hrushovski [15], by McCrudden [25], and by Tao [30].
In the direction of the intuition described earlier, Kemperman [18] showed in
1964 that $\mu(XY)\geq\mu(X)+\mu(Y)$ when $G$ is connected, unimodular and
noncompact. Even more important for us is the followin generalization to all
connected noncompact locally compact groups, which reads
$\frac{\nu(X)}{\nu(XY)}+\frac{\mu(Y)}{\mu(XY)}\leq 1.$
While applicable to all locally compact groups, Kemperman’s inequalities are
not sharp even for $\mathbb{R}^{2}$ giving a weaker conclusion than the
Brunn–Minkowski inequality. The most definite result toward the correct lower
bound was obtained by McCrudden [25] in 1969. In effect, he showed that when
$G$ is a unimodular solvable Lie group of dimension $d$, and $m$ is the
dimension of the maximal compact subgroup, we have
$\mu(XY)^{1/(d-m)}\geq\mu(X)^{1/(d-m)}+\mu(Y)^{1/(d-m)}.$
The above differs from McCrudden’s original statement in that $m$ was defined
using an inductive idea in [25]; the current form is more suitable to get the
later generalization and to show that it is indeed sharp. A number of special
cases of this result were rediscovered by Gromov [12], by Hrushovski [16], by
Leonardi and Mansou [22], and by Tao [30]. Sharpness for nilpotent groups was
essentially proven by Monti [28]; see also Tao [30].
### 1.2. Statement of main results
Suppose $G$ is Lie group with connected component $G_{0}$. Following Levi
decomposition (Fact E.5), we have an exact sequence of Lie groups
$1\to Q\to G_{0}\to S\to 1$
where $Q$ is solvable and $S$ is semisimple. It is known that the center
$Z(S)$ is a discrete abelian group of finite rank $h$; see Facts E.9 and E.10.
We call $h$ the helix dimension of $G$. As an example,
$\text{SL}_{2}(\mathbb{R})$ has helix dimension $0$ while its universal cover
has helix dimension $1$. If $h=0$, equivalently $S$ has finite center, we say
that $G$ is helix-free. Real linear algebraic groups and more generally, Nash
groups (equivalently, semialgebraic Lie groups or groups definable in the
field of real numbers) are helix free; see [1, Lemma 4.5] and the subsequent
discussion in the same paper. Our first main results is a generalization of
Brunn–Minkowski inequality to Lie groups whose exponent will be seen to be
sharp for helix-free Lie groups:
###### Theorem 1.1.
Suppose $G$ is a Lie group, $\mu$ is a left Haar measure, $\nu$ is a right
Haar measure, the dimension of $G$ is $d$, the maximal dimension of a compact
subgroup of $G$ is $m$, the helix dimension of $G$ is $h$, and $X,Y$ are
compact subsets of $G$ with positive measure. Then
(2)
$\frac{\nu(X)^{1/(d-m-h)}}{\nu(XY)^{1/(d-m-h)}}+\frac{\mu(Y)^{1/(d-m-h)}}{\mu(XY)^{1/(d-m-h)}}\leq
1;$
the left-hand-side is interpreted as $\max\\{\nu(X)/\nu(XY),\mu(Y)/\mu(XY)\\}$
if $d-m-h=0$. In particular, if $G$ is unimodular, then
$\mu(XY)^{\tfrac{1}{d-m-h}}\geq\mu(X)^{\tfrac{1}{d-m-h}}+\mu(Y)^{\tfrac{1}{d-m-h}}.$
Now consider an arbitrary locally compact group $G$. Using the Gleason–Yamabe
Theorem (Fact C.2), one can choose an open subgroup $G^{\prime}$ of $G$ and a
normal compact subgroup $H$ of $G^{\prime}$ such that $G^{\prime}/H$ is a Lie
group. It is shown in Proposition 2.8 that
$n=\dim(G^{\prime}/H)-\max\\{\dim(K):K\text{ is a compact subgroup of
}G^{\prime}/H\\}$
is independent of the choice of $G^{\prime}$ and $H$ satisfying the above
properties. We call $n$ the noncompact Lie dimension of $G$. Let $Q$ be the
radical (i.e, the maximal connected closed solvable normal subgroup, see Fact
E.4) of $G^{\prime}/H$. Note that $(G^{\prime}/H)_{0}/Q)$ has discrete center
$Z((G^{\prime}/H)_{0}/Q$ by Facts E.9 and E.10. We call
$h=\mathrm{rank}(Z((G^{\prime}/H)_{0}/Q))$
the helix dimension of $G$. We will also show that the helix dimension $h$ of
$G^{\prime}/H$ is independent of the choice of $G^{\prime}$ and $H$ in
Proposition 2.8. Our second main result reads:
###### Theorem 1.2.
Suppose $G$ is a locally compact group with noncompact Lie dimension $n$ and
helix dimension $h$, $\mu$ is a left Haar measure, $\nu$ is a right Haar
measure, and $X,Y$ are compact subsets of $G$ with positive measure. Then
$\frac{\nu(X)^{1/(n-h)}}{\nu(XY)^{1/(n-h)}}+\frac{\mu(Y)^{1/(n-h)}}{\mu(XY)^{1/(n-h)}}\leq
1;$
the left-hand-side is interpreted as $\max\\{\nu(X)/\nu(XY),\mu(Y)/\mu(XY)\\}$
when $n-h=0$. In particular, if $G$ is unimodular, then
$\mu(XY)^{\tfrac{1}{n-h}}\geq\mu(X)^{\tfrac{1}{n-h}}+\mu(Y)^{\tfrac{1}{n-h}}.$
When $G$ is as in Theorem 1.1, the noncompact Lie dimension $n$ is simply
$d-m$, so Theorem 1.2 is a generalization of Theorem 1.1. On the other hand,
Theorem 1.2 is equally applicable to totally disconnected locally compact
groups, which are the polar opposite of Lie groups.
Our last main result tells us that when $G$ is helix-free, the exponent
$1/(n-h)=1/n$ in Theorem 1.1 and Theorem 1.2 are sharp even when we assume
further that $X=Y$. As usual in the current setting, we write $X^{k}$ for the
$k$-fold product of $X$.
###### Theorem 1.3.
Suppose $G$ is a locally compact group with noncompact Lie dimension $n$,
$\mu$ is a left Haar measure, and $\nu$ is a right Haar measure. Then
1. (1)
When $n=0$, there is a compact set $X$ with positive left and right measure in
$G$ such that $\mu(X^{2})=\mu(X)$ and $\nu(X^{2})=\nu(X)$.
2. (2)
When $n>0$, for every $\varepsilon>0$, there is a compact set $X$ with
positive left and right measure in $G$ such that
$\frac{\nu(X)^{\frac{1}{n}-\varepsilon}}{\nu(X^{2})^{\frac{1}{n}-\varepsilon}}+\frac{\mu(X)^{\frac{1}{n}-\varepsilon}}{\mu(X^{2})^{\frac{1}{n}-\varepsilon}}>1.$
As a corollary, if $G$ is unimodular with $n>0$, for every
$\varepsilon^{\prime}>0$, there is a compact set $X$ in $G$ such that
$\mu(X^{2})<(2^{n}+\varepsilon^{\prime})\mu(X)$.
The upper bound given in Theorem 1.3 matches the lower bound given in Theorem
1.2 when the group is helix-free, that is a group has helix dimension $0$,
which essentially means the semisimple part of the group has finite center.
Hence, for these groups, our theorems resolve the problem of generalizing the
Brunn–Minkowski inequality, which was suggested by Henstock and Macbeath [13],
by Hrushovski [15], by McCrudden [24], and by Tao [30].
We believe that the exponent in Theorem 1.3 should be correct for all locally
compact groups, which is made precise by the following conjecture:
###### Conjecture 1.4 (Nonabelian Brunn–Minkowski Conjecture).
Suppose $G$ is a locally compact group with noncompact Lie dimension $n$,
$\mu$ is a left Haar measure, $\nu$ is a right Haar measure, and $X,Y$ are
compact subsets of $G$ with positive measure. Then
$\frac{\nu(X)^{1/n}}{\nu(XY)^{1/n}}+\frac{\mu(Y)^{1/n}}{\mu(XY)^{1/n}}\leq 1;$
the left-hand-side is interpreted as $\max\\{\nu(X)/\nu(XY),\mu(Y)/\mu(XY)\\}$
when $n=0$.
We remark that, the exponent in the inequality obtained in Theorem 1.2, if is
not sharp, still has the correct order of magnitude, as the helix dimension
$h$ of $G$ is always at most $n/3$, where $n$ is the noncompact Lie dimension
of $G$; see Corollary 2.15.
The next result shows that one can reduce Conjecture 1.4 to all simply
connected simple Lie groups. Unexpectedly, the hardest remaining cases are
what one might initially regard to be the simplest cases.
###### Theorem 1.5.
Suppose the nonabelian Brunn–Minkowski conjecture holds for all simply
connected simple Lie groups, then it holds for all locally compact groups.
In the statements of our main results, we require the sets $X$ and $Y$ to be
compact. The reason is that, when $X$ and $Y$ are just measurable, the set
$XY$ may not be measurable. We remark that by using the regularity property of
Haar measure, the conclusions in our main theorems still hold for measurable
$X$ and $Y$ if we replace $\mu(XY)$ and $\nu(XY)$ by inner Haar measures.
The results of this paper continue a line of work by the first two authors
[17] on small measure expansions in locally compact groups. Through
classifying groups $G$ and compact subsets $X$ and $Y$ of $G$ with nearly
minimal expansion, it is shown there that when $G$ is a simple compact Lie
group and $\mu(X)$ sufficiently small,
$\mu(X^{2})>(2+c)\mu(X)$
for a positive constant $c$. This can be seen as a continuous analog of the
expansion gap results. For noncompact simple Lie groups, Theorem 1.1 provides
a significant strengthening counterpart where we have $\mu(X^{2})\geq
4\mu(X)$. As we will see later, some of the techniques used in this paper are
further developments from techniques used in [17].
The equality for Theorems 1.1 and 1.2 can happen for $\mathbb{R}^{d}$, but
might be impossible for general $G$. In fact, from McCrudden’s result [26],
the equality cannot happen even when $G$ is the Heisenberg group. It would
also be interesting to understand when equality nearly happens and develop a
theory similar to that of Christ, Figalli, and Jerison [6, 7, 8] for
$\mathbb{R}^{d}$.
Like the Brunn–Minkowski inequality for $\mathbb{R}^{d}$, our results do not
rely on the normalization of Haar measures. However, by fixing a Haar measure
$\mu$ on a unimodular group $G$, it would be interesting to determine the
value of
$\min\\{\mu(XY):X,Y\subseteq G\text{ are
compact},\mu(X)=\alpha,\mu(Y)=\beta\\},$
for given $\alpha,\beta\in\mathbb{R}^{>0}$, and to classify the situations
where the equality happens. We do not pursue this question here.
### 1.3. Overview of the proof
In this subsection, we discuss the idea of the proof of the main results and
the organization of the paper. For expository purpose, we restrict our
attention to helix-free locally compact groups, where we can fully prove
Conjecture 1.4. The proof of the full versions of Theorems 1.1 and 1.2
requires a more involved discussion on the helix dimension, which is developed
in Section 2.
In the current situation, for all our three theorems, the exponent of the
inequalities are controlled by $n$ of $G$ instead of just its topological
dimension $d$ as in the simpler versions for $\mathbb{R}^{d}$. Recall that,
for a Lie group $G$, $n=d-m$ where $m$ the maximum dimension of a compact
subgroup of $G$. The proof of Theorem 1.3 explains the critical role of $m$:
Our construction is essentially a small neighborhood of a compact subgroup of
$G$ having maximal dimension, see Figure 1. One may then naturally conjecture
that the above is the best we can do. Theorems 1.1 and 1.2 confirm this
intuition for helix-free groups.
Figure 1. Let $G=\mathrm{SL}(2,\mathbb{R})$ (the open region bounded by the
outer torus), and let $K=\mathrm{SO}(2,\mathbb{R})$ be the maximal compact
subgroup of $G$. If we take $X$ to be a small closed neighborhood of $K$
(closed region bounded by the inner torus), Theorem 1.3 says when $X$ is
sufficiently small, $\mu_{G}(X^{2})$ will be very close to $4\mu_{G}(X)$
instead of $8\mu_{G}(X)$, although $G$ has topological dimension $3$.
To motivate our proofs of Theorems 1.1 and 1.2, we first recall some proofs of
the known cases of the Brunn–Minkowski inequality. Over $\mathbb{R}^{d}$, the
usual strategy is to induct on dimensions. This is generalized by McCrudden to
obtain the following “unimodular exponent splitting” result: Given an exact
sequence of _unimodular_ locally compact groups
$1\to H\to G\to G/H\to 1,$
if $H$ and $G/H$ satisfy Brunn–Minkowski inequalities with exponents $1/n_{1}$
and $1/n_{2}$, respectively, then the group $G$ satisfies a Brunn–Minkowski
inequality with exponent $1/(n_{1}+n_{2})$.
McCrudden’s proof of the above result can be seen as the following “spillover”
argument: For each $g$ in $G$, we call $X\cap gH$ a _fiber_ of $X$, and refer
to the size of $g^{-1}X\cap H$ in $H$ as its _length_. Let $\pi:G\to G/H$ be
the quotient map. We now partition $X$ and $Y$ each into $N$ parts. Suppose
$X=\bigcup_{i=1}^{N}X_{i}$ and $Y=\bigcup_{i=1}^{N}Y_{i}$, we require that the
images under $\pi$ of the $X_{i}$’s are pairwise disjoint, the shortest fiber-
length in each $X_{i}$ is at least the longest fiber-length in $X_{i-1}$, and
likewise for the $Y_{i}$’s.
The induction hypotheses, i.e., the Brunn–Minkowski inequalities, in $H$ and
$G/H$ give us a lower bound $l_{N}$ on fiber-lengths in $X_{N}Y_{N}$ and a
lower bound $w_{N}$ on the size of $\pi(X_{N}Y_{N})$ in $G/H$. Their product
$l_{N}w_{N}$ is a lower bound for $\mu(X_{N}Y_{N})$. Next we consider
$(X_{N-1}\cup X_{N})(Y_{N-1}\cup Y_{N})$. Again a lower bound $l_{N-1}$ on
fiber-lengths in this set and a lower bound $w_{N-1}$ on the size of its image
under $\pi$ can be obtained from the induction hypotheses on $H$ and $G/H$.
From our method, we have $l_{N-1}\leq l_{N}$ and $w_{N-1}\geq w_{N}$. The
$l_{N-1}w_{N-1}$ will be a weak lower bound for $\mu((X_{N-1}\cup
X_{N})(Y_{N-1}\cup Y_{N}))$ since the fibers in $X_{N}Y_{N}$ are
“exceptionally long”. Taking all of these into account, a stronger lower bound
is
$l_{N}w_{N}+l_{N-1}(w_{N-1}-w_{N}).$
Repeating the above process and taking the limit $N\rightarrow\infty$ we have
the “spillover” argument which enables McCrudden to obtain his result.
McCrudden applied this result to obtain the Brunn–Minkowski inequality for
unimodular solvable groups with sharp exponents. A simpler proof of his result
is given in Section 4 for completeness. In the proof of our main theorems, one
important ingredient will be an exponent splitting result (that is a
generalization of his).
McCrudden’s method completely stops working when one is looking to prove
Brunn–Minkowski for simple groups since there is no nontrivial closed normal
subgroup to induct from. Next we explain how we overcome this main difficulty.
Our method turns out to work also for semisimple groups in the same way and we
will explain it in this more general setting.
Let us assume $G$ is a connected semisimple Lie group with finite center
(hence helix-free and automatically unimodular) and think about how we can
prove the Brunn–Minkowski for it. One can consider the Iwasawa decomposition
$G=KAN$ where $K$ has a compact Lie algebra and $Q=AN$ is solvable and try to
connect the Brunn–Minkowski of $S$ to a similar property of $Q$. However, $Q$
may not be unimodular in general. Let $\Delta_{Q}$ be the modular function on
$G$. One can choose to compromise by choosing $Q^{\prime}=\ker(\Delta_{Q})$
that is unimodular and try to use the Brunn–Minkowski for $Q^{\prime}$ to
prove the Brunn–Minkowski on $G$. This is indeed a good direction to go but
along this direction one inevitably gives up on the sharp exponent $1/n$ and
can at best prove a weaker inequality with the worse exponent $1/(n-1)$.
Because of this, it is necessary to formulate an inequality for nonunimodular
groups that is a good analogue of (1). We propose the inequality (2), which
seems to be new in the literature. To prove (2) for $AN$, we need a
nonunimodular exponent splitting result for the exact sequence coming out from
the modular function. It turns out that the spillover method can also be used
to reduce the problem to the case where the modular function is almost
constant on $X$ and $Y$. We work this out in Section 5. In the next more
involved step in the same section, we obtain an approximate version of
McCrudden’s result, which involves another use of the spillover method, to
finish off the proof.
In the next crucial step, we prove that the Brunn–Minkowski for a semisimple
$G$ follows from (2) for the solvable $AN$. Our method was motivated by a
recent paper [17] by the first two authors, which characterizes nearly minimal
expansion sets. Over there, a key idea is to choose a fiber $f$ uniformly at
random in $Y$ and uses $Xf$ to estimate $XY$. For our current proof, we also
choose two fibers $f_{X}$ and $f_{Y}$ randomly from $X$ and $Y$, but with
respect to two carefully chosen probability measures $\mathrm{p}_{X}$ and
$\mathrm{p}_{Y}$ that are in general nonuniform. We show that by constructing
$\mathrm{p}_{X}$ and $\mathrm{p}_{Y}$ based on the structural information of
$X$ and $Y$, $\mu(XY)$ can be estimated by the expected size of $f_{X}f_{Y}$
in $AN$, and the latter is well controlled by the Brunn–Minkowski inequality
(2) for $AN$. This part is done in Section 6. It worth noting that in this
case our inequality matches the upper bound construction when the semisimple
group has a finite center.
With the above preparation, we can explain how we prove Brunn–Minkowski for a
general helix-free Lie group $G$. Using reductions proved in Sections 4, 5,
and 6, we can reduce the problem to the case where $G$ is unimodular and
connected. Such $G$ can be decomposed into a semi-direct product of a
unimodular solvable group $Q$ and a semisimple group $S$ via the Levi
decomposition. We already know how to handle $S$ from the discussion in
Section 6. McCrudden’s result can then be used to deal with $Q$ and to deduce
the desired inequality for $G$.
In many of our reductions, we have an exact sequence of groups $1\to H\to G\to
G/H\to 1$ and want to deduce the Brunn–Minkowski for $G$ from the
Brunn–Minkowski for $H$ and $G/H$. One tricky issue is that this inductive
method only gives sharp results if the sum of the noncompact Lie dimensions
and helix dimensions of $H$ and $G/H$ is equal to the noncompact Lie dimension
of $G$. Unfortunately this is not always true (see examples in page 15). With
this warning in mind, we must ensure the above property is always satisfied in
the whole reduction. Our discussion in Section 2 guarantees this.
In the remaining part, we discuss some new challenges in the proof of Theorem
1.2 for a helix-free locally compact group $G$. The Gleason–Yamabe Theorem
tells us that $G$ contains an open subgroup $G^{\prime}$ that has a Lie
quotient $G^{\prime}/H$ with $H$ compact. For the start, we need to handle the
nonuniqueness in the choice of $G^{\prime}$ and $H$ and make sure that every
choice gives the same desired result. This requires some nontrivial effort and
makes heavy use of the Gleason–Yamabe Theorem, and we prove it in Section 2.
The rest of the proof of Theorem 1.2 has two steps. In the first step, we
reduce the problem to unimodular groups. This is done with a similar strategy
as used in the proof of the Lie group case with the additional help of a
dichotomy result proved in Section 7. To motivate the second step, recall that
in the Lie group case we first reduce the problem to connected groups. In our
second step, unlike in the Lie group case, the identity component of our group
here may not be open. Hence the correct analogue is to reduce the situation to
open subgroups with a Lie quotient, which requires some additional results in
Section 4. The desired result then follows from the Lie group case.
## 2\. Noncompact Lie dimension and helix dimension
In this section, we show that noncompact Lie dimensions and helix dimensions
are well defined in locally compact groups and that they behave well in many
exact sequences. The latter is the nontrivial underlying reason that the lower
bound in Theorem 1.1 and Theorem 1.2 matches the upper bound in Theorem 1.3
for helix-free locally compact groups.
Throughout the section, all groups are locally compact, we will use various
definitions and facts from Appendices C, D, and E. The following lemma
discusses the behavior of Iwasawa decomposition under taking quotient by a
compact normal subgroup.
###### Lemma 2.1.
Suppose $G$ is a connected semisimple Lie group, $H$ is a (not necessarily
connected) compact subgroup of $G$. Then we have the following.
1. (1)
There is an Iwasawa decomposition $G=KAN$ such that $H\leq K$.
2. (2)
Assume further that $H$ is a normal subgroup of $G$, $G=KAN$ is an Iwasawa
decomposition such that $H\leq K$, $G^{\prime}=G/H$, and $\pi:G\to G^{\prime}$
is the quotient map. Then there is an Iwasawa decomposition
$G^{\prime}=K^{\prime}A^{\prime}N^{\prime}$ such that $\pi(K)=K^{\prime}$.
###### Proof.
We first prove (1). Let $Z(G)$ be the center of $G$, $G^{\prime}=G/Z(G)$ and
$\rho:G\to G^{\prime}$ be the quotient map, and $H^{\prime}=\rho(H)$. By Facts
E.9 and E.10, $\rho$ is a covering map and $G^{\prime}$ is centerless. Let
$\mathfrak{g}$ be the common Lie algebra of $G$ and $G^{\prime}$, and
$\mathrm{exp}:\mathfrak{g}\to G$ and $\mathrm{exp}^{\prime}:\mathfrak{g}\to
G^{\prime}$ be the exponential maps. Using Fact E.14.2 about Iwasawa
decomposition, it suffices to construct a Cartan involution $\tau$ of
$\mathfrak{g}$ such that if $\mathfrak{k}$ is the subalgebra of $\mathfrak{g}$
fixed by $\tau$ and $\mathrm{exp}(\mathfrak{k})=K$, then $H\leq K$. Take a
maximal compact subgroup $K^{\prime}$ of $G^{\prime}$ that contains
$H^{\prime}$. Let $\tau_{0}$ be an arbitrary Cartan involution of $G$ (this
exists because of Fact E.13). Let $\mathfrak{k}_{0}$ be the the subalgebra of
$\mathfrak{g}$ fixed by $\tau_{0}$, and
$K^{\prime}_{0}=\mathrm{exp}(\mathfrak{k}_{0})$ in $G^{\prime}$. Then by Fact
E.14.2 about Iwasawa decomposition and the earlier observation that
$G^{\prime}$ is centerless, $K^{\prime}_{0}$ is a maximal compact subgroup of
$G^{\prime}$. By Fact D.4.1 and the assumption that $G$ is connected, there is
an automorphism $\sigma^{\prime}$ of $G^{\prime}$ such that
$\sigma^{\prime}(K^{\prime}_{0})=K^{\prime}$. Let $\alpha$ be the automorphism
of $\mathfrak{g}$ obtain by taking the tangent map of $\sigma^{\prime}$, and
let
$\tau=\alpha\tau_{0}\alpha^{-1}\text{ and
}\mathfrak{k}=\alpha(\mathfrak{k}_{0})$
As every Cartan–Killing form is invariant under automorphisms of
$\mathfrak{g}$, we get that $\tau$ is a Cartan involution. It is also easy to
check that $\mathfrak{k}$ is the subalgebra of $\mathfrak{g}$ fixed by $\tau$.
Using the functoriality of the exponential function (Fact E.2), we get
$K^{\prime}=\mathrm{exp}(\mathfrak{k})$. Now set
$K=\mathrm{exp}(\mathfrak{k})$. By Fact E.14.2, we get an Iwasawa
decomposition $G=KAN$. Therefore, by the functoriality of the exponential
function (Fact E.2), $K^{\prime}=\rho(K)$. Now as $H^{\prime}\leq K^{\prime}$,
every element of $H$ is in $Z(G)K$. By Fact E.14.2 about Iwasawa
decomposition, we have $Z(G)\subseteq K$, so $H\leq K$ as desired.
We now prove (2). Set $K^{\prime}=\pi(K)$. Let $\mathfrak{g}$, $\mathfrak{h}$,
and $\mathfrak{k}$ be the Lie algebras of $G$, $H$, and $K$, and let
$\kappa_{\mathfrak{g}}$, $\kappa_{\mathfrak{h}}$, $\kappa_{\mathfrak{k}}$ be
the Cartan–Killing form of $\mathfrak{g}$, $\mathfrak{h}$, and $\mathfrak{k}$.
Then, $\mathfrak{g}^{\prime}=\mathfrak{g}/\mathfrak{h}$ is the Lie algebra of
$G^{\prime}$, and $\mathfrak{k}^{\prime}=\mathfrak{k}/\mathfrak{h}$ is the Lie
algebra of $K^{\prime}$ by Fact E.1. Let $\tau$ be a Cartan involution of
$\mathfrak{g}$ that fixes $\mathfrak{k}$. We will construct from this a Cartan
involution $\tau^{\prime}$ of $\mathfrak{g}^{\prime}$ which fixes
$\mathfrak{k}^{\prime}$. If we have done so, then using Fact E.14.2, we obtain
$A^{\prime}$ and $N^{\prime}$ such that
$G^{\prime}=K^{\prime}A^{\prime}N^{\prime}$ is an Iwasawa decomposition, which
completes the proof.
Now we construct $\tau^{\prime}$ as described earlier. As $\mathfrak{g}$ is
semisimple, the Lie algebras $\mathfrak{h}$ and $\mathfrak{k}$ are also
semisimple. With $\mathfrak{q}$ the orthogonal complement of $\mathfrak{k}$ in
$\mathfrak{g}$ with respect to $\kappa_{\mathfrak{g}}$ and $\mathfrak{c}$ the
orthogonal complement of $\mathfrak{h}$ in $\mathfrak{k}$ with respect to
$\kappa_{\mathfrak{k}}$, we have $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$
and $\mathfrak{k}=\mathfrak{h}\oplus\mathfrak{c}$ by Fact E.7. By the same
fact, with $\kappa_{\mathfrak{p}}$ and $\kappa_{\mathfrak{c}}$ the
Cartan–Killing forms of $\mathfrak{p}$ and $\mathfrak{c}$, we have
$\kappa_{\mathfrak{g}}=\kappa_{\mathfrak{k}}\oplus\kappa_{\mathfrak{p}}$ and
$\kappa_{\mathfrak{k}}=\kappa_{\mathfrak{h}}\oplus\kappa_{\mathfrak{c}}$. It
is then easy to see that every elements of $\mathfrak{c}\oplus\mathfrak{p}$ is
orthogonal to $\mathfrak{h}$ with respect to $\kappa_{\mathfrak{g}}$. A
dimension comparison gives us $\mathfrak{c}\oplus\mathfrak{p}=\mathfrak{d}$
with $\mathfrak{d}$ the orthogonal complement of $\mathfrak{h}$ in
$\mathfrak{g}$. In summary, we have
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}=\mathfrak{h}\oplus\mathfrak{c}\oplus\mathfrak{p}=\mathfrak{h}\oplus\mathfrak{d}\quad\text{and}\quad\kappa_{\mathfrak{g}}=\kappa_{\mathfrak{k}}\oplus\kappa_{\mathfrak{p}}=\kappa_{\mathfrak{h}}\oplus\kappa_{\mathfrak{c}}\oplus\kappa_{\mathfrak{p}}=\kappa_{\mathfrak{h}}\oplus\kappa_{\mathfrak{d}}.$
As a particular consequence, the quotient map from $\mathfrak{g}$ to
$\mathfrak{g}^{\prime}$ restricts to isomorphisms of Lie algebras from
$\mathfrak{d}$ to $\mathfrak{g}^{\prime}=\mathfrak{g}/\mathfrak{h}$ and from
$\mathfrak{c}$ to $\mathfrak{k}^{\prime}=\mathfrak{k}/\mathfrak{h}$. Since
$\mathfrak{h}$ is a subalgebra of $\mathfrak{k}$, $\tau$ fixes $\mathfrak{h}$.
As Cartan–Killing forms are invariant under automorphisms, $\tau$ restricts to
an endomorphism of $\mathfrak{d}$, which the the orthogonal complement of
$\mathfrak{h}$ in $\mathfrak{g}$ under $\kappa_{\mathfrak{g}}$. Therefore,
$\tau|_{\mathfrak{d}}$ is an involution of $\mathfrak{d}$. The bilinear form
$\mathfrak{d}\times\mathfrak{d}:(x,y)\mapsto-\kappa_{\mathfrak{d}}(x,\tau|_{\mathfrak{d}}(y))$
is positive definite as it is simply the restriction to $\mathfrak{d}$ of the
positive definite bilinear form
$\mathfrak{g}\times\mathfrak{g}:(x,y)\mapsto-\kappa_{\mathfrak{d}}(x,\tau(y))$.
Hence, $\tau|_{\mathfrak{d}}$ is a Cartan involution of $\mathfrak{d}$. It is
clear that the subalgebra of $\mathfrak{d}$ fixed by $\tau|_{\mathfrak{d}}$ is
$\mathfrak{c}$. Finally, let $\tau^{\prime}$ be the pushforward of
$\tau|_{\mathfrak{d}}$ under the quotient map from $\mathfrak{g}$ to
$\mathfrak{g}^{\prime}$. It is easy to see that $\tau^{\prime}$ satisfies the
desired requirement. ∎
The following lemma allows us to compute noncompact Lie dimensions for the
universal cover of a compact Lie group.
###### Lemma 2.2.
Suppose that $K$ is a covering group of a compact Lie group $K^{\prime}$ with
the covering map $\rho:K\to K^{\prime}$, and that $K$ and $K^{\prime}$ are
connected. If $\ker(\rho)$ is a discrete group of rank $h$, and $m$ is the
maximum dimension of a compact subgroup of $K$. Then $h=\dim(K)-m$.
###### Proof.
We first consider the case when $K$ is a solvable group. Then
$K^{\prime}\cong\mathbb{T}^{k}$ where $k$ is the dimension of $K$ by Fact
D.5.2. Recall that $K$ is a quotient of the universal cover of $K^{\prime}$,
which is $\mathbb{R}^{k}$. Hence,
$K\cong\mathbb{R}^{h}\times\mathbb{T}^{k-h}$. It is easy to see that the
maximum dimension of a compact subgroup of $K$ is $k-h$, which gives us the
desired conclusion in this case.
We now prove the statement of the Lemma. Let $Q_{K}$ be the radical of $K$,
$Q_{K^{\prime}}$ the radical of $K^{\prime}$, $S_{K}=K/Q_{K}$, and
$S_{K^{\prime}}=K^{\prime}/Q_{K^{\prime}}$. Note that $K$ and $K^{\prime}$
have the same Lie algebra $\mathfrak{k}$. By Fact E.4, $Q_{K}$ and
$Q_{K^{\prime}}$ have the same Lie algebra $\mathfrak{q}$, which is the
radical of $\mathfrak{k}$. Moreover, by the functoriality of the exponential
function (Fact E.2), $\rho$ restrict to a covering map from $Q_{K}$ to
$Q_{K^{\prime}}$ with kernel $\ker\rho\cap Q_{K}$. By Fact E.1, the Lie
algebras of $S_{K}$ and $S_{K^{\prime}}$ are both isomorphic to
$\mathfrak{k}/\mathfrak{q}$. Hence, $S_{K}$ is a connected semisimple Lie
group with compact Lie algebra. Using Fact E.11, we get $S_{K}$ is compact
with finite center $Z(S_{K})$. Let $\pi:K\to S_{K}$ be the quotient map. Note
that $\ker\rho$ is a subgroup of the center of $K$ by Fact D.6. Hence, the
image of $\pi|_{\ker\rho}$ is a subset of $Z(S_{K})$, which is finite. As a
consequence, $\ker{\rho}\cap Q_{K}$, which is the kernel of $\pi|_{\ker\rho}$,
has the same rank $h$ as $\ker\rho$. Let $m_{1}$ and $m_{2}$ be the maximum
dimensions of a compact subgroup of $Q_{K}$ and of $S_{K}$ respectively. Then
$m=m_{1}+m_{2}$ by Fact D.4.2. By the special case for the solvable group $K$
proven earlier, $h+m_{1}=\dim Q_{K}$. As $S_{K}$ is compact, $m_{2}=\dim
S_{K}$. Thus, $h+m=h+m_{1}+m_{2}=\dim(Q_{K})+\dim(S_{K})$ as desired. ∎
The following proposition links the noncompact Lie dimension and the helix
dimension.
###### Proposition 2.3.
Suppose $G$ is a connected semisimple Lie group of dimension $d$, $m$ is the
maximal dimension of a compact subgroup of $G$, $h$ is the helix dimension of
$G$, and $G=KAN$ is an Iwasawa decomposition of $G$. Then $h=\dim K-m$, or
equivalently, $d-m-h=\dim(AN)$.
###### Proof.
Let $Z(G)$ be the center of $G$. Then $Z(G)$ has rank $h$ by the definition.
By Fact E.14.2, we have $Z(G)$ is a subset of $K$. Let $G^{\prime}=G/Z(G)$,
and $K^{\prime}=K/Z(G)$. Using Lemma 2.1.2, we obtain $A^{\prime}$ and
$N^{\prime}$ such that $G^{\prime}=K^{\prime}A^{\prime}N^{\prime}$ is an
Iwasawa decomposition. Let $\rho:G\to G^{\prime}$ be the quotient map. The
group $Z(G)$ is discrete by Fact E.9, so $\rho$ and $\rho|_{K}$ are covering
maps.
Now, the maximum dimension of a compact subgroup of $G$ is the same as that of
$K$ by Lemma 2.1.1. Applying Lemma 2.2 to $K$, we have that $h=\dim K-m$. Note
that $d=\dim(K)+\dim(AN)$ by Fact E.14, so we also get $d-m-h=\dim(AN)$. ∎
The next lemma discusses the noncompact Lie dimensions and the helix
dimensions of a Lie group and its open subgroups.
###### Lemma 2.4.
Suppose $G$ is a Lie group, and $G^{\prime}$ is an open subgroup of $G$. Then
$G$ and $G^{\prime}$ have the same dimension, the same maximum dimension of a
compact subgroup, and the same helix dimension.
###### Proof.
It is clear that $G$ and $G^{\prime}$ have the same dimension. Any compact
subgroup of $G^{\prime}$ is a compact subgroup of $G$. If $K$ is a compact
subgroup of $G$, then $K\cap G^{\prime}$ is an open subgroup of $K$, hence
$K\cap G^{\prime}$ has the same dimension as $K$. Therefore the maximum
dimension of a compact subgroup of $G$ is the same as that of $G^{\prime}$.
Finally, note that $G$ and $G^{\prime}$ have the same identity component
$G_{0}$, and the helix dimension is defined using $G_{0}$. Thus, $G$ and
$G^{\prime}$ have the same helix dimension. ∎
The following Lemma tells us the behavior of radical under quotient by a
compact normal subgroup.
###### Lemma 2.5.
Suppose $G$ is a Lie group, $H$ is a compact normal subgroup of $G$,
$G^{\prime}=G/H$, $\pi:G\to G^{\prime}$ is the quotient map. Let $Q$ be the
radical of $G$, $S=G/Q$. Then we have the following:
1. (1)
with $Q^{\prime}=\pi(Q)$, and $S^{\prime}=G^{\prime}/Q^{\prime}$, we have $HQ$
is closed in $G$, $Q^{\prime}=HQ/H$, and
$S^{\prime}=G^{\prime}/(HQ/H)=(G/H)/(HQ/H))$ is canonically isomorphic as a
topological group to both $G/HQ$ and $(G/Q)/(HQ/Q)=S/(HQ/Q)$;
2. (2)
$Q^{\prime}$ is the radical of $G^{\prime}$;
###### Proof.
We prove (1). As $H$ is compact, we get $HQ$ is closed in $G$ by Lemma A.3.
Then $Q^{\prime}=HQ/H$, and $S^{\prime}=G^{\prime}/(HQ/H)=(G/H)/(HQ/H))$. The
remaining part of (1) is a consequence of the third isomorphism theorem (Fact
A.1.3).
We next prove (2). As $Q^{\prime}$ is a quotient of the solvable group $Q$, it
is solvable. Moreover, $Q^{\prime}$ is a connected closed normal subsgroup of
$G^{\prime}$ as $Q$ is a connected closed normal subgroup of $G$. By (1),
$G^{\prime}/Q^{\prime}$ is a quotient of the semisimple group $S$. Hence,
$G^{\prime}/Q^{\prime}$ is semisimple. Therefore, $Q^{\prime}$ is the maximal
connected solvable closed normal subgroup of $G^{\prime}$. In other words,
$Q^{\prime}$ is the radical of $G^{\prime}$. ∎
The next lemma says in a Lie group, taking quotient by a normal compact group
does not change the helix dimension. Doing so also does not change the
difference between the dimension and the dimension of a maximum compact
subgroup.
###### Lemma 2.6.
Suppose $G$ is a Lie group, $H$ is a compact normal subgroup of $G$, and
$G^{\prime}=G/H$. Let $d$, $m$, and $h$ be the dimension, the maximal
dimension of a compact subgroup, and the helix dimension of $G$, respectively.
Define $d^{\prime}$, $m^{\prime}$, and $h^{\prime}$ likewise for $G^{\prime}$.
Then:
1. (1)
$d=d^{\prime}+\dim(H)$ and $m=m^{\prime}+\dim(H)$;
2. (2)
$h=h^{\prime}$.
###### Proof.
We prove (1). Clearly, $d=d^{\prime}+\dim(H)$. If $K$ is a compact subgroup of
$G$ and $K^{\prime}=\pi(K)$, then $K^{\prime}$ is a compact subgroup of
$G^{\prime}$, then $\dim(K^{\prime})+\dim(H)=\dim(K)$. Conversely, if
$K^{\prime}$ is a compact subgroup of $G^{\prime}$, then
$K=\pi^{-1}(K^{\prime})$ is a compact subgroup of $G$ by Lemma A.4, and Lemma
2.2 that $\dim(K)=\dim(K^{\prime})+\dim(H)$. Therefore,
$m=m^{\prime}+\dim(H)$.
We now prove (2). First further assume that both $G$ and $G^{\prime}$ are
semisimple. Let $\pi:G\to G^{\prime}$ be the quotient map. Using Lemma 2.1.1,
we obtain an Iwasawa decomposition $G=KAN$ of $G$ such that $H\subseteq K$. By
Lemma 2.1.2, we obtain an Iwasawa decomposition
$G^{\prime}=K^{\prime}A^{\prime}N^{\prime}$ with $K^{\prime}=\pi(K)$,
$A^{\prime}=\pi(A)$, and $N^{\prime}=\pi_{N}$. Let $m_{K}$ be the maximum
dimension of a compact subgroup of $K$, and $m^{\prime}_{K}$ be the maximum
dimension of a compact subgroup of $K^{\prime}$. By Proposition 2.3,
$m_{K}+h=\dim(K)$, and $m_{K^{\prime}}+h^{\prime}=\dim(K^{\prime})$. Now, by
(1) applied to $K$, we have $m_{K}={m_{K}^{\prime}}+\dim(H)$. Therefore, we
get $h=h^{\prime}$.
Next, consider the case where $G$ is connected. Let $Q$ be the radical of $G$,
$S=G/Q$, $Q^{\prime}=\pi(Q)$, and $S^{\prime}=G^{\prime}/Q^{\prime}$. Then by
Lemma 2.5.2, $Q^{\prime}$ is the radical of $G^{\prime}$. Hence, it suffices
to show that $S$ and $S^{\prime}$ has the same helix dimension. By Lemma
2.5.1, $S^{\prime}$ is isomorphic as a topological group to $S/(HQ/Q)$. Note
that $HQ/Q$ is isomorphic as a topological group to $H/(H\cap Q)$ by the
second isomorphism theorem for Lie groups (Fact D.3.2). In particular, $HQ/Q$
is compact, and $S^{\prime}$ is the quotient of $S$ by a compact group.
Applying the known case for semisimple and connected groups, we get the
desired conclusion.
Finally, we address the general case. Let $G_{0}$ be the identity component of
$G$. Then $G_{0}$ is open by Fact D.2, and $G_{0}H/H$ is an open subgroup of
$G^{\prime}=G/H$. Hence, by Lemma 2.4, $G$ has the same helix dimension as
$G_{0}$, and $G^{\prime}$ has the same helix dimension as $G_{0}H/H$. By the
second isomorphism theorem (Fact A.1.2), $G_{0}H/H$ is isomorphic as a
topological group to $G_{0}/(G_{0}\cap H)$, which is a quotient of $G_{0}$ by
a compact subgroup. Thus, we get the desired conclusion for the general case
from the known case discussed above for connected groups. ∎
###### Lemma 2.7.
Suppose $G$ is an almost Lie group, $H_{1}$ and $H_{2}$ are closed normal
subgroup of $G$ such that $G/H_{1}$ and $G/H_{2}$ are Lie groups, and
$H=H_{1}\cap H_{2}$. Then $G/H$ is a Lie group.
###### Proof.
By Fact C.1, $G/H$ is an almost-Lie group. In light of Fact C.2.2, we want to
construct an open neighborhood $U$ of the identity in $G/H$ that contains no
nontrivial compact subgroup. Let $\pi:G\to G/H$, $\pi_{1}:G\to G/H_{1}$, and
$\pi_{2}:G\to G/H_{2}$ be the quotient maps. Using Fact A.1.3, we get
continuous surjective group homomorphisms $p_{1}:G/H\to G/H_{1}$ and
$p_{2}:G/H\to G/H_{2}$ such that
$\pi_{1}=p_{1}\circ\pi\quad\text{and}\quad\pi_{2}=p_{2}\circ\pi.$
As $G/H_{1}$ is a Lie group, we can use Fact C.2.2 to choose an open
neighborhood $U_{1}$ of the identity in $G/H_{1}$ such that $U_{1}$ contains
no nontrival compact subgroup of $G/H_{1}$. Choose an open neighborhood
$U_{2}$ of the identity in $G/H_{2}$ likewise, and set
$U=p_{1}^{-1}(U_{1})\cap p_{2}^{-1}(U_{2}).$
If $K\subseteq U$ is a compact subgroup of $G/H$, then $p_{1}(K)$ is a compact
subgroup of $U_{1}$. By our choice of $U_{1}$,
$p_{1}(L)=\\{\mathrm{id}_{G/H_{1}}\\}$, which implies that
$\pi_{1}^{-1}(p(K))=\pi^{-1}(K)$ is a subgroup of $H_{1}$. A similar argument
yields that $\pi_{2}^{-1}(p_{2}(K))=\pi^{-1}(K)$ is a subgroup of $H_{2}$.
Hence, $\pi^{-1}(K)$ must be a subgroup of $H=H_{1}\cap H_{2}$. It follows
that $K=\\{\mathrm{id}_{G/H}\\}$, which is the desired conclusion. ∎
Proposition 2.8 below ensures us the notion of noncompact Lie dimension and
helix dimension of a locally compact group as described in the introduction
are well defined.
###### Proposition 2.8.
Suppose $G^{\prime}$ is an open subgroup of $G$, and $H\vartriangleleft
G^{\prime}$ is compact such that $G^{\prime}/H$ is a Lie group with dimension
$d$, with maximum dimension of a compact subgroup $m$, and helix dimension
$h$. Then $d-m$ and $h$ are independent of the choice of $G^{\prime}$ and $H$.
###### Proof.
We first prove a simpler statement: If $G^{\prime}$ is an almost Lie subgroup
of $G$, $H$ is a compact subgroup of $G^{\prime}$, and we define $d$, $m$, and
$h$ as in the statement of the Proposition, then $d-m$ and $h$ are independent
of the choice of $H$. Let $H_{1}$ and $H_{2}$ be compact and normal subgroups
of $G$ such that both $G/H_{1}$ and $G/H_{2}$ are Lie groups. Then by Lemma
2.7, $G/(H_{1}\cap H_{2})$ is also a Lie group. Note that $G/H_{1}$ and
$G/H_{2}$ are quotients of $G/(H_{1}\cap H_{2})$ by compact subgroups by the
third isomorphism theorem (Fact A.1.3). Hence, it follows from Lemma 2.6 that
$G/H_{1}$ and $G/(H_{1}\cap H_{2})$ have the same difference between the
dimension and the maximum dimension of a compact subgroup, and the same helix
dimension. A similar statement holds for $G/H_{2}$ and $G/(H_{1}\cap H_{2})$.
This completes the proof of the simpler statement.
Now we show the statement of the proposition. Let $G^{\prime}_{1}$ and
$G^{\prime}_{2}$ be open subgroups of $G$, $H_{1}$ and $H_{2}$ are compact
normal subgroup of $G^{\prime}_{1}$ and $G^{\prime}_{2}$ respectively such
that $G^{\prime}_{1}/H_{1}$ and $G^{\prime}_{2}/H_{2}$ are Lie groups. Using
the Gleason–Yamabe Theorem (Fact C.2), we get an open subgroup $G^{\prime}$ of
$G_{1}\cap G_{2}$ which is an almost Lie group. Then $G^{\prime}$ is an open
subgroup of $G$. Note that $G^{\prime}\cap H_{1}$ and $G^{\prime}\cap H_{2}$
are compact subgroups of $G^{\prime}$. Then $G^{\prime}/(G^{\prime}\cap
H_{1})$ is an open subgroup of $G^{\prime}_{1}/H_{1}$. It follows from Lemma
2.6 that $G^{\prime}/H_{1}$ and $G^{\prime}/(G^{\prime}\cap H_{1})$ have the
same difference between the dimension and the maximum dimension and the same
helix dimension. A similar statement hold for $G^{\prime}/H_{2}$ and
$G^{\prime}/(G^{\prime}\cap H_{2})$. Thus, from the simpler statement we
proved in the preceding paragraph, $G_{1}^{\prime}/H_{1}$ and
$G_{2}^{\prime}/H_{2}$ have the same noncompact dimension and and the same
helix dimension. ∎
We have the following two corollaries.
###### Corollary 2.9.
If $H$ is an open subgroup of $G$, then $H$ has the same noncompact Lie
dimension and helix dimension as $G$.
###### Proof.
Proposition 2.8 implies that the noncompact Lie dimension and helix dimension
of a locally compact group is the same as its open almost-Lie subgroups, if
those exist. Hence, it suffices to show that there is a common almost-Lie open
subgroup of $G$ and $H$. This is an immediate consequence of the
Gleason–Yamabe Theorem (Fact C.2.1). ∎
###### Corollary 2.10.
If $H$ is a compact normal subgroup of $G$, then $G/H$ has the same noncompact
Lie dimension and helix dimension as $G$.
###### Proof.
Let $\pi$ be the projection from $G$ to $G/H$. If $G/H$ is a Lie group, then
from the definitions, $G$ has the same noncompact Lie dimension and helix
dimension as $G/H$. Hence, the conclusion holds in this special case.
Suppose there is a compact $K\vartriangleleft G/H$ such that $(G/H)/K$ is a
Lie group, then $(G/H)/K$ is isomorphic as topological group to
$G/\pi^{-1}(K)$ by the third isomorphism theorem (Fact A.1.3). By Lemma A.4,
$\pi^{-1}(K)$ is compact. Hence $(G/H)/K$ is a quotient of $G$ by a compact
normal subgroup, and we can use the previous case to get the desired
conclusion.
Now we treat the general situation. By the Gleason–Yamabe Theorem, we get an
almost-Lie open subgroup $G^{\prime}$ of $G$. Then $G^{\prime}H$ is an open
subgroup of $G$ and hence has the same noncompact Lie dimension and helix
dimension as $G$ by Corollary 2.9. By the second isomorphism theorem (Fact
A.1.2), we get that $G^{\prime}/(G^{\prime}\cap H)$ is isomorphic to
$G^{\prime}H/H$ which is an open subgroup of $G/H$. In particular,
$G^{\prime}/(G^{\prime}\cap H)$ has the same noncompact Lie dimension and
helix dimension as $G/H$ by Corollary 2.9. Note that
$G^{\prime}/(G^{\prime}\cap H)$ is an almost-Lie group by Fact C.1. Hence, we
can find $K$ such that $(G^{\prime}/(G^{\prime}\cap H))/K$ is a Lie group. We
are back to the earlier known situation in the second paragraph. ∎
We have the following lemma about the Iwasawa decompositions.
###### Lemma 2.11.
Suppose $1\to H\to G\overset{\pi\ }{\to}G/H\to 1$ is an exact sequence of
connected semisimple Lie groups. Then there are Iwasawa decompositions
$G=KAN$, $H=K_{1}A_{1}N_{1}$, and $G/H=K_{2}A_{2}N_{2}$ such that
$K_{1}=(K\cap H)_{0}$, and $K_{2}=\pi(K)$.
###### Proof.
Let $\mathfrak{g}$ and $\mathfrak{h}$ be the Lie algebras of $G$ and $H$, and
let $\kappa_{\mathfrak{g}}$ and $\kappa_{\mathfrak{h}}$ be the Cartan–Killing
form of $\mathfrak{g}$ and $\mathfrak{h}$. Then $\mathfrak{g}/\mathfrak{h}$ is
the Lie algebra of $G/H$, and $\mathfrak{g}$ ,$\mathfrak{h}$, and
$\mathfrak{g}/\mathfrak{h}$ are semisimple. By Fact E.7,
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{c}\text{ and
}\kappa_{\mathfrak{g}}=\kappa_{\mathfrak{h}}\oplus\kappa_{\mathfrak{c}}$
where $\kappa_{\mathfrak{c}}$ is the orthogonal complement of
$\kappa_{\mathfrak{h}}$ with respect to $\kappa_{\mathfrak{g}}$, and
$\kappa_{\mathfrak{c}}$ is the Cartan–Killing form of $\kappa_{\mathfrak{c}}$.
Therefore, the quotient map from $\mathfrak{g}$ to $\mathfrak{g}/\mathfrak{h}$
induces an isomorphism from $\mathfrak{c}$ to $\mathfrak{g}/\mathfrak{h}$, so
we can identify $\mathfrak{g}/\mathfrak{h}$ with $\mathfrak{c}$. Let
$\tau_{1}$ and $\tau_{2}$ be a Cartan involutions of $\mathfrak{h}$ and
$\mathfrak{c}$. Then $\tau=\tau_{1}\oplus\tau_{2}$ is an involution of
$\mathfrak{g}$. As $\tau_{1}$ and $\tau_{2}$ are Cartan involutions, the
bilinear forms
$\mathfrak{h}\times\mathfrak{h}:(x_{1},y_{1})\mapsto-\kappa_{\mathfrak{h}}(x_{1},\tau_{1}(y_{1}))$
and
$\mathfrak{c}\times\mathfrak{c}:(x_{2},y_{2})\mapsto-\kappa_{\mathfrak{c}}(x_{2},\tau_{2}(y_{2}))$
are positive definite. Hence, the bilinear from
$\mathfrak{g}\times\mathfrak{g}:(x,y)\mapsto-\kappa_{\mathfrak{g}}(x,\tau(y))$
is also positive definite. Therefore, $\tau$ is a Cartan involution of
$\mathfrak{g}$. Let $\mathfrak{k}$, $\mathfrak{k}_{1}$, and $\mathfrak{k}_{2}$
be the Lie subalgebras of $\mathfrak{g}$, $\mathfrak{h}$, and $\mathfrak{c}$
fixed by $\tau$, $\tau_{1}$, and $\tau_{2}$ respectively. It is easy to see
that $\mathfrak{k}=\mathfrak{k}_{1}\oplus\mathfrak{k}_{2}$. Let
$\mathrm{exp}:\mathfrak{g}\to G$, $\mathrm{exp}_{1}:\mathfrak{h}\to H$, and
$\mathrm{exp}_{2}:\mathfrak{c}\to G/H$ be the exponential maps, and set
$K=\mathrm{exp}(\mathfrak{k}),K_{1}=\mathrm{exp}_{1}(\mathfrak{k}_{1})\text{
and }K_{2}=\mathrm{exp}(\mathfrak{k}_{2}).$
From Fact E.14, we obtain Iwasawa decompositions $G=KAN$, $H=K_{1}A_{1}N_{1}$,
and $G/H=K_{2}A_{2}N_{2}$. By the functoriality of the exponential function
(Fact E.2), we get $K_{1}\leq K\cap H$, and $K_{2}=\pi(K)$. Since $K_{1}$ is
connected, by a dimension calculation we have $K_{1}=(K\cap H)_{0}$. ∎
In a short exact sequence of locally compact groups, one may hope that the
noncompact Lie dimension and the helix dimension of the middle term is the sum
of those of the outer terms. This is not true in general. For instance, in the
exact sequence
$1\to\mathbb{Z}\to\mathbb{R}\to\mathbb{R}/\mathbb{Z}\to 1,$
the noncompact Lie dimension of $\mathbb{R}$ is $1$, while both $\mathbb{Z}$
and $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ has noncompact Lie dimension $0$.
Another exmaple is the following. Let $H$ be the universal cover of
$\mathrm{SL}(2,\mathbb{R})$, and let
$G=(H\times\mathbb{R})/\\{(n,n):n\in\mathbb{Z}\\}$. Then we have the exact
sequence
$1\to H\to G\to\mathbb{T}\to 1,$
the helix dimension of $H$ is $1$, but the helix dimensions of $G$ and
$\mathbb{T}$ are $0$.
Nevertheless, we have the summability of noncompact Lie dimensions and helix
dimensions in many short exact sequences of interest:
###### Proposition 2.12.
Suppose $1\to H\to G\overset{\pi\ }{\to}G/H\to 1$ is an exact sequence of
connected Lie groups. Then we have the following:
1. (1)
If $n$, $n_{1}$, and $n_{2}$ are the noncompact Lie dimensions of $G$, $H$,
and $G/H$ respectively, then $n=n_{1}+n_{2}$;
2. (2)
If $G$ is moreover semisimple, and $h$, $h_{1}$, and $h_{2}$ are the helix
dimensions of $G$, $H$, and $G/H$ respectively, then $h=h_{1}+h_{2}$.
###### Proof.
We first prove (1). Let $m$ be the maximum dimension of a compact subgroup in
$G$. As $G$ is connected, $m$ is also the dimension of an arbitrary maximal
compact subgroup of $G$ by Fact D.4.1. Defining $m_{1}$ and $m_{2}$ likewise
for $H$ and $G/H$, we get similar conclusions for them from the connectedness
of $H$ and $G/H$. Let $K$ be a maximal compact subgroup of $G$. By Fact D.4.2,
$K\cap H$ is a maximal compact subgroup in $H$, and $\pi(K)$ is a maximal
compact subgroup in $G/H$. The kernel of $\pi|_{K}$ is isomorphic to $K\cap
H$, and the image is $\pi(K)$. Hence, $m=m_{1}+m_{2}$. This gives us (1)
recalling that $m+n=\dim(G)$, $m_{1}+n_{1}=\dim(H)$, $m_{2}+n_{2}=\dim(G/H)$,
and $\dim(G)=\dim(H)+\dim(G/H)$.
We now prove (2). Since $Z(G)\cap H\leq Z(H)$, and $\pi(Z(G))\leq Z(G/H)$, we
have $h\leq h_{1}+h_{2}$. It remains to show $h\geq h_{1}+h_{2}$. As $G$ is
semisimple, $H$ and $G/H$ are semisimple by Fact E.8. Take Iwasawa
decompositions $G=KAN$, $H=K_{1}A_{1}N_{1}$, and $G/H=K_{2}A_{2}N_{2}$ as in
Lemma 2.11. By the first isomorphism theorem for Lie groups (Fact D.3.1),
$1\to K\cap H\to K\to K_{2}\to 1$ is an exact sequence of Lie groups. We also
have an exact sequence
(3) $1\to K_{1}\to K\to K_{2}^{\prime}\to 1.$
As $K_{1}=(K\cap H)_{0}$, by the third isomorphism theorem, we have
$K_{2}=K/(K\cap H)=(K/K_{1})/((K\cap H)/K_{1})=K_{2}^{\prime}/((K\cap
H)/K_{1})$. Since $(K\cap H)/K_{1}$ is discrete, $K_{2}^{\prime}$ is a
covering group of $K_{2}$. Let $\phi:K_{2}^{\prime}\to K_{2}$ be the covering
map. Note that $\phi$ has discrete kernel, and $K_{2}$, $K_{2}^{\prime}$ have
the same dimension. Suppose $S$ is a compact subgroup of $K_{2}^{\prime}$ with
the maximum dimension. Then $\phi(S)$ is a compact subgroup of $K_{2}$, and
$S$ and $\phi(S)$ have the same dimension. This shows that the noncompact Lie
dimension of $K_{2}^{\prime}$ is at least the noncompact Lie dimension of
$K_{2}$. By (3) and Statement (1), the noncompact Lie dimension of $K$ is the
sum of noncompact Lie dimensions of $K_{1}$ and $K_{2}^{\prime}$, hence it is
at least the sum of noncompact Lie dimensions of $K_{1}$ and $K_{2}$. It then
follows from Proposition 2.3 that $h\geq h_{1}+h_{2}$. ∎
###### Lemma 2.13.
Suppose $1\to H\to G\overset{\pi\ }{\to}(\mathbb{R}^{>0},\times)\to 1$ is an
exact sequence of Lie groups, and $G$ is connected. Then $H$ is connected.
###### Proof.
Consider first the case when when $G$ and $H$ are Lie groups but are not
necessarily connected. Let $G_{0}$ and $H_{0}$ be the identity components of
$G$ and $H$ respectively. As Lie groups are locally path connected, $G_{0}$ is
open in $G$. Hence, $G_{0}$ and $G$ have the same noncompact Lie dimension by
Corollary 2.9. Likewise, $H_{0}$ has the same noncompact Lie dimension as $H$.
As $G_{0}$ is an open connected subgroup of $G$, the map $\pi|_{G_{0}}$ is
continuous and open. Hence, its image $\pi(G_{0})$ is an open connected
subgroup of $(\mathbb{R}^{>0},\times)$. Therefore,
$\pi(G_{0})=(\mathbb{R}^{>0},\times)$, and $\pi|_{G_{0}}$ is a quotient map by
the first isomorphism theorem (Fact A.1.1). The kernel of $\pi|_{G_{0}}$ is
$H\cap G_{0}$, so we get the exact sequence of Lie groups
$1\to H\cap G_{0}\to
G_{0}\overset{\pi|_{G_{0}}}{\to}(\mathbb{R}^{>0},\times)\to 1.$
We claim that $H_{0}=H\cap G_{0}$, which will bring us back to the known case
where both $G$ and $H$ are connected. The forward inclusion is immediate by
definition. By the third isomorphism theorem (Fact A.1.3), we get the exact
sequence of Lie groups
$1\to(H\cap G_{0})/H_{0}\to G_{0}/H_{0}\to(\mathbb{R}^{>0},\times)\to 1.$
The group $(H\cap G_{0})/H_{0}$ is discrete. Hence, $G_{0}/H_{0}$ is a Lie
group with dimension $1$. As $G_{0}$ is connected, the Lie group $G_{0}/H_{0}$
is also connected. Hence, $G_{0}/H_{0}$ is either isomorphic to $\mathbb{R}$
or $\mathbb{T}$. But since $G_{0}/H_{0}$ has $(\mathbb{R}^{>0},\times)$ as a
quotient, it cannot be compact, and therefore must be isomorphic to
$\mathbb{R}$. This implies that $(H\cap G_{0})/H_{0}$ is trivial, and hence
$H_{0}=H\cap G_{0}$. ∎
The next proposition gives us a summability result of noncompact Lie
dimensions along a short exact sequence of locally compact groups when the
quotient group is $(\mathbb{R}^{>0},\times)$.
###### Proposition 2.14.
Suppose $1\to H\to G\overset{\pi\ }{\to}(\mathbb{R}^{>0},\times)\to 1$ is an
exact sequence of locally compact groups. Then we have the following:
1. (1)
If $n$, $n_{1}$, and $n_{2}$ are the noncompact Lie dimensions of $G$, $H$,
and $(\mathbb{R}^{>0},\times)$ respectively, then $n=n_{1}+n_{2}=n_{1}+1$.
2. (2)
$G$ and $H$ have the same helix dimension.
###### Proof.
First, we consider the case when $G$ is a connected Lie group. Then by Lemma
2.13, $H$ is also connected. Hence, (1) for this case is a consequence of
Proposition 2.12.1.
We prove (2) for this special case. Let $Q$ be the radical of $G$. We claim
that $QH=G$, or equivalently, that $\pi(Q)=(\mathbb{R}^{>0},\times)$. Suppose
this is not true. Then $\pi(Q)$ is a connected subgroup of
$(\mathbb{R}^{>0},\times)$, so it must be $\\{1\\}$. Hence, $Q\subseteq H$.
Then $(\mathbb{R}^{>0},\times)=G/H$ which is isomorphic as a topological group
to $(G/Q)/(H/Q)$ by the third isomorphism theorem (Fact A.1.3). This is a
contradiction, because $(G/Q)/(H/Q)$ is semisimple as a quotient of the
semisimple group $G/Q$, while $(\mathbb{R}^{>0},\times)$ is solvable.
We next show that $Q\cap H$ is the radical of $H$. The radical of $H$ is a
characteristic closed subgroup of $H$ (by Fact E.4), hence a connected
solvable closed normal subgroup of $G$. Thus, the radical of $H$ is a subgroup
of $Q\cap H$. It is straightforward that $Q\cap H$ is solvable. We also have
that $Q\cap H$ is second countable as both $Q$ and $H$ are second countable.
From the preceding paragraph, $\pi(Q)=(\mathbb{R}^{>0},\times)$. Using the
first isomorphism theorem for Lie groups (Fact D.3.1), we have the exact
sequence
$1\to Q\cap H\to Q\to(\mathbb{R}^{>0},\times)\to 1.$
Applying Lemma 2.13, we learn that $Q\cap H$ is connected. This completes the
proof that $Q\cap H$ is the radical of $H$.
Note that $QH=G$ and $Q$ is a closed subgroup of $G$. Hence, by the second
isomorphism theorem for Lie groups (Fact D.3.2), $H/(Q\cap H)$ is isomorphic
as a topological group to $HQ/Q=G/Q$. Therefore $G$ and $H$ have the same
helix dimension.
Next, we address the slightly more general case where $G$ is a Lie group but
not necessarily connected. Let $G_{0}$ be the connected component of $G$. Then
$\pi(G_{0})$ is an open subgroup of $(\mathbb{R}^{>0},\times)$, so
$\pi(G_{0})=(\mathbb{R}^{>0},\times)$. By the first isomorphism theorem for
Lie groups (Fact D.3.1), we have the exact sequence
$1\to G_{0}\cap H\to G_{0}\to(\mathbb{R}^{>0},\times)\to 1.$
Applying Lemma 6.3 and the known case of the current lemma where the middle
term of the exact sequence is a connected Lie group, we obtain both (1) and
(2) for this more general case.
Using the Gleason–Yamabe theorem and a similar argument as in the preceding
paragraph, we can reduce (1) and (2) for general locally compact groups to the
case where we assume that $G$ is an almost-Lie group. Hence, there is a
compact normal subgroup $K$ of $G$ such that $G/K$ is a Lie group. As $K$ is
compact, $\pi(K)$ is a compact subgroup of $(\mathbb{R}^{>0},\times)$, so
$\pi(K)=\\{1\\}$. Hence $K\vartriangleleft H$. By the third isomorphism
theorem (Fact A.1.3), we have the exact sequence $1\to H/K\to
G/K\to(\mathbb{R}^{>0},\times)\to 1$. Applying Lemma 2.6 and the known case of
the current lemma where the middle term of the exact sequence is a Lie group,
we obtain both (1) and (2) for this remaining case. ∎
We discuss the relationship between the noncompact Lie dimension and helix
dimension of a locally compact group $G$.
###### Corollary 2.15.
Suppose $G$ has noncompact dimension $n$ and helix dimension $h$. Then we have
$h\leq n/3$.
###### Proof.
We first check the result for simple Lie groups. If $h=0$, then the statement
holds vacuously. Hence, using Fact E.12, it suffices to consider the case
where $h=1$. Let $G=KAN$ be an Iwasawa decomposition. Then by Proposition 2.2,
we have $n-1=\dim(AN)\geq 0$. Hence, $n>0$ and $G$ is not compact. From Fact
E.15, we have $\dim(AN)\geq 2$. Therefore $n\geq 3$. Hence, we get the desired
conclusion for simple Lie groups.
When $G$ is a connected semisimple Lie group which is not simple. Using
induction on dimension, we can assume we have proven the statement for all
connected semisimple Lie groups of smaller dimensions. Using Fact E.6, we get
an exact sequence of semisimple Lie groups $1\to H\to G\to G/H\to 1$ with
$0<\dim(H)<\dim(G)$. Replacing $H$ with its connected component if necessary,
we can arrange that $H$ is connected. The desired conclusion then follows from
Proposition 2.12.2.
For a general locally compact group $G$, from Proposition 2.8, we may assume
$G$ is a Lie group. Corollary 2.9 and Fact D.2 allow us to reduce the problem
to connected Lie groups. By Lemma 2.5 and Lemma 2.4, the radical of $G$ only
contributes the noncompact Lie dimension of $G$. Using Fact E.5 and
Proposition 2.12.1, we reduce the problem to connected semisimple Lie groups.
∎
## 3\. Proof of Theorem 1.3
The constructions given in this section are open sets (hence all have positive
measure), and the exact statement given in Theorem 1.3 (i.e., in the compact
sets case) follows by the inner regularity of Haar measure.
We first prove the theorem when $G$ is a unimodular Lie group.
###### Proof of Theorem 1.3, unimodular Lie group case.
Since $G$ is unimodular, without loss of generality we assume that $\mu=\nu$.
Let $d$ be the dimension of $G$. Let $K$ be the maximal compact subgroup of
$G$ and let $m=\dim K$. Hence $n=d-m$ is the noncompact Lie dimension of $G$.
If $n=0$, then the identity component $G_{0}$ of $G$ is compact. Taking
$X=G_{0}$, we have
$\mu(X)=\mu(X^{2}).$
Hence Theorem 1.3 holds in this case. In the rest of the proof we assume
$n>0$.
Since $K$ is closed, $G/K$ is a homogeneous (and smooth) manifold. Fix an
arbitrary $G$-invariant (smooth) Riemannian metric on $G/K$ (such a metric
exists by first finding a $K$-invariant Riemannian metric at $[\mathrm{id}]$
and then extend it onto the whole $G/K$ by the action of $G$). This metric
induces a volume measure $\mathrm{Vol}$ on $G/K$.
Let $\pi$ be the projection from $G$ to $G/K$. For any Borel subset $U$ of
$G/K$, $\pi^{-1}(U)$ is also Borel and hence $\mu$-measurable. For any $r>0$,
we use $B_{r}$ to denote the (open) $r$-ball around $[\mathrm{id}]$ on $G/K$
under the chosen metric and use $D_{r}$ to denote $\pi^{-1}(B_{r})$. We claim
that:
1. (i)
There exists a constant $b>0$ only depending on the metric on $G/K$ such that
as Borel measures $\pi_{*}(\mu)=b\cdot\mathrm{Vol}$, and
2. (ii)
For any $r>0$, $D_{r}\\!\cdot\\!D_{r}\subseteq D_{2r}$.
We postpone the proofs of claims (i) and (ii) to the end of this proof and
first show how they lead to Theorem 1.3. We can take $X$ to be $D_{\delta}$
for a sufficiently small $\delta>0$ (depending on $\varepsilon$) to be
determined. Then by (i),
$\mu(X)=\pi_{*}(\mu(B_{\delta}))=b\cdot\mathrm{Vol}(B_{\delta}).$
And by (ii), $X^{2}\subseteq D_{2\delta}$ and hence as before, we get
$\mu(X^{2})\leq\mu(D_{2\delta})=b\cdot\mathrm{Vol}(B_{2\delta})$. Note that
the invariant metric on $G/K$ is smooth and thus
$\lim_{\delta\rightarrow
0}\frac{\mathrm{Vol}(B_{2\delta})}{\mathrm{Vol}(B_{\delta})}=2^{n}.$
Hence a sufficiently small $\delta$ can guarantee
$\frac{\mu(X)^{\frac{1}{n}-\varepsilon}}{\mu(X^{2})^{\frac{1}{n}-\varepsilon}}>\frac{1}{2}$
and we have proved Theorem 1.3 in this special case.
It remains to prove claims (i) and (ii). To see claim (i), note that
$\mathrm{Vol}$ is $G$-invariant. We also see that $\pi_{*}(\mu)$ is
$G$-invariant because $\mu(\pi^{-1}(U))=\mu(g\pi^{-1}(U))=\mu(\pi^{-1}(gU))$
for any $g\in G$ and any Borel subset $U\subseteq G/K$. Since the
$G$-invariant Borel measure on $G/K$ is unique up to a scalar (see Theorem
8.36 in [20]), $\mathrm{Vol}$ has to be a scalar multiple of $\pi_{*}(\mu)$.
Finally we verify claim (ii). Taking arbitrary $g_{1},g_{2}\in D_{r}$ and it
suffices to show $g_{1}g_{2}\in D_{2r}$. By definition, there is a piecewise
smooth curve $\gamma_{j}$ connecting $[\mathrm{id}]$ and $[g_{j}]$ such that
the length of $\gamma_{j}$ is strictly smaller than $r$ (for $j=1,2$). Note
that by the invariance of the metric, $[g_{1}]\gamma_{2}$ must have the same
length as $\gamma_{2}$. Let $\gamma$ be the curve formed by
$[g_{1}]\gamma_{2}$ after $\gamma_{1}$. It is a curve connecting
$[\mathrm{id}]$ and $[g_{1}g_{2}]$ and by the reasoning above has two pieces
and each of them has length strictly smaller than $r$. Hence $\gamma$ has
length shorter than $2r$ and thus by definition $g_{1}g_{2}\in D_{2r}$. We
have successfully verified (ii). ∎
Running the above proof with a little bit of extra effort, we have the
following slightly stronger “stability” result. We will use it in the
generalization to the nonunimodular Lie group case.
###### Proposition 3.1.
Given any unimodular Lie group $G$, let $n$ be its noncompact Lie dimension.
Let $\tilde{\varepsilon}>0$ be fixed. Then there exists precompact open
subsets $X$ and $X_{1}$ with $\mu(X)>0$ such that the closure
$\overline{X}\subseteq X_{1}$ and
$\mu(X_{1}\\!\cdot\\!X)<(2+\tilde{\varepsilon})^{n}\mu(X)$.
###### Proof.
This proof is very similar to the proof of the unimodular Lie case of Theorem
1.3 we just did. We continue to use notations in that proof and take
$X=D_{\delta}=\pi^{-1}(B_{\delta})$ and
$X_{1}=D_{\delta_{1}}=\pi^{-1}(B_{\delta_{1}})$ where $0<\delta<\delta_{1}$
and both $\delta$ and $\delta_{1}$ to be determined.
We see that $X$ and $X_{1}$ are open because $X=\pi^{-1}(B_{\delta})$, etc.
$B_{\delta}$ and $B_{\delta_{1}}$ are precompact, by Lemma A.4, $X$ and
$X_{1}$ are also precompact. Moreover we have
$\overline{X}\subseteq\pi^{-1}(\overline{B_{\delta}})\subseteq\pi^{-1}(B_{\delta_{1}})=X_{1}$.
Now by the same reasoning as in the previous proof of Theorem 1.3 (unimodular
Lie case), we see that $X_{1}\\!\cdot\\!X\subseteq D_{\delta_{1}+\delta}$.
Now,
$\lim_{\delta\rightarrow
0}\frac{\mathrm{Vol}(B_{2\delta})}{\mathrm{Vol}(B_{\delta})}=2^{n},\quad\text{and}\quad\lim_{\delta_{1}\rightarrow\delta}\frac{\mathrm{Vol}(B_{2\delta})}{\mathrm{Vol}(B_{\delta_{1}+\delta})}=1.$
Hence we can take $\delta$ sufficiently small, and then $\delta_{1}$
sufficiently close to $\delta$, such that we have all good properties in the
last paragraph and
$\frac{\mu(X_{1}\\!\cdot\\!X)}{\mu(X)}\leq\frac{\mu(D_{\delta_{1}+\delta})}{\mu(D_{\delta})}=\frac{\mathrm{Vol}(B_{\delta_{1}+\delta})}{\mathrm{Vol}(B_{\delta})}<(2+\tilde{\varepsilon})^{n},$
which proves the proposition. ∎
Next we use Proposition 3.1 to prove Theorem 1.3 for general Lie groups.
###### Proof of Theorem 1.3, Lie group case.
We have already proved the theorem when $G$ is unimodular. In the rest of this
proof, we assume $G$ is nonunimodular. Let $G_{0}$ be the connected component
of $G$. Since $\mu_{G}|_{G_{0}}$ is a left Haar measure on $G_{0}$, and same
holds $\nu_{G}|_{G_{0}}$, we may assume without loss of generality that
$G=G_{0}$. As the only connected subgroups of $(\mathbb{R}^{>0},\times)$ is
itself and $\\{1\\}$, and $G$ is not unimodular, the modular function
$\Delta_{G}$ must be surjective. Hence, $\Delta_{G}$ is a quotient map by the
first isomorphism for Lie groups Fact D.3.1.
Let $H$ be the kernel of the modular function on $G$. By Proposition 2.14.1,
the noncompact Lie dimension of $H$ is $n-1$ where $n$ is the noncompact Lie
dimension of $G$. By Fact B.2.1, $H$ is unimodular. To avoid confusion, we
will always use $\mu_{G}$ and $\nu_{G}$ for $\mu$ and $\nu$ below and use
$\mu_{H}=\nu_{H}$ to denote a fixed Haar measure on $H$.
In light of Fact B.3, we can fix a Haar measure $\,\mathrm{d}r$ on the
multiplicative group $(\mathbb{R}^{>0},\times)=G/H$ such that for any Borel
function $f$ on $g$,
(4)
$\int_{G}f(x)\,\mathrm{d}\mu_{G}(x)=\int_{G/H}\int_{H}f(rh)\,\mathrm{d}\mu_{H}(h)\,\mathrm{d}r.$
Let $\mathfrak{g}$ and $\mathfrak{h}$ be the Lie algebras of $G$ and $H$,
respectively. We fix an element $Z\in\mathfrak{g}$ such that
$Z\notin\mathfrak{h}$. Note that $t\mapsto\Delta(\mathrm{exp}(tZ))$ is a
nontrivial continuous group homomorphism from $(\mathbb{R},+)$ to
$(\mathbb{R}^{>0},\times)$. As the only connected subset of
$(\mathbb{R}^{>0},\times)$ are points and intervals, this map must be
surjective, and hence an isomorphism by the first isomorphism for Lie groups
(Fact D.3). In light of the quotient integral formula (4), we can choose an
appropriate Haar measure $\,\mathrm{d}t$ on $\mathbb{R}$ such that for any
Borel subset $A$ of $G$, we have the Fubini-type measure formula
(5) $\mu_{G}(A)=\int_{\mathbb{R}}\mu_{H}((\mathrm{exp}(-tZ)A)\cap
H)\,\mathrm{d}t.$
Without loss of generality we assume $\,\mathrm{d}t$ is the standard Lebesgue
measure (otherwise we multiply $\mu_{G}$ by a constant).
With the preliminary discussions above, we now construct $X$ satisfying the
inequality in Theorem 1.3.
Before going to details of the construction, we first describe the intuition
behind it. We arrange our $X$ to live very close to $H$ so that $\mu$ and
$\nu$ are almost proportional on $X$ and $X^{2}$. We then realize that it
suffices to choose our $X$ to be like a thickened copy of the almost sharp
example of Theorem 1.3 for (the unimodular group) $H$.
More precisely, let $\tilde{\varepsilon}>0$ be a small number (depending on
$\varepsilon$) to be determined. let $\tilde{X}$ and $\tilde{X}_{1}$ be the
“$X$” and “$X_{1}$”, respectively, in Proposition 3.1 where we replace “$G$”
by “$H$”. We now take
$X=\\{\mathrm{exp}(tZ)h:t\in[0,\tilde{\varepsilon}],h\in\tilde{X}\\}$ and will
show that $X^{2}$ is reasonably small when $\tilde{\varepsilon}$ is small
enough.
By (5), we have
(6) $\mu_{G}(X^{2})=\int_{\mathbb{R}}\mu_{H}((\mathrm{exp}(-tZ)X^{2})\cap
H)\,\mathrm{d}t.$
Note that an arbitrary element in $X^{2}$ can be written as
$\displaystyle\ \mathrm{exp}(t_{1}Z)h_{1}\mathrm{exp}(t_{2}Z)h_{2}$
$\displaystyle=$ $\displaystyle\
\mathrm{exp}((t_{1}+t_{2})Z)(\mathrm{exp}(-t_{2}Z)h_{1}\mathrm{exp}(t_{2}Z)\\!\cdot\\!h_{2})\in\mathrm{exp}((t_{1}+t_{2})Z)H,$
where $t_{1},t_{2}\in[0,\tilde{\varepsilon}]$ and $h_{1},h_{2}\in H$. Hence
(6) is reduced to
(7)
$\mu_{G}(X^{2})=\int_{0}^{2\tilde{\varepsilon}}\mu_{H}((\mathrm{exp}(-tZ)X^{2})\cap
H)\,\mathrm{d}t$
and moreover for any $0\leq t_{0}\leq 2\tilde{\varepsilon}$, we see from the
above discussion that
$(\mathrm{exp}(-t_{0}Z)X^{2})\cap H=\bigcup_{0\leq
t_{1},t_{2}\leq\tilde{\varepsilon},t_{1}+t_{2}=t_{0}}(\mathrm{exp}(-t_{1}Z)\tilde{X}\mathrm{exp}(t_{1}Z))\\!\cdot\\!\tilde{X}.$
By Lemma D.7 and Proposition 3.1, when $\tilde{\varepsilon}$ is sufficiently
small, which we will always assume, we have the above union contained in
$\tilde{X}_{1}\\!\cdot\\!\tilde{X}$. Now by (7),
(8)
$\mu_{G}(X^{2})\leq\int_{0}^{2\tilde{\varepsilon}}\mu_{H}(\tilde{X}_{1}\\!\cdot\\!\tilde{X})\,\mathrm{d}t=2\tilde{\varepsilon}\mu_{H}(\tilde{X}_{1}\\!\cdot\\!\tilde{X}).$
On the other hand, by (5) we have
(9) $\mu_{G}(X)=\tilde{\varepsilon}\mu_{H}(\tilde{X}).$
Combining (8) and (9) and use the measure properties of $\tilde{X}$ and
$\tilde{X}_{1}$ guaranteed by Proposition 3.1, we have
(10)
$\frac{\mu_{G}(X)}{\mu_{G}(X^{2})}\geq\frac{\mu_{H}(\tilde{X})}{2\mu_{H}(\tilde{X}_{1}\\!\cdot\\!\tilde{X})}>\frac{1}{2(2+\tilde{\varepsilon})^{n-1}}.$
Recall that $\Delta(\mathrm{exp}(\cdot Z))$ is an isomorphism from
$(\mathbb{R},+)$ to $(\mathbb{R}^{>0},\times)$. Hence there exists a constant
$C>0$ only depending on $Z$ such that on the support of $X$ we have
$e^{-C\tilde{\varepsilon}}<\Delta<e^{C\tilde{\varepsilon}}$ and on the support
of $X^{2}$ we have
$e^{-2C\tilde{\varepsilon}}<\Delta<e^{2C\tilde{\varepsilon}}$. Thus by Fact
B.2.4, we have
$\frac{\nu_{G}(X)}{\mu_{G}(X)}>e^{-C\tilde{\varepsilon}}$
and
$\frac{\mu_{G}(X^{2})}{\nu_{G}(X^{2})}>e^{-2C\tilde{\varepsilon}}.$
Combining the above inequalities with (10), we have
(11)
$\frac{\nu_{G}(X)}{\nu_{G}(X^{2})}>\frac{e^{-3C\tilde{\varepsilon}}}{2(2+\tilde{\varepsilon})^{n-1}}.$
Hence for the $X$ we constructed,
(12)
$\frac{\nu_{G}(X)^{\frac{1}{n}-\varepsilon}}{\nu_{G}(X^{2})^{\frac{1}{n}-\varepsilon}}+\frac{\mu_{G}(X)^{\frac{1}{n}-\varepsilon}}{\mu_{G}(X^{2})^{\frac{1}{n}-\varepsilon}}>(1+e^{-3C\tilde{\varepsilon}(\frac{1}{n}-\varepsilon)})(2(2+\tilde{\varepsilon})^{n-1})^{-\frac{1}{n}+\varepsilon}.$
It suffices to take $\tilde{\varepsilon}$ small enough such that the right
hand side of (12) is $>1$. ∎
With the Gleason–Yamabe Theorem and the results developed in Section 2, we are
able to pass our Lie group constructions to general locally compact groups.
###### Proof of Theorem 1.3.
By Fact C.2, there is open subgroup $G^{\prime}$ of $G$ which is almost-Lie.
Since $\mu_{G}|_{G^{\prime}}$ is a left Haar measure on $G^{\prime}$, and same
holds $\nu_{G}|_{G^{\prime}}$, we may assume without loss of generality that
$G$ is almost-Lie.
With this assumption, there is a a short exact sequence $0\to H\to
G\xrightarrow{\pi}G/H\to 0$ where $H$ is a compact subgroup, and $G/H$ is a
Lie group. Let $X$ be a subset of $G/H$ such that
(13)
$\frac{\nu_{G/H}(X)^{\frac{1}{n}-\varepsilon}}{\nu_{G/H}(X^{2})^{\frac{1}{n}-\varepsilon}}+\frac{\mu_{G/H}(X)^{\frac{1}{n}-\varepsilon}}{\mu_{G/H}(X^{2})^{\frac{1}{n}-\varepsilon}}>1,$
where $n$ is the noncompact Lie dimension of $G/H$. Thus by the quotient
integral formula, we have
$\displaystyle\mu_{G}(\pi^{-1}(X))$
$\displaystyle=\int_{G/H}\mu_{H}(g^{-1}(\pi^{-1}(X))\cap
H)\,\mathrm{d}\mu_{G/H}(g)$
$\displaystyle=\int_{G/H}\mathbbm{1}_{X}(g)\,\mathrm{d}\mu_{G/H}(g)=\mu_{G/H}(X),$
and similarly $\nu_{G}(\pi^{-1}(X))=\nu_{G/H}(X)$. Observe that
$\pi^{-1}(X^{2})=\pi^{-1}(X)\cdot\pi^{-1}(X)$. Thus the desired conclusion
follows from (13) and Propositions 2.8. ∎
## 4\. Reduction to outer terms of certain short exact sequences
For $n\in\mathbb{Z}^{\geq 0}$ and $(x,y)\in\mathbb{R}^{2}$, we set
$\big{\|}(x,y)\big{\|}_{1/n}=\begin{cases}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|x|}^{1/n}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|y|}^{1/n})^{n}&\text{
if }n\neq 0,\\\
\max\\{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|x|},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|y|}\\}&\text{
if }n=0.\end{cases}$
We say that the group $G$ satisfies the Brunn–Minkowski inequality with
exponent $n$, abbreviated as BM($n$), if for all compact $X,Y\subseteq G$,
$\Bigg{\|}\Big{(}\frac{\nu(X)}{\nu(XY)},\frac{\mu(Y)}{\mu(XY)}\Big{)}\Bigg{\|}_{1/n}\leq
1.$
When $G$ is unimodular and $n\geq 1$, the above is equivalent to having the
inequality $\mu(XY)^{1/n}\geq\mu(X)^{1/n}+\mu(Y)^{1/n}.$ Note that
$\frac{\nu(X)}{\nu(XY)}\leq 1$ and $\frac{\mu(Y)}{\mu(XY)}\leq 1$. Hence,
every locally compact group $G$ satisfies the Brunn–Minkowski inequality with
exponent $n=0$. Moreover, if $n<n^{\prime}$ and $G$ satisfies the
Brunn–Minkowski inequality with exponent $n^{\prime}$, then it satisfies the
Brunn–Minkowski inequality with exponent $n$.
Given a function $f:X\to\mathbb{R}$, for every $t\in\mathbb{R}$, define the
_superlevel set_ of $f$
$L^{+}_{f}(t):=\\{x\in X:f(x)\geq t\\}.$
We will use this notation at various points in the later proofs. We use the
following simple consequence of Fubini concerning the superlevel sets:
###### Fact 4.1.
Let $f:G\to\mathbb{R}$ be a function. For every $r>0$,
$\int_{G}f^{r}(x)\,\mathrm{d}x=\int_{\mathbb{R}^{\geq
0}}rx^{r-1}L_{f}^{+}(x)\,\mathrm{d}x.$
The next proposition is the main result of this section. The current statement
of the proposition is proved by McCrudden as the main result in [25]. We give
a simpler (but essentially the same) proof here for the sake of the
completeness.
###### Proposition 4.2.
Let $G$ be a unimodular group, $n_{1},n_{2}\geq 0$ are integers, $H$ is a
closed normal subgroup of $G$ satisfying $\mathrm{BM}(n_{1})$, and the
quotient group $G/H$ is unimodular satisfying $\mathrm{BM}(n_{2})$. Then $G$
satisfies $\mathrm{BM}(n_{1}+n_{2})$.
###### Proof.
Suppose $\Omega$ is a compact subset of $G$. Let the “fiber length function”
$f_{\Omega}:G/H\to\mathbb{R}^{\geq 0}$ be a measurable function such that for
every $gH\in G/H$, $f_{\Omega}(gH)=\mu_{H}(g^{-1}\Omega\cap H)$. The case when
both $n_{1}=n_{2}=0$ holds trivially.
Now we split the proof into three cases.
Case 1. _When $n_{1}\geq 1$ and $n_{1}+n_{2}\geq 2$._
By the quotient integral formula (Fact B.3), we have
$\displaystyle\mu_{G}^{1/(n_{1}+n_{2})}(\Omega)$
$\displaystyle=\left(\int_{G/H}f_{\Omega}(x)\,\mathrm{d}\mu_{G/H}(x)\right)^{1/(n_{1}+n_{2})}$
(14)
$\displaystyle=\left(\int_{\mathbb{R}^{>0}}n_{1}t^{n_{1}-1}\mu_{G/H}\big{(}L^{+}_{f_{\Omega}}(t^{n_{1}})\big{)}\,\mathrm{d}t\right)^{1/(n_{1}+n_{2})}.$
Set $\alpha=\frac{n_{1}-1}{n_{1}+n_{2}-1}$,
$\beta=\frac{n_{2}}{n_{1}+n_{2}-1}$, $\gamma=n_{1}+n_{2}-1$, and
$F_{\Omega}(t)=t^{\alpha}\mu_{G/H}^{\beta/n_{2}}\left(L_{f_{\Omega}}^{+}(t^{n_{1}})\right),$
for compact set $\Omega$ in $G$ and $t>0$ (Note that $F_{\Omega}$ is well-
defined when $n_{2}=0$). Then (14) can be rewritten as
(15)
$n_{1}^{-1/(\gamma+1)}\mu_{G}^{1/(\gamma+1)}(\Omega)=\left(\int_{\mathbb{R}^{>0}}F_{\Omega}^{\gamma}(t)\,\mathrm{d}t\right)^{1/(\gamma+1)}$
Fix nonempty compact sets $X,Y\subseteq G$. By (15), we need to show that
(16)
$\left(\int_{\mathbb{R}^{>0}}F_{XY}^{\gamma}(t)\,\mathrm{d}t\right)^{1/(\gamma+1)}\geq\left(\int_{\mathbb{R}^{>0}}F_{X}^{\gamma}(t)\,\mathrm{d}t\right)^{1/(\gamma+1)}+\left(\int_{\mathbb{R}^{>0}}F_{Y}^{\gamma}(t)\,\mathrm{d}t\right)^{1/(\gamma+1)}$
We will do so in two steps. First, we will show the following convexity
property
(17) $F_{XY}(t_{1}+t_{2})\geq F_{X}(t_{1})+F_{Y}(t_{2}).$
For every $t_{1},t_{2}\in\mathbb{R}^{>0}$, since $H$ satisfies
$\mathrm{BM}(n_{1})$, by definition we have
$L^{+}_{f_{X}}(t_{1}^{n_{1}})L^{+}_{f_{Y}}(t_{2}^{n_{1}})\subseteq
L^{+}_{f_{XY}}\left(\big{(}t_{1}+t_{2}\big{)}^{n_{1}}\right).$
Also, since $G/H$ satisfies $\mathrm{BM}(n_{2})$, we have
(18)
$\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{X}}(t_{1}^{n_{1}})\right)+\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{Y}}(t_{2}^{n_{1}})\right)\leq\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{XY}}\left(\big{(}t_{1}+t_{2}\big{)}^{n_{1}}\right)\right).$
By Hölder’s inequality and (18), as well as the fact that $n_{1},n_{2}\geq 1$,
we obtain
$\displaystyle\,(t_{1}+t_{2})^{n_{1}-1}\left(\mu_{G/H}^{1/n_{2}}\big{(}L^{+}_{f_{XY}}((t_{1}+t_{2})^{n_{1}})\big{)}\right)^{n_{2}}$
$\displaystyle\geq$
$\displaystyle\,(t_{1}+t_{2})^{n_{1}-1}\left(\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{X}}(t_{1}^{n_{1}})\right)+\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{Y}}(t_{2}^{n_{1}})\right)\right)^{n_{2}}$
$\displaystyle=$
$\displaystyle\,\big{\|}\left(t_{1}^{\alpha},t_{2}^{\alpha}\right)\big{\|}_{1/\alpha}^{\gamma}\left\|\left(\mu_{G/H}^{\beta/n_{2}}\left(L^{+}_{f_{X}}(t_{1}^{n_{1}})\right),\mu_{G/H}^{\beta/n_{2}}\left(L^{+}_{f_{Y}}(t_{2}^{n_{1}})\right)\right)\right\|_{1/\beta}^{\gamma}$
$\displaystyle\geq$
$\displaystyle\,\left(t_{1}^{\alpha}\mu_{G/H}^{\beta/n_{2}}\left(L^{+}_{f_{X}}(t_{1}^{n_{1}})\right)+t_{2}^{\alpha}\mu_{G/H}^{\beta/n_{2}}\left(L^{+}_{f_{Y}}(t_{2}^{n_{1}})\right)\right)^{\gamma}.$
We remark that the above inequalities also make sense when $n_{2}=0$. In that
case $\|(a,b)\|_{1/n_{2}}$ is to be understood as $\max\\{a,b\\}$ for every
$a,b\in\mathbb{R}^{\geq 0}$. The first line of the above inequality is
$F_{XY}^{\gamma}(t_{1}+t_{2})$ and the last line is
$(F_{X}(t_{1})+F_{Y}(t_{2}))^{\gamma}$. So we finished the first step.
We now prove (16). By the above convexity property (17) and Kneser’s
inequality [21] for $\mathbb{R}$ (i.e. the Brunn–Minkowski inequality for
$\mathbb{R}$), we have
(19)
$\mu_{\mathbb{R}}\big{(}L^{+}_{F_{XY}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{1}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{2}})\big{)}\geq\mu_{\mathbb{R}}\big{(}L^{+}_{F_{X}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{1}})\big{)}+\mu_{\mathbb{R}}\big{(}L^{+}_{F_{Y}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{2}})\big{)}.$
Let $M_{X}=\mathrm{ess}\sup_{x}F_{X}(x)$,
$M_{Y}=\mathrm{ess}\sup_{x}F_{Y}(x)$. By Hölder’s inequality and (19), we have
$\displaystyle\int_{\mathbb{R}^{>0}}F^{\gamma}_{XY}(s)\,\mathrm{d}s$
$\displaystyle\geq\int_{0}^{M_{X}+M_{Y}}\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{\gamma-1}\mu_{\mathbb{R}}\big{(}L^{+}_{F_{XY}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\big{)}\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}$
$\displaystyle=(M_{X}+M_{Y})^{\gamma}\int_{0}^{1}\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{\gamma-1}\mu_{\mathbb{R}}\big{(}L^{+}_{F_{XY}}(M_{X}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}+M_{Y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\big{)}\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}$
$\displaystyle\geq(M_{X}+M_{Y})^{\gamma}\int_{0}^{1}\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{\gamma-1}\mu_{\mathbb{R}}\big{(}L^{+}_{F_{X}}(M_{X}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\big{)}\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}$
$\displaystyle\quad+(M_{X}+M_{Y})^{\gamma}\int_{0}^{1}\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{\gamma-1}\mu_{\mathbb{R}}\big{(}L^{+}_{F_{Y}}(M_{Y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\big{)}\,\mathrm{d}s$
(20)
$\displaystyle=(M_{X}+M_{Y})^{\gamma}\left(\frac{1}{M_{X}^{\gamma}}\int_{\mathbb{R}^{>0}}F^{\gamma}_{X}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}+\frac{1}{M_{Y}^{\gamma}}\int_{\mathbb{R}^{>0}}F^{\gamma}_{Y}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}\right).$
Finally, by (14), (20) and Hölder’s inequality,
$\displaystyle\,n_{1}^{-1/(\gamma+1)}\mu_{G}^{1/(\gamma+1)}(XY)$
$\displaystyle=$
$\displaystyle\,\left(\int_{\mathbb{R}^{>0}}F^{\gamma}_{XY}(t)\,\mathrm{d}t\right)^{1/(\gamma+1)}$
$\displaystyle\geq$
$\displaystyle\,\left(\big{(}(M_{X}^{\gamma})^{1/\gamma}+(M_{Y}^{\gamma})^{1/\gamma}\big{)}^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\gamma/(\gamma+1)}}\left(\frac{1}{M_{X}^{\gamma}}\int_{\mathbb{R}^{>0}}F^{\gamma}_{X}(t)\,\mathrm{d}t+\frac{1}{M_{Y}^{\gamma}}\int_{\mathbb{R}^{>0}}F^{\gamma}_{Y}(t)\,\mathrm{d}t\right)\right)^{1/(\gamma+1)}$
$\displaystyle\geq$
$\displaystyle\,\left(\int_{\mathbb{R}^{>0}}F_{X}^{\gamma}(t)\,\mathrm{d}t\right)^{1/(\gamma+1)}+\left(\int_{\mathbb{R}^{>0}}F_{Y}^{\gamma}(t)\,\mathrm{d}t\right)^{1/(\gamma+1)}$
$\displaystyle=$
$\displaystyle\,n_{1}^{-1/(\gamma+1)}\mu_{G}^{1/(\gamma+1)}(X)+n_{1}^{-1/(\gamma+1)}\mu_{G}^{1/(\gamma+1)}(Y),$
this proves the case when $n_{1}$ is at least $1$.
Case 2. _When $n_{1}=1$ and $n_{2}=0$._
In this case, the conclusion can be derived from (14) directly. In particular,
using the fact that $G/H$ satisfies $\mathrm{BM}(0)$ and $H$ satisfies
$\mathrm{BM}(1)$, we have
$\mu_{G/H}\left(L^{+}_{f_{XY}}\left(t_{1}+t_{2}\right)\right)\geq\max\left\\{\mu_{G/H}\left(L^{+}_{f_{X}}(t_{1})\right),\mu_{G/H}\left(L^{+}_{f_{Y}}(t_{2})\right)\right\\}.$
Let $N_{X}=\sup_{t}f_{X}(t)$ and $N_{Y}=\sup_{t}f_{Y}(t)$. Therefore, by
Hölder’s inequality,
$\displaystyle\mu_{G}(XY)$
$\displaystyle=\int_{\mathbb{R}^{>0}}\mu_{G/H}(L^{+}_{f_{XY}}(t))\,\mathrm{d}t$
$\displaystyle=\int_{\mathbb{R}^{>0}}(N_{X}+N_{Y})\mu_{G/H}(L^{+}_{f_{XY}}((N_{X}+N_{Y})t))\,\mathrm{d}t$
$\displaystyle\geq(N_{X}+N_{Y})\max\left\\{\int_{0}^{1}\mu_{G/H}(L^{+}_{f_{X}}(N_{X}t)\,\mathrm{d}t,\int_{0}^{1}\mu_{G/H}(L^{+}_{f_{Y}}(N_{Y}t)\,\mathrm{d}t\right\\}$
$\displaystyle\geq
N_{X}\int_{0}^{1}\mu_{G/H}(L^{+}_{f_{X}}(N_{X}t))\,\mathrm{d}t+N_{Y}\int_{0}^{1}\mu_{G/H}(L^{+}_{f_{Y}}(N_{Y}t))\,\mathrm{d}t$
$\displaystyle=\mu_{G}(X)+\mu_{G}(Y).$
Thus $G$ satisfies $\mathrm{BM}(1)$.
Case 3. _When $n_{1}=0$ and $n_{2}\geq 1$._
Applying Brunn–Minkowski inequality with exponent $0$ on $H$, and the fact
that $G/H$ satisfies $\mathrm{BM}(n_{2})$, we obtain
(21)
$\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{XY}}\left(\max\\{t_{1},t_{2}\\}\right)\right)\geq\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{X}}(t_{1})\right)+\mu_{G/H}^{1/n_{2}}\left(L^{+}_{f_{Y}}(t_{1})\right).$
Given a compact set $\Omega$ in $G$, we define
$E_{\Omega}(t)=\mu_{G/H}^{1/n_{2}}(L^{+}_{f_{\Omega}}(t)),t>0.$
Thus by (21), we have $E_{XY}(\max\\{a_{1},a_{2}\\})\geq
E_{X}(a_{1})+E_{Y}(a_{2})$ for all $a_{1},a_{2}$. This can be seen as a
“convexity property” for $E$, but the maximum operator insider the function
$E$ prevent us from using the same argument as used in Case 1 for $F$. On the
other hand, we observe that
(22)
$\mu_{\mathbb{R}}(L^{+}_{E_{XY}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}_{1}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}_{2}))\geq\max\\{\mu_{\mathbb{R}}(L^{+}_{E_{X}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}_{1})),\mu_{\mathbb{R}}(L^{+}_{E_{Y}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}_{2}))\\}.$
Now we consider $\mu_{G}(XY)$. We have
(23)
$\displaystyle\mu_{G}^{1/n_{2}}(XY)=\left(\int_{\mathbb{R}^{>0}}E_{XY}^{n_{2}}(s)\,\mathrm{d}s\right)^{1/n_{2}}=\left(\int_{\mathbb{R}^{>0}}n_{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{n_{2}-1}\mu_{\mathbb{R}}(L_{E_{XY}}^{+}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}))\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}\right)^{1/n_{2}}$
Let $P_{X}=\mathrm{ess}\sup_{t}E_{X}(t)$ and
$P_{Y}=\mathrm{ess}\sup_{t}E_{Y}(t)$. By (22) and (23) we see
$\displaystyle\,n_{2}^{-1/n_{2}}\mu_{G}^{1/n_{2}}(XY)$ $\displaystyle\geq$
$\displaystyle\,\left(\\!(P_{X}+P_{Y})^{n_{2}}\max\left\\{\int_{0}^{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{n_{2}-1}\mu_{\mathbb{R}}(L^{+}_{E_{X}}(P_{X}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s},\int_{0}^{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{n_{2}-1}\mu_{\mathbb{R}}(L^{+}_{E_{Y}}(P_{Y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s})\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}\right\\}\\!\right)^{1/n_{2}}$
$\displaystyle\geq$
$\displaystyle\,\left(P_{X}^{n_{2}}\int_{0}^{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{n_{2}-1}\mu_{\mathbb{R}}(L^{+}_{E_{X}}(P_{X}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}))\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}\right)^{1/n_{2}}+\left(P_{Y}^{n_{2}}\int_{0}^{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}^{n_{2}-1}\mu_{\mathbb{R}}(L^{+}_{E_{Y}}(P_{Y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}))\,\mathrm{d}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s}\right)^{1/n_{2}}$
$\displaystyle=$
$\displaystyle\,n_{2}^{-1/n_{2}}\mu_{G}^{1/n_{2}}(X)+n_{2}^{-1/n_{2}}\mu_{G}^{1/n_{2}}(Y).$
This proves the case when $n_{1}=0$, and hence finishes the proof of the
proposition. ∎
Using a similar technique as used in the proof of Proposition 4.2, we are able
to reduce the problem to open subgroups.
###### Proposition 4.3.
Let $G$ be a unimodular group, and let $G^{\prime}$ be an open subgroup of
$G$. Suppose $G^{\prime}$ satisfies $\mathrm{BM}(n)$ for some integer $n\geq
0$, then $G$ satisfies $\mathrm{BM}(n)$.
###### Proof.
When $n=0$, the conclusion follows from $\mu(XY)\geq\mu(Y)$. In the remaining
time we assume $n\geq 1$.
Let $\mu_{G}$ be a Haar measure on $G$, and let $\mu_{G^{\prime}}$ be the
restricted Haar measure of $\mu_{G}$ on $G^{\prime}$. By Fact A.1.1, for every
compact set $\Omega$ in $G$ we have
$\mu_{G}(\Omega)=\sum_{g\in G/G^{\prime}}\mu_{G^{\prime}}(g\Omega\cap
G^{\prime}).$
We similarly define $f_{\Omega}:G/G^{\prime}\to\mathbb{R}^{\geq 0}$ such that
$f_{\Omega}(g)=\mu_{G^{\prime}}(g^{-1}\Omega\cap G^{\prime})$.
Fix two compact sets $X,Y$ in $G$. Using the fact that $G^{\prime}$ satisfies
$\mathrm{BM}(n)$, we have
$\left|L^{+}_{f_{XY}}\left((t_{1}+t_{2})^{n}\right)\right|\geq\max\left\\{\left|L^{+}_{f_{X}}(t_{1}^{n})\right|,\left|L^{+}_{f_{Y}}(t_{2}^{n})\right|\right\\}$
because if $f_{X}(g_{1}),\ldots,f_{X}(g_{k})\geq t_{1}^{n}$ and
$f_{Y}(\tilde{g})\geq t_{2}^{n}$ we have
$f_{XY}(g_{1}\tilde{g}),\ldots,f_{XY}(g_{k}\tilde{g})\geq(t_{1}+t_{2})^{n}$ .
Let $N_{X}=\sup_{g}f_{X}(g)$ and $N_{Y}=\sup_{g}f_{Y}(g)$. By the above
inequality we deduce
$\displaystyle\ n^{-1/n}\mu_{G}^{1/n}(XY)$ $\displaystyle=$
$\displaystyle\,\left(\int_{\mathbb{R}^{>0}}t^{n-1}|L^{+}_{f_{XY}}(t^{n})|\,\mathrm{d}t\right)^{1/n}$
$\displaystyle\geq$
$\displaystyle\,\left(\\!(N_{X}+N_{Y})^{n}\max\left\\{\int_{0}^{1}t^{n-1}|L^{+}_{f_{X}}((N_{X}t)^{n}|\,\mathrm{d}t,\int_{0}^{1}t^{n-1}|L^{+}_{f_{Y}}((N_{Y}t)^{n}|\,\mathrm{d}t\right\\}\\!\right)^{1/n}$
$\displaystyle\geq$
$\displaystyle\,\left(N_{X}^{n}\int_{0}^{1}t^{n-1}|L^{+}_{f_{X}}((N_{X}t)^{n})|\,\mathrm{d}t\right)^{1/n}+\left(N_{Y}^{n}\int_{0}^{1}t^{n-1}|L^{+}_{f_{Y}}((N_{Y}t)^{n})|\,\mathrm{d}t\right)^{1/n}$
$\displaystyle=$
$\displaystyle\,n^{-1/n}\mu_{G}^{1/n}(X)+n^{-1/n}\mu_{G}^{1/n}(Y),$
Thus $G$ satisfies $\mathrm{BM}(n)$. ∎
## 5\. Reduction to unimodular subgroups
The main result of this section allows us to obtain a Brunn–Minkowski
inequality for a nonunimodular group from its certain unimodular normal
subgroup. We use $\mu_{\mathbb{R}^{\times}}$ to denote a Haar measure on the
multiplicative group $(\mathbb{R}^{>0},\times)$. The next lemma concerns the
case when the modular function on $X$ and on $Y$ are “sufficiently uniform”.
###### Lemma 5.1.
Suppose the modular function $\Delta_{G}:G\to(\mathbb{R}^{>0},\times)$ is a
quotient map of topological groups. Let $X,Y$ be compact subsets of $G$, and
parameters $a,b,\varepsilon>0$ and $n\geq 0$ an integer, such that for every
$x\in X$, $\Delta_{G}(x)\in[a,a+\varepsilon)$ and for every $y\in Y$,
$\Delta_{G}(y)\in[b,b+\varepsilon)$. Suppose $H=\ker(\Delta_{G})$ satisfying
$\mathrm{BM}(n)$. Then
$\frac{\nu_{G}(X)^{1/(n+1)}}{\nu_{G}(XY)^{1/(n+1)}}+\frac{\mu_{G}(Y)^{1/(n+1)}}{\mu_{G}(XY)^{1/(n+1)}}\leq
1+f(\varepsilon),$
where $f(\varepsilon)$ is an explicit function depending only on $a,b,n$ and
$\varepsilon$, and $f(\varepsilon)\to 0$ as $\varepsilon\to 0$. Moreover, this
convergence is uniform when $n$ is fixed and $a$ and $b$ vary over compact
sets.
###### Proof.
We first consider the case when $n\geq 1$. For every compact subset $\Omega$
of $G$, define two functions
$\ell_{\Omega},r_{\Omega}:(\mathbb{R}^{>0},\times)\to\mathbb{R}^{\geq 0}$ such
that
$\ell_{\Omega}(g)=\mu_{H}(g^{-1}\Omega\cap H),\text{ and
}r_{\Omega}(g)=\mu_{H}(\Omega g^{-1}\cap H).$
Note that given $g_{1},g_{2}$ in $G$, note that $(X\cap
Hg_{1})\\!\cdot\\!(Y\cap g_{2}H)$ lies in
$Hg_{1}g_{2}H=(g_{1}g_{2})(g_{1}g_{2})^{-1}H(g_{1}g_{2})H=H(g_{1}g_{2})H(g_{1}g_{2})^{-1}(g_{1}g_{2})$
since $H$ is normal. Now we fix Haar measures
$\mu_{H},\mu_{\mathbb{R}^{\times}}$ on $H$ and on $(\mathbb{R}^{>0},\times)$,
and these two measures will also uniquely determine a left Haar measure
$\mu_{G}$ on $G$ and a right Haar measure $\nu_{G}$ on $G$ via the quotient
integral formula.
For every compact sets $X_{1},X_{2}$ in $H$, and $g_{1},g_{2}$ in $G$, by the
above equality, $X_{1}g_{1}g_{2}X_{2}\subseteq g_{1}g_{2}H$. By Fact B.4.2 and
the fact that $H$ satisfies $\mathrm{BM}(n)$, we have
$\displaystyle\mu^{1/n}_{H}((g_{1}g_{2})^{-1}X_{1}g_{1}g_{2}X_{2})$
$\displaystyle\geq\mu^{1/n}_{H}((g_{1}g_{2})^{-1}X_{1}g_{1}g_{2})+\mu^{1/n}_{H}(X_{2})$
(24)
$\displaystyle=(\Delta_{G}(g_{1})\Delta_{G}(g_{2}))^{-1/n}\mu^{1/n}_{H}(X_{1})+\mu^{1/n}_{H}(X_{2}).$
In light of this, applying the Brunn–Minkowski inequality on
$(\mathbb{R}^{>0},\times)$, we get
$\displaystyle\mu_{\mathbb{R}^{\times}}\left(L^{+}_{\ell_{XY}}\left(\Big{(}\inf_{x\in
X,y\in
Y}(\Delta_{G}(x)\Delta_{G}(y))^{-1/n}t_{1}+t_{2}\Big{)}^{n}\right)\right)\geq\mu_{\mathbb{R}^{\times}}(L^{+}_{r_{X}}((t_{1})^{n}))+\mu_{\mathbb{R}^{\times}}(L^{+}_{\ell_{Y}}(t_{2}^{n})),$
and similarly for right Haar measure on $H$, we have
$\displaystyle\mu_{\mathbb{R}^{\times}}\left(L^{+}_{r_{XY}}\left(\Big{(}t_{1}+\inf_{x\in
X,y\in
Y}(\Delta_{G}(x)\Delta_{G}(y))^{1/n}t_{2}\Big{)}^{n}\right)\right)\geq\mu_{\mathbb{R}^{\times}}(L^{+}_{r_{X}}((t_{1})^{n}))+\mu_{\mathbb{R}^{\times}}(L^{+}_{\ell_{Y}}(t_{2}^{n})).$
Let $M_{X}=\sup_{x}\mu_{\mathbb{R}^{\times}}(L^{+}_{r_{X}}(x))$ and
$M_{Y}=\sup_{y}\mu_{\mathbb{R}^{\times}}(L^{+}_{\ell_{Y}}(y))$. By a change of
variables and then by the first inequality above, we have
$\displaystyle\mu_{G}(XY)$
$\displaystyle=\int_{\mathbb{R}^{\times}}\mu_{H}(g^{-1}XY\cap
H)\,\mathrm{d}\mu_{\mathbb{R}^{\times}}(g)$
$\displaystyle=\int_{\mathbb{R}^{>0}}nt^{n-1}\mu_{\mathbb{R}^{\times}}(L^{+}_{\ell_{XY}}(t^{n}))\,\mathrm{d}t$
$\displaystyle\geq\Big{\|}\Big{(}\frac{1}{(a+\varepsilon)(b+\varepsilon)}M_{X},M_{Y}\Big{)}\Big{\|}_{1/n}$
$\displaystyle\quad\cdot\int_{0}^{1}nt^{n-1}\mu_{\mathbb{R}^{\times}}\left(L_{\ell_{XY}}^{+}\left(\Big{(}\Big{(}\frac{1}{(a+\varepsilon)(b+\varepsilon)}M_{X}\Big{)}^{1/n}t+M_{Y}^{1/n}t\Big{)}^{n}\right)\right)\,\mathrm{d}t$
$\displaystyle\geq\Big{\|}\Big{(}\frac{1}{(a+\varepsilon)(b+\varepsilon)}M_{X},M_{Y}\Big{)}\Big{\|}_{1/n}\left(\frac{1}{M_{X}}\nu_{G}(X)+\frac{1}{M_{Y}}\mu_{G}(Y)\right).$
Thus by Hölder’s inequality, we get
(25)
$\displaystyle\mu_{G}^{1/(n+1)}(XY)\geq\left(\frac{1}{(a+\varepsilon)(b+\varepsilon)}\nu_{G}(X)\right)^{1/(n+1)}+\mu^{1/(n+1)}_{G}(Y).$
Similarly, for $\nu_{G}(XY)$ we have
$\displaystyle\nu_{G}(XY)$
$\displaystyle=\int_{\mathbb{R}^{\times}}\mu_{H}(XYg^{-1}\cap
H)\,\mathrm{d}\mu_{\mathbb{R}^{\times}}(g)$
$\displaystyle=\int_{\mathbb{R}^{>0}}nt^{n-1}\mu_{\mathbb{R}^{\times}}(L^{+}_{r_{XY}}(t^{n}))\,\mathrm{d}t$
$\displaystyle\geq\Big{\|}\Big{(}M_{X},abM_{Y}\Big{)}\Big{\|}_{1/n}\left(\frac{1}{M_{X}}\nu_{G}(X)+\frac{1}{M_{Y}}\mu_{G}(Y)\right),$
and we obtain
(26)
$\displaystyle\nu_{G}^{1/(n+1)}(XY)\geq\nu_{G}^{1/(n+1)}(X)+\big{(}ab\mu_{G}(Y)\big{)}^{1/(n+1)}.$
Therefore, combining (25) and (26), we conclude
$\displaystyle\,\frac{\nu_{G}^{1/(n+1)}(X)}{\nu_{G}^{1/(n+1)}(XY)}+\frac{\mu_{G}^{1/(n+1)}(Y)}{\mu_{G}^{1/(n+1)}(XY)}$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{1+(Cab)^{1/(n+1)}}+\frac{1}{1+\big{(}\frac{1}{C(a+\varepsilon)(b+\varepsilon)}\big{)}^{1/(n+1)}}$
$\displaystyle\leq$
$\displaystyle\,1+\frac{(C(ab+\varepsilon(a+b+\varepsilon)))^{1/(n+1)}-(Cab)^{1/(n+1)}}{(1+(Cab)^{1/(n+1)})(1+(C(a+\varepsilon)(b+\varepsilon))^{1/(n+1)})}.$
where $C=\mu_{G}(Y)/\nu_{G}(X)$.
Hence
$\displaystyle\frac{\nu_{G}^{1/(n+1)}(X)}{\nu_{G}^{1/(n+1)}(XY)}+\frac{\mu_{G}^{1/(n+1)}(Y)}{\mu_{G}^{1/(n+1)}(XY)}\leq
1+f(\varepsilon)$
where
$f(\varepsilon)=\sup_{r>0}\frac{(r(ab+\varepsilon(a+b+\varepsilon)))^{1/(n+1)}-(rab)^{1/(n+1)}}{(1+(rab)^{1/(n+1)})(1+(r(a+\varepsilon)(b+\varepsilon))^{1/(n+1)})}$
depends only on $a,b,n$ and $\varepsilon$ and we see
$\lim_{\varepsilon\rightarrow 0}f(\varepsilon)=0$ uniformly when $a,b$ taken
values in a compact set by an elementary computation.
The remaining case is when $n=0$. Note that in this case, inequality (5)
becomes
$\displaystyle\mu_{H}((g_{1}g_{2})^{-1}X_{1}g_{1}g_{2}X_{2})\geq\max\\{(\Delta_{G}(g_{1})\Delta_{G}(g_{2}))^{-1}\mu_{H}(X_{1}),\mu_{H}(X_{2})\\}.$
This implies for every $t_{1},t_{2}$,
$\displaystyle\mu_{\mathbb{R}^{\times}}\left(L^{+}_{\ell_{XY}}\max\left\\{\inf_{x\in
X,y\in
Y}(\Delta_{G}(x)\Delta_{G}(y))^{-1}t_{1},t_{2}\right\\}\right)\geq\mu_{\mathbb{R}^{\times}}(L^{+}_{r_{X}}(t_{1}))+\mu_{\mathbb{R}^{\times}}(L^{+}_{\ell_{Y}}(t_{2})).$
For any compact set $\Omega$ in $G$, define two functions
$\Phi_{\Omega},\Psi_{\Omega}:\mathbb{R}\to\mathbb{R}$, that
$\Phi_{\Omega}(t)=\mu_{\mathbb{R}^{\times}}(L^{+}_{\ell_{\Omega}}(t)),\quad\text{and}\quad\Psi_{\Omega}(t)=\mu_{\mathbb{R}^{\times}}(L^{+}_{r_{\Omega}}(t)).$
Thus we have
$\mu_{\mathbb{R}}(L^{+}_{\Phi_{XY}}(t_{1}+t_{2}))\geq\max\left\\{\inf_{x\in
X,y\in
Y}(\Delta_{G}(x)\Delta_{G}(y))^{-1}\mu_{\mathbb{R}}(L^{+}_{\Psi_{X}}(t_{1})),\mu_{\mathbb{R}}(L^{+}_{\Phi_{Y}}(t_{2}))\right\\}.$
Let $N_{X}=\sup_{x}\mu_{\mathbb{R}}(L^{+}_{\Psi_{X}}(x))$ and
$N_{Y}=\sup_{y}\mu_{\mathbb{R}}(L^{+}_{\Phi_{Y}}(y))$. By a change of
variable, for $\mu_{G}(XY)$ we have
$\displaystyle\mu_{G}(XY)$
$\displaystyle=\int_{\mathbb{R}^{>0}}\mu_{\mathbb{R}}(L^{+}_{\Phi_{XY}}(t))\,\mathrm{d}t$
$\displaystyle\geq(N_{X}+N_{Y})\max\left\\{\frac{1}{(a+\varepsilon)(b+\varepsilon)}\frac{\nu_{G}(X)}{N_{X}},\frac{\mu_{G}(Y)}{N_{Y}}\right\\}$
(27)
$\displaystyle\geq\frac{1}{(a+\varepsilon)(b+\varepsilon)}\nu_{G}(X)+\mu_{G}(Y).$
Similarly, for every $t_{1},t_{2}$ we also have
$\displaystyle\mu_{\mathbb{R}^{\times}}\left(L^{+}_{r_{XY}}\max\left\\{t_{1},\inf_{x\in
X,y\in
Y}\Delta_{G}(x)\Delta_{G}(y)t_{2}\right\\}\right)\geq\mu_{\mathbb{R}^{\times}}(L^{+}_{r_{X}}(t_{1}))+\mu_{\mathbb{R}^{\times}}(L^{+}_{\ell_{Y}}(t_{2})),$
which implies
$\mu_{\mathbb{R}}(L^{+}_{\Psi_{XY}}(t_{1}+t_{2}))\geq\max\left\\{\mu_{\mathbb{R}}(L^{+}_{\Psi_{X}}(t_{1})),\inf_{x\in
X,y\in
Y}\Delta_{G}(x)\Delta_{G}(y)\mu_{\mathbb{R}}(L^{+}_{\Phi_{Y}}(t_{2}))\right\\}.$
Therefore, for $\nu_{G}(XY)$ we get
$\nu_{G}(XY)\geq\nu_{G}(X)+ab\mu_{G}(Y).$
Together with (27), similarly as in the case when $n\geq 1$, we get
$\displaystyle\frac{\nu_{G}(X)}{\nu_{G}(XY)}+\frac{\mu_{G}(Y)}{\mu_{G}(XY)}$
$\displaystyle\leq\frac{1}{1+Cab}+\frac{1}{1+\frac{1}{C(a+\varepsilon)(b+\varepsilon)}}$
$\displaystyle\leq 1+\frac{\varepsilon
C(a+b+\varepsilon)}{(1+Cab)(1+C(a+\varepsilon)(b+\varepsilon))},$
where $C=\mu_{G}(Y)/\nu_{G}(X)$. The conclusion follows by taking
$f(\varepsilon)=\sup_{r>0}\frac{\varepsilon
r(a+b+\varepsilon)}{(1+rab)(1+r(a+\varepsilon)(b+\varepsilon))},$
and we can see that $f(\varepsilon)\to 0$ as $\varepsilon\to 0$ uniformly when
$a,b$ taken values in a compact set by elementary computations. ∎
The next proposition is the main result of the section. As we mentioned in the
introduction, the proof uses a discretized “spillover” method. We remark that
one can always make the proof continuous like what we did in Section 4, but we
give a discrete proof here since we believe this reflects our idea in a
clearer way.
###### Proposition 5.2.
Suppose $G$ is a locally compact group with $H=\ker(\Delta_{G})$ satisfying
$\mathrm{BM}(n)$. Suppose the map $\Delta_{G}:G\to(\mathbb{R}^{>0},\times)$ is
a quotient map of topological groups, then $G$ satisfies $\mathrm{BM}(n+1)$.
###### Proof.
Since $X$ and $Y$ are compact, there are $a_{1},a_{2},b_{1}$ and $b_{2}>0$,
such that
$a_{1}=\inf_{x\in X}\Delta_{G}(x),\ a_{2}=\sup_{x\in X}\Delta_{G}(x),\
b_{1}=\inf_{y\in Y}\Delta_{G}(y),\ b_{2}=\sup_{y\in Y}\Delta_{G}(y).$
We fix $\mu_{G}$ and $\nu_{G}$ as in the proof of Lemma 5.1, and let
$\varepsilon>0$ be a sufficient small number (depending on $a_{1},a_{2},b_{1}$
and $b_{2}$). Then by Fact B.3 and familiar properties of integrable functions
on $\mathbb{R}$, there is an $N>0$, such that we can partition $[a_{1},a_{2}]$
and $[b_{1},b_{2}]$ into $N$ subintervals, that is
$[a_{1},a_{2}]=\bigcup_{i=1}^{N}A_{i},\quad[b_{1},b_{2}]=\bigcup_{i=1}^{N}B_{i},$
such that each subinterval has length at most $\varepsilon$, and the
intersection of $X$ with $\bigcup_{g\in A_{i}}Hg$ has $\nu_{G}$-measure
$\nu_{G}(X)/N$, the intersection of $Y$ with $\bigcup_{g\in B_{i}}gH$ has
$\mu_{G}$-measure $\mu_{G}(Y)/N$, for every $1\leq i\leq N$.
Let $X_{i}=X\cap HA_{i}$ and let $Y_{i}=Y\cap B_{i}H$. Then
$\nu_{G}(X)=\sum_{i=1}^{N}\nu_{G}(X_{i})$ and
$\mu_{G}(Y)=\sum_{i=1}^{N}\mu_{G}(Y_{i})$. In particular, we have
$\mu_{G}(XY)\geq\sum_{i=1}^{N}\mu_{G}(X_{i}Y_{i})$ and
$\nu_{G}(XY)\geq\sum_{i=1}^{N}\nu_{G}(X_{i}Y_{i})$. Observe that given $1\leq
i,j\leq N$ and $i\neq j$, $X_{i}Y_{i}$ and $X_{j}Y_{j}$ are disjoint. Indeed,
the modulus of every element in $X_{i}Y_{i}$ lies in $A_{i}B_{i}$ and the
modulus of every element in $X_{j}Y_{j}$ lies in $A_{j}B_{j}$. But
$A_{i}B_{i}$ and $A_{j}B_{j}$ are disjoint subsets of $\mathbb{R}^{>0}$ when
$i\neq j$.
By Lemma 5.1, for every $1\leq i\leq N$, there is a function
$f_{i}(\varepsilon)$, such that $f_{i}(\varepsilon)\rightarrow 0$ when
$\varepsilon\rightarrow 0$ uniformly, and
$\frac{\nu_{G}^{1/(n+1)}(X_{i})}{\nu_{G}^{1/(n+1)}(X_{i}Y_{i})}+\frac{\mu_{G}^{1/(n+1)}(Y_{i})}{\mu_{G}^{1/(n+1)}(X_{i}Y_{i})}\leq
1+f_{i}(\varepsilon).$
Take $\tilde{f}(\varepsilon)=\sup_{i}f_{i}(\varepsilon)$, hence
$\tilde{f}(\varepsilon)\to 0$ as $\varepsilon\to 0$. Therefore, for every
$1\leq t\leq N$,
(28)
$\displaystyle\frac{\nu_{G}^{1/(n+1)}(X)}{\nu_{G}^{1/(n+1)}(XY)}+\frac{\mu_{G}^{1/(n+1)}(Y)}{\mu_{G}^{1/(n+1)}(XY)}\leq\left(\frac{N\nu_{G}(X_{t})}{\sum_{i=1}^{N}\nu_{G}(X_{i}Y_{i})}\right)^{\frac{1}{n+1}}+\left(\frac{N\mu_{G}(Y_{t})}{\sum_{i=1}^{N}\mu_{G}(X_{i}Y_{i})}\right)^{\frac{1}{n+1}}.$
Also by Hölder’s inequality, we observe that for every $t$,
(29)
$\displaystyle\left(\sum_{i=1}^{N}\left(\frac{\nu_{G}(X_{i})}{\nu_{G}(X_{i}Y_{i})}\right)^{\frac{1}{n+2}\cdot\frac{n+2}{n+1}}\right)^{\frac{n+1}{n+2}}\left(\sum_{i=1}^{N}\nu_{G}(X_{i}Y_{i})\right)^{\frac{1}{n+2}}\geq
N\nu_{G}^{\frac{1}{n+2}}(X_{t}).$
Averaging (28) over all $t$ and using inequality (29), we have
$\displaystyle\,\frac{\nu_{G}^{1/(n+1)}(X)}{\nu_{G}^{1/(n+1)}(XY)}+\frac{\mu_{G}^{1/(n+1)}(Y)}{\mu_{G}^{1/(n+1)}(XY)}$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{N}\sum_{i=1}^{N}\left(\frac{\nu_{G}(X_{i})}{\nu_{G}(X_{i}Y_{i})}\right)^{1/(n+1)}+\frac{1}{N}\sum_{i=1}^{N}\left(\frac{\mu_{G}(X_{i})}{\mu_{G}(X_{i}Y_{i})}\right)^{1/(n+1)}\leq
1+\tilde{f}(\varepsilon).$
The desired conclusion follows by taking $\varepsilon\to 0$. ∎
## 6\. Reduction to cocompact and codiscrete subgroups
The main results in this section will help us to reduce the problem to
cocompact subgroups or open normal subgroups. We make use of the following
integral formula, see [19, Proposition 5.26, Consequence 1].
###### Fact 6.1.
Let $G$ be a connected unimodular Lie group. Suppose $S,T$ are closed
subgroups of $G$, such that $G=ST$, and the intersection $S\cap T$ is compact.
Then there is a left Haar measure $\mu_{S}$ on $S$ and a right Haar measure
$\nu_{T}$ on $T$, such that
$\int_{G}f(x)\,\mathrm{d}\mu_{G}(x)=\int_{S\times
T}f(st)\,\mathrm{d}\mu_{S}(s)\,\mathrm{d}\nu_{T}(t),$
for every $f\in C_{c}(G)$.
The next proposition allows us to reduce the problem to closed cocompact
subgroups with the same noncomapct Lie dimension.
###### Proposition 6.2.
Suppose $G$ is connected unimodular Lie group, $H$ is a connected closed
subgroup of $G$ satisfying $\mathrm{BM}(n)$, $K$ is a connected unimodular
subgroup of $G$, such that $G=KH$ and $K\cap H$ is compact. Then $G$ satisfies
$\mathrm{BM}(n)$.
###### Proof.
We assume $n\geq 1$, otherwise the result is trivial. Note that both $G$ and
$K$ are unimodular. In light of this we will not be using $\nu_{G}$,
$\nu_{K}$, etc. and only use $\mu_{G}=\nu_{G}$ and $\mu_{K}=\nu_{K}$ below.
We fix a Haar measure $\mu_{K}$ on $K$, and a Haar measure $\mu_{G}$ on $G$.
These measures will also uniquely determine a left Haar measure $\mu_{H}$ and
a right Haar measure $\nu_{H}$ on $H$ such that we have the integral formula
in Fact 6.1 and another similar formula involving
$\,\mathrm{d}\mu_{H}(h)\,\mathrm{d}\mu_{K}(k)$. For a compact subset $\Omega$
of $G$, we define two functions $r_{\Omega},\ell_{\Omega}:K\to\mathbb{R}^{\geq
0}$, such that
$r_{\Omega}(k):=\nu_{H}(k\Omega\cap H),\quad\ell_{\Omega}(k):=\mu_{H}(\Omega
k\cap H),$
for every $k\in K$. We also define two bivariate functions
$R_{\Omega},L_{\Omega}:K\times K\to\mathbb{R}^{\geq 0}$ that for every
$k_{1},k_{2}$ in $K$,
$R_{\Omega}(k_{1},k_{2}):=\nu_{H}(k_{1}\Omega k_{2}\cap H),\quad
L_{\Omega}(k_{1},k_{2}):=\mu_{H}(k_{1}\Omega k_{2}\cap H).$
Thus Fact 6.1 gives us
$\mu_{G}(\Omega)=\int_{K}\nu_{H}(k^{-1}\Omega\cap
H)\,\mathrm{d}\mu_{K}(k)=\int_{K}\mu_{H}(\Omega k^{-1}\cap
H)\,\mathrm{d}\mu_{K}(k).$
We define two probability measures $\mathrm{p}_{X}$ and $\mathrm{p}_{Y}$ on
$K$ in the following way:
$\,\mathrm{d}\mathrm{p}_{X}=\frac{r_{X}\,\mathrm{d}\mu_{K}}{\mu_{G}(X)},\quad\,\mathrm{d}\mathrm{p}_{Y}=\frac{\ell_{Y}\,\mathrm{d}\mu_{K}}{\mu_{G}(Y)}.$
Now, we choose a left coset $k_{1}H$ of $H$ in $G$ randomly with respect to
the probability measure $\mathrm{p}_{X}$, and choose a right coset $Hk_{2}$ of
$H$ in $G$ randomly with respect to the probability measure $\mathrm{p}_{Y}$.
By the fact that $H$ satisfies $\mathrm{BM}(n)$, we get
$\left(\frac{r_{X}(k_{1})}{R_{XY}(k_{1},k_{2})}\right)^{1/n}+\left(\frac{\ell_{Y}(k_{2})}{L_{XY}(k_{1},k_{2})}\right)^{1/n}\leq
1.$
This implies
(30)
$\mathbb{E}_{\mathrm{p}_{X}(k_{1})}\mathbb{E}_{\mathrm{p}_{Y}(k_{2})}\left[\left(\frac{r_{X}(k_{1})}{R_{XY}(k_{1},k_{2})}\right)^{1/n}+\left(\frac{\ell_{Y}(k_{2})}{L_{XY}(k_{1},k_{2})}\right)^{1/n}\right]\leq
1.$
On the other hand, by Hölder’s inequality, Fact 6.1 and the fact that $G$ is
unimodular,
$\displaystyle\,\mathbb{E}_{\mathrm{p}_{X}(k_{1})}\left(\frac{r_{X}(k_{1})}{R_{XY}(k_{1},k_{2})}\right)^{\frac{1}{n}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{\mu_{G}(X)}\int_{K}r_{X}^{\frac{n+1}{n}}(k_{1})R^{-\frac{1}{n}}_{XY}(k_{1},k_{2})\,\mathrm{d}\mu_{K}(k_{1})$
$\displaystyle\geq$
$\displaystyle\,\frac{1}{\mu_{G}(X)}\left(\int_{K}r_{X}(k_{1})\,\mathrm{d}\mu_{K}(k_{1})\cdot\left(\int_{K}R_{XY}(k_{1},k_{2})\,\mathrm{d}\mu_{K}(k_{1})\right)^{-\frac{1}{n+1}}\right)^{\frac{n+1}{n}}$
$\displaystyle=$
$\displaystyle\,\left(\frac{\mu_{G}(X)}{\mu_{G}(XY)}\right)^{\frac{1}{n}}.$
We have a similar inequality concerning
$\mathbb{E}_{\mathrm{p}_{Y}(k_{2})}\left(\frac{\ell_{Y}(k_{2})}{L_{XY}(k_{1},k_{2})}\right)^{\frac{1}{n}}$.
Combining both inequalities with (30), we get
$\displaystyle\left(\frac{\mu_{G}(X)}{\mu_{G}(XY)}\right)^{\frac{1}{n}}+\left(\frac{\mu_{G}(Y)}{\mu_{G}(XY)}\right)^{\frac{1}{n}}\leq
1,$
and hence $G$ satisfies $\mathrm{BM}(n)$. ∎
Using the proportionated averaging trick in a similar fashion, the next result
allows us to reduce the problem to certain open subgroups.
###### Proposition 6.3.
Let $G$ be a locally compact group, and let $G^{\prime}$ be an open normal
unimodular subgroup of $G$. Suppose $G^{\prime}$ satisfies $\mathrm{BM}(n)$
for some integer $n\geq 1$. Then $G$ satisfies $\mathrm{BM}(n)$.
###### Proof.
Let $\mu_{G^{\prime}}$ be a left (and hence right) Haar measure on
$G^{\prime}$. By Fact B.4.2, there is a left Haar measure $\mu_{G}$ and a
right Haar measure $\nu_{G}$ on $G$, such that for every compact set $\Omega$
in $G$ we have
$\nu_{G}(\Omega)=\sum_{g\in
G/G^{\prime}}\Delta_{G}(g^{-1})\mu_{G^{\prime}}(g^{-1}\Omega\cap
G^{\prime}),\quad\mu_{G}(\Omega)=\sum_{g\in G^{\prime}\backslash
G}\Delta_{G}(g)\mu_{G^{\prime}}(\Omega g^{-1}\cap G^{\prime}).$
Now we fix two compact sets $X,Y$ in $G$. For every $g\in G/G^{\prime}$, let
$X_{g}=g^{-1}X\cap G^{\prime}$, and we similarly define $Y_{h}=Xh^{-1}\cap
G^{\prime}$ for every $h\in G^{\prime}\backslash G$. Since $G^{\prime}$
satisfies $\mathrm{BM}(n)$, we have that
(31)
$\left(\frac{\mu_{G^{\prime}}(X_{g})}{\mu_{G^{\prime}}(X_{g}Y_{h})}\right)^{1/n}+\left(\frac{\mu_{G^{\prime}}(Y_{h})}{\mu_{G^{\prime}}(X_{g}Y_{h})}\right)^{1/n}\leq
1.$
Now we choose $g$ from $G/G^{\prime}$ randomly with probability
$\mathrm{p}_{X}(g)=\frac{\Delta_{G}(g^{-1})\mu_{G^{\prime}}(X_{g})}{\nu_{G}(X)}$.
Therefore by Hölder’s inequality,
$\displaystyle\mathbb{E}_{\mathrm{p}_{X}(g)}\left(\frac{\mu_{G^{\prime}}(X_{g})}{\mu_{G^{\prime}}(X_{g}Y_{h})}\right)^{\frac{1}{n}}$
$\displaystyle=\frac{1}{\nu_{G}(X)}\sum_{g\in
G/G^{\prime}}\frac{\left(\mu_{G^{\prime}}(X_{g})\Delta_{G}(g^{-1})\right)^{\frac{n+1}{n}}}{\left(\mu_{G^{\prime}}(X_{g}Y_{h})\Delta_{G}(g^{-1})\right)^{\frac{1}{n}}}$
$\displaystyle\geq\left(\frac{\nu_{G}(X)}{\nu_{G}(XYh)}\right)^{\frac{1}{n}}=\left(\frac{\nu_{G}(X)}{\nu_{G}(XY)}\right)^{\frac{1}{n}}.$
Similarly, we choose $h$ from $G^{\prime}\backslash G$ randomly with
probability
$\mathrm{p}_{Y}(h)=\frac{\Delta_{G}(h)\mu_{G^{\prime}}(Y_{h})}{\mu_{G}(Y)}$.
Again using Hölder’s inequality, we conclude that
$\mathbb{E}_{\mathrm{p}_{Y}(h)}\left(\frac{\mu_{G^{\prime}}(Y_{h})}{\mu_{G^{\prime}}(X_{g}Y_{h})}\right)^{\frac{1}{n}}\geq\left(\frac{\mu_{G}(Y)}{\mu_{G}(XY)}\right)^{\frac{1}{n}}.$
Hence by (31),
$\displaystyle\,\left(\frac{\nu_{G}(X)}{\nu_{G}(XY)}\right)^{\frac{1}{n}}+\left(\frac{\mu_{G}(Y)}{\mu_{G}(XY)}\right)^{\frac{1}{n}}$
$\displaystyle\leq$
$\displaystyle\,\mathbb{E}_{\mathrm{p}_{X}(g)}\mathbb{E}_{\mathrm{p}_{Y}(h)}\left[\left(\frac{\mu_{G^{\prime}}(X_{g})}{\mu_{G^{\prime}}(X_{g}Y_{h})}\right)^{1/n}+\left(\frac{\mu_{G^{\prime}}(Y_{h})}{\mu_{G^{\prime}}(X_{g}Y_{h})}\right)^{1/n}\right]\leq
1,$
and thus $G$ also satisfies $\mathrm{BM}(n)$. ∎
## 7\. Proof of Theorems 1.1, 1.2, and 1.5
### 7.1. A dichotomy lemma
In this subsection, we prove a dichotomy result for the kernel of a continuous
homomorphism to $(\mathbb{R}^{>0},\times)$.
The following lemma records a fact on open maps between locally compact
groups.
###### Lemma 7.1.
Suppose $G,H$ are locally compact groups, $\phi:G\to H$ is a continuous and
surjective group homomorphism, and there is an open subgroup $G^{\prime}$ of
$G$ such that $\phi|_{G^{\prime}}$ is open. Then $\phi:G\to H$ is a quotient
map of locally compact groups.
###### Proof.
By the first isomorphism theorem (Fact A.1.1), it suffices to check that
$\phi$ is open. Suppose $U$ is an open subset of $G$. Then $U=\bigcup_{a\in
G}U\cap aG^{\prime}$. For each $a\in G$, we have
$\phi(U\cap aG^{\prime})=\phi(a)\phi|_{G^{\prime}}(a^{-1}U\cap G^{\prime}).$
As $\phi|_{G^{\prime}}$ is open, $\phi(U\cap aG^{\prime})$ is open for each
$a\in G$. Hence, $\phi(U)=\bigcup_{a\in G}\phi(U\cap aG^{\prime})$ is open in
$H$, which is the desired conclusion. ∎
In the next lemma we present our main dichotomy result.
###### Lemma 7.2.
If $G$ is a locally compact group, and $\pi:G\to(\mathbb{R}^{>0},\times)$ is a
continuous group homomorphism. Then exactly one of the following holds:
1. (1)
we have the short exact sequence of locally compact groups
$1\to\ker\pi\to G\overset{\pi\ }{\to}(\mathbb{R}^{>0},\times)\to 1;$
2. (2)
$\ker\pi$ is an open subgroup of $G$.
###### Proof.
It is easy to see that (1) and (2) are mutually disjoint, so we need to prove
that we are always either in (1) or (2). Consider first the case when $G$ is a
Lie group. Let $G_{0}$ be the identity component of $G$. Then $G_{0}$ is open
by Fact D.2. Hence $\pi(G_{0})$ is a connected subgroup of
$(\mathbb{R}^{>0},\times)$. As the only connected subsets of
$(\mathbb{R}^{>0},\times)$ are points and intervals, we deduce that
$\pi(G_{0})$ can only be $\\{1\\}$ or $(\mathbb{R}^{>0},\times)$. In the
former case, $\ker\pi$ is open as a union of translations of $G_{0}$. Now
suppose $\pi(G_{0})=(\mathbb{R}^{>0},\times)$. Since $G_{0}$ is a connected
Lie group. Using the first isomorphism theorem for Lie group (Fact D.3.1), we
get $\pi|_{G_{0}}$ is open. Applying Lemma 7.1, we get that $\pi$ is a
quotient map as desired.
We now deal with the general situation where $G$ is locally compact. Using the
Gleason–Yamabe Theorem (Fact C.2.1), we obtain an almost-Lie open subgroup
$G^{\prime}$ of $G$. Since $G^{\prime}$ is open, the natural embedding of
$i:G^{\prime}\to G$ induces a continuous homomorphism
$\pi|_{G^{\prime}}:G^{\prime}\to(\mathbb{R}^{>0},\times)$. Note that there is
a compact normal subgroup $H$ of $G^{\prime}$ such that $G^{\prime}/H$ is a
Lie group. Then $H\leq\ker(\pi|_{G^{\prime}})$ since $\pi|_{G^{\prime}}(H)$ is
a compact subgroup of $(\mathbb{R}^{>0},\times)$. Let $\phi:G^{\prime}\to
G^{\prime}/H$ be the quotient map. Hence the homomorphisms induce a continuous
group homomorphism $\psi$ from $G^{\prime}/H$ to $(\mathbb{R}^{>0},\times)$.
${G^{\prime}}$${G^{\prime}/H}$${G}$${(\mathbb{R}^{>0},\times)}$$\scriptstyle{\phi}$$\scriptstyle{\pi|_{G^{\prime}}}$$\scriptstyle{i}$$\scriptstyle{\psi}$$\scriptstyle{\pi}$
Note that the above diagram commutes. By the proven special case for Lie
groups, we then either have the exact sequence
$1\to\ker\psi\to G^{\prime}/H\to(\mathbb{R}^{>0},\times)\to 1$
or $\ker\psi$ is open in $G^{\prime}/H$. In the former case,
$\pi|_{G^{\prime}}$ is open as a composition of open maps. By Lemma 7.1, we
conclude that $\pi$ is a quotient map in this case. In the latter case,
$\ker(\pi|_{G^{\prime}})$ is open in $G^{\prime}$. Thus, here we have
$\ker\pi$ is open in $G$ because $\ker\pi$ is a union of translations of
$\ker(\pi|_{G^{\prime}})$. ∎
The modular function $\Delta_{G}:G\to(\mathbb{R}^{>0},\times)$ is a continuous
group homomorphism by Fact B.2.2, but generally not a quotient map. It is easy
to construct examples where $G/(\ker\Delta_{G})$ is discrete. The above
proposition claims that these are the only two possibilities, which will be
used in the later proofs.
### 7.2. Proofs of the main theorems
In this subsection, we prove Theorems 1.1 and 1.2. For the reader’s
convenience, Proposition 7.3 gathers together all the induction steps we can
do using the earlier results with the exception of Proposition 6.2, which will
be used in the proof of Theorem 1.1 directly.
###### Proposition 7.3.
Let $G$ be a locally compact group with noncompact Lie dimension $n$ and helix
dimension $h$. Let $\Delta_{G}:G\to(\mathbb{R}^{>0},\times)$ be the modular
function of $G$. Then $G$ satisfies $\mathrm{BM}(n-h)$ if one of the following
assumptions holds:
1. (1)
The locally compact group $\ker\Delta_{G}$ has noncompact Lie dimension
$n^{\prime}$ and helix dimension $h^{\prime}$, and $\ker\Delta_{G}$ satisfies
$\mathrm{BM}(n^{\prime}-h^{\prime})$.
2. (2)
$G$ is unimodular, $G^{\prime}$ is an open subgroup of $G$ such that
$G^{\prime}$ has noncompact Lie dimension $n^{\prime}$ and helix dimension
$h^{\prime}$ and satisfies $\mathrm{BM}(n^{\prime}-h^{\prime})$.
3. (3)
$G$ is unimodular, $H$ is a compact normal subgroup of $G$, the quotient $G/H$
has noncompact Lie dimension $n^{\prime}$ and helix dimension $h^{\prime}$ and
satisfies $\mathrm{BM}(n^{\prime}-h^{\prime})$.
4. (4)
There is an exact sequence of connected semisimple Lie groups
$1\to H\to G\to G/H\to 1$
such that $H$ has non compact Lie dimension $n_{1}$ and helix dimension
$h_{1}$, and satisfies $\mathrm{BM}(n_{1}-h_{1})$, and $G/H$ has noncompact
Lie dimension $n_{2}$ and helix dimension $h_{2}$, and satisfies
$\mathrm{BM}(n_{2}-h_{2})$.
5. (5)
There is an exact sequence of connected unimodular Lie groups
$1\to H\to G\to G/H\to 1$
such that $H$ has noncompact Lie dimension $n_{1}$ and helix dimension $0$,
and satisfies $\mathrm{BM}(n_{1})$, and $G/H$ has noncompact Lie dimension
$n_{2}$ and helix dimension $h_{2}$ with $h_{2}=h$, and satisfies
$\mathrm{BM}(n_{2}-h)$.
###### Proof.
We first prove (1). Note that by Fact B.2.1, $\ker\Delta_{G}$ is unimodular.
By Lemma 7.2, we either have the exact sequence of locally compact groups
$1\to\ker\Delta_{G}\to G\to(\mathbb{R}^{>0},\times)\to 1$
or $\ker\Delta_{G}$ is open in $G$. In the former case, by Proposition 2.14,
we have $n=n^{\prime}+1$ and $h=h^{\prime}$. Hence, in this case $G$ satisfies
$\mathrm{BM}(n-h)$ by Proposition 5.2. In the latter case, by Corollary 2.9,
$n=n^{\prime}$ and $h=h^{\prime}$. Here, we have $G$ satisfies
$\mathrm{BM}(n-h)$ by Proposition 6.3.
Next we prove (2). By Corollary 2.9, we have $n=n^{\prime}$ and
$h=h^{\prime}$. The desired conclusion then follows from Proposition 4.3.
We now prove (3). By Corollary 2.10, we have $n=n^{\prime}$ and
$h=h^{\prime}$. Also by Corollary 2.10, the compact group $H$ has noncompact
Lie dimension and helix dimension $0$. Hence, using Proposition 4.2, we obtain
the conclusion that we want.
We prove (4). By Proposition 2.12.1 and Proposition 2.12.2 respectively, we
have $n=n_{1}+n_{2}$ and $h=h_{1}+h_{2}$. Recall that semisimple groups are
unimodular. Using Proposition 4.2, we learn that $G$ satisfies
$\mathrm{BM}(n-h)$.
Finally, we prove (5). By Proposition 2.12.1, we have $n=n_{1}+n_{2}$. Since
the helix dimension of $H$ is $0$, and the helix dimension of $G/H$ is $h$, by
Proposition 4.2, $G$ satisfies $\mathrm{BM}(n-h)$. ∎
The following corollary says that when $G$ is a Lie group, we can further
reduce the problem to connected unimodular groups.
###### Corollary 7.4.
Let $G$ be a Lie group with noncompact Lie dimension $n$ and helix dimension
$h$. Let $\Delta_{G}:G\to(\mathbb{R}^{>0},\times)$ be the modular function of
$G$. Let $G^{\prime}=(\ker\Delta_{G})_{0}$ be the identity component of
$\ker\Delta_{G}$ with noncompact Lie dimension $n^{\prime}$ and helix
dimension $h^{\prime}$. Then $G^{\prime}$ is connected and unimodular, and if
$G^{\prime}$ satisfies $\mathrm{BM}(n^{\prime}-h^{\prime})$, $G$ satisfies
$\mathrm{BM}(n-h)$.
###### Proof.
Note that $(\ker\Delta_{G})_{0}$ is open in $\ker\Delta_{G}$ by Fact D.2. The
desired conclusion is then a consequence of Proposition 7.3.1 and Proposition
7.3.2. ∎
Now we are able to prove the main inequality (2) for Lie groups. As mentioned
earlier, the main strategy is induction on dimension.
###### Proof of Theorem 1.1.
Consider first the case where $G$ is a solvable Lie group. Using Corollary
7.4, we can also assume that $G$ is connected and unimodular. Recall that $d$
is the topological dimension of $G$. The case when $d=0$ or $1$ is trivial, as
every group satisfies $\mathrm{BM}(0)$, and the one dimensional solvable Lie
group is either $\mathbb{T}$ or $\mathbb{R}$ by Fact D.5.1. If $G$ is abelian,
then it is isomorphic to $\mathbb{T}^{m}\times\mathbb{R}^{d-m}$. We get a
desired conclusion applying Proposition 7.3.5 repeatedly. Otherwise, from the
solvability of $G$ we get the exact sequence
$1\to[G,G]\to G\to G/[G,G]\to 1$
with both $[G,G]$ and $G/[G,G]$ connected, solvable and having smaller
dimensions than $G$. Note that $G/[G,G]$ is abelian, and hence unimodular.
Applying Proposition 7.3.5, and the statement for of the theorem for abelian
Lie groups, we get desired conclusion for this case.
Consider next the case where $G$ is connected and semisimple. We may further
assume that $G$ is a connected simple Lie group, otherwise by Fact E.6, we can
always find a connected group $H\vartriangleleft G$ such that both $H$ and
$G/H$ are connected semisimple Lie groups with lower dimension; by Proposition
7.3.4, the Brunn–Minkowski inequality on $G$ can be obtained from the
Brunn–Minkowski inequalities on $H$ and $G/H$. Now we write $G=KAN$ as in Fact
E.14. We first consider the case when $G$ has a finite center, and then $K$ is
compact. Let $n$ be the noncompact Lie dimension of $G$. Hence, $n$ is the
dimension of the solvable Lie group $Q=AN$. Note that $A$ and $N$ are simply
connected by Fact E.14. Hence their noncompact Lie dimensions are the same as
their dimensions by Fact D.5.2. By Proposition 2.12.1 and Fact E.14, the
noncompact Lie dimension of $Q$ is $n$, and hence $Q$ satisfies
$\mathrm{BM}(n)$ from the solvable Lie case. We obtain the desired conclusion
for $G$ by applying Proposition 6.2.
Suppose the connected simple Lie group $G$ has a center of rank $h\geq 1$.
Apply Proposition 6.2 again, and we obtain an inequality (2) for $G$ with
exponent $\dim(AN)$. By Proposition 2.3, we have $\dim(AN)=n-h$. The desired
conclusion for the connected semisimple Lie groups follows similarly from Fact
E.6 and Proposition 7.3.4.
Finally, we show the statement for an arbitrary Lie group $G$. Using Corollary
7.4 again, we can assume that $G$ is connected and unimodular. Then by Fact
E.5 we obtain an exact sequence
$1\to Q\to G\to S\to 1,$
where $Q$ is a connected unimodular solvable group and $S$ is a connected
semisimple Lie group. We then apply Proposition 7.3.5 and the earlier two
cases to get the desired conclusion. ∎
Finally, we prove the inequality (2) for all locally compact groups.
###### Proof of Theorem 1.2.
By Proposition 7.3.1 we can assume that $G$ is unimodular. By the
Gleason–Yamabe Theorem (Fact C.2.1), $G$ has an almost-Lie open subgroup. Now
using Proposition 7.3.2, we can further assume that $G$ is a unimodular
almost-Lie group. Then we can choose a compact subgroup $K$ of $G$ such that
$G/K$ is a unimodular Lie group. The desired conclusion then follows from
Theorem 1.1 and Proposition 7.3.3. ∎
We briefly discuss Theorem 1.5, which is a consequence of the proof of Theorem
1.2.
###### Proof of Theorem 1.5.
Repeating the arguments in the proofs of Proposition 7.3, Corollary 7.4,
Theorem 1.1, Theorem 1.2, and Fact E.6 while ignoring the helix dimension, it
suffices to show the theorem when $G$ is a simple Lie group.
From the hypothesis, we already have the desired conclusion under the further
assumption that our simple Lie group $G$ is also simply connected. We now
consider the general case. If $G$ has finite center, the result is a special
case of Theorem 1.1. So suppose the center $Z(G)$ of $G$ is infinite. Let
$\widetilde{G}$ be the universal cover of $G$, $Z(\widetilde{G})$ its center,
and $\rho:\widetilde{G}\to G$ the covering map. Then $\ker\rho$ is a subgroup
of $Z(\widetilde{G})$ by Fact E.10. Using Fact E.12, the center
$Z(\widetilde{G})$ have rank at most $1$. By the earlier assumption, the
center $Z(G)$ also has rank at least $1$. Hence, by Fact E.10, both
$Z(\widetilde{G})$ and $Z(G)$ must have rank $1$, and $\ker\rho$ is finite.
Therefore, the desired conclusion for $G$ can be reduced to that of
$\widetilde{G}$ by taking the inverse image under $\rho$, which we already
know from the hypothesis. ∎
## Appendix A Some results about topological groups
This section gathers some facts about topological groups which is needed in
the proof. We begin with the three isomorphism theorems of topological groups.
Note that the third isomorphism theorem is almost the same as the familiar
result for groups, whereas first two isomorphism theorems require extra
assumptions; see [3, Proposition III.2.24], [3, Proposition III.4.1], and [3,
Proposition III.2.22] for details. For this fact, we do not need to assume
that $G$ is locally compact. The quotient $G/H$ is equipped with the quotient
topology (i.e., $X\subseteq G/H$ is open if and only if it inverse image under
the quotient map is open).
###### Fact A.1.
Suppose $H$ is a closed normal subgroup of $G$. Then we have the following:
1. (1)
_(First isomorphism theorem)_ Suppose $\phi:G\to Q$ is a continuous surjective
group homomorphism with $\ker\phi=H$. Then the exact sequence of groups
$1\to H\to G\to Q\to 1$
is an exact sequence of topological groups if and only if $\phi$ is open; the
former condition is equivalent to saying that $Q$ is canonically isomorphic to
$G/H$ as topological groups.
2. (2)
_(Second isomorphism theorem)_ Suppose $S$ is a closed subgroup of $G$ and $H$
is compact. Then $S/(S\cap H)$ is canonically isomorphic to the image of
$SH/H$ as topological groups. This is also equivalent to saying that we have
the exact sequence of topological groups
$1\to H\to SH\to S/(S\cap H)\to 1.$
3. (3)
_(Third isomorphism theorem)_ Suppose $S\leq G$ is closed, and $H\leq S$. Then
$S/H$ is a closed subgroup of $G/H$. If $S\vartriangleleft G$ is normal, then
$S/H$ is a normal subgroup of $G/H$, and we have the exact sequence of
topological groups
$1\to S/H\to G/H\to G/S\to 1;$
this is the same as saying that $(G/H)/(S/H)$ is is canonically isomorphic to
$G/S$ as topological groups.
We also need the following simple property of locally compact groups [9,
Theorem 6.7].
###### Fact A.2.
Closed subgroups and quotient groups of a locally compact groups are locally
compact.
The following lemma holds for all topological group.
###### Lemma A.3.
Suppose $X,Y\subseteq G$, $X$ is compact and $Y$ is closed. Then $XY$ is
closed.
###### Proof.
Let $a$ be in $G\setminus XY$. Then $X^{-1}a$ is compact and $X^{-1}a\cap
Y=\emptyset$. For each point $x\in X^{-1}a$, we choose an open neighborhood of
identity $U_{x}$ such that $xU^{2}_{x}\cap Y=\emptyset$. Then $(xU_{x})_{x\in
X^{-1}a}$ is an open cover of $X^{-1}a$. Using the fact that $X^{-1}a$ is
compact, we get a subcover $(U_{i})_{i=1}^{k}$. Set
$U=\bigcap_{i=1}^{k}U_{i}$. It is easy to check that $X^{-1}aU\cap
Y=\emptyset$. Then $aU\cap XY=\emptyset$, which implies that $XY$ is closed as
$a$ can be chosen arbitrarily. ∎
The next lemma records a simple fact of compact subgroups.
###### Lemma A.4.
If $H$ is a compact subgroup of $G$, then the quotient map $\pi:G\to G/H$ is a
proper map (i.e., the inverse image of compact subsets are compact).
###### Proof.
Let $\Omega$ be a compact subset of $G/H$. In particular $\Omega$ is closed.
Hence, $\pi^{-1}(\Omega)$ is closed, so it suffices to find a compact set
containing $\pi^{-1}(\Omega)$. Since $G$ is locally compact, we can find an
open covering $(U_{i})_{i\in I}$ of $\pi^{-1}(\Omega)$ such that $U_{i}$ has
compact closure $\overline{U_{i}}$ for each $i\in I$. Then $(\pi U_{i})_{i\in
I}$ is an open cover of $\Omega$ as $\pi$ is open. Using the assumption that
$\Omega$ is compact, we get a finite $I^{\prime}\subseteq I$ such that
$(\pi(U_{i}))_{i\in I^{\prime}}$ is an open cover of $\Omega$. Then
$\bigcup_{i\in I^{\prime}}\overline{U_{i}}H$ is a compact set containing
$\pi^{-1}(\Omega)$. ∎
## Appendix B Measures and the modular function
We say that a measure $\mu$ on the collection of Borel subsets of $G$ is a
left Haar measure if it satisfies the following properties:
1. (1)
(left-translation-invariant) $\mu(X)=\mu(aX)$ for all $a\in G$ and all
measurable sets $X\subseteq G$.
2. (2)
(inner and outer regular) $\mu(X)=\sup\mu(K)=\inf\mu(U)$ with $K$ ranging over
compact subsets of $X$ and $U$ ranging over open subsets of $G$ containing
$X$.
3. (3)
(compactly finite) $\mu$ takes finite measure on compact subsets of $G$.
The notion of a right Haar measure is obtained by making the obvious
modifications to the above definition. The following classical result by Haar
makes the above notions enduring features of locally compact group:
###### Fact B.1.
[9, Theorem 2.20] Up to multiplication by a positive constant, there is a
unique left Haar measure and of $G$. A similar statement holds for right Haar
measure.
Given a locally compact group $G$, and $\mu$ is a left Haar measure on $G$.
For every $x\in G$, recall that $\Delta_{G}:x\mapsto\mu_{x}/\mu$ is the
_modular function_ of $G$, where $\mu_{x}$ is a left Haar measure on $G$
defined by $\mu_{x}(A)=\mu(Ax)$, for every measurable set $A$. When the image
of $\Delta_{G}$ is always $1$, we say $G$ is _unimodular_. In general,
$\Delta_{G}(x)$ takes values in $\mathbb{R}^{>0}$. We use
$(\mathbb{R}^{>0},\times)$ to denote the multiplicative group of positive real
number together with the usual Euclidean topology. The next fact records some
basic properties of the modular function; See [9, Section 2.4].
###### Fact B.2.
Let $G$ be a locally compact group. Assuming $\mu$ is a left Haar measure and
$\nu$ is a right Haar measure.
1. (1)
Suppose $H$ is a normal closed subgroup of $G$, then $\Delta_{H}=\Delta_{G}$.
In particular, if $H=\ker\Delta_{G}$, then $H$ is unimodular.
2. (2)
The function $\Delta_{G}:G\to(\mathbb{R}^{>0},\times)$ is a continuous
homomorphism.
3. (3)
For every $x\in G$ and every measurable set $A$, we have
$\mu(Ax)=\Delta_{G}(x)\mu(A)$, and $\nu(xA)=\Delta_{G}^{-1}(x)\nu(A)$.
4. (4)
There is a constant $c$ such that
$\int_{G}f\,\mathrm{d}\mu=c\int_{G}f\Delta_{G}\,\mathrm{d}\nu$ for every $f\in
C_{c}(G)$.
We use the following integral formula [9, Theorem 2.49] in our proofs.
###### Fact B.3 (Quotient integral formula).
Let $G$ be a locally compact group, and let $H$ be a closed normal subgroup of
$G$. Given $\mu_{G}$, $\mu_{H}$ left Haar measures on $G$ and on $H$. Then
there is a unique left Haar measure $\mu_{G/H}$ on $G/H$, such that for every
$f\in C_{c}(G)$,
$\int_{G}f(x)\,\mathrm{d}\mu_{G}(x)=\int_{G/H}\int_{H}f(xh)\,\mathrm{d}\mu_{H}(h)\,\mathrm{d}\mu_{G/H}(x).$
The following fact is a consequence of a result about Haar measure on closed
subgroups and quotients [4, Proposition VII. 2.7.10].
###### Fact B.4.
Suppose $G$ is nonunimodular, and $\Delta_{G}:G\to(\mathbb{R}^{>0},\times)$ is
the modular function of $G$, then we have the following:
1. (1)
If $K\vartriangleleft G$ is a compact normal subgroup of $G$, $\Delta_{G/K}$
is the modular function of $G/K$, and $\pi:G\to G/K$ is the quotient map, then
we have $\Delta_{G}=\Delta_{G/K}\circ\pi$.
2. (2)
If $H\vartriangleleft G$ is a closed unimodular group, and $\mu_{H}$ is a Haar
measure on $H$. Suppose $G/H$ is unimodular, and $X$ is a compact subset of
$H$. Then for every $g\in G$, $\mu_{H}(gXg^{-1})=\Delta_{G}(g)\mu_{H}(X)$.
## Appendix C Almost-Lie groups and the Gleason–Yamabe Theorem
In our proof we need the solution of Hilbert’s 5th problem, which is known as
the Gleason–Yamabe Theorem [11, 31], to reduce the problem into Lie groups.
For convenience, we introduce the following terminology. A locally compact
group $G$ is an almost-Lie group if every open neighborhood $U$ of the
identity in $G$ contains a compact $H\vartriangleleft G$ such that $G/H$ is a
Lie group.
###### Lemma C.1.
Suppose $G$ is an almost-Lie group. Then every open subgroup of $G$ and every
quotient of $G$ by a closed normal subgroup is an almost-Lie group.
###### Proof.
We first show that every open subgroup of $G$ is almost-Lie. Let $S$ be an
open subgroup of $G$, and $U$ is an open neighborhood of identity in $S$. We
need to find a compact subgroup $K$ of $S$ such that $K\subseteq U$ and $S/K$
is a Lie group. Since $U$ is also a neighborhood of identity in $G$, $U$
contains a compact normal subgroup $K$ of $G$ such that $G/K$ is a Lie group.
Note that $K\vartriangleleft S$. As $S$ is open, $S/K$ is open in $G/K$ and
hence a Lie group as desired.
Next, suppose $H$ is a closed normal subgroup of $G$, and $\pi:G\to G/H$ is
the quotient map. If $U$ is an open neighborhood of the identity in $G/H$,
then $\pi^{-1}(U)$ is an open neighborhood of identity in $G$. Hence, we can
get a normal compact subgroup $K$ of $G$ such that $K\subseteq\pi^{-1}(U)$ and
that $G/K$ is a Lie group. Then $\pi(K)$ is a compact subgroup of $U$. With
$S=\pi^{-1}(\pi(K))$, we have $\pi(K)=S/H$. Since $K$ is normal in $G$ we have
$\pi(K)$ is normal in $G/H$ and thus $S$ is normal in $G$. Whence by the third
isomorphism theorem (Fact A.1.3), we conclude that $(G/H)/\pi(K)\cong G/S$. By
the third isomorphism theorem again, we have $G/S\cong(G/K)/(S/K)$, thus $G/S$
is a Lie group. ∎
We use the following strong version of the Gleason–Yamabe Theorem.
###### Fact C.2.
We have the following:
1. (1)
_(Gleason–Yamabe Theorem)_ Suppose $G$ is a locally compact group. Then there
is an open subgroup of $G$ which is an almost-Lie group.
2. (2)
An almost-Lie group $G$ is a Lie group if and only if there is an open
neighborhood $U$ of the identity in $G$ that contains no nontrivial compact
subgroup of $G$.
Fact C.2.2 is not officially part of the Gleason–Yamabe Theorem. However, the
forward direction is an easy fact about the no small subgroup property of Lie
groups, and the and backward direction is a direct consequence of Fact C.2.1.
## Appendix D Some results about Lie groups
In this section we gather some facts and lemmas about Lie groups and Lie
algebras. Throughout the paper, all the Lie groups are finite dimensional
second countable _real_ Lie groups.
###### Fact D.1.
Closed subgroups and quotient groups of Lie groups are Lie groups.
The identity component of a topological group $G$ is the connected component
containing the identity element. The identity component of a topological group
$G$ might not be open even if $G$ is locally compact. For instance, there are
nondiscrete totally disconnected locally compact groups. For these groups, the
identity component only consists of the identity element, and it is not open
because the topology is not discrete. Nevertheless, the following holds for
Lie groups [14, Proposition 9.1.15].
###### Fact D.2.
If $G$ is a Lie group, then the identity component of $G$ is open and is
contained in every open subgroups of $G$.
In Fact A.1, we introduce the three isomorphism theorems of topological
groups. When $G$ is a Lie group, we can weaken the assumption required for the
first two isomorphism theorems; see [2, Proposition 3.11.2, Proposition 3.31].
###### Fact D.3.
Suppose $G$ is a Lie group, and $H$ is a closed normal subgroup of $G$. Then
we have the following:
1. (1)
_(First isomorphism theorem for Lie groups)_ If $Q$ is a Lie group, $\phi:G\to
Q$ is a surjective and continuous group homomorphism, and $G$ has countably
many connected components. Then $Q$ is isomorphic as a topological group to
$G/H$.
2. (2)
_(Second isomorphism theorem for Lie groups)_ Suppose $G$ is a finite
dimensional Lie group, and $S$ is a closed subgroup of $G$, and $SH$ is a
closed subgroup of $G$. Then $S/(S\cap H)$ is canonically isomorphic to the
image of $SH/H$ as Lie groups. This is also equivalent to saying that we have
the exact sequence of Lie groups
$1\to H\to SH\to S/(S\cap H)\to 1.$
We also need the following fact about maximal compact subgroups consisting of
Theorem 14.1.3 (iii) and Theorem 14.3.13 (i) (a) of [14]:
###### Fact D.4.
Suppose $G$ is a Lie group with finitely many connected components. Then we
have the following:
1. (1)
All maximal compact subgroups of $G$ are conjugate.
2. (2)
If $0\to H\to G\overset{\pi\ }{\to}G/H\to 0$ is an exact sequence of connected
Lie groups, and $K$ is a maximal compact subgroup of $G$, then $K\cap H$ is a
maximal compact subgroup of $H$, and $\pi(K)$ is a maximal compact subgroup of
$G/H$.
We also use the following simple classification results for Lie groups.
###### Fact D.5.
Let $G$ be a connected Lie group.
1. (1)
If $G$ has dimension $1$, then it is isomorphic to either $\mathbb{R}$ or
$\mathbb{T}$ as topological groups.
2. (2)
If $G$ is a solvable group with dimension $d$, and the maximal compact
subgroups of $G$ have dimension $m$. Then $G$ is diffemorphic to
$\mathbb{T}^{m}\times\mathbb{R}^{d-m}$. Moreover, if $G$ is compact, then
$G\cong\mathbb{T}^{d}$.
We say that a topological group $G$ is a covering group of a topological group
$G$ with covering homomorphism $\rho$ if $\rho:G\to G^{\prime}$ is a
topological group homomorphism which is also a covering map. The following is
a consequence of [14, Theorem 9.5.4]:
###### Fact D.6.
Suppose that $G$ and $G^{\prime}$ are connected Lie groups and that $G$ is a
covering group of $G^{\prime}$ with covering homomorphism $\rho$. Then
$\ker\rho$ is a closed normal subgroup of the center $Z(G)$ of $G$.
We end this section with a lemma about conjugate actions on compact sets in
Lie groups.
###### Lemma D.7.
For a Lie group $G$ and a closed normal subgroup $H$, if a precompact
$A\subseteq H$ such that the closure of $A$ is in $B$ and $B$ is a relative
open subset in $H$, then the following holds: When $g\in G$ is sufficiently
close to $\mathrm{id}_{G}$, we have $gAg^{-1}\subseteq B$.
###### Proof.
We prove the lemma by contradiction. Assuming there exist sequences
$g_{n}\rightarrow\mathrm{id}$ and $\\{h_{n}\\}\subseteq A$ such that
$g_{n}h_{n}g_{n}^{-1}\notin B$. Since $A$ is precompact we may assume
$h_{n}\rightarrow h\in\overline{A}$. But then $g_{n}h_{n}g_{n}^{-1}\rightarrow
h\in\overline{A}$. This contradicts the fact that each $g_{n}h_{n}g_{n}^{-1}$
is in the closed set $H\setminus B$ that does not meet $\overline{A}$. Hence
the assumption is false and the conclusion holds. ∎
## Appendix E Solvable and Semisimple Lie groups
From [14, Section 9.1], there is a functor $\bm{\mathrm{L}}$ from the category
of Lie groups to the category of Lie algebras that assigns each Lie group $G$
to its Lie algebra $\bm{\mathrm{L}}(G)$ and a Lie group morphism $\phi:G\to H$
to its tangent morphism
$\bm{\mathrm{L}}(\phi):\bm{\mathrm{L}}(G)\to\bm{\mathrm{L}}(H)$ of Lie
algebras. We will adopt a more colloquial language in this paper, invoking
this functor implicitly.
###### Fact E.1.
Suppose $G$ and $H$ are Lie groups, and $\mathfrak{g}$ and $\mathfrak{h}$ are
their Lie algebras. If $H$ is a subgroup of $G$, then $\mathfrak{h}$ is a
subalgebra of $\mathfrak{g}$. If $H$ is a normal subgroup of $G$, then
$\mathfrak{h}$ is an ideal in $\mathfrak{g}$, and $\mathfrak{g}/\mathfrak{h}$
is canonically isomorphic to the Lie algebra of $G/H$.
Suppose $\mathfrak{g}$ is the Lie algebra of $G$. The exponential function
$\mathrm{exp}:\mathfrak{g}\to G$ is defined as in [14, Section 9.2]. We will
use the functoriality of the exponential function [14, Proposition 9.2.10]
###### Fact E.2.
Suppose $G$ and $H$ are Lie groups, $\phi:G\to H$ is a homomorphism of Lie
groups, $\mathfrak{g}$ and $\mathfrak{h}$ are the Lie algebras of $G$ and $H$,
$\alpha:\mathfrak{g}\to\mathfrak{h}$ is the tangent morphism of $\phi$, and
$\mathrm{exp}_{G}:\mathfrak{g}\to G$ and $\mathrm{exp}_{H}:\mathfrak{h}\to H$
are the exponential maps. Then
$\mathrm{exp}_{H}\circ\alpha=\mathrm{exp}_{G}\circ\phi$. In other words, the
following diagram commutes:
${G}$${H}$${\mathfrak{g}}$${\mathfrak{h}}$$\scriptstyle{\phi}$$\scriptstyle{\alpha}$$\scriptstyle{\mathrm{exp}_{G}}$$\scriptstyle{\mathrm{exp}_{H}}$
Suppose $\mathfrak{g}$ is a Lie algebra. The derived Lie algebra
$[\mathfrak{g},\mathfrak{g}]$ of $\mathfrak{g}$ is the subalgebra of
$\mathfrak{g}$ generated by the Lie brackets of the pairs of elements of
$\mathfrak{g}$. We say that $\mathfrak{g}$ is solvable if the derived sequence
$\mathfrak{g}\geq[\mathfrak{g},\mathfrak{g}]\geq[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]\geq\ldots$
eventually arrive at the $0$-algebra. A Lie group is solvable if its Lie
algebra is solvable. The following is a consequence of [14, Proposition
5.4.3]:
###### Fact E.3.
Every subalgebra and quotient algebra of a solvable Lie algebra is solvable.
Hence, every closed subgroup and quotient group of a solvable Lie group is
solvable.
The following is another consequence of [14, Proposition 5.4.3]:
###### Fact E.4.
Suppose $\mathfrak{g}$ is a Lie algebra. Then $\mathfrak{g}$ has a largest
solvable subalgebra $\mathfrak{q}$. If $G$ is a Lie group with Lie algebra
$\mathfrak{g}$ and $\mathrm{exp}:\mathfrak{g}\to G$ is the exponential map,
then $Q=\langle\mathrm{exp}(\mathfrak{q})\rangle$ is the largest closed
connected solvable subgroup of $G$. Hence, $Q$ is a characteristic subgroup of
$G$.
The subalgebra $\mathfrak{q}$ as in Fact E.4 is called the radical of
$\mathfrak{g}$, and the subgroup $Q$ as in Fact E.4 is called the radical of
$G$. A Lie algebra is semisimple if it has trivial radical. A lie group is
semisimple if its Lie algebra is semisimple, or equivalently, if it has
trivial radical. The following results follows from [14, Proposition 5.4.3]:
###### Fact E.5.
Let $G$ be a connected Lie group. Let $Q$ be the radical of $G$. Then $S=G/Q$
is a semisimple Lie group.
A Lie group is simple if its Lie algebra is simple. Note that a simple Lie
group needs not to be simple as a group. We use the following fact for simple
Lie groups.
###### Fact E.6.
A connected Lie group $G$ is a simple Lie group if and only if all its normal
proper subgroups are discrete, and contained in $Z(G)$.
Suppose $\mathfrak{g}$ is a finite dimensional Lie algebra. For
$x\in\mathfrak{g}$, let
$\text{ad}x:\mathfrak{g}\to\mathfrak{g},y\mapsto[x,y]$. Then ad is an
endomorphism of $\mathfrak{g}$. The Cartan–Killing form of
$\kappa_{\mathfrak{g}}:\mathfrak{g}\times\mathfrak{g}\to\mathbb{R}$ is given
by
$\kappa_{\mathfrak{g}}(x,y)=\text{tr}(\text{ad}x\ \text{ad}y).$
The Cartan–Killing form is invariant under an automorphism of $\mathfrak{g}$
as this corresponds to a change of basis. The following fact is from [14,
Lemma 5.5.8]
###### Fact E.7.
Suppose $\mathfrak{g}$ is a Lie algebra, $\kappa_{\mathfrak{g}}$ is the
Cartan–Killing form of $\mathfrak{g}$, and $\mathfrak{h}$ is an ideal of
$\mathfrak{g}$. Then the orthogonal space $\mathfrak{h}^{\perp}$ of
$\mathfrak{h}$ with respect to $\kappa_{\mathfrak{g}}$ is also an ideal. If
$\mathfrak{g}$ is semisimple, then
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{h}^{\perp}$ and
$\kappa_{\mathfrak{g}}=\kappa_{\mathfrak{h}}\oplus\kappa_{\mathfrak{h}^{\perp}}$
where $\kappa_{\mathfrak{h}}$ and $\kappa_{\mathfrak{h}^{\perp}}$ are the
Cartan–Killing form of $\mathfrak{h}$ and $\mathfrak{h}^{\perp}$.
The following fact follows from [14, Lemma 5.5.13]. It is also a consequence
of Fact E.7 and the alternative characterization of semisimple Lie algebras as
those whose Cartan–Killing form is nondegenerate.
###### Fact E.8.
Every ideal and quotient algebra of a semisimple Lie algebra is semisimple.
Hence, every normal subgroup and quotient group of a semisimple Lie group is
semisimple.
The first and second assertions in the following fact are immediate
consequences of Facts E.4, E.3, E.8
###### Fact E.9.
If $G$ is a connected semisimple Lie group, then its center $Z(G)$ is a
finitely generated discrete group, the quotient map $\rho:G\to G/Z(G)$ is a
covering map.
The following fact is a consequence of [14, Proposition 9.5.2 and Theorem
9.5.4].
###### Fact E.10.
If $G$ and $G^{\prime}$ are connected Lie groups, $\rho:G\to G^{\prime}$ is
covering map, $Z(G)$ and $Z(G^{\prime})$ are the centers of $G$ and
$G^{\prime}$. Then we have $\ker\rho\leq Z(G)$ and
$Z(G^{\prime})=Z(G)/\ker\rho$.
The first assertion in the following fact is known as Weyl’s theorem on Lie
groups with semisimple compact Lie algebra [14, Theorem 12.1.17].
###### Fact E.11.
If $G$ is a connected semisimple Lie group with compact Lie algebra, then $G$
is compact and $Z(G)$ is finite.
The following Fact is a consequence of Fact E.11 and the result in [29]. This
can also be proven directly using [14, Proposition 13.1.10 (ii)]; we thank
Jinpeng An for pointing this out to us.
###### Fact E.12.
If $G$ is a simply connected simple Lie group, then the center $Z(G)$ of $G$
has rank at most $1$.
Suppose $\mathfrak{g}$ is a finite dimensional Lie algebra with Cartan–Killing
form $\kappa_{\mathfrak{g}}$. A Lie algebra automorphism $\tau$ of
$\mathfrak{g}$ is a Cartan involution if $\tau^{2}=\mathrm{id}_{\mathfrak{g}}$
and $(x,y)\mapsto-\kappa_{\mathfrak{g}}(x,\tau(y))$ is a positive definite
bilinear form. The following fact is [14, Theorem 13.2.10]
###### Fact E.13.
Let $\mathfrak{g}$ be a semisimple Lie algebra. Then $\mathfrak{g}$ has a
Cartan involution $\tau$.
We refer the reader to [20, Section 6.4] for the full definition of Iwasawa
decomposition; we will need the following fact which is a consequence of [20,
Theorem 6.31, Theorem 6.46] and [14, Corollary 12.2.3].
###### Fact E.14 (Iwasawa decomposition).
Suppose $G$ is a connected semisimple Lie group with Lie algebra
$\mathfrak{g}$, $\tau$ is a Cartan’s involution of $\mathfrak{g}$,
$\mathfrak{k}$ the subalgebra of $\mathfrak{g}$ fixed by $\tau$, and
$\mathrm{exp}:\mathfrak{g}\to G$ is the exponential map. Then there is an
Iwasawa decomposition $G=KAN$ such that the following holds:
1. (1)
the multiplication map
$\Phi:K\times A\times N\rightarrow G:(k,a,n)\mapsto kan$
is a diffeomorphism.
2. (2)
$K=\mathrm{exp}(\mathfrak{k)}$ is a connected closed subgroup of $G$,
$Z(G)\subseteq K$, and $K$ is a maximal compact subgroup of $G$ if $Z(G)$ is
finite.
3. (3)
$A$ is an abelian closed subgroup of $G$, $N$ is a nilpotent closed subgroup
of $G$, and both $A$ and $N$ are simply connected.
4. (4)
$Q=AN$, we have that $Q$ is a solvable closed subgroup of $G$, and
$N\vartriangleleft Q$.
The following fact is a consequence of the definition of Iwasawa decomposition
in [20, Section 6.4].
###### Fact E.15.
If $G$ is a noncompact semisimple Lie group with Iwasawa decomposition
$G=KAN$, then $AN$ has dimension at least $2$.
## Acknowledgements
The authors would like to thank Ehud Hrushovski for introducing the authors to
this problem, József Balogh and Anand Pillay for helpful discussions, Richard
Gardner, Vitali Milman, and Rolf Schneider for several historical remarks, and
Guoxian Song for drawing the figure. Special thanks to Jinpeng An for
suggesting many useful references, and for carefully reading the paper and
pointing out an important error in the first version of the manuscript.
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|
# Implementing Admittance Relaying for Microgrid Protection
Arthur K. Barnes 1* — and Adam Mate 1* — Manuscript submitted: December 14,
2020. 1 The authors are with the Advanced Network Science Initiative at Los
Alamos National Laboratory, Los Alamos, NM 87544 USA. Email:{abarnes,
<EMAIL_ADDRESS>versions of one or more of the figures in this paper are
available online at https://ieeexplore.ieee.org.LANL ANSI LA-UR-20-30128.
###### Abstract
The rapid increase of distributed energy resources has led to the widespread
deployment of microgrids. These flexible and efficient local energy grids are
able to operate in both grid-connected mode and islanded mode; they are
interfaced to the main power system by a fast semiconductor switch and
commonly make use of inverter-interfaced generation. This paper focuses on
inverter interfaced microgrids, which present a challenge for protection as
they do not provide the high short-circuit current necessary for conventional
time-overcurrent protection. The application of admittance relaying for the
protection of inverter-interfaced microgrids is investigated as a potential
solution. The comparison of analytical and simulated results of performed four
experiments prove the suitability of admittance relaying for microgrids
protection.
###### Index Terms:
power system operation, admittance relaying, microgrid, distribution network,
protection.
## I Introduction
In their recent paper, McDermott et al. [1] evaluated issues with microgrid
protection. They highlighted the underlying difficulties, which include the
lack of fault current from inverter-interfaced generation [2], the varying
fault current between grid-connected and islanded modes [2], the potential for
normally-meshed operation [3] and unbalanced operation due to single-phase
loads [3]. A handful of possible solutions (with some severe limitations) have
been proposed for these challenges over the years. Dewadasa et al. [3, 4]
identified admittance protection for load protection; however, there is a
potential for loss of protection selectivity in the case of upstream line-
ground faults [1]. Kar et al. [5] identified differential protection based on
the discrete S-transform for line protection; however, there is a potential
for blinding the protection when fault contribution on either end of the line
is similar [1]. Barnes et al. [6] identified dynamic state estimation for the
protection of radial portions of microgrids; however, there is a potential for
sensitivity to initial conditions, particularly at the delta-connected load
models. Nevertheless, it is worth noting that not all microgrid designs
present these above discussed challenges; for example, microgrids could choose
to omit meshed operation.
This paper investigates the application of admittance relaying for the
protection of inverter-interfaced microgrids. Section II describes the
relevant background of admittance and pilot protection as solutions for line
and load protection, in addition to describing the protection schemes analyzed
and the behavior of inverter-interfaced microgrids under faults. Section III
derives sequence networks for the fault cases considered. Section IV presents
the transient model constructed to validate the impedances calculated from the
sequence networks. Section V compares analytical and simulated results from
the sequence networks and transient model. Finally, Section VI summarizes the
conclusions of this paper in terms of the suitability of admittance relaying
for microgrid protection.
## II Background
### II-A Admittance Protection
For protection, the same quantities are used as that of Dewadasa [3, 4, 7],
where line-ground faults are detected by:
$Z_{lg}^{1}=\frac{V_{ar}}{I_{ar}+kI_{ar}^{0}}$
and line-line faults are detected by:
$Z_{ll}^{1}=\frac{V_{cr}-V_{br}}{I_{br}-I_{cr}}$
where $Z_{lg}^{1}$ and $Z_{ll}^{1}$ are the measured positive-sequence
impedances (measured by the phase and ground distance relaying); $V_{ar}$,
$V_{br}$, and $V_{cr}$ are the measured phase-ground voltages; $I_{ar}$,
$I_{br}$, and $I_{cr}$ are the measured phase currents; and $k$ is related to
the ratio of positive- and negative-sequence line impedance:
$k=1-\frac{Z^{0}}{Z^{1}}$
Both $Z_{lg}^{1}$ and $Z_{ll}^{1}$ measure the positive-sequence impedance
between the relay and the fault, which allows them to accurately measure the
distance to the fault for practical lines of sufficient length with mutual
impedance between conductors.
Admittance protection on its own is suitable for load bus protection. For line
protection, as line lengths are low in microgrids, it may be necessary to use
pilot protection. This means that the likelihood of protection erroneously
determining if a fault is in- or out-of-zone is high [7].
### II-B Pilot Protection
The recommended pilot protection scheme is directional comparison blocking
(abbr. DCB). Typically in this scheme, a line is protected by two directional
distance relays on either end for line-line faults and directional overcurrent
relays for line-ground faults [7, 8, 9]. A schematic for one such relay in
oneline diagram from Fig. 1 is illustrated in Fig. 2. For inverter-interfaced
microgrid protection, as it is not possible to rely on the presence of fault
currents, it is necessary to use ground distance relaying instead of
overcurrent relaying [2].
Figure 1: Oneline diagram for a DCB scheme. Figure 2: Schematic for one relay
in a DCB scheme.
DCB operates with a bidirectional channel along the line by which each relay
can send a blocking signal preventing the other from operating. An important
feature of this scheme is that the channel is only transmitting if a relay
detects a fault and is actively sending the blocking signal. The scheme is
therefore biased towards dependability in that the breakers on either end of
the line can operate if the communication channel is inoperable. If a relay’s
directional element detects that the fault is external, it will send a
blocking signal to the relay on the other end of the line to prevent it from
operating. DCB is therefore suitable for transmission line taps or radial
portions of a microgrid where the relay on the receiving end of the line may
not experience fault current, causing an unblocking scheme could fail to
operate.
### II-C Inverter-Interfaced Microgrids
Generation with inverter front-ends in microgrids is expected to supply
single-phase loads in addition to balanced three-phase loads. Proportional-
resonant (abbr. PR) controllers offer better performance during unbalanced
operation and in the presence of load harmonics compared to proportional-
integral (abbr. PI) controllers in a rotating reference frame (e.g., DQ0) [10,
11]. PR controllers can operate in a stationary reference frame that can
either be the raw “A, B, C” coordinates or the “$\alpha$, $\beta$” coordinates
produced by the Clarke transformation [10]. In case the inverter supplies a
four-wire system, an additional controller for the “$\gamma$” coordinate,
corresponding to zero-sequence quantities, is necessary [12]; therefore it may
be preferable to remain in “A, B, C” coordinates. In this paper, a three-phase
inverter with no neutral connection is selected and a delta-grounded wye
transformer provides a source for zero-sequence current.
A major distinction between inverter-based generation and rotating machinery
is that the thermal time-constants for power electronic semiconductor modules
are very low, consequently inverters are unable to supply fault current for
the amount of time it takes for conventional protection to clear a fault. Two
main options are available for protecting inverters from overheating during
faults: instantaneous saturation and latching current limiters [12]. In an
inverter with an inner current loop and outer voltage or power loop, such as
that described in [11], the current limiters operate on the output of the
voltage loop, which acts as a reference to the current loop. In instantaneous
saturation, the current reference is simply bounded within an allowable range;
this introduces considerable voltage harmonics into the system, which can
complicate protection. Latched current limiters avoid harmonic injection
during faults, but can introduce a current discontinuity when switching the
current controller reference from the voltage controller output to the limited
current signal [12].
## III Equivalent Sequence Networks
The system considered is a two bus microgrid, illustrated in Fig. 3; the
system parameters are presented later in Section IV.
Figure 3: Oneline diagram for the considered two bus microgrid system.
Six cases are considered for deriving sequence networks:
1. 1.
Line-ground fault for an ideal voltage source with an upstream relay.
2. 2.
Line-ground fault for a current-limiting inverter with an upstream relay.
3. 3.
Line-ground fault with a downstream relay.
4. 4.
Line-line fault for an ideal voltage source with an upstream relay.
5. 5.
Line-line fault for a current-limiting inverter with an upstream relay.
6. 6.
Line-line fault with a downstream relay.
It is shown that the impedance calculations for downstream relays are not
affected by choice of an ideal voltage source or a current-limiting inverter.
### III-A Line-Ground Midpoint Faults
1) Ideal Voltage Source and Upstream Relay
To analyze the behavior of the considered microgrid during faults, it is
necessary to create the Thevenin equivalent circuit of the system from the
perspective of the protective relaying. The first step is to draw out the
relevant portions with all conductors depicted, as seen in Fig. 4; note that
the microgrid inverter and the delta winding of the transformer are not
illustrated. As mentioned earlier, the microgrid uses an inverter with a
three-phase H-bridge connected to a delta-wye transformer with a grounded wye.
Rather than using three H-bridges to provide neutral currents (as in [3]),
this system relies on the transformer grounding to do so.
Figure 4: Thevenin equivalent circuit of the microgrid.
Fig. 5 illustrates the equivalent sequence networks of the microgrid for the
case of a midpoint line-ground fault and ideal voltage source. The current
flowing into the fault is identical for each circuit:
$I_{f}^{1}=I_{f}^{2}=I_{f}^{0}=I_{f}/3$. This means that the sequence
equivalent circuits can be linked in series.
(a) Equivalent positive-sequence network (b) Equivalent negative-sequence
network (c) Equivalent zero-sequence network
Figure 5: Sequence networks for a midpoint line-ground fault with an ideal
voltage source.
For the positive-sequence network, the Thevenin equivalent impedance:
$Z_{eq1}=Z_{1M}^{1}||(Z_{M2}^{1}+Z_{L})$
where $||$ denotes the equivalent impedance of parallel circuit elements:
$Z_{x}||Z_{y}=\frac{Z_{x}Z_{y}}{Z_{x}+Z_{y}}$
The Thevenin equivalent voltage:
$V_{eq1}=V_{S}\cdot\frac{Z_{M2}^{1}+Z_{L}}{Z_{1M}^{1}+(Z_{M2}^{1}+Z_{L})}$
For the negative-sequence network, the Thevenin equivalent impedance is the
same as $Z_{eq1}$:
$Z_{eq2}=Z_{eq1}=Z_{1M}^{1}||(Z_{M2}^{1}+Z_{L}).$
Last, for the zero-sequence network, the Thevenin equivalent impedance:
$Z_{eq0}=Z_{1M}^{0}||(Z_{M2}^{0}+Z_{L}+3Z_{Lg}).$
This set of interconnected sequence networks can be simplified, as illustrated
in Fig. 6.
Figure 6: Simplified sequence network for a midpoint line-ground fault with an
ideal voltage source.
Given a protective relay at the load bus, the current flowing though the relay
can be determined by calculating the current flowing through the relay in the
positive-, negative-, and zero-sequence networks.
First, the impedance of the positive-sequence network downstream of the relay:
$Z_{d}=Z_{M2}^{1}+Z_{L}$
Next, the equivalent impedance of the negative-sequence network, zero-sequence
network, and fault:
$Z_{20}=Z_{eq2}+Z_{eq0}+3Z_{f}$
therefore the equivalent impedance downstream of the relay:
$Z_{20d}=Z_{20}||Z_{d}$
The measured positive-sequence current at the relay:
$I_{r}^{1}=\frac{V_{s}^{1}}{Z_{1M}+Z_{20d}}$
To calculate the negative- and zero-sequence currents through the relay, the
first step is to calculate the Norton equivalent of the voltage source:
$I_{sn}=\frac{V_{s}^{1}}{Z_{1M}^{1}}$
The equivalent impedance of the upstream and downstream portions of the
positive-sequence network in parallel:
$Z_{1d}=Z_{1M}^{1}||Z_{d}$
which produces the simplified Norton equivalent circuit, illustrated in Fig.
7.
Figure 7: Norton equivalent circuit for a midpoint line-ground fault with an
ideal voltage source.
Using a current divider and assuming that $Z_{1M}^{1}\ll Z_{L}$, the negative-
and zero-sequence current flowing through the relay:
$I_{r}^{2}=I_{r}^{0}=I_{sn}\frac{Z_{1d}}{Z_{20}+Z_{1d}}$
therefore the measured phase A current at the relay:
$I_{r}^{a}=I_{r}^{1}+I_{r}^{2}+I_{r}^{0}$
and the voltage at the relay:
$V_{r}^{a}=V_{s}^{1}-Z_{1M}I_{r}^{1}+Z_{1M}^{1}I_{r}^{2}+Z_{1M}^{0}I_{r}^{0}$
finally, the positive-sequence impedance can be calculated as:
$Z_{r}^{1}=\frac{V_{r}^{ag}}{I_{r}^{a}}$
The measured positive-sequence impedance at the relay, as fault current
varies, is illustrated in Fig. 8 (the fault resistance is varied from 3.68
$[\Omega$] to 1 [$k\Omega$] and the lower bound is selected for a fault
current twice that of the load current).
Figure 8: Measured positive-sequence impedance at an upstream relay for a
midpoint line-ground fault with an ideal voltage source.
2) Current-Limiting Inverter and Upstream Relay
The current-limiting inverter is modeled as an unbalanced voltage source for
the simplicity of calculation (as opposed to voltage sources on the unfaulted
phases and current sources on the faulted phases) [3]. This results in voltage
sources present on each of the sequence networks. The values of the inverter
negative- and zero-sequence voltages are determined by simulation (described
in Section IV) and are calculated to be roughly 60% of the positive-sequence
voltage. The simplified sequence network is illustrated in Fig. 9.
Figure 9: Simplified sequence network for a midpoint line-ground fault with a
current-limiting inverter.
The current through the relay can be calculated given:
$\displaystyle V_{1}=$ $\displaystyle V_{s}^{1}$ $\displaystyle Z_{1}=$
$\displaystyle Z_{1M}^{1}$ $\displaystyle V_{2}=$ $\displaystyle
V_{eq2}+V_{eq0}$ $\displaystyle Z_{2}=$ $\displaystyle Z_{eq2}+Z_{eq0}+3Z_{f}$
$\displaystyle Z_{d}=$ $\displaystyle Z_{M2}^{1}+Z_{L}$
Using parallel branch combinations:
$\displaystyle Z_{1d}$ $\displaystyle=Z_{1}||Z_{d}$ $\displaystyle Z_{2d}$
$\displaystyle=Z_{2}||Z_{d}$
and Norton equivalent sources:
$\displaystyle I_{1n}$ $\displaystyle=\frac{V_{1}}{Z_{1}}$ $\displaystyle
I_{2n}$ $\displaystyle=\frac{V_{2}}{Z_{2}}$
the positive- and negative-sequence currents through the relay are calculated
as follows:
First, the negative- and zero-sequence voltage sources are disabled. The
positive-sequence current, resulting from the positive-sequence voltage
source, measured flowing through the relay:
$I_{11}=\frac{V_{1}}{Z_{1}+Z_{2d}}$
Second, the Norton equivalent circuit (illustrated in Fig. 10), with the
negative- and zero-sequence voltage sources disabled, is used. The negative-
sequence current, resulting from the positive-sequence voltage source, flowing
through the relay:
$I_{21}\approx I_{1n}\frac{Z_{1d}}{Z_{2}+Z_{1d}}$
assuming that $Z_{1M}^{1}\ll Z_{d}$
Figure 10: Norton equivalent circuit for a line-ground fault, supplied by a
current-limiting inverter, with the negative-sequence and zero-sequence
voltage sources disabled.
Third, the Norton equivalent circuit (illustrated in Fig. 11), with the
positive-sequence voltage source disabled, is used. The positive-sequence
current, from the contributions of the negative- and zero-sequence voltage
sources, flowing through the relay:
$I_{12}=I_{2n}\frac{Z_{2d}}{Z_{1}+Z_{2d}}$
Figure 11: Norton equivalent circuit for a line-ground fault, supplied by a
current-limiting inverter, with positive-sequence voltage source disabled.
Finally, the negative-sequence current contribution from the negative- and
zero-sequence sources can be calculated:
$I_{22}\approx\frac{V_{2}}{Z_{2}+Z_{1d}}$
again assuming that $Z_{1M}^{1}\ll Z_{L}$
The measured current running through the relay:
$\displaystyle I_{r}^{1}$ $\displaystyle=I_{11}+I_{12}$ $\displaystyle
I_{r}^{2}$ $\displaystyle=I_{21}+I_{22}$ $\displaystyle I_{r}^{0}$
$\displaystyle=I_{r}^{2}$ $\displaystyle I_{r}^{a}$
$\displaystyle=I_{r}^{1}+I_{r}^{2}+I_{r}^{0}$
The measured positive-sequence impedance:
$Z_{r}=\frac{V_{r}^{ag}}{I_{r}^{a}}$
The measured positive-sequence impedance at the relay, as a function of the
fault resistance, is illustrated in Fig. 8.
Figure 12: Measured positive-sequence impedance at an upstream relay for a
midpoint line-ground fault with a current-limiting inverter.
3) Downstream Relay
By using the definition of sequence voltages [13] and Fig. 9, the measured
voltage across the relay:
$V_{r}^{ag}=Z_{d}^{1}I_{r}^{1}+Z_{d}^{1}I_{r}^{1}+Z_{d}^{0}I_{r}^{0}$
where
$\displaystyle Z_{d}^{1}=$ $\displaystyle Z_{M2}^{1}+Z_{L}$ $\displaystyle
Z_{d}^{0}=$ $\displaystyle Z_{M2}^{0}+Z_{L}+3Z_{Lg}$
The measured impedance by the relay:
$\displaystyle Z_{r}^{1}$
$\displaystyle=\frac{V_{r}^{ag}}{I_{r}^{a}+kI_{r}^{0}}$
$\displaystyle=\frac{Z_{d}^{1}[I_{r}^{1}+I_{r}^{2}+(Z_{d}^{0}/Z_{d}^{1})I_{r}^{0}]}{I_{r}^{a}+kI_{r}^{0}}$
It is therefore apparent that $Z_{r}=Z_{d}^{1}$ and relay is not going to
operate for line-ground faults when $k=1-Z_{d}^{0}/Z_{d}^{1}$, unless it
experiences load encroachment.
### III-B Line-Line Midpoint Faults
For line-line midpoint faults, to facilitate expressing the fault quantities
analytically, the analysis assumes that the fault impedance is small (but non-
zero).
1) Ideal Voltage Source and Upstream Relay
Fig. 13 illustrates the equivalent sequence networks of the microgrid for the
case of a midpoint line-line fault across phases B and C. It is apparent from
the figure that the currents flowing into the fault are equal in magnitude,
but with opposite signs: $I_{bf}$ = $-I_{cf}$. Furthermore, the line-ground
voltages at the fault location on phases B and C are equal: $V_{bf}$ =
$V_{cf}$.
Figure 13: Sequence networks for a midpoint line-line fault with an ideal
voltage source.
From the definition of sequence quantities, it can be shown that $V_{af}^{1}$
= $V_{af}^{2}$ and $I_{af}^{1}$ = $-I_{af}^{2}$. This implies that the
positive- and negative-sequence networks are connected in parallel via the
fault, while the zero-sequence network is isolated from the other two (as
shown in Fig. 13). Given that there is no source of zero-sequence voltage,
there is no zero-sequence current flowing through the system for this fault
type.
The connected positive- and negative-sequence networks can be simplified, as
illustrated in Fig. 14. As is the case for the line-ground fault, the upstream
impedance is $Z_{1}$ = $Z_{1M}^{1}$, the downstream impedance is $Z_{d}$ =
$Z_{M2}^{1}$ \+ $Z_{L}$, and the shunt impedance from the negative-sequence
network is $Z_{2}$ = $Z_{eq2}$, where $Z_{eq2}$ is the same as defined in
Section III-A-1.
Figure 14: Simplified sequence network for a midpoint line-line fault with an
ideal voltage source.
The equivalent downstream impedance from the relay:
$\displaystyle Z_{1d}=$ $\displaystyle Z_{1}||Z_{d}$ $\displaystyle Z_{2d}=$
$\displaystyle Z_{2}||Z_{d}$
The positive-sequence current, as a consequence of a line-line fault, flowing
through the relay:
$I_{ar}^{1}=\frac{V_{s}}{Z_{1}+Z_{2d}}$
The negative-sequence current flowing through the relay is $I_{r}^{2}\approx
I_{r}^{1}$ for $Z_{1M}^{1}\ll Z_{L}$. The measured impedance by the relay:
$Z_{r}=\frac{V_{br}-V_{cr}}{I_{br}-I_{cr}}=\frac{V_{ar}^{2}-V_{ar}^{1}}{I_{ar}^{1}+I_{ar}^{2}}$
For an upstream relay, where the difference $V_{br}-V_{cr}$ is very small
while $I_{br}$ and $I_{cr}$ are not, the measured fault impedance is near
zero.
2) Current-Limiting Inverter and Upstream Relay
In case a current-limiting inverter is used, although the zero-sequence
network remains isolated, the inverter now provides zero-sequence voltage;
consequently, this network can no longer be neglected in the fault analysis.
To calculate the measured fault impedance, first the interconnection of
sequence networks needs to be simplified (illustrated in Fig. 15).
Figure 15: Simplified sequence network for a midpoint line-line fault with a
current-limiting inverter.
First, similarly to Section III-A-2, the negative-sequence voltage source is
disabled. The positive-sequence current, resulting from the positive-sequence
voltage source, measured through the relay:
$I_{11}=\frac{V_{1}}{Z_{1}+Z_{2d}}$
Second, the Norton equivalent circuit (illustrated in Fig. 16), with the
negative-sequence voltage source disabled, is used. The negative-sequence
current, resulting from the positive-sequence voltage source, flowing through
the relay:
$I_{21}\approx I_{1n}\frac{Z_{1d}}{Z_{1d}+Z_{2}}$
assuming that $Z_{1M}^{1}\ll Z_{L}$.
Figure 16: Norton equivalent circuit for a line-line fault, supplied by a
current-limiting inverter, with the negative-sequence voltage source disabled.
Third, the positive-sequence voltage source is disabled. The negative-sequence
current flowing through the relay:
$I_{22}\approx\frac{V_{2}}{Z_{2}+Z_{1d}}$ (1)
assuming $Z_{1M}^{1}\ll Z_{L}$
Finally, the Norton equivalent circuit (illustrated in Fig. 17), with the
negative-sequence voltage source disabled, is used. The positive-sequence
current, from the negative-sequence voltage source, flowing through the relay:
$I_{12}\approx I_{2n}$
Figure 17: Norton equivalent circuit for line-line fault, supplied by a
current-limiting inverter, with positive-sequence voltage source disabled.
The measured impedance can be calculated similarly to Section III-A-2; as with
the ideal voltage source, it is near zero for neglible fault impedance.
3) Downstream Relay
By using the definition of sequence voltages [13] and Fig. 14, the measured
impedance by the relay:
$\displaystyle Z_{r}=$
$\displaystyle\frac{V_{r}^{b}-V_{r}^{c}}{I_{r}^{b}-I_{r}^{c}}=\frac{(\alpha^{2}-\alpha)V_{ar^{1}}+(\alpha-\alpha^{2})V_{ar}^{2}}{(\alpha^{2}-\alpha)I_{ar^{1}}+(\alpha-\alpha^{2})I_{ar^{2}}}$
$\displaystyle=$
$\displaystyle\frac{(\alpha^{2}+\alpha)(V_{a}r^{1}-V_{ar}^{2})}{(\alpha^{2}+\alpha)(I_{ar}^{1}-I_{ar}^{2})}=\frac{V_{a}r^{1}-V_{ar}^{2}}{I_{ar}^{1}-I_{ar}^{2}}$
Substituting in the downstream impedances:
$\displaystyle Z_{d}^{1}=$ $\displaystyle Z_{M2}^{1}+Z_{L}$ $\displaystyle
Z_{d}^{0}=$ $\displaystyle Z_{M2}^{1}+Z_{L}+3Z_{Lg}$
the measured impedance can be simplified:
$Z_{r}=\frac{Z_{d}^{1}I_{ar}^{1}-Z_{d}^{1}I_{ar}^{2}}{I_{ar}^{1}-I_{ar}^{2}}=\frac{Z_{d}^{1}(I_{ar}^{1}-I_{ar}^{2})}{I_{ar}^{1}-I_{ar}^{2}}=Z_{d}^{1}$
This relies on quantities $V_{br}$, $V_{cr}$, $I_{br}$, and $I_{cr}$ being
nonzero, which is the case for a practical fault with nonzero fault impedance.
## IV Transient Model
To validate the above calculated impedances, produced from the equivalent
sequence networks, the considered two bus microgrid system is modeled in the
MATLAB/Simulink® SimScape multi-physics simulation environment, using the
Specialized Power Systems library. This model is based on the design of an
inverter using a PR controller [10] presented in [11].
The microgrid is illustrated in Figs. 18 – 20, while the system parameters are
listed in Tables I – IV.
Figure 18: Transient microgrid model. Figure 19: Transient inverter model. Figure 20: Transient current limiter model. TABLE I: Global Parameters Name | Symbol | Value | Unit
---|---|---|---
Grid frequency | f0 | 60 | Hz
Line-line voltage | V | 480 | V
TABLE II: Synchronous Generator Parameters Subsystem | Symbol | Value
---|---|---
Voltage loop | kpv | 0.35
Voltage loop | krv | 400
Voltage loop | kvh5 | 4
Voltage loop | kvh7 | 20
Voltage loop | kvh11 | 11
Current loop | kpi | 0.7
Current loop | kri | 400
Current loop | kih5 | 30
Current loop | kih7 | 30
Current loop | kih11 | 30
TABLE III: Inverter Controller Parameters Subsystem | Symbol | Value
---|---|---
Voltage loop | kpv | 0.35
Voltage loop | krv | 400
Voltage loop | kvh5 | 4
Voltage loop | kvh7 | 20
Voltage loop | kvh11 | 11
Current loop | kpi | 0.7
Current loop | kri | 400
Current loop | kih5 | 30
Current loop | kih7 | 30
Current loop | kih11 | 30
TABLE IV: Hardware Parameters Name | Symbol | Value | Unit
---|---|---|---
Inverter rated power | P | 50 | kW
DC-bus voltage | Vdc | 1800 | V
Output filter inductance | L | 18 | $\mu$F
Output filter capacitance | C | 250 | nF
Maximum rms output current | Imax | 70 | A
Cable resistance | Rc | 39 | m$\Omega$
Cable inductance | Lc | 70.8 | $\mu$H
Load real power | Pd | 25 | kW
Load reactive power | Qd | 12.5 | kW
## V Results
Fig. 21 presents the simulation results for the four cases considered: (a)
line-ground fault with an upstream ground distance relay, (b) line-ground
fault with a downstream ground distance relay, (c) line-line fault with an
upstream phase distance relay, and (d) line-line fault with a downstream phase
distance relay. For these cases, only the current-limiting inverter is
considered and the ideal voltage source case is not modeled.
(a) Line-ground fault with an upstream ground distance relay
(b) Line-ground fault with a downstream ground distance relay
(c) Line-line fault with an upstream phase distance relay
(d) Line-line fault with a downstream phase distance relay
Figure 21: Measured impedances on different faults with different distance
relays for the four cases considered.
Figs. 21a and 21c illustrate that for both upstream relay cases, the measured
impedance varies over time from the load resistance towards the origin, while
remaining in the first quadrant. Figs. 12 and 21a highlight the correspondence
between the predicted and measured impedances. Fig. 21c confirms that the
measured impedance for a relay upstream of a line-line fault is near zero.
Figs. 21b and 21d proves that the measured impedance for downstream relays is
approximately the load impedance in both cases.
## VI Conclusions
Both the analytical and simulation results suggest that conventional relaying
quantities for phase and ground distance protection are suitable for the
protection of both lines and load buses in inverter-interfaced microgrids.
Although the case studies of this paper used midpoint faults, the measured
quantities change only a small amount for load bus faults as $Z_{1M}^{1}\ll
Z_{L}$. For the case of line protection, it is likely that pilot protection is
necessary rather than relying on zone-based distance protection. The low line
impedances of microgrids make it likely that other impedances (e.g., inverter
virtual impedance, load impedance, fault impedance) dominate, rendering it
difficult to distinguish in- or out-of-zone faults based purely on measured
impedance.
The results do not support the observations of Dewadasa et al. [3, 4] on
protection misperations for an upstream line-ground fault when a grounded-wye
load is present. While the performed case studies were not able to determine
why Dewadasa observed misoperations, from Fig. 5c it is apparent that an
inverter lacking a grounding source, causing $Z_{1M}^{0}$ to be open-
circuited, could result in downstream ground overcurrent protection without
directional relaying to trip. Inspection of Fig.5b confirms the claim of [3]
that negative-sequence current is suitable for use as a polarizing quantity
for directional relaying in line-ground faults.
## References
* [1] T. McDermott, B. Vyakaranam, R. Fan, P. T. Mana, T. Smith, Z. Li, J. Hambrick, and A. K. Barnes. Protective Relaying for Distribution and Microgrids Evolving from Radial to Bi-Directional Power Flow. In Proceedings of the 2018 Western Protective Relay Conference, Oct. 2018.
* [2] R. M. Tumilty, M. Brucoli, G. M. Burt, and T. C. Green. Approaches to Network Protection for Inverter Dominated Electrical Distribution Systems. In Proceedings of the 3rd IET International Conference on Power Electronics, Machines and Drives, 2006, pages 622–626, Apr. 2006.
* [3] J. M. Dewadasa, A. Ghosh, and G. Ledwich. Line Protection in Inverter Supplied Networks. In Proceedings of the 2008 Australasian Universities Power Engineering Conference, pages 1–6, Dec. 2008.
* [4] J. M. Dewadasa, A. Ghosh, and G. Ledwich. Distance Protection Solution for a Converter Controlled Microgrid. In Proceedings of the 15th National Power Systems Conference, 2008\.
* [5] S. Kar and S. R. Samantaray. Time-Frequency Transform-Based Differential Scheme for Microgrid Protection. IEEE IET Generation, Transmission & Distribution, 8(2):310–320, Feb. 2014.
* [6] A. K. Barnes and A. Mate. Dynamic State Estimation for Radial Microgrid Protection. In Proceedings of the 2021 IEEE/IAS 57th Industrial and Commercial Power Systems Technical Conference, pages 1–9, Apr. 2021.
* [7] J. L. Blackburn. Protective Relaying Principles and Applications. Taylor & Francis, 3rd edition edition, 2007.
* [8] S. Tamronglak. Analysis of Power System Disturbances due to Relay Hidden Failures. PhD thesis, Virginia Polytechnic Institute and State University, Apr. 1994\.
* [9] D. C. Elizondo. A Methodology to Assess and Rank the Effects of Hidden Failures in Protection Schemes based on Regions of Vulnerability and Index of Severity. PhD thesis, Virginia Polytechnic Institute and State University, Apr. 2003\.
* [10] R. Teodorescu, F. Blaabjerg, and M. Liserre. Proportional-Resonant Controllers. A New Breed of Controllers Suitable for Grid-Connected Voltage-Source Converters. Journal of Electrical Engineering, pages 1–6, 2003.
* [11] J. C. Vasquez, J. M. Guerrero, M. Savaghebi, J. Eloy-Garcia, and R. Teodorescu. Modeling, Analysis, and Design of Stationary-Reference-Frame Droop-Controlled Parallel Three-Phase Voltage Source Inverters. IEEE Transactions on Industrial Electronics, 60(4), Apr. 2013\.
* [12] N. Bottrell and T. C. Green. Comparison of Current-Limiting Strategies During Fault Ride-Through of Inverters to Prevent Latch-Up and Wind-Up. IEEE Transactions on Power Electronics, 29(7):3786–3797, Jul. 2014.
* [13] A. R. Bergen and V. Vittal. Power Systems Analysis. Pearson Education India Publishing, 2009.
|
11institutetext: Department of Natural and Mathematical Sciences, Özyeğin
University, 34794 İstanbul Turkey
# The variational method, backreactions, and the absorption probability in
Wald type problems
Koray Düztaş
(Received: date / Revised version: date)
###### Abstract
We argue that the variational method in Wald type thought experiments,
involves order of magnitude problems when one imposes the fact that $\delta M$
is inherently a first order quantity itself. One observes that the
contribution of the second order perturbations is actually of the fourth
order. Therefore backreactions have to be explicitly calculated. Here, we re-
consider the overspinning problem for Kerr-Newman black holes interacting with
test fields. We calculate the backreaction effects due to the induced increase
in the angular velocity of the event horizon, which brings a partial solution
to the overspinning problem. To bring an ultimate solution, we argue that the
absorption probability should be taken into account in Wald type problems
where black holes interact with test fields. This fundamentally alters the
course of the analysis of the thought experiments. Due to the fact that a
small fraction of the challenging modes is absorbed by the black holes,
overspinning is prevented for both nearly extremal and extremal cases. Some
extreme cases are easily fixed by backreaction effects. The arguments do not
apply to the generic overspinning by fermionic fields for which the absorption
probability is positive definite.
###### pacs:
04.20.DwSingularities and cosmic censorship
## 1 Introduction
One of the main unsolved problems in classical general relativity is the
validity of the cosmic censorship conjecture due to Penrose ccc . The
conjecture aims to circumvent the problems that could arise if a curvature
singularity is allowed to be in causal contact with distant observers. This is
achieved by forbidding the existence of naked singularities in a physical
universe. The gravitational collapse of a massive object should end up in a
black hole surrounded by an event horizon rather than a naked singularity, as
prescribed by Penrose and Hawking singtheo .
The natural question at this stage is whether the event horizon of a black
hole can be destroyed by test bodies or fields to expose the curvature
singularity lurking at the center. The possibility to destroy an event horizon
was first evaluated in a thought experiment constructed by Wald wald74 . Wald
started with an extremal Kerr-Newman black hole and attempted to increase the
charge and angular momentum beyond the extremal limit by sending in test
bodies from infinity. It turns out that the test bodies that could potentially
overcharge or overspin an extremal Kerr-Newman black hole are not absorbed by
the black hole. The event horizon is stable and the smooth structure of the
space-time is maintained excluding the black hole region inside the event
horizon. Later, Hubeny adapted an alternative approach to Wald type problems
where one starts with a nearly extremal black hole instead of an extremal one
hu . She showed that a nearly extremal Reissner-Nordström black hole can be
overcharged into a naked singularity by a test body. Jacobson and Sotiriou
applied an analogous analysis to show that nearly extremal Kerr black holes
can be overspun by test bodies js . Düztaş and Semiz derived the same result
for nearly extremal Kerr black holes interacting with test fields overspin .
In these works the overspinning and overcharging of nearly extremal black
holes are not quite generic, which suggests that they should be fixed by
employing backreaction effects. It was argued that the self force effects can
prevent the overcharging backhu and overspinning backjs of nearly extremal
black holes by test bodies. In a very recent work we showed that the
absorption of the test fields that could overspin nearly extremal black holes
is not allowed due to the increase in the angular velocity of the event
horizon before the absorption of the field kerrmog .
In literature there exist various attempts to overspin or overcharge black
holes with test bodies f1 ; saa ; gao ; siahaan ; magne ; dilat ; higher ; v1
; he ; wang ; gim ; jamil ; shay ; shay2 ; zeng , and fields semiz ; emccc ;
duztas ; toth ; natario ; duztas2 ; mode ; taub-nut ; kerrsen ; gwak5 ; gwak6
; hong ; yang ; bai . The effect of quantum tunnelling and particle creation
has also been incorporated in Wald type problems q1 ; q2 ; q3 ; q4 ; q5 ; q6 ;
q7 . In recent years, the possibility to destroy the event horizon in the
asymptotically anti-de Sitter cases has also become an active field of
research btz ; gwak1 ; gwak2 ; gwak3 ; gwak4 ; chen ; he2 . ( For a recent
review see ong )
In Wald type problems the backreaction effects are difficult to compute and
most of the time the results are restricted to order of magnitude estimates.
Recently Sorce and Wald designed a new type of gedanken experiment by adapting
a variational approach w2 . They derived an explicit expression for the second
order effects, so one does not have to explicitly compute self force or finite
size effects. Currently, the Sorce-Wald method is widely accepted among the
researchers working on Wald type problems. Recently we have also employed
Sorce-Wald method to test the stability of the event horizon for Martinez,
Teitelboim, Zanelli (MTZ) black holes mtz . In our analysis, we have imposed
the fact that $\delta M$ is inherently a first order quantity. We observed
that, imposing this fact causes order of magnitude problems to arise in the
method developed by Sorce and Wald. In section (2), we further scrutinize the
Sorce-Wald method by imposing the fact that $\delta M$ is inherently a first
order quantity itself, for test bodies and fields. Unfortunately, it turns out
that the order of magnitude problems do not pertain to the MTZ case. They are
also manifest in the case of Kerr-Newman black holes, for which the Sorce-Wald
method was developed. We present the details in section (2.1).
The order of magnitude problems in the Sorce-Wald method suggest that the
backreactions should be explicitly calculated. Based on this argument, we re-
visit the overspinning problem for nearly extremal and extremal Kerr-Newman
black holes interacting with test fields, in section (3). In a recent work we
showed that extremal Kerr-Newman black holes which satisfy $J^{2}/M^{4}<(1/3)$
can be overspun by scalar test fields generic . We argued that the
overspinning is not quite generic and it is prone to be fixed by backreaction
effects. In section (3.1), we show that nearly extremal black holes can also
be overspun and employ the backreaction effects based on Will’s argument that
the angular velocity of the event horizon increases before the absorption of
the test field will . The employment of this backreaction effect brings a
partial solution to the problem. The destruction of the event horizon can be
prevented for certain classes of nearly extremal and extremal black holes,
with a sufficiently large magnitude of angular momentum. In section (3.2), we
show that the same argument applies to extremal black holes. We analytically
derive the relevant magnitude of the initial value of angular momentum, for
both nearly extremal and extremal cases.
The interaction of test fields with black holes is actually a scattering
problem. The field is partially absorbed by the black hole and partially
reflected back to infinity. The fact that only a fraction of the incoming
field is absorbed by the black hole has been ignored in all the thought
experiments constructed so far, including the works of this author. In section
(4), we take the absorption probabilities of test fields into account, which
fundamentally changes the course of the analysis of the problem. We show that
a very small fraction of the challenging test fields are absorbed by the black
hole which has no practical effect on the mass and angular momentum parameters
of the space-time. In section (4.1) we evaluate the optimal perturbations with
the lowest possible energy relative to their angular momentum and charge. We
show that the absorption probability is zero for the optimal perturbations. In
that case, the test field is entirely reflected back to infinity. The space-
time parameters remain identically the same after the interaction with the
test field. In sections (4.2) and (4.3) we perturb nearly extremal and
extremal black holes with challenging modes with frequencies slightly larger
than the optimal perturbations. We show that the absorption probability is
very low for the challenging modes. It turns out that most of the energy and
angular momentum carried by the challenging modes is reflected back to
infinity. Still there exists a set of fine-tuned parameters that seem to be
capable of overspinning extremal and nearly extremal Kerr-Newman black holes.
In sections (4.2) and (4.3), we also show that, these anomalies are remedied
by backreaction effects due to the induced increase in the angular velocity of
the event horizon.
## 2 Sorce-Wald method
The Kerr-Newman metric describes a black hole surrounded by an event horizon
provided that the spacetime parameters satisfy the main inequality
$M^{2}\geq a^{2}+Q^{2}$ (1)
where $M$, $a\equiv J/M$, and $Q$ are respectively the mass, angular momentum
and charge parameters of the spacetime. In Wald type problems, one starts with
an extremal or nearly extremal black hole satisfying the main criterion (1)
and attempts to increase the angular momentum and/or charge parameters beyond
the extremal limit, by sending in test bodies or fields from infinity. The
main assumption in these thought experiments is that the interaction of the
black hole with test bodies and fields does not alter the background geometry
of the spacetime, but leads to perturbations in the mass, angular momentum,
and charge parameters. After a sufficiently long time, the spacetime is
supposed to settle down to a new Kerr-Newman solution with modified
parameters. Apparently the energy, angular momentum, and the charge of the
test body or field should be very small compared to the initial parameters of
the spacetime so that the assumption that the background geometry in the final
state is also a Kerr-Newman solution, is justified. Then, one can check if the
final parameters of the spacetime represent a Kerr-Newman black hole
satisfying the inequality (1) or a naked singularity which violates it. For
that purpose we prefer to define
$\delta_{\rm{fin}}\equiv
M_{\rm{fin}}^{2}-Q_{\rm{fin}}^{2}-\frac{J_{\rm{fin}}^{2}}{M_{\rm{fin}}^{2}}$
(2)
If the contribution of the second order terms are taken into account in
calculating $\delta_{\rm{fin}}$, one should also incorporate the effect of
backreactions which bring second order corrections to (2), so that the
calculation can be considered consistent. As we mentioned in the introduction,
the overcharging of Reissner-Nordstro̧m black holes hu , and the overspinning
of Kerr black holes js ; overspin can be fixed by backreaction effects backhu
; backjs ; kerrmog . The backreaction effects comprise finite size effects,
self interaction, gravitational radiation, the effect of black hole radiation,
induced increase in the angular velocity of the horizon and many more possible
effects pertaining to the specific problem. The Sorce-Wald method has been
developed to bring an ultimate solution to the problem of determining and
calculating the backreactions w2 . Sorce and Wald (SW) attempted obtain an
expression for the full second order correction $\delta^{2}M$ without having
to calculate the backreaction effects explicitly. To check whether the event
horizon can be destroyed SW first derive an expression for the minimum energy
of the incoming test body or field so that it is absorbed by the black hole.
$\delta M-\Omega_{H}\delta J-\Phi_{H}\delta Q\geq 0$ (3)
where $\Omega_{H}=a/(r_{+}^{2}+a^{2})$, $\Phi_{H}=(Qr_{+})/(r_{+}^{2}+a^{2})$,
and $r_{+}$ is the horizon radius. The condition (3) is well known in black
hole physics. The first derivation without assuming the validity of cosmic
censorship known to this author is by Needham in 1980 needham . The condition
(3) determines the lowest possible energy for a given combination of angular
momentum and charge that would allow the absorption of a test field. The
perturbations with the lowest possible energy are referred to as the optimal
perturbations. The perturbations that do not satisfy the condition (3) are not
absorbed by the black hole. If the absorption of these perturbations was
allowed, they would lead to a generic destruction of the event horizon since
they carry relatively large angular momentum and charge. However, the
condition (3) only applies to the perturbations that satisfy the null energy
condition. For fermionic fields there is no lower bound on the energy that
would prevent the absorption of the challenging modes. In generic we argued
that the absence of the lower bound for the energy of the fermionic fields
leads to a generic destruction of the event horizon.
Sorce and Wald proceed by parametrizing a nearly extremal black hole as
$M^{2}-Q^{2}-(J/M)^{2}=M^{2}\epsilon^{2}$ (4)
which is common in Wald type problems. The small parameter $\epsilon$
determines the closeness of the black hole to extremality. For $\epsilon\ll 1$
the black hole is very close to extremality in which case the effect of the
interactions with test bodies and fields become relevant. Next, Sorce and Wald
define the function:
$f(\lambda)=M(\lambda)^{2}-Q(\lambda)^{2}-J(\lambda)^{2}/M(\lambda)^{2}$ (5)
If $f(\lambda)<0$ the inequality (1) is violated and the event horizon cannot
exist. Next $f(\lambda)$ is expanded to second order in $\lambda$
$\displaystyle f(\lambda)$ $\displaystyle=$
$\displaystyle\left(M^{2}-Q^{2}-\frac{J^{2}}{M^{2}}\right)$ (6)
$\displaystyle+$ $\displaystyle 2\lambda\left(\frac{M^{4}+J^{2}}{2M^{3}}\delta
M-\frac{J}{M^{2}}\delta J-Q\delta Q\right)$ $\displaystyle+$
$\displaystyle\lambda^{2}\left[\frac{M^{4}+J^{2}}{2M^{3}}\delta^{2}M-\frac{J}{M^{2}}\delta^{2}J-Q\delta^{2}Q+\frac{4J}{M^{3}}\delta
J\delta M\right.$ $\displaystyle-$ $\displaystyle\left.\frac{1}{M^{2}}(\delta
J)^{2}+\left(\frac{M^{4}-3J^{2}}{M^{4}}\right)(\delta M)^{2}-(\delta
Q)^{2}\right]$
To avoid any confusion we refer to $\delta M,\delta J,\delta Q$ as first order
perturbations, and $\delta^{2}M,(\delta M)^{2},...$ terms as second order
perturbations. For the first order perturbations Needham’s condition (3)
implies that
$\displaystyle f(\lambda)$ $\displaystyle\geq$ $\displaystyle
M^{2}\epsilon^{2}+\frac{2}{M^{4}+J^{2}}\left((J^{2}-M^{4})Q\delta
Q-2JM^{2}\delta J)\right)\lambda\epsilon$ (7) $\displaystyle+$ $\displaystyle
O(\lambda^{2},\epsilon^{3},\epsilon^{2}\lambda)$
The equations (6) and (7) above, are the equations (119) and (120) in the
relevant paper of Sorce and Wald w2 . To derive (7), one imposes the condition
(3) that the test body or field is absorbed by the black hole, and expresses
$\delta M$ in terms of $\delta J$ and $\delta Q$. At this point Sorce and Wald
claim that neglecting the terms of order $O(\lambda^{2})$ it is possible to
make $f(\lambda)<0$. This statement aims to convince the readers that the
variational method reproduces the previous results that the nearly extremal
black holes can be overcharged or overspun when the second order terms are
neglected. The main claim of SW is that $f(\lambda)$ becomes positive again by
considering the effect of the terms that are second order in $\lambda$. Here
we show that these two results cannot be obtained by the method developed by
Sorce and Wald, when one does not ignore the fact that $\delta M$ is
inherently a first order quantity, itself.
### 2.1 Sorce-Wald method with the correct test body/field approximation
The small parameter $\lambda$ is introduced in (5) to ensure that the
variation of the function $f$ from its initial value $f(0)$ is small, in
accord with the test body/field approximation. However, in (6) the parameter
$\lambda$ explicitly multiplies the terms $\delta M,\delta J,\delta Q$. Since
$\delta M$ is inherently a first order quantity, the $\lambda\epsilon\delta M$
terms actually contribute to third order to $f(\lambda)$, while the
$\lambda^{2}\delta^{2}M$ terms contribute to fourth order. One cannot ignore
the fact that $\delta M$ is a small quantity itself and proceed as if $\delta
M\sim M$, as it was done in the derivation of Sorce and Wald.
To clarify the fact that $\delta M$ is a small quantity itself, we let
$\delta M=M\zeta$ (8)
where $\zeta$ parametrises the energy of the perturbation and the fact that
$\zeta\ll 1$ ensures that the test body/field approximation is not violated.
In principle the parameters determining the magnitude of the perturbations and
the closeness to extremality need not be equal. However, for numerical
calculations one can let $\epsilon\sim\zeta$. Imposing the fact that the
perturbations are small themselves (7) implies that
$f(\lambda)\sim O(\epsilon^{2})-O(\lambda\epsilon\zeta)$ (9)
which is valid to first order in $\lambda$. (Note that the term
$(J^{2}-M^{4})$ is negative.) It is easy to see that when one imposes the fact
that the first order perturbations are of the “first order” themselves, it is
not possible to make $f(\lambda)$ –defined by SW– negative for the first order
terms. The variational method does not reproduce the previous results due
Hubeny hu , Jacobson-Sotiriou js and Düztaş-Semiz overspin . For the first
order perturbations, the results of SW contradict with the previous results
when the fact that $\delta M$ is inherently a small quantity is taken into
account.
Though it is manifest in (9) that $f(\lambda)$ defined by SW, cannot be made
negative for the first order terms, it would be appropriate to elaborate on
this subject considering the fact that the SW method is widely accepted in
black hole physics. For simplicity let us consider a neutral body or a field
$(\delta Q=0)$, incident on a nearly extremal black hole. For the optimal
perturbations (3) implies that
$J\delta J=M\delta M(r_{+}^{2}+a^{2})$
Imposing the fact that $\delta M=M\zeta$ by the definition (8)
$\displaystyle J\delta J$ $\displaystyle=$ $\displaystyle
M^{2}\zeta\left[M^{2}(1+\epsilon)^{2}+\frac{J^{2}}{M^{2}}\right]$ (10)
$\displaystyle=$
$\displaystyle(M^{4}+J^{2})\zeta+O(\epsilon\zeta,\epsilon^{2}\zeta)$
where we have substituted $r_{+}=M(1+\epsilon)$ for a nearly extremal black
hole parametrized as (4). Now we substitute the expression for $J\delta J$
derived in (10) to the expression for $f(\lambda)$. For the optimal
perturbations one derives
$f(\lambda)=M^{2}\epsilon^{2}-4M^{2}\lambda\epsilon\zeta-O(\lambda\epsilon^{2}\zeta,\lambda\epsilon^{3}\zeta)$
(11)
Again it is manifest in (11) that $f(\lambda)$ defined by SW, cannot be made
negative for small $\lambda$, $\epsilon$ and $\zeta$. The claim that
$f(\lambda)$ can be made negative by the terms first order in $\lambda$
requires one to assume that $\delta M\sim M$, which apparently contradicts the
test body/field approximation.
The main claim of SW is that the negativeness of $f(\lambda)$ can be fixed by
the contribution of the terms that are second order in $\lambda$. Though the
fact that $f(\lambda)$ cannot be made negative by the first order terms
renders this claim irrelevant, it is necessary to evaluate the contribution of
the second order terms when the derivation is corrected by imposing $\delta
M=M\zeta$. To second order in $\lambda$ we have
$f(\lambda)\sim
O(\epsilon^{2})-O(\lambda\epsilon\zeta)+O(\lambda^{2}\zeta^{2})$ (12)
It is manifest in (12) that the contribution of the second order perturbations
vanishes in (6), as it becomes fourth order when multiplied by the square of
the small parameter $\lambda$. (Note that the leading term in (12) –which is
zeroth order in $\lambda$– is actually second order in $\epsilon$.) In that
respect it is not possible to incorporate the effect of the second order
perturbations into the analysis using the SW method. Moreover, when the
analysis is corrected by imposing the fact that $\delta M$ is a first order
quantity, even the first order perturbations $(\delta M,\delta J,\delta Q)$ do
not contribute to $f(\lambda)$ as one can observe in (11) and (12).
In the previous works by Hubeny, Jacobson-Sotiriou, and Düztaş-Semiz,
$\delta_{\rm{fin}}$ defined in (2) is made negative for nearly extremal black
holes which corresponds to making $f(\lambda)$ negative in the derivation of
SW hu ; js ; overspin . The numerical value of $\delta_{\rm{fin}}$ turns out
to be of the order $-M^{2}\epsilon^{2}$ which suggests that the destruction of
the event horizon can be fixed by the second order corrections due to the
backreaction effects. Later, these corrections were indeed achieved by
employing backreaction effects backhu ; backjs ; kerrmog . The SW method does
not reproduce any of these results when one imposes the fact that $\delta M$
is a first order quantity for test bodies and fields. One observes that the
terms first order in $\lambda$, contribute to $f(\lambda)$ to third order so
they cannot make $f(\lambda)$ negative. The terms that are second order in
$\lambda$, cannot fix anything since their contribution is of the fourth
order. Therefore the SW method cannot be used to evaluate the effect of second
order perturbations. Backreactions have to be explicitly calculated.
## 3 Re-visiting the over-spinning problem
In this section we re-visit the over-spinning problem for Kerr-Newman black
holes and explicitly calculate the backreaction effects, which supervenes on
the argument that they cannot be calculated using the SW method. We start by
attempting to overspin a nearly extremal Kerr-Newman black hole parametrised
as (4), by a neutral, scalar test field with frequency $\omega$ and azimuthal
wave number $m$. We adapt the parametrization (8) so that the field carries
energy $\delta M=M\zeta$, in accord with the test field approximation. At the
end of the interaction, the final parameters of the space-time satisfy:
$\displaystyle M_{\rm{fin}}$ $\displaystyle=$ $\displaystyle M+M\zeta$
$\displaystyle J_{\rm{fin}}$ $\displaystyle=$ $\displaystyle
J+\frac{m}{\omega}M\zeta$ $\displaystyle Q_{\rm{fin}}$ $\displaystyle=$
$\displaystyle Q$ (13)
where $M,J,Q$ are the initial parameters which satisfy (4). Now, we demand
that the black hole is overspun at the end of the interaction; i.e.
$\delta_{\rm{fin}}<0$
$\delta_{\rm{fin}}=(M+M\zeta)^{2}-Q^{2}-\frac{(J+\frac{m}{\omega}M\zeta)^{2}}{(M+M\zeta)^{2}}<0$
(14)
We can substitute $Q^{2}=M^{2}-J^{2}/M^{2}-M^{2}\epsilon^{2}$ by using (4).
Re-arranging (14), we get
$M^{2}\left(\zeta^{2}+2\zeta+\epsilon^{2}+\frac{J^{2}}{M^{4}}\right)<\frac{J+\frac{m}{\omega}M\zeta)^{2}}{M^{2}(1+\zeta)^{2}}$
(15)
We define the dimensionless parameter
$\alpha\equiv J/M^{2}$ (16)
Note that for a nearly extremal Kerr-Newman black hole parametrised as (4),
the sum $J^{2}/M^{2}+Q^{2}$ has a fixed value which is equal to
$M^{2}(1-\epsilon^{2})$ for a fixed mass $M$. However nearly extremal black
holes satisfying (4) may have different values of angular momentum and charge
keeping the sum $J^{2}/M^{2}+Q^{2}$ fixed. We use the dimensionless parameter
$\alpha$ to identify different Kerr-Newman black holes –with a fixed mass $M$–
that all satisfy (4). Also note that, substituting $\alpha\equiv J/M^{2}$ the
parametrisation (4) can be re-written in terms of the dimensionless variable
$\alpha$.
$1-\alpha^{2}-\frac{Q^{2}}{M^{2}}=\epsilon^{2}$ (17)
We proceed by taking the square root of both sides of (15). The condition
$\delta_{\rm{fin}}<0$ reduces to
$\omega<\omega_{\rm{max}}=\frac{m\zeta}{M\left[(1+\zeta)\sqrt{\zeta^{2}+2\zeta+\epsilon^{2}+\alpha^{2}}-\alpha\right]}$
(18)
We have considered the interaction of a nearly extremal black hole
parametrised as (4) with a test field carrying energy $\delta M=M\zeta$ and
angular momentum $\delta J=(m/\omega)\delta M$. Note that $\delta J$ is
inversely proportional to the frequency $\omega$. The equation (18) implies
that a test field with a frequency $\omega<\omega_{\rm{max}}$, will contribute
to the angular momentum parameter of the black hole with a magnitude
sufficiently larger than its contribution to the energy parameter so that the
final parameters of the black hole describe a naked singularity with
$\delta_{\rm{fin}}<0$. In that case we could conclude that the nearly extremal
Kerr-Newman black hole is overspun into a naked singularity. However, we
should also demand that the test field is absorbed by the black hole, i.e.
$\omega$ is larger than the limiting frequency for superradiance, which we
denote by $\omega_{\rm{sl}}$. For a nearly extremal black hole parametrised as
(4), which is perturbed by a neutral test field $(\delta Q=0)$, the
superradiance limit is given by
$\omega_{\rm{sl}}=\frac{ma}{r_{+}^{2}+a^{2}}=\frac{m}{M\left[\frac{(1+\epsilon)^{2}}{\alpha}+\alpha\right]}$
(19)
Kerr-Newman black holes with different values of $\alpha$ defined in (16),
have different superradiance limits. For lower values of $\alpha$ which
describe black holes with relatively low angular momentum, the superradiance
limit will also be low. In that case the absorption of the modes with
relatively low frequencies will be allowed. The test fields with frequency in
the range $\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$ simultaneously satisfy
the two conditions that the field is absorbed by the black hole and it
contributes to the angular momentum parameter with a sufficiently large
magnitude to overspin the black hole into a naked singularity. We can conclude
that the test fields with energy $\delta M=M\zeta$ and frequency in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$ can be used to overspin a nearly
extremal Kerr-Newman black hole into a naked singularity, provided that
$\omega_{\rm{sl}}<\omega_{\rm{max}}$ . Comparing (18) and (19) one observes
that the upper limit for the frequency of the incident field
$\omega_{\rm{max}}$ derived in (18), is larger than the superradiance limit
$\omega_{\rm{sl}}$ for any value of $\alpha$, in the relevant range $(0,1)$.
It turns out that every nearly extremal Kerr-Newman black hole satisfying (4)
can be overspun by neutral test fields, regardless of the specific value of
$\alpha$ defined in (16). The validity of this conclusion is limited to the
case, where one ignores the backreaction effects.
### 3.1 Backreactions for nearly extremal black holes
In this derivation we have not ignored the contribution of the second order
terms $(\delta M)^{2}$ and $(\delta J)^{2}$. Therefore we have to employ the
backreaction effects to test whether the destruction of the event horizon can
be fixed. Since, the Sorce-Wald method is invalid we have to explicitly
determine and calculate the backreaction effects. Backreaction effects will
bring second order corrections to $\delta_{\rm{fin}}$ which can in principle
restore the event horizon. The most legitimate type of backreaction effect for
an overspinning problem is the induced increase in the angular velocity of the
event horizon before the absorption of the test body/field occurs, which was
suggested by Will will . The induced increase in the angular velocity of the
event horizon leads to an increase in the superradiance limit. This implies
that the absorption of the challenging modes with relatively low frequencies
can be prevented. In a recent paper we have employed this backreaction effect
for the overspinning problem of Kerr-MOG black holes kerrmog .
We envisage a test field with angular momentum $\delta J$ incident on a black
hole with mass $M$. According to the estimate in will , the angular velocity
of the event horizon increases by an amount
$\Delta\omega=\frac{\delta J}{4M^{3}}$ (20)
The increase in the angular velocity of the event horizon results in an
increase in the superradiance limit, which will be modified as
$\omega_{\rm{sl}}^{\prime}=\omega_{\rm{sl}}+\Delta\omega$ (21)
In (18) we have derived the maximum value for the frequency of a test field
that could overspin a nearly extremal Kerr-newman black hole parametrised as
(4). We noted that the fields with frequency in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$ can lead to overspinning. However
as the test field is incident on the black hole the angular velocity of the
event horizon will increase, which will lead to a modification in the
superradiance limit given derived in (21). If the modified value of the
superradiance limit exceeds the frequency of the incoming field, the
absorption of the test field is prevented and the event horizon cannot be
destroyed. The test field will be scattered back to infinity with a larger
magnitude. Note that $\delta J$ and $\Delta\omega$ given in (20) are inversely
proportional to the frequency $\omega$. For that reason, if the modified value
of the superradiance limit exceeds the incoming frequency for
$\omega\simeq\omega_{\rm{max}}$, it will exceed the incoming frequency even
further for smaller values in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$, as we have argued in kerrmog .
Therefore it is critical to calculate $\Delta\omega$ for the frequencies
arbitrarily close to $\omega_{\rm{max}}$.
To calculate backreactions, we first envisage a test field with frequency
arbitrarily close to but slightly less than $\omega_{\rm{max}}$, incident on a
nearly extremal Kerr-Newman black hole parametrised as (4). The test field
carries energy $\delta M=M\zeta$ and angular momentum $\delta
J=(m/\omega)\delta M$, where $\omega\lesssim\omega_{\rm{max}}$. According to
the derivation in the previous section, this test field will lead to the
overspinning of the nearly extremal Kerr-Newman black hole. Now we incorporate
the backreaction effects due to the induced increase in the angular velocity
of the event horizon. We check if the modified value of the superradiance
limit derived in (21), exceeds the frequency of the incoming field. For
simplicity we let $\epsilon\simeq\zeta$ and substitute
$\omega=\omega_{\rm{max}}$ in (20) to calculate $\Delta\omega$.
$\Delta\omega=\frac{\left(\frac{m}{\omega}\right)M\epsilon}{4M^{3}}=\frac{\left[(1+\epsilon)\sqrt{2\epsilon^{2}+2\epsilon+\alpha^{2}}-\alpha\right]}{4M}$
(22)
For $\omega\simeq\omega_{\rm{max}}$, the increase in the superradiance limit
is given by $\Delta\omega$ in (22), which leads to the modified value of the
superradiance limit denoted by $\omega_{\rm{sl}}^{\prime}$ derived in (21).
Then, we need to compare $\omega_{\rm{sl}}^{\prime}$ and $\omega_{\rm{max}}$.
If $\omega_{\rm{sl}}^{\prime}$ is larger than $\omega_{\rm{max}}$, no net
absorption of the test field will occur and overspinning will be prevented.
It turns out that $\omega_{\rm{sl}}^{\prime}$ defined in (21) is indeed larger
than $\omega_{\rm{max}}$ provided that
$\alpha\gtrsim 0.50$ (23)
Note that the parameter $\alpha\equiv J/M^{2}$ defined in (16) is used to
distinguish different nearly extremal Kerr-Newman black holes that all satisfy
(4). These black holes may have different angular momentum and charge
parameters keeping the sum $(J^{2}/M^{2}+Q^{2})$ fixed. Without employing
bakreaction effects, one derives that every nearly extremal Kerr-Newman black
hole can be overspun by test fields, regardless of the specific value of
$\alpha$. This includes the Kerr limit $Q\to 0$,
$\alpha^{2}\to(1-\epsilon^{2})$. When one employs backreaction effects due to
the induced increase in the superradiance limit, it turns out that there are
two there are two possibilities depending on the specific value of $\alpha$.
If $\alpha<0.5$, the modified value of the superradiance limit
($\omega^{\prime}_{\rm{sl}}$), will still be smaller than $\omega_{\rm{max}}$.
The absorption of the modes in the range
$\omega^{\prime}_{\rm{sl}}<\omega<\omega_{\rm{max}}$ will lead to
overspinning. However if $\alpha>0.5$, the increase in the superradiance limit
will be sufficient to prevent the absorption of all challenging modes. Since
$\omega^{\prime}_{\rm{sl}}$ is larger than $\omega_{\rm{max}}$ for this class
of nearly extremal Kerr-Newman black holes, all the modes with
$\omega<\omega_{\rm{max}}$ that could potentially overspin the black hole will
be reflected back to infinity without any net absorption. Thus, the
backreaction effects will prevent overspinning. The argument is also valid in
the Kerr limit $Q\to 0$. For nearly extremal Kerr black holes, the parameter
$\alpha$ has a unique value which is equal to $\sqrt{1-\epsilon^{2}}$.
Manifestly, $\alpha>0.5$ for nearly extremal Kerr black holes and the
overspinning problem is fixed by employing backreaction effects.
It would be appropriate to give a numerical example to elucidate the subject.
For that purpose, let us consider a nearly extremal Kerr-Newman black hole
with $\alpha=0.5$ and $\epsilon=0.01$. Initially the parameters of this black
hole satisfy
$M^{2}-(J^{2}/M^{2})-Q^{2}=(0.0001)M^{2}$
or equivalently
$1-\alpha^{2}-(Q^{2}/M^{2})=0.0001$
For $\alpha=0.5$ the initial parameters of the black hole are given by
$J=0.5M^{2}$ and $Q^{2}=0.7499M^{2}$. Now we perturb this black hole with a
test field with energy $\delta M=M\zeta$ and angular momentum $\delta
J=(m/\omega)\delta M$. To choose the frequency of the test field we calculate
the critical values. We choose $m=1$ which is the mode with the highest
probability of absorption. We use equation (19) to calculate the superradiance
limit.
$\omega_{\rm{sl}}=0.39366(1/M)$
Letting $\zeta=\epsilon=0.01$, we can find the maximum value for the frequency
of the test field that could overspin the black hole, which is analytically
derived in (18)
$\omega_{\rm{max}}=0.399908(1/M)$
If we choose the frequency of the incoming field in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$, the test field with energy
$\delta M=0.01M$ will be absorbed by the black hole and it will lead to
overspinning. For example let us choose
$\omega=0.395(1/M)$
Using $\delta M=0.01M$ and we can calculate $\delta J$
$\delta J=(m/\omega)\delta M=0.025316M^{2}$
The final parameters of the black hole satisfy
$\delta_{\rm{fin}}=(M+\delta M)^{2}-\frac{(J+\delta J)^{2}}{(M+\delta
M)^{2}}-Q^{2}=-000319M^{2}$
Note that $Q^{2}=0.7499M^{2}$ for $\alpha=0.5$. The fact that
$\delta_{\rm{fin}}$ is negative imply that the final parameters of the black
hole describe a naked singularity. One can choose any value in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$, and verify that
$\delta_{\rm{fin}}$ is negative.
Now we employ the backreaction effects due to the induced increase in the
angular velocity of the horizon. We have argued that the increase in the
angular velocity of the horizon leads to an increase in the superradiance
limit, which will prevent the absorption of the challenging modes for a class
of nearly extremal Kerr-Newman black holes with $\alpha\gtrsim 0.5$. The
increase in the superradiance limit $(\Delta\omega)$ is analytically derived
in (22). Note that $(\Delta\omega)$ is inversely proportional to the frequency
of the incoming field so the minimum increase occurs for
$\omega\simeq\omega_{\rm{max}}$ considering the challenging modes. Using (22)
we calculate the minimum increase in the superradiance limit for this black
hole
$\Delta\omega=0.00625(1/M)$
With this increase the superradiance limit is modified as
$\omega_{\rm{sl}}^{\prime}=0.39991(1/M)$
which implies that the modes with $\omega<0.39991(1/M)$ will not be absorbed
by the black hole. Since this value is larger than $\omega_{\rm{max}}$, none
of the challenging modes will be absorbed by the black hole. Thus, the
increase in the superradiance limit prevents overspinning as no net absorption
of the challenging modes occur.
However for smaller values of $\alpha$, $\omega_{\rm{sl}}^{\prime}$ will still
be less than $\omega_{\rm{max}}$. Then, the frequencies in the range
$\omega_{\rm{sl}}^{\prime}<\omega<\omega_{\rm{max}}$ can be used to overspin
the nearly extremal black hole. The increase in the angular velocity of the
event horizon does not bring an ultimate solution to the overspinning problem
for Kerr-Newman black holes. However, backreactions is an open problem.
Various different forms of backreactions can be considered which can possibly
fix the overspinning problem.
### 3.2 Backreactions for extremal black holes
By definition, the initial parameters of extremal Kerr-Newman black holes
satisfy:
$M^{2}-Q^{2}-(J^{2})/(M^{2})=0$ (24)
or equivalently
$1-\alpha^{2}-(Q^{2}/M^{2})=0$ (25)
If an extremal Kerr-Newman black hole is perturbed by a neutral test field,
the limiting frequency for superradiance is
$\omega_{\rm{sl-
ex}}=\frac{ma}{r_{+}^{2}+a^{2}}=\frac{m}{M\left(\frac{1}{\alpha}+\alpha\right)}$
(26)
In a recent paper we have shown that $\delta_{\rm{fin}}$ becomes negative for
an extremal Kerr-Newman black hole if the frequency of the test field is less
than the maximum value generic
$\omega<\omega_{\rm{max-
ex}}=\frac{m\zeta}{M\left[(1+\zeta)\sqrt{\zeta^{2}+2\zeta+\alpha^{2}}-\alpha\right]}$
(27)
which is the $\epsilon\to 0$ limit of the value derived for nearly extremal
black holes in (18). Again the two conditions should be satisfied
simultaneously for overspinning to occur. The test field should be absorbed by
the black hole and its contribution to the angular momentum parameter should
be larger that the contribution to the mass parameter. In generic we have
shown that $\omega_{\rm{sl-ex}}$ is less than $\omega_{\rm{max-ex}}$ for a
class of extremal Kerr-Newman black holes which satisfy
$\alpha^{2}\equiv\frac{J^{2}}{M^{4}}<\frac{1}{3}\Rightarrow\alpha\lesssim
0.577$ (28)
This implies that the test fields with frequency in the range $\omega_{\rm{sl-
ex}}<\omega<\omega_{\rm{max-ex}}$ can be used to overspin the extremal Kerr-
Newman black holes which satisfy (28). The derivation is incomplete as it
ignores the contribution of the backreaction effects. As in the case of nearly
extremal black holes we calculate the induced increase in the angular velocity
of the event horizon before the absorption of the test field.
$\Delta\omega=\frac{\delta J}{4M^{3}}=\frac{(m/\omega)M\zeta}{4M^{3}}$ (29)
By substituting $\omega=\omega_{\rm{max-ex}}$ in (29) we derive that
$\Delta\omega=\frac{1}{4M}\left[(1+\zeta)\sqrt{\zeta^{2}+2\zeta+\alpha^{2}}-\alpha\right]$
(30)
We should add the induced increase in the angular momentum of the extremal
Kerr-Newman black hole derived in (30) to the superradiance limit (26). This
gives us the modified value of the superradiance limit. As we have argued in
the previous section, we demand that the modified value of the superradiance
limit is larger than $\omega_{\rm{max-ex}}$ derived in (27), so that the
absorption of all the challenging modes with $\omega<\omega_{\rm{max-ex}}$ is
prevented. For that purpose we let $\zeta=0.01$, and demand that
$\omega_{\rm{sl-ex}}+\Delta\omega>\omega_{\rm{max-ex}}$ (31)
One derives that (31) is satisfied, i.e. the modified value of superradiance
will exceed the maximum value of the frequency of the test field
$\omega_{\rm{max-ex}}$, provided that
$\alpha\gtrsim 0.31$ (32)
The dimensionless parameter $\alpha$ was introduced to distinguish extremal
Kerr-Newman black holes with different angular momentum and charge parameters
that all satisfy (24) and its dimensionless equivalent (25). In a recent paper
we derived that extremal Kerr-Newman black holes which satisfy $\alpha\lesssim
0.57$ can be overspun by test fields generic . The employment of backreaction
effects bring a further restriction to the class of extremal Kerr-Newman black
holes that can be overspun by test fields. The result (32) implies that the
absorption of all the challenging test fields will be prevented due to the
induced increase in the superradiance limit, provided that
$\alpha\equiv(J)/(M^{2})\gtrsim 0.31$. Let us elucidate the subject with a
numerical example. For that purpose let us consider an extremal Kerr-Newman
black hole with $\alpha=0.32$. The initial parameters of this black hole
satisfy $J=0.32M^{2}$ and $Q^{2}=0.8976M^{2}$ so that the black hole is
extremal. Our first claim is that the black hole can be overspun by test
fields if one ignores the backreaction effects. To verify this claim let us
first calculate the critical values $\omega_{\rm{sl-ex}}$ and
$\omega_{\rm{max-ex}}$ which were analytically derived in (26) and (27).
$\displaystyle\omega_{\rm{max-ex}}=0.298507(m/M)$
$\displaystyle\omega_{\rm{sl-ex}}=0.290276(m/M)$ (33)
We claim that if we choose a test field with frequency in the range
$\omega_{\rm{sl-ex}}<\omega<\omega_{\rm{max-ex}}$ and energy $\delta
M=M\zeta$, it will be absorbed by the extremal black hole and overspin it into
a naked singularity. Let us choose a test field with
$\omega=0.295(m/M);\quad\delta M=M\zeta=0.01M$ $\delta
J=\frac{m}{\omega}\delta M=0.033898M^{2}$
By definition $\delta_{\rm{in}}=0$ for an extremal black hole. Let us
calculate $\delta_{\rm{fin}}$.
$\delta_{\rm{fin}}=(M+\delta M)^{2}-\frac{(J+\delta J)^{2}}{(M+\delta
M)^{2}}-Q^{2}=-0.000276M^{2}$
The negative value of $\delta_{\rm{fin}}$ implies that the black hole is
overspun. However the fact that $\delta_{\rm{fin}}\sim\zeta^{2}$ indicates
that the overspinning can be fixed by backreaction effects.
Our second claim is that the overspinning of this extremal Kerr-Newman black
hole should be fixed by the induced increase in the superradiance limit since
$\alpha>0.31$. Using the analytical result (30), we calculate that, the
limiting frequency for superradiance to occur will increase by an amount
$\Delta\omega=0.008375(1/M)$ (34)
Then, for $m=1$ the modified value of the superradiance limit will be
$\omega^{\prime}_{\rm{sl-ex}}=\omega_{\rm{sl-ex}}+\Delta\omega=0.298651(1/M)$
(35)
Since the modified value of the superradiant limit exceeds $\omega_{\rm{max-
ex}}$, the test fields which could overspin the black hole will not be
absorbed by the black hole. In particular the test field that we have chosen
for our numerical example will not be absorbed, since
$\omega=0.295(1/M)<0.298651(1/M)$. However for smaller values of $\alpha$ the
modified value of the superradiance limit will still be less than
$\omega_{\rm{max-ex}}$. The induced increase in the angular velocity of the
event horizon brings further restrictions to the class of extremal Kerr-Newman
black holes which can be overspun by test fields, though it does not
completely fix the problem. As in the case of nearly extremal black holes, we
note that different type of backreaction effects can be employed to fix the
problem.
## 4 Absorption Probabilities in Wald type problems
As we stated in the introduction, the interaction of black holes with test
fields is actually a scattering problem. The test fields are partially
absorbed by the black hole, and partially reflected back to infinity. For
classical fields, t he transmission and reflection coefficients represent the
ratios of energies that are respectively, absorbed by the black hole and
scattered back to infinity. The conservation of energy implies that the sum of
the coefficients should be unity.
$\frac{\Phi_{\rm{reflected}}}{\Phi_{\rm{incident}}}+\frac{\Phi_{\rm{transmitted}}}{\Phi_{\rm{incident}}}=1$
(36)
Conventionally the relative flux
$(\Phi_{\rm{transmitted}})/(\Phi_{\rm{incident}})$ is interpreted as the
absorption probability of the incident field. This notion would be improper
for the cases of superradiant scattering. If the frequency of the incoming
field is lower than the superradiance limit $(\omega<\omega_{\rm{sl}})$, the
wave carries energy out of the black hole and the absorption probability will
be negative. The conservation of energy described by the equation (36)
continues to hold. Though the notion of a negative probability can be
improper, we shall continue adapt the conventional term “absorption
probability” for the relative flux
$(\Phi_{\rm{transmitted}})/(\Phi_{\rm{incident}})$, throughout this paper.
In all the previous Wald type problems to test the stability of event
horizons, the effect of the absorption probability of the test fields has been
ignored. The works of this author are not exceptions to this general attitude.
Ignoring the absorption probability corresponds to assuming that the
probability is of the order of unity if the field is absorbed by the black
hole. However the test fields are partially absorbed by the black hole and
partially reflected back to infinity. Only the transmitted part of the test
field contributes to the mass, angular momentum, and charge parameters of the
black hole. In this sense, the magnitude of the contribution is directly
proportional to the absorption probability
$(\Phi_{\rm{transmitted}})/(\Phi_{\rm{incident}})$. For the challenging modes,
the absorption probability approaches zero as the frequency becomes close to
the superradiance limit. This fundamentally alters the course of the analysis
for the interaction of test fields with extremal and nearly extremal black
holes.
In this work we incorporate the effect of absorption probabilities into Wald
type problems. For that purpose, we modify the contributions of a test field
to the mass, angular momentum, and charge parameters of the black hole taking
the absorption probabilities into consideration. If a test field carries
energy $M\zeta$ and the absorption probability of the field is $\Gamma$, the
energy absorbed by the black hole will be
$E_{\rm{abs}}=\Gamma(M\zeta)$ (37)
while the energy reflected back to infinity is
$E_{\rm{ref}}=(1-\Gamma)(M\zeta)$ (38)
The expressions (37) and (38) are direct consequences of the fact that only
the transmitted part of the test field contribute to the parameters of the
black hole. As we have mentioned above, in all the previous problems, the
field is assumed to be entirely absorbed by the black hole (the absorption
probability of the field is $\Gamma$ is assumed to be of the order of unity)
so that $\delta M=M\zeta$. After a sufficiently long time, the mass and
angular momentum parameters of the black hole will be modified as:
$\displaystyle M_{\rm{final}}$ $\displaystyle=$ $\displaystyle
M+E_{\rm{abs}}=M+\Gamma(M\zeta)$ $\displaystyle J_{\rm{final}}$
$\displaystyle=$ $\displaystyle
J+J_{\rm{abs}}=J+\frac{m}{\omega}\Gamma(M\zeta)$ (39)
where $E_{\rm{abs}}$ and $J_{\rm{abs}}$ are the energy and the angular
momentum absorbed by the black hole. The absorption probability $\Gamma$ can
be positive, negative or zero. The absorption probability $\Gamma$ appearing
in (39) should not be confused with the small parameter $\lambda$ introduced
by Sorce and Wald. $\Gamma$ is not necessarily a small parameter. It can be of
the order of unity for test fields with frequency $\omega\gg\omega_{\rm{sl}}$.
It is identically zero for the optimal perturbations with
$\omega=\omega_{\rm{sl}}$, and it is negative for the test fields in the
superradiant range $\omega<\omega_{\rm{sl}}$
The absorption probabilities $\Gamma_{s\omega lm}$ for the wave modes with
spin $s$, frequency $\omega$, spheroidal harmonic $l$ and azimuthal wave
number $m$, were first calculated by Page page . The absorption probability of
the wave depends on the parameters of the black hole such as the mass $M$, the
charge $Q$, the angular momentum $J$, the area of the event horizon $A$, the
surface gravity $\kappa$, the angular velocity of the event horizon $\Omega$
and the electrostatic potential of the event horizon $\Phi$. The parameters of
the black hole are not independent. In particular The area and the surface
gravity satisfy
$\kappa A=4\pi(r_{+}-M)$
where $(r_{+}-M)=\sqrt{M^{2}-a^{2}-Q^{2}}$ for a Kerr-Newman black hole. In
this work we are interested in the absorption probability of scalar fields
($s=0$) incident on Kerr-Newman black holes. The modes with $m=0$ do not
contribute to angular momentum, therefore we should consider the modes with
$m\geq 1$. For $l=1$ Page’s results imply that
$\Gamma_{0\omega
1m}=\frac{1}{9}\frac{A}{\pi}[M^{2}-(m^{2}-1)a^{2}-Q^{2}](\omega-m\Omega)\omega^{3}$
(40)
The factor $(\omega-m\Omega)$ implies that the absorption probability will be
negative if the incident field is in a superradiant mode
$(\omega<\omega_{\rm{sl}})$. In that case the mass of the black hole will
decrease. However, the angular momentum will decrease by a much larger
magnitude and $\delta_{\rm{fin}}$ will be positive. Therefore the modes in the
superradiant range $(\omega<\omega_{\rm{sl}})$ do not lead to overspinning.
### 4.1 Optimal perturbations
Optimal perturbations satisfy the inequality (3) at the lower limit so that
$\delta M=\Omega\delta J+\phi\delta Q$ (41)
This constitutes the lower limit to allow the absorption of a test body or
field. For neutral test fields with energy $\delta M$ and angular momentum
$\delta J=(m/\omega)\delta M$, (41) implies that the frequency of the test
field is equal to the superradiance limit $(\omega=\omega_{\rm{sl}}=m\Omega)$
for the optimal perturbations. Since the test bodies and fields with lower
energies than the optimal perturbations are not absorbed by the black holes,
they need not be considered for the overspinning and overcharging problems.
The optimal perturbations carry the lowest possible energy relative to their
angular momentum and charge. Thus, when one ignores absorption probabilities,
these modes appear to be more likely to lead to overspinning or overcharging
than any other mode that is absorbed by the black hole.
However when one incorporates the absorption probabilities into the problem,
the course of the analysis is fundamentally altered. It is manifest in (40)
that the absorption probability is zero for the optimal perturbations with
$\omega=\omega_{\rm{sl}}=m\Omega$. In that case the field is entirely
reflected back to infinity, with the same amplitude. No net absorption of the
field occurs. The final parameters of the spacetime given by (39) are
identically equal to the initial parameters.
$\displaystyle M_{\rm{final}}=M+\Gamma(M\zeta)=M$ $\displaystyle
J_{\rm{final}}=J+\frac{m}{\omega}\Gamma(M\zeta)=J$ (42)
Therefore the optimal perturbations do not challenge the stability of the
event horizon. Whether we start with an extremal or a nearly extremal black
hole, we end up with the same black hole surrounded by an event horizon. The
parameters of the black hole remain invariant after the interaction. The black
hole maintains its initial state. When absorption probability is taken into
account, the optimal perturbations become irrelevant for the overspinning and
overcharging problems.
### 4.2 Nearly extremal black holes and challenging modes
The modes that could potentially overspin black holes have frequencies close
to the superradiance limit. The absorption probabilities of these modes are
approach zero as one approaches the superradiance limit. By definition, a test
field carries a small amount of energy and angular momentum and only a small
fraction of its energy and angular momentum will be absorbed by the black hole
if its frequency is close to the superradiance limit. Therefore it seems to be
very difficult for a test field to drive a nearly extremal black hole to
extremality and beyond, when the absorption probability is taken into account.
To evaluate this quantitatively, let us consider a scalar field with frequency
$\omega=m\Omega(1+\xi)$ (43)
where the small parameter $\xi\ll 1$ assures that the frequency of the
incoming field is close to the superradiance limit $\omega_{\rm{sl}}=m\Omega$.
The scalar field is incident on a nearly extremal Kerr-Newman black hole
parametrised as (4), where $\epsilon\ll 1$ determines the closeness of the
black hole to extremality. The highest absorption probability occurs for
$m=1$. Substituting $\omega=\Omega(1+\xi)$ in (40)
$\displaystyle\Gamma_{0\omega 11}$ $\displaystyle=$
$\displaystyle\frac{8}{9}M^{2}(1+\epsilon)[M^{2}-Q^{2}](\Omega\xi)[\Omega(1+\xi)]^{3}$
(44) $\displaystyle\sim$ $\displaystyle\xi+O(\epsilon\xi)$
where we have used $A=8\pi M^{2}(1+\epsilon)$ for a nearly extremal black
hole. The leading term in the absorption probability of a challenging mode is
of the order of $\xi$. Now, we can re-evaluate the overspinning problem taking
the absorption probability into consideration. We start with a Kerr-Newman
black hole satisfying
$M^{2}-Q^{2}-\frac{J^{2}}{M^{2}}=M^{2}\epsilon^{2}$
which is perturbed by a test field with
$\displaystyle m=1;\quad\omega=m\Omega(1+\xi)$ $\displaystyle\delta
M=M\zeta;\quad\delta J=\frac{1}{\omega}\delta M;\quad\delta Q=0$
$\displaystyle\Gamma\sim\xi$ (45)
We should note that we choose the energy, frequency, and the azimuthal wave
number of the test field to challenge the stability of the event horizon. The
absorption probability of this test field is not a choice; it follows from
(40). After the interaction of the test field with the Kerr-Newman black hole,
the final parameters of the space-time will take the form
$\displaystyle M_{\rm{final}}$ $\displaystyle=$ $\displaystyle M+M\zeta\xi$
$\displaystyle J_{\rm{final}}$ $\displaystyle=$ $\displaystyle
J+\frac{1}{\omega}M\zeta\xi$ $\displaystyle Q_{\rm{final}}$ $\displaystyle=$
$\displaystyle Q$ (46)
We demand that the final parameters of the space-time describe a naked
singularity.
$(M+M\zeta\xi)^{2}-Q^{2}-\frac{\left(J+\frac{1}{\omega}M\zeta\xi\right)^{2}}{(M+M\zeta\xi)^{2}}<0$
(47)
As in the previous sections, we eliminate $Q$ from (47), and the define the
dimensionless variable $\alpha=J/M^{2}$. After some algebra one derives that
the condition (47) is equivalent to
$\omega<\omega_{\rm{max}}=\frac{\zeta\xi}{M\left[(1+\zeta\xi)\sqrt{2\zeta\xi+\epsilon^{2}+\alpha^{2}}-\alpha\right]}$
(48)
We have assumed that the frequency of the incoming field is slightly larger
than the superradiance limit by imposing $\Gamma=\xi$. Remember that the
superradiance limit for a neutral test field is
$\omega_{\rm{sl}}=\frac{m}{M\left[\frac{(1+\epsilon)^{2}}{\alpha}+\alpha\right]}$
The maximum value of the frequency of the incident field derived in (48), has
to be larger than the superradiance limit. Letting $\epsilon=\zeta=\xi=0.01$,
one derives that this will only be possible if
$\alpha\lesssim 0.01041$ (49)
We started with a nearly extremal Kerr-Newman black hole parametrised as (4).
We perturbed this black hole with a test field with energy $\delta M=M\zeta$.
For overspinning to occur the frequency of the test should be as small as
possible since the contribution to the angular momentum is inversely
proportional to the frequency. The test field should also be absorbed by the
black hole which entails that the frequency should be larger than the
superradiance limit. Therefore we the frequency should be slightly larger than
the superradiance limit ($\omega=m\Omega(1+\xi)$) for the test field. Using
(40) which follows from the seminal results by Page page , one can show that
the absorption probability for this field is of the order of $\xi$. The final
parameters of the black hole are given by (39), which implies that only the
fraction of the test field that is absorbed by the black hole contributes to
the mass, angular momentum, (and charge) parameters. We demand that the final
parameters of the black hole describe a naked singularity; i.e.
$\delta_{\rm{fin}}<0$. We derive that this demand is satisfied if the
frequency of the field is smaller than the maximum value derived in (48). This
value is larger than the superradiance limit for a class of Kerr-Newman black
holes with $\alpha=J/M^{2}\lesssim 0.01041$. To clarify the reader, we note
that for a nearly extremal Kerr-Newman black hole with mass $M=1$ and
$\alpha=0.01$, the angular momentum and charge parameters satisfy
$J^{2}=0.0001$ and $Q^{2}=0.9998$ with $\epsilon=0.01$. (See equation (17). )
For this class of nearly extremal Kerr-Newman black holes there exist modes
with positive absorption probability, that could increase the angular momentum
parameter beyond the extremal limit. This stems from the fact that the modes
with very low frequencies can have positive absorption probabilities for such
low values of $\alpha$. For these modes, the contribution to angular momentum
will be very large as it is inversely proportional to the frequency. However,
the induced increase in the angular velocity of the event horizon will also be
very large for these fields. The modified value of the superradiance limit,
which was analytically derived in (21), will considerably exceed the frequency
of the test field and its absorption will be prevented.
Let us clarify this argument with a quantitative examle. For that purpose we
consider a nearly extremal Kerr-Newman black hole with $\alpha\equiv
J/M^{2}=0.01$. For this black hole, we can use (48) and the general expression
for the superradiance limit to find that
$\displaystyle\omega_{\rm{max}}=0.009998\frac{1}{M}$
$\displaystyle\omega_{\rm{sl}}=0.009802\frac{1}{M}$ (50)
Let us consider a test field with
$\displaystyle m=1;\quad\omega=0.0099\frac{1}{M}\sim m\Omega(1+\xi)$
$\displaystyle\delta M=M\zeta=0.01M$ $\displaystyle\delta
J=\frac{m}{\omega}\delta M=1.0101M^{2}$ $\displaystyle\Gamma\sim\xi$ (51)
Without considering backreaction effects, one derives that the final
parameters of the black hole given in (46) describe a naked singularity rather
than a black hole. However, one can notice that $\delta J$ is too large for
this field; in particular it is even larger than $M^{2}$. Let us calculate the
modified value of the limiting frequency due to the induced increase in the
angular momentum of the horizon, for this mode. Using (20) and (21),
$\displaystyle\Delta\omega=\frac{\delta
J}{4M^{3}}=0.2525\left(\frac{1}{M}\right)$
$\displaystyle\omega_{\rm{sl}}^{\prime}=\omega_{\rm{sl}}+\Delta\omega=0.262302\left(\frac{1}{M}\right)$
(52)
The modified value of the superradiance limit is far larger than the frequency
of the incoming field, which assures that the test field will not be absorbed
by the black hole. Therefore the backreaction effects fix the overspinning
problem for nearly extremal Kerr-Newman black holes with $\alpha\equiv
J/M^{2}\lesssim 0.01041$. Moreover, one can argue that the challenging modes
for this class of Kerr-Newman black holes cannot be treated in the test field
approximation, since $\delta J\gtrsim M^{2}$. In this case these modes can be
excluded, and the overspinning problem becomes irrelevant. In either case, the
incorporation of the absorption probability brings an ultimate solution to the
overspinning problem for nearly extremal Kerr-Newman black holes.
### 4.3 Extremal black holes and challenging modes
We mentioned that the absorption probability is very low for the challenging
modes. Most of the energy and angular momentum carried by the test field is
reflected back to infinity. This leads one to conclude that it should be very
difficult for a test field to drive a nearly extremal black hole to
extremality and beyond. The situation is different for extremal black holes. A
tiny excess amount of angular momentum can lead to overspinning, since
$\delta_{\rm{in}}$ is equal to zero. In this section, we calculate the
possibility to overspin an extremal Kerr-Newman black hole by neutral test
fields by taking the absorption probabilities into consideration. By
definition, an extremal black hole satisfies (24) and its dimensionless
equivalent (25). The dimensionless parameter $\alpha\equiv J/M^{2}$ allows us
to distinguish extremal black holes with different parameters of charge and
angular momentum. As in the case of nearly extremal black holes we attempt to
destroy the horizon by sending in a test field with energy $\delta M=M\zeta$,
and frequency close to the superradiance limit, such that the absorption
probability takes the form $\Gamma\sim\xi$. Proceeding the same way as in the
previous section, we choose $m=1$ for the test field which maximizes the
absorption probability. We find that the event horizon will be destroyed if
the frequency of the test field satisfies
$\omega<\omega_{\rm{max-
ex}}=\frac{\zeta\xi}{M\left[(1+\zeta\xi)\sqrt{2\zeta\xi+\alpha^{2}}-\alpha\right]}$
(53)
which is the $\epsilon\to 0$ limit of the result found in (48). Remember that
the superradiance limit for a neutral test field incident on an extremal black
hole is
$\omega_{\rm{sl}}=\frac{1}{M\left[\frac{1}{\alpha}+\alpha\right]}$
where $m=1$. Letting $\zeta=\xi=0.01$, one observes that the upper limit
$\omega_{\rm{max-ex}}$ derived in (53), and the lower limit $\omega_{\rm{sl}}$
for the range of the frequencies that can lead to overspinning, almost
coincide in the range $0<\alpha<1$. The upper limit derived in (53) is
slightly larger than the superradiance limit provided that
$\alpha\lesssim 0.70707$ (54)
where the dimensionless parameter $\alpha$ defined in (16), distinguishes
extremal Kerr-Newman black holes with different angular momentum and charge
parameters, keeping the sum $(Q^{2}+J^{2}/M^{2})$ fixed. For extremal Kerr-
Newman black holes satisfying (54), there exists a narrow range of frequencies
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max-ex}}$ that can lead to overspinning.
However, this can easily be fixed by employing backreaction effects. The
induced increase in the angular momentum of the event horizon can be
calculated as
$\Delta\omega=\frac{\delta
J}{4M^{3}}=\frac{1}{M}\frac{(1+\zeta\xi)\sqrt{2\zeta\xi+\alpha^{2}}-\alpha}{4\xi}$
(55)
where we have used
$\delta J=\frac{m}{\omega}\delta M=\left(\frac{1}{\omega_{\rm{max-
ex}}}\right)M\zeta$
As we have argued previously it is critical to calculate $\Delta\omega$ for
the upper limit $\omega_{\rm{max-ex}}$. Substituting the superradiance limit
for extremal black holes and the induced increase in the superradiance limit
derived in (55), the modified value of the superradiance limit takes the form:
$\displaystyle\omega^{\prime}_{\rm{sl}}$ $\displaystyle=$
$\displaystyle\omega_{\rm{sl}}+\Delta\omega$ (56) $\displaystyle=$
$\displaystyle\frac{1}{M\left[\frac{1}{\alpha}+\alpha\right]}+\frac{1}{M}\frac{(1+\zeta\xi)\sqrt{2\zeta\xi+\alpha^{2}}-\alpha}{4\xi}$
To overspin an extremal Kerr-Newman black hole, the frequency of a test field
should be less than the maximum value derived in (53). A test field with such
a low frequency can have a positive absorption probability only for a class of
extremal black holes satisfying (54). We checked if the overspinning can be
fixed by the induced increase in the angular momentum of the black hole, which
modifies the superradiance limit. Due to the induced increase in the angular
momentum, the superradiance limit increases. The absorption of the modes with
frequencies lower than the modified value of the superradiance limit is
prevented, though their absorption probability appears to be positive when one
ignores backreaction effects based on the induced increase in the angular
momentum. If this modified value exceeds the maximum value derived in (53),
the absorption of all the challenging modes with low frequencies and positive
absorption probabilities will be prevented. One observes that the modified
value of the superradiance limit exceeds the maximum value of the frequency
that can lead to overspinning, for any value of $\alpha$ in the range
$0<\alpha<1$. To clarify this, we have plotted $\omega_{\rm{max-ex}}$,
$\omega_{\rm{sl}}$, and $\omega_{\rm{sl}}^{\prime}$ as a function of $\alpha$
in figure (1).
Figure 1: The superradiance limit $\omega_{\rm{sl}}$ and the upper limit
$\omega_{\rm{max-ex}}$ almost coincide in the range $0<\alpha<1$. The modified
value of superradiance $\omega_{\rm{sl}}^{\prime}$ exceeds the upper limit
$\omega_{\rm{max-ex}}$. (Here we let $M=1$)
The fact that the modified value of the superradiance limit exceeds the upper
limit of frequency $\omega_{\rm{max-ex}}$, indicates that the absorption of
the fine-tuned frequencies in the narrow range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max-ex}}$ will be prevented; i.e. the
event horizon cannot be destroyed. Taking absorption probabilities into
consideration fixes the overspinning problem for the extremal case as well as
the nearly extremal case.
## 5 Summary and conclusions
In this work we have re-considered the overspinning problem for Kerr-Newman
black holes. First, we have scrutinized the recent analysis by Sorce and Wald
where they employ a variational method and expand the field configurations to
second order in the small parameter $\lambda$. Sorce and Wald claim that they
have obtained an expression for the full second order correction $\delta^{2}M$
without having to calculate the backreaction effects explicitly. In a recent
paper we have also employed the method developed by Sorce and Wald for MTZ
black holes. In that work we have imposed that $\delta M$ is a first order
quantity itself, for test bodies and fields. We have noticed that imposing
this non-controversial fact leads to order of magnitude problems in the SW
method. Therefore we have concluded that it would be better to calculate the
backreaction effects explicitly. Here we argued that the order of magnitude
problems do not pertain to the case of MTZ black holes, they also appear in
the case of Kerr-Newman black holes. The argument is simple. Since $\delta M$
is inherently a first order quantity:
$\lambda\delta M\Rightarrow\mbox{second order, not first}$
$\lambda\epsilon\delta M\Rightarrow\mbox{third order, not second}$
$\lambda^{2}(\delta M)^{2}\Rightarrow\mbox{fourth order, not second}$
$\lambda^{2}\delta^{2}M\Rightarrow\mbox{fourth order, not second}$
Sorce and Wald claim that the function $f(\lambda)$ in the equation (6) –which
is the equation (119) in w2 – can be made negative for the terms first order
in $\lambda$ and the contribution of the terms second order in $\lambda$ makes
$f(\lambda)$ positive again. This implies that the previous results due to
Hubeny hu , Jacobson-Sotiriou js , and Düztaş-Semiz overspin are reproduced.
Nearly extremal black holes can be destroyed by test bodies and fields and the
destruction of the event horizon is fixed by employing backreaction effects.
We showed that these correct results cannot be reproduced by SW method. The
leading term in $f(\lambda)$ is of the second order $(M^{2}\epsilon^{2})$,
whereas the contribution of the second order perturbations are of the fourth
order $(\lambda^{2}\delta^{2}M)$. Therefore the effect of second order
corrections cannot be incorporated using SW method, unless one fallaciously
imposes $\delta M\sim M$. This assumption apparently contradicts the test
body/field approximation. For that reason, backreactions should be identified
and explicitly calculated for every specific problem.
Based on the argument that the SW method is invalid, we re-visited the
overspinning problem for extremal and nearly extremal Kerr-Newman black holes.
In a recent paper we had shown that there exists a class of extremal Kerr-
Newman black holes which can be overspun by neutral test fields generic . The
overspinning is possible if the extremal Kerr-Newman black hole satisfies
$\alpha^{2}\equiv J^{2}/M^{4}<1/3$. (The dimensionless parameter $\alpha$ is
introduced to distinguish black holes with different angular momentum and
charge parameters keeping the sum $(Q^{2}+J^{2}/M^{2})$ fixed. See equations
(17) and (25). ) There, we gave numerical examples and compared our results
with previous claims. Here we showed that nearly extremal Kerr-Newman black
holes can also be overspun independent of the value of $\alpha$. In our
analysis we do not ignore the contribution of the second order terms $(\delta
M)^{2}$ and $(\delta J)^{2}$ which drastically changes the results. Therefore
we need to calculate the backreaction effects to complete our analysis. In
this work, we employed the backreaction effects due to the increase in the
angular velocity of the horizon which was first suggested by Will will . The
induced increase in the angular velocity of the event horizon leads to an
increase in the superradiance limit. This prevents the absorption of the modes
that could potentially overspin the black hole. We showed that this
backreaction brings further restrictions to the classes of extremal and nearly
extremal black holes that can be overspun by test fields. Overspinning is
prevented for nearly extremal black holes with $\alpha\gtrsim 0.50$ and for
extremal black holes with $\alpha\gtrsim 0.31$. We noted that the effect of
backreactions is an open problem and there could be different sorts of
backreaction effects that could potentially bring a full solution to the
overspinning problem.
The results derived in this work can also be exploited to evaluate the
possibility to overspin Kerr black holes. In the limit $Q\to 0$, extremal Kerr
black holes are identified with $\alpha\equiv J/M^{2}=1$ whereas nearly
extremal ones are parametrised as $\alpha^{2}=1-\epsilon^{2}$. There is no
overspinning problem for extremal Kerr black holes even if one ignores the
backreaction effects, since $\alpha^{2}=1>(1/3)$. The backreaction effects due
to the increase in the superradiance limit fixes the overspinning problem for
nearly extremal Kerr black holes, since $\alpha=1-\epsilon^{2}>(0.50)$. These
findings are in accord with previous results on Kerr black holes.
In all the previous thought experiments –including the works of this author–
the absorption probability of test fields was ignored. Ignoring the
probability corresponds to assuming that it is of the order of unity. However,
the interaction of black holes with test fields is a scattering problem. A
fraction of the test field is absorbed by the black hole, while part of it is
reflected back to infinity. In section (4), we have incorporated the
absorption probabilities in the thought experiments to test whether the event
horizon can be destroyed. The fact that only a small fraction of the
challenging modes is absorbed by the black holes, fundamentally changes the
course of the analysis in favour of the cosmic censorship conjecture. We
calculated the absorption probability for test fields with frequency close to
the superradiance limit, using the seminal results by Page page . The
absorption probability of a test field with frequency $\omega=m\Omega(1+\xi)$
turns out to be of the order $O(\xi)$. The probability approaches zero for
optimal perturbations with frequency $\omega=m\Omega$, which implies that
these fields are entirely reflected back to infinity. The parameters of the
space-time remain invariant after the interaction with these test fields.
Hence, the event horizon cannot be destroyed. For the nearly extremal case, we
derived that there exists a class of Kerr-Newman black holes identified by
$\alpha\lesssim 0.01401$, which can be destroyed by test fields. Overspinning
occurs due to the fact that test fields with very low frequencies can be
absorbed by these black holes. The contribution of these test fields to the
angular momentum parameter will be large, since it is inversely proportional
to the frequency. However the induced increase in the angular momentum of the
event horizon is also large for these perturbations. Therefore the
overspinning is easily fixed by employing the backreaction effects. We noted
that one can also argue that these fields cannot be treated in the test field
approximation due to the large magnitude of $\delta J$. This argument would
render the overspinning problem irrelevant. For the case of extremal black
holes we derived that there exists a set of fine-tuned parameters that can
overspin Kerr-Newman black holes with $\alpha\lesssim 0.70707$. However, the
range of frequencies is very narrow and the overspinning problem is fixed by
employing backreaction effects. Both for extremal and nearly extremal Kerr-
Newman black holes, the ultimate solution to the overspinning problem follows
by the incorporation of the absorption probabilities into the analysis.
In a recent paper we argued that fermionic fields lead to a generic
destruction of the event horizon in the classical picture generic . Since the
fermionic fields do not obey the weak energy condition, one cannot find a
lower bound for the energy of a fermionic field similar to the condition (3).
The absorption probability is positive definite and it approaches zero only as
$\omega$ approaches zero, as confirmed by Page page . For that reason, the
arguments about the absorption probability of scalar fields and its effect on
the overspinning problem developed in this paper, do not apply to fermionic
fields.
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11institutetext: Kapteyn Astronomical Institute, University of Groningen,
Landleven 12, 9747 AD Groningen, The Netherlands
11email<EMAIL_ADDRESS>22institutetext: RIKEN Center for Computational
Science, 7-1-26 Minatojima-minami-machi, Chuo-ku, Kobe, Hyogo 650-0047, Japan
33institutetext: Department of Physics, School of Science, The University of
Tokyo, Bunkyo, Tokyo 113-0033, Japan 44institutetext: Research Center for the
Early Universe, Graduate School of Science, University of Tokyo, Bunkyo-ku,
Tokyo 113-0033, Japan 55institutetext: Kavli IPMU (WPI), UTIAS, The University
of Tokyo, Kashiwa, Chiba 277-8583, Japan 66institutetext: Astronomical
Institute, Tohoku University, Aoba, Sendai 980-8578, Japan
# $R$-process enhancements of Gaia-Enceladus in GALAH DR3
Tadafumi Matsuno 11 Yutaka Hirai 2266 Yuta Tarumi 33 Kenta Hotokezaka 4455
Masaomi Tanaka 66 Amina Helmi 11
###### Abstract
Context. The dominant site of production of $r$-process elements remains
unclear despite recent observations of a neutron star merger. Observational
constraints on the properties of the sites can be obtained by comparing
$r$-process abundances in different environments. The recent Gaia data
releases and large samples from high-resolution optical spectroscopic surveys
are enabling us to compare $r$-process element abundances between stars formed
in an accreted dwarf galaxy, Gaia-Enceladus, and those formed in the Milky
Way.
Aims. We aim to understand the origin of $r$-process elements in Gaia-
Enceladus.
Methods. We first construct a sample of stars to study Eu abundances without
being affected by the detection limit. We then kinematically select 76 Gaia-
Enceladus stars and 81 in-situ stars from the Galactic Archaeology with HERMES
(GALAH) DR3, of which 47 and 55 stars can be used to study Eu reliably.
Results. Gaia-Enceladus stars clearly show higher ratios of [Eu/Mg] than in-
situ stars. High [Eu/Mg] along with low [Mg/Fe] are also seen in relatively
massive satellite galaxies such as the LMC, Fornax, and Sagittarius dwarfs. On
the other hand, unlike these galaxies, Gaia-Enceladus does not show enhanced
[Ba/Eu] or [La/Eu] ratios suggesting a lack of significant $s$-process
contribution. From comparisons with simple chemical evolution models, we show
that the high [Eu/Mg] of Gaia-Enceladus can naturally be explained by
considering $r$-process enrichment by neutron-star mergers with delay time
distribution that follows a similar power-law as type Ia supernovae but with a
shorter minimum delay time.
## 1 Introduction
The observations of gravitational waves from a neutron star merger (NSM)
GW170817 and its electromagnetic counterparts (Abbott et al., 2017) provided
evidence that a copious amount of $r$-process elements are ejected in NSMs
(e.g., Kasen et al., 2017; Tanaka et al., 2017; Rosswog et al., 2018; Watson
et al., 2019). Despite the fact that the estimated amount of $r$-process
elements produced in GW170817 ($\sim 0.05M_{\odot}$) is sufficient to provide
all the $r$-process elements in the Milky Way, the enrichment history of
$r$-process elements in the Milky Way is still under debate (e.g., Matteucci
et al., 2014; Ishimaru et al., 2015; Shen et al., 2015; van de Voort et al.,
2015, 2020; Hotokezaka et al., 2018; Haynes & Kobayashi, 2019; Côté et al.,
2019).
One of the ways to tackle this problem is to investigate stars formed in
different environments, as we may expect these to have had different star
formation timescales and initial mass function (IMF) than the Milky Way. In
small systems such as the low-mass dwarf galaxies around the Milky Way, the
expected number of $r$-process enrichment events becomes less than one, which
enables one to estimate rate and yield of a single event (e.g., Hirai et al.,
2015, 2017; Beniamini et al., 2016; Safarzadeh & Scannapieco, 2017; Ojima et
al., 2018; Tarumi et al., 2020). For example, Ji et al. (2016) reported that
an ultra-faint dwarf galaxy (Reticulum II; $M_{\star}\sim
10^{3}\,\mathrm{M_{\odot}}$) contains a number of stars with enhanced
$r$-process abundance. From the fraction of ultra-faint dwarf galaxies with
enhanced $r$-process abundance, they estimate one $r$-process production event
per 1000$-$2000 supernovae. The observed [Eu/H] also provides an estimate on
the yield from a single event as $M_{\rm Eu}\sim
10^{-4.5}\,\mathrm{M_{\odot}}$. Tsujimoto et al. (2017) obtained a similar
yield from the observation of very metal-poor stars in the more massive
($M_{\star}\sim 10^{5}\,\mathrm{M_{\odot}}$) Draco dwarf spheroidal galaxy.
Observations of stars in other satellites of the Milky Way have shown that the
most massive dwarf galaxies ($M_{\star}>10^{7}\,\mathrm{M_{\odot}}$) tend to
have enhanced $r$-process abundances. For example, McWilliam et al. (2013)
have shown that [Eu/Mg] ratio is higher in the Sagittarius dwarf galaxy than
in the Milky Way. Together with the abundances of other elemental abundances,
they interpret this result as a consequence of a top-light IMF in Sagittarius
and the production of Eu by relatively low-mass supernovae compared to those
producing Mg. Lemasle et al. (2014) also reached a similar conclusion from the
high [Eu/Mg] ratio of Fornax dwarf galaxy. On the other hand, Skúladóttir &
Salvadori (2020) suggested that the high [Eu/Mg] values observed in these two
galaxies are due to the delay time of $r$-process production events, which is
consistent with NSMs as the production site. We note that Skúladóttir &
Salvadori (2020) also suggested the need of quick source for $r$-process
elements in addition to the delayed enrichments by NSMs to explain abundance
pattern in another dwarf galaxy, Sculptor.
Figure 1: A fraction of stars with Eu detection as a function of $\log g$,
[Fe/H], and $S/N$. The color and the size of the symbol respectively shows the
fraction of Eu detection and the total number of stars in each bin. Dashed
lines show $[\mathrm{Fe/H}]=-1.3$ and $\log g=1.9$.
Thanks to the recent data releases from the Gaia mission (Gaia Collaboration
et al., 2016, 2018, 2020), we are now able to carry out a similar exercise in
the Milky Way. Stars originated from the same accreted galaxy share similar
motions even long after the accretion event, creating substructures in the
distribution of stellar kinematics (e.g., Helmi & de Zeeuw, 2000; Gómez &
Helmi, 2010). Precise astrometry by the Gaia mission has enabled the
identification of such substructures from a large sample of stars with precise
kinematic data (e.g., Koppelman et al., 2018, 2019; Helmi et al., 2018;
Belokurov et al., 2018; Myeong et al., 2018; Yuan et al., 2019; Naidu et al.,
2020). The advantage of studying abundance patterns of these accreted stars is
that some of these are located in the proximity of the Sun. This makes it
possible to carry out detailed investigation of chemical abundances over many
elements from high signal-to-noise ratio, high-resolution spectroscopy.
The most prominent kinematic substructure seen in the Galactic halo is Gaia-
Enceladus a.k.a. Gaia-Sausage (Belokurov et al., 2018; Helmi et al., 2018),
which is now considered to be the debris from the last major merger that Milky
Way has experienced. Helmi et al. (2018) and Haywood et al. (2018) found that
chemically the stars correspond to the group of halo stars with low [Mg/Fe]
abundance ratios first discovered by Nissen & Schuster (2010, hereafter,
NS10). This low [Mg/Fe] is generally interpreted as a result of combined
effect of prolonged star formation of this population and delayed enrichment
of Fe by type Ia supernovae (SNe Ia) (NS10, Vincenzo et al., 2019). A second
also important population of stars with hot kinematics, has high [Mg/Fe] up to
high metallicity. This indicates the stars formed on a short timescale so that
their abundance ratio is predominantly determined by the yields of massive
stars. Their kinematics suggest that this high-Mg population corresponds to
the Milky Way’s disk that was present (or partly formed) during the merger
with Gaia-Enceladus (e.g., NS10; Schuster et al., 2012; McCarthy et al., 2012;
Helmi et al., 2018; Belokurov et al., 2019).
In the present study, we compare Eu abundances of stars in Gaia-Enceladus with
those of stars formed in the Milky Way (the in-situ stars having high-[Mg/Fe])
with the aim to obtain constraints on $r$-process enrichment processes.
Although Ishigaki et al. (2013), Fishlock et al. (2017) and Matsuno et al.
(2020) presented hints of Eu enhancements of the low-Mg halo stars, their
samples were of limited size. Thanks to the Gaia mission and the recent data
release from the optical high-resolution spectroscopic survey, the Galactic
Archaeology with HERMES (GALAH; De Silva et al., 2015), we can now study with
a larger sample analysed homogeneously. Gaia-Enceladus provides not only an
opportunity to study stars formed outside of the Milky Way in detail with
high-quality spectra, but also enables us to study the effect of the duration
of star formation. In comparison to the three massive satellite galaxies of
the Milky Way (namely Sagittarius, Fornax and the LMC) all of which have had
prolonged star formation history, star formation in Gaia-Enceladus was
truncated about $\sim 10\,\mathrm{Gyr}$ as a result of tidal disruption.
This paper is organised as follows. We firstly discuss the sample selection in
Section 2, move on to the results in Section 3, and finally provide
interpretation in Section 4.
## 2 Data
Figure 2: A portion of GALAH spectra around the Eu $6645\,\mathrm{\AA}$
absorption line for 13 stars with $1.7<\log g<1.9$,
$50<\texttt{snr\\_c3\\_iraf}<60$, and $-1.3<[\mathrm{Fe/H}]<-1.2$, of which 11
stars have Eu detection. The location of the Eu absorption line is indicated
by the vertical orange line. The detection of the Eu line is clear for the 11
objects. Figure 3: Kinematics of the halo stars in GALAH DR3. Gaia-Enceladus
(orange) and in-situ stars (blue) are selected in the $J_{R}-L_{z}$ plane (see
text). Figure 4: [Mg/Fe] abundance ratio of prograde (positive $L_{Z}$) stars
with $-1.2<[\mathrm{Fe/H}]<-0.8$ as a function of radial action ($J_{R}$).
Green squares show stars that satisfy the selection criteria a)-b), while grey
points are selected without the $\log g$ selection. The lower (upper) boundary
of $\sqrt{J_{R}}$ for Gaia-Enceladus (in-situ) stars selection is indicated.
This figure illustrates the selection in $J_{R}$ efficiently select Gaia-
Enceladus and in-situ stars with high purity.
We use chemical abundance from GALAH DR3 (De Silva et al., 2015; Buder et al.,
2020). The GALAH survey measures chemical abundance of stars from high-
resolution optical spectra ($R\sim 28000$) with typical signal-to-noise ratio
($S/N$) of 50. In the present study, we focus on five elements (Mg, Fe, Ba,
La, and Eu), for which GALAH wavelength coverage allows determination of
abundances. Following selections are imposed to discuss abundances of these
elements.
* a)
$\texttt{flag\\_sp}=0$ and $\texttt{flag\\_fe\\_h}=0$
* b)
$\log g<1.9$ and $\texttt{snr\\_c3\\_iraf}>50$
The first condition is to ensure that stellar parameters and metallicity are
measured reliably. When discussing elemental abundance ratios [X/Y], we
further limit the sample to those with flag_X_fe$=0$ and flag_Y_fe$=0$, which
mean that the abundances of these elements are actually measured.
The last condition is used to construct a sample that includes high fraction
of stars with Eu detection ($\texttt{flag\\_Eu\\_fe}=0$). Eu measurements in
GALAH rely on the Eu $6645\,\mathrm{\AA}$ line, which is not so strong to be
detected in high-gravity low-metallicity stars. Figure 1 shows how the
fraction of stars with Eu detection changes as a function of [Fe/H], surface
gravity ($\log g$), and the average $S/N$ in the CCD3 (snr_c3_iraf), where the
Eu line is located. It is clear in the figure that the fraction of Eu
detection decreases toward lower metallicity, higher gravity, and lower $S/N$.
The $\log g$ dependency is naturally expected since most of Eu are singly
ionized in the photospheres of F, G, and K type stars and since the line is
formed by singly-ionized Eu (Gray, 2008). From the inspection of Figure 1, we
conclude that the fraction of Eu detected stars remains high ($>70-80\%$) down
to [Fe/H]$\sim-1.3$ if we impose the condition b), which can be confirmed from
Figure 2, where spectra around the Eu $6645\,\mathrm{\AA}$ line are shown for
stars that are close to the selection boundaries. We caution against
interpreting Eu abundance below [Fe/H]$=-1.3$ since the obtained abundance
trend could be biased because of the large fraction of stars without Eu
detection. We note that the fractions for other elements (Mg, Ba, and La)
remain very high ($>95\%$) down to [Fe/H]$=-2.0$ if we adopt the selection
conditions a)-b).
Figure 5: Chemical abundances of ratios of Gaia-Enceladus and in-situ stars in
GALAH DR3. Typical uncertainties are shown in black symbols in the bottom. The
grey shaded region indicates the metallicity range where the fraction of stars
with Eu detection becomes small. Small blue and red triangles are high-Mg/low-
Mg populations from NS10, for which abundances are taken from NS10, NS11 and
Fishlock et al. (2017).
We further select stars based on their kinematics, which are also provided as
a GALAH DR3 value-added catalog (Buder et al., 2020), which is based on Gaia
data release 2 (Gaia Collaboration et al., 2018; Lindegren et al., 2018).
Although details of the calculation are described in Buder et al. (2020), we
note that they calculated kinematics assuming the Milky Way potential of
McMillan (2017). We firstly select stars satisfying
$\texttt{parallax\\_over\\_error}>5$, $\texttt{ruwe}<1.4$, and
$|\@vec{v}-\@vec{v}_{\rm LSR}|>180\,\mathrm{km\,s^{-1}}$. The first two
conditions are on the quality of astrometric measurements to ensure reliable
kinematic information, while the last condition on kinematics is to remove the
majority of disk stars. The kinematics of the selected stars are shown in
Figure 3. We note that our kinematic selection is not meant to exclusively
select halo stars. The high-Mg in-situ halo population is known to have the
identical chemical abundance at fixed metallicity as thick disk stars (NS10;
Nissen & Schuster, 2011, hereafter, NS11). It is indeed suggested to be heated
disk stars (e.g., McCarthy et al., 2012; Helmi et al., 2018; Belokurov et al.,
2019) hence having similar formation sites as thick disk stars. The inclusion
of some amount of thick disk stars in the sample allows us to have a large
sample of in-situ stars to which abundances of Gaia-Enceladus stars are
compared.
We use the radial action ($J_{R}$) and the angular momentum around the
$z$-axis of the Galaxy ($L_{z}$) since this $J_{R}-L_{z}$ plane enables a
clean selection of Gaia-Enceladus stars (Feuillet et al., 2020). The selection
for Gaia-Enceladus is taken from Feuillet et al. (2020) as
$-500\,\mathrm{kpc\,km\,s^{-1}}<L_{z}<500\,\mathrm{kpc\,km\,s^{-1}}$ and
$30\,\mathrm{kpc^{1/2}\,km^{1/2}\,s^{-1/2}}<\sqrt{J_{R}}$ (Figure 3).
Similarly in-situ stars are selected as $0\,\mathrm{kpc\,km\,s^{-1}}<L_{z}$
and $\sqrt{J_{R}}<15\,\mathrm{kpc^{1/2}\,km^{1/2}\,s^{-1/2}}$. In this way, we
have selected 76 and 81 stars as Gaia-Enceladus and in-situ stars, of which 60
and 61 stars have Eu detection, respectively. The numbers of stars at
[Fe/H]$>-1.3$, where we consider we can reliably interpret the measured Eu
abundance, are 47 and 58, of which 47 and 55 stars have Eu measurements.
The choice of lower (upper) boundary in $\sqrt{J_{R}}$ for Gaia-Enceladus (in-
situ) selections is justified in Figure 4, where [Mg/Fe] ratios of prograde
stars within $[\mathrm{Fe/H}]=-1.0\pm 0.2$ are shown as a function of
$\sqrt{J_{R}}$. Since the [Mg/Fe] difference between Gaia-Enceladus and in-
situ stars is clear in this metallicity range, these stars allow us to
investigate how well we are selecting Gaia-Enceladus / in-situ stars. It is
clear that below $\sqrt{J_{R}}=15\,\mathrm{kpc^{1/2}\,km^{1/2}\,s^{-1/2}}$,
almost all the stars have high [Mg/Fe], indicating high purity of our in-situ
selection. Similarly the figure also illustrates the absence of high [Mg/Fe]
at $\sqrt{J_{R}}>30\,\mathrm{kpc^{1/2}\,km^{1/2}\,s^{-1/2}}$, showing high
purity in the Gaia-Enceladus selection.
## 3 Results
Figure 6: Same as Figure 5, but for abundance ratios between Mg, Ba, La, and
Eu.
The obtained chemical abundance ratios are shown in Figures 5 and 6. It is
clear that the Gaia-Enceladus stars show lower [Mg/Fe] ratios at
[Fe/H]$\gtrsim-1.5$ (top left panel of Figure 5). This is consistent with
Helmi et al. (2018), Haywood et al. (2018), Mackereth et al. (2019), and Di
Matteo et al. (2019), who showed from APOGEE data that the low-$\alpha$ halo
population identified by NS10 corresponds to the debris from the relatively
massive accreted dwarf galaxy, Gaia-Enceladus.
We directly compare abundance ratios of Gaia-Enceladus and in-situ stars in
GALAH DR3 with the high-/low-Mg populations of NS10 in Figures 5 and 6. The
figure shows the difference in [Mg/Fe] between the two subsamples is similar
to that seen between the low-/high-Mg populations of NS10; the two populations
have different [Mg/Fe] by $0.1-0.2\,\mathrm{dex}$ at [Fe/H]$\sim-1.0$ and
merge toward lower metallicity around [Fe/H]$\sim-1.5$. There are systematic
offsets in [X/Fe] between the GALAH and NS10’s abundances for all the
elements. The amount of the offsets are $\sim 0.2\,\mathrm{dex}$ for Mg, La,
and Eu and $\sim 0.5\,\mathrm{dex}$ for Ba.
These offsets would be due to metallicity-dependent systematics present in
abundance analysis, such as those caused by non-LTE/3D effects, different
selection of absorption lines, and difference in the method of stellar
parameters (e.g., Jofré et al., 2019; Hinkel et al., 2016) 111Although the
reason for particularly large Ba abundance difference is unclear, we note that
the Ba lines are close to saturation (Buder, S. private communication), which
might make it harder to obtain Ba abundance precisely. . Since they act in a
similar manner in stars with similar metallicity and temperature, our
discussion is not affected by these systematics.
Figure 7: Abundance trends of Gaia-Enceladus and in-situ stars in comparison
with literature. The GALAH data are binned in metallicity and the weighted
average values are plotted. The number of stars in each metallicity bin is
between 5 to 33 and errorbars indicate the uncertainties in the estimated
average estimated from the bootstrap sampling. The comparison sample is from
Letarte et al. (2010) and Lemasle et al. (2014) for Fornax (values are
corrected with the corrigendum Letarte et al., 2018), from Bonifacio et al.
(2000) and McWilliam et al. (2013) for Sagittarius, and Van der Swaelmen et
al. (2013) for LMC.
We now proceed to discuss neutron-capture elements. The $s$-process elemental
abundances (Ba and La) do not show clear differences in [X/Fe] between Gaia-
Enceladus and in-situ stars (Figure 5), although the scatter in [Ba/Fe] is
relatively large. On the other hand, there is a tendency of Gaia-Enceladus
stars having higher value of [Eu/Fe]. Since Eu is an almost pure $r$-process
element, this result indicates that Gaia-Enceladus has enhanced $r$-process
element abundances compared to the in-situ population.
Although [X/Fe] is widely used when interpreting abundance ratios, Fe has at
least two multiple nucleosynthesis channels (SNe Ia and core-collapse
supernovae, CCSNe), which could complicate the interpretation. Since the
production of Mg is dominated by CCSNe unlike Fe, the [X/Mg] ratio provides us
with a way to infer the efficiency of the nucleosynthesis event that produces
the element X relative to CCSNe. Therefore, we compare [Eu/Mg] in the leftmost
panel of Figure 6. The Eu enhancement of Gaia-Enceladus stars becomes even
clearer in [Eu/Mg] than in [Eu/Fe]. This is because the large abundance of Fe
relative to Mg in Gaia-Enceladus obscures its Eu enhancement when the
comparison is made in [Eu/Fe].
Figure 6 also presents abundance ratios between $s$\- and $r$-process
elements. The ratios [Ba/Eu] and [La/Eu] increase when there are significant
enrichments by $s$-process typically from low-to-intermediate mass asymptotic
giant branch stars. Since low-to-intermediate mass stars evolves slowly, the
ratio increases with time. Gaia-Enceladus stars do not have higher [Ba/Eu] or
[La/Eu] ratios than in-situ stars. The absence of enhanced $s$-to-$r$
abundance ratio also supports $r$-process origin of Eu.
These results are in line with Ishigaki et al. (2013), Fishlock et al. (2017),
and Matsuno et al. (2020), who indicated high Eu abundance of their low-Mg
halo populations. We confirmed and strengthen their findings with a large
sample from the recent high-resolution spectroscopic survey and with the data
of stellar kinematics obtained from astrometric measurements by the Gaia
mission.
Figure 7 presents comparisons of abundance ratios with massive dwarf galaxies
that show Eu enhancements (LMC, Sagittarius and Fornax dwarf galaxies; Van der
Swaelmen et al., 2013; McWilliam et al., 2013; Lemasle et al., 2014). The
similarities between Gaia-Enceladus and these galaxies also lie in their
[Mg/Fe] ratios (the top right panel of Figure 7). All of the four systems have
lower [Mg/Fe] than the Milky Way in-situ stars. On the other hand, there are
difference in $s$-to-$r$ element abundance ratios ([Ba/Eu] and [La/Eu], again
in Figure 7). Gaia-Enceladus does not show the signs of significant
$s$-process contribution, which is seen in all the three surviving dwarf
galaxies as high values of [Ba/Eu] or [La/Eu], or increasing trends in these
ratios with metallicity (Van der Swaelmen et al., 2013; Letarte et al., 2010;
Lemasle et al., 2014; McWilliam et al., 2013).
## 4 Discussion & Conclusion
We will now discuss the possible origin of the high [Eu/Mg] ratios of Gaia-
Enceladus stars as well as those of surviving massive satellites galaxies. The
left panel of Figure 8 shows [Eu/Mg] and [Mg/Fe] ratios of the stars in these
systems. An anti-correlation is found in the two abundance ratios in the sense
that systems with lower [Mg/Fe] ratios have higher [Eu/Mg]. Gaia-Enceladus
provides unique data in this context since its stars are formed in
environments outside the Milky Way while the star formation is not so
prolonged compared to the surviving galaxies.
The high [Eu/Mg] ratio indicates that $r$-process elements are produced more
efficiently relative to Mg. There are two possibilities for the cause of high
[Eu/Mg]: an enhanced production of Eu or a suppressed production of Mg.
In the case of enhanced production of Eu, it would be likely due to the
combined effect of delayed production of $r$-process elements and prolonged
star formation of Gaia-Enceladus, which is a similar explanation as provided
by NS10 and Vincenzo et al. (2019) for the low-[Mg/Fe] ratio. If this is the
case, NSM would be a promising site for the source of $r$-process elements in
Gaia-Enceladus since it is expected to have a delay time.
The other possibility is suppressed Mg production as a result of top-light
IMF. Among CCSNe, more massive progenitors produce higher amount of Mg (e.g.,
Nomoto et al. 2006). Therefore, a lack of massive stars as a result of a top-
light IMF can lead to low [Mg/Fe]. Fernández-Alvar et al. (2018) indeed
suggested that top-light IMF could be a part of the reason of the low [Mg/Fe]
of Gaia-Enceladus. As we discuss later, since low-mass progenitors are
expected to produce more $r$-process elements through NSMs than massive stars,
the top-light IMF might also be able to explain the high [Eu/Mg] and the low
[Mg/Fe] of Gaia-Enceladus.
To test these two scenarios, we perform one-zone chemical evolution
calculations. From a comparison between the observed data and the models, we
show that high [Eu/Mg] and low [Mg/Fe] ratios are naturally explained by
chemical enrichments from NSMs and SNe Ia without modifying IMF.
Figure 8: (left) [Eu/Mg] and [Mg/Fe] of stars with $[\mathrm{Fe/H}]>-1.3$.
Symbols follow Figure 5 (for in-situ and Gaia-Enceladus stars) and Figure 7
(for LMC, Sagittarius, and Fornax). One-zone chemical evolution models are
shown with the thick black line (baseline model A: Eu from NSMs with the
standard Chabrier IMF), with the thick grey line (model B: a constant delay
time for $r$-process enrichments, which represents the scenario that Eu is
produced by CCSNe), and with the thin black line (model C: top-light IMF).
Models are shown with solid lines for $[\mathrm{Fe/H}]>-1.3$ and dashed line
for $-2.5<[\mathrm{Fe/H}]<-1.3$. Typical uncertainties in GALAH DR3 are shown
in the bottom right. The red arrows indicate how stars move in this figure
because of the uncertainties in [Mg/Fe]. (right) The same chemical evolution
models but as a function of [Fe/H]. The blue solid line shows the baseline
model A but shifted to higher metallicity by $0.5\,\mathrm{dex}$ to present a
track that mimics the fast chemical evolution of in-situ stars. Note that the
in-situ model completely overlaps in the left panel and that the model B
completely overlaps with the baseline model A in the [Mg/Fe]–[Fe/H] panel.
We firstly discuss our baseline model, where we adopt a widely-assumed IMF
from 0.1 to 100 $\mathrm{M_{\sun}}$ (Chabrier, 2003) and SNe Ia-like delay
time distribution for NSMs. The chemical evolution models adopt an initial gas
mass of 2$\leavevmode\nobreak\ \times\leavevmode\nobreak\
10^{9}\leavevmode\nobreak\ \mathrm{M_{\sun}}$ to make chemical abundances
similar to those found for Gaia-Enceladus stars. After 3 Gyr evolution, the
stellar mass of this model reaches 1$\leavevmode\nobreak\
\times\leavevmode\nobreak\ 10^{9}\leavevmode\nobreak\ \mathrm{M_{\sun}}$. Here
we assume the CCSN yield of Chieffi & Limongi (2004) from 13 to 35
$\mathrm{M_{\sun}}$ for the enrichment of Mg and Fe. We also adopt the yield
of Seitenzahl et al. (2013) computed in the N100 model of SNe Ia. SNe Ia
distribute Fe following a delay time distribution with a power-law index of
$-$1 (Maoz & Mannucci, 2012) and a minimum delay of 5$\leavevmode\nobreak\
\times\leavevmode\nobreak\ 10^{8}$ yr (Homma et al., 2015). For the enrichment
of Eu, we assume that all Eu comes from NSMs with a rate of 0.5% of stars from
8 to 20 $\mathrm{M_{\sun}}$. This rate is consistent with the recent
constraints (Pol et al., 2019). The yield of Eu is taken from Wanajo et al.
(2014). A delay time distribution is similar to that of SNe Ia but a minimum
delay is set to be 2$\leavevmode\nobreak\ \times\leavevmode\nobreak\
10^{7}\,\mathrm{yr}$ following the observations of short gamma-ray bursts
(Wanderman & Piran, 2015). Stellar lifetimes are taken from Portinari et al.
(1998). All these models are compiled using celib (Saitoh, 2017).
This baseline model is shown as the thick black lines in Figure 8 (model A).
The delay time of NSMs and that of SNe Ia respectively cause an increase in
[Eu/Mg] and a decrease in [Mg/Fe] with time. Since the minimum delay time is
shorter for NSMs, [Eu/Mg] starts increasing before [Mg/Fe] starts decreasing
(see the right two panels of Figure 8). This is the reason why we see the
vertical evolution in the left panel of Figure 8. Once SNe Ia start
contributing, the chemical evolution then proceeds toward the top left of that
panel. Note that the evolution in the left panel of Figure 8 does not depend
on the timescale of the evolution.
The relative positions of Gaia-Enceladus and in-situ stars in the left panel
of Figure 8 can be understood as the result of this chemical evolution.
Because of lower star formation efficiency, Gaia-Enceladus and in-situ stars
have different age metallicity relations in the sense that Gaia-Enceladus has
younger age at fixed metallicity than in-situ stars (Schuster et al., 2012;
Hawkins et al., 2014). Therefore, it allows more nucleosynthesis events with
delay time to enrich the system, which lowers [Mg/Fe] and elevates [Eu/Mg]
(see these values at $t=1\,\mathrm{and}\,3\,\mathrm{Gyr}$ marked in Figure 8).
It is also worth noting that the baseline model naturally explains [Mg/Fe] and
[Eu/Mg] of LMC, Sagittarius, and Fornax in a similar manner. Since these
galaxies have more prolonged star formation, they are more likely to be
enriched by delayed nucleosynthesis events such as SNe Ia and NSMs than Gaia-
Enceladus, which would result in even lower [Mg/Fe] and higher [Eu/Mg].
In addition, the delay times of the NSMs and SNe Ia might also help to enhance
their importance in the chemical evolution of dwarf galaxies. Galaxies blow
out copious amounts of metals through CCSNe-driven outflows (Springel &
Hernquist, 2003; Tumlinson et al., 2011). The metal fraction of an outflow may
be biased to elements produced by CCSNe since they explode while star
formation is ongoing, which would collectively heat up the interstellar medium
(ISM). On the other hand, elements produced in delayed sources such as type-Ia
SNe and NSM accumulate in the ISM with a higher efficiency. Dwarf galaxies
might have lost a larger fraction of $\alpha$-elements due to their shallower
potential compared to the Milky Way. Therefore, it could be possible that
[Mg/Fe] and [Eu/Mg] change more rapidly in dwarf galaxies once SNe Ia and NSMs
start to operate. If we take this effect into account, the chemical evolution
model track in the left panel of Figure 8 would be extended to the upper left,
allowing the model to reproduce the [Eu/Mg] and [Mg/Fe] of the dwarf galaxies.
In Model B we consider the case in which Eu is synthesized in CCSNe driven by
the magneto-rotational instability (e.g., Winteler et al., 2012; Nishimura et
al., 2015) or collapsars (Siegel et al., 2019). We assume a constant delay
time of $2\times 10^{7}\,\mathrm{yr}$ for the $r$-process production events
instead of the distribution with a power-law index of $-$1 adopted in the
baseline model. Note that, since the $r$-process yields from these CCSNe are
uncertain, we assume the same yield as in model A. As shown in the left panel
of Figure 8, model B (in grey) predicts almost constant [Eu/Mg], indicating
that the [Eu/Mg] ratio does not differ even if systems have different star
formation efficiency. Thus, the model B does not provide an explanation for
the higher [Eu/Mg] values of systems with lower [Mg/Fe] than in-situ stars.
In order to explain the high [Eu/Mg] abundance with the delay time of
$r$-process enrichments, it is also necessary to have a short minimum delay
time ($<$ a few Gyr). This is because Gaia-Enceladus is estimated to have been
accreted and have stopped star formation about 10 Gyr ago (Helmi et al., 2018;
Gallart et al., 2019; Chaplin et al., 2020; Belokurov et al., 2019; Bonaca et
al., 2020). No star formation should take place after the disruption, which
sets an upper limit on the minimum delay time. Note that GW170817 took place
in an S0-type galaxy and its delay time has been estimated as $1$ – $10\,{\rm
Gyr}$ (Blanchard et al., 2017; Levan et al., 2017) and therefore NSMs that
have the same delay time to GW170817 might not be able to enrich Gaia-
Enceladus. However, Beniamini & Piran (2019) study the delay time distribution
of NSMs based on Galactic binary pulsars and find that at least $40\%$ of NSMs
have a delay time less than $1\,{\rm Gyr}$. Moreover, the observed redshift
distribution of short GRBs indicates a minimum delay time of a few tens of Myr
(Wanderman & Piran, 2015; D’Avanzo et al., 2014). These studies at least
indicate that the minimum delay time of NSMs should be shorter than that of
SNe Ia (Strolger et al., 2020). If we consider that the low-[Mg/Fe] of Gaia-
Enceladus is due to the delay time of SNe Ia, there is no difficulty in
explaining the high [Eu/Mg] with the delay time of NSMs.
This scenario with the baseline model is at first sight similar to that
suggested by Skúladóttir & Salvadori (2020) for the Sagittarius and Fornax
dwarf galaxies. However, their scenario would not be directly applicable to
Gaia-Enceladus. They used high $s$-to-$r$ process abundance ratios ([Ba/Eu],
[La/Eu]; Figure 7) as evidence of prolonged star formation activity of
Sagittarius and Fornax. Gaia-Enceladus, on the other hand, has no sign of
significant $s$-process contribution, which indicates that the star formation
did not last long as in Fornax or Sagittarius. Skúladóttir & Salvadori (2020)
obtained a minimum time delay of $4\,\mathrm{Gyr}$ from the absence of Eu
enhancements in Sculptor dwarf galaxy. Note however that a source with delay
time of $4\,\mathrm{Gyr}$ would not be able to enrich Gaia-Enceladus.
An additional chemical evolution model is shown in Figure 8, which assumes a
top-light IMF (the Chabrier IMF from 0.1 to 15 $M_{\sun}$; Model C), and which
produces high [Eu/Mg] and low [Mg/Fe]. The reason of the high [Eu/Mg] in this
model is that Eu is preferentially produced by lower mass progenitors than
those that produce significant amounts of Mg. Since the event rate and yields
of NSMs do not strongly depend on the initial mass of the progenitor stars,
the more abundant lower mass stars contribute more to the production of Eu
than more massive stars. Additionally, while we assume that the fraction of
NSMs do not depend on the progenitor mass, supernova explosions of more
massive stars are more likely to destroy the binary system, which would
decrease binary neutron star systems originated from more massive stars
(Hills, 1983).
The possibility of a top light IMF was suggested for Sagittarius (McWilliam et
al., 2013) and for Fornax (Lemasle et al., 2014) as an explanation for their
low [Mg/Fe] and high [Eu/Mg], although they considered supernova explosions of
low mass progenitors as the sites of $r$-process nucleosynthesis. The model C
calculation confirms that, if the IMF is top-light in Gaia-Enceladus and in
the massive satellites, it is possible to explain their lower [Mg/Fe] and
higher [Eu/Mg] ratios at high metallicity in comparison to the in-situ stars,
which would have standard IMF. However, an additional complication arises in
this scenario, namely the [Mg/Fe] of Gaia-Enceladus stars at low metallicity
([Fe/H]$\lesssim-1.5$), is the same as that of in-situ stars. Since the
[Mg/Fe] ratio is always lower in a top light IMF than for a standard IMF, this
would require the IMF of Gaia-Enceladus to change as the metallicity
increases.
Another important feature in the top-light IMF model is the shallow slope in
[Eu/Mg]–[Mg/Fe] at high metallicity. Because of the lack of most massive
stars, which evolve faster, the delay time in NSMs is less important in this
chemical evolution model. As a result, the [Eu/Mg] ratio does not increase
significantly compared to the decrease in [Mg/Fe]. Constraining this slope
from precise abundance measurements might enable one to estimate the IMF. We
refrain from interpreting the observed slope in the current data set since the
spread in [Eu/Mg]–[Mg/Fe] is not significantly larger than the measurement
uncertainty for neither of Gaia-Enceladus or in-situ stars.
In conclusion, we consider that the baseline model A provides the most
reasonable explanation for the high [Eu/Mg] and low [Mg/Fe] values of Gaia-
Enceladus and other massive satellite galaxies. While the baseline model A was
computed for Gaia-Enceladus, we here comment on the expected evolution of in-
situ stars using a similar model. Since the in-situ star formation proceeds on
a shorter timescale, the metallicity would be higher than that of Gaia-
Enceladus at the same age. Although the in-situ track shown in the left panel
of Figure 8 would be similar to that of Gaia-Enceladus, in-situ track would
not be extended toward top left as Gaia-Enceladus (see the values at $t=1$ and
$3\leavevmode\nobreak\ \mathrm{Gyr}$). The tracks in the right panels would be
shifted to higher metallicity (the blue line in the right panels). As a result
of the flat [Mg/Fe] evolution at low metallicity, [Mg/Fe] ratios are expected
to be identical between Gaia-Enceladus and in-situ stars up to
$[\mathrm{Fe/H}]\sim-1.5$, when Gaia-Enceladus starts experiencing enrichments
by SNe Ia and consequently a decrease in [Mg/Fe]. This is indeed consistent
with the observations. The [Eu/Mg] of the in-situ stars are expected to be
lower than in Gaia-Enceladus down to even lower metallicity because of the
increasing trend of [Eu/Mg] at low metallicity, which reflects the power-law
delay time distribution of NSMs. We note that if a change in the IMF would be
the reason of the lower [Mg/Fe] of Gaia-Enceladus at
$[\mathrm{Fe/H}]\gtrsim-1.5$, the higher [Eu/Mg] of Gaia-Enceladus stars
should only appear at the same metallicity range since the high [Eu/Mg] should
also be triggered by the same reason.
Therefore, the [Eu/Mg] of in-situ stars and Gaia-Enceladus stars at lower
metallicity are expected to be useful to disentangle further different
scenarios. Unfortunately, we cannot explore the Eu abundance of such low
metallicity stars with the current data set. This is because of the weakness
of the Eu $6645\,\mathrm{\AA}$ line, which prevent us from investigating the
Eu abundance trend below [Fe/H]$\sim-1.3$ (see Section 2). The Eu abundance of
stars with lower metallicity can however be studied by analysing stronger Eu
lines in bluer wavelengths (e.g., Eu $4129\,\mathrm{\AA}$).
We compared the chemical evolution models with observed trends of [Mg/Fe] and
[Eu/Mg] as a function of [Fe/H] in the right panel of Figure 8. The difference
between the baseline and in-situ models are qualitatively in good agreement
with the observed difference between Gaia-Enceladus and in-situ stars,
although the models do not fully reproduce the observed values of the
abundance ratios for each population or the amount of the difference between
them. The disagreements could be results of uncertainties in modelling star
formation (e.g., star formation efficiency, star formation history), gas
inflow/outflow, nucleosynthesis processes (e.g., yields, delay time
distribution of SNe Ia/NSMs). Our conclusion is not affected by these
uncertainties; as long as Gaia-Enceladus has lower star formation efficiency
than in-situ stars, its higher [Eu/Mg] is a natural consequence of $r$-process
enrichments by the NSMs with delay time.
Characterizing the Eu abundance in an accreted system is also an important
step to uncover the accretion history of the Milky Way. While substructures in
the kinematics of stars enable one to identify candidates of past accretion
signatures, additional information is necessary to relate each substructure to
individual accretion events. This is because a single accretion event can
produce multiple kinematic streams and because different accretion events may
overlap in phase-space. The idea of chemical tagging is to use chemical
abundance of stars to group stars according to their origins (Freeman & Bland-
Hawthorn, 2002). Our results of different [Eu/Mg] ratios between Gaia-
Enceladus and in-situ stars indicate that having Eu abundance of stars clearly
benefits the chemical tagging. Since abundance differences between galaxies
can be small, adding an independent chemical dimension is an important step to
make chemical tagging work. During the preparation of this manuscript, Aguado
et al. (2020) suggested Eu enhancements for Gaia-Enceladus from their high-
resolution observations of stars, which is consistent with our study. They
also suggested a similar Eu enhancement in Sequoia, another kinematic
substructure in the Milky Way, supporting the effectiveness of Eu abundance in
understanding the Milky Way accretion history.
###### Acknowledgements.
This research has been supported by a Spinoza Grant from the Dutch Research
Council (NWO), MEXT and JSPS KAKENHI Grant Numbers 20K14532, 19H01933,
17H06363, 19H00694, and 20H00158. YH has been supported by the Special
Postdoctoral Researchers (SPDR) program at RIKEN.
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|
# The Observed Rate of Binary Black Hole Mergers can be Entirely Explained by
Globular Clusters
Carl L. Rodriguez McWilliams Center for Cosmology and Department of Physics,
Carnegie Mellon University, Pittsburgh, PA 15213, USA Kyle Kremer TAPIR,
California Institute of Technology, Pasadena, CA 91125, USA The Observatories
of the Carnegie Institution for Science, Pasadena, CA 91101, USA Sourav
Chatterjee Tata Institute of Fundamental Research, Homi Bhabha Road, Navy
Nagar, Colaba, Mumbai 400005, India Giacomo Fragione Center for
Interdisciplinary Exploration & Research in Astrophysics (CIERA) and
Department of Physics & Astronomy, Northwestern University, Evanston, IL
60208, USA Abraham Loeb Astronomy Department, Harvard University, 60 Garden
Street, Cambridge, MA 02138 Frederic A. Rasio Center for Interdisciplinary
Exploration & Research in Astrophysics (CIERA) and Department of Physics &
Astronomy, Northwestern University, Evanston, IL 60208, USA Newlin C.
Weatherford Center for Interdisciplinary Exploration & Research in
Astrophysics (CIERA) and Department of Physics & Astronomy, Northwestern
University, Evanston, IL 60208, USA Claire S. Ye Center for Interdisciplinary
Exploration & Research in Astrophysics (CIERA) and Department of Physics &
Astronomy, Northwestern University, Evanston, IL 60208, USA
###### Abstract
Since the first signal in 2015, the gravitational-wave detections of merging
binary black holes (BBHs) by the LIGO and Virgo collaborations (LVC) have
completely transformed our understanding of the lives and deaths of compact
object binaries, and have motivated an enormous amount of theoretical work on
the astrophysical origin of these objects. We show that the phenomenological
fit to the redshift-dependent merger rate of BBHs from Abbott et al. (2020) is
consistent with a purely dynamical origin for these objects, and that the
current merger rate of BBHs from the LVC could be explained entirely with
globular clusters alone. While this does not prove that globular clusters are
the dominant formation channel, we emphasize that many formation scenarios
could contribute a significant fraction of the current LVC rate, and that any
analysis that assumes a single (or dominant) mechanism for producing BBH
mergers is implicitly using a specious astrophysical prior.
binary black holes — gravitational waves — mergers — globular clusters
## 1
Over the past 5 years, the observed rate of BBH mergers in the local universe
has been significantly constrained. After the detections of GW150914,
GW151012, and GW151226 in the first observing run, initial estimates put the
volumetric merger rate between 9 to $240~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ at 90%
confidence at $z=0$ (Abbott et al., 2016). Merger rates $\gtrsim
100~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ were largely understood to be explicable by
isolated binary evolution (e.g., Belczynski et al., 2016), with dynamical
assembly (e.g., Rodriguez et al., 2016) only contributing a small fraction of
the population. But as more BBH mergers have been detected, the measured local
merger rate has decreased by an order of magnitude, while sufficient numbers
of higher-redshift mergers allow the slope of the rate to be observed. By
fitting the observed detections to a phenomenological model of the form
$R(z)=R_{0}(1+z)^{\kappa}$, an analysis of the latest GW transient catalog
(GWTC-2, Abbott et al., 2020) suggests an increasing merger rate with a local
value of $19^{+16}_{-9}~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z=0$.
Figure 1: The merger rate of BBHs from GCs compared to the latest LVC results.
The left panel shows the predictions from Rodriguez & Loeb (2018) and updated
predictions from Kremer et al. (2020) using the same cosmological model. The
right panel shows the fits to the models from Kremer et al. (2020) broken down
according to cluster virial radius, assuming that all clusters are born with
initial radii of 0.5, 1, 2, or 4 pc. The total prediction on the left is the
average of these four values. In blue, we show the median, 50%, and 90%
intervals of the phenomenological merger rate fit from GWTC-2 (Abbott et al.,
2020).
There have been many theoretical models of the merger rate of BBHs from GCs,
both before and after the first detection of GWs (e.g. Portegies Zwart &
Mcmillan, 2000; Rodriguez et al., 2015) with the majority predicting a
volumetric merger rate at $z=0$ of $\sim 10~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$.
Using the Cluster Monte Carlo code (CMC), several groups have studied the BBH
merger rate from GCs and the unique properties of their sources. The analysis
of the BBH merger rate from GCs initially predicted a merger rate anywhere
from 2 to $20~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z=0$, based on the luminosity
function and comoving spatial density of observed GCs in the local universe
(Rodriguez et al., 2016). Of course, this initial analysis ignored the
contributions of GCs that were disrupted before the present day, which can
significantly increase the contribution to the BBH merger rate (e.g., Fragione
& Kocsis, 2018). In Rodriguez et al. (2018), we combined a cosmological model
for GC formation (El-Badry et al., 2019) with star-by-star CMC models of GCs
to estimate the BBH merger rate as a function of cosmological redshift.
In the left panel of Figure 1, we show the predictions from Rodriguez et al.
(2018) and updated predictions using the same cosmological model and fitting
procedure111For the updated model we have added an additional correction
accounting for the formation of central-massive BHs in GCs, from Antonini et
al. (2019); see Kremer et al. (2019) for details. This decreases the BBH
merger rate from the densest GCs by $\sim 10\%$ with newer GC models (covering
a wider range of parameters) from Kremer et al. (2020). When compared to the
allowed range and cosmological evolution of the BBH merger rate from GWTC-2
(Abbott et al., 2020), it is obvious that _the entire phenomenological BBH
merger rate can potentially be explained by GCs alone_. At $z=0$ the original
fits predict a merger rate of $15~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$, while the
newer fits from Kremer et al. (2020) predict a merger rate of
$17~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$. In both cases, the merger rate increases as
a function of redshift. Comparing the ratio of the rate at $z=1$ to $z=0$, we
find that $R(1)/R(0)=3.1$ for the original fits from Rodriguez et al. (2018),
and $R(1)/R(0)=3.2$ for the Kremer et al. (2020) models. Both ratios are
consistent with the phenomenological value of $R(1)/R(0)=2.5^{+7.8}_{-1.9}$
from Abbott et al. (2020).
The primary improvement in the models of Kremer et al. (2020) is the extent of
the grid, with the newer grid containing clusters with higher initial masses,
more compact initial virial radii (as small as 0.5 pc), and a wider range of
metallicities. All together, they capture nearly the complete spectrum of
present-day dense star clusters in the Milky Way. We have assumed in the left
panel of Figure 1 that clusters of different initial radii contribute equally
to the total rate. To make this more explicit, we show what the merger rate
would look like from GCs assuming all cluster were born with virial radii of
0.5, 1, 2, or 4 pc in the right panel of Figure 1. The four values clearly
span the 90% region of allowed merger rates from Abbott et al. (2020). As
pointed out in Kremer et al. (2020), initial concentrations of clusters
directly control the slope of the merger rate, with the contributions from
clusters with 0.5, 1, 2, and 4 pc having ratios of $R(1)/R(0)=4.2$, 2.8, 2.1,
and 2.6, respectively. Given that many young clusters in the local universe
are observed to have initial effective radii between 1 and 2 pc (e.g.
Scheepmaker et al., 2007), this suggests remarkably good agreement with
current LVC findings.
Of course, there are many such dynamical scenarios for forming merging BBHs ,
many of which are consistent with the 90% uncertainties from the LVC’s
phenomenological model. But while the measured merger rates can now be
explained by a single formation channel alone, there is no reason to believe
that is the case. Several studies of the current LVC BBH catalog have
suggested that multiple formation scenarios likely operate in producing BBH
mergers (e.g., Wong et al., 2020; Zevin et al., 2020) with the latter
suggesting that no single channel likely contributes more than 70% to the
total population. This note is meant to emphasize this point: given that GCs
alone can naturally explain the most up-to-date BBH merger rate from the LVC,
it is no longer reasonable to assume that dynamical processes constitute a
subdominant fraction of the full merger rate.
## References
* Abbott et al. (2016) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, PRX, 6, 041015
* Abbott et al. (2020) Abbott, R., Abbott, T. D., Abraham, S., et al. 2020, arXiv e-prints, arXiv:2010.14533
* Antonini et al. (2019) Antonini, F., Gieles, M., & Gualandris, A. 2019, MNRAS, 486, 5008
* Belczynski et al. (2016) Belczynski, K., Heger, A., Gladysz, W., et al. 2016, Astronomy & Astrophysics, 594, A97
* El-Badry et al. (2019) El-Badry, K., Quataert, E., Weisz, D. R., Choksi, N., & Boylan-Kolchin, M. 2019, MNRAS, 482, 4528
* Fragione & Kocsis (2018) Fragione, G., & Kocsis, B. 2018, Phys. Rev. Lett., 121, 161103
* Kremer et al. (2019) Kremer, K., Lu, W., Rodriguez, C. L., Lachat, M., & Rasio, F. A. 2019, ApJ, 881, 75
* Kremer et al. (2020) Kremer, K., Ye, C. S., Rui, N. Z., et al. 2020, ApJS, 247, 48
* Portegies Zwart & Mcmillan (2000) Portegies Zwart, S. F., & Mcmillan, S. L. W. 2000, ApJ, 528, 17
* Rodriguez et al. (2018) Rodriguez, C. L., Amaro-Seoane, P., Chatterjee, S., & Rasio, F. A. 2018, Physical Review Letters, 120, 151101
* Rodriguez et al. (2016) Rodriguez, C. L., Chatterjee, S., & Rasio, F. A. 2016, Phys. Rev. D, 93, 084029
* Rodriguez & Loeb (2018) Rodriguez, C. L., & Loeb, A. 2018, ApJ, 866, L5
* Rodriguez et al. (2015) Rodriguez, C. L., Morscher, M., Pattabiraman, B., et al. 2015, Physical Review Letters, 115, 051101
* Scheepmaker et al. (2007) Scheepmaker, R. A., Haas, M. R., Gieles, M., et al. 2007, A&A, 469, 925
* Wong et al. (2020) Wong, K. W. K., Breivik, K., Kremer, K., & Callister, T. 2020, arXiv e-prints, arXiv:2011.03564
* Zevin et al. (2020) Zevin, M., Bavera, S. S., Berry, C. P. L., et al. 2020, arXiv e-prints, arXiv:2011.10057
|
# On Monte-Carlo methods in convex stochastic optimization
Daniel Bartl Department of Mathematics, Vienna University, Austria
<EMAIL_ADDRESS>and Shahar Mendelson Mathematical Sciences
Institute, The Australian National University Canberra, Australia
<EMAIL_ADDRESS>
###### Abstract.
We develop a novel procedure for estimating the optimizer of general convex
stochastic optimization problems of the form
$\min_{x\in\mathcal{X}}\mathbf{E}[F(x,\xi)]$, when the given data is a finite
independent sample selected according to $\xi$. The procedure is based on a
median-of-means tournament, and is the first procedure that exhibits the
optimal statistical performance in heavy tailed situations: we recover the
asymptotic rates dictated by the central limit theorem in a non-asymptotic
manner once the sample size exceeds some explicitly computable threshold.
Additionally, our results apply in the high-dimensional setup, as the
threshold sample size exhibits the optimal dependence on the dimension (up to
a logarithmic factor). The general setting allows us to recover recent results
on multivariate mean estimation and linear regression in heavy-tailed
situations and to prove the first sharp, non-asymptotic results for the
portfolio optimization problem.
###### Key words and phrases:
stochastic optimization, sample-path optimization, stochastic counterpart
method, finite sample / non-asymptotic concentration inequality
###### 2010 Mathematics Subject Classification:
90C15,90B50,62J02
###### Contents
1. 1 Introduction and appetizer
2. 2 Main results
1. 2.1 Difficulties caused by non-Gaussian tails
2. 2.2 What to expect when tails are Gaussian
3. 2.3 Recovering Gaussian rates without sub-Gaussian tails
4. 2.4 The procedure
5. 2.5 Related literature
3. 3 Applications
1. 3.1 Multivariate mean estimation
2. 3.2 Linear regression
3. 3.3 Ridge regression
4. 3.4 Portfolio optimization
4. 4 On the smallest singular value of general random matrix ensembles
5. 5 Proofs of the main results
1. 5.1 Estimation error, the quadratic term
2. 5.2 Estimation error, the multiplier term
3. 5.3 Prediction error
4. 5.4 Proof under a deterministic lower bound of the Hessian
6. 6 Proofs for the portfolio optimization problem
1. 6.1 The proof of Corollary 3.7
2. 6.2 The proof of Corollary 2.10
7. 7 Concluding remarks
## 1\. Introduction and appetizer
_Stochastic optimization_ is widely used as a way of solving certain problems
numerically. It appears in diverse areas of mathematics, with a generic convex
stochastic optimization problem taking the following form: One is given a
random variable $\xi$ whose range is a measurable space $\Xi$, a convex set of
actions $\mathcal{X}\subseteq\mathbb{R}^{d}$, and a function
$F\colon\mathcal{X}\times\Xi\to\mathbb{R}$ that is convex in its first
argument. The objective is to solve the optimization problem
(SO) $\displaystyle\min_{x\in\mathcal{X}}f(x)\quad\text{where}\quad
f(x):=\mathbf{E}[F(x,\xi)].$
In typical situations, however, one does not have access to the function $f$
directly. Rather, the information one is given is the set of values
$F(\cdot,\xi_{i})_{i=1}^{N}$, where $(\xi_{i})_{i=1}^{N}$ is an independent
_sample_ , selected according to $\xi$ and of cardinality $N$. This type of
random data is natural, for example, if the distribution of $\xi$ is not known
and the only possibility is to sample it; or when an exact computation of $f$
is unfeasible and one relies on Monte-Carlo methods to evaluate it instead. We
refer the reader to Shapiro, Dentcheva, and Ruszczyński [35] or to Kim,
Pasupathy, and Henderson [12] for introductions on such aspects of stochastic
optimization.
Regardless of the reason why one uses a random sample, the fundamental
question remains unchanged: to what degree (SO) can be recovered when the
given data is a random sample?
To be more accurate, assume that (SO) admits a unique optimal action
$x^{\ast}$ and denote by $\widehat{x}_{N}^{\ast}$ a candidate for the optimal
action that is selected using some sample-based procedure. Given a prescribed
error $r>0$, one seeks to bound the probability that the _estimation error_
$\|\widehat{x}_{N}^{\ast}-x^{\ast}\|$ or the _prediction error_ / _optimality
gap_ $f(\widehat{x}_{N}^{\ast})-f(x^{\ast})$ exceeds $r$ in terms of the
sample size $N$.
It should be stressed that the norm $\|\cdot\|$ need not be the Euclidean
norm. Rather, the right choice of $\|\cdot\|$ turns out to be a natural
Hilbertian structure endowed by the Hessian of $f$. The reason for that is
clarified in what follows.
The typical approach used to produce $\widehat{x}_{N}^{\ast}$ is called
_sample average approximation_ (SAA) and is denoted in what follows by
$\widehat{x}^{\text{SAA}}_{N}$. The choice is very natural:
$\widehat{x}^{\text{SAA}}_{N}$ is a minimizer of the empirical mean
$\widehat{f}_{N}:=\frac{1}{N}\sum_{i=1}^{N}F(\cdot,\xi_{i})$.
Asymptotic properties of the SAA solution have been thoroughly investigated.
Roughly and inaccurately put, the SAA solution behaves asymptotically as one
would expect based on the central limit theorem. However, as we shall explain
immediately, these asymptotic results can be misleading. In fact, unless one
imposes _highly restrictive integrability assumptions_ , when given a finite
sample the SAA solution behaves poorly: it exhibits drastically weaker rates
than what one may expect based on the asymptotic behaviour.
In contrast to the SAA solution, we propose a _novel procedure_ that aims at
selecting the optimal action when given a finite random sample. This procedure
exhibits the best possible performance regarding the estimation and prediction
errors, and it does so in completely heavy tailed situations. Our results are
based on the methods developed by G. Lugosi and the second named author in
[18, 19, 20].
Before explaining what we mean by “highly restrictive integrability
assumptions” and indicating the very poor behaviour of SAA in their absence,
let us interject with an example of an important convex stochastic
optimization problem. This example will accompany us throughout the article
and will help concretize the results and assumptions of this article.
###### Example 1.1 (Portfolio optimization).
Modern portfolio theory was initiated by Markowitz [21] and is among the
central optimization problems in mathematical finance. Without going into
details, the _portfolio optimization problem_ has three ingredients:
* •
a $d$-dimensional random vector $X$, which is interpreted as (discounted)
future prices of some stocks/goods;
* •
a random variable $Y$, which is interpreted as the (random) future payoff;
* •
a concave, increasing utility function $U\colon\mathbb{R}\to\mathbb{R}$.
We assume that the stocks $X$ are available for buying and selling today at
the prices $\pi$ so that a trading strategy $x\in\mathbb{R}^{d}$ bears the
cost $\langle\pi,x\rangle:=\sum_{i=1}^{d}\pi_{i}x_{i}$ and gives the (random,
future) payoff $\langle X,x\rangle$. After trading according to the strategy
$x$, the investor’s terminal wealth is $Y+\langle X-\pi,x\rangle$ and her goal
is to maximize the expected utility, namely
$\max_{x\in\mathcal{X}}\mathbf{E}\big{[}U\big{(}Y+\langle
X-\pi,x\rangle\big{)}\big{]},$
where $\mathcal{X}\subseteq\mathbb{R}^{d}$ is closed and convex. (For instance
$\mathcal{X}=[0,\infty)^{d}$ corresponds to short-selling constraints.) We
refer, e.g., to Föllmer and Schied [7, Chapter 3] or to Shapiro, Dentcheva,
and Ruszczynski [35, Section 1.4] for a more elaborate introduction to the
portfolio optimization problem.
Setting $F(x,\xi):=\ell(-Y-\langle X-\pi,x\rangle)$ with $\xi:=(X,Y)$ and
$\ell:=-U(-\,\cdot\,)$, the portfolio optimization problem is indeed a special
instance of (SO).
It is important to stress that the portfolio optimization problem highlights
the natural presence of _heavy tails_ : for one, even if the input $X$ is
light-tailed (e.g. Gaussian, as is the case in the Bachelier model), the
composition with the utility function $U$ can render the problem heavy-tailed.
This is particularly true for the exponential utility function
$U=-\exp(-\,\cdot\,)$, which is arguably (among) the most important utility
functions. Additionally, $X$ itself is often heavy tailed; for instance, in
the famous Black-Scholes model $X$ is log-normal.
Before explaining the new procedure, it is worthwhile to outline what is known
on the statistical performance of the sample average approximation method. As
we already mentioned, the vast majority of known results are of an
_asymptotic_ nature. To ease notation we assume here and for the rest of this
section that $\nabla^{2}f(x^{\ast})=\mathrm{Id}$ (recall that $x^{\ast}$ is
the unique minimizer of $f$).
One can show that under suitable regularity and mild integrability
assumptions, $\sqrt{N}(\widehat{x}^{\mathrm{SAA}}_{N}-x^{\ast})$ is
asymptotically a multivariate Gaussian with zero mean and covariance matrix
$\mathrm{\mathbf{C}ov}[\nabla F(x^{\ast},\xi)]$, see e.g. [35, Chapter 5]. In
particular, if we set
$\sigma^{2}:=\lambda_{\max}(\mathrm{\mathbf{C}ov}[\nabla F(x^{\ast},\xi)])$
to be the largest eigenvalue of the covariance matrix of the gradient of $F$
at $x^{\ast}$ and denote by $\|\cdot\|_{2}$ the Euclidean norm, it follows
that
$\displaystyle\mathbf{P}\Big{[}\|\widehat{x}^{\mathrm{SAA}}_{N}-x^{\ast}\|_{2}\geq
r\Big{]}\approx
2\exp\Big{(}-cN\frac{r^{2}}{\sigma^{2}}\Big{)}\quad\text{asymptotically as
}N\to\infty.$
This error rate is often used to calculate the minimal sample size $N$
required to guarantee that the estimation error is below the wanted threshold
$r$ with some prescribed confidence $1-\delta$. However, as we shall explain
in Section 2.1 below, unless (restrictive) integrability assumptions are
imposed, the asymptomatic exponential decay really does hold _only
asymptotically_. Indeed, it may very well be possible that the _non-
asymptotic_ (or, finite sample) rate
$\displaystyle\mathbf{P}\Big{[}\|\widehat{x}^{\mathrm{SAA}}_{N}-x^{\ast}\|_{2}\geq
r\Big{]}\leq\frac{c\sigma^{2}}{Nr^{2}}\quad\text{for all }N\geq 1$
cannot be improved, meaning that the finite sample rate decays linearly in $N$
rather than exponentially.
The meaning of this significant gap between asymptotic and finite sample
behaviour is that the asymptotic estimate is misleading: while it suggests
that $\frac{\sigma^{2}}{r^{2}}\cdot\log(\frac{2}{\delta})$ samples are enough
to guarantee a confidence of $1-\delta$, one actually needs
$\frac{\sigma^{2}}{r^{2}}\cdot\frac{1}{\delta}$ samples. For small values of
$\delta$ this gap in the required sample size is significant.
_Our contribution_ is the following. We construct a procedure that estimates
the optimal action $\widehat{x}_{N}^{\ast}$ — but it is not SAA. Recalling
that $\nabla^{2}f(x^{\ast})=\mathrm{Id}$ throughout this section for
notational simplicity, given a finite sample, the procedure recovers the
optimal Gaussian rate under modest integrability assumptions: for (small)
$r>0$ we have
$\displaystyle\mathbf{P}\Big{[}\|\widehat{x}^{\ast}_{N}-x^{\ast}\|_{2}\geq
r\Big{]}\leq 2\exp\Big{(}-CN\frac{r^{2}}{\sigma^{2}}\Big{)}\quad\text{whenever
}N\geq N_{0}(r),$
where $N_{0}(r)$ can be controlled explicitly and depends only on certain low-
order moments of the random variables involved (see Theorem 2.9 for the
precise statement). As a matter of fact, in typical situations
$N_{0}(r)\lesssim\max\Big{\\{}d\log(d),\frac{\mathrm{trace}(\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)])}{r^{2}}\Big{\\}}.$
It is worthwhile mentioning that the prediction error is one order of
magnitude smaller than the estimation error, namely, for $N\geq N_{0}(r)$
$\mathbf{P}\Big{[}f(\widehat{x}_{N}^{\ast})\geq f(x^{\ast})+r^{2}\Big{]}\leq
2\exp\Big{(}-CN\frac{r^{2}}{\sigma^{2}}\Big{)}.$
Although this is a trivial consequence of the first order condition for
optimality if $\mathcal{X}=\mathbb{R}^{d}$, it is far less obvious if
$\mathcal{X}$ is a proper subset of $\mathbb{R}^{d}$ and $x^{\ast}$ lies on
the boundary of $\mathcal{X}$.
###### Remark 1.2.
While our procedure showcases the possibility of drastically improving the
statistical properties of the SAA, this improvement does not come for free. A
major advantage of SAA is its computational simplicity, and unfortunately, our
procedure is the outcome of a (rather complex) tournament that takes place
between the actions in $\mathcal{X}$. As a result, its computational cost is
quite high (see Section 2.4). Tournament based procedures are used in other
natural statistical problems and there are ongoing attempts to obtain
alternative procedures that maintain the tournament’s optimal statistical
performance in a computationally feasible way—see, for example [6, 11].
Finding procedures that do the same for stochastic optimization is a
worthwhile challenge.
Plan of the article. We begin Section 2 with a detailed explanation of the
devastating effect heavy-tailed random variables can have on SAA; we then
formulate our main result (Theorem 2.9), discuss its application to the
portfolio optimization problem, and survey related literature. Section 3
contains a description of several other applications of our main result. In
Section 4 we lay the groundwork for the proof of Theorem 2.9 by establishing a
high probability lower bound on the smallest singular value of a general
random matrix ensemble (see Theorem 4.4 and Corollary 4.5)—a result that is of
independent interest. This lower bound will be used in Section 5, where we
prove our main result. Proofs related to the portfolio optimization problem
are presented in Section 6. Finally, Section 7 contains two concluding
remarks.
## 2\. Main results
### 2.1. Difficulties caused by non-Gaussian tails
Let us revisit our claim that heavy tails drastically change the statistical
performance of the sample average approximation. As a starting point, consider
the more basic problem of estimating the mean $\mu:=\mathbf{E}[\xi]$ of a one-
dimensional, square integrable random variable $\xi$. It should be noted that
by setting $F(x,\xi):=\frac{1}{2}x^{2}-\xi x$, one-dimensional mean estimation
becomes a stochastic optimization problem.
Following the SAA approach, the corresponding estimator for the mean is
$\widehat{\mu}_{N}:=\frac{1}{N}\sum_{i=1}^{N}\xi_{i}$. The central limit
theorem then guarantees that
(2.1) $\displaystyle\mathbf{P}[|\widehat{\mu}_{N}-\mu|\geq r]\leq
2\exp\Big{(}-N\frac{r^{2}}{2\sigma^{2}}\Big{)}\quad\text{asymptotically as
}N\to\infty,$
where $\sigma^{2}:=\mathrm{\mathbf{V}ar}[\xi]$ denotes the variance of $\xi$.
On the other hand, invoking Markov’s inequality to bound the probability in
(2.1) for finite $N$ only implies that
(2.2) $\displaystyle\mathbf{P}[|\widehat{\mu}_{N}-\mu|\geq
r]\leq\frac{\sigma^{2}}{Nr^{2}}\quad\text{for every }N\geq 1$
and that dictates a much slower rate than the bound obtained in (2.1). It
should be stressed that the weaker bound is not caused by looseness in
Markov’s inequality; in fact, there are examples where (2.2) is sharp (up to a
constant). Indeed, let $r$ and $N$ such that $Nr^{2}\geq 1$, and let $\xi$ be
the symmetric random variable taking the values $\pm Nr$ with probability
$\frac{1}{2(Nr)^{2}}$ and $0$ with probability $1-\frac{1}{(Nr)^{2}}$. Then
$\mu=0$ and $\sigma^{2}=1$. Moreover, given a sample of cardinality $N$, a
straightforward computation shows that there is an absolute constant $C$ such
that the following holds: with probability at least $\frac{C}{Nr^{2}}$,
exactly one of the sample points is nonzero. On that event we clearly have
$|\widehat{\mu}_{N}-\mu|=r$, showing that the estimate in Markov’s inequality
(2.2) is sharp (up to the absolute multiplicative constant $C$).
In order to improve (2.2) one needs to impose a stronger integrability
assumption and, eventually, one can show that (2.1) holds for all $N\geq 1$ if
and only if $\xi$ has sub-Gaussian tails in the sense that
$\mathbf{P}[|\xi-\mathbf{E}[\xi]|\geq t]$ is at most of the order
$\exp(-c\frac{t^{2}}{\sigma^{2}})$ for $t\geq c^{\prime}\sigma$.
At this point, sub-Gaussian tails seem unavoidable if one’s goal is to have
finite sample estimates that match the asymptotic ones (as dictated by the
central limit theorem). There is, however, one important possibility that so
far has been neglected: we are free to come up with an alternative estimator
instead of the empirical average. To explain, at an intuitive level, how this
might be a way out of our predicament, note that the non-optimal performance
of the empirical mean stems from the fact that, in the presence of heavy
tails, some of the observations will have untypically large values. These
observations, while few in numbers, offset the empirical mean from its true
counterpart, and the hope is that getting rid of those outliers would lead to
a better statistical performance. The so-called _median-of-means_ estimator is
a simple yet powerful estimator that does just that. It goes back at least to
Nemirovsky and Yudin [27].
Partition the sample $\\{1,\dots,N\\}=\cup_{j=1}^{n}I_{j}$ into $n$ disjoint
blocks $I_{j}$ of cardinality $m:=3\frac{\sigma^{2}}{r^{2}}$ (w.l.o.g. assume
that $n$ and $m$ are integers). By (2.2), we have that
$\mathbf{P}[|\widehat{\mu}_{I_{j}}-\mu|\geq
r]\leq\frac{1}{3}\quad\text{where}\quad\widehat{\mu}_{I_{j}}:=\frac{1}{m}\sum_{i\in
I_{j}}\xi_{i}$
and a basic Binomial calculation reveals that the probability that the
majority of the $n$ blocks satisfy $|\widehat{\mu}_{I_{j}}-\mu|\geq r$ is of
the order of $\exp(-cn)$. Since $n=N\frac{r^{2}}{3\sigma^{2}}$, we conclude
that
$\mathbf{P}\Big{[}\Big{|}\mathop{\mathrm{median}}_{j=1,\dots,n}\,\widehat{\mu}_{I_{j}}-\mu\Big{|}\geq
r\Big{]}\leq 2\exp\Big{(}-CN\frac{r^{2}}{\sigma^{2}}\Big{)}\quad\text{for all
}N\geq 1.$
In other words, the median-of-means estimator exhibits the best possible
performance (2.1) (up to a multiplicative constant) under the sole assumption
that $\xi$ has a finite second moment.
Appealing as this sounds, it is important to stress that the median is a one-
dimensional object and has no simple vector-valued analogue. In fact, the
question of an optimal multivariate mean estimation procedure, assuming only
that the vector has a finite mean and covariance, remained open until it was
resolved recently in [19]. In contrast, stochastic optimization is a multi-
dimensional problem, and just like multivariate mean estimation, simply
minimizing the functional $\mathop{\mathrm{median}}_{j}\frac{1}{m}\sum_{i\in
I_{j}}F(\cdot,\xi_{i})$ does not lead to an optimal estimator. What has a
better chance of success is the _tournament procedure_ which happens to be a
powerful extension of the idea of median-of-means. We will explain the
procedure in Section 2.4 below.
### 2.2. What to expect when tails are Gaussian
Ignoring for a second the difficulties caused by non-Gaussian tails, let us
explain the kind of result one could hope for in general convex stochastic
optimization problems and how the underlying dimension $d$ enters the picture.
This will serve as our benchmark in what follows.
To that end, consider the case where $F$ is a _quadratic function_ defined on
the whole of $\mathbb{R}^{d}$, that is,
(2.3) $\displaystyle F(x,\xi):=\langle b,x\rangle+\frac{1}{2}\langle
Ax,x\rangle\qquad\text{for }x\in\mathcal{X}:=\mathbb{R}^{d},$
where $b=b(\xi)$ is a $d$-dimensional Gaussian vector with zero mean and
$A=A(\xi)$ is a random positive definite symmetric $(d\times d)$-matrix
specified in what follows. Although this example appears to be very special,
it should not be considered as a toy example: by a second order Taylor
expansion, every convex function is approximately a quadratic function, at
least locally, around the minimizer.
By (2.3), it follows that
$\nabla^{2}f(x^{\ast})=\mathbf{E}[A],\quad\quad\nabla
F(x^{\ast},\xi)=b\quad\text{and}\quad\nabla^{2}F(x^{\ast},\xi)=A,$
and we assume throughout that $\mathbf{E}[A]$ is non-degenerate (i.e.
$\mathbf{E}[A]$ has full rank). Setting
$\|z\|:=\langle\nabla^{2}f(x^{\ast})z,z\rangle^{\frac{1}{2}}$ for
$z\in\mathbb{R}^{d}$ and recalling that $b$ has zero mean, it is evident that
$f=\frac{1}{2}\|\cdot\|^{2}$; thus, the optimal action is given by
$x^{\ast}=0$.
###### Remark 2.1.
To get a clearer picture of this setup, it might help the reader to first
consider the case $\nabla^{2}f(x^{\ast})=\mathrm{Id}$, and then $\|\cdot\|$ is
just the Euclidean norm.
The advantage of using the quadratic function considered here is that the
sample average approximation optimizer has a particularly simple form:
$\widehat{x}_{N}^{\text{SAA}}$ is any element satisfying
(2.4)
$\displaystyle\Big{(}\frac{1}{N}\sum_{i=1}^{N}A_{i}\Big{)}\widehat{x}_{N}^{\text{SAA}}=\frac{1}{N}\sum_{i=1}^{N}b_{i}.$
To explain the statistical behavior of $\widehat{x}_{N}^{\text{SAA}}$, let us
first focus on the gradient and assume for the sake of simplicity that the
_Hessian is deterministic_ , that is, $A=\mathbf{E}[A]$. In that case,
$\nabla^{2}f(x^{\ast})\widehat{x}_{N}^{\text{SAA}}$ is Gaussian with mean
$x^{\ast}=0$ and covariance $\frac{1}{N}\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)]$. A straightforward computation (noting that
$\|\cdot\|=\|\nabla^{2}f(x^{\ast})^{\frac{1}{2}}\cdot\|_{2}$) reveals the
following: for the estimation error
$\|\widehat{x}_{N}^{\text{SAA}}-x^{\ast}\|$ to be smaller than $r$ with
constant probability (say, with probability at least $\frac{1}{2}$), it is
necessary to have a sample size of cardinality larger than $N\geq
N_{\mathrm{G}}(r)$, where
(2.5) $\displaystyle N_{\mathrm{G}}(r)$
$\displaystyle:=\frac{1}{r^{2}}\mathop{\mathrm{trace}}\Big{(}\nabla^{2}f(x^{\ast})^{-1}\cdot\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)]\Big{)}.$
On the other hand, for large $N$, it follows from the concentration of a
Lipschitz function of the Gaussian vector that the probability that the
estimation error exceeds $r$ is of the order
$2\exp(-cN\frac{r^{2}}{\sigma^{2}})$. And the variance parameter is
(2.6) $\displaystyle\sigma^{2}$
$\displaystyle:=\lambda_{\max}\Big{(}\nabla^{2}f(x^{\ast})^{-1}\cdot\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)]\Big{)}.$
To summarize, when the quadratic function has a deterministic Hessian, the
minimal sample size needed to guarantee that the estimation error does not
exceed $r$ with constant probability is $N_{\mathrm{G}}(r)$, whereas the
correct variance parameter (namely $\sigma^{2}$) dictates the high-probability
regime. Note that the latter, of course, matches the variance parameter of
[35, Chapter 5], as stated in Section 1.
In a next step, still within the setting of the quadratic function (2.3), let
us remove the assumption that the Hessian is deterministic. In that case, if
one wishes to make any statement regarding the estimation (or, prediction)
error, the empirical Hessian $\frac{1}{N}\sum_{i=1}^{N}A_{i}$ on the left hand
side of (2.4) must not be degenerate. One can readily verify that, unless some
specific assumptions are made, the empirical Hessian is singular with
probability 1 whenever $N<d$. Thus, $N\geq d$ is another restriction on the
minimal sample size (though, at this point, it is far from obvious that a
sample of size $d$ or proportional to $d$ would suffice to guarantee a non-
degenerate Hessian with, say, constant probability).
Following these observations one can make a very _optimistic guess_ on the
estimate one can hope to obtain: that there exists a procedure
$\widehat{x}^{N}_{\ast}$ such that for
$N\geq\max\\{N_{\mathrm{G}}(r),d\\},$
with probability at least
$1-2\exp\Big{(}-cN\frac{r^{2}}{\sigma^{2}}\Big{)},$
we have that
$\|\widehat{x}_{N}^{\ast}-x^{\ast}\|\leq r.$
###### Remark 2.2.
This is indeed an optimist guess, and is “very gaussian”. The minimal sample
size $\max\\{N_{\mathrm{G}}(r),d\\}$ is the result of rather trivial
obstructions; their removal is necessary if the estimation error is to have
any chance of being smaller than $r$ with _constant_ probability. Furthermore,
the variance term $\sigma^{2}$, which dictates the _high_ probability regime
(for large $N$), is effectively one dimensional: it corresponds to the worst
direction of the gradient (w.r.t. the norm $\|\cdot\|$).
Before we proceed with the main (affirmative) result of this article, let us
conclude this section with a comment regarding the relation between the
estimation error and the prediction error / the optimality gap, in a more
general setup than the simple example we presented previously.
If $\mathcal{X}=\mathbb{R}^{d}$, a Taylor expansion and the first order
condition for optimality of $x^{\ast}$ immediately implies that
$\displaystyle f(x)-f(x^{\ast})$
$\displaystyle=\frac{1}{2}\langle\nabla^{2}f(y)(x-x^{\ast}),x-x^{\ast}\rangle$
where $y$ is some mid-point between $x^{\ast}$ and $x$. In particular, setting
$\|B\|_{\mathrm{op}}:=\sup_{z\in\mathbb{R}^{d}\text{ s.t.\ }\|z\|\leq
1}\langle Bz,z\rangle$ to be the operator norm111 Note that
$\|\cdot\|_{\mathrm{op}}$ is indeed the operator norm from
$(\mathbb{R}^{d},\|\cdot\|)$ to $(\mathbb{R}^{d},\|\cdot\|_{\ast})$, where
$\|\cdot\|_{\ast}$ is the dual norm of $\|\cdot\|$, i.e.
$\|y\|_{\ast}:=\sup_{\|x\|\leq 1}\langle x,y\rangle$. of a positive semi-
definite $(d\times d)$-matrix $B$, and
(2.7) $\displaystyle c_{\mathrm{H}}:=\sup_{y\in\mathcal{X}\text{ s.t.\
}\|y-x^{\ast}\|<1}\,\,\frac{1}{2}\|\nabla^{2}f(y)\|_{\mathrm{op}}$
it is clear that
$f(x)-f(x^{\ast})\leq c_{\mathrm{H}}r^{2}\quad\text{whenever
}\|x-x^{\ast}\|\leq r$
and $r<1$. Thus, one can make another _highly optimistic guess_ : that there
exists a procedure $\widehat{x}_{N}^{\ast}$ for which the prediction error /
optimality gap is smaller than the estimation error by at least one order of
magnitude.
What makes this guess optimistic (and nontrivial), is that the above argument
crucially relies on the fact that $\mathcal{X}=\mathbb{R}^{d}$; or, more
generally, that $x^{\ast}$ lies in the interior of $\mathcal{X}$. That need
not be the case.
### 2.3. Recovering Gaussian rates without sub-Gaussian tails
This section contains the main result of the article, formulated in Theorem
2.9 below. It provides affirmative answers to the optimistic guesses made in
the previous section (under some mild assumptions, of course). The assumptions
might appear technical at first glance, and to help the reader put them in
context, each assumption will be explained in the case of portfolio
optimization, and in a heuristic manner; the detailed analysis will be
presented in Section 6.
###### Assumption 2.3 (Convexity and coercivity).
The following hold:
1. (a)
$\mathcal{X}\subseteq\mathbb{R}^{d}$ is closed and convex;
2. (b)
$x\mapsto F(x,\gamma)$ is convex and twice continuously differentiable222 If
$\mathcal{X}$ is not open, we mean by “continuously differentiable” that there
is a continuous function $\nabla F(\cdot,\gamma)$ such that
$F(y,\gamma)-F(x,\gamma)=\int_{0}^{1}\langle\nabla
F(x+t(y-x),\gamma),y-x\rangle\,dt$. A similar notion holds for $\nabla^{2}F$
for the “twice continuously differentiable”. for every $\gamma\in\Xi$;
3. (c)
$F(x,\xi)$, and $\nabla^{2}F(x,\xi)$ are integrable and $\nabla F(x,\xi)$ is
square integrable for every $x\in\mathcal{X}$.
Further, there exists an _optimal action_ $x^{\ast}\in\mathcal{X}$ that
satisfies $f(x^{\ast})=\inf_{x\in\mathcal{X}}f(x)$, and the seminorm induced
by the Hessian of $f$ at $x^{\ast}$ given by
$\|z\|:=\langle\nabla^{2}f(x^{\ast})z,z\rangle^{\frac{1}{2}}\quad\text{for
}z\in\mathbb{R}^{d}$
is a true norm (i.e. $\|z\|=0$ implies $z=0$).
###### Remark 2.4.
While $f$ clearly inherits convexity from $F$, it is not clear a priori that
$f$ is twice continuously differentiable (in the same sense as $F$ if
$\mathcal{X}$ is not open). This follows once Assumption 2.7 below is imposed,
as we shall explain in the beginning of Section 5.
Note that the minimizer $x^{\ast}$ in Assumption 2.3 is unique, as $\|\cdot\|$
is a norm. In fact, the latter relates to a standard assumption in stochastic
optimization—the so called _quadratic growth condition_ : that there is a
constant $\kappa>0$ such that
$\displaystyle f(x)\geq f(x^{\ast})+\kappa\|x-x^{\ast}\|_{2}^{2}$
for all $x$ close to $x^{\ast}$. Indeed, denoting
$\tilde{\kappa}:=\lambda_{\min}(\nabla^{2}f(x^{\ast}))$, the smallest
eigenvalue of the Hessian of $f$ at $x^{\ast}$, we have that
$\tilde{\kappa}>0$ whenever $\|\cdot\|$ is true norm. Moreover, a Taylor
expansion shows that the quadratic growth condition holds with constant
$\kappa=\tilde{\kappa}$ for all $x$ in an infinitesimal neighbourhood of
$x^{\ast}$ (or with constant $\kappa=\frac{\tilde{\kappa}}{2}$ in a
sufficiently small neighbourhood). Conversely, at least when $x^{\ast}$ lies
in the interior of $\mathcal{X}$, the quadratic growth condition readily
implies $\tilde{\kappa}\geq\kappa$ and in particular that $\|\cdot\|$ is a
true norm.
> Let us now give an intuitive interpretation of Assumption 2.3 in the context
> of the portfolio optimization example. To ease notation, we shall assume
> that $\pi=\mathbf{E}[X]=0$ and that $\mathrm{\mathbf{C}ov}[X]=\mathrm{Id}$.
> The convexity and differentiability parts of the assumption are clear, and
> it is straightforward to verify that
>
> $\displaystyle\nabla F(x,\xi)$ $\displaystyle=-\ell^{\prime}(-Y-\langle
> X,x\rangle)\cdot X,$ $\displaystyle\nabla^{2}F(x,\xi)$
> $\displaystyle=\ell^{\prime\prime}(-Y-\langle X,x\rangle)\cdot X\otimes X.$
>
> In particular
>
> $\|z\|^{2}=\mathbf{E}[\ell^{\prime\prime}(-Y-\langle
> X,x^{\ast}\rangle)\cdot\langle X,z\rangle^{2}].$
>
> Ignoring the $\ell^{\prime\prime}$-term inside the expectation for the
> moment, this would imply that $\|\cdot\|=\|\cdot\|_{2}$. In general, when
> accounting for the $\ell^{\prime\prime}$-term, a minor integrability
> assumption will be used to guarantee that $\|\cdot\|$ and $\|\cdot\|_{2}$
> are equivalent norms.
In Section 2.2 we argued that unless the Hessian has a specific form, the
empirical Hessian is singular whenever $N\leq d$. However, without further
assumptions, believing that a sample of cardinality $d$ is enough to guarantee
a non-degenerate empirical Hessian (say with constant probability) is too
optimistic—certainly in the general setting we are interested in here.
Degeneracy can happen even in dimension $d=1$: if $\nabla^{2}F(x^{\ast},\xi)$
takes the value $\frac{1}{\varepsilon}$ with probability $\varepsilon>0$ and
is zero otherwise, the endowed norm $\|\cdot\|$ is simply the absolute
value—regardless of the choice of $\varepsilon$. However, with probability
$(1-\varepsilon)^{N}$, all observations in a sample of size $N$ are zero, and
for small $\varepsilon$, e.g. $\varepsilon=\frac{1}{N^{2}}$, that probability
converges to 1 as $N\to\infty$.
As it happens, the following modest integrability assumption on the Hessian
prevents such behavior.
###### Assumption 2.5 (Integrability of the Hessian).
There is a constant $L$ such that
$\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle^{2}]\leq L$
for every $z\in\mathbb{R}^{d}$ with $\|z\|=1$.
Another way of formulating Assumption 2.5 is in the sense of _norm-
equivalence_ : for every $z\in\mathbb{R}^{d}$, we have that
(2.8)
$\displaystyle\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle^{2}]^{\frac{1}{4}}\leq
L^{\frac{1}{4}}\cdot\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle]^{\frac{1}{2}},$
i.e. the $L_{4}$-norm and the $L_{2}$-norm of the forms
$\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle^{\frac{1}{2}}$ are
equivalent333Note that the reverse inequality to (2.8) is trivially true with
constant $1$, by Hölder’s inequality.. Therefore the constant $L$ pertains to
the worst direction $z\in\mathbb{R}^{d}$ (and not e.g. the average over
different directions). As such, $L$ typically does not depend on the dimension
$d$.
Under Assumption 2.5 one can prove a lower bound on the smallest singular
value of the empirical Hessian whenever $N\gtrsim d\log(d)$.
> To check Assumption 2.5 in the portfolio optimization example, note that
>
>
> $\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle^{2}]=\mathbf{E}[\ell^{\prime\prime}(-Y-\langle
> X,x^{\ast}\rangle)^{2}\cdot\langle X,z\rangle^{4}].$
>
> Thus, Assumption 2.5 is a simple consequence of Hölder’s inequality and a
> minor integrability condition.
###### Remark 2.6.
Let us stress that Assumption 2.5 is just a tractable way of ensuring that our
argument works; it could be replaced by the more general assumption that
$\nabla^{2}F(x^{\ast},\xi)$ satisfies a so-called _stable lower bound_ , see
Remark 7.2. The stable lower bound and its role in obtaining lower bounds on
the smallest singular values of rather general random matrix ensembles is
described in Theorem 4.4 and in Corollary 4.1.
There is another reason why the minimal sample should be at least a (large
constant) multiple of $d$, namely, because $F$ need not be quadratic. In the
example in Section 2.2 $F$ was a quadratic function, and as a result the
Hessian did not depend on the action $x$. In general, when invoking a second
order Taylor expansion, the Hessian does depend on some mid-point
$x^{\ast}+t(x-x^{\ast})$. At the same time, Assumption 2.5 only takes into
account the Hessian at the optimizer; therefore, some continuity assumption is
needed if one is to control the deviation from quadratic, which is governed by
$\displaystyle\mathcal{E}_{\mathrm{H}}(x):=\sup_{t\in[0,1]}\Big{|}\Big{\langle}\Big{(}\nabla^{2}F(x^{\ast}\\!\\!+\\!t(x\\!-\\!x^{\ast}),\xi)-\nabla^{2}F(x^{\ast},\xi)\Big{)}(x-x^{\ast}),x-x^{\ast}\Big{\rangle}\Big{|}$
for $x\in\mathcal{X}$.
Note that $\mathcal{E}_{\mathrm{H}}(x)$ is likely to be of order
$\|x-x^{\ast}\|^{3}$ under a suitable Lipschitz condition on the Hessian.
Assumption 2.7 is there to ensure that $\mathcal{E}_{\mathrm{H}}(x)$ is
sufficiently small.
###### Assumption 2.7 (Continuity of the Hessian).
There exists a radius $r_{0}\in(0,1)$ such that the following hold.
1. (a)
There is a Hölder coefficient $\alpha\in(0,1]$ and a measurable function
$K\colon\Xi\to[0,\infty)$ such that $\mathbf{E}[K(\xi)]<\infty$ and for all
$x,y\in\mathcal{X}$ with $\|x-x^{\ast}\|,\|y-x^{\ast}\|\leq r_{0}$, we have
that
$\displaystyle\Big{\|}\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi)\Big{\|}_{\mathrm{op}}$
$\displaystyle\leq\|x-y\|^{\alpha}\cdot K(\xi).$
2. (b)
For some given constant $c_{1}$ (which will be specified in Theorem 2.9 and
depends only on the parameter $L$ of Assumption 2.5) and for all
$x\in\mathcal{X}$ with $\|x-x^{\ast}\|\leq r_{0}$, we have that
$\displaystyle\mathbf{P}\Big{[}\mathcal{E}_{\mathrm{H}}(x)\leq\frac{\|x-x^{\ast}\|^{2}}{8}\Big{]}$
$\displaystyle\geq(1-c_{1}).$
Assumption 2.7 implies that, setting
(2.9) $\displaystyle N_{\mathrm{H},\mathcal{E}}$
$\displaystyle:=\frac{d}{\alpha}\log\Big{(}r_{0}^{\alpha}\mathbf{E}[K(\xi)]+2\Big{)},$
whenever $N\gtrsim N_{\mathrm{H},\mathcal{E}}$, the error caused by replacing
$F$ by its quadratic approximation does not distort the outcome by too much.
###### Remark 2.8.
At a first glance it might seem as if part (b) of Assumption 2.7 follows from
part (a). It is true that
$\mathcal{E}_{\mathrm{H}}(x)\leq\|x-x^{\ast}\|^{2+\alpha}\cdot K(\xi)$, but
there is one important difference: in typical situations,
$\mathcal{E}_{\mathrm{H}}$ does not depend on the dimension $d$, while $K$
does (we shall see this phenomenon in the portfolio optimization problem). In
particular, estimating $\mathcal{E}_{\mathrm{H}}$ using $K(\xi)$ will
unnecessarily force the threshold radius $r_{0}$ to depend on the dimension,
which is something we wish to avoid. On the other hand, the dimension-
dependent term $\mathbf{E}[K(\xi)]$ only appears through a logarithmic factor
in the minimal sample size.
> Returning to the portfolio optimization problem, let us, for the sake of a
> simplified exposition, pretend that $\ell^{\prime\prime}$ is $1$-Lipschitz
> continuous, and recall that $\|\cdot\|_{2}$ and $\|\cdot\|$ are equivalent
> norms. Then
>
> (2.10)
> $\displaystyle\begin{split}&\big{|}\big{\langle}(\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi))z,z\big{\rangle}\big{|}\\\
> &=|\ell^{\prime\prime}(-Y-\langle X,x\rangle)-\ell^{\prime\prime}(-Y-\langle
> X,y\rangle)|\cdot\langle X,z\rangle^{2}\\\ &\leq|\langle
> X,x-y\rangle|\cdot\langle X,z\rangle^{2}.\end{split}$
>
> Thus $\mathcal{E}_{\mathrm{H}}(x)\leq|\langle X,x-x^{\ast}\rangle|^{3}$ and,
> under some mild integrability assumption, the latter term behaves like
> $\|x-x^{\ast}\|_{2}^{3}$ on average. Markov’s inequality and the fact that
> $\|\cdot\|$ and $\|\cdot\|_{2}$ are equivalent norms imply that
>
>
> $\mathbf{P}\Big{[}\mathcal{E}_{\mathrm{H}}(x)>\frac{1}{8}\|x-x^{\ast}\|^{2}\Big{]}\leq\frac{8\mathbf{E}[\mathcal{E}_{\mathrm{H}}(x)]}{\|x-x^{\ast}\|^{2}}\leq\frac{c8\mathbf{E}[\mathcal{E}_{\mathrm{H}}(x)]}{\|x-x^{\ast}\|_{2}^{2}}$
>
> is of order $\|x-x^{\ast}\|_{2}$.
>
> On the other hand, (2.10) implies that
>
> $\|\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi)\|_{\mathrm{op}}\leq
> K(\xi)\cdot\|x-y\|_{2},$
>
> for $K(\xi):=\|X\|_{2}^{3}$ which typically scales like $d^{\frac{3}{2}}$.
With all the definitions set in place let us turn to the formulation of our
main result. Recall that $N_{\mathrm{G}}(r)$, $\sigma^{2}$, $c_{\mathrm{H}}$,
and $N_{\mathrm{H},\mathcal{E}}$ were defined in (2.5), (2.6), (2.7), and
(2.9) respectively.
###### Theorem 2.9 (Estimation and prediction error).
There are constants $c_{1},c_{2},c_{3}$ depending only on $L$ such that the
following holds. Assume that Assumptions 2.3, 2.5, 2.7 hold, let
$r\in(0,r_{0})$ and consider
$N\geq c_{2}\max\\{d\log(2d),N_{\mathrm{H},\mathcal{E}},N_{\mathrm{G}}(r)\\}.$
Then there is a procedure $\widehat{x}_{N}^{\ast}$ such that, with probability
at least
$1-2\exp\Big{(}-c_{3}N\min\Big{\\{}1,\frac{r^{2}}{\sigma^{2}}\Big{\\}}\Big{)},$
we have that
$\displaystyle\|\widehat{x}_{N}^{\ast}-x^{\ast}\|$ $\displaystyle\leq r,$
$\displaystyle\ f(\widehat{x}_{N}^{\ast})$ $\displaystyle\leq
f(x^{\ast})+2c_{\mathrm{H}}\cdot r^{2}.$
The procedure is described in Section 2.4.
Theorem 2.9 implies that our procedure recovers (up to multiplicative
constants) the optimal asymptotic rates for the sample average approximation
[35, Chapter 5] in a non-asymptotic fashion and when the random variables
involved can be heavy tailed.
> To complete the heuristics pertaining to the portfolio optimization problem,
> one has to compute $N_{\mathrm{G}}(r),\sigma^{2}$ and $c_{\mathrm{H}}$.
> Again, for the sake of simplicity we shall ignore $\ell^{\prime}$ and
> $\ell^{\prime\prime}$ at every appearance (keeping in mind that some minor
> integrability assumptions guarantee that this simplification does not shift
> the results by too much from the truth).
>
> In this case, $\mathrm{\mathbf{C}ov}[\nabla F(x^{\ast},\xi)]=\mathrm{Id}$
> and $\nabla^{2}f(x^{\ast})=\mathrm{Id}$, and in particular
>
> $\sigma^{2}=\lambda_{\max}(\mathrm{Id})=1\ \ \text{and}\ \
> N_{\mathrm{G}}(r)=\frac{\mathrm{trace}(\mathrm{Id})}{r^{2}}=\frac{d}{r^{2}}.$
>
> Moreover, ignoring $\ell^{\prime\prime}$ also clearly implies that
> $c_{\mathrm{H}}=1$.
In Corollary 3.7 we specify all the assumptions that are needed to make this
heuristic argument hold; but for now let us state a particularly simple case
which is of interest in its own right: the exponential portfolio optimization
in the Bachelier model.
Recall that in this model $U(\cdot)=-\exp(-\,\cdot)$ is the exponential
utility function and $X$ is zero-mean Gaussian. We assume that the covariance
matrix of $X$ is non-degenerate and that both $Y$ and $U(2Y)$ are integrable.
Under these assumptions, we shall see that there exists a unique optimal
action $x^{\ast}\in\mathcal{X}$. Set
$\displaystyle\bar{\sigma}^{2}$
$\displaystyle:=\mathbf{E}\big{[}\exp(-Y-\langle
X,x^{\ast}\rangle)^{2}\big{]}^{\frac{1}{2}},$
and assume that Assumption 2.5 holds true. While the latter assumption can be
verified via some integrability conditions (we shall see this in context of
the general portfolio optimization problem in Lemma 6.2), the obtained bounds
may fail to be sharp. To showcase that Assumption 2.5 can sometimes be easily
verified by other means, consider for a moment $Y=\langle
X,\tilde{x}\rangle+W$ for some $\tilde{x}\in\mathcal{X}$ and $W$ that is
independent of $X$. Then $x^{\ast}=\tilde{x}$ and Gaussian norm equivalence
(i.e. there is an absolute constant $C$ such that $\mathbf{E}[\langle
X,z\rangle^{4}]^{\frac{1}{4}}\leq C\mathbf{E}[\langle
X,z\rangle^{2}]^{\frac{1}{2}}$ for every $z\in\mathbb{R}^{d}$) together with
independence of $X$ and $W$ readily implies that Assumption 2.5 is satisfied
with
$L=\frac{C^{4}\mathbf{E}[\exp(W)^{2}]}{\mathbf{E}[\exp(W)]^{2}}.$
###### Corollary 2.10 (Exponential portfolio optimization).
Under the above assumptions, there are constants $c_{1},c_{2},c_{3},c_{4}>0$
depending only on $L$ and $\mathbf{E}[|Y+\langle X,x^{\ast}\rangle|]$ such
that the following holds. For
$r\in(0,\min\\{1,\frac{c_{1}}{\bar{\sigma}^{2}}\\})$ and
$N\geq
c_{2}\max\Big{\\{}\frac{d\bar{\sigma}^{2}}{r^{2}},\,d^{\frac{3}{2}}\,\Big{\\}},$
there is a procedure $\widehat{x}_{N}^{\ast}$ such that, with probability at
least
$1-2\exp\Big{(}-c_{3}N\min\Big{\\{}1,\frac{r^{2}}{\bar{\sigma}^{2}}\Big{\\}}\Big{)},$
we have that
$\displaystyle\|\widehat{x}_{N}^{\ast}-x^{\ast}\|$ $\displaystyle\leq r,$
$\displaystyle u(\widehat{x}_{N}^{\ast})$ $\displaystyle\geq
u(x^{\ast})-c_{4}r^{2}.$
###### Remark 2.11.
The origin of the (somewhat unfamiliar) term $d^{\frac{3}{2}}$ appearing in
the minimal sample size in Corollary 2.10 is $N_{\mathrm{H},\mathcal{E}}$.
However, as our previous heuristics indicate, that term should be of order
$d\log(d)$.
As it happens, the source of this difference is the exponential utility
function. Indeed, the heuristic presentation was based on a simplifying
assumption: that $\ell^{\prime\prime\prime}$ was bounded by 1. That allowed us
to conclude that $\mathbf{E}[K(\xi)]$ was of the order $d^{3}$, resulting in
$N_{\mathrm{H},\mathcal{E}}$ of order $d\log(d^{3})=3d\log(d)$. Here, however,
$\ell^{\prime\prime\prime}$ is the exponential function; the term
$\mathbf{E}[K(\xi)]$ is actually of order $\exp(\sqrt{d})$ resulting in
$N_{\mathrm{H},\mathcal{E}}$ that is of order
$d\log(\exp(\sqrt{d}))=d^{\frac{3}{2}}$.
It should be stressed that although the term $d^{\frac{3}{2}}$ in the minimal
sample size of Corollary 2.10 could be off by a factor of $\sqrt{d}$,
Corollary 2.10 is, to the best of our knowledge, the first non-asymptotic
estimate for the exponential portfolio optimization problem.
In some of the examples we will present later, the Hessian additionally
satisfies a deterministic lower bound:
###### Assumption 2.12 (Deterministic lower bound of the Hessian).
There is $r_{0}\in(0,1)$ and $\varepsilon>0$ such that
$\nabla^{2}F(x,\xi)\succeq\varepsilon\nabla^{2}f(x^{\ast})$
for all $x\in\mathcal{X}$ with $\|x-x^{\ast}\|\leq r_{0}$.
Moreover, $\nabla f(x)=\mathbf{E}[\nabla F(x,\xi)]$ and
$\nabla^{2}f(x)=\mathbf{E}[\nabla^{2}F(x,\xi)]$ for all $x\in\mathcal{X}$ with
$\|x-x^{\ast}\|<r_{0}$.
When Assumption 2.12 holds true, the two Assumptions 2.5 and 2.7 that were
imposed to control the smallest singular value of the Hessian are not needed,
and Theorem 2.9 can be simplified as follows.
###### Theorem 2.13 (Estimation and prediction error, simplified).
There are constants $c_{1},c_{2}$ depending only on $\varepsilon$ such that
the following holds. Assume that Assumptions 2.3 and 2.12 hold, let
$r\in(0,r_{0})$, and consider
$N\geq c_{1}N_{\mathrm{G}}(r).$
Then there is a procedure $\widehat{x}_{N}^{\ast}$ such that, with probability
at least
$1-2\exp\Big{(}-c_{2}N\min\Big{\\{}1,\frac{r^{2}}{\sigma^{2}}\Big{\\}}\Big{)},$
we have that
$\displaystyle\|\widehat{x}_{N}^{\ast}-x^{\ast}\|$ $\displaystyle\leq r,$
$\displaystyle\ f(\widehat{x}_{N}^{\ast})$ $\displaystyle\leq
f(x^{\ast})+2c_{\mathrm{H}}\cdot r^{2}.$
Before presenting the procedure $\widehat{x}_{N}^{\ast}$ in detail, let us
compare the outcome of Theorem 2.9 with the current state of the art. Our
focus is on recent non-asymptotic results by Oliveira and Thompson [28, 29]; a
general literature review will be presented in Section 2.5.
For a clearer comparison, let us restate Theorem 2.9 (pertaining to the
prediction error) as follows:
given some fixed confidence level $\delta\in(0,1)$, small $r>0$, and
$N\geq c_{2}\max\\{d\log(2d),N_{\mathrm{H},\mathcal{E}},N_{\mathrm{G}}(r)\\},$
the prediction error is bounded by $r^{2}$ with probability at least
$1-\delta$, whenever the sample $N$ satisfies
(2.11) $\displaystyle
N\gtrsim\frac{\sigma^{2}}{r^{2}}\log\Big{(}\frac{2}{\delta}\Big{)}$
In contrast, the main result of Oliveira and Thompson [29, Theorem 3]
regarding general convex stochastic optimization problems is the following.
There is a random variable $\widehat{\Sigma}_{N}$ (which we shall not define
here) that satisfies
$\widehat{\Sigma}_{N}^{2}\gtrsim d\cdot\Big{(}\mathbf{E}[\|\nabla
F(x^{\ast},\xi)\|^{2}]+\frac{1}{N}\sum_{i=1}^{N}\|\nabla
F(x^{\ast},\xi_{i})\|^{2}\Big{)}$
and in order to guarantee that the prediction error is bounded by $r^{2}$ with
probability at least $1-\delta$ one should have
(2.12)
$\mathbf{P}\Big{[}N\geq\frac{\widehat{\Sigma}_{N}^{2}}{r^{2}}\log\Big{(}\frac{2}{\delta}\Big{)}\Big{]}\geq
1-\delta.$
There are two _major_ differences between these two results. The first one,
which should not come as a great surprise following the discussion in Section
2.1, is that $\widehat{\Sigma}_{N}^{2}$ does not concentrate around its mean
with high probability in heavy tailed situations. Thus, (2.12) forces $N$ to
grow like $(\frac{1}{\delta})^{\frac{1}{p}}$ for some power $p>1$ depending on
the integrability of $\|\nabla F(x^{\ast},\xi)\|$. For small $\delta$ (i.e. if
one is interested in high confidence) this is in stark contrast to the order
$\log(\frac{1}{\delta})$ in (2.11). The second difference is the dependence of
$\widehat{\Sigma}_{N}^{2}$ on the dimension $d$. Indeed, even if we neglect
the integrability issues and replace $\widehat{\Sigma}_{N}^{2}$ by its mean,
what we end up with is not the correct variance parameter (which appears in
the central limit theorem). For instance, if $\|\cdot\|$ is the Euclidean
norm, then
$\widehat{\Sigma}_{N}^{2}\gtrsim
d\cdot\mathrm{trace}(\mathrm{\mathbf{C}ov}[\nabla F(x^{\ast},\xi)]),$
which is at least $d$ times (possibly even $d^{2}$ times) larger than the true
variance parameter $\sigma^{2}=\lambda_{\max}(\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)])$. In high-dimensional problems such as the portfolio
optimization problem, where $d$ is the number of stocks and is likely to be a
large number, this difference is significant. As a result, even for moderate
confidence levels $\delta$, the required sample size jumps up by a factor of
$d$ (or even $d^{2}$) from what we would expect.
### 2.4. The procedure
As we already explained previously, to have any hope of optimal performance,
the procedure $\widehat{x}_{N}^{\ast}$ _cannot_ be the sample average
approximation. Instead, $\widehat{x}_{N}^{\ast}$ will be determined through
median-of-mean tournaments conducted between all $x\in\mathcal{X}$, following
the method introduced in [20].
The first phase of the procedure returns a set of candidates, each with a
small estimation error.
1. (Step 1)
For some (small) tuning parameter $\theta$ to be specified later, set
$n:=\theta
N\min\Big{\\{}1,\frac{r^{2}}{\sigma^{2}}\Big{\\}}\quad\text{and}\quad
m:=\frac{N}{n}.$
Without loss of generality assume that $m$ and $n$ are integers. Partition
$\\{1,\dots,N\\}=\bigcup_{j=1}^{n}I_{j}$
into $n$ disjoint blocks $I_{j}$, each of equal cardinality $|I_{j}|=m$.
2. (Step 2)
For every $x\in\mathcal{X}$, compute the empirical mean of $F(x,\xi)$ on the
$j$-th block
$\widehat{f}_{I_{j}}(x):=\frac{1}{m}\sum_{i\in I_{j}}F(x,\xi_{i}).$
We then say that $x\in\mathcal{X}$ _defeats_ $y\in\mathcal{X}$ _on the $j$-th
block_ if $\widehat{f}_{I_{j}}(x)<\widehat{f}_{I_{j}}(y)$, and that $x$ _wins
the match against_ $y$ if
$\widehat{f}_{I_{j}}(x)<\widehat{f}_{I_{j}}(y)\quad\text{on more than
}\frac{n}{2}\text{ blocks}\ j,$
i.e. if $x$ defeats $y$ on a majority of the blocks.
Denote by $\tilde{\mathcal{X}}_{N}^{\ast}\subseteq\mathcal{X}$ the set of
_champions_ , i.e.
$\tilde{\mathcal{X}}_{N}^{\ast}:=\Big{\\{}x\in\mathcal{X}:\begin{array}[]{l}x\text{
wins the match against every}\\\ y\in\mathcal{X}\text{ that satisfies
}\|x-y\|\geq r\end{array}\Big{\\}}.$
The following proposition shows that elements in
$\tilde{\mathcal{X}}_{N}^{\ast}$ satisfy the part of Theorem 2.9 pertaining to
the estimation error.
###### Proposition 2.14 (Estimation error).
In the setting of Theorem 2.9, with probability at least $1-2\exp(-c_{3}n)$,
we have that $x^{\ast}\in\tilde{\mathcal{X}}_{N}^{\ast}$ and every
$x\in\tilde{\mathcal{X}}_{N}^{\ast}$ satisfies $\|x-x^{\ast}\|\leq r$.
As we already explained, if $\mathcal{X}=\mathbb{R}^{d}$ or, more generally,
if $x^{\ast}$ lies in the interior of $\mathcal{X}$, the first order condition
for optimality immediately implies that $f(x)\leq
f(x^{\ast})+c_{\mathrm{H}}r^{2}$ for every $x\in\mathcal{X}$ with
$\|x-x^{\ast}\|\leq r$. In particular, in that case, it follows from
Proposition 2.14 that any choice
$\widehat{x}_{N}^{\ast}\in\tilde{\mathcal{X}}_{N}^{\ast}$ satisfies the
assertion of Theorem 2.9.
However, in general, if one wishes to find
$x\in\tilde{\mathcal{X}}_{N}^{\ast}$ with a small prediction error, one
requires an additional procedure, which we describe now.
To simplify notation, assume without loss of generality that the set
$\tilde{\mathcal{X}}_{N}^{\ast}$ has already been determined, and that we can
run an additional second procedure, for which we are given a new (independent)
sample $F(\cdot,\xi_{i})_{i=N+1}^{2N}$.
Again partition $\\{N+1,\dots,2N\\}$ into $n$ disjoint blocks $I_{j}^{\prime}$
of cardinality $|I_{j}^{\prime}|=m$ with the same $n$ and $m$, and denote by
$\widehat{f}_{I_{j}^{\prime}}(\cdot)$ the empirical mean on the block
$I_{j}^{\prime}$.
1. (Step 3)
We say that $x\in\mathcal{X}$ _wins its home match_ against $y\in\mathcal{X}$
if
$\widehat{f}_{I_{j}^{\prime}}(x)\leq\widehat{f}_{I_{j}^{\prime}}(y)+\frac{c_{\mathrm{H}}r^{2}}{4}\quad\text{
on more than }\frac{n}{2}\text{ blocks }j.$
Denote by $\widehat{\mathcal{X}}_{N}^{\ast}$ the _winners_ , i.e.,
$\widehat{\mathcal{X}}_{N}^{\ast}:=\Big{\\{}x\in\tilde{\mathcal{X}}_{N}^{\ast}:\begin{array}[]{l}x\text{
wins its home match }\\\ \text{against every
}y\in\tilde{\mathcal{X}}_{N}^{\ast}\end{array}\Big{\\}}.$
In light of Proposition 2.14, the crucial advantage here is that, with high
probability, matches are only carried out between competitors that are close
to $x^{\ast}$. The following proposition shows that any
$\widehat{x}_{N}^{\ast}\in\widehat{\mathcal{X}}_{N}^{\ast}$ satisfies the
requirements in Theorem 2.9:
###### Proposition 2.15 (Prediction error).
In the setting of Theorem 2.9, with probability at least $1-2\exp(-c_{3}n)$,
we have that $x^{\ast}\in\widehat{\mathcal{X}}_{N}^{\ast}$ and every
$x\in\widehat{\mathcal{X}}_{N}^{\ast}$ satisfies
$f(x)<f(x^{\ast})+2c_{\mathrm{H}}r^{2}$.
### 2.5. Related literature
Stochastic optimization and the statistical properties of the sample average
approximation method have been studied intensively for several decades; it is
therefore impossible to mention every single contribution. Instead, we refer
to Kim, Pasupathy, and Henderson [12], Shapiro, Dentcheva, and Ruszczyński
[35], Kleywegt, Shapiro, Homem-de-Mello [13], Shapiro [34], or Homem-de-Mello
and Bayraksan [10]. As we already mentioned, the statistical analysis in these
works is always of asymptotic nature. Let us also refer to Banholzer, Fliege,
and Werner [2] for a recent study of asymptotic almost sure convergence rates
and an up-to date review on the asymptotic convergence analysis, and to
Bertsimas, Gupta, and Kallus [4] who raise concerns that most statistical
results for sample average approximation are of asymptotic nature.
Shifting to a non-asymptotic analysis of the performance of the SAA, the
available literature gets considerably less diverse. An early reference here
is Pflug [30, 31], who relies on Talagrand’s deviation inequality for the
supremum of Gaussian processes (which is obviously suitable only in very
special, light-tailed scenarios). Similar methods were adapted by Römisch [32]
and Vogel [41] and (for the error of the value—i.e.
$\min_{x\in\mathcal{X}}f(x)$) by Guigues, Juditsky, and Nemirovski in [9]. The
two recent papers by Oliveira and Thompson [28, 29] which have been mentioned
at the end of Section 2 contain results that are closest to ours.
The portfolio optimization problem is often listed as a prime example of a
stochastic optimization problem, see e.g. [35, Section 1.4]. As such, and with
the exception of perhaps more applied studies like [8, 42, 44], existing
estimates on the problem have been derived as applications of general
results—much like in our work.
There are other (optimization) problems in mathematical finance that have been
analyzed via sampling, such as the estimation of risk measures. We refer the
reader to Weber [43] (which relies on large deviation methods) for an early
reference and to [3] for a more recent review. We believe that our methods can
be applied to those type of problems as well, and defer it to future work.
We should mention that the statistical analysis of the SAA has close ties to
the statistical analysis of problems in high-dimensional statistics, such as
linear regression. But despite obvious similarities, the two fields have grown
adrift. The current focus in statistical learning literature is on non-
asymptotic statements that hold under increasingly relaxed assumptions. Among
the outcomes of this approach were [20, 23, 24]—alternatives to the sample
average approximation in statistical learning problems that recover the
Gaussian rates in completely heavy tailed situations. We believe that pursuing
the same direction in the context of stochastic optimization would lead to
intriguing questions. Indeed, there are many variants of (convex) stochastic
optimization problems that are not covered in this article. For example, among
these questions is _chance constrained stochastic optimization_ , where the
optimization takes place only over actions $x\in\mathcal{X}$ satisfying
probabilistic constraints of the form $\mathbf{E}[G(x,\xi)]\leq 0$. We are
confident that our methods can be adapted to these settings as well, though we
shall leave this for future work.
## 3\. Applications
Before continuing with the portfolio optimization problem (in its general
form), we present three of Theorem 2.9.
### 3.1. Multivariate mean estimation
As a first application of Theorem 2.9, let us return to the problem of mean
estimation—this time for a square integrable $d$-dimensional random vector
$\xi$. Mean estimation is a stochastic optimization problem because
$\mathbf{E}[\xi]=\mathrm{argmin}\big{\\{}\mathbf{E}[\|x-\xi\|_{2}^{2}]:x\in\mathbb{R}^{d}\big{\\}}.$
This suggests that one should set
$F(x,\xi):=\frac{1}{2}\|x-\xi\|_{2}^{2}\quad\text{for
}x\in\mathcal{X}:=\mathbb{R}^{d}$
so that $x^{\ast}=\mathbf{E}[\xi]$. (The factor $\frac{1}{2}$ has only a
normalizing purpose to make the computations below clearer.) As an intimidate
consequence of Theorem 2.13 we obtain the following.
###### Corollary 3.1 (Multivariate mean estimation).
There are absolute constants $C_{1},C_{2}$ such that the following holds. Let
$r>0$ and let
$N\geq
C_{1}\frac{\mathop{\mathrm{trace}}(\mathrm{\mathbf{C}ov}[\xi])}{r^{2}}.$
Then there is a procedure $\widehat{x}_{N}^{\ast}$ such that, with probability
at least
$1-2\exp\Big{(}-C_{2}N\min\Big{\\{}1,\frac{r^{2}}{\lambda_{\mathrm{max}}(\mathrm{\mathbf{C}ov}[\xi])}\Big{\\}}\Big{)},$
we have that
$\displaystyle\big{\|}\widehat{x}_{N}^{\ast}-\mathbf{E}[\xi]\big{\|}_{2}$
$\displaystyle\leq r.$
Let us once again mention that the question of finding an estimator of the
mean of a heavy tailed random vector that exhibits Gaussian rates remained
open until very recently: it was settled in [19, Theorem 1]; see also [17] for
a recent survey. Corollary 3.1 recovers [19, Theorem 1].
###### Proof of Corollary 3.1.
For every $x\in\mathcal{X}$, we have that
$\displaystyle\nabla F(x,\xi)$ $\displaystyle=x-\xi,$
$\displaystyle\nabla^{2}F(x,\xi)$ $\displaystyle=\mathrm{Id}.$
In particular $\|\cdot\|=\|\cdot\|_{2}$ and Assumption 2.3 is clearly
satisfied. Actually, as the Hessian is deterministic and independent of the
action $x$, we are in the setting of Theorem 2.13 and it remains to compute
the parameters $N_{\mathrm{G}}(r)$ and $\sigma^{2}$. To that end, it suffices
to note that $\nabla^{2}f(x^{\ast})^{-1}=\mathrm{Id}$ and
$\mathrm{\mathbf{C}ov}[\nabla F(x^{\ast},\xi)]=\mathrm{\mathbf{C}ov}[\xi]$;
hence
$\displaystyle N_{\mathrm{G}}(r)$
$\displaystyle=\frac{1}{r^{2}}\mathop{\mathrm{trace}}(\mathrm{\mathbf{C}ov}[\xi]),$
$\displaystyle\sigma^{2}$
$\displaystyle=\lambda_{\mathrm{max}}(\mathrm{\mathbf{C}ov}[\xi]).$
The proof therefore follows from Theorem 2.13. ∎
### 3.2. Linear regression
Linear regression is one of the fundamental problems studied in statistics.
Given a one-dimensional random variable $Y$ and a $d$-dimensional random
vector $X$, the task is to find the best possible forecast of $Y$ based on
linear combinations of $X$. To be more precise, one seeks the minimizer of
$x\mapsto\mathbf{E}[(\langle X,x\rangle-Y)^{2}]$ over $x\in\mathbb{R}^{d}$ (or
a subset thereof). This problem clearly falls within the scope of this article
by considering
$F(x,\xi):=\frac{1}{2}(\langle X,x\rangle-Y)^{2}\quad\text{where }\xi:=(X,Y)$
with $\mathcal{X}\subseteq\mathbb{R}^{d}$. (The purpose of the factor
$\frac{1}{2}$ is again only for convenience.) In order to lighten notation, we
shall impose a standard assumption on $X$: that it is centred and isotropic.
The latter means that its covariance matrix is the identity.
###### Assumption 3.2.
The set $\mathcal{X}\subseteq\mathbb{R}^{d}$ is closed and convex, $X$ is
centred and isotropic, and there is a constant $L_{X}$ such that
(3.1) $\displaystyle\mathbf{E}[\langle X,x\rangle^{4}]^{\frac{1}{4}}$
$\displaystyle\leq L_{X}\mathbf{E}[\langle X,x\rangle^{2}]^{\frac{1}{2}}$
for every $x\in\mathbb{R}^{d}$. Further,
$\bar{\sigma}^{2}:=\mathbf{E}[(\langle
X,x^{\ast}\rangle-Y)^{4}]^{\frac{1}{2}}$
is finite.
The assumption (3.1), which means that the $L_{4}$ and $L_{2}$ norms of linear
forms of $X$ are equivalent, is a typical assumption made in high-dimensional
statistics and it is not too restrictive.
###### Corollary 3.3 (Linear Regression).
If Assumption 3.2 is satisfied, there are constants $c_{1}$ and $c_{2}$
depending only on $L_{X}$ such that the following holds. Let $r>0$ and
$N\geq
c_{1}\max\Big{\\{}d\log(2d)\,,\,\frac{d\cdot\bar{\sigma}^{2}}{r^{2}}\Big{\\}}.$
Then there is a procedure $\widehat{x}_{N}^{\ast}$ such that, with probability
at least
$1-2\exp\Big{(}-c_{2}N\min\Big{\\{}1,\frac{r^{2}}{\bar{\sigma}^{2}}\Big{\\}}\Big{)},$
we have that
$\displaystyle\|\widehat{x}_{N}^{\ast}-x^{\ast}\|_{2}$ $\displaystyle\leq r,$
$\displaystyle\mathbf{E}_{X,Y}[(\langle
X,\widehat{x}_{N}^{\ast}\rangle-Y)^{2}]$ $\displaystyle\leq\mathbf{E}[(\langle
X,x^{\ast}\rangle-Y)^{2}]+2r^{2}.$
Here, $\mathbf{E}_{X,Y}[\cdot]$ denotes the expectation taken only over $X$
and $Y$ (and, of course, not the sample $(X_{i},Y_{i})_{i=1}^{N}$ used for the
computation of $\widehat{x}_{N}^{\ast}$).
Compared to the benchmark result on linear regression in a heavy tailed
scenario [20], Corollary 3.3 has an additional $\log(2d)$ term in the estimate
on the sample size. This term is merely an artifact of the generality of our
main result. Its origin lies in the matrix-Bernstein inequality, which we use
in the process of bounding the smallest singular value of a general random
matrix ensemble (namely $\nabla^{2}F(x^{\ast},\xi)$); that extra factor is not
needed in this example and can be easily removed.
###### Proof of Corollary 3.3.
For every $x\in\mathcal{X}$, we have that
$\displaystyle\nabla F(x,\xi)$ $\displaystyle=(\langle X,x\rangle-Y)X,$
$\displaystyle\nabla^{2}F(x,\xi)$ $\displaystyle=X\otimes X.$
As $X$ is isotropic, this implies $\nabla^{2}f(x^{\ast})=\mathrm{Id}$ and in
particular $\|\cdot\|=\|\cdot\|_{2}$. Also, as $(\langle X,x^{\ast}\rangle-Y)$
and $X$ are both in $L_{4}$ by assumption, the Cauchy-Schwartz inequality
shows that $\nabla F(x^{\ast},\xi)$ is square integrable. In particular,
Assumption 2.3 is satisfied. Employing the norm equivalence of $X$ (3.1), we
obtain
$\displaystyle\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle^{2}]$
$\displaystyle=\mathbf{E}[\langle X,z\rangle^{4}]$ $\displaystyle\leq
L_{X}^{4}\|z\|^{4}$
for every $z\in\mathbb{R}^{d}$; thus Assumption 2.5 is satisfied with
$L=L_{X}^{4}$. Finally, as the Hessian is independent of the action $x$,
Assumption 2.7 is clearly satisfied with $\alpha=1$ , $K\equiv 0$ and an
arbitrary $r_{0}$; in particular, $N_{\mathrm{H},\mathcal{E}}\leq d$.
It remains to compute the parameters $N_{\mathrm{G}}(r),\sigma^{2}$, and
$c_{\mathrm{H}}$. Using once again that the Hessian is independent of the
action $x$, it is evident that $c_{\mathrm{H}}=1$. Turning to $\sigma^{2}$ and
$N_{\mathrm{G}}(r)$, recall that $\nabla^{2}f(x^{\ast})=\mathrm{Id}$, and let
us estimate the largest eigenvalue and the trace of
$\mathrm{\mathbf{C}ov}[\nabla F(x^{\ast},\xi)]$.
For every $z\in\mathbb{R}^{d}$, the Cauchy-Schwartz inequality together with
(3.1) imply
(3.2) $\displaystyle\begin{split}\langle\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)]z,z\rangle&=\mathrm{\mathbf{V}ar}[(\langle
X,x^{\ast}\rangle-Y)\langle X,z\rangle]\\\
&\leq\mathbf{E}\big{[}\big{(}(\langle X,x^{\ast}\rangle-Y)\langle
X,z\rangle\big{)}^{2}\big{]}\\\ &\leq\mathbf{E}[(\langle
X,x^{\ast}\rangle-Y)^{4}]^{\frac{1}{2}}\mathbf{E}[\langle
X,z\rangle^{4}]^{\frac{1}{2}}\\\
&\leq\bar{\sigma}^{2}L_{X}^{2}\|z\|_{2}^{2}.\end{split}$
This clearly implies $\sigma^{2}\leq L_{X}^{2}\bar{\sigma}^{2}$ and, taking
the standard Euclidean basis $z=e_{i}$ in (3.2), we conclude
$\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)]_{ii}\leq\bar{\sigma}^{2}L_{X}^{2}$, hence
$\displaystyle N_{\mathrm{G}}(r)$
$\displaystyle=\frac{1}{r^{2}}\sum_{i=1}^{d}\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)]_{ii}\leq\frac{\bar{\sigma}^{2}L_{X}^{2}d}{r^{2}}.$
The proof now follows from Theorem 2.9. ∎
### 3.3. Ridge regression
A popular modification of linear regression is _ridge regression_ (also known
as _weight decay regression_ and _$\ell_{2}$ -regularized regression_ in the
machine learning community). The idea is that one penalizes a large Euclidean
norm of $x$ by setting
$F(x,\xi):=(\langle X,x\rangle-Y)^{2}+\|x\|_{2}^{2}$
where $\xi:=(X,Y)$ and $x\in\mathcal{X}\subseteq\mathbb{R}^{d}$. The aim of
this sort of penalization is to counteract _over-fitting_.
In contrast to the estimate we obtain in linear regression, the factor
$d\log(2d)$ in the minimal sample is not needed here:
###### Corollary 3.4.
If Assumption 3.2 is satisfied, there are absolute constants $C_{1}$ and
$C_{2}$ such that the following holds. Let $r>0$ and set
$N\geq C_{1}\frac{d\cdot\bar{\sigma}^{2}}{r^{2}}.$
Then there is a procedure $\widehat{x}_{N}^{\ast}$ such that, with probability
at least
$1-2\exp\Big{(}-C_{2}N\min\Big{\\{}1,\frac{r^{2}}{\bar{\sigma}^{2}}\Big{\\}}\Big{)},$
satisfies that
$\displaystyle\|\widehat{x}_{N}^{\ast}-x^{\ast}\|_{2}$ $\displaystyle\leq r,$
$\displaystyle\mathbf{E}_{X,Y}[(\langle
X,\widehat{x}_{N}^{\ast}\rangle-Y)^{2}]+\|\widehat{x}_{N}^{\ast}\|_{2}^{2}$
$\displaystyle\leq\mathbf{E}[(\langle
X,x^{\ast}\rangle-Y)^{2}]+\|x^{\ast}\|_{2}^{2}+2r^{2}.$
###### Proof.
For every $x\in\mathcal{X}$, we have that
$\displaystyle\nabla^{2}F(x,\xi)$ $\displaystyle=2X\otimes X+2\mathrm{Id},$
$\displaystyle\nabla^{2}f(x)$ $\displaystyle=4\mathrm{Id};$
hence we are in the setting of Theorem 2.13. All that remains is to estimate
the parameters $N_{\mathrm{G}}(r)$ and $\sigma^{2}$, and this can be done
exactly as in the proof of Corollary 3.3. ∎
### 3.4. Portfolio optimization
As a final example, we address the portfolio optimization problem in its
general form, that is, we apply Theorem 2.9 to the problem
$F(x,\xi):=\ell(V_{x})\quad\text{where }\begin{array}[]{l}V_{x}:=-Y-\langle
X,x\rangle,\\\ \hskip 6.00006pt\xi:=(X,Y),\end{array}$
for $x\in\mathcal{X}\subseteq\mathbb{R}^{d}$ that is closed and convex.
Moreover, $\ell\colon\mathbb{R}\to\mathbb{R}$ is strictly convex, increasing,
bounded from below, and we assume that it is three times continuously
differentiable.
We shall impose the following two assumptions on the zero-mean random vector
$X$. Firstly, assume that $X$ satisfies a (directional) $L_{6}$-$L_{2}$ norm
equivalence, i.e. there is a constant $L_{X}$ such that
(3.3) $\displaystyle\mathbf{E}[\langle X,z\rangle^{6}]^{\frac{1}{6}}\leq
L_{X}\cdot\mathbf{E}[\langle X,z\rangle^{2}]^{\frac{1}{2}}<\infty$
for all $z\in\mathbb{R}^{d}$. In addition, we assume that the following no-
arbitrage condition holds
(3.4) $\displaystyle\mathbf{P}[\langle X,z\rangle<0]>0\quad\text{for all
}z\in\mathbb{R}^{d}\setminus\\{0\\}.$
###### Remark 3.5.
The classical no-arbitrage condition used in mathematical finance reads as
follows: for every $z\in\mathbb{R}^{d}$, we have that
$\mathbf{P}[\langle X,z\rangle<0]=0\text{ implies that }\mathbf{P}[\langle
X,z\rangle>0]=0;$
i.e. it is not possible to make profit without taking any risk. Under this
condition, it is well-known that one can decompose $\mathbb{R}^{d}=V\oplus
V^{\perp}$ into an orthogonal sum such that $\mathbf{P}[\langle
X,v\rangle<0]>0$ for all $v\in V\setminus\\{0\\}$ and $\mathbf{P}[\langle
X,w\rangle=0]=1$ for all $w\in V^{\perp}$, see e.g. [7, Section 1.3]. In
particular, replacing $\mathcal{X}$ by $\mathcal{X}\cap V$ viewed as a subset
of $\mathbb{R}^{\mathrm{dim}(V)}$ does not affect the outcome of the portfolio
optimization problem but guarantees that (3.4) holds.
Moreover, we shall assume that part (c) of Assumption 2.3, pertaining the
integrability of $F,\nabla F,\nabla^{2}F$, is satisfied. By Hölder’s
inequality and (3.3), this is the case if, for example,
$\ell(V_{x}),\ell^{\prime}(V_{x})^{4},\ell^{\prime\prime}(V_{x})^{2}$ are
integrable for every $x\in\mathcal{X}$. Then $f$ is real-valued, and standard
arguments building on the no-arbitrage condition, show that $f$ is strictly
convex and coercive. In particular, a unique optimal action
$x^{\ast}\in\mathcal{X}$ exists; see e.g. [7, Section 3.1].
Finally, denoting by $\mathcal{B}_{1}^{\ast}$ the ball of radius 1 w.r.t. the
norm $\|\cdot\|$ centered at $x^{\ast}$ and restricted to $\mathcal{X}$, we
assume that
$\displaystyle\bar{\sigma}^{2}$
$\displaystyle:=\mathbf{E}[(\tfrac{\ell^{\prime}(V_{x^{\ast}})^{2}}{\ell^{\prime\prime}(V_{x^{\ast}})})^{2}]^{\frac{1}{2}},$
$\displaystyle v_{1}$ $\displaystyle:=\mathbf{E}[|V_{x^{\ast}}|],$
$\displaystyle v_{2}$
$\displaystyle:=\mathbf{E}[\ell^{\prime\prime}(V_{x^{\ast}})^{6}]^{\frac{1}{6}},$
$\displaystyle v_{K}$
$\displaystyle:=\mathbf{E}[\sup\nolimits_{x\in\mathcal{B}_{1}^{\ast}}\ell^{\prime\prime\prime}(V_{x})^{2}]^{\frac{1}{2}},$
$\displaystyle v_{\mathcal{E}_{\mathrm{H}}}$
$\displaystyle:=\sup\nolimits_{x\in\mathcal{B}_{1}^{\ast}}\mathbf{E}[\sup\nolimits_{t\in[0,1]}\ell^{\prime\prime\prime}(V_{x^{\ast}+t(x-x^{\ast})})^{2}]^{\frac{1}{2}}$
are all finite.
###### Remark 3.6.
If, for instance, $\ell^{\prime\prime\prime}$ is non-negative and increasing,
the terms $v_{K}$ and $v_{\mathcal{E}_{\mathrm{H}}}$ can be simplified as
$\displaystyle\sup_{x\in\mathcal{B}_{1}^{\ast}}\ell^{\prime\prime\prime}(V_{x})$
$\displaystyle=\ell^{\prime\prime\prime}(V_{x^{\ast}}+\|X\|_{\ast}),$
$\displaystyle\sup_{t\in[0,1]}\ell^{\prime\prime\prime}(V_{x^{\ast}+t(x-x^{\ast})})$
$\displaystyle\leq\ell^{\prime\prime\prime}(V_{x^{\ast}}+|\langle
X,x-x^{\ast}\rangle|)$
where $\|\cdot\|_{\ast}:=\sup_{x\in\mathbb{R}^{d}\text{ s.t.\ }\|x\|\leq
1}\langle x,\cdot\rangle$ denotes the dual norm of $\|\cdot\|$. (Note that if
$\|\cdot\|$ is (equivalent to) the Euclidean norm, then its dual norm is
(equivalent to) the Euclidean norm too).
Under these assumptions, we obtain the following:
###### Corollary 3.7 (Portfolio optimization).
There are constants $c_{1},c_{2},c_{3},c_{4}$ depending only on
$L_{X},v_{1},v_{2}$ such that the following holds. For
$r\in(0,\min\\{1,\frac{c_{1}}{v_{\mathcal{E}_{\mathrm{H}}}}\\})$ and
$N\geq
c_{1}\max\Big{\\{}\frac{d\cdot\bar{\sigma}^{2}}{r^{2}},\,d\log(d(v_{K}+2))\,\Big{\\}},$
there is a procedure $\widehat{x}_{N}^{\ast}$ such that, with probability at
least
$1-2\exp\Big{(}-c_{2}N\min\Big{\\{}1,\frac{r^{2}}{\bar{\sigma}^{2}}\Big{\\}}\Big{)},$
we have that
$\displaystyle\|\widehat{x}_{N}^{\ast}-x^{\ast}\|$ $\displaystyle\leq r,$
$\displaystyle u(\widehat{x}_{N}^{\ast})$ $\displaystyle\geq
u(x^{\ast})-c_{4}r^{2}.$
We postpone the proof of Corollary 3.7; it will be presented in Section 6,
where we also show how to recover the estimate in the exponential portfolio
optimization problem (i.e. Corollary 2.10).
## 4\. On the smallest singular value of general random matrix ensembles
In the course of the analysis needed in the proof of Theorem 2.9, a crucial
ingredient is that sampling can exhibit that
$\langle\nabla^{2}F(x^{\ast},\xi)(x-x^{\ast}),x-x^{\ast}\rangle$ is
sufficiently large. Put differently, it is essential to derive a suitable
lower bound on the smallest singular value of the empirical random matrix of
$\nabla^{2}F(x^{\ast},\xi)$.
Results of this type have been studied extensively [1, 14, 33, 36, 39, 45],
however (to the best of our knowledge) the focus was on random matrix
ensembles that had some additional special structure, like iid rows/columns.
Unfortunately such special structure need not exist in our setting.
In this section, let $A$ be a real, square integrable, positive-semidefinite
$(d\times d)$-random matrix and let $(A_{i})_{i\geq 1}$ be independent copies
of $A$. (in the context of this article, the case that interests us is
$A=\nabla^{2}F(x^{\ast},\xi)$.) Denote its expectation by
$\mathbb{A}:=\mathbf{E}[A]$, the semi-norm endowed by $\mathbb{A}$ is
$\|x\|:=\langle\mathbb{A}x,x\rangle^{\frac{1}{2}}=\mathbf{E}[\langle
Ax,x\rangle]^{\frac{1}{2}}$
and the corresponding unit sphere is
$S:=\\{x\in\mathbb{R}^{d}:\|x\|=1\\}.$
To simplify the presentation, assume that $\|\cdot\|$ is a true norm, i.e.
$\|x\|=0$ implies that $x=0$. Next, for any $(d\times d)$-matrix $B$, its
operator norm is given by
$\|B\|_{\mathrm{op}}:=\max_{x,y\in S}\langle Bx,y\rangle.$
Before stating the main result of this section (Theorem 4.4) let us begin with
one of its outcomes.
###### Corollary 4.1.
Assume that there is a constant $L>0$ such that
(4.1) $\displaystyle\mathbf{E}[\langle Ax,x\rangle^{2}]^{\frac{1}{4}}\leq
L\quad\text{for all }x\in S.$
Then there are constants $c_{1}$ and $c_{2}$ that depend only on $L$ such that
the following holds. Let $\gamma\in(0,1)$ and assume that
$N\geq c_{1}\frac{d}{\gamma^{2}}\log\Big{(}\frac{2d}{\gamma}\Big{)}.$
Then, with probability at least
$1-2\exp\big{(}-c_{2}N\gamma^{2}\big{)},$
we have that
$\lambda_{\min}\Big{(}\frac{1}{N}\sum_{i=1}^{N}A_{i}\Big{)}\geq(1-\gamma)\lambda_{\min}(\mathbf{E}[A]).$
(Here, of course, $\lambda_{\min}$ denotes the smallest singular value.)
As the results of this section will be used in the proof of Theorem 2.9 for
the construction of a median-of-means tournament, we also need a median-of-
means version of Corollary 4.1. To that end, as before, let $N=nm$ for two
integers $n$ and $m$, and consider a partition of $\\{1,\dots,N\\}$ into $n$
disjoint blocks $I_{j}$, each one of cardinality $|I_{j}|=m$. Throughout this
article, we make the _notational convention_ that the letter $j$ always refers
blocks, i.e. $j$ is always an element of $\\{1,\dots,n\\}$. In addition,
$0<C,C_{0},C_{1},\dots$ denote absolute constants independent of all
parameters. They are allowed to change their values from line to line.
We often encounter the so-called _Rademacher random variables_
$(\varepsilon_{i})_{i\geq 1}$, which are independent, symmetric random signs
(i.e. $\mathbf{P}[\varepsilon_{i}=\pm 1]=\frac{1}{2}$) that are also
independent of all the other random variables that appear in the analysis; in
particular, they are independent of $(A_{i})_{i=1}^{N}$.
We have already seen in the introduction that the smallest empirical singular
value of a random matrix cannot be bounded (even with constant probability)
without imposing some of assumption. While an integrability assumption as in
(4.1) will do the job, the following definition from [25] contains the essence
of what is actually needed.
###### Definition 4.2 (Stable lower bound).
A set $H\subseteq L_{2}$ of real-valued functions satisfies a _stable lower
bound_ with parameters $(m,\gamma,l,k)$ if the following holds: for every
$h\in H$ and independent copies $(h_{i})_{i\geq 1}$ of $h$, with probability
at least $1-2\exp(-k)$, for all $J\subseteq\\{1,\dots,m\\}$ with $|J|\leq l$,
we have that
$\frac{1}{m}\sum_{i\in\\{1,\dots,m\\}\setminus
J}h_{i}^{2}\geq(1-\gamma)\mathbf{E}[h^{2}].$
We say that a (symmetric, positive semi-definite, $d\times d$) random matrix
$A$ satisfies a stable lower bound with parameters $(m,\gamma,l,k)$ if
$H:=\\{\langle Ax,x\rangle^{\frac{1}{2}}:x\in S\\}$ does (with the same
parameters).
The stable lower bound can be seen as an extension of the _small ball
property_. Recall that a set $H$ is said to satisfy a small ball property with
parameters $(\kappa,\delta)$ if
$\mathbf{P}\big{[}h^{2}\geq\kappa^{2}\cdot\mathbf{E}[h^{2}]\big{]}\geq\delta\quad\text{for
all }h\in H.$
This condition is used frequently in problems involving a quadratic term (see,
e.g. [15, 22, 26]).
###### Remark 4.3.
Let $H=\\{\langle Ax,x\rangle^{\frac{1}{2}}:x\in S\\}$. Then the following
hold.
1. (i)
If $H$ satisfies the small ball property with constants $(\kappa,\delta)$,
then $H$ satisfies a stable lower bound with parameters
$(m,\gamma,k,l)=\Big{(}m,1-\frac{\delta\kappa^{2}}{2},s_{1}\delta
m,s_{2}\delta m\Big{)}$
for every $m$, where $s_{1},s_{2}>0$ are absolute constants.
2. (ii)
If $H$ is bounded in $L_{p}$ for some $p>2$, that is,
$\mathbf{E}[|h|^{p}]^{\frac{1}{p}}\leq L\quad\text{for all }h\in H$
where $L$ is a fixed constant, then $H$ satisfies a stable lower bound with
parameters
$(m,\gamma,k,l)=\Big{(}m,\gamma,s_{1}m\gamma^{\frac{p}{p-2}},s_{2}m\gamma^{\max\\{\frac{p}{p-2},2\\}}\Big{)}$
where $s_{1},s_{2}>0$ are constants depending only on $p$ and $L$.
The first statement is an immediate consequence of a Binomial concentration
inequality and second statement can be found in [25, Section 2.1] together
with a more thorough discussion and analysis of stable lower bounds.
The following theorem is the main result of this section. To formulate it,
recall that for a random matrix $A$, $\mathbf{E}[A]$ is denoted by
$\mathbb{A}$.
###### Theorem 4.4.
There are absolute constants $C_{1}$ and $C_{2}$ such that the following
holds. Fix $\gamma,\tau\in(0,1)$ and assume that
1. (a)
the random matrix $A$ satisfies a stable lower bound with parameters
$(m,\frac{\gamma}{2},2l,k)$ were $k\geq\max\\{4,2\log(\frac{4}{\tau})\\}$,
2. (b)
the sample size satisfies
$N\geq
C_{1}\max\Big{\\{}\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}},\frac{dm}{\tau
k}\log\Big{(}\frac{\log(3d)\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{\gamma\tau\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}\Big{)}\Big{\\}}.$
Then, with probability at least
$1-2\exp\Big{(}-C_{2}N\tau\min\Big{\\{}\frac{l}{m},\frac{k}{m}\Big{\\}}\Big{)},$
for every $x\in\mathbb{R}^{d}$ and every $J_{j}\subseteq I_{j}$ with
$|J_{j}|\leq l$ for all $j$, we have that
$\Big{|}\Big{\\{}j:\frac{1}{m}\sum_{i\in I_{j}\setminus J_{j}}\langle
A_{i}x,x\rangle\geq(1-\gamma)\|x\|^{2}\Big{\\}}\Big{|}\geq(1-\tau)n.$
The formulation in Theorem 4.4 is tailer-made for the median-of-means analysis
which is presented in the next section (as part of the proof of Theorem 2.9).
It ensures a stability property that is stronger than a mere lower bound on
the quadratic form (and therefore, a lower estimate on the smallest singular
value). Rather, it gives a useful lower bound even if a proportion of the
sample is arbitrarily modified or removed. This will be useful in the proof of
Theorem 2.9 when passing from the Hessian evaluated at the optimizer to the
Hessian evaluated at mid-points. In addition, Theorem 4.4 provides an almost
isometric lower bound (i.e. when $\gamma$ is close to zero) while the analysis
for Theorem 2.9 actually only requires an isomorphic lower bound.
Finally, let us formulate the following immediate consequence of Theorem 4.4.
###### Corollary 4.5.
There are absolute constants $C_{1}$ and $C_{2}$ such that the following
holds. Fix $\gamma\in(0,1)$ and assume that
1. (a)
the random matrix $A$ satisfies a stable lower bound with parameters
$(N,\frac{\gamma}{2},2l,k)$ for some $k\geq 4$,
2. (b)
the sample size satisfies
$N\geq
C_{1}\max\Big{\\{}\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}},\frac{dN}{k}\log\Big{(}\frac{\log(3d)\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{\gamma\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}\Big{)}\Big{\\}}.$
Then, with probability at least
$1-2\exp\Big{(}-C_{2}N\min\Big{\\{}\frac{l}{N},\frac{k}{N}\Big{\\}}\Big{)},$
we have that
$\lambda_{\min}\Big{(}\frac{1}{N}\sum_{i=1}^{N}A_{i}\Big{)}\geq(1-\gamma)\lambda_{\min}(\mathbf{E}[A]).$
###### Proof.
Applying Theorem 4.4 with $n=1$ (hence $m=N$), $\tau=0.6$, and
$J_{j}=\emptyset$ yields the following: with probability at least
$1-2\exp(-C_{2}N\min\\{\frac{l}{N},\frac{k}{N}\\})$, for every
$x\in\mathbb{R}^{d}$, we have that
$\displaystyle\Big{\langle}\frac{1}{N}\sum_{i=1}^{N}A_{i}x,x\Big{\rangle}$
$\displaystyle\geq(1-\gamma)\langle\mathbf{E}[A]x,x\rangle.$
A twofold application of the extremal expression
$\lambda_{\min}(\cdot)=\min_{x\in\mathbb{R}^{d}\text{ s.t.\
}\|x\|_{2}=1}\langle\,\cdot\,x,x\rangle$ of the smallest singular value
completes the proof. ∎
In order to recover Corollary 4.1 from Corollary 4.5 (and later also to apply
Theorem 4.4 in the proof of Theorem 2.9) we need two simple observations,
showing that under a $L_{4}-L_{2}$ norm equivalence, the estimate on the
sample size in Corollary 4.5 and Theorem 4.4 can be simplified.
###### Lemma 4.6.
We have that
(4.2) $\displaystyle
1\leq\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}\leq\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]\leq
d\cdot\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}.$
###### Proof.
We start with the final inequality in (4.2). Recall that
$\mathbb{A}=\mathbf{E}[A]$ and note that the relation
$\|\cdot\|=\|\mathbb{A}^{\frac{1}{2}}\cdot\|_{2}$ transfers to the operator
norm in the following sense: for any $(d\times d)$-matrix $B$, we have that
(4.3)
$\displaystyle\begin{split}\|B\|_{\mathrm{op}}&=\|\mathbb{A}^{-\frac{1}{2}}B\mathbb{A}^{-\frac{1}{2}}\|_{\mathrm{op}_{2}}\quad\text{where}\\\
\|B\|_{\mathrm{op}_{2}}&:=\max_{x,y\in\mathbb{R}^{d}\text{ s.t.\
}\|x\|_{2},\|y\|_{2}\leq 1}\langle Bx,y\rangle.\end{split}$
Indeed, $\\{x\in\mathbb{R}^{d}:\langle\mathbb{A}x,x\rangle\leq 1\\}$ is the
ellipsoid $\mathbb{A}^{-\frac{1}{2}}B_{2}^{d}$ (where $B_{2}^{d}$ is the unit
ball w.r.t. the Euclidean norm).
The norm $\|\cdot\|_{\mathrm{op}_{2}}$ is the usual spectral norm which, for
positive semidefinite matrices, equals the largest singular value
$\lambda_{\max}(\cdot)$.
Setting $F:=\mathbb{A}^{-\frac{1}{2}}A\mathbb{A}^{-\frac{1}{2}}$ which is
positive semidefinite, we thus have
$\displaystyle\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]$
$\displaystyle=\mathbf{E}[\|F^{2}\|_{\mathrm{op}_{2}}]$
$\displaystyle=\mathbf{E}[\lambda_{\max}(F^{2})]\leq\sum_{i=1}^{d}\mathbf{E}[(F^{2})_{ii}],$
where the last inequality follows by bounding the largest singular value by
the trace. On the other hand, for every $i=1,\dots,d$, taking the Euclidean
unit vector $x=y=e_{i}$ in (4.3) shows that
$\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}=\|\mathbf{E}[F^{2}]\|_{\mathrm{op}_{2}}\geq\mathbf{E}[(F^{2})_{ii}]$
and hence the last inequality of the lemma follows.
The second inequality in (4.2) is trivial and the first inequality follows
from (4.3) and Jensen’s inequality for matrices (which states that
$\mathbf{E}[F^{2}]\succeq\mathbf{E}[F]^{2}=\mathrm{Id}$). This completes the
proof. ∎
###### Lemma 4.7.
Assume that there is a constant $L$ such that $\mathbf{E}[\langle
Ax,x\rangle^{2}]^{\frac{1}{4}}\leq L$ for all $x\in S$. Then
$\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}\leq dL^{4}.$
###### Proof.
Recall that $F:=\mathbb{A}^{-\frac{1}{2}}A\mathbb{A}^{-\frac{1}{2}}$ and that,
by (4.3),
$\displaystyle\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}=\|\mathbf{E}[F^{2}]\|_{\mathrm{op}_{2}}=\max_{x\in\mathbb{R}^{d}\text{
s.t.\ }\|x\|_{2}\leq 1}\mathbf{E}[\langle F^{2}x,x\rangle].$
Moreover, for every $x\in\mathbb{R}^{d}$, we have
$\displaystyle\langle F^{2}x,x\rangle$ $\displaystyle=\langle
FF^{\frac{1}{2}}x,F^{\frac{1}{2}}x\rangle$
$\displaystyle\leq\|F\|_{\mathrm{op}_{2}}\|F^{\frac{1}{2}}x\|_{2}^{2},$
which, in combination with the Cauchy-Schwartz inequality, yields that
$\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}\leq\max_{x\in\mathbb{R}^{d}\text{
s.t.\ }\|x\|_{2}\leq
1}\mathbf{E}[\|F\|_{\mathrm{op}_{2}}^{2}]^{\frac{1}{2}}\mathbf{E}[\|F^{\frac{1}{2}}x\|_{2}^{4}]^{\frac{1}{2}}.$
At this point, recall that
$\\{x\in\mathbb{R}^{d}:\langle\mathbb{A}x,x\rangle\leq 1\\}$ is the ellipsoid
$\mathbb{A}^{-\frac{1}{2}}B_{2}^{d}$, and note that an equivalent formulation
of (4.1) is
$\displaystyle\mathbf{E}[\|F^{\frac{1}{2}}x\|_{2}^{4}]^{\frac{1}{2}}$
$\displaystyle=\mathbf{E}[\langle Fx,x\rangle^{2}]^{\frac{1}{2}}$
$\displaystyle\leq L^{2}\mathbf{E}[\langle Fx,x\rangle]$
for every $x\in\mathbb{R}^{d}$, and that $\mathbf{E}[\langle Fx,x\rangle]=1$
for every $x\in\mathbb{R}^{d}$ with $\|x\|_{2}=1$. Hence, bounding
$\|F\|_{\mathrm{op}_{2}}=\lambda_{\max}(F)$ by the trace of $F$,
$\displaystyle\mathbf{E}[\|F\|_{\mathrm{op}_{2}}^{2}]^{\frac{1}{2}}$
$\displaystyle\leq\sum_{i=1}^{d}\mathbf{E}[(F_{ii})^{2}]^{\frac{1}{2}}$
$\displaystyle=\sum_{i=1}^{d}\mathbf{E}[\langle
Fe_{i},e_{i}\rangle^{2}]^{\frac{1}{2}}\leq dL^{2}.$
In conclusion $\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}\leq dL^{4}$, as
claimed. ∎
###### Proof of Corollary 4.1.
By Remark 4.3 (ii), the random matrix $A$ satisfies a stable lower bound with
parameters $(N,\frac{\gamma}{2},s_{1}N\gamma^{2},s_{2}N\gamma^{2})$ for
constants $s_{1}$ and $s_{2}$ depending only on $L$. The proof therefore
follows from Corollary 4.5 together with Lemma 4.6 and Lemma 4.7. ∎
Throughout this article, we often aim to prove statements that are supposed to
hold with high probability, uniformly over (uncountable) sets; for example
that for all $x\in S$, most coordinates of $(\langle
A_{i}x,x\rangle)_{i=1}^{N}$ are suitably large. The proofs of such statements
follow (in principle) a recurring scheme: in a first step, we prove that a
slightly stronger variant of the statement holds with high probability for a
single element (i.e. Lemma 4.8 below). By a trivial union bound, this allows
to extend the validity of the statement to a finite set of high cardinality,
and we shall choose such a set with an extra feature: it approximates the
original set in a suitable sense (i.e. Lemma 4.10 below). It then remains to
show that the oscillations caused by passing from the approximating set to the
whole set do not distort the outcome by too much (i.e. Lemma 4.14 below). From
now on, we shall continue under the same assumptions used in the formulation
of Theorem 4.4 without mentioning these assumptions at every instance.
###### Lemma 4.8.
There is an absolute constant $C$ such that the following holds. For every
$x\in S$, with probability at least $1-2\exp(-CN\tau\frac{k}{m})$, for all
choices $J_{j}\subseteq I_{j}$ with $|J_{j}|\leq 2l$, we have that
$\displaystyle\Big{|}\Big{\\{}j:\frac{1}{m}\sum_{i\in I_{j}\setminus
J_{j}}\langle A_{i}x,x\rangle\geq
1-\frac{\gamma}{2}\Big{\\}}\Big{|}\geq\Big{(}1-\frac{\tau}{2}\Big{)}n.$
In particular, for a set $\bar{S}\subseteq S$ satisfying
$\log(|\bar{S}|)\leq\frac{1}{2}C\tau N\frac{k}{m}$, the above statement holds
uniformly over $x\in\bar{S}$.
###### Proof.
The proof follows from an application of Bennett’s inequality and is
essentially the same as the proof of [25, Lemma 4.3]. For completeness, we
sketch the argument. Fix some $x\in S$ and, for every $j$, set
$\delta(j):=\begin{cases}0&\text{if }\frac{1}{m}\sum_{i\in I_{j}\setminus
J_{j}}\langle A_{i}x,x\rangle\geq 1-\frac{\gamma}{2}\text{ for all
}J_{j}\subseteq I_{j}\text{ with }|J_{j}|\leq 2l,\\\
1&\text{otherwise.}\end{cases}$
By definition of the stable lower bound, we have that
$\delta:=\mathbf{P}[\delta(j)=1]\leq 2\exp(-k)\leq\frac{3}{4}.$
If $\delta=0$, there is nothing to prove, so assume otherwise. Now make use of
Bennett’s inequality [5, Theorem 2.9]: for every $u\geq 2$ with probability at
least $1-2\exp(-C\delta n\cdot u\log(u))$, we have that
$|\\{j:\delta(j)=1\\}|\leq u\delta n.$
Apply this to
$u:=\frac{\tau}{2\delta}\geq\exp\Big{(}\frac{k}{2}\Big{)}\geq 2$
(where the second inequality follows as
$\tau\geq\frac{1}{4}\exp(-\frac{k}{2})$ by assumption) and observe that
$\displaystyle C\delta n\cdot u\log(u)$ $\displaystyle=\frac{1}{2}C\tau
n\log\Big{(}\frac{\tau}{2\delta}\Big{)}$ $\displaystyle\geq\frac{C\tau
nk}{4}=\frac{C\tau Nk}{4m}.$
This completes the first part of the proof. The “in particular” part is a
consequence of the union bound. ∎
In a next step we choose a subset of $S$ of high cardinality which “covers”
$S$ w.r.t. the natural metric.
###### Definition 4.9.
Let $(S,d)$ be a metric space and let $\rho>0$. A set $\bar{S}\subset S$ is a
$\rho$-cover of $S$ with respect to the metric $d$ if for every $s\in S$ there
is $\bar{s}\in\bar{S}$ such that $d(s,\bar{s})\leq\rho$.
For now, let $C$ be the absolute constant from Lemma 4.8 and set $C_{1}$ to be
the constant from assumption (b) in Theorem 4.4. Recall that we have the
freedom to choose $C_{1}$ as we see fit, and set $C_{0}$ to be a constant
specified in what follows.
###### Lemma 4.10.
Let
$\rho:=\frac{C_{0}\gamma\tau\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}{\log(3d)\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}.$
Then there is a $\rho$-cover $\bar{S}\subseteq S$ (w.r.t. the norm
$\|\cdot\|$) of log-cardinality at most $\frac{1}{2}C\tau N\frac{k}{m}$.
###### Proof.
By a simple volumetric argument (see e.g. [38, Exercise 2.2.14]) there is a
set
$\bar{S}_{2}\subseteq S_{2}:=\\{x\in\mathbb{R}^{d}:\|x\|_{2}=1\\}$
such that, for every $x\in S_{2}$ there is $y=y(x)\in\bar{S}_{2}$ with
$\|x-y\|_{2}\leq\rho$ with cardinality
$\displaystyle\log(|\bar{S}_{2}|)$ $\displaystyle\leq
d\log\Big{(}\frac{6}{\rho}\Big{)}$ $\displaystyle\leq
C_{3}d\log\Big{(}\frac{\log(3d)\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{\gamma\tau\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}\Big{)},$
where $C_{3}$ depends only on $C_{0}$. By assumption (b) on the sample size in
Theorem 4.4 we conclude that $\log(|\bar{S}_{2}|)\leq\frac{1}{2}C\tau
N\frac{k}{m}$ once the constant $C_{1}$ (in assumption (b)) is chosen
sufficiently large. Finally, the relation
$\|\cdot\|=\|\mathbb{A}^{\frac{1}{2}}\cdot\|_{2}$ readily implies that the set
$\bar{S}:=\\{\mathbb{A}^{-\frac{1}{2}}y:y\in\bar{S}_{2}\\}$
satisfies the statement of the lemma. ∎
The final step in the proof consist of showing that the transition from a
$\rho$-cover $\bar{S}$ to the whole $S$ does not distort the wanted outcome by
too much. To that end, we fix from now on the $\rho$-cover $\bar{S}$ of Lemma
4.10 and denote by $y=y(x)\in\bar{S}$ the element satisfying
$\|x-y\|\leq\rho$. Also, for every $x\in S$, set
$\Delta(x):=\langle Ax,x\rangle-\langle Ay,y\rangle.$
###### Remark 4.11.
The following two preliminary lemmas are stated here to maintain a
chronological order within the proofs. However, it might be helpful to skip to
(the proof of) Lemma 4.14 where their role is clarified and return here
afterwards.
###### Lemma 4.12.
We have that
$\mathbf{E}[|\Delta(x)|]\leq 3\rho\quad\text{for every }x\in S.$
###### Proof.
Fix some $x\in S$. As $A$ is symmetric and positive semidefinite, it follows
from the triangle inequality that
$\big{|}\langle Ax,x\rangle^{\frac{1}{2}}-\langle
Ay,y\rangle^{\frac{1}{2}}\big{|}\leq\langle A(x-y),x-y\rangle^{\frac{1}{2}}.$
Combined with the fact that $|a^{2}-b^{2}|\leq 2|a-b|\max\\{a,b\\}$ for
$a,b\geq 0$, we have
$\displaystyle|\Delta(x)|$ $\displaystyle\leq 2\langle
A(x-y),x-y\rangle^{\frac{1}{2}}\cdot\max\big{\\{}\langle
Ax,x\rangle^{\frac{1}{2}},\langle Ay,y\rangle^{\frac{1}{2}}\big{\\}},$
and by the Cauchy-Schwartz inequality
$\displaystyle\mathbf{E}[|\Delta(x)|]^{2}$ $\displaystyle\leq
4\mathbf{E}[\langle A(x-y),x-y\rangle]\cdot\big{(}\mathbf{E}[\langle
Ax,x\rangle]+\mathbf{E}[\langle Ay,y\rangle]\big{)}$
$\displaystyle=4\|x-y\|^{2}\cdot(\|x\|^{2}+\|y\|^{2})$
$\displaystyle=8\|x-y\|^{2},$
where the second equality follows by definition of the norm $\|\cdot\|$ and
the final one holds because $x,y\in S$. ∎
###### Lemma 4.13.
Let $C_{T}$ be the absolute constant in Talagrand’s concentration inequality
[37]. Moreover, set $b:=\gamma\frac{m}{l}$, let $(A_{i})_{i=1}^{N}$ be
independent copies of $A$ and set $(\varepsilon_{i})_{i\geq 1}$ to be
independent Rademacher random variables that are independent of
$(A_{i})_{i=1}^{N}$. Then we have that
$\mathbf{E}\Big{[}\sup_{x\in
S}\Big{|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}(|\Delta_{i}(x)|\wedge
b)\Big{|}\Big{]}\leq\frac{\gamma\tau}{C_{T}16}$
once $C_{0}$ (the absolute constant of Lemma 4.10) is small enough.
###### Proof.
As a preliminary step, we invoke the contraction inequality for Bernoulli
processes [16, Corollary 3.17] conditionally on $(A_{i})_{i=1}^{N}$, and
applied to the $1$-Lipschitz map $t\mapsto|t|\wedge b$ which passes through
the origin. It follows that
$\displaystyle\mathbf{E}\Big{[}\sup_{x\in
S}\Big{|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}(|\Delta_{i}(x)|\wedge
b)\Big{|}\Big{]}$ $\displaystyle\leq 2\mathbf{E}\Big{[}\sup_{x\in
S}\Big{|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}\Delta_{i}(x)\Big{|}\Big{]}.$
Rewriting
$\Delta_{i}(x)=\langle A_{i}(x+y),x-y\rangle$
we have that
$\displaystyle\Big{|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}\Delta_{i}(x)\Big{|}$
$\displaystyle=\Big{|}\Big{\langle}\Big{(}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}A_{i}\Big{)}(x+y),x-y\Big{\rangle}\Big{|}$
$\displaystyle\leq
2\rho\Big{\|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}A_{i}\Big{\|}_{\mathrm{op}},$
because $\|x+y\|\leq 2$ and $\|x-y\|\leq\rho$. Next, set
$F_{i}:=\varepsilon_{i}\mathbb{A}^{-\frac{1}{2}}A_{i}\mathbb{A}^{-\frac{1}{2}}$
for every $i$ and recall the relation between $\|\cdot\|_{\mathrm{op}}$ and
the spectral norm $\|\cdot\|_{\mathrm{op}_{2}}$ stated in (4.3). The Matrix-
Bernstein inequality [40, Theorem I] implies that
$\displaystyle\mathbf{E}\Big{[}\Big{\|}\sum_{i=1}^{N}\varepsilon_{i}A_{i}\Big{\|}_{\mathrm{op}}\Big{]}$
$\displaystyle=\mathbf{E}\Big{[}\Big{\|}\sum_{i=1}^{N}F_{i}\Big{\|}_{\mathrm{op}_{2}}\Big{]}$
$\displaystyle\leq\Big{(}C(d)N\|\mathbf{E}[F^{2}]\|_{\mathrm{op}_{2}}\Big{)}^{\frac{1}{2}}+C(d)\mathbf{E}\Big{[}\max_{1\leq
i\leq N}\|F_{i}\|_{\mathrm{op}_{2}}^{2}\Big{]}^{\frac{1}{2}}$
where
$\displaystyle C(d)$ $\displaystyle=4(1+2\log(2d))\leq 22\log(2d).$
Further, estimating the maximum by the sum and using that
$\|F\|_{\mathrm{op}_{2}}^{2}=\|F^{2}\|_{\mathrm{op}_{2}}$, we trivially have
$\displaystyle\mathbf{E}\Big{[}\max_{1\leq i\leq
N}\|F_{i}\|_{\mathrm{op}_{2}}^{2}\Big{]}$ $\displaystyle\leq
N\mathbf{E}[\|F\|_{\mathrm{op}_{2}}^{2}]$
$\displaystyle=N\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}].$
Putting everything together, we therefore obtain
$\displaystyle\mathbf{E}\Big{[}\sup_{x\in
S}\Big{|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}(|\Delta_{i}(x)|\wedge
b)\Big{|}\Big{]}$ $\displaystyle\leq 4\rho
C(d)\Big{(}\Big{(}\frac{\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}{N}\Big{)}^{\frac{1}{2}}+\Big{(}\frac{\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{N}\Big{)}^{\frac{1}{2}}\Big{)}$
$\displaystyle\leq 4\rho
C(d)\Big{(}1+\Big{(}\frac{\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}\Big{)}^{\frac{1}{2}}\Big{)},$
where the second inequality follows from assumption (b) in Theorem 4.4: that
on the sample size satisfies
$N\geq\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}$. By Lemma 4.6, the
expectation of the operator norm is always larger than the operator norm of
the expectation, hence
$\displaystyle\mathbf{E}\Big{[}\sup_{x\in
S}\Big{|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}(|\Delta_{i}(x)|\wedge
b)\Big{|}\Big{]}$ $\displaystyle\leq 8\rho
C(d)\frac{\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}.$
Recalling the value of $\rho$ from Lemma 4.10 shows that the latter term is at
most $\frac{\gamma\tau}{C_{T}16}$. This completes the proof. ∎
###### Lemma 4.14.
There exists an absolute constant $C$ such that the following holds. For every
$x\in S$ and every $j$, let
$J_{j}^{\ast}(x):=\\{\text{largest }l\text{ coordinates of
}(|\Delta_{i}(x)|)_{i\in I_{j}}\\}\subseteq I_{j}.$
Then, with probability at least $1-2\exp(-C_{2}N\tau\frac{l}{m})$, we have
that
(4.4) $\displaystyle\sup_{x\in S}\Big{|}\Big{\\{}j:\frac{1}{m}\sum_{i\in
I_{j}\setminus
J_{j}^{\ast}(x)}|\Delta_{i}(x)|>\frac{\gamma}{2}\Big{\\}}\Big{|}\leq\frac{\tau
n}{2}.$
###### Proof.
Set $b:=\gamma\frac{m}{l}$. We claim that, for every $x\in S$ and every $j$,
(4.5) $\displaystyle\begin{split}\frac{1}{m}\sum_{i\in I_{j}\setminus
J_{j}^{\ast}(x)}|\Delta_{i}(x)|&>\frac{\gamma}{2}\quad\text{implies}\quad\\\
\frac{1}{m}\sum_{i\in I_{j}}(|\Delta_{i}(x)|\wedge
b)&>\frac{\gamma}{2}.\end{split}$
Indeed, if $|\Delta_{i}(x)|\leq b$ for $i\in I_{j}\setminus J_{j}^{\ast}(x)$,
the second sum in (4.5) is trivially at least as big as the first one.
Otherwise, if there is $i_{0}\in I_{j}\setminus J_{j}^{\ast}(x)$ for which
$|\Delta_{i_{0}}(x)|>b$, then by definition of $J_{j}^{\ast}(x)$, there are
least $l$ coordinates $i\in I_{j}$ for which $|\Delta_{i}(x)|>b$. In
particular, the second sum in (4.5) is at least $\frac{1}{m}lb=\gamma$, and
(4.5) holds. Therefore, it suffices to show that $R\leq\frac{1}{4}\gamma\tau$
holds with high probability, where
$R:=\sup_{x\in S}\frac{1}{N}\sum_{i=1}^{N}(|\Delta_{i}(x)|\wedge b).$
To that end, Talagrand’s concentration inequality for bounded empirical
processes [37] (see also [5]): there is an absolute constant $C_{T}$ such that
$\displaystyle\mathbf{P}\Big{[}R\leq
R_{1}+C_{T}\big{(}R_{2}+R_{3}+R_{4}\big{)}\Big{]}\geq 1-2\exp(-u)$
for every $u\geq 0$, where
$\displaystyle R_{1}$ $\displaystyle:=\sup_{x\in
S}\mathbf{E}[|\Delta(x)|\wedge b],$ $\displaystyle R_{2}$
$\displaystyle:=\sup_{x\in S}\mathbf{E}[(|\Delta(x)|\wedge
b)^{2}]^{\frac{1}{2}}\cdot\Big{(}\frac{u}{N}\Big{)}^{\frac{1}{2}},$
$\displaystyle R_{3}$ $\displaystyle:=\sup_{x\in S}\||\Delta(x)|\wedge
b\|_{L^{\infty}}\cdot\frac{u}{N},$ $\displaystyle R_{4}$
$\displaystyle:=\mathbf{E}\Big{[}\sup_{x\in
S}\Big{|}\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}(|\Delta_{i}(x)|\wedge
b)\Big{|}\Big{]}.$
Thus, to conclude the proof, let us show that for $u=CN\tau\frac{l}{m}$, the
sum $R_{1}+C_{T}(R_{2}+R_{3}+R_{4})$ is smaller than $\frac{1}{4}\gamma\tau$.
We now proceed to bound $R_{1},\dots,R_{4}$. First, recall that $C_{0}$ is the
absolute constant of Lemma 4.10 which we are still able to choose as small as
we want, and note that
$\rho=\frac{C_{0}\gamma\tau\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}{\log(3d)\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}\leq\frac{C_{0}\gamma\tau}{\log(3)}\leq
C_{0}\gamma\tau,$
where the first inequality holds by Lemma 4.6 and the second one by absorbing
$\frac{1}{\log(3)}$ into $C_{0}$.
By Lemma 4.12 we have that $\mathbf{E}[|\Delta(x)|]\leq 3\rho$ for every $x\in
S$; thus
$R_{1}\leq 3\rho\leq\frac{\gamma\tau}{16}$
as soon as $C_{0}<\frac{1}{48}$.
For the terms $R_{2}$ and $R_{3}$, which involve $u=CN\tau\frac{l}{m}$, note
that
$\mathbf{E}[(|\Delta(x)|\wedge b)^{2}]\leq 3\rho b\quad\text{for every }x\in
S.$
Indeed, this follows from the trivial estimate $(|\Delta(x)|\wedge
b)^{2}\leq|\Delta(x)|b$ and Lemma 4.12 once again. Recalling that
$b=\gamma\frac{m}{l}$, we therefore have
$\displaystyle R_{2}$ $\displaystyle\leq\Big{(}\frac{3\rho
bu}{N}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\big{(}3C_{0}C\gamma^{2}\tau^{2}\big{)}^{\frac{1}{2}}\leq\frac{\gamma\tau}{C_{T}16}$
once $C_{0}$ is small enough. Moreover,
$R_{3}\leq\frac{bu}{N}=C\gamma\tau\leq\frac{\gamma\tau}{C_{T}16}$
provided that $C$ is small enough.
Finally, by Lemma 4.13, we have $R_{4}\leq\frac{\gamma\tau}{C_{T}16}$. This
completes the proof ∎
###### Proof of Theorem 4.4.
The statement of Theorem 4.4 is clearly homogeneous in $x\in\mathbb{R}^{d}$,
hence it suffices to restrict to $x\in S$. The proof follows from a
combination of Lemma 4.8 and Lemma 4.14. Indeed, using the notation of Lemma
4.14, for every $x\in S$, $y=y(x)$, and every $J_{j}\subseteq I_{j}$ with
$|J_{j}|\leq l$, we write
$\displaystyle\frac{1}{m}\sum_{i\in I_{j}\setminus J_{j}}\langle
A_{i}x,x\rangle$ $\displaystyle\geq\frac{1}{m}\sum_{i\in
I_{j}\setminus(J_{j}\cup J_{j}^{\ast}(x))}\langle A_{i}x,x\rangle$ (4.6)
$\displaystyle\geq\frac{1}{m}\sum_{i\in I_{j}\setminus(J_{j}\cup
J_{j}^{\ast}(x))}\langle A_{i}y,y\rangle-\frac{1}{m}\sum_{i\in
I_{j}\setminus(J_{j}\cup J_{j}^{\ast}(x))}|\Delta_{i}(x)|.$
By Lemma 4.8, with probability at least $1-2\exp(-C\tau N\frac{k}{m})$, for
every $x$, $y$, and sets $J_{j}$ as above, we have that
$\frac{1}{m}\sum_{i\in I_{j}\setminus(J_{j}\cup J_{j}^{\ast}(x))}\langle
A_{i}y,y\rangle\geq 1-\frac{\gamma}{2}\quad\text{on at least
}\Big{(}1-\frac{\tau}{2}\Big{)}n\text{ blocks }.$
Next, by Lemma 4.14, with probability at least $1-2\exp(-C\tau N\frac{l}{m})$,
for every $x$, $y$, and sets $J_{j}$ as above, we have that
$\frac{1}{m}\sum_{i\in I_{j}\setminus(J_{j}\cup
J_{j}^{\ast}(x))}|\Delta_{i}(x)|\leq\frac{\gamma}{2}\quad\text{on at least
}\Big{(}1-\frac{\tau}{2}\Big{)}n\text{ blocks }.$
Taking the intersection of the two high probability events yields the claim. ∎
## 5\. Proofs of the main results
In addition to the _notational conventions_ already explained in Section 4,
set $c,c_{0},c_{1},\dots$ to be constants that may depend on $L$ (the
parameter appearing in Assumption 2.5). As before, these constants may change
their values from line to line. Moreover, $0<\theta<\tau<\frac{1}{4}$ are
constants that may depend on $L$ as well. For the sake of a clearer
presentation, rather than stating the explicit values of $\theta$ and $\tau$
now, we collect constraints on their values along the way.
Next recall that
$n=\theta
N\min\Big{\\{}1,\frac{r^{2}}{\sigma^{2}}\Big{\\}}\quad\text{and}\quad
m=\frac{N}{n},$
where we assume without loss of generality that $m$ and $n$ are integers.
Thus, $N=nm$ and $m\geq\frac{1}{\theta}$. Finally, set
$\displaystyle\mathcal{B}_{r}^{\ast}$
$\displaystyle:=\\{x\in\mathcal{X}:\|x-x^{\ast}\|\leq r\\},$
$\displaystyle\mathcal{S}_{r}^{\ast}$
$\displaystyle:=\\{x\in\mathcal{X}:\|x-x^{\ast}\|=r\\}$
to be the ball and sphere of radius $r$ around $x^{\ast}$ restricted to
$\mathcal{X}$, respectively. Recall the constant $r_{0}$ of Assumption 2.7 and
assume throughout that $r\leq r_{0}$.
The proof of Theorem 2.9 relies on the following decomposition: for every $j$
and every $x\in\mathcal{X}$, a Taylor expansion implies that
$\displaystyle\widehat{f}_{I_{j}}(x)-\widehat{f}_{I_{j}}(x^{\ast})$
$\displaystyle=\frac{1}{m}\sum_{i\in I_{j}}\langle\nabla
F(x^{\ast},\xi_{i}),x-x^{\ast}\rangle+\frac{1}{2}\frac{1}{m}\sum_{i\in
I_{j}}\langle\nabla^{2}F(z_{i},\xi_{i})(x-x^{\ast}),x-x^{\ast}\rangle$
$\displaystyle=:M_{x,x^{\ast}}(j)+\frac{1}{2}Q_{x,x^{\ast}}(j),$
where $z_{i}$ are midpoints between $x$ and $x^{\ast}$ (and each $z_{i}$ may
depend on $\xi_{i}$). For obvious reasons we call $Q_{x,x^{\ast}}$ the
_quadratic term_ and $M_{x,x^{\ast}}$ the _multiplier term_.
We start with the proof of the estimation error, formulated in Proposition
2.14. With the decomposition of
$\widehat{f}_{I_{j}}(x)-\widehat{f}_{I_{j}}(x^{\ast})$ into a multiplier and a
quadratic term at hand, the _strategy of the proof_ that $x^{\ast}$ defeats
any competitor $x\in\mathcal{S}_{r}^{\ast}$ on the $j$-th block (i.e. that
$\widehat{f}_{I_{j}}(x)-\widehat{f}_{I_{j}}(x^{\ast})>0$), consists of showing
that the quadratic term is likely to be positive (of order $r^{2}$) and the
multiplier term is likely not to be too negative:
###### Lemma 5.1.
There is a constant $c$ such that the following holds. With probability at
least $1-2\exp(-c\tau^{2}n)$, for every $x\in\mathcal{S}_{r}^{\ast}$, we have
that
(5.1)
$\displaystyle\Big{|}\Big{\\{}j:\frac{1}{2}Q_{x,x^{\ast}}(j)\geq\frac{r^{2}}{8}\Big{\\}}\Big{|}$
$\displaystyle\geq(1-\tau)n,$ (5.2)
$\displaystyle\Big{|}\Big{\\{}j:M_{x,x^{\ast}}(j)\geq-\frac{r^{2}}{16}\Big{\\}}\Big{|}$
$\displaystyle\geq(1-\tau)n.$
Let us show that Lemma 5.1 implies Proposition 2.14:
###### Proof of Proposition 2.14.
On the high probability event from Lemma 5.1, for every
$x\in\mathcal{S}_{r}^{\ast}$, we have that
$\widehat{f}_{I_{j}}(x)-\widehat{f}_{I_{j}}(x^{\ast})\geq\frac{r^{2}}{16}>0\quad\text{on
at least }(1-2\tau)n\text{ blocks }j.$
Therefore, as $\tau<\frac{1}{4}$, on that event, $x^{\ast}$ wins the match
against every $x\in\mathcal{S}_{r}^{\ast}$.
The extension to all $x\in\mathcal{X}$ with $\|x-x^{\ast}\|\geq r$ is a simple
consequence of convexity. Indeed, let $x\in\mathcal{X}$ with
$\|x-x^{\ast}\|\geq r$ and set
$y:=x^{\ast}+\frac{r}{\|x-x^{\ast}\|}(x-x^{\ast})\in\mathcal{S}_{r}^{\ast}.$
Note now that (for every sample)
$\widehat{f}_{I_{j}}(\cdot)-\widehat{f}_{I_{j}}(x^{\ast})$ is a convex
function which equals zero at $x^{\ast}$. Hence, if this function is strictly
positive in $y$, then, by convexity, it is also strictly positive on
$\\{x^{\ast}+t(y-x^{\ast}):t\geq 1\\}\cap\mathcal{X},$
which is the subset of the ray that originates from $x^{\ast}$ and passed
through $y$, consisting of the points that are “beyond” $y$. Taking
$t=\frac{1}{r}\|x-x^{\ast}\|$, we see that $x^{\ast}$ defeats $x$ (at least)
on the same blocks on which it defeats $y$. In conclusion, on the high
probability event of the lemma, $x^{\ast}$ wins the match against $x$. ∎
###### Remark 5.2.
By Assumption 2.3, the functions $F$, $\nabla F$ and $\nabla^{2}F$ are so-
called Carathéodory functions and therefore are jointly measurable. Moreover,
by Assumption 2.7 and the dominated convergence theorem, one can readily
verify that $f$ is twice continuously differentiable near $x^{\ast}$ with
$\nabla f(x)=\mathbf{E}[\nabla
F(x,\xi)]\quad\text{and}\quad\nabla^{2}f(x)=\mathbf{E}[\nabla^{2}F(x,\xi)]$
for all $x\in\mathcal{X}$ with $\|x-x^{\ast}\|<r_{0}$. In particular, from
this we get that
$\|\cdot\|=\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)\cdot,\cdot\rangle]^{\frac{1}{2}}$.
### 5.1. Estimation error, the quadratic term
This subsection contains the proof of (5.1) from Lemma 5.1: we show that the
quadratic term is likely to be at least of order $r^{2}$. The proof relies on
the results of Section 4 and the strategy is the following. In a first step,
we ignore the fact that the Hessian in the definition of $Q_{x,x^{\ast}}$ is
evaluated at a midpoint between $x^{\ast}$ and $x$, considering instead the
Hessian evaluated at the optimizer $x^{\ast}$. We employ the median-of-mean-
type lower bound on the smallest singular value of the random matrix
$\nabla^{2}F(x^{\ast},\xi)$ established in Section 4, which is summarized in
the following lemma.
###### Lemma 5.3.
There are constants $s_{1},c>0$ depending only on $L$ such that the following
holds. With probability at least $1-2\exp(-c\tau N)$, for every
$x\in\mathcal{S}_{r}^{\ast}$ and every choices of subsets $J_{j}\subseteq
I_{j}$ with $|J_{j}|\leq s_{1}m$, we have that
(5.3) $\displaystyle\Big{|}\Big{\\{}j:\frac{1}{m}\sum_{i\in I_{j}\setminus
J_{j}}\langle\nabla^{2}F(x^{\ast},\xi_{i})(x-x^{\ast}),x-x^{\ast}\rangle\geq\frac{r^{2}}{2}\Big{\\}}\Big{|}$
$\displaystyle\geq\Big{(}1-\frac{\tau}{2}\Big{)}n.$
###### Proof.
We apply Theorem 4.4 with $A:=\nabla^{2}F(x^{\ast},\xi)$. By Remark 4.3, the
random matrix $A$ satisfies a stable lower bound with parameters
$\Big{(}m,\frac{\gamma}{2},2l,k\Big{)}=\Big{(}m,\frac{1}{4},2s_{1}m,s_{2}m\Big{)}$
for constants $s_{1},s_{2}$ that depend only on $L$. Moreover, once $\theta$
is sufficiently small, we have that
$k=s_{2}m\geq\frac{s_{2}}{\theta}\geq\max\Big{\\{}4,2\log\Big{(}\frac{4}{\tau}\Big{)}\Big{\\}},$
showing that assumption (a) in Theorem 4.4 is satisfied. Lemma 4.7 and Lemma
4.6 imply that
$\displaystyle\max\Big{\\{}\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}},\frac{dm}{\tau
k}\log\Big{(}\frac{\log(3d)\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{\gamma\tau\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}\Big{)}\Big{\\}}$
$\displaystyle\leq\max\Big{\\{}c_{0}d,\frac{ds_{2}}{\tau}\log\Big{(}\frac{\log(3d)2d}{\tau}\Big{)}\Big{\\}}$
$\displaystyle\leq c_{0}d\log(2d)$
for a constant $c_{0}$ that depends only $L$ and $\tau$. And, as $\tau$
depends only on $L$, $c_{0}$ actually only depends only on $L$.
Recall that Theorem 2.9 has the requirement that $N\geq c_{2}d\log(2d)$ for a
constant $c_{2}$ that may depend on $L$ and which we are free to choose to be
as large as we went. Doing so shows that assumption (b) of Theorem 4.4 holds
as well.
Setting $c:=C_{2}\min\\{s_{1},s_{2}\\}$ where $C_{2}$ is the constant from
Theorem 4.4, it follows from that theorem that (5.3) holds with probability at
least $1-2\exp(-c\tau N)$. ∎
From now on, we fix the constant $s_{1}$ from Lemma 5.3. As $s_{1}$ depends
only on $L$, all constants $\theta,\tau,c,c_{1},\dots$ which are allowed to
depend on $L$ may also depend on $s_{1}$.
In a next step, we show that replacing the Hessian at a midpoint with the
Hessian at $x^{\ast}$ does not come at a high cost. Clearly, in Lemma 5.3 we
may arbitrarily modify / delete $s_{1}m$ coordinates from each block $j$, and
we shall argue in the following that the errors
$\mathcal{E}_{\mathrm{H},i}(x)=\sup_{t\in[0,1]}\Big{|}\Big{\langle}\Big{(}\nabla^{2}F(x^{\ast}+t(x-x^{\ast}),\xi_{i})-\nabla^{2}F(x^{\ast},\xi_{i})\Big{)}(x-x^{\ast}),x-x^{\ast}\Big{\rangle}\Big{|}$
are well behaved on the remaining blocks. As before, the proof has two
components. In the first one, we analyze what happens for a single
$x\in\mathcal{S}_{r}^{\ast}$. Here the error is governed by the probability
that $\mathcal{E}_{\mathrm{H}}(x)$ is large, where we recall that $r<r_{0}$
and by Assumption 2.7
$\sup_{x\in\mathcal{S}_{r}^{\ast}}\mathbf{P}\Big{[}\mathcal{E}_{\mathrm{H}}(x)\geq\frac{r^{2}}{8}\Big{]}\leq
c_{1}$
for a constant $c_{1}$ which we are free to choose depending on $L$—hence,
$c_{1}$ may depend on $s_{1}$.
###### Lemma 5.4.
There is a constant $c$ such that, for every $x\in\mathcal{S}_{r}^{\ast}$ and
every $j$, with probability at least $1-2\exp(-cm)$, we have that
$\displaystyle\Big{|}\Big{\\{}i\in
I_{j}:\mathcal{E}_{\mathrm{H},i}(x)\leq\frac{r^{2}}{8}\Big{\\}}\Big{|}$
$\displaystyle\geq\Big{(}1-\frac{s_{1}}{2}\Big{)}m.$
###### Proof.
Fix some $x\in\mathcal{S}_{r}^{\ast}$. Setting $c_{1}:=\frac{s_{1}}{4}$,
Assumption 2.7 implies that
$\delta:=\mathbf{P}\Big{[}\mathcal{E}_{\mathrm{H}}(x)\geq\frac{r^{2}}{8}\Big{]}\leq\frac{s_{1}}{4}.$
If $\delta=0$ there is nothing to prove, so assume otherwise and apply the
Binomial concentration inequality [5, Corollary 2.11]: there is an absolute
constant $C>0$ such that, for every $u\geq 0$, with probability at least
$1-2\exp(-Cm\delta\min\\{u,u^{2}\\})$, we have that
$\Big{|}\Big{\\{}i\in
I_{j}:\mathcal{E}_{\mathrm{H},i}(x)\geq\frac{r^{2}}{8}\Big{\\}}\Big{|}\leq
m\delta(1+u).$
Applying this to $u:=\frac{s_{1}}{2\delta}-1\geq 1$ completes the proof (with
$c=\frac{Cs_{1}}{4}$). ∎
Next, let us show that most blocks have many indices $i$ for which the errors
$\mathcal{E}_{\mathrm{H},i}(x)$ are small.
###### Lemma 5.5.
There is a constant $c$ such that, for every $x\in\mathcal{S}_{r}^{\ast}$,
with probability at least $1-2\exp(-c\tau N)$, we have that
$\displaystyle\Big{|}\Big{\\{}j:\big{|}\big{\\{}i\in
I_{j}:\mathcal{E}_{\mathrm{H},i}(x)\leq\tfrac{r^{2}}{8}\big{\\}}\big{|}\geq\big{(}1-\tfrac{s_{1}}{2}\big{)}m\Big{\\}}\Big{|}$
$\displaystyle\geq\Big{(}1-\frac{\tau}{4}\Big{)}n.$
In particular, the statement holds uniformly over sets
$\bar{\mathcal{S}}_{r}^{\ast}\subseteq\mathcal{S}_{r}^{\ast}$ of cardinality
at most $\log(|\bar{\mathcal{S}}_{r}^{\ast}|)\leq\frac{1}{2}c\tau N$.
###### Proof.
Fix $x\in\mathcal{S}_{r}^{\ast}$ and, for every $j$, set
$\delta(j):=\begin{cases}1,&\text{if }|\\{i\in
I_{j}:\mathcal{E}_{\mathrm{H},i}(x)\leq\frac{1}{8}r^{2}\\}|\geq(1-\frac{1}{2}s_{1})m\\\
0,&\text{otherwise}.\end{cases}$
By Lemma 5.4 we have that
$\delta:=\mathbf{P}[\delta(j)=1]\leq 2\exp(-c_{0}m)$
for a constant $c_{0}$.
Now recall that $m\geq\frac{1}{\theta}$ and therefore $\delta\leq\frac{3}{4}$
once $\theta$ is small enough (i.e.
$\theta\leq\frac{1}{c_{0}}\log(\frac{8}{3})$ suffices). By Bennett’s
inequality, exactly as in the proof of Lemma 4.8, we have that for every
$u\geq 2$, with probability at least $1-2\exp(-C\delta nu\log(u))$,
$|\\{j:\delta(j)=1\\}|\leq u\delta n.$
To complete the proof, set $u:=\frac{\tau}{4\delta}$ so that
$u\geq\exp(\frac{c_{0}m}{2})\geq 2$, again once $\theta$ is small enough. ∎
The final step is to extend the outcome of the lemma from the net
$\bar{\mathcal{S}}_{r}^{\ast}$ to the whole $\mathcal{S}_{r}^{\ast}$. To that
end, recall that by Assumption 2.7
$\|\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi)\|_{\mathrm{op}}\leq\|x-y\|^{\alpha}\cdot
K(\xi)$
for all $x,y\in\mathcal{B}_{r}^{\ast}$.
###### Remark 5.6.
In what follows we shall, from time to time, divide by $\mathbf{E}[K(\xi)]$;
therefore, we shall assume without loss of generality that
$\mathbf{E}[K(\xi)]>0$. Note that if $\mathbf{E}[K(\xi)]=0$ then the Hessian
is constant in $\mathcal{B}_{r}^{\ast}$ and there is nothing to prove:
$\mathcal{E}_{\mathrm{H}}(x)=0$ for every $x\in\mathcal{B}_{r}^{\ast}$.
###### Lemma 5.7.
Let $c_{0}$ be the constant of Lemma 5.5 and let $C_{0}$ be an absolute
constant to be specified later. Set
$\rho:=\Big{(}\frac{C_{0}\tau
s_{1}}{r_{0}^{\alpha}\mathbf{E}[K(\xi)]}\Big{)}^{\frac{1}{\alpha}}.$
Then $\mathcal{S}_{r}^{\ast}$ contains a $\rho r$-cover with respect to the
norm $\|\cdot\|$, which is denoted by $\bar{\mathcal{S}}_{r}^{\ast}$, and
$\log(|\bar{\mathcal{S}}_{r}^{\ast}|)\leq\frac{1}{2}c_{0}\tau N.$
###### Proof.
If $\rho\geq 1$ there is nothing to prove; thus we may assume without loss of
generality that $0<\rho<1$. By a standard volumetric argument (see e.g. [38,
Exercise 2.2.14]) there is a $\rho r$ cover
$\bar{\mathcal{S}}_{r}^{\ast}\subseteq\mathcal{S}_{r}^{\ast}$, and
$\displaystyle\log(|\bar{\mathcal{S}}_{r}^{\ast}|)$ $\displaystyle\leq
d\log\Big{(}\frac{12}{\rho}\Big{)}$
$\displaystyle=\frac{d}{\alpha}\log\Big{(}\frac{r_{0}^{\alpha}\mathbf{E}[K(\xi)]}{C_{0}\tau
s_{1}}\Big{)}.$
By assumption we have
$N\geq c_{2}N_{\mathrm{H},\mathcal{E}}\equiv
c_{2}\frac{d}{\alpha}\log\Big{(}r_{0}^{\alpha}\mathbf{E}[K(\xi)]+2\Big{)}$
for a constant $c_{2}$ which we are free to choose large enough. Then
$\log(|\bar{\mathcal{S}}_{r}^{\ast}|)\leq\frac{1}{2}c_{0}\tau N$, as claimed.
∎
###### Lemma 5.8.
Let $\bar{\mathcal{S}}_{r}^{\ast}$ be as in Lemma 5.7, and for every
$x\in\mathcal{S}_{r}^{\ast}$ set $y=y(x)$ to be the nearest point to $x$ in
$\bar{\mathcal{S}}_{r}^{\ast}$.
There is an absolute constant $C$ such that the following holds. With
probability at least $1-2\exp(-C\tau^{2}n)$, for every
$x\in\mathcal{S}_{r}^{\ast}$, we have that
$\Big{|}\Big{\\{}j:\big{|}\big{\\{}i\in
I_{j}:|\mathcal{E}_{\mathrm{H},i}(x)-\mathcal{E}_{\mathrm{H},i}(y)|\geq\tfrac{r^{2}}{8}\big{\\}}\big{|}\leq\tfrac{s_{1}m}{2}\Big{\\}}\Big{|}\geq\Big{(}1-\frac{\tau}{4}\Big{)}n.$
###### Proof.
Let
$\Psi:=\sup_{x\in\mathcal{S}_{r}^{\ast}}\frac{1}{n}\sum_{j=1}^{n}1_{|\\{i\in
I_{j}:|\mathcal{E}_{\mathrm{H},i}(x)-\mathcal{E}_{\mathrm{H},i}(y)|\geq\frac{1}{8}r^{2}\\}|\geq\frac{1}{2}s_{1}m}$
and observe that it suffices to show that
$\mathbf{P}[\Psi\geq\frac{\tau}{4}]\leq 2\exp(-C\tau^{2}n)$. By the bounded
differences inequality [5, Theorem 6.2], we have that for every $u\geq 0$,
$\mathbf{P}[\Psi\geq\mathbf{E}[\Psi]+u]\leq 2\exp(-Cnu^{2}),$
and setting $u:=\frac{\tau}{8}$, all that remains to show is that
$\mathbf{E}[\Psi]\leq\frac{\tau}{8}$.
Note that
$\displaystyle 1_{|\\{i\in
I_{j}:|\mathcal{E}_{\mathrm{H},i}(x)-\mathcal{E}_{\mathrm{H},i}(y)|\geq\frac{1}{8}r^{2}\\}|\geq\frac{1}{2}s_{1}m}$
$\displaystyle\leq\frac{16}{r^{2}s_{1}}\frac{1}{m}\sum_{i\in
I_{j}}|\mathcal{E}_{\mathrm{H},i}(x)-\mathcal{E}_{\mathrm{H},i}(y)|$
for every $j$; hence,
(5.4) $\displaystyle\Psi$
$\displaystyle\leq\frac{16}{r^{2}s_{1}}\sup_{x\in\mathcal{S}_{r}^{\ast}}\frac{1}{N}\sum_{i=1}^{N}|\mathcal{E}_{\mathrm{H},i}(x)-\mathcal{E}_{\mathrm{H},i}(y)|.$
To control the difference of the $\mathcal{E}$’s, for simpler notation, for
$(t,z)\in[0,1]\times\mathcal{S}_{r}^{\ast}$ set
$A^{tz}_{i}:=\nabla^{2}F(x^{\ast}+t(z-x^{\ast}),\xi_{i})-\nabla^{2}F(x^{\ast},\xi_{i}),$
and observe that
$\mathcal{E}_{\mathrm{H},i}(z)=\sup_{t\in[0,1]}|\langle
A^{tz}_{i}(z-x^{\ast}),z-x^{\ast}\rangle|$
for every $z\in\mathcal{S}_{r}^{\ast}$. Now, for every $t\in[0,1]$ and every
$x\in\mathcal{S}_{r}^{\ast}$ (and $y=y(x)$),
$\displaystyle\big{|}\langle A^{tx}_{i}(x-x^{\ast}),x-x^{\ast}\rangle-\langle
A^{ty}_{i}(y-x^{\ast}),y-x^{\ast}\rangle\big{|}$ $\displaystyle=\big{|}\langle
A^{tx}_{i}(x-x^{\ast}+y-x^{\ast}),x-y\rangle-\langle(A^{ty}_{i}-A^{tx}_{i})(y-x^{\ast}),y-x^{\ast}\rangle\big{|}$
$\displaystyle\leq 2\rho
r^{2}\|A^{tx}_{i}\|_{\mathrm{op}}+r^{2}\|A^{ty}_{i}-A^{tx}_{i}\|_{\mathrm{op}},$
where the last inequality holds by definition of the operator norm and by
noting that $\|x-y\|\leq\rho r$, $\|x-x^{\ast}+y-x^{\ast}\|\leq 2r$, and
$\|y-x^{\ast}\|\leq r$. Invoking the subadditivity of “$\sup_{t}(\cdot)$” and
the triangle inequality,
$\displaystyle|\mathcal{E}_{\mathrm{H}_{i}}(x)-\mathcal{E}_{\mathrm{H}_{i}}(x)|$
$\displaystyle\leq 2\rho
r^{2}\sup_{t\in[0,1]}\|A^{tx}_{i}\|_{\mathrm{op}}+r^{2}\sup_{t\in[0,1]}\|A^{ty}_{i}-A^{tx}_{i}\|_{\mathrm{op}}$
$\displaystyle\leq 2\rho
r^{2}r^{\alpha}K(\xi_{i})+r^{2}(r\rho)^{\alpha}K(\xi_{i})$ (5.5)
$\displaystyle\leq 3r^{2}(\rho r)^{\alpha}K(\xi_{i})$
where the second inequality holds by Assumption 2.7 on the continuity of the
Hessian and as we may assume without loss of generality that $\rho\leq 1$ (and
hence $\rho\leq\rho^{\alpha}$).
Plugging (5.5) into (5.4) implies that
$\displaystyle\Psi$ $\displaystyle\leq\frac{16}{r^{2}s_{1}}\cdot 3r^{2}(\rho
r)^{\alpha}\frac{1}{N}\sum_{i=1}^{N}K(\xi_{i})$
and therefore
$\displaystyle\mathbf{E}[\Psi]$
$\displaystyle\leq\frac{48\rho^{\alpha}r_{0}^{\alpha}\mathbf{E}[K(\xi)]}{s_{1}}.$
Setting
$\rho=\Big{(}\frac{C_{0}\tau
s_{1}}{r_{0}^{\alpha}\mathbf{E}[K(\xi)]}\Big{)}^{\frac{1}{\alpha}}$
just as in Lemma 5.7, and recalling that we are free to choose $C_{0}$ small
enough, it follows that $\mathbf{E}[\Psi]\leq\frac{\tau}{8}$, as required. ∎
We are now ready for the
###### Proof of Proposition 5.1, quadratic part.
For every $x\in\mathcal{S}_{r}^{\ast}$ and $j$, set
$J_{j}^{\ast}(x):=\\{\text{largest $s_{1}m$ coordinates of
}(\mathcal{E}_{\mathrm{H},i}(x))_{i\in I_{j}}\\}\subseteq I_{j}.$
For every $j$, recalling the definition of $\mathcal{E}_{\mathrm{H}}(x)$ and
the fact that $\nabla^{2}F(z,\xi)$ is positive semidefinite, it follows that
$\displaystyle Q_{x,x^{\ast}}(j)$ $\displaystyle\geq\frac{1}{m}\sum_{i\in
I_{j}\setminus
J_{j}^{\ast}(x)}\inf_{t\in[0,1]}\big{\langle}\nabla^{2}F(x^{\ast}+t(x-x^{\ast}),\xi_{i})(x-x^{\ast}),x-x^{\ast}\big{\rangle}$
$\displaystyle\geq\frac{1}{m}\sum_{i\in I_{j}\setminus
J_{j}^{\ast}(x)}\big{\langle}\nabla^{2}F(x^{\ast},\xi_{i})(x-x^{\ast}),x-x^{\ast}\big{\rangle}-\frac{1}{m}\sum_{i\in
I_{j}\setminus J_{j}^{\ast}(x)}\mathcal{E}_{\mathrm{H},i}(x)$
$\displaystyle=:A_{x}(j)+B_{x}(j)$
As $|J_{j}^{\ast}(x)|\leq s_{1}m$ for every $x\in\mathcal{S}_{r}^{\ast}$ by
definition, Lemma 5.3 implies that, with probability at least $1-2\exp(-c\tau
n)$, for every $x\in\mathcal{S}_{r}^{\ast}$, we have that
$A_{x}(j)\geq\frac{r^{2}}{2}\quad\text{on more than
}\Big{(}1-\frac{\tau}{2}\Big{)}n\text{ of the blocks }j.$
Moreover, combining Lemma 5.5 and Lemma 5.8 implies that, with probability at
least $1-2\exp(-c\tau^{2}n)$, for every $x\in\mathcal{S}_{r}^{\ast}$, we have
that
$\displaystyle\Big{|}\Big{\\{}j:\big{|}\big{\\{}i\in
I_{j}:\mathcal{E}_{\mathrm{H},i}(x)\leq\tfrac{r^{2}}{4}\big{\\}}\big{|}\geq(1-s_{1})m\Big{\\}}\Big{|}$
$\displaystyle\geq\Big{(}1-\frac{\tau}{2}\Big{)}n,$
and on that event clearly
$B_{x}(j)\geq-\frac{r^{2}}{4}\quad\text{on more than
}\Big{(}1-\frac{\tau}{2}\Big{)}n\text{ of the blocks }j.$
In particular, combining the estimates on $A_{x}(j)$ and $B_{x}(j)$ gives:
with probability at least $1-2\exp(-c\tau^{2}n)$, for every
$x\in\mathcal{S}_{r}^{\ast}$, we have that
$Q_{x,x^{\ast}}(j)\geq\frac{r^{2}}{4}\quad\text{on more than }(1-\tau)n\text{
of the blocks }j.$
This completes the proof. ∎
### 5.2. Estimation error, the multiplier term
This subsection contains the proof of (5.2) from Lemma 5.1, stating that the
multiplier term
$M_{x,x^{\ast}}(j)=\frac{1}{m}\sum_{i\in I_{j}}\langle\nabla
F(x^{\ast},\xi_{i}),x-x^{\ast}\rangle$
is likely to be at most of order $r^{2}$. To ease notation, set
$\displaystyle\mathbb{H}$
$\displaystyle:=\nabla^{2}f(x^{\ast})=\mathbf{E}[\nabla^{2}F(x^{\ast},\xi)],$
$\displaystyle\mathbb{G}$ $\displaystyle:=\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)],$
ı.e., $\mathbb{H}$ is the Hessian of $f$ at $x^{\ast}$ and $\mathbb{G}$ is the
covariance matrix of the gradient of $F(\cdot,\xi)$ at $x^{\ast}$. In
particular, a straightforward computation shows that
(5.6) $\displaystyle N_{\mathrm{G}}(r)$
$\displaystyle=\frac{1}{r^{2}}\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G})=\frac{1}{r^{2}}\mathrm{trace}(\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}}),$
(5.7)
$\displaystyle\begin{split}\sigma^{2}&=\lambda_{\max}(\mathbb{H}^{-1}\mathbb{G})\\\
&=\lambda_{\max}(\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}})=\|\mathbb{G}\|_{\mathrm{op}}.\end{split}$
As before, we analyze what happens for a single $x\in\mathcal{S}_{r}^{\ast}$;
the high probability estimate we obtain allows us to control a net in
$\mathcal{S}_{r}^{\ast}$; we then show that passing from the net to the entire
set $\mathcal{S}_{r}^{\ast}$ does not distort the outcome by too much.
###### Lemma 5.9.
There is an absolute constant $C$ such that, for every
$x\in\mathcal{S}_{r}^{\ast}$, with probability at least
$1-2\exp(-C\tau^{2}n)$, we have that
$\Big{|}\Big{\\{}j:M_{x,x^{\ast}}(j)\geq-\frac{r^{2}}{32}\Big{\\}}|\geq\Big{(}1-\frac{\tau}{2}\Big{)}n.$
In particular, the statement holds uniformly over a set
$\bar{\mathcal{S}}_{r}^{\ast}\subseteq\mathcal{S}_{r}^{\ast}$ of cardinality
at most
$\log(\frac{1}{2}|\bar{\mathcal{S}}_{r}^{\ast}|)\leq\frac{1}{2}C\tau^{2}n$.
###### Proof.
Fix some $x\in\mathcal{S}_{r}^{\ast}$ and define for $1\leq i\leq N$,
$\displaystyle U_{i}$ $\displaystyle:=\langle\nabla
F(x^{\ast},\xi_{i})-\mathbf{E}[\nabla F(x^{\ast},\xi)],x-x^{\ast}\rangle$
$\displaystyle=\langle\nabla
F(x^{\ast},\xi_{i}),x-x^{\ast}\rangle-\langle\nabla
f(x^{\ast}),x-x^{\ast}\rangle.$
If $x^{\ast}$ lies in the interior of $\mathcal{X}$, the first order condition
for optimality implies that $\langle\nabla f(x^{\ast}),x-x^{\ast}\rangle$
equals zero. In general, the first order condition implies that this term is
non-negative. In either case, we get that
$\displaystyle\Big{|}\frac{1}{m}\sum_{i\in I_{j}}U_{i}\Big{|}$
$\displaystyle\leq\frac{r^{2}}{32}\quad\text{implies}\quad
M_{x,x^{\ast}}(j)\geq-\frac{r^{2}}{32}$
and we shall show that the former happens on most blocks.
To that end, consider first a single block $j$. As the $U_{i}$’s are i.i.d.
zero mean random variables, Markov’s inequality together with the Cauchy-
Schwartz inequality implies that
(5.8) $\displaystyle\begin{split}\mathbf{P}\Big{[}\Big{|}\frac{1}{m}\sum_{i\in
I_{j}}U_{i}\Big{|}\geq\frac{r^{2}}{32}\Big{]}&\leq\frac{32}{r^{2}}\mathbf{E}\Big{[}\Big{|}\frac{1}{m}\sum_{i\in
I_{j}}U_{i}\Big{|}\Big{]}\\\
&\leq\frac{32}{r^{2}}\Big{(}\frac{\mathbf{E}[|U_{1}|^{2}]}{m}\Big{)}^{\frac{1}{2}}.\end{split}$
By the definition of $\sigma$ (or rather, by the alternative expression in
(5.7)) and as $\|x-x^{\ast}\|=r$, we have that
(5.9)
$\displaystyle\begin{split}\mathbf{E}[|U_{1}|^{2}]&=\langle\mathbb{G}(x-x^{\ast}),x-x^{\ast}\rangle\\\
&\leq r^{2}\sigma^{2}.\end{split}$
Combining (5.8) and (5.9) and using that $m\geq\tfrac{\sigma^{2}}{\theta
r^{2}}$, we conclude that
$\displaystyle\mathbf{P}\Big{[}\Big{|}\frac{1}{m}\sum_{i\in
I_{j}}U_{i}\Big{|}\geq\frac{r^{2}}{32}\Big{]}$
$\displaystyle\leq\frac{32r\sigma}{r^{2}\sqrt{m}}$ $\displaystyle\leq
32\sqrt{\theta}\leq\frac{\tau}{4}$
as soon as $\theta$ is sufficiently small.
The claim now follows from a Binomial estimate—just as in the proof of Lemma
5.4: the probability that a single block $j$ has the wanted property is at
least $1-\frac{\tau}{4}$ (by the above); therefore, the probability that the
number of desirable blocks $j$ is smaller than the mean (which is at least
$(1-\frac{\tau}{4})n$) by more than $\frac{\tau}{4}n$ is at most
$2\exp(-Cn\tau^{2})$. ∎
###### Lemma 5.10.
Let $C_{0}$ be the absolute constant of Lemma 5.9. Then there is an absolute
constant $C_{1}$ and a set
$\bar{\mathcal{S}}_{r}^{\ast}\subseteq\mathcal{S}_{r}^{\ast}$ with cardinality
$\log(\frac{1}{2}|\bar{\mathcal{S}}_{r}^{\ast}|)\leq\frac{1}{2}C_{0}n\tau^{2}$
such that the following holds. Let
$\rho:=\Big{(}\frac{\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G})}{C_{1}\tau^{2}n}\Big{)}^{\frac{1}{2}}.$
For every $x\in\mathcal{S}_{r}^{\ast}$ there is
$y=y(x)\in\bar{\mathcal{S}}_{r}^{\ast}$ with
(5.10) $\displaystyle\langle\mathbb{G}(x-y),x-y\rangle^{\frac{1}{2}}$
$\displaystyle\leq 2\rho r\quad\text{and },$ (5.11)
$\displaystyle\langle\nabla f(x^{\ast}),x-y\rangle$ $\displaystyle\geq 0.$
###### Proof.
As a first step, we ignore (5.11) and construct a set
$\tilde{\mathcal{S}}_{r}^{\ast}$ with log-cardinality satisfying
$\log(\frac{1}{2}|\tilde{\mathcal{S}}_{r}^{\ast}|)\leq\frac{1}{2}C_{0}n\tau^{2}$
such that (5.10) holds with $\rho r$ instead of $2\rho r$. To that end,
observe that covering the sphere
$\\{x\in\mathbb{R}^{d}:\langle\mathbb{H}x,x\rangle=1\\}$ w.r.t. to the norm
endowed by $\mathbb{G}$ is equivalent to covering the Euclidean sphere
$\\{x\in\mathbb{R}^{d}:\langle x,x\rangle=1\\}$ w.r.t. the norm endowed by
$\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}}$. Hence,
denoting by $\mathcal{G}$ the standard Gaussian vector in $\mathbb{R}^{d}$,
the dual Sudakov inequality (see e.g. [16, Theorem 3.18]) guarantees the
existence of a $\rho r$ cover of
$\tilde{\mathcal{S}}_{r}^{\ast}\subseteq\mathcal{S}_{r}^{\ast}$ with respect
to the norm $\langle\mathbb{G}\cdot,\cdot\rangle^{\frac{1}{2}}$ such that
$\log\Big{(}\frac{|\tilde{\mathcal{S}}_{r}^{\ast}|}{2}\Big{)}\leq
C_{2}\Big{(}\frac{\mathbf{E}[\langle\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}}\mathcal{G},\mathcal{G}\rangle^{\frac{1}{2}}]}{\rho}\Big{)}^{2};$
thus, for every $x\in\mathcal{S}^{\ast}_{r}$ there is
$y=y(x)\in\tilde{\mathcal{S}}_{r}^{\ast}$ with
$\langle\mathbb{G}(y-x),y-x\rangle^{\frac{1}{2}}\leq\rho r$. Observe that
$\displaystyle\mathbf{E}[\langle\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}}\mathcal{G},\mathcal{G}\rangle^{\frac{1}{2}}]^{2}$
$\displaystyle=\mathbf{E}[\|(\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}})^{\frac{1}{2}}\mathcal{G}\|_{2}]^{2}$
$\displaystyle\leq\mathbf{E}[\|(\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}})^{\frac{1}{2}}\mathcal{G}\|_{2}^{2}]$
$\displaystyle=\mathop{\mathrm{trace}}(\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}})=\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G}),$
(where the last equality was already observed in (5.6)). Thus,
$\displaystyle\begin{split}\log\Big{(}\frac{|\tilde{\mathcal{S}}_{r}^{\ast}|}{2}\Big{)}&\leq\frac{C_{2}\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G})}{\rho^{2}}\\\
&=C_{2}C_{1}\tau^{2}n\leq\frac{1}{2}C_{0}\tau^{2}n,\end{split}$
once $C_{1}$ is sufficiency small.
Next, we modify $\tilde{\mathcal{S}}_{r}^{\ast}$, ensuring that both equations
(5.10) and (5.11) hold: for every $z\in\tilde{\mathcal{S}}_{r}^{\ast}$, pick
some
$y(z)\in\mathop{\mathrm{argmin}}\Big{\\{}\langle\nabla
f(x^{\ast}),y\rangle:y\in\mathcal{S}_{r}^{\ast}\text{ s.t.\
}\langle\mathbb{G}(y-z),(y-z)\rangle^{\frac{1}{2}}\leq\rho r\Big{\\}};$
thus, $y(z)$ is the minimizer of $\langle\nabla f(x^{\ast}),y\rangle$ with the
$\rho r$-ball (with respect to the norm
$\langle\mathbb{G}\cdot,\cdot\rangle^{\frac{1}{2}}$) centred at $z$.
It is straightforward to verify that the set
$\bar{\mathcal{S}}_{r}^{\ast}:=\\{y(z):z\in\tilde{\mathcal{S}}_{r}^{\ast}\\}$
satisfies the statement of the lemma. ∎
###### Lemma 5.11.
Let $\bar{\mathcal{S}}_{r}^{\ast}$ and $y=y(x)$ be as in Lemma 5.10. There is
an absolute constant $C$ such that the following holds. With probability at
least $1-2\exp(-C\tau^{2}n)$, for every $x\in\mathcal{S}_{r}^{\ast}$, we have
that
$\Big{|}\Big{\\{}j:M_{x,x^{\ast}}(j)\geq
M_{y,x^{\ast}}(j)-\frac{r^{2}}{32}\Big{\\}}\Big{|}\geq\Big{(}1-\frac{\tau}{2}\Big{)}n.$
###### Proof.
For every $x\in\mathcal{X}$ and $j$, set
$\displaystyle\Delta_{x}(j)$
$\displaystyle:=M_{x,x^{\ast}}(j)-M_{y,x^{\ast}}(j)$
$\displaystyle\bar{\Delta}_{x}(j)$
$\displaystyle:=\Delta_{x}(j)-\mathbf{E}[\Delta_{x}(j)]$
$\displaystyle=\frac{1}{m}\sum_{i\in I_{j}}\langle\nabla
F(x^{\ast},\xi_{i})-\mathbf{E}[\nabla F(x^{\ast},\xi)],x-y\rangle.$
Recalling that $\mathbf{E}[\Delta_{x}(j)]\geq 0$ by Lemma 5.10 and setting
$\Psi:=\sup_{x\in\mathcal{S}_{r}^{\ast}}\frac{1}{n}\sum_{j=1}^{n}1_{\\{|\bar{\Delta}_{x}(j)|\geq\frac{1}{32}r^{2}\\}},$
the statement of the lemma therefore follows if $\Psi\leq\frac{\tau}{2}$ holds
with probability at least $1-2\exp(-C\tau^{2}n)$.
To that end, we once more rely on the bounded difference inequality [5,
Theorem 6.2]: for every $u\geq 0$, we have that
$\mathbf{P}[\Psi\geq\mathbf{E}[\Psi]+u]\leq 2\exp(-Cnu^{2}).$
Setting $u:=\frac{\tau}{4}$, all that is left is to show that
$\mathbf{E}[\Psi]\leq\frac{\tau}{4}$.
First, observe that $1_{|a|\geq b}\leq\frac{1}{b}|a|$. Hence,
$\displaystyle\Psi$
$\displaystyle\leq\frac{32}{r^{2}}\sup_{x\in\mathcal{S}_{r}^{\ast}}\frac{1}{n}\sum_{j=1}^{n}|\bar{\Delta}_{x}(j)|$
$\displaystyle\leq\frac{32}{r^{2}}\Big{(}\sup_{x\in\mathcal{S}_{r}^{\ast}}\mathbf{E}[|\bar{\Delta}_{x}(1)|]+\sup_{x\in\mathcal{S}_{r}^{\ast}}\frac{1}{n}\sum_{j=1}^{n}\Big{|}\bar{\Delta}_{x}(j)\big{|}-\mathbf{E}[|\bar{\Delta}_{x}(j)|]\Big{|}\Big{)}$
$\displaystyle=:\frac{32}{r^{2}}\Big{(}R_{1}+R_{2}\Big{)}.$
In particular $\mathbf{E}[\Psi]\leq\frac{32}{r^{2}}(R_{1}+\mathbf{E}[R_{2}])$.
To estimate $R_{1}$, recall that $\bar{\Delta}_{x}(j)$ is a sum of
independent, zero mean random variables. Thus,
$\displaystyle\mathbf{E}[|\bar{\Delta}_{x}(1)|]$
$\displaystyle\leq\Big{(}\frac{\langle\mathbb{G}(x-y),x-y\rangle}{m}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq
2r\Big{(}\frac{\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G})}{C_{1}\tau^{2}nm}\Big{)}^{\frac{1}{2}}$
for every $x\in\mathcal{S}_{r}^{\ast}$, where the second inequality follows
from Lemma 5.10. Recalling that, by our assumptions,
$nm=N\geq
c_{2}N_{\mathrm{G}}(r)\equiv\frac{c_{2}\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G})}{r^{2}},$
we conclude that
$\mathbf{E}[|\bar{\Delta}_{x}(1)|]\leq\frac{2r^{2}}{\tau\sqrt{C_{1}c_{2}}}$.
Therefore,
$R_{1}\leq\frac{2r^{2}}{\tau\sqrt{C_{1}c_{2}}}.$
Turning to $R_{2}$, by symmetrization, the contraction theorem for Rademacher
processes, and de-symmetrization (see e.g. [5, Section 11.3]), it is evident
that
$\displaystyle\mathbf{E}[R_{2}]$ $\displaystyle\leq
2\mathbf{E}\Big{[}\sup_{x\in\mathcal{S}_{r}^{\ast}}\Big{|}\frac{1}{n}\sum_{j=1}^{n}\varepsilon_{j}|\bar{\Delta}_{x}(j)|\Big{|}\Big{]}$
$\displaystyle\leq
4\mathbf{E}\Big{[}\sup_{x\in\mathcal{S}_{r}^{\ast}}\Big{|}\frac{1}{n}\sum_{j=1}^{n}\varepsilon_{j}\bar{\Delta}_{x}(j)\Big{|}\Big{]}$
$\displaystyle\leq
8\mathbf{E}\Big{[}\sup_{x\in\mathcal{S}_{r}^{\ast}}\Big{|}\frac{1}{n}\sum_{j=1}^{n}\bar{\Delta}_{x}(j)\Big{|}\Big{]}.$
Moreover, by the definition of $\bar{\Delta}_{x}(j)$,
(5.12) $\displaystyle\mathbf{E}[R_{2}]$ $\displaystyle\leq
8\mathbf{E}\Big{[}\sup_{x\in\mathcal{S}_{r}^{\ast}}\Big{|}\frac{1}{N}\sum_{i=1}^{N}\big{\langle}\nabla
F(x^{\ast},\xi_{i})-\mathbf{E}[\nabla
F(x^{\ast},\xi)],x-y\big{\rangle}\Big{|}\Big{]}.$
Recall that $\|\cdot\|=\|\mathbb{H}^{\frac{1}{2}}\cdot\|_{2}$ and that
$\|x-y\|\leq 2r$. Therefore, for any $z\in\mathbb{R}^{d}$,
$\displaystyle\sup_{x\in\mathcal{S}_{r}^{\ast}}|\langle z,x-y\rangle|$
$\displaystyle=\sup_{x\in\mathcal{S}_{r}^{\ast}}|\langle\mathbb{H}^{-\frac{1}{2}}z,\mathbb{H}^{\frac{1}{2}}(x-y)\rangle|$
$\displaystyle\leq 2r\|\mathbb{H}^{-\frac{1}{2}}z\|_{2}.$
Together with linearity of $z\mapsto\langle z,x-y\rangle$ and (5.12),
$\displaystyle\mathbf{E}[R_{2}]$ $\displaystyle\leq
16r\mathbf{E}\Big{[}\Big{\|}\frac{1}{N}\sum_{i=1}^{N}\mathbb{H}^{-\frac{1}{2}}\Big{(}\nabla
F(x^{\ast},\xi_{i})-\mathbf{E}[\nabla
F(x^{\ast},\xi)]\Big{)}\Big{\|}_{2}\Big{]}$ $\displaystyle\leq
16r\Big{(}\frac{\mathrm{trace}(\mathrm{\mathbf{C}ov}[\mathbb{H}^{-\frac{1}{2}}\nabla
F(x^{\ast},\xi)])}{N}\Big{)}^{\frac{1}{2}}.$
It remains to observe that
$\displaystyle\mathrm{trace}(\mathrm{\mathbf{C}ov}[\mathbb{H}^{-\frac{1}{2}}\nabla
F(x^{\ast},\xi)])$
$\displaystyle=\mathrm{trace}(\mathbb{H}^{-\frac{1}{2}}\mathbb{G}\mathbb{H}^{-\frac{1}{2}})$
$\displaystyle=\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G}).$
Since $N\geq
c_{2}N_{\mathrm{G}}(r)\equiv\frac{c_{2}}{r^{2}}\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G})$,
it is evident that
$\mathbf{E}[R_{2}]\leq\frac{16r^{2}}{\sqrt{c_{2}}},$
and combining the two estimates,
$\displaystyle\mathbf{E}[\Psi]$
$\displaystyle\leq\frac{32}{r^{2}}\Big{(}\frac{2r^{2}}{\tau\sqrt{C_{1}c_{2}}}+\frac{16r^{2}}{\sqrt{c_{2}}}\Big{)}\leq\frac{\tau}{4},$
where the last inequality holds as soon as $c_{2}$ is large enough. ∎
###### Proof of Proposition 5.1, multiplier part.
Let $\bar{\mathcal{S}}_{r}^{\ast}$ be the cover defined in Lemma 5.10. By
Lemma 5.9, with probability at least $1-2\exp(-c\tau^{2}n)$, for every
$y\in\bar{\mathcal{S}}_{r}^{\ast}$, we have that
$M_{y,x^{\ast}}(j)\geq-\frac{r^{2}}{32}\ \quad\text{on more than
}\Big{(}1-\frac{\tau}{2}\Big{)}n\text{ blocks }j.$
Moreover, by Lemma 5.11, with probability at least $1-2\exp(-c\tau^{2}n)$, for
every $x\in\mathcal{S}_{r}^{\ast}$ there is $y\in\bar{\mathcal{S}}_{r}^{\ast}$
such that
$M_{x,x^{\ast}}(j)\geq M_{y,x^{\ast}}(i)-\frac{r^{2}}{32}\ \quad\text{on more
than }\Big{(}1-\frac{\tau}{2}\Big{)}n\text{ blocks }j.$
Combining the two estimates, it follows that, with probability at least
$1-2\exp(-c\tau^{2}n)$, for every $x\in\mathcal{S}_{r}^{\ast}$,
$M_{x,x^{\ast}}(j)\geq-\frac{r^{2}}{16}\quad\text{on more than
}(1-\tau)n\text{ blocks }j,$
which is exactly what we wanted to show. ∎
### 5.3. Prediction error
In this section we shall prove Proposition 2.15, dealing with the prediction
error. To that end, let
$\mathcal{U}_{r}^{\ast}:=\\{x\in\mathcal{B}_{r}^{\ast}:f(x)\geq
f(x^{\ast})+2c_{\mathrm{H}}r^{2}\\}$
be the set of all $x\in\mathcal{B}_{r}^{\ast}$ that are in an “unfavorable
position”. Let us stress again that if $x^{\ast}$ lies in the interior of
$\mathcal{X}$, then $\mathcal{U}_{r}^{\ast}$ is empty and the estimate on the
prediction error holds automatically. We therefore assume that
$\mathcal{U}_{r}^{\ast}$ is not empty.
The proof of Proposition 2.15 relies on the convexity of $F$: for any
$x,y\in\mathcal{X}$ and $j$, we have that
$\displaystyle\widehat{f}_{I_{j}^{\prime}}(x)-\widehat{f}_{I_{j}^{\prime}}(y)$
$\displaystyle\geq\frac{1}{m}\sum_{i\in I_{j}^{\prime}}\langle\nabla
F(y,\xi_{i}),x-y\rangle=:M_{x,y}^{\prime}(j).$
In particular, this implies that
1. (i)
$x^{\ast}$ wins its home match against $x$ if
$M_{x,x^{\ast}}^{\prime}(j)\geq-\frac{c_{\mathrm{H}}r^{2}}{4}\text{ on more
than }\frac{n}{2}\text{ blocks }j,$
2. (ii)
$x$ does not win its home match against $x^{\ast}$ if
$M_{x,x^{\ast}}^{\prime}(j)>\frac{c_{\mathrm{H}}r^{2}}{4}\text{ on more than
}\frac{n}{2}\text{ blocks }j,$
Thus, Proposition 2.15 is a consequence of the following lemma, which we shall
prove in this section.
###### Lemma 5.12.
There is a constant $c$ such that the following holds. With probability at
least $1-2\exp(-c\tau^{2}n)$, for every $x\in\mathcal{B}_{r}^{\ast}$ and every
$y\in\mathcal{U}_{r}^{\ast}$ we have that
(5.13)
$\displaystyle\Big{|}\Big{\\{}j:M_{x,x^{\ast}}^{\prime}(j)\geq-\frac{c_{\mathrm{H}}r^{2}}{4}\Big{\\}}\Big{|}$
$\displaystyle\geq(1-2\tau)n,$ (5.14)
$\displaystyle\Big{|}\Big{\\{}j:M_{y,x^{\ast}}^{\prime}(j)>\frac{c_{\mathrm{H}}r^{2}}{4}\Big{\\}}\Big{|}$
$\displaystyle\geq(1-2\tau)n.$
Just as in the analysis of the estimation error, Proposition 2.15 is an easy
consequence of Lemma 5.12:
###### Proof of Proposition 2.15.
By Proposition 2.14 we have that with probability at least
$1-2\exp(-c\tau^{2}n)$, $x^{\ast}\in\tilde{\mathcal{X}}_{N}^{\ast}$ and
$\tilde{\mathcal{X}}_{N}^{\ast}\subseteq\mathcal{B}_{r}^{\ast}$. In the
following we argue conditionally on that high probability event, and recall
that $\tau<\frac{1}{4}$. By Lemma 5.12, with probability at least
$1-2\exp(-c\tau^{2}n)$, we have that $x^{\ast}$ wins its home match against
every competitor in $\mathcal{B}_{r}^{\ast}$ (in particular, against every
competitor in $\tilde{\mathcal{X}}_{N}^{\ast}$). Moreover, on the same event,
every element in $\tilde{\mathcal{X}}_{N}^{\ast}$ that is in an unfavorable
position loses its home match against $x^{\ast}$. Thus,
$x^{\ast}\in\widehat{\mathcal{X}}_{N}^{\ast}$ and
$\widehat{\mathcal{X}}_{N}^{\ast}\subseteq\mathcal{B}_{r}^{\ast}\setminus\mathcal{U}_{r}^{\ast}$,
which is exactly what we wanted to show. ∎
The first part of Lemma 5.12 (namely (5.13)) is an immediate consequence of
Lemma 5.1. Thus, in the following, we focus on the second part of Lemma 5.12
(namely (5.14)), dealing with $\mathcal{U}_{r}^{\ast}$. Our starting point is
the following observation:
###### Lemma 5.13.
Let $x\in\mathcal{U}_{r}^{\ast}$. Then we have that
$\langle\nabla f(x^{\ast}),x-x^{\ast}\rangle\geq c_{\mathrm{H}}r^{2}.$
###### Proof.
Since $x\in\mathcal{U}_{r}^{\ast}$ we have that $2c_{\mathrm{H}}r^{2}\leq
f(x)-f(x^{\ast})$. A Taylor expansion around $x^{\ast}$ shows that there is a
midpoint $z$ such that
$\displaystyle f(x)-f(x^{\ast})$ $\displaystyle=\langle\nabla
f(x^{\ast}),x-x^{\ast}\rangle+\frac{1}{2}\langle\nabla^{2}f(z)(x-x^{\ast}),x-x^{\ast}\rangle$
$\displaystyle\leq\langle\nabla
f(x^{\ast}),x-x^{\ast}\rangle+c_{\mathrm{H}}r^{2}.$
The proof clearly follows. ∎
Thanks to Lemma 5.13, the proof of the second part of Lemma 5.12 (namely
(5.14)) follows the same path as the proof of the bound on the multiplier term
in the context of the estimation error. Thus, we shall only sketch the
argument for the sake of completeness.
###### Lemma 5.14.
There is an absolute constant $C$ such that, for every
$x\in\mathcal{U}_{r}^{\ast}$, with probability at least
$1-2\exp(-C\tau^{2}n)$, we have that
(5.15)
$\displaystyle\Big{|}\Big{\\{}j:M_{x,x^{\ast}}^{\prime}(j)\geq\frac{c_{\mathrm{H}}r^{2}}{2}\Big{\\}}\Big{|}\geq(1-\tau)n.$
In particular, the statement holds uniformly for sets
$\bar{\mathcal{U}}_{r}^{\ast}\subseteq\mathcal{U}_{r}^{\ast}$ whose
cardinality satisfies
$\log(\frac{1}{2}|\bar{\mathcal{U}}_{r}^{\ast}|)\leq\frac{1}{2}C\tau^{2}n$.
###### Proof.
Fix $x\in\mathcal{U}_{r}^{\ast}$. By Lemma 5.13, we have
$\mathbf{E}[\langle\nabla F(x^{\ast},\xi),x-x^{\ast}\rangle]\geq
c_{\mathrm{H}}r^{2}.$
Thus, exactly as in the proof of Lemma 5.9, we conclude that
$\mathbf{P}\Big{[}M_{x,x^{\ast}}^{\prime}(j)\leq\frac{c_{\mathrm{H}}r^{2}}{2}\Big{]}\leq\frac{2\sqrt{\theta}}{c_{\mathrm{H}}}\leq
2\sqrt{\theta}\leq\frac{\tau}{2}.$
(as long as $\theta$ is small enough, and the second inequality holds because
$c_{\mathrm{H}}\geq 1$). A Binomial estimate (just as in the proof of Lemma
5.9) can be used to show that (5.15) holds with probability at least
$1-2\exp(-C\tau^{2}n)$. ∎
###### Lemma 5.15.
Let $C_{0}$ be the absolute constant of Lemma 5.14. Then there is an absolute
constant $C_{1}$ and a set
$\bar{\mathcal{U}}_{r}^{\ast}\subseteq\mathcal{U}_{r}^{\ast}$ whose
cardinality satisfies
$\log(\frac{1}{2}|\bar{\mathcal{U}}_{r}^{\ast}|)\leq\frac{1}{2}C_{0}n\tau^{2}$,
such that the following holds.
Let
$\rho:=\Big{(}\frac{\mathrm{trace}(\mathbb{H}^{-1}\mathbb{G})}{C_{1}\tau^{2}n}\Big{)}^{\frac{1}{2}}$
For every $x\in\mathcal{U}_{r}^{\ast}$ there is
$y=y(x)\in\bar{\mathcal{U}}_{r}^{\ast}$ with
(5.16) $\displaystyle\langle\mathbb{G}(x-y),x-y\rangle^{\frac{1}{2}}$
$\displaystyle\leq 2\rho r\quad\text{and},$ (5.17) $\displaystyle\langle\nabla
f(x^{\ast}),x-y\rangle$ $\displaystyle\geq 0.$
###### Proof.
Just as in Lemma 5.10, we can construct a set
$\tilde{\mathcal{B}}_{r}^{\ast}\subseteq\mathcal{B}_{r}^{\ast}$ satisfying
(5.16) with $2\rho r$ replaced by $\rho r$. The modification of that set is
again similar: for $z\in\tilde{\mathcal{B}}_{r}^{\ast}$, define
$y(z)\in\mathrm{argmin}\Big{\\{}\langle\nabla
f(x^{\ast}),y\rangle:y\in\mathcal{U}_{r}^{\ast}\text{ s.t.\
}\langle\mathbb{G}(y-z),y-z\rangle^{\frac{1}{2}}\leq\rho r\Big{\\}}.$
with the convention $y(z):=y_{0}$ for some fixed
$y_{0}\in\mathcal{U}_{r}^{\ast}$ if the above set is empty. Then
$\bar{\mathcal{U}}_{r}^{\ast}:=\\{y(z):z\in\tilde{\mathcal{B}}_{r}^{\ast}\\}$
satisfies the statement of the lemma. ∎
Finally, fix the set $\bar{\mathcal{U}}_{r}^{\ast}$ of Lemma 5.15 and for
$x\in\mathcal{U}_{r}^{\ast}$ denote by $y=y(x)\in\bar{\mathcal{U}}_{r}^{\ast}$
the best approximation in the cover constructed in Lemma 5.15.
###### Lemma 5.16.
There is an absolute constant $C$ such that the following holds. With
probability at least $1-2\exp(-C\tau^{2}n)$, for every
$x\in\mathcal{U}_{r}^{\ast}$, we have that
$\Big{|}\Big{\\{}j:M_{x,x^{\ast}}^{\prime}(j)\geq
M_{y,x^{\ast}}^{\prime}(j)-\frac{c_{\mathrm{H}}r^{2}}{8}\Big{\\}}\Big{|}\geq(1-\tau)n.$
###### Proof.
Recall that $c_{\mathrm{H}}\geq 1$ by its definition. The claim follows
(without any modification) just as in the proof of Lemma 5.11. ∎
###### Proof of Lemma 5.12.
Let us prove (5.13). Note that Lemma 5.1 (without any modifications in the
proof) yields the following: with probability at least $1-2\exp(-c\tau^{2}n)$,
for all $x\in\mathcal{B}_{r}^{\ast}$, we have that
$M_{x,x^{\ast}}^{\prime}(j)\geq-\frac{c_{\mathrm{H}}r^{2}}{4}\text{ on more
than }(1-\tau)n\text{ blocks }j.$
In particular, this holds for
$\mathcal{U}_{r}^{\ast}\subseteq\mathcal{B}_{r}^{\ast}$.
As for (5.14), a combination of Lemma 5.14 and Lemma 5.16 shows that with
probability at least $1-2\exp(-c\tau^{2}n)$, for every
$x\in\mathcal{U}_{r}^{\ast}$, we have that
$M_{x,x^{\ast}}^{\prime}(j)>\frac{c_{\mathrm{H}}r^{2}}{4}\quad\text{on more
than }(1-2\tau)n\text{ blocks }j.$
This completes the proof. ∎
### 5.4. Proof under a deterministic lower bound of the Hessian
To conclude this section, let us prove Theorem 2.13. There are only very few
modifications needed in the proof of our main results, as we explain in what
follows.
In the proof of Theorem 2.9, the only place where the requirement
$N\geq c_{2}\max\\{d\log(2d),N_{\mathrm{H},\mathcal{E}}\\}$
was used, was for the statement of Lemma 5.1 pertaining to the quadratic term,
namely (5.1). However, Assumption 2.12 clearly implies that, with probability
1, for every $x\in\mathcal{S}_{r}^{\ast}$, we have that
$\Big{|}\Big{\\{}j:\frac{1}{2}Q_{x,x^{\ast}}(j)\geq\frac{r^{2}\varepsilon}{2}\Big{\\}}\Big{|}=n.$
Thus, the only modification that is needed, is to prove the part of Lemma 5.1
pertaining to the multiplier term (namely (5.2)) with $\frac{\varepsilon
r^{2}}{4}$ instead of $\frac{r^{2}}{16}$. Inspecting the proof, one readily
sees that this is possible once the constants $\theta,\tau,c,c_{1},\dots$ are
allowed to depend on $\varepsilon$.
## 6\. Proofs for the portfolio optimization problem
### 6.1. The proof of Corollary 3.7
The proof builds on several lemmas stated below, making the heuristic
computations explained in the introduction rigorous. Throughout, we work under
the assumptions made in Section 3.4. Note that
$\displaystyle\nabla F(x,\xi)$ $\displaystyle=-\ell^{\prime}(V_{x})\cdot X,$
$\displaystyle\nabla^{2}F(x,\xi)$
$\displaystyle=\ell^{\prime\prime}(V_{x})\cdot X\otimes X$
where we recall $V_{x}=-Y-\langle X,x\rangle$ for $x\in\mathcal{X}$. Finally,
denote by
$\|\cdot\|_{X}:=\mathbf{E}[\langle
X,\cdot\rangle^{2}]^{\frac{1}{2}}=\langle\mathrm{\mathbf{C}ov}[X]\cdot,\cdot\rangle^{\frac{1}{2}}=\|\mathrm{\mathbf{C}ov}[X]^{\frac{1}{2}}\cdot\|_{2}$
the norm endowed by $X$. Note that $\|\cdot\|_{X}$ is indeed a norm by the no-
arbitrage condition.
###### Lemma 6.1.
There is a constant $c_{1}>0$ depending on $L_{X},v_{1}$ and a constant
$c_{2}>0$ depending on $L_{X},v_{1},v_{2}$ such that
$c_{1}\|\cdot\|_{X}\leq\|\cdot\|\leq c_{2}\|\cdot\|_{X}.$
In particular, $\|\cdot\|$ is a true norm.
###### Proof.
We start with the second inequality. Hölder’s inequality (with exponent
$\frac{3}{2}$ and conjugate exponent $3$) implies that
$\displaystyle\|z\|^{2}$
$\displaystyle=\mathbf{E}[\ell^{\prime\prime}(V_{x^{\ast}})\langle
X,z\rangle^{2}]$
$\displaystyle\leq\mathbf{E}[\ell^{\prime\prime}(V_{x^{\ast}})^{\frac{3}{2}}]^{\frac{2}{3}}\mathbf{E}[\langle
X,z\rangle^{6}]^{\frac{1}{3}}$
for every $z\in\mathbb{R}^{d}$. The first term is bounded by $v_{2}$ and norm
equivalence of $X$ (see (3.3)) implies that the second term is bounded by
$L_{X}^{2}\|z\|_{X}^{2}$.
We continue with the first inequality. The Paley-Zygmund inequality together
with norm equivalence of $X$ implies that
(6.1) $\displaystyle\mathbf{P}\Big{[}\langle
X,z\rangle^{2}\geq\frac{1}{2}\|z\|_{X}^{2}\Big{]}$
$\displaystyle\geq\frac{(1-\frac{1}{2})^{2}\mathbf{E}[\langle
X,z\rangle^{2}]^{2}}{\mathbf{E}[\langle
X,z\rangle^{4}]}\geq\frac{1}{4L^{4}_{X}}$
for every $z\in\mathbb{R}^{d}\setminus\\{0\\}$. Moreover, setting
$\varepsilon:=\min\\{\ell^{\prime\prime}(u):|u|\leq
8L_{X}^{4}\mathbf{E}[|V_{x^{\ast}}|]\\},$
an application of Markov’s inequality shows that
$\mathbf{P}[\ell^{\prime\prime}(V_{x^{\ast}})<\varepsilon]\leq\frac{1}{8L_{X}^{4}}.$
In combination with (6.1), we conclude that
$\displaystyle\|z\|^{2}$
$\displaystyle\equiv\mathbf{E}[\ell^{\prime\prime}(V_{x^{\ast}})\langle
X,z\rangle^{2}]\geq\frac{\varepsilon\|z\|_{X}^{2}}{16L_{X}^{4}}.$
Finally, as noted previously, the no-arbitrage condition (3.4) immediately
implies that $\|\cdot\|_{X}$ is a true norm. Hence, by the above, $\|\cdot\|$
is a true norm as well. This completes the proof. ∎
###### Lemma 6.2.
Assumption 2.5 is satisfied with a constant $L$ depending on
$L_{X},v_{1},v_{2}$.
###### Proof.
An application of Hölder’s inequality (with exponents $3,\frac{3}{2}$) implies
that
$\displaystyle\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle^{2}]$
$\displaystyle=\mathbf{E}[\ell^{\prime\prime}(V_{x^{\ast}})^{2}\langle
X,z\rangle^{4}]$
$\displaystyle\leq\mathbf{E}[\ell^{\prime\prime}(V_{x^{\ast}})^{6}]^{\frac{1}{3}}\mathbf{E}[\langle
X,z\rangle^{6}]^{\frac{2}{3}}$
for every $z\in\mathbb{R}^{d}$. The first term is $v_{2}^{2}$ by definition
and, by norm equivalence of $X$ (see (3.3)) and Lemma 6.1, the second term is
bounded uniformly in $\\{z\in\mathbb{R}^{d}:\|z\|\leq 1\\}$ by a constant
depending on $L_{X},v_{1}$. ∎
###### Lemma 6.3.
Let $L$ be the parameter from Assumption 2.5. Then
$\displaystyle\sigma^{2}\leq\sqrt{L}\cdot\bar{\sigma}^{2}\quad\text{and}\quad
N_{\mathrm{G}}(r)\leq\sqrt{L}\cdot\frac{\bar{\sigma}^{2}\cdot d}{r^{2}}.$
###### Proof.
We make the preliminary claim that
$\mathrm{\mathbf{C}ov}[\nabla
F(x^{\ast},\xi)]\preceq\bar{\sigma}^{2}\sqrt{L}\nabla^{2}f(x^{\ast}).$
Indeed, for every $z\in\mathbb{R}^{d}$,
$\displaystyle\langle\mathrm{\mathbf{C}ov}[\nabla F(x^{\ast},\xi)]z,z\rangle$
$\displaystyle\leq\mathbf{E}[\ell^{\prime}(V_{x^{\ast}})^{2}\langle
X,z\rangle^{2}]$
$\displaystyle=\mathbf{E}\Big{[}\frac{\ell^{\prime}(V_{x^{\ast}})^{2}}{\ell^{\prime\prime}(V_{x^{\ast}})}\cdot\ell^{\prime\prime}(V_{x^{\ast}})\langle
X,z\rangle^{2}\Big{]}$
$\displaystyle\leq\bar{\sigma}^{2}\mathbf{E}[(\ell^{\prime\prime}(V_{x^{\ast}})\langle
X,z\rangle^{2})^{2}]^{\frac{1}{2}},$
where the last step follows from Hölder’s inequality. Moreover, by Assumption
2.5,
$\displaystyle\mathbf{E}[(\ell^{\prime\prime}(V_{x^{\ast}})\langle
X,z\rangle^{2})^{2}]^{\frac{1}{2}}$
$\displaystyle=\mathbf{E}[\langle\nabla^{2}F(x^{\ast},\xi)z,z\rangle^{2}]^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{L}\|z\|^{2}$
$\displaystyle=\sqrt{L}\langle\nabla^{2}f(x^{\ast})z,z\rangle$
hence the preliminary claim follows.
As both the largest singular value and the trace are monotone w.r.t. the
positive semidefinite order, the statement of the lemma follows. ∎
We need the following simple auxiliary lemma. Recall that $\|\cdot\|_{\ast}$
denotes the dual norm of $\|\cdot\|$.
###### Lemma 6.4.
There is a constant $c$ depending on $L_{X},v_{1}$ such that
$\mathbf{E}[\|X\|_{\ast}^{6}]\leq cd^{3}$.
###### Proof.
By Lemma 6.1 we have $\|\cdot\|_{X}\leq c_{1}\|\cdot\|$ for a constant $c_{1}$
depending on $L_{X},v_{1}$. This immediately implies that
$\|\cdot\|_{\ast}\leq\frac{1}{c_{1}}\|\cdot\|_{X,\ast}$ where
$\|\cdot\|_{X,\ast}$ is the dual norm of $\|\cdot\|_{X}$. As the norm
$\|\cdot\|_{X}$ is endowed by $\mathrm{\mathbf{C}ov}[X]$, its dual norm
$\|\cdot\|_{X,\ast}$ is endowed by $\mathrm{\mathbf{C}ov}[X]^{-1}$, whence
$\|X\|_{X,\ast}=\|Y\|_{2}\quad\text{for
}Y:=\mathrm{\mathbf{C}ov}[X]^{-\frac{1}{2}}X.$
Applying Hölder’s inequality (in its version for three random variables, with
exponents $3,3,3$) shows that
$\mathbf{E}[\|Y\|_{2}^{6}]=\sum_{i,j,k=1}^{d}\mathbf{E}[Y_{i}^{2}Y_{j}^{2}Y_{k}^{2}]\leq\sum_{i,j,k=1}^{d}\mathbf{E}[Y_{i}^{6}]^{\frac{1}{3}}\mathbf{E}[Y_{j}^{6}]^{\frac{1}{3}}\mathbf{E}[Y_{k}^{6}]^{\frac{1}{3}}.$
It remains to note that $Y$ satisfies the same norm equivalence as $X$ does,
and therefore, denoting by $e_{i}$ the $i$-th standard Euclidean unit vector,
$\displaystyle\mathbf{E}[Y_{i}^{6}]^{\frac{1}{3}}$
$\displaystyle=\mathbf{E}[\langle Y,e_{i}\rangle^{6}]^{\frac{1}{3}}$
$\displaystyle\leq L_{X}^{2}\mathbf{E}[\langle Y,e_{i}\rangle^{2}]$
$\displaystyle=L_{X}^{2}\mathrm{\mathbf{C}ov}[Y]_{ii}=L_{X}^{2}.$
Combining everything completes the proof. ∎
###### Lemma 6.5.
There is a constant $c>0$ depending on $L_{X},v_{1}$ such that, for every
$x,y\in\mathcal{B}_{1}^{\ast}$, we have that
(6.2)
$\displaystyle\mathbf{P}\Big{[}\mathcal{E}_{\mathrm{H}}(x)\leq\frac{\|x-x^{\ast}\|^{2}}{8}\Big{]}$
$\displaystyle\leq cv_{\mathcal{E}_{\mathrm{H}}}\cdot\|x-x^{\ast}\|,$ (6.3)
$\displaystyle\Big{\|}\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi)\Big{\|}_{\mathrm{op}}$
$\displaystyle\leq\|x-y\|\cdot K(\xi)$
where $K(\xi)$ satisfies $\mathbf{E}[K(\xi)]\leq cv_{K}d^{\frac{3}{2}}$.
###### Proof.
As a preliminary observation, note that a Taylor expansion gives
(6.4)
$\displaystyle\begin{split}&\langle(\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi))z,z\rangle\\\
&=\ell^{\prime\prime\prime}(V_{x+t(y-x)})\langle X,x-y\rangle\langle
X,z\rangle^{2}\end{split}$
for every $z\in\mathbb{R}^{d}$, where $t\in[0,1]$ is some number depending on
$x,y,\xi$.
We start by proving (6.2). For every $x\in\mathcal{B}_{1}^{\ast}$, by (6.4)
and definition of $\mathcal{E}_{\mathrm{H}}$, we have that
$\displaystyle\mathcal{E}_{\mathrm{H}}(x)$
$\displaystyle\equiv\sup_{t\in[0,1]}\big{|}\big{\langle}\big{(}\nabla^{2}F(x^{\ast}+t(x-x^{\ast}),\xi)-\nabla^{2}F(x^{\ast},\xi)\big{)}(x-x^{\ast}),x-x^{\ast}\big{\rangle}\big{|}$
$\displaystyle\leq\sup_{t\in[0,1]}|\ell^{\prime\prime\prime}(V_{x^{\ast}+t(x-x^{\ast})})||\langle
X,x-x^{\ast}\rangle|^{3}.$
In particular, applying Hölder’s inequality, the norm equivalence of $X$ from
(3.3), and Lemma 6.1, we obtain
$\displaystyle\mathbf{E}[\mathcal{E}_{\mathrm{H}}(x)]$ $\displaystyle\leq
v_{\mathcal{E}_{\mathrm{H}}}\mathbf{E}[\langle
X,x-x^{\ast}\rangle^{6}]^{\frac{1}{2}}$ $\displaystyle\leq
v_{\mathcal{E}_{\mathrm{H}}}L_{X}^{3}\|x-x^{\ast}\|_{X}^{3}$
$\displaystyle\leq c_{1}v_{\mathcal{E}_{\mathrm{H}}}\|x-x^{\ast}\|^{3}$
for a constant $c_{1}$ depending on $L_{X},v_{1}$. The claim (6.2) therefore
follows from Markov’s inequality.
We now prove (6.3). The definition of the operator norm together with the
inequality $|\langle X,x-y\rangle|\leq\|X\|_{\ast}\|x-y\|$ (which holds by
definition of the dual norm $\|\cdot\|_{\ast}$) shows that (6.4) implies
$\displaystyle\|\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi)\|_{\mathrm{op}}$
$\displaystyle\leq\sup_{\tilde{x},\tilde{y}\in\mathcal{B}_{1}^{\ast}\text{ and
}t\in[0,1]}|\ell^{\prime\prime\prime}(V_{\tilde{x}+t(\tilde{y}-\tilde{x})})|\cdot\|x-y\|\cdot\|X\|_{\ast}^{3}$
$\displaystyle=:K(\xi)\cdot\|x-y\|.$
This proves (6.3).
It remains to control the expectation of $K(\xi)$. For every
$\tilde{x},\tilde{y}\in\mathcal{B}_{1}^{\ast}$ and $t\in[0,1]$ we have
$\tilde{x}+t(\tilde{y}-\tilde{x})\in\mathcal{B}_{1}^{\ast}$ by convexity of
$\mathcal{X}$, whence
$K(\xi)\leq\sup_{x\in\mathcal{B}_{1}^{\ast}}|\ell^{\prime\prime\prime}(V_{x})|\|X\|_{\ast}^{3}$.
Therefore Hölder’s inequality and the definition of $v_{K}$ imply that
$\displaystyle\mathbf{E}[K(\xi)]$ $\displaystyle\leq
v_{K}\mathbf{E}[\|X\|_{\ast}^{6}]^{\frac{1}{2}}.$
By Lemma 6.4, the last term is bounded by $c_{2}d^{\frac{3}{2}}$ for a
constant $c_{2}$ depending on $L_{X},v_{1}$. This completes the proof. ∎
###### Lemma 6.6.
There is a constant $c$ depending on $L_{X},v_{1}$ such that
$c_{\mathrm{H}}\leq cv_{\mathcal{E}_{\mathrm{H}}}$.
###### Proof.
We need to show that $\|\mathbf{E}[\nabla^{2}F(x,\xi)]\|_{\mathrm{op}}\leq
cv_{\mathcal{E}_{\mathrm{H}}}$ for every $x\in\mathcal{B}_{1}^{\ast}$. Fix
such $x$. From the Taylor expansion (6.4) we get
$\nabla^{2}F(x,\xi)=\nabla^{2}F(x^{\ast},\xi)+\ell^{\prime\prime\prime}(V_{x^{\ast}+t(x-x^{\ast})})\langle
X,x-x^{\ast}\rangle X\otimes X$
for some $t\in[0,1]$ which depends on $x$ and $\xi$.
The expectation of the first term equals $\nabla^{2}f(x^{\ast})$, and the
operator norm of the latter equals 1. To estimate the operator norm of the
expectation of the second term, let $z\in\mathbb{R}^{d}$ with $\|z\|\leq 1$.
Then Hölder’s inequality (in its version for three random variables, with
exponents $2,6,3$) implies
$\displaystyle\mathbf{E}[\ell^{\prime\prime\prime}(V_{x^{\ast}+t(x-x^{\ast})})\langle
X,x-x^{\ast}\rangle\langle X,z\rangle^{2}]$ $\displaystyle\leq
v_{\mathcal{E}_{\mathrm{H}}}\cdot\mathbf{E}[\langle
X,x-x^{\ast}\rangle^{6}]^{\frac{1}{6}}\cdot\mathbf{E}[\langle
X,z\rangle^{6}]^{\frac{1}{3}}$ $\displaystyle\leq
v_{\mathcal{E}_{\mathrm{H}}}\cdot L_{X}\|x-x^{\ast}\|_{X}\cdot
L_{X}^{2}\|z\|_{X}^{2},$
where the last inequality follows from norm equivalence of $X$ from (3.3).
Finally, recalling that $\|x-x^{\ast}\|\leq 1$ and $\|z\|\leq 1$, the proof is
completed by an application of Lemma 6.1 which states that $\|\cdot\|_{X}\leq
c_{1}\|\cdot\|$ for a constant depending on $L_{X},v_{1}$. ∎
We are now ready for the
###### Proof of Corollary 3.7.
Regarding Assumption 2.3: convexity and differentiability are clearly
satisfied, and integrability holds by assumption. Moreover, by Lemma 6.1,
$\|\cdot\|$ is a true norm.
Assumption 2.5 follows from Lemma 6.2.
Lemma 6.5 shows that Assumption 2.7 is satisfied for $\alpha=1$ and
$r_{0}:=\min\Big{\\{}1,\frac{c_{0}}{v_{\mathcal{E}_{\mathrm{H}}}}\Big{\\}},$
where $c_{0}$ is a constant depending on the constant $c$ of Lemma 6.5 and the
parameter $L$ of Assumption 2.5; hence $c_{0}$ depends on $L_{X},v_{1},v_{2}$.
Moreover, Lemma 6.5 also gives $N_{\mathcal{E},\mathrm{H}}\leq
c_{1}d\log(dv_{K}+2)$ for a constant $c_{1}$ depending on $L_{X},v_{1}$.
The parameters $N_{\mathrm{G}}(r)$, $\sigma^{2}$, and $c_{\mathrm{H}}$ are
bounded in Lemma 6.3 and Lemma 6.6, respectively. Combing everything completes
the proof. ∎
### 6.2. The proof of Corollary 2.10
First note that the $\bar{\sigma}$ of the Corollary 2.10 and the
$\bar{\sigma}$ of Corollary 3.7 coincide. Moreover, as $X$ is Gaussian with
non-degenerate covariance matrix, the no-arbitrage condition (3.4) readily
follows. Further recall the well-known Gaussian norm equivalence
$\mathbf{E}[\langle X,z\rangle^{p}]^{\frac{1}{p}}\leq
C\sqrt{p}\cdot\mathbf{E}[\langle X,z\rangle^{2}]^{\frac{1}{2}}$
for every $z\in\mathbb{R}^{d}$ and every $p\geq 2$, where $C$ is an absolute
constant. In particular (3.3) holds and our assumption that $U(2Y)$ is
integrable implies part (c) of Assumption 2.3. Along the same lines as
Gaussian norm equivalence, the following holds.
###### Lemma 6.7.
There is a constant $c$ depending on $L$ and
$v_{1}\equiv\mathbf{E}[|V_{x^{\ast}}|]$ such that
$\displaystyle\mathbf{E}[\|X\|_{\ast}^{12}]$ $\displaystyle\leq cd^{6},$
$\displaystyle\mathbf{E}[\exp(4\|X\|_{\ast})]$
$\displaystyle\leq\exp(c\sqrt{d}),$ $\displaystyle\mathbf{E}[\langle
X,x-x^{\ast}\rangle^{12}]$ $\displaystyle\leq c\|x-x^{\ast}\|^{12},$
$\displaystyle\mathbf{E}[\exp(4|\langle X,x-x^{\ast}\rangle|)]$
$\displaystyle\leq c$
for every $x\in\mathcal{B}_{1}^{\ast}$.
###### Proof.
The two statements involving $\|X\|_{\ast}$ follow from similar arguments as
given in Lemma 6.4, noting that $\mathrm{\mathbf{C}ov}[X]^{-\frac{1}{2}}X$ is
standard Gaussian. The two statements involving $\langle X,x-x^{\ast}\rangle$
follow from Gaussian norm equivalence and the bound $\|\cdot\|_{X}\leq
c\|\cdot\|$ from Lemma 6.1 for a constant $c$ depending on $L,v_{1}$. ∎
###### Proof of Corollary 2.10.
The only modifications needed pertain to Lemma 6.5 and Lemma 6.6, where terms
can be simplified due to special features of the exponential function.
We start with Lemma 6.5. As $\ell^{\prime\prime\prime}=\exp$ is increasing and
as $\exp(a+b)=\exp(a)\exp(b)$ for $a,b\in\mathbb{R}$, we may use Remark 3.6 to
get
$\displaystyle\mathcal{E}_{\mathrm{H}}(x)$
$\displaystyle\leq\exp(V_{x^{\ast}})\cdot\exp(|\langle
X,x-x^{\ast}\rangle|)\cdot|\langle X,x-x^{\ast}\rangle|^{3},$
$\displaystyle\|\nabla^{2}F(x,\xi)-\nabla^{2}F(y,\xi)\|_{\mathrm{op}}$
$\displaystyle\leq\exp(V_{x^{\ast}})\cdot\exp(\|X\|_{\ast})\cdot\|X\|_{\ast}^{3}\|x-y\|$
$\displaystyle=:K(\xi)\|x-y\|.$
It remains to bound the expectation all terms. To that end, applying Hölder’s
inequality (in its version for three random variable, with exponents $2,4,4$)
and Lemma 6.7 gives
(6.5)
$\displaystyle\begin{split}\mathbf{E}[\mathcal{E}_{\mathrm{H}}(x)]&\leq\mathbf{E}[\exp(2V_{x^{\ast}})]^{\frac{1}{2}}\mathbf{E}[\exp(4|\langle
X,x-x^{\ast}\rangle|)]^{\frac{1}{4}}\mathbf{E}[\langle
X,x-x^{\ast}\rangle^{12}]^{\frac{1}{4}}\\\
&\leq\bar{\sigma}^{2}c_{1}\|x-x^{\ast}\|^{3},\end{split}$ (6.6)
$\displaystyle\begin{split}\mathbf{E}[K(\xi)]&\leq\mathbf{E}[\exp(2V_{x^{\ast}})]^{\frac{1}{2}}\mathbf{E}[\exp(4\|X\|_{\ast})]^{\frac{1}{4}}\mathbf{E}[\|X\|_{\ast}^{12}]^{\frac{1}{4}}\\\
&\leq\bar{\sigma}^{2}\exp(c_{1}\sqrt{d}),\end{split}$
for a constant $c_{1}$ depending on $L,v_{1}$.
In particular, (6.5) shows that Assumption 2.7 is satisfied for
$r_{0}:=\min\Big{\\{}1,\frac{c_{2}}{\bar{\sigma}^{2}}\Big{\\}}$
where $c_{2}$ is a constant depending on $L,v_{1}$. In combination with (6.6),
this implies that
$\displaystyle N_{\mathcal{E},\mathrm{H}}$ $\displaystyle\leq
d\log(r_{0}\bar{\sigma}^{2}\exp(c_{1}\sqrt{d}))$ $\displaystyle\leq
d\log(c_{2}\exp(c_{1}\sqrt{d}))\leq c_{3}d^{\frac{3}{2}}$
for a constant $c_{3}$ depending on $L,v_{1}$.
Similar (but simpler) arguments show that $c_{\mathrm{H}}$ can be bounded in
terms of $L,v_{1}$. This completes the proof. ∎
## 7\. Concluding remarks
###### Remark 7.1 (Dependence of the procedure on parameters).
All the parameters (e.g., $\sigma^{2}$, $\|\cdot\|$, $L$, $N_{\mathrm{G}}$
etc. ) that are required in the formulation of Theorem 2.9 (the only
parameters needed in the definition of the procedure $\widehat{x}_{N}^{\ast}$
are $c_{\mathrm{H}}$, $L$, $\sigma$ and $r$) depend on the unknown optimal
action $x^{\ast}$.
While an a priori knowledge of the parameters seems unrealistic, there are
various ways around this problem. It should be stressed that any type of
estimate on these parameters suffices to ensure the procedure performs well.
For example, finding some $\hat{\sigma}^{\prime}$ such that
$\frac{1}{2}\hat{\sigma}^{\prime}\leq\sigma\leq 2\hat{\sigma}^{\prime}$ is
enough for our purposes, and estimating $\sigma$ within a constant
multiplicative factor is a considerably simpler task than the ones we have to
deal with in the analysis of the procedure $\widehat{x}_{N}^{\ast}$.
Alternatively, one can replace the parameters with the (local) worst-case
scenario; for example, instead of $\sigma^{2}$, to consider
$\bar{\sigma}^{2}:=\sup_{x\text{ close to }x^{\ast}}\sigma^{2}(x)$
where $\sigma^{2}(x)$ is defined just as $\sigma^{2}$ but with gradient and
Hessian evaluated at $x$ rather than at $x^{\ast}$. Finally, it is much
simpler to test whether a solution is a good one than producing a candidate.
Therefore, one can increase the sample size and test the candidates that are
produced. Once the sample size passes the critical threshold from Theorem 2.9,
a good candidate will be identified.
All of these are standard methods and there are plenty of other alternatives
to tackle such issues. We shall not pursue these aspects further in this
article.
###### Remark 7.2 (On the integrability of the Hessian).
In the course of the proof of Theorem 2.9, the only place Assumption 2.5 was
used was in Lemma 5.3, and there it was used twice: firstly, by Remark 4.3,
Assumption 2.5 guarantees the existence of three constants
$s_{0},s_{1},s_{2}>0$ depending only on $L$, such that, for every $m\geq 1$,
the random matrix
(7.3) $\displaystyle\nabla^{2}F(x^{\ast},\xi)\begin{array}[]{l}\text{satisfies
a stable lower bound with}\\\ \text{parameters
}(m,s_{0},2s_{1}m,s_{2}m).\end{array}$
(In fact, one can choose $s_{0}=\frac{1}{4}$ as we did for notational
purposes). On the other hand, by Lemma 4.6 and Lemma 4.7, Assumption 2.5
implies that the minimal sample size from Theorem 4.4,
$N_{\mathrm{H}}:=\max\Big{\\{}\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}},\log\Big{(}\log(3d)\frac{\mathbf{E}[\|A\mathbb{A}^{-1}A\|_{\mathrm{op}}]}{\|\mathbf{E}[A\mathbb{A}^{-1}A]\|_{\mathrm{op}}}\Big{)}\Big{\\}}$
(where $A:=\nabla^{2}F(x^{\ast},\xi)$ and $\mathbb{A}:=\nabla^{2}f(x^{\ast})$)
can be bounded by $c\cdot d\log(2d)$ for a constant $c$ depending only on $L$.
In particular, we see that Theorem 2.9 remains valid if Assumption 2.5 is
replaced by assumption (7.3) together with the requirement that the sample
size exceeds $c_{2}N_{\mathrm{H}}$ (in that case the constants
$c_{1},c_{2},c_{3}$ appearing in Theorem 2.9 depend on $s_{0},s_{1},s_{2}$).
Acknowledgements: Daniel Bartl is grateful for financial support through the
Vienna Science and Technology Fund (WWTF) project MA16-021 and the Austrian
Science Fund (FWF) project P28661.
Part of this work was conducted while Shahar Mendelson was visiting the
Faculty of Mathematics, University of Vienna, and the Erwin Schrödinger
Institute, Vienna. He would also like to thank Jungo Connectivity for its
support.
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|
# On the derivation of the Khmaladze transforms
Leigh A Roberts School of Economics and Finance,
Victoria University of Wellington,
Wellington, New Zealand
###### Abstract
Some 40 years ago Khmaladze introduced a transform which greatly facilitated
the distribution free goodness of fit testing of statistical hypotheses. In
the last decade, he has published a related transform, broadly offering an
alternative means to the same end.
The aim of this paper is to derive these transforms using relatively
elementary means, making some simplifications, but losing little in the way of
generality. In this way it is hoped to make these transforms more accessible
and more widely used in statistical practice. We also propose a change of name
of the second transform to the Khmaladze rotation, in order to better reflect
its nature.
###### keywords:
Khmaladze transform, distribution free , goodness of fit, linear algebraic
derivation, projection, reflection
††journal: ArXiv
###### Contents
1. 1 Introduction
2. 2 The second Khmaladze transform, or the Khmaladze rotation
1. 2.1 The statistical setup
2. 2.2 Rotating from one $\chi^{2}$ test to another
1. 2.2.1 Projection of chi square statistics
2. 2.2.2 From a projection operator to a reflection operator
3. 2.3 From the chi-squared statistic to the empirical process
1. 2.3.1 Brownian Motion in time $P(x)$
2. 2.3.2 The Brownian Bridge in time $P(x)$
3. 2.3.3 Projection from BM to BB in time $P(x)$
4. 2.4 From point parametric to function parametric form for the stochastic processes
1. 2.4.1 Projection operators in the primal and dual spaces
2. 2.4.2 Covariance of $v^{q}_{P}(\phi)$ and $v^{q}_{P}(\widetilde{\phi})$ for $q=q_{0}$
5. 2.5 Rotation/reflection operator in the functional (dual) space
1. 2.5.1 Rotation from one stochastic process to another when parameters are known
6. 2.6 Estimating parameters
1. 2.6.1 The score function
2. 2.6.2 Reassembling the jigsaw
3. 2.6.3 Covariance of $v^{q}_{P}(\phi)$ and $v^{q}_{P}(\widetilde{\phi})$ for general $q$
4. 2.6.4 For consistency with chi squared when df = $n-K-1$
7. 2.7 Rotation operators in the general case
1. 2.7.1 The rotation operator as a succession of reflections
2. 2.7.2 Rotation from one empirical process to another
8. 2.8 Higher dimensional Khmadadze rotations, and the colour-blind problem
1. 2.8.1 The Khmadadze rotation in two dimensions
2. 2.8.2 The colour-blind problem
3. 3 The Khmadadze transform
1. 3.1 The linear regression model
2. 3.2 Choice of regressand and regressors
3. 3.3 KT1 through linear regression
## 1 Introduction
The generic goodness of fit problem lies in separating possible outcomes of a
statistical experiment into a finite number of cells, noting expected and
observed frequencies for each cell, and testing for smallness of differences
between them. The classic test for this is the chi squared test, which is
distribution free, in the sense that the distribution of the chi squared
statistic does not depend on the distribution generating the data, provided
only that the hypothetical distribution be fully specified.
The vital contribution made by Khmaladze some 40 years ago was to allow
distribution free goodness of fit tests for compound hypotheses [Khmaladze,
1979, 1981]. Using the Khmaladze Transform (hereafter KT, or occasionally the
‘first’ transform, or ‘KT1’), one could test for whether the data could
plausibly arise from a given distributional family. This more or less
corresponds to what is done in practice: only rarely would a statistician wish
to test for a specific parametric value within the distributional family of
interest, which was the only possibility available before the introduction of
the KT.
Papers utilising the KT are cited in Li [2009], Koul & Swordson [2011] and Kim
[2016], i.a. It is however clear that the uptake by the statistical community
of such an important conceptual advance in goodness of fit testing has been
slow.
Within the last decade, Khmaladze has published another, and ostensibly
simpler, transform to test goodness of fit in a distribution free manner
[Khmaladze, 2013a, 2016]. The new transform has tentatively been labelled as
the second Khmaladze transform, generally referred to below as the second
transform or ‘KT2’. We suggest that this second transform be relabelled as the
‘Khmaladze rotation’.
Adoption of the second transform also appears to be slow. Of the papers
Dumitrescu & Khmaladze [2019], Kennedy [2018], Khmaladze [2017], Nguyen
[2017a, b] and Roberts [2019], only Kennedy applies the KT2 to real data. He
is also the only one of these authors to suggest a wider use of the second KT
to choose an optimal model amongst competing models.
Both transforms involve projections of the empirical process. The first
transform projects onto a Brownian motion (BM), obtained essentially by
regressing the incremental empirical distribution function (EDF) on the
‘future’; what this cryptic description really means is that the goodness of
fit test assumes that all the data is to hand, viz. that the statistical
experiment is complete, when the goodness of fit test is carried out. The
second transform utilises the same types of projections to change from one
empirical process to another, without losing any statistical information. The
test of goodness of fit can then be carried out in one statistical framework
or the other, whichever is more convenient.
The purpose of this paper is not to apply these transforms to data analysis,
but to spread the word about these elegant and potentially very useful
transforms, by explaining their genesis more simply and intuitively. The idea
is to discretise the empirical processes being tested, so that operators and
operands in functional space become matrices and vectors. We then develop the
ideas underlying the transforms in $N-$dimensional space, hoping that the
straightforward linear algebraic approach will make the underlying reasoning
transparent. The loss of generality is in fact slight, and such discretisation
of the underlying spaces, operators and operands is no more than one would
impose for computing purposes.
Our starting point is to decompose the chi squared statistic. Mimicking the
approach taken in Khmaladze [2013a], we project a standard normal vector onto
a vector having the covariance structure of the chi squared components; then
we consider rotation or reflection from one chi squared statistic to another.
This already gives the two elements underlying the second transform, viz.
projection and rotation/reflection. The idea of projection was not new: the
fitting of linear regression underpinning the first transform is a projection
of asymptotically normal regressors to model the increment of the empirical
process. It is the rotation, or more properly reflection, from one chi squared
statistic to another which defines the underlying rationale of the second
transform, and differentiates it from the first transform. Our basic task is
to find the projection and rotation operators of the second transform, moving
from one (discretised) empirical process to another.
Following the definitions of the projection and reflection operators applied
to the chi squared statistic, we then make the ostensibly slight adjustment to
operate on the empirical process, essentially the numerator of the chi squared
statistic. Starting with Brownian Motion (BM) in a discrete time $P$, we
recast operators as matrices and functional operands as vectors, possibly
semi-infinite in length. We project the BM to a Brownian bridge (BB) in time
$P$, and rotate from one BB to another, in time $R$ say. Allowing for
parameters to be estimated is allowed for by further projections, most easily
seen by changing from point parametrisation to functional parametrisation of
the empirical process.
The resulting $q-$projected BMs may be rotated from one empirical process to
another, say from time $P$ to time $R$. We are still working with discrete
distributions; but within that limitation, we have proved the validity of the
KT2, whereby a goodness of fit test may be effected either in the $P$ space or
the $R$ space as convenient, and no statistical information is lost in
rotation from one to the other.
A further short section extends the second transform to higher dimensional
distributions $P$, and illustrates the outworking for the colour blind
problem, for which we largely follow the first part of Dumitrescu & Khmaladze
[2019].
Khmaladze has largely adopted this discretised approach to elucidate and prove
the original transform in the framework of a simple mortality investigation
[Khmaladze, 2013b, ch. 7]. In the final section of this paper we flesh out his
development and provide additional comments.
## 2 The second Khmaladze transform, or the Khmaladze rotation
### 2.1 The statistical setup
Given a random variable X with distribution function $F(x)$, it is not
essential but notationally convenient to suppose that the support of $X$ is
bounded away from minus infinity: suppose there exists a finite number $M_{F}$
such that $X>M_{F}$ with probability one. Then we define a grid of points
$M_{F}=x_{1}<x_{2}<\ldots<x_{N}<x_{N+1}=\infty$, and let the $j$th cell be
defined by $X\in[x_{j},x_{j+1})$ for $1\leq j\leq N$, with associated
probabilities $\int_{[x_{j},x_{j+1})}dF(x)=p_{j}$. Should there be an atom of
$X$ at the grid point $x_{j}$, then the saltus at $x_{j}$ will be included in
$p_{j}$ but not in $p_{j-1}$. Noting that $\sum_{j=1}^{N}p_{j}=1$, we further
define the approximating discrete distribution function $P(x)$, with atoms at
$\\{x_{j}\\}_{j=1}^{N}$.
In simpler terms, the non-decreasing step function $P:\mathbb{R}\to[0,1]$, is
piecewise constant apart from steps occurring at points
$x_{1},x_{2},\ldots,x_{N}$. The step or saltus at $x_{j}$ is to be $p_{j}>0$;
the function $P$ is to be continuous from the right; $P(x)=0$ when $x<x_{1}$;
and $P(x)=1$ when $x\geq x_{N}$. The point of supposing the existence of such
a lower bound as $M_{F}$ is merely for the convenience of not regarding
$-\infty$ as a possible value for the corresponding putative random variable:
it is not an essential step.
In like vein, suppose a random variable $Y$ with distribution function $G(y)$,
similarly bounded away from minus infinity, with a grid
$M_{G}=y_{1}<y_{2}<\ldots<y_{N}<y_{N+1}=\infty$, and let the $j$th cell be
defined by $Y\in[y_{j},y_{j+1})$ for $1\leq j\leq N$, with probabilities
$\int_{[y_{j},y_{j+1})}dG(y)=r_{j}$.
The aim of the second Khmaladze transform or Khmaladze rotation is to rotate
the stochastic process with distribution function $F$ to that with
distribution function $G$. Our simplified approach is to approximate $F$ and
$G$ by the discrete distributions $P$ and $R$ respectively, and rotate from
$P$ to $R$.
This latter is more easily visualised than the full rotation from $F$ to $G$,
because all operations may be effected by standard linear algebraic
procedures. Resulting vectors and matrices are approximations to the integral
operators and functional vectors occurring in the full rotation. The
outworkings below will be set out for finite dimension $N$, but would be
essentially unchanged if $N$ were countably infinite, with vectors of semi-
infinite length.
### 2.2 Rotating from one $\chi^{2}$ test to another
#### 2.2.1 Projection of chi square statistics
More or less following Khmaladze [2013a], we normalise the observed
frequencies and sum to obtain the conventional chi-squared statistic. For a
sample size $n$ and a total number $N$ of cells, define
$Y^{\prime}_{j}=\frac{\nu_{j}-np_{j}}{\sqrt{np_{j}}}\quad\mbox{for}\ 1\leq
j\leq N$ (1)
with $\nu_{j}$ the observed frequency in the $j$th cell, and $p_{j}$ the
probability of a data point falling in the $j$th cell.
Assuming that $p_{j}>0$ for all $j$, we set
$Y^{\prime}=\begin{pmatrix}Y^{\prime}_{1}&Y^{\prime}_{2}&\ldots&Y^{\prime}_{N}\end{pmatrix}^{T}$,
$p=\begin{pmatrix}p_{1}&p_{2}&\ldots&p_{N}\end{pmatrix}^{T}$ and
$\sqrt{p}=\begin{pmatrix}\sqrt{p}_{1}&\sqrt{p}_{2}&\ldots&\sqrt{p}_{N}\end{pmatrix}^{T}$.
Note that $\sqrt{p}^{T}\sqrt{p}=\sum_{j=1}^{n}p_{j}=1$. Summing the squares of
the statistics in (1) produces the conventional chi-squared statistic
$\chi^{2}=\sum_{j=1}^{n}\frac{(O_{j}-E_{j})^{2}}{E_{j}}=\sum_{j=1}^{n}{Y^{\prime}_{j}}^{2}={Y^{\prime}}^{T}Y^{\prime}$
(2)
Chibisov [1971], for example, discusses the distribution of the chi-squared
statistic when cell boundaries are determined in light of the data; but we
assume boundaries to be fixed independently of the data.
The frequencies $\nu_{j}$ have a multinomial distribution, with covariance
structure given by $\mbox{Var}(\nu_{j})=np_{j}(1-p_{j})$,
$\mbox{Cov}(\nu_{j},\nu_{k})=-np_{j}p_{k}$ for $j\neq k$. The covariance
matrix of $Y^{\prime}$ reduces to
$E\
Y^{\prime}{Y^{\prime}}^{T}=\begin{pmatrix}1-p_{1}&-\sqrt{p}_{1}\sqrt{p}_{2}&\ldots&-\sqrt{p}_{1}\sqrt{p}_{N}\\\
-\sqrt{p}_{2}\sqrt{p}_{1}&1-p_{2}&\ldots&-\sqrt{p}_{2}\sqrt{p}_{N}\\\
&&\ldots&\\\
-\sqrt{p}_{N}\sqrt{p}_{1}&-\sqrt{p}_{N}\sqrt{p}_{2}&\ldots&1-p_{N}\\\
\end{pmatrix}=I-\sqrt{p}\sqrt{p}^{T}$ (3)
$\sqrt{r}$$\sqrt{p}$$Z$$\pi_{\sqrt{r}}Z$$\pi_{\sqrt{p}}Z$ Figure 1:
Projections of $Z$ perpendicular to the unit vectors $\sqrt{p}$ and $\sqrt{r}$
Define $\pi_{a}(b)$ to be a projection from $b$ perpendicular to $a$, as
illustrated in Figure 1. First note that
$\pi_{\sqrt{p}}=I-\sqrt{p}\sqrt{p}^{T}\qquad\qquad\pi_{\sqrt{p}}^{T}=\pi_{\sqrt{p}}\qquad\qquad\pi_{\sqrt{p}}^{2}=\pi_{\sqrt{p}}$
(4)
Setting $Z\sim{\cal N}(0,I)$, define the Gaussian vector $Y$
$Y=Z-\sqrt{p}{\sqrt{p}}^{T}Z=\pi_{\sqrt{p}}Z\qquad\qquad\sqrt{p}^{T}\,Y=0\qquad\qquad
Y^{T}=Z^{T}\pi_{\sqrt{p}}$
It is well known that the chi-squared statistic in (2) has the
$\chi^{2}_{n-1}$ limiting distribution, with mean $n-1$. For completeness, and
for comparison with later work, we note that the asymptotic limiting statistic
has the same mean:
$E\ Y^{T}Y=E\ Z^{T}\pi_{\sqrt{p}}\pi_{\sqrt{p}}Z=E\ Z^{T}\pi_{\sqrt{p}}Z=E\
Z^{T}\left(I-\sqrt{p}\sqrt{p}^{T}\right)Z$ $=E\ Z^{T}Z-E\
Z^{T}\sqrt{p}\sqrt{p}^{T}Z=E\ Z^{T}Z-E\ \sqrt{p}^{T}ZZ^{T}\sqrt{p}=E\
Z^{T}Z-\sqrt{p}^{T}\sqrt{p}=n-1$ (5)
We are however more concerned with the covariance of $Y$:
$E\ YY^{T}=E\
\pi_{\sqrt{p}}ZZ^{T}\pi_{\sqrt{p}}=\pi_{\sqrt{p}}E(ZZ^{T})\pi_{\sqrt{p}}=\pi_{\sqrt{p}}^{2}=\pi_{\sqrt{p}}$
(6)
or
$E\ YY^{T}=\begin{pmatrix}1&0&\ldots&0\\\ 0&1&\ldots&0\\\ &&\ldots&\\\
0&0&\ldots&1\end{pmatrix}-\begin{pmatrix}\sqrt{p}_{1}\\\ \sqrt{p}_{2}\\\
\ldots\\\
\sqrt{p}_{N}\end{pmatrix}\begin{pmatrix}\sqrt{p}_{1}&\sqrt{p}_{2}&\ldots&\sqrt{p}_{N}\end{pmatrix}$
$=\begin{pmatrix}1-p_{1}&-\sqrt{p}_{1}\sqrt{p}_{2}&\ldots&-\sqrt{p}_{1}\sqrt{p}_{N}\\\
-\sqrt{p}_{2}\sqrt{p}_{1}&1-p_{2}&\ldots&-\sqrt{p}_{2}\sqrt{p}_{N}\\\
&&\ldots&\\\
-\sqrt{p}_{N}\sqrt{p}_{1}&-\sqrt{p}_{N}\sqrt{p}_{2}&\ldots&1-p_{N}\end{pmatrix}$
in agreement with the covariance of $Y^{\prime}$ in (3). As $n\to\infty$,
$Y^{\prime}$ tends weakly to $Y$, i.e., the (cumulative) distribution function
of $Y^{\prime}$ tends to that of $Y$, at all points of continuity of the
latter. In simpler terms, we may asymptotically approximate $Y^{\prime}$ by
the normally distributed $Y$. This is certainly so if the infimum of
$\\{p_{j}:1\leq j\leq N\\}$ is bounded away from zero.
#### 2.2.2 From a projection operator to a reflection operator
Given vectors of unit length $\sqrt{p}$ and $\sqrt{r}$, the reflection
operator $U_{\sqrt{p},\sqrt{r}}=U_{0}$ is defined as
$U_{\sqrt{p},\sqrt{r}}=U_{0}=I-c_{0}(\sqrt{p}-\sqrt{r})(\sqrt{p}-\sqrt{r})^{T}$
in which the constant $c_{0}$ is given by
$c_{0}=\frac{2}{\parallel\sqrt{p}-\sqrt{r}\parallel^{2}}=\frac{1}{1-<\sqrt{p},\sqrt{r}>}$
and in turn the norm is given by $\|s\|^{2}=\ <s,s>\ =s^{T}s$ and the inner
product as $<s,t>\ =s^{T}t$. The reflection operator swaps $\sqrt{p}$ and
$\sqrt{r}$ around, while leaving unchanged all vectors orthogonal to both of
them. We note that $U_{0}^{T}=U_{0}$, $U_{0}^{2}=I$ and
$U_{0}\pi_{\sqrt{p}}U_{0}=\pi_{\sqrt{r}}$.
The covariance of $U_{0}Y$ is
$\mbox{cov}(U_{0}Y)=E\,U_{0}Y(U_{0}Y)^{T}=U_{0}\,E\,YY^{T}\,U_{0}=U_{0}\pi_{\sqrt{p}}U_{0}=\pi_{\sqrt{r}}=I-\sqrt{r}\sqrt{r}^{T}$
Consider a second stochastic vector of length $N$ defined by
$T=\pi_{\sqrt{r}}Z_{1}$ where $Z_{1}\sim{\cal N}(0,I)$ and $Z,Z_{1}$ are
independent.
We may consider $T^{\,\prime}$ to be defined from a normalised multinomial
variate as in (1), with $p_{j}$ replaced by $r_{j}$; again we may
asymptotically approximate $T^{\,\prime}$ by the normally distributed $T$.
Thus $T$ and $U_{0}Y$ have the same distribution, since they are Gaussian with
identical means and covariance.
So given two stochastic processes and a common number $N$ of cells with cell
probabilities $p$ and $r$, we have a means of rotating from one process to
another, or at least from one chi-squared test to another. The projections of
$Z$ and $Z_{1}$ to give $Y$ and $T$ respectively are not reversible, and so
lose information. In contrast, the transform from $Y$ to $T$ (we are working
asymptotically, and assume normality) is reversible. The goodness of fit test
may be more conveniently carried out for one process than the other, and there
is no loss of statistical information in rotating from one stochastic process
to another.
### 2.3 From the chi-squared statistic to the empirical process
We have normalised the multinomial numerator in (1) in order to define the
rotation $U_{0}$ between the stochastic processes $Y^{\prime}$ and
$T^{\,\prime}$, or rather between their asyptotic limits $Y$ and $T$.
Empirical processes in practice assume the form of observed minus expected
frequencies, and our first task is to remove the standard deviation in the
denominator from (1), although we retain the factor of $\sqrt{n}$ needed for
sensible scaling of the empirical process.
Accordingly we define $N\times N$ diagonal matrices $D_{p}$ containing
$\\{p_{j}\\}_{1}^{N}$, in the given order, along the diagonal: that is,
$D_{p}=\mbox{diag}(p)$. The square root of $D_{p}$ is unambiguously defined,
with elements $\sqrt{p_{j}}$ down the diagonal, denoted either by
$D_{\sqrt{p}}$ or $D_{p}^{1/2}$. Thus the vector $D_{p}^{1/2}Y^{\prime}$
contains elements $(\nu_{j}-np_{j})/\sqrt{n}$, for $j=1,\ldots,n$, although we
generally work instead with the limiting Gaussian vector $D_{p}^{1/2}Y$. The
diagonal $N\times N$ matrix $D_{r}$ is defined analogously for probabilities
$r$.
Define a lower triangular $N\times N$ matrix $J$ by setting all elements on
and below the diagonal to unity, and the elements above the diagonal to zero.
This ‘accumulating’ matrix has the effect of cumulating a column vector: the
$j$th element of $Y$, for instance, is $Y_{j}$, while the $j$th element of
$JY$ is $\sum_{k=1}^{j}Y_{k}$. Further, let $j_{k}^{T}$ denote the $k$th row
of the accumulation matrix $J$, so that the column vector $j_{k}$ consists of
$k$ unities followed by $N-k$ zeroes.
A caution is in order here. A common convention is to denote random variables
by capital letters, with possible or realised (sample) values denoted by the
corresponding small letter. We do not necessarily keep to this convention
here.
#### 2.3.1 Brownian Motion in time $P(x)$
Consider again the vector $Z\sim{\cal N}(0,I)$. The covariance matrix of
$D_{p}^{1/2}Z$ is given by
$\mbox{Cov}\left(D_{\sqrt{p}}Z\right)=E\,D_{\sqrt{p}}ZZ^{T}D_{\sqrt{p}}=D_{p}$
The elements of the vector $D_{p}^{1/2}Z$ are increments of Brownian motion
(BM) in the time $P(x)$. Integrating or summing that process yields BM in time
$P(x)$. The stochastic vector $JD_{p}^{1/2}Z$ has the distribution of a BM
with respect to the time $P(x)$, or in the time $P(x)$; and its covariance
matrix is $J\,D_{p}\,J^{T}$.
We define $\vec{\Delta w_{P}}=D_{p}^{1/2}Z$, the vectorised increments of BM
in time $P$; we further set $\vec{w_{P}}=JD_{p}^{1/2}Z$, the vectorised BM in
time $P$, or the vectorised $P$ BM.
Suppose $x\in[x_{k},x_{k+1})$. The conventional expression of BM in time $P$,
evaluated at time $x$, would then be
$w_{P}(x)=\sum_{j\leq k}\sqrt{p_{j}}\,Z_{j}=j_{k}^{T}\,D_{p}^{1/2}\,Z$ (7)
which is normally distributed, with mean zero and variance
$\int_{(-\infty,x]}dP(y)=\sum_{j\leq k}p_{j}$.
Further assume that $x^{\,\prime}\in[x_{l},x_{l+1})$, where $k\leq l$. The
covariance of $w_{P}(x)$ and $w_{P}(x^{\,\prime})$ is given by
$\mbox{Cov}\left(w_{P}(x),w_{P}(x^{\,\prime})\right)=\mbox{Cov}\left(j_{k}^{T}\,D_{p}^{1/2}\,Z,j_{l}^{T}\,D_{p}^{1/2}\,Z\right)$
$=j_{k}^{T}\,D_{p}^{1/2}\,EZZ^{T}\,D_{p}^{1/2}\,j_{l}=j_{k}^{T}\,D_{p}\,j_{l}=\sum_{j\leq
k}p_{j}=\int_{-\infty}^{{\scriptsize\mbox{min}}(x,x^{\,\prime})}dP(y)=P\left(\mbox{min}(x,x^{\,\prime})\right)$
which is the standard expression for the covariance of BM in time $P(x)$ for
any distribution function $P(x)$.
Now suppose that $x\in[x_{k},x_{k+1})$ and $x+\Delta x\in[x_{m},x_{m+1})$,
where $k\leq m$. Conventional increments of $w_{P}(x)$ are given by
$\Delta w_{P}(x)=w_{P}(x+\Delta x)-w_{P}(x)=\sum_{k<j\leq
m}\sqrt{p_{j}}\,Z_{j}=(j_{m}-j_{k})^{T}\,D_{p}^{1/2}\,Z$
where $\Delta w_{P}(x)$ is again normally distributed, with mean zero and
variance $\int_{x}^{x+\Delta x}dP(y)=\sum_{k<j\leq m}p_{j}$.
#### 2.3.2 The Brownian Bridge in time $P(x)$
In the last section, we rescaled the standard normal column vector $Z$ and
interpreted the elements of $D_{p}^{1/2}Z$ as increments of BM in time $P(x)$.
In like manner, the column vector $D_{p}^{1/2}Y$ contains increments of the
Brownian Bridge (BB) process $v_{P}(x)$ in time $P(x)$, so that the BB process
in time $P(x)$ is $JD_{p}^{1/2}Y$.
As previously, we define $\vec{\Delta v_{P}}=D_{p}^{1/2}Y$, the vectorised
increments of the BB in time $P$; and $\vec{v_{P}}=JD_{p}^{1/2}Y$, the
vectorised BB in time $P$.
By analogy with (7) above, and still assuming that $x_{k}\leq x<x_{k+1}$, we
have that
$v_{P}(x)=\sum_{j\leq k}\sqrt{p_{j}}\,Y_{j}=j_{k}^{T}\,D_{p}^{1/2}\,Y$
which is normally distributed, with mean zero; but the covariance structure of
the $P$ BB is now more complicated.
Again supposing that $x^{\,\prime}\in[x_{l},x_{l+1})$, with $k\leq l$, the
covariance of $v_{P}(x)$ and $v_{P}(x^{\,\prime})$ is given by
$\mbox{Cov}\left(v_{P}(x),v_{P}(x^{\,\prime})\right)=\mbox{Cov}\left(j_{k}^{T}\,D_{p}^{1/2}\,Y,j_{l}^{T}\,D_{p}^{1/2}\,Y\right)$
(8)
$=j_{k}^{T}\,D_{p}^{1/2}\,EYY^{T}\,D_{p}^{1/2}\,j_{l}=j_{k}^{T}\,D_{p}^{1/2}\left(I-\sqrt{p}\sqrt{p}^{T}\right)D_{p}^{1/2}\,j_{l}$
$=j_{k}^{T}\,D_{p}\,j_{l}-j_{k}^{T}\,pp^{T}\,j_{l}=\sum_{j\leq
k}p_{j}-\sum_{j\leq k}p_{j}\sum_{j\leq l}p_{j}$
$=\int_{-\infty}^{{\scriptsize\mbox{min}}(x,x^{\,\prime})}dP(y)-\int_{-\infty}^{x}dP(y)\int_{-\infty}^{x^{\,\prime}}dP(y)=P\left(\mbox{min}(x,x^{\,\prime})\right)-P\left(x\right)P\left(x^{\,\prime}\right)$
which is the standard expression for the covariance of a BB in time $P(x)$, or
a $P$ BB for short, for any distribution function $P(x)$.
When $x+\Delta x\in[x_{m},x_{m+1})$ the increments of $v_{P}(x)$ are given by
$\Delta v_{P}(x)=v_{P}(x+\Delta x)-v_{P}(x)=\sum_{k<j\leq
m}\sqrt{p_{j}}\,Y_{j}=(j_{m}-j_{k})^{T}\,D_{p}^{1/2}\,Y$
where $\Delta v_{P}(x)$ is again normally distributed, with mean zero. The
variance of $\Delta v_{P}(x)$ is
$\mbox{Var}(\Delta
v_{P}(x))=(j_{m}-j_{k})^{T}\,D_{p}^{1/2}\,EYY^{T}\,D_{p}^{1/2}(j_{m}-j_{k})$
$=(j_{m}-j_{k})^{T}\,D_{p}^{1/2}\left(I-\sqrt{p}\sqrt{p}^{T}\right)D_{p}^{1/2}(j_{m}-j_{k})$
$=(j_{m}-j_{k})^{T}\,D_{p}(j_{m}-j_{k})-(j_{m}-j_{k})^{T}\,pp^{T}\,(j_{m}-j_{k})$
$=\sum_{k<j\leq m}p_{j}-\left(\sum_{k<j\leq m}p_{j}\right)^{2}$ (9)
#### 2.3.3 Projection from BM to BB in time $P(x)$
Let $q_{0}$ be a column vector of length $N$ in which every element is unity.
$Y=\pi_{\sqrt{p}}Z=Z-\sqrt{p}{\sqrt{p}}^{T}Z=\left(I-D_{p}^{1/2}q_{0}q_{0}^{T}D_{p}^{1/2}\right)Z$
$D_{p}^{1/2}Y=\left(I-D_{p}q_{0}q_{0}^{T}\right)D_{p}^{1/2}Z=\Pi_{P}^{q_{0}}D_{p}^{1/2}Z$
$\vec{\Delta v_{P}}=\left(I-D_{p}q_{0}q_{0}^{T}\right)\vec{\Delta
w_{P}}=\Pi_{P}^{q_{0}}\,\vec{\Delta w_{P}}$
Working in the primal space with $v_{P}$ and $w_{P}$, the projection operator
$\Pi_{P}^{q_{0}}$ projects $P$ BM $\vec{w_{P}}$ onto $q_{0}-$projected $P$ BM,
which is just the $P$ BB $\vec{v_{P}}$ [Khmaladze, 2016]. Abbreviating for the
moment by setting $\Pi_{P}^{q_{0}}=\Pi$ (to be generalised later in (25) on p.
25), we have
$\Pi_{P}^{q_{0}}=\Pi=I-D_{p}q_{0}q_{0}^{T}\qquad\Pi^{2}=\Pi\qquad\Pi
D_{p}\Pi^{T}=D_{p}-D_{p}q_{0}q_{0}^{T}D_{p}=\Pi D_{p}=D_{p}\Pi^{T}$ (10)
Noting that $q_{0}^{T}D_{p}q_{0}=1$, proofs are as follows
$\Pi^{2}=\left(I-D_{p}q_{0}q_{0}^{T}\right)\left(I-D_{p}q_{0}q_{0}^{T}\right)=I-2D_{p}q_{0}q_{0}^{T}+D_{p}q_{0}q_{0}^{T}D_{p}q_{0}q_{0}^{T}=I-D_{p}q_{0}q_{0}^{T}=\Pi$
$\Pi
D_{p}\Pi^{T}=\left(I-D_{p}q_{0}q_{0}^{T}\right)D_{p}\left(I-q_{0}q_{0}^{T}D_{p}\right)$
$=D_{p}-2D_{p}q_{0}q_{0}^{T}D_{p}+D_{p}q_{0}q_{0}^{T}D_{p}q_{0}q_{0}^{T}D_{p}=D_{p}-D_{p}q_{0}q_{0}^{T}D_{p}$
### 2.4 From point parametric to function parametric form for the stochastic
processes
Now we change to function parametric form for the BM and BB.
Let $\Phi=\\{\phi\\}$ be a family of functions $\phi(x)\in L^{2}_{P}$, i.e.
functions $\phi$ such that $\int\phi(x)^{2}dP(x)<\infty$. We set
$\vec{\phi}=\begin{pmatrix}\phi_{1}&\phi_{2}&\ldots&\phi_{N}\end{pmatrix}^{T}$,
where $\phi_{j}=\phi(x_{j})$. In an abuse of notation we shall often write
$\phi$ for $\vec{\phi}$, since there seems little likelihood of confusing the
function $\phi(x)$ and the vector of its non-zero values at the atoms of $P$.
The finiteness condition for the norm of members of $\Phi$ reduces to
$\|\phi\|^{2}_{P}=\int\phi(x)^{2}dP(x)=\vec{\phi}^{\
T}\,D_{p}\,\vec{\phi}=\phi^{\ T}\,D_{p}\,\phi<\infty$, which ceases to be
vacuous if $N$ is allowed to be countably infinite, given that the values
$\phi_{j}$ are finite.
Similarly we define a family $\Psi$ of functions $\psi(y)\in L^{2}_{R}$, with
vector of values
$\vec{\psi}=\begin{pmatrix}\psi_{1}&\psi_{2}&\ldots&\psi_{N}\end{pmatrix}^{T}$,
where $\psi_{j}=\psi(y_{j})$. Again writing $\psi$ for $\vec{\psi}$, we are
imposing the analogous constraint on the norm, viz.
$\|\psi\|^{2}_{R}=\int\psi(x)^{2}dR(x)=\vec{\psi}^{\
T}\,D_{R}\,\vec{\psi}=\psi^{\ T}\,D_{R}\,\psi<\infty$.
#### 2.4.1 Projection operators in the primal and dual spaces
Writing $\vec{\phi}^{\ T}\vec{\Delta v_{P}}=\phi^{\ T}\vec{\Delta v_{P}}$
then, we have
$\phi^{\ T}\vec{\Delta v_{P}}=\phi^{\ T}\Pi_{P}^{q_{0}}\vec{\Delta
w_{P}}=\phi^{\ T}\vec{\Delta w_{P}}-\phi^{\ T}D_{p}q_{0}q_{0}^{T}\vec{\Delta
w_{P}}$ $=\phi^{\ T}\Pi\,\vec{\Delta
w_{P}}=\left(\Pi^{T}\phi\right)^{T}\vec{\Delta w_{P}}$ (11)
so that $\Pi=I-D_{p}q_{0}q_{0}^{T}$ is the projection operator in the primal
space, acting on $\vec{\Delta w_{P}}$ to produce $\vec{\Delta v_{P}}$; and
$\Pi^{T}=I-q_{0}q_{0}^{T}D_{p}$ is the projection operator in the dual space,
acting on the functions $\phi\in\Phi$. Equation (11) may be rewritten in more
conventional fashion as
$\int_{-\infty}^{\infty}\phi(x)dv_{P}(x)=\int_{-\infty}^{\infty}\phi(x)dw_{P}(x)-\int_{-\infty}^{\infty}\phi(x)q_{0}(x)dP(x)\int_{-\infty}^{\infty}q_{0}(x)dw_{P}(x)$
or more succinctly as
$v_{P}^{q_{0}}(\phi)=v_{P}(\phi)=w_{P}(\phi)\ -<\phi,q_{0}>_{P}\ w_{P}(q_{0})$
(12)
in which $v_{P}(\phi)$ is $q_{0}$-projected $P$ BM [Khmaladze, 2016], or
$q_{0}$-projected BM in time $P(x)$; and where
$\int_{-\infty}^{\infty}\phi(x)q_{0}(x)dP(x)=\int_{-\infty}^{\infty}\phi(x)dP(x)=\
<\phi,q_{0}>_{P}$
To retrieve the point parametric version $v_{P}(x)$, set
$\phi(s)=\phi_{t}(s)=\mathds{1}_{\\{s<t\\}}$ = the Heaviside function,
starting at 1 and dropping to zero at $t$. From (12)
$\int_{-\infty}^{x}dv_{P}(y)=\int_{-\infty}^{x}dw_{P}(y)-\int_{-\infty}^{x}dP(y)\int_{-\infty}^{\infty}q_{0}(x)dw_{P}(x)$
$v_{P}^{q_{0}}(x)=v_{P}(x)=w_{P}(x)-P(x)w_{P}(\infty)$
#### 2.4.2 Covariance of $v^{q}_{P}(\phi)$ and $v^{q}_{P}(\widetilde{\phi})$
for $q=q_{0}$
$\mbox{Cov}\left(v_{P}(\phi),v_{P}(\widetilde{\phi})\right)=\mbox{Cov}\left(\phi^{\
T}\,\vec{\Delta v_{P}},{{\widetilde{\phi}}}\,^{\,T}\,\vec{\Delta
v_{P}}\right)=\mbox{Cov}\left(\phi^{\ T}\Pi\,\vec{\Delta
w_{P}},{{\widetilde{\phi}}}\,^{\,T}\Pi\,\vec{\Delta w_{P}}\right)$ $=E\
\phi^{\ T}\Pi\,\vec{\Delta w_{P}}\,\vec{\Delta
w_{P}}^{\,T}\Pi^{T}{{\widetilde{\phi}}}=\phi^{\ T}\Pi
D_{p}\Pi^{T}{{\widetilde{\phi}}}$
$\mbox{Cov}\left(v_{P}(\phi),v_{P}(\widetilde{\phi})\right)=\phi^{\
T}\left(D_{p}-D_{p}q_{0}q_{0}^{T}D_{p}\right){{\widetilde{\phi}}}$ (13)
More conventionally perhaps, this would be expressed as
$Cov\left(v_{P}(\phi),v_{P}(\widetilde{\phi})\right)=\int\phi\,\widetilde{\phi}\,dP-\left(\int\phi\,dP\right)\left(\int\widetilde{\phi}\,dP\right)$
(14)
Analogously for the BM with functional parametrisation
$\mbox{Cov}\left(w_{P}(\phi),w_{P}(\widetilde{\phi})\right)=\mbox{Cov}\left(\phi^{\
T}\,\vec{\Delta w_{P}},{{\widetilde{\phi}}}\,^{\,T}\,\vec{\Delta
w_{P}}\right)$ $=E\ \phi^{\ T}\,\vec{\Delta w_{P}}\,\vec{\Delta
w_{P}}^{T}{{\widetilde{\phi}}}=\phi^{\
T}D_{p}\,{{\widetilde{\phi}}}=\int\phi\,\widetilde{\phi}\,dP$
### 2.5 Rotation/reflection operator in the functional (dual) space
For rotation from one stochastic process to another, we operate on functions
in the dual space. For $\xi$ and $\eta$ functions in $\Phi$ of unit $P$-norm,
i.e. $\int\xi^{2}dP=\ <\xi,\xi>_{P}\ =\|\xi\|^{2}=\vec{\xi}^{\
T}D_{P}\,\vec{\xi}=\xi^{\ T}D_{P}\,\xi=1$, and analogously for $\eta$, define
the involution
$U_{\xi,\eta}=I-c(\xi-\eta)(\xi-\eta)^{T}D_{p}$
in which
$c=\frac{2}{\|\xi-\eta\|_{P}^{2}}=\frac{1}{1-<\xi,\eta>_{P}}$
and in turn $<\xi,\eta>_{P}\ =\xi^{\ T}D_{P}\,\eta$. Then $U_{\xi,\eta}$ swaps
$\xi$ and $\eta$ around, and leaves unchanged any vector orthogonal (or rather
$P-$orthogonal) to $\xi$ and $\eta$. That is,
$<\xi,\zeta>_{P}\ =\ <\eta,\zeta>_{P}\ =0\quad\Rightarrow\quad
U_{\xi,\eta}\zeta=\zeta$
The corresponding rotation operator in the primal space is $U_{\xi,\eta}^{T}$.
Also $U_{\xi,\eta}$ preserves the $P-$norm, i.e.,
$U_{\xi,\eta}^{T}D_{p}U_{\xi,\eta}=D_{p}$: for any $N-$vector $\zeta$,
$\|U_{\xi,\eta}\zeta\|^{2}=\zeta^{T}U_{\xi,\eta}^{T}D_{p}U_{\xi,\eta}\zeta=\zeta^{T}D_{p}\zeta=\|\zeta\|^{2}$.
#### 2.5.1 Rotation from one stochastic process to another when parameters
are known
We set $s_{0}=q_{0}$, intending $q_{0}$ to be associated with the source $P$
distribution, and $s_{0}$ to be associated with the target $R$ distribution.
When we allow parameters to be estimated, the vectors $q_{j}$ and $s_{j}$ for
$0<j\leq K$ will become the (normalised) score functions ($N-$vectors) for the
$K$ dimensional parameter $\theta$ in the respective spaces. For the moment
the parameter $\theta$ is assumed known, requiring the use of $q_{0}$ and
$s_{0}$ alone to define the rotation/reflection. In the more general case, we
shall require a sequence of reflections to define the desired rotation, as set
out in §2.7.1 on p. 2.7.1.
Define $L=D_{r}^{1/2}\,D_{p}^{-1/2}$, so that $LD_{p}L=D_{r}$. Note that for
$\psi\in L^{2}_{R}$, $L\vec{\psi}\in L^{2}_{P}$: i.e., writing $L\psi$ for
$L\vec{\psi}$,
$\left(L\psi\right)^{T}D_{p}L\psi=\psi^{\ T}LD_{p}L\psi=\psi^{\
T}D_{r}\psi<\infty$
Set $U=U_{q_{0},Ls_{0}}$, so that $q_{0}=ULs_{0}$. Then the rotation from one
stochastic process to another is given by
$v^{s_{0}}_{R}(\psi\,)=v^{q_{0}}_{P}(UL\psi)$ (15)
Proof is by showing commonality of covariances. From (13),
$Cov\left(v^{q_{0}}_{P}(UL\psi),v^{q_{0}}_{P}(UL\widetilde{\psi})\right)=\psi^{T}L^{T}U^{T}\left(D_{p}-D_{p}q_{0}q_{0}^{T}D_{p}\right)UL\widetilde{\psi}$
$=\psi^{T}L^{T}D_{p}L\widetilde{\psi}-\psi^{T}L^{T}U^{T}D_{p}ULs_{0}\,s_{0}^{T}L^{T}U^{T}D_{p}UL\widetilde{\psi}$
$=\psi^{T}L^{T}D_{p}L\widetilde{\psi}-\psi^{T}L^{T}D_{p}Ls_{0}\,s_{0}^{T}L^{T}D_{p}L\widetilde{\psi}$
$=\psi^{T}D_{r}\widetilde{\psi}-\psi^{T}D_{r}s_{0}\,s_{0}^{T}D_{r}\widetilde{\psi}=\psi^{T}\left(D_{r}-D_{r}s_{0}s_{0}^{T}D_{r}\right)\widetilde{\psi}=Cov\left(v^{s_{0}}_{R}(\psi),v^{s_{0}}_{R}(\widetilde{\psi})\right)$
### 2.6 Estimating parameters
#### 2.6.1 The score function
Recalling the definition of $Y^{\prime}$ from (1) on p. 1, we define an
analogous statistic $\widehat{Y}_{j}^{\prime}$ when the parameter is
estimated:
$Y_{j}^{\prime}=\frac{\nu_{j}-np_{j}}{\sqrt{np_{j}}}\qquad\qquad\widehat{Y}_{j}^{\prime}=\frac{\nu_{j}-n\widehat{p}_{j}}{\sqrt{n\widehat{p}_{j}}}$
Estimating $K$ unknown parameters by minimising chi squared [Cramer, 1946,
§30.3],
$\widehat{Y}^{\prime}=Y^{\prime}-B(B^{T}B)^{-1}B^{T}Y^{\prime}+o_{P}(1)$ (16)
$B$ is an $N\times K$ matrix, with rows $j=1,2,\ldots,N$ and columns
$k=1,2,\ldots,K$.
$B=(B_{jk})=\left(\frac{1}{\sqrt{p}_{j}}\frac{\partial
p_{j}}{\partial\theta_{k}}\right)=D_{p}^{-1/2}\left(\frac{\partial
p_{j}}{\partial\theta_{k}}\right)$ $=D_{p}^{1/2}\left(\frac{\partial
p_{j}\big{/}\partial\theta_{k}}{p_{j}}\right)=D_{p}^{1/2}\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{K}\end{pmatrix}$
(17)
in which $Q_{k}$ is the column $N-$vector with elements $\\{\frac{\partial
p_{j}/\partial\theta_{k}}{p_{j}}\\}_{j=1}^{N}$ for $k=1,\ldots,K$; in other
words, $\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{K}\end{pmatrix}$ is the (non-
normalised) score function. To normalise this function, we note that
$B^{T}B=\begin{pmatrix}Q_{1}^{T}\\\ Q_{2}^{T}\\\ \ldots\\\
Q_{K}^{T}\end{pmatrix}D_{P}\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{K}\end{pmatrix}=\Gamma$
(18)
where $\Gamma$ is the information matrix. We may now define normalised score
functions as
$\begin{pmatrix}q_{1}&q_{2}&\ldots&q_{K}\end{pmatrix}=\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{K}\end{pmatrix}\Gamma^{-1/2}$
From (18) then we have
$\Gamma^{-1/2}\begin{pmatrix}Q_{1}^{T}\\\ Q_{2}^{T}\\\ \ldots\\\
Q_{K}^{T}\end{pmatrix}D_{p}\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{K}\end{pmatrix}\Gamma^{-1/2}=I$
$\begin{pmatrix}q_{1}^{T}\\\ q_{2}^{T}\\\ \ldots\\\
q_{K}^{T}\end{pmatrix}D_{p}\begin{pmatrix}q_{1}&q_{2}&\ldots&q_{K}\end{pmatrix}=I$
So, with $\delta_{jk}$ denoting the Kronecker delta,
$q_{j}^{T}D_{p}q_{k}=\delta_{jk}\qquad\mbox{for}\qquad 1\leq j\leq K,1\leq
k\leq K$
but in fact more is true: letting the heavy dot denote
$\frac{\partial}{\partial\theta_{k}}$, and noting that $\sum p_{j}=1$, yields
$\sum\overset{\bullet}{p}_{j}=0=\sum\frac{\overset{\bullet}{p}_{j}}{p_{j}}\,p_{j}=Q_{k}^{T}D_{p}\,q_{0}=q_{0}^{T}D_{p}Q_{k}$
This is true for every $k$, so
$q_{0}^{T}D_{p}\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{K}\end{pmatrix}\Gamma^{-1/2}=q_{0}^{T}D_{p}\begin{pmatrix}q_{1}&q_{2}&\ldots&q_{K}\end{pmatrix}=0$
Finally then we may write
$q_{j}^{T}D_{p}q_{k}=\delta_{jk}\qquad\mbox{for}\qquad 0\leq j\leq K,0\leq
k\leq K$ (19)
#### 2.6.2 Reassembling the jigsaw
Fleshing out (16) we find
$B(B^{T}B)^{-1}B^{T}=D_{p}^{1/2}\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{K}\end{pmatrix}\Gamma^{-1}\begin{pmatrix}Q_{1}^{T}\\\
Q_{2}^{T}\\\ \ldots\\\ Q_{K}^{T}\end{pmatrix}D_{p}^{1/2}$
$=D_{p}^{1/2}\begin{pmatrix}q_{1}&q_{2}&\ldots&q_{K}\end{pmatrix}\begin{pmatrix}q_{1}^{T}\\\
q_{2}^{T}\\\ \ldots\\\ q_{K}^{T}\end{pmatrix}D_{p}^{1/2}$
$=\sum_{k=1}^{K}D_{p}^{1/2}q_{k}q_{k}^{T}D_{p}^{1/2}$ (20)
$Y=Z-\sqrt{p}{\sqrt{p}}^{T}Z=\left(I-D_{p}^{1/2}q_{0}q_{0}^{T}D_{p}^{1/2}\right)Z=\pi_{\sqrt{p}}\,Z$
(21)
From (16) we define the Gaussian $N-$vector
$\widehat{Y}=Y-B(B^{T}B)^{-1}B^{T}Y$
and from (21)
$\widehat{Y}=Z-\sqrt{p}{\sqrt{p}}^{T}Z-B(B^{T}B)^{-1}B^{T}Z-B(B^{T}B)^{-1}B^{T}\sqrt{p}{\sqrt{p}}^{T}Z+o_{P}(1)$
(22)
but
$B^{T}\sqrt{p}=\begin{pmatrix}Q_{1}^{T}\\\ Q_{2}^{T}\\\ \ldots\\\
Q_{K}^{T}\end{pmatrix}D_{p}^{1/2}\sqrt{p}=\begin{pmatrix}Q_{1}^{T}\\\
Q_{2}^{T}\\\ \ldots\\\ Q_{K}^{T}\end{pmatrix}D_{p}q_{0}=\begin{pmatrix}0\\\
0\\\ \ldots\\\ 0\end{pmatrix}$
so, from (22),
$\widehat{Y}=Z-\sqrt{p}{\sqrt{p}}^{T}Z-B(B^{T}B)^{-1}B^{T}Z+o_{P}(1)$ (23)
Applying (20) and (21), and disregarding the residual in (23),
$\widehat{Y}=\left[I-D_{p}^{1/2}\sum_{k=0}^{K}q_{k}q_{k}^{T}D_{p}^{1/2}\right]Z=\widehat{\pi}_{\sqrt{p}}Z$
(24)
Again, in analogy with (4) and (6) on pp. 4 and 6, and utilising (19),
$\widehat{\pi}_{\sqrt{p}}^{T}=\widehat{\pi}_{\sqrt{p}}\qquad\widehat{\pi}_{\sqrt{p}}^{2}=\widehat{\pi}_{\sqrt{p}}\qquad
E\ \widehat{Y}\widehat{Y}^{T}=\widehat{\pi}_{\sqrt{p}}$
From (24) we have that
$D_{p}^{1/2}\widehat{Y}=\left[I-D_{p}\sum_{k=0}^{K}q_{k}q_{k}^{T}\right]D_{p}^{1/2}Z$
Eschewing the clumsy notation $\vec{\Delta\widehat{v}_{P}}$ in favour of the
simpler $\Delta\widehat{v}_{P}$, we rewrite the last equation as
$\Delta\widehat{v}_{P}=D_{p}^{1/2}\widehat{Y}=\left[I-D_{p}\sum_{k=0}^{K}q_{k}q_{k}^{T}\right]\vec{\Delta
w_{P}}=\Pi_{P}^{q}\,\vec{\Delta w_{P}}$
in which $\Pi_{P}^{q}$ is defined as shown, extending the previously defined
$\Pi_{P}^{q_{0}}$ in (10) on p. 10, and where
$q=\left(q_{0},q_{1},\ldots,q_{K}\right)$.
$Cov(\Delta\widehat{v}_{P})=cov(D_{p}^{1/2}\widehat{Y})=D_{p}^{1/2}\widehat{\pi}_{\sqrt{p}}D_{p}^{1/2}$
Now we use the abbreviation $\Pi$ for $\Pi_{P}^{q}$ rather than
$\Pi_{P}^{q_{0}}$. Proofs of the relations in (25) are similar to those in
(10) on p. 10; and again utilising (19):
$\Pi_{P}^{q}=\Pi=I-D_{p}\sum_{k=0}^{K}q_{k}q_{k}^{T}\qquad\qquad\Pi^{2}=\Pi\qquad\qquad\Pi
D_{p}\Pi^{T}=\Pi D_{p}=D_{p}\Pi^{T}$ (25)
Thus
$Cov(\Delta\widehat{v}_{P})=D_{p}^{1/2}\widehat{\pi}_{\sqrt{p}}D_{p}^{1/2}=\Pi^{q}_{P}D_{p}{\Pi^{q}_{P}}^{T}=\Pi^{q}_{P}D_{p}=D_{p}{\Pi^{q}_{P}}^{T}$
Changing to function parametric form yields
$\phi^{T}\Delta\widehat{v}_{P}=\phi^{T}\left(I-D_{p}\sum_{k=0}^{K}q_{k}q_{k}^{T}\right)\vec{\Delta
w_{P}}=\phi^{T}\Pi^{q}_{P}\vec{\Delta w_{P}}=\vec{\Delta
w_{P}}^{T}{\Pi^{q}_{P}}^{T}\phi$
Once again, as in (11) on p. 11, when operating on functions in the dual
space, the projection operator becomes the transpose of the operator
$\Pi^{q}_{P}$ in the primal space.
The $q$ projected $P$ BM is
$v^{q}_{P}(\phi)=\widehat{v}_{P}(\phi)=\phi^{T}\Pi^{q}_{P}\vec{\Delta
w_{P}}=\phi^{T}\left(I-D_{p}\sum_{k=0}^{K}q_{k}q_{k}^{T}\right)\vec{\Delta
w_{P}}$ $v^{q}_{P}(\phi)=w_{P}(\phi)\ -\sum_{k=0}^{K}<\phi,q_{k}>_{P}\
w_{P}(q_{k})$
which becomes, in more conventional form,
$\int_{-\infty}^{\infty}\phi(x)dv^{q}_{P}(x)=\int_{-\infty}^{\infty}\phi(x)dw_{P}(x)-\sum_{k=0}^{K}\int_{-\infty}^{\infty}\phi(x)q_{k}(x)dP(x)\int_{-\infty}^{\infty}q_{k}(x)dw_{P}(x)$
The point parametric version is
$v^{q}_{F}(x)=\widehat{v}_{P}(x)=w_{P}(x)-\sum_{k=0}^{K}\int_{-\infty}^{x}q_{k}(y)dP(y)\int_{-\infty}^{\infty}q_{k}(x)dw_{P}(x)$
#### 2.6.3 Covariance of $v^{q}_{P}(\phi)$ and $v^{q}_{P}(\widetilde{\phi})$
for general $q$
$Cov(v^{q}_{P}(\phi),v^{q}_{P}(\widetilde{\phi}))=E\,\phi^{T}\Pi^{q}_{P}\vec{\Delta
w_{P}}\,\vec{\Delta
w_{P}}^{T}\,{\Pi^{q}_{P}}^{T}\widetilde{\phi}=\phi^{T}\Pi^{q}_{P}D_{p}{\Pi^{q}_{P}}^{T}\widetilde{\phi}$
$=\phi^{T}\left(D_{p}-D_{p}\sum_{k=0}^{K}q_{k}q_{k}^{T}D_{p}\right)\widetilde{\phi}=\phi^{T}\,D_{p}^{1/2}\,\widehat{\pi}_{\sqrt{p}}\,D_{p}^{1/2}\,\widetilde{\phi}$
#### 2.6.4 For consistency with chi squared when df = $n-K-1$
For comparison with the mean of the chi-squared distribution when parameters
were assumed known, as in (5) on p. 5, we have
$E\ {\widehat{Y}}^{T}\,\widehat{Y}=E\
X^{T}\widehat{\pi}_{\sqrt{p}}^{\,T}\,\widehat{\pi}_{\sqrt{p}}X=E\
X^{T}\widehat{\pi}_{\sqrt{p}}X=E\left[X^{T}X-\sum_{k=0}^{K}X^{T}D_{p}^{1/2}q_{k}q_{k}^{T}D_{p}^{1/2}X\right]$
$=E\left[X^{T}X-\sum_{k=0}^{K}q_{k}^{T}D_{p}^{1/2}XX^{T}D_{p}^{1/2}q_{k}\right]=I-\sum_{k=0}^{K}q_{k}^{T}D_{p}q_{k}=n-(K+1)$
### 2.7 Rotation operators in the general case
#### 2.7.1 The rotation operator as a succession of reflections
We require a matrix $V_{K}$ with the properties that $V_{K}Ls_{k}=q_{k}$ for
$0\leq k\leq K$, and $V_{K}^{T}D_{p}V_{K}=D_{p}$; i.e., the linear map sends
score functions to score functions, with an adjustment to allow for the
different functional spaces $L^{2}_{P}$ and $L^{2}_{R}$; and the transform
also preserves the $P$-norm.
Khmaladze’s method for finding a suitable $V_{K}$ is adumbrated in Khmaladze
[2013a], Nguyen [2017a] and Kennedy [2018], i.a.; but the methodology is set
out more fully in Roberts [2019], which we follow.
$V_{0}=W_{0}=U_{q_{0},Ls_{0}}\qquad\widetilde{Ls_{1}}=W_{0}Ls_{1}\qquad
W_{1}=U_{q_{1},\widetilde{Ls_{1}}}\qquad V_{1}=W_{1}\,W_{0}$
$\widetilde{Ls_{2}}=W_{1}W_{0}Ls_{2}=V_{1}Ls_{2}\qquad
W_{2}=U_{q_{2},\widetilde{Ls_{2}}}\qquad V_{2}=W_{2}\,W_{1}\,W_{0}$
and so on. More formally we have the recursion
$\widetilde{Ls_{j}}=V_{j-1}Ls_{j}\qquad
W_{j}=U_{q_{j},\widetilde{Ls_{j}}}\qquad
V_{j}=\prod_{k=0}^{j}W_{k}\qquad\qquad\mbox{for}\qquad j\geq 1$
Then Roberts [2019] shows that $V_{K}Ls_{k}=q_{k}$ for $0\leq k\leq K$; and it
is straightforward to show that $V_{K}^{T}D_{p}V_{K}=D_{p}$.
#### 2.7.2 Rotation from one empirical process to another
The rotation from one stochastic process to another is given by
$v^{s}_{R}(\psi)=v^{q}_{P}(V_{K}L\psi)$ (26)
Proof proceeds as in (15) on p. 15, again utilising (19).
$Cov\left(v^{q}_{P}(V_{K}L\psi),v^{q}_{P}(V_{K}L\widetilde{\psi})\right)=\psi^{T}L^{T}V_{K}^{T}\left(D_{p}-\sum_{k=0}^{K}D_{p}q_{k}q_{k}^{T}D_{p}\right)V_{K}L\widetilde{\psi}$
$=\psi^{T}L^{T}D_{p}L\widetilde{\psi}-\sum_{k=0}^{K}\psi^{T}L^{T}V_{K}^{T}D_{p}V_{K}Ls_{k}\,s_{k}^{T}L^{T}V_{K}^{T}D_{p}V_{K}L\widetilde{\psi}$
$=\psi^{T}L^{T}D_{p}L\widetilde{\psi}-\sum_{k=0}^{K}\psi^{T}L^{T}D_{p}Ls_{k}\,s_{k}^{T}L^{T}D_{p}L\widetilde{\psi}$
$=\psi^{T}D_{r}\widetilde{\psi}-\sum_{k=0}^{K}\psi^{T}D_{r}s_{k}\,s_{k}^{T}D_{r}\widetilde{\psi}=\psi^{T}\left(D_{r}-\sum_{k=0}^{K}D_{r}s_{k}s_{k}^{T}D_{r}\right)\widetilde{\psi}=Cov\left(v^{s}_{R}(\psi),v^{s}_{R}(\widetilde{\psi})\right)$
### 2.8 Higher dimensional Khmadadze rotations, and the colour-blind problem
We consider firstly how to fit the two-dimensional Khmaladze rotation into the
linear algebraic framework detailed above, and comment briefly on its three-
dimensional cousin. Largely taking our cue from Dumitrescu & Khmaladze [2019],
we then symmetrise the underlying Borel sets to look at the ‘colour-blind’
problem.
#### 2.8.1 The Khmadadze rotation in two dimensions
Consider a sample of $n$ realisations of independent and identically
distributed pairs $(X_{j},Y_{j})$, with distribution function $H(x,y)$, and
marginal distribution functions $F(x),G(y)$. We approximate $F$ and $G$ by
discrete distributions $P$ and $R$ respectively, along the lines of §2.1 on p.
2.1. This time however, we are not rotating from $F$ to $G$; rather both
distributions are necessary to build the framework for our analysis.
The EDF is given by
$H_{n}(x,y)=\frac{1}{n}\sum_{j=1}^{n}\mathds{1}_{\\{x-X_{j}\geq
0\\}}\mathds{1}_{\\{y-Y_{j}\geq 0\\}}$
and the basic empirical process as
$v_{n}(x,y)=H_{n}(x,y)-H(x,y)$
Rather than mapping the one dimensional BM onto a BB, we now use the 2
dimensional BM $w_{H}(x,y)$ in ‘time’ $H(x,y)$, projecting it (or rather tying
it down at the edges) to obtain a Brownian ‘pillow’ [Dumitrescu & Khmaladze,
2019, §2.1]. Writing $w$ for $w_{H}$ we have:
$v(x,y)=w(x,y)-x\,w(\infty,y)-y\,w(x,\infty)+x\,y\,w(\infty,\infty)$
in which we may ‘anchor’ the BM at the origin; or more generally at the lower
end of the supports $(M_{F},M_{G})$ as in §2.1.
The analysis of the preceding sections involving the Khmaladze rotation
proceeds via
$v_{n}(\phi)=\int\phi(x,y)dv_{n}(x,y)\qquad\mbox{and}\qquad
v(\phi)=\int\phi(x,y)dv(x,y)$
Now define the rectangle $R(a,b)$ in the plane as
$R(a,b)=\\{(x,y):x\leq a,y\leq b\\}$
Then we set $\phi_{a,b}(x,y)=\mathds{1}_{\\{(x,y)\in R(a,b)\\}}$; and the
family $\Phi$ is generated by all such functions, for $M_{F}<a,M_{G}<b$.
The central result (26) on p. 26 remains valid, even if the vectors of length
$N$ in the previous development are now vectors of length $N^{2}$ as we
vectorise functions and variables over the plane.
Moving to higher dimensions is straightforward, conceptually at least. The
Brownian pillow, or its equivalent, in three dimensions assumes the form
$z(x,y,z)=w(x,y,z)-z\,w(x,y,\infty)-y\,w(x,\infty,z)-x\,w(\infty,y,z)$
$+y\,z\,w(x,\infty,\infty)+x\,z\,w(\infty,y,\infty)+x\,y\,w(\infty,\infty,z)-x\,y\,z\,w(\infty,\infty,\infty)$
Changing notation in anticipation of our work in the next section, set
$R(a_{1},a_{2},a_{3})=\\{(x,y,z):x\leq a_{1},y\leq a_{2},z\leq a_{3}\\}$ (27)
and again the family $\Phi$ is generated by all functions of the form
$\phi_{a_{1},a_{2},a_{3}}(x,y,z)=\mathds{1}_{\\{(x,y,z)\in
R(a_{1},a_{2},a_{3})\\}}$.
The extension to higher dimensions is feasible in principle, but how useful
the Khmaladze rotation will be in higher dimensions remains to be seen.
#### 2.8.2 The colour-blind problem
In a recent article Dumitrescu & Khmaladze [2019] have reconsidered the colour
blind problem, in which pairs of observed coloured items are unable to be
distinguished by a colour blind observer. Further background is available in
Parsadanishvili [1982] and Parsadanishvili & Khmaladze [1982].
To be precise, suppose that weights $X_{r}$ and $X_{g}$ of red and green balls
have distribution functions $P_{r}(x)$ and $P_{g}(x)$ respectively, and the
$i$th data point consists of a pair $\left(X_{r}^{(i)},X_{g}^{(i)}\right)$,
where the random variables are independent from data point to data point,
although the weights of the red and green balls need not be independent. Then
Dumitrescu & Khmaladze [2019], projecting the empirical process and utilising
the properties of the Brownian pillow, consider the possibilities for
statistical inference available to a colour blind person, able to measure the
maximum and minimum weights of each pair, but without any means of knowing
whether that maximum or minimum comes from the red or the green ball.
We symmetrise the rectangles $R(a,b)$ by defining
$S(a,b)=S(b,a)=R(a,b)\cup R(b,a)$
Now we set $\phi_{a,b}(x,y)=\mathds{1}_{\\{(x,y)\in S(a,b)\\}}$; the function
$\phi$ is symmetric in $x$ and $y$, necessarily so by virtue of the fact that
the colour blind person cannot distinguish between the colours.
The derivation of (26) proceeds as previously, but the vectors now have length
$N(N+1)/2$.
In three dimensions, and retaining the notation in (27), we have
$S(a_{1},a_{2},a_{3})=\cup_{\sigma\in S_{3}}R(a_{\sigma 1,\sigma 2,\sigma 3})$
where $S_{3}$ is the symmetric group on 3 symbols. Again extension to higher
dimensions, and more than two colours, is feasible conceptually; but its
utility in practice, and especially in the context of the Khmaladze rotation,
remains to be seen.
## 3 The Khmadadze transform
Khmaladze [2013b, ch. 7] has derived the KT in the simple setting of a
mortality investigation of $n$ independent and identically distributed lives.
After a short recapitulation of least squares regression, we shall follow his
treatment closely, clarifying some points as we go.
### 3.1 The linear regression model
The model is
$y_{t}=\begin{pmatrix}x_{1t}&x_{2t}\end{pmatrix}\begin{pmatrix}a_{t}\\\
b_{t}\end{pmatrix}+\epsilon_{t}$
Centre this to obtain
$y_{t}-Ey_{t}=\begin{pmatrix}x_{1t}-Ex_{1t}&x_{2t}-Ex_{2t}\end{pmatrix}\begin{pmatrix}a_{t}\\\
b_{t}\end{pmatrix}+\epsilon_{t}$
from which
$\begin{pmatrix}x_{1t}-Ex_{1t}\\\
x_{2t}-Ex_{2t}\end{pmatrix}(y_{t}-Ey_{t})=\begin{pmatrix}x_{1t}-Ex_{1t}\\\
x_{2t}-Ex_{2t}\end{pmatrix}\begin{pmatrix}x_{1t}-Ex_{1t}&x_{2t}-Ex_{2t}\end{pmatrix}\begin{pmatrix}a_{t}\\\
b_{t}\end{pmatrix}+\begin{pmatrix}x_{1t}-Ex_{1t}\\\
x_{2t}-Ex_{2t}\end{pmatrix}\epsilon_{t}$
Assuming the covariates and the residual are uncorrelated, taking expectations
yields
$E\,\begin{pmatrix}x_{1t}-Ex_{1t}\\\
x_{2t}-Ex_{2t}\end{pmatrix}(y_{t}-Ey_{t})=\mbox{Covar}(x_{1t},x_{2t})\begin{pmatrix}a_{t}\\\
b_{t}\end{pmatrix}$
in which $\mbox{Covar}(x,y)$ stands for the covariance matrix, while
$\mbox{Cov}(x,y)$ stands for the covariance of $x$ and $y$, the off diagonal
elements of the covariance matrix. We have now
$\begin{pmatrix}\widehat{a}_{t}\\\
\widehat{b}_{t}\end{pmatrix}=\mbox{Covar}(x_{1t},x_{2t})^{-1}E\,\begin{pmatrix}x_{1t}-Ex_{1t}\\\
x_{2t}-Ex_{2t}\end{pmatrix}(y_{t}-Ey_{t})$
$=\mbox{Covar}(x_{1t},x_{2t})^{-1}E\,\begin{pmatrix}x_{1t}-Ex_{1t}\\\
x_{2t}-Ex_{2t}\end{pmatrix}y_{t}$
Finally the predicted value of the regressand is
$\widehat{y}_{t}=\begin{pmatrix}x_{1t}&x_{2t}\end{pmatrix}\mbox{Covar}(x_{1t},x_{2t})^{-1}\begin{pmatrix}E\,(x_{1t}-Ex_{1t})y_{t}\\\
E\,(x_{2t}-Ex_{2t})y_{t}\end{pmatrix}$ (28)
In econometrics, the linear predictor from regression models can seem
unrelated to the last equation, although the difference is more of form than
of substance. When the parameters are not time varying, it is usual to stack
the regressand and regressors, with the model expressed as
$Y=X\beta+\epsilon$, with $Y$ a column vector and $X$ a matrix, whence the
predicted value of $Y$ is $\widehat{Y}=X(X^{T}X)^{-1}X^{T}Y$, with the moments
being estimated from the sample. The projection from $Y$ to $\widehat{Y}$ in
(16) on p. 16, for instance, is of this form.
The regression sought in the case of KT1 is not amenable to the stacking of
variables in this way because the distributions of the random variables
$x_{1t}$ and $x_{2t}$ depend on $t$, as do the coefficients.
Instead of (28), the formula for the predicted value of $y_{t}$ is often taken
to be
$\widehat{y}_{t}=\begin{pmatrix}x_{1t}&x_{2t}\end{pmatrix}\mbox{Covar}(x_{1t},x_{2t})^{-1}\begin{pmatrix}E\,x_{1t}y_{t}\\\
E\,x_{2t}y_{t}\end{pmatrix}$ (29)
which is the formula used, for example in Koul & Swordson [2011].
### 3.2 Choice of regressand and regressors
Following [Khmaladze, 2013b, ch. 7], we place the derivation of KT1 in the
context of a mortality investigation. There are $n$ people in the sample, and
we are investigating the duration of life, so that the variable $x$ in §2.2 is
time. The intention is to predict $\nu_{l}$, the number of deaths in the $l$th
cell, from a linear regression model.
The empirical distribution function (EDF) is defined as
$\widehat{F}_{n}(x)=\frac{1}{n}\mathds{1}_{\\{x_{j}\leq x\\}}$, with increment
$\Delta\widehat{F}_{n}(x)=\widehat{F}_{n}(x+\Delta x)-\widehat{F}_{n}(x)$,
which becomes in our previous notation
$\Delta\widehat{F}_{n}(x_{j})=\nu_{j}/n$. Then $\widehat{F}(x_{l})$ is the
actual proportion of the sample dead by time $x_{l}$, and
$F_{\theta}(x_{l})=E\,\widehat{F}(x_{l})$ the expected proportion.
Recalling the definition of the score function $Q_{1}$ in (17) on p. 17, and
assuming there is but one unknown parameter, so that $K=1$, we have
$\qquad Q_{1}=\begin{pmatrix}Q_{11}&Q_{12}&\ldots&Q_{1N}\end{pmatrix}^{T}$
To predict $\nu_{l}$ we would think of using its expected value $p_{l}$, as
well as its derivative
$\overset{\bullet}{p}_{l}=(\overset{\bullet}{p}_{l}/p_{l})\times p_{l}$. We
recast these as their sample counterparts $\nu_{l}$ and $Q_{1l}\nu_{l}$, and
sum over the future cells to produce Khmaladze’s regressors
$\frac{1}{n}\sum_{j=l}^{N}\nu_{j}$ and
$\frac{1}{n}\sum_{j=l}^{N}Q_{1j}(\hat{\theta})\nu_{j}$. The MLE $\hat{\theta}$
used in the second regressor is calculated from the entire sample – see (33)
on p. 33 below. The scaling arises because the regressand is to be the
increment in the EDF, and $\frac{1}{n}\nu_{l}=\Delta\widehat{F}_{n}(x_{l})$.
The first regressor is an obvious enough choice, in that
$\frac{1}{n}\sum_{j=l}^{N}\nu_{j}=1-\widehat{F}(x_{l})$ is the proportion of
survivors at time $x_{k}$; and the more surviving at time $x_{l}$, i.e. the
higher the exposed to risk at time $x_{l}$, the greater the expected number of
deaths within the period $[x_{l},x_{l+1})$.
### 3.3 KT1 through linear regression
Once all the data is available, i.e. everyone in the sample has died, we are
testing the goodness of fit of a particular distribution of duration of life,
or rather a given family of distributions $F_{\theta}(x)$ depending on an
unknown parameter $\theta$. We assume $F_{\theta}(x)$ to be continuous, with
corresponding non-normalised score function denoted by
$h(x,\theta)=\frac{\partial}{\partial\theta}\,f_{\theta}(x)\big{/}f_{\theta}(x)$.
We pretend to be partway through the cohort dying off, and try to predict the
number dying in the next short interval, given firstly the number surviving at
the moment, secondly the future times of death of the current survivors, and
thirdly parameter estimates obtained from the entire sample. The ‘present’
time is $t=x_{l}$, and we wish to predict $\nu_{l}$, the number dying before
time $x_{l+1}$.
The key result we need is the following, taken from Khmaladze [2013b, p. 79]
$\mbox{Cov}\left(\int g_{1}(x)\,d\widehat{F}_{n}(x),\int
g_{2}(x)\,d\widehat{F}_{n}(x)\right)$ $=E\ \int
g_{1}(x)\left(d\widehat{F}_{n}(x)-dF_{\theta}(x)\right)\int
g_{2}(x)\left(d\widehat{F}_{n}(x)-dF_{\theta}(x)\right)$
$=\frac{1}{n}\left(\int g_{1}(x)g_{2}(x)dF_{\theta}(x)-\int
g_{1}(x)dF_{\theta}(x)\int g_{2}(x)dF_{\theta}(x)\right)$ (30)
This result should be compared with (8), (9) and (14) on pp. 8, 9 and 14
respectively; and see also Khmaladze [2013b, p. 46].
In the present context,
$y_{t}=\int_{t}^{t+\Delta
t}d\widehat{F}_{n}(x)=\int_{x_{l}}^{x_{l+1}}d\widehat{F}_{n}(x)\qquad\qquad
x_{1t}=\int_{t}^{\infty}d\widehat{F}_{n}(x)\qquad$
$x_{2t}=\int_{t}^{\infty}h(x,\theta)d\widehat{F}_{n}(x)\qquad\qquad
x_{2t}^{*}=\int_{t}^{\infty}\left[h(x,\theta)-E_{\theta}^{t}\right]d\widehat{F}_{n}(x)$
in which $E_{\theta}^{t}$ is the expectation of $h$ conditional upon surviving
until time t:
$E_{\theta}^{t}=\frac{\int_{t}^{\infty}h(x,\theta)dF_{\theta}(x)}{1-F_{\theta}(t)}$
Expected values of $x_{2t}$ and $x_{2t}^{*}$ are given by
$E\,x_{2t}=\int_{t}^{\infty}h(x,\theta)dF_{\theta}(x)=E_{\theta}^{t}[1-F_{\theta}(t)]$
and
$E\,x_{2t}^{*}=\int_{t}^{\infty}\left[h(x,\theta)-E_{\theta}^{t}\right]dF_{\theta}(x)=E_{\theta}^{t}\left[1-F_{\theta}(t)\right]-E_{\theta}^{t}\left[1-F_{\theta}(t)\right]=0$
(31)
From (30) and (31) we have that
$\mbox{Cov}(x_{1t},x_{2t}^{*})=\mbox{Cov}\left(\int_{t}^{\infty}d\widehat{F}_{n}(x),\int_{t}^{\infty}\left[h(x,\theta)-E_{\theta}^{t}\right]d\widehat{F}_{n}(x)\right)$
$=\frac{1}{n}\left(\int_{t}^{\infty}\left[h(x,\theta)-E_{\theta}^{t}\right]dF_{\theta}(x)-\left(1-F_{\theta}(t)\right)\int_{t}^{\infty}\left[h(x,\theta)-E_{\theta}^{t}\right]dF_{\theta}(x)\right)$
$=\frac{1}{n}F_{\theta}(t)\int_{t}^{\infty}\left[h(x,\theta)-E_{\theta}^{t}\right]dF_{\theta}(x)=0$
The vanishing of $\mbox{Cov}(x_{1t},x_{2t}^{*})$ simplifies calculations
substantially, since the covariance matrix to be inverted in (28) reduces to a
diagonal matrix.
Along the same lines we have
$\mbox{Cov}(x_{1t},x_{2t})=\mbox{Cov}\left(\int_{t}^{\infty}d\widehat{F}_{n}(x),\int_{t}^{\infty}h(x,\theta)d\widehat{F}_{n}(x)\right)$
$=\frac{1}{n}\left(\int_{t}^{\infty}h(x,\theta)dF_{\theta}(x)-[1-F_{\theta}(t)]\int_{t}^{\infty}h(x,\theta)dF_{\theta}(x)\right)$
$=\frac{1}{n}F_{\theta}(t)\int_{t}^{\infty}h(x,\theta)dF_{\theta}(x)$
and
$\mbox{Cov}(y_{t},x_{2t})=\mbox{Cov}\left(\int_{t}^{t+\Delta
t}d\widehat{F}_{n}(x),\int_{t}^{\infty}h(x,\theta)d\widehat{F}_{n}(x)\right)$
$=\frac{1}{n}\left(\int_{t}^{t+\Delta
t}h(x,\theta)dF_{\theta}(x)-[F_{\theta}(t+\Delta
t)-F_{\theta}(t)]\int_{t}^{\infty}h(x,\theta)dF_{\theta}(x)\right)$
$\approx\frac{1}{n}\
p_{l}\left(h(x_{l},\theta)-E_{\theta}^{t}[1-F_{\theta}(t)]\right)$
and
$\mbox{Cov}(y_{t},x_{1t})=\mbox{Cov}\left(\int_{t}^{t+\Delta
t}d\widehat{F}_{n}(x),\int_{t}^{\infty}d\widehat{F}_{n}(x)\right)$
$=\frac{1}{n}\left(\int_{t}^{t+\Delta t}dF_{\theta}(x)-[F_{\theta}(t+\Delta
t)-F_{\theta}(t)]\int_{t}^{\infty}dF_{\theta}(x)\right)$ $\approx\frac{1}{n}\
p_{l}\left(1-[1-F_{\theta}(t)]\right)$
The prediction from (28) on p. 28 becomes
$\frac{1}{n}\widehat{\nu}_{l}=\frac{1}{n}\,p_{l}\left[\int_{t}^{\infty}\begin{pmatrix}1&h(x_{l},\theta)\end{pmatrix}d\widehat{F}_{n}(x)\right]\mbox{Covar}(x_{1t},x_{2t})^{-1}\begin{pmatrix}1-[1-F_{\theta}(t)]\\\
h(x_{l},\theta)-E_{\theta}^{t}[1-F_{\theta}(t)]\end{pmatrix}$
The prediction from (29) on p. 29 becomes
$\frac{1}{n}\widehat{\nu}_{l}=\frac{1}{n}\,p_{l}\left[\int_{t}^{\infty}\begin{pmatrix}1&h(x_{l},\theta)\end{pmatrix}d\widehat{F}_{n}(x)\right]\mbox{Covar}(x_{1t},x_{2t})^{-1}\begin{pmatrix}1\\\
h(x_{l},\theta)\end{pmatrix}$ (32)
Apart from scaling and replacing $p_{l}$ by $dF_{\theta}(t)$, the predicted
value of $d\widehat{F}_{n}(x)$ as given in (32) agrees with that given in Koul
& Swordson [2011] in their expression for $dw_{n\theta}(x)$, equal to
$\sqrt{n}(y-\widehat{y})$ in our notation. Should there be further covariates,
say $K$ in all, the vector $(1,h)$ would be extended to
$(1,h_{1},h_{2},\ldots,h_{K})$.
The increment $dw_{n\theta}(x)$ is in the nature of a BM, because it is
uncorrelated with the past. More precisely, $dw_{n\theta}(x)$ is uncorrelated
with the future, by the nature of the regression that
$\mbox{Cov}(x_{jt},y_{t}-\widehat{y})=0$ for $j=1,2$; and the past and future
are mirror images of each other.
This is clear for $x_{1t}$, since the number of survivors is the sample size
minus the number who have died so far. As for the score function, we recall
the following simple properties.
$\sum
p_{j}=1\qquad\sum\overset{\bullet}{p}_{j}=0\qquad\sum\frac{\overset{\bullet}{p}_{j}}{p_{j}}\,p_{j}=0\qquad\sum
Q_{1j}(\theta)p_{j}=0$
From the last of these relations, we recall that the MLE of $\theta$, say
$\hat{\theta}$, is given by
$\sum Q_{1j}(\hat{\theta})\nu_{j}=0$ (33)
which is the sample counterpart of
$\int h(x,\theta)dF_{\theta}(x)=0$
The variable $x_{2t}$ reflects the future, but it equally reflects the past.
## Acknowledgement
Thanks to Estate Khmaladze for comments on an early draft of this paper.
Responsibility for the contents naturally remains with the author.
## References
* Chibisov [1971] Chibisov, D. M. (1971). Certain chi-square type tests for continuous distributions. Theory of Probability and its Applications, 16, 1–22.
* Cramer [1946] Cramer, H. (1946). Mathematical Methods of Statistics. Princeton University Press.
* Dumitrescu & Khmaladze [2019] Dumitrescu, L., & Khmaladze, E. V. (2019). Asymptotic hypotheses testing for the colour blind problem. Electronic Journal of Statistics, 13, 4573–4595.
* Kennedy [2018] Kennedy, A. P. (2018). Analysis and Prediction of High Frequency Foreign Exchange Data. Master’s thesis School of Mathematics and Statistics, Victoria University, Wellington, New Zealand.
* Khmaladze [1979] Khmaladze, E. V. (1979). The use of $\omega^{2}$ tests for testing parametric hypotheses. Theory of Probability and its Applications, 24, 283–301.
* Khmaladze [1981] Khmaladze, E. V. (1981). Martingale approach in the theory of goodness-of-fit tests. Theory of Probability and its Applications, 26, 240–257.
* Khmaladze [2013a] Khmaladze, E. V. (2013a). Note on distribution free testing for discrete distributions. Annals of Statistics, 41, 2979–2993.
* Khmaladze [2013b] Khmaladze, E. V. (2013b). Statistical methods with applications to demography and life insurance. CRC Press.
* Khmaladze [2016] Khmaladze, E. V. (2016). Unitary transformations, empirical processes and distribution free testing. Bernoulli, 22, 563–588.
* Khmaladze [2017] Khmaladze, E. V. (2017). Distribution free testing for conditional distributions given covariates. Statistics & Probability Letters, 129, 348–354.
* Kim [2016] Kim, J. (2016). Goodness-of-fit test: Khmaladze transformation vs empirical likelihood. ArXiv:1602.05885v2 [stat.AP], 22 April 2016.
* Koul & Swordson [2011] Koul, H. L., & Swordson, E. (2011). Khmaladze transformation. In International Encyclopedia of Statistical Science (pp. 715–718). Springer.
* Li [2009] Li, B. (2009). Asymptotically Distribution-Free Goodness-of-Fit Testing: A Unifying View. Econometric Reviews, 28, 632–657.
* Nguyen [2017a] Nguyen, T. T. M. (2017a). Asymptotic methods of testing statistical hypotheses. Ph.D. thesis School of Mathematics and Statistics, Victoria University, Wellington, New Zealand.
* Nguyen [2017b] Nguyen, T. T. M. (2017b). A new approach to distribution free tests in contingency tables. Metrika, 80, 153–170.
* Parsadanishvili [1982] Parsadanishvili, E. G. (1982). Empirical and rank empirical fields and the colorblind problem. Theory of Probability and its Applications, 27, 883–885. Summary of presentation on 25 May at the Steklov Institute, Moscow.
* Parsadanishvili & Khmaladze [1982] Parsadanishvili, E. G., & Khmaladze, E. V. (1982). The testing of statistical hypotheses on unidentifiable objects. Theory of Probability and its Applications, 27, 175–182.
* Roberts [2019] Roberts, L. A. (2019). Distribution free goodness of fit testing of grouped Bernoulli trials. Statistics & Probability Letters, 150, 47–53.
|
# A NOTE ON AN OPEN CONJECTURE IN RATIONAL DYNAMICAL SYSTEM
Zeraoulia Rafik
<EMAIL_ADDRESS>
_not_ university de Batna.2
###### Abstract
This note is an attempt with the open conjecture $8$ proposed by the authors
of[1] which states :
Assume $\alpha,\beta,\lambda\in[0,\infty)$. Then every positive solution of
the difference equation :
$z_{n+1}=\frac{\alpha+z_{n}\beta+z_{n-1}\lambda}{z_{n-2}},\quad n=0,1,\ldots$
is bounded if and only if $\beta=\lambda$.
We will use a construction of sub-energy function and properties of Todd’s
difference equation to disprove that conjecture in general.
_K_ eywords Difference equation; boundedness properties $\cdot$ Todd’s
equation $\cdot$ sub-energy function
MSC:39A10; 39A22
## 1 Introduction
A major project in the field of rational difference equations [3] has been to
determine the boundedness properties of all third-order equations of the form
:
$x_{n+1}=\frac{\alpha+\beta x_{n}+\gamma x_{n-1}+\delta
x_{n-2}}{A+Bx_{n}+Cx_{n-1}+Dx_{n-1}}$ (1)
with nonnegative parameters $\alpha,\beta,\gamma,\delta,A,B,C$ and $D$ , one
wishes to show that either the solutions remain bounded for all positive
initial conditions, or there exist positive initial conditions so that the
solutions are unbounded .Dynamics of Third-Order Rational Difference Equations
with Open Problems and Conjectures [4] treats the large class of difference
equations described by Equation (1),Some open problems related to (1) in which
the boundedness properties were not known was recently solved in [2], By the
following assumption $\delta=A=B=C=0$ with the variable change
$x_{n}\to\dfrac{x_{n}}{D}$ , with $\alpha\geq 0,\beta>0,\gamma>0,D>0$ equation
(1) reduces to the following form :
$x_{n+1}=\frac{D\alpha+x_{n}\beta+x_{n-1}\gamma}{x_{n-2}},\quad
n=0,1,\ldots,D\alpha=\alpha^{\prime}$ (2)
It is shown in a paper by Lugo and Palladino [5] that there exist unbounded
solutions of (2)in the case that $0\leq\alpha<1$ and
$0<\beta<\frac{1}{3}$.Ying Sue Huang and Peter M. Knopf showed in [3] for
$\alpha^{\prime}\geq 0,\beta>0$ and if $\beta\neq 1$ there exist positive
initial conditions such that the solutions are unbounded except for the case
$\alpha^{\prime}=0$ and $\beta>1$, Question related to Boundedness of
solutions of (2) in the case $\beta=\gamma$ is the folllowing conjecture which
it is proposed as eight open conjecture in this paper [1] by G. LADAS, G. LUGO
AND F. J. PALLADINO, In our present paper we disprove in general the only if
part of the conjecture 8 in [1] using sub-energy function and some properties
of Todd’s difference equation [6] and [7]
## 2 Conjecture
Assume $\alpha,\beta,\lambda\in[0,\infty)$. Then every positive solution of
the difference equation :
$z_{n+1}=\frac{\alpha+z_{n}\beta+z_{n-1}\lambda}{z_{n-2}},\quad n=0,1,\ldots$
(3)
is bounded if and only if $\beta=\lambda$
## 3 Proof
Suppose that $\beta=\lambda>0$. Let $x_{n}:=z_{n}/\beta$ and
$c:=\alpha/\beta^{2}$. Then the dynamics (3) can be rewritten as
$\qquad x_{n+1}=\frac{c+x_{n}+x_{n-1}}{x_{n-2}}$ (4)
(say for $n=2,3,\dots$), just with one parameter $c\geq 0$ , The dynamic (4)is
exactly the Todd’s difference equation ,In this case the equation is generally
referred to by the cognomen “Todd’s equation” and possesses the invariant :
$\displaystyle(c+x_{n}+x_{n-1}+x_{n-2})\biggl{(}1+\dfrac{1}{x_{n}}\biggr{)}\biggl{(}1+\dfrac{1}{x_{n-1}}\biggr{)}\biggl{(}1+\dfrac{1}{x_{n-2}}\biggr{)}=\text{constant}$
(5)
The invariants of difference equations play an important role in understanding
the stability and qualitative behavior of their solutions. To be more precise,
if the invariant is a bounded mainfold [8], then the solution is also bounded,
Recently Hirota et al [9] found two conserved quantities $H_{n}^{1}$ and
$H_{n}^{2}$ for the third- order Lyness equation , Note that Lyness equation
is a special case of equation (4) such that $c=1$ ,The two quantities are
independents and One of the conserved quantities is the same form as that of
(5) ,Both of two conserved quantities formula were derived from discretization
of an anharmonic oscillator namely using its equation of its motion see the
first equation here [9], we may consider those conserved quantities as
conserved sub- energy of anharmonic oscillator this means that (5) present a
sub energy function of that anharmonic oscillator , To prove the "if" part of
the conjecture it would be enough to construct for each nonnegative $c$, a
"sub-energy" function [12] $f_{c}\colon(0,\infty)^{3}\to\mathbb{R}$ such that
:
$\qquad f_{c}(x_{0},x_{1},x_{2})\to\infty\quad\text{as}\quad
x_{0}+x_{1}+x_{2}\to\infty$ (6)
Note that the sub-energy function is the invariant of the third difference
equation ,namely the dynamics (4) , if we assume that :
$f_{c}(x_{n},x_{n-1},x_{n-2})=\displaystyle(c+x_{n}+x_{n-1}+x_{n-2})\biggl{(}1+\dfrac{1}{x_{n}}\biggr{)}\biggl{(}1+\dfrac{1}{x_{n-1}}\biggr{)}\biggl{(}1+\dfrac{1}{x_{n-2}}\biggr{)}=\text{constant},n\geq
0$ (7)
then the condition (6) is satisfied in (7) .see Lemma2 in ([14].p.4) .For RHS
of (7) see also Theorem2.1 in ([6]p.31) ,And Since the invariant of the
dynamic of (4) is constant then $f_{c}$ could be referred to as the
conservation of energy along the path of the dynamical system.For some natural
$k$ and all $x=(x_{0},x_{1},x_{2})\in(0,\infty)^{3}$ one has the "sub-energy"
inequality $f_{c}(T^{k}x)\leq f_{c}(x)$, where $Tx:=(x_{1},x_{2},x_{3})$, with
$x_{3}=\frac{c+x_{2}+x_{1}}{x_{0}}$, according to the dynamics. Of course,
$T^{k}$ is the $k$th power of the operator $T$. For $k=1$, the sub-energy
inequality is the functional inequality
$\qquad f_{c}\Big{(}x_{1},x_{2},\frac{c+x_{2}+x_{1}}{x_{0}}\Big{)}\leq
f_{c}(x_{0},x_{1},x_{2})\quad\text{for all positive $x_{0},x_{1},x_{2}$, }$
(8)
To construct a sub-energy function, one might want to start with some easy
function $f_{c,0}$ such that $f_{c,0}(x_{0},x_{1},x_{2})\to\infty$ as
$x_{0}+x_{1}+x_{2}\to\infty$, and then consider something like
$f_{c,0}\vee(f_{c,0}\circ T^{k})\vee(f_{c,0}\circ T^{2k})\vee\dots$,Inequality
(8) can be obviously restated in the following more symmetric form:
$\qquad x_{0}x_{3}=c+x_{1}+x_{2}\implies f_{c}(x_{1},x_{2},x_{3})\leq
f_{c}(x_{0},x_{1},x_{2})$ (9)
for all positive real $x_{0},x_{1},x_{2},x_{3}$. The condition
$x_{0}+x_{1}+x_{2}\to\infty$ in (6) can be replaced by any one of the
following (stronger) conditions: (i) $x_{0}\to\infty$ or (ii) $x_{1}\to\infty$
or (iii) $x_{2}\to\infty$; this of course will replace condition (6) by a
weaker condition, which will make it easier to construct a sub-energy function
$f_{c}$ ,Here are details: Suppose that (8) holds for some function $f_{c}$
such that $f_{c}(x_{0},x_{1},x_{2})\to\infty$ as $x_{0}\to\infty$. Suppose
that, nonetheless, a positive sequence $(x_{0},x_{1},\dots)$ satisfying
condition (4) is unbounded, so that, as $k\to\infty$, one has
$x_{n_{k}}\to\infty$ for some sequence $(n_{k})$ of natural numbers. Then
$f_{c}(x_{n_{k}},x_{1+n_{k}},x_{2+n_{k}})\to\infty$ as $k\to\infty$. This
contradicts (6), which implies, by induction, that
$f_{c}(x_{n},x_{1+n},x_{2+n})\leq f_{c}(x_{0},x_{1},x_{2})$ for all natural
$n$. Quite similarly one can do with (ii) $x_{1}\to\infty$ or (iii)
$x_{2}\to\infty$ in place of (i) $x_{0}\to\infty$.
Also, instead of the dynamics of the triples $(x_{n},x_{1+n},x_{2+n})$ one can
consider the corresponding dynamics (in $n$) of the consecutive $m$-tuples
$(x_{n},\dots,x_{m-1+n})$ for any fixed natural $m$.
Also, instead of inequality $f_{c}(x_{1},x_{2},x_{3})\leq
f_{c}(x_{0},x_{1},x_{2})$ in (6), one may consider a weaker inequality like
$f_{c}(x_{2},x_{3},x_{4})\leq f_{c}(x_{0},x_{1},x_{2})\vee
f_{c}(x_{1},x_{2},x_{3})$ for all positive $x_{0},\dots,x_{4}$ satisfying
condition (4), Thanks to the invariant of Todd’s difference equation (7) which
it is defined in our case to be a sub-energy function such that it is easy to
see that the "if" part of the conjecture would follow since the sub-energy
$f_{c}$ is always found .In ([6], p.35) Authors showed that every positive
solution of dynamics (4) using invariant are bounded and persist this result
is the affirmation that invariant must be a constant sub-energy function which
it is always found for all positive initial conditions [15] One can try to do
the "only if" part in a similar manner. Suppose that $0<\beta\neq\lambda>0$.
Let $u_{n}:=z_{n}/\sqrt{\beta\lambda}$, $c:=\alpha/(\beta\lambda)$, and
$a:=\sqrt{\beta/\lambda}\neq 1$. Then the dynamics (4) can be rewritten as:
$\qquad u_{n+1}=\frac{c+au_{n}+u_{n-1}/a}{u_{n-2}},$ (10)
just with two parameters, $c\geq 0$ and $a>0$. Suppose one can construct, for
each pair $(c,a)\in[0,\infty)\times\big{(}(0,\infty)\setminus\\{1\\}\big{)}$
and some $\rho=\rho_{c,a}\in(1,\infty)$, a "$\rho$-super-energy" function
$g=g_{a,c;\rho}\colon(0,\infty)^{3}\to(0,\infty)$ such that $g$ is bounded on
each bounded subset of $(0,\infty)^{3}$ and
$\qquad
g\Big{(}u_{1},u_{2},\frac{c+au_{2}+u_{1}/a}{u_{0}}\Big{)}\geq\rho\,g(u_{0},u_{1},u_{2})\quad\text{for
all positive $u_{0},u_{1},u_{2}$.}$ (11)
Then, by induction,
$g(u_{n},u_{1+n},u_{2+n})\geq\rho^{n}g(u_{0},u_{1},u_{2})\to\infty$ as
$n\to\infty$, for any sequence $(u_{n})$ satisfying (10). Therefore and
because $g$ is bounded on each bounded subset of $(0,\infty)^{3}$, it would
follow that the sequence $(u_{n})$ is unbounded.
For any pair $(c,a)\in[0,\infty)\times(0,\infty))$ and any
$\rho\in(1,\infty)$, there is no "$\rho$-super-energy" function
$g\colon(0,\infty)^{3}\to(0,\infty)$. This follows because the point
$(u_{a,c},u_{a,c},u_{a,c})$ with
$u_{a,c}:=\dfrac{1+a^{2}+\sqrt{a^{4}+4a^{2}c+2a^{2}+1}}{2a}$ is a fixed point
(in fact, the only fixed point) of the map $T$ given by the formula
$T(u_{0},u_{1},u_{2})=\Big{(}u_{1},u_{2},\dfrac{c+au_{2}+u_{1}/a}{u_{0}}\Big{)}$.
(If $a\neq 1$, then this point is the only fixed point [13] of the map $T^{2}$
as well.)
This also disproves, in general, the "only if" part of the conjecture defined
in (3)
However, One may now try to amend this conjecture by excluding the initial
point $(u_{a,c},u_{a,c},u_{a,c})$. Then, accordingly, the definition of a
"$\rho$-super-energy" function would have it defined on a subset (say $S$) of
the set $(0,\infty)^{3}\setminus\\{(u_{a,c},u_{a,c},u_{a,c})\\}$, instead of
$(0,\infty)^{3}$; such a subset may be allowed to depend on the choice of the
initial point $(u_{0},u_{1},u_{2})$, say on its distance from the fixed point
$(u_{a,c},u_{a,c},u_{a,c})$, and one would then have to also prove that $S$ is
invariant under the map $T$ , M. R. S. KULENOVIC showed in ([10] .p.4 ) that
the construction of lyaponov function is possible for third-order
generalizations of Lyness’ equation ,namely Todd’s equation which it is given
by :
$V(x,y,z)=I(x,y,z)-I(p,p,p)=I(x,y,z)-\dfrac{(p+1)^{4}}{p^{2}}$ (12)
Where $p$ is the equilibrum of Todd’s equation or the dynamics defined in
$(\ref{eq_2})$ Consequently, $p$ is stable, Here $I(x,y,z)$ is the invariant
of todd’s equation which it is defined in (7), Assume
$S=D_{p}=(0,\infty)^{3}\setminus\\{(u_{a,c},u_{a,c},u_{a,c})\\}$ is the
neighborhood of $p$. Since the lyaponove function of the dynamics
$(\ref{eq_7})$ exists [12] and well defined this means that all the following
three conditions are satisfied .See ([11].p.177):
* •
1) $V(p)=I(p)-I(p)=0$
* •
2) $V(S)=I(S)-I(p)>0,\text{for}x\in S$
* •
3) $(VT)(S)=I(T(S))-I(p)=I(S)-I(p)=V(S)$
we use the fact that $I$ is an invariant implies $S$ is invariant under the
map $T$ using both of conditions $2$) and $3$)
Conclusion We disproved the only if part of the titled conjecture using both
of sub energy function and boundedness of Todd’s difference equation then the
conjecture is true only for the if part but not in general ,Existence of sub
energy function implies strongly energy conservation along path of dynamical
system which indicate the stability and boundedness of dynamics ,Conversely
the dynamics would be chaotic .
Acknowledgements The Author thanks Iosif Pinelis from Department of
Mathematical Sciences Michigan Technological University for his help to
contribute the best to this paper .
Funding
No funding supported this research .
Data Availability
The data used to support the findings of this study are available from the
corresponding author upon request
Competing interor the ests The author declare that they have no competing
interests.
Authors’ contributions
Author contributed to the writing of the present article. He also read and
approved the final manuscript
Authors’ information (Optional) University of Batna2.Algeria ,. 53, Route de
Constantine. Fésdis, Batna 05078.Departement of mathematics
High school Hamla3. Benflis Taher
## References
* [1] G. LADAS, G. LUGO AND F. J. PALLADINO OPEN PROBLEMS AND CONJECTURES ON RATIONAL SYSTEMS IN THREE DIMENSIONS, In SARAJEVO JOURNAL OF MATHEMATICS Vol.8 (21) (2012), 311–321
* [2] Y.S. Huang and P.M. Knopf On the boundedness of solutions of a class of third-order rational difference equations, In J. Difference Equ. Appl. 24(10) (2018), pp. 1541–1587
* [3] Ying Sue Huang and Peter M. Knopf Boundedness properties of the generalized Todd’s difference equation, In Journal of Difference Equations and Applications, 25:5, 676-707, 10.1080/10236198.2019.1619711
* [4] E. Camouzis and G. Ladas Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, In Chapman and Hall/CRC, Boca Raton, FL, 2008
* [5] G. Lugo and F.J. Palladino Unboundedness for some classes of rational difference equations, In Int. J. Differ. Equ. 4(1) (2009), pp. 97–113.
* [6] E.A. Grove and G. Ladas Periodicity in nonlinear difference equations, Revista Cubo May 2002, 195–230, In Revista Cubo May 2002, 195–230.
* [7] M.R.S. Kulenovic and G. Ladas Dynamics of Second Order Rational Difference Equations With Open Problems and Conjectures, In Chapman and Hall/CRC Press,
* [8] M.GAO and Y.KATO Some invariants for k’th -order Lyness equation, In Material Engineering, Graduate School of Engineering Hiroshima University, Higashi-Hiroshima, 739-8527, Japan, Applied Mathematics Letters 17 (2004) 1183-1189
* [9] R. Hirota, K. Kimura and H. Yahagi How to find the conserved quantities of nonlinear discrete equations, J. Phys. In A: Math. Gen. 34, 10377-10386, (2001).
* [10] M. R. S. KULENOVIC Invariants and Related Liapunov Functions for Difference Equations , In Department of Mathematics University of Rhode Island Kingston, RI 02881, U,S.A. (Received and accepted January 2000) ,Applied Mathematics Letters 13 (2000) 1-8
* [11] W.G. Kelley and A.C. Peterson Difference Equations, In Academic Press, (1991)
* [12] V. Z. GRINES, M. K. NOSKOVA, AND O. V. POCHINKA THE CONSTRUCTION OF AN ENERGY FUNCTION FOR THREE-DIMENSIONAL CASCADES WITH A TWO-DIMENSIONAL EXPANDING ATTRACTOR, In Article electronically published on November 18, 2015 Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 76 (2015), vyp. 2)
* [13] Senada Kalabušić, Emin Bešo, Naida Mujić and Esmir Pilav Stability analysis of a certain class of difference equations by using KAM theory, In Advances in Difference Equations volume 2019, Article number: 209 (2019)
* [14] G. Bastien and M. Rogalski Results and Conjectures about Order q Lyness’ Difference Equation , with a Particular Study of the Case q = 3 In Advances in Difference Equations Volume 2009, Article ID 134749, 36 pages doi:10.1155/2009/134749
* [15] ANNA CIMA, ARMENGOL GASULL and VÍCTOR MAÑOSA DYNAMICS OF SOME RATIONAL DISCRETE DYNAMICAL SYSTEMS VIA INVARIANTS, In AInternational Journal of Bifurcation and Chaos VOL. 16, NO. 03,https://doi.org/10.1142/S0218127406015027
|
11institutetext: Ruhr University Bochum, Faculty of Physics and Astronomy,
Astronomical Institute, Universitätsstr. 150, 44801 Bochum, Germany
11email<EMAIL_ADDRESS>22institutetext: European Southern Observatory,
Karl-Schwarzschild-Str. 2, 85748, Garching, Germany
22email<EMAIL_ADDRESS>33institutetext: Univ. Lyon, ENS de Lyon, Univ. Lyon
1, CNRS, Centre de Recherche Astrophysique de Lyon, UMR5574, 69007, Lyon,
France
33email<EMAIL_ADDRESS>44institutetext: Cosmic Dawn Center (DAWN),
Denmark 55institutetext: Niels Bohr Institute, University of Copenhagen,
Lyngbyvej 2, DK-2100 Copenhagen, Denmark 66institutetext: Department of
Physics and Astronomy, York University, 4700, Keele Street, Toronto, Ontario,
ON MJ3 1P3, Canada
# Low Surface Brightness Galaxies in $z>1$ Galaxy Clusters:
HST approaches the Progenitors of Local Ultra Diffuse Galaxies
Aisha Bachmann 112As part of the 1st ESO Summer Research Programme2As part of
the 1st ESO Summer Research Programme Remco F. J. van der Burg 22 Jérémy
Fensch 3322 Gabriel Brammer 4455 Adam Muzzin 66
(Submitted 9 December 2020; accepted 15 January 2021 )
Ultra Diffuse Galaxies (UDGs), a type of large Low Surface Brightness (LSB)
galaxies with particularly large effective radii ($r_{\mathrm{eff}}>1.5$ kpc),
are now routinely studied in the local ($z$¡0.1) universe. While they are
found to be abundant in clusters, groups, and in the field, their formation
mechanisms remain elusive and an active topic of debate. New insights may be
found by studying their counterparts at higher redshifts ($z$¿1.0), even
though cosmological surface brightness dimming makes them particularly
difficult to detect and study there. This work uses the deepest Hubble Space
Telescope (HST) imaging stacks of $z$ ¿ 1 clusters, namely: SPT-CL J2106-5844
and MOO J1014+0038. These two clusters, at $z$=1.13 and $z$=1.23, were
monitored as part of the HST See-Change program. Compared to the Hubble
Extreme Deep Field (XDF) as reference field, we find statistical over-
densities of large LSB galaxies in both clusters. Based on stellar population
modelling and assuming no size evolution, we find that the faintest sources we
can detect are about as bright as expected for the progenitors of the
brightest local UDGs. We find that the LSBs we detect in SPT-CL J2106-5844 and
MOO J1014-5844 already have old stellar populations that place them on the red
sequence. Correcting for incompleteness, and based on an extrapolation of
local scaling relations, we estimate that distant UDGs are relatively under-
abundant compared to local UDGs by a factor $\sim$3\. A plausible explanation
for the implied increase with time would be a significant size growth of these
galaxies in the last $\sim$ 8 Gyr, as also suggested by hydrodynamical
simulations.
###### Key Words.:
galaxies: clusters: general – galaxies: dwarfs – galaxies: formation
## 1 Introduction
Dwarf galaxies are showing a vast range of properties in size and luminosity.
Twenty particularly large (~10 kpc) dwarfs with low surface brightness (LSB)
were discovered through extensive photometric studies of the Virgo cluster by
Sandage & Binggeli (1984). An additional 27 examples were found in the Virgo
cluster (Impey et al., 1988) as well as in the Fornax cluster (Ferguson &
Sandage, 1988). While these objects were found in high density environments,
similar low surface brightness objects were also discovered in lower density
environments such as the field (Dalcanton et al., 1997; Román et al., 2019).
More recently, after discovering 47 similar LSB objects (reff
~$3-10^{\prime\prime}$ , or reff ¿ 1.5 kpc, and $\mu(g,0)=24-26$ mag arcsec-2)
in the Coma cluster, the term ”Ultra Diffuse Galaxies” (UDGs) was introduced
for these objects (van Dokkum et al., 2015a). Due to their projected density,
and their spatial coincidence with the Coma cluster, van Dokkum et al. (2015a)
concluded these objects to be a part of the Coma cluster, a statement that was
confirmed by a follow-up spectroscopic study (van Dokkum et al., 2015b).
Even though it is still a challenge to obtain redshift measurements of sizable
samples of UDGs, their overdensity in galaxy clusters allows us to study their
properties in such over-dense environments. An approximately linear dependence
between the abundance of UDGs in galaxy clusters, and the cluster mass was
found (van der Burg et al., 2017; Janssens et al., 2017; Román & Trujillo,
2017). UDGs within clusters appear to be found on the red sequence (van Dokkum
et al., 2015a; van der Burg et al., 2016), while UDG-like galaxies found in
the field appear to be typically bluer (Leisman et al., 2017; Prole et al.,
2019).
Important open questions surrounding the study of UDGs are related to their
origin, for which different theories have been proposed in the literature. van
Dokkum et al. (2015a) suggest that (some) UDGs may have formed in haloes with
masses similar to the Milky Way, but “failed” to form an L* galaxy. UDGs may
also be the extremes in a continuous distribution in dwarf galaxy properties,
having acquired their expanded sizes due to internal (Amorisco & Loeb, 2016;
Di Cintio et al., 2017), or external (Bennet et al., 2018) processes.
Given the suggested low dark-matter content of some field UDGs (van Dokkum et
al., 2018), they may also have formed as tidal dwarf galaxies (Bennet et al.,
2018; Fensch et al., 2019). Since these formation processes happen over
different time scales, observing the evolving properties of UDGs may help
distinguish between different scenarios. To this end, we search for LSB
galaxies in the two galaxy clusters SPTCL-2106-5844 ($z$ = 1.13) and
MOO-1014+0038 ($z$ = 1.23). Using deep HST image stacks for those clusters, we
reach the spatial resolution required to measure their sizes. We discuss our
results in the context of the UDGs found in the local Universe.
This paper is organized as follows. Section 2 provides an overview of the data
we used. Section 3 describes how we select our sample of UDGs. In Sect. 4 we
discuss our results regarding abundance and colour of the sample, and we
summarize in Sect. 5. We adopt the $\Lambda$CDM cosmology with $\Omega_{m}$ =
0.3, $\Omega_{\Lambda}$ = 0.7 and H0 = 70 km s-1 Mpc-1. At the redshift of our
clusters, 1 arcsec corresponds to ~8.2 - 8.4 kpc. For stellar masses we assume
the Initial Mass Function (IMF) from Chabrier (2003). All magnitudes we quote
are in the AB magnitude system.
## 2 Data
We are making use of HST photometry taken as part of the See Change program
(HST GO 13677, 14327; PI: Perlmutter), which targeted galaxy clusters in the
range $1.13<z<1.75$. The main goal of their program was to find high-$z$
supernovae (SN) type Ia, and to use these to constrain the expansion rate of
the universe. The survey strategy has therefore been to take data with a
$\sim$ monthly cadence (e.g. Williams et al., 2020). Rather than using the
individual exposures, we use the image stacks, which reach a combined exposure
time from 3.99 to 5.13 hours in F140W per cluster we examined. In this work we
focus on the two lowest-$z$ clusters from their sample.
The first cluster we analyze is SPT-CL J2106-5844 (hereafter SPTCL-2106) at
redshift $z$=1.13. It was discovered with the South Pole Telescope (SPT),
thanks to its strong Sunyaev-Zel’dovich effect (SZ) signal, yielding a mass
estimate of M200 = (1.27 $\pm$ 0.21) $\times$ $10^{15}$ M⊙ (Foley et al.,
2011). The second cluster is MOO J1014+0038 (MOO-1014) at redshift z = 1.23,
discovered by the Massive and Distant Clusters of WISE Survey (MaDCoWS) based
on a rich overdensity of galaxies (Gonzalez et al., 2019). It has a mass
estimate of M200 = (5.6 $\pm$ 0.6) $\times$ $10^{14}$ Msun and a strong SZ
signature (Brodwin et al., 2015). RGB images of the two clusters are shown in
Figs. 6 & 7.
Figure 1: RGB (red = F140W, green = F105W, blue = F814W) images of members of
the studied clusters. Top: Two spectroscopically-confirmed bright members of
SPTCL-2106 (on the left) and three LSB galaxies that are likely members of
SPTCL-2106 (based on a reference field comparison, on the right). Bottom:
Three LSB galaxies that are likely members of the cluster MOO-1014 (based on a
reference field comparison).
To create the full-depth mosaics of both clusters we start by aligning each of
the multiple SN monitoring “visits” first internally to a catalog of sources
detected in a single F140W visit and then globally to sources matched in the
GAIA DR2 catalog (Gaia Collaboration et al., 2018). We use the AstroDrizzle
software package (Gonzaga & et al., 2012) to identify and mask cosmic rays and
bad pixels in the aligned individual exposures and perform source detection on
the final combined F140W (WFC3/IR) mosaic generated with 60 mas pixels. We
further use F814W (WFC3/UVIS) stacks to provide basic colour information (or
limits on the inferred colour based on the F814W detection limit). The
combination of these filters bridge the 4000Å break at the redshifts of our
clusters, hence providing clues on stellar populations and a way to help
assess the sample purity.
### 2.1 Reference field
To estimate the level of contamination of the sample by foreground and
background objects, we require a field survey with the same filter bands and
image depth. We therefore utilise the
data stacks taken in the Hubble eXtreme Deep
Field111https://archive.stsci.edu/prepds/xdf/ (XDF). These deep stacks are
composed of data from 19 different HST programs covering the Hubble Ultra Deep
Field from 2002 to 2012. For details on the data reduction we refer to
Illingworth et al. (2013).
To ensure similar source detection limits as for the clusters, we add
artificial noise so that the recovered fraction of simulated sources were
similar between the reference and cluster fields (see Sect. 3.3).
Figure 2: Left: Recovery fractions for the simulated sources in SPTCL-2106.
Middle: Same for MOO-1014. Right: Same for the XDF reference field, with noise
level matched to the cluster fields. The colourbar shows the recovery
fractions. While the detection limits are comparable between the panels, the
XDF shows overall a better recovery fraction for brighter targets than in the
clusters due to a higher source crowding in the cluster fields. Also
highlighted is the regime of objects with a surface brightness from 24.0 to
26.5 mag arcsec-2 in F140W and a radius from 1.5 to 7.0 kpc.
## 3 Analysis
### 3.1 Source detection
Sources are detected by running SExtractor (Bertin & Arnouts, 1996) on the
F140W images of the clusters and the XDF. The parameters used to ensure
optimal detection of faint and extended sources in the clusters can be found
in Appendix B, and an identical setup was used for the XDF images. We only
consider sources in the relatively central regions of the cluster stacks,
where exposure time is nearly uniform (cf. Figs. 6 & 7). Examples of detected
objects in each of the two clusters are shown Fig. 1.
### 3.2 Structural parameters
For the detected sources, structural parameters such as magnitude, radius,
ellipticity and Sérsic index of the detected objects are determined using
GALFIT (Peng et al., 2002) on the F140W image after masking neighbouring
objects that are detected by SExtractor. GALFIT is also allowed to
simultaneously fit a constant value to the sky background to improve the
overall fit. In order to measure reliable colours, the F814W fluxes of the
detected objects are measured on the corresponding stack by forcing GALFIT to
use the morphological parameters obtained from the F140W stack, and only fit
the flux normalisation.
To estimate the measurement uncertainty, the GALFIT models are injected on a
hundred different random locations in the corresponding cluster, requiring
that there were no prior detections, and measured exactly as before. In this
way the 1-$\sigma$-uncertainties for size and magnitude in F140W, and
magnitude in F814W, are obtained. We note that GALFIT measured a slightly
lower flux (by about 0.2 mag) and smaller size (by about 0.3 kpc) than the
simulated inputs. In the following, we correct for this small bias.
### 3.3 Image simulations
To assess the completeness of our UDG progenitor selection, we perform all
processing steps also on a range of image simulations. For this we inject
objects with Sérsic profiles on random locations in the HST stacks. We choose
a constant Sérsic-$n$ parameter of unity, which corresponds to typical light
profiles of UDGs measured in the local Universe (e.g. van Dokkum et al.,
2015a; Koda et al., 2015; van der Burg et al., 2016). Sizes are drawn
uniformly between 0$\aas@@fstack{\prime\prime}$1 and
1$\aas@@fstack{\prime\prime}$0, and ellipticities $f$, defined as $f=1-b/a$
with b/a the axis ratio, uniformly between 0.0 and 0.2. Each step was
performed identically on the cluster- and on the reference field image.
Fig. 2 shows the recovered fractions of the inserted objects, for the two
clusters and the reference field. While the detection limits in the different
panels looks similar overall, we note that there is a substantial difference
between the clusters and the reference field. Even for relatively bright
sources, the recovery fraction is lower in the cluster than in the field. This
is expected given the relatively high number of large and bright sources
crowding the cluster stacks. These recovery fractions are used to determine
the limits of our analysis, and to correct the detected sources for
incompleteness, both due to limiting depth and due to crowding/obscuration.
### 3.4 Sample selection
While the definition of UDGs is rather arbitrary, we initially filter the
sample by using a definition similar to that used in the Local Universe:
* •
surface brightness in F140W $\geq$ 24.0 mag arcsec-2 (i.e., not corrected for
surface brightness dimming),
* •
effective radius between 1.5 and 7.0 kpc,
* •
Sérsic index $\leq$ 4.0 to increase the sample purity in favour of sources
reminiscent of UDGs in the local universe,
* •
distance between SExtractor detection and GALFIT position of measurement ¡ 3.0
pixels to ensure that both consider the same object.
A fainter limit for the surface brightness is not included as one purpose of
the study is to test the detection limits of LSB galaxies with the available
data. After detecting sources in direction of the cluster we assume that each
source is at the redshift of the cluster to infer their physical parameters.
This is a valid assumption since we subsequently perform a statistical
subtraction of fore- and background objects, thereby making the same
assumption for sources detected in the reference field.
After discarding failed detections with a by-eye scan we find 95 objects (10
discarded) in SPTCL-2106 (70 in reference field, 6 discarded) and 111 objects
(10 discarded) in MOO-1014 (72 in reference field, 9 discarded). We note that
the number of objects in the reference field depends on the assumed angular
diameter distance, and thus the redshift of the cluster. We conservatively
only consider sources that would have had a detection probability of at least
50$\%$ in the reference field for both cluster and reference field objects and
only count the objects above this limit to ensure a reasonable completeness
correction. For the remaining sources we apply a completeness correction that
is thus at most a factor 2 based on the determined recovery fractions (see
Sect. 3.3). We also apply a correction based on the different sky area covered
by the cluster and reference field. We are left with a statistical count of 99
$\pm$ 10 objects in SPTCL-2106 (67 $\pm$ 8 in reference field) and 90 $\pm$ 10
in MOO-1014 (70 $\pm$ 8 in reference field).
Colour-Magnitude diagrams for all the detected sources falling within the
50$\%$-limit (see Fig. 8) in both clusters and the reference field are shown
in the Appendix C. We subtract the reference field galaxies from the nearest
cluster object in regards of colour (F814W-F140W) and magnitude (F140W). For
this we use the colour and magnitude measurements for each cluster object as
coordinates and subtracted the reference field object from the nearest cluster
object. The relative weights, which include corrections for incompleteness and
different covered sky areas, are taken into account. For more details on the
subtraction of the reference field we refer to van der Burg et al. (2016).
After the subtraction we are left with a statistical count of 32 $\pm$ 13 in
SPTCL-2106 and 20 $\pm$ 13 in MOO-1014.
Figure 3: A selection by size and apparent magnitude, in F140W, of our LSB
galaxies. The panels show the different clusters. Red: LSB galaxies detected
following our selection criteria and with a weight higher than 0.5. Blue and
Grey: the samples by van Dokkum et al. (2015a) and Mobasher et al. (2001),
both shifted to our observed redshift by accounting for an E+K correction (see
Sect. 4). We plot curves of constant surface brightness,
$\mathrm{\mu(r_{eff,circ}})$ = 24.0, 26.0 mag arcsec-2 evolved from the Coma
cluster redshift to the high-$z$ clusters. Average errorbars for our sample
are shown. Figure 4: Same es Fig. 3 but for both clusters combined and with
the sample by van Dokkum et al. (2015a) with the size evolution taken out.
Cyan: the sample by van Dokkum et al. (2015a), where the suggested size
evolution since $z\sim 1$ (cf. Sect. 4.3) is taken out. Figure 5: Colour-
Magnitude diagrams for the sample of LSB candidates in the clusters after
correcting for fore- and background interlopers and incompleteness. The
colourbar gives the residual weight of the data points (as detailed in Sect.
3.4). The left panel shows the cluster SPTCL-2106 and the right panel the
cluster MOO-1014. Additionally shown in the left panel are the positions in
the colour-magnitude diagram of spectroscopically-confirmed members of
SPTCL-2106, and an extrapolation of the red sequence as defined by the
confirmed members of SPTCL-2106, ignoring some of the data points that may be
part of a bluer/starforming cluster population to guide the reader’s eye. This
line has a slope of -0.12.
## 4 Results and Discussion
### 4.1 Comparison with local UDGs
To be able to compare our LSB candidates to likely progenitors of UDGs studied
in the local universe, we evolve local UDGs back to the redshift of our
clusters following a simple stellar population model. The model assumes a
passively-evolving stellar population that was formed at
$z_{\mathrm{form}}=1.5$, which is in line with intermediately old ages of UDGs
measured locally (Ferré-Mateu et al., 2018; Ruiz-Lara et al., 2018; Fensch et
al., 2019). It is based on stellar population synthesis models from Bruzual &
Charlot (2003), a star formation history $\mathrm{SFR}\propto e^{-t/\tau}$
with a short e-folding time of $\tau$ = 10 Myr, a Chabrier (2003) initial mass
function and no dust extinction. We use the magnitudes, physical sizes and
filters of van Dokkum et al. (2015a) as anchor point, and estimate how those
UDGs would appear at the redshift of our cluster as observed through the
WFC3/F140W filter. This estimate accounts, by construction, for surface
brightness dimming. The 47 UDG candidates are shown in Fig. 3. Additionally we
evolved the dwarf and giant galaxies found by Mobasher et al. (2001) in the
Coma cluster back to the redshift of our clusters in the same way and plotted
them as well in Fig. 3.
Figure 3 shows that the LSB samples of our clusters lie in the area between
the samples by van Dokkum et al. (2015a) and Mobasher et al. (2001), making
them fainter than the progenitors of normal dwarf and giant galaxies in the
Coma cluster, and almost as faint as the expected progenitors of the UDGs
studied by van Dokkum et al. (2015a). We can see a small overlap between our
samples and the compared objects from both other studies. As a reference, we
also plot the curves of constant surface brightness,
$\mu(r,r_{\mathrm{eff,circ}})=24.0,26.0$ mag arcsec-2, which is a common
selection boundary for UDGs in the local universe, also evolved to the
redshifts of our clusters. This indicates that only half of the objects we are
able to detect in both clusters could classify as progenitors of the brightest
UDGs known in the local universe, based on this evolution model.
We note that, while projecting the local Coma galaxies back to higher
redshift, we have only evolved their fluxes and ignored any potential size
evolution.However, numerical simulations suggest that a typical UDG may see
its radius increase with age from around 2.5 - 3 kpc at $z$ = 1 to 4 - 5 kpc
at $z$ = 0 (Martin et al., 2019; Wright et al., 2021). Accounting for such an
expansion would bring the data points from van Dokkum et al. (2015a), when
evolved to the redshift of our clusters, downward, and thus closer to our data
points. A possible size evolution is described in Sect. 4.3 and the sample by
van Dokkum et al. (2015a) effected by it is plotted in Fig. 4.
### 4.2 Colour
Figure 5 shows the colours and magnitudes of the sample of LSB galaxies. The
residual weight shows the number of galaxies we expect in the observed cluster
area per detected object after accounting for the subtraction of the reference
field. Also shown in Fig. 5 are the positions in the colour-magnitude diagram
of galaxies which were spectroscopically confirmed as members of SPTCL-2106
(by the GOGREEN collaboration, Balogh et al., 2021). The colours were measured
in the same filter bands and following an identical method as for the LSB
candidates. Ignoring some of the data points that may be part of a bluer/star-
forming cluster population, we note that the bulk of the LSB galaxies lie on
an extended red-sequence, shown as a pink-dashed line, as defined by the
brighter cluster galaxies. We note that the slope of this estimate red-
sequence, -0.12, is consistent with $z\sim 1$ estimates by e.g. Bell et al.
(2004). No similar population of bright cluster members has been
spectroscopically identified for MOO-1014, so that we plot the same red-
sequence estimate for this cluster, ignoring the small redshift difference
between the two clusters. The LSB galaxies of SPTCL-2106 and MOO-1014 show
colours that are consistent with the red sequence, suggesting that these
galaxies are likely quenched and have thus already stopped forming stars.
### 4.3 Abundance comparison with local universe UDGs
To put the measured abundance of LSB galaxies in our clusters into context, we
compare it to the abundance of UDGs in local clusters, within projected
$R<R_{200}$. For this, we have to make assumptions regarding the underlying
magnitude and size distribution of dwarf galaxies, and how these evolve with
redshift. We assume a flat magnitude distribution for different size bins
(consistent with what is observed in the Coma cluster, by Danieli & van
Dokkum, 2019), and the same size distribution as measured for UDGs in local
clusters (for radii between 1.5 and 7.0 kpc van der Burg et al., 2016). Based
on Fig. 2 we can also assume that our samples in the range with surface
brightness from 24.0 to 26.5 mag in F140W and a radius from 1.5 to 7.0 kpc are
mostly complete for the parameter range studied of both clusters.
The available HST imaging does not allow us to probe radii out to $R_{200}$,
but only to radii corresponding to $\sim 0.35\cdot R_{200}$ for SPTCL-2106 and
$\sim 0.50\cdot R_{200}$ for MOO-1014. To correct for the missing area, we
assume that LSBs approximately trace the overall matter distribution in the
cluster, which is described by an NFW (Navarro et al., 1997) profile with
concentration $c_{200}$=3 (e.g. Duffy et al., 2008). Integrating this profile
along the line-of-sight indicates that we probe a fraction of $\sim 0.4\pm
0.1$ of the LSB population in SPTCL-2106 and $\sim 0.55\pm 0.1$ in MOO-1014.
Based on the assumed magnitude and size distribution222We considered
uncertainties in the assumed size distribution (as measured in van der Burg et
al., 2016) and magnitude distribution (as measured in Danieli & van Dokkum,
2019), finding that these affect our estimated number of high-z UDGs by at
most 14%, hence not impacting our conclusions., and after applying the needed
correction for missed area, we would estimate a total number of 80 $\pm$ 38
UDGs in SPTCL-2106 and 36 $\pm$ 25 UDGs in MOO-1014. Studies of local clusters
suggest an abundance of $\sim$100-200 in clusters of this mass, being three
times higher than our best estimate for the clusters we study. This thus
implies a substantial increase in the UDG abundance with time since $z\sim 1$.
A possible explanation for this implied evolution is that we are assuming the
UDG progenitors to be of the same size as local Universe UDGs. If their
progenitors were actually smaller at $z$ = 1 (as suggested by several
simulations, cf. Martin et al., 2019; Wright et al., 2021) they would not fall
into our selection criteria and thus be missed in the current analysis.
Assuming the size distribution of UDGs in the local universe, as described by
the power law shown in Fig. 7 of van der Burg et al. (2016), we find that a
size growth by a factor $\sim$1.4 of all galaxies may have boosted the number
of galaxies classified as UDGs by a factor $\sim$3 since $z\sim 1$.
Hydrodynamical simulations by Martin et al. (2019) would predict a slightly
larger size growth by a factor $\sim$1.8. This suggests that size evolution is
sufficient to explain the observed underabundance of UDGs.
## 5 Summary and outlook
This paper studies the abundance of LSB galaxies in two $z>1$ clusters,
SPTCL-2106 and MOO-1014, down to the detection limit of the available deep HST
imaging data. We correct for background objects by comparing the detections
with those measured in the XDF as reference field. Simulations are run to
estimate completeness limits, and to tailor the depth of the reference field
to the cluster imaging. We summarise our main conclusions as follows:
* •
Within the parameter space we defined, we find a statistical overdensity of 32
$\pm$ 13 LSB galaxies in SPTCL-2106 and 20 $\pm$ 13 in MOO-1014.
* •
We find the colours of those LSB galaxies in SPTCL-2106 and MOO-1014 to be
consistent with an extension of the red sequence, as defined by
spectroscopically identified brighter cluster members. This suggests that the
LSB galaxies in both clusters are already evolving passively.
* •
Based on a simple stellar population evolution model, we compare our detected
LSB galaxies with the expected progenitors of local UDGs in the Coma cluster.
This suggests that the faintest sources we can detect approximate the expected
progenitors of local UDGs.
* •
Based on an extrapolation, motivated by local scaling relations, we estimate
an overall abundance of 80 $\pm$ 38 UDGs in SPTCL-2106 and 36 $\pm$ 25 UDGs in
MOO-1014. We note that this is about three times lower than the abundance of
UDGs in local galaxy clusters having similar masses.
* •
One way to interpret the implied evolution is by assuming a substantial size
growth of dwarf galaxies since $z\sim 1$, which would then increase the
numbers of those that classify as UDGs. As we discuss, this is qualitatively
consistent with results from hydrodynamical simulations.
We stress that this study uses the deepest data available for galaxy clusters
at $z$ ¿ 1 that still allow to spatial resolve galaxies of 1-2 kpc, and thus
reaches the limit of current instrumentation. For further studies on the
existence and properties of distant UDGs, data with higher spatial resolution
and depth is needed, being within reach of the next generation of telescopes.
###### Acknowledgements.
We thank the anonymous referee for their useful comments that substantially
clarified the paper. AB acknowledges a 6-week ESO summer studentship during
which a substantial part of this research was done.
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## Appendix A RGB images of the clusters
Figure 6: RGB (red = F140W, green = F105W, blue = F814W) image of the cluster
SPTCL-2106. Figure 7: RGB (red = F140W, green = F105W, blue = F814W) image of
the cluster MOO-1014.
## Appendix B SExtractor parameters
Table 1: SExtractor parameters used. All other parameters were left to their defaults. Parameter | Value
---|---
DETECT_MINAREA | 7
DETECT_THRESH | 1.1
ANALYSIS_THRESH | 1.1
BACK_TYPE | MANUAL
BACK_VALUE | 0
FILTER_TYPE | GAUSSIAN
FILTER | default
## Appendix C Additional Figures
Figure 8: Colour-Magnitude diagrams for the sample of LSB candidates in the
clusters and the reference field that fall within the 50% detection limit.
These are the raw numbers, without accounting for incompleteness, or
background interlopers. The left panel shows the cluster SPTCL-2106 and the
right panel shows the cluster MOO-1014. Additionally shown in the left panel
are the positions in the colour-magnitude diagram of spectroscopically-
confirmed members of SPTCL-2106 and the same extrapolation of the red-sequence
as described in Fig. 5.
|
# Know the enemy: 2D Fermi liquids
Sankar Das Sarma Yunxiang Liao Condensed Matter Theory Center and Joint
Quantum Institute,
Department of Physics, University of Maryland, College Park, MD 20742, USA
###### Abstract
We describe an analytical theory investigating the regime of validity of the
Fermi liquid theory in interacting, via the long-range Coulomb coupling, two-
dimensional Fermi systems comparing it with the corresponding 3D systems. We
find that the 2D Fermi liquid theory and 2D quasiparticles are robust up to
high energies and temperatures of the order of Fermi energy above the Fermi
surface, very similar to the corresponding three-dimensional situation. We
calculate the phase diagram in the frequency-temperature space separating the
collisionless ballistic regime and the collision-dominated hydrodynamic regime
for 2D and 3D interacting electron systems. We also provide the temperature
corrections up to third order for the renormalized effective mass, and comment
on the validity of 2D Wiedemann-Franz law and 2D Kadawoki-Woods relation.
###### keywords:
Fermi liquid , electron self-energy , random phase approximation
††journal: Annals of Physics
###### Contents
1. 1 Introduction
2. 2 Theory
1. 2.1 General formulas for electron self-energy
1. 2.1.1 Keldysh approach to interacting electrons
2. 2.1.2 RPA dynamically screened interaction
3. 2.1.3 RPA self-energy
2. 2.2 2D electron self-energy
1. 2.2.1 Momentum integration
2. 2.2.2 Frequency integration
3. 2.3 3D electron self-energy
4. 2.4 Discussion of the analytical results
3. 3 Results
1. 3.1 Applicability of the FL theory
2. 3.2 Effective mass
3. 3.3 Hydrodynamic and ballistic regimes
4. 4 Wiedemann-Franz (WF) and Kadowaki-Woods (KW) relations for 2D interacting systems
1. 4.1 WF Law
2. 4.2 KW law
5. 5 Conclusion
6. A Derivation of self-energy formulas using Matsubara technique
7. B Integrals involving hyperbolic functions
## 1 Introduction
In a famous talk at the 1989 Kathmandu Summer School, Phil Anderson questioned
the validity of the Fermi liquid theory and the applicability of the
quasiparticle concept to high-temperature cuprate superconductors specifically
and to 2D interacting electron systems generally [1]. One of his lectures,
which was also quoted in his published lecture note, has the memorable slogan
“Know the enemy”, alluding specifically to the Fermi liquid theory as the
enemy. Anderson was the first person vehemently and tirelessly pushing the
idea that the 2D physics of cuprate superconductors is beyond the Fermi
liquid-BCS paradigm, and represents new physics, which is now commonly
referred to as non-Fermi liquids, a terminology virtually unknown in 1989.
The basic challenges Anderson posed were simple: (1) Is it possible that
interactions destroy the 2D Fermi surface just as they do in one dimension
leading to a Luttinger liquid? (2) Is it possible that cuprates represent a
new emergent form of superconductivity which simply cannot be explained and
understood using the highly successful Fermi liquid-BCS formalism of electron
pairing around the Fermi surface leading to a superconducting instability of
Copper pairs condensing into a BCS ground state?
Amazingly, there is still no answer to the second question even after 30 years
and many thousands of theoretical papers as there is no consensus on the
accepted theory of cuprate superconductivity. In fact, even the precise
mechanism for cuprate superconductivity is still actively debated in the
theoretical community, and Anderson himself worked on developing an
appropriate theory for the cuprates for the rest of his life. It is, however,
a great testimonial to Anderson’s early insight that most theorists working on
cuprate superconductivity accept that its explanation most likely lies outside
the standard Fermi liquid-BCS paradigm. Our current work is on the continuum
2D interacting Coulombic electron system whereas most work on cuprates uses
lattice models with short-range interactions.
But the answer to Anderson’s first question is definitively known. Two
dimensional interacting Fermi systems are Fermi liquids similar to 3D
interacting Fermi systems, and not non-Fermi liquids like interacting one
dimensional Luttinger liquids. In fact, Anderson’s trenchant questioning of
the nature of interacting 2D electron systems (“Know the enemy”) led to many
theoretical developments establishing conclusively, and in fact, even
rigorously, that interacting 2D electron systems are indeed normal Fermi
liquids with well-defined renormalized Fermi surfaces very much like 3D normal
metals and normal He-3 [2, 3, 4]. The basic idea, which is now established
with reasonable mathematical rigor, is rather simple and essentially the same
as it is for 3D interacting Fermi systems. The imaginary part of the
interacting 2D self-energy at energy $E$ goes as $(E-E_{\mathsf{F}})^{2}$ up
to some logarithmic corrections, with $E_{\mathsf{F}}$ being the Fermi energy,
and this holds to all orders in perturbation theory. Thus, the single-particle
spectral function is a delta function at the Fermi surface, guaranteeing the
existence of an interacting Fermi surface and low-energy quasiparticles with
one to one correspondence to the noninteracting Fermi gas. This behavior is
similar to 3D systems within logarithmic accuracy, and thus 2D systems are
similar to 3D systems. This is very different from interacting 1D systems,
which within the same perturbation theory gives an imaginary self-energy going
as $(E-E_{F})(\ln(E-E_{F}))^{1/2}$ [5]. Although a perturbation theory is
neither meaningful nor valid for a 1D system, it is clear that the Fermi
surface cannot exist in 1D already based on this simple perturbative argument.
On the other hand, the perturbative analysis applies in 2D and shows that
Fermi surface exists in the 2D interacting systems just as in 3D.
In the current work, we extend Anderson’s first question, trying to understand
‘the enemy’ better. The question we ask is the extent to which the Fermi
liquid theory applies in 2D electron liquids in the presence of long-range
Coulomb interactions. In particular, how far in energy from the Fermi surface
and how high in temperature can we go and still find well-defined
quasiparticles in 2D interacting systems? What is the regime of applicability
of the concept of 2D Fermi liquids? We answer these questions analytically,
both in 2D and 3D comparing the two situations, using a many-body perturbation
theory which is exact for the Coulomb-interacting system in the high-density
limit.
We also calculate the temperature dependent effective mass renormalization,
comment on the 2D Wiedemann-Franz law and Kadowaki-Woods relation, and
estimate the hydrodynamic regime in Fermi systems interacting through long-
range Coulomb interactions.
Our work is completely analytical involving expansions in inverse density,
energy, and temperature on an equal footing. The theory itself uses the well-
established leading-order dynamical screening or random phase approximation
(RPA) for the self-energy, which is exact in the high-density limit.
## 2 Theory
In this section, we derive the self-energy for an electron system with long-
range Coulomb interactions in both 2D and 3D [6, 7, 8, 9]. In particular, we
provide the analytical expressions for both the real and imaginary parts of
the on-shell self-energy, for arbitrary energy-to-temperature ratio
$\varepsilon/T$. Here, $\varepsilon(=E-E_{\mathsf{F}})$ is the quasiparticle
energy measured from the Fermi surface and $T$ is the temperature. The result
is valid to the leading order in $r_{s}$ and to several orders in
$\varepsilon/E_{\mathsf{F}}$ and $T/T_{\mathsf{F}}$. Here, $r_{s}$ is the
standard dimensionless Coulomb coupling parameter (the ratio of the
interparticle separation to the effective Bohr radius), and $T_{\mathsf{F}}$
is the noninteracting Fermi temperature. As usual, $r_{s}$ is the many body
perturbation parameter (or the density-dependent effective fine structure
constant) for the theory which is strictly valid only for $r_{s}\ll 1$, but in
practice works well empirically for metallic electron densities $(r_{s}\sim
3-6)$ [10]. In Sec. 2.1, we start with reviewing the derivation of general
self-energy formulas which are expressed as integrals involving RPA
dynamically screened interaction and are used to extract the explicit self-
energy expressions. Sec. 2.2 is devoted to the detailed calculation of the 2D
electron self-energy, which is presented before in Ref. [9] and reviewed here
for completeness. In Sec. 2.3, we provide the analytical expressions for the
3D electron self-energy when energy $\varepsilon$ and temperature $T$ are
arbitrary with respect to each other, which, to the best of our knowledge, are
new for long-range Coulomb interactions. See Refs. [11, 12, 13] for the
results of self-energy for the case of short-range interactions in both 2D and
3D. We also present the subleading terms in $\varepsilon/E_{\mathsf{F}}$
($T/T_{\mathsf{F}}$) for the 3D self-energy in the $\varepsilon\gg T$
($T\gg\varepsilon$) limit.
### 2.1 General formulas for electron self-energy
#### 2.1.1 Keldysh approach to interacting electrons
In the framework of Keldysh formalism (see Ref. [14] for a review), we now
rederive the general formulas for self-energy which are then used to obtain
the analytic expressions presented in Secs. 2.2 and 2.3. An alternative
derivation employing Matsubara technique (see for example Refs. [6, 7]) will
be reviewed in Appendix A.
We start with the partition function $Z$ of a $d-$dimensional electron system
with Coulomb interactions on a Keldysh contour which runs from $t=-\infty$ to
$t=\infty$ and back to $t=-\infty$. The system is assumed to be in the thermal
equilibrium and noninteracting at the distant past $t=-\infty$, after which
the interactions are then adiabatically switched on. In terms of coherent
state functional integral, the partition function can be expressed as
$\displaystyle\begin{aligned}
Z=&\int\mathcal{D}\left(\bar{\psi},\psi\right)\,\exp\left(iS_{0}+iS_{\mathsf{int}}\right),\end{aligned}$
(2.1)
where $S_{0}$ and $S_{\mathsf{int}}$, which stand for the free and interacting
parts of the action, respectively, are given by
$\displaystyle\begin{aligned}
S_{0}=\,&\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}dt^{\prime}\int
d^{d}\bm{\mathrm{r}}\int
d^{d}\bm{\mathrm{r}}^{\prime}\bar{\psi}(\bm{\mathrm{r}},t)\;\hat{G}_{0}^{-1}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})\;\psi(\bm{\mathrm{r^{\prime}}},t^{\prime}),\\\
S_{\mathsf{int}}=\,&-{\frac{1}{2}}\,\sum_{a=\pm}\zeta_{a}\int_{-\infty}^{\infty}dt\int
d^{d}\bm{\mathrm{r}}\int
d^{d}\bm{\mathrm{r}}^{\prime}\,\bar{\psi}_{\sigma}^{a}(\bm{\mathrm{r}},t)\bar{\psi}_{\sigma^{\prime}}^{a}(\bm{\mathrm{r}}^{\prime},t)V(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime})\psi_{\sigma^{\prime}}^{a}(\bm{\mathrm{r}}^{\prime},t)\psi_{\sigma}^{a}(\bm{\mathrm{r}},t).\end{aligned}$
(2.2)
Here $\psi^{+}$($\bar{\psi}^{+}$) and $\psi^{-}$($\bar{\psi}^{-}$) represent
Grassmann fields residue on the forward and backward paths of the Keldysh
contour, respectively. They are also labeled by a spin index
$\sigma=\uparrow,\downarrow$. The overall sign factor $\zeta_{a}$ takes the
value of $1$ ($-1$) for $a=+$ ($-$). $V(\bm{\mathrm{r}})=e^{2}/r$ is the bare
Coulomb interaction potential and its Fourier transform is
$V(\bm{\mathrm{q}})=2\pi e^{2}/q$ ($V(\bm{\mathrm{q}})=4\pi e^{2}/q^{2}$) in
2D (3D). In the present paper, we work in the units of
$\hbar=k_{\mathsf{B}}=1$, and use $E_{\mathsf{F}}$ and $T_{F}$
interchangeably. $G_{0}$ is the noninteracting contour ordered electron
Green’s function, and has the following structure in Keldysh space:
$\displaystyle\begin{aligned}
\hat{G}_{0}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})=-i\braket{\psi(\bm{\mathrm{r}},t)\bar{\psi}(\bm{\mathrm{r}}^{\prime},t^{\prime})}_{0}=\begin{bmatrix}G_{0}^{(\mathrm{T})}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})&G_{0}^{(<)}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
G_{0}^{(>)}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})&G_{0}^{(\bar{\mathrm{T}})}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})\end{bmatrix},\end{aligned}$
(2.3)
with $G_{0}^{(\mathrm{T})}$, $G_{0}^{(\bar{\mathrm{T}})}$, $G_{0}^{(<)}$ and
$G_{0}^{(>)}$ being the time-ordered, anti-time-ordered, lesser, and greater
noninteracting electron Green’s function, respectively. The angular bracket
with subscript $0$ indicates functional integration over Grassmann fields
$\psi$ and $\bar{\psi}$ with weight $e^{iS_{0}}$ (Eq. 2.2).
With the help of a real bosonic field $\phi$, we can perform a Hubbard-
Stratonovich transformation to decouple the interactions
$\displaystyle\begin{aligned}
&e^{iS_{\mathsf{int}}}=\int\mathcal{D}\phi\exp\left[\frac{i}{2}\sum_{a=\pm}\zeta_{a}\int\limits_{\bm{\mathrm{q}},\omega}\phi_{a}(\bm{\mathrm{q}},\omega)V^{-1}(\bm{\mathrm{q}})\phi_{a}(-\bm{\mathrm{q}},-\omega)-i\sum_{a=\pm}\zeta_{a}\int\limits_{\varepsilon,\bm{\mathrm{k}}}\int\limits_{\omega,\bm{\mathrm{q}}}\phi^{a}(\bm{\mathrm{q}},\omega)\bar{\psi}^{a}_{\sigma}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\psi^{a}_{\sigma}(\bm{\mathrm{k}},\varepsilon)\right].\end{aligned}$
(2.4)
Here we have used the short-hand notations $\int_{\bm{\mathrm{k}}}\equiv\int
d^{d}\bm{\mathrm{k}}/(2\pi)^{d}$ and
$\int_{\omega}\equiv\int_{-\infty}^{\infty}d\omega/(2\pi)$. Introducing the
classical and quantum components of the bosonic field $\phi$ :
$\displaystyle\phi_{\mathsf{cl}}=\left(\phi_{+}+\phi_{-}\right)/\sqrt{2},\qquad\phi_{\mathsf{q}}=\left(\phi_{+}-\phi_{-}\right)/\sqrt{2},$
(2.5)
we rewrite Eq. 2.4 as
$\displaystyle\begin{aligned}
e^{iS_{\mathsf{int}}}=\,\int\mathcal{D}\phi\,\exp&\left[i\int\limits_{\bm{\mathrm{q}},\omega}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)V^{-1}(\bm{\mathrm{q}})\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega)-\frac{i}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\varepsilon,\bm{\mathrm{q}},\omega}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)\,\bar{\psi}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{\tau}^{3}\psi(\bm{\mathrm{k}},\varepsilon)\right.\\\
&\left.-\frac{i}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\varepsilon,\bm{\mathrm{q}},\omega}\phi_{\mathsf{q}}(\bm{\mathrm{q}},\omega)\,\bar{\psi}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\,\psi(\bm{\mathrm{k}},\varepsilon)\right],\end{aligned}$
(2.6)
where $\hat{\tau}^{i}$ denotes the Pauli matrix in the Keldysh space.
It is usually convenient to apply the Keldysh rotation to the fermionic fields
$\psi$ and $\bar{\psi}$:
$\displaystyle\begin{aligned}
\psi(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\frac{\hat{\tau}^{3}+\hat{\tau}^{1}}{\sqrt{2}}\psi(\bm{\mathrm{k}},\varepsilon),\qquad\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\,\frac{\hat{1}-i\hat{\tau}^{2}}{\sqrt{2}},\end{aligned}$
(2.7)
after which the bare electron Green’s function $\hat{G}_{0}$ assumes the
following form in the Keldysh space
$\displaystyle\begin{aligned}
\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)=\,\begin{bmatrix}G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)&G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)\\\
0&G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\end{bmatrix}.\end{aligned}$ (2.8)
In the present paper, we use superscripts $``R"$, $``A"$ and $``K"$ to denote
the retarded, advanced and Keldysh components respectively. The three
components of the bare electron Green’s function
$\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)$ are connected by the fluctuation-
dissipation theorem (FDT):
$\displaystyle
G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)=\left[G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)-G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\right]\tanh\left(\frac{\varepsilon}{2T}\right),$
(2.9)
and are given by, respectively,
$\displaystyle\begin{aligned}
&G_{0}^{(R/A)}(\bm{\mathrm{k}},\varepsilon)=\left(\varepsilon-\xi_{\bm{\mathrm{k}}}\pm
i\eta\right)^{-1}\\!\\!\\!\\!\\!\\!\\!,\\\
&G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)=-2\pi
i\delta\left(\varepsilon-\xi_{\bm{\mathrm{k}}}\right)\tanh\left(\frac{\varepsilon}{2T}\right).\end{aligned}$
(2.10)
Here $\eta$ is a positive infinitesimal, and $\xi_{\bm{\mathrm{k}}}\equiv
k^{2}/2m-\mu$, with $\mu$ being the chemical potential and $m$ the bare mass.
We note that $G_{0}^{(R/A)}(\bm{\mathrm{k}},\varepsilon)$ contains information
about the spectrum only, while $G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)$
knows also about the distribution function.
We then apply another transformation to the fermionic fields $\psi$ and
$\bar{\psi}$
$\displaystyle\begin{aligned}
\psi(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\hat{M}_{F}(\varepsilon)\,\psi(\bm{\mathrm{k}},\varepsilon),\qquad\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\,\hat{M}_{F}(\varepsilon),\end{aligned}$
(2.11)
where $\hat{M}_{F}(\varepsilon)$ is distribution function dependent and is
defined as, in Keldysh space,
$\displaystyle\begin{aligned}
\hat{M}_{F}(\varepsilon)=\begin{bmatrix}1&\tanh\left(\varepsilon/2T\right)\\\
0&-1\end{bmatrix}.\end{aligned}$ (2.12)
The advantage of this transformation is that the noninteracting electron
Green’s function becomes diagonal and acquires the form
$\displaystyle\begin{aligned}
\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)=\,\begin{bmatrix}G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)&0\\\
0&G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\end{bmatrix}.\end{aligned}$ (2.13)
After this transformation, the partition function can now be expressed as
$\displaystyle\begin{aligned}
&Z=\,\int\mathcal{D}\left(\bar{\psi},\psi\right)\mathcal{D}\phi\,\exp\left(iS_{\psi}+iS_{\phi}+iS_{c}\right),\\\
&S_{\phi}=\,\int\limits_{\bm{\mathrm{q}},\omega}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)V^{-1}(q)\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega),\\\
&S_{\psi}=\,\int\limits_{\bm{\mathrm{k}},\varepsilon}\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\left(\varepsilon-\xi_{\bm{\mathrm{k}}}+i\eta\hat{\tau}^{3}\right)\psi(\bm{\mathrm{k}},\varepsilon),\\\
&S_{c}=\,-\frac{1}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\bm{\mathrm{k}}^{\prime},\varepsilon,\varepsilon^{\prime}}\phi_{\mathsf{cl}}(\bm{\mathrm{k}}-\bm{\mathrm{k}}^{\prime},\varepsilon-\varepsilon^{\prime})\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\hat{M}_{F}(\varepsilon)\hat{M}_{F}(\varepsilon^{\prime})\psi(\bm{\mathrm{k}}^{\prime},\varepsilon^{\prime})\,\\\
&\qquad-\frac{1}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\bm{\mathrm{k}}^{\prime},\varepsilon,\varepsilon^{\prime}}\phi_{\mathsf{q}}(\bm{\mathrm{k}}-\bm{\mathrm{k}}^{\prime},\varepsilon-\varepsilon^{\prime})\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\hat{M}_{F}(\varepsilon)\hat{\tau}^{1}\hat{M}_{F}(\varepsilon^{\prime})\psi(\bm{\mathrm{k}}^{\prime},\varepsilon^{\prime}).\end{aligned}$
(2.14)
#### 2.1.2 RPA dynamically screened interaction
To proceed, we first integrate out the fermionic fields $\psi$ and
$\bar{\psi}$ to get an effective theory of the bosonic field $\phi$:
$\displaystyle\begin{aligned}
&Z=\int\mathcal{D}\phi\,\exp\left(iS_{\phi}+\ln\braket{\exp(iS_{c})}_{\psi}\right).\end{aligned}$
(2.15)
Here the angular bracket with a subscript $\psi$ is used to indicate the
functional averaging over the fermionic fields $\psi$ and $\bar{\psi}$ with
the weight $\exp\left(iS_{\psi}\right)$, so $\braket{\exp(iS_{c})}_{\psi}$
represents
$\displaystyle\begin{aligned}
&\braket{\exp(iS_{c})}_{\psi}=\,\int\mathcal{D}\left(\bar{\psi},\psi\right)\exp\left(iS_{\psi}+iS_{c}\right).\end{aligned}$
(2.16)
Within the framework of the RPA approximation, we approximate
$\braket{\exp(iS_{c})}_{\psi}$ by the leading order term in the cumulant
expansion:
$\displaystyle\begin{aligned}
&\ln\left\langle\exp(iS_{c})\right\rangle_{\psi}\approx\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}.\end{aligned}$
(2.17)
The effective action $\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}$
can be expressed in terms of the polarization operator $\hat{\Pi}$ which can
be considered as the self-energy of the bosonic field
$\phi=(\phi_{\mathsf{cl}},\phi_{\mathsf{q}})^{T}$:
$\displaystyle\begin{aligned}
\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}=-\frac{i}{2}&\int_{\bm{\mathrm{q}},\omega}\phi^{\mathsf{T}}(-\bm{\mathrm{q}},-\omega)\hat{\Pi}(\bm{\mathrm{q}},\omega)\phi(\bm{\mathrm{q}},\omega),\\\
\Pi^{ab}(\bm{\mathrm{q}},\omega)=\,-i\int\limits_{\bm{\mathrm{k}},\varepsilon}&\operatorname{Tr}\left\\{\left[\frac{1+\zeta_{a}}{2}+\frac{1-\zeta_{a}}{2}\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\right.\\\
&\qquad\left.\times\left[\frac{1+\zeta_{b}}{2}+\frac{1-\zeta_{b}}{2}\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon)\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)\hat{M}_{F}(\varepsilon)\right\\}.\end{aligned}$
(2.18)
Utilizing the causality relation
$\int_{\bm{\mathrm{k}},\varepsilon}G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)=0,$
and inserting the explicit expression for $\hat{M}_{F}(\varepsilon)$ (Eq.
2.12) into Eq. 2.18, one can verify that the polarization operator $\hat{\Pi}$
acquires the following triangular structure in the Keldysh space:
$\displaystyle\begin{aligned}
&\hat{\Pi}(\bm{\mathrm{q}},\omega)=\,\begin{bmatrix}0&\Pi^{(A)}(\bm{\mathrm{q}},\omega)\\\
\Pi^{(R)}(\bm{\mathrm{q}},\omega)&\Pi^{(K)}(\bm{\mathrm{q}},\omega)\end{bmatrix},\end{aligned}$
(2.19)
and its components are connected by the FDT for bosons
$\displaystyle\begin{aligned}
&\Pi^{(K)}(\bm{\mathrm{q}},\omega)=\,\left[\Pi^{(R)}(\bm{\mathrm{q}},\omega)-\Pi^{(A)}(\bm{\mathrm{q}},\omega)\right]\coth\left(\frac{\omega}{2T}\right).\end{aligned}$
(2.20)
One can also show that the retarded (advanced) component
$\Pi^{(R/A)}(\bm{\mathrm{q}},\omega)$ is given by
$\displaystyle\begin{aligned}
\Pi^{(R)}(\bm{\mathrm{q}},\omega)=\,\left(\Pi^{(A)}(\bm{\mathrm{q}},\omega)\right)^{*}=\,&-i\int\limits_{\bm{\mathrm{k}},\varepsilon}\left[G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)+G_{0}^{(K)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\right].\end{aligned}$
(2.21)
Employing the explicit form of the bare electron Green’s function (Eq. 2.10),
Eq. 2.21 can be reduced to the following integral representing the non-
interacting polarization function or the irreducible polarizability (“bare
bubble”):
$\displaystyle\begin{aligned}
\Pi^{(R)}(\bm{\mathrm{q}},\omega)=\,&\int\limits_{\bm{\mathrm{k}}}\frac{\tanh(\xi_{\bm{\mathrm{k}}+\bm{\mathrm{q}}}/2T)-\tanh(\xi_{\bm{\mathrm{k}}}/2T)}{\omega+\xi_{\bm{\mathrm{k}}}-\xi_{\bm{\mathrm{k}}+\bm{\mathrm{q}}}+i\eta},\end{aligned}$
(2.22)
whose result at zero temperature has been found by Stern [15] and Lindhard
[16] in dimensions $d=2$ and $d=3$, respectively. In terms of the
dimensionless parameters $u\equiv\omega/v_{\mathsf{F}}q$ and $x\equiv
q/2k_{\mathsf{F}}$, the zero temperature polarization operator takes the
following form in 2D
$\displaystyle\begin{aligned}
\operatorname{Re}\Pi_{0}^{(R)}(u>0,x)=&-\nu\left\\{1-\frac{\operatorname{sgn}{(x+u)}}{2x}\sqrt{\left(x+u\right)^{2}-1}\,\Theta\left(x+u-1\right)\right.\\\
&\left.\qquad\quad-\frac{\operatorname{sgn}{(x-u)}}{2x}\sqrt{\left(x-u\right)^{2}-1}\,\Theta\left(\lvert
x-u\rvert-1\right)\right\\},\\\
\operatorname{Im}\Pi_{0}^{(R)}(u>0,x)=&-\nu\left\\{-\frac{1}{2x}\sqrt{1-\left(x+u\right)^{2}}\Theta\left(1-(x+u)\right)+\frac{1}{2x}\sqrt{1-\left(x-u\right)^{2}}\Theta\left(1-\lvert
x-u\rvert\right)\right\\},\end{aligned}$ (2.23)
and in 3D it is given by
$\displaystyle\begin{aligned}
&\operatorname{Re}\Pi_{0}^{(R)}(u>0,x)=-\nu\left\\{\frac{1}{2}+\frac{1}{8x}\left[1-\left(x-u\right)^{2}\right]\ln\bigg{\lvert}\frac{x-u+1}{x-u-1}\bigg{\rvert}+\frac{1}{8x}\left[1-\left(x+u\right)^{2}\right]\ln\bigg{\lvert}\frac{x+u+1}{x+u-1}\bigg{\rvert}\right\\},\\\
&\operatorname{Im}\Pi_{0}^{(R)}(u>0,x)=-\pi\nu\left\\{\frac{u}{2}\Theta\left(\lvert
1-x\rvert-u\right)+\frac{1}{8x}\left[1-\left(x-u\right)^{2}\right]\Theta(1+x-u)\Theta\left(u-\lvert
1-x\rvert\right)\right\\}.\end{aligned}$ (2.24)
Here $\nu$ represents the density of states at Fermi level and is given by
$\nu=m/\pi$ and $\nu=mk_{\mathsf{F}}/\pi^{2}$ in 2D (Eq. 2.23) and 3D (Eq.
LABEL:eq:pi-3), respectively. We note that Eqs. 2.23 and LABEL:eq:pi-3 give
the forms of the polarization operator with frequency $\omega\geq 0$. The
expressions for $\omega<0$ can be deduced from these equations with the help
of
$\operatorname{Im}\Pi^{(R)}(\bm{\mathrm{q}},\omega)=-\operatorname{Im}\Pi^{(R)}(\bm{\mathrm{q}},-\omega)$
and
$\operatorname{Re}\Pi^{(R)}(\bm{\mathrm{q}},\omega)=\operatorname{Re}\Pi^{(R)}(\bm{\mathrm{q}},-\omega)$.
Once the polarization operator is known, it is straightforward to deduce
$\hat{D}(\bm{\mathrm{q}},\omega)$, the interaction dressed Green’s function
for the bosonic field $\phi$, also known as the RPA dynamically screened
interaction potential, from the Dyson equation
$\displaystyle\begin{aligned}
\hat{D}(\bm{\mathrm{q}},\omega)\equiv\,&-i\left\langle\begin{bmatrix}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)\\\
\phi_{\mathsf{q}}(\bm{\mathrm{q}},\omega)\end{bmatrix}\begin{bmatrix}\phi_{\mathsf{cl}}(-\bm{\mathrm{q}},-\omega)&\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega)\end{bmatrix}\right\rangle=\left[\hat{D}_{0}(\bm{\mathrm{q}},\omega)-\hat{\Pi}(\bm{\mathrm{q}},\omega)\right]^{-1}\\!\\!\\!\\!\\!\\!\\!.\end{aligned}$
(2.25)
Here $\hat{D}_{0}$ is the bare bosonic Green’s function arising from the free
action $S_{\phi}$ (Eq. 2.14), and takes the form
$\displaystyle\begin{aligned}
\hat{D}_{0}(\bm{\mathrm{q}},\omega)=\begin{bmatrix}0&V(\bm{\mathrm{q}})\\\
V(\bm{\mathrm{q}})&0\end{bmatrix}.\end{aligned}$ (2.26)
From Eq. 2.25, one can see that $\hat{D}$ acquires the same structure in
Keldysh space as $\Pi^{-1}$:
$\displaystyle\begin{aligned}
\hat{D}(\bm{\mathrm{q}},\omega)=\,&\begin{bmatrix}D^{(K)}(\bm{\mathrm{q}},\omega)&D^{(R)}(\bm{\mathrm{q}},\omega)\\\
D^{(A)}(\bm{\mathrm{q}},\omega)&0\end{bmatrix},\end{aligned}$ (2.27)
with components given by
$\displaystyle\begin{aligned}
D^{(R)}(\bm{\mathrm{q}},\omega)=\,&\left[D^{(A)}(\bm{\mathrm{q}},\omega)\right]^{*}=\,\left[V^{-1}(q)-\Pi^{(R)}(\bm{\mathrm{q}},\omega)\right]^{-1}\\!\\!\\!\\!\\!\\!\\!,\end{aligned}$
(2.28a) $\displaystyle\begin{aligned}
D^{(K)}(\bm{\mathrm{q}},\omega)=\,&\left[D^{(R)}(\bm{\mathrm{q}},\omega)-D^{(A)}(\bm{\mathrm{q}},\omega)\right]\coth\left(\frac{\omega}{2T}\right).\end{aligned}$
(2.28b)
A diagrammatic representation of the Dyson equation (Eq. 2.28a) for the
bosonic field $\phi$ is depicted in Fig. 1(a) where the bare (dressed) bosonic
propagator $D_{0}$ ($D$), i.e., the bare (dynamically screened) Coulomb
interaction, is represented by the red wavy line with a open (solid) dot in
the middle. The black bare bubble corresponds to the polarization operator,
with each line representing a bare electron Green’s function $G_{0}$ in Eq.
2.21.
Figure 1: A diagrammatic representation of (a) the Dyson equation for the
dynamically screened RPA interaction (Eq. 2.28a) and (b) the RPA electron
self-energy. The dynamically screened (bare) interaction can be considered as
the dressed (bare) propagator $D$ ($D_{0}$) of the bosonic field $\phi$
introduced to decouple the interactions, and is represented diagrammatically
by the red wavy line with a solid (open) dot in the middle. The polarization
operator $\Pi$ can be considered as the self-energy for bosonic field $\phi$
and is shown by the black bubble where each line represents a bare electron
Green’s function $G_{0}$. To the leading order in the dynamically screened
interaction, the electron self-energy is given by the diagram in panel (b).
#### 2.1.3 RPA self-energy
We now return to Eq. 2.14, the partition function before the fermionic fields
$\psi$ and $\bar{\psi}$ are integrated out. To the leading order in RPA
interaction, the electron self-energy can be obtained from $(iS_{c})^{2}$ the
square of the action which couples the fermionic and bosonic fields (Eq.
2.14):
$\displaystyle\begin{aligned}
-i\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)=&-\frac{1}{2}\int\limits_{\bm{\mathrm{q}},\omega}\left\langle\hat{M}_{F}(\varepsilon)\left[\phi_{\mathsf{cl}}(-\bm{\mathrm{q}},-\omega)+\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega)\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon+\omega)\psi_{\sigma}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right.\\\
&\left.\times\bar{\psi}_{\sigma}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\left[\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)+\phi_{\mathsf{q}}(\bm{\mathrm{q}},\omega)\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon)\right\rangle.\end{aligned}$
(2.29)
The angular bracket here represents averaging with respect to the weight:
$\displaystyle\begin{aligned}
\exp\left(iS_{\psi}+iS_{\phi}+\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}\right)=\exp\left(i\int\limits_{\bm{\mathrm{k}},\varepsilon}\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)\psi(\bm{\mathrm{k}},\varepsilon)+\frac{i}{2}\int\limits_{\bm{\mathrm{q}},\omega}\phi(\bm{\mathrm{q}},\omega)\hat{D}(\bm{\mathrm{q}},\omega)\phi(-\bm{\mathrm{q}},-\omega)\right).\end{aligned}$
(2.30)
Employing Wick’s theorem, we then equate
$-i\braket{\phi(\bm{\mathrm{q}},\omega)\phi(-\bm{\mathrm{q}},-\omega)}$ to the
RPA dynamically screened interaction $\hat{D}(\bm{\mathrm{q}},\omega)$ and
$-i\braket{\psi(\bm{\mathrm{k}},\varepsilon)\bar{\psi}(-\bm{\mathrm{k}},-\varepsilon)}$
to the bare electron Green’s function
$\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)$ in Eq. 2.29, and obtain
$\displaystyle\begin{aligned}
\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)=&\frac{i}{2}\int_{\bm{\mathrm{q}},\omega}D^{(K)}(-\bm{\mathrm{q}},-\omega)\hat{M}_{F}(\varepsilon)\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\hat{M}_{F}(\varepsilon)\\\
+\,&\frac{i}{2}\int_{\bm{\mathrm{q}},\omega}D^{(R)}(-\bm{\mathrm{q}},-\omega)\hat{M}_{F}(\varepsilon)\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\hat{\tau}^{1}\hat{M}_{F}(\varepsilon)\\\
+\,&\frac{i}{2}\int_{\bm{\mathrm{q}},\omega}D^{(A)}(-\bm{\mathrm{q}},-\omega)\hat{M}_{F}(\varepsilon)\hat{\tau}^{1}\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\hat{M}_{F}(\varepsilon).\end{aligned}$
(2.31)
$\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)$ is shown diagrammatically by Fig.
1(b) where the black solid line and the red wavy line with a closed dot
represent, respectively, the bare electron Green’s function $G_{0}$ and the
RPA dynamically screened interaction $D$. Inserting Eqs. 2.12 and 2.13 into
the equation above and using the causality relation
$\int\limits_{\bm{\mathrm{q}},\omega}D^{(R)}(\bm{\mathrm{q}},\omega)G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)=0,$
one can prove that $\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)$ is diagonal in
the Keldysh space
$\displaystyle\begin{aligned}
&\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)=\,\begin{bmatrix}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)&0\\\
0&\Sigma^{(A)}(\bm{\mathrm{k}},\varepsilon)\end{bmatrix},\end{aligned}$ (2.32)
and its retarded (advanced) component is given by
$\displaystyle\begin{aligned}
&\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,\left[\Sigma^{(A)}(\bm{\mathrm{k}},\varepsilon)\right]^{*}=\frac{i}{2}\int\limits_{\bm{\mathrm{q}},\omega}\left\\{D^{(K)}(\bm{\mathrm{q}},\omega)G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)+D^{(A)}(\bm{\mathrm{q}},\omega)G_{0}^{(K)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right\\}.\end{aligned}$
(2.33)
With the help of Kramers-Krönig relation,
$\displaystyle
f^{(R)}(\bm{\mathrm{k}},\varepsilon)=\int_{-\infty}^{\infty}\dfrac{d\varepsilon^{\prime}}{\pi}\dfrac{\operatorname{Im}f^{(R)}(\bm{\mathrm{k}},\varepsilon^{\prime})}{\varepsilon^{\prime}-\varepsilon-i\eta},$
(2.34)
one can rewrite Eq. 2.33 as
$\displaystyle\begin{aligned}
&\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,-2\int\limits_{\bm{\mathrm{q}},\omega,\omega^{\prime}}\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\operatorname{Im}G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega^{\prime})\frac{1}{\omega^{\prime}-\omega-i\eta}\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\varepsilon+\omega^{\prime}}{2T}\right)\right].\end{aligned}$
(2.35)
In two dimensions, after inserting the explicit expression for the bare
electron Green’s function (Eq. 2.10) and integrating over the angular
direction of the momentum $\bm{\mathrm{q}}$, Eq. 2.35 reduces to
$\displaystyle\begin{aligned}
\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{m}{\pi
k}\int\limits_{\omega,\omega^{\prime}}\int_{0}^{\infty}dq\Theta\left(1-\left|\frac{m\omega^{\prime}}{kq}\right|\right)\dfrac{\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left(\frac{m\omega^{\prime}}{kq}\right)^{2}}}\\\
&\times\dfrac{1}{\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}-\omega-i\eta}\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\varepsilon+\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}}{2T}\right)\right],\end{aligned}$
(2.36)
where for simplicity we have defined
$\displaystyle\begin{aligned}
\Delta\varepsilon\equiv\varepsilon-\xi_{\bm{\mathrm{k}}}.\end{aligned}$ (2.37)
From Eq. 2.36, one then find that the imaginary part of electron self-energy
is given by the following integral
$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,\frac{m}{4\pi^{2}k}\int_{-\infty}^{\infty}d\omega\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\omega+\varepsilon}{2T}\right)\right]\int_{q_{-}(\omega)}^{q_{+}(\omega)}dq\dfrac{\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left[\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right]^{2}}},\end{aligned}$
(2.38)
where
$\displaystyle q_{\pm}(\omega)=\left|\pm
k+\sqrt{k^{2}+2m\left(\omega+\Delta\varepsilon\right)}\right|.$ (2.39)
For any $q$ within the regime $q_{-}(\omega)\leq q\leq q_{+}(\omega)$,
$\left|\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right|\leq
1$ is always satisfied.
Applying the Kramers-Krönig relation (Eq. 2.34), Eq. 2.36 can be rewritten as
$\displaystyle\begin{aligned}
\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{m}{4\pi^{2}k}\int_{0}^{\infty}dq\int_{-\infty}^{\infty}d\omega^{\prime}\tanh\left(\frac{\varepsilon+\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}}{2T}\right)\dfrac{\Theta\left(1-\left|\frac{m\omega^{\prime}}{kq}\right|\right)}{\sqrt{1-\left(\frac{m\omega^{\prime}}{kq}\right)^{2}}}D^{(A)}(\bm{\mathrm{q}},\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m})\\\
+&\frac{m}{4\pi^{3}k}\int_{-\infty}^{\infty}d\omega\coth\left(\frac{\omega}{2T}\right)\int_{0}^{\infty}dq\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\int_{-\frac{kq}{m}}^{\frac{kq}{m}}d\omega^{\prime}\dfrac{1}{\sqrt{1-\left(\frac{m\omega^{\prime}}{kq}\right)^{2}}}\dfrac{1}{\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}-\omega-i\eta}.\end{aligned}$
(2.40)
Taking the real part of this equation, one has
$\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{m}{4\pi^{2}k}\int_{-\infty}^{\infty}d\omega\tanh\left(\frac{\varepsilon+\omega}{2T}\right)\int_{q_{-}(\omega)}^{q_{+}(\omega)}dq\dfrac{\operatorname{Re}D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left[\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right]^{2}}}\\\
-&\frac{m}{4\pi^{2}k}\int_{-\infty}^{\infty}d\omega\coth\left(\frac{\omega}{2T}\right)\left(\int_{0}^{q_{-}(\omega)}dq+\int_{q_{+}(\omega)}^{\infty}dq\right)\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\frac{\operatorname{sgn}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)}{\sqrt{\left[\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right]^{2}-1}}.\end{aligned}$
(2.41)
Proceeding in a way analogous to the one outlined above for the 2D self-energy
formulas Eqs. 2.38 and 2.41, one can prove that, in 3D, the imaginary and real
parts of electron self-energy are given by the following integrals
$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{1}{8\pi^{2}}\frac{m}{k}\int_{-\infty}^{\infty}d\omega\int_{q_{-}(\omega)}^{q_{+}(\omega)}dqq\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\varepsilon+\omega}{2T}\right)\right],\\\
\operatorname{Re}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{1}{8\pi^{2}}\frac{m}{k}\int_{-\infty}^{\infty}d\omega\int_{q_{-}(\omega)}^{q_{+}(\omega)}dqq\operatorname{Re}D^{(R)}(\bm{\mathrm{q}},\omega)\tanh\left(\frac{\omega+\varepsilon}{2T}\right)\\\
&+\frac{1}{8\pi^{3}}\frac{m}{k}\int_{0}^{\infty}dqq\int_{-\infty}^{\infty}d\omega\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\coth\left(\frac{\omega}{2T}\right)\ln\left[\left|\dfrac{\omega-\frac{kq}{m}+\Delta\varepsilon-\frac{q^{2}}{2m}}{\omega+\frac{kq}{m}+\Delta\varepsilon-\frac{q^{2}}{2m}}\right|\right].\end{aligned}$
(2.42)
### 2.2 2D electron self-energy
The self-energy formulas Eqs. 2.38, 2.41 and 2.42 derived in the previous
section are written as two-variables integrals involving the RPA dynamically
screened interaction $D^{(R)}(\bm{\mathrm{q}},\omega)$ which is given by
another integral for the polarization operator
$\Pi^{(R)}(\bm{\mathrm{q}},\omega)$ (Eq. 2.22). In this section, we will use
these formulas to calculate the on-shell ($\varepsilon=\xi_{\bm{\mathrm{k}}}$)
electron self-energy close to the Fermi surface ($k=k_{\mathsf{F}}$) in 2D. We
will work in the regime where $r_{s}^{3/2}\ll\Delta/E_{\mathsf{F}}\ll r_{s}\ll
1$ with $\Delta=\left\\{|\varepsilon|,T\right\\}$, and evaluate the result to
the leading order in $r_{s}$ and to several orders in
$\Delta/(E_{\mathsf{F}}r_{s})$. Note that the theoretical problem is extremely
subtle and challenging since it involves expansions in three distinct ‘small’
parameters: the dimensionless Coulomb coupling $r_{s}$, temperature $T/T_{F}$,
and energy $\varepsilon/E_{F}$. It turns out that the actual small parameters
for temperature (energy) expansions are in fact $T/T_{\mathsf{F}}r_{s}$
($\varepsilon/E_{\mathsf{F}}r_{s}$), but we allow $T$ and $\varepsilon$ to be
comparable in our theory, which considerably complicates the calculations.
Using Eq. 2.38 and 2.41 and setting $\Delta\varepsilon=0$, we find that the
real and imaginary parts of the on-shell self-energy in 2D are given by
$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\varepsilon)=\,&\int_{0}^{\infty}\frac{d\omega}{2\pi}\left[2\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\omega+\varepsilon}{2T}\right)-\tanh\left(\frac{\omega-\varepsilon}{2T}\right)\right]\operatorname{Im}I(\omega),\end{aligned}$
(2.43a) $\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(\varepsilon)=\,\int_{0}^{\infty}\frac{d\omega}{2\pi}\left[\tanh\left(\frac{\omega+\varepsilon}{2T}\right)-\tanh\left(\frac{\omega-\varepsilon}{2T}\right)\right]\operatorname{Re}I(\omega),\end{aligned}$
(2.43b)
where
$\displaystyle I(\omega)\equiv\frac{m}{2\pi
k}\int_{q_{-}(\omega)}^{q_{+}(\omega)}dq\dfrac{D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left[\frac{m}{kq}\left(\omega-\frac{q^{2}}{2m}\right)\right]^{2}}}.$
(2.44)
Here we have neglected the second integral in Eq. 2.41 which vanishes to the
leading order in $r_{s}$.
#### 2.2.1 Momentum integration
We will now evaluate $I(\omega)$ defined in Eq. 2.44 by performing the
momentum integration. For convenience, we introduce
$\displaystyle\begin{aligned}
\alpha\equiv\frac{r_{s}}{\sqrt{2}},\quad\delta&\equiv\frac{\omega}{4E_{\mathsf{F}}},\quad
x\equiv\frac{q}{2k_{\mathsf{F}}},\quad
x_{\pm}(\delta)\equiv\frac{q_{\pm}(\omega)}{2k_{\mathsf{F}}}.\end{aligned}$
(2.45)
Here $x_{\pm}(\delta)$ is a solution to the equation $(x-\delta/x)^{2}=1$, and
is given by
$\displaystyle x_{-}(\delta)$
$\displaystyle=\frac{1}{2}\Big{|}1-\sqrt{1+4\delta}\,\Big{|},\quad
x_{+}(\delta)=\frac{1}{2}\Big{(}1+\sqrt{1+4\delta}\Big{)}.$ (2.46)
Using the newly introduced variables, Eq. 2.44 can then be rewritten as
$\displaystyle I(\delta)=$
$\displaystyle\frac{m}{\pi}\int_{x_{-}(\delta)}^{x_{+}(\delta)}dx\;D^{(R)}(x,\delta)\;\Bigg{[}1-\left(x-\frac{\delta}{x}\right)^{2}\Bigg{]}^{-1/2}\\!\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\frac{m}{\pi}\int_{x_{-}(\delta)}^{x_{+}(\delta)}dx\;D^{(R)}(x,\delta)\;x\Big{[}\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)\Big{]}^{-1/2}\\!\\!\\!\\!\\!\\!\\!,$
(2.47)
where the retarded RPA interaction $D^{(R)}(x,\delta)$ (Eq. 2.28a) is given by
$D^{(R)}(x,\delta)=\nu^{-1}\dfrac{\left(\frac{x}{\alpha}-\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)\right)+i\left(\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)\right)}{\left(\frac{x}{\alpha}-\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)\right)^{2}+\left(\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)\right)^{2}}.$
(2.48)
Here we have approximated the polarization bubble by the zero-temperature
result $\Pi_{0}$ (Eq. 2.23) whose real and imaginary parts, in terms of the
new variables, take the forms
$\displaystyle\begin{aligned}
\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)=&-1+\frac{1}{2x^{2}}\operatorname{sgn}\bigg{(}1-\frac{\delta}{x^{2}}\bigg{)}\operatorname{Re}\sqrt{-\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)}\\\
&+\frac{1}{2x^{2}}\operatorname{sgn}\bigg{(}1+\frac{\delta}{x^{2}}\bigg{)}\operatorname{Re}\sqrt{4\delta
x^{2}-\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)},\end{aligned}$
(2.49a) $\displaystyle\begin{aligned}
\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)=&-\frac{1}{2x^{2}}\operatorname{Re}\sqrt{\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)}+\frac{1}{2x^{2}}\operatorname{Re}\sqrt{\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)-4\delta
x^{2}}.\end{aligned}$ (2.49b)
It is clear that the main contribution to the integral in Eq. 2.43 is from
$\omega$ of the order of $|\varepsilon|$ or $T$ due to the presence of the
thermal factor which involves tanh and coth functions. As mentioned earlier,
we consider the regime where $r_{s}^{3/2}\ll\Delta/E_{\mathsf{F}}\ll r_{s}\ll
1$, with $\Delta=\left\\{|\varepsilon|,T\right\\}$, and therefore will
calculate $I(\delta)$ for $\alpha^{3/2}\ll|\delta|\ll\alpha\ll 1$, to the
leading order in $\alpha$ and several orders in $|\delta|/\alpha$.
To proceed, we divide the interval of integration in Eq. 2.47 into three
subintervals: $(x_{-},l_{-})$, $(l_{-},l_{+})$ and $(l_{+},x_{+})$. Here
$l_{\pm}$ can take arbitrary value as long as it satisfies $x_{-}\ll
l_{-}\ll\alpha\ll l_{+}\ll x_{+}$. Within these subregions, we then expand the
integrand using different small parameters, such as $x/\alpha$, $\delta/x$,
$x/x_{+}$ and $\alpha/x$. We note that these variables are small in some
subregions but become comparable to or larger than $1$ in other subregions. In
other words, it is not possible to use a single small parameter for the entire
integration interval. This lack of a single unique small parameter in the
theory has prevented the regime of $\varepsilon\sim T$, while both being
small, being studied in the theoretical literature at all in spite of the long
history of electronic many body field theories [6, 7, 12].
We set the real part of the polarization operator to
$\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)=-1,$ (2.50)
which is valid as long as $x$ lies within the region $x\in(a_{-},a_{+})$.
$a_{\pm}$ is defined as $a_{-}=x_{-}[1+O(|\delta|^{1-b})]$ and
$a_{+}=x_{+}[1-O(|\delta|)]$ where $b\in(0,1)$ is arbitrary. Furthermore, for
$x\in(a_{-},l_{+})$, $\operatorname{Im}\Pi_{0}^{(R)}$ (Eq. 2.49b) can be
approximated as
$\displaystyle\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(\sqrt{z^{2}+x_{-}^{2}},\delta)=$
$\displaystyle-\frac{zx_{+}}{2(z^{2}+x_{-}^{2})}\Big{[}1-\sqrt{1-\frac{4\delta(z^{2}+x_{-}^{2})}{z^{2}x_{+}^{2}}}+O\Big{(}\frac{l_{+}^{2}}{x_{+}^{2}}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle-\frac{zx_{+}}{2(z^{2}+x_{-}^{2})}\frac{4\delta(z^{2}+x_{-}^{2})}{2z^{2}x_{+}^{2}}\Big{[}1+O\big{(}|\delta|^{b}\big{)}\Big{]}=-\frac{\delta}{z}\Big{[}1+O\big{(}|\delta|^{b}\big{)}\Big{]},$
(2.51)
where $z^{2}=x^{2}-x_{-}^{2}$. For $x\in(l_{+},a_{+})$, all we need to know is
that
$\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)=O\Big{(}|\delta|^{1/2},\frac{\delta}{l_{+}}\Big{)}.$
(2.52)
We will now show that the contribution to $I(\delta)$ from $x$ outside the
region $x\in(a_{-},a_{+})$ is of higher order in $\delta$ and therefore can be
ignored. Let us first calculate $I(x_{-},a_{-})$, i.e., the contribution to
$I(\delta)$ from $x_{-}<x<a_{-}$. Throughout this section, $I(x_{1},x_{2})$ is
used to denote the contribution to $I(\delta)$ from the integration over the
interval $x\in(x_{1},x_{2})$. Applying a change of variables
$z^{2}=x^{2}-x_{-}^{2}$, one obtains
$\displaystyle
I(x_{-},a_{-})=\int_{0}^{O(|\delta|^{\frac{3-b}{2}})}dz\;D^{(R)}(\sqrt{z^{2}+x_{-}^{2}},\delta)\,\Big{(}x_{+}^{2}-z^{2}-x_{-}^{2}\Big{)}^{-1/2}=O(|\delta|^{\frac{3-b}{2}}).$
(2.53)
Here we have used the fact that
$|D^{(R)}(x,\delta)|\leq\nu^{-1}\alpha\,O(\alpha^{-1})$ inside the region
$x\in(x_{-},a_{-})$. Similarly, for the region $a_{+}<x<x_{+}$,
$|\nu^{-1}\Pi_{0}^{(R)}(x,\delta)|=O(|\delta|^{1/2})$ which leads to
$|D^{(R)}(x,\delta)|=\nu^{-1}O(\alpha)$. Applying the transformation
$u^{2}=x_{+}^{2}-x^{2}$, we find the contribution to $I(\delta)$ from this
region
$\displaystyle
I(a_{+},x_{+})=\int_{0}^{O(|\delta|^{1/2})}du\;D^{(R)}(\sqrt{x_{+}^{2}-u^{2}},\delta)\,\Big{(}x_{+}^{2}-u^{2}-x_{-}^{2}\Big{)}^{-1/2}=\alpha\;O(|\delta|^{1/2}).$
(2.54)
We have proved that the contribution from outside the regime
$x\in(a_{-},a_{+})$ can be ignored, and will now evaluate
$\operatorname{Im}I(\delta)$ by carrying out the integration in Eq. 2.47
separately for the three subintervals: $(a_{-},l_{-})$, $(l_{-},l_{+})$, and
$(l_{+},a_{+})$. Within the first region $x\in(a_{-},l_{-})$, the integrand
can be expanded in terms of the small parameter $x/\alpha\ll 1$. Employing the
transformation $z^{2}=x^{2}-\delta^{2}$, and inserting Eqs. 2.50 and 2.2.1
into Eq. 2.47, one obtains
$\displaystyle\operatorname{Im}I(a_{-},l_{-})=-\int_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}dz\;\frac{\delta/z}{(\frac{\sqrt{z^{2}+\delta^{2}}}{\alpha}+1)^{2}+(\delta/z)^{2}}$
$\displaystyle=-\int_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}dz\;\frac{\delta}{z}\bigg{(}1+\Big{(}\frac{\delta}{z}\Big{)}^{2}\bigg{)}^{-1}\bigg{[}1-\frac{2\sqrt{z^{2}+\delta^{2}}}{\alpha}\bigg{(}1+\Big{(}\frac{\delta}{z}\Big{)}^{2}\bigg{)}^{-1}\bigg{]}$
$\displaystyle=-{\alpha}\bigg{[}\frac{\delta}{\alpha}\ln\left(\frac{l_{-}}{|\delta|}\right)+4\frac{\delta^{2}}{\alpha^{2}}\operatorname{sgn}\delta-2\frac{l_{-}\delta}{\alpha^{2}}\bigg{]}.$
(2.55)
For $x$ lying inside the region $(l_{-},l_{+})$, we instead expand in terms of
$|\delta|/x\ll 1$ and $x\ll 1$, which leads to
$\displaystyle\operatorname{Im}I(l_{-},l_{+})=-\int_{l_{-}}^{l_{+}}dx\;\frac{\delta\alpha^{2}}{x(x+\alpha)^{2}}=-\alpha\bigg{[}-\frac{\delta}{\alpha}+\frac{\delta}{\alpha}\ln\left(\frac{\alpha}{l_{-}}\right)+2\frac{l_{-}\delta}{\alpha^{2}}\bigg{]}.$
(2.56)
Finally, the contribution from $x\in(l_{+},a_{+})$, i.e.,
$\operatorname{Im}I(l_{+},a_{+})$, is negligible as the integrand is of the
order of $\alpha$. Combining contributions from all subregions, we find the
imaginary part of $I(\delta)$
$\operatorname{Im}I(\delta)=-\alpha\left\\{\frac{\delta}{\alpha}\left[-\\!1+\ln\left(\frac{\alpha}{|\delta|}\right)\right]+4\frac{\delta^{2}}{\alpha^{2}}\operatorname{sgn}\delta\right\\}.$
(2.57)
We then turn to the calculation of the real part of $I(\delta)$ by considering
the three subregions separately as in the case of
$\operatorname{Im}I(\delta)$. For $a_{-}\leq x\leq l_{-}$, we use $x/\alpha\ll
1$ as the small expansion parameter, and approximate
$\operatorname{Re}D^{(R)}$ by
$\displaystyle\operatorname{Re}D^{(R)}(x,\delta)=\nu^{-1}\frac{1+\frac{x}{\alpha}}{(1+\frac{x}{\alpha})^{2}+\frac{\delta^{2}}{z^{2}}+O(|\delta|^{\frac{3b-1}{2}})}=\nu^{-1}\frac{1+\frac{x}{\alpha}}{1+\frac{\delta^{2}}{z^{2}}}\bigg{[}1-\frac{\frac{2x}{\alpha}+\frac{x^{2}}{\alpha^{2}}}{1+\frac{\delta^{2}}{z^{2}}}+\bigg{(}\frac{\frac{2x}{\alpha}}{1+\frac{\delta^{2}}{z^{2}}}\bigg{)}^{2}+O\Big{(}\frac{x^{3}}{\alpha^{3}}\Big{)}\bigg{]},$
(2.58)
where $z^{2}=x^{2}-x_{-}^{2}$. This leads to
$\displaystyle\operatorname{Re}I(a_{-},l_{-})=\nu\int_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}dz\;\operatorname{Re}D^{(R)}(\sqrt{z^{2}+x_{-}^{2}},\delta)\,\Big{[}1+O(l_{-}^{2})\Big{]}$
$\displaystyle=\alpha\bigg{[}\frac{z}{\alpha}-\frac{z(5\delta^{2}+z^{2})}{2\alpha^{2}\sqrt{\delta^{2}+z^{2}}}-\frac{\delta}{\alpha}\arctan\left(\frac{z}{\delta}\right)+\frac{5\delta^{2}}{2\alpha^{2}}\ln\big{(}z+\sqrt{\delta^{2}+z^{2}}\big{)}\bigg{]}\bigg{|}_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}$
$\displaystyle=\alpha\bigg{[}-\frac{\pi|\delta|}{2\alpha}+\frac{l_{-}}{\alpha}-\frac{7\delta^{2}}{4\alpha^{2}}+\frac{5\delta^{2}}{2\alpha^{2}}\ln\left(\frac{2l_{-}}{|\delta|}\right)+\frac{\delta^{2}}{2\alpha
l_{-}}-\frac{l_{-}^{2}}{2\alpha^{2}}+O\Big{(}\frac{l_{-}^{3}}{\alpha^{3}},\;\frac{\delta^{3}}{l_{-}^{3}},\;\frac{|\delta|^{\frac{3-b}{2}}}{\alpha},\;|\delta|^{\frac{3b-1}{2}}\Big{)}\bigg{]}.$
(2.59)
On the other hand, for the region $x\in(l_{-},l_{+})$, we expand in terms of
$|\delta|/x\ll 1$ and $x\ll 1$ instead of $x/\alpha$. Substituting
$\operatorname{Re}D^{(R)}(x,\delta)=\nu^{-1}\frac{1+\frac{x}{\alpha}}{(1+\frac{x}{\alpha})^{2}+\frac{\delta^{2}}{x^{2}}+O(|\delta|^{\frac{3b-1}{2}},\,\frac{\delta^{4}}{x^{4}})},$
(2.60)
into Eq. 2.47, one obtains
$\displaystyle\operatorname{Re}I(l_{-},l_{+})=\nu\int_{l_{-}}^{l_{+}}dx\;\operatorname{Re}D^{(R)}(x,\delta)\,\bigg{[}1+\frac{\delta^{2}}{2x^{2}}+O\Big{(}\frac{\delta^{3}}{x^{3}},x\Big{)}\bigg{]}$
$\displaystyle=\alpha\bigg{[}-\frac{l_{-}}{\alpha}-\frac{5\delta^{2}}{2\alpha^{2}}+\frac{5\delta^{2}}{2\alpha^{2}}\ln\left(\frac{\alpha}{l_{-}}\right)-\frac{\delta^{2}}{2\alpha
l_{-}}+\frac{l_{-}^{2}}{2\alpha^{2}}+\ln\left(\frac{l_{+}}{\alpha}\right)+O\Big{(}\frac{l_{-}^{3}}{\alpha^{3}},\;\frac{\delta^{3}}{l_{-}^{3}},\;\frac{\alpha}{l_{+}},\;l_{+}\Big{)}\bigg{]}.$
(2.61)
For the last region $x\in(l_{+},a_{+})$, $\alpha/x\ll 1$ now becomes the small
expansion parameter. Keeping only the leading order term in $\alpha/x\ll 1$,
we have
$D^{(R)}(x,\delta)=\nu^{-1}\Big{[}\frac{\alpha}{x}+O\Big{(}\frac{\alpha^{2}}{x^{2}}\Big{)}\Big{]},$
(2.62)
which leads to
$\displaystyle\operatorname{Re}I(l_{+},a_{+})=\int_{l_{+}}^{1+O(\delta)}dx\;\Big{[}\frac{\alpha}{x}+O\Big{(}\frac{\alpha^{2}}{x^{2}}\Big{)}\Big{]}(1-x^{2})^{-1/2}=\alpha\Big{[}\ln\left(\frac{2}{l_{+}}\right)+O\Big{(}\frac{\alpha}{l_{+}},\;l_{+}\Big{)}\Big{]}.$
(2.63)
Combining everything, we arrive at
$\displaystyle\operatorname{Re}I(\delta)=\alpha\bigg{[}\ln\left(\frac{2}{\alpha}\right)-\frac{\pi|\delta|}{2\alpha}+\frac{\delta^{2}}{\alpha^{2}}\Big{(}-\frac{17}{4}+\frac{5}{2}\ln\left(\frac{2\alpha}{|\delta|}\right)\Big{)}+O\Big{(}\frac{\delta^{3}}{\alpha^{3}},\;\alpha,\;|\delta|^{\frac{1-b}{2}},\;|\delta|^{\frac{3b-1}{2}}\Big{)}\bigg{]}.$
(2.64)
#### 2.2.2 Frequency integration
The calculation presented in the previous section shows that
$\displaystyle\begin{aligned}
\operatorname{Im}I(\omega)=\,&-\left\\{\frac{1}{4}\frac{|\omega|}{E_{\mathsf{F}}}\left[\ln\left(\frac{2\sqrt{2}r_{s}E_{\mathsf{F}}}{|\omega|}\right)-1\right]+\frac{1}{2\sqrt{2}r_{s}}\left(\frac{\omega}{E_{\mathsf{F}}}\right)^{2}\right\\}\operatorname{sgn}\omega,\end{aligned}$
(2.65a) $\displaystyle\begin{aligned}
\operatorname{Re}I(\omega)=\frac{r_{s}}{\sqrt{2}}\left[\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)-\frac{\pi}{4\sqrt{2}r_{s}}\frac{|\omega|}{E_{\mathsf{F}}}+\frac{5}{16r_{s}^{2}}\frac{\omega^{2}}{E_{\mathsf{F}}^{2}}\ln\left(\frac{4\sqrt{2}r_{s}E_{\mathsf{F}}}{|\omega|}\right)-\frac{17}{32r_{s}^{2}}\frac{\omega^{2}}{E_{\mathsf{F}}^{2}}\right],\end{aligned}$
(2.65b)
where we have transformed back to the original variables. These expressions
can be used to extract the electron self-energy on the mass shell by
substitution using Eq. 2.43 and carrying out the frequency integrations.
The frequency integrations in Eq. 2.43 involve hyperbolic functions $\tanh(x)$
and $\coth(x)$, and are of the form
$\displaystyle\begin{aligned}
I_{1}(a)=&\int_{0}^{\infty}dxf(x)\left[2\coth(x)-\tanh(x+a)-\tanh(x-a)\right],\\\
I_{2}(a)=&\int_{0}^{\infty}dxf(x)\left[\tanh(x+a)-\tanh(x-a)\right].\end{aligned}$
(2.66)
To carry out such integrals, one may take advantage of the following equations
which express $\tanh(x)$ and $\coth(x)$ as infinite exponential series:
$\displaystyle\begin{aligned}
&\tanh(x)=\,1+2\sum_{k=1}^{\infty}(-1)^{k}e^{-2kx},\qquad\coth(x)=\,1+2\sum_{k=1}^{\infty}e^{-2kx},\qquad
x>0.\end{aligned}$ (2.67)
In Appendix. B, we use the equations above and provide the analytical results
for integrals of the form Eq. 2.66 for various functions $f(x)$.
Using Eq. B.2 in Appendix B, we obtain the analytical expression for the
imaginary part of the 2D self-energy on the mass shell:
$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)=&-\frac{1}{8\pi}\left(\pi^{2}+\frac{\varepsilon^{2}}{T^{2}}\right)\frac{T^{2}}{E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{T}\right)\\\
&-\left\\{-\frac{\pi}{24}\left(6-\gamma_{E}-\ln\left(\frac{2}{\pi^{2}}\right)-24\ln\mathrm{A}\right)-\frac{\left(2-\gamma_{E}-\ln
2\right)}{8\pi}\frac{\varepsilon^{2}}{T^{2}}\right.\\\
&\left.\qquad+\frac{1}{4\pi}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\varepsilon/T})+\partial_{s}\operatorname{Li}_{s}(-e^{\varepsilon/T})\right]\bigg{\lvert}_{s=2}\right\\}\frac{T^{2}}{E_{\mathsf{F}}}\\\
&-\frac{\sqrt{2}}{\pi}\left[\zeta(3)-\frac{1}{2}\operatorname{Li}_{3}(-e^{\varepsilon/T})-\frac{1}{2}\operatorname{Li}_{3}(-e^{-\varepsilon/T})\right]\frac{T^{3}}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.68)
Here $\gamma_{E}\approx 0.577216$ is the Euler’s constant, and
$\mathrm{A}\approx 1.28243$ is the Glaisher’s constant.
$\operatorname{Li}_{s}(z)=\sum_{k=1}^{\infty}{z^{k}}/{k^{s}}$ stands for the
polylogarithm function, and $\zeta(z)=\sum_{k=1}^{\infty}{1}/{k^{z}}$
represents the Riemann zeta function. We emphasize that this result is valid
for arbitrary value of $\varepsilon/T$, as long as $|\varepsilon|,T\ll
E_{\mathsf{F}}$ condition is satisfied.
One can derive from Eq. 2.68 the asymptotic behaviors of
$\operatorname{Im}\Sigma^{(R)}$ in the low-energy limit $|\varepsilon|\ll T$
as well as the low-temperature limit $T\ll|\varepsilon|$. With the help of
$\displaystyle\begin{aligned}
\partial_{s}\operatorname{Li}_{s}(-1)|_{s=2}=\,-\frac{\pi^{2}}{12}\left(\gamma_{E}+\ln\left(4\pi\right)-12\ln\mathrm{A}\right),\qquad\operatorname{Li}_{3}(-1)=-\frac{3}{4}\zeta(3),\end{aligned}$
(2.69)
we find that, for $|\varepsilon|\ll T$, $\operatorname{Im}\Sigma^{(R)}$
assumes the form
$\displaystyle\begin{aligned} &\operatorname{Im}\Sigma^{(R)}(|\varepsilon|\ll
T)=\,-\frac{\pi}{8}\frac{T^{2}}{E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{T}\right)+\frac{\pi}{24}\left(6+\ln\left(2\pi^{3}\right)-36\ln\mathrm{A}\right)\frac{T^{2}}{E_{\mathsf{F}}}-\frac{7\zeta(3)}{2\sqrt{2}\pi}\frac{T^{3}}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.70)
Similarly for $T\ll|\varepsilon|$, we have
$\displaystyle\begin{aligned} &\operatorname{Im}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,-\frac{\varepsilon^{2}}{8\pi
E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{|\varepsilon|}\right)-\left(\ln
4-1\right)\frac{\varepsilon^{2}}{16\pi
E_{\mathsf{F}}}-\frac{1}{6\sqrt{2}\pi}\frac{|\varepsilon|^{3}}{r_{s}E_{\mathsf{F}}^{2}},\end{aligned}$
(2.71)
where we have used
$\displaystyle\begin{aligned}
&\lim\limits_{x\rightarrow\infty}\operatorname{Li}_{s}(-e^{x})=-\frac{x^{s}}{\Gamma\left(s+1\right)},\quad\lim\limits_{x\rightarrow\infty}\partial_{s}\operatorname{Li}_{s}(-e^{x})=-\frac{x^{s}\ln
x}{\Gamma\left(s+1\right)}+\frac{\Gamma^{\prime}\left(s+1\right)}{\Gamma^{2}\left(s+1\right)}x^{s}.\end{aligned}$
(2.72)
In a way analogous to the one that leads to the expression of the imaginary
part of self-energy, we evaluate the integration in Eq. 2.43 using Eq. B.2 in
Appendix B, and derive the real part of the on-shell self-energy for arbitrary
$\varepsilon/T$ in 2D:
$\displaystyle\begin{aligned}
&\operatorname{Re}\Sigma^{(R)}(\varepsilon,T)=\,\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)\varepsilon-\frac{1}{8}\frac{T}{\varepsilon}\left[\operatorname{Li}_{2}(-e^{-\frac{\varepsilon}{T}})-\operatorname{Li}_{2}(-e^{\frac{\varepsilon}{T}})\right]\frac{T\varepsilon}{E_{\mathsf{F}}}+\frac{5}{48\sqrt{2}\pi}\left(\pi^{2}+\frac{\varepsilon^{2}}{T^{2}}\right)\frac{T^{2}\varepsilon}{r_{s}E_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}E_{\mathsf{F}}}{T}\right)\\\
&-\left\\{\frac{1}{96\sqrt{2}\pi}\left(32-10\gamma_{E}-25\ln
2\right)\left(\frac{\varepsilon^{2}}{T^{2}}+\pi^{2}\right)+\frac{5}{8\sqrt{2}\pi}\frac{T}{\varepsilon}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\frac{\varepsilon}{T}})-\partial_{s}\operatorname{Li}_{s}(-e^{\frac{\varepsilon}{T}})\right]\bigg{\lvert}_{s=3}\right\\}\frac{T^{2}\varepsilon}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.73)
In the limit of $|\varepsilon|/T\ll 1$, one has
$\displaystyle\begin{aligned}
&\operatorname{Li}_{2}(-e^{-\frac{\varepsilon}{T}})-\operatorname{Li}_{2}(-e^{\frac{\varepsilon}{T}})=2\ln
2\frac{\varepsilon}{T}+O\left((\frac{\varepsilon}{T})^{2}\right),\\\
&\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\frac{\varepsilon}{T}})-\partial_{s}\operatorname{Li}_{s}(-e^{\frac{\varepsilon}{T}})\right]\bigg{\lvert}_{s=3}=2\partial_{s}\partial_{z}\operatorname{Li}_{s}(z)|_{s=3,z=-1}\frac{\varepsilon}{T}+O\left((\frac{\varepsilon}{T})^{2}\right),\end{aligned}$
(2.74)
where $\partial_{s}\partial_{z}\operatorname{Li}_{s}(z)|_{s=3,z=-1}$ can be
further simplified to $(\pi^{2}\ln 2+6\zeta^{\prime}(2))/12$. Substitution of
Eq. 2.74 into Eq. 2.73 leads to the asymptotic expression for the real part of
self-energy in the limit of $|\varepsilon|\ll T$,
$\displaystyle\begin{aligned} \operatorname{Re}\Sigma^{(R)}(|\varepsilon|\ll
T)=\,&\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)\varepsilon-\frac{\ln
2}{4}\frac{T\varepsilon}{E_{\mathsf{F}}}+\frac{5\pi}{48\sqrt{2}r_{s}}\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}E_{\mathsf{F}}}{T}\right)\\\
&+\left[-\frac{\pi}{96\sqrt{2}}\left(32-10\gamma_{E}-25\ln
2\right)-\frac{5}{8\sqrt{2}\pi}\left(\zeta^{\prime}(2)+\frac{\pi^{2}}{6}\ln
2\right)\right]\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}r_{s}}.\end{aligned}$
(2.75)
The expression for $\operatorname{Re}\Sigma^{(R)}$ in the low-temperature
limit $T\ll|\varepsilon|$ can also be extracted from Eq. 2.73. With the help
of Eq. 2.72, we arrive at the result
$\displaystyle\begin{aligned} &\operatorname{Re}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)\varepsilon-\frac{1}{16}\frac{\varepsilon|\varepsilon|}{E_{\mathsf{F}}}+\frac{5}{48\sqrt{2}\pi}\frac{\varepsilon^{3}}{r_{s}E_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}E_{\mathsf{F}}}{|\varepsilon|}\right)+\frac{-41+75\ln
2}{288\sqrt{2}\pi}\frac{\varepsilon^{3}}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.76)
### 2.3 3D electron self-energy
Starting from the general formulas Eq. 2.42, one can derive the 3D electron
self-energy in a way analogous to the one presented in Sec. 2.2 in the case of
2D. In this section, without giving the details of the calculation, we present
directly the explicit expressions for the 3D electron self-energy on the mass
shell, which are valid to the leading order in $r_{s}$, and to several orders
in $\varepsilon/E_{\mathsf{F}}$ and $T/E_{\mathsf{F}}$. The 3D calculations
follow the general procedure given above in depth for the corresponding 2D
theory. To simplify the calculation, we consider a range of temperature
(energy) slightly different from the one in the 2D case:
$r_{s}\ll{T}/{T_{\mathsf{F}}}\ll\sqrt{r_{s}}$
($r_{s}\ll{\varepsilon}/{E_{\mathsf{F}}}\ll\sqrt{r_{s}}$).
In 3D, for arbitrary energy-to-temperature ratio, the imaginary part of the
on-shell self-energy acquires the form of
$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)=\,&-\frac{1}{32}(\frac{\pi^{4}}{12})^{1/3}\sqrt{r_{s}}\frac{1}{E_{\mathsf{F}}}\left(\pi^{2}T^{2}+\varepsilon^{2}\right)-\frac{\pi}{8}C_{0}\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left[\zeta(3)-\frac{1}{2}\left(\operatorname{Li}_{3}(-e^{-\frac{\varepsilon}{T}})+\operatorname{Li}_{3}(-e^{\frac{\varepsilon}{T}})\right)\right],\end{aligned}$
(2.77)
where $C_{0}$ is a constant defined as
$\displaystyle
C_{0}=\int_{0}^{1}dy\frac{1}{y^{2}}\left[\dfrac{1}{\left(1-\frac{y}{2}\ln\left(\dfrac{1+y}{1-y}\right)\right)^{2}+\left(\frac{\pi}{2}y\right)^{2}}-1\right]-1\approx-1.5326.$
(2.78)
From the equation above, one can find $\operatorname{Im}\Sigma^{(R)}$ in both
the low-energy limit and the low-temperature limit:
$\displaystyle\begin{aligned} \operatorname{Im}\Sigma^{(R)}(|\varepsilon|\ll
T)=\,&-\frac{1}{32}(\frac{\pi^{4}}{12})^{1/3}\pi^{2}\sqrt{r_{s}}\frac{T^{2}}{E_{\mathsf{F}}}-\frac{7\pi}{32}\zeta(3)C_{0}\frac{T^{3}}{E_{\mathsf{F}}^{2}},\end{aligned}$
(2.79a) $\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,&-\frac{1}{32}(\frac{\pi^{4}}{12})^{1/3}\sqrt{r_{s}}\frac{\varepsilon^{2}}{E_{\mathsf{F}}}-\frac{\pi}{96}C_{0}\frac{|\varepsilon|^{3}}{E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.79b)
Similarly, we evaluate the real part of the 3D on-shell self-energy in the
case where $\varepsilon$ and $T$ are arbitrary with respect to each other, and
find
$\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(\varepsilon,T)=\,&\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)\varepsilon-\frac{4-\pi^{2}}{192}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{T}{E_{\mathsf{F}}}\right)\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left(\frac{\varepsilon^{3}}{T^{3}}+\pi^{2}\frac{\varepsilon}{T}\right)\\\
&-\left[\frac{4-\pi^{2}}{384}\left(3-2\gamma_{E}-2\ln
2\right)-\frac{2}{3}C_{1}\right]\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left(\frac{\varepsilon^{3}}{T^{3}}+\pi^{2}\frac{\varepsilon}{T}\right)\\\
&-\frac{4-\pi^{2}}{32}\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\frac{\varepsilon}{T}})-\partial_{s}\operatorname{Li}_{s}(-e^{\frac{\varepsilon}{T}})\right]\bigg{|}_{s=3},\end{aligned}$
(2.80)
where
$\displaystyle\begin{aligned}
C_{1}\equiv\frac{1}{32}\int_{0}^{1}dy\frac{1}{y^{3}}\left[\dfrac{1-\frac{y}{2}\ln\left(\dfrac{1+y}{1-y}\right)}{\left(1-\frac{y}{2}\ln\left(\dfrac{1+y}{1-y}\right)\right)^{2}+\left(\frac{\pi}{2}y\right)^{2}}-\left(1+\frac{4-\pi^{2}}{4}y^{2}\right)\right]+\frac{-16+3\pi^{2}-4(4-\pi^{2})\ln
2}{512}\approx 0.059.\end{aligned}$ (2.81)
In the limits where $|\varepsilon|/T\ll 1$ and $|\varepsilon|/T\gg 1$, Eq.
2.80 reduces to, respectively,
$\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(T\gg|\varepsilon|)=\,&\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)\varepsilon-\frac{4-\pi^{2}}{192}\pi^{2}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{T}{E_{\mathsf{F}}}\right)\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}}\\\
&-\left[\frac{4-\pi^{2}}{384}\left(3-2\gamma_{E}-2\ln
2\right)-\frac{2}{3}C_{1}+\frac{4-\pi^{2}}{32\pi^{2}}\left(\frac{\pi^{2}}{6}\ln
2+\zeta^{\prime}(2)\right)\right]\pi^{2}\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}},\end{aligned}$
(2.82a) $\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,&\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)\varepsilon-\frac{4-\pi^{2}}{192}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{|\varepsilon|}{E_{\mathsf{F}}}\right)\frac{\varepsilon^{3}}{E_{\mathsf{F}}^{2}}\\\
&+\left[\frac{4-\pi^{2}}{576}\left(1+3\ln
2\right)+\frac{2}{3}C_{1}\right]\frac{\varepsilon^{3}}{E_{\mathsf{F}}^{2}}.\\\
\end{aligned}$ (2.82b)
### 2.4 Discussion of the analytical results
It may be worthwhile to summarize our analytical findings for the self-energy
at low temperatures and energies (and to the leading-order in $r_{s}$).
Both in 2D and 3D, $\operatorname{Im}\Sigma^{(R)}$ goes as $T^{2}$ for
$\varepsilon=0$ and as $\varepsilon^{2}$ for $T=0$ (with an additional log
correction in 2D) in the leading order, as is already well-known. This
establishes the perturbative stability of the Fermi surface, implying that
both 3D and 2D interacting systems are Fermi liquids, in contrast to
interacting 1D fermions. The subleading terms in the 3D
$\operatorname{Im}\Sigma^{(R)}$ are $O(T^{3})$ for $\varepsilon=0$ and
$O(\varepsilon^{3})$ for $T=0$. In 2D, however, the corresponding subleading
terms are $O(T^{2})$ and $O(\varepsilon^{2})$, respectively since the leading
order terms go as $O(T^{2}\ln T)$ and $O(\varepsilon^{2}\ln\varepsilon)$. The
next order terms in 2D are cubic, as expected.
When energy and temperature are comparable, the results for
$\operatorname{Im}\Sigma^{(R)}$ are complicated, involving logarithmic
integrals in $\exp(-\varepsilon/T)$ in addition to powers of $\varepsilon$ and
$T$ along with log factors in 2D. In 3D (Eq. 2.77),
$\operatorname{Im}\Sigma^{(R)}$ goes as $(\varepsilon^{2}+\pi^{2}T^{2})$ plus
term involving logarithmic integrals. In 2D (Eq. 2.69),
$\operatorname{Im}\Sigma^{(R)}$ for small, but comparable, $\varepsilon$ and
$T$, goes as $(\varepsilon^{2}+\pi^{2}T^{2})\ln(T/E_{\mathsf{F}})$ with
additional terms involving $O(T^{2})$, $O(\varepsilon^{2})$, and logarithmic
integrals.
The analytical behavior of $\operatorname{Re}\Sigma^{(R)}$ is as follows.
In 3D, for $\varepsilon\ll T$, $\operatorname{Re}\Sigma^{(R)}$ goes as
$O(\varepsilon)+O(\varepsilon T^{2}\ln T)+O(\varepsilon T^{2})$, whereas for
$\varepsilon\gg T$, it goes as
$O(\varepsilon)+O(\varepsilon^{3}\ln\varepsilon)+O(\varepsilon^{3})$. In 2D,
for $\varepsilon\ll T$, $\operatorname{Re}\Sigma^{(R)}$ goes as
$O(\varepsilon)+O(\varepsilon T)+O(\varepsilon T^{2}\ln T)+O(\varepsilon
T^{2})$, whereas for $\varepsilon\gg T$, it goes as
$O(\varepsilon)+O(\varepsilon^{2})+O(\varepsilon^{3}\ln\varepsilon)+O(\varepsilon^{3})$.
When energy and temperature are comparable, but both small, the behavior of
$\operatorname{Re}\Sigma^{(R)}$ is complicated with the appearance of
logarithmic integrals similar to the situation for
$\operatorname{Im}\Sigma^{(R)}$ discussed above. In 3D,
$\operatorname{Re}\Sigma^{(R)}$ then goes as
$O(\varepsilon)+O([\varepsilon^{3}+\pi^{2}\varepsilon T^{2}]\ln
T)+O(\varepsilon^{3}+\pi^{2}\varepsilon T^{2})$ plus terms involving
logarithmic integrals as in Eq. 2.80. In 2D, for general small values of
energy and temperature, $\operatorname{Re}\Sigma^{(R)}$ behaves as
$O(\varepsilon)+O([\pi^{2}T^{2}\varepsilon+\varepsilon^{3}]\ln
T)+O(\pi^{2}T^{2}\varepsilon+\varepsilon^{3})$ plus several terms involving
logarithmic integrals as shown in Eq. 2.73.
The presence of various logarithmic terms and combinations of powers of $T$
and $\varepsilon$ along with logarithmic integrals made the calculation of the
self-energy a challenge for arbitrary (but small) energy and temperature even
in the $r_{s}\ll 1$ limit for the last 60 years [6], which we finally managed
to resolve [9].
The $O(\varepsilon^{2})$ or $O(\varepsilon^{2}\ln\varepsilon)$ asymptotic
behavior of $\operatorname{Im}\Sigma^{(R)}$ in 3D or 2D respectively assures
the existence of a Fermi surface at $\varepsilon=0$ since
$\operatorname{Re}\Sigma^{(R)}$ always goes as $O(\varepsilon)$. The question
we address in the rest of this paper is what happens at finite temperature and
energy where $\operatorname{Im}\Sigma^{(R)}$ in principle could be larger than
energy or temperature itself, indicating that the quasiparticle picture fails
there.
Our goal is to ascertain the domain of validity of the Fermi liquid theory and
the associated quasiparticle picture by comparing the quasiparticle energy
$\varepsilon$ with the quasiparticle damping defined by the magnitude of
$\operatorname{Im}\Sigma^{(R)}(\varepsilon)$, both in 3D and 2D, contrasting
the two cases.
## 3 Results
### 3.1 Applicability of the FL theory
The Fermi liquid theory describes interacting Fermi systems at low
temperatures and excitation energies in terms of quasiparticles - long-lived
elementary excitations which are adiabatically connected to the excitations of
noninteracting systems. Therefore it relies on the existence of well-defined
quasiparticles, whose damping rates, determined by the imaginary part of the
self-energy, should be small compared with their energies. More specifically,
the quasiparticle description is valid when the magnitude of the imaginary
part of the self-energy $|\operatorname{Im}\Sigma^{(R)}(\varepsilon)|$ is
small compared with $\varepsilon$. To the leading order,
$\operatorname{Im}\Sigma^{(R)}(\varepsilon)$ scales as
$\varepsilon^{2}\ln\varepsilon$ and $\varepsilon^{2}$, respectively, in 2D and
3D (see Refs. [8, 17, 18, 19, 20, 21], as well as Eqs. 2.71 and 2.79b).
Therefore, at sufficiently low energy $\varepsilon$, the criterion
$|\operatorname{Im}\Sigma^{(R)}(\varepsilon)|<\varepsilon$ is satisfied and
the quasiparticle is well defined. However, at higher energy or temperature,
this might no longer be the case. In this section, we use the previously
obtained analytical expressions for the electron self-energy to determine the
regime in which the quasiparticle description is applicable (invalid) - called
Fermi-liquid (FL) or non-Fermi liquid (NFL) regime in the following. We
compute the on-shell $\operatorname{Im}\Sigma^{(R)}(\varepsilon)/\varepsilon$
at zero temperature as well as $\operatorname{Im}\Sigma^{(R)}(T)/T$ at the
Fermi level (i.e. $\varepsilon=0$), in an attempt to determine the crossover
energy $\varepsilon_{c}$ and crossover temperature $T_{c}$ below which the
quasiparticles are well defined. $\varepsilon_{c}$ and $T_{c}$ separate the FL
and NFL regimes, and are determined by the conditions:
$\displaystyle\begin{aligned}
-\operatorname{Im}\Sigma^{(R)}(\varepsilon_{c},T=0)/\varepsilon_{c}=1,\qquad-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T_{c})/T_{c}=1.\end{aligned}$
(3.1)
Of course, the precise magnitudes of $\varepsilon_{c}$ and $T_{c}$ will depend
on the expressions for the self-energy we use, and the leading-order and sub-
leading-order theories may give different results, but our interest is in
understanding the qualitative trends being mindful of the fact that the
analytical theory is meaningful only up to energy/temperature where the
subleading terms are smaller than the leading terms and the constraint
$\varepsilon,T<E_{\mathsf{F}}$ applies. Strictly speaking, the analytical
expressions for the electron self-energy presented in the previous section are
derived within RPA, and are valid in the high-density, low-energy and low-
temperature regime. However, it has been found that the RPA approximation
works reasonably well even outside the high density regime. For example, in
Ref. [10], it has been shown that, for an electron gas at metallic densities,
the effective mass, Pauli spin susceptibility and compressibility obtained
from RPA are in good agreement with the experiment. We note that
$\varepsilon/E_{\mathsf{F}}r_{s}$ and $T/E_{\mathsf{F}}r_{s}$
($\varepsilon/E_{\mathsf{F}}\sqrt{r_{s}}$ and $T/E_{\mathsf{F}}\sqrt{r_{s}}$)
are used, apart from $r_{s}$, as the small expansion parameters in the
calculation of the 2D (3D) self-energy. Larger $r_{s}$ therefore means wider
energy and temperature ranges of applicability (for not too large value of
$r_{s}$). For these reasons, in this section, we assume that the previously
obtained analytical expressions for the self-energy apply to arbitrary
$r_{s}$, $\varepsilon/E_{\mathsf{F}}$ and $T/E_{\mathsf{F}}$, and use them to
estimate when the quasiparticle description breaks down. Our goal is to
determine the energy/temperature regime where the Fermi liquid theory applies
within our approximations.
Figure 2: (a) $-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon$ as a
function of $\varepsilon/E_{\mathsf{F}}$ at $T=0$ in 2D; (b)
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/T$ as a function of
$T/E_{\mathsf{F}}$ at $\varepsilon=0$ in 2D; (c) and (d) are same as (a) and
(b), respectively, but in $d=3$ dimensions. (a) [(c)] is obtained using the
zero temperature self-energy expression Eq. 2.71 (Eq. 2.79b) up to the
$\varepsilon^{2}$ term, while (b) [(d)] is from zero energy self-energy
expression Eq. 2.70 (Eq. 2.79a) up to the $T^{2}$ term. In all four panels, we
consider $r_{s}=0.1,0.5,1.0,2.0,5.0,10.0$ represented by solid lines with
different colors. The horizontal dashed lines take the value of $1$ and
separate the FL and NFL regimes where the quasiparticle description is
applicable and invalid, respectively.
In Fig. 2(a) (Fig. 2(c)), the ratio
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon$ at $T=0$ is
plotted as a function of the dimensionless energy $\varepsilon/E_{\mathsf{F}}$
for 2D (3D) electron systems for various interaction parameter $r_{s}$ values.
For Fig. 2(a), we apply Eq. 2.71, the analytical expression for zero
temperature $\operatorname{Im}\Sigma^{(R)}$ in 2D, and retain the leading
order $\varepsilon^{2}\ln\varepsilon$ term as well as the subleading
$\varepsilon^{2}$ term, while for Fig. 2(c), only the leading order
$\varepsilon^{2}$ term in the zero temperature expression for the imaginary
part of the 3D self-energy (Eq. 2.79b) is used. Figs. 2(b) and 2(d) are
plotted in a similar way. Instead of
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$, we plot
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ as a function of
$T/E_{\mathsf{F}}$ for the same set of $r_{s}$ values for both 2D (Fig. 2(b))
and 3D (Fig. 2(d)) systems. For these two figures, we apply Eq. 2.70 and Eq.
2.79a, which give the zero-energy form of $\operatorname{Im}\Sigma^{(R)}$ for
$d=2$ and $d=3$, respectively, and neglect the highest order $T^{3}$ terms in
both equations.
In Figs. 2(a) and 2(b), which depict the 2D case, almost all curves lie below
the horizontal dashed line with the value of $1$ (except for the one
corresponds to $r_{s}=10$ in Fig. 2(b)). This means that the conditions
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon<1$ and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T<1$ always hold in the
observed regime. However, as $\varepsilon/E_{\mathsf{F}}$ $(T/E_{\mathsf{F}})$
increases, the self-energy expression used to plot Fig. 2(a) (Fig. 2(b)) is no
longer valid, and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
$\left(-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T\right)$ becomes
negative for large enough $\varepsilon$ ($T$) for almost all $r_{s}$
considered. This implies that the corresponding approximation breaks down at
high energy, but quasiparticles remain well-defined up to the energy cut off
where the theory is valid. In Fig. 2(b), the purple curve which represents the
case of $r_{s}=10$ intersects with $-\operatorname{Im}\Sigma^{(R)}/T=1$ line
(dashed line) at $T_{c}(r_{s}=10)\approx 1.25E_{\mathsf{F}}$, above which the
quasiparticle is no longer well defined (for this value of $r_{s}$). If we
smoothly extrapolate the 2D results in Figs. 2(a) and 2(b) from their low
energy monotonic behavior (before the pathological maxima induced by the
failure of the expansion), then the results for different $r_{s}$ values all
smoothly cross the dashed line (i.e. unity) for increasing
$\varepsilon/E_{\mathsf{F}}$ and $T/T_{\mathsf{F}}$ with decreasing $r_{s}$,
implying that the Fermi liquid regime is larger for lower $r_{s}$, which is
understandable since lower $r_{s}$ indicates weaker interactions.
In 3D (Figs. 2(c) and 2(d)),
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
($-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$) from the leading order
zero temperature (zero energy) self-energy expression remains positive for all
$\varepsilon$ ($T$). Furthermore, for each $r_{s}$, there exists a
$\varepsilon_{c}$ ($T_{c}$) above which the ratio
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
$\left(-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T\right)$ exceeds $1$,
signaling the break down of the quasiparticle description. Substituting the
leading order terms in Eq. 2.79b and 2.79a into Eq. 3.1, we have
$\displaystyle\begin{aligned}
\frac{\varepsilon_{c}^{(3D)}}{E_{\mathsf{F}}}=\,32(\frac{12}{\pi^{4}})^{1/3}\frac{1}{\sqrt{r_{s}}},\qquad\frac{T_{c}^{(3D)}}{E_{\mathsf{F}}}=\,32(\frac{12}{\pi^{10}})^{1/3}\frac{1}{\sqrt{r_{s}}}.\end{aligned}$
(3.2)
From this result, one can see that $\varepsilon_{c}^{(3D)}/E_{\mathsf{F}}$ and
$T_{c}^{(3D)}/E_{\mathsf{F}}$ have a $r_{s}$-dependence of $1/\sqrt{r_{s}}$
and decay with increasing $r_{s}$, in accordance with Figs. 2(c) and 2(d). The
regime of Fermi liquid validity shrinks in energy with increasing $r_{s}$
consistent with increasing interaction strength in the system. We emphasize
that this result is only an approximation from the leading order 3D self-
energy expressions valid for a small range of temperatures and energies.
Figure 3: Plots of
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$ in the
$(r_{s},\varepsilon/E_{\mathsf{F}})$-plane (left panels), and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ in the
$(r_{s},T/E_{\mathsf{F}})$-plane (right panels) in 2D (upper panels) and 3D
(lower panels). For panels (a)-(d), Eqs. 2.71, 2.70, 2.79b and 2.79a are used,
in respective order, and the highest order $T^{3}$ or $\varepsilon^{3}$ terms
in these equations are neglected. In the left (right) panels, the FL regime
where $0<-\operatorname{Im}\Sigma(\varepsilon,T=0)/\varepsilon<1$
($0<-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T<1$) is indicated by the
blue area, whereas the NFL regime where
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon>1$
($-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T>1$) is represented by the
yellow area. Their boundary is indicated by red contour line which corresponds
to $\varepsilon_{c}$ ($T_{c}$) defined in Eq. 3.1. In the region colored in
orange, which is separated from the rest by green contour with the value of
zero, the approximated self-energy expressions are no longer valid and lead to
a positive value of $\operatorname{Im}\Sigma^{(R)}$.
Using the same formulas for the imaginary part of the self-energy (given by an
expansion to order $\varepsilon^{2}$ for the zero temperature case and to
order $T^{2}$ for the zero energy case), in Fig 3, we show contour plots of
$\operatorname{Im}\Sigma^{(R)}/\varepsilon$ at $T=0$ in the
$(r_{s},\varepsilon/E_{\mathsf{F}})$-plane (left panels), as well as
$-\operatorname{Im}\Sigma^{(R)}/T$ at $\varepsilon=0$ in the
$(r_{s},T/E_{\mathsf{F}})$-plane (right panels) for $d=2$ (upper panels) and
$d=3$ (lower panels). In the left (right) panels, the blue areas are used to
indicate the FL regimes where $0<-\operatorname{Im}\Sigma^{(R)}/\varepsilon<1$
($0<-\operatorname{Im}\Sigma^{(R)}/T<1$), whereas the yellow areas correspond
to the NFL regimes where $-\operatorname{Im}\Sigma^{(R)}/\varepsilon>1$
($-\operatorname{Im}\Sigma^{(R)}/T>1$). They are separated by the red contour
lines which take the value of one and correspond to $\varepsilon_{c}$
($T_{c}$) defined by Eq. 3.1. As shown in Figs. 3(c) and 3(d),
$\varepsilon_{c}^{(3D)}/E_{\mathsf{F}}$ and $T_{c}^{(3D)}/E_{\mathsf{F}}$
decreases monotonically with increasing $r_{s}$ in 3D, consistent with Eq.
3.2. In Figs. 3(a) and 3(b), there exist regimes where the self-energy
expressions are no longer valid and the resulting 2D
$\operatorname{Im}\Sigma^{(R)}$ is positive. We use the orange areas to
indicate such regimes, where the current self-energy expressions are unable to
tell if the quasiparticle is well defined or not. Furthermore, using the
current approximation for $\operatorname{Im}\Sigma^{(R)}$, the solution to Eq.
3.1 \- $\varepsilon_{c}^{(2D)}$ (or $T_{c}^{(2D)}$ for small $r_{s}$) does not
exist (see Figs. 3(a) and 3(b)) . In this case,
$-\operatorname{Im}\Sigma^{(R)}/\varepsilon<1$
($-\operatorname{Im}\Sigma^{(R)}/T<1$) always holds in the region where this
formula is applicable. We mention that if we use extrapolations of the 2D
self-energy from their low energy behavior so that the pathological behavior
of the imaginary self-energy decreasing at higher energies and temperatures is
eliminated (so that the orange region in the figures disappears), then the 2D
results in Fig. 3 look qualitatively the same as the 3D results with the
orange regions in Fig. 3(a) and 3(b) mostly becoming blue.
Figure 4: The regimes (green areas) where the applicable criterion -
$E^{3}(T^{3})$ term must be smaller than $E^{2}(T^{2})$ term - is satisfied
for self-energy expressions Eq. 2.71 (panel a), Eq. 2.70 (panel b), Eq. 2.79b
(panel c) and Eq. 2.79a (panel d).
As explained earlier, the self-energy expressions used to obtain Figs. 2 and 3
are applicable to a small range of energies and temperatures because of the
leading order nature of the analytical expansion. We now retain the $T^{3}$
and $\varepsilon^{3}$ terms in the self-energy formulas Eqs. 2.71, 2.70, 2.79b
and 2.79a to evaluate $\operatorname{Im}\Sigma^{(R)}/\varepsilon$ and
$\operatorname{Im}\Sigma^{(R)}/T$ and to analyze the applicable regime of the
FL quasiparticle description. We note that these expressions have a wider
range of applicability compared with the one used in Figs. 2 and 3, but are
only valid when the $T^{3}$ ($\varepsilon^{3}$) term is smaller than the
$T^{2}$ ($\varepsilon^{2}$) term. We find that the leading upper bounds for
the regimes satisfying this condition for the 2D and 3D cases are given by,
respectively,
$\displaystyle\begin{aligned}
\frac{\varepsilon_{3}^{(2D)}}{E_{\mathsf{F}}}=\,&\frac{3\sqrt{2}\left(\ln
4-1\right)}{8}r_{s},\qquad&\frac{T_{3}^{(2D)}}{E_{\mathsf{F}}}=\,&\frac{\pi^{2}\left(6+\ln
2\pi^{3}-36\ln\mathrm{A}\right)}{42\sqrt{2}\zeta(3)}r_{s},\\\
\frac{\varepsilon_{3}^{(3D)}}{E_{\mathsf{F}}}=\,&\frac{(\frac{9\pi}{4})^{1/3}}{C_{0}}\sqrt{r_{s}},\qquad&\frac{T_{3}^{(3D)}}{E_{\mathsf{F}}}=\,&\frac{(\frac{\pi^{7}}{12})^{1/3}}{7\zeta(3)C_{0}}\sqrt{r_{s}}.\end{aligned}$
(3.3)
We note that, in 2D, within the applicability range
$\varepsilon<\varepsilon_{3}^{(2D)}$ ($T<T_{3}^{(2D)}$), the leading order
$\varepsilon^{2}\ln\varepsilon$ ($T^{2}\ln T$) term is also larger than
$\varepsilon^{2}$ ($T^{2}$) term. $\varepsilon_{3}$ (left panels) and $T_{3}$
(right panels) are plotted by the dashed green curves in Fig. 4 for 2D (upper
panels) and 3D (lower panels) systems. The green areas in this figure
represent the regimes where $E^{3}(T^{3})$ term in the self-energy expressions
is always smaller than $E^{2}(T^{2})$ term.
Figure 5: The same as Fig. 2 but retaining the highest order $T^{3}$ and
$\varepsilon^{3}$ terms in Eqs. 2.71, 2.70, 2.79b and 2.79a. Figure 6: The
same as Fig. 3 but using the self-energy expressions which are valid up to
$\varepsilon^{3}$ and $T^{3}$ (i.e. keeping all terms in Eqs. 2.71, 2.70,
2.79b and 2.79a).
In Figs. 5 and 6, we replot
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$ and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ as in Figs. 2 and 3, but
with the self-energy expressions which include the $T^{3}(\varepsilon^{3})$
term. As in Fig. 2, different curves in Fig. 6 correspond to different values
of $r_{s}$. For each curve, we use a vertical dotted line of the same color to
indicate the upper limit $\varepsilon_{3}$ or $T_{3}$ (Eq. 3.3) of the
applicable range for the corresponding self-energy expression. We notice that,
for the 2D case, these curves exhibit nonmonotonic dependence on $r_{s}$,
which will disappear if we incorporate the $r_{s}$ dependence of the Fermi
energy ($E\sim r_{s}^{-2}$) and plot in fixed units instead (see Figs. 7(a)
and 7(b)). In the observed regime, for the 3D case, the ratios
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$ and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ are always smaller than
one and become negative (similar to what happens for 2D results in Fig. 2)
outside the applicable ranges of the self-energy expressions (see Figs. 5(c)
and 5(d)), whereas for the 2D case these ratios exceed one at higher energies
and temperatures (see Figs. 5(a) and 5(b)).
In Fig. 6, the blue, yellow and orange areas correspond to, respectively, the
regimes where the ratio
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
($-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$) lies within $(0,1)$,
stays above one, and remains negative (i.e. theoretically inapplicable), as in
Fig. 3. We also add a green area in each panel to indicate the region where
the corresponding self-energy expression is applicable (see Eq. 3.3). In Figs.
6(a) and 6(b) which are associated with the 2D case, we find both FL and NFL
regimes, indicated by the blue and yellow areas, respectively. The red contour
lines take the value of one and represent $\varepsilon_{c}^{(2D)}$ (Fig. 6
(a)) or $T_{c}^{(2D)}$ (Fig. 6 (b)). As one can see from these figures, over
some range of $r_{s}$ values, $\varepsilon_{c}^{(2D)}/E_{\mathsf{F}}$
($T_{c}^{(2D)}/E_{\mathsf{F}}$) grows with increasing $r_{s}$ which seems
contrary to a naive expectation that the quasiparticle description breaks down
at lower energy and temperature for larger interaction strength. However,
taking into account the $r_{s}$ dependence of $E_{\mathsf{F}}$, we find that
$\varepsilon_{c}^{(2D)}$ and $T_{c}^{(2D)}$ in fixed units decay with
increasing $r_{s}$ as expected. See Figs. 7(c) and 7(d) where we replot Figs.
6(a) and 6(b) in fixed units. We also note that $\varepsilon_{c}^{(2D)}$ and
$T_{c}^{(2D)}$ stay outside the applicable regime of the self-energy
expression (green area), and are therefore not reliable. This result shows
that within the applicable regime, the FL quasiparticle description is valid.
In the lower panels of Fig. 6 which depict the 3D cases, the yellow areas
disappear. Instead, one finds orange areas which correspond to the regimes
where the associated self-energy expressions are invalid and yield
$\operatorname{Im}\Sigma^{(R)}>0$. As before, we have no information about
whether the quasiparticle is well defined or not in the orange areas. The
current analytical theory is simply inapplicable in the orange areas. It is,
in principle, possible that the Fermi liquid theory remains valid with well-
defined quasiparticles for all energies and temperatures since our analytical
theory cannot access arbitrary quasiparticle energy and temperature well above
$E_{\mathsf{F}}$.
Figure 7: The same as Figs. 5(a)-(b) and 6(a)-(b) but plotted in fixed units.
Note that the nonmonotonic $r_{s}$ dependence in Fig. 5(a)-(b) disappears
after using the fixed units.
One compelling conclusion of the results presented in Figs. 2 -7 is that the
Fermi liquid theory and the quasiparticle picture are extremely robust in both
2D and 3D interacting systems, generically remaining valid at least up to an
energy $E_{\mathsf{F}}$ above the Fermi level. In general, the regime of this
robustness decreases with increasing $r_{s}$, but even for $r_{s}\sim 10$,
where our RPA theory is suspect, the quasiparticles remain well-defined (i.e.,
the magnitude of $\operatorname{Im}\Sigma^{(R)}$ is less than $\varepsilon$
and/or $T$) up to an energy $E_{\mathsf{F}}$ above the Fermi level. The
stability is quantitatively slightly weaker in 2D than in 3D, but the
difference is not significant enough to draw any conclusion.
### 3.2 Effective mass
As mentioned earlier, in the Fermi liquid theory, an interacting electron
system is composed of quasiparticles whose effective mass is different from
the bare mass of electron but is renormalized by the electron-electron
interactions. The effective mass of the quasiparticle is a fundamental
parameter in the Fermi liquid theory, and has been extensively studied before.
See for example, Refs. [8, 10, 22, 23, 24, 25, 26] for electron systems with
Coulomb interactions and Refs. [11, 12] for the case of the short-range
interactions. In this section, we use the analytical expressions for the real
part of the self-energy provided in Sec. 2 to rederive the effective mass for
Coulomb interactions. We also provide a higher order $T^{2}$ correction to the
previous result [25] in both 2D and 3D.
To the leading order in dynamically screened interactions, the effective mass
can be obtained from the real part of the self-energy using the following
formula [25]:
$\displaystyle\begin{aligned}
\frac{m^{*}(T)}{m}=\dfrac{1-\frac{\partial}{\partial\varepsilon}\operatorname{Re}\Sigma^{(R)}(\varepsilon,\xi_{\bm{\mathrm{k}}})}{1+\frac{\partial}{\partial\xi_{\bm{\mathrm{k}}}}\operatorname{Re}\Sigma^{(R)}(\varepsilon,\xi_{\bm{\mathrm{k}}})}\bigg{|}_{\varepsilon=\xi_{\bm{\mathrm{k}}}=0}\approx
1-\left(\frac{\partial}{\partial\varepsilon}+\frac{\partial}{\partial\xi_{\bm{\mathrm{k}}}}\right)\operatorname{Re}\Sigma^{(R)}(\varepsilon,\xi_{\bm{\mathrm{k}}})\bigg{|}_{\varepsilon=\xi_{\bm{\mathrm{k}}}=0}.\end{aligned}$
(3.4)
Here $m^{*}$ and $m$ denote the effective mass and bare mass, respectively.
Inserting the low energy limit ($\varepsilon\ll T$) expression for the real
part of the on-shell ($\varepsilon=\xi_{\bm{\mathrm{k}}}$) self-energy (Eq.
2.75) into the equation above, we obtain the effective mass for 2D systems
$\displaystyle\begin{aligned}
\frac{m^{*}(T)}{m}=\,&1-\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)+\frac{\ln
2}{4}\frac{T}{T_{\mathsf{F}}}-\frac{5\pi}{48\sqrt{2}r_{s}}\frac{T^{2}}{T_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}T_{\mathsf{F}}}{T}\right)\\\
&-\left[-\frac{\pi}{96\sqrt{2}}\left(32-10\gamma_{E}-25\ln
2\right)-\frac{5}{8\sqrt{2}\pi}\left(\zeta^{\prime}(2)+\frac{\pi^{2}}{6}\ln
2\right)\right]\frac{T^{2}}{T_{\mathsf{F}}^{2}r_{s}}.\end{aligned}$ (3.5)
This expression is valid for quasiparticles with $\varepsilon\ll T$ in the
regime of $r_{s}\ll 1$ and $r_{s}^{3/2}\ll T/E_{\mathsf{F}}\ll r_{s}$.
Similarly, applying Eq. 2.82a, we find that the 3D effective mass takes the
following form for $r_{s}\ll{T}/{T_{\mathsf{F}}}\ll\sqrt{r_{s}}$:
$\displaystyle\begin{aligned}
\frac{m^{*}(T)}{m}=\,&1-\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)+\frac{4-\pi^{2}}{192}\pi^{2}\frac{T^{2}}{E_{\mathsf{F}}^{2}}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{T}{E_{\mathsf{F}}}\right)\\\
&+\left[\frac{4-\pi^{2}}{384}\left(3-2\gamma_{E}-2\ln
2\right)-\frac{2}{3}C_{1}+\frac{4-\pi^{2}}{32\pi^{2}}\left(\frac{\pi^{2}}{6}\ln
2+\zeta^{\prime}(2)\right)\right]\pi^{2}\frac{T^{2}}{E_{\mathsf{F}}^{2}}.\end{aligned}$
(3.6)
We note that the applicable temperature range of this result is different from
the one in Ref. [25]. Our current results include both leading and subleading
temperature corrections to the renormalized quasiparticle effective mass in
both 2D and 3D.
We emphasize that, in 2D, the leading order temperature correction to the
effective mass is linear in $T$ and independent of $r_{s}$ [24, 25]. This
linear-in-$T$ correction has also been found for 2D systems with short-range
interactions [11, 12]. By contrast, for 3D systems, the leading order
temperature correction is of the order of $T^{2}\ln T$, much smaller compared
with its 2D counterpart. The corresponding coefficient is also
$r_{s}$-independent. The appearance of the linear-in-$T$ leading order
effective mass renormalization in 2D compared with the leading order $T^{2}\ln
T$ renormalization in 3D implies much stronger interaction effects in 2D
compared with 3D. But, this stronger effective 2D interaction does not imply
any failure of the Fermi liquid theory.
### 3.3 Hydrodynamic and ballistic regimes
Hydrodynamics is a useful approach to describe physical processes of
interacting systems at time and length scales that are much larger compared
with the ones associated with local equilibration, provided that the dominant
microscopic inter-particle scattering process is momentum-conserving. Electron
liquids can be described hydrodynamically if the dominant scattering process
is the electron-electron collision which is responsible for local
equilibration. In this case, the electron system reaches local equilibrium at
the time scale of the electron-electron inelastic scattering time, which is
much shorter than the other time scales in the problem, and therefore obeys
hydrodynamics. By contrast, if electron-impurity and/or electron-phonon
scattering is stronger than the electron-electron scattering so that momentum
conservation is not preserved, hydrodynamics does not apply. We assume here
that the electron-impurity and electron-phonon scattering are negligible in
the system under consideration.
The main condition for hydrodynamics to be applicable to an electron liquid
[27, 28, 29] is that the electron-electron scattering time
$\tau_{\varepsilon}$ is much smaller than the time scale or inverse frequency
of the physical process being studied ($\varepsilon^{-1}$):
$\displaystyle\begin{aligned}
\tau_{\varepsilon}\ll\varepsilon^{-1}.\end{aligned}$ (3.7)
Equivalently, the mean free path $l=v_{\mathsf{F}}\tau_{\varepsilon}$
associated with electron-electron collisions should be much smaller compared
with the external length scale $q^{-1}$:
$\displaystyle l\ll q^{-1}.$ (3.8)
Here energy $\varepsilon$ can be considered as an external frequency for an
experiment probing the response of the system to an applied field, and $q$ is
the corresponding external wavevector which is related to external energy
$\varepsilon$ by $\varepsilon=qv_{\mathsf{F}}$. The collision-dominated regime
where Eq. 3.7 is satisfied is known as the hydrodynamic regime. On the other
hand, the collisionless regime where the hydrodynamics condition Eq. 3.7 no
longer applies is usually called the ballistic regime.
As mentioned earlier, the electron-electron inelastic scattering rate can be
extracted from the imaginary part of the self-energy:
$\displaystyle\begin{aligned}
\tau^{-1}_{\varepsilon}=-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T).\end{aligned}$
(3.9)
We then insert the previously obtained formulas for
$\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)$ for arbitrary $\varepsilon/T$
to investigate the hydrodynamic and ballistic regimes. As in Sec. 3.1, we
assume the self-energy formulas are applicable even for low densities, high
temperatures and high energies.
Figure 8: Boundaries separating the hydrodynamic regimes where
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon>1$ and the
ballistic regimes where
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon<1$ for 2D (upper
panels) and 3D (lower panels) electron systems for
$r_{s}=0.1,0.5,1.0,2.0,5.0,10.0$. The left panels are obtained using all terms
in the self-energy expressions Eqs. 2.68 and 2.77 (which are applicable to
arbitrary $\varepsilon/T$), while for the right panels, the highest order
terms in these equations are neglected. In the
$(\varepsilon/E_{\mathsf{F}},T/E_{\mathsf{F}})$-plane, the hydrodynamics
regimes lie in the low energy (high temperature) part, whereas the ballistic
regime correspond to the high energy (low temperature) part. The inset of
panel (b) zooms in the lower-left (low energy and temperature) part to clearly
show the behavior of $r_{s}=0.1$ case.
Using Eq. 3.7 and Eq. 3.9, it is straightforward to see that the hydrodynamic
regime should follow
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon>1$, whereas the
ballistic regime satisfies
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon<1$. In Fig. 8, in
the $(\varepsilon/E_{\mathsf{F}},T/E_{\mathsf{F}})$-plane, we depict contours
given by the following condition for various $r_{s}$
$\displaystyle\begin{aligned}
-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon=1.\end{aligned}$
(3.10)
These contours divide the plane into two parts: the low energy (high
temperature) part corresponds to the hydrodynamic regime, while the high
energy (low temperature) part is associated with the ballistic regime. We
consider both 2D and 3D cases which are shown in the upper and lower panels of
Fig. 8, respectively. For Fig. 8(a) (Fig. 8(c)), all terms in the 2D (3D)
self-energy expression Eq. 2.68 (Eq. 2.77) are used, whereas for Fig. 8(b)
(Fig. 8(d)), the highest order term in that equation is ignored. We note that
the boundaries of the hydrodynamic and ballistic regimes obtained from
different self-energy approximations are quite distinct from each other. Both
approximations are in fact only applicable to sufficiently low energies and
temperatures, and the results for $\varepsilon/E_{\mathsf{F}}>1$ or
$T/E_{\mathsf{F}}>1$ in Fig. 8 are therefore not reliable.
We emphasize that if electron-impurity and/or electron-phonon scattering
effects are present, then one must also compare the electron-electron
scattering rates with these other scattering rates, and hydrodynamics would
apply only if electron-electron scattering is the dominant scattering process.
In real materials it is a challenge to create conditions for the hydrodynamic
regime since electron-impurity and electron-phonon scattering tend to dominate
at low and high temperatures, respectively.
## 4 Wiedemann-Franz (WF) and Kadowaki-Woods (KW) relations for 2D
interacting systems
Given our analytical results for the 2D inelastic scattering rates for
electron-electron interactions, we can briefly comment on the repercussions
for the well-known Wiedemann-Franz (WF) and Kadowaki-Woods (KW) relations in
2D Fermi liquids.
The WF law [30] states that the ratio $L=\kappa/(\sigma T)$, where $\sigma$
$(\kappa)$ is the electronic electrical (thermal) conductivity, is universal
in metals (i.e. Fermi liquids), and the KW law [31] states that the ratio
$K=\mathcal{A}/\gamma^{2}$ is universal, where $\mathcal{A}$ $(\gamma)$ is the
coefficient of the $T^{2}$ term (the linear $T$ term) in the temperature
dependence of the resistivity (specific heat). Both of these universalities
are obeyed rather widely in metals (for the KW law, the metal must show a
dominant $T^{2}$ dependence in its resistivity limiting it to the so-called
strongly correlated materials such as transition metals, heavy fermion
compounds and metallic oxides).
Both of these quantities involve the electronic dc conductivity $(\sigma)$ or
the resistivity ($\rho=1/\sigma$), which is operationally defined by the
following Drude formula:
$\displaystyle\sigma=\frac{ne^{2}\tau}{m^{*}}.$ (4.1)
Here, $n$ is the 2D carrier density and $m^{*}$ is the effective mass, and the
key quantity is the transport relaxation time $\tau$ which depends on the
details of the scattering process. In ordinary metals and doped
semiconductors, transport is mostly limited by electron-impurity (at lower
temperatures) and electron-phonon (at higher temperatures) scattering of the
carriers. This is consistent with the observed temperature-independence
(arising from impurity scattering) and linear-in-$T$ temperature dependence
(arising from phonon scattering in the equipartition regime) of the electrical
resistivity in simple metals and doped semiconductors at low and high
temperatures, respectively. Consideration of electron-impurity and electron-
phonon interactions is beyond the scope of the current work - see, e.g., Refs.
[32, 33, 34].
Our focus in the current work is electron-electron interaction, which, by
virtue of the Galilean invariance in a continuum system, cannot affect the
resistivity since electron-electron scattering usually conserves total
momentum of the system (and transport is a momentum relaxation phenomenon). It
is, however, often claimed in the literature that a hallmark of a Fermi liquid
is the manifestation of a dominant $T^{2}$ temperature dependence of the
resistivity, with the temperature-dependent part of the resistivity going
primarily as
$\displaystyle\rho(T)=1/\sigma(T)=\mathcal{A}T^{2}.$ (4.2)
Although no simple metal or 2D doped semiconductor shows such a $T^{2}$
temperature dependence in the resistivity, many strongly correlated metals
(e.g. transition metals, heavy fermion compounds, various metallic oxides) do.
Since the intraband electron-electron scattering cannot contribute to the
resistivity by virtue of its explicit momentum conserving nature, the
widespread assumption is that the $T^{2}$ resistivity manifesting in many
narrow band metals arises from electron-electron scattering involving umklapp
or interband scattering processes in a Fermi liquid which do not conserve
total momentum. We accept this assumption uncritically, and consider the
consequences for such a $T^{2}$ resistivity arising from electron-electron
scattering for the WF and KW relations using our theory for the inelastic
electron-electron interaction induced scattering rate.
This will involve the assumption that our calculated imaginary part of the on-
shell electron self-energy at the Fermi energy is indeed the appropriate
scattering for the resistivity so that we can make the identification that our
calculated imaginary part of the temperature-dependent electron self-energy at
the Fermi energy defines the transport scattering relaxation time $\tau$
entering the electrical resistivity through:
$\displaystyle 1/\tau=-2\operatorname{Im}\Sigma^{(R)}(0,T).$ (4.3)
Using our analytical results for $\operatorname{Im}\Sigma^{(R)}$ (Eq. 2.70),
we get:
$\displaystyle\frac{1}{2\tau}=\frac{\pi}{8}\frac{T^{2}}{E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{T}\right)-\frac{\pi}{24}\left(6+\ln
2\pi^{3}-36\ln\mathrm{A}\right)\frac{T^{2}}{E_{\mathsf{F}}}+\frac{7\zeta(3)}{2\sqrt{2}\pi}\frac{T^{3}}{r_{s}E_{\mathsf{F}}^{2}}.$
(4.4)
With this expression for $\tau$ along with the Drude formula for the dc
resistivity, we now discuss below the WF and KW relations individually for 2D
systems.
### 4.1 WF Law
For the WF law, which is a statement on the ratio of the electrical and
thermal conductivity, we can write [35]
$\displaystyle L=\frac{\kappa}{\sigma
T}=\frac{\pi^{2}}{3}\frac{\tau}{\tau+\tau_{i}}.$ (4.5)
Here, $\tau_{i}$ is the temperature-independent momentum relaxation time due
to electron-impurity elastic scattering which must dominate at $T=0$, where
$\tau(T)$ above, arising from inelastic electron-electron scattering, becomes
infinite. Since $1/\tau\sim-T^{2}\ln(T/T_{F})$, we conclude that the relevant
WF law for 2D strongly correlated systems becomes:
$\displaystyle L\sim\dfrac{L_{0}}{1+B(T/T_{F})^{2}\ln(T/T_{F})}.$ (4.6)
Here the nonuniversal materials and sample dependent constant $B$ [36] depends
both on the strength of the impurity scattering and the prefactor of the
$T^{2}\ln T$ term in our imaginary self-energy, and $L_{0}$ is the ideal
Lorenz number for the WF law which must be recovered at $T=0$ where the
electron-electron scattering vanishes. The important qualitative result,
however, is that the WF law will be strongly violated if the electron-electron
scattering induced inelastic scattering is strong in the system, and at low
temperatures this violation, or equivalently the suppression of the Lorenz
number from its ideal WF value, will follow a
$[1+B(T/T_{F})^{2}\ln(T/T_{F})]^{-1}$ dependence in a Fermi liquid 2D metal
(with $B$ being a materials-dependent nonuniversal number). This violation of
the WF law in 2D metals should be observable for strongly interacting systems
at low enough temperatures, where the electron-phonon interaction is
negligible (but electron-electron interaction is still significant). Note that
we can further refine the violation of the WF law by adding the next-to-
leading-order temperature dependent terms in the imaginary part of the self-
energy (Eq. 2.70). We note that the $\ln T$ term in above 2D equations is most
likely absent in the transport coefficients, but this is beyond the scope of
the current work.
### 4.2 KW law
For the KW law, we need an expression for the 2D electronic specific heat,
which in the leading-order Fermi liquid theory can be written as:
$\displaystyle C_{e}=C(m^{*}T),$ (4.7)
where $C$ depends only on universal constants, and $m^{*}$ is the carrier
effective mass. Thus, $\gamma=Cm^{*}$ for the 2D Fermi liquid. Note that in
2D, carrier density or Fermi energy does not enter the expression for the
electronic specific heat by virtue of the constancy of the 2D density of
states. For the KW relation, we need the ratio $K$ of the $T^{2}$ term in the
resistivity with the linear-in-$T$ term in the specific heat, which is given
by:
$\displaystyle K=\mathcal{A}/(Cm^{*})^{2}$ (4.8)
with $\mathcal{A}$ given by:
$\displaystyle\mathcal{A}=\frac{m^{*}}{ne^{2}}\frac{1}{\tau}\frac{1}{T^{2}}=\mathcal{A}^{\prime}\frac{m^{*}}{n}\frac{1}{E_{F}}\ln\left(\frac{T_{F}}{T}\right),$
(4.9)
where $\mathcal{A}^{\prime}$ depends on universal constants. Remembering that
in 2D Fermi systems, $E_{\mathsf{F}}\sim(n/m^{*})$, we get, combining
everything, for the KW ratio:
$\displaystyle K=\frac{B^{\prime}}{n^{2}}\ln\left(\frac{T_{F}}{T}\right),$
(4.10)
where $B^{\prime}$ is universal within the free electron type Fermi liquid
theories. (Again, the $\ln T$ is most likely absent, but this is beyond the
scope of the current work.) We note that within the free electron type band
structure model for Fermi liquids, the Kadowaki-Woods ratio goes as $n^{-2}$
in 2D in contrast to the corresponding $n^{-7/3}$ dependence in 3D systems.
This precise difference by a factor of $n^{-1/3}$ can be understood as arising
from the dimensional difference between 2D and 3D resistivity definitions: in
2D the resistivity is simply measured in ohms whereas in 3D it is measured in
ohm.cm, so the two units must differ by a unit of length, which is precisely
what $n^{-1/3}$ is for a 3D carrier density $n$. It may be worthwhile to
emphasize that the KW relation is by no means universal either in 2D or in 3D
by virtue of the strong density dependence inherent in the KW ratio. It is
only when different materials have similar effective carrier densities, one
can talk about a universal KW relation, and even then it is rather a dubious
universality since the origin of the $T^{2}$ resistivity term may differ from
system to system.
## 5 Conclusion
We have analytically investigated the domain of validity of the Fermi liquid
theory and the quasiparticle picture in Coulomb interacting continuum 2D and
3D electron liquids. Using the leading-order dynamical screening
approximation, which is exact in the high-density limit, we calculate exact
expressions for the real and imaginary parts of the electron self-energy in
expansions in temperature and energy measured from the Fermi surface. Using
the calculated imaginary part of the self-energy, we estimate the
$r_{s}$-dependent crossover energy and temperature scales above which well-
defined quasiparticles no longer exist. We find that in both 2D and 3D systems
the quasiparticle picture remains valid at energies and temperatures
comparable to Fermi energies or above, implying a robust validity of Fermi
liquid theory, not only in 3D, but also in 2D. In general, the energy scales
for the validity of the Fermi liquid theory decreases with increasing $r_{s}$,
but the theory being exact only for small $r_{s}$, this finding, although
intuitively appealing, is only tentative. Recent work, which calculates both
the on-shell and the off-shell self-energy numerically without making
$\varepsilon\ll E_{\mathsf{F}}$ approximation, has also concluded that the
regime of validity of the 2D Fermi liquid theory and quasiparticle concept is
wide and applies to very high energies [37].
We emphasize that the applicability of the theory for $r_{s}>1$ is irrelevant
to our work since we are theoretically addressing a matter of principle: how
high in $\varepsilon$ ($T$) does the theory still remain valid? All we need is
the controlled nature of the theory, which is true for $r_{s}\ll 1$, in the
weak coupling perturbative RG sense. We mention that RPA itself is a
controlled approximation giving exact results for small $r_{s}$ although it is
known to work very well empirically at metallic densities (with $r_{s}\sim
4-6$). The reason is a likely cancellation of higher-order diagrams, but no
systematic theory exists for $r_{s}>1$ because of the failure of the
$r_{s}$-expansion. Quantum Monte Carlo calculations can be used in some
situations to study the ground state properties, but are inapplicable to the
question of the validity of the Fermi liquid theory (we are asking in this
work) which must study the quasiparticle excitations that cannot be done by
Monte Carlo techniques. What we show here is that the Fermi liquid theory
applies at very high energies away from the Fermi surface as well as at very
high temperatures not only at small $r_{s}$, but also at intermediate to large
$r_{s}$. Our results for $r_{s}>1$ are obviously not exact, but as long as
there is no ground state strong-coupling quantum phase transition, the theory
should remain valid qualitatively at higher $r_{s}$.
We also provide 2D and 3D ‘phase diagrams’ for temperature and frequency,
where the electron liquid crosses over from the collisionless ballistic regime
to the collision dominated hydrodynamic regime neglecting effects of impurity
and phonon scattering, and obtained general expressions for 2D Wiedemann-Franz
and Kadowaki-Woods relations, using our calculated self-energy expressions. We
have also provided effective mass renormalization results for 2D and 3D
systems including leading and subleading temperature corrections, extending
existing results in the literature.
## Acknowledgement
This work is supported by the Laboratory for Physical Sciences (SDS) and the
Simons Foundation “Ultra-Quantum Matter” Research Collaboration (YL).
## Appendix A Derivation of self-energy formulas using Matsubara technique
For the sake of completeness, in this Appendix we briefly review an
alternative derivation of the self-energy formula Eq. 2.33 using the Matsubara
instead of Keldysh technique. For more details, see for example Refs. [6, 7].
As explained in earlier sections, for electrons interacting via Coulomb
interactions, in the high density limit $r_{s}\ll 1$, the electron self-energy
is given by the RPA self-energy represented by the diagram in Fig. 1(b). In
the Matsubara formalism, the black sold line represents the noninteracting
Matsubara Green’s function for electrons, which acquires the form
$\displaystyle
G_{0}(\bm{\mathrm{k}},i\varepsilon_{n})=\left(i\varepsilon_{n}-\xi_{\bm{\mathrm{k}}}\right)^{-1}.$
(A.1)
Here $\varepsilon_{n}=2\pi(n+1/2)T$ is the fermionic Matsubara frequency. The
red wavy line with a solid dot stands for the dynamically screened RPA
interaction $D$ defined diagrammatically by the Dyson equation shown in Fig.
1(a). In particular, $D$ is related to the bare interaction $V$ (red wavy line
with a open dot) and the polarization bubble $\Pi$ (black bubble) through the
following equation:
$\displaystyle\begin{aligned}
D(\bm{\mathrm{q}},i\omega_{m})=\left[V^{-1}(\bm{\mathrm{q}})-\Pi(\bm{\mathrm{q}},i\omega_{m})\right]^{-1},\end{aligned}$
(A.2)
with $\omega_{m}=2\pi mT$ being the bosonic Matsubara frequency.
$\Pi(\bm{\mathrm{q}},i\omega_{m})$ is given by a product of two bare electron
Green’s functions:
$\displaystyle\begin{aligned}
\Pi(\bm{\mathrm{q}},i\omega_{m})=2T\sum_{\varepsilon_{n}}\int\frac{d^{d}\bm{\mathrm{k}}}{(2\pi)^{d}}G_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},i\varepsilon_{n}+i\omega_{m})G_{0}(\bm{\mathrm{k}},i\varepsilon_{n}),\end{aligned}$
(A.3)
where the overall factor of two arises from the summation over spin indices.
It is then straightforward to see that the RPA self-energy depicted in Fig.
1(b) evaluates to
$\displaystyle\begin{aligned}
\Sigma(\bm{\mathrm{k}},i\varepsilon_{n})=-T\sum_{\omega_{m}}\int\frac{d^{d}\bm{\mathrm{q}}}{(2\pi)^{d}}D(\bm{\mathrm{q}},i\omega_{m})G_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},i\varepsilon_{n}+i\omega_{m}).\end{aligned}$
(A.4)
After analytical continuation, the equation above then leads to the retarded
self-energy
$\displaystyle\begin{aligned}
\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\frac{i}{2}\int\frac{d^{d}\bm{\mathrm{q}}}{(2\pi)^{d}}\int\frac{d\omega}{2\pi}&\left\\{\left[D^{(R)}(\bm{\mathrm{q}},\omega)-D^{(A)}(\bm{\mathrm{q}},\omega)\right]G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\coth\left(\frac{\omega}{2T}\right)\right.\\\
&\left.+D^{(A)}(\bm{\mathrm{q}},\omega)\left[G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)-G_{0}^{(A)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right]\tanh\left(\frac{\omega+\varepsilon}{2T}\right)\right\\}.\end{aligned}$
(A.5)
With the help of the FDT Eqs. 2.9 and 2.28b, one can prove that this equation
is equivalent to the self-energy formula Eq. 2.33 derived in the Keldysh
formalism. Similarly, by analytical continuation, Eq. A.3 can be transformed
to
$\displaystyle\begin{aligned}
\Pi^{(R)}(\bm{\mathrm{q}},\omega)=-i\int\frac{d^{d}\bm{\mathrm{k}}}{(2\pi)^{d}}\int\frac{d\varepsilon}{2\pi}&\left\\{G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\left[G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)-G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\right]\tanh\left(\frac{\varepsilon}{2T}\right)\right.\\\
&\left.+\left[G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)-G_{0}^{(A)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right]G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\tanh\left(\frac{\varepsilon+\omega}{2T}\right)\right\\},\end{aligned}$
(A.6)
which is equivalent to the retarded polarization operator formula Eq. 2.21
derived in Sec. 2.1.
## Appendix B Integrals involving hyperbolic functions
In this Appendix, we provide analytical results for integrals of the form
$I_{1,2}(a)$ in Eq. 2.66, which are needed for the calculation of electron
self-energy with arbitrary value of $\varepsilon/T$. The results are obtained
with the help of the exponential series expansion of the hyperbolic functions
(Eq. 2.67). In particular, $I_{1,2}(a)$ can be rewritten as
$\displaystyle\begin{aligned}
I_{1}(a)=&2\sum_{k=1}^{\infty}\left[2-(-1)^{k}e^{-2ka}-(-1)^{k}e^{2ka}\right]\int_{0}^{\infty}dxf(x)e^{-2kx}\\\
&+4\sum_{k=1}^{\infty}\int_{0}^{a}dxf(x)(-1)^{k}\cosh\left({2kx-2ka}\right)+2\int_{0}^{a}dxf(x),\end{aligned}$
(B.1a) $\displaystyle\begin{aligned}
I_{2}(a)=\,&2\sum_{k=1}^{\infty}(-1)^{k}\left[e^{-2ka}-e^{2ka}\right]\int_{0}^{\infty}dxf(x)e^{-2kx}\\\
&+4\sum_{k=1}^{\infty}\int_{0}^{a}dxf(x)(-1)^{k}\cosh\left({2kx-2ka}\right)+2\int_{0}^{a}dxf(x).\end{aligned}$
(B.1b)
For both $I_{1}(a)$ and $I_{2}(a)$, the last two terms cancel with each other
due to the fact that $\sum_{k=1}^{\infty}(-1)^{k}\cosh(2kx-2ka)=-1/2$, and we
are left with the first terms in both Eq. B.1a and Eq. B.1b. By inserting the
result of the integral $\int_{0}^{\infty}dxf(x)e^{-2kx}$ into Eq. B.1, one is
able to evaluate $I_{1,2}(a)$ for various $f(x)$:
$\displaystyle\begin{aligned}
&\int_{0}^{\infty}dx\left[2\coth\left(x\right)-\tanh\left(x+a\right)-\tanh\left(x-a\right)\right]x=\,\frac{\pi^{2}}{4}+a^{2},\\\
&\int_{0}^{\infty}dx\left[2\coth\left(x\right)-\tanh\left(x+a\right)-\tanh\left(x-a\right)\right]x\ln
x\\\ &=\,\left(1-\gamma_{E}-\ln
2\right)a^{2}+\frac{\pi^{2}}{12}\left(3-\gamma_{E}-\ln\frac{2}{\pi^{2}}-24\ln\mathrm{A}\right)-\frac{1}{2}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-2a})+\partial_{s}\operatorname{Li}_{s}(-e^{2a})\right]\bigg{\lvert}_{s=2},\\\
&\int_{0}^{\infty}dx\left[2\coth\left(x\right)-\tanh\left(x+a\right)-\tanh\left(x-a\right)\right]x^{2}=\,\zeta(3)-\frac{1}{2}\left[\operatorname{Li}_{3}(-e^{-2a})+\operatorname{Li}_{3}(-e^{2a})\right],\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]=\,2a,\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]x=\,\frac{1}{2}\left[\operatorname{Li}_{2}(-e^{-2a})-\operatorname{Li}_{2}(-e^{2a})\right],\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]x^{2}=\,\frac{2}{3}a^{3}+\frac{\pi^{2}}{6}a,\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]x^{2}\ln
x=\,\left(3-2\gamma_{E}-\ln
4\right)\left(\frac{1}{3}a^{3}+\frac{\pi^{2}}{12}a\right)+\frac{1}{2}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-2a})-\partial_{s}\operatorname{Li}_{s}(-e^{2a})\right]\bigg{\lvert}_{s=3}.\end{aligned}$
(B.2)
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|
# Charming ALPs
Adrian Carmona<EMAIL_ADDRESS>CAFPE and Departamento de Física Teórica y del
Cosmos,
Universidad de Granada, E18071 Granada, Spain Christiane Scherb cscherb@uni-
mainz.de PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical
Physics,
Johannes Gutenberg University, 55099 Mainz, Germany Pedro Schwaller
<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence & Mainz Institute
for Theoretical Physics,
Johannes Gutenberg University, 55099 Mainz, Germany
###### Abstract
Axion-like particles (ALPs) are ubiquitous in models of new physics explaining
some of the most pressing puzzles of the Standard Model. However, until
relatively recently, little attention has been paid to its interplay with
flavour. In this work, we study in detail the phenomenology of ALPs that
exclusively interact with up-type quarks at the tree-level, which arise in
some well-motivated ultra-violet completions such as QCD-like dark sectors or
Froggatt-Nielsen type models of flavour. Our study is performed in the low-
energy effective theory to highlight the key features of these scenarios in a
model independent way. We derive all the existing constraints on these models
and demonstrate how upcoming experiments at fixed-target facilities and the
LHC can probe regions of the parameter space which are currently not excluded
by cosmological and astrophysical bounds. We also emphasize how a future
measurement of the currently unavailable meson decay $D\to\pi+\rm{invisible}$
could complement these upcoming searches.
††preprint: MITP-21-003
## I Introduction
One of the outstanding open questions in particle physics is the nature of
dark matter (DM) and whether it is part of a larger dark sector that we yet
have to discover. Most realistic models require some form of non-gravitational
interaction between us and the dark sector in order to satisfy cosmological
constraints. These ”portals” then offer the possibility to probe the dark
sector through laboratory experiments or astrophysical observations.
An intriguing possibility is that the portal to the dark sector is flavour
sensitive or even connected to an ultra-violet (UV) theory of flavour. Simple
flavoured dark matter scenarios have received a lot of attention recently [1,
2, 3, 4], since they allow probes of dark sector physics using low energy
flavour observables. For strongly coupled dark sectors, a flavoured portal
imprints the Standard Model (SM) flavour structure on the dark sector, leading
to a variety of new phenomena [5, 6].
So far only couplings to down-type quarks were considered in this context,
therefore it is natural to ask what new aspects arise if the portal couples to
up-type quarks instead. 111Simple DM models which dominantly couple to up-type
quarks were also studied e.g. in [7, 8, 9]. The key feature in this case is
the emergence of a light pseudo Nambu-Goldstone boson (pNGB) which dominantly
couples to up-type quarks, and which we therefore dub a charming ALPs.
Our main goal in this work is to develop the effective theory of the charming
ALP and its phenomenological profile, independently of its embedding in
different UV scenarios. Besides QCD-like dark sectors [10, 11, 12, 12, 5, 13],
these particles can arise e.g. in specific Froggatt-Nielsen (FN) models [14]
where only right-handed (RH) up-quarks have non-zero charges. The different UV
completions provide some guidance for the structure of the effective
couplings, which we use to define benchmark scenarios for the charming ALP.
We will study the phenomenology of these different benchmark models through
their low-energy effective field theory (EFT), taking into account the effects
of QCD confinement for low enough ALP masses. A lot of work has been done in
this arena, see e.g [15, 16, 17, 18, 19, 20, 21] for the study of ALP collider
signatures, [22, 23, 24] and [25, 26, 27] for the study of flavour-changing
neutral currents (FCNCs) in the quark and lepton sector, respectively, as well
as [28, 29, 30, 31] for the calculation of the one-loop running. It is worth
to mention also [32, 33] regarding CP-violating probes of ALPs.
The work is organized as follows: We introduce the effective Lagrangian
describing the charming ALPs and motivate briefly the four particular
benchmarks models studied in this work in section II. In section III we
examine the different flavour constraints relevant for charming ALPs. More
specifically, we consider $D-\bar{D}$ mixing, exotic decays of $D$, $B$ and
$K$ mesons as well as the decay $J/\psi\to a\gamma$. We discuss the bounds
arising from astrophysical observables and cosmology in section IV, describing
also the different ALP decay channels. In section V we study the different
collider probes on the models at hand, including upcoming fix-target
experiments as well as those at LHC forward detectors. We combine all these
different bounds and discuss the resulting constraints on the parameter space
of the models in section VI. We conclude in section VII. Finally, we present
in some detail the particular UV completions considered in this work in
appendices A and B.
## II Charming ALP EFT
We consider a general ALP, which we will denote $a$, with flavour-violating
couplings to RH up-quarks. The most general EFT describing such a system is
given by the following Lagrangian [34, 35]
$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}a)(\partial^{\mu}a)-\frac{m_{a}^{2}}{2}a^{2}+\frac{\partial_{\mu}a}{f_{a}}\left[(c_{u_{R}})_{ij}\bar{u}_{Ri}\gamma^{\mu}u_{Rj}+c_{H}H^{\dagger}i\overleftrightarrow{D_{\mu}}H\right]$
$\displaystyle-\frac{a}{f_{a}}\left[c_{g}\frac{g_{3}^{2}}{32\pi^{2}}G_{\mu\nu}^{a}\tilde{G}^{\mu\nu
a}+c_{W}\frac{g_{2}^{2}}{32\pi^{2}}W_{\mu\nu}^{I}\tilde{W}^{\mu\nu
I}+c_{B}\frac{g_{1}^{2}}{32\pi^{2}}B_{\mu\nu}\tilde{B}^{\mu\nu}\right],$ (1)
where $g_{1},g_{2}$ and $g_{3}$ are the gauge couplings of $U(1)_{Y}$,
$SU(2)_{L}$ and $SU(3)$, respectively, whereas $B_{\mu\nu}$,
$W_{\mu\nu}^{I},\,I=1,2,3,$ and $G_{\mu\nu}^{a},\,a=1,\ldots,8,$ are their
corresponding field-strength tensors. Furthermore,
$\tilde{B}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}B^{\alpha\beta},\ldots,$
denote their corresponding duals, while $H$ stands for the SM Higgs doublet.
The Wilson coefficients (WCs) $c_{g},c_{W},c_{B}$ and $c_{H}\in\mathbb{R}$,
whereas $c_{u_{R}}$ is a hermitian matrix. In order to write down the above
Lagrangian we have assumed that $a$ is the pNGB of the spontaneous breaking of
some global $U(1)$ symmetry, which is softly broken and may be anomalous. We
have also assumed that the couplings to leptons, SM quark doublets and RH
down-type quarks vanish. Here, contrary to the QCD axion case, we will treat
$m_{a}$ and $f_{a}$ as independent parameters.
Using field redefinitions, we can trade the operator
$\mathcal{O}_{H}=(\partial^{\mu}a/f_{a})H^{\dagger}i\overleftrightarrow{D_{\mu}}H$
by the flavour-blind and chirality conserving one (see e.g. [34, 16])
$\displaystyle\frac{\partial_{\mu}a}{f_{a}}\left[\frac{1}{3}\bar{q}_{Li}\gamma^{\mu}q_{Li}+\frac{4}{3}\bar{u}_{Ri}\gamma^{\mu}u_{Ri}-\frac{2}{3}\bar{d}_{Ri}\gamma^{\mu}d_{Ri}\right.$
$\displaystyle\left.-\bar{l}_{Li}\gamma^{\mu}l_{Li}-2\bar{e}_{Ri}\gamma^{\mu}e_{Ri}\right].$
(2)
Together with
$\displaystyle\mathcal{O}_{W}=\frac{a}{f_{a}}\frac{g_{2}^{2}}{32\pi^{2}}W_{\mu\nu}^{I}\tilde{W}^{\mu\nu
I}$ (3)
this operator induces flavour-changing neutral currents (FCNCs) at one-loop,
which have been studied in [24] in the framework of $B$ and $K$-meson decays.
Here we will just assume that both WCs are small enough so that the leading
flavour-violating effects are parametrized by $c_{u_{R}}$. Furthermore, after
integrating by parts and using equations of motion, one can express this
chirality-conserving operator as a function of
$\displaystyle-i\frac{a}{f_{a}}\bar{q}_{Lk}\tilde{H}u_{Rj}\left(Y_{u}\right)_{ks}(c_{u_{R}})_{sj}+\mathrm{h.c.}$
(4)
plus some extra contributions to the anomalous terms in (1), where
$\tilde{H}=i\sigma^{2}H^{\ast}$ and $Y_{u}$ is the up Yukawa matrix,
$\displaystyle-\bar{q}_{Lk}\tilde{H}u_{Rj}\left(Y_{u}\right)_{kj}+\mathrm{h.c.}\,.$
(5)
Note that, without any loss of generality, we can always choose a basis where
$\displaystyle Y_{u}=\lambda_{u},\qquad Y_{d}=\tilde{V}\lambda_{d},$ (6)
with $\lambda_{u,d}$ diagonal matrices with real and positive entries and
$\tilde{V}$ a unitary matrix. In the case where there is no extra contribution
to the fermion masses, $\tilde{V}$ is just the CKM mixing matrix $V$ and
$\lambda_{u}=\sqrt{2}\mathcal{M}_{u}/v$,
$\lambda_{d}=\sqrt{2}\mathcal{M}_{d}/v$, where
$\mathcal{M}_{u}=\mathrm{diag}(m_{u},m_{c},m_{t})$,
$\mathcal{M}_{d}=\mathrm{diag}(m_{d},m_{s},m_{b})$, and $v=246$ GeV the Higgs
vacuum expectation value. In this basis, the RH up-quarks do not need to be
rotated to diagonalize the mass matrices generated after electroweak symmetry
breaking. Indeed, one could just take
$\displaystyle U_{L}^{d}=\tilde{V},\quad
U_{R}^{d}=U_{L}^{u}=U_{R}^{u}=\mathbbm{1}.$ (7)
Henceforth, we will assume that the above EFT Lagrangian is defined in such a
basis. Then, if we denote the WC of the operators in (4) by $\mathcal{C}$, one
has that
$\displaystyle\mathcal{C}_{ij}=(\lambda_{u})_{ii}(c_{u_{R}})_{ij}.$ (8)
For small ALP masses, $m_{a}\lesssim 1$ GeV, $a$ will mostly decay to hadrons.
These decays will proceed through the following Lagrangian [34, 35, 36, 37]
$\displaystyle\mathcal{L}_{\rm aChPT}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}a)(\partial^{\mu}a)-\frac{m_{a}^{2}}{2}a^{2}-\frac{a}{f_{a}}\frac{e^{2}}{32\pi^{2}}c_{\gamma}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{f_{\pi}^{2}}{4}\mathrm{Tr}\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}\right)$
$\displaystyle+\frac{f_{\pi}^{2}B_{0}}{2}\mathrm{Tr}\left(\hat{m}_{q}(a)U^{\dagger}+U\hat{m}_{q}^{\dagger}(a)\right)+i\frac{f_{\pi}^{2}}{2}\frac{\partial_{\mu}a}{f_{a}}\mathrm{Tr}\left[(\hat{c}+\varkappa_{q}c_{g})\left(UD^{\mu}U^{\dagger}\right)\right],$
(9)
where $B_{0}$ is a constant, $f_{\pi}\approx 93\,\mathrm{MeV}$ is the pion
decay constant and $m_{q}$ is the quark mass matrix
$m_{q}=\mathrm{diag}(m_{u},m_{d},m_{s})$. In the above equation,
$\displaystyle U(\Pi)=\mathrm{exp}\left(2i\Pi/f_{\pi}\right),$ (10)
where
$\displaystyle\Pi=\varphi^{a}\frac{\lambda^{a}}{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{1}{\sqrt{2}}\pi^{0}+\frac{\eta_{8}}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{\eta_{8}}{\sqrt{6}}&K^{0}\\\
K^{-}&K^{0}&-\frac{2}{\sqrt{6}}\eta_{8}\end{pmatrix}$ (11)
is the Goldstone matrix describing the spontaneous symmetry breaking
$SU(3)_{L}\otimes SU(3)_{R}\to SU(3)_{V}$ of QCD. On the other hand,
$\varkappa_{q}=m_{q}^{-1}/\mathrm{Tr}(m_{q}^{-1})$,
$\hat{c}=\mathrm{diag}((c_{u_{R}})_{11},0,0)$, and
$\displaystyle\hat{m}_{q}(a)$
$\displaystyle=\mathrm{exp}\left(-i\varkappa_{q}c_{g}\frac{a}{2f_{a}}\right)m_{q}\left(-i\varkappa_{q}c_{g}\frac{a}{2f_{a}}\right),$
(12) $\displaystyle D_{\mu}U$
$\displaystyle=\partial_{\mu}U+ieA_{\mu}\Big{[}Q_{q},U\Big{]},$ (13)
$\displaystyle c_{\gamma}$
$\displaystyle=c_{W}+c_{B}-2\,N_{c}c_{g}\mathrm{Tr}\left(\varkappa_{q}Q_{q}^{2}\right),$
(14)
with $e=g_{2}g_{1}/\sqrt{g_{1}^{2}+g_{2}^{2}}$ the electric charge and
$Q_{q}=1/3\,\mathrm{diag}(2,-1,-1)$. One should note that, in order to get the
above Lagrangian, we had to get rid of the gluon coupling by the following
chiral transformation
$\displaystyle
q\to\mathrm{exp}\Big{(}-i\frac{a}{2f_{a}}c_{g}\varkappa_{q}(1+\gamma_{5})\Big{)}q.$
(15)
Note that this Lagrangian gives an irreducible contribution to the ALP mass
$\displaystyle m_{a\,\textrm{QCD}}^{2}$
$\displaystyle=c_{g}\frac{m_{\pi}^{2}f_{\pi}^{2}}{(m_{d}+m_{u})f_{a}^{2}}\frac{m_{u}m_{d}m_{s}}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}$
$\displaystyle+\mathcal{O}\left(\frac{m_{\pi}^{2}f_{\pi}^{4}}{f_{a}^{4}}\right),$
(16)
where
$\displaystyle
m_{\pi}^{2}=B_{0}(m_{u}+m_{d})+\mathcal{O}\left(\frac{m_{\pi}^{2}f_{\pi}^{4}}{f_{a}^{4}}\right).$
(17)
Kinetic mixing arising from the last term in eq. (9) induces a mass mixing
between the different neutral pions. In particular, we obtain
$\displaystyle\pi$
$\displaystyle\to\pi-\frac{f_{\pi}}{f_{a}}\frac{m_{a}^{2}}{m_{a}^{2}-m_{\pi}^{2}}\left(\mathcal{K}_{\pi}-\frac{\mathcal{K}_{\eta}\delta_{I}m_{\pi}^{2}}{\sqrt{3}(m_{a}^{2}-m_{\eta}^{2})}\right)a$
$\displaystyle-\frac{\delta_{I}m_{\pi}^{2}}{\sqrt{3}(m_{\eta}^{2}-m_{\pi}^{2})}\eta_{8}+\mathcal{O}(f_{\pi}^{2}/f_{a}^{2})+\mathcal{O}(\delta_{I}^{2}),$
(18)
where
$\displaystyle\delta_{I}=\frac{m_{d}-m_{u}}{m_{d}+m_{u}}\approx\frac{1}{3},\qquad
m_{\eta}^{2}=\frac{m_{d}+m_{u}+4m_{s}}{3(m_{u}+m_{d})}m_{\pi}^{2}$ (19)
and
$\displaystyle\mathcal{K}_{\pi}$
$\displaystyle=c_{g}\frac{m_{s}(m_{d}-m_{u})}{2(m_{s}m_{u}+m_{d}m_{u}+m_{d}m_{s})}+\frac{(c_{u_{R}})_{11}}{2},$
(20) $\displaystyle\mathcal{K}_{\eta}$
$\displaystyle=c_{g}\frac{m_{s}(m_{d}+m_{u})-2m_{d}m_{u}}{2\sqrt{3}((m_{s}m_{u}+m_{d}m_{u}+m_{d}m_{s}))}+\frac{(c_{u_{R}})_{11}}{2\sqrt{3}}.$
(21)
As already mentioned, we are interested in scenarios where the ALP only
interacts with RH up-quarks, and it is likely to mediate flavour-changing
processes. In order to explore how the different experimental constraints are
intertwined and the best way to probe these models, we will consider four
different benchmarks. In practice, each of these scenarios corresponds to a
particular choice of the matrix $c_{u_{R}}$ defined above.
The first two benchmarks are motivated by theories of ’dark QCD’, where the SM
is extended with a confining dark sector, composed of $n_{d}$ dark flavours
transforming under $SU(N_{d})$. These models constitute a particular UV
completion of the scenarios we have in mind, where light pNGB bosons do only
interact at leading order with the SM RH up-quarks. Indeed, this is a natural
outcome when both sectors are mediated by a scalar, bifundamental of both
confining groups, with hypercharge $-2/3$. We refer the reader to appendix A
and references therein for more details. At the end of the day, after
integrating out the heavy scalar mediator, and assuming confinement in the
dark QCD group, we obtain almost degenerate, parametrically light scalars with
a Lagrangian along the lines of (1). When studying phenomena where the
interplay of the different scalars is not relevant, one can examine the
phenomenological impact of the different degrees of freedom separately. In
particular, focusing on the ’diagonal’ dark pions $\pi_{D_{3}}$ and
$\pi_{D_{8}}$, we obtain
$\displaystyle c_{u_{R}}^{(3)}$
$\displaystyle=\frac{-\kappa_{0}^{2}}{4}\begin{pmatrix}9c_{12}^{2}-4s_{12}^{2}&13c_{12}s_{12}&0\\\
13c_{12}s_{12}&9s_{12}^{2}-4c_{12}^{2}&0\\\ 0&0&0\end{pmatrix},$ (22)
$\displaystyle c_{u_{R}}^{(8)}$
$\displaystyle=\frac{-\kappa_{0}^{2}}{4\sqrt{3}}\begin{pmatrix}4s_{12}^{2}+9c_{12}^{2}&5c_{12}s_{12}&0\\\
5c_{12}s_{12}&4c_{12}^{2}+9s_{12}^{2}&0\\\ 0&0&-2\end{pmatrix},$ (23)
as well as $c_{H}=0$ and $c_{g}=c_{W}=c_{B}=0$ at tree level. In the equation
above $\kappa_{0}\in\mathbb{R}^{+}$ and $c_{12}=\cos\theta_{12}$,
$s_{12}=\sin\theta_{12}$, with $\theta_{12}\in[0,\pi]$. For the sake of
concreteness, following [5], we fix $\theta_{12}=0.022$.
There is another class of models that can naturally UV complete these
scenarios. It is the case of FN models where only RH up-quarks have non-zero
charges. Such models are attractive because they naturally result in enhanced
Yukawa couplings, while still being in agreement with existing flavour bounds.
They where already considered in a slightly different context [38, 39, 40] but
without paying attention to the phenomenology of the light scalar degree of
freedom, _the flavon_. We refer the reader to appendix B and references
therein for more details. One can reproduce the hierarchical up-quark masses
with RH up-quark charges $n_{u}=(2,1,0)$ under the $U(1)$ flavour group,
assuming that the vev of the scalar breaking such symmetry is $\epsilon\sim
m_{c}/m_{t}$ times its UV cutoff. At the end of the day, this setup leads to
an ALP Lagrangian along the lines of equation (1) with
$\displaystyle c_{u_{R}}\sim\begin{pmatrix}2&3\epsilon&3\epsilon^{2}\\\
3\epsilon&1&\epsilon\\\ 3\epsilon^{2}&\epsilon&\epsilon^{2}\end{pmatrix},$
(24)
whereas $c_{H}=0$. The concrete values of the anomalous couplings
$c_{g},c_{W},c_{B}$ depend on the specific UV completion of the FN model and
may all be zero, which is the case we will consider in the following. We
define the FN-motivated benchmark with $c_{u_{R}}$ given as above, whereas the
other WCs are zero.
Finally, we also consider an infra-red (IR) motivated scenario, where
$(c_{u_{R}})=1$, $\forall i,j$ at the scale $f_{a}$. This is representative of
the anarchic limit, where no hierarchies are present in $c_{u_{R}}$ and all
the entries are of the same order. Similarly to the previous cases, we assume
that the anomalous gauge couplings and $c_{H}$ are negligible.
## III Flavour constraints
The presence of flavour-changing ALP couplings to RH up-quarks will induce
several flavour violating processes, constraining significantly the parameter
space. In particular, we will consider
* •
the $\Delta F=2$ process of $D-\bar{D}$ mixing displayed in fig. 1 and
* •
$\Delta F=1$ processes like the exotic decays of $D$, $B$ and $K$ mesons (see
fig. 2).
As shown in fig. 2, in the models at hand, exotic $D$ meson decays are tree
level processes while $B$ and $K$ decays can only happen at one loop. In
addition, ALPs also contribute to radiative $J/\psi$ decays (c.f. fig. 3).
Figure 1: Parton level diagrams for ALP-mediated $D-\bar{D}$ mixing.
Figure 2: Parton level diagrams for exotic $D$, $K$ and $B$ meson decays
involving ALPs.
### III.1 $D-\bar{D}$ mixing
The effective Hamiltonian relevant for $D$ meson mixing reads [41]
$\displaystyle\mathcal{H}_{\rm eff}^{\Delta
C=2}=\sum_{i=1}^{5}C_{i}\mathcal{O}_{i}+\sum_{i=1}^{3}\tilde{C}_{i}\tilde{\mathcal{O}}_{i},$
(25)
where
$\displaystyle\mathcal{O}_{1}$
$\displaystyle=(\bar{c}_{L}^{\alpha}\gamma^{\mu}u_{L}^{\alpha})(\bar{c}_{L}^{\beta}\gamma_{\mu}u_{L}^{\beta}),$
(26) $\displaystyle\mathcal{O}_{2}$
$\displaystyle=(\bar{c}_{R}^{\alpha}u_{L}^{\alpha})(\bar{c}_{R}^{\beta}u_{L}^{\beta}),\quad\mathcal{O}_{3}=(\bar{c}_{R}^{\alpha}u_{L}^{\beta})(\bar{c}_{R}^{\beta}u_{L}^{\alpha}),$
(27) $\displaystyle\mathcal{O}_{4}$
$\displaystyle=(\bar{c}_{R}^{\alpha}u_{L}^{\alpha})(\bar{c}_{L}^{\beta}u_{R}^{\beta}),\quad\mathcal{O}_{5}=(\bar{c}_{R}^{\alpha}u_{L}^{\beta})(\bar{c}_{L}^{\beta}u_{R}^{\alpha}),$
(28)
and $\tilde{\mathcal{O}}_{1,2,3}$ are obtained from $\mathcal{O}_{i}$ after
exchanging both chiralities, i.e., $L\leftrightarrow R$. Henceforth, we will
just be focusing on the new physics contribution to these WCs, i.e.,
$C_{i}=C_{i}^{\rm NP}$ and $\tilde{C}_{i}=\tilde{C}_{i}^{\rm NP}$. Depending
on the specific mass of the ALP, such contributions will involve either short
or long-distance physics. The first case occurs when integrating out a heavy
enough ALP, $m_{a}\gg m_{c}$, whereas the second one is the consequence of
applying naively the operator product expansion (OPE) in powers of $\sim
1/m_{c}$ to the $D-\bar{D}$ system, when $m_{a}\ll m_{c}$. In the first case,
one obtains
$\displaystyle\tilde{C}_{2}$
$\displaystyle=\frac{(c_{u_{R}})_{21}^{2}}{2m_{a}^{2}}\frac{m_{c}^{2}}{f_{a}^{2}},\
C_{2}=\tilde{C}_{2}\frac{m_{u}^{2}}{m_{c}^{2}},\
C_{4}=-2\tilde{C_{2}}\frac{m_{u}}{m_{c}},$ (29)
and zero elsewhere, while in the second one
$\displaystyle\tilde{C}_{2}$
$\displaystyle=-\frac{(c_{u_{R}})_{21}^{2}}{2f_{a}^{2}},\
C_{2}=\tilde{C}_{2}\frac{m_{u}^{2}}{m_{c}^{2}},\
C_{4}=-2\tilde{C_{2}}\frac{m_{u}}{m_{c}},$ (30)
with all other WCs vanishing. In general,
$\displaystyle 2m_{D}M_{12}^{\rm NP}$
$\displaystyle=\sum_{i=1}^{5}C_{i}(\mu)\langle
D^{0}|\mathcal{O}_{i}|\bar{D}^{0}\rangle(\mu)$
$\displaystyle+\sum_{i=1}^{3}\tilde{C}_{i}(\mu)\langle
D^{0}|\tilde{\mathcal{O}}_{i}|\bar{D}^{0}\rangle(\mu),$ (31)
where $\langle D^{0}|\tilde{\mathcal{O}}_{i}|\bar{D}^{0}\rangle=\langle
D^{0}|\mathcal{O}_{i}|\bar{D}^{0}\rangle$, for $i=1,2,3$, due to parity
conservation of QCD and $m_{D}=1.865\,$GeV [42]. For small ALP masses, we get
[43]
$\displaystyle|M_{12}^{\rm
NP}|=\frac{1}{2m_{D}}\frac{(c_{u_{R}})_{21}^{2}}{2f_{a}^{2}}\,(0.1561\,\mathrm{GeV}^{4}),$
(32)
where $\langle\mathcal{O}_{2}\rangle=-0.1561\,\mathrm{GeV}^{4}$ at $\mu=3$
GeV, see [43], and we have neglected $\mathcal{O}(m_{u}/m_{c})$ corrections.
However, when studying short-distance physics, it is also necessary to run the
different WCs from the scale of integration $\Lambda=m_{a}$, to the scale
$\mu\sim m_{c}\sim 3\,$GeV. Neglecting $\mathcal{O}(m_{u}/m_{c})$ effects at
the UV, we obtain [44]
$\displaystyle\tilde{C}_{2}(\mu)$
$\displaystyle=\left[r(\mu,\Lambda)^{\frac{1-\sqrt{241}}{6}}\left(\frac{1}{2}-\frac{52}{\sqrt{241}}\right)\right.$
$\displaystyle\left.+\,r(\mu,\Lambda)^{\frac{1+\sqrt{241}}{6}}\left(\frac{1}{2}+\frac{52}{\sqrt{241}}\right)\right]\tilde{C}_{2}(\Lambda),$
(33) $\displaystyle\tilde{C}_{3}(\mu)$
$\displaystyle=\frac{705}{32\sqrt{241}}\left[r(\mu,\Lambda)^{\frac{1-\sqrt{241}}{6}}-r(\mu,\Lambda)^{\frac{1+\sqrt{241}}{6}}\right]\tilde{C}_{2}(\Lambda),$
where
$\displaystyle
r(\mu,\Lambda)=\left(\frac{\alpha_{s}(\Lambda)}{\alpha_{s}(m_{t})}\right)^{2/7}\left(\frac{\alpha_{s}(m_{t})}{\alpha_{s}(m_{b})}\right)^{6/23}\left(\frac{\alpha_{s}(m_{b})}{\alpha_{s}(\mu)}\right)^{6/25}.$
(34)
Due to the running, to evaluate $M_{12}$ we also need
$\langle\mathcal{O}_{3}\rangle$ at $\mu=3\,$GeV, which reads
$0.0464\,\mathrm{GeV}^{4}$ [43]. As an example, for $\Lambda=m_{a}=2\,$TeV and
$\mu=3\,$GeV, we obtain
$\displaystyle|M_{12}^{\rm
NP}|=\frac{1}{2m_{D}}\frac{(c_{u_{R}})_{21}^{2}}{2f_{a}^{2}}\,(8.95\cdot
10^{-9}\,\mathrm{GeV}^{4}).$ (35)
We demand that the new physics contribution to $x_{12}=2|M_{12}|/\Gamma$ does
not exceed its upper bound at 95% confidence level (CL) [45], i.e.,
$\displaystyle x_{12}^{\rm NP}=\frac{2|M_{12}^{\rm NP}|}{\Gamma_{D}}<0.63\cdot
10^{-2},$ (36)
where $\Gamma_{D}=1.60497\cdot 10^{-12}\,\mathrm{GeV}$ [42].
### III.2 Exotic D, B and K decays
We study $\Delta F=1$ decays of the form $M\to Na$ with
$M=D^{\pm,0},B^{\pm,0},K^{\pm,0}$ and $N=\pi^{\pm,0},K^{\pm,0}$. The
corresponding parton level Feynman diagram are shown in fig. 2. The associated
matrix element can be decomposed as
$\displaystyle\langle
N(p^{\prime})|\bar{q}_{i}\gamma_{\mu}q_{j}|M(p)\rangle=(p+p^{\prime})_{\mu}f_{+}^{MN}(k^{2})+k_{\mu}f_{-}^{MN}(k^{2})$
with $k_{\mu}=(p-p^{\prime})_{\mu}$ the momentum transfer and $q_{i}$ and
$q_{j}$ the relevant quarks for the decay at the parton level. The scalar form
factor is then defined as
$\displaystyle
f_{0}^{MN}(k^{2})=f_{+}^{MN}(k^{2})+\frac{k^{2}}{m_{M}^{2}-m_{N}^{2}}f_{-}^{MN}(k^{2})$
(38)
and the resulting decay width is given by
$\displaystyle\Gamma(M\to Na)=\frac{m_{M}^{3}|\varkappa_{MN}|^{2}}{64\pi
f_{a}^{2}}\left(1-\frac{m_{N}^{2}}{m_{M}^{2}}\right)^{2}(f_{0}^{MN}(m_{a}^{2}))^{2}$
$\displaystyle\times\sqrt{\left(1-\frac{(m_{N}+m_{a})^{2}}{m_{M}^{2}}\right)\left(1-\frac{(m_{N}-m_{a})^{2}}{m_{M}^{2}}\right)},$
(39)
where $\varkappa_{MN}$ is defined by
$\displaystyle\mathcal{L}\supset\varkappa_{MN}\frac{\partial^{\mu}a}{2f_{a}}\bar{q}_{i}\gamma_{\mu}q_{j}+\mathrm{h.c.}.$
(40)
In the model at hand and neglecting small isospin-breaking effects, the exotic
$D$ meson decay $D^{\pm,0}\to\pi^{\pm,0}a$ is induced by the dimension-5
operator
$\displaystyle\mathcal{L}\supset\left(c_{u_{R}}\right)_{ij}\frac{\partial_{\mu}a}{f_{a}}\left(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j}\right).$
(41)
The width for such decay channel can be read from equation (39) by simply
replacing $\varkappa_{MN}$ with $(c_{u_{R}})_{12}$, $m_{M}=m_{D}$,
$m_{N}=m_{\pi}$ and using $f_{0}^{D\pi}(m_{a}^{2})$ from [46].
On the other hand, the one-loop running of $c_{u_{R}}$ from the UV scale
$f_{a}$ to the IR scale $\mu$ will generate a term 222The one-loop running
also generates a non-zero WC for $\mathcal{O}_{H}$, but since such operator is
flavour-blind it does not contribute to any of these $\Delta F=1$ processes.
It will be relevant though for the astrophysical constraints, see below.
$\displaystyle(c_{q_{L}})_{ij}\frac{\partial_{\mu}a}{f_{a}}\left(\bar{q}_{Li}\gamma^{\mu}q_{Lj}\right)$
(42)
at low energies. Indeed, one obtains [28, 29, 30, 31]
$\displaystyle 16\pi^{2}\frac{d{c_{q_{L}}}}{d\ln\mu}$
$\displaystyle=-\lambda_{u}c_{u_{R}}\lambda_{u}\Rightarrow$ $\displaystyle
c_{q_{L}}$
$\displaystyle=\frac{\lambda_{u}c_{u_{R}}\lambda_{u}}{32\pi^{2}}\ln\left(\frac{f_{a}^{2}}{\mu^{2}}\right).$
(43)
After EWSB, this operator leads to
$\displaystyle\frac{\partial_{\mu}a}{f_{a}}\left[(c_{u_{L}})_{ij}\bar{u}_{Li}\gamma^{\mu}u_{Lj}+(c_{d_{L}})_{ij}\bar{d}_{Li}\gamma^{\mu}d_{Lj}\right],$
(44)
where $c_{u_{L}}=c_{q_{L}}$ and $c_{d_{L}}=V^{\dagger}c_{q_{L}}V$. More
explicitly
$\displaystyle(c_{d_{L}})_{ij}=\frac{1}{16\pi^{2}v^{2}}V_{ri}^{\ast}(\mathcal{M}_{u})_{rr}(c_{u_{R}})_{rs}(\mathcal{M}_{u})_{ss}V_{sj}\ln\left(\frac{f_{a}^{2}}{\mu^{2}}\right).$
(45)
This WC is responsible for the exotic decays $B\to Ka$, $B\to\pi a$ and
$K\to\pi a$, where $B=B^{\pm,0},K=K^{\pm,0}$ and $\pi=\pi^{\pm,0}$. 333The
form factors we use are computed in the isospin preserving limit so we do not
make a difference between e.g. $B^{\pm}\to K^{\pm}a$ and $B^{0}\to K^{0}a$.
The corresponding expressions can be read from equation (39) after taking
$\mu\sim m_{t}$ and
$(B\to Ka)$, | $\varkappa_{MN}=(c_{u_{d_{L}}})_{32}$, | $m_{N}=m_{B}$, | $m_{M}=m_{K}$, | $f_{0}^{MN}(m_{a}^{2})=f_{0}^{BK}(m_{a}^{2})$ | [47],
---|---|---|---|---|---
$(B\to\pi a)$, | $\varkappa_{MN}=(c_{u_{d_{L}}})_{31}$, | $m_{N}=m_{B}$, | $m_{M}=m_{\pi}$, | $f_{0}^{MN}(m_{a}^{2})=f_{0}^{B\pi}(m_{a}^{2})$ | [48],
$(K\to\pi a)$, | $\varkappa_{MN}=(c_{u_{d_{L}}})_{21}$, | $m_{N}=m_{K}$, | $m_{M}=m_{\pi}$, | $f_{0}^{MN}(m_{a}^{2})=f_{0}^{K\pi}(m_{a}^{2})$ | [49].
There are no constraints on the branching ratio
$\mathrm{Br}(D\to\pi+\mathrm{invisible})$ to date. However, there are
measurements of the three-body meson decay
$D^{+}\to(\tau^{+}\to\pi^{+}\nu)\bar{\nu}$ [50, 51]. Since these analyses show
the event distribution as a function of the missing mass squared $M_{\rm
miss}^{2}$ (which would correspond to $m_{a}^{2}$ in the ALP case), one could
recast them to constrain the branching ratio $D^{+}\to\pi^{+}a$. 444 See [52]
for the study of the long-distance lepton-mediated contributions to $B$ and
$D$-meson semi-invisible decays. This was done e.g. in [29] for the massless
axion by concentrating on the bins with $M_{\rm miss}^{2}\leq
0.05\,\rm{GeV}^{2}$. Since, as we will see, the total width of the ALP in all
the benchmarks under consideration is very small, one can safely produce a
similar bound for different values of $m_{a}$ by comparing the observed number
of events with the predicted background for every bin having $M_{\rm
miss}^{2}\geq 0$. More precisely, we derive 90% CL on $\mathrm{BR}(D\to\pi a)$
by using the TLimit class of ROOT [53], which implements the CLs method [54]
and allows to include systematic errors in the background and signal. The
bounds arising from [50] turn out to be stronger than those resulting from the
use of the more recent experimental analysis in [51].
Regarding exotic meson decays involving down quarks, there are several
analysis focused on a massless axion $X^{0}$, like $K^{+}\to\pi^{+}X^{0}$ [55]
and $B^{\pm}\to\pi^{\pm}X^{0}$, $B^{\pm}\to K^{\pm}X^{0}$ [56]. There are no
searches for a massive ALP $a$, with the exception of the recent analysis in
[57], where bounds on $K^{+}\to\pi^{+}a$ as a function of $m_{a}$ were
presented. Similarly to the $D^{\pm}\to\pi^{\pm}a$ case, we fill this gap by
recasting existing searches on three-body decays, where the relevant kinematic
information is provided. In particular, we derive constraints on $B\to Ka$ and
$B\to\pi a$ by recasting the searches performed in [58] and [59],
respectively. More specifically, we set 90% CL on $B\to Ka$ by combining the
observed number of events and the predicted background for $B^{+}\to
K^{+}\nu\bar{\nu}$ and $B^{0}\to K^{0}\nu\bar{\nu}$ for every
$s_{B}=k^{2}/m_{B}^{2}=m_{a}^{2}/m_{B}^{2}$ bin in figure 5 of [58] with the
CLs method. Finally, for the case of $B\to\pi a$, we derive 90% CL limits on
$B\to\pi a$ with the CLs method by comparing the observed number of events
with the predicted background for every
$p_{\pi}\equiv\sqrt{\vec{p}_{\pi}^{\,2}}$ bin in the right panel of figure 4
in [59] (which is in one-to-one correspondence with the ALP mass via
$m_{a}^{2}=m_{B}^{2}+m_{\pi}^{2}-2m_{B}\sqrt{m_{\pi}^{2}+\vec{p}_{\pi}^{\,2}}$
).
On the other hand, it is expected that Belle II will be sensitive to the SM
$\mathrm{Br}(B^{\pm}\to K^{\pm}\nu\bar{\nu})=(4.0\pm 0.5)\times 10^{-6}$ [60]
at 10% accuracy with 50 ab-1 of data [61], whereas NA62 will measure the
branching ratio $\mathrm{Br}(K^{\pm}\to\pi^{\pm}\nu\bar{\nu})$ to within 10%
of its SM value $\mathrm{Br}(K^{\pm}\to\pi^{\pm}\nu\bar{\nu})=(8.4\pm
4.1)\times 10^{-11}$ [62, 63]. Regardless of whether these collaborations
publish limits directly on a two-body decay or a recast of the three-body
decay analysis is needed, such numbers represent a great improvement with
respect to current bounds. 555See e.g. [64] for a two-body interpretation of
the NA62 prospects.
### III.3 Radiative $J/\psi$ decays
Figure 3: Parton level diagram for the decay $J/\psi\to a\gamma$.
Diagonal ALP couplings to charm quarks are strongly constrained by charmonium
decays like $J/\psi\to a\gamma$, as first proposed by [65] and later studied
by many others, see e.g. [66, 67, 68, 69, 70, 71]. The parton-level diagram
prompting such decay is shown in figure 3. In order to absorb some of the QCD
uncertainties of the calculation, it is convenient to normalize the
corresponding branching ratio by the one of $J/\psi\to\mu^{+}\mu^{-}$, which
is accurately measured $\mathrm{Br}(J/\psi\to\mu^{-}\mu^{+})=5.973$ % [72].
One then obtains
$\displaystyle\frac{\mathrm{Br}(J/\psi\to
a\gamma)}{\mathrm{Br}(J/\psi\to\mu^{-}\mu^{+})}=$
$\displaystyle\frac{G_{F}m_{c}^{2}v^{2}}{\sqrt{2}\pi\alpha_{\rm
em}}\left(\frac{(c_{u_{R}})_{22}}{f_{a}}\right)^{2}\times$
$\displaystyle\left(1-\frac{m_{a}^{2}}{m_{J/\psi}^{2}}\right)F,$ (46)
where $F\sim\mathcal{O}(1/2)$ is a correction factor accounting for QCD
effects [73, 74], contributions related to bound-state formation [75, 76] as
well as relativistic corrections [77]. For the sake of concreteness, we will
assume that $F=1/2$ henceforth. This leads to
$\displaystyle\mathrm{Br}(J/\psi\to
a\gamma)=(1.05\,\mathrm{GeV}^{2})\,\left(\frac{(c_{u_{R}})_{22}}{f_{a}}\right)^{2}\left(1-\frac{m_{a}^{2}}{m_{J/\psi}^{2}}\right).$
(47)
This decay has been searched for by the CLEO collaboration [78], which we will
use to constrain the benchmark models at hand.
## IV Astrophysical and cosmological bounds
### IV.1 Bounds from supernova SN1987a
The observed neutrino burst due to the core-collapse supernova SN1987a can
impose constraints on the ALP parameter space. Since neutrino emission
constitutes the main cooling mechanism for the proto-neutron star resulting
from the collapse, a too large ALP emission could compete with this cooling
mechanism and eventually conflict the observed amount of neutrinos. Following
[79], we will impose that the ALP luminosity in the proto-neutron star $L_{a}$
does not exceed the neutrino one $L_{\nu}$, i.e., $L_{a}\leq L_{\nu}=3\cdot
10^{52}\,\rm{erg}/s$. 666One should note, however, that the authors of ref.
[80] have cast some serious doubts on supernova cooling bounds for ALPs.
The ALP luminosity in the proto-neutron star is given by [81, 82]
$\displaystyle L_{a}=\int_{r\leq
R_{\nu}}dV\,\int_{m_{a}}^{\infty}d\omega\,\left(\frac{dP_{a}}{dVd\omega}\right)\,e^{-\tau},$
(48)
where $\omega$ is the ALP energy and $R_{\nu}\sim\mathcal{O}(40\,\rm{km})$ is
the radius of the neutrinosphere, beyond which neutrinos free stream until
arriving to the Earth, and we have taken into account the probability
$e^{-\tau}$ for an ALP produced within the neutrinosphere to reach $R_{\rm
far}\sim\mathcal{O}(100-1000\,\rm{km})$, after which neutrinos are not
produced efficiently. If this is not the case, ALPs being produced within the
neutrinosphere get ’trapped’ due to their large couplings and their energy is
eventually converted back into neutrinos. Such probability is computed with
the help of the optical depth $\tau=\tau(m_{a},\omega,r,R_{\rm far})$, for
which we will take $R_{\rm far}=100\,\rm{km}$ [81]. In the above expression,
$dP_{a}/dVd\omega$ is the ALP differential power. For the charming ALPs
considered here, the main channel will be the bremsstrahlung process $N+N\to
N+N+a$, since the sole tree-level couplings are those to up-type quarks and
therefore to nucleons. Such differential power is given by [79, 83]
$\displaystyle\frac{dP_{a}}{dVd\omega}=\frac{1}{2\pi^{2}}\omega^{3}\Gamma_{a}e^{-\omega/T}\beta^{2},$
(49)
where $T$ is the temperature as a function of the radius, $\beta$ is a phase
space factor $\beta=\sqrt{1-m_{a}^{2}/\omega^{2}}$ and $\Gamma_{a}$ is the ALP
absorption width. The latter is given by
$\displaystyle\Gamma_{a}=\Gamma_{a}^{pp}+\Gamma_{a}^{nn}+\Gamma_{a}^{pn}+\Gamma_{a}^{np}$
(50)
with [82]
$\displaystyle\Gamma_{a}^{NN^{\prime}}=\frac{c_{aNN}^{2}Y_{N}Y_{N^{\prime}}}{4f_{a}^{2}}\frac{\omega}{2}\frac{n_{B}^{2}\sigma_{np\pi}}{\omega^{2}}\gamma_{\rm
f}\gamma_{\rm p}\gamma_{\rm h},\quad N^{(\prime)}=n,p.$ (51)
In the above equation, $c_{aNN}$ is the ALP-nucleon coupling, which reads (see
appendix C for more details)
$\displaystyle c_{app}$ $\displaystyle=(c_{u_{R}})_{11}\left(0.75\pm
0.03\right),$ (52) $\displaystyle c_{ann}$
$\displaystyle=(c_{u_{R}})_{11}\left(-0.51\pm 0.03\right),$ (53)
while $Y_{N^{(\prime)}}$ is the mass fraction of the nucleon $N^{(\prime)}$,
$n_{B}$ is the baryon density, $n_{B}=\rho/m_{N}$, and $\sigma_{np\pi}$ is
given by
$\displaystyle\sigma_{np\pi}=4\alpha_{\pi}^{2}\sqrt{\pi T/m_{N}^{5}},$ (54)
with $\alpha_{\pi}\approx 15$. For concreteness we take $Y_{p}=0.3$ and
$Y_{n}=1-Y_{p}=0.7$. Moreover [84],
$\displaystyle 1/\gamma_{\rm
f}=1+\left(n_{B}\sigma_{np\pi}/(2\omega)\right)^{2},$ (55)
while we use $\gamma_{\rm p}=s(n_{B},Y_{N},\omega/T,m_{\pi}/T)$ with $s$ given
by eq. (49) of [85]. 777Note that at the end of the day, $s$ is divided by an
extra factor $(1-\exp(-x))$ in order to preserve the detailed balance more
explicitly. On the other hand, following [64], we assume that [86]
$\displaystyle\gamma_{\rm
h}=-0.0726502\ln(\rho)+10^{10}/\rho^{0.9395710}+2.5558616,$ (56)
where the density $\rho$ is expressed in $\rm{g}\,\rm{cm}^{-3}$. Similarly to
[64], we assume for $\rho(r)$ and $T(r)$ and the ”fiducial” profiles of [81]
$\displaystyle\rho(r)=\rho_{c}\times\left\\{\begin{array}[]{ll}1+k_{\rho}(1-r/R_{c})&r<R_{c}\\\
(r/R_{c})^{-\nu}&r\geq R_{c}\end{array}\right.,$ (59) $\displaystyle
T(r)=T_{c}\times\left\\{\begin{array}[]{ll}1+k_{T}(1-r/R_{c})&r<R_{c}\\\
(r/R_{c})^{-\nu/3}&r\geq R_{c}\end{array}\right.,$ (62)
with $k_{\rho}=0.2$, $k_{T}=-0.5$, $\nu=5$, $R_{c}=10\,\rm{km}$, $T_{c}=30$
MeV and $\rho_{c}=3\cdot 10^{14}\,\rm{g}/\rm{cm}^{3}$. We define $R_{\nu}$ as
the distance at which the temperature is $3\,$MeV, obtaining
$R_{\nu}=39.81\,\rm{km}$. Finally, for the optical depth we take [64]
$\displaystyle\tau=(R_{\rm
far}-R_{\nu})\beta^{-1}\Gamma_{a}(R_{\nu})+\beta^{-1}\int_{r}^{R_{\nu}}d\tilde{r}\,\Gamma_{a}(\tilde{r}).$
(63)
### IV.2 Bounds from red giant burst
The one-loop running of the dimension-5 effective Lagrangian will generate ALP
couplings to electrons at low energy. Such couplings face very strong
astrophysical bounds for small values of $m_{a}$. This effect can be
particularly relevant when the ALP couples to the top quark, since it will
contribute significantly to the running [28, 29, 30, 31]:
$\displaystyle 16\pi^{2}\frac{dc_{H}}{d\ln\mu}$
$\displaystyle=-6\mathrm{Tr}\left(\lambda_{u}c_{u_{R}}\lambda_{u}\right)\Rightarrow$
$\displaystyle c_{H}$
$\displaystyle=\frac{3}{8\pi^{2}v^{2}}\mathrm{Tr}\left(\mathcal{M}_{u}c_{u_{R}}\mathcal{M}_{u}\right)\ln\left(\frac{f_{a}^{2}}{\mu^{2}}\right),$
(64)
which leads after field redefinition to
$\displaystyle-
c_{H}\frac{\partial_{\mu}a}{f_{a}}\left(\bar{l}_{Li}\gamma^{\mu}l_{Li}+2\bar{e}_{Ri}\gamma^{\mu}e_{Ri}\right).$
(65)
After EWSB, these operators lead to
$\displaystyle\mathcal{L}\supset\frac{ic_{H}a}{f_{a}}m_{\ell}(\bar{\ell}\gamma_{5}\ell)=i\,a\,g_{a\ell\ell}(\bar{\ell}\gamma_{5}\ell),\quad\ell=e,\mu,\tau.$
(66)
More explicitly, at $\mu\sim m_{t}$,
$\displaystyle g_{aee}=\frac{c_{H}m_{e}}{f_{a}}=\frac{3m_{e}}{8\pi
v^{2}f_{a}}\ln\left(\frac{f_{a}^{2}}{m_{t}^{2}}\right)\sum_{i=1}^{3}\left(\mathcal{M}_{u}\right)_{ii}(c_{u_{R}})_{ii}.$
(67)
This coupling is bounded by data from red giant bursts [83, 87, 88, 89],
$g_{aee}\lesssim 1.6\cdot 10^{-13}$, for ALP masses below the temperature of
the red giant.
### IV.3 ALP lifetime, branching ratios and cosmological bounds
Cosmological bounds are very sensitive to the total decay width and the
different branching ratios of the ALP, that we discuss in the following. We do
this both for small values of the ALP mass, where QCD is confined and one can
use chiral perturbation theory, as well as for larger values where the
dominant ALP decays can be computed using quark-hadron duality [90, 91].
Following [5], we determine the energy scale separating both pictures by
demanding that the total decay width to hadrons or SM quarks is of the same
order in both regimes, which leads to $\sim 1$ GeV for the benchmark models at
hand.
For small masses, $m_{a}\lesssim 1\,$GeV, the ALP decays to two photons via
the mixing (18) and the subsequent decay $\pi\to\gamma\gamma$, plus one-loop
contributions coming from the integration of heavy quarks. This leads to [18,
21]
$\displaystyle\Gamma(a\to\gamma\gamma)$ $\displaystyle\approx\frac{\alpha_{\rm
em}^{2}m_{a}^{3}}{(4\pi)^{3}f_{a}^{2}}\left|\sum_{i=2}^{3}\frac{4}{3}(c_{u_{R}})_{ii}B_{1}(\tau_{i})\right.$
$\displaystyle\left.-\frac{m_{a}^{2}}{2(m_{\pi}^{2}-m_{a}^{2})}(c_{u_{R}})_{11}\right|^{2},$
(68)
where $B_{1}(\tau_{i})=1-\tau_{i}f^{2}(\tau)$,
$\tau_{i}=4(\mathcal{M}_{u})_{ii}^{2}/m_{a}^{2}$, and
$f(\tau_{i})=\left\\{\begin{array}[]{cc}\arcsin(1/\sqrt{\tau_{i}})&\tau_{i}\geq
1\\\
\frac{\pi}{2}+\frac{i}{2}\ln\left(\frac{1+\sqrt{1-\tau_{i}}}{1-\sqrt{1-\tau_{i}}}\right)&\tau_{i}<1\end{array}.\right.$
(69)
This decay channel will be the only one present, whenever the one-loop lepton
decays $a\to\ell^{+}\ell^{-}$ are kinematically closed. The leptonic decay
widths can be written as
$\displaystyle\Gamma(a\to\ell^{+}\ell^{-})=\frac{m_{a}m_{\ell}^{2}}{8\pi
f_{a}^{2}}\sqrt{1-\frac{4m_{\ell}^{2}}{m_{a}^{2}}}|c_{H}|^{2}.$ (70)
The diphoton final state will dominate over the $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ decays in models where $(c_{u_{R}})_{33}$ is absent or
negligible. Otherwise, once $a\to e^{+}e^{-}$ and $a\to\mu^{+}\mu^{-}$ open
up, they will become the main ALP decay channel, at least for large values of
$f_{a}$ leading to log-enhanced $g_{a\ell\ell}$ couplings. At any rate, the
$m_{a}^{3}$ dependence of the diphoton decay will make
$\Gamma(a\to\gamma\gamma)$ increase faster than the dilepton decay width with
increasing values of $m_{a}$. In some cases, this can turn such decay channel
into the leading one, once more, for larger ALP masses before $a\to 3\pi$
opens kinematically. Such decay channel will always dominate the
$a\to\gamma\gamma$ final state in our models, with its decay width reading
[18]
$\displaystyle\Gamma(a\to\pi^{a}\pi^{b}\pi^{0})$
$\displaystyle=\frac{\pi}{12}\frac{m_{a}m_{\pi}^{4}}{f_{a}^{2}f_{\pi}^{2}}\left[\frac{(c_{u_{R}})_{11}}{32\pi^{2}}\right]^{2}g_{ab}\left(\frac{m_{\pi}^{2}}{m_{a}^{2}}\right),$
(71)
with
$\displaystyle g_{00}(x)$
$\displaystyle=\frac{2}{(1-x)^{2}}\int_{4x}^{(1-\sqrt{x})^{2}}dz\sqrt{1-\frac{4x}{z}}\lambda^{1/2}(1,z,x),$
(72) $\displaystyle g_{+-}(x)$
$\displaystyle=\frac{12}{(1-x)^{2}}\int_{4x}^{(1-\sqrt{x})^{2}}dz\sqrt{1-\frac{4x}{z}}(z-x)^{2}$
$\displaystyle\times\lambda^{1/2}(1,z,x),$ (73)
and $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$. However, when
$(c_{u_{R}})_{33}$ is sizeable and/or $f_{a}$ high enough to have significant
$g_{a\ell\ell}$ couplings, $a\to\mu^{+}\mu^{-}$ can be the dominant decay
channel in this region of ALP masses.
For masses $m_{a}\gtrsim 1\,$GeV, equation (9) becomes invalid and the
dominant ALP decays into hadrons can be computed using quark-hadron duality.
The ALP will then decay into gluons and quarks. While the first decay is loop
induced, the tree-level decay into quarks will be suppressed by the light
Yukawa couplings. Close to threshold, $a\to\bar{q}^{(\prime)}q$ will be the
leading decay channel, whereas for larger values of $m_{a}$ $a\to gg$ will
take over. In this regime, the $a\to\gamma\gamma$ decay will always be sub-
leading and read
$\displaystyle\Gamma(a\to\gamma\gamma)$ $\displaystyle\approx\frac{\alpha_{\rm
em}^{2}m_{a}^{3}}{(4\pi)^{3}f_{a}^{2}}\left|\sum_{i=1}^{3}\frac{4}{3}(c_{u_{R}})_{ii}B_{1}(\tau_{i})\right|^{2}.$
(74)
Figure 4: Branching ratios of the ALP as a function of its mass, $m_{a}$, for
the different benchmark models. The top panels correspond to the dark-QCD
inspired case with $a=\pi_{D_{3}}$ (top left) and $a=\pi_{D_{8}}$ (top right),
whereas the bottom panels show the anarchic scenario (bottom left) and the FN
motivated benchmark (bottom right), respectively. In all cases, we have
assumed $f_{a}=10^{4}$ TeV. Moreover, for the dark-QCD motivated scenarios
illustrated in the top panels we have taken $\kappa_{0}=1$. We illustrate with
a gray narrow band around 1 GeV, the matching between the calculations
performed using chiral perturbation theory and with quark-hadron duality.
We show in figure 4 the different branching ratios for the models at hand for
$f_{a}=10^{4}$ TeV, assuming that $\kappa_{0}=1$ in the dark-QCD motivated
benchmarks. We can see that in the cases where the ALP coupling to top quarks
is absent or suppressed (top-left and bottom-right plots), $a\to\gamma\gamma$
is the leading decay mode for most of the region $m_{a}\lesssim 1$ GeV until
$a\to 3\pi$ is kinematically allowed. In the cases where this coupling is
present (top-right and bottom-left plots), and for this choice of $f_{a}$,
$a\to e^{+}e^{-}$ becomes the leading channel until $a\to\mu^{+}\mu^{-}$ opens
up. For ALP masses $\gtrsim 1$ GeV, decays into hadrons are by far the
dominant channels, with $a\to gg$ leading at large masses.
Now that we have computed the different branching ratios and lifetimes, we can
evaluate the impact of the cosmological constraints. It is important to stress
that most of them are derived assuming that $a$ only interacts with photons.
However, as discussed in [92], one can apply these cosmological bounds to the
more general case where other couplings are present. At the end of the day, we
will be able to recast the limits from [92, 93, 94] by using the ALP lifetime
$1/\Gamma$, with $\Gamma$ the ALP total decay width. These bounds include the
possible impact on $N_{\rm eff}$, potential distortions of the cosmic
microwave-background spectrum as well as modifications of the predicted big-
bang nucleosynthesis, see [92, 93, 94].
Experiment | distance from IP | length of decay volume | radius/opening angle | $N_{D}$
---|---|---|---|---
FASER | 480 m | 1.5 m | 0.1 m | $1.1\times 10^{15}$
FASER2 | 480 m | 5 m | 1 m | $2.2\times 10^{16}$
MATHUSLA | 68 m downstream, | 100 m | 25 m high | $2.2\times 10^{16}$
| 60 m above | | |
NA62 | 80 m | 65 m | $\theta_{\rm max}=0.05$ | $2\times 10^{15}$
SHiP | 60 m | 50 m | 2.5 m | $6.8\times 10^{17}$
CHARM | 480 m | 35 m | $0.0068<\theta<0.0126$ | $4.08\times 10^{15}$
Table 1: Detector parameters for the different fixed-target experiments and
LHC forward detectors considered.
Indeed, the bounds can directly be applied when the decay to lepton pairs is
dominant, i.e. when there is a sizeable coupling of the ALP with the top
quark, provided one interprets $1/\Gamma$ as the total lifetime. 888When
$a\to\mu^{+}\mu^{-}$ dominates, this slightly over-estimates the excluded
region, since the subsequent decay of the muon also heats the neutrino bath,
which reduces the impact on $N_{\rm eff}$. In the cases where $a\to 3\pi$
dominates, bounds from 4He overproduction – the dominant constraint in this
region – will still hold regardless of the changes in the branching ratios,
since only a minimal amount of charged pions is enough for this bound to
apply. For even larger masses, ALP decays into hadrons will eventually make
its lifetime shorter than a second, making nucleosynthesis constraints
harmless. Therefore, even for these masses, we can apply the corresponding
bounds if we interpret $\tau$ as the total lifetime.
## V Collider probes
### V.1 Fixed target experiments
The main production mode for charming ALPs at fixed-target experiments is the
decay of $D$ mesons. We consider NA62 [95] and the proposed SHiP experiment
[96] as possible detection experiments. We also consider the bounds imposed by
the CHARM experiment in [97]. The geometrical outlines of the experiments are
listed in table 1.
NA62 operating in beam dump mode, meaning the target is lifted so that the 400
GeV proton beam hits the Cu collimator located 20 m downstream, can be used to
search for hidden sector particles [98]. A short run in beam dump mode in
November 2016 provided useful information about the backgrounds. It was found
that an upstream veto in front of the decay volume could reduce the background
to nearly zero [99]. The layout of the SHiP detector is proposed with the aim
of reducing the beam-induced backgrounds to 0.1 events [100, 96], so that 3
decay events correspond to the expected exclusion region at over 95% CL.
The total number of dark pions decaying inside the respective decay volume is
$\displaystyle N_{a}=N_{D}\cdot\mathrm{Br}(D\to\pi a)\cdot\varepsilon_{\rm
geom}\cdot F_{\rm decay}\,,$ (75)
with $\varepsilon_{\rm geom}$ the geometric acceptance, defined as the
fraction of ALPs with lab frame momentum at the acceptance angle of the
respective detector and $F_{\rm decay}$ the fraction of ALPs that decay inside
the decay volume of the respective detector. $F_{\rm decay}$ and
$\varepsilon_{\rm geom}$ are calculated following [5]. The $D$ meson momentum
distribution for SHiP is taken from [101]. The same distribution is used for
the NA62 case as the proton beam is the same.
The CHARM experiment searched for ALPs decaying into pairs of photons,
electrons and muons. In [97] no events where found, so that we set a bound at
90% CL at $N_{\rm obs}=2.3$ events [102]. We assume that for our model the
main production channel for ALPs is the decay $D^{\pm}\to\pi^{\pm}a$. CHARM
also has a 400 GeV proton beam, so we again use the same momentum distribution
for the $D$ mesons. The number of protons on target is $2.4\times 10^{18}$
[97] and such a proton beam has a probability of $1.7\times 10^{-3}$ to
produce a pair of $c$ quarks [103], leading to $4.08\times 10^{15}$ produced
$D$ mesons. Multiplying eq. (75) by $\sum_{i=\gamma,e,\mu}Br(a\to ii)$ to take
into account that in [97] only the $\gamma\gamma$, $ee$ and $\mu\mu$ final
states were searched for we use the same procedure as for NA62 and SHiP to
impose the 90% CL bound from CHARM.
### V.2 LHC forward detectors
FASER [104] and the proposed MATHUSLA detector [105, 106] are designed to
detect long-lived particles produced in proton-proton collisions at LHC.
Hadron collisions have the advantage of additional production modes, such as
production via gluon fusion. The detector parameters are given in table 1. We
focus again on the production via $D$ meson decays. The meson momentum
distribution was simulated using FONLL with CTEQ6.6 [107]. As for SHiP and
NA62 the number of dark pions decaying inside the decay volume can be
calculated using eq. (75). For LHC run-3 $N_{D}=1.1\times 10^{15}$, which
increases by a factor 20 at the High Luminosity (HL)-LHC. FASER will operate
at LHC run-3, while FASER2 and MATHUSLA are under consideration for the HL-
LHC. Following the procedure as described for SHiP and NA62 the number of dark
pions decaying inside FASERs, FASER2s and MATHUSLAs decay volume is
calculated. At least three events must decay inside the respective decay
volume for a discovery.
## VI Results
Figure 5: Experimental constraints and expected bounds on $1/f_{a}$ as a
function of $m_{a}$ for the dark-QCD inspired models. Left panels correspond
to the case $a=\pi_{D}^{3}$, while right panels illustrate the scenario
$a=\pi_{D}^{8}$. Light green and light red areas correspond to the expected
constraints coming from Belle II and NA62, respectively. We show by a red line
the constraints arising from the recast of the three-body decay
$D^{+}\to(\tau^{+}\to\pi^{+}\bar{\nu})\nu$, whereas the impact of a direct
measurement of $\mathrm{Br}(D\to\pi a)$ is represented by black lines, with
values going from $10^{-1}$ to $10^{-8}$, each one a decade smaller. Dashed
lines correspond to different fix-target experiments and collider probes of
the model, see the main text for more details. Lower panels zoom in the
regions where upcoming experiments are sensitive. In order to evaluate the
logarithms coming from the one-loop running we further assume $\kappa_{0}=1$.
The resulting constraints on the parameter space of the four benchmarks models
under consideration are displayed in figures 5 and 6. More specifically, we
show in figure 5 the different bounds for the dark-QCD motivated benchmarks,
with left panels corresponding to the case $a=\pi_{D_{3}}$ and the right ones
to $a=\pi_{D_{8}}$, respectively. For these two benchmarks we further assume
$\kappa_{0}=1$ in order to evaluate the logarithms coming from the one-loop
running, which only depend on $f_{a}$. Still, different choices of
$\kappa_{0}\sim\mathcal{O}(1)$ will not significantly affect the resulting
bounds shown in the figure. On the other hand, we show in figure 6 the
corresponding bounds for the anarchic (left panels) and FN (right panels)
scenarios.
Figure 6: Same as figure 5 but for the anarchic and the FN inspired models,
left and right panels, respectively. Lower panels zoom in the regions where
upcoming experiments are sensitive.
In both figures, we represent in dark yellow the region of the parameter space
excluded by $D$ meson mixing, whereas the bounds from exotic meson decays
$B\to Ka$, $B\to\pi a$ and $K\to\pi a$ are displayed in green, blue and red,
respectively. Moreover, for the sake of illustration, one can roughly
interpret the expected bounds on $B\to K\nu\bar{\nu}$ from Belle II as
sensitivity to $B\to Ka$, which we show in light green. Similarly, one could
do the same thing in the case of $K\to\pi\nu\bar{\nu}$ from NA62, which is
represented in light red. A proper recast of these two upcoming experimental
results will most likely result in slightly weaker bounds. On the other hand,
we show the limits arising from the recast of
$D^{+}\to(\tau^{+}\to\pi^{+}\nu)\bar{\nu}$ as a red line. Finally, we show in
purple the bounds resulting from $J/\psi\to a\gamma$ searches by the CLEO
collaboration. The cosmological bounds discussed previously are shown in gray
in both figures, with the constraints arising from red giant bursts and from
SN1987a exhibited in lilac and light blue, respectively.
The region where the ALP mass lies above the $D$ and $B$ meson masses is only
weakly constrained by flavour observables, and is open for probes at the
energy frontier, i.e. the LHC and future colliders. Below the $D$ meson mass,
some viable regions remain which can be probed by the upcoming or proposed
collider experiments discussed in section V. The projected bounds from SHiP
and NA62 are represented by dark red and orange dashed contours, respectively.
Above these lines more than three events are expected. Furthermore, the
discovery lines for FASER with LHC run-3 and FASER2 at HL-LHC are displayed in
dashed light and dark blue, respectively, whereas the detection line for
MATHUSLA is pictured as a turquoise dashed contour line.
In order to appreciate better the region which can be probed by these upcoming
experiments, we show in the lower panels of figures 5 and 6 a zoomed-in view
of the area which may be reached. While FASER will mainly validate the
constraints from the CHARM experiment shown in light yellow, FASER2 at the HL-
LHC as well as NA62 will probe new regions of parameter space, with SHiP and
MATHUSLA covering the remaining unexplored areas below the charm mass
threshold.
A possible measurement of $\mathrm{Br}(D\to\pi+\rm{invisible})$ could provide
a complementary test of these parts of the parameter space, and might be
crucial to probe the region close to the charm mass at relatively large
coupling. In the FN inspired model as well as the $\pi_{D}^{3}$ scenario this
region of parameter space is not otherwise accessible, while in the
$\pi_{D}^{8}$ scenario it will be probed by Belle II, and in the anarchic case
it is already excluded. Similarly, the low mass region of the $\pi_{D}^{3}$
scenario is only accessible via $\mathrm{Br}(D\to\pi+\rm{invisible})$ and
future NA62 measurements. To display the discovery potential of such a
measurement, we show black contour lines corresponding to values of
$\mathrm{Br}(D^{\pm}\to\pi^{\pm}\rm{invisible})\in\\{10^{-8},\,10^{-7},\,10^{-6},\,10^{-5},\,10^{-4},\,10^{-3},\,10^{-2},\,10^{-1}\\}$.
This demonstrates that providing an experimental measurement of the exotic
meson decay $D\to\pi a$ is paramount to leave no stone unturned in the quest
for well motivated extensions of the SM that feature charming ALPs.
## VII Conclusions
In this work we have presented several examples of models featuring charming
ALPs, i.e., light pNGBs having off-diagonal couplings with SM up-quarks, and
studied in detail the phenomenology associated with their low-energy EFTs.
More specifically, we have studied the constraints arising from flavour
experiments, astrophysics and cosmology as well as planned fixed-target and
collider experiments in four benchmark models. We have shown that such
scenarios have still a large unexplored parameter space. We have also
demonstrated how future collider and fixed-target experiments can probe these
models and that they could be perfectly complemented by the measurement of the
exotic decay $D\to\pi+\rm{invisible}$, which is currently unavailable. We thus
encourage our experimental colleagues to proceed with such measurement. In the
absence of dedicated searches, we have also derived bounds on the parameter
space of the models by recasting three-body meson decays like
$D^{+}\to(\tau^{+}\to\pi^{+}\nu)\bar{\nu}$ or $B\to K/\pi\,\nu\bar{\nu}$.
The scenarios considered here can be the low-energy EFT of several compelling
UV completions. We have presented two of them: the case of a QCD-like dark
sector interacting with the SM via a heavy scalar mediator with hypercharge
$-2/3$, and a FN model of flavour where only RH up-quarks and a heavy scalar
have non-zero charges with respect to an spontaneously broken global $U(1)$
symmetry. Some of the phenomenology studied here may change when considering
the whole picture in the dark-QCD case, since there might be a non-trivial
interplay between the complete set of pNGBs in some regions of the parameter
space. In the present work, we have focused on the phenomenological aspects
expected to hold when singling out one of such light states. The complete
dark-QCD model will be studied in an upcoming work, where a detailed study of
the full charming dark sector including the possible connection with dark
matter will be presented. A particularly interesting aspect of such scenarios
to be studied is the collider phenomenology involving rare top decays $t\to
ca$ as well as the phenomenology of ’charming’ emerging jets.
A final region of parameter space that was left unexplored in our study is
that of very small ALP masses and couplings, i.e. the lower left regions of
our figures. There the charming ALP would be a stable dark matter (DM)
candidate. The freeze-out of such a DM candidate is excluded for $m_{a}\gtrsim
100$ eV due to DM overproduction [92] and in the whole parameter region if
structure formation bounds are also taken into account [108]. On the other
hand, if the charming ALP DM is produced via a ”freeze-in” mechanism, a region
of parameter space remains viable [108] in principle. A more detailed
exploration of this region of the charming ALP is left for future work.
###### Acknowledgements.
We thank Felix Kahlhöfer and Fatih Ertas for useful feedback. AC thanks Mikael
Chala, Jorge Martin-Camalich, Matthias Neubert and Robert Ziegler for fruitful
discussions. AC acknowledges funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 754446 and UGR Research and Knowledge Transfer Found – Athenea3i.
Work in Mainz was supported by the Cluster of Excellence Precision Physics,
Fundamental Interactions, and Structure of Matter (PRISMA+ EXC 2118/1) funded
by the German Research Foundation (DFG) within the German Excellence Strategy
(Project ID 39083149), and by grant 05H18UMCA1 of the German Federal Ministry
for Education and Research (BMBF).
## Appendix A A dark QCD UV completion
One particular class of theories which can provide a UV completion to the
charming ALP EFT is what can be collectively denoted by ’dark QCD’, see e.g.
[11, 12, 5]. In these scenarios, one assumes that the SM is extended with a
new dark QCD-like gauge group $SU(N_{d})_{d}$ with $n_{d}$ Dirac fermions,
$Q_{\alpha},\,\alpha=1,\ldots,n_{d}$, singlets of the SM gauge group and
transforming in the fundamental representation of $SU(N_{d})_{d}$. For
concreteness one can assume $N_{d}=3=n_{d}$, which allows in particular the
QCD-like sector to confine at a scale $\Lambda_{d\rm QCD}$. Both sectors talk
to each other through the coupling to a heavy scalar mediator, $\mathcal{X}$,
transforming as a $(\mathbf{3},\bar{\mathbf{3}})$ under $SU(3)\otimes
SU(3)_{d}$, which is also charged under $SU(2)_{L}\otimes U(1)_{Y}$ as
$\mathbf{1}_{-2/3}$. Such scalar mediator is naturally heavy, also in
agreement with LHC bounds. A similar setup was presented first in [11] but
with a different assignment of hypercharge, $Y=1/3$, such that only couplings
to RH down-like quarks were allowed. It was shown in [12, 12, 5] that this
scenario lead to _emerging jets_ and its flavour phenomenology was studied in
[5]. Here, we focus on a different hypercharge assignment, which leads to a
distinct phenomenology. The structure of the model is sketched in figure 7.
Figure 7: Schematic illustration of the high-energy and low-energy regimes of
the dark QCD UV completion.
The Lagrangian of the dark sector reads
$\displaystyle\mathcal{L}_{D}$
$\displaystyle=-\frac{1}{4}\mathcal{G}_{d}^{\mu\nu,a}\mathcal{G}^{d,a}_{\mu\nu}+\bar{Q}_{\alpha}i\cancel{D}Q_{\alpha}-m_{Q\alpha,\beta}\bar{Q}_{\alpha}Q_{\beta}$
$\displaystyle+|D_{\mu}\mathcal{X}|^{2}-m_{\mathcal{X}}^{2}|\mathcal{X}|^{2}-\left[\kappa_{\alpha
i}\bar{u}_{Ri}\mathcal{X}Q_{\alpha}+\mathrm{h.c.}\right],$ (76)
where we use greek (latin) indices to denote the flavour indices in the dark
(visible) QCD sector and we do not need to explicitly write the corresponding
covariant derivatives. Similarly, we did not make explicit color or dark color
indices with the exception of $\mathcal{G}_{\mu\nu}^{a}$, $a=1,\ldots,8$, the
field-strength tensor for the dark QCD. In general, the coupling matrix
$\kappa_{\alpha i}$ can be expressed as
$\displaystyle\kappa=VDU$ (77)
where both $U$ and $V$ are $3\times 3$ unitary matrices and $D$ is a $3\times
3$ diagonal matrix. In the case where
$m_{Q\alpha\beta}=m_{Q}\delta_{\alpha\beta}$, there is a $U(3)_{d}$ flavour
symmetry in the dark sector, which can be used to rotate away $V$. We will
assume that this is the case henceforth. The matrix $U$ can be further
expressed as the following product
$\displaystyle U=U_{23}U_{13}U_{12},$ (78)
where $U_{ij}$ are unitary rotations between the flavours $i$ and $j$. For
example, $U_{12}$ reads
$\displaystyle U_{12}=\begin{pmatrix}c_{12}&s_{12}e^{-i\delta_{12}}&0\\\
-s_{12}e^{-i\delta_{12}}&c_{12}&0\\\ 0&0&1\end{pmatrix},$ (79)
where $c_{12}=\cos\theta_{12}$, $s_{12}=\sin\theta_{12}$. Moreover, following
[2, 5] one can express $D$ as
$\displaystyle
D=\mathrm{diag}\left(\kappa_{0}+\kappa_{1},\kappa_{0}+\kappa_{2},\kappa_{0}-(\kappa_{1}+\kappa_{2})\right),$
(80)
where $\kappa_{0}\geq|\kappa_{1}+\kappa_{2}|$. For the sake of concreteness,
we will consider the benchmark model defined by
$\displaystyle\kappa_{1}=\kappa_{0}/2,\quad\kappa_{2}=0,\quad\theta_{13}=\theta_{23}=0,\quad\delta_{ij}=0,\,\forall
i,j.$ (81)
We also assume that all this happens in a basis where the SM up Yukawa matrix
is diagonal.
If the mass of scalar mediator is much heavier than $\Lambda_{d\rm QCD}$ and
$\Lambda_{\rm QCD}$, one gets in addition to the SM Lagrangian
$\displaystyle\mathcal{L}_{\mathrm{eff}}$
$\displaystyle=-\frac{1}{4}\mathcal{G}_{d}^{\mu\nu,a}\mathcal{G}^{d,a}_{\mu\nu}+\bar{Q}_{\alpha}i\cancel{D}Q_{\alpha}-m_{Q}\bar{Q}_{\alpha}Q_{\alpha}$
$\displaystyle-\frac{\kappa_{\alpha i}\kappa^{\ast}_{\beta
j}}{2m_{\mathcal{X}}^{2}}\left(\bar{Q}_{\beta}\gamma_{\mu}P_{L}Q_{\alpha}\right)(\bar{u}_{Ri}\gamma^{\mu}u_{Rj}),$
(82)
after integrating out $\mathcal{X}$ and using Fierz identities. Similarly to
QCD, in the limit $m_{Q}\to 0$ and $m_{\mathcal{X}}\to\infty$, the dark sector
features a global dark chiral symmetry $SU(3)_{dL}\otimes SU(3)_{dR}$, which
we assume is spontaneously broken to its diagonal group $SU(3)_{dV}$ by the
dark QCD condensate
$\langle\bar{Q}_{\alpha}Q_{\beta}\rangle\propto\delta_{\alpha\beta}\Lambda_{d\rm
QCD}^{3}$. This spontaneous symmetry breaking delivers 8 Nambu-Goldstone
bosons, $\pi_{D_{1}},\ldots,\pi_{D_{8}}$, which will become pNGBs once we
switch on $m_{Q}$. We will consider the case where $m_{Q}\ll\Lambda_{d\rm
QCD}$ so that the pNGBs are parametrically lighter than the rest of particles
in the spectrum. With the exception of the lightest baryonic bound states
carrying a conserved dark baryon number, the rest of the particles of the
spectrum will undergo fast decays to dark pions. Therefore, it is a reasonable
approximation to just consider such dark pions. In the case at hand where
$n_{d}=3=N_{d}$, one can write down an effective theory for the resulting
eight pNGBs along the lines of the QCD case of pions and kaons. Using the
basis of Gell-Mann matrices, $\lambda^{a}$, $a=1,\ldots,8$, one can write
$\displaystyle\Pi_{D}$ $\displaystyle=\pi_{D_{a}}\frac{\lambda^{a}}{2}$ (83)
$\displaystyle=\frac{1}{2}\begin{pmatrix}\pi_{D_{3}}+\frac{\pi_{D_{8}}}{\sqrt{3}}&\pi_{D_{1}}-i\pi_{D_{2}}&\pi_{D_{4}}-i\pi_{D_{5}}\\\
\pi_{D_{1}}+i\pi_{D_{2}}&-\pi_{D_{3}}+\frac{\pi_{D_{8}}}{\sqrt{3}}&\pi_{D_{6}}-i\pi_{D_{7}}\\\
\pi_{D_{4}}+i\pi_{D_{5}}&\pi_{D_{6}}+i\pi_{D_{7}}&-\frac{2}{\sqrt{3}}\pi_{D_{8}}\end{pmatrix}.$
The corresponding Goldstone matrix transforming non-linearly under
$SU(3)_{dL}\otimes SU(3)_{dR}$ and linearly under $SU(3)_{dV}$ can be written
as
$\displaystyle
U_{D}(\Pi_{D})=\mathrm{exp}\left(\frac{2i}{f_{d}}\Pi_{D}\right),$ (84)
where $f_{d}$ is the dark pion decay constant, in principle a free parameter.
In analogy to chiral perturbation theory (ChPT) for QCD, the Lagrangian
describing the dark pions in the absence of interaction with the SM is given
by
$\displaystyle\mathcal{L}_{d\rm ChPT}$
$\displaystyle=\frac{f_{d}^{2}}{4}\mathrm{Tr}\left(\partial_{\mu}U_{D}\partial^{\mu}U^{\dagger}_{D}\right)$
$\displaystyle+\frac{f_{d}^{2}B_{D}}{2}m_{Q}\mathrm{Tr}\left(U_{D}^{\dagger}+U_{D}\right),$
(85)
where $B_{d}$ is a constant related to the dark pion mass. More precisely
$\displaystyle m_{\pi_{D_{a}}}^{2}=m_{\pi_{D}}^{2}=2m_{Q}B_{d}.$ (86)
As one can see, they are all degenerate in mass, but small splittings will be
induced by their interactions with the SM. Such radiative corrections will
define new mass eigenstates
$\displaystyle\pi^{(1,2)}_{D}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\pi_{D_{1}}-i\pi_{D_{2}}\right),$ (87)
$\displaystyle\pi^{(1,3)}_{D}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\pi_{D_{3}}-i\pi_{D_{4}}\right),$ (88)
$\displaystyle\pi^{(2,3)}_{D}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\pi_{D_{6}}-i\pi_{D_{7}}\right),$ (89)
with $\pi_{D_{3}}$ and $\pi_{D_{8}}$ unchanged.
If the dark pions are light enough, $m_{\pi_{D}}\lesssim 4\pi f_{\pi}$, their
decays are better described by ChPT for the SM quarks. The part containing
only SM fields is described by
$\displaystyle\mathcal{L}_{\rm
ChPT}=\frac{f_{\pi}^{2}}{4}\mathrm{Tr}\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}\right)+\frac{f_{\pi}^{2}B_{0}}{2}\mathrm{Tr}\left(m_{q}U^{\dagger}+Um_{q}^{\dagger}\right),$
(90)
whereas the interaction with the dark QCD sector is described by
$\displaystyle\mathcal{L}_{\rm ChPT}^{\rm mix}$
$\displaystyle=-\frac{f_{d}^{2}f_{\pi}^{2}}{2m_{\mathcal{X}}^{2}}\kappa_{\alpha
i}\kappa^{\ast}_{\beta
j}\mathrm{Tr}\left(c_{\beta\alpha}U_{D}^{\dagger}\left(\partial_{\mu}U_{D}\right)\right)\times$
$\displaystyle\mathrm{Tr}\left(c_{ij}U\left(\partial^{\mu}U\right)^{\dagger}\right),$
(91)
where the projection matrices $c_{\alpha\beta}$ and $c_{ij}$ are defined as
$\displaystyle
c_{\alpha\beta}^{mn}=\delta_{\alpha}^{m}\delta_{n}^{\beta},\quad\alpha,\beta=1,2,3,\quad
c_{ij}^{mn}=\delta_{i}^{m}\delta_{j}^{n},\quad i,j=1,$ (92)
being zero otherwise. For larger dark pion masses, one can use quark-hadron
duality [90, 91], with the relevant Lagrangian being
$\displaystyle\mathcal{L}_{\rm
mix}=i\frac{f_{d}^{2}}{2m_{\mathcal{X}}^{2}}\kappa_{\alpha
i}\kappa^{\ast}_{\beta
j}\mathrm{Tr}\left(c_{\beta\alpha}U_{D}^{\dagger}\left(\partial_{\mu}U_{D}\right)\right)(\bar{u}_{Ri}\gamma^{\mu}u_{Rj}).$
(93)
For the benchmark model at hand, only $\pi_{D_{3}},\pi_{D_{8}}$ and
$\pi_{D}^{(1,2)}$ decay at tree-level. The rest of dark pions, which decay
through loop-induced processes, are therefore long-lived. For this reason, we
focus on the first dark pions, and in particular, on the two real ones
$\pi_{D_{3}}$ and $\pi_{D_{8}}$. 999At the phenomenological level, the main
difference between the couplings of these two pions is the presence of a tree-
level coupling with the RH top. The rest of pions will interpolate between
these two scenarios or be too-long lived. In particular, the mixing terms in
eqs. (91) and (93) involving $\pi_{D_{3}}$ and $\pi_{D_{8}}$ read
$\displaystyle\mathcal{L}_{\rm ChPT}^{\rm mix}$
$\displaystyle\supset-\frac{f_{d}f_{\pi}^{2}}{2m_{\mathcal{X}}^{2}}\sum_{a=3,8}\sum_{\alpha\beta}\kappa_{\alpha
i}\kappa^{\ast}_{\beta
j}\left(\lambda^{a}\right)_{\alpha\beta}\partial_{\mu}\pi_{D_{a}}\times$
$\displaystyle\mathrm{Tr}\left(c_{ij}U\left(\partial^{\mu}U\right)^{\dagger}\right)$
(94)
and
$\displaystyle\mathcal{L}_{\rm
mix}\supset-\frac{f_{d}}{2m_{\mathcal{X}}^{2}}\sum_{a=3,8}\sum_{\alpha\beta}\kappa_{\alpha
i}\kappa^{\ast}_{\beta
j}\left(\lambda^{a}\right)_{\alpha\beta}\partial_{\mu}\pi_{D_{a}}(\bar{u}_{Ri}\gamma^{\mu}u_{Rj}),$
(95)
respectively. Comparing these equations with the mixing terms present in eqs.
(1) and (9) for $a=\pi_{D_{3}}$ and $\pi_{D_{8}}$, respectively, we obtain
$f_{a}=m_{\mathcal{X}}^{2}/f_{d}$ and
$\displaystyle(c_{u_{R}}^{(a)})_{ij}=-\sum_{\alpha\beta}\kappa_{\alpha
i}\kappa^{\ast}_{\beta j}\left(\lambda^{a}\right)_{\alpha\beta},\qquad a=3,8.$
(96)
More explicitly,
$\displaystyle c_{u_{R}}^{(3)}$
$\displaystyle=\frac{\kappa_{0}^{2}}{4}\begin{pmatrix}4s_{12}^{2}-9c_{12}^{2}&-13c_{12}s_{12}&0\\\
-13c_{12}s_{12}&4c_{12}^{2}-9s_{12}^{2}&0\\\ 0&0&0\end{pmatrix},$ (97)
$\displaystyle c_{u_{R}}^{(8)}$
$\displaystyle=\frac{-\kappa_{0}^{2}}{4\sqrt{3}}\begin{pmatrix}4s_{12}^{2}+9c_{12}^{2}&5c_{12}s_{12}&0\\\
5c_{12}s_{12}&4c_{12}^{2}+9s_{12}^{2}&0\\\ 0&0&-2\end{pmatrix}.$ (98)
## Appendix B A Froggatt-Nielsen UV completion
Another motivation for the ALPs considered here corresponds to what is
generically known by the name of flavons or familions (see [109, 110, 111,
112, 113, 87] and e.g. [114, 115, 116, 117, 118, 119, 120, 121, 122, 123] for
more recent implementations), pNGBs of some spontaneously broken flavour
symmetry, which may be anomalous, and that generically feature flavour-
violating couplings to quarks or leptons. One particular setup leading to the
scenario we have in mind, i.e., the effective Lagrangian (1) in addition to
$c_{H}=0$, is given by FN models when only RH up-quarks have non-zero charges.
Specifically, one considers a global $U(1)$ flavour symmetry, spontaneously
broken by the vacuum expectation value of some extra scalar $\langle
S\rangle=f_{a}$, where
$\displaystyle S=\frac{1}{\sqrt{2}}(f_{a}+s)e^{ia/f_{a}},$ (99)
and has charge $-1$ under this new $U(1)$. If only $u_{Ri}$ are charged under
such global symmetry, with charges $n_{i}^{u}$, Yukawa couplings for up-quarks
will be higher-dimensional
$\displaystyle\mathcal{L}\supset-(y_{u})_{ij}\left(\frac{S}{\Lambda}\right)^{n_{j}^{u}}q_{Li}\tilde{H}u_{Rj}+\mathrm{h.c.},$
(100)
where one typically assumes that $f_{a}<\Lambda$. At the end of the day, such
term in the Lagrangian will generate interactions like (4)
$\displaystyle-\frac{ia}{f_{a}}\bar{q}_{Li}\tilde{H}u_{Rj}n_{j}^{u}=-\frac{ia}{f_{a}}\bar{q}_{Li}\tilde{H}u_{Rj}(Y_{u})_{ij}n_{j}^{u},$
(101)
where $Y_{u}$ is the effective up Yukawa matrix,
$\displaystyle(Y_{u})_{ij}=(y_{u})_{ij}\left(\frac{f_{a}}{\Lambda}\right)^{n_{j}^{u}}.$
(102)
If we assume that $(y_{u})_{ij}=\mathcal{O}(1)$ and take
$f_{a}/\Lambda=\epsilon\sim m_{c}/m_{t}$, we can get the correct up quark
masses by choosing $n_{u}=(2,1,0)$ since
$\displaystyle(m_{u},m_{c},m_{t})\sim\frac{1}{\sqrt{2}}v\,(\epsilon^{2},\epsilon,1)$
(103)
and
$\displaystyle\frac{m_{u}}{m_{t}}\sim\epsilon^{2},\qquad\frac{m_{c}}{m_{t}}\sim\epsilon.$
(104)
In this case, we can diagonalize $Y_{u}$ by making
$\displaystyle u_{R}\to U_{R}^{u}u_{R},\qquad u_{L}\to U_{L}^{u}u_{L},$ (105)
with
$\displaystyle U_{R}^{u}\sim\begin{pmatrix}1&\epsilon&\epsilon^{2}\\\
\epsilon&1&\epsilon\\\
\epsilon^{2}&\epsilon&1\end{pmatrix},\qquad(U_{L}^{u})_{ij}\sim\mathcal{O}(1).$
(106)
This leads, after going to the basis where $Y_{u}$ is diagonal to
$\displaystyle c_{u_{R}}\sim\begin{pmatrix}2&3\epsilon&3\epsilon^{2}\\\
3\epsilon&1&\epsilon\\\ 3\epsilon^{2}&\epsilon&\epsilon^{2}\end{pmatrix}.$
(107)
This sentence is here to fix the layout.
## Appendix C ALP couplings to nucleons
The leading order ALP couplings with nucleons can be read from the following
Lagrangian [124, 125, 34, 126]
$\displaystyle\mathcal{L}_{\rm int}$
$\displaystyle=\left(\frac{\partial_{\mu}a}{4f_{a}}\right)\Big{\\{}\mathrm{Tr}\left((\hat{c}+\varkappa_{q}c_{g})\lambda^{a}\right)\left(F\,\mathrm{Tr}\left(\bar{B}\gamma^{\mu}\gamma_{5}\left[\lambda^{a},B\right]\right)+D\,\mathrm{Tr}\left(\bar{B}\,\gamma^{\mu}\gamma_{5}\left\\{\lambda^{a},B\right\\}\right)\right)$
$\displaystyle+$
$\displaystyle\frac{1}{3}\mathrm{Tr}\left(\hat{c}+\varkappa_{q}c_{g}\right)\,S\,\mathrm{Tr}\left(\bar{B}\,\gamma^{\mu}\gamma_{5}B\right)+\mathrm{Tr}\left((\hat{c}+\varkappa_{q}c_{g})\lambda^{a}\right)\mathrm{Tr}\left(\bar{B}\,\gamma^{\mu}\left[\lambda^{a},B\right]\right)\Big{\\}}$
(108)
where $B$ is the baryon matrix
$\displaystyle
B=\frac{1}{\sqrt{2}}B^{a}\lambda^{a}=\begin{pmatrix}\frac{\Sigma^{0}}{\sqrt{2}}+\frac{\Lambda}{\sqrt{6}}&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{\Sigma^{0}}{\sqrt{2}}+\frac{\Lambda}{\sqrt{6}}&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2\Lambda}{\sqrt{6}}\end{pmatrix},$ (109)
and the axial-vector coupling constants $F$ and $D$ are defined by
$\displaystyle\langle
B^{\prime}_{i}|J_{j}^{(8)}|B_{k}\rangle=if_{ijk}F+d_{ijk}D,$ (110)
with $J_{j}^{(8)}$ the weak axial-vector hadronic current, transforming as an
$SU(3)$ octet, $f_{ijk}$ the totally antisymmetric structure constants of
$SU(3)$ and $d_{ijk}$ the totally symmetric ones. On the other hand, $S$ is
defined by the singlet current which can be renormalized independently. $D$
and $F$ can be determined by hyperon semileptonic decays [127], leading to
$F=0.463\pm 0.008$, $D=0.804\pm 0.008$. On the other hand $S\approx 0.13\pm
0.2$ [128]. We are particularly interested in the $a\bar{N}N$ couplings, with
$N=p,n$, which read
$\displaystyle\mathcal{L}_{\rm
int}\supset\frac{1}{12}\frac{\partial_{\mu}a}{f_{a}}\left(c_{u_{R}}\right)_{11}\Big{(}\big{[}S-4D\big{]}(\bar{n}\gamma^{\mu}\gamma_{5}n)+\big{[}2D+6F+S\big{]}(\bar{p}\gamma^{\mu}\gamma_{5}p)+6\bar{p}\gamma^{\mu}p\Big{)}$
(111)
The last term in the equation above is a total derivative which can be
neglected. In this case
$\displaystyle\mathcal{L}_{aNN}=\sum_{N=p,n}\frac{\partial_{\mu}a}{2f_{a}}c_{aNN}\bar{N}\gamma^{\mu}\gamma_{5}N$
(112)
with
$\displaystyle c_{app}$
$\displaystyle=(c_{u_{R}})_{11}\left(F+\frac{1}{3}D+\frac{1}{6}S\right)=(c_{u_{R}})_{11}\left(0.75\pm
0.03\right),$ (113) $\displaystyle c_{ann}$
$\displaystyle=(c_{u_{R}})_{11}\left(\frac{1}{6}S-\frac{2}{3}D\right)=(c_{u_{R}})_{11}\left(-0.51\pm
0.03\right).$ (114)
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|
# Disorder-induced topology in quench dynamics
Hsiu-Chuan Hsu<EMAIL_ADDRESS>Graduate Institute of Applied Physics,
National Chengchi University, Taipei 11605, Taiwan Department of Computer
Science, National Chengchi University, Taipei 11605, Taiwan Pok-Man Chiu
Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
Po-Yao Chang<EMAIL_ADDRESS>Department of Physics, National Tsing
Hua University, Hsinchu 30013, Taiwan
###### Abstract
We study the effect of strong disorder on topology and entanglement in quench
dynamics. Although disorder-induced topological phases have been well studied
in equilibrium, the disorder-induced topology in quench dynamics has not been
explored. In this work, we predict a disorder-induced topology of post-quench
states characterized by the quantized dynamical Chern number and the crossings
in the entanglement spectrum in $(1+1)$ dimensions. The dynamical Chern number
undergoes transitions from zero to unity, and back to zero when increasing the
disorder strength. The boundaries between different dynamical Chern numbers
are determined by delocalized critical points in the post-quench Hamiltonian
with the strong disorder. An experimental realization in quantum walks is
discussed.
## I Introduction
Topological phases of matter out-of-equilibrium and their phase transitions
have attracted much theoretical and experimental interest. Their topological
and non-equilibrium features have been demonstrated in various systems
including ultracold-atomic gases Foster _et al._ (2014); Plekhanov _et al._
(2017); Cooper _et al._ (2019); Salerno _et al._ (2019), quantum optics
Rechtsman _et al._ (2013); Wang _et al._ (2019a); Ozawa _et al._ (2019),
superconducting qubits Kyriienko and Sørensen (2018); Malz and Smith (2021),
and condensed matter systems Kitagawa _et al._ (2011); Ezawa (2013); Kundu
_et al._ (2014); Gulácsi and Dóra (2015); Farrell and Pereg-Barnea (2015);
Takasan _et al._ (2017); Owerre (2018); Lubatsch and Frank (2019); Oka and
Kitamura (2019). Among these, the topological Floquet systems have been widely
studied Kitagawa _et al._ (2010a); Lindner _et al._ (2011); Jiang _et al._
(2011). These systems exhibit protected boundary states which are robust in
the presence of disorder. More recently, topological phases in dynamical
quench systems are proposed Yang _et al._ (2018); Gong and Ueda (2018); Chang
(2018); Zhu _et al._ (2020); Hu and Zhao (2020). For example, for a trivial
state under a sudden quench by the Su-Schrieffer-Heeger (SSH) model, the
topology of the post-quench state is characterized by the dynamical Chern
numbers Yang _et al._ (2018); Chang (2018), the quantization of which
describes a skyrmion texture of the post-quench pseudospin in the momentum-
time space Wang _et al._ (2019b). The topology in quench dynamics has been
shown experimentally in photonic quantum walks Cardano _et al._ (2017); Wang
_et al._ (2019b); Xu _et al._ (2019) and superconducting qubits Flurin _et
al._ (2017); Guo _et al._ (2019). Moreover, the entanglement spectrum
provides an additional probe of the topology. The robustness of crossings in
the entanglement spectrum of the post-quench states indicates the nontrivial
topology in quench dynamics Gong and Ueda (2018); Chang (2018).
Besides the topological structures that emerge in quench dynamics, non-trivial
topology can arise from disordered systems in equilibrium. In the strong
disorder regime, an unexpected topological phase with extensive boundary
states is stabilized by the strong disorder. This phase is termed the
“topological Anderson insulator” Li _et al._ (2009); Jiang _et al._ (2009);
Groth _et al._ (2009); Guo _et al._ (2010); Hsu and Chen (2020) and the
transition between trivial and non-trivial phases is described by the
delocalization criticality Mondragon-Shem _et al._ (2014). A generalization
of the topological Anderson insulator to Floquet systems is proposed Titum
_et al._ (2015, 2016, 2017); Liu _et al._ (2020). The strong disorder drives
trivial Floquet systems into topological phases that host chiral edge modes
coexisting with the localized bulk states in two-dimensional lattices. The
transition also links to delocalization Wauters _et al._ (2019). In
constrast, quench Anderson disorder was studied theoretically in simple
lattice models Rahmani and Vishveshwara (2018); Lundgren _et al._ (2019). It
has been shown that in the strong disorder regime, where the Anderson
localization sets in, there is no sharp transition in the quench dynamics
Rahmani and Vishveshwara (2018).
Although there are extensive studies in disorder-induced topology in Floquet
systems, the effect of disorder on topology in quench dynamics is less
discussed. It is shown that the crossings in the entanglement spectrum are
robust against weak disorder and interactions Gong and Ueda (2018). However,
it has not been known whether disorder could induce topology in quench
dynamics.
In this work, we demonstrate the strong disorder-induced topology in quench
dynamics. We consider a quench protocol described by a trivial initial state
(a fully pseudospin-polarized state) under a sudden quench by the SSH
Hamiltonian in the presence of strong disorder. The topology of the post-
quench state is characterized by the dynamical Chern number which is
zero/unity when the SSH model is trivial/non-trivial. We start at the clean
limit where the post-quench state is trivial. When the disorder strength is
above the critical value, the post-quench state has a quantized dynamical
Chern number. The entanglement spectrum of the post-quench states shows robust
crossings which indicate the disorder-induced topology in quench dynamics. The
post-quench SSH Hamiltonian in this strong disorder regime has a disorder-
induced winding number. The phase boundaries coincide with the transitions
between vanishing and quantized dynamical Chern numbers. Our results
demonstrate that the disorder-induced topology in quench dynamics in $(1+1)$
dimensions is directly related to the topological Anderson insulator.
## II The post-quench Hamiltonian
We consider an eigenstate $|\Psi_{0}\rangle$ of a pre-quench Hamiltonian
$H_{\rm pre}$ at $t=0$ under a sudden quench by a post-quench Hamiltonian
$H_{\rm post}$, and the post-quench state is $|\Psi(t)\rangle=\exp[-iH_{\rm
post}t]|\Psi_{0}\rangle$. We consider $H_{\rm post}=H_{0}+H_{U}$, with
$\displaystyle H_{0}$
$\displaystyle=\sum_{x=1}^{N_{x}}J_{0}c^{\dagger}_{x,a}c_{x,b}+J_{1}c^{\dagger}_{x+1,a}c_{x,b}+{\rm
h.c.},$ $\displaystyle H_{U}$
$\displaystyle=\sum_{x=1}^{N_{x}}U_{1x}c^{\dagger}_{x,a}c_{x,b}+U_{2x}c^{\dagger}_{x,a}c_{x+1,b}+{\rm
h.c.},$ (1)
where $H_{0}$ is the SSH Hamiltonian and $H_{U}$ is the time-reversal and
particle-hole symmetry preserving disorder. Here $x$ is the label of the unit-
cell, $N_{x}$ is the total number of the unit-cell.
$c^{\dagger}_{xa(b)},c_{xa(b)}$ are the creation and annihilation operators on
sublattices $a,b$ on the $x$-th unit-cell. $J_{0(1)}$ denotes the
intracell(intercell) coupling, and $U_{1(2)x}$ is the random
intracell(intercell) coupling strength given by the random number in the
uniform distribution $\left[-W_{1(2)}/2,W_{1(2)}/2\right]$. We choose the
disorder strengths $W_{1}=2W_{2}=W_{0}$. The post-quench Hamiltonian $H_{\rm
post}$ has the time-reversal symmetry $T:c_{xa(b)}\to c_{xa(b)}$, $i\to-i$,
and the particle-hole symmetry $C:c_{xa(b)}\to c_{xb(a)}$, $i\to-i$. I.e., it
belongs to the BDI symmetry class, $T^{2}=C^{2}=1$.
The topology of the post-quench Hamiltonian $H_{\rm post}$ in the presence of
strong disorder is characterized by the winding number $W$ and the phase
diagram is shown in Fig. 1(a). In the clean limit with
$(J_{0}/J_{1},W_{0}/J_{1})=(1.1,0)$, the winding number is zero. When the
disorder strength increases, the winding number becomes unity when
$W_{0}/J_{1}\gtrsim 1.7$ and is back to zero when $W_{0}/J_{1}\gtrsim 3.6$
[the white dash in Fig. 1(a)]. This behavior demonstrates the disorder-induced
quantized winding number in the post-quench Hamiltonian and is referred to a
topological Anderson insulator Meier _et al._ (2018). The phase boundaries of
the trivial and the topological Anderson insulating phases are obtained by the
divergence of the localization length $\lambda$ Mondragon-Shem _et al._
(2014) [see App. A],
$\displaystyle\frac{1}{\lambda}=\bigg{|}\ln\left[\frac{\big{|}2J_{1}+W_{1}\big{|}^{\frac{J_{1}}{W_{1}}+\frac{1}{2}}\big{|}2J_{0}-W_{2}\big{|}^{\frac{J_{0}}{W_{2}}-\frac{1}{2}}}{\big{|}2J_{1}-W_{1}\big{|}^{\frac{J_{1}}{W_{1}}-\frac{1}{2}}\big{|}2J_{0}+W_{2}\big{|}^{\frac{J_{0}}{W_{2}}+\frac{1}{2}}}\right]\bigg{|}.$
(2)
## III The quench protocol
In the clean limit, the post-quench Hamiltonian is diagonalized in the
momentum space $H_{\rm post}=\sum_{k}\psi^{\dagger}_{k}\mathcal{H}_{\rm
post}(k)\psi_{k}$ with $\psi_{k}=(c_{ka},c_{kb})^{\rm T}$ with eigenenergies
$\pm|E(k)|$. Since each single-particle state does not interact with each
other, the single-particle state evolves individually
$|\psi(k,t)\rangle=e^{-i\mathcal{H}_{\rm post}(k)t}|\psi_{0}(k)\rangle$, where
$|\psi_{0}(k)\rangle$ is the single-particle ground state of the pre-quenched
single-particle Hamiltonian $\mathcal{H}_{\rm pre}(k)$. For each individual
post-quench single-particle state, the period of the dynamics is
$T_{k}=2\pi/|E(k)|$. The set of single-particle states $|\psi(k,t)\rangle$
have a corresponding momentum-time manifold $k\in[0,2\pi]$,
$t_{k}\in[0,T_{k}]$ which is a momentum-time torus. This torus is distorted
because different $k$ has different circumference $T_{k}$. Since the
deformation of the distorted torus to a ordinary torus (same circumference)
does not change the topology, one can rescale the period of the dynamics to be
$T_{k}=2\pi$ The rescaling of the period is equivalent to flattening the post-
quench Hamiltonian, $\mathcal{H}^{F}(k)=\mathcal{H}_{\rm post}(k)/|E_{(k)}|$.
We focus on the flattened Hamiltonian which allows us to construct the
effective Hamiltonian $\mathcal{H}_{\rm
eff}(k,t)=e^{-i\mathcal{H}^{F}(k)t}\mathcal{H}_{\rm
pre}(k)e^{i\mathcal{H}^{F}(k)t}$ for analyzing the topological property of the
post-quench dynamics [see App. B].
### III.1 Different pre-quench Hamiltonians
The post-quench state has two inputs, the pre-quench Hamiltonian
$\mathcal{H}_{\rm pre}(k)$ and the post-quench Hamiltonian
$\mathcal{H}_{0}(k)$. If the pre-quench and the post-quench Hamiltonians are
in the same symmetry class (BDI), the topology of the post-quench state is
characterized by the dynamical Chern number in the half of the Brillouin zone
(BZ), $k\in[0,\pi]$ and $t\in\left[0,\pi\right]$ Yang _et al._ (2018).
However, the dynamical Chern number is vanishing in the full BZ,
$k\in[0,2\pi]$ and $t\in\left[0,\pi\right]$. To study the disorder-induced
topology in quench dynamics, the real-space formalism is needed and requires
the information of the full BZ. Since the dynamical Chern number vanishes in
the full BZ and $t\in\left[0,\pi\right]$, no disorder-induced topology can
happen in this quench protocol. On the other hand, if the pre-quench
Hamiltonian $\mathcal{H}_{\rm pre}(k)=-\sigma_{z}$ which is not in the same
symmetry class as the post-quench Hamiltonian, the dynamical Chern number is
quantized in the full BZ, $t\in\left[0,\pi/2\right]$ Chang (2018) [see App.
B]. This pre-quench Hamiltonian allows us to formulate the dynamical Chern
number in real-space and study the disorder-induced topology.
In this case, the single-particle state is fully pseudospin polarized and the
real-space expression is $|\psi_{i}\rangle=\left(1,\>0\right)^{\rm
T}\otimes|i\rangle$, where $\left(1,\>0\right)^{\rm T}$ denotes one particle
at the sublattice $a$, $|i\rangle=(0,\cdots,1,\cdots,0)^{\rm T}$ denotes the
only non-vanishing $i$-th element with $i$ being the site label $i=1\dots
N_{x}$. The post-quench Hamiltonian in the presence of the disorder can be
flattened by using the projectors,
$\mathcal{H}^{F}=|\psi_{+}\rangle\langle\psi_{+}|-|\psi_{-}\rangle\langle\psi_{-}|$,
where $|\psi_{\pm}\rangle$ are the eigenstates of $\mathcal{H}$ with
positive/negative energies.
### III.2 Berry phase and dynamical Chern number
To determine the dynamical Chern number in the real space, we compute the
Berry phase with the twisted boundary condition Niu _et al._ (1985); Qi _et
al._ (2006) by the overlap matrix Gresch _et al._ (2017); Kuno (2019); Bonini
_et al._ (2020). The overlap matrix at a $t$ is defined as
$M_{ij}^{\ell}(t)=\langle\psi^{\theta_{\ell}}_{i}(t)|\psi^{\theta_{\ell+1}}_{j}(t)\rangle$,
where
$|\psi_{i}^{\theta_{\ell}}(t)\rangle=\exp[-i{\mathcal{H}^{\theta_{\ell}}_{\rm
post}}t]|\psi_{i}\rangle$, $i$ is the index of the single-particle state, and
$\mathcal{H}^{\theta_{\ell}}_{\rm post}$ is the flattened post-quench
Hamiltonian with twisted boundary phase ${\theta_{\ell}}=\frac{2\pi\ell}{L}$
Kuno (2019), where $L$ is the number of mesh points and $l=1,\cdots,L$. The
Berry phase is given by $\gamma(t)={\rm
Im}\left[\ln\det\prod_{\ell=1}^{L}M^{\ell}(t)\right]$. The Berry phase as a
function of $t$ has no jump when the post-quench state is trivial [Fig.1(b)
blue dots]. In contrast, when the post-quench state is topological, the Berry
phase flow has $2\pi$ jumps at $t=\pi/4$ as shown by the red dots in Fig.1(b).
The Wannier center flow also shows similar behavior which we demonstrate in
the App. C.
The dynamical Chern number is obtained by integrating the time derivative of
the Berry phase $C_{\rm
dyn}=\frac{1}{2\pi}\int^{\pi/2}_{0}dt\frac{\partial\gamma(t)}{\partial t}$.
Since $t=\pi/2$ is the time taken for the pseudospin to precess from the north
pole to the south pole, the integration is equivalent to counting the numbers
of the pseudospin
$\hat{n}_{i}(t)=\langle\psi_{i}(t)|\vec{\sigma}|\psi_{i}(t)\rangle$ wrapping
around the entire Bloch sphere Chang (2018). The disorder-induced dynamical
Chern number is shown in the red dots in Fig. 2(a). In the weak disorder
limit, $W_{0}/J_{1}\lesssim 2.2$ and $J_{0}/J_{1}=1.1$, the dynamical Chern
number vanishes. While increasing the disorder strength $W_{0}$, the dynamical
Chern number is quantized with negligible fluctuations in the region
$2.2\lesssim W_{0}\lesssim 3.2$. This behavior demonstrates that the disorder
drives the trivial post-quench state to be topological, and we refer it to the
disorder-induced topology in quench dynamics.
The phase boundaries of the zero and unity dynamical Chern numbers coincide
with the phase boundaries of the post-quench Hamiltonian obtained from the
divergence of the localization length [the white dashed line in Fig. 1 (a) and
the blue dots in Fig. 2]. It was demonstrated that in the clean limit, the
topology of the quench dynamics is related to that of the post-quench
Hamiltonian Chang (2018); Gong and Ueda (2018). Here, we observe that the
relation is still held for the disorder-induced topology.
Figure 1: (a) The phase diagram of the post-quench Hamiltonian
$H=H_{0}+H_{U}$.The white dashed line denotes $J_{0}=1.1$. (b) The time-
dependent Berry phase in the clean limit. The blue dots are for the trivial
post-quench state ($J_{0}/J_{1}=1.1$). The red dots are for the topological
post-quench state with a Berry phase flow from $t=0$ to $\pi/2$
($J_{0}/J_{1}=0.5$). Figure 2: The disorder-average dynamical Chern number
and the localization length for the post-quench Hamiltonian. The error bar is
the standard deviation. There are more than $20$ disorder realizations for
each data point. The parameters are $J_{0}/J_{1}=1.1,N_{x}=400$.
### III.3 Entanglement spectrum
The entanglement spectrum provides the additional information of the topology
induced by disorder in quench dynamics. It is shown that the crossings in the
entanglement spectrum reveal the topological properties in both the
equilibrium systems Li and Haldane (2008); Pollmann _et al._ (2010);
Fidkowski (2010); Turner _et al._ (2010); Peschel and Chung (2011); Hughes
_et al._ (2011); Chang _et al._ (2014) and out-of-equilibrium systems Gong
and Ueda (2018); Chang (2018); McGinley and Cooper (2018); Pastori _et al._
(2020). The presence/absence of the robust crossings in the entanglement
spectrum indicates the post-quench state is topological/trivial. To compute
the entanglement properties, the system is bipartite spatially into $A$ and
$B$ subsystems, where the post-quench many-body state is expressed as
$|\Psi(t)\rangle=\sum_{i,j}C_{ij}(t)|A_{i}\rangle|B_{j}\rangle$ with
$|A(B)_{i}\rangle$ being the local basis in subsystem $A(B)$. We can compute
the reduced density matrix $\rho_{A}(t)={\rm
Tr}_{B}|\Psi(t)\rangle\langle\Psi(t)|=\frac{1}{N}e^{-H_{A}(t)}$, where
$H_{A}(t)$ is referred to the entanglement Hamiltonian, $N$ is the
normalization constant, and the spectrum of $H_{A}(t)$ is the entanglement
spectrum.
In free-fermion systems, the eigenvalues of the reduced density matrix can be
obtained from the correlation matrix $C_{\bf
x,x^{\prime}}(t)=\langle\Psi(t)|c^{\dagger}_{\bf x}c_{\bf
x^{\prime}}|\Psi(t)\rangle=\sum_{i}|\psi_{i}({\bf
x^{\prime}},t)\rangle\langle\psi_{i}({\bf x},t)|$, where $|\psi_{i}({\bf
x},t)\rangle$ is the postquench single-particle state [see App. D]. The
spectrum $\xi(t)$ of the correlation matrix $C_{\bf x,x^{\prime}}(t)$ with
$x,x^{\prime}$ being restricted in $A$ is related to the entanglement spectrum
$\epsilon(t)$ by $\xi(t)=1/(1+e^{\epsilon(t)})$ Peschel (2003). For
simplicity, we refer $\xi(t)$ to the entanglement spectrum.
In the clean limit at $J_{0}/J_{1}=1.1$ [Fig. 3(a)], the post-quench state is
trivial and no crossings in the entanglement spectrum $\xi(t)$. When the
disorder strength is above the critical values, the entanglement spectrum
$\xi(t)$ of the post-quench state shows a crossing at $t=\pi/4$ [Fig. 3(b)].
The existence of the crossings in the entanglement spectrum agrees with the
non-vanishing dynamical Chern number of the post-quench state. We demonstrate
the non-vanishing dynamical Chern number and the crossings in the entanglement
spectrum for other parameters in App. E.
Figure 3: The entanglement spectrum of the postquench state with the
bipartition $l_{A}=l_{B}=N_{x}/2$, where $l_{A(B)}$ is the length of the
subsystem $A(B)$ and $N_{x}$ is the length of the total system. The parameters
are $J_{0}/J_{1}=1.1$. (a) $W_{0}=0$ (clean limit). (b) $W_{0}=3$. There are
$100$ disorder realizations for each data point.
## IV Experimental realization
Discrete-time quantum walks are great platforms for simulating the topological
phases of matter Kitagawa _et al._ (2010b); Cardano _et al._ (2017); Wang
_et al._ (2018), quantum quenches Wang _et al._ (2019b); Xu _et al._ (2019),
and disorder phenomena Obuse and Kawakami (2011); Zeng and Yong (2017); Kumar
_et al._ (2018). Following Ref. Wang _et al._ (2019b), the discrete-time
evolution operator for a one-dimensional lattice with single photons can be
engineered by the cascaded half-wave plates and beam displacers. The Hilbert
space is spanned by the polarization states
$\\{|P_{+}\rangle,|P_{-}\rangle\\}$ and the position state $|x\rangle$ with
$x\in\mathbb{Z}$. The corresponding evolution operator for each time step is
$U=R(\phi_{1}/2)SR(\phi_{2})SR(\phi_{1}/2)$, where $R(\phi)$ rotate the
polarization by $\phi$ with respect to $y$-axis, and $S$ is the shift operator
$S=\sum_{x}|x-1\rangle\langle x|\otimes|P_{+}\rangle\langle
P_{+}|+|x+1\rangle\langle x|\otimes|P_{-}\rangle\langle P_{-}|$. The
polarization angle $\phi_{1(2)}(x)$ is spatially dependent and disorder can be
introduced by choosing different $\phi_{1(2)}(x)$ for different position $x$.
Figure 4: The post-quench psuedospin texture in the momentum-time space
without disorder with $J_{0}/J_{1}=0.5$ for (a)flattened Hamiltonian, (b) non-
flattened Hamiltonian. The post-quench psuedospin forms the Skyrmion texture
in the momentum-time domain, $k\in[0,2\pi]$, $t\in[0,\pi/2]$ in (a) and
$t\in[0,T_{k}]$ in (b) where $T_{k}=\pi/(2E(k))$ is shown by the green line.
(c) The long-time average of $\eta$ for the non-flattened Hamiltonian with
$J_{0}/J_{1}=0.5$. Two curves almost overlap. The error bars are the standard
deviation for $400$ disorder realization. $N_{x}=31$.
In a translation-invariant case, the unitary operator can be diagonalized in
the momentum space and the effective Hamiltonian describing the pre/post-
quench system has the form $\mathcal{H}_{\rm eff}(k)=-i\ln U(k)$. It is shown
that this quantum walk protocol Wang _et al._ (2019b) can simulate a sudden
quench between $\mathcal{H}^{\rm i}_{\rm eff}(k)$ and $\mathcal{H}^{\rm
f}_{\rm eff}(k)$ of the SSH model. Here $\mathcal{H}^{\rm i}_{\rm eff}(k)$ and
$\mathcal{H}^{\rm f}_{\rm eff}(k)$ are referred to the pre-quench and the
post-quench Hamiltonians. The topology of the post-quench state can be
extrapolated from the post-quench pseudospin ${\bf n}(k,t)={\rm
Tr}[\rho(k,t)\bm{\sigma}]$ with
$\rho(k,t)=|\psi_{k}(t)\rangle\langle\psi_{k}(t)|$. The post-quench pseudospin
forms the Skyrmion texture in the momentum-time domain when the post-quench
state has non-vanishing dynamical Chern number [Fig. 4(a)]. The Skyrmion
texture can be understood as the pseudospin pointing along the $+(-)z$
direction at $t=0(\pi/2)$ and rotating clockwise as a function of $k$ on the
$x-y$ plane. In the experimental setup, the Hamiltonian is non-flatten and the
period of dynamics of each momentum is $T_{k}=\pi/(2E(k))$. Nevertheless, the
Skyrmion texture of the pseudospin can be observed in the momentum-time domain
$k\in[0,2\pi]$, $t_{k}\in[0,T_{k}]$ [Fig. 4(b)] and was measured
experimentally in the quantum walk setup Wang _et al._ (2019b).
In the presence of disorder, the momentum is no longer a good quantum number
and the momentum-dependent period is not well-defined. For the non-flattened
post-quench Hamiltoinan, we propose to measure the long-time average of the
pseudospins
$\overline{\langle\sigma_{i}\rangle_{T}}=\frac{1}{T}\int_{0}^{T}dt\overline{\langle\sigma_{i}\rangle}$,
where $\langle\sigma_{i}\rangle={\rm
Tr}\left[\rho^{\prime}(\tilde{k},t)\sigma_{i}\right]$ and
$\displaystyle\rho^{\prime}(\tilde{k},t)=\overline{\frac{1}{2}\sum_{i=0}^{3}\sum_{x_{1},x_{2}}e^{-i\tilde{k}(x_{1}-x_{2})}\langle\psi_{x_{1}}(t)|\sigma_{i}|\psi_{x_{2}}(t)\rangle\sigma_{i}}.$
(3)
$\rho^{\prime}(\tilde{k},t)$ is the disorder-averaged density matrix in the
pseudomomentum-time space, where $\overline{\cdots}$ denotes the disorder
average. Here $\tilde{k}$ is referred to the pseudomomentum, which indicates
that the momentum is no longer a good quantum number in disordered systems.
Since the $x,y-$ components of the Skyrmion texture shows a $2\pi$ winding as
a function of the pseudomomemtum $\tilde{k}$, one can monitor the in-plane
pseudospin texture to detect the nontrivial topology by defining
$\displaystyle\eta={\rm
Im}\log\left[\overline{\langle\sigma_{x}\rangle}_{T}+i\overline{\langle\sigma_{y}\rangle}_{T}\right].$
(4)
If the the post-quench state is topological, $\eta$ shows a $2\pi$ difference
in $\tilde{k}=0$ to $2\pi$.
We numerically show that $\eta$ can detect the topology of the post-quench
state in Fig. 4 (c) and 5. The time taken for the average is $T=\pi/E_{min}$,
where $E_{min}$ is the minimum absolute eigenenergy of the post-quench
Hamiltonian in the clean limit. This average time $T$ is the largest time-
scale in the system. First, we demonstrate the topology of post-quench state
is robust in the weak disorder region. As shown in Fig. 4 (c), the in-plane
pseudospin angle $\eta$ exhibits a $2\pi$ winding in the clean limit
$W_{0}/J_{1}=0$ and the weak disorder region $W_{0}/J_{1}=1$ for the parameter
$J_{0}/J_{1}=0.5$. Next, we consider the disorder-induced topology for the
parameter $J_{0}/J_{1}=1.1$. As we demonstrate previously, the post-quench
state is topological for the disorder strength $1.7\lesssim
W_{0}/J_{1}\lesssim 3.6$. As shown in Fig. 5 (a), $\eta$ does not has a $2\pi$
winding at $W_{0}/J_{1}=1$ and $W_{0}/J_{1}=6$, but exhibits a $2\pi$ winding
at $W_{0}/J_{1}=3$, reflecting the disorder-induced topology. In contrast, for
the parameter $J_{0}/J_{1}=1.5$ which does not exhibit the disorder-induced
topology, $\eta$ does not show a $2\pi$ winding with different strong disorder
strengths as shown in Fig. 5 (b).
Figure 5: The long-time avarage of $\eta$ for the post-quench psuedospin in
the pseudomomentum space given by non-flattened Hamiltonian with (a)
$\frac{J_{0}}{J_{1}}=1.1$, and (b) $\frac{J_{0}}{J_{1}}=1.5$. The error bars
are the standard deviations for $400$ disorder realizations. $N_{x}=31$.
## V Conclusion
We predict the disorder-induced topology in quench dynamics in (1+1)
dimensions. The topology is characterized by the dynamical Chern number and
crossings in the entanglement spectrum. We show the boundaries between trivial
and nontrivial post-quench states are identified by delocalized critical
points in the post-quench Hamiltonian. The quantized dynamical Chern number in
$(1+1)$ dimensions corresponds to the winding number of the one-dimensional
topological Anderson insulating phase of the SSH model. Finally, we propose
this phenomenon can be realized in quantum walk experiments.
###### Acknowledgements.
The authors thank Ching-Hao Chang and Chao-Cheng Kaun for hosting the workshop
of quantum materials at Research Center for Applied Sciences, Academia Sinica,
where the work was partially initiated. H.C.H. was supported by the Ministry
of Science and Technology (MOST) in Taiwan, MOST 108-2112-M-004-002-MY2.
P.-Y.C. was supported by the Young Scholar Fellowship Program under MOST. This
work was supported by the MOST under grant No. 110-2636-M-007-007.
## Appendix A Localization length
When electrons are localized, the wave function exponentially decays with
length, i.e. $\phi_{L}\propto e^{-L/\lambda}$, where
$\phi_{L}=\sum_{n=1}^{L}(\phi_{na},\phi_{nb})^{T}c_{n}^{\dagger}$ is the
eigenstate of the Hamiltonian $H=H_{o}+H_{U}$ with length $n$, $\phi_{na/b}$
are the coefficients for sublattice $a/b$ at site $n$ and $\lambda$ is the
localization length. The Schrodinger equation for zero eigenenergy state
becomes
$\displaystyle(J_{0}+U_{1n})\phi_{nb}+(J_{1}+U_{2n})\phi_{n-1,b}=0,$ (5)
$\displaystyle(J_{0}+U_{1n})\phi_{na}+(J_{1}+U_{2n})\phi_{n+1,a}=0.$ (6)
The above equations give the ratio of coefficients between the first and the
last site,
$\big{|}\phi_{La}\big{|}=\prod_{n=1}^{L}\big{|}\frac{J_{1}+U_{2n}}{J_{0}+U_{1n}}\phi_{1a}\big{|}$
and
$\big{|}\phi_{Lb}\big{|}=\prod_{n=1}^{L}\big{|}\frac{J_{0}+U_{1n}}{J_{1}+U_{2n}}\phi_{1b}\big{|}$
for each sublattice, respectively. The final localization length for the
system is the minimum of that of the sublattices. Thus, the localization
length is given by
$\displaystyle\frac{1}{\lambda}=\frac{1}{L}\ln\prod_{n=1}^{L}\big{|}\frac{J_{1}+U_{2n}}{J_{0}+U_{1n}}\big{|}.$
(7)
The equation can be solved analytically Mondragon-Shem _et al._ (2014).
Another approach to calculate the localization length is via Green’s function.
The localization length $\lambda$ is defined by
$\displaystyle\frac{2}{\lambda}=-\lim_{L\rightarrow\infty}\frac{1}{L}\mathrm{Tr}\ln|G_{1,L}|^{2},$
(8)
where $n$ is the total number of sites of the one-dimensional Hamiltonian,
$G_{1,L}$ is the propagator connecting the first and the last slice of the
system MacKinnon and Kramer (1983). $G_{1,n}$ is computed with the iterative
Green’s function method MacKinnon and Kramer (1983); Kramer and MacKinnon
(1993); Lewenkopf and Mucciolo (2013) by computing the onsite Green’s function
$G_{n,n}=\left(E-h_{n}-U_{f}G_{n-1,n-1}U_{b}\right)$ and
$G_{1,n}=G_{1,n-1}U_{b}G_{n,n}$ recursively till $n$ is large enough for
convergence, where $h_{n}=(J_{0}+U_{1,n})\sigma_{x}$,
$U_{f(b)}=(J_{1}+U_{2n})(\sigma_{x}+(-)i\sigma_{y})/2$ and $U_{1(2)n}$ are
defined in the main text. Within this method, the Hamiltonian is constructed
in a slicing scheme, i.e.
$\displaystyle H_{N}=\sum_{i=1}^{N}\left(|i\rangle h_{i}\langle i|+|i\rangle
U_{b}\langle i+1|+|i+1\rangle U_{f}\langle i|\right)$ (9)
for the system with $N$ slices, where $|i\rangle$ is the state for the $i$-th
slice, $U_{f(b)}$ is the forward (backward) hopping matrices between the
neighboring slices, and
To calculate the Greens function for the system with $N+1$ slices, the
Hamiltonian for $N+1$ slices is
$\displaystyle H_{N+1}=H_{N}+|N+1\rangle h_{N+1}\langle N+1|+H^{\prime},$ (10)
where $h_{N+1}$ is the Hamiltonian for $N+1$-th slices, the hopping matrix
$H^{\prime}=|N\rangle U_{b}\langle N+1|+|N+1\rangle U_{f}\langle N|$ between
the $N-$th and $N+1-$th slice is treated as a perturbing term to
$H_{N}+|N+1\rangle h_{N+1}\langle N+1|$. According to Dyson equation, the
perturbed Greens function is given by $G_{N+1}=G_{o}+G_{o}H^{\prime}G_{N+1}$,
where $G_{o}=G_{N}+|N+1\rangle\left(E-h_{N+1}\right)^{-1}\langle N+1|$.
Substitute $H^{\prime}$ into the Dyson equation, one obtains the Greens
function for $N+1$ slices ($G_{N+1}$) in which the submatrices are given by
$\displaystyle\langle N+1|G_{N+1}|N+1\rangle$
$\displaystyle=\left(E-h_{N+1}-U_{f}\langle N|G_{N}|N\rangle
U_{b}\right)^{-1},$ (11) $\displaystyle\langle 1|G_{N+1}|N+1\rangle$
$\displaystyle=\langle 1|G_{N}|N\rangle U_{b}\langle N+1|G_{N+1}|N+1\rangle.$
(12)
Eqs. (11) and (12) are the main iterative equations for obtaining the
localization length shown in Fig. 8(a).
## Appendix B Symmetry analysis and topological classification
The flattened Hamiltonian formalism allows us to construct the effective
Hamiltonian,
$\displaystyle\mathcal{H}_{\rm
eff}(k,t)=e^{-i\mathcal{H}^{F}_{0}t}\mathcal{H}_{\rm
pre}(k)e^{i\mathcal{H}^{F}_{0}t}.$ (13)
The topological invariants can be classified according to the symmetries of
the effective Hamiltonian. For the pre-quench Hamiltonian $\mathcal{H}_{\rm
pre}(k)=-\sigma_{z}$ and the post-quench Hamiltonian
$\mathcal{H}_{0}(k)=h_{x}(k)\sigma_{x}+h_{y}(k)\sigma_{y}$, one has the
effective Hamiltonian
$\displaystyle\mathcal{H}_{\rm eff}(k,t)$ $\displaystyle=$
$\displaystyle\frac{h_{y}(k)\sin
2t}{\sqrt{h_{x}(k)^{2}+h_{y}(k)^{2}}}\sigma_{x}$ (14)
$\displaystyle-\frac{h_{x}(k)\sin
2t}{\sqrt{h_{x}(k)^{2}+h_{y}(k)^{2}}}\sigma_{y}+\cos 2t\sigma_{z}.$
The effective Hamiltonian breaks the particle-hole symmetry explicitly, but
preserves the time-reversal symmetry $\mathcal{TH}_{\rm
eff}(k,t)\mathcal{T}^{-1}=\mathcal{H}_{\rm eff}(-k,-t)$, and the additional
two two-fold symmetries $\sigma_{z}\mathcal{H}_{\rm
eff}(k,t)\sigma_{z}=\mathcal{H}_{\rm eff}(k,-t)$, $\sigma_{x}\mathcal{H}_{\rm
eff}(k,t)\sigma_{x}=-\mathcal{H}_{\rm eff}(-k,t)$. These two additional
symmetries together with the time-reversal symmetry lead to a $\mathbb{Z}$
classification in $(1+1)$ dimensions. The former two-fold symmetry acts like
the reflection symmetry in the time domain. There are two fixed points $t=0$
and $\pi/2$ such that $[\sigma_{z},\mathcal{H}_{\rm
eff}(k,0)]=[\sigma_{z},\mathcal{H}_{\rm eff}(k,\pi/2)]=0$. The dynamical Chern
number in this effective Hamiltonian is quantized in the half of the momentum-
time space $k\in[0,2\pi]$, $t\in[0,\pi/2]$ Chiu _et al._ (2013, 2016);
Morimoto and Furusaki (2013); Shiozaki and Sato (2014).
The effective Hamiltonian has the following symmetries
$\displaystyle\mathcal{TH}_{\rm eff}(k,t)\mathcal{T}^{-1}=\mathcal{H}_{\rm
eff}(-k,-t),$ $\displaystyle\mathcal{R}_{t}{H}_{\rm
eff}(k,t)\mathcal{R}^{-1}_{t}=\mathcal{H}_{\rm eff}(k,-t),$
$\displaystyle\mathcal{M}_{x}\mathcal{H}_{\rm
eff}(k,t)\mathcal{M}_{x}^{-1}=-\mathcal{H}_{\rm eff}(-k,t),$ (15)
where $\mathcal{T}^{2}=\mathcal{R}_{t}^{2}=\mathcal{M}_{x}^{2}=1$,
$\\{\mathcal{R}_{t},\mathcal{M}_{x}\\}=0$, and
$[\mathcal{T},\mathcal{R}_{t}]=[\mathcal{T},\mathcal{M}_{x}]=0$.
The effective Hamiltonian can be expressed in terms of the effective massive
Dirac Hamiltonian $\mathcal{H}_{\rm
eff}(k,t)=k\gamma_{1}+t\gamma_{2}+M_{0}\gamma_{0}$, with
$\\{\gamma_{i},\gamma_{j}\\}=0$ ($i=0,1,2$). We construct the minimal
effective Dirac Hamiltonian in terms of the tensor product form of the Pauli
matrices
$\displaystyle\gamma_{1}=\sigma_{x}\otimes\sigma_{x},\quad\gamma_{2}=\sigma_{y}\otimes\mathbb{I}_{2\times
2},\quad\gamma_{0}=\sigma_{z}\otimes\mathbb{I}_{2\times 2},$
$\displaystyle\mathcal{T}=\mathbb{I}_{2\times
2}\otimes\sigma_{z}\mathcal{K},\quad\mathcal{R}_{t}=\sigma_{z}\otimes\sigma_{z},\quad\mathcal{M}_{x}=\sigma_{x}\otimes\mathbb{I}_{2\times
2}.$ (16)
One can check the only allowed mass term which preserving all the symmetries
is the $\gamma_{0}$. For the $\mathbb{Z}$ classification, we need to make
copies of the original effective Hamiltonian. For simplicity, we just make one
copy. The double Hamiltonian is $\mathcal{H}_{\rm
eff}(k,t)=k\gamma_{1}\otimes\mathbb{I}_{2\times
2}+t\gamma_{2}\otimes\mathbb{I}_{2\times
2}+M_{0}\gamma_{0}\otimes\mathbb{I}_{2\times 2}$, for which there are no other
symmetry-preserving mass terms. This indicates that different phases are not
adiabatically connected in this system. On the other hand, we can flip one
momentum of the copy and construct the double Hamiltonian, $\mathcal{H}_{\rm
eff}(k,t)=k\gamma_{1}\otimes\sigma_{z}+t\gamma_{2}\otimes\mathbb{I}_{2\times
2}+M_{0}\gamma_{0}\otimes\mathbb{I}_{2\times 2}$. There is another symmetry
allowed mass term (anti-commute with $\gamma_{0}\otimes\mathbb{I}_{2\times
2}\otimes\mathbb{I}_{2\times 2}$),
$M_{1}=\sigma_{y}\otimes\sigma_{y}\otimes\sigma_{y}$. This indicates the
systems are all in the same phase. We conclude from the above analysis that
the system belongs to a $\mathbb{Z}$ classification. Similar classification
schemes can be found in Refs. [Chiu _et al._ , 2013, 2016; Morimoto and
Furusaki, 2013; Shiozaki and Sato, 2014].
## Appendix C Wannier center with disorders
In translational invariant systems, the Wannier orbits are constructed from
the Bloch states $u_{n{\bf k}}({\bf r})$, $w_{n}({\bf
r-R})=\frac{1}{\Omega}\int d{\bf k}e^{i{\bf k\cdot(r-R)}}u_{n{\bf k}}({\bf
r})$, with $\Omega$ being the volume of the system, $\bf R$ is the position of
the unit-cell, and $\bf r$ is the local position of the Wannier orbits within
the unit-cell. In an insulator, these Wannier orbits are localized states and
are the eigenstates of the projected position operator $X_{P}=PXP$, where $P$
is the projector to the occupied states which are well defined in an
insulator.
To construct the Wannier orbits without using Bloch states, we first write
down the Hamiltonian in the real space $\mathcal{H}_{IJ}$, where $I(J)$
includes the band indices and positions. The spectrum can exhibit a gap and
the corresponding occupied states $|\psi_{\alpha I}\rangle$ are well defined.
Here $\alpha$ is the eigen-energy index. The corresponding projectors are
$P_{IJ}=\sum_{\alpha\in{\rm occ.}}|\psi_{\alpha I}\rangle\langle\psi_{\alpha
J}|$. The position operator can be defined by as a diagonal matrix ${\rm
diag}(1,\cdots,1,2,\cdots 2,\cdots,N,\cdots,N)$, where $N$ is the total number
of sites and at each site there are $L$ bands. The projected position operator
can be constructed as usual $X_{P}=PXP$ Kivelson (1982).
Since the Wannier orbits are the eigenstates of the $X_{P}$, we can find
diagonalize the $X_{P}$ and get the set of eigenstates. If the set of the
eigenstates are localized states, then these states are the Wannier orbits and
the corresponding eigenvalues are the position of the Wannier states. The
Wannier center of a localized state in $M$-th site can be defined as
$wc_{M}=|\langle w_{M}|X_{P}|w_{M}\rangle-M|$. We have $0<\langle
x_{M}\rangle<1$. We can further define the average Wannier center
$\overline{wc}=\frac{1}{N}\sum_{m=1}^{N}wc_{M}$. In the presence of the chiral
symmetry in one dimension gapped systems, the average Wannier center can have
two values $\overline{wc}=0$ and $0.5$. The former corresponds to a trivial
phase and the latter is the topological phase. Although in the quench setup,
the effective Hamiltonian does not have the chiral symmetry, we observe the
Wannier center of the topological post-quench state reaches
$\overline{wc}=0.5$ [Fig. 6(b)]. On the other hand, for the trivial post-
quench state, the Wannier center is below $0.5$ [Fig. 6(a)].
Figure 6: The Wannier center as a function of $t$. (a) Disorder-free
Hamiltonian $(\frac{J_{0}}{J_{1}},\frac{W_{0}}{J_{1}})=(1.1,0)$, (b)
disordered Hamiltonian $(\frac{J_{0}}{J_{1}},\frac{W_{0}}{J_{1}})=(1.1,3)$.
There are $100$ disorder realizations.
## Appendix D Correlation function formalism in quench setups
We consider an initial state contains $N$ particles. Each single-particle
state we denote by $|\phi_{\alpha}({\bf x})\rangle,\alpha=1,\cdots,N$, ${\bf
x}$ is the internal degrees of freedom, including position, spin, and the
band. We require these single-particle states are orthonormal, $\sum_{\bf
x}\langle\phi_{\alpha}({\bf x})|\phi_{\beta}({\bf
x})\rangle=\delta_{\alpha,\beta}$. The N-particle initial state can be
expressed as the Slater determinant of these single-particle state,
$\displaystyle|\Psi_{0}\rangle={\rm Det}[|\phi_{i}({\bf x}_{j})\rangle],\quad
i,j=1,\cdots,N.$ (17)
We consider an unitary evolution of this initial state $|\Psi_{0}\rangle$ by a
static Hamiltonian $H=\sum_{\bf x,x^{\prime}}\mathcal{H}_{\bf
x,x^{\prime}}c^{\dagger}_{\bf x}c_{\bf x^{\prime}}$, where $c^{(\dagger)}_{\bf
x}$ is the annihilation (creation) operator. Each single-particle state under
this evolution is $|\phi_{\alpha}({\bf x},t)\rangle=\sum_{\bf
x^{\prime}}\exp[-i\mathcal{H}_{\bf x,x^{\prime}}t]|\phi_{\alpha}({\bf
x}^{\prime})\rangle,\alpha=1,\cdots,N$. The post-quench N-particle state is
$\displaystyle|\Psi(t)\rangle=e^{-iHt}|\Psi_{0}\rangle={\rm
Det}[|\phi_{i}({\bf x}_{j},t)\rangle]=\prod_{i}d_{i}^{\dagger}(t)|0\rangle,$
(18)
where
$\displaystyle d_{i}^{\dagger}(t)$
$\displaystyle=e^{-iHt}d_{i}^{\dagger}e^{iHt}=e^{-iHt}\sum_{\bf y}V_{i{\bf
y}}c^{\dagger}_{\bf y}e^{iHt}$ $\displaystyle=\sum_{\bf x,y}V_{i{\bf y}}U_{\bf
y,x}(t)c^{\dagger}_{\bf x}$ (19)
with $U_{\bf y,x}(t)=e^{-i\mathcal{H}_{\bf y,x}t}$ and $V_{i{\bf y}}$ being an
unitary matrix that rotates $d_{i}^{\dagger}$ to $c_{\bf y}^{\dagger}$.
The post-quench single-particle state is
$\displaystyle d_{i}^{\dagger}(t)|0\rangle=\sum_{\bf x,y}V_{i{\bf y}}U_{\bf
y,x}(t)c^{\dagger}_{\bf x}|0\rangle=\sum_{\bf x}|\phi_{i}({\bf x},t)\rangle.$
(20)
The correlation function constructed from the N-particle post-quench state is
$\displaystyle C_{\bf x,x^{\prime}}(t)$
$\displaystyle=\langle\Psi(t)|c^{\dagger}_{\bf x}c_{\bf
x^{\prime}}|\Psi(t)\rangle=\langle
0|\prod_{\alpha}d_{\alpha}e^{iHt}c^{\dagger}_{\bf x}c_{\bf
x^{\prime}}e^{-iHt}\prod_{\beta}d^{\dagger}_{\beta}|0\rangle\rangle$
$\displaystyle=\langle 0|\prod_{\alpha}d_{\alpha}[\sum_{{\bf y},i}d_{i}U_{\bf
x,y}(t)V_{{\bf y}i}]^{\dagger}[\sum_{{\bf y^{\prime}},j}U_{\bf
x^{\prime},y^{\prime}}(t)V_{{\bf
y^{\prime}}j}d_{i}]\prod_{\beta}d^{\dagger}_{\beta}||0\rangle$
$\displaystyle=\sum_{i}[\sum_{\bf y}U_{\bf x,y}(t)V_{{\bf
y}i}]^{\dagger}[\sum_{\bf y^{\prime}}U_{\bf x^{\prime},y^{\prime}}(t)V_{{\bf
y^{\prime}}i}]$ $\displaystyle=\sum_{i}|\phi_{i}({\bf
x^{\prime}},t)\rangle\langle\phi_{i}({\bf x},t)|.$ (21)
The correlation matrix can be used for computing the entanglement spectrum.
The existence of the crossings in the entanglement spectrum can detect the
topology of the post-quench state as demonstrated in several examples.
## Appendix E Other parameters for the disorder-induced topology in quench
dynamics
In the clean limit, the dynamical Chern numbers are calculated for $0\leq
J_{0}\leq 2$ and $J_{1}=1$ of the SSH Hamiltonian $H_{o}$. The results are
shown in Fig. 7. For $J_{0}>1$, the static Hamiltonian becomes trivial and the
dynamical Chern number is zero.
Figure 7: The dynamical Chern number (DCN) for the static Hamiltonian in the
clean limit. The parameters are $J_{0}=1.1,J_{1}=1,N_{x}=400,L=20$. Here $L$
is the number of mesh points for the twisted boundary condition.
We consider the case with vanishing intercell disorder $W_{2}=0$. We find for
$2.2\lesssim W_{1}\lesssim 3.8$, the dynamical Chern number is close to an
integer with vanishing fluctuations as shown in Fig. 8(a). The phase
boundaries, where the dynamical Chern number is close to half-integer, are at
$W_{0}=1.5,4.9$. The localization length $\lambda$ also indicates delocalized
transitions at the same values of $W_{0}$ [Fig. 8 (a)]. The entanglement
spectrum has a crossing at $t=\pi/4$ when the post-quench state has integer
dynamical Chern number $W_{1}=3$ [Fig. 8(b)].
Figure 8: (color online) (a) The disorder averaged mean dynamical Chern number
and localization length obtained from Eq. (8) for the quench Hamiltonian. The
error bar is the standard deviation. The parameters are
$J_{0}=1.1,J_{1}=1,N_{x}=100,L=400$. (b) The entanglement spectrum of the
post-quench state with $W_{1}=3$. The parameters are
$J_{0}=1.1,J_{1}=1,N_{x}=400$. There are more than $50$ disorder realizations
for each data point.
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|
# Stiefel Liquids:
Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
Liujun Zou Yin-Chen He Chong Wang Perimeter Institute for Theoretical
Physics, Waterloo, Ontario, Canada N2L 2Y5
###### Abstract
We propose a new type of quantum liquids, dubbed Stiefel liquids, based on
$2+1$ dimensional nonlinear sigma models on target space $SO(N)/SO(4)$,
supplemented with Wess-Zumino-Witten terms. We argue that the Stiefel liquids
form a class of critical quantum liquids with extraordinary properties, such
as large emergent symmetries, a cascade structure, and nontrivial quantum
anomalies. We show that the well known deconfined quantum critical point and
$U(1)$ Dirac spin liquid are unified as two special examples of Stiefel
liquids, with $N=5$ and $N=6$, respectively. Furthermore, we conjecture that
Stiefel liquids with $N>6$ are non-Lagrangian, in the sense that under
renormalization group they flow to infrared (conformally invariant) fixed
points that cannot be described by any renormalizable continuum Lagrangian.
Such non-Lagrangian states are beyond the paradigm of parton gauge mean-field
theory familiar in the study of exotic quantum liquids in condensed matter
physics. The intrinsic absence of (conventional or parton-like) mean-field
construction also means that, within the traditional approaches, it will be
difficult to decide whether a non-Lagrangian state can actually emerge from a
specific UV system (such as a lattice spin system). For this purpose we
hypothesize that a quantum state is emergible from a lattice system if its
quantum anomalies match with the constraints from the (generalized) Lieb-
Schultz-Mattis theorems. Based on this hypothesis, we find that some of the
non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum
spin systems, for example, on triangular or Kagome lattice, through the
intertwinement between non-coplanar magnetic orders and valence-bond-solid
orders.
###### Contents
1. I Introduction
2. II Summary of results
3. III Review of background
1. III.1 Deconfined quantum critical point
2. III.2 U(1) Dirac spin liquid
4. IV Stiefel liquids: generality
1. IV.1 Wess-Zumino-Witten sigma model on Stiefel manifold $SO(N)/SO(4)$
2. IV.2 Symmetries
3. IV.3 Cascade structure of the SLs
4. IV.4 Possible fixed points at strong coupling
5. V $N=6$: Dirac spin liquids
1. V.1 Deriving $k=1$ WZW model from QED3
2. V.2 General $k$: $U(k)$ QCD with $N_{f}=4$
6. VI Quantum anomaly of SL(N,k)
1. VI.1 $SO(N)\times SO(N-4)$ on orientable manifolds
2. VI.2 Charge conjugation
3. VI.3 Unorientable manifolds
4. VI.4 Anomaly for the faithful $I^{(N)}$ symmetry from monopole characteristics
5. VI.5 Semion topological order from time-reversal breaking
7. VII Possible lattice realizations for $N>6$
1. VII.1 General strategy
1. VII.1.1 Anomaly matching
2. VII.1.2 Dynamical stability
2. VII.2 List of LSM-like anomalies in (2+1)d
3. VII.3 Warm-up: anomaly-matching for DQCP
4. VII.4 $N=7$: intertwining non-coplanar magnets with valence-bond solids
1. VII.4.1 Triangular lattice
2. VII.4.2 Kagome lattice
8. VIII Discussion
9. A More on the proposed WZW action
10. B A gauge theory description of SL(N=5,k)
11. C Full quantum anomaly of the DQCP
12. D WZW models on the Grassmannian manifold $\frac{G(2N)}{G(N)\times G(N)}$
13. E Explicit homomorphism between the $su(4)$ and $so(6)$ generators
14. F $I^{(N)}$ anomalies of SL(N)
1. F.1 The case with an even $N$
1. F.1.1 The case with $N=2\ ({\rm mod\ }4)$
2. F.1.2 The case with $N=0\ ({\rm mod\ }4)$
2. F.2 The case with an odd $N$
15. G Explicit calculations of the $U(1)$ DSL
16. H More on the LSM constraints
17. I Anomaly matching of the $U(1)$ DSL on a triangular lattice
## I Introduction
The richness of quantum phases and phase transitions never ceases to surprise
us. Over the years many interesting many-body states have been discovered or
proposed in various systems, such as different symmetry-breaking orders,
topological orders, and even exotic quantum criticality. One lesson Senthil
and Fisher (2006) we have learnt is that the vicinity of several competing (or
intertwining Fradkin _et al._ (2015)) orders may be a natural venue to look
for exotic quantum criticality. For example, it is proposed that the
deconfined quantum critical point (DQCP) may arise as a transition between a
Neel antiferromagnet (AF) and a valance bond solid (VBS) Senthil _et al._
(2004a, b), and a $U(1)$ Dirac spin liquid (DSL) may arise in the vicinity of
various intertwined orders Affleck and Marston (1988); Wen and Lee (1996);
Hastings (2000); Hermele _et al._ (2005, 2008); Song _et al._ (2020, 2019).
The physics of DQCP and $U(1)$ DSL, which will be discussed in more detail
later, have shed important light on the study of quantum matter. So a natural
question is
* •
Can one find more intriguing quantum criticality, possibly again around the
vicinity of some competing orders?
On the conceptual front, by now quantum states with low-energy excitations
described by well-defined quasiparticles and/or quasistrings, are relatively
well understood. These include Landau symmetry-breaking orders, various types
of topological phases and conventional Fermi liquids. Such states are
tractable because at sufficiently low energies they become weakly coupled and
admit simple effective descriptions, even if the system may be complicated at
the lattice scale. In contrast, understanding quantum states that remain
strongly coupled even at the lowest energies – and therefore do not admit
descriptions in terms of quasiparticles – remains a great challenge,
especially in dimensions greater than $(1+1)$ due to the lack of exact
analytic results. The widely held mentality, when dealing with such states, is
to start from a non-interacting mean-field theory, and introduce fluctuations
that are weak at some energy scale. The fluctuations may grow under
renormalization group (RG) flow, in which case the low-energy theory will
eventually become strongly coupled and describe the non-quasiparticle
dynamics. Here the mean-field theory can be formulated in terms of the
original physical degrees of freedom (DOFs), like spins, as is done for Landau
symmetry-breaking orders. It can also be formulated in terms of more
interesting objects called partons, which are “fractions” of local DOFs –
examples include composite bosons/fermions in fractional quantum Hall effects
and spinons in spin liquids. Fluctuations on top of a parton mean-field theory
typically lead to a gauge theory, which forms the theoretical basis of a large
number of exotic quantum phases in modern condensed matter physics Wen (2004).
Most (if not all) states in condensed matter physics are understood within the
mean-field mentality. In fact, this mentality is so deeply rooted in condensed
matter physics that very often a state can be considered “understood” only if
a mean-field picture is obtained.
Although most (if not all) states theoretically studied in condensed matter
physics can be described by a mean field plus some weak fluctuations at some
scale, a priori, there is no reason to assume that all non-quasiparticle
states admit some mean-field descriptions. One may therefore wonder if there
is an approach that can complement the mean-field theory, and whether one can
use this approach to study quantum phases or phase transitions that cannot be
described by any mean-field theory plus weak fluctuations. This question can
also be formulated in the realm of quantum field theories. The universal
properties of a field theory are characterized by a fixed point under RG, and
such a fixed point usually allows a description in terms of a weakly-coupled
renormalizable continuum Lagrangian at certain energy scale. Such a
renormalizable-Lagrangian description of a fixed point is essentially the
field-theoretic version of the mean-field description of a quantum phase or
phase transition. In this language, a mean field formulated in terms of
partons corresponds to a gauge theory that is renormalizable, i.e. weakly
coupled at the UV scale111We should note that our identification of “mean-
field theory” in condensed matter and “renormalizable Lagrangian” in field
theory is sometimes loose. For example, the Sachdev-Ye-Kitaev model Sachdev
and Ye (1993); Kitaev (2015) has a renormalizable Lagrangian, but the
corresponding “mean field” in our definition would have a trivial zero
Hamiltonian, which is not a useful mean field.. So one may similarly wonder if
there are interesting RG fixed points that are intrinsically non-
renormalizable, i.e., cannot be described by any weakly-coupled renormalizable
continuum Lagrangian at any scale, a property sometimes refered to as “non-
Lagrangian”. Some examples of such “non-Lagrangian” theories have been
discussed in the string theory and supersymmetric field theory literature over
the years (see, for example, Refs. García-Etxebarria and Regalado (2016); Beem
_et al._ (2016); Gukov (2017) for some recent exploration and Ref. Heckman and
Rudelius (2019) for a review), but it is not clear whether those examples
could be directly relevant in the context of condensed matter physics. In
particular, we are interested in non-supersymmetric theories realizable in
relatively low dimensions such as $(2+1)$. If such non-Lagrangian theories can
be identified, they also enrich our understanding of the landscape of quantum
field theories in an intriguing way.
So the following important questions arise:
* •
In condensed matter systems, are there exotic quantum phases and phase
transitions beyond the paradigm of mean-field $+$ weak fluctuations?
Equivalently, are there non-Lagrangian RG fixed points relevant in condensed
matter physics?
* •
If the answers to the above questions are yes, how can we tell in which
systems such RG fixed points can emerge? How can we predict the physical
properties of these states?
Besides the conceptual importance, these questions may also be practically
relevant. If the quantum phases and phase transitions envisioned above do
exist, they should be included as candidate theories for many of the elusive
experimental and numerical systems, for example in spin liquid physics Savary
and Balents (2017); Zhou _et al._ (2017); Broholm _et al._ (2020).
In this paper we focus on critical quantum states (phases or phase
transitions) that are effectively described by some conformal field theories
(CFTs) at low energies. We also focus on bosonic systems such as spin models.
We shall first look for inspirations from two well known exotic quantum
critical states: the DQCP and the $U(1)$ DSL – we review these two states in
Sec. III. The effective theory of DQCP is usually formulated in terms of some
gauge theories that flow to strong coupling in the IR. However, there is a
non-renormalizable description of DQCP based purely on local (gauge invariant)
DOFs, formulated as a non-linear sigma model (NLSM) supplemented with a
topological Wess-Zumino-Witten (WZW) term Tanaka and Hu (2005); Senthil and
Fisher (2006); Nahum _et al._ (2015a); Wang _et al._ (2017). 222Known
$(2+1)$-d non-supersymmetric Lorentz-invariant renormalizable Lagrangians all
only involve critical bosons/soft spins, Dirac/Majorana fermions, gauge
fields, and their combinations. In particular, a NLSM where the DOF lives in a
manifold and bears some constraints is non-renormalizable. This description
comes with some virtues such as the locality of all DOFs and a manifest
emergent $SO(5)$ symmetry. The fact that this description is strongly coupled
in both UV and IR is usually viewed as a drawback. However, since we are now
aiming to study “non-Lagrangian” theories that do not have weak-coupling
descriptions anyway, it seems natural to try to turn this “bug” into a
“feature”, by generalizing the NLSM construction to some “non-Lagrangian”
critical states. To achieve this, it turns out to be useful to consider the
$U(1)$ DSL, which is known to be closely related to the DQCP Wang _et al._
(2017); Song _et al._ (2020, 2019). If we can extend the NLSM construction to
the $U(1)$ DSL, we may then “extrapolate” the two theories to obtain an entire
series of theories, some of which could possibly go beyond any mean-field $+$
weak fluctuations description.
With these motivations, we study a special type of $(2+1)$-d quantum many-body
states, each labeled by two integers $(N,k)$, with $N\geqslant 5$ and $k\neq
0$. Their effective theories are formulated purely in terms of local DOF,
described by a NLSM defined on a target space $SO(N)/SO(4)$, supplemented with
a WZW term at level $k$ Wess and Zumino (1971); Witten (1983). The manifold
$SO(N)/SO(4)$ is known as a Stiefel manifold (e.g., see Ref. Nakahara (2003)),
so we dub these states “Stiefel liquids” (SLs), and we refer to an SL labeled
by $(N,k)$ as SL(N,k), and SL(N,k=1) may also be simply written as SL(N).
These SLs have many interesting properties, such as a large emergent symmetry.
Furthermore, there is a cascade structure among them: for each $k$, an SL with
a smaller $N$ can be obtained from an SL with a larger $N$ by appropriately
perturbing the latter and focusing on the resulting low-energy sector.
We propose that these SLs form cascades of extraordinary critical quantum
liquids. In fact, SL(5) is precisely the effective field theory for the DQCP
mentioned above. Furthermore, we will argue that SL(6) describes the $U(1)$
DSL discussed above, and is thus a dual description of the latter purely based
on local DOFs. Due to the cascade structure, SL(N>6) can be viewed as
extrapolating theories of the DQCP and $U(1)$ DSL. We will argue that SL(N>6)
can flow to conformally invariant RG fixed points in the IR. Furthermore, they
appear to have no obvious(renormalizable) gauge-theoretic description, and we
conjecture that they are in fact non-Lagrangian. We provide various reasonable
arguments to support this conjecture, although a rigorous proof is currently
lacking, and it is unclear if such a proof is possible at all. However, as we
argue, even if this conjecture can be disproved, one has to necessarily invoke
novel ingredients of renormalizable field theories that have not been
appreciated so far, and in this way new general insights can still be gained.
SL(5) and SL(6), namely, the DQCP and the $U(1)$ DSL, can both emerge in some
lattice spin systems. The standard way to establish the emergibility of these
states is to construct their corresponding mean-field theory on the lattice,
based on the parton trick. For their non-Lagrangian counterparts with $N>6$,
we do not have any known mean-field construction, and some alternative
approach has to be adopted. In Sec. VII, we propose an approach, which is
complimentary to the traditional mean-field approach, to study in which
systems they may emerge. This approach is based on the hypothesis that a
quantum state described by some effective field theory is emergible from a
lattice system if and only if the quantum anomalies of the field theory match
that of the lattice system.
Quantum anomalies, in particular, ’t Hooft anomalies, were originally
introduced as an obstruction to consistently coupling a system with certain
global symmetry to a gauge field corresponding to this symmetry Hooft (1980),
and recently it has been realized that they are also related to whether this
symmetry can be realized in an on-site fashion Chen _et al._ (2011). That is,
the anomaly detects the structure of locality and/or the interplay between
symmetry and locality of a system. Furthermore, quantum anomaly is an RG
invariant, and it is powerful in constraining the IR fate of a system based on
its UV information, in that a theory with a nontrivial anomaly is forbidden to
have a symmetric short-range entangled ground state Hooft (1980). For a
lattice system, the ’t Hooft anomaly is intimately related Cheng _et al._
(2016); Jian _et al._ (2018a); Cho _et al._ (2017); Metlitski and Thorngren
(2018) to the Lieb-Schutz-Mattis-type theorems Lieb _et al._ (1961); Oshikawa
(2000); Hastings (2004); Po _et al._ (2017) for quantum matters on lattice
systems.
In this paper, matching the anomalies of two seemingly different theories
motivates us to propose that these two theories can emerge in the same
physical setup. In addition, anomaly-based considerations also enable us to
make specific concrete predictions of a system, without referring to the
details of its theory (such as its Hamiltonian). For example, we propose that
SL(7) can be realized in spin-$1/2$ triangular and kagome lattice systems, and
we are able to predict some of its detailed physical properties, such as the
crystal momenta of gapless modes in this realization. One interesting
observation is that SL(7) can naturally arise in the vicinity of competing
non-coplanar magnetic order and VBS, which is a natural generalization of that
SL(5) (DQCP) can naturally arise in the vicinity of competing collinear
magnetic order and VBS, and SL(6) ($U(1)$ DSL) can naturally arise in the
vicinity of competing non-collinear but coplanar magnetic order and VBS. We
illustrate these in Fig. 1.
Figure 1: The Stiefel liquid out of intertwined orders in quantum magnets. (a)
$(N=5,k=1)$ SL is the widely studied deconfined phase transition, which can
arise from the intertwinement of collinear magnetic order (e.g. Neel state)
and valence bond solid. (b) $(N=6,k=1)$ SL is the widely studied $U(1)$ Dirac
spin liquid, which can arise from the intertwinement of non-collinear magnetic
order (but coplanar) and valence bond solid. (c) $(N=7,k=1)$ SL is a new
critical quantum liquid, which can arise from the intertwinement of non-
coplanar magnetic order and valence bond solid. On triangular lattice the non-
coplanar order is known as the tetrahedral order. On kagome lattice the non-
coplanar order is known as the cuboctahedral order, in which the
magnetizations on the three sublattices are
$S_{A}=-\bm{Q}_{3}\cos(n\pi)-\bm{Q}_{1}\cos[(m+n)\pi]$,
$S_{B}=\bm{Q}_{3}\cos(n\pi)-\bm{Q}_{2}\cos(m\pi)$,
$S_{C}=\bm{Q}_{2}\cos(m\pi)+\bm{Q}_{1}\cos[(m+n)\pi]$ with $Q_{1,2,3}$ being
orthogonal to each other.
Thinking more broadly, the absence of mean-field constructions or
renormalizable continuum Lagrangians forces us to focus on more universal
aspects of the critical states. The natural goal here is to obtain an
intrinsic characterization of universal many-body physics: a complete
characterization of the universal properties of a many-body system, without
explicitly referring to any Hamiltonian, Lagrangian, or wave function. This
goal is also motivated by the observation that, although often useful, a
Hamiltonian/Lagrangian/wave function is just a specific UV regularization of
the universal physics of the underlying quantum phase or phase transition.
Therefore, it is conceptually and aesthetically desirable to find such an
intrinsic characterization. (Of course, to obtain non-universal details of a
many-body system, its Hamiltonian/Lagrangian/wave function is needed.) In
fact, such an intrinsic characterization has been (partly) achieved in various
systems, such as CFTs in $(1+1)$-d Di Francesco _et al._ (1996), a large
class of gapped phases in various dimensions Kitaev (2006); Etingof _et al._
(2009); Chen _et al._ (2013a); Barkeshli _et al._ (2019); Lan _et al._
(2016, 2017a, 2017b, 2018); Gaiotto and Johnson-Freyd (2019a); Lan and Wen
(2019); Gaiotto and Johnson-Freyd (2019b); Kong _et al._ (2020); Johnson-
Freyd (2020), and symmetry-enriched $U(1)$ quantum spin liquids in $(3+1)$-d
Wang and Senthil (2013, 2016); Zou _et al._ (2018); Zou (2018); Hsin and
Turzillo (2020); Ning _et al._ (2020). The present work can be viewed as a
small step toward this ambitious goal for more complicated critical states of
matter.
## II Summary of results
The main results of this paper are summarized below. This part also serves as
a map for this paper.
1. 1.
We propose a class of exotic $(2+1)$-d quantum many-body states dubbed Stiefel
liquids (SLs), each indexed by two integers, $N\geqslant 5$ and $k\neq 0$. The
effective theory of an SL with index $(N,k)$, SL(N,k), is formulated as a
nonlinear sigma model (NLSM) on target manifold $SO(N)/SO(4)$, supplemented
with a Wess-Zumino-Witten (WZW) term at level $k$. The target manifold is also
known as Stifel manifold $V_{N,N-4}$ (or simply $V_{N}$ in this paper), hence
the name Stiefel liquid. The Stiefel manifold can be parameterized using an
$N\times(N-4)$ matrix $n_{ji}$ satisfying $n^{T}n=I_{N-4}$, where $I_{N-4}$ is
the $(N-4)$-dimensional identity matrix. The NLSM is defined using the action
$\displaystyle\qquad S^{(N,k)}[n]=\frac{1}{2g}\int d^{2+1}x{\rm
Tr}(\partial_{\mu}n^{T}\partial^{\mu}n)+k\cdot S^{(N)}_{\rm{WZW}}.$
The first term is a standard kinetic term, and the WZW term is well defined
since $\pi_{4}(V_{N})=\mathbb{Z}$ and $\pi_{i}(V_{N})$ are trivial for $i<4$.
The detailed form of the WZW term will be discussed in Sec. IV.1. As we will
mainly be interested in $k=1$, we also use SL(N) to denote SL(N,1).
The above NLSM is non-renormalizable, so its dynamics at strong coupling ($g$
not small) is not clearly defined on face value. We can nevertheless argue, as
we do in Sec. IV.4, that for each $k\neq 0$ there is a critical $N_{c}(k)$,
such that for $N>N_{c}(k)$ the theory can flow to a stable CFT fixed point at
strong coupling. The stable fixed point is separated from the spontaneous
symmetry breaking phase in the weak-coupling regime (small $g$) by a critical
point. Those stable fixed points represent the critical Stiefel liquids which
are the main focus of this paper. We argue, based on existing numerical
results, that for $k=1$ it is likely that $5<N_{c}<6$.
2. 2.
In Sec. IV.2 we carefully discuss the symmetries of the Stiefel liquids. It
turns out that the SL(N,k) theory has a rather nontrivial symmetry group. The
continous symmetry is $SO(N)\times SO(N-4)$ if $N$ is odd, and for $N$ even it
is $(SO(N)\times SO(N-4))/\mathbb{Z}_{2}$. An SL also has discrete
$\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$ symmetries. These discrete
symmetries turn out to act nontrivially in the $SO(N)$ and $SO(N-4)$ internal
spaces, so they form semi-direct products ($\rtimes$) with the continuous
symmetries – we discuss this carefully in Sec. IV.2. The SL theory also has
the standard Poincaré symmetry group. The full symmetry is therefore the
Poincaré plus
$\displaystyle(SO(N)\times
SO(N-4))\rtimes(\mathbb{Z}_{2}^{\mathcal{C}}\times\mathbb{Z}_{2}^{\mathcal{T}}),$
$\displaystyle\hskip 5.0ptN=2n+1$ $\displaystyle\left(\frac{SO(N)\times
SO(N-4)}{\mathbb{Z}_{2}}\right)\rtimes(\mathbb{Z}_{2}^{\mathcal{C}}\times\mathbb{Z}_{2}^{\mathcal{T}}),$
$\displaystyle\hskip 5.0ptN=2n$
3. 3.
In Sec. IV.3 we discuss a cascade structure of the SLs: for each $k$, by
appropriately breaking the symmetry of SL(N,k) with a larger $N$, the low-
energy sector of the resulting theory is another SL with the same $k$ but a
smaller $N$. In particular, if we condense the first column of the $n_{ji}$
matrix to a fixed unit vector, say $n_{j1}=(1,0,0...)^{T}$, we break the $\sim
SO(N)\times SO(N-4)$ symmetry of SL(N,k) down to $\sim SO(N-1)\times SO(N-5)$
and obtain the SL(N-1,k) theory.
4. 4.
The SL(5) theory turns out to be nothing but the sigma-model description of
the DQCP. In Appendix B, we extend this correspondence to general $k>0$ and
propose that SL(5,k) can be described by a $USp(2k)$ gauge theory with
$N_{f}=2$ flavors of fermions – the case with $k=1$ and $USp(2)=SU(2)$ is a
familiar result.
5. 5.
In Sec. V, we argue that SL(6,k) is equivalent to the $U(k)$ DSL, i.e., 4
flavors of gapless Dirac fermions coupled to a $U(k)$ gauge field. The field
$n_{ji}$ on the Stiefel manifold $SO(6)/SO(4)$ corresponds to the monopoles in
the $U(k)$ gauge theory. In particular, for $k=1$ this gives an SL description
of the familiar $U(1)$ Dirac spin liquid. In Appendix G, we further support
this correspondence by explicitly matching the anomalies of the $U(1)$ Dirac
spin liquid with the SL(6) theory. In particular, we verify that various probe
monopoles in the $U(1)$ DSL theory do have the nontrivial properties required
by the anomalies.
6. 6.
In Sec. IV.1, we conjecture that SL(N) with $N>6$ is non-Lagrangian, i.e., its
conformally invariant fixed point has no description in terms of a weakly-
coupled renormalizable continuum Lagrangian at any scale. Equivalently, SL(N)
with $N>6$ cannot be accessed using any mean-field theory plus weak
fluctuations. In particular, they are beyond the standard parton (mean-field)
construction and gauge theory. We further elaborate on the arguments
underlying this conjecture in Sec. VIII.
7. 7.
In Sec. VI, we analyze the quantum anomalies of the SLs. One way to
characterize the anomaly is to view our $(2+1)$-d system as the boundary of a
$(3+1)$-d bulk, and the bulk has a nontrivial topological response when
coupled to background gauge fields. We can couple the SL(N,k) theory to a
background gauge field with gauge symmetry $(SO(N)\times
SO(N-4))\rtimes(\mathbb{Z}_{2}^{\mathcal{C}}\times\mathbb{Z}_{2}^{\mathcal{T}})$,
which turns out (as we discuss carefully in Sec. VI) to be equivalent to
coupling to an $O(N)\times O(N-4)$ gauge field, with the following restriction
on the gauge bundles:
$w_{1}^{O(N)}+w_{1}^{O(N-4)}+w_{1}^{TM}=0\hskip 5.0pt({\rm{mod}}\hskip
2.0pt2).$
The three terms are the first Stiefel-Whitney (SW) classes of the $O(N)$,
$O(N-4)$ and $(3+1)$-d spacetime tangent bundles, respectively. The anomaly is
then given by the bulk topological response (see Sec. VI for the derivation)
$\displaystyle
ik\pi\int_{X_{4}}\left\\{w_{4}^{O(N)}+w_{4}^{O(N-4)}+\left[w_{2}^{O(N-4)}+\left(w_{1}^{O(N-4)}\right)^{2}\right]\left(w_{2}^{O(N)}+w_{2}^{O(N-4)}\right)+\left(w_{1}^{O(N-4)}\right)^{4}\right\\}.$
Here $w_{1}$, $w_{2}$ and $w_{4}$ are the first, second and fourth SW classes
of the corresponding bundles, respectively, and all the products involved are
cup products. This anomaly is $\mathbb{Z}_{2}$-classified and only SLs with
odd $k$ are nontrivial. The above bulk response gives the complete anomaly of
SL(N) with an odd $N$. The case with an even $N$ is more complicated, and the
above result is just a partial characterization in this case. To improve this
result, in Sec. VI.4 we characterize the anomaly for the even-$N$ case by
studying the monopoles corresponding to the global symmetry of the theory.
This characterization, although still partial, contains physics beyond that in
the above bulk action. In particular, SL(N,k) with an even $N$ and $k=2$ (mod
$4$) turns out to be also anomalous, which cannot be detected by the above
bulk action.
The above anomaly of SL(N,k) with odd $k$ cannot be saturated by a gapped
topological order – the IR theory has to be either gapless or symmetry-
breaking. In Sec. VI.5, we show that if time-reversal and space reflection
symmetries are broken, the anomalies of SL(N) can be realized by a semion (or
anti-semion) topological order, for all $N\geqslant 5$.
8. 8.
In Sec. VII we discuss possible lattice realizations of Stiefel liquids with
$N>6$, which as we discussed are likely non-Lagrangian. The intrinsic absence
of a mean-field construction for these states makes it challenging to decide
whether a Stiefel liquid, say with $N=7$, can emerge out of a lattice system.
We therefore propose an approach based on anomaly matching: we hypothesize
that a state (like SL(7)) is emergible from a lattice system if and only if
the ’t Hooft anomalies of the state match that of the lattice system. The
anomalies of the lattice system come from the generalized Lieb-Schultz-Mattis
theorems. The necessity of this condition is actually known, so we only
hypothesize the sufficiency part.
We then find that SL(7), the simplest non-Lagrangian SL, can indeed be
realized on lattice spin systems if the microscopic physical symmetries are
properly implemented in the low-energy theory. Here the “microscopic
symmetries” include the $SO(3)$ rotation, time-reversal and lattice
symmetries. Specifically, we identify two realizations of the SL(7) theory on
triangular and Kagome lattices, respectively, both with one spin-$1/2$ moment
sitting on each lattice site. Many observable properties of these specific
realization of SL(7) are discussed. In particular, we find that this state can
naturally arise in the vicinity of competing non-coplanar magnetic order and
valance-bond solid (VBS). The corresponding non-coplaner magnetic orders are
known as the tetrahedral order on triangular lattice and the cuboctahedral
order on Kagome lattice, respectively. These realizations of the SL(7) state
are very natural generalizations of the realizations of SL(5) (DQCP) and SL(6)
($U(1)$ DSL), since the DQCP arises in the vicinity of competing collinear
magnetic order and VBS, and the $U(1)$ DSL arises in the vicinity of competing
coplanar magnetic orders and VBS (as illustrated in Fig. 1).
We finish with some discussions in Sec. VIII. Various appendices contain
additional details, as well as some general results.
Before proceeding, we will first briefly review the physics of the deconfined
quantum critical point and the $U(1)$ Dirac spin liquid in Sec. III.
## III Review of background
In this section, we first review some aspects of the DQCP and $U(1)$ DSL,
which partly motivate the present work.
### III.1 Deconfined quantum critical point
The classic DQCP was proposed as a critical theory for a quantum phase
transition between a Neel AF and a VBS on a square lattice Senthil _et al._
(2004a, b). Because the symmetry respected by either of these two phases is
not a subgroup of the symmetry of the other, such a transition is considered
to be beyond the Landau-Ginzburg-Wilson-Fisher paradiam if it is continuous.
The original formulation of the DQCP is in terms of two flavors of bosons
coupled to a dynamical $U(1)$ gauge field, and over the years many dual
formulations have been proposed Senthil and Fisher (2006); Wang _et al._
(2017).
The formulation that is most relevant for our purpose is written in terms of a
5-component unit vector $\bm{n}$, whose first 3 and last 2 components can be
thought of as the order parameters of the AF and the VBS, respectively. So
this is a formulation directly based on local DOFs. The low-energy effective
theory at the DQCP is a NLSM with a WZW term:
$\displaystyle S^{(5,k)}=S_{0}+k\cdot S^{(5)}_{\rm WZW}$ (1)
The meaning of the superscripts “$(5,k)$” will be clear later. For the DQCP,
$k=1$, and this seemingly redundant factor is inserted for later convenience.
The first term $S_{0}=\int d^{3}x\frac{1}{2g}(\partial_{\mu}\bm{n})^{2}$ is
the action of the usual NLSM. To define the WZW term, $S_{\rm WZW}^{(5)}$, one
first needs to add one more dimension to the physical spacetime and extend the
unit vector $\bm{n}$ into this extra dimension. We denote the coordinate of
this extra dimension by $u$, and the extended unit vector by $\bm{n}^{e}$,
such that $\bm{n}^{e}(x,y,t,u=0)=\bm{n}(x,y,t)$ and
$\bm{n}^{e}(x,y,t,u=1)=\bm{n}^{r}$, with $n^{r}$ being an arbitrary fixed
reference vector, which, for example, can be taken to be
$\bm{n}^{r}=(1,0,0,0,0)^{T}$. For notational brevity, in the following we will
drop the superscript “e” in the extended vector and simply write it as
$\bm{n}$, and the meaning of $\bm{n}$ should be clear from the context. In
terms of the extended vector, the WZW term is
$\displaystyle S_{\rm WZW}^{(5)}=\frac{2\pi}{\Omega_{4}}\int_{0}^{1}du\int
d^{3}x\det(\tilde{n})$ (2)
where $\Omega_{4}=8\pi^{2}/3$ is the volume of $S^{4}$ with unit radius, and
$\tilde{n}$ is a 5-by-5 matrix defined as
$\tilde{n}\equiv(n,\partial_{x}n,\partial_{y}n,\partial_{t}n,\partial_{u}n)$
Namely, the first column of $\tilde{n}$ is $\bm{n}$, and its last 4 columns
are derivatives of $\bm{n}$ arranged in the above way. More explicitly,
$\det(\tilde{n})=\epsilon^{i_{1}i_{2}i_{3}i_{4}i_{5}}n_{i_{1}}\partial_{x}n_{i_{2}}\partial_{y}n_{i_{3}}\partial_{t}n_{i_{4}}\partial_{u}n_{i_{5}}$
Geometrically, the WZW term (apart from the factor of $2\pi$) is the ratio of
the volume swept by $\bm{n}$ (as its coordinates vary) and $\Omega_{4}$, the
total volume of $S^{4}$ with unit radius. Physically, the WZW term intertwines
the Neel and VBS orders Senthil and Fisher (2006); Wang _et al._ (2017) (see
also earlier related works Read and Sachdev (1989, 1990)).
The theory Eq. (1) enjoys an $I^{(5)}=SO(5)$ symmetry, under which $\bm{n}$
transforms in its vector representation. The purpose for the notation
$I^{(5)}$ will be clear later. It is useful to imagine enlarging the $SO(5)$
symmetry group into $O(5)$. Due to the WZW term, the improper $Z_{2}$ rotation
of this $O(5)$ group is not a symmetry of Eq. (1), but when it combines with
the reversal of a space or time coordinate, it becomes the reflection or time
reversal symmetry. We denote this symmetry as $O(5)^{T}$.
The Neel-VBS transition is driven by a rank-$2$ anisotropy term
$\lambda(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}-n_{4}^{2}-n_{5}^{2})$, with $\lambda<0$
favoring the Neel order and $\lambda>0$ favoring the VBS order. At weak
coupling the sigma model orders spontaneously and the Neel-VBS transition
driven by the anisotropy will be first order. The DQCP, as a continuous Neel-
VBS transition, then requires a nontrivial fixed point at strong coupling. The
strong coupling dynamics, strictly speaking, is not well defined just from the
sigma model Lagrangian since the theory is not renormalizable. Nevertheless,
if such a strong-coupling fixed point exists, several nontrivial properties of
this fixed point can be readily inferred:
1. 1.
The theory has the full $O(5)^{T}$ symmetry.
2. 2.
Local operators that transform trivially under $O(5)^{T}$ must be RG
irrelevant.
3. 3.
The theory has a ’t Hooft anomaly, characterized by the topological action of
a $(3+1)$-d symmetry-protected topological phase (SPT) whose boundary can host
the DQCP:
$\displaystyle\mathcal{Z}^{(5)}_{\rm
bulk}=\exp\left(i\pi\int_{X_{4}}w_{4}^{O(5)}\right)$ (3)
where $X_{4}$ is the $4$ dimensional spacetime manifold that the bulk SPT
lives in, and $w_{4}^{O(5)}$ is the fourth Stiefel-Whitney (SW) class of the
probe $O(5)^{T}$ gauge bundle that couples to the SPT. This topological
response theory is subject to a constraint, $w_{1}^{O(5)}=w_{1}^{TM}\ ({\rm
mod\ }2)$, with $w_{1}^{TM}$ the first SW class of the tangent bundle of
$X_{4}$. This constraint guarantees that the orientation-reversal symmetry
(i.e., reflection and time reversal) is accompanied by an improper $Z_{2}$
rotation of the $O(5)$. The $SO(5)$ anomaly has been derived in Ref. Wang _et
al._ (2017), and in Appendix C we extend it to the full $O(5)^{T}$.
Below we collect some further results on the anomaly that are relevant to the
present paper.
1. 1.
The anomaly is $\mathbb{Z}_{2}$ classified, i.e., they disappear if two copies
of this theory are stacked together. This means that the anomalies of the
theory described by Eq. (1) remains the same if $k$ is changed by any even
integer.
2. 2.
In many cases, the anomaly of a $(2+1)$-d theory can be realized by a
symmetric gapped topological order, but the anomaly of the DQCP cannot, if the
system preserves both the $SO(5)$ and an orientation-reversal symmetry. In
other words, due to this anomaly, as long as the system preserves the $SO(5)$
symmetry together with any orientation-reversal symmetry, it has to be
gapless. This is an example of symmetry-enforced gaplessness Wang and Senthil
(2014).
3. 3.
A physical way to characterize the anomaly of this theory is to gauge the
$SO(5)$ symmetry, and examine the structure of the monopoles of the resulting
$SO(5)$ gauge field. An $SO(5)$ monopole can be obtained by embedding a $U(1)$
monopole into one of the generators of the $SO(5)$ gauge group Wu and Yang
(1975), and the field configuration of this monopole breaks the $SO(5)$ into
$SO(3)\times SO(2)$. Then it is meaningful to ask what representation this
$SO(5)$ monopole carries under the remaining $SO(3)\times SO(2)$. It turns out
that it carries a spinor representation under $SO(3)$ and no charge under
$SO(2)$ (up to binding the original matter fields in the fundamental
representation of the $SO(5)$).
Finally, we comment on the current status of the numerical studies on the
actual IR dynamics of the DQCP. The emergence of the $SO(5)$ symmetry that
rotates between Neel and VBS orders has been observed numerically Nahum _et
al._ (2015a). On the other hand, the seemingly continuous transition Sandvik
(2007); Jiang _et al._ (2008); Melko and Kaul (2008); Charrier _et al._
(2008); Motrunich and Vishwanath (2008); Kuklov _et al._ (2008); Chen _et
al._ (2009); Lou _et al._ (2009); Banerjee _et al._ (2010); Charrier and
Alet (2010); Sandvik (2010); Bartosch (2013); Harada _et al._ (2013); Chen
_et al._ (2013b); Nahum _et al._ (2015b); Sreejith and Powell (2015); Shao
_et al._ (2016); Liu _et al._ (2019); Li _et al._ (2019); Sandvik and Zhao
(2020) shows some puzzling features, including unconventional finite-size
behavior Sandvik (2007); Nahum _et al._ (2015b); Shao _et al._ (2016) and
measured critical exponents that violate bounds from conformal bootstrap
Nakayama and Ohtsuki (2016); Poland _et al._ (2019). One plausible
explanation is that the DQCP is pseudo-critical, i.e., it is not a truly
continuous phase transition, but its correlation length is very large. A
universal mechanism for such pseudo-criticality based on the notion of complex
fixed points Wang _et al._ (2017); Gorbenko _et al._ (2018) have been
proposed. In terms of the WZW sigma model Eq. (1), this means that the
hypothesized strong-coupling fixed point does not really exist and the theory
flows all the way to the weakly coupled, first order transition regime.
However, there is a region, around some nontrivial coupling strength $g^{*}$,
in which the RG flow is slow (also known as “walking”Kaplan _et al._ (2009)).
As a result the system behaves almost like a critical point up to some large
length scale. A theory for the walking behavior in the sigma model has been
put forward in Refs. Ma and Wang (2020); Nahum (2020).
### III.2 U(1) Dirac spin liquid
The $U(1)$ DSL was introduced as a critical quantum liquid that can emerge in
certain spin systems Affleck and Marston (1988); Wen and Lee (1996), and its
contemporary standard model is formulated in terms of 4 flavors of gapless
Dirac fermions minimally coupled to a dynamical $U(1)$ gauge field, with the
Lagrangian
$\displaystyle\mathcal{L}=\sum_{i=1}^{4}\bar{\psi}_{i}i\not{D}_{a}\psi_{i}+\frac{1}{4e^{2}}f_{\mu\nu}f^{\mu\nu}$
(4)
where $\not{D}_{a}$ is the covariant derivative of the Dirac fermions, $\psi$,
which are coupled to the dynamical $U(1)$ gauge field, $a$, whose field
strength is $f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$. The
Dirac fermion $\psi$ is not a local (gauge invariant) excitation here. Naively
the simplest local operators are fermion biliners like
$\bar{\psi}_{i}\psi_{j}$. It turns out that the most important local operators
are the monopole operators Borokhov _et al._ (2002); Dyer _et al._ (2013) –
these are operators that insert $U(1)$ gauge flux, in units of $2\pi$, into
the system.
The symmetries of the DSL are discussed in detail in Refs. Borokhov _et al._
(2002); Dyer _et al._ (2013); Song _et al._ (2020). In particular, it has an
$I^{(6)}=(SO(6)\times U(1)_{\rm top})/Z_{2}$ symmetry, and the purpose for the
notation $I^{(6)}$ will be clear later. The Dirac fermions transform under a
flavor $SU(4)$ which is the spinor group of the $SO(6)$. The fermion bilinears
$\bar{\psi}_{i}\psi_{j}$ form a singlet $\oplus$ an adjoint representation
under $SO(6)$. The $U(1)_{\rm top}$ corresponds to the conservation of gauge
flux, with conserved current
$j_{\mu}=\epsilon_{\mu\nu\lambda}\partial^{\nu}a^{\lambda}/(2\pi)$ (the
subscript “top” is due to the fact that this current conservation does not
rely on the detailed equations of motion and is therefore “topological”). By
definition only monopole operators are charged under the $U(1)_{\rm top}$. It
turns out Borokhov _et al._ (2002) that the most fundamental monopoles also
transform as a vector under the $SO(6)$. More concretely, the monopole can be
represented by a 6-component complex bosonic field $\Phi$, such that the
$SO(6)$ rotates the components of $\Phi$, and the $U(1)_{\rm top}$ acts by
multiplying $\Phi$ by a phase factor.
Besides $I^{(6)}$, this theory also enjoys discrete charge conjugation
$\mathcal{C}$, reflection $\mathcal{R}$, and time reversal $\mathcal{T}$
symmetries. To describe the actions of these discrete symmetries, it is useful
to imagine enlarging the $SO(6)$ and $U(1)_{\rm top}$ symmetries to $O(6)$ and
$O(2)$, respectively. Then it turns out Song _et al._ (2020) that the
improper $Z_{2}$ rotation of neither $O(6)$ nor $O(2)$ is a symmetry of the
DSL, but the $\mathcal{C}$ symmetry can be viewed as the combination of these
two improper $Z_{2}$ rotations. The $\mathcal{R}$ and $\mathcal{T}$ symmetries
can be viewed as a combination of spacetime orientation reversal and the
improper $Z_{2}$ rotation of either $O(6)$ or $O(2)$ (but not both).
The $\Phi$ operators are the most fundamental local operators in the theory,
in the sense that any other local operator can be built up using the $\Phi$’s.
Let us look at some examples. The $SU(4)$-singlet mass operator
$\bar{\psi}\psi$ is identified as
$\bar{\psi}_{i}\psi_{i}\sim
i\epsilon^{abcdef}(\Phi_{a}^{\dagger}\Phi_{b}-\Phi_{a}\Phi_{b}^{\dagger})(\Phi_{c}^{\dagger}\Phi_{d}-\Phi_{c}\Phi_{d}^{\dagger})(\Phi_{e}^{\dagger}\Phi_{f}-\Phi_{e}\Phi_{f}^{\dagger})$
where $a,b,c,d,e,f=1,2,3,4,5,6$. The $SU(4)$-adjoint mass operator (i.e.,
$\bar{\psi}_{i}\psi_{j}-\bar{\psi}_{k}\psi_{k}\delta_{ij}/4$) is identified as
the rank-2 antisymmetric tensor of $\Phi$ that is neutral under the $U(1)_{\rm
top}$, i.e., $i(\Phi_{a}^{\dagger}\Phi_{b}-\Phi_{b}^{\dagger}\Phi_{a})$. One
can construct more of such identifications of operators. Here two operators
are identified if they transform identically under all global symmetries
(including both continuous and discrete symmetries).
The quantum anomalies of the $U(1)$ DSL have been partly analyzed in Ref. Song
_et al._ (2020) and more recently in Ref. Calvera and Wang (2021), and we will
study them further in this paper.
The following facts about the nearby phases of the $U(1)$ DSL will be
extremely useful for our later developments (see Refs. Wang _et al._ (2017);
Song _et al._ (2020, 2019) for details).
1. 1.
By condensing one component of $\Phi$ in the $U(1)$ DSL, the resulting state
has the same symmetries and anomalies as the DQCP. It may even be possible
that the theory indeed flows to DQCP once the monopole perturbation is turned
on.
2. 2.
Because of the above, just like the DQCP, the $U(1)$ DSL also enjoys symmetry-
enforced gaplessness. One way to gap it out is to turn on an $SU(4)$-singlet
mass of the Dirac fermions, which will drive the system into a semion (or
anti-semion) topological order that breaks the orientation-reversal
symmetries.
3. 3.
By turning on a proper $SU(4)$-adjoint mass of the Dirac fermions, a
condensate of $\Phi$ will be automatically induced, such that the remaining
continuous symmetry of the system is $(SO(4)\times SO(2))/Z_{2}$, where the
$SO(4)\subset SO(6)\subset I^{(6)}$ acts only on 4 components of $\Phi$, and
the $SO(2)$ is a combination of $U(1)_{\rm top}$ and the $SO(2)\subset
SO(6)\subset I^{(6)}$ acting on the other 2 components of $\Phi$. There are
also remaining discrete $\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$
symmetries. The anomalies are completely removed in this case. For $U(1)$ DSLs
realized on lattice spin systems, this “chiral symmetry breaking” is the
mechanism for realizing conventional Landau symmetry-breaking orders from the
DSL – examples include the coplanar ($120^{\circ}$) magnetic orders on
triangular and Kagome lattices and various VBS orders Song _et al._ (2019).
The $U(1)$ DSL is likely to be realized in spin-1/2 Heisenberg magnets on
kagome and triangular lattices Ran _et al._ (2007); Iqbal _et al._ (2016);
He _et al._ (2017); Hu _et al._ (2019). Furthermore, lattice Monte Carlo
simulations support that the gauge theory Eq. (4) indeed flows to a CFT
Karthik and Narayanan (2016a, b).
## IV Stiefel liquids: generality
In this section we introduce the general theory of SLs and discuss some of
their interesting properties. Recall that each Stiefel liquid will be labeled
by two integers $(N,k)$, with $N\geqslant 5$ and $k\neq 0$. We will denote a
SL corresponding to $(N,k)$ by SL(N,k). Since we will mostly focus on the case
with $k=1$, we will also use the shorthand SL(N) to denote SL(N,k=1).
### IV.1 Wess-Zumino-Witten sigma model on Stiefel manifold $SO(N)/SO(4)$
The DOF of SL(N,k) is characterized by an $N$-by-$(N-4)$ real matrix, denoted
by $n$, such that the columns of $n$ are orthonormal, i.e., $n^{T}n=I_{N-4}$,
with $I_{N-4}$ the $(N-4)$-dimensional identity matrix. In mathematical terms,
this matrix $n$ defines an $(N-4)$-frame in the $N$-dimensional Euclidean
space. These $(N-4)$-frames live in a manifold $V_{N}\equiv SO(N)/SO(4)$,
known as a Stiefel manifold Nakahara (2003). Taking the Stiefel manifold as
the target space, a NLSM with the following action can be defined in any
dimension:
$\displaystyle S_{0}[n]=\frac{1}{2g}\int d^{d+1}x{\rm
Tr}(\partial_{\mu}n^{T}\partial^{\mu}n)$ (5)
where the $n$ in the square braket indicates the dependence of the action on
the configuration of $n$.
It is known that the homotopy groups of $V_{N}$ with any $N\geqslant 5$
satisfy that $\pi_{n}(V_{N})=0$ for $n<4$, and $\pi_{4}(V_{N})=\mathbb{Z}$, so
a WZW term based on a closed 4-form on $V_{N}$ can be defined for any
$N\geqslant 5$ in three spacetime dimensions Wess and Zumino (1971); Witten
(1983); Hull and Spence (1991). To define this WZW term, we will first add one
more dimension to the physical spacetime and extend the matrix $n$ into this
extra dimension. Denote the coordinate of the extra dimension by $u$, and the
extended matrix by $n^{e}$, such that $n^{e}(x,y,t,u=0)=n(x,y,t)$ and
$n^{e}(x,y,t,u=1)=n^{r}$, with $n^{r}$ a fixed reference matrix with entries
$(n^{r})_{ji}=\delta_{ji}$, where $(\cdot)_{ji}$ represents the entry in the
$j$th row and $i$th column of the relevant matrix. For notational brevity, in
the following we will drop the superscript “e” in the extended matrix and
simply write it as $n$, and the meaning of the matrix $n$ should be clear from
the context.
To the best of our knowledge, the expression for such a WZW term or closed
4-form on $V_{N}$ with $N\geqslant 6$ is unavailable in the previous
literature. We propose that the WZW term on $V_{N}$ is given by the following
(real-time) action:
$\displaystyle S_{{\rm
WZW}}^{(N)}[n]=\frac{2\pi}{\Omega_{4}}\int_{0}^{1}du\int
d^{3}x\sum_{i,i^{\prime}=1}^{N-4}\det\left(\tilde{n}_{(ii^{\prime})}\right)$
(6)
where the $N$-by-$N$ matrix $\tilde{n}_{(ii^{\prime})}$ is given by
$\displaystyle\tilde{n}_{(ii^{\prime})}=(n,\partial_{x}n_{i},\partial_{y}n_{i},\partial_{t}n_{i^{\prime}},\partial_{u}n_{i^{\prime}})$
(7)
where $n_{i}$ represents the $i$th column of $n$ (the repeated indices $i$ and
$i^{\prime}$ are not summed over in the right hand side of Eq. (7)). That is,
the first $N-4$ columns of $\tilde{n}_{ii^{\prime}}$ are just $n$, and its
last 4 columns are derivatives of the columns of $n$ arranged in the above
way. More explicitly,
$\displaystyle\det(\tilde{n}_{(ii^{\prime})})=\frac{1}{(N-4)!}\epsilon^{i_{1}i_{2}\cdots
i_{N-4}}\epsilon^{j_{1}j_{2}\cdots j_{N}}n_{j_{1}i_{1}}n_{j_{2}i_{2}}\cdots
n_{j_{N-4}i_{N-4}}\partial_{x}n_{j_{N-3}i}\partial_{y}n_{j_{N-2}i}\partial_{t}n_{j_{N-1}i^{\prime}}\partial_{u}n_{j_{N}i^{\prime}}$
(8)
where the $\epsilon$’s are the fully anti-symmetric symbols with rank $N-4$
and $N$, respectively. More comments on the mathematical aspects of this
action are given in Appendix A.
Taken together, the effective action of SL(N) is given by
$\displaystyle S^{(N)}[n]=S_{0}[n]+S_{{\rm WZW}}^{(N)}[n]$ (9)
The effective action of SL(N,k) is the level-$k$ generalization of Eq. (9):
$\displaystyle S^{(N,k)}=S_{0}+k\cdot S^{(N)}_{\rm WZW}.$ (10)
We remark that the $(2+1)$-d WZW-NLSM is non-renormalizable at strong
couplings, so these theories should be defined with an explicit UV
regularization. However, a symmetry-preserving local UV regularization should
not affect the quantum anomalies of the theory. As for the IR dynamics of the
theory, strictly speaking, it depends on the specific UV regularization, which
is similar to the situation where a quantum phase or phase transition is
described by a lattice Hamiltonian. In Sec. IV.4, we will argue that there
should exist UV regularizations under which $S^{(N,k)}$ flows to a conformally
invariant fixed point under RG if $N$ is larger than a $k$-dependent critical
value, $N_{c}(k)$, and thus describes a critical quantum liquid. If
$N<N_{c}(k)$, $S^{(N,k)}$ does not flow to a nontrivial CFT; instead, its most
likely fate is to flow to a Goldstone phase. In general, $N_{c}(k)$ increases
with $k$, and we will argue that $N_{c}(1)<6$.
Notice when $N=5$, $n$ is just a column vector that can be identified as the
vector $\bm{n}$ in Sec. III.1, and Eq. (9) is precisely Eq. (1). So SL(5) is
precisely the DQCP. Since the DQCP, or SL(5), can be described by
(renormalizable) gauge theories Wang _et al._ (2017), it is natural to ask if
other SLs can also be reformulated in terms of a renormalizable field theory,
such as a gauge theory. In appendix B, we provide an alternative description
of SL(5,k) in terms of a $USp(2k)$ gauge theory333Refs. Lee and Sachdev (2015)
and Ippoliti _et al._ (2018) argue that SL(5,k) with $k=1,2$ can also arise
in certain fermionic systems, such as monolayer and bilayer graphenes. We note
that the $USp(2k)$ gauge theories in Appendix. B are purely bosonic theories,
and all Stiefel liquids are also fundamentally bosonic theories and do not
need to involve fermions in any intrinsic way., and in Sec. V we provide a
$U(k)$-gauge-theoretic description of SL(6,k).
However, we cannot find any gauge-theoretic formulation for SL(N) with $N>6$.
In fact, due to their delicate symmetry structure (discussed below) and
intricate anomaly properties (discussed in Sec. VI), we conjecture that the
conformally invariant fixed points corresponding to SL(N>6) are non-
Lagrangian, i.e., they have no description in terms of a weakly-coupled
renormalizable continuum Lagrangian at any scale. In Sec. VIII, we present
more detailed arguments supporting this conjecture. As for SL(N,k) with $N>6$
and $k>1$, it may also be non-Lagrangian if it flows to a CFT.
Below we discuss the basic physical properties of the SLs in more detail.
### IV.2 Symmetries
In addition to the Poincaré symmetry, the actions in Eqs. (9) and (10) are
invariant under an $SO(N)$ transformation, which acts as:
$\displaystyle n\rightarrow Ln,L\in SO(N),$ (11)
and another $SO(N-4)$ transformation, which acts as
$\displaystyle n\rightarrow nR,R\in SO(N-4).$ (12)
Notice that for even $N$, the two $\mathbb{Z}_{2}$ centers $L=-I_{N}$ and
$R=-I_{N-4}$ act identically. So $S^{(N)}$ and $S^{(N,k)}$ have a continuous
symmetry group $I^{(N)}$, where $I^{(N)}=(SO(N)\times SO(N-4))/Z_{2}$ for even
$N$ and $I^{(N)}=SO(N)\times SO(N-4)$ for odd $N$. Recall that $I^{(5)}$ and
$I^{(6)}$ were already introduced in Sec. III.
Besides this continuous symmetry, $S^{(N)}$ and $S^{(N,k)}$ also have discrete
charge conjugation, reflection, and time reversal symmetries, i.e.,
$\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$. These symmetries can be
combined with elements of $I^{(N)}$ to be redefined, and we will utilize this
freedom to redefine these symmetries whenever useful later in the paper.
A particular implementation of these discrete symmetries for $N\geqslant 6$ is
$\displaystyle\begin{split}&\mathcal{C}:n_{ji}\rightarrow(-1)^{f_{ji}}n_{ji}\\\
&\mathcal{R}:n_{ji}\rightarrow\left\\{\begin{array}[]{lr}n_{ji},&j\leqslant
N-1\\\ -n_{ji},&j=N\end{array}\right.\\\
&\mathcal{T}:n_{ji}\rightarrow\left\\{\begin{array}[]{lr}n_{ji},&j\leqslant
N-1\\\ -n_{ji},&j=N\end{array}\right.\end{split}$ (13)
with $f_{ji}=1$ if $(j=N\ \&\ i<N-4)$ or $(j<N\ \&\ i=N-4)$, and $f_{ji}=0$
otherwise. Notice that $\mathcal{R}$ and $\mathcal{T}$ also need to flip a
spatial or temporal coordinate, respectively.
Another useful way to characterize these symmetries for $N\geqslant 6$ is to
imagine enlarging the $SO(N)$ and $SO(N-4)$ in $I^{(N)}$ to $O(N)$ and
$O(N-4)$, respectively. Then the improper rotation of neither the $O(N)$ nor
the $O(N-4)$ is a symmetry due to the WZW term, but the combination of these
two improper rotations is the $\mathcal{C}$ symmetry. Also, $\mathcal{R}$ and
$\mathcal{T}$ can be viewed as an improper rotation of either $O(N)$ or
$O(N-4)$ combined with a reversal of the appropriate spacetime coordinate.
When combined with the continuous symmetries, the full symmetry group (besides
the Poincaré) can be written as
$\displaystyle(SO(N)\times
SO(N-4))\rtimes(\mathbb{Z}_{2}^{\mathcal{C}}\times\mathbb{Z}_{2}^{\mathcal{T}}),$
$\displaystyle\hskip 5.0ptN=2n+1,$ $\displaystyle\left(\frac{SO(N)\times
SO(N-4)}{\mathbb{Z}_{2}}\right)\rtimes(\mathbb{Z}_{2}^{\mathcal{C}}\times\mathbb{Z}_{2}^{\mathcal{T}}),$
$\displaystyle\hskip 5.0ptN=2n,$ (14)
where $\mathbb{Z}_{2}^{\mathcal{C}}$ and $\mathbb{Z}_{2}^{\mathcal{T}}$ are
generated by $\mathcal{C}$ and $\mathcal{T}$, respectively. The semi-direct
product $\rtimes$ comes from nontrivial relations like Eq. (13). We do not
need to list $\mathbb{Z}_{2}^{\mathcal{R}}$ above since it is related to
$\mathbb{Z}_{2}^{\mathcal{T}}$ through a Lorentz rotation.
For $N=5$, the matrix $n$ contains only a single column, and we can suppress
its column index and denote it by $n_{j}$, with $j=1,2,\cdots,5$. In this
case, the above $\mathcal{C}$ symmetry does not exist444However, some elements
of the $I^{(5)}=SO(5)$ group may be interpreted as a charge conjugation
symmetry in certain gauge-theoretic formulation of this theory Wang _et al._
(2017)., and we take the actions of the $\mathcal{R}$ and $\mathcal{T}$
symmetries to be
$\displaystyle\begin{split}&\mathcal{R}:n_{1,2,3,4}\rightarrow
n_{1,2,3,4},n_{5}\rightarrow-n_{5}\\\ &\mathcal{T}:n_{1,2,3,4}\rightarrow
n_{1,2,3,4},n_{5}\rightarrow-n_{5}\end{split}$ (15)
which is analogous to the case with $N\geqslant 6$.
### IV.3 Cascade structure of the SLs
Now we discuss the relations between different SLs.
As is common for WZW theories, SL(N,k) with $k>1$ can be viewed as $k$ copies
of SL(N) that have strong “ferromagnetic” couplings between them. More
precisely, the action of $k$ copies of decoupled SL(N) is
$\sum_{i=1}^{k}S^{(N)}[n_{i}]$. A strong ferromagnetic coupling, $-\sum_{i\neq
j}g_{ij}\int d^{3}x{\rm Tr}(n_{i}^{T}n_{j})$ with $g_{ij}>0$, has the tendency
of identifying different $n_{i}$’s as a single matrix, $n$. Focusing on the
low-energy sector that contains only $n$, the total action takes the form of
Eq. (10). That is, we can get SL(N,k) by appropriately coupling $k$ copies of
SL(N). Notice that this coupling does not break the symmetries discussed in
Sec. IV.2, as expected.
SL(N,k) and SL(N,-k) almost have identical properties, because one of them can
be obtained from the other by an improper $Z_{2}$ operation of either the
$O(N)$ or $O(N-4)$. But they have opposite quantum anomalies, since a composed
system of SL(N,k) and SL(N,-k) can be turned into a trivial state by a strong
ferromagnetic coupling. Because of the similarity between SL(N,±k), in this
paper we will mostly focus on the case with $k>0$.
Next, we remark on an interesting and important specific property of the SLs:
the cascade structure.
Suppose we start with SL(N,k) described by $S^{(N,k)}$ with $N\geqslant 6$,
and fix, say, $n_{11}=n_{22}=\cdots=n_{mm}=1$ with $m\leqslant N-5$, while
allowing the other components of $n$ to fluctuate under the orthonormal
constraint, $n^{T}n=I_{N-4}$. Now fluctuations of the entries in the first $m$
rows and the first $m$ columns of $n$ are frozen, while fluctuations of the
other entries remain at low energies. The $I^{(N)}$ symmetry of $S^{(N,k)}$ is
explicitly broken to $I^{(N,m)}$, where $I^{(N,m)}=[SO(N-m)\times
SO(N-4-m)\times SO(m)]/Z_{2}$ if both $N$ and $m$ are even, and
$I^{(N,m)}=SO(N-m)\times SO(N-4-m)\times SO(m)$ if at least one of $N$ and $m$
is odd. Here the $SO(N-m)\subset SO(N)\subseteq I^{(N)}$ acts on the last
$N-m$ rows of $n$, $SO(N-4-m)\subset SO(N-4)\subset I^{(N)}$ acts on the last
$N-4-m$ columns of $n$, and $SO(m)$ is a combination of the $SO(m)\subset
SO(N)\subseteq I^{(N)}$ acting on the first $m$ rows of $n$ and the
$SO(m)\subset SO(N-4)\subset I^{(N)}$ acting on the first $m$ columns of $n$.
To focus on the low-energy fluctuations, we can define a reduced
$(N-m)$-by-$(N-4-m)$ matrix $n_{\rm red}$, by removing the first $m$ rows and
first $m$ columns of $n$. This reduced matrix still satisfies an orthonormal
condition, $n_{\rm red}^{T}n_{\rm red}=I_{N-4-m}$. In addition,
$S^{(N,k)}[n]=S^{(N-m,k)}[n_{\rm red}]$. That is, the low-energy dynamics of
this symmetry-broken descendent of SL(N,k) is effectively identical to that of
SL(N-m,k) (see Fig. 2). We remark that within the low-energy sector, the
remaining continuous symmetry is $I^{(N-m)}$, which is generally smaller than
$I^{(N,m)}$. Physically, this is because some elements of $I^{(N,m)}$ only act
on the frozen DOFs of this symmetry-broken descendent, but not on its low-
energy DOFs, as discussed above. Also notice that the discrete $\mathcal{R}$
and $\mathcal{T}$ symmetries defined in Eqs. (13) and (15) are preserved for
all $m\leqslant N-5$, and the $\mathcal{C}$ symmetry defined in Eq. (13) is
preserved for $m<N-5$ and broken when $m=N-5$.
Therefore, SL(N,k) for each $k$ form a cascade of theories, where the ones
with a smaller $N$ can be obtained from the ones with a larger $N$ by
appropriately perturbing the latter and focusing on the resulting low-energy
sector.
If $m=N-4$ in the above, then all entries of $n$ are fixed, the remaining
continuous symmetry is $(SO(4)\times SO(N-4))/Z_{2}$ for even $N$ and
$SO(4)\times SO(N-4)$ for odd $N$, there is no longer any low-energy DOF in
the system, and the resulting state is no longer a SL. In particular, the WZW
term is now completely removed. Note that the $\mathcal{R}$ and $\mathcal{T}$
symmetries in Eq. (13) are still preserved, but the $\mathcal{C}$ symmetry
defined there is broken. However, the following $\mathcal{C}^{\prime}$
symmetry, which is a combination of that $\mathcal{C}$ and an element in
$I^{(N)}$, is preserved:
$\displaystyle\mathcal{C}^{\prime}:n_{ji}\rightarrow(-1)^{f^{\prime}_{ji}}n_{ji}$
(16)
where $f^{\prime}_{ji}=1$ if $(j=1\ \&\ i>1)$ or $(j>1\ \&\ i=1)$, and
$f^{\prime}_{ji}=0$ otherwise.
Since the WZW term is expected to capture the quantum anomalies of the SL,
from the above cascade structure we see that the anomalies of a SL with a
smaller $N$ should be contained in the anomalies of a SL with a larger $N$.
Furthermore, by explicitly breaking the symmetries of SL(N,k) via tuning up
$m$ from 0 to $N-4$ as above, its anomalies are gradually removed, and also
localized to the low-energy sector defined by $n_{\rm red}$. For example, if
the above procedure of symmetry breaking is applied to SL(N) with $m=N-5$, the
anomaly of the system is reduced to that of SL(5), given by Eq. (3). If this
procedure is applied with $m=N-4$ instead, the anomaly of the system is
completely removed. As another application, this cascade structure also
implies that SL(N) with any $N\geqslant 6$ have symmetry-enforced gaplessness,
similar to that of SL(5), which is reviewed in Sec. III.
The cascade structure among the SLs remarked here will be repeatedly used
later, and it plays an important role in simplifying the discussion of the
anomaly.
Figure 2: The cascade structure of Stiefel liquids.
### IV.4 Possible fixed points at strong coupling
Now we argue that for sufficiently large $N$ and small $k$, with certain mild
physical assumptions, the action $S^{(N,k)}$ in Eq. (10) can flow to a
conformally invariant fixed point under RG, so that these SLs can be a
critical quantum liquid.
We will explore $S^{(N,k)}$ piece by piece, and we start by completely
ignoring the WZW term in Eq. (10) and only considering $S_{0}$. When $N$ is
suffiently large, $S_{0}$ is expected to be able to describe an order-disorder
transition of the matrix $n$.
One way to see this is to look at the saddle point corresponding to the
$S_{0}$. The constraint $n^{T}n=I_{N-4}$ can be eliminated by introducing an
$(N-4)$-by-$(N-4)$-matrix-valued Lagrangian multiplier $\alpha$, such that the
(Euclidean) path integral becomes
$\displaystyle\begin{split}\mathcal{Z}_{0}=&\int\mathcal{D}n\mathcal{D}\alpha\exp\Big{\\{}\int
d^{3}x\left(-\frac{1}{2g}\right)\cdot\big{[}\partial_{\mu}n_{ji}\partial^{\mu}n_{ji}\\\
&\qquad\qquad\qquad\quad-i\alpha_{ij}(n_{kj}n_{ki}-\delta_{ij})\big{]}\Big{\\}}\\\
=&\int\mathcal{D}n\mathcal{D}\alpha\exp\left\\{\int\frac{d^{3}x}{2g}\left[n_{kj}\left(\delta_{ij}\square+i\alpha_{ij}\right)n_{ki}\right]\right\\}\\\
&\qquad\qquad\qquad\cdot\exp\left(-i\int\frac{d^{3}x}{2g}{\rm
Tr}(\alpha)\right)\\\ =&\int\mathcal{D}\alpha\exp\left[-\frac{N}{2}{\rm
tr}\ln\left(\delta_{ij}\square+i\alpha_{ij}\right)\right]\\\
&\qquad\quad\cdot\exp\left(-i\int d^{3}x\frac{{\rm
Tr}(\alpha)}{2g}\right)\end{split}$ (17)
where $\square\equiv\partial_{\mu}\partial^{\mu}$. The Lagrangian multiplier
$\alpha$ is introduced in the first step. In the second step we integrate by
parts. In the last step we integrate out $n$ and ignore a constant
multiplicative factor to the path integral, where the factor of $N$ appears
because the index $k$ in the middle line runs from 1 to $N$, and trace is
taken over both the matrix indices and the spacetime coordinates.
When $N\gg 1$, we expect that the remaining path integral will be dominated by
the configuration of $\alpha$ that satisfies the saddle-point equation.
Assuming an ansatz to the saddle-point equation with
$\alpha_{ij}=i\Delta^{2}\delta_{ij}$, where $\Delta$ is a number (not a
matrix), the saddle-point equation becomes
$\displaystyle\frac{1}{2g}=\frac{N}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{k^{2}+\Delta^{2}}$
(18)
Taking the limit $N\rightarrow\infty$ while $gN$ is fixed, we find that the
above equation has a solution with nonzero $\Delta$ if $g$ is larger than
certain critical value, $g_{0}\sim\mathcal{O}(1/(N\Lambda))$, where $\Lambda$
is a UV cutoff. In this case, since $\Delta\neq 0$, the matrix $n$ acquires a
gap, and the system is in the disordered phase. On the other hand, when
$g<g_{0}$, the system is in the ordered phase. Note that the structure of this
saddle-point equation is the same as the usual $O(N)$ vector NLSM Peskin and
Schroeder (1995), despite that $n$ is a matrix.
The above results suggest that the beta function for $S_{0}$ at sufficiently
large $N$ is
$\displaystyle\beta(\tilde{g})=-\tilde{g}+\beta_{0}(\tilde{g})$ (19)
where $\tilde{g}\equiv g\Lambda$ is the dimensionless coupling. The first term
comes from the engineering dimension of $g$, and the second term,
$\beta_{0}(\tilde{g})$, represents loop corrections. The precise form of
$\beta_{0}(\tilde{g})$ is hard to obtain even at large $N$. In fact, as
$N\to\infty$ this model approaches the $SO(N)$ sigma model, for which even the
leading correction to the beta function requires summing over all planar
diagrams (see, for example, Ref. Wegner (1989) for the $3$-loop result). For
us, what is important is that $\beta(\tilde{g})=0$ at
$\tilde{g}=\tilde{g}_{0}\equiv g_{0}\Lambda\sim\mathcal{O}(1/N)$.
Physically, the above discussion suggests that for the NLSM defined by
$S_{0}$, there is an attractive fixed point at $\tilde{g}=0$, corresponding to
the ordered phase where the symmetry is broken spontaneously. There is also a
repulsive fixed point at $\tilde{g}=\tilde{g}_{0}\sim\mathcal{O}(1/N)$,
corresponding to the order-disorder transition.
Next, we include the WZW term and consider the full $S^{(N,k)}$. We will view
the WZW term as a perturbation to $S_{0}$, and it is expected to contribute a
term of the following form at the leading order to the beta function 555There
is a class of “leading-order contributions” at large $N$, each of the form
$-C_{V}k^{V}\tilde{g}^{2V+1}$, with $V\geqslant 2$ an integer and
$C_{V}\sim\mathcal{O}(1)$. Notice in the limit considered in the next
paragraph, i.e., $N\rightarrow\infty$, $k\rightarrow\infty$ and $k/N^{2}$
fixed, all these contributions are at $\mathcal{O}(1/N)$ if
$\tilde{g}\sim\mathcal{O}(1/N)$.:
$\displaystyle\delta\beta(\tilde{g})=-Ck^{2}\tilde{g}^{5}$ (20)
where $C$ is of order $1$. It is expected that $C>0$, because the WZW term
yields a phase factor in the path integral and induces destructive
interference of the paths, which tends to prevent $\tilde{g}$ from flowing to
a large value, just like the effect of Haldane’s phase in spin chains Haldane
(1983, 1983); Affleck and Haldane (1987). To further understand the effect of
this contribution to the beta function, let us first consider cases with
$k\sim O(1)$, which are the main focus of this paper. At large $N$, the term
Eq. (20) is negligible when $\tilde{g}\sim 1/N$, so it does not affect the
order-disorder transition significantly. The WZW could affect the nature of
the disordered state. For example, for odd $k$ we know that the disordered
state cannot be a trivially gapped phase because of the nontrivial ’t Hooft
anomaly – it has to be either gapless or spontaneously break some other
symmetry. It is hard to tell exactly what happens in the disordered regime
since we no longer have analytic control.
It then turns out to be useful to consider a different limit with
$N\rightarrow\infty$, $k\rightarrow\infty$ and $k/N^{2}=\alpha$ with
$\alpha\sim O(1)$ fixed 666Instead of looking at this specific limit, one may
also think in the following more general and heuristic way. When
$k\sim\mathcal{O}(1)$, there are the attractive fixed point corresponding to
the Goldstone phase, and the repulsive fixed point corresponding to the order-
disorder transition. When $k$ is very large (for a fixed $N$), there should
only be a single fixed point, i.e., the attractive one corresponding to the
Goldstone phase. As $k$ increases from $\mathcal{O}(1)$ to a large value, for
the repulsive fixed point to disappear, it should collide with another fixed
point and annihilate, schematically as shown in Fig. 3 (a). This observation
implies the existence of another interacting attractive fixed point for small
enough $k$. According to the forms of Eqs. (19) and (20), the value of $k$
below which the interacting attractive fixed point exists scales with $N$ as
$N^{2}$, and the coupling $\tilde{g}$ for this fixed point is
$\tilde{g}\sim\mathcal{O}(1/N)$. This additional interacting attractive fixed
point corresponds to the critical Stiefel liquid.. Fig. 3 (a) illustrates
several different scenarios. If $\alpha$ is smaller than some critical value
$\alpha_{c}$, then the repulsive fixed point at $\tilde{g}=\tilde{g}_{0}$ will
be shifted to a larger value, and another attractive fixed point at
$\tilde{g}\sim\mathcal{O}(1/N)$ emerges777One may wonder at this new fixed
point if there will be relevant perturbations not controlled by $g$. However,
if the order-disorder transition has only one relevant singlet operator (i.e.,
the tuning of $g$), as naturally expected, this new fixed point should have no
relevant perturbation once the full symmetry of $S^{(N,k)}$ is preserved,
i.e., this fixed point is indeed attractive. Otherwise, there will be
additional fixed points besides the ones in Fig. 3 (a), which appears to be
unnatural.. Because this attractive fixed point is still weakly coupled at
large $N$, we expect it to describe a symmetry-preserving critical state,
rather than a gapped or Goldstone state. The repulsive fixed point then
describes the transition from the critical phase to the symmetry breaking
phase. On the other hand, if $\alpha>\alpha_{c}$, the attractive and repulsive
fixed points will collide and annihilate with each other, and the only fixed
point left will be the weakly-coupled one with spontaneously broken symmetry.
In other words, with a smaller $k$, there is a stronger tendency for the WZW-
NLSM to have an attractive fixed point at finite coupling. Although this
conclusion is drawn in the $k/N^{2}\sim O(1)$ regime, it seems natural to
assume that this trend is qualitatively true even for $k\sim O(1)$. Namely, we
expect that $S^{(N,k)}$ can describe a critical quantum liquid even for $k=1$,
if $N\gg 1$.888Note that this nicely corroborates our results that $S^{(5,k)}$
and $S^{(6,k)}$ have dual descriptions based on $USp(2k)$ and $U(k)$ gauge
theories, since these gauge theories are generally expected to have stronger
tendency to be critical (unstable) for small (large) $k$. More generally, we
propose that for each $k\neq 0$, there exists an integer $N_{c}(k)$ that
increases as $k$ increases, such that when $N\geqslant N_{c}(k)$, $S^{(N,k)}$
can flow to a conformally invariant attractive fixed point, corresponding to
SL(N,k). The precise form of $N_{c}(k)$ is unknown at this stage (see Fig. 3
(b)).
We can gain more confidence about the above arguments, or conjectures, from
WZW sigma models on Grassmannian manifolds such as $U(2N)/U(N)\times U(N)$.
Our arguments can be equally applied in those cases and we conclude that a
strong-coupling attractive fixed point should occur for sufficiently large $N$
(see also Ref. Bi _et al._ (2016)). Unlike the Stiefel WZW models, however,
the Grassmannian WZW models are naturally related to various non-Abelian gauge
theories (QCD) in $(2+1)$-d Komargodski and Seiberg (2018), which we review in
Appendix D. The rank $N$ in the WZW model corresponds to flavor number in
those QCD theories, and it is well known that a large-flavor QCD does flow to
a nontrivial fixed point in $(2+1)$-d. This serves as a nontrivial check of
our arguments on the existence of strong coupling fixed points.
Also notice that if $\alpha$ happens to be barely above $\alpha_{c}$ (as in
Fig. 3), the RG flow will be slow near the fixed-points-collision region. This
gives a mechanism for the “walking” of the coupling constant Kaplan _et al._
(2009) and pseudo-critical behavior Wang _et al._ (2017); Gorbenko _et al._
(2018) of the system.
Recall that SL(5,1) is just the DQCP. Later we will argue that SL(6,1) is in
fact the $U(1)$ DSL. As reviewed in Sec. III, these two states are likely
pseudo-critical and critical, respectively. This implies that for $k=1$,
$5<N_{c}<6$, so we propose that $S^{(N,k)}$ for all $N\geqslant 6$ and $k=1$
can still flow to a conformally invariant fixed point and describe a critical
state999Note that this proposal is consistent with the symmetry-enforced
gaplessness of these theories, as required by their anomalies. See Sec. IV.3
for more details..
Figure 3: Fixed point structure and schematic phase diagram of SL(N,k). In
(a), different fixed point structures for different values of $\alpha$ are
shown. There is always an attractive fixed point, represented by the red
circle, which corresponds to the symmetry-broken state. The structure of the
other fixed points depends on the relation between $\alpha$ and
$\alpha_{c}\sim\mathcal{O}(1)$. The precise value of $\alpha_{c}$ depends on
$\beta_{0}$ and $C$. If $\alpha<\alpha_{c}$, there are two other fixed points,
a repulsive one represented by the blue circle, corresponding to an order-
disorder transition, and an attractive one, represented by the yellow circle,
corresponding to a stable critical quantum liquid, i.e., the SL. As $\alpha$
increases and approaches $\alpha_{c}$, the blue and yellow fixed points
approach each other and collide when $\alpha=\alpha_{c}$. When
$\alpha>\alpha_{c}$, the original blue and yellow fixed points disappear
(become complex fixed points Wang _et al._ (2017); Gorbenko _et al._
(2018)). We would like to take $\alpha$ below $\alpha_{c}$. In (b), a
schematic phase diagram of SL(N,k) is shown. For the $(N,k)$ in the critical
regime, it is possible to tune the parameters of the system to yield a
critical quantum liquid, while in the symmetry-breaking regime this is not
possible. The yellow star represents $(N,k)=(6,1)$, i.e., the $U(1)$ DSL, and
the black star represents $(N,k)=(5,1)$, the DQCP. The precise boundary
between the critical and symmetry-breaking regimes is currently undetermined.
To summarize this section, we have proposed the existence of SLs, whose
effective field theories can be directly formulated in terms of local DOF,
given in Eqs. (9) and (10). These SLs have an interesting symmetry structure
discussed in Sec. IV.2, and they are interrelated with a special cascade
structure discussed in Sec. IV.3. We argue that SL(N) with $N\geqslant 6$ can
flow to a CFT fixed point under RG. Furthermore, we conjecture that SL(N) with
$N>6$ are all non-Lagrangian. Putting these compactly, SLs are cascades of
extraordinary critical quantum liquids.
## V $N=6$: Dirac spin liquids
According to the previous discussion, it is readily seen that the SL(N=5) is
special in many ways among all SLs. For example, its continuous symmetry
$I^{(5)}=SO(5)$, which is qualitatively different from $I^{(N)}$ with
$N\geqslant 6$, in the sense that the former has only one connected component,
while the latter has two. In this section, we focus on the more nontrivial
case, i.e., SL(N=6,k), and we argue that SL(N=6,k) has a dual description in
terms of $U(k)$ DSLs, i.e., 4 flavors of gapless Dirac fermions coupled to a
$U(k)$ gauge theory.
### V.1 Deriving $k=1$ WZW model from QED3
We begin with deriving the Stiefel WZW sigma model, $S^{(N=6)}$ in Eq. (9),
from QED3 with $N_{f}=4$. The derivation is similar in spirit to that in Ref.
Senthil and Fisher (2006), although somewhat more complicated in detail.
We start from the QED3 Lagrangian:
$\sum_{\alpha=1}^{4}\bar{\psi}_{\alpha}i\not{D}_{a}\psi_{\alpha}-\frac{1}{2\pi}A_{\rm
top}da$ (21)
where $a$ is the dynamical $U(1)$ gauge field and $\psi$ is the Dirac fermion.
Compared to Eq. (4), here the Maxwell term of $a$ is suppressed, because it is
unimportant for the current discussion. Also, the last term,
$-\frac{1}{2\pi}adA_{\rm
top}\equiv-\frac{1}{2\pi}\epsilon_{\mu\nu\lambda}A^{\mu}_{\rm
top}\partial^{\nu}a^{\lambda}$, is introduced to keep track of the conserved
flux of $a$ by introducing the probe $U(1)_{\rm top}$ gauge field $A_{\rm
top}$. The subscript “top” is due to the fact that the conservation of the
current corresponding to this $U(1)$ symmetry,
$j_{\mu}=\epsilon_{\mu\nu\lambda}\partial^{\nu}a^{\lambda}/(2\pi)$, does not
rely on the equations of motion of the theory. For later convenience, here we
introduce the following notation of the generators of the $SU(4)$ flavor
symmetry of this theory:
$\sigma_{ab}\equiv\frac{1}{2}\sigma_{a}\otimes\sigma_{b}$, with $a,\
b=0,1,2,3$ but $a$ and $b$ not simultaneously zero. Here $\sigma_{0}=I_{2}$
and $\sigma_{1,2,3}$ are the standard Pauli matrices.
We now consider dynamically breaking the $SU(4)$ flavor symmetry down to
$(SU(2)\times SU(2)\times U(1))/Z_{2}$, which is believed to be the most
likely symmetry breaking pattern for this theory Pisarski (1984); Vafa and
Witten (1984a, b); Polychronakos (1988); Pisarski (1991). This introduces an
order parameter $\mathcal{P}$, defined in the complex Grassmannian manifold
$U(4)/\left(U(2)\times U(2)\right)$, that couples to the Dirac fermions as an
$SU(4)$-adjoint mass:
$m\mathcal{P}_{\alpha\beta}\bar{\psi}_{\alpha}\psi_{\beta},$ (22)
where $m$ is the coupling strength, which physically means the magnitude of
the mass. Now we can formally integrate out the Dirac fermions and obtain an
effective theory in terms of the $\mathcal{P}$ and $a$ fields. One can expand
in $1/m$ and obtain a $U(4)/\left(U(2)\times U(2)\right)$ sigma model for the
$\mathcal{P}$ fields. As shown in Ref. Jian _et al._ (2018b), this sigma
model comes with a WZW term with coefficient $k=1$. Note that this WZW term is
well defined because the target Grassmannian manifold has $\pi_{4}=\mathbb{Z}$
and $\pi_{3}=0$.
However, this Grassmannian WZW theory is not the end of the story: the $U(1)$
gauge field is still present and we expect nontrivial couplings between the
gauge field and the $\mathcal{P}$ field. The most important coupling is
$a\cdot j^{Sk},$ (23)
where $j^{Sk}$ is the Skyrmion current of the $\mathcal{P}$ field. The
Skyrmion current is well defined because the target Grassmannian manifold has
$\pi_{2}=\mathbb{Z}$. The existence of this coupling is due to that the
elementary Skyrmion is a fermion that carries unit gauge charge, which is
nothing but the original Dirac fermion, $\psi$. To see this, let us build up a
simple type of Skyrmion by first considering a mass term that only couples to
the first two flavors of Dirac fermions, $\psi_{\alpha=1,2}$, i.e., a mass of
the form $\bar{\psi}N_{i}\sigma_{i}\otimes(I_{2}+\sigma_{3})\psi$, where
$i=1,2,3$ and hence $\bm{N}\in S^{2}$. This $\bm{N}$ field can then form a
standard Skyrmion configuration in space. Now this mass can be extended to an
allowed configuration for the $\mathcal{P}$ field, by adding a constant mass
for the other two Dirac fermions, say,
$\bar{\psi}\sigma_{3}\otimes(I_{2}-\sigma_{3})\psi$, which would not change
the topological properties of the skyrmion. However, it is well known that
this $\bm{N}$ field has a Hopf term with $\theta=\pi$ in its effective theory,
and the Skyrmion is a fermion with gauge charge $1$ Abanov and Wiegmann (2000,
2001). Since this is the minimum gauge charge of the theory, we conclude that
the elementary Skyrmion in $\mathcal{P}$ field also has gauge charge $1$ and
is a fermion, i.e., it is the original Dirac fermion $\psi$. This justifies
the coupling in Eq. (23).
So the sigma model should be written as
$\displaystyle\begin{split}S=S_{G0}[P]&+S_{\rm G-WZW}[\mathcal{P}]\\\ &+\int
d^{3}x\left(a\cdot j^{Sk}-\frac{1}{2\pi}A_{\rm top}da\right)\end{split}$ (24)
where $S_{G0}[P]$ is the NLSM on the Grassmannian manifold without any
topological term, and $S_{\rm G-WZW}[\mathcal{P}]$ is the WZW action on this
Grassmannian manifold. The precise expressions of $S_{G0}[P]$ and $S_{\rm
G-WZW}[P]$ are unimportant for our purpose.
Next, notice that the complex Grassmannian $U(4)/(U(2)\times U(2))$ is
equivalent to the real Grassmannian $SO(6)/(SO(4)\times SO(2))$, since
$SU(4)\sim SO(6)$, $SU(2)\times SU(2)\sim SO(4)$ and $U(n)\sim U(1)\times
SU(n)$ (all up to some discrete quotients, which do not change the conclusion
here). The real Grassmannian $SO(6)/(SO(4)\times SO(2))$ can be rewritten as
follows: introduce two orthonormal $SO(6)$ vectors $n_{1}$ and $n_{2}$, and
rewrite the matrix field
$\mathcal{P}=-2i\left(n_{1}^{T}T_{ab}n_{2}\right)\cdot\sigma_{ab}$, where
$T_{ab}$ is the $SO(6)$ generator that corresponds to $\sigma_{ab}$ (see
Appendix E). The fact that $T_{ab}^{T}=-T_{ab}$ means that this rewriting
introduces an $SO(2)$ gauge redundancy that rotates between $n_{1}$ and
$n_{2}$, i.e., there is a dynamical $SO(2)$ gauge field, denoted by $b$, that
couples to $n_{1}$ and $n_{2}$. Notice that $n=(n_{1},n_{2})$ lives on nothing
but the Stiefel manifold $SO(6)/SO(4)$. We have therefore rewritten the
Grassmannian sigma model, $S_{G0}[\mathcal{P}]$, in terms of an $SO(2)=U(1)$
gauge field–$b$, coupled to an order parameter that lives on the Stiefel
manifold $SO(6)/SO(4)$, i.e., $S_{0}[n,b]$, the NLSM in Eq. (5) with $n$
minimally coupled to $b$. So what do the Grassmannian WZW term and the
Skyrmion current become in this representation? Mathematically, our rewriting
of the Grassmannian in terms of an $SO(2)$ gauge theory coupled to a Stiefel
manifold corresponds to the fibration
$\displaystyle SO(2)\to\frac{SO(6)}{SO(4)}\to\frac{SO(6)}{SO(4)\times SO(2)}.$
The long exact sequence from this fibration leads to two isomorphisms:
$\displaystyle p:$ $\displaystyle\hskip
5.0pt\pi_{4}\left(\frac{SO(6)}{SO(4)\times
SO(2)}\right)\to\pi_{4}\left(\frac{SO(6)}{SO(4)}\right),$
$\displaystyle\beta:$ $\displaystyle\hskip
5.0pt\pi_{2}\left(\frac{SO(6)}{SO(4)\times SO(2)}\right)\to\pi_{1}(SO(2)).$
(25)
This means that any nontrivial winding in $\pi_{4}$ and $\pi_{2}$ of the
Grassmannian should be fully encoded in the corresponding homotopy groups of
the Stiefel and $SO(2)$, respectively. Since the Grassmannian WZW term comes
from $\pi_{4}\left(SO(6)/\left(SO(4)\times SO(2)\right)\right)$, it should
simply become the WZW term of the Stiefel manifold (which comes from
$\pi_{4}(SO(6)/SO(4))$), given by Eq. (6). The Grassmannian Skyrmion current
comes from $\pi_{2}\left(SO(6)/\left(SO(4)\times SO(2)\right)\right)$, so it
should simply become the flux current of the $SO(2)$ gauge theory (which comes
from $\pi_{1}(SO(2))$):
$j^{Sk}_{\mu}=\frac{1}{2\pi}\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{\lambda}.$
(26)
The complete theory in Eq. (24) can now be written as
$\displaystyle S=$ $\displaystyle S_{0}[n,b]+S^{(6)}_{\rm WZW}[n,b]$ (27)
$\displaystyle+\int d^{3}x\left(\frac{1}{2\pi}adb-\frac{1}{2\pi}A_{\rm
top}da\right).$
Now integrating out the $a$ gauge field, the $b$ gauge field will be set to
$b=A$. The only IR degrees of freedom left is the Stiefel field $n$ with the
action of WZW model at $k=1$. The $n$ fields couple to the probe gauge field
$A$ as charge-$1$ fields, which leads to the interpretation that they
correspond to the monopole operators in the original QED3. This completes our
derivation.
In passing, we mention that in Appendix G we also explicitly derive some
properties regarding the quantum anomalies of the $U(1)$ DSL and show that
they match with that of SL(6). This provides further evidence for the
equivalence between the $U(1)$ DSL and SL(6).
### V.2 General $k$: $U(k)$ QCD with $N_{f}=4$
The above derivation can be generalized quite readily to
$U(k)=\left(U(1)\times SU(k)\right)/\mathbb{Z}_{k}$ gauge theories with
$N_{f}=4$ fundamental Dirac fermions. Denote the $U(k)$ gauge field that is
minimally coupled to the Dirac fermions by $\mathbf{a}=a+\tilde{a}\mathbf{1}$,
where $a$ is an $SU(k)$ gauge field and $\tilde{a}$ is a $U(1)$ gauge field.
Note that now the minimal local monopole carries $2\pi$ flux of ${\rm
Tr}(\mathbf{a})=k\tilde{a}$, so the coupling of the theory to $A_{\rm top}$
takes the form $-\frac{1}{2\pi}A_{\rm top}d{\rm Tr}(\mathbf{a})$. Also notice
that the Dirac fermion carries charge $1/k$ under ${\rm Tr}(\mathbf{a})$.
We can now proceed with the same arguments as in the QED case. First we
introduce a mass operator on Grassmannian $SO(6)/(SO(4)\times SO(2))$, which
is a color singlet and $SU(4)$ adjoint. Then we integrate out all Dirac
fermions. This gives a Grassmannian sigma model with a WZW term, which is at
level $k$ because of the color multiplicity of the Dirac fermions. There is
also a Skyrmion term like Eq. (23) from each color, but with $a$ therein
replaced by ${\rm Tr}(\mathbf{a})$, and with a coefficient $1/k$, since the
Dirac fermions carry gauge charge $1/k$ under ${\rm Tr}(\mathbf{a})$. Summing
over all colors gives precisely ${\rm Tr}(\mathbf{a})\cdot j^{Sk}$. The
remaining $U(k)$ gauge field splits into an $SU(k)$ and $U(1)$ part. The
$SU(k)$ gauge field now does not couple to any IR degrees of freedom, so we
expect it to flow to strong coupling and eventually confine (and be gapped).
The $U(1)$ part can be analyzed in the same way as we did for QED. At the end
of the day we again obtain a Stiefel WZW sigma model, now at level $k$.
Therefore, we propose that SL(6,k) and a $U(k)$ DSL are dual. In passing, we
note that a $U(k)$ DSL is proposed to arise in a spin-$k/2$ system Calvera and
Wang (2020).
## VI Quantum anomaly of SL(N,k)
The above derivation of the effective theory of SL(6,k) from Dirac spin
liquids should really be viewed as at the kinematic level, i.e., this
derivation does not guarantee that these two theories have identical IR
dynamics. In fact, rigorously showing that two theories have identical IR
dynamics is generally formidably challenging.
Now we investigate the kinematic aspects of the SLs in greater detail, by
analyzing the quantum anomaly of SL(N) for general $N\geqslant 5$. The results
for SL(N,k) can be readily obtained from those for SL(N) by viewing the former
as $k$ copies of the latter. The anomaly takes the form of an invertible
topological response term in $(3+1)$-d spacetime, characterizing an SPT phase
whose boundary can host our physical SL(N). In principle, the anomaly can be
calculated directly from the action of $S^{(N)}$, but in practice this appears
to be quite complicated. Instead, we will show in this section that the
anomaly of SL(N) is essentially fixed by the phase diagram, or more precisely,
by the cascade structure discussed in Sec. IV.3.
Below we will first treat the continuous symmetry of SL(N) to be $SO(N)\times
SO(N-4)$, and derive the topological response function corresponding to the
full anomaly associated with the entire symmetry of the theory, i.e.,
including both this continuous and the discrete symmetries. For $N$ odd, this
treatment is faithful and complete. For $N$ even, the symmetry $SO(N)\times
SO(N-4)$ is larger than the faithful $I^{(N)}=(SO(N)\times SO(N-4))/Z_{2}$
symmetry possessed by SL(N). Physically, one can think of this symmetry
enlargement as from some trivially gapped local DOFs that, e.g., transform as
an $SO(N)$ vector and $SO(N-4)$ singlet. The risk of working with the larger
$SO(N)\times SO(N-4)$ symmetry is that we may miss some anomalies that are
nontrivial only for the original $I^{(N)}$ group. In the later part of this
section, by analyzing the properties of the monopoles of the $I^{(N)}$
symmetry, we will derive the anomaly associated with the faithful $I^{(N)}$
symmetry for all $N\geqslant 5$. However, there we will not explicitly derive
the anomaly associated with the discrete symmetries, which is left for future
work.
The final result with the continuous symmetry taken to be the enlarged
$SO(N)\times SO(N-4)$ is given by Eq. (35). Some of the physical implications
of this anomaly can be read off from Table 5. If the continuous symmetry is
taken to be the faithful $I^{(N)}$, for even $N$ the monopole corresponding to
the $I^{(N)}$ symmetry of SL(N) has the structure given by root 3 of Eq. (44).
One important implication of this improved characterization of the anomaly is
that for even $N$, SL(N,2) is still anomalous, and SL(N,4) has no
$I^{(N)}$-anomaly. This is to be contrasted from Eq. (35), which implies that
SL(N,2) is anomaly-free for any $N$.
To derive Eq. (35), we take the following three steps. We first put the system
on an orientable manifold and also ignore the $\mathcal{C}$ symmetry. Next, we
include the $\mathcal{C}$ symmetry, but still stay on an orientable manifold.
This restriction to orientable manifolds means that we are not considering the
full anomaly associated with the orientation-reversal (time-reversal and
reflection) symmetries. Finally, we put the theory on a possibly unorientable
manifold (still with $\mathcal{C}$ taken into account), in order to fully
characterize the anomaly associated with all symmetries. Recall that from the
discussion in Sec. IV.2, it is sufficient to consider $\mathcal{C}$ and
$\mathcal{T}$, and the results will already capture anomalies associated with
$\mathcal{R}$.
### VI.1 $SO(N)\times SO(N-4)$ on orientable manifolds
We first consider orientable 4-manifolds $X_{4}$, with vanishing first
Stiefel-Whitney (SW) class $w_{1}^{TM}=0$ (mod $2$). We shall also neglect
charge conjugation symmetry for now. A general response term in $4d$ takes the
form
$\displaystyle S_{\rm bulk}=$ $\displaystyle
i\pi\int_{X_{4}}(a_{1}w_{4}^{SO(N)}+a_{2}w_{4}^{SO(N-4)}$ (28)
$\displaystyle+a_{3}w_{2}^{SO(N)}w_{2}^{SO(N-4)}+a_{4}w_{2}^{SO(N)}w_{2}^{SO(N)}$
$\displaystyle+a_{5}w_{2}^{SO(N-4)}w_{2}^{SO(N-4)}+a_{6}w_{2}^{TM}w_{2}^{TM}),$
where $a_{1,2,3,4,5,6}\in\\{0,1\\}$ are unknowns, $w_{2}\in
H^{2}(X_{4},\mathbb{Z}_{2})$ and $w_{4}\in H^{4}(X_{4},\mathbb{Z}_{2})$ are
the second and fourth SW classes of the corresponding bundles ($SO(N)$,
$SO(N-4)$ and tangent bundles), respectively. The products among the SW
classes here and below all refer to the cup product. The physical meanings of
various of these topological response terms are given in Table 5.
We now try to fix the unknown coefficients in the above expression. We do not
attempt to directly gauge the SL(N) and compute the anomaly. Instead, we shall
use two simple facts due to the cascade structure among the SLs discussed in
Sec. IV.3:
1. 1.
If the $SO(N)\times SO(N-4)$ symmetry is broken to $SO(5)\subset SO(N)$, the
anomaly becomes simply $i\pi\int w_{4}^{SO(5)}$. Namely, if we set
$w_{2}^{SO(N-4)}$ and $w_{4}^{SO(N-4)}$ to trivial,
$w_{2}^{SO(N)}=w_{2}^{SO(5)}$ and $w_{4}^{SO(N)}=w_{4}^{SO(5)}$, the anomaly
term should become just become $w_{4}^{SO(5)}$. This comes from the fact (as
reviewed in Sec. III.1) that SL(5) corresponds to the DQCP and has the simple
$w_{4}$ anomaly.
2. 2.
If the $SO(N)\times SO(N-4)$ symmetry is broken to $SO(4)\times
SO(N-4)^{\prime}$, where $SO(4)\subset SO(N)$ and $SO(N-4)^{\prime}$ is a
combination of $SO(N-4)\subset SO(N)$ and the original $SO(N-4)$, then there
is no anomaly left since the theory admits a simple ordered state (see also
Sec. IV.1 for more details). This means that if we set
$w_{2}^{SO(N)}=w_{2}^{SO(4)}+w_{2}^{SO(N-4)^{\prime}}$ and
$w_{2}^{SO(N-4)}=w_{2}^{SO(N-4)^{\prime}}$, the anomaly should vanish.
It turns out that the above two conditions unambiguously fix $S_{\rm bulk}$ to
be
$\displaystyle S_{\rm bulk}=$ $\displaystyle
i\pi\int_{X_{4}}(w_{4}^{SO(N)}+w_{4}^{SO(N-4)}+w_{2}^{SO(N)}w_{2}^{SO(N-4)}$
(29) $\displaystyle+w_{2}^{SO(N-4)}w_{2}^{SO(N-4)}).$
To show this, we use the facts that (a) if $SO(N)$ symmetry is broken to
$SO(N-m)\times SO(m)$, then
$w_{4}^{SO(N)}=w_{4}^{SO(N-m)}+w_{4}^{SO(m)}+w_{2}^{SO(N-m)}w_{2}^{SO(m)}$,
and (b) $w_{4}^{SO(n)}=0$ for $n\leqslant 4$.
### VI.2 Charge conjugation
We now consider the $\mathcal{C}$ symmetry, which is a $\mathbb{Z}_{2}$
symmetry that is improper (orientation-reversing) in both the $SO(N)$ and
$SO(N-4)$ spaces. Upon including this improper $\mathbb{Z}_{2}$ symmetry, the
probe $SO(N)\times SO(N-4)$ gauge field will be enhanced to an $O(N)\times
O(N-4)$ bundle, with the restriction
$w_{1}^{O(N)}=w_{1}^{O(N-4)}\hskip 5.0pt({\rm{mod}}\hskip 2.0pt2),$ (30)
where $w_{1}$ is the first SW class of the corresponding bundle. This equation
states that an improper rotation in the $O(N)$ space is also improper in
$O(N-4)$.
To study the anomaly, we utilize the simple fact that $O(N)\subset SO(N+1)$.
So the anomaly of the $O(N)\times O(N-4)$ bundle in the $S^{(N)}$ theory is
completely fixed by the known anomaly of the $SO(N+1)\times SO(N-3)$ bundle in
the $S^{(N+1)}$ theory.
To be concrete, let us start from the $S^{(N+1)}$ theory, and condense the
component $\langle n_{11}\rangle=1$ – as discussed in Sec. IV.1, this leads to
the $S^{(N)}$ theory. The condensate breaks the $SO(N+1)\times SO(N-3)$
symmetry down to $(SO(N)\times SO(N-4))\rtimes\mathbb{Z}_{2}^{\mathcal{C}}$,
which is a subgroup of $O(N)\times O(N-4)$. Now starting from the
$SO(N+1)\times SO(N-3)$ anomaly in Eq. (29), we can obtain the anomaly
associated with the $O(N)\times O(N-4)$ bundle as follows. First we split the
bundles $SO(N+1)\to O(1)^{A}\times O(N)$ and $SO(N-3)\to O(1)^{B}\times
O(N-4)$ (remember $O(1)\sim\mathbb{Z}_{2}$), with the condition that
$w_{1}^{O(1)^{A}}=w_{1}^{O(N)}=w_{1}^{O(1)^{B}}=w_{1}^{O(N-4)}\hskip
5.0pt({\rm{mod}}\hskip 2.0pt2).$ (31)
The first and last equal signs come from the $SO(N+1)\times SO(N-3)$ “parent”
group, and the second equal sign comes from the fact that the condensate
$\langle n_{11}\rangle$ forces the identification of $O(1)^{A}$ and
$O(1)^{B}$. This gives rise to the restriction Eq. (30). In the following we
shall denote the common $w_{1}$ of these bundles to be simply $w_{1}$. Now
take the terms in Eq. (29) and apply Whitney product formula:
$\displaystyle w_{4}^{SO(N+1)}$ $\displaystyle\to$ $\displaystyle
w_{4}^{O(N)}+w_{1}w_{3}^{O(N)},$ $\displaystyle w_{4}^{SO(N-3)}$
$\displaystyle\to$ $\displaystyle w_{4}^{O(N-4)}+w_{1}w_{3}^{O(N-4)},$
$\displaystyle w_{2}^{SO(N+1)}$ $\displaystyle\to$ $\displaystyle
w_{2}^{O(N)}+w_{1}^{2},$ $\displaystyle w_{2}^{SO(N-3)}$ $\displaystyle\to$
$\displaystyle w_{2}^{O(N-4)}+w_{1}^{2}.$ (32)
From the Wu formula we have $\int w_{1}^{O(N)}w_{3}^{O(N)}=\int{\rm
Sq}^{1}(w_{3}^{(O(N))})=\int w_{1}^{TM}w_{3}^{O(N)}=0$ (mod $2$) on orientable
manifolds, where ${\rm Sq}^{1}$ is the Steenrod square operation. After some
algebra we obtain the following anomaly:
$\displaystyle S_{\rm bulk}$ $\displaystyle=$ $\displaystyle
i\pi\int_{X_{4}}(w_{4}^{O(N)}+w_{4}^{O(N-4)}+w_{2}^{O(N)}w_{2}^{O(N-4)}$ (33)
$\displaystyle+$
$\displaystyle(w_{2}^{O(N-4)})^{2}+w_{1}^{2}(w_{2}^{O(N)}+w_{2}^{O(N-4)})).$
In particular, for $N=6$, the above anomaly agrees with an explicit
computation for the $U(1)$ Dirac spin liquid in Ref. Calvera and Wang (2021).
This further strengthens the support for the equivalence between SL(6) and the
$U(1)$ DSL.
### VI.3 Unorientable manifolds
We now consider the anomaly on possibly unorientable manifolds. As discussed
in Sec. IV.1, an orientation-reversing symmetry (such as time-reversal) should
also be orientation-reversing in either the $SO(N)$ or $SO(N-4)$ space (but
not both). This means that we should again consider an $O(N)\times O(N-4)$
bundle as we did for charge conjugation, but now the restriction Eq. (30) is
modified:
$w_{1}^{O(N)}+w_{1}^{O(N-4)}+w_{1}^{TM}=0\hskip 5.0pt({\rm{mod}}\hskip
2.0pt2).$ (34)
So it is now meaningful to ask which $w_{1}$’s participate in the anomaly
terms like Eq. (33) – from Eq. (34) there are two linearly independent ones.
We now again take advantage of two facts due to the cascade structure among
the SLs, as discussed in Sec. IV.1:
1. 1.
If we reduce the theory to $S^{(5)}$ through a set of condensation, so that
the $O(N-4)$ gauge symmetry is completely broken and $O(N)$ is broken to
$O(5)$, the resulting theory is known to have the anomaly $i\pi\int
w_{4}^{O(5)}$, with the restriction $w_{1}^{O(5)}=w_{1}^{TM}$ (mod $2$).
2. 2.
We can enter a completely ordered phase by condensing the first $N-4$ rows of
the order parameter. This leaves behind an $O(4)\times O(N-4)^{\prime}$ bundle
($O(N-4)^{\prime}$ being a combination of an $O(N-4)\subset O(N)$ and the
original $O(N-4)$) with the restriction $w_{1}^{O(4)}+w_{1}^{TM}=0$ (mod $2$).
The anomaly should completely vanish for this bundle.
One can check that there is only a single anomaly that satisfies the above two
conditions, and reduces to Eq. (33) on orientable manifolds101010On orientable
manifolds, $\int(w_{1}^{O(N-4)})^{4}=\int{\rm Sq}^{1}(w_{1}^{O(N-4)})^{3}=\int
w_{1}^{TM}(w_{1}^{O(N-4)})^{3}=0$, so the orientable anomaly Eq. (33) is
reproduced.:
$\displaystyle S_{\rm bulk}=$ $\displaystyle
i\pi\int_{X_{4}}(w_{4}^{O(N)}+w_{4}^{O(N-4)}+w_{2}^{O(N)}w_{2}^{O(N-4)}$ (35)
$\displaystyle+(w_{2}^{O(N-4)})^{2}+(w_{1}^{O(N-4)})^{4}$
$\displaystyle+(w_{1}^{O(N-4)})^{2}(w_{2}^{O(N)}+w_{2}^{O(N-4)})).$
It is relatively easy to see that this anomaly satisfies condition (1). To
verify condition (2), the derivation goes as follows. We split the $O(N)$
bundle to $O(4)\times O(N-4)$ and identify the $O(N-4)\subset O(N)$ with the
original $O(N-4)$. The SW classes of $O(N)$ split according to Whitney product
formula. This leads to the following anomaly for the $O(4)\times O(N-4)$
bundle:
$\displaystyle
i\pi\int_{X_{4}}(w_{1}^{O(4)}w_{3}^{O(N-4)}+w_{1}^{O(N-4)}w_{3}^{O(4)}$
$\displaystyle+w_{1}^{O(4)}w_{1}^{O(N-4)}w_{2}^{O(N-4)}+(w_{1}^{O(N-4)})^{4}$
$\displaystyle+(w_{1}^{O(N-4)})^{2}w_{2}^{O(4)}+(w_{1}^{O(N-4)})^{3}w_{1}^{O(4)}),$
(36)
with the restriction $w_{1}^{O(4)}=w_{1}^{TM}$ (mod $2$). We now notice that
these SW classes are not completely independent. There are several useful (mod
$2$) relations, valid when integrated over $X_{4}$:
$\displaystyle 0$ $\displaystyle=$ $\displaystyle{\rm
Sq}^{1}(w_{1}^{O(N-4)}w_{2}^{O(4)})+w_{1}^{TM}w_{1}^{O(N-4)}w_{2}^{O(4)}$
$\displaystyle=$
$\displaystyle(w_{1}^{O(N-4)})^{2}w_{2}^{O(4)}+w_{1}^{O(N-4)}w_{1}^{O(4)}w_{2}^{O(4)}$
$\displaystyle+w_{1}^{O(N-4)}w_{3}^{O(4)}+w_{1}^{TM}w_{1}^{O(N-4)}w_{2}^{O(4)}$
$\displaystyle=$
$\displaystyle(w_{1}^{O(N-4)})^{2}w_{2}^{O(4)}+w_{1}^{O(N-4)}w_{3}^{O(4)};$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle{\rm Sq}^{1}\cdot{\rm
Sq}^{1}(w_{2}^{O(N-4)})$ $\displaystyle=$ $\displaystyle{\rm
Sq}^{1}(w_{3}^{O(N-4)}+w_{1}^{O(N-4)}w_{2}^{O(N-4)})$ $\displaystyle=$
$\displaystyle
w_{1}^{O(4)}w_{3}^{O(N-4)}+w_{1}^{O(4)}w_{1}^{O(N-4)}w_{2}^{O(N-4)};$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle(w_{1}^{O(N-4)})^{4}+{\rm
Sq}^{1}[(w_{1}^{O(N-4)})^{3}]$ (37) $\displaystyle=$
$\displaystyle(w_{1}^{O(N-4)})^{4}+w_{1}^{O(4)}(w_{1}^{O(N-4)})^{3}.$
These relations come from several properties of the Steenrod square ${\rm
Sq}^{1}$: ${\rm Sq}^{1}x=w_{1}^{TM}x$ for $x\in H^{3}(X_{4},\mathbb{Z}_{2})$,
${\rm Sq}^{1}w_{1}^{O(n)}=(w_{1}^{O(n)})^{2}$, ${\rm
Sq}^{1}w_{2}^{O(n)}=w_{1}^{O(n)}w_{2}^{O(n)}+w_{3}^{O(n)}$, ${\rm
Sq}^{1}(x\cup y)=({\rm Sq}^{1}x)\cup y+x\cup{\rm Sq}^{1}y$, ${\rm
Sq}^{1}\cdot{\rm Sq}^{1}=0$ as well as the (mod $2$) restriction
$w_{1}^{O(4)}+w_{1}^{TM}=0$. The remnant anomaly Eq. (VI.3) vanishes upon
plugging in these relations, as promised. Furthermore, one can check that Eq.
(35) is the unique anomaly that satisfies the above properties due to the
cascade structure and reduces to Eq. (33) on orientable manifolds.
We therefore conclude that Eq. (35), together with the restriction Eq. (34),
forms the complete anomaly of our theory.
### VI.4 Anomaly for the faithful $I^{(N)}$ symmetry from monopole
characteristics
In the above we have derived the anomaly of the SLs by taking its continuous
symmetry to be $SO(N)\times SO(N-4)$. As alluded before, this treatment is
complete for odd $N$. For even $N$, this symmetry is larger than the faithful
$I^{(N)}$ symmetry, and we may miss some anomalies by just looking at the
enlarged symmetry. In this subsection, we will derive the anomaly associated
with the faithful $I^{(N)}$ symmetry for even $N$. We will see that the
$I^{(N)}$ anomaly of the SLs can still be unambiguously pinned down from the
cascade structure. Interestingly, although the analysis in the previous
subsections indicates that SL(N,2) is anomaly-free, here we find that for even
$N$, if the faithful $I^{(N)}$ symmetry is properly taken into account,
SL(N,4) is anomaly-free, but SL(N,2) is still anomalous. In the following
discussion we will mainly focus on anomalies that involve the continuous
symmetries, and we leave the full anomaly (for example, on unorientable
manifolds) to future works.
Our approach is to consider the $(3+1)$-d SPT whose boundary can host the SL,
gauge the $I^{(N)}$ symmetry of this SPT, and use the properties of the
$I^{(N)}$ monopoles as a characterization of the SPT. The bulk-boundary
correspondence due to anomaly-inflow indicates that this is also a
characterization of anomaly of the SL. This approach is a generalization of
the one used in the study of symmetry-enriched $U(1)$ quantum spin liquids
Wang and Senthil (2013, 2016); Zou _et al._ (2018); Zou (2018). Note that
since the properties of the monopoles capture the properties of the ’t Hooft
lines of the corresponding $I^{(N)}$ gauge theory, the discussion here can be
equivalently phrased in terms of the ’t Hooft lines. However, we will use the
language of the monopoles. Here we will focus on the case with an even $N$,
and in Appendix F.2 we apply this approach to odd $N$ to reproduce the results
obtained before.
To start, let us ask what is the fundamental monopole of an $I^{(N)}$ gauge
theory, where by “fundamental” we mean that all dyonic excitations can be
viewed as a bound state of certain numbers of such a fundamental monopole and
the pure gauge charge. Naively, one might expect that there are two types of
fundamental monopoles: the $SO(N)$ monopole and the $SO(N-4)$ monopole.
However, due to the locking of the $Z_{2}$ centers of the $SO(N)$ and
$SO(N-4)$ symmetries, those are not the fundamental monopole. Instead, the
fundamental monopole can be viewed as a bound state of half of an $SO(N)$
monopole and half of an $SO(N-4)$ monopole. More explicitly, denote the field
configuration of a unit $U(1)$ monopole by $A_{U(1)}$, whose precise
expression is unimportant, and a particular realization is given in Ref. Wu
and Yang (1975). Write the $SO(N)$ and $SO(N-4)$ gauge fields as
$A^{SO(N)}=A_{a}^{L}T_{a}^{L}$ and $A^{SO(N-4)}=A_{a}^{R}T_{a}^{R}$, with
$\\{T_{a}^{L}\\}$ and $\\{T_{a}^{R}\\}$ the generators of $SO(N)$ and
$SO(N-4)$, respectively. For example, $T_{12}^{L}$ generates the $SO(N)$
rotations in the $(1,2)$-plane, $T_{34}^{R}$ generates the $SO(N-4)$ rotations
in the $(3,4)$-plane, etc. A fundamental $I^{(N)}$ monopole can be realized by
the following field configuration:
$\displaystyle\begin{split}&A_{12}^{L}=A_{34}^{L}=A_{56}^{L}=\cdots
A_{N-1,N}^{L}\\\ =&A_{12}^{R}=A_{34}^{R}=A_{56}^{R}=\cdots
A_{N-5,N-4}^{R}=\frac{A_{U(1)}}{2}\end{split}$ (38)
That is, this $I^{(N)}$ monopole is obtained by embedding half-$U(1)$
monopoles into the maximal Abelian subgroup of $I^{(N)}$. This configuration
breaks the continuous $I^{(N)}$ symmetry to $(SO(2)^{N-2})/Z_{2}$. So it is
convenient to denote a general excitation in this $I^{(N)}$ gauge theory by
the following excitation matrix:
$\displaystyle\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}=\left(\begin{array}[]{cccc|cccc}q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
m_{12}^{L}&m_{34}^{L}&\cdots&m_{N-1,N}^{L}&m_{12}^{R}&m_{34}^{R}&\cdots&m_{N-5,N-4}^{R}\end{array}\right)_{s}$
(43)
where the first (second) row represents the electric (magnetic) charges of
this excitation under $A_{ij}^{L,R}$, $s=0\ ({\rm mod\ }2)$ ($s=1\ ({\rm mod\
}2)$) represents that this excitation is a boson (fermion), and the vertical
line separates the charges related to the original $SO(N)$ and $SO(N-4)$
subgroups of $I^{(N)}$. The fundamental monopole has
$\bm{m}=(\frac{1}{2},\frac{1}{2},\cdots,\frac{1}{2})$, and its $\bm{q}$ and
$s$ will characterize the corresponding SPT.
There are multiple constraints on the possible excitation matrices that a
consistent theory should satisfy. For instance, a pure gauge charge should
have an excitation matrix such that $\bm{m}=0$, and all entries of $\bm{q}$
are integers that sum up to an even integer, such as
$\bm{q}=(1,0,0,\cdots,0,1)$. Another important constraint is the Dirac
quantization condition, which in this case states that two excitations with
excitation matrices $\left(\begin{array}[]{c}\bm{q}_{1}\\\
\bm{m}_{1}\end{array}\right)_{s_{1}}$ and
$\left(\begin{array}[]{c}\bm{q}_{2}\\\ \bm{m}_{2}\end{array}\right)_{s_{2}}$
should satisfy
$\bm{q}_{1}\cdot\bm{m}_{2}-\bm{q}_{2}\cdot\bm{m}_{1}\in\mathbb{Z}$. As a
sanity check, consider a fundamental monopole with
$\bm{m}=(\frac{1}{2},\frac{1}{2},\cdots,\frac{1}{2})$ as above, and an
elementary pure gauge charge with $\bm{m}=0$ and $\bm{q}=(1,0,0,\cdots,0,1)$,
we see that the Dirac quantization condition is indeed satisfied. This
actually explains why the above configuration of the fundamental monopole is
valid in this theory. Other constraints come from the $\mathcal{C}$,
$\mathcal{R}$ and $\mathcal{T}$ symmetries, as well as the fact that the
original theory has an $I^{(N)}$ gauge structure (see Appendix F for more
details).
Taking all these constraints into account, as shown in Appendix F, there are
only very few classes of distinct types of SPTs. In particular, if $N=2\ ({\rm
mod\ }4)$, the structures of fundamental monopoles can be classified as
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}$, where the three
generators, or “roots”, are given by
$\displaystyle\begin{split}&{\rm root\
}1:\left(\begin{array}[]{cccc|cccc}0&0&\cdots&0&0&0&\cdots&0\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{f}\\\
&{\rm root\ }2:\left(\begin{array}[]{cccc|cccc}0&0&\cdots&0&0&0&\cdots&1\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}\\\
&{\rm root\
}3:\left(\begin{array}[]{cccc|cccc}\frac{1}{4}&\frac{1}{4}&\cdots&\frac{1}{4}&-\frac{1}{4}&-\frac{1}{4}&\cdots&-\frac{1}{4}\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}\end{split}$
(44)
For $N=0\ ({\rm mod\ }4)$, the structures of the fundamental monopoles can be
classified as
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}$,
i.e., it has one more $\mathbb{Z}_{2}$ factor compared to the case with $N=2\
({\rm mod\ }4)$. The fundamental monopoles in Eq. (44) are still the roots for
the first $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ factor, and
the root for the additional $\mathbb{Z}_{2}$ factor has the following
fundamental monopole:
$\displaystyle{\rm root\
}4:\left(\begin{array}[]{ccc|cccc}0&\cdots&0&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}$
(47)
It is useful to derive the properties of the $SO(N)$ and $SO(N-4)$ monopoles
from these fundamental $I^{(N)}$ monopoles. The results are listed in Table 1,
and the details of the derivation can be found in Appendix F. It is
interesting to notice that the results for $N=2\ ({\rm mod\ }4)$, $N=0\ ({\rm
mod\ }8)$ and $N=4\ ({\rm mod\ }8)$ are all different. It is known that the
spinor representations of $SO(N)$ in these three classes are complex, real,
and pseudoreal, respectively Zee (2016), which may be related to the
difference of the monopoles in SLs with different $N$.
| $SO(N)$ monopole | $SO(N-4)$ monopole
---|---|---
root 1 | (singlet, singlet, boson) | (singlet, singlet, boson)
root 2 | (singlet, singlet, fermion) | (singlet, singlet, fermion)
root 3 | (spinor, spinor, boson) | (spinor, spinor, fermion)
root 4 with $N=0\ ({\rm mod\ }8)$ | (singlet, singlet, fermion) | (singlet, vector, boson)
root 4 with $N=4\ ({\rm mod\ }8)$ | (singlet, singlet, boson) | (singlet, vector, fermion)
Table 1: Properties of the $SO(N)$ and $SO(N-4)$ monopoles of the root states
for even $N$. The first three roots apply to all even $N$, and root 4 only
applies to the case with $N$ an integral multiple of 4. The $SO(N)$ monopole
breaks the $I^{(N)}$ symmetry to $(SO(2)\times SO(N-2)\times SO(N-4))/Z_{2}$,
and it always has no charge under the $SO(2)$. Its three corresponding entries
represent its representation under the $SO(N-2)$, its representation under the
$SO(N-4)$, and its statistics, respectively. The $SO(N-4)$ monopole breaks the
$I^{(N)}$ symmetry to $(SO(N)\times SO(N-6)\times SO(2))/Z_{2}$, and it always
has no charge under the $SO(2)$. Its three corresponding entries represent its
representation under the $SO(N)$, its representation under the $SO(N-6)$, and
its statistics, respectively. For the case with $N=6$, the second entry does
not exist for its $SO(N-4)$ monopole. Notice these properties are determined
up to attaching pure gauge charges.
The above discussion implies the existence of various $I^{(N)}$-SPTs, and thus
also of the $I^{(N)}$-anomalies. Which of the anomalies are compatible with
the cascade structure of the SLs, in particular, the two conditions in Sec.
VI.3? It is relatively easy to examine the first condition. Note that the
$SO(N)$ monopole breaks the $SO(N)$ symmetry to $SO(N-2)$. To satisfy the
first constraint, this $SO(N)$ monopole should carry a spinor representation
of the remaining $SO(N-2)$, which means that root 3 or its inverse must be
involved in the anomaly of the SL. It is a bit more complicated to examine the
second condition, and we leave the details to Appendix F. The result is that
only root 3 or its inverse can satisfy both constraints. Therefore, we
conclude that for even $N$ the $I^{(N)}$-anomalies of SL(N,±1) are those of
root 3 and its inverse, respectively.
In passing, we mention that the monopole properties of the $U(1)$ DSL are
explicitly derived in Appendix G, which agree with that of SL(6). This match
provides further evidence for our statement in Sec. V that the $U(1)$ DSL and
SL(6) are actually equivalent.
### VI.5 Semion topological order from time-reversal breaking
It is well known Vishwanath and Senthil (2013) that non-perturbative anomalies
(such as those in this work) can sometimes be satisfied by gapped topological
orders in dimension $d\geqslant(2+1)$. However, it is also known that for the
$N=5$ theory (the deconfined criticality) the $w_{4}^{SO(5)}$ anomaly cannot
be matched by a gapped topological order if time-reversal symmetry is not
broken Wang _et al._ (2017). This statement can be easily generalized to
arbitrary $N\geqslant 5$ using similar arguments as in Ref. Wang _et al._
(2017): consider an $SO(N)$ monopole, represented as a unit $SO(2)\subset
SO(N)$ monopole in the first two components. The $w_{4}^{SO(N)}$ anomaly
requires the monopole to carry spinor representation for the remaining
$SO(N-2)$. For a gapped topologically ordered state, this condition can be
satisfied only by attaching a gapped anyon excitation to the monopole, with
the anyon carrying spinor representation under $SO(N-2)$. But an anyon should
in general carry irrep under the entire $SO(N)$, which means that the
$SO(N-2)$ spinor anyon should also carry $SO(2)$ charge $q=1/2$. This leads to
a nontrivial Hall conductance for the $SO(2)$, which necessarily breaks time-
reversal symmetry. For $N\geqslant 9$ the same argument applies to the
$SO(N-4)$ symmetry since there is also a $w_{4}^{SO(N-4)}$ anomaly.
If time-reversal is broken, either explicitly or spontaneously, then a gapped
topological order becomes possible. For DQCP ($N=5$) and $U(1)$ Dirac spin
liquid ($N=6$), it is known that the simplest topological order that satisfies
the anomaly is a semion (or anti-semion) topological order, with only one
nontrivial abelian anyon $s$ with exchange statistical phase $e^{i\pi/2}$ (or
$e^{-i\pi/2}$ for anti-semion $\bar{s}$) – basically each semion sees another
semion as a $\pi$-flux. We now argue that for any $N\geqslant 5$, the anomaly
can be matched by a semion topological order, in which the semion $s$ carries
spinor representation under both $SO(N)$ and $SO(N-4)$ (for $N=6$ “spinor rep”
for $SO(2)$ here means charge $1/2$). For simplicity we shall only consider
the $SO(N)\times SO(N-4)$ symmetry below, neglecting the charge conjugation
symmetry.
Consider an $SO(2)\subset SO(N)$ monopole. Since the semion carries charge
$\pm 1/2$ under this $SO(2)$, a semion sees a “bare” monopole as a $\pi$-flux.
To make the monopole local, one has to attach a semion to the bare monopole to
neutralize its mutual statistics with other semions. Since a semion carries
spinor rep under both $SO(N)$ and $SO(N-4)$, the monopole now also carries
spinor rep under $SO(N-2)\subset SO(N)$ and $SO(N-4)$. Using the same
reasoning, an $SO(N-4)$ monopole will also carry spinor rep under $SO(N)$ and
$SO(N-6)$ (the latter only if $N\geqslant 9$). These features match exactly
with the general anomaly (without time-reversal and charge conjugation):
$i\pi\int_{X_{4}}(w_{4}^{SO(N)}+w_{4}^{SO(N-4)}+w_{2}^{SO(N)}\cup
w_{2}^{SO(N-4)}).$ (48)
Notice that compared to Eq. (29), the $(w_{2}^{SO(N-4)})^{2}$ term is missing
from the above anomaly. This is because the $(w_{2})^{2}$ term is equivalent
to the standard $\Theta$-term for the $SO(N-4)$ gauge field. In the absence of
time-reversal symmetry the $\Theta$ angle can be continuously tuned to zero
and does not count as a nontrivial anomaly.
## VII Possible lattice realizations for $N>6$
In the above we have conjectured, based on various evidence, that Stiefel
liquids with $N>6$ and $k=1$ exist as an exotic type of critical quantum field
theories. As quantum field theories they are interesting because of the
possibility that they may be non-Lagrangian, which means that they cannot be
UV completed by any weakly-coupled renormalizable continuum Lagrangian. We now
discuss their relevance to condensed matter physics. Realizing a SL with $N>6$
in a condensed matter system will be particularly interesting, because it may
represent a critical quantum state that has no “mean-field” description, not
even one with partons, at any scale. In contrast, most correlated states
theoretically constructed so far, at least for non-disordered phases at
equilibrium, do admit some “mean-field” description at some energy scale
(typically in the UV). Therefore, our Stiefel liquid states, if realized, will
be an example beyond existing paradigms of quantum phases.
To be concrete, we shall discuss possible realizations of SL(N>6) in two
dimensional lattice spin systems. In Sec. VII.1 we discuss necessary
conditions for a SL to be “emergible”, that is, realizable in some local
Hamiltonian systems. The two important conditions to be discussed are (1)
anomaly matching and (2) dynamical stability. Subsequently we will discuss
some concrete examples relevant to SL(7). We propose that on a triangular
lattice, SL(7) can naturally arise as a competition (or intertwinement)
between a tetrahedral magnetic order and the 12-site VBS order, and on a
kagome lattice, SL(7) can naturally arise as a competition (or intertwinement)
between a cuboctahedral magnetic order and a VBS order.
### VII.1 General strategy
#### VII.1.1 Anomaly matching
Given an effective IR field theory (such as our Stiefel liquids), an important
question for condensed matter physicists is whether it can be realized as the
low-energy theory of some lattice local Hamiltonian system. In general, it is
very hard to definitively answer such questions, since the space of all local
Hamiltonians has infinite dimensions (corresponding to infinitely many tuning
parameters), and the vast majority of those Hamiltonians are not analytically
solvable.
A new approach to this type of questions has emerged in recent years based on
Lieb-Schultz-Mattis (LSM) type of theorems and ’t Hooft anomaly matching Lieb
_et al._ (1961); Oshikawa (2000); Hastings (2004); Po _et al._ (2017); Cheng
_et al._ (2016); Jian _et al._ (2018a); Cho _et al._ (2017); Huang _et al._
(2017). It has been well known, since Lieb-Schultz-Mattis, that certain
structures of the lattice Hilbert space can forbid a trivial (symmetric and
short-range entangled) ground state. For example, if a lattice spin system has
an odd number of $S=1/2$ moments per unit cell, then as long as the $SO(3)$
spin rotation and lattice translation symmetries are unbroken, the ground
state must either be gapless or topologically ordered. More recently, it has
been appreciated that such LSM constraints are equivalent to certain ’t Hooft
anomaly matching conditions111111In this paper we assume that the constraints
on the IR physics from the UV information are not associated with filling
factors relevant for systems with $U(1)$ and translation symmetries, otherwise
there can be further subtleties. See, for example, Ref. Song _et al._ (2021);
Else _et al._ (2021) for more details.. Again we use the example with a
spin-$1/2$ per unit cell, and now focus on $(2+1)$-d. If we try to couple the
system to background gauge fields of the $SO(3)$ spin rotation symmetry and
the $T_{x}$, $T_{y}$ translation symmetries (each with a group structure
$\mathbb{Z}$) Thorngren and Else (2018), the coupling should be anomalous,
with an anomaly term in one higher dimension:
$i\pi\int_{X_{4}}w_{2}^{SO(3)}xy,$ (49)
where $x,y\in H^{1}(X_{4},\mathbb{Z})$ are the integer-valued gauge fields
corresponding to the two translation symmetries. There may also be other
anomalies involving lattice rotations, reflections and time-reversal,
depending on the type of the lattice (we will discuss a concrete example in
Sec. VII.2). The LSM-like constraints state that the IR theories that emerge
out of such lattice systems should also match the above anomalies, since
anomalies are invariant under RG flow. This immediately rules out short-range
entangled symmetric ground states, since there would be no IR degrees of
freedom to match the anomaly. This also rules out conventional, Landau-
Ginzburg-Wilson-Fisher type of theories since those theories do not carry any
anomaly. The Stiefel liquids studied in this work do carry nontrivial
anomalies, and it is natural to expect that they can match the LSM anomalies
and emerge in certain situations.
When a critical field theory emerges out of a lattice system in the IR limit,
the local operators in the IR field theory can all be viewed as some coarse-
grained versions of lattice operators. One way to characterize this coarse-
graining is to utilize the fact that operators with low scaling dimensions are
characterized by their symmetry properties. For example, in the Ising model
the lattice spin, $S_{z}$ coarse-grains to the continuum real scalar field
$\phi$ in the Wilson-Fisher theory, because both operators are odd under the
global $\mathbb{Z}_{2}$ symmetry. More systematically, this coarse-graining is
described by an embedding of the symmetries at the lattice scale, $G_{UV}$, to
the symmetries of the IR theory, $G_{IR}$121212Here we assume that $G_{IR}$
contains only $0$-form symmetries, which is likely the case for our Stiefel
liquids. When $G_{IR}$ contains higher-form symmetries (such as in topological
orders), the argument can be readily generalized. In this paper, we also
assume that the many-body Hilbert space of the lattice system can be viewed as
a tensor product of local Hilbert spaces on different lattice sites. If this
is not the case, we expect that our approach still applies, as long as the
appropriate LSM constraints are used (see Ref. Kobayashi _et al._ (2019) for
some of such examples of lattice systems and LSM constraints).. Typically
$G_{UV}$ includes on-site symmetries like spin-rotation and time-reversal, as
well as lattice symmetries like translations and rotations. For the Stiefel
liquids, $G_{IR}$ symmetries include $SO(N)$, $SO(N-4)$, $\mathcal{C,R,T}$ as
well as the emergent Poincaré symmetry. More formally, this embedding is
characterized by a group homomorphism
$\varphi:G_{UV}\to G_{IR}.$ (50)
As a simple example, when the $\mathbb{Z}_{2}$ Wilson-Fisher theory is
realized from the Ising model near criticality, the lattice Ising
$\mathbb{Z}_{2}$ symmetry is mapped under $\varphi$ to the $\mathbb{Z}_{2}$
symmetry of the Wilson-Fisher theory. If a Stiefel liquid is realized out of a
spin system, both $G_{UV}$ and $G_{IR}$ will be more complicated than the
Ising-Wilson-Fisher theory, and in general there are multiple nontrivial group
homomorphisms $\varphi$ between $G_{UV}$ and $G_{IR}$. The natural question
is: which $\varphi$, if any, is physically legitimate? The LSM anomaly-
matching conditions provide a strong constraint: the IR theory contains an
anomaly $w[G_{IR}]$, as described in detail in Sec. VI. We can now pullback
the IR anomaly using $\varphi$, and obtain the corresponding anomaly for the
UV symmetry:
$w[G_{UV}]=\varphi^{*}w[G_{IR}].$ (51)
The requirement on $\varphi$ is that for the IR anomaly discussed in Sec. VI,
such as Eq. (35), the pullback yields exactly the LSM anomalies, such as Eq.
(52) and its various generalizations.
Anomaly-matching has thus been established as a necessary condition for a low-
energy theory to be emergible. Here we shall go one step further and
conjecture that it is also sufficient. This conjecture can be phrased as
Hypothesis of emergibility: a low-energy theory is emergible out of a lattice
system if and only if its anomaly matches with that from the lattice LSM-like
theorems.
Although there is no proof to this statement, there is also no known counter
example 131313There are systems with “SPT-LSM constraints” that allow a
symmetric short-ranged entangled ground state, but these symmetric short-range
entangled ground state must be a nontrivial SPT in certain sense Lu (2017);
Yang _et al._ (2018); Else and Thorngren (2020); Jiang _et al._ (2019). We
believe that these systems also satisfy the hypothesis, as long as both the UV
and IR anomalies are properly accounted for. Furthermore, the low-energy
theories we will consider below are all gapless, and stacking it with an SPT
is not expected to change its low-energy dynamics (at least in the bulk). So
we will ignore the subtlety associated with the SPT-LSM constraints in this
paper and leave it for future work.. An indirect piece of supporting evidence
of this conjecture is the existence of “featureless Mott insulators”: in
certain systems such as the half-filled honeycomb lattice, the ground state is
guaranteed to be gapless within free-fermion band theory, but there is no LSM-
like constraint, so one may expect that with strong interactions a trivial
state can emerge. Indeed, trivial states have been theoretically constructed
in various such systems Kimchi _et al._ (2013); Jian and Zaletel (2016); Kim
_et al._ (2016); Latimer and Wang (2021).
Once we make the above conjecture, the task of finding emergible Stiefel
liquids on certain lattice systems becomes the task of finding appropriate
homomorphism, $\varphi:G_{UV}\to G_{IR}$, that pulls back the IR anomaly to
the LSM anomaly.
#### VII.1.2 Dynamical stability
The above hypothesis of emergibility based on anomaly matching only concerns
about whether the IR theory can emerge at all, but does not make any statement
about the stability of this IR theory, even if it is emergible. In order for
the IR theory to be stable, we require that all $G_{UV}$-symmetry-allowed
local perturbations to this IR theory are RG irrelevant.
For Stiefel liquids, although we have argued in Sec. IV.4 that all
$G_{IR}$-symmetry-allowed local perturbations are RG irrelevant, since
$G_{UV}$ is much smaller than $G_{IR}$, in general there will be operators
that are nontrivial under $G_{IR}$ but trivial under $G_{UV}$, which may be RG
relevant. Therefore, to avoid discussing unstable states (which are
practically hard to access), we should look for $\varphi$ that only allows a
small number of relevant perturbations. However, we do not know the accurate
scaling dimensions of various operators in Stiefel liquids, although some
guesswork can be done based various trends at $N=5$ and $N=6$, which have been
numerically measured for DQCP and DSL, respectively. In general, we expect
operators in sufficiently high-rank representations of either $SO(N)$ or
$SO(N-4)$ to be irrelevant, but the “critical rank” is hard to determine. In
fact, even for the $U(1)$ DSL (SL(6)), it is still not entirely clear whether
various rank-$2$ operators are irrelevant or not. These are represented in the
QED3 theory as various fermion quartic interactions and $4\pi$ monopoles. We
will therefore consider embeddings $\varphi$ that disallow low-rank operators
(such as vectors) as much as possible.
More specifically, in the examples to be discussed in this section, two types
of operators will be symmetry-disallowed: (1) the $n$ operators, which are
vectors under both $SO(N)$ and $SO(N-4)$ – these are believed to be the most
relevant operators based on experience with DQCP and DSL; and (2) the
conserved currents of either $SO(N)$ or $SO(N-4)$, these operators have
scaling dimension $2$ and will be relevant if symmetry-allowed.
In the following we will construct some illuminating examples of lattice
realizations of Stiefel liquids, characterized by the embeddings $\varphi$.
These are by no means the only ways to realize Stiefel liquids on lattice
systems. Instead, our goal is to illustrate the possibility of realizing
Stiefel liquids in some lattice systems. We also note that all these
realizations have an $SO(3)$ spin rotational symmetry that is embeded into the
$SO(N)$ subgroup of $G_{IR}$, so it suffices to use Eq. (35) to characterize
the anomaly of a SL, for any $N$.
### VII.2 List of LSM-like anomalies in (2+1)d
Here we list LSM-like anomalies that can arise in a two dimensional lattice
spin system, with on-site $SO(3)$ and time-reversal symmetries as well as
lattice symmetries. For simplicity, the lattice symmetry we consider will only
include discrete translations, rotations and reflections.
First, if there is a odd number of $S=1/2$ moments per unit cell, there is the
aforementioned anomaly involving $SO(3)$ and translations:
$S_{tr-LSM}=i\pi\int_{X_{4}}w_{2}^{SO(3)}xy,$ (52)
where $x,y\in H^{1}(X_{4},\mathbb{Z})$ are the $T_{a_{1}},T_{a_{2}}$
translation gauge fields. Likewise, if each spin-$1/2$ moment is also a
Kramers doublet ($\mathcal{T}^{2}=-1$), then there should be another anomaly
$S_{\mathcal{T}-LSM}=i\pi\int_{X_{4}}t^{2}xy,$ (53)
where $t\in H^{1}(X_{4},\mathbb{Z}_{2})$ is the gauge field for time-reversal
symmetry.
Next, if the location of each spin-$1/2$ moment is also an inversion ($C_{2}$
rotation) center, there is another anomaly:
$S_{I-LSM}=i\pi\int_{X_{4}}c^{2}(w_{2}^{SO(3)}+t^{2}),$ (54)
where $c\in H^{1}(X_{4},\mathbb{Z}_{2})$ is a $\mathbb{Z}_{2}$ gauge field
associated with the $C_{2}$ rotation symmetry. The $\mathbb{Z}_{2}$ nature of
the topology of $SO(3)$ and $\mathcal{T}$ guarantees that other types of
rotations like $C_{3}$ will not contribute to anomaly.
Now consider the reflection symmetry $\mathcal{R}_{y}$, we should also examine
the reflection axis (the line that is invariant under reflection): we view the
reflection axis as a $(1+1)$-d system, with a translation symmetry $T_{a_{1}}$
that commutes with reflection and possibly a $C_{2}$ rotation that acts like
one dimensional inversion on the axis. If decorated on the reflection axis is
a spin-$1/2$ chain, it will have its own LSM-like anomalies. We will then have
the following anomalies
$i\pi\int_{X_{4}}r(w_{2}^{SO(3)}+t^{2})(x+c),$ (55)
where $r\in H^{1}(X_{4},\mathbb{Z}_{2})$ is the gauge field for reflection.
The fact that both time-reversal and reflection changes the space-time
orientation means that we also have the following restriction:
$r+t=w_{1}^{TM}\hskip 10.0pt({\rm{mod}}\hskip 2.0pt2).$ (56)
If the lattice system has all the above features, the LSM anomalies add up to:
$\displaystyle i\pi\int_{X^{4}}(w_{2}^{SO(3)}+t^{2})[xy+c^{2}+r(x+c)].$ (57)
Notice that there are also other LSM constraints that are not explicitly
described above, but are nevertheless contained in this anomaly formula. For
example, there can be analogous LSM constraints associated with the reflection
symmetry $\mathcal{R}_{x}$ (the reflection symmetry with invariant axis
perpendicular to that of $\mathcal{R}_{y}$). In Appendix H, we derive some of
these other LSM constraints from Eq. (57). We also stress that in a given
lattice system, there may be multiple different, say, $C_{2}$ rotation
symmetries. Some of them have rotation centers hosting an odd number of
spin-1/2’s, but others do not. When the above formula is applied to the
former, the contributions from Eq. (54) should be nontrivial, while if it is
applied to the latter, Eq. (54) should vanish. In general, the anomalies
associated with all these different $C_{2}$ centers should be specified
separately. But on $C_{6}$-symmetric lattices such as triangular and Kagome,
one typically only need to specify one inversion center and the others will be
determined by symmetries. For example, consider triangular lattice. There are
four inversion centers per unit cell: the $C_{6}$ center (lattice site) and
three $C_{2}$ centers (bond center) that are related to each other through
$C_{6}$ rotations. The parity of $2S$ ($S$ being the spin moment) of the
entire unit cell is given by the sum of the parity on each inversion center.
This means that if we have anomaly $aw_{2}^{SO(3)}xy$ and
$bw_{2}^{SO(3)}c^{2}$, where $a,b\in\\{0,1\\}$ and $c$ probes the site-
centered inversion, then the anomaly associated with the bond-centered
inversion (probed by $c^{\prime}$) will be given by
$(a+b)w_{2}^{SO(3)}(c^{\prime})^{2}$. For this reason we will only focus on
one inversion center in these lattices.
The above situations (summarized in Eq. (57)) happen for a variety of $2d$
lattice system, including square, triangular and Kagome lattices 141414In
Appendix H, an alternative expression for the LSM anomaly on a square lattice
is given. We have checked that all the following statements about states on a
square lattice hold if either the expression here or the alternative one there
is used.. These are common playgrounds for studying frustrated quantum
magnetism, and we will see some examples next.
### VII.3 Warm-up: anomaly-matching for DQCP
Given the variety of anomaly terms presented above, it is rather nontrivial
for a theory to exactly match all the anomalies. Below we show how this works
for the DQCP on square lattice. In appendix I, we show this for the slightly
more complicated case of $U(1)$ DSL on triangular lattice. Since these states
admit explicit parton mean-field constructions on the lattices, we expect them
to be emergible and the anomaly-matching should go through. So these exercises
serve as a benchmark for the anomaly-matching approach.
Figure 4: Square lattice and the relevant symmetries. Each filled red circle
represents an odd number of spin-1/2’s. The $C_{2}$ rotation is around a site
that hosts the spins, and the dashed line is the reflection axis of $R_{y}$.
As we reviewed in Sec. III.1, the DQCP corresponds to $S^{(5)}$, with IR
anomaly
$i\pi\int_{X_{4}}w_{4}^{O(5)},$ (58)
together with the restriction $w_{1}^{O(5)}=w_{1}^{TM}$.
We represent the microscopic symmetry implementations by their actions on the
$n$ field, which for DQCP is simply a $5$-component vector. The $SO(3)$ spin
rotation is implemented as
$n\to\left(\begin{array}[]{cc}SO^{s}(3)&0\\\ 0&I_{2}\end{array}\right)n.$ (59)
Time-reversal symmetry acts as
$n\to\left(\begin{array}[]{cc}-I_{3}&0\\\ 0&I_{2}\end{array}\right)n,\hskip
10.0pti\to-i.$ (60)
On square lattice the translation symmetries $T_{x},T_{y}$ are implemented as
(see Fig. 4)
$\displaystyle T_{x}:$ $\displaystyle
n\to\left(\begin{array}[]{ccccc}-1&0&0&0&0\\\ 0&-1&0&0&0\\\ 0&0&-1&0&0\\\
0&0&0&-1&0\\\ 0&0&0&0&1\end{array}\right)n,$ (66) $\displaystyle T_{y}:$
$\displaystyle n\to\left(\begin{array}[]{ccccc}-1&0&0&0&0\\\ 0&-1&0&0&0\\\
0&0&-1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&-1\end{array}\right)n.$ (72)
As for the lattice rotation, since only the site-centered inversion ($C_{2}$)
participates in the anomaly, we should just focus on it:
$C_{2}:n\to\left(\begin{array}[]{cc}I_{3}&0\\\ 0&-I_{2}\end{array}\right)n.$
(73)
Finally, for reflection $\mathcal{R}_{y}$:
$n\to\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&-1\end{array}\right)n.$ (74)
We can now pull back the $w_{4}^{O(5)}$ anomaly to the physical symmetries.
The calculation proceeds as follows. First, since none of the microscopic
symmetries mixes $n_{1,2,3}$ with $n_{4,5}$, we can decompose the $O(5)$
bundle into $O(3)\times O(2)$, where
$\displaystyle w_{1}^{O(3)}$ $\displaystyle=$ $\displaystyle t+x+y,$
$\displaystyle w_{1}^{O(2)}$ $\displaystyle=$ $\displaystyle r+x+y,$
$\displaystyle w_{2}^{O(3)}$ $\displaystyle=$ $\displaystyle
w_{2}^{SO(3)}+t^{2},$ $\displaystyle w_{2}^{O(2)}$ $\displaystyle=$
$\displaystyle xy+c^{2}+xr+cr,$ $\displaystyle w_{3}^{O(3)}$ $\displaystyle=$
$\displaystyle{\rm Sq}^{1}(w_{2}^{O(3)})+w_{1}^{O(3)}w_{2}^{O(3)}$ (75)
$\displaystyle=$ $\displaystyle{\rm
Sq}^{1}(w_{2}^{SO(3)})+(t+x+y)(w_{2}^{SO(3)}+t^{2}),$
and all higher SW classes vanish. Notice that we have used the fact that $x,y$
live in $H^{1}(X_{4},\mathbb{Z})$ so $x^{2}=y^{2}=0$. We also set
$cx=cy=ry=0$, based on the physical understanding that non-commuting
crystalline symmetries do not simultaneous contribute to the same anomaly term
– this can be seen, for example, from the dimension-reduction approach Song
_et al._ (2017). We can then write the $w_{4}^{O(5)}$ using Whitney product
formula:
$\displaystyle\begin{split}w_{4}^{O(5)}=&\sum_{i}w_{i}^{O(3)}w_{4-i}^{O(2)}\\\
=&(w_{2}^{SO(3)}+t^{2})(xy+c^{2}+xr+cr)\\\ &+r({\rm
Sq}^{1}(w_{2}^{SO(3)})+tw_{2}^{SO(3)}+t^{3})\\\ &+(x+y)[{\rm
Sq}^{1}(w_{2}^{SO(3)})+w_{1}^{TM}(w_{2}^{SO(3)}+t^{2})]\end{split}$ (76)
where we have used the constraint $r+t=w_{1}^{TM}$ to obtain the last line.
The first line above is exactly what we expect from LSM constraints from Eqs.
(52)-(55), so our task now is to show that the last two lines vanish. This
follows from the following relations:
$\displaystyle r{\rm Sq}^{1}(w_{2}^{SO(3)})$ $\displaystyle=$ $\displaystyle
w_{1}^{TM}rw_{2}^{SO(3)}+r^{2}w_{2}^{SO(3)}$ $\displaystyle=$ $\displaystyle
trw_{2}^{SO(3)},$ $\displaystyle rt^{3}$ $\displaystyle=$ $\displaystyle
w_{1}^{TM}t^{3}+t^{4}=0,$ $\displaystyle w_{1}^{TM}(x+y)w_{2}^{SO(3)}$
$\displaystyle=$ $\displaystyle{\rm Sq}^{1}[(x+y)w_{2}^{SO(3)}]$
$\displaystyle=$ $\displaystyle(x+y){\rm Sq}^{1}(w_{2}^{SO(3)}),$
$\displaystyle w_{1}^{TM}(x+y)t^{2}$ $\displaystyle=$ $\displaystyle{\rm
Sq}^{1}[(x+y)t^{2}]=0.$ (77)
### VII.4 $N=7$: intertwining non-coplanar magnets with valence-bond solids
In spin systems, as we reviewed in Sec. III, the DQCP (SL(N=5)) naturally
describes the competition (or intertwining) between collinear magnetic and
valence-bond solid (VBS) orders, while the $U(1)$ Dirac spin liquid (SL(N=6))
naturally describes the intertwining between coplanar magnetic and VBS orders.
The natural extension to the intertwining between non-coplanar magnetic and
VBS orders is the $N=7$ Stiefel liquid state. Below we discuss two lattice
realizations of the SL(7) theory, one on triangular lattice and one on Kagome
lattice.
#### VII.4.1 Triangular lattice
We consider a triangular lattice with an odd number of half-integer spins per
site, so the LSM anomalies are given by Eqs. (52), (53), (54) and (55). These
conditions impose strong constraints on the allowed lattice realizations of
the $N=7$ SL theory. We now describe an embedding of the microscopic
symmetries to the $N=7$ SL theory that matches the anomaly. We specify the
embedding by the symmetry actions on the $SO(7)/SO(4)$ field $n_{ji}$, where
the $SO(7)$ symmetry acts on the left and the $SO(3)$ symmetry acts on the
right.
Figure 5: Triangular lattice and the relevant symmetries. Each filled red
circle represents an odd number of spin-1/2’s. The $C_{6}$ rotation is around
a site that host the spins, and the dashed line is the reflection axis of
$R_{y}$.
First, the on-site spin rotation $SO^{s}(3)$ symmetry acts as an $SO^{s}(3)$
subgroup of $SO(7)$:
$n\to\left(\begin{array}[]{cc}SO^{s}(3)&0\\\ 0&I_{4}\end{array}\right)n.$ (78)
Next we specify time-reversal symmetry as
$n\to\left(\begin{array}[]{cc}-I_{3}&0\\\ 0&I_{4}\end{array}\right)n,\hskip
10.0pti\to-i.$ (79)
For translations along the three unit vectors $T_{\bm{a}_{1}}$,
$T_{\bm{a}_{2}}$ and $T_{-\bm{a}_{1}-\bm{a}_{2}}$ (see Fig. 5), we have (we
shall use $\sigma_{\mu}^{i,j}$ to denote the $\mu$-th Pauli matrix acting on
the $i,j$ indices)
$\displaystyle T_{\bm{a}_{1}}:$ $\displaystyle
n\to\left(\begin{array}[]{ccc}I_{3}&0&0\\\
0&\exp\left(i\frac{2\pi}{3}\sigma_{y}^{4,5}\right)&0\\\
0&0&\exp\left(-i\frac{2\pi}{3}\sigma_{y}^{6,7}\right)\end{array}\right)n\left(\begin{array}[]{ccc}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{array}\right),$ (86) $\displaystyle T_{\bm{a}_{2}}:$
$\displaystyle n\to\left(\begin{array}[]{ccc}I_{3}&0&0\\\
0&\exp\left(i\frac{2\pi}{3}\sigma_{y}^{4,5}\right)&0\\\
0&0&\exp\left(-i\frac{2\pi}{3}\sigma_{y}^{6,7}\right)\end{array}\right)n\left(\begin{array}[]{ccc}-1&0&0\\\
0&1&0\\\ 0&0&-1\end{array}\right),$ (93) $\displaystyle
T_{-\bm{a}_{1}-\bm{a}_{2}}:$ $\displaystyle
n\to\left(\begin{array}[]{ccc}I_{3}&0&0\\\
0&\exp\left(i\frac{2\pi}{3}\sigma_{y}^{4,5}\right)&0\\\
0&0&\exp\left(-i\frac{2\pi}{3}\sigma_{y}^{6,7}\right)\end{array}\right)n\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&-1\end{array}\right).$ (100)
The $C_{6}=C_{2}\times C_{3}$ rotation is implemented as
$C_{6}:n\to\left(\begin{array}[]{ccccc}I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&-1&0&0\\\ 0&0&0&1&0\\\
0&0&0&0&-1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right).$ (101)
Finally, the reflection $\mathcal{R}_{y}$ (preserving $\bm{a}_{1}$ but
exchanging $\bm{a}_{2}$ with $-\bm{a}_{1}-\bm{a}_{2}$) acts as
$\mathcal{R}_{y}:n\to\left(\begin{array}[]{ccccc}I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\
0&0&0&0&-1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&1\\\ \end{array}\right).$ (102)
The symmetry actions are chosen so that all components of the $n_{ji}$ field
are nontrivial under some symmetry action. This makes the state somewhat
stable since the most fundamental fields are not allowed by symmetry as
perturbations. One can check that the conserved currents of the $SO(7)\times
SO(3)$ symmetry are also forbidden, which is important since these operators
have scaling dimension $2$ and are relevant. Whether the theory is actually a
stable phase depends on the (yet unknown) relevance or irrelevance of
composite operators like $n_{ji}n_{j^{\prime}i^{\prime}}$.
One can check that the symmetry actions are consistent with the group algebra
and indeed gives a homomorphism. One can also check the anomaly-matching
conditions as follows. First, we pull back the SW classes to the physical
symmetries:
$\displaystyle w_{1}^{O(7)}$ $\displaystyle=$ $\displaystyle t,$
$\displaystyle w_{2}^{O(7)}$ $\displaystyle=$ $\displaystyle
w_{2}^{SO(3)}+t^{2}+c^{2}+r^{2}+rc,$ $\displaystyle w_{4}^{O(7)}$
$\displaystyle=$
$\displaystyle(w_{2}^{SO(3)}+t^{2})(c^{2}+r^{2}+rc)+tcr(c+r),$ $\displaystyle
w_{1}^{O(3)}$ $\displaystyle=$ $\displaystyle r,$ $\displaystyle w_{2}^{O(3)}$
$\displaystyle=$ $\displaystyle xy+xr.$ (103)
Notice that the restriction $w_{1}^{TM}=w_{1}^{O(7)}+w_{1}^{O(3)}=r+t$ is
satisfied. Now plugging these into the IR anomaly Eq. (35), we obtain
$\displaystyle S_{\rm bulk}$ $\displaystyle=$ $\displaystyle
i\pi\int_{X_{4}}((w_{2}^{SO(3)}+t^{2})(xy+c^{2}+xr+rc)$ (104)
$\displaystyle+(c^{2}+r^{2}+rc)(r^{2}+xy+xr)+r^{4}$
$\displaystyle+tcr(c+r)+r^{2}(xy+xr)).$
We now examine the relations among these terms. Again, we set $ry=cx=cy=0$,
since the crystalline symmetries involved in each product do not commute. So
the second and third lines of the above anomaly become
$\displaystyle c^{2}r^{2}+cr^{3}+tcr(c+r)$ (105) $\displaystyle=$
$\displaystyle(r+t)cr(c+r)$ $\displaystyle=$ $\displaystyle w_{1}^{TM}cr(c+r)$
$\displaystyle=$ $\displaystyle{\rm Sq}^{1}(c^{2}r+cr^{2})$ $\displaystyle=$
$\displaystyle 0.$
So only the first line of Eq. (104) remains, which is exactly what is required
as discussed in Sec. VII.2.
How do we think of this $N=7$ SL theory on lattice? We can interpret the
$n_{ji}$ field as a collection of fluctuating order parameters, whose nature
is decided by their symmetry properties. The first three rows of the field
$n_{ji}$ ($1\leqslant j\leqslant 3$), being a triplet in $SO^{s}(3)$,
describes magnetic fluctuations at the three $\bm{M}$ points ($(\pi,0)$,
$(0,\pi)$ and $(\pi,\pi)$) in the Brillouin zone. A possible pattern of
symmetry-breaking order is
$\langle n\rangle\sim\left(\begin{array}[]{c}O_{3\times 3}\\\ 0_{4\times
3}\end{array}\right),$ (106)
where $O_{3\times 3}$ is a $3\times 3$ orthogonal matrix and $0_{4\times 3}$
is a zero-matrix. This describes a non-coplanar magnetic order, with the
expectation of the spin operator $S_{i}$ ($i=1,2,3$) on the site
$\bm{r}=n\bm{a}_{1}+m\bm{a}_{2}$ ($n,m\in\mathbb{Z}$)
$\langle S_{i}\rangle\sim
O_{i1}\cos[(n+m)\pi]+O_{i2}\cos(n\pi)+O_{i3}\cos(m\pi),$ (107)
where the three vectors $O_{i1},O_{i2},O_{i3}$ are by construction
orthonormal. This non-coplanar magnetic order is also known as the tetrahedral
order on triangular lattice.
The theory can also form a VBS order by condensing $n_{ji}$ with $4\leqslant
j\leqslant 7$. By Eq. (86) this VBS order has momentum $\bm{K}+\bm{M}$, which
is the same as the commonly studied $12$-site VBS on triangular lattice.
We therefore conclude that the SL(7) theory can naturally arise as a
competition (or intertwinement) between the tetrahedral magnetic order and the
$12$-site VBS order on triangular lattice. The tetrahedral order is known,
numerically, to arise in the $J_{1}-J_{2}-J_{\chi}$ model Gong _et al._
(2017), where $J_{1}$ and $J_{2}$ are the nearest and next-nearest neighbor
Heisenberg couplings, respectively, and $J_{\chi}$ is the spin chirality
$\bm{S}_{i}\cdot(\bm{S}_{j}\times\bm{S}_{k})$. This model, however, explicitly
breaks the time-reversal and reflection symmetries due to the chirality term.
Since time-reversal breaking perturbations are relevant for Dirac spin liquids
(SL(6)), it may also be relevant for SL(7). Therefore, to search for SL(7), it
may be useful to find a time-reversal invariant lattice Hamiltonian that
realizes the tetrahedral order, and study the effect of various perturbations
on top of it. We also note that the tetrahedral order may be realized in
higher-spin systems with additional $(\bm{S}_{i}\cdot\bm{S}_{j})^{2}$ coupling
Yu and Kivelson (2020). So it is also interesting to explore whether SL
physics can arise in those systems. A smoking-gun signature for the SL(7)
state is that both the non-coplanar magnetic and VBS order parameters (i.e.,
the $21$ matrix elements of $n$) are critical with identical critical
exponents, which is a consequence of the emergent $SO(7)\times SO(3)$ global
symmetry. Similar physics has been numerically confirmed for the SL(5) (i.e.,
DQCP) Nahum _et al._ (2015a).
#### VII.4.2 Kagome lattice
We now consider a Kagome lattice with spin-$1/2$ per site. The Kagome lattice
has the same lattice symmetries as the triangular. There are three spin-$1/2$
moments in each unit cell, so the LSM anomaly involving translation symmetries
is identical to the triangular case. The only essential difference with the
triangular lattice is the lack of spin moment at the $C_{6}$ rotation center.
So instead of Eq. (57), the full LSM anomaly on Kagome is
$i\pi\int_{X^{4}}(w_{2}^{SO(3)}+t^{2})(xy+rx).$ (108)
Figure 6: Kagome lattice and the relevant symmetries. Each filled red circle
represents an odd number of spin-1/2’s. In this case the rotation center of
$C_{6}$ does not host any spin. Again, the dashed line is the reflection axis
of $R_{y}$.
We now describe a symmetry embedding (a lattice realization) of the SL(7)
theory on Kagome lattice. The spin rotation and time-reversal act on the first
three rows of $n$, in the same way as the triangular realization following Eq.
(78) and (79). The translation symmetries act as (see Fig. 6)
$\displaystyle T_{\bm{a}_{1}}:$ $\displaystyle n\to
n\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right),$ (112)
$\displaystyle T_{\bm{a}_{2}}:$ $\displaystyle n\to
n\left(\begin{array}[]{ccc}-1&0&0\\\ 0&1&0\\\ 0&0&-1\end{array}\right),$ (116)
$\displaystyle T_{-\bm{a}_{1}-\bm{a}_{2}}:$ $\displaystyle n\to
n\left(\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\ 0&0&-1\end{array}\right).$ (120)
The $C_{6}$ rotation acts as
$C_{6}:n\to\left(\begin{array}[]{ccccc}-I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\
0&0&0&0&1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right).$ (121)
Finally, the $\mathcal{R}_{y}$ reflection acts the same way as the triangular
case:
$\mathcal{R}_{y}:n\to\left(\begin{array}[]{ccccc}I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\
0&0&0&0&-1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&1\\\ \end{array}\right).$ (122)
We can now go through the anomaly calculation in a similar way as our earlier
examples. We shall omit the intermediate steps here and simply state that the
final result is indeed the required anomaly in Eq. (108).
Similar to the triangular case, we can again interpret this SL(7) theory as a
result of competition between non-coplanar magnetic and VBS orders, now both
at momenta $\bm{M}$ from Eq. (112). The non-coplanar magnetic order also has a
$C_{6}$ angular momentum $l=1$. This is known as the cuboctahedral order (more
precisely, the cuboc1 order in the language of Ref. Messio _et al._ (2011)).
This order has been numerically found for the $J_{1}-J_{2}-J_{3}$ Heisenberg
model in certain regimes Gong _et al._ (2015). Our results motivate further
exploration of the phase diagram near the cuboctahedral order, and see if the
SL(7) state could be realized. Again, just like the realization on the
triangular lattice, a smoking-gun signature of the SL(7) state is that the
non-coplanar magnetic and VBS order parameters are critical and have identical
critical exponent, due to the emergent $SO(7)\times SO(3)$ symmetry.
## VIII Discussion
In this paper, based on a nonlinear sigma model defined on a Stiefel manifold,
$SO(N)/SO(4)$, supplemented with a Wess-Zumino-Witten (WZW) term, we have put
forward the theory of Stiefel liquids, which are a family of critical quantum
liquids that have many extraordinary properties. For example, they have a
large emergent symmetry, a cascade structure, and nontrivial quantum
anomalies. Some of these Stiefel liquids are argued to be dual to the well
known deconfined quantum critical point and $U(1)$ Dirac spin liquid, and
others are conjectured to be non-Lagrangian, i.e., its corresponding RG fixed
point cannot be described by any weakly-coupled mean-field theory at any
scale, which, in particular, means that these states are beyond parton (mean-
field) construction widely used in the study of exotic quantum phases and
phase transitions.
We make some comments on why the “non-Lagrangian” conjecture for the
$N\geqslant 7$ Stiefel liquids may be reasonable. The most obvious gauge
theory candidates for such WZW fixed points are some kinds of QCD3 with
gapless Dirac fermions coupled to some gauge fields. However, as we review in
Appendix D, typical QCD3 correspond to WZW theories defined on Grassmannian
manifolds $G(2N)/G(N)\times G(N)$, where $G$ can be $U,SU,USp,SO$. In fact,
this is also why Stiefel liquids with $N=5,6$ do have gauge theory
descriptions, since the corresponding Stiefel manifolds in these two cases
also happen to be some kinds of Grassmannian:
$SO(5)/SO(4)=USp(4)/(USp(2)\times USp(2))$ and $SO(6)/SO(4)=SU(4)/(SU(2)\times
SU(2))$. For $SO(N\geqslant 7)/SO(4)$, we do not have such identification, so
the corresponding Stiefel liquids are not captured by some simple QCD3. We
also note the special role played by the $SO(N-4)$ symmetry in Stiefel
liquids. For $N=6$, this $SO(2)$ symmetry is realized in the gauge theory as
the flux conservation symmetry of the dynamical $U(1)$ gauge field. It is not
clear how this $SO(2)$ flux conservation symmetry could be generalized to
higher $SO(N-4)$ in different gauge theories. Besides these constraints from
symmetries, the intricate anomaly structures of the Stiefel liquids discussed
in Sec. VI also impose further nontrivial constraints on its possible
renormalizable-Lagrangian description. One possibility is that the $N\geqslant
7$ Stiefel liquids can be realized by gauge theories with significantly lower
symmetries in the UV Lagrangians, and the full IR symmetries emerge through
some nontrivial dynamics. This scenario will be hard to rule out, and if true,
it will likely shed new light on the dynamics of $(2+1)$-d gauge theories.
We mention that the fixed points of some quantum loop models were also
proposed to be non-Lagrangian Freedman _et al._ (2004, 2005); Dai and Nahum
(2020). The nature of such loop quantum criticality appears to be very
different from those studied in this paper. For example, they are not Lorentz
invariant.
Note that although the most commonly used parton approach uses canonical
bosonic/fermionic partons to construct a (non-interacting) mean field, there
are also some parton approaches that use other types of partonic DOFs, in
particular, ones that are subject to some constraints and are strongly
fluctuating at all scales (see, e.g., Ref. Xu and Sachdev (2010)). One may
wonder if the latter constrained-parton-based approach can lead to a
construction to the non-Lagrangian Stiefel liquids. Although it is not ruled
out, we believe this approach is difficult, and even if it can be achieved,
novel ideas are still needed to make it work. This is because: i) As far as we
know, all states constructed with constrained partons ultimately fall into the
paradigm of mean field plus weak fluctuations, but these (conjectured) non-
Lagrangian states are beyond this paradigm. ii) More technically, in such an
approach, one often (if not always) encounters Dirac fermions coupled to sigma
fields (i.e., bosonic fields subject to some constraints), but these sigma
fields live in certain Grassmannian manifold, which is difficult to be
converted into a Stiefel manifold relevant here, unless some nontrivial
mathematical facts can be used, in a way similar to the case of Stiefel
liquids with $N=6$.
In Sec. VII, we have proposed an approach based on the hypothesis of
emergibility, which is complementary to the conventional parton approach, to
study quantum phases and phase transitions. This approach is benchmarked with
some known examples, and then applied to predict that spin-1/2 triangular and
Kagome lattices can host one of the non-Lagrangian Stiefel liquids. This
approach can also predict some detailed properties of such lattice
realizations of these Stiefel liquids, such as the quantum numbers of various
critical order parameters with identical scaling exponents.
It may be useful to comment on the parton approach and compare it with our
anomaly-based approach. The parton approach is explicit, concrete, and
relatively easy to manipulate, and it has led to tremendous success and deep
insights in the study of strongly-correlated quantum matter. However, this
parton approach also has some drawbacks. More specifically, there are two
common treatments of a parton construction, leading to a projected wave
function and an effective gauge theory, respectively. The wave-function-based
treatment starts with an enlarged Hilbert space of the partons, and performs a
projection of a valid ground state of these partons, in order to return to the
physical Hilbert space. Although the projected wave function is indeed in the
physical Hilbert space, a priori, it may not describe any ground state of a
local Hamiltonian in the physical Hilbert space, and it is unclear what
universal properties it exhibits – these have to be checked case by case, say,
using numerical calculations. On the other hand, from the effective gauge
theory, it is more analytically tractable to deduce what ground state it
describes and what universal properties it possesses. However, a priori, it is
unclear whether such an effective gauge theory can really emerge from the
physical Hilbert space, although it is often assumed so without any rigorous
analytically controlled justification151515We note that sometimes such an
effective gauge theory is phrased in terms of a lattice gauge theory, where
the partons hop on the lattice sites and are coupled to the gauge fields
living on the lattice links. In such an interpretation, the physical Hilbert
space is often manifestly just a gauge-fluctuation-free subspace of the
Hilbert space of the lattice gauge theory, where certain hard gauge
constraints are imposed.. Our anomaly-based approach may be more abstract by
the contemporary standards, but it is closer to the intrinsic characterization
of the universal many-body physics discussed in the introduction, and it
directly hinges on the emergibility: states that fail to satisfy the anomaly-
matching condition are necessarily not emergible, although at this stage we
cannot rigorously prove that states that do satisfy the anomaly-matching
condition must be emergible. We also note that the parton approach is in fact
also essentially an attempt to verify anomaly-matching between the IR theory
and the microscopic setup, but through an explicit mean-field-like
construction.
Let us also comment on the significance of non-Lagrangian, or intrinsically
non-renormalizable, theories specifically in condensed matter physics. Since
most condensed matter systems do have some natural UV cutoffs, the concept of
renormalizability should not play a fundamental role in condensed matter
physics. This means that we should have a theoretical framework that can
handle both Lagrangian and non-Lagrangian theories – there should be no
intrinsic difference between the two. However, the fact is that not only we do
not have many tools to analyze non-Lagrangian theories, we did not even have
many serious examples of non-Lagrangian theory prior to this work. From this
perspective, what is really surprising is how difficult it was to find such
non-Lagrangian examples – this is perhaps rooted in our heavy reliance on
perturbative quantum field theories in the past. The examples found here may
also require and inspire us to develop new tools to analyze strongly
correlated systems in more intrinsic manners. One example is the problem of
“emergibility”, which was the focus of the later half of our paper: the
intrinsic non-renormalizability, or lack of mean-field construction, forced us
to further develop the anomaly-matching approach which may become useful for
future works on strongly correlated systems in general. Therefore, even though
renormalizability per se is not of fundamental importance in condensed matter
physics, being able to go beyond renormalizable theories is. Our work on
(likely) non-Lagrangian quantum criticality represents a step toward this
ambitious goal.
We finish this paper with some interesting open questions that we leave for
future work.
1. 1.
Although we have given a derivation of the effective theory of some of the
Stiefel liquids based on the gauge theoretic descriptions of Dirac spin
liquids, it is desirable to give a more explicit derivation. For example, it
is desirable to explicitly derive the expression of Skyrmion current in Eq.
(23) in terms of the $\mathcal{P}$ field. Also, it is nice to show how the WZW
term on the Grassmannian manifold reduces to that on the Stiefel manifold.
2. 2.
The quantum anomalies of the Stiefel liquids labeled with an even $N$ have not
been fully pinned down. It is interesting to finish this anomaly analysis. For
the purpose of studying Stiefel liquids, using their cascade structure may be
sufficient to fully determine their anomalies, just as what we have done in
this paper. However, we note that deriving the full quantum anomalies directly
based on the WZW action is also an intriguing theoretical challenge.
3. 3.
Although we have argued that the Stiefel liquids labeled by $N\geqslant 6$ can
flow to a conformally invariant fixed point under RG, we have not been able to
establish this with a controlled analysis. It is well motivated to find ways
to systematically study the IR dynamics of the Stiefel liquids. For example,
it is natural to ask if the Stiefel liquid fixed points predicted in this work
can be found in numerical conformal bootstrap. The large emergent symmetry
group $SO(N)\times SO(N-4)$ and the likely irrelevance of singlet operators
may provide some reasonable starting point for such investigation. Also, given
that the Stiefel liquids are described by a matrix model, it is interesting to
explore if they have any holographic dual.
4. 4.
The conjecture that the Stiefel liquids with $N>6$ are non-Lagrangian has not
been proved. It is important to prove or falsify it. We expect that the method
to prove or falsify it will necessarily bring in useful general insights.
Also, even if it is falsified, these Stiefel liquids are still interesting.
Relatedly, although a wave function is not strictly necessary for the
intrinsic characterization of the universal physics of a many-body system, it
may be interesting and useful to find a wave function for these Stiefel
liquids.
5. 5.
It is natural to ask whether similar WZW models on target manifolds other than
the Stiefel can lead to interesting fixed points. The most natural manifolds
are the Grassmannians, which have been studied Bi _et al._ (2016) and related
to various gauge theories Komargodski and Seiberg (2018) (see also Appendix
D). It will be interesting to either better understand the Grassmannian
theories, or to contemplate on theories based on other types of manifolds.
6. 6.
The hypothesis of emergibility has not been proved rigorously, and it is
important to prove or falsify it. If it can be proved, or at least further
justified, it can be applied to other cases to study other exotic quantum
phases and phase transitions. We expect this approach to give rise to many
more interesting results and novel insights in the future. If it will be
falsified, it is still very useful to find the correct general rules that
govern the emergibility of a given low-energy theory for a system.
7. 7.
Perhaps the most important question is whether some of the critical Stiefel
liquids can be realized in real materials. We have suggested that the $N=7$
Stiefel liquid may arise near certain non-coplanar magnetic orders.
Numerically those non-coplanar orders can arise in some relatively simple
lattice spin models Gong _et al._ (2015); Yu and Kivelson (2020); Gong _et
al._ (2017). It will be fascinating to explore further in the phase diagram of
those systems and see if the Stiefel liquid can indeed be found. This may
provide valuable guidance towards ultimate experimental realizations.
8. 8.
We have seen that the theories of the deconfined quantum critical point and
the $U(1)$ Dirac spin liquid can be formulated in terms of local DOFs. It may
be interesting to see if other exotic non-quasiparticle critical quantum
liquids can also be formulated in a similar way, and such a new formulation
may bring in new insights. For example, can the theory of a Fermi surface
coupled to a $U(1)$ gauge field in $(2+1)$-d be formulated purely in terms of
local DOFs? It is likely that such a formulation will explicitly involve the
infinitely many collective excitations around the Fermi surface, and the ideas
from Ref. Mross and Senthil (2011) may be useful.
###### Acknowledgements.
We thank Maissam Barkeshli, Zhen Bi, Vladimir Calvera, Davide Gaiotto, Meng
Guo, Chao-Ming Jian, Theo Johnson-Freyd, Steve Kivelson, John McGreevy, Subir
Sachdev, Cenke Xu, Weicheng Ye and Yi-Zhuang You for illuminating discussions.
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Industry Canada and
by the Province of Ontario through the Ministry of Colleges and Universities.
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## Appendix A More on the proposed WZW action
In the main text a WZW action for the $(2+1)$-d system of our interest is
proposed in Eq. (6). In this appendix we present more details on its
mathematical aspects.
For any even integer $d\geqslant 0$, we can define a WZW term for a $d+1$
(spacetime) dimensional system on a Stiefel manifold $V_{N,N-(d+2)}\equiv
SO(N)/SO(d+2)$, where $N\geqslant d+3$. An element on this Stiefel manifold
can be parameterized by an $N$-by-$\left(N-(d+2)\right)$ matrix, $n$, such
that its columns are orthonormal, i.e., $n^{T}n=I_{N-(d+2)}$. The
corresponding WZW action is
$\displaystyle S_{\rm
WZW}^{(N,d)}[n]=\frac{2\pi}{\Omega_{d+2}}\int_{0}^{1}du\int
d^{d+1}x\sum_{k_{1},k_{2},\cdots,k_{\frac{d+2}{2}}=1}^{N-(d+2)}\det(\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})})$
(123)
where $\Omega_{d+2}$ is the volume of $S^{d+2}$ with unit radius, and the
$N$-by-$N$ matrix $\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})}$ is
given by
$\displaystyle\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})}=(n,\partial_{x_{1}}n_{k_{1}},\partial_{x_{2}}n_{k_{1}},\partial_{x_{3}}n_{k_{2}},\partial_{x_{4}}n_{k_{2}},\cdots,\partial_{x_{d+1}}n_{k_{\frac{d+2}{2}}},\partial_{u}n_{k_{\frac{d+2}{2}}})$
(124)
where $x_{1,2,\cdots,x_{d+1}}$ is the coordinate of the physical spacetime,
$n_{k_{i}}$ is the $k_{i}$th column of $n$ (note that the repeated subscripts
$k_{i}$’s are not summed over in the right hand side of (124)). That is, the
first $N-(d+2)$ columns of
$\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})}$ is just $n$, and its last
$d+2$ columns are derivatives of the columns of $n$ arranged in the above way.
More explicitly,
$\displaystyle\begin{split}\det&(\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})})=\frac{1}{(N-(d+2))!}\epsilon^{i_{1}i_{2}\cdots
i_{N-(d+2)}}\epsilon^{j_{1}j_{2}\cdots
j_{N}}n_{j_{1}i_{1}}n_{j_{2}i_{2}}\cdots n_{j_{N-(d+2)}i_{N-(d+2)}}\\\
&\cdot\partial_{x_{1}}n_{j_{N-(d+2)+1}k_{1}}\partial_{x_{2}}n_{j_{N-(d+2)+2}k_{1}}\partial_{x_{3}}n_{j_{N-(d+2)+3}k_{2}}\partial_{x_{4}}n_{j_{N-(d+2)+4}k_{2}}\cdots\partial_{x_{d+1}}n_{j_{N-1}k_{\frac{d+2}{2}}}\partial_{u}n_{j_{N}k_{\frac{d+2}{2}}}\end{split}$
(125)
where the $\epsilon$’s are the fully anti-symmetric symbols with rank
$N-(d+2)$ and $N$, respectively.
It is straightforward to see that the WZW term presented in the main text is
precisely the special case of the above one with $d=2$, and in the main text
we denote $V_{N,N-4}$ by $V_{N}$. Also, it is clear that such a term can be
defined only if $d$ is even, and it is interesting to compare this observation
with the fact that the form of the homotopy groups of the Stiefel manifold
$V_{N,N-(d+2)}$ is qualitatively different for even $d$ and odd $d$. That is,
the first nontrivial homotopy group of $V_{N,N-(d+2)}$ is
$\displaystyle\pi_{d+2}V_{N,N-(d+2)}=\left\\{\begin{array}[]{lr}\mathbb{Z},&d+2\
{\rm even\ or\ }N=d+3\\\ \mathbb{Z}_{2},&d+2\ {\rm odd\ and\
}N>d+3\end{array}\right.$ (128)
The validity of the above WZW term requires it to be the integral of the
pullback of a closed $(d+2)$-form on $V_{N,N-(d+2)}$. The $(d+2)$-form on
$V_{N,N-(d+2)}$ that this WZW term is associated with is
$\displaystyle\begin{split}\omega&=\frac{1}{(N-(d+2))!}\epsilon^{i_{1}i_{2}\cdots
i_{N-(d+2)}}\epsilon^{j_{1}j_{2}\cdots
j_{N}}n_{j_{1}i_{1}}n_{j_{2}i_{2}}\cdots n_{j_{N-(d+2)}i_{N-(d+2)}}\\\
&\quad\cdot dn_{j_{N-(d+2)+1}k_{1}}\wedge dn_{j_{N-(d+2)+2}k_{1}}\wedge
dn_{j_{N-(d+2)+3}k_{2}}\wedge dn_{j_{N-(d+2)+4}k_{2}}\cdots\wedge
dn_{j_{N-1}k_{\frac{d+2}{2}}}\wedge dn_{j_{N}k_{\frac{d+2}{2}}}\end{split}$
(129)
where the repeated subscripts $k_{i}$’s are summed over. It can be shown that
the above form is indeed closed 161616We thank Vladimir Calvera for giving a
mathematical proof to the closedness..
It remains to fix the normalization factor in front of the WZW term. We start
with two observations:
1. 1.
For $N=d+3$ Eq. (123) is the familiar WZW term on $S^{d+2}$ with the correct
normalization factor.
2. 2.
For $N>d+3$, if we fix the first column of $n$ to a constant, say
$n_{1}=(1,0,0...)^{T}$, the proposed WZW term for $V_{N,N-(d+2)}$ becomes that
for $V_{N-1,(N-1)-(d+2)}$.
Mathematically, fixing the first column of $n$ describes an inclusion map
$i:V_{N-1,N-d-3}\to V_{N,N-d-2}$:
$\displaystyle n_{(N-1)\times(N-d-3)}\to\left(\begin{array}[]{cc}1&0\\\
0&n_{(N-1)\times(N-d-3)}\end{array}\right).$ (132)
This map then induces a homomorphism between the homotopy groups
$\pi_{d+2}(V_{N-1,N-d-3})\to\pi_{d+2}(V_{N,N-d-2})$. Based on the two
observations made above, the normalization factor in Eq. (123) will be
justified if the homomorphism
$\pi_{d+2}(V_{N-1,N-d-3})\to\pi_{d+2}(V_{N,N-d-2})$ induced by $i$ is an
isomorphism. The last statement can be proved using the long exact sequence of
homotopy groups associated with the fibration
$\displaystyle V_{N-1,N-d-3}\to V_{N,N-d-2}\to S^{N-1}.$ (133)
A pictorial consequence of the above argument is that a “generator” of
$\pi_{d+2}(V_{N,N-(d+2)})$ is given by fixing the entries of the first
$N-(d+2)-1$ columns of $n$ to be $n_{ji}=\delta_{ji}$, and letting the last
column, which now lives on $S^{d+2}$ with unit radius, wrap around the
$S^{d+2}$ once.
Some properties of the WZW term for the case with $d=2$ are discussed in Sec.
IV.1, and with minor modifications many of them also apply appropriately to
the case with a general even $d$.
## Appendix B A gauge theory description of SL(N=5,k)
In this appendix, we show that SL(N=5,k) has a natural gauge-theoretic
description, i.e., a QCD3 theory with $N_{f}=2$ flavor of fermions interacting
with a $USp(2k)$ gauge field. A special case of $k=1$, i.e., SL(5) or the
DQCP, was already discussed in Ref. Wang _et al._ (2017).
The global symmetry of the $N_{f}=2$ $USp(2k)$ QCD3 theory is
$USp(2N_{f})=USp(4)\cong SO(5)$, which is identical to SL(5,k). The fermions
are in the fundamental representation of the $USp(2k)$ gauge group and
$USp(4)$ global symmetry (i.e., spinor representation of $SO(5)$). The gauge
invariant operators with lowest scaling dimensions shall be the fermion mass
terms, which are $SO(5)$ vector and $SO(5)$ singlet. The $SO(5)$ vector mass
can be identified as the order parameter field $n$ of the NLSM. To see the
relation more explicitly, we can couple the $SO(5)$ vector mass to a $SO(5)$
bosonic vector $n_{i}$, and then integrate out fermions. This will yield an
$SO(5)$ NLSM, and the original $USp(2k)$ gauge field is expected to just
confine since it does not couple to any low-energy degrees of freedom.
Moreover, due to the Abanov-Wiegmann mechanism Abanov and Wiegmann (2000),
integrating out fermions also yields a level-$k$ WZW term of the $SO(5)$
vector $n_{i}$. The level-$k$ comes from the fact that there are $k$ copies of
fermions as the gauge group is $USp(2k)$. Therefore, we have derived that the
$N_{f}=2$ $USp(2k)$ QCD3 theory is dual to SL(N=5,k).
From the gauge-theoretic description of SL(N=5,k), we can gain some intuition
about the properties of SL with a general $(N,k)$:
1. 1.
Stability. It is clear that the larger $k$ is, it is more likely that the QCD3
theory will confine. Naturally, we expect that this feature will also hold for
$N>5$: for a given $N$ there exists a critical $k_{c}$, such that the SL can
flow into a critical phase when $k\leqslant k_{c}$.
2. 2.
Neighboring topological order. By turning on a time-reversal-breaking singlet
mass, the SL(N=5,k) will become the $USp(2k)_{\mp 1}^{s}$ 171717The
superscript $s$ refers to the fact that the gauge field is a spin gauge field.
TQFT, which is dual to the $USp(2)_{\pm k}=SU(2)_{\pm k}$ TQFT. $SU(2)_{1}$ is
just the semion topological order discussed in the main text. In Sec. VI.5, we
have argued that SL(N,1) with a general $N\geqslant 5$ can flow to the
$SU(2)_{\pm 1}$ TQFT under a time-reversal-breaking deformation. So it is
possible that the SL(N>5,k) will also flow to the $SU(2)_{\pm k}$ TQFT under
appropriate time-reversal-breaking perturbation.
3. 3.
Higgs descendent. As shown in Sec. IV B of Ref. Wang _et al._ (2017), for the
case with $k=1$, adding a flavor-singlet Higgs field can break the gauge
structure to $U(k)$ with $k=1$, and the resulting theory is equivalent to a
QED3 coupled to 4 gapless Dirac fermions, with one of the six monopole
operators added to the Lagrangian, i.e., schematically we have $USp(2)+{\rm
Higgs}=U(1)+{\rm monopole}$. This is how the DQCP is related to the $U(1)$
DSL. The argument there can actually be generalized to any $k$ to show that
$USp(2k)+{\rm Higgs}=U(k)+{\rm monopole}$. This is nicely compatible with the
cascade structure of the SLs, and our results that SL(5,k) and SL(6,k) can
also be described by $USp(2k)$ gauge theory and $U(k)$ gauge theory,
respectively.
4. 4.
Microscopic realization. For a spin-$k/2$ system, there is a natural parton
construction for the SL(N=5,k).
We first fractionalize spin operators into partons,
$S^{i}=-\frac{1}{4}\textrm{Tr}(X^{\dagger}X\sigma^{i}),$ (134)
with $X$ being a $2k\times 2$ matrix,
$X=\left(\begin{matrix}\psi_{1}^{\dagger}&\cdots&\psi^{\dagger}_{k}&\psi^{\dagger}_{k+1}&\cdots&\psi^{\dagger}_{2k}\\\
\psi_{k+1}&\cdots&\psi_{2k}&-\psi_{1}&\cdots&-\psi_{k}\end{matrix}\right)^{T}.$
(135)
Here $\psi_{i}^{\dagger}$ is the fermion creation operator. $X$ satisfies the
reality condition $X^{*}=\Omega^{c}X\Omega^{s}$ with
$\Omega^{c}=\left(\begin{matrix}0&I_{k}\\\
-I_{k}&0\end{matrix}\right),\quad\Omega^{S}=\left(\begin{matrix}0&1\\\
-1&0\end{matrix}\right).$ (136)
Here $I_{k}$ is a $k\times k$ identity matrix. So the spin operators can be
written as
$S^{i}=-\frac{1}{4}\textrm{Tr}(\Omega^{s}X^{T}\Omega^{c}X\sigma^{i})$. It is
apparent that this parton decomposition has a $USp(2k)$ gauge invariance,
namely, the spin operators are invariant under a $USp(2k)$ (left) rotation of
$X$, $RX$, as $R^{T}\Omega^{c}R=\Omega^{c}$. On the other hand, the $SO(3)$
spin rotation acts as the $USp(2)\cong SU(2)$ (right) rotation of $X$. The
local contraint of the parton construction is,
$X\Omega^{s}X^{T}=-\Omega^{c},$ (137)
or equivalently, $\psi_{i}^{\dagger}\psi_{i}=\psi_{i+k}^{\dagger}\psi_{i+k}$
for $i=1,\cdots,k$. One can further show that the spin operators defined in
Eq. (134) satisfy $[S^{i},S^{j}]=i\varepsilon^{ijk}S^{k}$ and
$\sum_{i}(S^{i})^{2}=\frac{k}{2}\left(\frac{k}{2}+1\right)$.
Therefore, the above parton construction has an emergent $USp(2k)$ gauge
structure, and the fermions $(\psi_{1},\cdots,\psi_{k})$ form a $USp(2k)$
fundamental. Putting $\psi_{i}$ fermions into a band structure with two Dirac
cones, we will get the $N_{f}=2$ $USp(2k)$ QCD3, or equivalently, SL(5,k).
So it is natural to look for SL(5,k) in a spin-$k/2$ system. Furthermore, as
proposed in Sec. V, SL(6,k) is dual to the $N_{f}=2$ $U(k)$ QCD3, so it is
also possible that SL(6,k) can emerge in a spin-$k/2$ system. This is
discussed in detail recently Calvera and Wang (2020). These observations lead
us to further conjecturing that it is also true SL(N>6,k) can also emerge in a
spin-$k/2$ system. Indeed, in Sec. VII.4, we have argued that SL(7) can emerge
in a spin-1/2 system.
## Appendix C Full quantum anomaly of the DQCP
The quantum anomaly of the DQCP, or equivalently, SL(5), was partially
analyzed in Ref. Wang _et al._ (2017), and an anomaly associated with the
$SO(5)$ symmetry was found, which is described by a $(3+1)$-d topological
response function, $i\pi\int_{M}w_{4}^{SO(5)}$, where $M$ is the closed
manifold in which the $(3+1)$-d bulk corresponding to the DQCP lives, and
$w_{4}^{SO(5)}$ is the fourth Stiefel-Whitney (SW) class of the $SO(5)$ gauge
bundle that couples to this bulk.
Besides the $SO(5)$ symmetry, the DQCP also enjoys a time reversal symmetry,
$\mathcal{T}$, which also contributes to the anomaly. To understand the full
anomaly associated with both the $SO(5)$ and $\mathcal{T}$ symmetries, it is
useful to enlarge the $SO(5)$ symmetry to $O(5)$ by including the improper
$Z_{2}$ rotation. Then the action of the time reversal symmetry is a
combination of this improper $Z_{2}$ rotation and a flip of the time
coordinate. In this appendix, we show that the full anomaly of the DQCP is
described by a $(3+1)$-d topological response function
$\displaystyle S=i\pi\int_{M}w_{4}^{O(5)}$ (138)
with a constraint $w_{1}^{O(5)}=w_{1}^{TM}\ ({\rm mod\ }2)$, where $TM$
denotes the tangent bundle of $M$. This constraint simply indicates that the
improper $Z_{2}$ rotation of the $O(5)$ symmetry is accompanied with a flip of
the time coordinate. For notational brevity, in the following we will suppress
the superscript “$TM$” of a SW class of $TM$.
To obtain the above result, let us look at the most general form that the
anomaly can take:
$\displaystyle
S=i\pi\int_{M}\left[a_{1}w_{4}^{O(5)}+a_{2}[w_{2}^{O(5)}]^{2}+a_{3}w_{1}^{2}w_{2}^{O(5)}+a_{4}w_{1}^{4}+a_{5}w_{2}^{2}\right]$
(139)
where we have used the (mod 2) relations $w_{1}^{O(5)}=w_{1}$,
$w_{1}w_{3}=w_{1}^{4}+w_{2}^{2}+w_{4}=0$, $w_{1}w_{2}=0$,
$[w_{2}^{O(5)}]^{2}=(w_{2}+w_{1}^{2})w_{2}^{O(5)}$, ${\rm
Sq}^{1}(w_{3}^{O(5)})=w_{1}w_{3}^{O(5)}$, ${\rm Sq}^{1}\cdot{\rm
Sq}^{1}(w_{2}^{O(5)})=0$, and $w_{3}^{O(5)}={\rm
Sq}^{1}(w_{2}^{O(5)})+w_{1}^{O(5)}w_{2}^{O(5)}$ to remove some terms.
The above topological response function must satisfy the following known
properties of the DQCP Vishwanath and Senthil (2013); Kapustin (2014); Wang
_et al._ (2017); Bi _et al._ (2015), which help us to deduce the values of
the $a$’s unambiguously:
1. 1.
As mentioned above, if only the $SO(5)$ symmetry is considered, the anomaly is
described by $i\pi\int_{M}w_{4}^{SO(5)}$.
In this case, $w_{4}^{O(5)}=w_{4}^{SO(5)}$, $w_{1}=0$, and
$w_{2}^{O(5)}=w_{2}^{SO(5)}$. So Eq. (139) becomes
$S=i\pi\int_{M}\left[a_{1}w_{4}^{SO(5)}+a_{2}[w_{2}^{SO(5)}]^{2}+a_{5}w_{2}^{2}\right]$,
which implies that $a_{1}=1$ and $a_{2}=a_{5}=0$.
2. 2.
Ignoring the $SO(5)$ symmetry and implementing $\mathcal{T}$ by
$n\rightarrow-n$, there is an anomaly associated with $\mathcal{T}$, described
by $i\pi\int_{M}w_{1}^{4}$. The corresponding bulk is known as $eTmT$ Wang and
Senthil (2013).
In this case, $w_{4}^{O(5)}=w_{1}^{4}$ and $w_{2}^{O(5)}=0$. So Eq. (139)
becomes $S=i\pi\int_{M}(1+a_{4})w_{1}^{4}$, which implies that $a_{4}=0$.
3. 3.
Suppose the full symmetry is broken to $SO(2)\times\mathcal{T}$, where the
$SO(2)$ rotates the first 2 components of $n$ and $\mathcal{T}$ flips its last
3 components, the anomaly is described by
$S_{3}=i\pi\int_{M}w_{1}^{2}w_{2}^{SO(2)}$. The corresponding bulk is known as
$eCmT$ Wang and Senthil (2013); Metlitski _et al._ (2013).
In this case, $w_{4}^{O(5)}=w_{1}^{2}w_{2}^{SO(2)}$ and
$w_{2}^{O(5)}=w_{2}^{SO(2)}+w_{1}^{2}$. So Eq. (139) becomes
$S=i\pi\int_{M}\left[(1+a_{3})w_{1}^{2}w_{2}^{SO(2)}+a_{3}w_{1}^{4}\right]$,
which implies that $a_{3}=0$.
In summary, the above three conditions imply that $a_{1}=1$ and
$a_{2}=a_{3}=a_{4}=a_{5}=0$. So the full anomaly of the DQCP is described by
Eq. (138).
## Appendix D WZW models on the Grassmannian manifold
$\frac{G(2N)}{G(N)\times G(N)}$
The Grassmannian manifolds $\frac{G(2N)}{G(N)\times G(N)}$ (with
$G=U,SU,SO,USp$) also have $\pi_{4}(\frac{G(2N)}{G(N)\times G(N)})=\mathbb{Z}$
and $\pi_{3}(\frac{G(2N)}{G(N)\times G(N)})=0$, so one can define $2+1$-d WZW
models on these Grassmannians. Using the argument in Sec. IV.4 we can obtain a
similar phase diagram (at least for large $N$). Namely, as one tunes the
coupling constant of NLSM, there are three fixed points: 1) an attractive
fixed point of a spontaneous-symmetry-breaking phase, with the ground state
manifold being $\frac{G(2N)}{G(N)\times G(N)}$; 2) a repulsive fixed point of
order-disorder transition; 3) an attractive fixed point of a critical quantum
liquid. The last attractive fixed point is the Grassmannian version of our
proposed SLs. Interestingly, the Grassmannian WZW models have simple
candidates of renormalizable Lagrangian descriptions, i.e., Dirac fermions
coupled to non-Abelian gauge fields Komargodski and Seiberg (2018). More
concretely, we have
1. 1.
The QCD3 theory with $N_{f}=2N$ Dirac fermions coupled to a $SU(k)$ gauge
field is a UV completion of the $\frac{U(2N)}{U(N)\times U(N)}$ NLSM model
with a level $k$ WZW term.
2. 2.
The QCD3 theory with $N_{f}=2N$ Majorana fermions coupled to a $SO(k)$ gauge
field is a UV completion of the $\frac{SO(2N)}{SO(N)\times SO(N)}$ NLSM model
with a level $k$ WZW term.
3. 3.
The QCD3 theory with $N_{f}=2N$ Dirac fermions coupled to a $USp(2k)$ gauge
field is a UV completion of the $\frac{USp(4N)}{USp(2N)\times USp(2N)}$ NLSM
model with a level $k$ WZW term.
4. 4.
The QCD3 theory with $N_{f}=2N$ Dirac fermions coupled to a $U(k)$ gauge field
is a UV completion of the $\frac{SU(2N)}{SU(N)\times SU(N)}$ NLSM model with a
level $k$ WZW term.181818This one is relatively new and will be discussed more
carefully elsewhere.
One may expect to see this correspondence by using the trick that appears
several times in the paper. We first couple the color-singlet fermion mass of
the QCD3 theory to a bosonic field that lives on the Grassmannian
$\frac{G(2N)}{G(N)\times G(N)}$, and then integrate out fermions. This will
give a NLSM of the bosonic field and will also generate a WZW term Abanov and
Wiegmann (2000). At last, the gauge field will confine by itself without doing
anything to the Grassmannian $\frac{G(2N)}{G(N)\times G(N)}$ WZW models. The
level $k$ (instead of 1) comes from the color multiplicity of the gauge field.
It is worth mentioning if one couples the fermion mass to a field living on
$\frac{G(2N)}{G(2N-M)\times G(M)}$ with $M\neq N$, integrating out Dirac
fermions will generate a Chern-Simons term for the gauge field as well. In
this case the gauge field will not confine, and one ends up with the
$\frac{G(2N)}{G(2N-M)\times G(M)}$ WZW model coupled to a Chern-Simons gauge
field.
We can also compare the global symmetry of the gauge theories and the
Grassmannian WZW models. The simplest one is the last case, where both the
gauge theory and the Grassmannian WZW models have an explicit $USp(4N)$ global
symmetry. For the first case, the $SU(k)$ gauge theory has an explicit
$SU(2N)\times U(1)$ symmetry, and the $U(1)$ symmetry is carried by the baryon
operator. The $\frac{U(2N)}{U(N)\times U(N)}$ WZW model has an explicit
$SU(2N)$ symmetry, which acts directly on the NLSM field. The nontrivial part
is the $U(1)$ symmetry, which comes from the topological property of the
manifold $\pi_{2}(\frac{U(2N)}{U(N)\times U(N)})=\mathbb{Z}$. The operator
charged under this topological $U(1)$ symmetry is the Skyrmion creation
operator, which is fermionic (or bosonic) if the level $k$ of the WZW term is
odd (or even) Komargodski and Seiberg (2018). This nicely matches the
statistics of the baryon operator of the $SU(k)$ gauge theory. Similarly, for
the second case one can also match the global symmetry by using
$\pi_{2}(\frac{SO(2N)}{SO(N)\times SO(N)})=\mathbb{Z}_{2}$ for $N>2$.
The identification of Grassmannian WZW models as the QCD3 theories further
corroborates the existence of SLs as critical quantum liquids, as discussed in
Sec. IV.4.
## Appendix E Explicit homomorphism between the $su(4)$ and $so(6)$
generators
To be self-contained, in this appendix we present the explicit homomorphism
between the $su(4)$ and $so(6)$ generators that is used in this paper.
Recall that the we write the $su(4)$ generators as
$\sigma_{ab}\equiv\frac{1}{2}\sigma_{a}\otimes\sigma_{b}$, with $a,\
b=0,1,2,3$ but $a$ and $b$ not simultaneously zero. Here $\sigma_{0}=I_{2}$
and $\sigma_{1,2,3}$ are the standard Pauli matrices. The correspondence
between the $su(4)$ and $so(6)$ generators are given as follows
$\displaystyle\begin{split}&\sigma_{01}\leftrightarrow T_{16},\
\sigma_{02}\leftrightarrow T_{62},\ \sigma_{03}\leftrightarrow T_{12},\\\
\sigma_{10}\leftrightarrow T_{54},\ &\sigma_{11}\leftrightarrow T_{32},\
\sigma_{12}\leftrightarrow T_{31},\ \sigma_{13}\leftrightarrow T_{63},\\\
\sigma_{20}\leftrightarrow T_{53},\ &\sigma_{21}\leftrightarrow T_{24},\
\sigma_{22}\leftrightarrow T_{14},\ \sigma_{23}\leftrightarrow T_{46},\\\
\sigma_{30}\leftrightarrow T_{34},\ &\sigma_{31}\leftrightarrow T_{25},\
\sigma_{32}\leftrightarrow T_{51},\ \sigma_{33}\leftrightarrow
T_{56}\end{split}$ (140)
where $(T_{ij})_{kl}=i(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$ is a
6-by-6 matrix, which generates rotations on the $(i,j)$-plane. One can
explicitly check that the above correspondence is indeed a homomorphism
between the $su(4)$ and $so(6)$ algebras.
## Appendix F $I^{(N)}$ anomalies of SL(N)
In this appendix, we present the details of the monopole-based approach to the
anomalies associated with the $I^{(N)}$ symmetry, where $I^{(N)}=(SO(N)\times
SO(N-4))/Z_{2}$ for even $N$ and $I^{(N)}=SO(N)\times SO(N-4)$ for odd $N$.
Our strategy is to consider $(3+1)$-d bosonic $I^{(N)}$-SPTs that are also
compatible with the discrete $\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$
symmetries, gauge the $I^{(N)}$ symmetry, and use the statistics and quantum
numbers of the fundamental $I^{(N)}$-monopoles of the resulting gauge theory
to characterize the SPT we start with. This also gives us a characterization
and classification of the $I^{(N)}$-anomalies of the $(2+1)$-d theories. Note
that this is not a full classification of the anomalies associated with both
$I^{(N)}$ and the discrete symmetries, and such a full classification is
expected to be a more refined version of the one presented here.
The main results are as follows.
1. 1.
If $N=2\ ({\rm mod\ }4)$, the SPTs or anomalies are classified into a
$\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{4}$ structure. The fundamental monopoles
of the root states are given by (148).
2. 2.
If $N=0\ ({\rm mod\ }4)$, the SPTs or anomalies are classified into a
$\mathbb{Z}_{2}^{3}\times\mathbb{Z}_{4}$ structure, i.e., it has one more
$\mathbb{Z}_{2}$ factor compared to the case with $N=2\ ({\rm mod\ }4)$. In
addition to fundamental monopoles of the types given in (148), this additional
$\mathbb{Z}_{2}$ factor corresponds to one more possible type of the
fundamental monopole, given in (151).
3. 3.
If $N$ is odd, the SPTs or anomalies are classified into a $Z_{2}^{5}$
structure. The fundamental monopoles of root states are listed in Table 5.
4. 4.
In all these cases, the statistics and quantum numbers of the $SO(N)$ and
$SO(N-4)$ monopoles of the root states are derived, given by Table 2 for even
$N$ and Table 5 for odd $N$.
5. 5.
In all these cases, we identify anomalies corresponding to theories that are
compatible with the cascade structure of the SLs, as discussed in Sec. IV.3.
In particular, two conditions need to be satisfied:
1. (a)
If the symmetry is broken to $SO(5)$, we can consider the $SO(5)$ monopole of
the resulting theory. An $SO(5)$ monopole breaks the $SO(5)$ symmetry to
$SO(2)\times SO(3)$. For a SL, the $SO(5)$ monopole of the resulting theory
should carry no charge under the $SO(2)$ but a spinor representation under
$SO(3)$.
2. (b)
If the symmetry of a SL is broken to $(SO(4)\times SO(N-4))/Z_{2}$, the
resulting theory should have no anomaly.
For even $N$, we find that only root 3 in (148) and its inverse satisfy both
conditions. For odd $N$, there is a single anomaly class that satisfies both
conditions, as discussed at the end of Appendix F.2.
### F.1 The case with an even $N$
We start the discussion with the case with an even $N$, as in this case
results not captured by Eq. (35) may arise.
As in the main text, we write the $SO(N)$ and $SO(N-4)$ gauge fields as
$A^{SO(N)}=A_{a}^{L}T_{a}^{L}$ and $A^{SO(N-4)}=A_{a}^{R}T_{a}^{R}$,
respectively, where $\\{T^{L}\\}$ and $\\{T^{L}\\}$ form the generators of
$SO(N)$ and $SO(N-4)$, respectively. For even $N$, the field configuration of
a fundamental monopole will be taken as
$\displaystyle A_{12}^{L}=A_{34}^{L}=A_{56}^{L}=\cdots
A_{N-1,N}^{L}=A_{12}^{R}=A_{34}^{R}=A_{56}^{R}=\cdots
A_{N-5,N-4}^{R}=\frac{A_{U(1)}}{2}$ (141)
where $A_{ij}^{L}$ ($A_{ij}^{R}$) is the gauge field corresponding to the
generator associated with rotations on the $(i,j)$-plane of $SO(N)$
($SO(N-4)$) symmetry, and $A_{U(1)}$ is the field configuration of a unit
monopole in a $U(1)$ gauge theory, which can taken to be of the form in Ref.
Wu and Yang (1975). Namely, this monopole is obtained by embedding many
half-$U(1)$-monopoles into the maximal Abelian group of $I^{(N)}$. The
configuration of such a monopole breaks the gauge symmetry from $I^{(N)}$ to
$(SO(2)^{N-2})/Z_{2}$. So it is convenient to denote a general excitation in
this $I^{(N)}$ gauge theory by the following excitation matrix:
$\displaystyle\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}=\left(\begin{array}[]{cccc|cccc}q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
m_{12}^{L}&m_{34}^{L}&\cdots&m_{N-1,N}^{L}&m_{12}^{R}&m_{34}^{R}&\cdots&m_{N-5,N-4}^{R}\end{array}\right)_{s}$
(146)
where the first (second) row represents the electric (magnetic) charges of
this excitation under $A_{ij}^{L,R}$, $s=0\ ({\rm mod\ }2)$ ($s=1\ ({\rm mod\
}2)$) represents that this excitation is a boson (fermion), and the vertical
line separates the charges related to the original $SO(N)$ and $SO(N-4)$
subgroups of $I^{(N)}$. The above fundamental monopole has
$\bm{m}=(\frac{1}{2},\frac{1}{2},\cdots,\frac{1}{2})$, and its $\bm{q}$ and
$s$ will characterize the corresponding SPT. Because the statistics of any
excitation can be unambiguously determined by its $\bm{q}$ and the statistics
of the fundamental monopole, later we will sometimes suppress the subscript
related to the statistics of this excitation.
In such a theory, the structures of the possible excitations are constrained
by the following conditions.
1. 1.
The pure gauge charges are built up with bosons in the bifundamental
representation of $I^{(N)}$. That is, if $\bm{m}=0$, then all entries of
$\bm{q}$ are integers that add up to an even integer. An example of the
elementary pure gauge charge has $\bm{m}=0$ and $\bm{q}=(1,0,0,\cdots,0,0,1)$.
2. 2.
The Dirac quantization condition for two excitations
$\left(\begin{array}[]{c}\bm{q}_{1}\\\ \bm{m}_{1}\end{array}\right)$ and
$\left(\begin{array}[]{c}\bm{q}_{2}\\\ \bm{m}_{2}\end{array}\right)$:
$\bm{q}_{1}\cdot\bm{m}_{2}-\bm{q}_{2}\cdot\bm{m}_{1}\in\mathbb{Z}$.
3. 3.
If an excitation exists, its $\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$
partners also exist. We take the actions of these discrete symmetries on the
excitation given by (146) to be
$\displaystyle\begin{split}&\mathcal{C}:\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}\rightarrow\left(\begin{array}[]{cccc|cccc}-q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&-q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
-m_{12}^{L}&m_{34}^{L}&\cdots&m_{N-1,N}^{L}&-m_{12}^{R}&m_{34}^{R}&\cdots&m_{N-5,N-4}^{R}\end{array}\right)_{s}\\\
&\mathcal{R}:\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}\rightarrow\left(\begin{array}[]{cccc|cccc}-q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
m_{12}^{L}&-m_{34}^{L}&\cdots&-m_{N-1,N}^{L}&-m_{12}^{R}&-m_{34}^{R}&\cdots&-m_{N-5,N-4}^{R}\end{array}\right)_{s}\\\
&\mathcal{T}:\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}\rightarrow\left(\begin{array}[]{cccc|cccc}q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&-q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
-m_{12}^{L}&-m_{34}^{L}&\cdots&-m_{N-1,N}^{L}&m_{12}^{R}&-m_{34}^{R}&\cdots&-m_{N-5,N-4}^{R}\end{array}\right)_{s}\end{split}$
(147)
4. 4.
The remaining $(SO(2)^{N-2})/Z_{2}$ has a normalizer subgroup in $I^{(N)}$. If
an excitation exists, its partners under the actions of the normalizer
subgroup also exist.
These are necessary conditions for a theory to be consistent, and we believe
they are also sufficient.
The above conditions impose strong constraints on the possible $\bm{q}$ of a
fundamental monopole. First, there is an element in the normalizer subgroup
associated with the remaining $(SO(2)^{N-2})/Z_{2}$, whose action is to
exchange the first two columns of the excitation matrix. Applying this
operation to the fundamental monopole yields a normalizer-partner of it. The
bound state of this normalizer-partner and the anti-particle of the
fundamental monopole has all entries in the excitation matrix being $0$,
except that the first two entries in the first row are
$\pm(q_{12}^{L}-q_{34}^{L})$. Because this is a pure gauge charge,
$q_{12}^{L}=q_{34}^{L}\ ({\rm mod\ }1)$. Similarly, it is easy to see that the
first $N/2$ entries in $\bm{q}$ of the fundamental monopoles are all equal mod
1, and the last $(N-4)/2$ entries in $\bm{q}$ are also all equal mod 1.
Second, it is always possible to attach the fundamental monopole with some
pure gauge charge, such that all its $\bm{q}$-entries are in the interval
$(-1,1]$. These two observations imply that we can always write a fundamental
monopole as
$\left(\begin{array}[]{cccc|cccc}q^{L}&q^{L}&\cdots&q^{L}&q^{R}&q^{R}&\cdots&q^{R}\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{s}$
if $N=2\ ({\rm mod\ }4)$, while if $N=0\ ({\rm mod\ }4)$, besides this
possibility, there is one more possible type:
$\left(\begin{array}[]{cccc|cccc}q^{L}&q^{L}&\cdots&q^{L}&q^{R}&q^{R}&\cdots&q^{R}+1\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{s}$.
#### F.1.1 The case with $N=2\ ({\rm mod\ }4)$
Now let us focus on the case with $N=2\ ({\rm mod\ }4)$. The case with $N=6$
needs some special treatment, so we defer the discussion on it for a moment.
If $N>6$, the 4-particle bound state of the fundamental monopole and its
$\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$ parnters is a pure gauge charge
with excitation matrix
$\left(\begin{array}[]{ccccc|ccccc}0&4q^{L}&4q^{L}&\cdots&4q^{L}&0&4q^{R}&4q^{R}&\cdots&4q^{R}\\\
0&0&0&\cdots&0&0&0&0&\cdots&0\end{array}\right)$. This implies that
$4q^{L,R}\in\mathbb{Z}$. The Dirac quantization condition on the fundamental
monopole and its $\mathcal{T}$-partner implies
$\frac{Nq^{L}+(N-8)q^{R}}{2}\in\mathbb{Z}$. So $q^{L}+q^{R}\in\mathbb{Z}$. Now
it is straightforward to see that there are only two elementary possibilities
of $(q^{L},q^{R})$, i.e., $(q^{L},q^{R})=(0,1)$ and
$(q^{L},q^{R})=(\frac{1}{4},-\frac{1}{4})$, and these possibilities are
elementary in the sense that all other possibilities can be obtained from them
by forming bound states of them and/or attaching pure gauge charges, i.e.,
they can be taken as the monopoles of the root states. So far we have not
considered the statistics of the fundamental monopole, and it can actually be
either bosonic or fermionic. These results suggest a
$\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{4}$ classification of the fundamental
monopoles, and the roots can be taken to be
$\displaystyle\begin{split}&{\rm root\
}1:\left(\begin{array}[]{cccc|cccc}0&0&\cdots&0&0&0&\cdots&0\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{f}\\\
&{\rm root\ }2:\left(\begin{array}[]{cccc|cccc}0&0&\cdots&0&0&0&\cdots&1\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}\\\
&{\rm root\
}3:\left(\begin{array}[]{cccc|cccc}\frac{1}{4}&\frac{1}{4}&\cdots&\frac{1}{4}&-\frac{1}{4}&-\frac{1}{4}&\cdots&-\frac{1}{4}\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}\end{split}$
(148)
Notice that in writing root 2, we have attached pure gauge charge to it. One
can check that these roots satisfy all 4 conditions listed at the beginning of
this subsection.
The above results also apply to the case with $N=6$, but the argument needs to
be slightly modified. For $N=6$, these 4 excitations exist:
$\left(\begin{array}[]{ccc|c}-q^{L}&-q^{L}&-q^{L}&q^{R}\\\
\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}\end{array}\right)_{s}$,
$\left(\begin{array}[]{ccc|c}q^{L}&-q^{L}&q^{L}&q^{R}\\\
-\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\end{array}\right)_{s}$,
$\left(\begin{array}[]{ccc|c}-q^{L}&-q^{L}&q^{L}&q^{R}\\\
-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{array}\right)_{s}$, and
$\left(\begin{array}[]{ccc|c}q^{L}&-q^{L}&-q^{L}&q^{R}\\\
\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\end{array}\right)$. The
4-particle bound state of these 4 excitations is
$\left(\begin{array}[]{ccc|c}0&-4q^{L}&0&4q^{R}\\\
0&0&0&0\end{array}\right)_{b}$, which means in this case we also have
$4q^{L,R}\in\mathbb{Z}$. Furthermore, the Dirac quantization condition for a
fundamental monopole and its $\mathcal{T}$-partner gives
$-3q^{L}+q^{R}\in\mathbb{Z}$. These conditions still suggest a
$Z_{2}^{2}\times Z_{4}$ classification, and the roots can still be taken as
the ones in (148).
It is useful to derive the structure of the $SO(N)$ and $SO(N-4)$ monopoles
from these fundamental monopoles. An $SO(N)$ monopole can be viewed as the
2-particle bound state of the fundamental monopole and its $\mathcal{R}$
partner:
$\left(\begin{array}[]{cccc|ccc}0&2q^{L}&\cdots&2q^{L}&2q^{R}&\cdots&2q^{R}\\\
1&0&\cdots&0&0&\cdots&0\end{array}\right)_{q_{L}+q_{R}}$. The $SO(N-4)$
monopole can be viewed as the 2-particle bound state of the fundamental
monopole and its $\mathcal{T}$ partner:
$\left(\begin{array}[]{ccc|cccc}2q^{L}&\cdots&2q^{L}&0&2q^{R}&\cdots&2q^{R}\\\
0&\cdots&0&1&0&\cdots&0\end{array}\right)_{q_{L}-3q_{R}}$. In physical terms,
the properties of these monopoles are summarized in Table 2.
| $SO(N)$ monopole | $SO(N-4)$ monopole
---|---|---
root 1 | (singlet, singlet, boson) | (singlet, singlet, boson)
root 2 | (singlet, singlet, fermion) | (singlet, singlet, fermion)
root 3 | (spinor, spinor, boson) | (spinor, spinor, fermion)
root 4 with $N=0\ ({\rm mod\ }8)$ | (singlet, singlet, fermion) | (singlet, vector, boson)
root 4 with $N=4\ ({\rm mod\ }8)$ | (singlet, singlet, boson) | (singlet, vector, fermion)
Table 2: Properties of the $SO(N)$ and $SO(N-4)$ monopoles of the root states
for even $N$. The first three roots apply to all even $N$, and root 4 only
applies to the case with $N$ an integral multiple of 4. The $SO(N)$ monopole
breaks the $I^{(N)}$ symmetry to $(SO(2)\times SO(N-2)\times SO(N-4))/Z_{2}$,
and it always has no charge under the $SO(2)$. Its three corresponding entries
represent its representation under the $SO(N-2)$, its representation under the
$SO(N-4)$, and its statistics, respectively. The $SO(N-4)$ monopole breaks the
$I^{(N)}$ symmetry to $(SO(N)\times SO(N-6)\times SO(2))/Z_{2}$, and it always
has no charge under the $SO(2)$. Its three corresponding entries represent its
representation under the $SO(N)$, its representation under the $SO(N-6)$, and
its statistics, respectively. For the case with $N=6$, the second entry does
not exist for its $SO(N-4)$ monopole. Notice these properties are determined
up to attaching pure gauge charges.
Let us check which anomalies correspond to states that satisfy the two
conditions listed at the beginning of this appendix. In order for the first
condition to be satisfied, according to Table 2, the state must contains the
anomaly corresponding to root 3 or its inverse.
It is a bit more complicated to check the second condition, but it actually
suffices to check a weaker condition: if the remaining $SO(N-4)$ symmetry is
further broken to $SO(2)^{\frac{N}{2}-2}$, the system is anomaly-free. To this
end, let us condense
$\left(\begin{array}[]{cccccc|cccc}1&1&\cdots&1&0&0&-1&-1&\cdots&-1\\\
0&0&\cdots&0&0&0&0&0&\cdots&0\end{array}\right)_{b}$. This condensate breaks
the $I^{(N)}$ symmetry into $(SO(2)^{\frac{N}{2}-2}\times SO(4))/Z_{2}$, and
the gauge fields corresponding to the remaining $\frac{N}{2}-2$ $SO(2)$
symmetries can be taken as
$A_{12}^{\prime}=\frac{1}{2}(A_{12}^{L}+A_{12}^{R}),A_{34}^{\prime}=\frac{1}{2}(A_{34}^{L}+A_{34}^{R}),\cdots,A_{N-5,N-4}^{\prime}=\frac{1}{2}(A_{N-5,N-4}^{L}+A_{N-5,N-4}^{R})$,
and for the $SO(4)\backsimeq\frac{SU(2)\times SU(2)}{Z_{2}}$, we denote the
gauge fields corresponding to these two $SU(2)$ subgroups by $B_{1,2}$, which
together form the $SO(4)$ gauge field.
The fundamental monopole remains deconfined in this condensate and becomes the
fundamental monopole of the resulting theory. In terms of the remaining
symmetries, it should be written as
$\left(\begin{array}[]{cccc|cc}q^{L}+q^{R}&q^{L}+q^{R}&\cdots&q^{L}+q^{R}&q^{L}&q^{L}\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{array}\right)_{s}$.
By adding to the theory a proper theta-term of the $SO(4)$ gauge field,
$\theta\epsilon_{\mu\nu\lambda\rho}{\rm
Tr}(\partial^{\mu}B_{1}^{\nu}\partial^{\nu}B_{1}^{\rho}-\partial^{\mu}B_{2}^{\nu}\partial^{\lambda}B_{2}^{\rho})$,
which preserves all remaining symmetries (including the remaining
$\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$ symmetries), this fundamental
monopole can be converted to
$\left(\begin{array}[]{cccc|cc}q^{L}+q^{R}&q^{L}+q^{R}&\cdots&q^{L}+q^{R}&0&0\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{array}\right)_{s}$.
In order for the theory to be anomaly-free, this monopole should be a boson
with trivial projective quantum numbers. This means $s=0\ ({\rm mod\ }2)$ and
$q^{L}+q^{R}=0\ ({\rm mod\ }2)$. Therefore, only root 3 and multiple copies of
it satisfy this condition.
To satisfy both conditions, the only possibilities are root 3 and its inverse.
Because the SLs with $(N,\pm 1)$ satisfy both conditions, we conclude that the
anomalies of these SLs are precisely the same as root 3 and its inverse.
Notice that from this analysis we cannot determine which of $(N,\pm 1)$
corresponds to root 3, and which corresponds to its inverse.
Now that we have identified the SLs with $(N,\pm 1)$ as states that realize
the anomalies of root 3 and its inverse, it may also be worth mentioning which
states realize the anomalies of the other 2 roots.
The $I^{(N)}$-anomaly of root 1 can be realized by a $Z_{2}$ topological order
(TO). Denote the $Z_{2}$ charge and flux by $e$ and $m$, respectively, the
symmetry actions on these topological sectors are given in Table 3.
| $e$ | $m$
---|---|---
$I^{(N)}$ | (singlet, vector) | (singlet, vector)
$\mathcal{C}$ | $e$ | $m$
$\mathcal{R}$ | $e$ | $m$
$\mathcal{T}$ | $e$ | $m$
Table 3: Projective symmetry actions on the topological sectors of the $Z_{2}$
TO that realizes the $I^{(N)}$ anomaly of root 1. In the row corresponding to
$I^{(N)}$, the two entries represent the representations of this excitation
under the $SO(N)$ and $SO(N-4)$ subgroups of $I^{(N)}$, respectively. When
$N=6$, the vector representation of the $SO(2)\subset I^{(6)}$ means charge 1
under $SO(2)$. Their partners under $\mathcal{C}$, $\mathcal{R}$ and
$\mathcal{T}$ are shown as above. Notice one can further specify data like
$T^{2}$ for $e$ and $m$, but it is unnecessary for our purpose.
To see that this $Z_{2}$ TO realizes the $I^{(N)}$-anomaly of root 1, consider
threading a fundamental monopole through the system, which leaves a flux.
Because of the above symmetry assignment, when an $e$ or $m$ circles around
this flux, it acquires a phase factor $-1$, no matter how far it is away from
the flux. Since this monopole threading process is local for the $(2+1)$-d
system, this $-1$ phase factor has to be cancelled by requiring that the flux
also trap an anyon that has $-1$ mutual braiding with both $e$ and $m$. This
anyon is $\epsilon$, the fermionic bound state of $e$ and $m$. Furthermore,
time reversal symmetry ensures that no polarization charge is induced around
this flux. Then the composite of this flux and $\epsilon$ is a fermion in the
trivial representation of $I^{(N)}$. Therefore, the fundamental monopole of
this $Z_{2}$ TO has precisely the structure as in root 1.
This $Z_{2}$ TO can be explicitly constructed using the layer construction in
Ref. Wang and Senthil (2013), and it is an analog of $eCmC$, the surface state
of a $(3+1)$-d bosonic topological insulator protected by $U(1)$ charge
conservation and time reversal Vishwanath and Senthil (2013); Wang and Senthil
(2013).
The $I^{(N)}$-anomaly of root 2 can be realized by a $Z_{4}\times Z_{2}$ TO.
Denote the $Z_{4}$ charge and flux by $e_{1}$ and $m_{1}$, and the $Z_{2}$
charge and flux by $e_{2}$ and $m_{2}$, respectively. We take the convention
such that the mutual statistics between $e_{1}$ and $m_{1}$ is $i$ if $N=6\
({\rm mod\ }8)$, and $-i$ if $N=2\ ({\rm mod\ }8)$. The symmetry actions on
these topological sectors are given in Table 4.
| $e_{1}$ | $m_{1}$ | $e_{2}$ | $m_{2}$ | $\epsilon_{2}\equiv e_{2}m_{2}$
---|---|---|---|---|---
$I^{(N)}$ | (singlet, fund.) | (singlet, anti-fund.) | (anti-fund., anti-fund.) | (fund., fund.) | (singlet, singlet)
$\mathcal{C}$ | $e_{1}^{-1}$ | $m_{1}^{-1}$ | $e_{2}^{-1}$ | $m_{2}^{-1}$ | $\epsilon_{2}$
$\mathcal{R}$ | $e_{1}$ | $m_{1}^{-1}e_{1}^{-2}\epsilon_{2}$ | $e_{2}^{-1}e_{1}^{-2}$ | $m_{2}^{-1}e_{1}^{2}$ | $\epsilon_{2}$
$\mathcal{T}$ | $e_{1}^{-1}$ | $m_{1}e_{1}^{2}\epsilon_{2}$ | $e_{2}e_{1}^{2}$ | $m_{2}e_{1}^{-2}$ | $\epsilon_{2}$
Table 4: Projective symmetry actions on the topological sectors of the
$Z_{4}\times Z_{2}$ TO that realizes the $I^{(N)}$ anomaly of root 2. In the
row corresponding to $I^{(N)}$, the two entries in each parenthesis represent
the representations of this excitation under the associated spin group of the
correponding $SO(N)$ and $SO(N-4)$ subgroups of $I^{(N)}$, where “fund.”
(“anti-fund.”) represents fundamental (anti-fundamental) representation of the
relevant spin group (when $N=6$, the fund. (anti-fund.) in the second entry in
the parenthesis means charge $1/2$ ($-1/2$) under $SO(2)\subset I^{(6)}$).
Their partners under $\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$ are shown
as above. Notice one can further specify data like $T^{2}$ for various anyons,
but it is unnecessary for our purpose.
To see that this theory realizes the $I^{(N)}$-anomaly of root 2, we can again
consider threading a fundamental monopole through the system and use a similar
argument as before. One can check that the flux left by the fundamental
monopole will trap an anyon $e_{1}m_{1}^{-1}$, and this fundamental monopole
is indeed a boson in the vector representation under $SO(N-4)$, so we conclude
that this theory realizes the $I^{(6)}$-anomaly of root 2.
We believe this $Z_{4}\times Z_{2}$ TO with the above symmetry implementation
is a consistent theory, although we do not have an explicit construction for
it.
#### F.1.2 The case with $N=0\ ({\rm mod\ }4)$
Next we turn to the case with $N=0\ ({\rm mod\ }4)$. From similar analysis as
before, now the constraints we obtain are $4q^{L,R}\in\mathbb{Z}$ and
$2(q^{L}+q^{R})\in\mathbb{Z}$. The three roots in (148) still satisfy these
constraints, but we find one additional root 191919This root is absent in the
case with $N=2\ ({\rm mod\ }4)$ because $q_{L}+q_{R}\in\mathbb{Z}$ in that
case, which is violated here.:
$\displaystyle{\rm root\
}4:\left(\begin{array}[]{ccc|cccc}0&\cdots&0&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\\\
\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}$
(151)
Similar as before, we can also derive the properties of the $SO(N)$ and
$SO(N-4)$ monopoles for this root, and the results are given in Table 2. From
this Table, we observe that there are not only differences in the anomalies
for the cases with $N=2\ ({\rm mod\ }4)$ and $N=0\ ({\rm mod\ }4)$, but also
differences in the anomalies for the cases with $N=0\ ({\rm mod\ }8)$ and
$N=4\ ({\rm mod\ }8)$. It is known that the spinor representations of $SO(N)$
in these three cases are different, i.e., they are complex, real, and
pseudoreal for $N=2\ ({\rm mod\ }4)$, $N=0\ ({\rm mod\ }8)$ and $N=4\ ({\rm
mod\ }8)$, respectively Zee (2016). Our analysis suggests a connection between
these two results.
Lastly, analogous arguments as before indicate that only root 3 and its
inverse satisfy both conditions discussed at the beginning of this appendix.
Because SLs with $(N,\pm 1)$ also satisfy those two conditions, we conclude
that these SLs have the same anomaly as root 3 and its inverse.
### F.2 The case with an odd $N$
Now we turn to the case with odd $N$, so $I^{(N)}=SO(N)\times SO(N-4)$. Notice
this analysis also applies to the case with an even $N$ if we add to the
system bosonic DOF in the vector representation of $SO(N)$.
In this case, the fundamental monopoles are simply the usual $SO(N)$ and
$SO(N-4)$ monopoles, so these two monopoles will characterize the $I^{(N)}$
SPTs and anomalies.
An $SO(N)$ monopole breaks $I^{(N)}$ into $SO(2)\times SO(N-2)\times SO(N-4)$.
This monopole can be denoted as
$\left(\begin{array}[]{cc|c}q_{N}&r_{N}^{L}&r_{N}^{R}\\\
1&0&0\end{array}\right)_{s_{N}}$, where $q_{N}$ represents the fractional
charge under the remaining $SO(2)$, $r_{N}^{L}$ represents the projective
quantum number under the remaining $SO(N-2)$, $r_{N}^{R}$ represents the
projective quantum number under $SO(N-4)$, and $s_{N}$ is the statistics of
this monopole. Similarly, an $SO(N-4)$ monopole can be denoted by
$\left(\begin{array}[]{c|cc}r_{N-4}^{L}&q_{N-4}&r_{N-4}^{R}\\\
0&1&0\end{array}\right)_{s_{N-4}}$. Just as in the usual $(3+1)$-d bosonic
topological insulator, time reversal symmetry and the bosonic statistics of
the pure gauge charge require that $q_{N}=q_{N-4}=0$ Vishwanath and Senthil
(2013). Furthermore, the Dirac quantization condition requires that whenever
the $SO(N)$ monopole carries a spinor representation under $SO(N-4)$, the
$SO(N-4)$ monopole must also carry a spinor representation under $SO(N)$, and
vice versa. These are all the constraints, and we get a $\mathbb{Z}_{2}^{5}$
classification of the $I^{(N)}$ SPTs or anomalies, and the structures of the
monopoles of the 5 root states are in Table 5.
| $SO(N)$ monopole | $SO(N-4)$ monopole | topological response function
---|---|---|---
root 1 | (singlet, singlet, fermion) | (singlet, singlet, boson) | $(w_{2}^{SO(N)})^{2}$
root 2 | (singlet, singlet, boson) | (singlet, singlet, fermion) | $(w_{2}^{SO(N-4)})^{2}$
root 3 | (spinor, singlet, boson) | (singlet, singlet, boson) | $w_{4}^{SO(N)}$
root 4 | (singlet, singlet, boson) | (singlet, spinor, boson) | $w_{4}^{SO(N-4)}$
root 5 | (singlet, spinor, boson) | (spinor, singlet, boson) | $w_{2}^{SO(N)}w_{2}^{SO(N-4)}$
Table 5: Properties of the $SO(N)$ and $SO(N-4)$ monopoles of the root states
for odd $N$. The $SO(N)$ monopole breaks the $I^{(N)}$ symmetry to
$SO(2)\times SO(N-2)\times SO(N-4)$, and it always has no fractional charge
under the $SO(2)$. Its three corresponding entries represent its
representation under the $SO(N-2)$, its representation under the $SO(N-4)$,
and its statistics, respectively. The $SO(N-4)$ monopole breaks the $I^{(N)}$
symmetry to $SO(N)\times SO(N-6)\times SO(2)$, and it always has no charge
under the $SO(2)$. Its three corresponding entries represent its
representation under the $SO(N)$, its representation under the $SO(N-6)$, and
its statistics, respectively. The last column lists the topological response
function corresponding to these root states.
Using similar arguments as before, one can show that only one of the
$2^{5}=32$ anomaly classes satisfies both conditions discussed at the
beginning of this appendix, which is (root 2)$\otimes$(root 3)$\otimes$(root
4)$\otimes$(root 5), i.e., a composed system made of root 2, root 3, root 4
and root 5. This anomaly class is identified with the one for the SLs, and
this result agrees with Eq. (35).
## Appendix G Explicit calculations of the $U(1)$ DSL
In this appendix, we explicitly derive the properties of the fundamental
$I^{(6)}$-monopole in a $U(1)$ DSL, whose effective theory is given by Eq.
(4). We will see that this fundamental $I^{(6)}$-monopole has precisely the
structure of root 3 in Eq. (44), which further strengthens our proposal that
the $U(1)$ DSL and SL(6) are equivalent. Our results also agree with a more
formal calculation in Ref. Calvera and Wang (2021).
Recall that we can take the Dirac fermions in a DSL to be in either the
fundamental or anti-fundamental representation of $SU(4)$, and take the
monopole of $a$ to have either charge $1$ or $-1$ under $U(1)_{\rm top}$, so
there are 4 different choices of the symmetry implementation. To be general,
we will consider the 4 cases together by introducing parameters $\zeta$ and
$\xi$, such that $\zeta=1$ ($\zeta=-1$) if the Dirac fermions are in the
fundamental (anti-fundamental) representation of $SU(4)$, and $\xi=\pm 1$ if
the monopole of $a$ carries charge $\pm 1$ under $U(1)_{\rm top}$202020Since
we can redefine the theory through a charge conjugation
$\tilde{\psi}=\psi^{\dagger}$, $\tilde{a}=-a$, the two signs $\zeta$ and $\xi$
can be flipped simultaneously without physical effect. Only the product
$\zeta\xi$ will eventually matter in the following discussions..
Next we will calculate the $q^{L}$, $q^{R}$ and $s$ of the fundamental
$I^{(6)}$-monopole for a DSL. In particular, we will thread a flux
corresponding to the fundamental $I^{(6)}$-monopole in Eq. (38), which has a
$\pi$-flux for $A_{12}$, $A_{34}$, $A_{56}$ and $A_{\rm top}$.
It is useful to denote the 4 flavors of Dirac fermions by $\psi_{\uparrow+}$,
$\psi_{\uparrow-}$, $\psi_{\downarrow+}$, and $\psi_{\downarrow-}$,
respectively. This notation is motivated by the lattice realizations of a DSL,
where $\uparrow$ and $\downarrow$ represent two physical spins, and $+$ and
$-$ represent the two valleys. According to the homomorphism between $su(4)$
and $so(6)$ in Appendix E, the charges under $(A_{12},A_{34},A_{56})$ carried
by $\psi_{\uparrow+}$, $\psi_{\uparrow-}$, $\psi_{\downarrow+}$ and
$\psi_{\downarrow-}$ are respectively $\zeta(1/2,1/2,1/2)$,
$\zeta(1/2,-1/2,-1/2)$, $\zeta(-1/2,1/2,-1/2)$ and $\zeta(-1/2,-1/2,1/2)$.
Note that the Dirac fermions are neutral under $A_{\rm top}$.
When the flux specified above is thread, $\psi_{\uparrow+}$ sees a total
$3\zeta\pi/2$ flux, while the other 3 flavors of Dirac fermions see a total
$-\zeta\pi/2$ flux. To construct a gauge invariant state corresponding to the
local fundamental $I^{(6)}$-monopole, one can consider a state where the
internal gauge field, $a$, has a flux of $\zeta\pi/2$, such that at the end
$\psi_{\uparrow+}$ sees a $2\zeta\pi$ flux and contributes a zero mode in this
flux background, and the other 3 flavors see no flux and contribute no zero
mode. Because there is a single zero mode in the background of a flux with
magnitude $2\pi$, no matter it is occupied or not, we get a bosonic state, so
$s=b$ for the fundamental $I^{(6)}$ monopole in Eq. (38). To determine
$q^{L,R}$ for this fundamental monopole, we need to determine whether this
zero mode is occupied or not.
The usual way to do this is to demand that the zero modes are half-filled.
However, that works only if the theory has a symmetry that preserves the flux
but flips the charge. In the present case, there is no such a symmetry, so a
different approach should be taken. To proceed, we will regularize the DSL as
follows. First, to preserve the $SU(4)$ flavor symmetry, for each flavor we
add to the system a gapped Dirac fermion, which contributes to the effective
action a term $-\pi\eta/2$ when combined with the original gapless Dirac
fermion with the same flavor, where $\eta$ is the $\eta$-invariant of the
Dirac operator corresponding to each flavor of gapless and gapped Dirac
fermions Witten (2016). These $\eta$-invariants will generally break the
$\mathcal{T}$ symmetry. To maintain the $\mathcal{T}$ symmetry, next we put
the system on the boundary of a $(3+1)$-d bulk that contributes an appropriate
theta-term to the bulk partition function, such that the combined partition
function of the boundary and bulk is $\mathcal{T}$ invariant. This particular
regularization of the theory should suffice to yield $q^{L,R}$ of the
fundamental monopole in Eq. (38), up to attaching local DOF.
With this regularization, the effective action212121More precisely the
partition function is $Z=|\det(\not{D})|\exp(iS_{\rm{eff}})$ in Euclidean
signature Witten (2016). is
$\displaystyle S_{\rm eff}=$
$\displaystyle\sum_{i=1}^{4}\left[-\frac{\pi}{2}\eta(a_{i})+\frac{1}{2}\frac{1}{4\pi}\int
d^{3}xa_{i}da_{i}\right]+\frac{\xi}{2\pi}\int d^{3}xA_{\rm top}da$ (152)
with
$\displaystyle\begin{split}&a_{1}=a+\frac{\zeta(A_{12}+A_{34}+A_{56})}{2}\\\
&a_{2}=a+\frac{\zeta(A_{12}-A_{34}-A_{56})}{2}\\\
&a_{3}=a+\frac{\zeta(-A_{12}+A_{34}-A_{56})}{2}\\\
&a_{4}=a+\frac{\zeta(-A_{12}-A_{34}+A_{56})}{2}\end{split}$ (153)
and $i=1,2,3,4$ correspond to contributions from $\uparrow+$, $\uparrow-$,
$\downarrow+$, $\downarrow-$, respectively. Notice that the second term in the
above effective action comes from reducing the theta-term of the $(3+1)$-d
bulk to the boundary. Furthermore, the coefficient of the CS term of the
dynamical gauge field $a$ is $1/(2\pi)$, which is well defined in $(2+1)$-d
and implies that the $(3+1)$-d bulk does not really need a dynamical gauge
field, consistent with the general expectation that a theory with a ’t Hooft
anomaly can live on the boundary of a short-range entangled bulk (see, e.g.,
Ref. Ning _et al._ (2020) for examples of theories that have more severe
anomalies than a ’t Hooft anomaly and thus can only live on the boundary of a
long-range entangled bulk). Also notice that all dependence on the metric of
the spacetime manifold of the system is suppressed, which will not affect our
following analysis.
This effective action can be used to read off the resulting charges under
various gauge fields when the flux is thread:
$\displaystyle\begin{split}&Q_{a}=N_{\psi_{\uparrow+}}+\frac{B_{a_{1}}+B_{a_{2}}+B_{a_{3}}+B_{a_{4}}}{4\pi}+\frac{\xi
B_{\rm top}}{2\pi}=N_{\psi_{\uparrow+}}+\frac{B_{a}}{\pi}+\frac{\xi B_{\rm
top}}{2\pi}\\\
&Q_{12}=\zeta\left(\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{a_{1}}+B_{a_{2}}-B_{a_{3}}-B_{a_{4}}}{8\pi}\right)=\zeta\cdot\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{12}}{4\pi}\\\
&Q_{34}=\zeta\left(\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{a_{1}}-B_{a_{2}}+B_{a_{3}}-B_{a_{4}}}{8\pi}\right)=\zeta\cdot\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{34}}{4\pi}\\\
&Q_{56}=\zeta\left(\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{a_{1}}-B_{a_{2}}-B_{a_{3}}+B_{a_{4}}}{8\pi}\right)=\zeta\cdot\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{56}}{4\pi}\\\
&Q_{\rm top}=\frac{\xi B_{a}}{2\pi}\end{split}$ (154)
where $Q_{(\cdot)}$ and $B_{(\cdot)}$ represent the charge and flux under the
corresponding gauge field, respectively, and $N_{\psi_{\uparrow+}}$ determines
whether the zero mode contributed by $\psi_{\uparrow+}$ is occupied, i.e., if
the flux seen by the fermion is positive (negative), then
$N_{\psi_{\uparrow+}}=0$ means it is occupied (unoccupied).
According to the previous discussion, now we have $B_{12}=B_{34}=B_{56}=B_{\rm
top}=\pi$ and $B_{a}=\zeta\pi/2$. To be gauge invariant, $Q_{a}=0$, which
means $N_{\psi_{\uparrow+}}=-(\zeta+\xi)/2$. Substituting this into the rest
of the equations yields $Q_{12}=Q_{34}=Q_{56}=-\frac{\zeta\xi}{4}$ and $Q_{\rm
top}=\frac{\zeta\xi}{4}$. That is, if $(\zeta,\xi)=(1,-1)$ or
$(\zeta,\xi)=(-1,1)$, $q^{L}=-q^{R}=1/4$, corresponding to root 3 in Eq. (44).
If $(\zeta,\xi)=(1,1)$ or $(\zeta,\xi)=(-1,-1)$, $q^{L}=-q^{R}=-1/4$,
corresponding to the inverse of root 3 in Eq. (44). Therefore, the $I^{(N)}$
anomaly of the $U(1)$ DSL is indeed identical to that of the SL(6,±1), which
even further strengthens our proposal that they are dual.
## Appendix H More on the LSM constraints
In the main text, we have used some physical arguments to propose that Eq.
(57) describes the complete set of LSM constraints for various lattice
systems. In this appendix, we extract some LSM contraints from Eq. (57) that
were not used in deriving Eq. (57). We will also give an alternative
expression for the LSM anomaly on a square lattice.
For convenience, we copy Eq. (57):
$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})[xy+c^{2}+r(x+c)]$ (155)
where $w_{2}^{SO(3)}$ is the second SW class of the $SO(3)$ gauge field
corresponding to the spin rotational symmetry, $t$ is the gauge field
corresponding to the time reversal symmetry, $x$ and $y$ are the gauge field
corresponding to translation along $T_{1}$ and $T_{2}$, respectively, $c$ is
the gauge field corresponding to $C_{2}$ site-centered lattice rotation, and
$r$ is the gauge field corresponding to the reflection symmetry $R_{y}$. There
is a constraint $r+t=w_{1}^{TM}\ ({\rm mod\ }2)$. This anomaly polynomial
should be viewed as a topological response function of the system under the
various gauge fields.
Physically, we expect that there will be LSM anomalies associated with
reflection symmetry $R_{x}$. To read them off, we need to design a
configuration of the gauge field corresponding to $R_{x}$ in Eq. (155).
Although the gauge field corresponding to $R_{x}$ does not explicitly appear
in Eq. (155), because $R_{x}$ is a combination of $C_{2}$ and $R_{y}$, we can
still have a gauge connection of $R_{x}$ by writing $c=c_{0}+r$. This means
that whenever there is a gauge connection corresponding to $R_{y}$, a gauge
connection corresponding to $C_{2}$ is also induced. Therefore, now $r$
actually represents a gauge connection corresponding to $R_{x}$, and $c_{0}$
is the gauge connection for pure $C_{2}$ rotation. Substituting $c=c_{0}+r$
into Eq. (155) yields
$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})[xy+c_{0}^{2}+r(x+c_{0})]$ (156)
The physical meaning of this new anomaly polynomial can be understood by
looking at various sub-symmetries of the system. For example, ignoring
translation symmetries, i.e., setting $x=y=0$, it becomes $S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})(c_{0}^{2}+rc_{0})$. The first term
in the second parenthesis, $c_{0}^{2}$, physically means that there is an LSM
anomaly if there is an odd number of spin-1/2’s at the $C_{2}$ center, just as
Eq. (54). The other term, $rc_{0}$, represents an LSM anomaly associated with
$R_{x}$ and $C_{2}$, if there is a $C_{2}$ center at the $R_{x}$-invariant
line and this $C_{2}$ center hosts an odd number of spin-1/2’s.
As another example, we can also ignore the $C_{2}$ symmetry by setting
$c_{0}=0$, and consider translations. On a triangular lattice, the
$R_{x}$-invariant line has a translation symmetry generated by
$T_{1}T_{2}^{2}$. The gauge field corresponding to such a translation symmetry
can be obtained by writing $y=y_{0}+2x$. Similar as above, now $x$ represents
the gauge field corresponding to $T_{1}T_{2}^{2}$, and $y_{0}$ represents the
gauge field corresponding to $T_{2}$. Substituting $c_{0}=0$ and $y=y_{0}+2x$
into Eq. (155) yields $S_{\rm
LSM}=i\pi\int_{X^{4}}(w_{2}^{SO(3)}+t^{2})(xy_{0}+rx)$. Now the first term in
the second parenthesis, $xy_{0}$, represents an LSM anomaly associated with
having an odd number of spin-1/2’s in each unit cell corresponding to the
translations $T_{2}$ and $T_{1}T_{2}^{2}$. The second term, $rx$, represents
an LSM anomaly associated with $R_{x}$ and $T_{1}T_{2}^{2}$, if there is an
odd number of spin-1/2’s in each unit cell of $T_{1}T_{2}^{2}$.
Similarly, we can also obtain the LSM anomaly associated with $R_{x}$ and
$R_{y}$. To do so, we can set $x=y=0$ and $r=c+r_{0}$ in Eq. (155). The first
of these two conditions amounts to ignoring translation symmetries, while the
second condition means that now $c$ is really a gauge field corresponding to
the $R_{x}$ symmetry, and $r_{0}$ is the gauge field corresponding to the
$R_{y}$ symmetry. Substituting these conditions into Eq. (155) yields $S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})cr_{0}$. This represents an LSM
anomaly if there is an odd number of spin-1/2’s at the intersecting point of
the reflection axes of $R_{x}$ and $R_{y}$. This LSM anomaly of course encodes
the one associated with the $C_{2}$ center. To see it formally, one way is to
further restrict $c=r_{0}$, which turns both $c$ and $r_{0}$ into the gauge
field corresponding to the $C_{2}$ rotation. Then this anomaly polynomial
becomes Eq. (54).
The above discussion motivates us to write the LSM anomaly on a square lattice
in terms of gauge fields corresponds to $T_{1,2}$, $R_{x,y}$, $SO(3)$ and
$\mathcal{T}$ symmetries. Denote the gauge fields corresponding to $R_{x,y}$
by $r_{x,y}$. Using an argument similar to that in the main text, the LSM
anomaly on a square lattice can be written as
$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})(x+r_{x})(y+r_{y})$ (157)
with a constraint $t+r_{x}+r_{y}=w_{1}^{TM}\ ({\rm mod\ }2)$. Note that this
expression is manifestly $C_{4}$ rotationally invariant.
To reproduce the Eq. (155), we want to write this anomaly in terms of gauge
fields of $R_{y}$, $C_{2}$, $T_{1,2}$, $SO(3)$ and $\mathcal{T}$. So we let
$r_{x}=c$ and $r_{y}=c+r$, then $c$ is the gauge field for $C_{2}$ and $r$ is
the gauge field for $R_{y}$. Then the above LSM anomaly precisely recover Eq.
(155), after using that $cx=cy=0$. It is interesting to note that in order to
derive Eq. (157) from Eq. (155), one needs to first replace the latter by
$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})[xy+c^{2}+r(x+c)+cy]$ (158)
Because $cy=0$, this expression should be equivalent to Eq. (155). Then by
setting $c=r_{x}$ and $r=r_{y}+r_{x}$ and using that $r_{x}x=r_{y}y=0$, this
anomaly becomes Eq. (157).
All the above results are consistent with the physical expectations. The
method employed above can also be readily applied to other situations to
extract other LSM anomalies.
## Appendix I Anomaly matching of the $U(1)$ DSL on a triangular lattice
In this appendix we show that the $U(1)$ DSL on a triangular lattice indeed
has the correct LSM anomaly.
The symmetries we will focus on are $SO^{s}(3)$ spin rotation, time reversal
$\mathcal{T}$, translations $T_{\bm{a}_{1},\bm{a}_{2}}$, site-centered $C_{2}$
rotation, and reflection $R_{y}$ that keeps $\bm{a}_{1}$ invariant. Their
actions on the $U(1)$ DSL on a triangular lattice are Song _et al._ (2020,
2019):
$\displaystyle\begin{split}&SO^{s}(3):\
n\rightarrow\left(\begin{array}[]{cc}I_{3}&\\\
&SO^{s}(3)\end{array}\right)n,\\\ &\mathcal{T}:\
n\rightarrow\left(\begin{array}[]{cc}I_{3}&\\\ &-I_{3}\end{array}\right)n,\\\
&T_{\bm{a}_{1}}:n\rightarrow\left(\begin{array}[]{cccc}-1&&&\\\ &1&&\\\
&&-1&\\\
&&&I_{3}\end{array}\right)n\exp\left(i\frac{2\pi}{3}\sigma_{y}\right),\\\
&T_{\bm{a}_{2}}:n\rightarrow\left(\begin{array}[]{cccc}1&&&\\\ &-1&&\\\
&&-1&\\\
&&&I_{3}\end{array}\right)n\exp\left(i\frac{2\pi}{3}\sigma_{y}\right),\\\
&C_{2}:n\rightarrow\left(\begin{array}[]{cc}I_{3}&\\\
&-I_{3}\end{array}\right)n\sigma_{z},\\\
&R_{y}:n\rightarrow\left(\begin{array}[]{cccc}&&-1&\\\ &1&&\\\ -1&&&\\\
&&&I_{3}\end{array}\right)n\end{split}$ (159)
From these symmetry actions, we get
$\displaystyle\begin{split}&w_{1}^{O(6)}=t+r+c,\ w_{1}^{O(2)}=c,\\\
&w_{2}^{O(6)}=xy+xr+rt+rc+c^{2}+w_{2}^{SO^{s}(3)}+t^{2},\ w_{2}^{O(2)}=0,\\\
&w_{4}^{O(6)}=(w_{2}^{SO^{s}(3)}+t^{2})(xy+xr+rc)+rc^{2}(t+c)\end{split}$
(160)
with the meanings of these symbols identical as those in the main text.
Substituting these expressions into Eq. (35) and performing some algebraic
manipulations yield
$\displaystyle S_{\rm
bulk}=i\pi\int_{M}[xy+c^{2}+r(x+c)]\left(w_{2}^{SO^{s}(3)}+t^{2}\right)=S_{\rm
LSM}$ (161)
which shows that the $U(1)$ DSL on a triangular lattice indeed has the correct
LSM anomaly.
|
# Non-Halo Structures and their Effects on Gravitational Lensing
T. R. G. Richardson1,2,3,4, J. Stücker3, R. E. Angulo 3,5, O. Hahn2,6,7
1 MAUCA — Master of Astrophysics, Université Côte d’Azur & Observatoire de la
Côte d’Azur, Parc Valrose, 06100 Nice (France)
2 Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Blvd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France.
3 Donostia International Physics Centre (DIPC), Paseo Manuel de Lardizabal 4,
20018 Donostia-San Sebastian, Spain.
4 Laboratoire Univers et Théories (LUTh), Observatoire de Paris - PSL, CNRS,
Université Paris Sciences Lettres, 5 Plc. Jules Janssen, 92190 Meudon, France
5 IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain.
6 Department of Astrophysics, University of Vienna, Türkenschanzstraße 17,
1180 Vienna, Austria
7 Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,
1090 Vienna, Austria E-mail<EMAIL_ADDRESS>(TR)
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Anomalies in the flux-ratios of the images of quadruply-lensed quasars have
been used to constrain the nature of dark matter. Assuming these lensing
perturbations are caused by dark matter haloes, it is possible to constrain
the mass of a hypothetical Warm Dark Matter (WDM) particle to be
$m_{\chi}>5.2$ keV. However, the assumption that perturbations are only caused
by DM haloes might not be correct as other structures, such as filaments and
pancakes, exist and make up a significant fraction of the mass in the
universe, ranging between 5$\%$ – 50$\%$ depending on the dark matter model.
Using novel fragmentation-free simulations of 1 and 3keV WDM cosmologies we
study these “non-halo” structures and estimate their impact on flux-ratio
observations. We find that these structures display sharp density gradients
with short correlation lengths, and can contribute more to the lensing signal
than all haloes up to the half-mode mass combined, thus reducing the
differences expected among WDM models. We estimate that this becomes
especially important for any flux-ratio based constraint sensitive to haloes
of mass $M\sim 10^{8}M_{\odot}$. We conclude that accounting for all types
structures in strong-lensing observations is required to improve the accuracy
of current and future constraints.
###### keywords:
large-scale structure of Universe – dark matter – gravitational lensing:
strong – methods: numerical.
††pubyear: 2020††pagerange: Non-Halo Structures and their Effects on
Gravitational Lensing–B
## 1 Introduction
One of the biggest puzzles of contemporary cosmology and particle physics is
the nature of the dark matter (DM). DM dominates in mass by about five to one
over ordinary matter (Planck Collaboration et al., 2018), it is required to
successfully reproduce many observations of the universe (e.g. Tegmark et al.,
2004; de Blok et al., 2008; Markevitch et al., 2004), and it is an essential
ingredient in cosmological $N$-body simulations that successfully predict the
structure of the universe (e.g. Kuhlen et al., 2012; Frenk & White, 2012, for
reviews).
Despite the cosmological evidence, there is no sign of a potential DM particle
at at the Large Hadron Collider, nor at direct detection experiments (e.g.
LUX, Akerib et al. 2017, XENON1T Aprile et al. 2018). This has made
traditional candidates for DM increasingly less popular, while others such as
axions (e.g. Sikivie, 2008; Marsh, 2016, for reviews) or primordial black
holes (e.g. Carr & Kühnel, 2020, for a review) are enjoying renewed interest.
Various competing DM models predict different features on cosmological scales,
which opens up the opportunity to distinguish them observationally. For
instance, sterile neutrino warm dark matter (WDM), with masses $\gtrsim
3\,{\text{keV}}$ (e.g. Boyarsky et al., 2019, for a review), or ultra-light
axion-like particles with masses of $\sim 10^{-20}\,{\text{eV}}$ forming a
condensate of ‘fuzzy’ DM (FDM, e.g. Niemeyer, 2020, for a review), are in
agreement with all large-scale structure data but predict smooth rather than
clumpy cosmic structure below a particle-mass-dependent scale. These
differences are expected to affect various cosmological observables thus it
becomes possible to constrain DM candidates and their properties.
There are currently three main venues to constrain DM with astrophysical
observations: i) The number and properties of Milky Way satellites; although
these are found to be severely affected by astrophysical processes such as gas
cooling and supernova explosions (e.g. Dekel & Silk, 1986; Ogiya & Mori, 2011;
Pontzen & Governato, 2012; Zolotov et al., 2012; Arraki et al., 2014), these
galaxies are still sensitive to the amount of primordial small scale
structure. ii) The amplitude of small-scale clustering of gas as measured by
the Lyman-$\alpha$ forest. This method has put strong constraints on both WDM
and FDM models down to scales that are now quite degenerate with astrophysical
processes (e.g. Narayanan et al., 2000; Viel et al., 2013; Iršič et al., 2017;
Kobayashi et al., 2017). iii) Perturbations in strong gravitational lensing
which are becoming increasingly competitive in constraining the nature of DM
thanks to recent advances in modelling (Vegetti et al., 2018; Gilman et al.,
2019; Hsueh et al., 2020; Gilman et al., 2020). We refer to Enzi et al. (2020)
for recent constraints and comparisons of these various methods.
In this paper, we will focus on constraints obtained with observations of
light fluxes of strongly-lensed quasars. The images of multiply-lensed high-
redshift quasars originate from different light paths and have potentially
encountered different structures which produce secondary lensing effects. This
leads to anomalies in the flux-ratios of the images which cannot be explained
by the main lens alone. These “anomalies” are found to be sensitive to even
very small DM structures, and can therefore be used to constrain the warmth of
DM. Recent analyses of quadruply-lensed quasars have found that DM has to be
colder than a thermal relic mass of $m_{\chi}<5.2$ keV to explain the measured
perturbations in the lensing signal (Gilman et al., 2019; Hsueh et al., 2020;
Gilman et al., 2020).
In recent studies the amount of perturbation is directly linked to the
abundance and concentration of DM haloes, implicitly neglecting any density
fluctuation outside of haloes (Gilman et al., 2019; Hsueh et al., 2020; Gilman
et al., 2020). However, FDM or WDM cosmologies are not completely devoid of
small-scale structure outside haloes. Since haloes form by triaxial collapse
(Zel’Dovich, 1970; Shandarin & Zeldovich, 1989), only partially collapsed non-
halo structures must exist: one-dimensionally collapsed ‘pancakes’ and two-
dimensionally collapsed ‘filaments’. Pancakes and filaments typically have
lower densities than haloes, which is often the primary motivation for
neglecting them. However, early and modern DM simulations have shown that
these structures contain high-contrast caustics (Buchert, 1989; Shandarin &
Zeldovich, 1989; Angulo et al., 2013) which create sharp high-density edges in
the density field outside haloes. In CDM, the same structures exist, but are
typically fragmented into even smaller substructures (e.g. Bond et al., 1996).
As the precision of observations is increasing, it is important to review all
the underlying assumptions in data analyses. Specifically, here we address the
question of whether it is correct to assume that in a non-cold DM universe the
only source of significant density fluctuations are collapsed haloes. In other
words, can filaments and pancakes in a warm DM universe cause flux-ratio
anomalies comparable with those of low-mass haloes in a colder cosmology?
We are able to address this question thanks to a new generation of
cosmological simulations (Hahn & Angulo, 2016; Stücker et al., 2020), which,
for the first time, simulate nonlinear structure with high precision and
devoid of artificial fragmentation. Using the simulated density fields, we
will create mock strong lensing-observations mimicking a quadruply-lensed
high-$z$ quasar. First, we will study an idealized case where we align a WDM-
filament with the lens geometry and show that filament could cause a relevant
perturbation. Afterwards, we will create more realistic mocks of random
projections of all (non-halo) structures in such WDM simulations to estimate
their contribution to the total number of lensing perturbations. We will show
that non-halo density fluctuation can indeed cause significant lensing
anomalies, comparable in amplitude to the joint effect of all haloes below the
half-mode mass of the corresponding WDM cosmology.
The paper is organised as follows: in §2 we present our $N$-body followed by
our gravitational lensing simulations in §3. In §4 we study a single filament
extracted from our simulations. In §5 we create a set of mock strong lensing
observations and study the statistics of these measurements. Finally, in §6 we
discuss our results and present our conclusions.
## 2 Simulations of WDM structure formation
In this section we describe our WDM numerical simulations. We first focus on
the differences between CDM and WDM initial conditions, and then discuss our
simulation technique. We refer to Stücker et al. (2020) and Stücker et al. (in
prep.) for specifics on our set of simulations.
Table 1: Parameters used in the simulations and throughout this work. The last line indicates the fraction of mass which was found to be outside of haloes at $z=0$. Parameter | 1 keV Sim. | 3 keV Sim.
---|---|---
$h$ | 0.679 | $-$
$\Omega_{\text{m}}$ | $0.3051$ | $-$
$\Omega_{\Lambda}$ | $0.6949$ | $-$
$\Omega_{\text{K}}$ | $0$ | $-$
$\sigma_{8}$ | $0.8154$ | $-$
$M_{\text{hm}}$ | $2.5\cdot 10^{10}h^{-1}{\rm M}_{\odot}$ | $5.7\cdot 10^{8}h^{-1}{\rm M}_{\odot}$
$L_{\text{box}}$ | $20h^{-1}$Mpc | $-$
$N_{\text{tracer}}$ | $512^{3}$ | $-$
$m_{\text{tracer}}$ | $5.0\cdot 10^{6}h^{-1}{\rm M}_{\odot}$ | $-$
$f_{\rm{non-halo}}$ | 45.7% | 34.8%
### 2.1 Initial conditions and simulation set-up
The main difference between CDM and WDM is that the thermal velocities of the
latter led it to free-stream out of small-scale perturbations in the early
Universe, effectively suppressing their growth. As the universe expands, these
initial velocities decay adiabatically and gravitationally-induced velocities
grow. Hence, a very good approximation is to consider WDM as a cold system
with a UV-truncated perturbation spectrum.
To compute these initial fluctuation spectra for our WDM simulations, we use
the parameterisation of the WDM transfer function by Bode et al. (2001) (but
see also Viel et al., 2005, for an alternative parameterisation), where the
matter density power spectrum for WDM is a low-pass filtered version of the
CDM spectrum:
$P_{\text{WDM}}(k)=\left(1+(\alpha k)^{-2}\right)^{-10}P_{\text{CDM}}(k)$ (1)
with
$\alpha=\frac{0.05}{h\,{\text{Mpc}}^{-1}}\left(\frac{\Omega_{\chi}}{.4}\right)^{0.15}\left(\frac{h}{.65}\right)^{1.3}\left(\frac{1\text{keV}}{m_{\chi}}\right)^{1.15}\left(\frac{1.5}{g_{\chi}}\right)^{0.29}$
(2)
where $g_{\chi}=1.5$, $m_{\chi}$ is the mass of the DM particle in units of
keV, and $\Omega_{\chi}$ is the mean DM density in units of the critical
density of the Universe.
Here we will simulate cosmological structures in two WDM cases with
$m_{\chi}=1\,{\text{keV}}$ and $3\,{\text{keV}}$ thermal relic WDM particles,
as well as a CDM case. We will refer to these simulations using their
respective thermal relic DM mass, i.e. the ‘1 keV simulation’ or ‘3 keV
simulation’.
We note that the parameterization of the cut-off that we are using here is
slightly different than the one that is most commonly used in the literature
nowadays from Schneider et al. (2012). This is so because these are
simulations from a larger suite of simulations which explicitly triangulates
the parameter space of possible cut-off functions as will be presented in
(Stücker, in prep.). If our models are matched to the parameterization of
Schneider et al. (2012) by matching at the half-mode mass, they correspond to
slightly warmer cosmologies of $0.82$ keV and $2.6$ keV. However, this does
not affect our conclusions, as we will mostly focus on qualitative
considerations.
Our simulations consist of a cosmological volume of linear size
$L_{\text{box}}=20\,h^{-1}$Mpc. We generated both CDM and WDM initial
conditions based on the same Gaussian noise field (implying that they share
the same large-scale structure) using the Music
software111https://www-n.oca.eu/ohahn/MUSIC/ (Hahn & Abel, 2011, 2013). We use
the cosmological parameters listed in Table 1.
As we discussed above, here we assume that both CDM and WDM evolve as a
collisionless fluid under their self-gravity and are thus governed by Vlasov-
Poisson dynamics (e.g. Peebles, 1980). In the case of CDM, the evolution can
be followed by $N$-body techniques. However, WDM simulations are significantly
more challenging numerically, as we will discuss next.
### 2.2 Fragmentation-free WDM simulations
#### Artificial fragmentation.
$N$-body simulations have been very successful in predicting the non-linear
evolution of cosmic structure from initial CDM perturbation spectra. However,
already early simulations of a UV-truncated WDM perturbation spectrum showed
that the same method performs significantly worse in this case, producing
large amounts of spurious small-scale clumps instead of smooth structures
(e.g. Wang & White, 2007) – an effect which has been termed ‘artificial
fragmentation’.
#### Simulation method.
Recently, a new class of methods based on tessellations of the cold phase
space distribution function (cf. Abel et al., 2012; Shandarin et al., 2012)
has been developed by Hahn & Abel (2013), where the full three-dimensional
hyper-surface of the cold distribution function is evolved. In this approach,
the $N$-body particles serve as vertices of three-dimensional simplicial
elements of the distribution function whose volume determines the DM density
everywhere in space, without coarse graining.
These methods have demonstrated to overcome the artificial fragmentation
problem. However, in regions of strong mixing inside virialised structures,
the number of vertices has to be increased to guarantee the tessellation still
approximates well the distribution function. Adaptive refinement approaches to
solve this problem have been proposed by Hahn & Angulo (2016) and by Sousbie &
Colombi (2016). Since the number of required vertices can become exceedingly
large (cf. Sousbie & Colombi, 2016, due to phase and chaotic mixing) inside of
haloes (particularly so if high force resolution is used), most recently
Stücker et al. (2020) have proposed a hybrid tessellation–$N$-body approach
that resorts to the $N$-body method in regions where three-dimensional
collapse has occurred based on a dynamical classification, and uses
tessellations in the dynamically simpler voids, pancakes and filaments.
This dynamical classification divides sheet tracing particles into four
classes, voids, pancakes, filaments and haloes respectively corresponding to
0, 1, 2 and 3 collapsed axes, only the latter of the four having released
N-body particles. This allows to trace and separate the different structures
present in a simulation, a feature used in later sections.
#### Resolution employed.
In our analysis, we use simulations based on this new approach proposed by
Stücker et al. (2020). Specifically, our simulations use $256^{3}$ particles
to reconstruct the density field through interpolation of the phase space
distribution function (where applicable), but additionally trace $512^{3}$
normal $N$-body particles which are used to reconstruct the density field
where the interpolation fails (i.e. mostly in three-dimensionally collapsed
structures) and can be used to infer other properties, such as the halo mass
function. A more in-depth analysis of these simulations will be presented in
forthcoming work (Stücker et al. 2021, in prep).
#### Mass density field.
With similar interpolation techniques to those used to compute forces one can,
in a post-processing step, recover a density field with much higher sampling
than the original output (e.g. Abel et al., 2012; Hahn & Angulo, 2016). Other
than generating detailed visualisations (e.g. Kaehler et al., 2012), this
feature also allows to recover small scale features that would not be visible
from the initial tracer particles and to reduce the discreteness noise in
lensing simulations (e.g. Angulo et al., 2014). In later sections we will
refer to the use of this technique as ‘resampling the density field’. Based on
high-quality density fields generated in this way, we perform simulations of
the strong gravitational lensing effect, which we describe next.
## 3 Gravitational lensing simulations
In this section, we give a brief account of gravitational lensing theory, we
then describe the technical details of our lensing simulations, and finish by
discussing the measurements of simulated flux ratios of multiply-lensed
sources.
### 3.1 Theory of gravitational lensing
We now recap the main equations of gravitational lensing and refer to one of
the many reviews and textbooks for details (e.g. Schneider et al., 1992;
Bartelmann & Schneider, 2001; Dodelson, 2003).
Let $D_{\text{d}}$ and $D_{\text{s}}$ be the angular diameter distance from
the observer to the deflector (i.e. the gravitational lens) and the source,
respectively, and $D_{\text{ds}}$ that between deflector and source. In the
‘flat lens approximation’ (Bartelmann & Schneider, 2001) the total ray
deflection angle $\boldsymbol{\alpha}$ is related to the position of the image
$\boldsymbol{\theta}$ and the position of the source $\boldsymbol{\beta}$ via:
$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}.$ (3)
A prominent feature of this lens equation is that a single position in the
source plane can map to several positions in the image plane, which originates
multiple images in strong lensing.
The deflection angle, $\boldsymbol{\alpha}$, is the gradient of the “lensing
potential” $\boldsymbol{\alpha}=\boldsymbol{\nabla}\psi$ which is given by a
two dimensional Poisson equation:
$\nabla^{2}\psi=2\kappa.$ (4)
where $\kappa$ is the “normalised surface density” defined as:
$\kappa=\frac{\Sigma(\boldsymbol{\theta})-\bar{\Sigma}}{\Sigma_{\text{cr}}}$
(5)
and $\Sigma(\boldsymbol{\theta})$ is the projected surface density,
$\Sigma_{\text{cr}}=\frac{c^{2}}{4\pi
G}\frac{D_{\text{s}}}{D_{\text{d}}D_{\text{ds}}}$ is the critical surface
density, $c$ is the speed of light, $G$ is the gravitational constant, and
$\bar{\Sigma}=\int_{0}^{z_{\text{bg}}}\rho_{\text{m}}dz$ is the mean surface
density.
Finally, the distortion matrix A is defined as the Jacobian of the mapping
between the source and image planes:
$\textbf{{A}}=\left|\frac{\partial\boldsymbol{\beta}}{\partial\boldsymbol{\theta}}\right|\quad\textrm{i.e.}\quad{A}_{ij}=\left|\partial_{j}\beta_{i}\right|=\left|\delta_{ij}-\partial_{i}\partial_{j}\psi\right|$
(6)
where $\partial_{i}$ is the partial derivative with respect to the $i$-th
coordinate of $\boldsymbol{\theta}$, and $\delta_{ij}$ is the Kronecker delta
symbol. We chose to clarify the indexing convention due to the presence of
numerical indices, represented by upper indices, that appear in later
sections.
The inverse of the determinant of this matrix defines the magnification
$\mu=\frac{1}{\det\left[\textbf{{A}}\right]}$ (7)
the curves in the image plane where $\det\left[\textbf{{A}}\right]=0$ are
called critical curves and limit the different regions where image
replications are formed. The projections of these curves onto the source plane
are called caustics and separate in which parts of the source plane are
multiply imaged (Note that at these curves the magnification becomes infinite,
but, in practice, astrophysical sources have a finite extent and so infinite
magnification is never achieved.)
### 3.2 Multiply-lensed quasar simulations
We now discuss how to numerically obtain deflection angles and distortion
matrices from a simulated density field. We start by defining a grid in the
image plane where we compute the normalised surface density and convergence
field $\kappa$. We then express the lensing potential as a convolution
$\psi=g*2\kappa.$ (8)
where $g$ is the Green’s function of the two-dimensional Laplace operator
$g(\boldsymbol{\theta}):=\frac{1}{2\pi}\ln(\|\boldsymbol{\theta}\|)$. We note
that we use the regularised integration kernel of Hejlesen et al. (2013) for
solving the 2D Poisson equation avoiding the problem created by the
singularity $\boldsymbol{\theta}=\boldsymbol{0}$ (c.f. Appendix (A) for
details).
The deflection angles, distortion matrices, and magnification can be obtained
by exploiting the differentiation property of convolution 222The
differentiation property of convolution is
$\partial_{i}(u(\mn@boldsymbol{x})*v(\mn@boldsymbol{x}))=\partial_{i}u(\mn@boldsymbol{x})*v(\mn@boldsymbol{x})=u(\mn@boldsymbol{x})*\partial_{i}v(\mn@boldsymbol{x})$
(9) where $u(\mn@boldsymbol{x})$, $v(\mn@boldsymbol{x})$ are generic
distributions and $\partial_{i}$ is the derivative with respect to the
$i^{\text{th}}$ component of $\mn@boldsymbol{x}$, such that we express the
different quantities with respect to analytical derivatives of the Green’s
function. , which gives:
$\displaystyle\alpha_{i}$ $\displaystyle=$
$\displaystyle\partial_{i}g*2\kappa$ (10) $\displaystyle\mathsf{A}_{ij}$
$\displaystyle=$ $\displaystyle\delta_{ij}-\partial_{i}\partial_{j}g*2\kappa,$
(11)
where $\delta_{ij}$ is the Kronecker delta symbol and A defines the
magnification. We have thus obtained all the quantities needed for our
application.
#### Force splitting.
To efficiently incorporate the small and large-scale environment of the lens,
we split the lensing potential as a sum of long and short range contributions,
$\psi=\psi_{s}+\psi_{\ell}$, where:
$\psi_{\ell}=(2\kappa*g)*h_{\ell}\qquad\textrm{,}\qquad\psi_{\text{s}}=(2\kappa*g)*h_{\text{s}},$
(12)
and
$\hat{h}_{\ell}:=\exp\left\\{-8\pi^{2}k^{2}\ell^{2}\right\\}\qquad\textrm{,}\qquad\hat{h}_{\text{s}}:=1-\hat{h}_{\ell},$
(13)
where $\ell$ is the splitting length-scale and the hat denotes the Fourier
transform.
Operationally, we compute $\psi_{\ell}$ on a mesh with periodic boundary
conditions that covers the full volume, and $\psi_{\text{s}}$ on a much finer
mesh with non-periodic boundary conditions that covers only a small region
around the lens and where the large-scale solution is interpolated linearly
onto it.
### 3.3 Flux ratio measurements
We now describe how we simulate quadruply-lensed quasars. We will first
describe these observations, then we will present the methods used to simulate
these systems using the previously presented implementation to solve the
lensing equation.
Figure 1: Distorted image produced by the reference lens model described in §
3.3, showing the position of critical curves and detected multiple images.
#### Lens model.
We set up a simulation of a typical quadrupally lensed quasar system. The main
deflector for these simulations is composed of a single projected elliptical
NFW profile (Golse & Kneib, 2002) characterised by its mass $M_{200}=4\times
10^{13}{\rm M}_{\odot}$, its concentration $c_{200}=8$, its eccentricity
$\epsilon=0.05$ and its orientation angle with respect to the main axes
$\lambda=\pi/4$, the lens is placed at redshift $z_{l}=0.29$ and the
background source is placed at $z_{bg}=1.71$ mimicking the redshift
configuration of PG 1115+080 (Weymann et al., 1980; Chiba et al., 2005). In
this configuration the 4 images of a quasar placed exactly at the centre form
at a radius $\theta_{\text{E}}\simeq 1$ ″as can be seen in Fig. 1, in the lens
plane this corresponds to $\theta_{\text{E}}D_{\text{d}}\simeq 3h^{-1}$kpc. In
later paragraphs we refer to this distribution as our reference lens model.
We have checked the sensitivity of our results to the details of our adopted
configuration by considering three different angular separations between the
lens and the lensed quasar. These alternative configurations produced
different absolute values for the flux-ratios, but all of them would give
similar conclusions about the relative contributions of haloes and non-haloes
– which we are focusing on in this article. However, for simplicity we will
restrict to the presentation of one configuration in this article.
#### Source model.
We assume a point-like source and model the flux-ratios by the ratios of the
magnifications at the image locations. (Note that Fig. 1 instead uses a disc
of radius $0.01$″and constant intensity for purpose of visualization.)
#### Image segmentation.
The alignment of the source and the deflector gives rise to five images, four
located near the outer critical curve and a a fifth located close to the
centre. The central image is heavily demagnified and often impossible to
detect in typical observations, while the four outer images are strongly
magnified and easily visible. In Fig. 1 we show the result of the same
simulation where we have labeled the multiple images.
In each realisation, we measure the magnification at the location of each
image $\theta$, where $\theta$ is inferred by solving numerically:
$||\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta})-\boldsymbol{\beta_{0}}||^{2}=0$
(14)
using a two-dimensional root finding algorithm. The four different initial
angles $\theta$ for solving this equation are found by using a projection of a
slightly extended source into the image plane and using the center of each
group (compare Figure 1). We employ scipy.optimize.root333scipy.optimize.root
documentation (Kelley, 1995; Virtanen et al., 2020) readily available in
Python, using the Krylov approximation for the inverse Jacobian. Formally, as
explained above, this equation admits 5 solutions. We therefore introduce an
intermediate step where we use a cluster finding algorithm to locate the
multiple images. We decided to use scipy.ndimage.label444scipy.ndimage.label
documentation (Weaver, 1985; Virtanen et al., 2020) readily available in
Python. Using as starting point the mean position of pixels belonging to a
replicated image and repeating the minimisation for each image we ensure that
we measure the magnification for all images and that the root finding
algorithm doesn’t converge twice, or more, to the same image.
#### Flux ratios.
The main observable is the fluxes of the multiply imaged quasar. Under the
hypothesis that the source has a small angular size with respect to the lens,
we then estimate the flux of each image, $F_{k}$, as:
$F_{k}=\mu_{k}\,F_{\text{int}}$ (15)
where $F_{\text{int}}$ is the intrinsic flux of the quasar and $\mu_{k}$ is
the magnification measured at the position of the image. Since the intrinsic
flux is unknown, it is not possible to recover the magnification of a single
image. However, since all the images originate from the same background
quasar, flux ratios remove the intrinsic contribution while retaining the
information in the magnifications. These ratios are defined with respect to
the brightest image produced by the model presented below, corresponding to
image 4 in Fig. 1.
## 4 The Density field in WDM
Figure 2: Top panel: Column density of a filament from the 1 keV simulation in
logarithmic units of $\rho_{\text{m}}h^{-1}$Mpc, seen side on. Bottom panels:
Same filament but projected along its axis for increasing projection depths.
All panels are normalised to the same colour scale and the column density
increases monotonically as the projection depth increases. We can observe the
diagonal structure in all projections, this structure is the pancake within
which the filament is embedded. In the right most panel the haloes at the end
of the filament are visible, it is these haloes that generate the high density
tail of the corresponding black curve of Fig. 3. Figure 3: Column density
distribution functions, $\phi(\Sigma)$, with $\Sigma$ in units of
$\rho_{\text{m}}h^{-1}$Mpc for increasing projection depths.The high-density
tail (likely caused by caustics) stops adding up coherently around
$10^{2}\rho_{\text{m}}h^{-1}$Mpc . The black curve is produced by the presence
of high-mass haloes at the end of the filament as can be seen in Fig. 2. The
two dashed lines indicate the column density distributions of typical (CDM)
NFW haloes with masses of $10^{9}M_{\odot}$ and $10^{11}M_{\odot}$. We see
that the filament achieves peak column densities comparable with a
$10^{9}M_{\odot}$ halo, while at the same time having much more total mass
associated with intermediate density levels as well.
In this section we will provide an initial exploration of the relative
importance of halo versus non-halo structures in WDM simulations, and examine
whether a single filament could create a lensing signal that is comparable
with a halo at the WDM cutoff scale.
### 4.1 Haloes
Haloes of WDM universes have already been investigated in numerous other
studies (eg. Bode et al., 2001; Schneider et al., 2012; Angulo et al., 2013)
and are reasonably well fitted by the two-parameter NFW profile (Navarro et
al., 1996; Lovell et al., 2014; Bose et al., 2016). However, the abundance is
heavily suppressed on small scales as a consequence of the dampening of the
primordial power spectrum.
Using some simplifying assumptions, one can estimate the mass-scale below
which the transfer function is suppressed by a factor of 2 with respect to the
CDM counterpart. This characteristic “half mode mass” (Viel et al., 2005;
Schneider et al., 2012) is:
$\displaystyle M_{\text{hm}}$
$\displaystyle=\frac{4\pi}{3}\rho_{\text{m}}\left(\frac{\lambda_{\text{hm}}}{2}\right)^{3}$
(16) $\displaystyle\lambda_{\text{hm}}$
$\displaystyle=2\pi\lambda_{\text{s}}^{\text{eff}}(2^{1/5}-1)^{-1/2}\simeq
16.29\lambda^{\text{eff}}_{\text{s}}$ (17)
with the effective free streaming scale
$\lambda_{\text{s}}^{\text{eff}}=\alpha$ defined in Eq. (2). We provide the
value of the half-mode mass for our 1 keV and 3 keV simulations in Table (1).
Free streaming and the following suppression of small perturbations affects
not only the abundance of small haloes, but also the density structure of
haloes close to the half mode mass (Bose et al., 2016; Ludlow et al., 2016).
Specifically, the concentration parameter of haloes of mass $M$ is modified as
(Bose et al., 2016):
$\frac{c_{\text{WDM}}(M,z)}{c_{\text{CDM}}(M,z)}=(1+z)^{\beta(z)}\left(1+60\frac{M_{\text{hm}}}{M}\right)^{-0.17}$
(18)
where $\beta(z)=0.026z-0.04$ and $z$ is the redshift.
Here we will employ this model as a reference against which to compare non-
halo structures both when it comes to their structure and the strong lensing
signal they produce.
### 4.2 The density structure of a filament
A large fraction of the total mass of the universe is expected to reside
beyond haloes in other structures like filaments, pancakes and voids.
Estimates of the fraction of mass outside of haloes depend strongly on the
free streaming scale of DM, but range between $5\%$ in very cold and $50\%$ in
very warm scenarios (e.g. Angulo & White, 2010; Buehlmann & Hahn, 2019). Thus,
we would like to quantify if such uncollapsed mass could create an observable
signature.
As an initial toy example, we selected a relatively long and straight (3.8
Mpc) filament from our 1 keV simulation. In addition, this filament does not
contain any major halo or substructures. For this task we employ DisPerSE
555http://www2.iap.fr/users/sousbie/web/html/indexd41d.html (Sousbie, 2011)
which is an automatic structure identificator based on Morse theory. As
discussed in §3, the main quantity relevant for lensing is the projected
density. Thus, as a best-case scenario, we will project the mass distribution
along the primary axis of the filament. This will inform us if there is at
least a small chance that a filament could generate a significant perturbation
to a lensing signal.
In Fig. 2 we show our selected filament. We display density projections
orthogonal to the filament’s primary axis (top panel) and along its primary
axis (bottom panels) with varying projection depths from 0.1 $h^{-1}$Mpc to 3
$h^{-1}$Mpc. We use the sheet resampling technique to increase the effective
mass resolution by a factor of $64^{3}$ to $m\approx 20h^{-1}M_{\odot}$. This
is possible because the filament is still dynamically simple enough to be
reconstructed by the interpolation algorithms described in §2.
In the top panel it can be seen that the filament exhibits several caustics
which were formed by shell-crossing events and collapse along the filament
minor axis. In the bottom left panel we show the slice orthogonal to the
filament’s axis which reveals a very rich structure. There is, for instance, a
coherent structure going from the top-left to the bottom-right which
corresponds to a pancake that this filament is embedded in. It can also be
seen that the filament’s internal structure is very different from the typical
almost-isotropic inner structure of haloes. Instead its density structure is
governed by multiple density peaks, sharp caustic edges, and different
overlapping streams.
In Fig. 3 we provide the distribution of column densities for varying
projection depths (matching those shown in the bottom panel of Fig. 2). It
becomes clear that column densities in filaments can reach orders of
$100\rho_{\text{m}}\,h^{-1}$Mpc666The value of $\rho_{\text{m}}h^{-1}$Mpc is
$8.33\cdot 10^{10}h^{2}{\rm M}_{\odot}\text{Mpc}^{-2}$ at redshift $z=0$.. In
comparison the average surface density of massive haloes of e.g.
$M_{200}=10^{11}{\rm M}_{\odot}$ and $c_{200}=11$ on a disc of radius
$r=3h^{-1}$kpc, the pertinent length for our reference lens model, is of the
order $2\cdot 10^{3}\rho_{\text{m}}h^{-1}$Mpc. On the other hand low mass
haloes of e.g. $M_{200}=10^{9}{\rm M}_{\odot}$ and $c=15$ reach only an
average surface density of the order $10^{2}\rho_{\text{m}}h^{-1}$Mpc
comparable to the filament.
An aspect of the filament projections worth noting is that even in this case,
where we purposely aligned the filament with the line of sight, the high
density features that we see in very thin slices do not add up very coherently
and, instead, they get smeared out in the thicker projections. Therefore, the
high column density tail does not get enhanced much beyond the $0.5h^{-1}$Mpc
projection.
The toy example analysed in this section shows that filaments can indeed have
significant densities with steep gradients, where the density can change by
one or two orders of magnitude in a very small region. These sharp steps could
potentially cause perturbations of a small enough coherence scale to be
relevant in the flux-ratio measurements. To quantitatively estimate their
relevance, however, not only the column densities, but their spatial pattern
needs to be considered.
### 4.3 A strong gravitational lens perturbed by a line of sight filament
To estimate quantitatively the effect a WDM filament can have as a lensing
perturber, we create a mock lensing observation using the setup discussed in §
3.3.
Our lensing simulations are performed on a grid of side length
$L=0.1h^{-1}$Mpc $\sim 23$ ″and $N=1024$ grid points per dimension for the
fine grid, $L=1h^{-1}$Mpc $\sim 230$ ″and $N=4096$ grid points per dimension
for the coarse grid and the splitting length $\ell=0.92$ ″.
We perturb our lensing simulations by adding the mass field corresponding to
the $2h^{-1}$Mpc deep projection of the filament shown in Fig. 2. We add the
filament’s projected density with varying offsets with respect to the main
deflectors center – on a path that takes it across the strongly lensed region.
For each offset, we measure the magnification ratios of all quasar images.
Each position is marked by its distance, $d$, from the centre of the strong
lensing region. The sign of $d$ is such that when the perturbation is
approaching the centre of the strong lensing region the sign is negative and
inversely when it is moving away the sign is positive. For reference the
radius at which the multiple images form is approximately $3$ to $4$
$h^{-1}$kpc as measured in the lens plane.
Figure 4: Relative change in flux ratio measurements in the presence of a
perturbation, $f$ with respect to the unperturbed flux ratios $f_{0}$. The
abscissas represent the offset of the perturbation with respect to the centre
of lens plane in physical scale as measured in the lens plane. In this
coordinate system, the multiple images of the quasar form at $~{}4h^{-1}$kpc.
Each line represents a different quasar image and are coloured according to
the labelling of the images in Fig. 1, the flux ratios are measure with
respect to image 4 (red image). Top Panel: The lens is perturbed with a
filament aligned with the line of sight for 2 $h^{-1}$Mpc. bottom panel: The
lens is perturbed by a small 1 keV halo of mass $M_{200}=10^{9}{\rm
M}_{\odot}$ and $c_{200}=5.82$. In both cases the perturbation follows the
same path through the lens. One can observe that the filament is able to
produce a considerable effect, similar to that produced by the halo. Figure 5:
Exerts from the three large projections of the simulation volume. Top, we
project only haloes as spherical NFW profiles. Bottom, we project only non-
halo structures. Centre, we project both haloes and non-haloes simultaneously.
This image has a width of $8h^{-1}$Mpc and a height of $2h^{-1}$Mpc. The small
white square in the bottom right shows the size of the individual lines of
sight to scale. Figure 6: Cumulative histograms showing the fraction of
summary statistics $S$ larger than $S_{i}$. The different colours show the
statistics when modeling only haloes with masses $M_{200}<10^{10}M_{\odot}$
(orange), only non-halo material (green) and haloes up to $M_{\text{hm}}$
(blue), we recall $M_{\text{hm}}(1\text{keV})=3.4\cdot 10^{10}h^{-1}{\rm
M}_{\odot}$ and $M_{\text{hm}}(3\text{keV})=7.7\cdot 10^{8}h^{-1}{\rm
M}_{\odot}$. Top panel measurements using the 1 keV simulation, middle panel
measurements from the 3 keV simulation and bottom panel shows some of the same
curves again, but focusing on a comparison between the two different warmths
of DM. The black dotted lines show the effects of CDM haloes below specific
mass cuts. A comparison of the the green curves to the left most dotted curve
shows that non-halo structures from the WDM universes have a comparable effect
on flux-ratios to CDM haloes with masses $M<10^{8}{\rm M}_{\odot}$.
For comparison, we repeat the same procedure, but using as a perturber a
projected spherical NFW halo (Golse & Kneib, 2002) with mass $M_{200}=1\times
10^{9}{\rm M}_{\odot}$ and concentration $c_{\text{WDM}}=5.82$. This
concentration corresponds to a typical halo at that mass, as given by the
combined mass concentration relations from Ludlow et al. (2016) and Bose et
al. (2016).
In Fig. 4, we plot the difference of the flux ratios measured in presence of a
perturbation, $f$, with respect to those in the case of the unperturbed lens,
$f_{0}$. For the halo, the chosen path brings the centre of the perturbing
halo close to two of the images, we see that the closer the structure is to
the image the stronger the impact on the measurement.
This figure shows that the filament can cause a perturbation larger than that
of the $10^{9}h^{-1}{\rm M}_{\odot}$ halo – even by an order of magnitude for
some separations, and reaching the order of $10$ percent compared to the
unperturbed case. The precise shape and amplitude, however, depends in a
complicated manner on the alignment of the structures and can hardly be
summarized in a simplified model. It will be the subject of the next section
to test the impact of such non-halo structures in a three-dimensional
cosmological context.
## 5 Flux ratio anomalies
We have seen in the previous section that material outside haloes can in
principle create flux-ratio perturbations comparable to small mass haloes. We
will now perform a more quantitative study to see whether such lensing
perturbations are likely or not.
### 5.1 Flux-ratio perturbations from random lines of sight
We use our reference lens model, for which all the lensing quantities have
analytical solutions, and perturb it according to mass distributions extracted
from our WDM simulations along random lines of sight.
We consider deep projections ($8\times 8\times 80\,h^{-3}$Mpc3) of the
periodic simulation volume choosing a viewing angle that avoids replications
of the same material in the projected volume and consider the projected volume
only in the single lensing plane approximation. We note that given the size of
our simulations $d=20h^{-1}\text{Mpc}$ we cannot create projection depths
comparable to the distance to typical strong lenses $\sim 1\text{Gpc}$.
Therefore we do not attempt to quantify the absolute effect of the non-halo
structures of the full line of sight. Instead we only consider the effects of
the non-halo structures relative to the effect of haloes inferred from the
same regions. We have checked for projection depths of $40\,h^{-3}$Mpc3 and
$160\,h^{-3}$Mpc3 that the relative contributions of haloes and non-haloes
stays roughly the same. We also see no reasons that the relative contributions
should be affected by the simplified assumption of a single-plane
approximation. In this restricted context one could consider our single lens
as one lens within many in a multiplane formalism.
We construct density fields with two different kinds of projections:
* •
The first projection considers _only haloes_. To reduce numerical noise, we
replace each simulated halo by a spherical NFW profile with a concentration
that has been fitted to the profile. Beyond the virial radius, we then fade
the profile with a cubic spline step function. We select objects with masses
$M_{200}<10^{10}{\rm M}_{\odot}$ which are those expected not to host a
detectable galaxy that can be directly inserted into a lens model (Gilman et
al., 2020). For comparison, we also consider a case projecting only haloes
below the expected $M_{\text{hm}}$ of the respective DM model.
* •
The second projection considers _only non-halo_ material. This is done by
selecting mass elements that are not part of a halo according to the release
criterion from Stücker et al. (2020) (c.f. their Eq. 17), and resampling them
to a $\sim 20\,{\rm M}_{\odot}$ mass resolution with the sheet interpolation
method. Note that due to this high resolution resampling our lensing mocks do
not suffer from discreteness effects like those inferred from traditional SPH
/ CIC density estimation techniques.
In Fig. 5 we show an example of the halo projection (without mass cut for this
figure), the non-halo projection and the sum of both.
Within these large projections ($80h^{-1}$Mpc deep) we then sample 1000 random
lines of sight with a side length $L_{\text{c}}\simeq 160h^{-1}$kpc. The size
of these cut-outs is chosen to be large compared to the radius at which the
multiple images form ($\theta_{\text{E}}D_{\text{d}}=3h^{-1}$kpc with respect
to the centre of the lens plane). The small square in the bottom right of Fig.
5 displays the size of these regions.
These cut-outs are then used to compute the gravitational lensing effect on
our fine grids $(N=1024)$ while the large scale contribution is calculated
with the full large projection using $N=8182$ grid points per dimension and a
splitting length $\ell=1.84$ ″. As such, the fine grid is calculated out to
$50\,\theta_{\text{E}}$ while the coarse grid is calculated out to
$2900\,\theta_{\text{E}}$ to capture the influence of the large scale
environment.
Using the linearity of Eq. (4) we then add this perturbation to the analytical
reference lens model. With a simulated quasar placed at redshift $z_{bg}=1.71$
and sky coordinates that compensate for the mean deflection generated by the
large scale environment, which ensures the quasar is quadruply lensed, we
measure the magnification of each image and calculate the three flux ratios,
$f_{i}$.
For each line of sight, we can quantify the perturbations to the flux ratios
using the following summary statistic:
$S:=\sqrt{\sum_{i=1}^{3}(f_{i}-f_{\text{ref}(i)})^{2}},$ (19)
where $f_{\text{ref}(i)}$ is a flux ratio for image $i$ as measured with the
reference unperturbed model, note this is the same statistic as used by Gilman
et al. (2019). The larger the value of $S$, the larger is the expected
perturbation to the main lensed images. We remind the reader that in our case
the absolute values of $S$ are about an order of magnitude smaller than of
typical lenses, since we have a quite short line of sight of $80$Mpc$/h$.
However, the relative comparison between the $S$-distribution of haloes and
non-halo structures should be unaffected by this since both scale with the
length of the line of sight.
In Fig. 6 we show the cumulative distributions for $S$ from our $1000$ lines
of sight. These distributions can be interpreted as the probability of
observing a flux ratio anomaly higher than a given level. In all panels,
different colours correspond to different types of structures. The _top_ and
_central_ panels differ by the warmth of the DM, while the _bottom_ panel
compares both models using different line styles. Additionally we added four
lines to indicate the flux-ratio perturbations that would be measured from CDM
haloes with different mass-cuts in the same projected volumes.
First of all we see that haloes from cosmologies with different warmth (solid
versus dashed orange lines in the last panel), indeed produce different
statistics for the flux-ratio perturbations. This is the effect that is used
to constrain the warmth of DM from flux-ratio observations (eg. Hsueh et al.,
2020; Gilman et al., 2020).
Second, we concentrate on the curves produced from the 3 keV simulation. We
observe that the typical perturbations caused by non-halo structures (green
lines) are smaller by a factor of a few compared to those produced by haloes
of $10^{10}\,M_{\odot}$ (orange lines). However, they are approximately 50%
more common than fluctuations produced by haloes at and below the half-mode
mass (shown by the blue lines). The situation is qualitatively similar in the
warmer 1keV case where non-halo material also contributes more than haloes
below the half-mode mass for moderate values of $S$, although they become
subdominant at higher $S$.
We now compare these perturbations to those produced by CDM haloes below
various mass thresholds, indicated by black dotted lines in Fig. 6. We see
that anomalies produced by non-halo structures are of similar abundance as
those caused by CDM haloes with masses $M<10^{8}{\rm M}_{\odot}$. Note that
the effect expected for WDM haloes in the respective cosmology would had been
much smaller owing to their lower concentrations and abundance.
When comparing the “non-halo” contribution in the WDM simulations (green lines
in the last panel) to $S$, we see that it decreases slightly, by $20-30\%$,
between the 1keV and 3keV models. Since the fraction of total mass outside of
haloes, $f_{\rm{non-halo}}$, is smaller in colder models, more mass is
collapsed into small haloes. As mentioned in Table 1, for the 1keV simulation
we have $f_{\rm{non-halo}}=46\%$ and for the 3keV universe $f_{\rm{non-
halo}}=35\%$. The ratio between these numbers is roughly consistent with the
shift in $S$. In simple excursion set models, the fraction of mass outside
haloes changes very slowly with the cut-off scale. Even for very cold DM
models, such as a 100GeV neutralino, about $5\%-20\%$ of mass is expected be
to reside outside of haloes at $z=0$. This fraction increases significantly at
higher redshifts (e.g. Angulo & White, 2010). Therefore, we speculate that the
“non-halo” contribution could shift from the $3$keV case to slightly, but not
significantly, smaller values for colder dark matter candidates.
These results suggest that non-halo material is not only dense enough to cause
lensing perturbations, but that it commonly does so in warm DM scenarios.
Furthermore, this shows that the often made assumption that non-halo
structures can be neglected for the modeling of flux-ratio perturbations,
holds only within certain limits. Constraints that are sensitive to haloes
with masses around $10^{8}M_{\odot}/h$ or less, could incur a systematic error
and provide inaccurate results. For instance, flux-ratio based evidence of the
existence of haloes below $10^{8}{\rm M}_{\odot}$ could had also been
originated by non-halo structures in a warmer cosmology. The quantitative
impact on specific observational constraints, however, most likely depends on
details of the modelling and observational setup. However, our results
indicate that the contribution of non-halo material must be assessed
carefully, and ideally, analysis pipelines should be tested in realistic
simulated mock observations.
## 6 Conclusions
The main question of this study was: _can the material outside of haloes –
which resides for example in filaments or pancakes – cause relevant effects in
strong gravitational lenses?_ In particular, we have focused on lensing
systems where a quasar is quadruply lensed. In such systems the ratios of
fluxes from the different images can be used to constrain the line-of-sight
structure and thereby the DM warmth (Gilman et al., 2019; Hsueh et al., 2020;
Gilman et al., 2020). So far all studies have only modeled the effect of
haloes, but not any other structures in the density field. However, for
example the caustic structure of a filament in a WDM universe can create
density variations on very small scales – even when no small haloes are
expected at these length scales. It is therefore important to understand the
effect of non-halo structures qualitatively and quantitatively.
In a first qualitative part of this study we have shown the caustic structure
and the sharp density edges that exist in projections of a warm DM filament
without substructure. Further we have confirmed that such a filament creates a
relevant perturbation when aligned with the line of sight, and found that its
effect is comparable that of a halo of $10^{9}{\rm M}_{\odot}$. This shows
that, at least in principle, such a filament could cause significant effects
in flux-ratio observations.
In a second – more quantitative – investigation we have created a large number
of mock lensing observations from random projections of two state-of-the-art
WDM simulations. From these measurements we have found that the flux-ratio
perturbations created by non-halo structures can be larger than those caused
by all haloes up to the half mode mass of the corresponding cosmology.
Because of the numerical requirements, we only simulated warm dark matter
models here which are already excluded by current constraints. However, we
argue that the relative importance of the non-halo structures (in comparison
to haloes around the half-mode mass) becomes even larger for colder DM
candidates (e.g. $m_{\chi}\sim 5$ keV). We speculate that the effect of non-
halo structures is roughly proportional to the total fraction of mass outside
of haloes, and therefore it should only exhibit a moderate decrease when
considering colder models. On the other hand, perturbations from haloes around
the half-mode mass decrease rapidly with decreasing half-mode mass. Even in
cold dark matter cosmologies a significant fraction of mass resides outside of
haloes and might have an impact on flux-ratio lensing observations.
Although a precise quantification of the effect on recent constraints of the
warmth of DM would require mock simulations mimicking details of the
observations and analyses, we can argue in general that Non-halo structures
have an effect on flux-ratio measurements which is comparable to line-of-sight
CDM haloes with masses of order $10^{8}M_{\odot}/h$ (or slightly more or less
depending on the warmth of the DM model). Therefore, these “non-halo”
structures should be considered in any flux-ratio analysis where the main
constraining power comes from haloes of $M\sim 10^{8}M_{\odot}/h$ or less. If
the constraining power comes from haloes with $M\gg 10^{8}M_{\odot}/h$, the
non-halo structures might still have a quantitative, but likely subdominant
impact.
We have made a variety of simplifications when estimating the perturbations of
flux-ratios. These include that we only modeled a single base-lens; we could
only project over a relatively short line of sight ($80h^{-1}$Mpc); we worked
in the single-lens approximation, we did not model baryonic effects; and we
only used simulations of two quite warm DM models incompatible with current
constraints. These simplifications prevent us from making an absolute estimate
of the frequency of anomalies caused by non-halo material, however, we expect
the relative contribution with respect to that of halos to be largely
unaffected by our assumptions, and thus we argue for the robustness of our
conclusions.
Despite these simplifications, our results call for a careful estimate of the
impact of non-halo material in current DM mass constraints. For example, an
estimate of how the inclusion of non-halo structures affects the posterior
distribution of the DM warmth like the ones presented in Gilman et al. (2019).
To achieve that, it would be necessary to implement a similar pipeline to
Gilman et al. (2019) that operates on actually observed systems and further it
would be required to have a larger variety of WDM simulations which also cover
a much larger volume as such that lines of sight with a depth of order Gpc can
be modeled. We leave such more sophisticated investigations to future studies.
## Acknowledgements
We acknowledge insightful comments on the manuscript from Simona Vegetti,
Carlo Giocoli, and Giulia Despali. T. R. thanks the Observatoire de la Côte
d’Azur, the DIPC, and the Erasmus+ program for making this joint research work
possible. J.S., R.A. and O.H. acknowledge funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programmes: Grant agreement No. 716151 (BACCO) for J.S. and R.A.; and grant
agreement No. 679145 (COSMO-SIMS) for O.H. The authors thankfully acknowledge
the computer resources at MareNostrumIV and technical support provided by the
Barcelona Supercomputing Center (RES-AECT-2019-3-0015).
## Data Availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
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## Appendix A Numerical implementations
In this section we discuss the specifics of the numerical implementations used
to solve the lens equation.
#### Computation of the lensing potential
As said in § (3) the gravitational lensing potential is defined by Eq. (4), a
poisson equation which can be solved by a convolution with the appropriate
Green’s function, see Eq. (8). The analytical Green’s function for the two-
dimensional Laplace operator presenting a singularity at the origin, which
prevents convergence, we instead use the regularised integration kernels
$g_{m}=-\frac{1}{2\pi}\left[\ln(\theta)-Q_{m}\left(\frac{\theta}{\epsilon}\right)\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)+\frac{1}{2}E_{1}\left(\frac{\theta^{2}}{2\epsilon^{2}}\right)\right]$
(20)
proposed by Hejlesen et al. (2013). Where $\epsilon$ is a smoothing parameter
set to 1.5 times the grid spacing $\delta$, $Q_{m}$ is a polynomial setting
the order $m\in\mathbb{N}$ of the kernel and $E_{1}$ is the exponential
integral distribution. This particular function has a finite value at
$\theta=0$,
$g_{m}(0)=\frac{1}{2\pi}\left[\frac{\gamma}{2}-\ln\left(\sqrt{2}\epsilon\right)+Q_{m}(0)\right]$
(21)
where $\gamma=0.5772156649$ is Euler’s constant.
After replacing the green’s function eq. (8) can be discretised and solved by
multiplication in Fourier space. When numerically evaluating eq. 8 using a
discrete Fourier transform (DFT), implicitly periodic boundary conditions are
assumed. One can account for isolated boundary conditions by zero padding. The
solution being to pad out $\kappa$ to twice its original size, assigning zero
to all the newly created cells. One then defines $g_{m}^{(ij)}(\theta^{(ij)})$
on this new grid and calculates the components of the FT of the lensing
potential as
$\hat{\psi}^{(qp)}=2\hat{g}_{m}^{(qp)}\hat{\kappa}^{(qp)},$ (22)
where the hat symbolises the Fourier transform and the upper $(qp)$ index
corresponds to the Fourier space position $\boldsymbol{k}^{(qp)}$. Finally we
recover the lensing potential by inverting the FT and removing the padded
area.
#### Computation of the deflection angles
During this work we have identified two methods of computing quantities that
are derivatives of the lensing potential, such as the deflection angles
$\boldsymbol{\alpha}^{(ij)}$ which allow to calculate the coordinates in the
source plane, $\boldsymbol{\beta}^{(ij)}$, where each grid point maps to. The
simplest manner is to use finite differences. To second order the components
of the deflection angles will be.
$\begin{cases}\alpha_{1}^{(ij)}=\frac{\psi^{(i+1,j)}-\psi^{(i-1,j)}}{2\Delta}\\\
\alpha_{2}^{(ij)}=\frac{\psi^{(i,j+1)}-\psi^{(i,j-1)}}{2\Delta}\end{cases}$
(23)
The second method is to use the differentiation property of the convolution,
see Eq. (9), such that we express the different quantities with respect to
analytical derivatives of the Green’s function. This gives the deflection
angle components,
$\alpha_{i}=\partial_{i}g_{m}*2\kappa$ (24)
using
$\displaystyle\begin{split}\partial_{i}g_{m}=-\frac{1}{2\pi}&\left\\{\frac{\theta_{i}}{\theta^{2}}\left[1-\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)\right]\right.\\\
&\left.+\frac{\theta_{i}}{\epsilon^{2}}\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)\left[Q_{m}\left(\frac{\theta}{\epsilon}\right)-\frac{\epsilon}{\theta}Q^{\prime}_{m}\left(\frac{\theta}{\epsilon}\right)\right]\right\\}\end{split}$
(25)
for the derivative of the Green’s function, with
$\partial_{i}g_{m}|_{\theta=0}=0$. We refer to this method as the spectral
method.
#### Computation of the Distortion matrix
Similarly, one can define the components of the distortion matrix A either
through finite differences or using analytical derivatives,
$\textbf{{A}}_{ij}=\delta_{ij}-\partial_{i}\partial_{j}g_{m}*2\kappa$ (26)
where $\delta_{ij}$ is the Kronecker delta symbol. For which we require two
expressions for the different possible combinations of derivatives. Leading
to,
$\displaystyle\begin{split}\partial_{i}^{2}g_{m}=&-\frac{1}{2\pi}\left\\{\frac{(-1)^{i}(\theta_{2}^{2}-\theta_{1}^{2})}{\theta^{4}}\left[1-\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)\right]\right.\\\
&+\frac{1}{\epsilon^{2}}\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)\left(\left(1-\frac{\theta_{i}}{\epsilon^{2}}\right)\left[Q_{m}\left(\frac{\theta}{\epsilon}\right)-\frac{\epsilon}{\theta}Q^{\prime}_{m}\left(\frac{\theta}{\epsilon}\right)\right]\right.\\\
&\left.\left.\frac{\theta_{i}^{2}}{\theta^{2}}\left[\left(\frac{\theta^{2}+\epsilon^{2}}{\theta\epsilon}\right)Q_{m}^{\prime}\left(\frac{\theta}{\epsilon}\right)-\frac{\epsilon}{\theta}Q^{\prime\prime}_{m}\left(\frac{\theta}{\epsilon}\right)-1\right]\right)\right\\}\end{split}$
(27)
for the diagonal terms, which evaluated at the origin yields
$\partial_{i}^{2}g_{m}|_{\theta=0}=\frac{1}{2\pi\epsilon^{2}}\left(\frac{1}{2}+Q_{m}(0)-Q_{m}^{\prime\prime}(0)\right)$,
and
$\displaystyle\begin{split}\partial_{i}\partial_{j}g_{m}=&-\frac{1}{2\pi}\left\\{\frac{\theta_{i}\theta_{j}}{\theta^{4}}\left[-\frac{\theta^{2}}{\epsilon^{2}}\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)+2\left(1-\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)\right)\right]\right.\\\
&-\frac{\theta_{i}\theta_{j}}{\epsilon^{2}}\exp\left(-\frac{\theta^{2}}{2\epsilon^{2}}\right)\left(-\frac{1}{\epsilon^{2}}\left[Q_{m}\left(\frac{\theta}{\epsilon}\right)-\frac{\epsilon}{\theta}Q_{m}^{\prime}\left(\frac{\theta}{\epsilon}\right)\right]\right.\\\
&\left.\left.+\frac{1}{\theta^{2}}\left[\frac{\theta^{2}+\epsilon^{2}}{\theta\epsilon}Q_{m}^{\prime}\left(\frac{\theta}{\epsilon}\right)-Q_{m}^{\prime\prime}\left(\frac{\theta}{\epsilon}\right)\right]\right)\right\\}\end{split}$
(28)
for the cross terms which yields
$\partial_{i}\partial_{j}g_{m}|_{\theta=0}=0$.
The convergence of both these schemes is discussed in App. (B) where one can
see that the spectral scheme is overall more accurate and presents a faster
convergence rate. On the basis of these tests we privilege the use of the
spectral scheme.
## Appendix B Numerical Convergence
Figure 7: RMSD of the determinant of distortion matrix with respect to the
analytical solution for increasing resolution. The black curve corresponds to
the FD scheme and the purple curve corresponds to the spectral method. We
observe that the spectral method is overall more accurate and converges at a
faster rate than the FD scheme.
Here we test the convergence of the numerical implementations discussed in
App. (A). To do so we set up a simple test problem to which we can find an
analytical solution, a Gaussian surface density field
$\kappa=\frac{K_{m}}{2\pi\sigma^{2}}\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)$
(29)
where $\theta=\sqrt{\theta_{1}^{2}+\theta_{2}^{2}}$ is a radial coordinate,
$K_{m}$ is the physical mass of the profile in units of the critical surface
density and $\sigma$ is its width. We solve Eq. (4) to obtain an analytical
solution for the lensing potential
$\psi=\frac{K_{m}}{4\pi}\left[\log\left(\frac{\theta^{4}}{4\sigma^{4}}\right)-2E_{i}\left(\frac{\theta^{2}}{2\sigma^{2}}\right)\right]+C$
(30)
where $E_{i}$ is the exponential integral and $C$ is a constant gauge term.
The main output of this code being the deflection angles and magnification it
is of interest to study them directly. As we have a strong radial symmetry to
our problem the solution is fully described by the norm $\alpha$ of the
deflection angle.
$\alpha=\frac{K_{m}}{\pi\theta}\left[1-\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)\right].$
(31)
We finally give the three independent components of the distortion tensor A
for $r<r_{d}$
$\begin{cases}\textbf{{A}}_{11}=1-\frac{K_{m}}{\pi}\left[\frac{\theta_{2}^{2}-\theta_{1}^{2}}{r^{4}}\left(1-\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)\right)+\frac{\theta_{1}^{2}}{\theta^{2}\sigma^{2}}\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)\right]\\\
\textbf{{A}}_{22}=1-\frac{K_{m}}{\pi}\left[\frac{\theta_{1}^{2}-\theta_{2}^{2}}{\theta^{4}}\left(1-\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)\right)+\frac{\theta_{2}^{2}}{\theta^{2}\sigma^{2}}\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)\right]\\\
\textbf{{A}}_{12}=\frac{K_{m}}{\pi}\frac{\theta_{1}\theta_{2}}{\theta^{4}}\left[2\left(1-\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)\right)-\frac{\theta^{2}}{\sigma^{2}}\exp\left(-\frac{\theta^{2}}{2\sigma^{2}}\right)\right]\end{cases}$
(32)
With this solution we now compute the Root Mean Square Deviation (RMSD)
between the numerical solution and the analytical solution, imposing in the
numerical case that the total mass of the profile is conserved. We repeat this
while increasing the resolution. The result of this is reproduced in Fig. 7
along with a power law fit giving an indication of the convergence rate.
|
# Continuum approach to real time dynamics of 1+1D gauge field theory:
out of horizon correlations of the Schwinger model
Ivan Kukuljan Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Str. 1,
DE-85748 Garching, Germany Munich Center for Quantum Science and Technology,
Schellingstr. 4, DE-80799 München, Germany
###### Abstract
We develop a truncated Hamiltonian method to study nonequilibrium real time
dynamics in the Schwinger model - the quantum electrodynamics in D=1+1. This
is a purely continuum method that captures reliably the invariance under local
and global gauge transformations and does not require a discretisation of
space-time. We use it to study a phenomenon that is expected not to be
tractable using lattice methods: we show that the 1+1D quantum electrodynamics
admits the dynamical horizon violation effect which was recently discovered in
the case of the sine-Gordon model. Following a quench of the model,
oscillatory long-range correlations develop, manifestly violating the horizon
bound. We find that the oscillation frequencies of the out-of-horizon
correlations correspond to twice the masses of the mesons of the model
suggesting that the effect is mediated through correlated meson pairs. We also
report on the cluster violation in the massive version of the model,
previously known in the massless Schwinger model. The results presented here
reveal a novel nonequilibrium phenomenon in 1+1D quantum electrodynamics and
make a first step towards establishing that the horizon violation effect is
present in gauge field theory.
###### pacs:
03.70.+k,11.15.-q,11.10.Ef
_Introduction. -_ Computing real time dynamics of an interacting many-body
quantum system is a notoriously difficult problem. It has been currently
getting an overwhelming amount of attention due to the fast developing field
of nonequilibrium physics both in high energy Kamenev (2011); Berges (2004);
Berges et al. (2004); Calzetta and Hu (2008); Grozdanov and Polonyi (2015a,
b); Calabrese and Cardy (2016); Bernard and Doyon (2016); Glorioso and Liu
(2018) and condensed matter physics Husmann et al. (2015); Vasseur and Moore
(2016); Medenjak et al. (2017) on one side and renewed interest in chaos and
information scrambling on the other side Sekino and Susskind (2008); Kitaev
(2014); Maldacena et al. (2016); Polchinski and Rosenhaus (2016); Jahnke
(2019). It is also becoming a matter of increased experimental importance
Langen et al. (2015); Bloch et al. (2008); Bernien et al. (2017); Madan et al.
(2018) . The set of tools to deal with the problem has been greatly enriched
by developments and new insights in integrability theory LeClair and Mussardo
(1999); Essler and Fagotti (2016); Caux (2016), holography Maldacena (1999);
Aharony et al. (2000); Casalderrey-Solana et al. (2014); Zaanen et al. (2015);
Liu and Sonner (2018) and numerical algorithms such as density matrix
renormalisation group (DMRG) White (1992); Schollwöck (2011), tensor networks
(TNS) Cirac and Verstraete (2009); Orús (2014); Bridgeman and Chubb (2017) and
lattice gauge theory Bender et al. (2020); Emonts et al. (2020). Although in
the present time, there is an abundance of excellent numerical methods
available for discrete systems, the methods for the real time evolution
directly in the continuum remain scarce and less developed.
A powerful class of algorithms are the truncated Hamiltonian methods (THM)
Yurov and Zomolodchikov (1990); James et al. (2018); Yurov and Zomolodchikov
(1991); Lässig et al. (1991); Feverati et al. (1998); Bajnok et al. (2001,
2002); Hogervorst et al. (2015); Rychkov and Vitale (2015). They are numerical
methods for quantum field theories (QFT) that work in the continuum and do not
require a discretisation of space-time. They can be applied to a wide set of
tasks like computing spectra Yurov and Zomolodchikov (1990, 1991); Lässig et
al. (1991); Feverati et al. (1998); Bajnok et al. (2001, 2002); Hogervorst et
al. (2015); Elias-Miró et al. (2017); Rychkov and Vitale (2015); Elias-Miró
and Hardy (2020) and level spacing statistics Brandino et al. (2010); Srdinšek
et al. (2020), studying symmetry breaking Rychkov and Vitale (2015),
correlation functions Kukuljan et al. (2018); Kukuljan et al. (2020a), real
time dynamics Rakovszky et al. (2016); Kukuljan et al. (2018); Hódsági et al.
(2018); Horváth et al. (2019); Kukuljan et al. (2020a) and also gauge field
theories Konik et al. (2015); Azaria et al. (2016). The class of methods
originates from the truncated conformal space approach (TCSA) introduced by
Yurov and Zamolodchikov Yurov and Zomolodchikov (1990). A QFT model on a
compact domain is regarded as point along the renormalisation group (RG) flow
from the ultra violet (UV) fixed point generated by a relevant perturbation.
The conformal field theory (CFT) algebraic machinery is used to represent the
Hamiltonian as a matrix in the basis of the UV fixed point CFT Hilbert space.
Finally, an energy cutoff is introduced to obtain a finite matrix which
enables numerical computation that indeed efficiently captures nonperturbative
effects. More broadly, instead of CFT, any solvable QFT can be used as the
starting point for the expansion.
One of the central properties of quantum physics out of equilibrium is the
horizon effect introduced by Cardy and Calabrese Calabrese and Cardy (2006,
2007); Iglói and Rieger (2000). A quantum system is initially prepared in a
short range correlated nonequilibrium state, $\left\langle
O(x)O(y)\right\rangle\propto e^{-\left|x-y\right|/\xi}$ with a local
observable $O$, the correlation length $\xi$, and let to evolve dynamically
for $t>0$ \- a protocol commonly termed a quantum quench. The horizon bound
states that the connected correlations following the quench spread within the
horizon: $\left|\left\langle
O(t,x)O(t,y)\right\rangle\right|<\kappa\,e^{-\text{max}\left\\{(\left|x-y\right|-2ct)/\xi_{h},0\right\\}}$
for some constant $\kappa$, where $\xi_{h}$ is called the horizon thickness
and $c$ is the maximal velocity of the theory - speed of light in QFT and the
Lieb-Robinson (LR) velocity in discrete systems Lieb and Robinson (1972). The
intuition is that correlations spread by pairs of entangled particles created
in initially correlated region $\left|x-y\right|\lesssim\xi$ and traveling to
opposite directions. This bound has been rigorously proven in CFT Calabrese
and Cardy (2006, 2007); Cardy (2016) and demonstrated, analytically and
numerically in a large set of interacting systems Calabrese and Cardy (2005);
Chiara et al. (2006); Burrell and Osborne (2007); Fagotti and Calabrese
(2008); Läuchli and Kollath (2008); Eisler and Peschel (2008); Manmana et al.
(2009); Calabrese et al. (2011); Iglói et al. (2012); Calabrese et al. (2012a,
b); Ganahl et al. (2012); Essler et al. (2012); Bardarson et al. (2012); Kim
and Huse (2013); Hauke and Tagliacozzo (2013); Schachenmayer et al. (2013);
Richerme et al. (2014); Carleo et al. (2014); Nezhadhaghighi and Rajabpour
(2014); Bonnes et al. (2014); Collura et al. (2014); Krutitsky et al. (2014);
Bucciantini et al. (2014); Kormos et al. (2014); Vosk and Altman (2014);
Rajabpour and Sotiriadis (2015); Buyskikh et al. (2016); Altman and Vosk
(2015); Fagotti and Collura (2015); Bertini and Fagotti (2016); Cardy (2016);
Castro-Alvaredo et al. (2016); Bertini et al. (2016); Bertini and Fagotti
(2016); Zhao et al. (2016); Pitsios et al. (2017); Kormos et al. (2017) as
well as observed in experiments Cheneau et al. (2012); Jurcevic et al. (2014);
Langen et al. (2013). It has therefore been believed to be a universal
property of quantum physics.
Figure 1: Dynamical horizon violation as found in the sine-Gordon model
Kukuljan et al. (2020a). The system is prepared in the ground state of a gaped
Hamiltonian $H_{0}$ with short range correlations $\propto
e^{-\left|x-y\right|/\xi}$. At time $t=0$ the Hamiltonian is quenched to $H$.
This generates cluster violating 4-point correlations of solitons and
antisolitons, eq. (2) (here symbolically pictured using classical solitons),
which are not observable at $t=0$ but result in oscillating out-of-horizon
correlations of local observables $\left\langle O(-x)O(x)\right\rangle$ at
later times. The horizon is depicted here with gray color and the horizon
violating correlations with red. Asymptotically, the latter oscillate with a
frequency 4 times the soliton mass $M$, respectively twice the breather masses
in the attractive regime.
In a recent publication together with Sotiriadis and Takács Kukuljan et al.
(2020a), we have demonstrated that the horizon bound can be violated in QFT
with nontrivial topological properties. We have proved this in the case of the
sine-Gordon (SG) field theory, a prototypical example of strongly correlated
QFT
$\mathcal{L}_{SG}=\frac{1}{2}(\partial_{\mu}\Phi)(\partial^{\mu}\Phi)+\frac{\mu^{2}}{\beta^{2}}\cos(\beta\Phi)$
(1)
Starting from short range correlated states, SG dynamics within a short time
generates infinite range correlations oscillating in time and clearly
violating the horizon bound. The mechanism is the following: Quenches in the
SG model create cluster violating four-body correlations between solitons
($S$) and anti-solitons ($A$), the topological excitations of the theory,
written schematically:
$\displaystyle\lim\limits_{|x-y|\rightarrow\infty}\left<A(x)S(x+a)A(y)S(y+b)\right>$
$\displaystyle\hskip
56.9055pt\neq\left<A(x)S(x+a)\right>\left<A(y)S(y+b)\right>$ (2)
The dynamics of the model then converts these solitonic correlations into two-
point correlations of local bosonic fields
$\left\langle\Phi(t,x)\Phi(t,y)\right\rangle$,
$\left\langle\Pi(t,x)\Pi(t,y)\right\rangle$ and
$\left\langle\partial_{x}\Phi(t,x)\partial_{y}\Phi(t,y)\right\rangle$. There
is no violation of relativistic causality involved because the cluster
violating correlations (2) are created by a quench, a global simultaneous
event and not by the unitary dynamics of the model which is strictly causal.
The mechanism suggests that the horizon violation should be found in any QFTs
with nontrivial field topologies, an important class of them being gauge field
theories. The results presented in this Letter represent the first steps
towards establishing that.
As a consequence of the Lieb-Robinson bound Prosen (2014); Bravyi et al.
(2006); Lieb and Robinson (1972) and the Araki theorem Araki (1969); Kliesch
et al. (2014), the horizon violation is expected not to be present in short-
range interacting discrete systems with finite local Hilbert space dimension
and is likely a genuinely field theoretical phenomenon. Therefore discretising
a model and simulating using DMRG or TNS Kogut and Susskind (1975); Buyens et
al. (2015, 2017); Hebenstreit et al. (2013); Spitz and Berges (2019);
Notarnicola et al. (2020); Chanda et al. (2020); Magnifico et al. (2020);
Bender et al. (2020); Emonts et al. (2020) is not an option so methods working
directly in the continuum are needed and THM seem to be the best class of
methods for the task.
_The Schwinger model. -_ We focus here on the simplest example of a gauge
field theory, the 1+1D quantum electrodynamics (QED), i.e. the (massive)
Schwinger model:
$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\Psi}\left(i\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}A_{\mu}-m\right)\Psi,$
with $\Psi=\left(\Psi_{-},\Psi_{+}\right)^{T}$ the Dirac fermion, $m$ the
electron mass and $e$ the electric charge. As a consequence of invariance
under large gauge transformations, the model has infinitely degenerate vacuum
states, the $\theta$ vacua for a parameter $\theta\in\left[0,2\pi\right)$ that
enters the bosonised form of the Hamiltonian and plays the physical role of
the constant background electric field Coleman et al. (1975); Coleman (1976).
The Schwinger model thus has two physical parameters, the ratio $m/e$ and
$\theta$.
The massless $m=0$ version of the model was solved exactly by Schwinger
Schwinger (1962) and has a gap of $e/\sqrt{\pi}$ corresponding to a meson, a
bound state of a fermion and an antifermion. The full massive $m>0$ version of
the model is not integrable and has a rich phase diagram where the number of
mesons depends on the values of the parameters $m/e$ and $\theta$ Coleman et
al. (1975); Coleman (1976); Kogut and Susskind (1975); Banks et al. (1976);
Crewther and Hamer (1980); Hamer et al. (1982); Adam (1997); Gutsfeld et al.
(1999); Gattringer et al. (1999); Sriganesh et al. (2000); Giusti et al.
(2001); Byrnes et al. (2002); Christian et al. (2006); Cichy et al. (2013);
Bañuls et al. (2013); Buyens et al. (2015); Buyens (2016). The Schwinger model
displays confinement and has been extensively studied for pair creation and
string breaking Coleman et al. (1975); Nakanishi (1978); Nakawaki (1980);
Gross et al. (1996); Hosotani et al. (1996); Cooper et al. (2006); Chu and
Vachaspati (2010); Hebenstreit et al. (2013); Klco et al. (2018); Zache et al.
(2019); Spitz and Berges (2019); Notarnicola et al. (2020); Chanda et al.
(2020); Magnifico et al. (2020); Gold et al. (2020).
Finally, it is known that due to the vacuum degeneracy, the massless version
of the Schwinger model exhibits cluster violation of correlators of chiral
fermion densities
$\rho_{\pm}(x)=N\left[\bar{\psi}(x)\frac{1\pm\gamma^{5}}{2}\psi(x)\right]$,
Lowenstein and Swieca (1971); Ferrari and Montalbano (1994); Abdalla et al.
(2001),
$\left\langle\rho_{-}(x_{1})\cdots\rho_{-}(x_{n})\rho_{+}(y_{1})\cdots\rho_{+}(y_{n})\right\rangle,$
(3)
closely related to the correlators from eq. (2). This makes the model a good
candidate for the horizon violation. The cluster violation is also intimately
related to confinement of gauge theories Lowdon (2016, 2017, 2018a, 2018b).
Here we study general quenches of the massive Schwinger model and focus on the
spreading of the current-current correlators:
$\displaystyle C_{\mu}(t,x,y)$ $\displaystyle=\left\langle
J^{\mu}(t,x)J^{\mu}(t,y)\right\rangle.$ (4)
We prepare the system in the ground state of the model with the prequench
values of the parameters $m_{0}/e_{0}$, $\theta_{0}$ and at time $t=0$ switch
the parameters to their postquench values $m/e$, $\theta$.
_The method. -_ We implement a THM for the Schwinger model in finite volume
$L$ with anti-periodic boundary conditions (Neveu-Schwarz sector). We
elimination the gauge redundancy of degrees of freedom alongside with the
bosonisation of the model Iso and Murayama (1990).
Choosing the Weyl (time) gauge, $A_{t}=0$, and defining $A\equiv A_{x}$, the
Hamiltonian of the model is
$H=\int_{0}^{L}dx\left(\frac{1}{2}\dot{A}^{2}-\bar{\Psi}\left[\gamma^{1}(i\partial_{x}-eA)-m\right]\Psi\right).$
Expanding the fermion currents
$J_{\sigma}(x)=\Psi_{\sigma}^{\dagger}(x)\Psi_{\sigma}(x)=\frac{1}{L}\left[Q_{\sigma}-\sigma\sum_{n>0}\sqrt{n}\left(b_{\sigma,n}e^{-\sigma
in\frac{2\pi}{L}x}+b_{\sigma,n}^{\dagger}e^{\sigma
in\frac{2\pi}{L}x}\right)\right]$, with the chirality $\sigma=\pm$, its modes
obey bosonic canonical commutation relations. Further defining the
$N_{\sigma}$ vacua as
$\left|0;N_{-}\right\rangle\equiv\prod_{n=N_{-}}^{\infty}c_{-,n}^{\dagger}\left|0\right\rangle$,
$\left|0;N_{+}\right\rangle\equiv\prod_{n=-\infty}^{N_{+}-1}c_{+,n}^{\dagger}\left|0\right\rangle$,
with $c_{\sigma,n}$ the fermion mode operators, the Hilbert space spanned by
bosonic modes $b_{\sigma,n}^{\dagger}$ on top of
$\left|0;N_{-}\right\rangle\otimes\left|0;N_{+}\right\rangle$ is equivalent to
the Hilbert space spanned by $c_{\sigma,n}^{\dagger}$ acting on top of
$\left|0\right\rangle$. This is the foundation for the bosonisation of the
model. Because of the invariance under large gauge transformations, the true
vacua of the system are the infinitely degenerate $\theta$ vacua
$\left|\theta\right\rangle=\sum_{N\in\mathbb{Z}}e^{-iN\theta}\left|0;N\right\rangle$
for $\theta\in[0,2\pi)$. Gauge invariance further implies that the only mode
of the EM potential $A$ that is not fixed by the Gauss law is the zero mode
$\alpha=\frac{1}{L}\int_{0}^{L}dx\,A(x)$ along with its its dual
$i\partial_{\alpha}=\int_{0}^{L}dx\,\dot{A}(x)$.
By setting
$B_{0}=\sqrt{\frac{1}{2ML}}\left(-\sqrt{\pi}\left\\{Q_{+}-Q_{-}\right\\}+\frac{\partial}{\partial\alpha}\right)$,
the part of the Hamiltonian involving the zero modes transforms into a
harmonic oscillator with the mass $M=\frac{e}{\sqrt{\pi}}$. Complemented with
a Bogoliubov transform of the nonzero momentum modes into massive bosonic
modes:
$B_{\sigma,n}=\frac{1}{2}\left(\frac{\sqrt{E_{n}}}{\sqrt{k_{n}}}+\frac{\sqrt{k_{n}}}{\sqrt{E_{n}}}\right)b_{\sigma,n}-\frac{1}{2}\left(\frac{\sqrt{E_{n}}}{\sqrt{k_{n}}}-\frac{\sqrt{k_{n}}}{\sqrt{E_{n}}}\right)b_{-\sigma,n}^{\dagger}$,
with $k_{n}=\frac{2\pi n}{L}$ and $E_{n}=\sqrt{M^{2}+k_{n}^{2}}$, the massless
part of the Hamiltonian is transformed into the Hamiltonian of a massive free
boson with the mass $M$. The mass term of the Hamiltonian is written in the
bosonic form using the bosonisation relation
$\Psi_{\sigma}(x)=F_{\sigma}\frac{1}{\sqrt{L}}:\negmedspace e^{-\sigma
i\left(\sqrt{4\pi}\Phi_{\sigma}(x)-\frac{\pi}{L}x\right)}\negmedspace:$ with
$\partial_{x}\Phi_{\sigma}(x)=\sqrt{\pi}J_{\sigma}(x)+\frac{\sigma
e}{2\sqrt{\pi}}\,A(x)$ the chiral boson field and $F_{\sigma}$ the Klein
factor. Then using
$F_{\sigma}^{\dagger}F_{-\sigma}\left|\theta\right\rangle=e^{\sigma
i\theta}\left|\theta\right\rangle$, the Schwinger model Hamiltonian takes the
bosonised form
$\displaystyle H$ $\displaystyle=H_{M}+U,$ (5) $\displaystyle H_{M}$
$\displaystyle=M\left(B_{0}^{\dagger}B_{0}\right)+\sum_{n>0}E_{n}\left(B_{+,n}^{\dagger}B_{+,n}+B_{-,n}^{\dagger}B_{-,n}\right),$
$\displaystyle U$
$\displaystyle=-\frac{mM}{2\pi}e^{\gamma}\int_{0}^{L}dx\,:\negmedspace\cos\left(\sqrt{4\pi}\Phi(x)+\theta\right)\negmedspace:_{M}.$
with $\Phi(x)=\Phi_{-}+\Phi_{+}$ with Bogoliubov transformed modes,
$:\bullet:_{M}$ denotes normal ordering w.r.t. the mass $M$ and $\gamma$ is
the Euler-Mascheroni constant.
The form of the Hamiltonian (5) offers a natural THM splitting into the
massive free part and the cosine potential. To implement the numerical method,
the cosine potential and the observables, are represented as matrices in the
Hilbert space of the free part - the Fock space generated by applying the
$B_{\sigma,n}^{\dagger}$ modes on the $\theta$ vacuum. Finally, an energy
cutoff $\left\langle\Psi\right|H_{M}\left|\Psi\right\rangle\leq
E_{\text{cut}}$ is imposed on the states $\left|\Psi\right\rangle$ of the THM
Hilbert space. Momentum conservation implied by translation invariance and the
decoupling of the $B_{0}$ mode from the rest of the modes are used to further
reduce the dimension of the Hilbert space by diagonalising each sector
separately. We use the Hilbert spaces with up to 20 000 states per sector. The
full details of the method can be found in the Supplemental Material Sup .
Figure 2: The THM spectrum the Schwinger model at $m/e=0.125$ in dependence of
the system size $L$ in the 0, 1 and 2 sectors of the total momentum. The
spectral lines are compared with the $L\rightarrow\infty$ results of the MPS
computations Bañuls et al. (2013) for the vector and the scalar particles and
the TNS Buyens (2016) for the heavy vector particle. On top of the spectrum,
the dominant frequency of the oscillations of the out-of-horizon correlations
are plotted. Figure 3: Left: Time dependent $\left\langle
J^{x}(t,x)J^{x}(t,y)\right\rangle$ and $\left\langle
J^{t}(t,x)J^{t}(t,y)\right\rangle$ correlations for different type of quenches
in the Schwinger model (initial correlations subtracted): 1.) Quench in $m/e$
with $m_{0}=0$, $m=0.125$, $\theta_{0}=\theta=0$; 2.) Quench in $\theta$ with
$\theta_{0}=\frac{\pi}{4}$, $\theta=0$, $m_{0}=m=0.125$. Both with
$e_{0}=e=1$, $L=40$. Upper right: Frequency spectrum of the out-of-horizon
component of the correlations (mass quenches to $m=0.25$, $e=1$, $L=47.5$)
compared to meson masses (full lines) and twice the values of meson masses
(dashed lines). Lower right: Cluster violation in the massive Schwinger model
at $m=0.125$, $\theta=0$, $e=1$, $L=40$.
_Results. -_ Our THM implementation of the Schwinger model recovers the
results from the literature for the meson masses and gives a region of highly
dense states above them, referred to as the continuum in the
$L\rightarrow\infty$ limit (fig. 2). This serves as a sanity check of the
method. We are able to get the masses of the vector meson precisely, while our
THM method seems to be slightly less precise for the scalar meson mass. We
have been able to simulate large system sizes $L\gg\frac{1}{M}$ where the
finite size effects are exponentially suppressed.
The results shown in fig. 3 indeed confirm that the Schwinger model exhibits
the horizon violation effect - the correlation functions $C_{x}(t,x,y)$ are
nonzero and oscillating for $|x-y|>2t$. The effect is found in quenches in
both $e/m$ and $\theta$ as well as in quenches to and from the massless
Schwinger model. The sign of the out-of-horizon correlations changes depending
whether the quenched parameter is increased or decreased. As is expected for
periodic boundary conditions, the effect is present in the $C_{x}$ and not
present in the $C_{t}$ channel.
To shed light on the origin of the effect, we study the clustering properties
of correlators of chiral densities (3) (fig. 3, lower right), more
specifically, its component
$\left\langle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\psi_{-\sigma}^{\dagger}(y)\psi_{\sigma}(y)\right\rangle$.
We find that the correlator violates clustering - when $x$ and $y$ are far
apart, the correlator does not cluster into
$\left\langle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\right\rangle\left\langle\psi_{-\sigma}^{\dagger}(y)\psi_{\sigma}(y)\right\rangle$.
In case of the massless Schwinger model, this clustering violation is well
known and can be computed analytically Lowenstein and Swieca (1971); Ferrari
and Montalbano (1994); Abdalla et al. (2001), in case of the massive version
of the model, this is to our knowledge a new result. Interestingly, in the
massless case, the normal ordered version of the correlator does not exhibit
the clustering violation while in the massive model, even the normal ordered
correlator violates clustering. We expect that similarly as in the SG model
Kukuljan et al. (2020a), the nonlinear postquench dynamics rotates the initial
clustering violation from such chiral correlators into the local nonchiral
observables. We note that in case of the ground states of the massive model,
we observe numerically a tiny clustering violation also in the $C_{x}$
correlators which is two orders of magnitude smaller than the cluster
violation of $\left\langle\rho_{\sigma}\rho_{-\sigma}\right\rangle$. We
expect, however, that this is not a physical fact but an artifact of the THM
truncation. Such tiny artifacts are common in derivative fields but do not
falsely produce the horizon violation effect, as was for example verified in
case of Klein-Gordon dynamics in the first version of Kukuljan et al. (2020a,
b). As well as that, our THM simulation of the Schwinger model displays the
horizon violation in the quenches starting from the massless model, where
there are no such artifacts in the initial state. So we expect that the effect
originates fully from the cluster violation of the chiral terms.
Fig. 2 shows how the dominant frequencies of the oscillations compare to the
spectrum of the model. Due simulation times limited to $t\leq L/4$, we are
only able to see a few oscillations. Therefore, the frequencies have
considerable error bars ($\Delta\omega\approx 2\pi/L$ \- half a frequency bin)
and the values of the possible discrete frequencies move with $L$ resulting in
a chainsaw pattern. The error bars compare to both the scalar meson mass and
twice the vector meson mass. Based on the mechanism of the effect in the SG
model Kukuljan et al. (2020a), it is expected that the frequencies correspond
to twice the mass of the lightest meson. This is supported by computations at
higher values of $m/e$, where those masses can be better discriminated
(Fourier spectrum in the upper right of fig. 3). This suggest that the horizon
violation is mediated through correlated vector meson pairs entangled by the
quench. In some cases even subdominant peaks appear close to twice the masses
of heavier mesons in the frequency spectra, suggesting that they could also be
contributing to the effect.
_Discussion. -_ We stress again that the observed phenomenon is in no
contradiction with relativistic causality as guaranteed by the Lorentz
invariance of the model the micro causality of the fields. Rather, the
violation of horizon can be likely traced back to the cluster violation of
chiral fermion fields as in the SG model Kukuljan et al. (2020a).
Using the simplest representative, we have hereby demonstrated that the
horizon violation occurs in gauge field theory. In the future, it would be
interesting to explore higher gauge theories like SU(2) or SU(3) or study the
Wess–Zumino–Witten models. It would be of crucial importance to answer whether
the effect is present also in $D>1+1$. There, gauge fields are dynamical, so
the physics could be drastically different. Further analytical approaches
should be found to get a better understanding of the effect in the Schwinger
model.
Figure 4: Decay of the anisotropic initial condition or a $\theta$ term in a
toy universe as a quench that generates long range correlations through the
horizon violation effect. Long range correla-tions are the price that the toy
universe has to pay for the initial anisotropy.
The horizon violation presented in this work is a novel phenomenon in 1+1D
quantum-electrodynamics. It is reasonable to expect that it could have
interesting physical implications, in particular if it turns out that the
effect is present also in higher dimensions. In condensed matter physics,
phase transitions are an ubiquitous phenomenon and could serve as a trigger
for horizon violation generating quenches. Here, already the $D=1+1$ case
could be an interesting candidate since at the present day there are numerous
experiments available for probing 1+1D physics Husmann et al. (2015). An
especially important class are ultra cold atoms in atom chips, where one
dimensional QFTs are directly realised and correlation functions can be
measured both in equilibrium states and nonequilibrium dynamics Schweigler et
al. (2017). In cosmology, there several candidates for quenches like the end
of inflation, the QCD and the electroweak transitions and topological symmetry
breaking in grand unified theories Weinberg (2008); Boyanovsky et al. (2006);
Gleiser (1998); Hindmarsh et al. (2020). Consider also the following example
illustrated in fig. 4: a toy universe is created with an anisotropic initial
condition - a nonzero background electric field. This is a possibility since
the zero background field case is a special, fine-tuned, value. In $D=1+1$ the
background electric field is stable while in $D=1+3$, it decays through the
electric breakdown of the vacuum Coleman (1976). The rapid decay of the
background electric field would serve as a quench that causes a horizon
violation effect in the QED degrees of freedom as we have seen here in the
$\theta_{0}\neq 0\rightarrow\theta=0$ quenches. This transforms the initial
anisotropy of the toy universe into long range correlations. Similarly, in a
higher gauge theory the effect could be triggered by a decay of the theta term
which is linked in some models with the cosmological constant Yokoyama (2002);
Jaikumar and Mazumdar (2003). It would be interesting to explore the possible
predictions for traces of this effect in the cosmic microwave background.
Finally, it would be interesting to use THM to explore the confinement and
string breaking phenomena in the Schwinger model and to use THM
implementations Azaria et al. (2016) to study dynamics of higher gauge
theories.
###### Acknowledgements.
This work was supported by the Max-Planck-Harvard Research Center for Quantum
Optics (MPHQ). The author wants to thank Mari Carmen Bañuls, Peter Lowdon,
Jernej Fesel Kamenik, Miha Nemevšek and Sašo Grozdanov for useful discussions.
Special thanks to Spyros Sotiriadis for many of our valuable discussions and
Gabor Takács for useful discussions and feedback to the first version of the
manuscript that helped improve this work.
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Supplemental Material
## Appendix A Details of the THM for the Schwinger model
Here we discuss the details of the truncated Hamiltonian method (THM)
implementation of the Schwinger model defined on an interval of length $L$
with anti-periodic boundary conditions.
### A.1 Bosonisation
The Schwinger model, the quantum electrodynamics in $D=1+1$, is defined by the
Lagrangian density
$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\bar{\Psi}(i\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}A_{\mu}-m)\Psi$
(6)
where $\Psi=\left(\begin{array}[]{c}\Psi_{-}\\\ \Psi_{+}\end{array}\right)$ is
the Dirac fermion field, $A_{\mu}$ the electromagnetic (EM) potential,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ the EM tensor, $m$ is
the electron mass and $e$ the electric charge. Choosing the Weyl (time) gauge,
$A_{0}=0$, defining $A\equiv A_{x}$, the Hamiltonian density takes the form:
$\displaystyle\mathcal{H}=\mathcal{H}_{\text{EM}}+\mathcal{H}_{\text{F}}+\mathcal{H}_{m},$
$\displaystyle\mathcal{H}_{\text{EM}}=\frac{1}{2}\dot{A}^{2},\hskip
42.67912pt$
$\displaystyle\mathcal{H}_{\text{F}}=-\bar{\Psi}\gamma^{1}(i\partial_{x}-eA)\Psi,\hskip
42.67912pt\mathcal{H}_{m}=m\bar{\Psi}\Psi.$ (7)
We use here the metric $g=\text{diag}(1,-1)$ and the gamma matrices
$\gamma^{0}=-\sigma_{1}$, $\gamma^{1}=i\sigma_{2}$.
In order to treat a gauge field theory in Hamiltonian formalism, one has to
remove the redundancy in the degrees of freedom coming from gauge invariance.
For the Schwinger model this goes hand in hand with bosonisation. Bosonisation
is an exact duality between fermionic and bosonic theories in 1+1D
relativistic QFT, developed by Tomonaga, Mattis, Lieb, Mandelstam, Coleman,
Haldane and others Tomonaga (1950); Mattis and Lieb (1965); Schotte and
Schotte (1969); Mattis (1974); Luther and Peschel (1974); Mandelstam (1975);
Coleman et al. (1975); Haldane (1981). We bosonise the Schwinger model here
using the operatorial (constructive) approach of Iso and Murayama Iso and
Murayama (1990).
The eigenfunctions and eigenvalues of the massless fermion part of the
Hamiltonian
$\mathcal{H}_{\text{F}}(x)=\sum_{\sigma=\pm}\sigma\Psi^{\dagger}(x)(i\partial_{x}-eA)\Psi(x)$
are $(i\partial_{x}-eA)\psi_{n}=\epsilon_{n}\psi_{n}$:
$\displaystyle\psi_{n}(x)$
$\displaystyle=\frac{1}{\sqrt{L}}e^{-i(\epsilon_{n}x+e\int_{0}^{x}dx\,A)},$
$\displaystyle\epsilon_{n}$
$\displaystyle=\frac{2\pi}{L}\left(n+\frac{1}{2}\delta_{b}-\frac{e\alpha
L}{2\pi}\right).$ (8)
The eigenfunctions satisfy periodic boundary conditions (Ramond sector) for
$\delta_{b}=0$ and anti-periodic (Neveu-Schwarz sector) for $\delta_{b}=1$.
The fermion bosonises to a boson with periodic boundary conditions for both
values of $\delta_{b}$ reflecting the fact that the bosonisation is an
equivalence between bosons and fermions up to $Z_{2}$. We will derive the
equations for general $\delta_{b}$ and in the end take $\delta_{b}=1$, the
Neveu-Schwarz sector of the fermion, for the THM study. We quantise the
fermion field by expansion
$\Psi(x)=\sum_{n\in\mathbb{Z}}c_{-,n}\psi_{n}(x)\left(\begin{array}[]{c}1\\\
0\end{array}\right)+c_{+,n}\psi_{n}(x)\left(\begin{array}[]{c}0\\\
1\end{array}\right)$ (9)
with the canonical anticommutation relation
$\left\\{c_{\sigma,n},c_{\rho,m}^{\dagger}\right\\}=\delta_{\sigma,\rho}\delta_{n,m}$.
Then $H_{\text{F}}=H_{+}+H_{-}$ with
$H_{\sigma}=\sigma\sum_{n\in\mathbb{Z}}\epsilon_{n}c_{\sigma,n}^{\dagger}c_{\sigma,n}$.
We expand the EM potential and its conjugate dual as:
$\displaystyle A(x)$ $\displaystyle=\alpha+\sum_{n\neq
0}A_{n}e^{i\frac{2\pi}{L}nx},$ $\displaystyle\dot{A}(x)$
$\displaystyle=i\frac{\delta}{\delta
A(x)}=\frac{i}{L}\left(\frac{\partial}{\partial\alpha}+\sum_{n\neq
0}\frac{\delta}{\delta A_{n}}e^{-i\frac{2\pi}{L}nx}\right),$ (10)
and we shall see that as a consequence of the gauge invariance only the zero
modes of the EM field $\alpha$ and $i\frac{\partial}{\partial\alpha}$ are
dynamical Manton (1985); Iso and Murayama (1990).
Expanding the fermion currents
$J_{\sigma}(x)=\Psi_{\sigma}^{\dagger}(x)\Psi_{\sigma}(x)=\frac{1}{L}\left[Q_{\sigma}-\sigma\sum_{n>0}\sqrt{n}\left(b_{\sigma,n}e^{-\sigma
in\frac{2\pi}{L}x}+b_{\sigma,n}^{\dagger}e^{\sigma
in\frac{2\pi}{L}x}\right)\right],$ (11)
its modes
$b_{\sigma,n}=-\frac{\sigma}{\sqrt{n}}\sum_{k\in\mathbb{Z}}c_{\sigma,k}^{\dagger}c_{\sigma,k+\sigma
n}$ obey canonical commutation relations
$\left[b_{\sigma,n},b_{\rho,m}^{\dagger}\right]=\delta_{\sigma,\rho}\delta_{n,m}$.
Further defining the $N_{\sigma}$ vacua as
$\left|0;N_{-}\right\rangle\equiv\prod_{n=N_{-}}^{\infty}c_{-,n}^{\dagger}\left|0\right\rangle,\hskip
42.67912pt\left|0;N_{+}\right\rangle\equiv\prod_{n=-\infty}^{N_{+}-1}c_{+,n}^{\dagger}\left|0\right\rangle$
(12)
it can be shown that the Hilbert space spanned by excitations with all
possible combinations of $b_{\sigma,n}^{\dagger}$ on top of
$\left|0;N_{-}\right\rangle\otimes\left|0;N_{+}\right\rangle$ is equivalent to
the Hilbert space spanned by all the possible combinations of
$c_{\sigma,n}^{\dagger}$ on top of $\left|0\right\rangle$. This is the core of
bosonisation.
The fermion number operators take the following expectation values on the
$N_{\sigma}$ vacua,
$\left\langle
Q_{\sigma}\right\rangle_{N_{\sigma}}=\sigma\left(N_{\sigma}-\frac{e\alpha
L}{2\pi}+\frac{1}{2}(\delta_{b}-1)\right),$ (13)
as can be shown by regularisation by Hurwitz zeta resummation. Similarly,
$\displaystyle\left\langle
H_{\sigma}\right\rangle_{N_{\sigma}}=\frac{2\pi}{L}\left[\frac{1}{2}\left\langle
Q_{\sigma}\right\rangle_{N_{\sigma}}^{2}-\frac{1}{24}\right].$ (14)
for $H=\int_{0}^{L}dx\,\mathcal{H}$.
#### Gauge invariance. -
The fermionic Hilbert space combined with the Hilbert space generated by the
modes of the EM modes display a redundancy of degrees of freedom as is
characteristic for gauge invariant theories and we have to eliminate this
redundancy. QED is invariant under the transformations
$A_{\mu(x)}\rightarrow A_{\mu}(x)-\partial_{\mu}\lambda(x),\hskip
42.67912pt\psi(x)\rightarrow e^{ie\lambda(x)}\psi(x).$ (15)
For systems defined on the circle (and other topologies with nontrivial
homotopic groups), the gauge transformations can be divided into small gauge
transformations where both $\lambda(x)$ and $e^{ie\lambda(x)}$ are single
valued and large gauge transformations where $e^{ie\lambda(x)}$ is single
valued but $\lambda(x)$ is not. Mathematically speaking, small gauge
transformations are homotopic to the identity of the Lie group and large gauge
transformations are not.
Let’s begin with small gauge transformations. As a consequence of the Dirac
conjecture Gràcia and Pons (1988); Kashiwa and Takahashi (1994) these are
represented in the Hilbert space by the operator
$U(\lambda)=\exp(-i\int_{0}^{L}dx\,G(x)\lambda(x))$ with the Gauss law
generator
$G(x)=\partial_{x}\left(-i\frac{\delta}{\delta A(x)}\right)-eJ_{0}(x)$ (16)
with $J_{0}=J_{+}+J_{-}$. Requiring that the physical states are invariant
under $G$ using the expansions (11), (10) gives
$\displaystyle Q\left|\text{physical state}\right\rangle$ $\displaystyle=0$
$\displaystyle\left\\{\frac{\delta}{\delta
A_{n}}+\frac{eL}{\sqrt{n}2\pi}\left(b_{-,n}^{\dagger}-b_{+,n}\right)\right\\}\left|\text{physical
state}\right\rangle$ $\displaystyle=0$
$\displaystyle\left\\{\frac{\delta}{\delta
A_{-n}}+\frac{eL}{\sqrt{n}2\pi}\left(b_{+,|n|}^{\dagger}-b_{-,|n|}\right)\right\\}\left|\text{physical
state}\right\rangle$ $\displaystyle=0$ (17)
For $Q=Q_{+}+Q_{-}$. Taking into account (13), the first constraint means that
$N_{-}=N_{+}=N$ in physical states and we can define the $N$ vacua as
$\left|0;N\right\rangle\equiv\left|0;N_{-}\right\rangle\otimes\left|0;N_{+}\right\rangle.$
(18)
The second and the third constrain mean that all the nonzero momentum modes of
the EM field are fixed by gauge invariance and the only dynamical modes of the
EM field are the zero modes $\alpha$ and $i\frac{\partial}{\partial\alpha}$
Manton (1985); Iso and Murayama (1990). It is also easy to see that under
small gauge transformations the wave functions (8) transform as
$\psi_{n}(x)\rightarrow e^{ie\lambda(x)-ie\lambda(0)}\psi_{n}(x)$ and thus
$c_{\sigma,n}\rightarrow e^{ie\lambda(0)}c_{\sigma,n}$. It is clear that the
currents and its momentum modes, including the charges are invariant under all
gauge transformations.
The homotopy grout of $U(1)$ symmetry is $\pi_{1}\left(U(1)\right)=\mathbb{Z}$
and large gauge transformations are generated by
$\lambda(x)=\frac{2\pi}{eL}wx,\hskip 28.45274ptw\in\mathbb{Z}$ (19)
The wave functions $\psi_{n}(x)$ are invariant under those thus the fermion
operators transform as $c_{n}\rightarrow c_{\sigma,n+w}$. Consequently, the
$N$ vacua transform as
$\left|0;N\right\rangle\rightarrow\left|0;N+w\right\rangle$. The large gauge
transformations commute with the Hamiltonian so they can be diagonalised in
the same basis. The eigenstates of large gauge transformations are the
$\theta$ vacua
$\left|\theta\right\rangle=\sum_{N\in\mathbb{Z}}e^{-iN\theta}\left|0;N\right\rangle,\hskip
28.45274pt\theta\in[0,2\pi)$ (20)
which form a continuous degenerate family of ground states of the fermionic
parts of the Schwinger model Hamiltonian. A ground state of the full
Hamiltonian is obtained as a tensor product of the ground state of the EM part
with a $\theta$ vacuum.
#### Hamiltonian. -
Following from the gauge invariance constraints (17) we have
$H_{\text{EM}}=-\frac{1}{2L}\left[\left(\frac{\partial}{\partial\alpha}\right)^{2}+2\left(\frac{eL}{2\pi}\right)^{2}\sum_{n>0}\frac{1}{n}\left(b_{-,n}^{\dagger}-b_{+,n}\right)\left(b_{+,n}^{\dagger}-b_{-,n}\right)\right].$
(21)
Taking into account that
$\left[H_{\text{F}},b_{\sigma,n}^{\dagger}\right]=\frac{2\pi}{L}nb_{\sigma,n}^{\dagger}$
and deducing its zero mode content from (14), the Hamiltonian $H_{\text{F}}$
can only take the form
$H_{\text{F}}=\frac{2\pi}{L}\sum_{\sigma=\pm}\left[\frac{1}{2}Q_{\sigma}^{2}-\frac{1}{24}+\sum_{n>0}nb_{\sigma,n}^{\dagger}b_{\sigma,n}\right].$
(22)
We can then split the massless part of the Schwinger model Hamiltonian into a
part with zero modes and a part with nonzero momentum modes:
$\displaystyle H_{\text{EM}}+H_{\text{F}}$
$\displaystyle=H_{0}+\sum_{n>0}H_{n}-\frac{2\pi}{12L}$ $\displaystyle H_{0}$
$\displaystyle=\frac{2\pi}{L}\left(\frac{Q^{2}+Q_{5}^{2}}{4}\right)-\frac{1}{2L}\left(\frac{\partial}{\partial\alpha}\right)^{2}$
$\displaystyle H_{n}$
$\displaystyle=\frac{2\pi}{L}n\left(b_{+,n}^{\dagger}b_{+,n}+b_{-,n}^{\dagger}b_{-,n}\right)-\frac{e^{2}L}{4\pi^{2}n}\left(b_{+,n}^{\dagger}-b_{-,n}\right)\left(b_{-,n}^{\dagger}-b_{+,n}\right).$
(23)
with $Q_{5}=Q_{+}-Q_{-}$ which takes the value
$Q_{5}=2N-\frac{ecL}{\pi}+\delta_{b}-1$ on physical states and we keep in mind
that $Q=0$ on physical states.
The zero mode Hamiltonian $H_{0}$ is the Hamiltonian of a massive harmonic
oscillator with mass
$M=\frac{e}{\sqrt{\pi}}$ (24)
and can be written in the canonical form as
$H_{0}=M\left(B_{0}^{\dagger}B_{0}+\frac{1}{2}\right)$ (25)
with
$B_{0}=\sqrt{\frac{1}{2ML}}\left(-\sqrt{\pi}Q_{5}+\frac{\partial}{\partial\alpha}\right)$,
$B_{0}^{\dagger}=\sqrt{\frac{1}{2ML}}\left(-\sqrt{\pi}Q_{5}-\frac{\partial}{\partial\alpha}\right)$.
The nonzero momentum Hamiltonians $H_{n}$ can be diagonalised with a
Bogoliubov transformation
$\displaystyle B_{\sigma,n}$
$\displaystyle=\cosh(t_{n})b_{\sigma,n}+\sinh(t_{n})b_{-\sigma,n}^{\dagger}$
$\displaystyle\cosh(t_{n})$
$\displaystyle=\frac{1}{2}\left(\frac{\sqrt{E_{n}}}{\sqrt{k_{n}}}+\frac{\sqrt{k_{n}}}{\sqrt{E_{n}}}\right)$
$\displaystyle\sinh(t_{n})$
$\displaystyle=-\frac{1}{2}\left(\frac{\sqrt{E_{n}}}{\sqrt{k_{n}}}-\frac{\sqrt{k_{n}}}{\sqrt{E_{n}}}\right)$
(26)
with $k_{n}=\frac{2\pi n}{L}$ and $E_{n}=\sqrt{M^{2}+k_{n}^{2}}$. Then
$H_{n}=E_{n}\left(B_{+,n}^{\dagger}B_{+,n}+B_{-,n}^{\dagger}B_{-,n}+1\right)$
(27)
and $H_{\text{EM}}+H_{\text{F}}$ becomes the Hamiltonian of the free massive
boson with the mass $M$. This reproduces the Schwinger’s result that the QED
in $D=1+1$ is gaped even if the bare mass of the fermion is zero. The
Bogoliubov operator of this transformation,
$U_{n}b_{\sigma,n}U_{n}^{\dagger}=B_{\sigma,n}$, is the squeezing operator
$U_{n}=\exp\left[-t_{n}\left(B_{+,n}^{\dagger}B_{-,n}^{\dagger}-B_{+,n}B_{-,n}\right)\right]$
meaning that the vacua annihilated by the massive modes $B_{\sigma,n}$ are the
squeezed coherent $\theta$ vacua
$\left|\theta\right\rangle_{M}=\left(\prod_{n>0}U_{n}\right)\left|\theta\right\rangle.$
(28)
It remains to treat the mass term in the Hamiltonian, $H_{m}$. We can express
it in terms of the bosonic momentum modes of the currents, $b_{\sigma,n}$
using the relation
$\Psi_{\sigma}(x)=F_{\sigma}\frac{1}{\sqrt{L}}:\negmedspace e^{-\sigma
i\left(\sqrt{4\pi}\Phi_{\sigma}(x)-\frac{\pi}{L}\delta_{b}x\right)}\negmedspace:$
(29)
with
$\Phi_{\sigma}(x)=\frac{1}{\sqrt{4\pi}}\left\\{\frac{2\pi}{L}Q_{\sigma}x-i\sum_{n>0}\frac{1}{\sqrt{n}}\left(b_{\sigma,n}e^{-\sigma
in\frac{2\pi}{L}x}-b_{\sigma,n}^{\dagger}e^{\sigma
in\frac{2\pi}{L}x}\right)+\sigma
e\int_{0}^{x}dx^{\prime}\,A(x^{\prime})\right\\}$ (30)
and with the normal ordering with respect to the modes $b_{\sigma,n}$, which
is the bosonisation relation for a fermion coupled to the EM field. Here, the
term with $\delta_{b}$ is to assure that the fermion field satisfies the
correct boundary conditions and $F_{\sigma}$ are the Klein factors satisfying
$\displaystyle\left[F_{\sigma},A_{m}\right]=\left[F_{\sigma},\frac{\delta}{\delta
A_{m}}\right]=0,$ $\displaystyle\hskip
42.67912pt\left[F_{\sigma},b_{\rho,m}\right]=\left[F_{\sigma},b_{\rho,m}^{\dagger}\right]=0,$
$\displaystyle\left[Q_{\sigma},F_{\rho}^{\dagger}\right]=\delta_{\sigma,\rho}F_{\rho}^{\dagger},$
$\displaystyle\hskip
42.67912pt\left[Q_{\sigma},F_{\rho}\right]=-\delta_{\sigma,\rho}F_{\rho},$
$\displaystyle\left\\{F_{\sigma}^{\dagger},F_{\rho}\right\\}=2\delta_{\sigma,\rho},$
$\displaystyle\hskip 42.67912ptF_{\sigma}^{\dagger}F_{\sigma}=1.$ (31)
Since a function of $b_{\sigma,n}$ and $b_{\sigma,n}^{\dagger}$ can never
alter the fermion number, the Klein factors make sure that $\Psi_{\sigma}(x)$
as defined above has the true fermionic character. Some authors prefer to use
exponentials of the zero modes of the compactified massless boson field in
place of the Klein factors and the two conventions are fully equivalent. In
particular, it can be shown that
$\left\\{\Psi_{\sigma}(x),\Psi_{\rho}(x)\right\\}=\delta_{\sigma,\rho}\delta(x-y)$.
Using the relations $\left[\frac{\delta}{\delta
A_{n}},b_{+,n}^{\dagger}\right]=\left[\frac{\delta}{\delta
A_{-n}},b_{+,n}\right]=\left[\frac{\delta}{\delta
A_{-n}},b_{-,n}^{\dagger}\right]=\left[\frac{\delta}{\delta
A_{n}},b_{-,n}\right]=\frac{eL}{2\pi\sqrt{n}}$ which follow from (17) it’s
easy to see that $\left[\frac{\delta}{\delta A(x)},\Psi_{\sigma}(y)\right]=0$.
Finally, considering that $F_{\sigma}\rightarrow e^{ie\lambda(0)}F_{\sigma}$
under gauge transformations, if follows that $\Psi_{\sigma}(x)$ transforms as
the fermion field. We also have, as follows from the second line of (31) that
$F_{\sigma}^{\dagger}F_{-\sigma}\left|0;N\right\rangle=\left|0;N+\sigma\right\rangle$
and thus:
$F_{\sigma}^{\dagger}F_{-\sigma}\left|\theta\right\rangle_{M}=e^{\sigma
i\theta}\left|\theta\right\rangle_{M}.$ (32)
Using the definition (30) we can also read out the fermionisation relation for
the fermion field coupled to the EM field, the inverse of the bosonisation
relation:
$\partial_{x}\Phi_{\sigma}(x)=\sqrt{\pi}J_{\sigma}(x)+\frac{\sigma
e}{2\sqrt{\pi}}\,A(x).$ (33)
We can use the bosonisation relation (29) to express the mass term in the
Hamiltonian as
$H_{m}=-m\frac{1}{L}e^{\sum_{n>0}\frac{1}{n}\left(1-\frac{k_{n}}{E_{n}}\right)}\int_{0}^{L}dx\,\sum_{\sigma=\pm}e^{\sigma
i\frac{2\pi}{L}(1-\delta_{b})x}:\negmedspace e^{\sigma
i\sqrt{4\pi}\Phi(x)}\negmedspace:_{M}F_{\sigma}^{\dagger}F_{-\sigma},$ (34)
where we have defined $\Phi(x)\equiv\Phi_{+}(x)+\Phi_{-}(x)$ and
$:\negmedspace\bullet\negmedspace:_{M}$ denotes normal ordering with respect
to the massive modes $B_{\sigma,n}$. The prefactor $e^{\sigma
i\frac{2\pi}{L}x}$ comes from commuting $F_{-\sigma}$ past
$e^{i\sigma\frac{2\pi}{L}Q_{-\sigma}x}$ using (31). The prefactor
$e^{\sum_{n>0}\frac{1}{n}\left(1-\frac{k_{n}}{E_{n}}\right)}$ comes from
substituting the Bogoliubov transform (26) into $\Phi_{\sigma}(x)$ and then
rearranging the expression for $H_{m}$ into the normal ordered form w.r.t.
$M$. In the $L\rightarrow\infty$ limit these prefactors take the value
$\frac{ML}{4\pi}e^{\gamma}$ where $\gamma=0.5772\ldots$ is the Euler-
Mascheroni constant.
Finally, putting all the terms together, with the $L\rightarrow\infty$
expression for the prefactor in the mass term, the Schwinger model Hamiltonian
takes the form
$\displaystyle H$
$\displaystyle=M\left(B_{0}^{\dagger}B_{0}\right)+\sum_{n>0}E_{n}\left(B_{+,n}^{\dagger}B_{+,n}+B_{-,n}^{\dagger}B_{-,n}\right)+\text{const}$
$\displaystyle\hskip
113.81102pt-\frac{mM}{4\pi}e^{\gamma}\int_{0}^{L}dx\,\sum_{\sigma=\pm}e^{\sigma
i\frac{2\pi}{L}(1-\delta_{b})x}:\negmedspace e^{\sigma
i\sqrt{4\pi}\Phi(x)}\negmedspace:_{M}F_{\sigma}^{\dagger}F_{-\sigma}$
$\displaystyle"\negmedspace=\negmedspace"\,\int_{0}^{L}\left[\frac{1}{2}\left(\Pi^{2}+(\partial_{x}\Phi)^{2}+M^{2}\Phi^{2}\right)-\frac{mM}{2\pi}e^{\gamma}:\negmedspace\cos\left(\sqrt{4\pi}\Phi(x)+\theta+\frac{2\pi}{L}(1-\delta_{b})x\right)\negmedspace:_{M}\right]$
(35)
where $\text{const}=\sum_{n>0}E_{n}+\frac{1}{2}M$ only affects the ground
state energy and will be irrelevant to us. The last "equality" is to be
understood only up to the details of the modes captured through the above
bosonisation procedure and has taken into account (32) and the fact that all
the physical states are created on top of the $\theta$ vacuum. The parameter
$\theta$ thus appears in the Hamiltonian and plays the role of the constant
background electric field as first pointed out by Coleman Coleman et al.
(1975); Coleman (1976). As is manifest in the first line, the zero mode
$B_{0}$ does not enter in the cosine term and is a harmonic oscillator
decoupled from the other degrees of freedom. For our THM implementation, we
choose $\delta_{b}=1$, the anti-periodic boundary conditions for the fermion,
the Neveu-Schwarz sector.
#### Hilbert space. -
As has been made explicit in the above discussion, the Hilbert space of the
Schwinger model after eliminating the gauge redundancy takes the form of the
tensor product of the Hilbert space of the zero modes with the Hilbert space
generated by all the possible bosonic excitations on top of the theta vacuum.
All together we can write any state in the Hilbert space in the form
$\left|\vec{r}\right\rangle\equiv\frac{1}{N_{\vec{r}}}\left(B_{0}^{\dagger}\right)^{r_{0}}\prod_{n=1}^{\infty}\left(B_{-,n}^{\dagger}\right)^{r_{-,n}}\left(B_{+,n}^{\dagger}\right)^{r_{+,n}}\left|0\right\rangle_{0}\otimes\left|\theta\right\rangle_{M}$
(36)
where $\vec{r}\equiv(r_{0},r_{-,1},r_{-,2},\ldots,r_{+,1},r_{+,2},\ldots)$ is
a vector of occupation numbers and $\left|0\right\rangle_{0}$ is the vacuum of
the $B_{0}$ mode. The normalisation is
$N_{\vec{r}}^{2}=(r_{0}!)\prod_{k=1}^{\infty}(r_{k,-}!)(r_{k,+}!)$.
### A.2 Truncated Hamiltonian method
The truncated Hamiltonian method (THM) consists of splitting the Hamiltonian
into an analytically solvable and an unsolvable part, the perturbing
potential. Then, expressing the perturbing operator and the observables as
matrices in the eigenbasis of the solvable part. Finally, an energy cutoff is
introduced which renders the matrices finite and enables numerical
diagonalisation which is the key to nonperturbative treatment of a strong
interaction with the THM. The above procedure of eliminating the redundant
degrees of freedom and bosonising the model suggests a natural splitting of
the Hamiltonian (35) into the quadratic part $H_{\text{EM}}+H_{\text{F}}$ and
the cosine potential $H_{m}$.
In the following we first list the matrix elements in the Hilbert space of the
quadratic part of the Hamiltonian for of all the required operators and then
discuss how to implement the THM.
#### Matrix elements. -
The matrix elements are computed between general states of the Hilbert space
$\left|\vec{r}\right\rangle$ and $\left|\vec{r}^{\prime}\right\rangle$,
defined in eq. (36)). The required matrix elements are:
Boson mode operators:
$\displaystyle\left<\vec{r}^{\prime}\right|B_{\sigma,n}^{\dagger}\left|\vec{r}\right>$
$\displaystyle=\left(\prod_{\rho,k\neq\sigma,n}\delta_{r^{\prime}_{\rho,k},r_{\rho,k}}\right)\sqrt{(r_{\sigma,n}+1)}\,\delta_{r^{\prime}_{\sigma,n}-1,r_{\sigma,n}}$
(37)
$\displaystyle\left<\vec{r}^{\prime}\right|B_{\sigma,n}\left|\vec{r}\right>$
$\displaystyle=\left(\prod_{\rho,k\neq\sigma,n}\delta_{r^{\prime}_{\rho,k},r_{\rho,k}}\right)\sqrt{r_{\sigma,n}}\,\delta_{r^{\prime}_{\sigma,n}+1,r_{\sigma,n}}$
(38)
Boson number operator:
$\displaystyle\left<\vec{r}^{\prime}\right|B_{\sigma,n}^{\dagger}B_{\sigma,n}\left|\vec{r}\right>$
$\displaystyle=r_{\sigma,n}\delta_{\vec{r}^{\prime},\vec{r}}$ (39)
Vertex operator: To implement the cosine potential we need the matrix elements
$\displaystyle\left<\vec{r}^{\prime}\right|:\negmedspace e^{\rho
i\sqrt{4\pi}\Phi(x)}\negmedspace:_{M}F_{\rho}^{\dagger}F_{-\rho}\left|\vec{r}\right>=$
$\displaystyle=e^{\rho
i\theta}\left\langle\vec{r}^{\prime}\right|\prod_{\sigma=\pm}\prod_{n=1}^{\infty}e^{-\rho\sqrt{\frac{2\pi}{L}}\frac{1}{\sqrt{E_{n}}}B_{\sigma,n}^{\dagger}e^{i\sigma
k_{n}x}}e^{\rho\sqrt{\frac{2\pi}{L}}\frac{1}{\sqrt{E_{n}}}B_{\sigma,n}e^{-i\sigma
k_{n}x}}\left|\vec{r}\right\rangle$ $\displaystyle=e^{\rho
i\theta}\delta_{r^{\prime}_{0},r_{0}}\prod_{\sigma=\pm}\prod_{n=1}^{\infty}\frac{1}{\sqrt{r^{\prime}_{\sigma,n}!r_{\sigma,n}!}}e^{i\sigma
k_{n}x(r^{\prime}_{\sigma,n}-r_{\sigma,n})}\cdot$ $\displaystyle\hskip
56.9055pt\cdot\sum_{j_{\sigma,n}^{\prime}=0}^{\infty}\sum_{j_{\sigma,n}=0}^{\infty}\frac{(-1)^{j^{\prime}_{\sigma,n}}}{j_{\sigma,n}!j^{\prime}_{\sigma,n}!}\left(\sqrt{\frac{2\pi}{L}}\frac{\rho}{\sqrt{E_{k}}}\right)^{j_{\sigma,n}+j^{\prime}_{\sigma,n}}\left\langle\left(B_{\sigma,n}\right)^{r^{\prime}_{\sigma,n}}\left(B_{\sigma,n}^{\dagger}\right)^{j^{\prime}_{\sigma,n}}\left(B_{\sigma,n}\right)^{j_{\sigma,n}}\left(B_{\sigma,n}^{\dagger}\right)^{r_{\sigma,n}}\right\rangle$
(40)
with
$\displaystyle\left\langle\left(B_{\sigma,n}\right)^{r^{\prime}_{\sigma,n}}\left(B_{\sigma,n}^{\dagger}\right)^{j^{\prime}_{\sigma,n}}\left(B_{\sigma,n}\right)^{j_{\sigma,n}}\left(B_{\sigma,n}^{\dagger}\right)^{r_{\sigma,n}}\right\rangle$
$\displaystyle=\left(\begin{array}[]{c}r^{\prime}_{\sigma,n}\\\
j^{\prime}_{\sigma,n}\end{array}\right)\left(\begin{array}[]{c}r_{\sigma,n}\\\
j_{\sigma,n}\end{array}\right)j^{\prime}_{\sigma,n}!j_{\sigma,n}!(r_{\sigma,n}-j_{\sigma,n})!\delta_{r^{\prime}_{\sigma,n}-j^{\prime}_{\sigma,n},r_{\sigma,n}-j_{\sigma,n}}\Theta(r_{\sigma,n}\geq
j_{\sigma,n})$ (45)
In the first line we have substituted in the Bogoliubov transformation (26)
and used (32) to evaluate the expectation value of the Klein factors. The last
equality follows by a power expansion. Upon the integration
$\int_{0}^{L}dx\,\left<\vec{r}^{\prime}\right|:\negmedspace e^{\rho
i\sqrt{4\pi}\Phi(x)}\negmedspace:_{M}F_{\rho}^{\dagger}F_{-\rho}\left|\vec{r}\right>$,
the factor $\prod_{\sigma=\pm}\prod_{n=1}^{\infty}e^{i\sigma
k_{n}x(r^{\prime}_{\sigma,n}-r_{\sigma,n})}$ gives the momentum conservation
$\delta\left(\sum_{\sigma=\pm}\sigma\sum_{n=1}^{\infty}n(r^{\prime}_{\sigma,n}-r_{\sigma,n})\right)$.
This is a manifestation of translation invariance and means that we can
diagonalise different total momentum sectors separately and compute the
dynamics only in the sector where the initial state resides, the total
momentum zero sector,
$\sum_{\sigma=\pm}\sigma\sum_{n=1}^{\infty}nr_{\sigma,n}=0$. The expression
for the matrix elements of the vertex operator is a product of terms
corresponding to the two chiralities which is another property that
facilitates the implementation. Furthermore, it is clear that the vertex
operator does not mix different sectors of the $B_{0}$ mode, so that the
Hamiltonian can be diagonalised in each sector separately.
Observables:
In the zero sector of the total momentum where the quench dynamics resides,
only those quadratic terms of bosonic modes in $C_{\mu}(t,x,y)$ give nonzero
contributions which preserve the momentum. Thus, the expectation values are:
$\displaystyle\left\langle\Psi\right|:\negmedspace
J^{0}(x)J^{0}(y)\negmedspace:\left|\Psi\right\rangle$
$\displaystyle=\frac{1}{\pi
L}\sum_{n=1}^{\infty}\frac{k_{n}^{2}}{E_{n}}\cos\left(k_{n}(x-y)\right)\left(\sum_{\sigma=\pm}\left\langle
B_{\sigma,n}^{\dagger}B_{\sigma,n}\right\rangle_{\Psi}-\left\langle
B_{-,n}B_{+,n}\right\rangle_{\Psi}-\left\langle
B_{-,n}^{\dagger}B_{+,n}^{\dagger}\right\rangle_{\Psi}\right)$
$\displaystyle\left\langle\Psi\right|:\negmedspace
J^{1}(x)J^{1}(y)\negmedspace:\left|\Psi\right\rangle$
$\displaystyle=\frac{1}{\pi
L}\sum_{n=1}^{\infty}E_{k}\cos\left(k_{n}(x-y)\right)\left(\sum_{\sigma=\pm}\left\langle
B_{\sigma,n}^{\dagger}B_{\sigma,n}\right\rangle_{\Psi}+\left\langle
B_{-,n}B_{+,n}\right\rangle_{\Psi}+\left\langle
B_{-,n}^{\dagger}B_{+,n}^{\dagger}\right\rangle_{\Psi}\right)$ (46)
where we used the mode expansion of the currents (11), expressed the charges
in terms of the $B_{0}$ modes using the equations below (25), abbreviated
$\left\langle\bullet\right\rangle_{\Psi}\equiv\left\langle\Psi\right|\bullet\left|\Psi\right\rangle$
and dropped the diverging
$\sum_{n=1}^{\infty}\frac{k_{n}^{2}}{E_{n}}\cos\left(k_{n}(x-y)\right)$ and
$\sum_{n=0}^{\infty}E_{n}\cos\left(k_{n}(x-y)\right)$ by normal ordering. The
expectation values of the quadratic terms on a state can be computed using the
matrix elements (37), (38) and (39).
To study the cluster violation of the correlators of chiral fermion fields
$\left\langle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\psi_{-\sigma}^{\dagger}(y)\psi_{\sigma}(y)\right\rangle$,
one can use:
* •
To get $\left\langle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\right\rangle$,
we bosonise using (29), substitute the Bogoliubov transformed operators (26)
and normal order with respect to the massive modes:
$\displaystyle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)=\frac{1}{L}e^{\sum_{n>0}\frac{1}{n}\left(1-\frac{k_{n}}{E_{n}}\right)}e^{\sigma
i\frac{2\pi}{L}(1-\delta_{b})x}:\negmedspace e^{\sigma
i\sqrt{4\pi}\Phi(x)}\negmedspace:_{M}F_{\sigma}^{\dagger}F_{-\sigma}$ (47)
The matrix elements are given by eq. (40) and notice that in the total
momentum zero sector, the factor
$\prod_{\sigma=\pm}\prod_{n=1}^{\infty}e^{i\sigma
k_{n}x(r^{\prime}_{\sigma,n}-r_{\sigma,n})}$ becomes just the identity,
reflecting the translation invariance.
* •
To get
$\left\langle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\psi_{-\sigma}^{\dagger}(y)\psi_{\sigma}(y)\right\rangle$,
we use eq. (47) and its conjugate with the Klein operator algebra (31). Normal
ordering the whole expression gives:
$\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\psi_{-\sigma}^{\dagger}(y)\psi_{\sigma}(y)=\frac{1}{L^{2}}\left(e^{\sum_{n>0}\frac{1}{n}\left(1-\frac{k_{n}}{E_{n}}\right)}\right)^{2}e^{\frac{2\pi}{L}\sum_{n>0}\frac{2}{E_{n}}\cos\left(\frac{2\pi}{L}n(x-y)\right)}:\negmedspace
e^{\sigma i\sqrt{4\pi}\left(\Phi(x)-\Phi(y)\right)}\negmedspace:_{M}$ (48)
Recall that
$\lim\limits_{L\rightarrow\infty}e^{\sum_{n>0}\frac{1}{n}\left(1-\frac{k_{n}}{E_{n}}\right)}=\frac{ML}{4\pi}e^{\gamma}$
where $\gamma$ is the Euler-Mascheroni constant so that the explicit $L$
dependence cancels out. In fact, we use this limiting expression to get the
results closer to the thermodynamic limit. The required matrix element is
given by
$\displaystyle\left<\vec{r}^{\prime}\right|:\negmedspace e^{\rho
i\sqrt{4\pi}\left(\Phi(x)-\Phi(y)\right)}\negmedspace:_{M}\left|\vec{r}\right>=$
$\displaystyle=\delta_{r^{\prime}_{0},r_{0}}\prod_{\sigma=\pm}\prod_{n=1}^{\infty}\frac{1}{\sqrt{r^{\prime}_{\sigma,n}!r_{\sigma,n}!}}\sum_{j_{\sigma,n}^{\prime}=0}^{\infty}\sum_{j_{\sigma,n}=0}^{\infty}\frac{(-1)^{j^{\prime}_{\sigma,n}}}{j_{\sigma,n}!j^{\prime}_{\sigma,n}!}\left(\sqrt{\frac{2\pi}{L}}\frac{\rho}{\sqrt{E_{k}}}\right)^{j_{\sigma,n}+j^{\prime}_{\sigma,n}}\cdot$
$\displaystyle\hskip 56.9055pt\cdot\left(e^{i\sigma k_{n}x}-e^{i\sigma
k_{n}y}\right)^{j^{\prime}_{\sigma,n}}\left(e^{-i\sigma k_{n}x}-e^{-i\sigma
k_{n}y}\right)^{j_{\sigma,n}}\left\langle\left(B_{\sigma,n}\right)^{r^{\prime}_{\sigma,n}}\left(B_{\sigma,n}^{\dagger}\right)^{j^{\prime}_{\sigma,n}}\left(B_{\sigma,n}\right)^{j_{\sigma,n}}\left(B_{\sigma,n}^{\dagger}\right)^{r_{\sigma,n}}\right\rangle$
(49)
and the expectation value of the boson operators is given by eq. (45).
#### Truncation. -
We preform the THM truncation by choosing a value for the cutoff energy
$E_{\text{cut}}$ and keeping only those states of the Hilbert space
$\left|\vec{r}\right\rangle$ for which
$\left\langle\vec{r}\right|H_{\text{EM}}+H_{\text{F}}\left|\vec{r}\right\rangle\leq
E_{\text{cut}}$. This results in a better converging code than for example if
truncating by keeping a fixed number of momentum modes. The truncation
criterium depends on the charge $e$ and the system size $L$ (as
$E_{n}=\sqrt{k_{n}^{2}+\frac{e^{2}}{\pi}}=\frac{2\pi}{L}\sqrt{n^{2}+\frac{L^{2}}{(2\pi)^{2}}\frac{e^{2}}{\pi}}$)
and for fixed $E_{\text{cut}}$ the number of states in the THM Hilbert space
decreases with increasing $e$ and $L$. Therefore, in practice the truncation
is done by choosing a desired number of states in the THM Hilbert space and
then for a given $e$ and $L$ finding $E_{\text{cut}}$ that gives us a Hilbert
space size closest to the desired one. In that way we can assure that results
obtained at different $e$ and $L$ are achieved with comparable Hilbert space
sizes.
The size of the Hilbert space that has to be kept in the computer’s memory can
be reduced by taking into account the symmetries of the model. Since the zero
mode $B_{0}$ is decoupled from the rest of the modes, we can diagonalise the
Hamiltonian in each of it’s sectors separately. In particular, for real time
dynamics following quenches it is enough to keep the $\left\langle
B_{0}^{\dagger}B_{0}\right\rangle=0$ sector where the initial states, the
ground states, reside. Furthermore, because of the translation invariance of
the model, the ground states are in the the zero total momentum sector
($p_{\text{tot}}=\sum_{\sigma=\pm}\sigma\sum_{n=1}^{\infty}k_{n}\left\langle
B_{\sigma,n}^{\dagger}B_{\sigma,n}\right\rangle=0$) of the Hilbert space,
which drastically reduces the number of states that have to be kept in the
computer’s memory in order to compute the quench dynamics. We do, however,
have to diagonalise the Hamiltonian also in the sectors with other values of
the total momentum in order to compute the full spectrum of the model (excited
states). For the results presented in this Letter, we use up to 20 000 states
per sector.
In case of truncated conformal space approach (TCSA) methods, where the
expansion is around a CFT, the renormalisation group theory guarantees that
for relevant perturbing operators, the cut-away high energy part of the
Hilbert space is only very weakly coupled to the low energy part and therefore
does not modify the low energy physics that one studies with such methods
Elias-Miró et al. (2017); James et al. (2018). In a more general expansion
like we use here, we cannot directly rely on the RG theory and have to
establish convergence by extensive tests. We have therefore tested that all
our results have converged with the THM cutoff. We have also tested that the
scalar particle mass computed with our method agrees with matrix product
states (MPS) and tensor network (TN) computations with a discretised version
of the Schwinger model Bañuls et al. (2013); Buyens et al. (2015); Buyens
(2016) (fig. 2 in the main text) and that in the $e\rightarrow 0$ limit we
recover the spectrum of the sine-Gordon model.
#### Quench protocol. -
In order to study the quench dynamics, one takes for the initial state the
ground state $\left|\Psi\right\rangle$ of the prequench Hamiltonian
$H(m_{0}/e_{0},\theta_{0},L)$ which can be found by numerical diagonalisation
of the Hamiltonian. At $t=0$, the parameters are quenched to the postquench
values $H(m/e,\theta,L)$. The dynamics is computed using the numerical
exponentiation of the postquench Hamiltonian:
$\left|\Psi(t)\right\rangle=e^{-itH}\left|\Psi\right\rangle.$ (50)
Finally, correlators are computed as expectation values on these states
$C_{\mu}(t,x,y)=\left\langle J^{\mu}(t,x)J^{\mu}(t,y)\right\rangle.$ (51)
## Appendix B Further results
Here we list some further results adding more detail to those presented in
fig. 3 in the main text.
The effect is found in quenches of either of the parameters of the system,
$e/m$ and $\theta$ as well as in quenches to and from the massless Schwinger
model. The sign of the out-of-horizon correlations changes depending whether
the quenched parameter is increased or decreased. Fig 5 gives an overview of
these observations.
Figure 5: Time dependent $\left\langle J^{x}(t,x)J^{x}(t,y)\right\rangle$ and
$\left\langle J^{t}(t,x)J^{t}(t,y)\right\rangle$ correlations for different
type of quenches in the Schwinger model (initial correlations subtracted): 1.)
Quench in $m/e$ with $m_{0}=0.25$, $m=0.125$, $\theta_{0}=\theta=0$; 2.)
Quench in $\theta$ with $\theta_{0}=\frac{\pi}{4}$, $\theta=0$,
$m_{0}=m=0.125$; 3.) Quench from the massless Schwinger model with $m_{0}=0$,
$m=0.125$, $\theta_{0}=\theta=0$; 4.) Quench to the massless model $m_{0}=0$,
$m=0.125$, $\theta_{0}=\theta=0$. All with $e_{0}=e=1$, $L=40$.
Fig. 6: The correlator
$\left\langle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\psi_{-\sigma}^{\dagger}(y)\psi_{\sigma}(y)\right\rangle$
exhibits clustering. While the clustering is restored by normal ordering for
the massless model, it is violated in the massive model even for the normal
ordered correlator. The magnitude of the correlators depends on both $e/m$ and
$\theta$.
Figure 6: Cluster violation of the
$\left\langle\psi_{\sigma}^{\dagger}(x)\psi_{-\sigma}(x)\psi_{-\sigma}^{\dagger}(y)\psi_{\sigma}(y)\right\rangle$
correlator at different values of the parameters: 1.) $m=0$, $\theta=0$; 2.)
$m=0.125$, $\theta=0$; 3.) $m=0.125$, $\theta=\pi/4$. All with $e_{0}=e=1$,
$L=40$.
Fig 7: In quenches to the special value of the parameter $\theta=\pi$, to the
mass above the Ising transition point, the horizon dynamics is strongly
suppressed, resembling the confined dynamics observed in Kormos et al. (2017).
Note that here we are plotting the $C_{t}$ correlator for which there is no
horizon violation effect.
Figure 7: Suppression of the horizon spreading in quenches to the $\theta=\pi$
line above the Ising phase transition point. Here, $\theta_{0}=0$,
$\theta=\pi$, $m_{0}=m=0.5615$, $e_{0}=e=1$, $L=40$.
|
††thanks<EMAIL_ADDRESS>P. K. F. and M. S. have
contributed equally.
# Randomizing multi-product formulas for improved Hamiltonian simulation
P. K. Faehrmann Dahlem Center for Complex Quantum Systems, Freie Universität
Berlin, 14195 Berlin, Germany M. Steudtner Dahlem Center for Complex Quantum
Systems, Freie Universität Berlin, 14195 Berlin, Germany R. Kueng Institute
for Integrated Circuits, Johannes Kepler University Linz, Austria M.
Kieferová Centre for Quantum Computation and Communication Technology, Centre
for Quantum Software and Information, University of Technology Sydney, NSW
2007, Australia J. Eisert Dahlem Center for Complex Quantum Systems, Freie
Universität Berlin, 14195 Berlin, Germany Helmholtz-Zentrum Berlin für
Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
(August 27, 2024)
###### Abstract
Quantum simulation, the simulation of quantum processes on quantum computers,
suggests a path forward for the efficient simulation of problems in condensed-
matter physics, quantum chemistry and materials science. While the majority of
quantum simulation algorithms is deterministic, a recent surge of ideas has
shown that randomization can greatly benefit algorithmic performance. In this
work, we introduce a scheme for quantum simulation that unites the advantages
of randomized compiling on the one hand and higher-order multi-product
formulas as they are used for example in linear-combination-of-unitaries (LCU)
algorithms or quantum error mitigation on the other hand. In doing so, we
propose a framework of randomized sampling that is expected to be useful for
programmable quantum simulators and present two new multi-product formula
algorithms tailored to it. Our framework greatly reduces the circuit depth by
circumventing the need for oblivious amplitude amplification required by the
implementation of multi-product formulas using standard LCU methods, rendering
it especially useful for near-term quantum computing. Our algorithms achieve a
simulation error that shrinks exponentially with the circuit depth. To
corroborate their functioning, we prove rigorous performance bounds as well as
the concentration of the randomized sampling procedure. We demonstrate the
functioning of the approach for several physically meaningful examples of
Hamiltonians, including fermionic systems and the Sachdev–Ye–Kitaev model, for
which the method provides a favorable scaling in effort.
## I Introduction
The simulation of quantum processes on quantum computers is one of the most
eagerly anticipated use-cases for quantum computing. The ability to simulate a
system’s time evolution promises to provide insights into the dynamics of
interacting quantum systems in situations where approximate classical
simulation methods fail and constitutes one of the cornerstones of quantum
technologies [1].
This work aims at substantially improving algorithms that simulate the
dynamics of expectation values of observables. The need for developing such a
machinery stems from the observation that state-of-the-art quantum devices are
still rather limited in their realizable circuit depths and control. We
therefore assume only access to a quantum-oracle machine which implements
single-qubit state preparation, controlled time evolution and quantum
measurements and strive for minimizing the required depth of suitable quantum
algorithms. Such a setting explicitly allows for the use of product formulas,
which are the earliest algorithms proposed for the simulation of time-
independent local Hamiltonians [2].
Such Trotter-Suzuki methods, as they are called, have evolved from comparably
simple prescriptions for local Hamiltonians to sophisticated schemes able to
capture more general sparse time-independent Hamiltonians [3, 4, 5] as well as
time-dependent Hamiltonians [6, 7] and open quantum systems [8, 9]. In spite
of their relative simplicity, product formulas are still at the forefront of
Hamiltonian simulation techniques. A numerical study has shown that product
formulas can in practice outperform more complex techniques [10] and their
complexity is better than initial estimates [11] suggest. In fact, product
formulas are nearly optimal for lattice model simulations [12].
Multi-product formulas introduced in the _linear-combinations-of-unitaries
(LCU)_ [13] approach have been built upon previous results of Trotter-Suzuki
methods and have recently been improved [14]. The idea of linearly combining
unitaries has led to quantum algorithms with an exponential speedup in
precision [15, 16, 17, 18] which have been recently gaining popularity.
Besides quantum simulation in the LCU framework, the mathematical construct of
multi-product formulas (see Fig. 1) can also be used in quantum error
mitigation [19]. Although these methods have optimal asymptotic error scaling,
they inherently require deep quantum circuits for their implementation and are
thus not suitable for medium-term applications. It is therefore crucial to
find algorithms that require shorter circuits and fewer digital gates.
Multi-product formulasQuantum error mitigation [19]Linear-combination-of-
unitaries [13, 14]Randomized sampling [here] Figure 1: Multi-product formulas,
although introduced with use in the LCU framework in mind, can also be used in
different frameworks, such as quantum error mitigation or randomized sampling,
which is the main contribution of this work.
Recently, yet a new element has been introduced to aid this search: the
element of _classical randomness_. The idea of randomization in Hamiltonian
simulation has heralded a renaissance of product formula methods [20, 21, 22,
23, 24]. For single steps of such techniques, a rigorous understanding has
recently been reached [24]. Randomized algorithms can also be considered when
one is only interested in estimating expectation values. For such a task we do
not need to prepare the time-evolved state (from which the observable will be
measured) perfectly. Instead, a lower-effort randomized algorithm can be used
such that the correct expectation value is obtained only after averaging the
measurement outcomes.
The aims of this work are twofold: First, we strive for combining the
advantages of higher-order multi-product formulas with those of schemes of
randomized compiling, to create a novel framework in which multi-product
formulas can be put to good use, which we dub “randomized sampling”.
Specifically, we adopt the above scenario of computing expectation values of
observables and propose to sample product formulas from multi-product
formulas, by which we circumvent the need to use the LCU framework and its
corresponding methods such as block encodings. Instead, we implement multi-
product formulas on average through random sampling. This results in a notable
reduction in algorithmic depth at the expense of additional circuit
evaluations. In this way, we see how notions of randomized compiling and
higher-order multi-product formulas – when suitably brought together – allow
for more resource-efficient notions of quantum simulation amenable to
programmable devices. Second, we develop alternative multi-product formulas
tailored to this new framework, which promise to outperform the accuracy of
the multi-product formulas introduced by Childs and Wiebe in Ref. [13] in the
regime of short simulation times. We show that using a quantum device limited
to the aforementioned operations can already yield substantial improvements to
fully analog approaches. It is therefore especially useful when algorithms
with a pure focus on noisy-intermediate-scale-quantum devices [25], such as
variational quantum algorithms, reach their limits, but fully digital
algorithms are not yet feasible.
The remainder of this work is organized as follows: After a short review of
multi-product formulas in Section II, we present the framework of randomized
sampling and a short summary of our main results in Section III. Section IV
then contains a detailed analysis of the proposed randomized sampling
framework and multi-product formulas and gives the proofs to the results
discussed in the previous section. To conclude, we compare the performance of
our formulas to that of Trotter-Suzuki and Childs and Wiebe type multi-product
formulas [13] in Section V, discussing both their error bounds and actual
performance on a number of physically plausible and interesting Hamiltonian
models of strongly correlated quantum systems, for which we find a favorable
performance over known schemes of quantum simulation.
## II Multi-product formulas
Multi-product formulas constitute the main ingredient of our work, and hence
we will briefly review the underlying ideas, starting with product formulas.
The goal of quantum simulations is to approximate the quantum dynamics of a
complex quantum system described by a many-body Hamiltonian decomposed as
$H=\sum_{k=1}^{L}h_{k}$ (1)
composed of Hermitian Hamiltonian terms $\\{h_{k}\\}$ of neither necessarily
small nor geometrically local support defined on a quantum lattice equipped
with a Hilbert space ${\cal H}=(\mathbb{C}^{d})^{\otimes n}$.
Here, $n$ is the number of degrees of freedom of finite local dimension $d$.
To access the Hamiltonian, we assume to have oracles $\mathbb{O}_{k}(t)$
implementing time evolutions under each term in the Hamiltonian
$\mathbb{O}_{k}(t)\ket{\psi}=e^{-\mathrm{i}h_{k}t}\ket{\psi}$ (2)
for any $k\in[1,L]$ and $t\in\mathbb{R}$. In digital quantum simulation, these
oracles are built from Clifford gates and phase rotations, but we do not
assume that such a decomposition is available to us, as the oracles could be
implemented by the time evolution of a programmable device. We only require to
have control over the implementation, i.e., that we can apply the oracle
$\mathbb{O}_{k}(t)$ depending on the state of a subsystem encoded as a qubit
as
$\displaystyle\ket{0}\\!\\!\bra{0}\otimes\mathbb{I}+\ket{1}\\!\\!\bra{1}\otimes\mathbb{O}_{k}(t)\,.$
(3)
Now, quantum simulation aims at making reliable predictions of the expectation
values of observables $O$ (which in the ideal case are local with a small
support on the lattice, but again, this is not a necessity) at later times
$T>0$
$\langle O(T)\rangle\coloneqq{\rm tr}(U(T)\rho U^{\dagger}(T)O)$ (4)
for initial quantum states $\rho$, where $U$ is the time evolution operator.
In practice, this commonly means to establish ways to approximate the
corresponding time evolution operator
$U(T)=\mathrm{e}^{-\mathrm{i}HT}\,.$ (5)
Although it is worth stressing that a-priori information about the time $T$
and properties of the initial state $\rho$, as well as features of locality of
the underlying Hamiltonian $H$ can be exploited to come up with highly
specialized approximations, we do not assume any underlying structure or
limitations in this work.
A simple, yet effective approach to approximating the time evolution operator
is that of using product formulas, which make use of $N$ consecutive
evolutions of individual Hamiltonian terms $h_{k_{j}}$ by associated short
time intervals $\alpha_{j}t$, defined with real coefficients $\alpha_{j}$ such
that $\sum_{j}\alpha_{j}=1$. Partial backwards evolutions are explicitly
allowed, i.e. $\alpha_{j}<0$ for some $j$, even though $t>0$. In general, such
a formula is defined as a product of time evolutions
$\mathrm{e}^{-\mathrm{i}Ht}\approx\prod_{j=1}^{N}\mathrm{e}^{-\mathrm{i}\alpha_{j}\,h_{k_{j}}t}\,$
(6)
where each single term evolution can be implemented via $\mathbb{O}_{k}(t)$.
Here, we distinguish between a direct evolution by time $t$ and a repeated
evolution of short time slices $t/r<1$ such that
$\mathrm{e}^{-\mathrm{i}Ht}\approx\left(\prod_{j=1}^{N}\mathrm{e}^{-\mathrm{i}\alpha_{j}\,h_{k_{j}}\frac{t}{r}}\right)^{r}\,.$
(7)
The most accurate known formulas of this type are _Trotter-Suzuki formulas_
[26]. They are recursively defined for any positive integer $\chi$ and any
time $t$ by
$\displaystyle S_{1}(t)=\prod_{k=1}^{L}\mathrm{e}^{-\mathrm{i}h_{k}t}\,,$ (8)
$\displaystyle
S_{2}(t)=\prod_{k=1}^{L}\mathrm{e}^{-\mathrm{i}h_{k}t/2}\prod_{k=L}^{1}\mathrm{e}^{-\mathrm{i}h_{k}t/2}\,,$
(9) $\displaystyle
S_{2\chi}(t)=\Big{(}S_{2\chi-2}(s_{2\chi-2}t)\Big{)}^{2}S_{2\chi-2}\Big{(}[1-4s_{2\chi-2}]t\Big{)}\Big{(}S_{2\chi-2}(s_{2\chi-2}t)\Big{)}^{2}$
(10)
with $s_{2\chi}\coloneqq(4-4^{1/(2\chi+1)})^{-1}$ for any positive integer
$\chi$. This specific choice of $s_{2\chi}$ ensures that the Taylor series of
$S_{2\chi}(t)$ matches that of the actual time evolution $U(t)$ up to
$\mathcal{O}(t^{2\chi+1})$, which makes it a good approximation for $t\ll 1$.
It is important to stress that constructing an $\mathcal{O}(t^{2\chi+1})$
approximation requires the application of $2\cdot 5^{\chi-1}(L-1)+1$
individual oracles $\mathbb{O}_{k}(t)$, where $L$ is again the number of
Hamiltonian terms. More generally, we would represent them by oracle calls and
we will refer to the number of exponentials $N_{\rm exp}$. Using repetitions
as in Eq. (7), the number of expoentials of the algorithm is of
$\mathcal{O}(rL5^{\chi-1})$ to approximate the actual time evolution $U(T)$ up
to an accuracy of $\varepsilon\in\mathcal{O}((T/r)^{2\chi+1})$ in operator
distance. The exponential dependence of the circuit depth on $\chi$ can pose a
challenge for real-world implementations, and especially so within the NISQ
regime. Altogether, the number of oracle calls needed to simulate the
evolution up to an error $\varepsilon\leq 1$ is upper bounded by
$N_{\rm
oracle}\leq\frac{2L5^{2\chi}\left(L\tau\right)^{1+1/2\chi}}{\varepsilon^{1/2\chi}},$
(11)
where $\tau:=\sum_{k=1}^{L}\lVert h_{k}\rVert T$ and $\lVert\cdot\rVert$
denotes the operator norm [4]. An alternative complexity bound using
commutator relations of the individual Hamiltonian terms can be found in Ref.
[11].
A known technique to decrease the number of required short time evolutions,
i.e., oracle calls, is the use of multi-product formulas [27, 28]. While the
Trotter-Suzuki approximation cancels erroneous contributions of higher-order
terms by adding backwards evolutions, multi-product formulas achieve the same
cancellations by superposing different product formulas. Conventionally, one
employs multi-product formulas that describe the same time evolution (up to a
fixed order of $t$), but whose erroneous higher-order contributions are of
different strength and can thus be made to cancel. This is achieved by
approximating the time evolution to the same Trotter-Suzuki order, but
considering product formulas different in the number of time slices [27, 29].
The Childs and Wiebe multi-product formula discussed in Ref. [13] is of the
form
$M_{K,2\chi}(t)\coloneqq\sum_{q=1}^{K+1}C_{q}S_{2\chi}(t/\ell_{q})^{\ell_{q}},$
(12)
where $K$ is an integer defining a cutoff and $\\{\ell_{q}\\}$ a set of
pairwise different integers. The coefficients $\\{C_{q}\\}$ are determined via
$\left(\begin{matrix}1&1&\,&\cdots&\,&1\\\
\ell_{1}^{-2\chi}&\ell_{2}^{-2\chi}&&\cdots&&\ell_{K+1}^{-2\chi}\\\
\ell_{1}^{-2\chi-2}&\ell_{2}^{-2\chi-2}&&\cdots&&\ell_{K+1}^{-2\chi-2}\\\
\vdots&\vdots&&\ddots&&\vdots\\\
\ell_{1}^{-2(K+\chi-1)}&\ell_{2}^{-2(K+\chi-1)}&&\cdots&&\ell_{K+1}^{-2(K+\chi-1)}\end{matrix}\right)\left(\begin{matrix}C_{1}\\\
C_{2}\\\ C_{3}\\\ C_{4}\\\ \vdots\\\ C_{K+1}\\\
\end{matrix}\right)=\left(\begin{matrix}1\\\ 0\\\ 0\\\ 0\\\ \vdots\\\
0\end{matrix}\right),$ (13)
ensuring the error terms in the multi-product formula vanish up to
$\mathcal{O}(t^{2(K+\chi)})$, resulting in
$\left\lVert
M_{K,\chi}-U(t)\right\rVert=\mathcal{O}\left(t^{2(K+\chi)+1}\right).$ (14)
Unlike Childs and Wiebe, we will henceforth use the simplest version achieved
by setting $\ell_{q}=q$ for all $q$.
In Definitions 2 and 3 and Section IV, we employ a different approach for
multi-product formulas and develop two techniques whose errors scale with
$\mathcal{O}(t^{2\chi R+1})$, and where $R$ is comparable to $K+1$.
Multi-product formulas were firstly used for quantum simulation by Childs and
Wiebe [13], who developed the _linear-combinations-of-unitaries (LCU)_
approach to directly implement multi-product formulas on a quantum system.
Note that sums of unitaries are not inherently physical operations since the
unitary group is not closed under addition. Childs and Wiebe use a non-
deterministic approach to implement multi-product formulas that can lead to
large algorithmic depth. Berry et al. [15] have implemented LCU for a
truncated Taylor series nearly perfectly but their use of oblivious amplitude
amplification requires an additional register and a complex state preparation
procedure. Recently, Ref. [14] has improved the condition number of multi-
product formulas and thus extended the use of LCU by amplitude amplification.
Using _randomized sampling_ such as proposed in Algorithm 1, we can circumvent
the need for these potentially deep circuits. Randomized compiling of multi-
product formulas entails sampling among the individual terms of Eq. (12)
requiring the implementation of just twice the number of oracle calls, rather
than the entire linear combination and the overhead for its implementation.
## III Randomized sampling using multi-product formulas
LCU-type algorithms, despite having a finite failure probability, are
deterministic when successfully applied. Apart from such deterministic
implementations, there is a recent interest in randomized algorithms [21, 22,
23, 24]. While these novel results improve the performance of product formulas
by randomizing the order of the short time evolution, we propose the
alternative setting of randomized compiling. With the goal of reducing the
circuit depth, we introduce the setting of randomized sampling: Given an
observable $O$ and a quantum system in the initial state $\rho$, our goal is
to find the expectation value of the observable after a time evolution
$U=\mathrm{e}^{-\mathrm{i}Ht}$ governed by the Hamiltonian
$H=\sum_{k=1}^{L}h_{k}$. That is, we wish to compute
$\langle O\rangle={\rm tr}(OU\rho U^{\dagger})\,.$ (15)
In the following, we describe a randomized algorithm for such a task, give
convergence guarantees and present novel multi-product formulas suited for
this framework. Note that we will require one part of our system to be encoded
as a qubit, while the rest acts as the simulator.
Figure 2: Quantum circuit for randomized sampling of some observable $O$
according to Algorithm 1. When $V_{\bullet}$ and $V_{\circ}$ are chosen from
$\set{V_{1},\ldots,V_{M}}$ with probabilities $\set{p_{1},\ldots,p_{M}}$ such
that $\sum_{k=1}^{M}p_{k}V_{k}=V$, the expectation value of the measurement
outcome is equal to $\operatorname{tr}({OV\rho V^{\dagger}})$.
###### Algorithm 1 (Randomized sampling).
Given an observable $O$ and an ensemble
$\mathcal{V}=\set{(V_{k},p_{k})}_{k=1}^{M}$ of $M$ unitaries $\\{V_{k}\\}$ and
corresponding probabilities $\\{p_{k}\\}$, we consider $N$ independent
repetitions of the following protocol:
1. 1.
Prepare $\ket{+}\\!\\!\bra{+}\otimes\rho\,$.
2. 2.
Sample $V_{\circ}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}\mathcal{V}$ and
apply the anti-controlled unitary $\ket{0}\\!\\!\bra{0}\otimes
V_{\circ}+\ket{1}\\!\\!\bra{1}\otimes\mathbb{I}\,$.
3. 3.
Sample $V_{\bullet}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}\mathcal{V}$ and
apply the controlled unitary
$\ket{0}\\!\\!\bra{0}\otimes\mathbb{I}+\ket{1}\\!\\!\bra{1}\otimes
V_{\bullet}\,$.
4. 4.
Perform a single shot of the POVM measurement associated with the observable
$X\otimes O\,$.
5. 5.
Store the measurement outcomes $\\{o_{j}\\}$.
The corresponding quantum circuit is shown in Fig. 2. As we show in more
detail in Section IV, we find the following result for infinitely many runs of
the algorithm:
###### Corollary 1 (Sample mean convergence).
Let $O$ be an observable with $\lVert O\rVert\leq 1$, let $\rho$ be an initial
state and let $\mathcal{V}=\set{(V_{k},p_{k})}_{k=1}^{M}$ be an ensemble of
unitaries $\set{V_{i}}$ and corresponding probabilities $\set{p_{i}}$, such
that
$\mathbb{E}_{\mathcal{V}}[V_{k}]=\sum_{k=1}^{M}p_{k}V_{k}=V\,.$ (16)
Then, the sample mean of Algorithm 1 converges to the expectation value of the
random measurement outcomes $\\{o_{j}\\}_{j}$, given by
$\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^{N}o_{j}=\mathrm{tr}\left(OV\rho
V^{\dagger}\right)\,.$ (17)
Furthermore, since an infinite number of measurements is infeasible, we give
an expression for the number $N$ of repetitions of Algorithm 1 sufficient for
achieving a desired accuracy and confidence for the goal of randomized
sampling. Using Hoeffding’s inequality [30], we can formulate the following
result:
###### Theorem 1 (Randomized implementation of sums of unitaries).
Let $O$ be an observable with $\lVert O\rVert\leq 1$, let $\rho$ an initial
state and let $\mathcal{V}=\set{(V_{k},p_{k})}_{k=1}^{M}$ be an ensemble of
unitaries $\set{V_{k}}$ and corresponding probabilities $\set{p_{k}}$, such
that
$\mathbb{E}_{\mathcal{V}}[V_{k}]=\sum_{k=1}^{M}p_{k}V_{k}=V\,.$ (18)
Then, for a fixed accuracy $\varepsilon\in(0,1)$ and confidence
$\delta\in(0,1)$, a total of
$N\geq\frac{2\log\left(2/\delta\right)}{\varepsilon^{2}}$ (19)
repetitions of Algorithm 1 suffice to accurately approximate the expectation
value $\mathrm{tr}\left(OV\rho V^{\dagger}\right)$ using the sampling mean
estimator, i.e.,
$\left\lvert\frac{1}{N}\sum_{j=1}^{N}o_{j}-\mathrm{tr}\left(OV\rho
V^{\dagger}\right)\right\rvert\leq\varepsilon$ (20)
with probability at least $1-\delta$.
So far, this framework is fairly general: Given a set $\mathcal{V}$, the
algorithm samples from the linear transform with the unitary $V$. For quantum
simulation, $V$ (a decent approximation to $U$) and $\mathcal{V}$ still have
to be found. This is where multi-product formulas come into play. Multi-
product formulas approximate $U$ by a weighted sum of product formulas, and so
the featured product formulas and their weights could make up the set
$\mathcal{V}$. The problem is that we require $\\{p_{k}\\}$ to form a
probability distribution, i. e., $p_{k}>0$ and $\sum_{k=1}^{M}p_{k}=1$. This
disqualifies us from identifying the sets $\\{p_{k}\\}$ with $\\{C_{q}\\}$
straightforwardly. While $\sum_{q}C_{q}=1$ holds for any multi-product formula
per construction (see the first component of Eq. (13)), the sign of some
$C_{q}$ will be negative [31]. We thus have to absorb the signs of these
negative $C_{q}$ into their unitaries $V_{q}$, but then we find that the
operation is not normalized as $\sum_{q}\lvert C_{q}\rvert>1$. At this point
we introduce the quantity $\Xi\coloneqq\sum_{q}|C_{q}|$, that we call
resolution factor. The resolution is practically used to define a probability
distribution via $p_{k}=\left|C_{k}\right|/\Xi$ such that Algorithm 1 samples
from
$\frac{1}{\Xi^{2}}\mathrm{tr}(OU\rho U^{\dagger})\,.$ (21)
As the time evolution is now approximated by $\Xi V$ rather than $V$, $\Xi$
factors into the number of required circuit evaluations. With $\Xi>1$
requiring to run the algorithm more often to achieve an error comparable with
the situation where $\Xi=1$, this resolution factor can be regarded as a
penalty. However, especially in near-term and medium-term applications, a
deeper circuit may pose a bottleneck much tighter than a slightly higher
number of circuit evaluations does.
We can make the following guarantees for approximating time evolutions in
randomized sampling with a nontrivial resolution and a finite number of
measurements:
###### Theorem 2 (Approximating unitaries with a finite number of
measurements,).
Let $U$ be a unitary, $O$ an observable with $\lVert O\rVert\leq 1$, $\rho$
some initial state and let $\mathcal{V}=\set{(V_{k},p_{k})}_{k=1}^{M}$ be an
ensemble of unitaries $\set{V_{k}}$ with a corresponding probability
distribution $\set{p_{k}}$ such that
$\mathbb{E}_{\mathcal{V}}[V_{k}]=\sum_{k=1}^{M}p_{k}V_{k}=V$. Let there be a
constant factor $\Xi\in\mathbb{R}_{+}$ such that
$\lVert\Xi V-U\rVert\leq\varepsilon/3\,.$ (22)
Then, it is sufficient to run
$N\geq 8\ln\left(2/\delta\right)\left(\frac{\Xi}{\varepsilon}\right)^{2}$ (23)
repetitions of Algorithm 1 to achieve
$\left\lvert\frac{1}{N}\sum_{j=1}^{N}o_{j}-\mathrm{tr}(OU\rho
U^{\dagger})\right\rvert\leq(1+\Xi)\,\varepsilon\,,$ (24)
with probability at least $1-\delta$.
The multi-product formulas of Childs and Wiebe can easily be adapted to this
randomized sampling scheme. Since we can ignore constraints encountered in LCU
techniques, our sole interest lies in a low algorithmic depth and a moderate
resolution and so one can set $\ell_{q}=q$ for all $q=1,\dots,K+1$ in Eq. (12)
and (13). However, those multi-product formulas make relatively little use of
the Trotter-Suzuki order $2\chi$ of its components. In fact the only reason
not to minimize the depth by using $S_{2}(\cdot)$ blocks is the high
resolution factor. To ease the impact of the resolution at higher orders of
the time evolution, we present a family of novel multi-product formulas with
an improved scaling in the Trotter-Suzuki order. While Childs and Wiebe’s
approach manipulates higher-order error terms, we employ a construction to
modulate the entire Taylor expansion.
###### Definition 1 (Linear combination of time evolution operators).
Let $\chi\geq 1$ and $R\geq 1$ be integers and $S_{2\chi}(t)$ a product
formula approximation to $U(t)$ with $\left\lVert
S_{2\chi}(t)-U(t)\right\rVert\in\mathcal{O}(t^{2\chi+1})$. Then, for any
$t\in\mathbb{R}$ and arbitrary
$\boldsymbol{b}=(b_{0},\,b_{1},\,\dots\,,\,b_{2\chi R})^{\top}$,
$\boldsymbol{\nu}=(\nu_{0},\,\nu_{2},\,\dots\,,\,\nu_{2\chi
R})^{\top}\in\mathbb{R}^{2\chi R+1}$ , we define the multi-product formula
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu},\boldsymbol{b},t)$ as
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu},\boldsymbol{b},t)\coloneqq\sum_{q=0}^{2\chi
R}C_{q}(\boldsymbol{\nu},\boldsymbol{b})\,S_{2\chi}\\!\left(b_{q}t\right),$
(25)
where
$\left(\begin{matrix}1&1&\,&\cdots&\,&1\\\ b_{0}&b_{1}&&\cdots&&b_{2\chi R}\\\
b_{0}^{2}&b_{1}^{2}&&\cdots&&b_{2\chi R}^{2}\\\
b_{0}^{3}&b_{1}^{3}&&\cdots&&b_{2\chi R}^{3}\\\
\vdots&\vdots&&\ddots&&\vdots\\\ \,b_{0}^{2\chi R}&\,b_{1}^{2\chi
R}&&\cdots&&\,b_{2\chi R}^{2\chi
R}\end{matrix}\right)\left(\begin{matrix}C_{0}\\\ C_{1}\\\ C_{2}\\\ C_{3}\\\
\vdots\\\ C_{2\chi R}\end{matrix}\right)=\left(\begin{matrix}\nu_{0}\\\
\nu_{1}\\\ \nu_{2}\\\ \nu_{3}\\\ \vdots\\\ \nu_{2\chi R}\end{matrix}\right)$
(26)
for some $(C_{0},\,C_{1},\,\dots\,,\,C_{2\chi R})^{\top}\in\mathbb{R}^{2\chi
R+1}$.
Here, the coefficients $\\{C_{q}\\}$ are related to the parameters
$\boldsymbol{b}$ and $\boldsymbol{\nu}$ via the inverse of the corresponding
Vandermonde matrix in Eq. (26) as is discussed in detail in Section IV. The
parameters $\boldsymbol{\nu}$ will have an important role in the following
construction, whereas $\boldsymbol{b}$ are tuned to strike a balance between
$\Xi$ and $\varepsilon$. Note that a similar construction has recently been
used to approximate energy measurements in the fully analog setting [32]. We
can now turn towards the definition of the first new multi-product formula.
###### Definition 2 (Matching multi-product formula).
Let $\chi\geq 1$ and $R\geq 1$ be integers and
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu},\boldsymbol{b},t)$ as defined in
Definition 1. Then, for any $t\in\mathbb{R}$, we define the multi-product
formula $\widetilde{M}_{2\chi,R}^{(\mathrm{m})}(t)$ as
$\widetilde{M}_{2\chi,R}^{(\mathrm{m})}(t)\coloneqq\prod_{r=1}^{R}\mathcal{L}_{2\chi,R}\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t\right)$
(27)
with $\nu_{0}^{(r)}=1$ and $\nu_{k}^{(r)}=0$ for $k>2\chi$ and all $r=1\dots
R$. The remaining $\left\\{\nu_{q}^{(r)}\right\\}$ are fixed by
$\sum_{k_{1}+k_{2}+\ldots+k_{R}=k}\frac{\nu_{k_{1}}^{(1)}\nu_{k_{2}}^{(2)}\cdots\nu_{k_{R}}^{(R)}}{k_{1}!\,k_{2}!\,\cdots\,k_{R}!}=\frac{1}{k!}$
(28)
for all $0\leq k\leq 2\chi R$, whereas the parameters
$\\{\boldsymbol{b}^{(r)}\\}$ can be chosen arbitrarily.
Multiplying the coefficients $C_{q}$ of the individual building blocks
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t)$ of the
matching multi-product formula results in a resolution factor of
$\Xi^{(\mathrm{m})}\coloneqq\prod_{r}\left(\sum_{q}\left\lvert
C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\right\rvert\right).$
(29)
For the time evolution to work, the set of vectors
$\left\\{\boldsymbol{\nu}^{(r)}\right\\}$ has to be found satisfying the
constraint in Eq. (28) for different $R$ and $2\chi$. When such a
decomposition is unavailable, we can employ a second version of the multi-
product formula in which the $\left\\{\boldsymbol{\nu}^{(r)}\right\\}$ are
already determined at the cost of a higher $\Xi$.
###### Definition 3 (Closed-form multi-product formula).
Let $\chi\geq 1$ and $R\geq 1$ be integers and
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu},\boldsymbol{b},t)$ as defined in
Definition 1. Then for any $t\in\mathbb{R}$, we define the multi-product
formula $\widetilde{M}_{2\chi,R}^{(\mathrm{cf})}(t)$ as
$\widetilde{M}_{2\chi,R}^{(\mathrm{cf})}(t)\coloneqq\sum_{r=1}^{R}\left(\mathcal{L}_{2\chi,R}\\!\left(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)},t\right)\right)^{r-1}\mathcal{L}_{2\chi,R}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t\right)$
(30)
with
$\displaystyle\nu^{(0)}_{k}$
$\displaystyle=\begin{cases}1,&\text{for}\;k=2\chi\\\
0,&\text{else},\end{cases}$ (31) $\displaystyle\nu^{(1)}_{k}$
$\displaystyle=\begin{cases}1,&\text{for}\;k\leq 2\chi\\\
0,&\text{else},\end{cases}$ (32) $\displaystyle\nu^{(n)}_{k}$
$\displaystyle=\begin{cases}\frac{k!\left((2\chi)!\right)^{n-1}}{(2\chi(n-1)+k)!},&\text{for}\;0<k\leq
2\chi\\\ 0,&\text{else},\end{cases}$ (33)
for all $\;0\leq k\leq 2\chi R\;$ and $\;1<n\leq R$. The parameters
$\\{\boldsymbol{b}^{(r)}\\}$ can be chosen arbitrarily.
Multiplying the coefficients of the individual
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t)$,
results in a resolution factor for the closed-form multi-product formula of
$\Xi^{(\mathrm{cf})}:=\sum_{r}\left(\sum_{q}\left\lvert
C_{q}\\!\left(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)}\\!\right)\right\rvert\right)^{r-1}\left(\sum_{q}\left\lvert
C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\right\rvert\right).$
(34)
For these multi-product formulas, we can prove the following two results:
###### Corollary 2 (Low depth unitary approximation).
Let $\widetilde{M}_{2\chi,R}(t)$ be a multi-product formula constructed
according to Definition 2 or 3. We then find that
$\left\lVert
U(t)-\widetilde{M}_{2\chi,R}(t)\right\rVert=\mathcal{O}\left(t^{2\chi
R+1}\right).$ (35)
###### Theorem 3 (Error bound for custom multi-product formulas).
For $\widetilde{M}_{2\chi,R}(t)$ being a multi-product formula constructed
according to Definitions 2 or 3, we have
$\left\lVert U(t)-\widetilde{M}_{2\chi,R}(t)\right\rVert\leq\left(1+\zeta
g_{\chi}^{2\chi R+1}\right)\frac{\left(\Lambda t\right)^{2\chi
R+1}}{\left(2\chi R+1\right)!}\,,$ (36)
with $g_{\chi}\ \coloneqq\frac{4\chi}{5}\left(\frac{5}{3}\right)^{\chi-1}$,
$\Lambda\coloneqq\sum\limits_{k}\left\lVert h_{k}\right\rVert$ and
corresponding
$\displaystyle\zeta^{(\mathrm{m})}$
$\displaystyle\coloneqq\sum_{q_{1}=0}^{2\chi R}\sum_{q_{2}=0}^{2\chi
R}\cdots\sum_{q_{r}=0}^{2\chi R}\left\lvert
C_{q_{1}}^{(1)}C_{q_{2}}^{(2)}\cdots C_{q_{r}}^{(r)}\right\rvert\left(\lvert
b_{q_{1}}^{(1)}\rvert+\ldots+\lvert b_{q_{r}}^{(r)}\rvert\right)$ (37)
$\displaystyle\zeta^{(\mathrm{cf})}$
$\displaystyle\coloneqq\sum_{r=1}^{R}\sum_{q_{1}=0}^{2\chi
R}\sum_{q_{2}=0}\cdots\sum_{q_{r}=0}^{2\chi R}\left\lvert
C_{q_{1}}^{(0)}C_{q_{2}}^{(0)}\cdots
C_{q_{r-1}}^{(0)}C_{q_{r}}^{(r)}\right\rvert\left(\lvert
b_{q_{1}}^{(0)}\rvert+\ldots+\lvert b_{q_{r-1}}^{(0)}\rvert+\lvert
b_{q_{r}}^{(r)}\rvert\right),$ (38)
for matching and closed-form multi-product formulas respectively.
Consequently, these formulas can be used for randomized sampling via Theorem
2. A numerical comparison of the error bounds of all presented formulas can be
found in Fig. 4, while an overview of circuit depths, error scaling and error
bounds is given in Table 1.
The matching and closed-form multi-product formulas both rely on products of
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t)$. For
Algorithm 1, these products can either be expanded and the resulting operators
and coefficients be identified with the sets $\\{V_{k}\\}$, $\\{p_{k}\\}$, or
sets of
$(V_{\circ}^{(1)},V_{\bullet}^{(1)}),\dots,(V_{\circ}^{(R)},V_{\bullet}^{(R)})$
be drawn and applied as in Fig. 3. For the matching multi-product formula, all
$V_{\circ}^{(r)}$ and $V_{\bullet}^{(r)}$ are drawn independently from each
other while for the closed-form multi-product formula, the set from which
$V_{\circ/\bullet}^{(r+1)}$ is drawn depends on the set from which
$V_{\circ/\bullet}^{(r)}$ has been sampled.
Figure 3: Quantum circuit for randomized sampling with a multi-product formula
$\prod_{r=1}^{R}\left(\sum_{k}p^{(r)}_{k}V^{(r)}_{k}\right)$. The unitaries
$V_{\circ}^{(r)},V_{\bullet}^{(r)}$ are drawn independently at random from
$\\{V^{(r)}_{k}\\}_{k}$ according to the probabilities $\\{p^{(r)}_{k}\\}_{k}$
for all $r=1\dots R$, such that the circuit samples from $\mathbb{E}(\langle
O\rangle)$.
Furthermore, since the above framework randomizes only over product formulas,
we can also think of a doubly-stochastic version in which the short-term
evolutions comprising the product formulas are sampled randomly as well. In
that sense, it is allowed to construct the multi-product formulas with
building blocks of
$\displaystyle\widehat{S}_{2}(t)=\frac{1}{2}\left(S_{1}^{\phantom{\dagger}}(t)+S_{1}^{\dagger}(-t)\right)\,,$
(39)
rather than (9). This is not possible for Childs and Wiebe’s multi-product
formulas, which require symmetric building blocks.
## IV Proofs
In this section, we provide proofs of the statements presented above. In
Section IV.1, we are proving our statements regarding the errors and
uncertainty of the randomized sampling procedure. We then turn our attention
to the matching and closed-form multi-product formula, providing the intuition
behind their construction and verifying their error scaling in Section IV.2.
We finally analyze the upper bounds for their error (which is the error of the
average time evolution they describe) in Section IV.3.
### IV.1 Randomized sampling
Let us start by following Algorithm 1 step by step. The controlled and anti-
controlled applications of $V_{\circ}$ and $V_{\bullet}$ are defined as
$\displaystyle\mathrm{\overline{C}}V_{\circ}$
$\displaystyle\coloneqq\ket{0}\\!\\!\bra{0}\otimes
V_{\circ}+\ket{1}\\!\\!\bra{1}\otimes\mathbb{I},$ (40)
$\displaystyle\mathrm{C}V_{\bullet}$
$\displaystyle\coloneqq\ket{0}\\!\\!\bra{0}\otimes\mathbb{I}+\ket{1}\\!\\!\bra{1}\otimes
V_{\bullet}\,,$ (41)
and the initial state $\rho$ will be transformed to
$\displaystyle\widetilde{\rho}$
$\displaystyle=\mathrm{C}V_{\bullet}\mathrm{\overline{C}}V_{\circ}\,\left(\left|+\right\rangle\\!\\!\left\langle+\right|\otimes\rho\right)\,(\mathrm{\overline{C}}V_{\circ})^{\dagger}(\mathrm{C}V_{\bullet})^{\dagger}$
(42) $\displaystyle=\frac{1}{2}\left(\ket{0}\\!\\!\bra{0}\otimes
V^{\phantom{\dagger}}_{\circ}\rho
V_{\circ}^{\dagger}+\ket{0}\\!\\!\bra{1}\otimes
V^{\phantom{\dagger}}_{\circ}\rho
V_{\bullet}^{\dagger}+\ket{1}\\!\\!\bra{0}\otimes
V^{\phantom{\dagger}}_{\bullet}\rho
V_{\circ}^{\dagger}+\ket{1}\\!\\!\bra{1}\otimes
V^{\phantom{\dagger}}_{\bullet}\rho V_{\bullet}^{\dagger}\right)$ (43)
after the third step of the protocol. The expectation value for the succeeding
measurement of $X\otimes O$ is then given by
$\mathrm{tr}\left(X\otimes
O\widetilde{\rho}\right)=\frac{1}{2}\mathrm{tr}\left(O(V^{\phantom{\dagger}}_{\circ}\rho
V_{\bullet}^{\dagger}+V^{\phantom{\dagger}}_{\bullet}\rho
V_{\circ}^{\dagger})\right).$ (44)
We now use that $V_{\circ}$ and $V_{\bullet}$ are sampled independently from
the same ensemble
$V_{\circ},V_{\bullet}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{V}=\set{(V_{k},p_{k})}_{k=1}^{M}$ (45)
such that
$\mathbb{E}_{\mathcal{V}}[V_{\circ}]=\mathbb{E}_{\mathcal{V}}[V_{\bullet}]=\sum_{k=1}^{M}p_{k}V_{k}=V\,.$
(46)
Then, we can combine the quantum average form Eq. (44) with the classical
average from Eq. (46) to conclude that the expectation value of the random
single-shot measurement outcome $o$ is given by
$\displaystyle\mathbb{E}[o]$
$\displaystyle=\mathbb{E}_{\mathcal{V}}\left[\mathrm{tr}\left(X\otimes
O\widetilde{\rho}\right)\right]=\mathrm{tr}\left(OV\rho V^{\dagger}\right),$
(47)
which proves Corollary 1.
For real world applications we will never achieve a perfect expectation value.
It is therefore vital to inspect the single-shot behavior of Algorithm 1 and
give an estimate for the number of iterations required to achieve the desired
precision. Note that
$\displaystyle O=\sum_{j=1}^{l}\omega_{j}P_{j}$ (48)
is an observable with eigenvalues $\lvert\omega_{j}\rvert\leq 1$ for $\lVert
O\rVert\leq 1$ and $\\{P_{j}\\}$ is a set of projectors. We thus find that
according to Born’s rule, the POVM measurement defined by $X\otimes O$ has
$2l$ possible measurement outcomes $o_{j}$ given by $\omega_{j}\in[-1,1]$.
Consequently, by applying Hoeffding’s inequality, we can prove the validity of
Theorem 1.
###### Proof of Theorem 1.
Since the single-shot measurement outcomes $\\{o_{j}\\}$ obtained from $N$
iterations of Algorithm 1 are individually bounded by the interval $[-1,1]$,
Hoeffding’s inequality states
$\mathrm{Prob}\left(\left\lvert\frac{1}{N}\sum_{j=1}^{N}o_{j}-\mathrm{tr}\left(OV\rho
V^{\dagger}\right)\right\rvert\geq\varepsilon\right)\leq
2\exp{\left(-\frac{N\varepsilon^{2}}{2}\right)}.$ (49)
For a fixed accuracy $\varepsilon\in(0,1)$ and confidence $\delta\in(0,1)$, it
is therefore sufficient to perform
$N\geq\frac{2\ln\left(2/\delta\right)}{\varepsilon^{2}}$ (50)
repetitions of Algorithm 1 to achieve the desired accuracy and confidence,
concluding the proof. ∎
Given recent developments, for example in Ref. [33], one might wonder whether
substantial improvements are possible by using a more refined estimator, most
notably median of means. Unfortunately, this is not very realistic. For
observables that obey $O^{2}=\mathbb{I}$, such as local and global Pauli
observables, the variance of a single-shot outcome $o_{j}\in\left[-1,1\right]$
becomes
$\mathrm{Var}\left[o_{j}\right]=1-\mathbb{E}\left[o_{j}\right]=\mathcal{O}(1)$.
In this case, the variance is of the same order as the magnitude and it is
impossible to (asymptotically) improve over Hoeffding’s inequality (asymptotic
normality) [34]. Now, we additionally assume that $V$ times the resolution
$\Xi$ approximates the time evolution operator $U$, i.e.,
$\lVert\Xi V-U\rVert\leq\hat{\varepsilon}\,.$ (51)
Furthermore, we will use the following result.
###### Lemma 1 (Closeness of expectation values).
Let $U$ be unitary and $V$ and $\Xi$ as given above such that Eq. (51) holds.
Furthermore, fix a state $\rho$ and observable $O$ with $\lVert O\rVert\leq
1$. Then,
$\left\lvert\mathrm{tr}\left(OU\rho
U^{\dagger}\right)-\Xi^{2}\mathrm{tr}\left(OV\rho
V^{\dagger}\right)\right\rvert\leq 3\hat{\varepsilon}\,.$ (52)
###### Proof.
We begin by defining $\widetilde{U}$ as the difference between the exact and
approximated time evolution
$\Xi V=\Xi\sum_{k=1}^{M}p_{k}V_{k}=U+\widetilde{U}.$ (53)
According to Eq. (51) we find that
$\lVert\widetilde{U}\rVert\leq\hat{\varepsilon}$. Consequently,
$\displaystyle\left\lvert\mathrm{tr}\left(OU\rho
U^{\dagger}\right)-\Xi^{2}\mathrm{tr}\left(OV\rho
V^{\dagger}\right)\right\rvert$
$\displaystyle=\left\lvert\mathrm{tr}\left(\widetilde{U}^{\dagger}OU\rho\right)+\mathrm{tr}\left(U^{\dagger}O\widetilde{U}\rho\right)+\mathrm{tr}\left(\widetilde{U}^{\dagger}O\widetilde{U}\rho\right)\right\rvert$
$\displaystyle\leq 2\lVert\widetilde{U}\rVert\lVert U\rVert\lVert
O\rVert+\lVert\widetilde{U}\rVert^{2}\lVert O\rVert$ (54) $\displaystyle\leq
2\hat{\varepsilon}+\hat{\varepsilon}^{2}$ (55) $\displaystyle\leq
3\hat{\varepsilon}.$ (56)
∎
By combining the insights from the previous discussion, we can now tackle
Theorem 2:
###### Proof of Theorem 2.
Add and subtract $\mathrm{tr}\left(OV\rho V^{\dagger}\right)$ to find
$\left\lvert\frac{\Xi^{2}}{N}\sum_{j=1}^{N}o_{j}-\mathrm{tr}\left(OU\rho
U^{\dagger}\right)\right\rvert\leq\Xi^{2}\left\lvert\frac{1}{N}\sum_{j=1}^{N}o_{j}-\mathrm{tr}\left(OV\rho
V^{\dagger}\right)\right\rvert+\left\lvert\vphantom{\sum_{j=1}^{N}}\Xi^{2}\mathrm{tr}\left(OV\rho
V^{\dagger}\right)-\mathrm{tr}\left(OU\rho U^{\dagger}\right)\right\rvert.$
(57)
By applying Theorem 1 with accuracy $\varepsilon/\Xi$ to the first term and
using Lemma 1 with $\varepsilon=3\hat{\varepsilon}$, we find that this upper
bound is given by $(1+\Xi)\varepsilon$. ∎
### IV.2 Construction of the new multi-product formulas
In Definitions 2 and 3, we propose two alternatives to Childs and Wiebe’s
multi-product formulas, which reduce the number of circuit evaluations
required for a randomized implementation and have improved error scaling. In
the following, we will motivate their construction and stress the advantages
they provide. First of all, note that every approximation of the exact time
evolution $\widetilde{U}(t)\approx\mathrm{e}^{-\mathrm{i}tH}$ can be written
as a sum of operators $\hat{A}_{k}$ such that
$\displaystyle\widetilde{U}(t)=\sum_{k=0}^{\infty}\hat{A}_{k}\,t^{k}\,,$ (58)
with $\hat{A}_{0}=\mathbb{I}$. For any approximated time evolution with an
error of $\mathcal{O}(t^{m+1})$, we find that $\hat{A}_{k}=(iH)^{k}/{k!}$ for
all $k\leq m$. In other words, the Taylor expansion of the exact time
evolution and its approximation has the same Taylor expansion for the first
$m$ non-trivial terms.
For the standard Trotter-Suzuki formula with $\widetilde{U}(t)=S_{2\chi}(t)$,
we have
$\hat{A}_{k}=(iH)^{k}/{k!}$ (59)
for all $k\leq 2\chi$. For $k>2\chi$, these operators $\hat{A}_{k}$ resemble
uncontrolled, erroneous operators. In Definition 1, we propose a superposition
of approximations with differently scaled times $S_{2\chi}(b_{n}t)$,
introducing controllable modulation parameters $\nu_{k}$ up to a Taylor order
of $2\chi R$. Its Taylor expansion is given by
$\displaystyle\mathcal{L}_{2\chi,R}(\boldsymbol{\nu},\boldsymbol{b},t)$
$\displaystyle=\sum_{q=0}^{2\chi
R}C_{q}(\boldsymbol{\nu},\boldsymbol{b})\,S_{2\chi}(b_{q}t)$ (60)
$\displaystyle=\sum_{k=0}^{\infty}\left(\sum_{q=0}^{2\chi
R}C^{\phantom{k}}_{q}\\!(\boldsymbol{\nu},\boldsymbol{b})\,b_{q}^{k}\right)\hat{A}_{k}t^{k}$
(61) $\displaystyle=\sum_{k=0}^{2\chi
R}\nu_{k}\,\hat{A}_{k}t^{k}\;+\;\mathcal{O}\left(t^{2\chi R+1}\right),$ (62)
where $C_{q},b_{q}\in\mathbb{R}$ and
$\nu_{k}=(\sum_{q}C^{\phantom{k}}_{q}b_{q}^{k})$ is fixed via the linear
transformation of the coefficients
$\boldsymbol{C}=(C_{0},C_{1},\dots,C_{2\chi R})^{\top}$
with the Vandermonde matrix $B_{j,k}=b_{k}^{j-1}$,
$B\boldsymbol{C}=\boldsymbol{\nu}$ as in Eq. (26). The condition number of the
Vandermonde matrix as the product of its Hilbert-Schmidt norm and the norm of
the pseudo-inverse can be bounded from above and below by explicit expressions
involving the vector defining the Vandermonde matrix [35]. Choosing the vector
$\boldsymbol{b}$, we can calculate the coefficients $\boldsymbol{C}$ for a
fixed solution vector $\boldsymbol{\nu}$ using the inverse Vandermonde matrix
$B^{-1}$, which is found exactly to be [36, 37]
$\displaystyle\left(B^{-1}\right)_{j,k}=\frac{(-1)^{k-1}}{\prod\limits_{m\,\in\,\boldsymbol{\mu}(j)}\left(b_{m}-b_{j}\right)}\sum_{\scriptsize\quad\begin{matrix}\boldsymbol{a}\in\mathbb{F}_{2}^{{}^{2\chi
R}}\\\ |\boldsymbol{a}|=2\chi R-k\end{matrix}}\prod_{i=0}^{2\chi
R-1}\left(b_{\mu_{i}(j)}\right)^{a_{i}}$ (63)
$\displaystyle\text{with}\quad\boldsymbol{\mu}(j)=(0,1,\ldots,j-1,j+1,\ldots,2\chi
R)\,,$
where the sum runs over all binary strings
$\boldsymbol{a}=(a_{0}\,a_{1}\,\dots\,a_{2\chi R-1})$ of length $2\chi R$ and
Hamming weight $2\chi R-k$. However, we have found that for numerical
purposes, a matrix inversion of $B$ clearly outmatches the analytical
computation of $B^{-1}$ in terms of runtime.
Using Eq. (62), we can now manipulate the Taylor expansion of a Trotter-Suzuki
block $S_{2\chi}(t)$ by replacing it with some
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu},\boldsymbol{b},t)$. The key insight
here is that while the operators $\hat{A}_{k}$ are only correct for Taylor
orders $k\leq 2\chi$, we gain control of the prefactors up to order $2\chi R$.
Consequently, we can eliminate all erroneous $\hat{A}_{k}$ for $2\chi<k\leq
2\chi R$ by setting the corresponding $\nu_{k}$ to zero. This allows us to
construct the matching and closed-form multi-product formulas using blocks of
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu},\boldsymbol{b},t)$ with different
$\boldsymbol{\nu}$ (and possibly $\boldsymbol{b}$).
#### IV.2.1 Matching multi-product formula
The first version of our proposed multi-product formula builds upon the
multiplication of $R$ multi-product building blocks
$\mathcal{L}_{2\chi,R}(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t)$ that
according to Eq. (62) can be written as
$\mathcal{L}_{2\chi,R}\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t\right)=\sum_{k=0}^{2\chi}\frac{\nu^{(r)}_{k}}{k!}\left(-\mathrm{i}Ht\right)^{k}+\sum_{k=2\chi+1}^{2\chi
R}\nu_{k}\hat{A}_{k}+\mathcal{O}\left(t^{2\chi R+1}\right)\,,$ (64)
where we can eliminate the second sum by setting $\nu_{k}=0$ for $2\chi<k\leq
2\chi R$. Their product now yields
$\prod_{r=1}^{R}\mathcal{L}_{2\chi,R}\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t\right)=\sum_{k=0}^{2\chi
R}\mu_{k}\left(-\mathrm{i}Ht\right)^{k}+\mathcal{O}(t^{2\chi R+1})$ (65)
with
$\mu_{k}\,=\sum_{i_{1}+i_{2}+\ldots+i_{R}=k}\frac{\nu_{i_{1}}^{(1)}\nu_{i_{2}}^{(2)}\cdots\nu_{i_{R}}^{(R)}}{i_{1}!i_{2}!\cdots
i_{R}!}\,.$ (66)
To mimic the exact time evolution up to $\mathcal{O}(t^{2\chi R+1})$, we
require $\mu_{k}=1/k!$, leading to Definition 2.
#### IV.2.2 Closed-form multi-product formula
The closed-form version of the presented multi-product formula sums products
of building blocks $\mathcal{L}$ . To motivate the specific construction, it
is useful to take a look at the specific case of $2\chi=4$ and $R=3$ for some
choice of $\boldsymbol{b}$ and $t$ using the shorthand
$\mathcal{L}\left(\begin{smallmatrix}\nu_{0}\\\ :\\\ \nu_{2\chi
R}\end{smallmatrix}\right):=\mathcal{L}(\boldsymbol{\nu},\boldsymbol{b},t),$
(67)
to get
$\widetilde{M}^{(\mathrm{cf})}_{4,3}=\mathcal{L}\left(\begin{matrix}1\\\
\boldsymbol{1}\\\ \boldsymbol{1}\\\ \boldsymbol{1}\\\ \boldsymbol{1}\\\ 0\\\
0\\\ \vdots\end{matrix}\right)+4!\,\mathcal{L}\left(\begin{matrix}0\\\
\boldsymbol{0}\\\ \boldsymbol{0}\\\ \boldsymbol{0}\\\ \boldsymbol{1}\\\ 0\\\
0\\\ \vdots\end{matrix}\right)\mathcal{L}\left(\begin{matrix}0\\\
\boldsymbol{1!/5!}\\\ \boldsymbol{2!/6!}\\\ \boldsymbol{3!/7!}\\\
\boldsymbol{4!/8!}\\\ 0\\\ 0\\\
\vdots\end{matrix}\right)+(4!)^{2}\mathcal{L}\left(\begin{matrix}0\\\
\boldsymbol{0}\\\ \boldsymbol{0}\\\ \boldsymbol{0}\\\ \boldsymbol{1}\\\ 0\\\
0\\\ \vdots\end{matrix}\right)\mathcal{L}\left(\begin{matrix}0\\\
\boldsymbol{0}\\\ \boldsymbol{0}\\\ \boldsymbol{0}\\\ \boldsymbol{1}\\\ 0\\\
0\\\ \vdots\end{matrix}\right)\mathcal{L}\left(\begin{matrix}0\\\
\boldsymbol{1!/9!}\\\ \boldsymbol{2!/10!}\\\ \boldsymbol{3!/11!}\\\
\boldsymbol{4!/12!}\\\ 0\\\ 0\\\ \vdots\end{matrix}\right)\,,$ (68)
where the orders $1,\dots,2\chi$ have been visually highlighted for clarity.
In this example the eleventh order in $t$ is revealed by multiplying the
corresponding terms from the Taylor expansion of $\mathcal{L}$ in Eq. (60),
with the corresponding coefficients in (68)
$(4!)^{2}\times\frac{(-\mathrm{i}Ht)^{4}}{4!}\times\frac{(-\mathrm{i}Ht)^{4}}{4!}\times
3!/11!\times\frac{(-\mathrm{i}Ht)^{3}}{3!}=\frac{(-\mathrm{i}Ht)^{11}}{11!}\,.$
(69)
The first term of Eq. (68) takes care of the zeroth and first $2\chi$ order of
$U$. The second term multiplies all terms of orders $t^{1}\dots t^{2\chi}$
with $t^{2\chi}$ and so takes care of the next $2\chi$ terms of the expansion.
The third term multiplies with a $\mathcal{O}(t^{2\chi})$ term twice, taking
care of the subsequent $2\chi$ terms – a pattern emerges. In a general setting
(with arbitrary $2\chi$ and $R$) we can write this sum as
$\displaystyle\widetilde{M}_{2\chi,R}^{(\mathrm{cf})}(t)$
$\displaystyle\coloneqq\sum_{r=1}^{R}\left(\mathcal{L}_{2\chi,R}\left(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)},t\right)\right)^{r-1}\mathcal{L}_{2\chi,R}\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t\right)$
$\displaystyle=\sum_{r=1}^{R}\left(\sum_{j=0}^{2\chi
R}\nu_{j}^{(0)}\frac{\left(-\mathrm{i}Ht\right)^{j}}{j!}+\mathcal{O}\left(t^{2\chi
R+1}\right)\right)^{r-1}\left(\sum_{k=0}^{2\chi
R}\nu_{k}^{(r)}\frac{\left(-\mathrm{i}Ht\right)^{k}}{k!}+\mathcal{O}\left(t^{2\chi
R+1}\right)\right)\,$ (70)
$\displaystyle=\sum_{r=1}^{R}\left(\frac{(-\mathrm{i}Ht)^{2\chi}}{(2\chi)!}\right)^{r-1}\sum_{k=0}^{2\chi
R}\nu_{k}^{(r)}\frac{\left(-\mathrm{i}Ht\right)^{k}}{k!}\;+\;\mathcal{O}\left(t^{2\chi
R+1}\right)$ (71)
where we have used Eq. (62) in Eq. (70) and $\nu_{j}^{(0)}=\delta_{2\chi,j}$
from Definition 3 to collapse the first factor into
$(-\mathrm{i}Ht)^{2\chi(r-1)}/(2\chi)!$ in Eq. (71). Considering also that
$\nu_{j}^{(0)}=1$ for $0\leq j\leq 2\chi$, the $r=1$ term (that we recognize
are the first $2\chi$ orders of the time evolution) can be separated from the
sum. Discarding all terms that vanish due to $\nu_{k}^{(r)}=0$ we rewrite Eq.
(71) to
$\displaystyle\widetilde{M}_{2\chi,R}^{(\mathrm{cf})}(t)\;$
$\displaystyle=\;\sum_{j=0}^{2\chi}\frac{(-\mathrm{i}Ht)^{j}}{j!}\;+\;\sum_{r=2}^{R}\sum_{k=1}^{2\chi}\nu_{k}^{(r)}\frac{\left(-\mathrm{i}Ht\right)^{2\chi(r-1)+k}}{\left((2\chi)!\right)^{r-1}k!}\;+\;\mathcal{O}\left(t^{2\chi
R+1}\right)\,,$ (72)
for which we consult Definition 3, a last time resolving the remaining
$\nu_{k}^{(r)}$. This leaves us with the correct time evolution up to order
$2\chi R+1$, thus proving the definition.
### IV.3 Error of the averaged operators
Following upon the insights from Section IV.2 we find Corollary 2 already
proven by Eq. (64) and Eq. (72) as long as $\widetilde{M}_{2\chi,R}$ is
constructed according to Definition 2 or 3. The proof of Theorem 3 requires a
more involved error analysis.
###### Proof of Theorem 3.
We first need to bound the remainder terms of the Taylor series expansions of
$U(t)$ and $\widetilde{M}^{(\mathrm{m})}$. In the following,
$\mathrm{\textbf{R}}_{\ell}(f)$ denotes the remainder term of the Taylor
series of an operator-valued function $f$ truncated at $\ell$-th order in $t$.
We thus find that
$\displaystyle\left\lVert\vphantom{\prod_{r}^{R}}U(t)-\widetilde{M}^{(\mathrm{m})}_{2\chi,R}\right\rVert$
$\displaystyle=\left\lVert\mathrm{e}^{-\mathrm{i}tH}-\prod_{r=1}^{R}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\,S_{2\chi}\left(b^{(r)}_{q}t\right)\right]\right\rVert$
$\displaystyle\leq\left\lVert\vphantom{\mathrm{\textbf{R}}_{2\chi
R}\left(\prod_{r=1}^{R}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\,S_{2\chi}\left(b^{(r)}_{q}t\right)\right]\right)}\mathrm{\textbf{R}}_{2\chi
R}\left(\mathrm{e}^{-\mathrm{i}tH}\right)\right\rVert+\left\lVert\mathrm{\textbf{R}}_{2\chi
R}\left(\prod_{r=1}^{R}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\,S_{2\chi}\left(b^{(r)}_{q}t\right)\right]\right)\right\rVert.$
(73)
Moving forward, we employ some recently established results on the theory of
Trotter errors. Specifically, we make use of the ‘Trotter error with 1-norm
scaling’ lemma of Ref. [11]. Building upon these insights in this fresh
context, we find that the exponential remainder of a product formula such as
in Eq. (6) can be bounded by
$\displaystyle\left\lVert\mathrm{\textbf{R}}_{\ell}\left(\prod_{j=1}^{N}\mathrm{e}^{-\mathrm{i}\alpha_{j}h_{k_{j}}t}\right)\right\rVert$
$\displaystyle=\frac{t^{\ell+1}}{(\ell+1)!}\left\lVert\left(\frac{\partial}{\partial
t}\right)^{\ell}\prod_{j=1}^{N}\mathrm{e}^{-\mathrm{i}\alpha_{j}h_{k_{j}}t}\right\rVert$
(74)
$\displaystyle=\frac{t^{\ell+1}}{(\ell+1)!}\left\lVert\sum_{x_{1}+\dots+x_{N}=\ell+1}\frac{(\ell+1)!}{x_{1}!\cdots
x_{N}!}\prod_{j=1}^{N}\left(\frac{\partial}{\partial
t}\right)^{x_{n}}\mathrm{e}^{-\mathrm{i}\alpha_{j}h_{k_{j}}t}\right\rVert$
$\displaystyle=\frac{t^{\ell+1}}{(\ell+1)!}\left\lVert\sum_{x_{1}+\dots+x_{N}=\ell+1}\frac{(\ell+1)!}{x_{1}!\cdots
x_{N}!}\prod_{j=1}^{N}\left(-\mathrm{i}\alpha_{j}\,h_{k_{j}}\,t\right)^{x_{j}}\mathrm{e}^{-\mathrm{i}\alpha_{j}h_{k_{j}}t}\right\rVert$
$\displaystyle\leq\frac{t^{\ell+1}}{(\ell+1)!}\sum_{x_{1}+\dots+x_{N}=\ell+1}\frac{(\ell+1)!}{x_{1}!\cdots
x_{N}!}\prod_{j=1}^{N}\left\lVert\alpha_{j}\,h_{k_{j}}\,t\right\rVert^{x_{j}}\cdot\underbrace{\left\lVert\mathrm{e}^{-\mathrm{i}\alpha_{j}h_{k_{j}}t}\right\rVert}_{=1}$
$\displaystyle=\frac{\left(\sum_{j=1}^{N}\left\lVert\alpha_{j}\,h_{k_{j}}\right\rVert
t\right)^{\ell+1}}{(\ell+1)!}\,.$
This bound can now be applied to (73). For the first term we obtain
$\left\lVert\mathrm{\textbf{R}}_{2\chi
R}\left(\mathrm{e}^{-\mathrm{i}tH}\right)\right\rVert\leq\frac{\left(\Lambda
t\right)^{2\chi R+1}}{(2\chi R+1)!},$ (75)
while the second term can be bounded by
$\displaystyle\left\lVert\mathrm{\textbf{R}}_{2\chi
R}\left(\prod_{r=1}^{R}\left[\sum_{q=0}^{2\chi
R}C_{q}^{(r)}S_{2\chi}\left(b^{(r)}_{q}t\right)\right]\right)\right\rVert$
$\displaystyle\leq\sum_{q_{1}=0}^{2\chi R}\cdots\sum_{q_{r}=0}^{2\chi
R}\left\lvert C_{q_{1}}^{(1)}\cdots
C_{q_{r}}^{(r)}\right\rvert\left\lVert\mathrm{\textbf{R}}_{2\chi
R}\left(S_{2\chi}\left(b^{(1)}_{q_{1}}t\right)\cdots
S_{2\chi}\left(b^{(r)}_{q_{r}}t\right)\right)\right\rVert$
$\displaystyle\leq\frac{\left(g_{\chi}\Lambda t\right)^{2\chi R+1}}{(2\chi
R+1)!}\sum_{q_{1}=0}^{2\chi R}\cdots\sum_{q_{r}=0}^{2\chi R}\left\lvert
C_{q_{1}}^{(1)}\cdots C_{q_{r}}^{(r)}\right\rvert\left(\lvert
b_{q_{1}}^{(1)}\rvert+\ldots+\lvert b_{q_{r}}^{(r)}\rvert\right),$ (76)
where we have introduced the factor $g_{\chi}$ from the Trotter-Suzuki
decomposition [13], which is defined as
$g_{\chi}\coloneqq\frac{4\chi}{5}\left(\frac{5}{3}\right)^{\chi-1}.$ (77)
At the same time, we have defined $\Lambda\coloneqq\sum_{k}\lVert
h_{k}\rVert$. Consequently, we can bound the error of the matching multi-
product formula via
$\left\lVert
U(t)-\widetilde{M}^{(\mathrm{m})}_{2\chi,R}(t)\right\rVert\leq\left(1+\zeta^{(\mathrm{m})}g_{\chi}^{2\chi
R+1}\right)\frac{\left(\Lambda t\right)^{2\chi R+1}}{\left(2\chi
R+1\right)!}\,,$ (78)
with
$\zeta^{(\mathrm{m})}\coloneqq\sum_{q_{1}=0}^{2\chi R}\sum_{q_{2}=0}^{2\chi
R}\cdots\sum_{q_{r}=0}^{2\chi R}\left\lvert
C_{q_{1}}^{(1)}C_{q_{2}}^{(2)}\cdots C_{q_{r}}^{(r)}\right\rvert\left(\lvert
b_{q_{1}}^{(1)}\rvert+\ldots+\lvert b_{q_{r}}^{(r)}\rvert\right),$ (79)
concluding the proof.
The error analysis of the closed-form multi-product formula follows along
similar lines: Again, we bound the remainder terms of the Taylor series
expansions of $U(t)$ and $\widetilde{M}^{(\mathrm{cf})}$, finding
$\displaystyle\left\lVert
U(t)-\widetilde{M}^{(\mathrm{cf})}_{2\chi,R}\right\rVert=\left\lVert\mathrm{e}^{-\mathrm{i}tH}-\sum_{r=1}^{R}\left(\mathcal{L}_{2\chi,R}\left(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)},t\right)\right)^{r-1}\mathcal{L}_{2\chi,R}\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)},t\right)\right\rVert$
$\displaystyle\leq\left\lVert\vphantom{\mathrm{\textbf{R}}_{2\chi
R}\left(\sum_{r=1}^{R}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\,S_{2\chi}\left(b^{(r)}_{q}t\right)\right]^{r-1}\left[\sum_{q=0}^{2\chi
R}C_{q}^{(r)}\,S_{2\chi}\left(b_{q}t\right)\right]\right)}\mathrm{\textbf{R}}_{2\chi
R}\left(\mathrm{e}^{-\mathrm{i}tH}\right)\right\rVert$
$\displaystyle\quad+\;\left\lVert\mathrm{\textbf{R}}_{2\chi
R}\left(\sum_{r=1}^{R}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)}\\!\right)\,S_{2\chi}\\!\left(b_{q}^{(0)}t\right)\right]^{r-1}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\,\,S_{2\chi}\\!\left(b_{q}^{(r)}t\right)\right]\right)\right\rVert\,,$
(80)
where the second term can be bounded by
$\displaystyle\left\lVert\mathrm{\textbf{R}}_{2\chi
R}\left(\sum_{r=1}^{R}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)}\\!\right)\,S_{2\chi}\\!\left(b_{q}^{(0)}t\right)\right]^{r-1}\left[\sum_{q=0}^{2\chi
R}C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\\!\right)\,\,S_{2\chi}\\!\left(b_{q}^{(r)}t\right)\right]\right)\right\rVert$
$\displaystyle\;\leq\;\sum_{r=1}^{R}\sum_{q_{1}=0}^{2\chi
R}\cdots\sum_{q_{r}=0}^{2\chi R}\left\lvert
C_{q_{1}}(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)})\cdots
C_{q_{r-1}}(\boldsymbol{\nu}^{(0)},\boldsymbol{b}^{(0)})\,C_{q_{r}}(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)})\right\rvert$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\left\lVert\mathrm{\textbf{R}}_{2\chi
R}\left(S_{2\chi}\left(b_{q_{1}}^{(0)}\right)\cdots
S_{2\chi}\left(b_{q_{r-1}}^{(0)}\right)\,S_{2\chi}\left(b_{q_{r}}^{(r)}\right)\right)\right\rVert$
(81)
Defining
$\displaystyle\zeta^{(\mathrm{cf})}$ $\displaystyle\coloneqq$
$\displaystyle\sum_{r=1}^{R}\sum_{q_{1}=0}^{2\chi
R}\sum_{q_{2}=0}\cdots\sum_{q_{r}=0}^{2\chi R}\left\lvert
C_{q_{1}}^{(0)}C_{q_{2}}^{(0)}\cdots
C_{q_{r-1}}^{(0)}C_{q_{r}}^{(r)}\right\rvert\left(\lvert
b_{q_{1}}^{(0)}\rvert+\ldots+\lvert b_{q_{r-1}}^{(0)}\rvert+\lvert
b_{q_{r}}^{(r)}\rvert\right),$ (82) $\displaystyle\Lambda$
$\displaystyle\coloneqq$ $\displaystyle\sum_{k}\lVert h_{k}\rVert\,,$ (83)
we find
$\left\lVert
U(t)-\widetilde{M}^{(\mathrm{cf})}_{2\chi,R}(t)\right\rVert\leq\left(1+\zeta^{(\mathrm{cf})}g_{\chi}^{2\chi
R+1}\right)\frac{\left(\Lambda t\right)^{2\chi R+1}}{\left(2\chi
R+1\right)!}\,,$ (84)
concluding the proof. ∎
## V Comparison and numerical validation
Algorithm | Trotter-Suzuki [26] | Childs & Wiebe [13] | Novel multi-product formulas [here]
---|---|---|---
Formula | $S_{2\chi}(t/r)^{r}$ | $\sum_{q=1}^{K+1}C_{q}S_{2\chi}(t/q)^{q}$ | $\prod_{r=1}^{R}\sum_{q=0}^{2\chi R}C_{q}\\!\left(\boldsymbol{\nu}^{(r)},\boldsymbol{b}^{(r)}\right)\;S_{2\chi}\\!\left(b_{q}^{(r)}t\right)$
Max. depth | $r\cdot 2\cdot 5^{\chi-1}$ | $(K+1)\cdot 2\cdot 5^{\chi-1}$ | $R\cdot 2\cdot 5^{\chi-1}$
Error scaling | $\mathcal{O}((t/r)^{2\chi+1})$ | $\mathcal{O}(t^{2(\chi+K)+1})$ | $\mathcal{O}(t^{2\chi R+1})$
Error bound | $2\frac{\left(g_{\chi}\Lambda t/r\right)^{2\chi+1}}{(2\chi+1)!}$ | | $\left(1+g_{\chi}^{2(\chi+K)+1}\sum\limits_{q}\left|C_{q}\right|\right)$
---
$\times\;\frac{\left(\Lambda t\right)^{2(\chi+K)+1}}{(2(\chi+K)+1)!}$
$\left(1+g_{\chi}^{2\chi R+1}\zeta\right)\frac{\left(\Lambda t\right)^{2\chi
R+1}}{\left(2\chi R+1\right)!}$
Table 1: Comparison of the standard Trotter-Suzuki formula, the multi-product
formula used by Childs and Wiebe and the closed-form / matching multi-product
formula introduced in this work. Here, $g_{\chi}=(5/3)^{\chi-1}4\chi/5$
originates in the exponential tail of the Trotter-Suzuki terms and
$\Lambda=\sum_{k}\left\lVert h_{k}\right\rVert$. The formula dependent
$\zeta^{(\mathrm{m})}$ and $\zeta^{(\mathrm{cf})}$ are defined and derived in
equations (79) and (82), respectively, and depend on the used $C_{q}$ and
$b_{q}$. The maximum circuit depth is given for their implementation using the
randomized sampling framework introduced in Algorithm 1. Figure 4: Comparing
error bounds for Trotter-Suzuki product formulas as well as several multi-
product formulas for different $\tau=\Lambda t$. The formulas for these bounds
are given in Table 1. We have chosen $2\chi=4$ and $r=R=K+1=3$ to fix the
depth of all methods to be 30 oracle calls. The parameters
$\\{\boldsymbol{b}^{(r)}\\}$ have been numerically optimized with an initial
guess of $\boldsymbol{b}=\set{1,-1,2,-2,\ldots,7}$. We find the resolution
factors for the three multi-product formulas to be
$\Xi^{(\mathrm{CW})}\\!\approx 3.13$, $\Xi^{(\mathrm{m})}\\!\approx 1.22$ and
$\Xi^{(\mathrm{cf})}\\!\approx 1.36$. In the black box optimization, we have
used objective functions $\propto(\Xi^{\mathrm{(m)}})^{20}$ and
$\propto(\Xi^{\mathrm{(cf)}})^{10}$ to ensure reasonable resolution factors.
We will now compare the Trotter-Suzuki product formula algorithm and multi-
product formulas in the randomized sampling framework. To make comparisons as
fair as possible, we will assume that each algorithm uses at most $R$-fold
sequences of $S_{2\chi}(\cdot)$ blocks. Product formulas may use repetitions
as in Eq. (7) achieving an error of $\mathcal{O}((t/R)^{2\chi+1})$, which is a
reliable way to approximate time evolutions for longer times $R\gg\tau>1$.
However, repetitions improve the accuracy of shorter time evolutions only
minimally. The situation is different if $R$ is an integer power of five,
which allows us to build the next higher Trotter-Suzuki order and approximate
the time evolution up to a leading order of $2(\chi+\log_{5}R)+1$ in $t$. This
means improving the leading power of $t$ comes at an exponential cost for the
circuit depth. The situation can be remedied by sampling from a multi-product
formula. Remarkably, the statement for sampling observable eigenvalues with
product formulas is very similar to Theorem 2.
This allows us to disregard the sampling error in the comparison, and only
think about asymptotic limits. The main difference between product and multi-
product formulas is that Trotter-Suzuki algorithms do not have a resolution
factor. This is equivalent to $\Xi=1$ with consequences for sampling
complexity and error bounds. This advantage is, however, quickly outweighed as
Childs and Wiebe’s product formula clearly delivers an improved approximation
that is exact up to a leading order of $2(\chi+R)-1$ in $t$. Modifying the
leading power of $t$ is now possible by adding exponentially fewer terms. Note
that they require the same circuit depth for $R=K+1$ in the randomized
sampling framework due to the final term in Eq. (12). With matching and
closed-form multi-product formulas, the approximation can be further improved
to leading order $2\chi R+1$ in $t$. This means that only one additional
$S_{2\chi}(\cdot)$ block improves the order by $2\chi$ rather than $2$ as for
multi-product formulas of the prior art. While the scaling in $t$ of the novel
multi-product formulas is superior to those of Childs and Wiebe, their
theoretical error bound is not as tight, leading to a crossover of error
bounds at $\sum|h_{k}|t<1$.
We have plotted error bounds of all formulas for fixed circuit depth between
all methods with respect to one particular optimization of the multi-product
formula parameters in Fig. 4. Optimizing the set of parameters
$\\{\boldsymbol{b}^{(r)}\\}$, we tend to achieve noticeably lower resolution
factors than obtained using Childs and Wiebe’s multi-product formula. We
generally find the matching multi-product formula to have a smaller resolution
factor than the closed-form multi-product formula. While the matching multi-
product formula has a better resolution, it requires an additional layer of
classical optimization.
It is important to note that since we did not find useful bounds relating
$\boldsymbol{b}$ to any of the resolution factors, further improvements of the
bounds in Fig. 4 seem possible. In the absence of relations $\Xi(b)$, it is
necessary to numerically optimize all $\boldsymbol{b}^{(r)}$ with respect to a
chosen loss function. We found that global optimization using basin hopping
combined with Nelder-Mead optimization yields the best results, since the
optimization landscape exhibits a large number of local minima. The
optimization is also sensitive to the initial guess for those parameters, with
an equal spread of positive and negative integers, i.e.,
$\boldsymbol{b}_{\text{init}}=(1,-1,2,-2,3,-3,\ldots,\chi R+1)$, leading to
better results. It is also fruitful to vary the loss function used for
optimization. While using the error bound for a fixed $\tau$ yields the lowest
error, the corresponding resolution factors are too large to be practical.
Amplifying the impact of the resolution factor by using $\Xi^{p}$ for some
power $p$ in the loss function yielded better results, though other useful
loss functions could be explored further. We have also found that bounding
each parameter $b$ by the total maximum to lead to more robust results.
To corroborate the functioning of the newly proposed multi-product formulas
beyond theoretical bounds, we also compare the actual performance with that of
the conventional multi-product formula and Trotter-Suzuki. Here, we compare
the actual operator distance to the ideal time evolution for the following
five physically plausible and meaningful Hamiltonians, with the results shown
in Fig. 5:
Figure 5: Numerical comparison of Trotter-Suzuki product formulas, Childs and
Wiebe’s multi-product formula and the multi-product formulas proposed in this
work for different $\tau=\sum_{k}||h_{k}||t$. Here, we approximate the time
evolution operator and plot the operator distance between the approximation
and the ideal evolution operator for a number of physically plausible and
interesting local, but not necessarily geometrically local, Hamiltonians,
defined in equations (87), (86), (88) and (89). The simulated distances
(thick, dashed lines) are much smaller than the theoretical bounds (thin,
solid lines), but do have remarkably similar features overall.
First, we consider a standard _Heisenberg Hamiltonian_ with periodic boundary
conditions, described by
$H_{\mathrm{Heisenberg}}=-\sum_{\langle
i,j\rangle}\left(X_{i}X_{j}+Y_{i}Y_{j}+Z_{i}Z_{j}\right)+2\sum_{i}X_{i}.$ (85)
This is a Hamiltonian that plays an important role in condensed matter
physics, as a prototypical model capturing ferromagnetism. Trotter-Suzuki and
Childs-Wiebe formulas perform extremely well, as the Hamiltonian comprises
many commuting terms. Since their performance guarantees rely on nested
commutators [11], this behavior is to be expected and demonstrates the
superior performance of these well studies formulas for lattice Hamiltonians,
which is not reached by the newly proposed multi-product formulas. However,
they are less optimal for Hamiltonians with fewer commuting terms. Motivated
by these findings, we now turn to investigating a Hamiltonian comprising
mutually anti-commuting terms, defined as
$H_{\mathrm{anti}}=\sum_{i=0}^{6}Z^{\otimes i}\otimes(X+Y)+Z^{\otimes 7}.$
(86)
Anti-commuting terms play an important role when using quantum simulation
algorithms to investigate _fermionic quantum models_. Here, we find a notable
and in fact significant advantage of the newly proposed multi-product formulas
over Trotter-Suzuki and Childs-Wiebe already for quite large parameters of
$\tau=\sum|h_{k}|t$. Consequently, we expect our formulas to work well and
exceed previous methods for fermionic systems, which once transformed into
qubit Hamiltonians by virtue of the Jordan-Wigner transformation, will have
fewer commuting terms than standard lattice spin Hamiltonians.
This behavior is confirmed and further corroborated by our results of the
_Sachdev–Ye–Kitaev (SYK)_ model as defined in Ref. [38], whose Hamiltonian is
given by
$H_{\mathrm{SYK}}=\frac{1}{4\cdot
4!}\sum_{p,q,r,s=0}^{N-1}J_{p,q,r,s}\gamma_{p}\gamma_{q}\gamma_{r}\gamma_{s},$
(87)
where $N$ is the number of Majorana fermion mode operators $\gamma_{p}$ and
the $J_{p,q,r,s}$ are real-valued scalars drawn randomly from a normal
distribution with variance $\sigma^{2}=3!/N^{3}$. This is an intricate local,
but not geometrically local, model that is believed to provide insights into
instances of strongly correlated quantum materials. It is used in the study of
scrambling dynamics and has a close relation with discrete models that capture
aspects of holography in the black hole context.
Again invoking the Jordan-Wigner transformation, $N$ Majorana fermion mode
operators can be mapped onto $N/2$ qubits. For our simulations, we thus chose
$N=10$ and $N=14$, leading to a five and seven qubit model, respectively.
Here, we again find a notable and in instances substantial advantage for large
$\tau$ and even for large times $t$. This is a physically highly plausible and
interesting model for which our new simulation methods fare well.
As a further fermionic system, we look at the spinful Hubbard model on two by
two sites, defined as [39]
$H_{\mathrm{Hub}}=-t\sum_{\langle
i,j\rangle,\sigma}(a^{\dagger}_{i,\sigma}a_{j,\sigma}+a^{\dagger}_{j,\sigma}a_{i,\sigma})+U\sum_{i}a^{\dagger}_{i,\uparrow}a_{i,\uparrow}a^{\dagger}_{i,\downarrow}a_{i,\downarrow}-\mu\sum_{i}\sum_{\sigma}a^{\dagger}_{i,\sigma}a_{i,\sigma}-h\sum_{i}(a^{\dagger}_{i,\uparrow}a_{i,\uparrow}-a^{\dagger}_{i,\downarrow}a_{i,\downarrow}),$
(88)
with spin $\sigma$, tunneling amplitude $t=2$, Coulomb potential $U=2$,
magnetic field $h=0.5$ and chemical potential $\mu=0.25$. Also, $a$ and
$a^{\dagger}$ represent fermionic annihilation and creation operators. These
are comparably small system sizes, but already show the substantial potential
of the proposed simulation method.
Finally, as a last family of examples, in order to gauge the performance of
our methods for larger system sizes, we investigate a system of 200 non-
interacting (“free”) fermions with nearest neighbor interactions and periodic
boundary conditions
$H_{\mathrm{ff}}(h)=\sum_{i,j}h_{i,j}a^{\dagger}_{i}a^{\dagger}_{j},$ (89)
with $h_{i,j}=1$ if the respective fermions are nearest neighbors and
$h_{i,j}=0$ otherwise. Note that for gauging the performance for free
fermions, we do not compare the time evolution operator
$U(t)=\mathrm{e}^{-\mathrm{i}Ht}$ to its approximation, but the Greens
function propagator $G(t)=\mathrm{e}^{-\mathrm{i}ht}$ to its approximation.
All comparisons are done in the randomized sampling framework for a Trotter-
Suzuki order of $2\chi=4$, the number of repetitions $R=3$ and corresponding
parameters for all other algorithms, ensuring an equal depth measured in the
number of the required oracle calls. Although the straightforward comparison
of their bounds in Fig. 4 suggests an advantage of our multi-product formulas
at about $\tau=0.1$, we find that this is the case for much larger $\tau$
already on actual systems as shown in Fig. 5. Note also that since the
Hamiltonians we consider are not necessarily geometrically local, known
classical simulation techniques will be heavily challenged even for comparably
short simulation times
While the performance of the simple multi-product formula and Trotter-Suzuki
is also much better than their bounds indicate, we find that the presented
bounds for our proposed multi-product formulas are comparably looser. It is
also worth to note that performance of the matching version is slightly better
than that of the closed-form version, although it comes with the penalty of an
additional, classical optimization loop. We also find that advantages of our
multi-product formulas are visible even for $\tau>1$; in the case of the SYK
model, this holds true even for actual simulation times $t>1$. Additionally,
we find that this advantage can also be maintained for larger system sizes, as
indicated by their performance on the model of free fermions. The presented
numerical studies therefore provide strong arguments for the functioning of
our proposed multi-product formulas and their advantage in the presented
regimes. It is also important to note that the corresponding resolution
factors required for the randomized sampling scheme of
$\Xi^{\mathrm{(cf)}}\approx 1.36$ and $\Xi^{\mathrm{(m)}}\approx 1.22$ are
significantly better than the $\Xi^{\mathrm{(CW)}}\approx 3.1$ of the multi-
product formula proposed by Childs and Wiebe in Ref. [13].
## VI Discussion and conclusion
In this work, we have brought together two main ingredients of methods of
quantum simulation. The results are substantially more resource efficient ways
of performing short-time Hamiltonian simulation. These are on the one hand
higher-order multi-product formulas [13, 14], on the other an element which
has long been underappreciated but recently been of high interest: This is the
element of _randomness_ [20, 21, 22, 23, 24, 24]. Overcoming the prejudice
that the time evolution has to be completed in each run of a compilation, we
have introduced a novel framework for implementing multi-product formulas,
with the goal of estimating expectation values of time evolved observables.
Concretely, we have proposed a randomized sampling approach that focuses on
the time evolution of the observable, not the state. When implementing multi-
product formulas in a randomized fashion rather than via block encodings in
the LCU framework, we can circumvent the need for additional amplitude
amplification or post-selection. The results presented here have been obtained
by only requiring access to a quantum-oracle machine which implements single-
qubit state preparation, controlled time evolution and quantum measurements.
They are thus especially relevant for pre-digital settings, in which NISQ
algorithms reach their limits but where full-fledged, digital, long-time
evolution algorithms are not yet available. Consequently, this work may be
seen as targeting a regime in between the digital and analog setting, where we
have some form of parametric control over a simulator system allowing us to
compile the target time evolution with sequences of the simulator’s time
evolutions [40, 41]. This programmable regime then constitutes a departure
from the analog setting with relatively little control over the simulator, and
is not as strict as digital simulation, where the control over the quantum
system is strong enough to fashion its interactions into quantum gates.
Within this randomized sampling framework, we have proposed two new multi-
product formulas. These schemes have been equipped with full rigorous
performance guarantees. Furthermore, we have included a detailed estimation of
the number of circuit evaluations that are required, a vital metric for
randomized approaches. Comparing error bounds of these newly introduced
algorithms with Trotter-Suzuki product formulas and Childs and Wiebe’s multi-
product formula, we find that they outperform the latter for a fixed circuit
depth in a practically relevant regime. While their performance for long
simulation times would scale exponential in time, the base of this
exponential, the resolution factor in our case, while always strictly larger
than one, could be optimized further and even for our toy example was in the
range of $1.2-1.4$. Also, since we do not solely consider geometrically local
Hamiltonians, even comparably short simulation times are a highly difficult
task for known classical simulation techniques.
Benchmarking them on five different Hamiltonians, all of which physically well
motivated and each interesting in its own right, we have found that this
advantage can be expected already at comparably large simulation times. While
lattice Hamiltonians with many commuting terms are most likely best
approximated using Trotter-Suzuki or Childs-Wiebe formulas, the newly proposed
multi-product formulas show a clear advantage for fermionic Hamiltonians and
those with a small number of commuting term. This insight points to the
direction that there might not be a universally optimal quantum simulation
algorithm for digital quantum simulation. Instead, some algorithms could be
better suited to capture the specifics of a given local Hamiltonian model. The
downside of the proposed methods is that more measurements are required to
reach the desired precision through the resolution factor. This resolution
factor can be optimized using a classical black-box optimization, which is de-
facto a requirement for the functioning of the proposed multi-product
formulas.
The present work is essentially bridging the gap between analog and (perhaps
error-corrected), fully digital quantum technology: Not only do we expect
there to be other randomized sampling schemes in digital quantum simulation,
but once one is able to replace the element of randomness with block
encodings, one can switch from these expectation value based algorithm to
algorithms based on quantum phase estimation. Overall, the method introduced
gives rise to a less resource demanding way of performing Hamiltonian
simulation, while also remaining conceptually and technologically simpler than
for instance qubitization [18], bringing such ideas to an extent closer to the
realm of near-term quantum computing.
Looking ahead, it remains an open problem to relate the parameters
$\boldsymbol{b}$ in Definition 1 to the resolution factor in a way that would
eliminate the need for black-box optimization. So far, we can only connect the
two quantities analytically, and that involves the complicated product with
the Vandermonde matrix (26). Alternatively, one could improve the optimization
rather than replacing it. We have used only simple optimizers and loss
functions, and expect possible improvements for more involved loss functions
and optimization algorithms. Furthermore, the presented constructions are just
two of the plethora of new multi-product formulas that could be constructed
with Definition 1 and might exhibit better error bounds and resolution.
## VII Acknowledgments
This work has been supported by the DFG (CRC 183 project B01 and A03, EI
519/21-1). This work has also received funding from the European Unions
Horizon 2020 research and innovation program under grant agreement No. 817482
(PASQuanS), specifically dedicated to programmable quantum simulators. It has
also been supported by the BMWi (PlanQK) and the BMBF (DAQC on notions of
digital-analog quantum simulation and FermiQP on fermionic quantum
processors).
M. K. acknowledges funding from ARC Centre of Excellence for Quantum
Computation and Communication Technology (CQC2T), project number CE170100012.
The authors endorse Scientific CO2nduct [42] and provide a CO2 emission table
in the appendix.
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## Appendix A CO2 emission table
Numerical simulations |
---|---
Total Kernel Hours [$\mathrm{h}$] | $\approx 800$
Thermal Design Power Per Kernel [$\mathrm{W}$] | 5.75
Total Energy Consumption Simulations [$\mathrm{kWh}$] | 4.6
Average Emission Of CO2 In Germany [$\mathrm{kg/kWh}$] | 0.56
Total CO2-Emission For Numerical Simulations [$\mathrm{kg}$] | 2.6
Were The Emissions Offset? | Yes
Transport |
Total CO2-Emission For Transport [$\mathrm{kg}$] | 0
Total CO2-Emission [$\mathrm{kg}$] | 2.6
|
# Combined Constraints on First Generation Leptoquarks
Andreas Crivellin<EMAIL_ADDRESS>CERN Theory Division, CH–1211
Geneva 23, Switzerland Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Paul Scherrer Institut,
CH–5232 Villigen PSI, Switzerland Dario Müller<EMAIL_ADDRESS>Physik-
Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich,
Switzerland Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland Luc
Schnell<EMAIL_ADDRESS>Departement Physik, ETH Zürich, Otto-Stern-
Weg 1, CH-8093 Zürich, Switzerland Département de Physique, École
Polytechnique, Route de Saclay, FR-91128 Palaiseau Cedex, France
###### Abstract
In this article we perform a combined analysis of low energy precision
constraints and LHC searches for leptoquarks which couple to first generation
fermions. Considering all ten leptoquark representations, five scalar and five
vector ones, we study at the precision frontier the constraints from
$K\to\pi\nu\nu$, $K\to\pi e^{+}e^{-}$, $K^{0}-\bar{K}^{0}$ and
$D^{0}-\bar{D}^{0}$ mixing, as well as from experiments searching for parity
violation (APV and QWEAK). We include LHC searches for $s$-channel single
resonant production, pair production and Drell-Yan-like signatures of
leptoquarks. Interestingly, we find that the recent non-resonant di-lepton
analysis of ATLAS provides stronger bounds than the resonant searches recasted
so far to constrain $t$-channel production of leptoquarks. Taking into account
all these bounds, we observe that none of the leptoquark representations can
address the so-called “Cabibbo angle anomaly” via a direct contribution to
super-allowed beta decays.
Leptoquarks
###### pacs:
13.20.He,13.25.Es,13.35.Dx,14.80.Sv
††preprint: CERN-TH-2021-012, PSI-PR-21-01, ZU-TH 01/21
## I Introduction
Leptoquarks (LQs) were first proposed in the context of the Pati-Salam model
Pati and Salam (1974) and $SU(5)$ Grand Unified theories Georgi and Glashow
(1974); Dimopoulos et al. (1980) but later on also postulated in composite
models with quark and lepton substructure Schrempp and Schrempp (1985), the
strong coupling version of the Standard Model (SM) Abbott and Farhi (1981),
horizontal symmetry theories Banks and Rabinovici (1979), extended technicolor
Lane (1993) as well as in $SO(10)$ Senjanovic and Sokorac (1983), $SU(15)$
Frampton and Lee (1990), superstring-inspired E6 models Witten (1985) and the
R-parity violating MSSM (see e.g. Ref. Barbier et al. (2005) for a review).
With the HERA excess Adloff et al. (1997); Breitweg et al. (1997) they came
into the focus of the high energy community Dreiner and Morawitz (1997);
Kalinowski et al. (1997); Kunszt and Stirling (1997); Altarelli et al. (1997);
Hewett and Rizzo (1997); Plehn et al. (1997) but after its disappearance the
interest in LQ decreased.
Within recent years LQs experienced a revival, mainly due to the so-called
“flavor anomalies”. These are discrepancies between measurements and the SM
predictions which point towards lepton flavor universality (LFU) violating new
physics (NP) in $R(D^{(*)})$ Lees et al. (2012, 2013); Aaij et al. (2015,
2018a, 2018b); Abdesselam et al. (2019), $b\to s\ell^{+}\ell^{-}$ Khachatryan
et al. (2015); Aaij et al. (2016); Abdesselam et al. (2016); Aaij et al.
(2017, 2019, 2020) and in the anomalous magnetic moment (AMM) of the muon
($a_{\mu}$) Bennett et al. (2006), with a significance of $>\\!3\,\sigma$
Amhis et al. (2017); Murgui et al. (2019); Shi et al. (2019); Blanke et al.
(2019); Kumbhakar et al. (2020), $>\\!5\sigma$ Capdevila et al. (2018);
Altmannshofer et al. (2017a); Algueró et al. (2019); Alok et al. (2019);
Ciuchini et al. (2019); Aebischer et al. (2020a); Arbey et al. (2019); Kumar
et al. (2019a) and $>\\!3\,\sigma$ Aoyama et al. (2020), respectively. In this
context, it has been shown that LQs can explain $b\to s\ell^{+}\ell^{-}$ data
Alonso et al. (2015); Calibbi et al. (2015); Hiller et al. (2016);
Bhattacharya et al. (2017); Buttazzo et al. (2017); Barbieri et al. (2016,
2017); Calibbi et al. (2018); Crivellin et al. (2018a); Bordone et al.
(2018a); Kumar et al. (2019b); Crivellin et al. (2019); Crivellin and
Saturnino (2019a); Cornella et al. (2019); Bordone et al. (2020); Bernigaud et
al. (2020); Aebischer et al. (2019); Fuentes-Martín et al. (2020); Popov et
al. (2019); Fajfer and Košnik (2016); Blanke and Crivellin (2018); de Medeiros
Varzielas and Talbert (2019); de Medeiros Varzielas and Hiller (2015);
Crivellin et al. (2020a); Saad (2020); Saad and Thapa (2020); Gherardi et al.
(2020a); Da Rold and Lamagna (2020), $R(D^{(*)})$ Alonso et al. (2015);
Calibbi et al. (2015); Fajfer and Košnik (2016); Bhattacharya et al. (2017);
Buttazzo et al. (2017); Barbieri et al. (2016, 2017); Calibbi et al. (2018);
Bordone et al. (2018b, a); Kumar et al. (2019b); Biswas et al. (2020);
Crivellin et al. (2019); Blanke and Crivellin (2018); Heeck and Teresi (2018);
de Medeiros Varzielas and Talbert (2019); Cornella et al. (2019); Bordone et
al. (2020); Sahoo and Mohanta (2015); Chen et al. (2016); Dey et al. (2018);
Bečirević and Sumensari (2017); Chauhan et al. (2018); Bečirević et al.
(2018); Popov et al. (2019); Fajfer et al. (2012); Deshpande and Menon (2013);
Freytsis et al. (2015); Bauer and Neubert (2016); Li et al. (2016); Zhu et al.
(2016); Popov and White (2017); Deshpande and He (2017); Bečirević et al.
(2016); Cai et al. (2017); Altmannshofer et al. (2017b); Kamali et al. (2018);
Mandal et al. (2019); Azatov et al. (2018); Zhu et al. (2018); Angelescu et
al. (2018); Kim et al. (2019); Aydemir et al. (2020); Crivellin and Saturnino
(2019b); Yan et al. (2019); Crivellin et al. (2017); Marzocca (2018); Bigaran
et al. (2019); Crivellin et al. (2020a); Saad (2020); Bhupal Dev et al.
(2020); Saad and Thapa (2020); Altmannshofer et al. (2020); Fuentes-Martín and
Stangl (2020); Gherardi et al. (2020a); Da Rold and Lamagna (2020) and/or
$a_{\mu}$ Bauer and Neubert (2016); Djouadi et al. (1990); Chakraverty et al.
(2001); Cheung (2001); Popov and White (2017); Chen et al. (2016); Biggio et
al. (2016); Davidson et al. (1994); Couture and Konig (1996); Mahanta (2001);
Queiroz et al. (2015); Coluccio Leskow et al. (2017); Chen et al. (2017); Das
et al. (2016); Crivellin et al. (2017); Cai et al. (2017); Crivellin et al.
(2018b); Kowalska et al. (2019); Doršner et al. (2020a); Crivellin et al.
(2020a); Delle Rose et al. (2020); Saad (2020); Bigaran and Volkas (2020);
Doršner et al. (2020b); Fuentes-Martín and Stangl (2020); Gherardi et al.
(2020a); Babu et al. (2020); Crivellin et al. (2020b), making them prime
candidates for extending the SM with new particles.
Therefore, the investigation of LQ effects (in observables other than the
flavor anomalies) is very well motivated. Complementary to direct LHC searches
Kramer et al. (1997, 2005); Faroughy et al. (2017); Greljo and Marzocca
(2017); Doršner et al. (2017); Cerri et al. (2019); Bandyopadhyay and Mandal
(2018); Hiller et al. (2018); Faber et al. (2020); Schmaltz and Zhong (2019);
Chandak et al. (2019); Allanach et al. (2020); Buonocore et al. (2020a);
Borschensky et al. (2020), leptonic observables Crivellin et al. (2020c) and
oblique electroweak (EW) parameters as well as Higgs couplings to gauge bosons
Keith and Ma (1997); Doršner et al. (2016); Bhaskar et al. (2020); Zhang et
al. (2019); Gherardi et al. (2020b); Crivellin et al. (2020d) can be used to
test LQs indirectly. Furthermore, if the LQs couple to first generation
fermions particularly many low energy precision probes can be affected Shanker
(1982a, b); Leurer (1994a, b); Davidson et al. (1994). Also beta decays can
receive a tree-level effect from LQs which is interesting in the context of
the so-called “Cabibbo-Angle Anomaly” Grossman et al. (2020); Seng et al.
(2020), where a (apparent) deficit in first row CKM unitarity can be
reconciled via NP effects Belfatto et al. (2020); Coutinho et al. (2020);
Crivellin and Hoferichter (2020); Capdevila et al. (2020); Crivellin et al.
(2020e); Kirk (2020); Alok et al. (2020); Crivellin et al. (2020f, g). Since a
destructive effect w.r.t the purely left-handed SM amplitude is required by
data, $SU(2)_{L}$ gauge invariance also leads to effects in rare Kaon decays
and/or $D^{0}-\bar{D}^{0}$ in LQ models Bobeth and Buras (2018); Doršner et
al. (2020c) which are complementary to LHC bounds. Therefore, it is
interesting to investigate if it is possible to account for the Cabibbo angle
anomaly once all other (relevant) available constraints are taken into
account.
In this article we perform a complete analysis of all ten LQ representations,
assuming only couplings to first-generation (weak-eigenstate) fermions to
determine the combined allowed regions in parameter space. For this purpose,
we define our setup and conventions in Sec. II and perform the matching on the
relevant operators of the SM effective field theory (SMEFT). In Sec. III we
calculate how the SMEFT coefficients are related to experimental constraints,
perform the phenomenological analysis in Sec. IV and conclude in Sec. V.
## II Setup and Matching
LQs have first been classified systematically in Ref. Buchmuller et al. (1987)
into 10 possible representations under the SM gauge group: five scalar and
five vector ones, as listed in Table 1. The conventions are chosen such that
the electric charge $Q$ is given by $Q=\frac{1}{2}Y+T_{3}$, where $Y$ is the
hypercharge and $T_{3}$ the third component of the weak isospin. These
representations allow for couplings to SM quarks and leptons as given in Table
2. Here we did not consider couplings to two quarks, which, together with the
couplings in Table 2, would lead to proton decay. Note that such couplings can
be avoided (to all orders in perturbation theory) by assigning baryon and/or
lepton number to the LQs. In the following, we denote the LQ masses according
to their representation and use small $m$ for the scalar LQs and capital $M$
for the vector LQs.
Field | $\Phi_{1}$ | $\tilde{\Phi}_{1}$ | $\Phi_{2}$ | $\tilde{\Phi}_{2}$ | $\Phi_{3}$ | $V_{1}$ | $\tilde{V}_{1}$ | $V_{2}$ | $\tilde{V}_{2}$ | $V_{3}$
---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{c}$ | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3
$SU(2)_{L}$ | 1 | 1 | 2 | 2 | 3 | 1 | 1 | 2 | 2 | 3
$U(1)_{Y}$ | $-\frac{2}{3}$ | $-\frac{8}{3}$ | $\frac{7}{3}$ | $\frac{1}{3}$ | $-\frac{2}{3}$ | $\frac{4}{3}$ | $\frac{10}{3}$ | $-\frac{5}{3}$ | $\frac{1}{3}$ | $\frac{4}{3}$
Table 1: The ten possible representations of scalar and vector LQs under the
SM gauge group.
### II.1 Matching
We now perform the tree-level matching of our ten LQ representations on
$SU(2)_{L}$ gauge invariant dimension-six four-fermion operators using the
basis of Ref. Grzadkowski et al. (2010) and find
$\begin{array}[]{*{20}{c}}{}\hfil&\vline&{C_{\ell q}^{(1)}}&{C_{\ell
q}^{(3)}}&{{C_{qe}}}&{{C_{\ell u}}}&{{C_{\ell d}}}&{{C_{eu}}}&{{C_{ed}}}\\\
\hline\cr{{\Phi_{1}}}&\vline&{\dfrac{{|\lambda_{1}^{L}{|^{2}}}}{{4m_{1}^{2}}}}&{-\dfrac{{|\lambda_{1}^{L}{|^{2}}}}{{4m_{1}^{2}}}}&*&*&*&{\dfrac{{|\lambda_{1}^{R}{|^{2}}}}{{2m_{1}^{2}}}}&*\\\
{{{\tilde{\Phi}}_{1}}}&\vline&*&*&*&*&*&*&{\dfrac{{|{{\tilde{\lambda}}_{1}}{|^{2}}}}{{2\tilde{m}_{1}^{2}}}}\\\
{{\Phi_{2}}}&\vline&*&*&{-\dfrac{{|\lambda_{2}^{LR}{|^{2}}}}{{2m_{2}^{2}}}}&{-\dfrac{{|\lambda_{2}^{RL}{|^{2}}}}{{2m_{2}^{2}}}}&*&*&*\\\
{{{\tilde{\Phi}}_{2}}}&\vline&*&*&*&*&{-\dfrac{{|{{\tilde{\lambda}}_{2}}{|^{2}}}}{{2\tilde{m}_{2}^{2}}}}&*&*\\\
{{\Phi_{3}}}&\vline&{\dfrac{{3|\lambda_{3}^{2}|}}{{4m_{3}^{2}}}}&{\dfrac{{|\lambda_{3}^{2}|}}{{4m_{3}^{2}}}}&*&*&*&*&*\\\
{V_{1}}&\vline&{-\dfrac{{|\kappa_{1}^{L}{|^{2}}}}{{2M_{1}^{2}}}}&{-\dfrac{{|\kappa_{1}^{L}{|^{2}}}}{{2M_{1}^{2}}}}&*&*&*&*&{-\dfrac{{|\kappa_{1}^{R}{|^{2}}}}{{M_{1}^{2}}}}\\\
{\tilde{V}_{1}}&\vline&*&*&*&*&*&{-\dfrac{{|{{\tilde{\kappa}}_{1}}{|^{2}}}}{{\tilde{M}_{1}^{2}}}}&*\\\
{V_{2}}&\vline&*&*&{\dfrac{{|\kappa_{2}^{LR}{|^{2}}}}{{M_{2}^{2}}}}&*&{\dfrac{{|\kappa_{2}^{RL}{|^{2}}}}{{M_{2}^{2}}}}&*&*\\\
{\tilde{V}_{2}}&\vline&*&*&*&{\dfrac{{|{{\tilde{\kappa}}_{2}}{|^{2}}}}{{\tilde{M}_{2}^{2}}}}&*&*&*\\\
{V_{3}}&\vline&{-\dfrac{{3\left|{\kappa_{3}^{2}}\right|}}{{2M_{3}^{2}}}}&{\dfrac{{|\kappa_{3}^{2}|}}{{2M_{3}^{2}}}}&*&*&*&*&*\end{array}$
(12)
in agreement with Ref. Alonso et al. (2015); Doršner et al. (2016); Mandal and
Pich (2019); Gherardi et al. (2020b)
| $L$ | $e$
---|---|---
${\bar{Q}}$ | ${\kappa_{1}^{L}{\gamma_{\mu}}V_{1}^{\mu}+\kappa_{3}{\gamma_{\mu}}\left({\tau\cdot V_{3}^{\mu}}\right)}$ | ${\lambda_{2}^{LR}{\Phi_{2}}}$
${\bar{d}}$ | ${\tilde{\lambda}_{2}\tilde{\Phi}_{2}^{T}i{\tau_{2}}}$ | ${\kappa_{1}^{R}{\gamma_{\mu}}V_{1}^{\mu}}$
${\bar{u}}$ | ${\lambda_{2}^{RL}\Phi_{2}^{T}i{\tau_{2}}}$ | ${\tilde{\kappa}_{1}{\gamma_{\mu}}\tilde{V}_{1}^{\mu}}$
${\bar{Q}^{c}}$ | ${\lambda_{3}i{\tau_{2}}{{\left({\tau\cdot{\Phi_{3}}}\right)}^{\dagger}}+\lambda_{1}^{L}i{\tau_{2}}\Phi_{1}^{\dagger}}$ | $\kappa_{2}^{LR}{\gamma_{\mu}}{V_{2}^{\mu{\dagger}}}$
${\bar{d}^{c}}$ | ${\kappa_{2}^{RL}{\gamma_{\mu}}V_{2}^{\mu{\dagger}}}$ | ${\tilde{\lambda}_{1}\tilde{\Phi}_{1}^{\dagger}}$
${\bar{u}^{c}}$ | $\tilde{\kappa}_{2}{{\gamma_{\mu}}\tilde{V}_{2}^{\mu{\dagger}}}$ | ${\lambda_{1}^{R}\Phi_{1}^{\dagger}}$
Table 2: Interaction terms of the LQ representations listed in Table 1, where
$Q$ and $L$ represent the left-handed quark and lepton $SU(2)_{L}$ doublets,
$e$, $d$ and $u$ the right-handed $SU(2)_{L}$ singlets, the superscript $c$
stands for charge conjugation and $\tau_{i}$ are the Pauli matrices.
For simplicity, we do not include flavor indices, since we will only consider
couplings to first generation fermions (in the weak basis). Furthermore, we
assume that $\Phi_{1}$, $\Phi_{2}$, $V_{1}$ and $V_{2}$ possess only one of
the two possible couplings at the same time. Therefore, no scalar or tensor
operators are generated, where the former ones are very stringently
constrained from $\pi\to e\nu$.
Let us now consider the one-loop matching on four-quark operators Aebischer et
al. (2020b) involving only left-handed fields:
$\displaystyle Q_{qq}^{(1)}$
$\displaystyle=\big{[}\bar{Q}\gamma^{\mu}Q\big{]}\big{[}\bar{Q}\gamma_{\mu}Q\big{]}\,,$
(20) $\displaystyle Q_{qq}^{(3)}$
$\displaystyle=\big{[}\bar{Q}{\tau^{I}}\gamma^{\mu}Q\big{]}\big{[}\bar{Q}{\tau}^{I}\gamma^{\mu}Q\big{]}\,,$
(21)
where the color indices are contracted within each bilinear and the Wilson
coefficients are given by
$\displaystyle\Phi_{1}:$ $\displaystyle C_{qq}^{(1)}$
$\displaystyle=\frac{-|\lambda_{1}^{L}|^{4}}{256\pi^{2}m_{1}^{2}}\,,$
$\displaystyle
C_{qq}^{(3)}=\frac{-|\lambda_{1}^{L}|^{4}}{256\pi^{2}m_{1}^{2}}\,,$ (22a)
$\displaystyle\Phi_{2}:$ $\displaystyle C_{qq}^{(1)}$
$\displaystyle=\frac{-|\lambda_{2}^{LR}|^{4}}{128\pi^{2}m_{2}^{2}}\,,$ (22b)
$\displaystyle\Phi_{3}:$ $\displaystyle C_{qq}^{(1)}$
$\displaystyle=\frac{-9|\lambda_{3}|^{4}}{256\pi^{2}m_{3}^{2}}\,,$
$\displaystyle C_{qq}^{(3)}=\frac{-|\lambda_{3}|^{4}}{256\pi^{2}m_{3}^{2}}\,,$
(22c) $\displaystyle V_{1}:$ $\displaystyle C_{qq}^{(1)}$
$\displaystyle=\frac{-|\kappa_{1}^{L}|^{4}}{32\pi^{2}M_{1}^{2}}\,,$ (22d)
$\displaystyle V_{2}:$ $\displaystyle C_{qq}^{(1)}$
$\displaystyle=\frac{-|\kappa_{2}^{LR}|^{4}}{32\pi^{2}M_{2}^{2}}\,,$ (22e)
$\displaystyle V_{3}:$ $\displaystyle C_{qq}^{(1)}$
$\displaystyle=\frac{-3|\kappa_{3}|^{4}}{32\pi^{2}M_{3}^{2}}\,,$
$\displaystyle C_{qq}^{(3)}=\frac{-|\kappa_{3}|^{4}}{16\pi^{2}M_{3}^{2}}\,.$
(22f)
Due to $SU(2)_{L}$, these operators will necessarily give rise to
$K^{0}-\bar{K}^{0}$ and/or $D^{0}-\bar{D}^{0}$ mixing after electroweak
symmetry breaking. For the vector LQs we calculated the diagrams in Feynman
gauge, i.e. neglecting Goldstone contributions. In this way a finite result is
obtained and the estimate is conservative in the sense that the NP
contribution obtained is smaller than (the finite part of) the one in unitary
gauge where large logarithms involving the cut-off appear Barbieri et al.
(2017).
### II.2 Electroweak Symmetry Breaking
For left-handed quarks “first generation” is only well defined in the
interaction basis as after electroweak symmetry breaking non-diagonal mass
matrices for the quarks are generated. In order to work in the physical basis
with diagonal mass terms, we have to rotate the quark fields111The same is
true for charged leptons. However, in the limit of vanishing neutrino masses
all rotation necessary to diagonalize the charged lepton mass matrix are
unphysical since they can be absorbed into a field redefinition.
$\displaystyle\begin{aligned} d_{L,f}\to U_{fi}^{d_{L}}d_{L,i}\,,\\\
d_{R,f}\to U_{fi}^{d_{R}}d_{R,i}\,,\\\ u_{L,f}\to U_{fi}^{u_{L}}u_{L,i}\,,\\\
u_{R,f}\to U_{fi}^{u_{R}}u_{R,i}\,,\end{aligned}$ (23)
with the unitary matrices $U^{u_{L,R}}$ and $U^{d_{L,R}}$. While the right-
handed rotations can be absorbed by a re-definition of the couplings and are
thus unphysical, the left-handed ones form the Cabibbo-Kobayashi-Maskawa (CKM)
matrix
$\displaystyle V_{fi}\equiv U^{u_{L}*}_{jf}U^{d_{L}}_{ji}\,.$ (24)
As we want to study first generation LQs (defined in the weak basis), and
flavor violating effects involving first and second quark generation quarks
are most stringently constrained, we can focus on the $2\times 2$ sector which
is related to the relatively large Cabibbo angle $\theta_{c}\approx 0.22$. We
can thus parameterize the matrices in Eq. (23) as
$\displaystyle\begin{aligned}
U^{uL}&=\begin{pmatrix}\cos(\alpha)&\sin(\alpha)\\\
-\sin(\alpha)&\cos(\alpha)\end{pmatrix},\;\\\
U^{dL}&=\begin{pmatrix}\cos(\beta)&\sin(\beta)\\\
-\sin(\beta)&\cos(\beta)\end{pmatrix}\,.\end{aligned}$ (25)
Using Eq. (24) this yields
$\displaystyle V$
$\displaystyle=\begin{pmatrix}\cos({\beta-\alpha})&\sin({\beta-\alpha})\\\
-\sin({\beta-\alpha})&\cos({\beta-\alpha})\end{pmatrix}$ (26a)
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}\begin{pmatrix}\cos(\theta_{c})&\sin(\theta_{c})\\\
-\sin(\theta_{c})&\cos(\theta_{c})\end{pmatrix}\,.$ (26b)
Hence, we can write
$\displaystyle U^{uL}$
$\displaystyle=\begin{pmatrix}\cos({\beta-{\theta_{c}}})&\sin({\beta-{\theta_{c}}})\\\
-\sin({\beta-{\theta_{c}}})&\cos({\beta-{\theta_{c}}})\end{pmatrix}\,.$ (27)
If $\beta=0$, we work in the so-called down basis where no CKM elements appear
in flavor changing neutral currents (FCNCs) with down-type quarks. On the
other hand, if we choose $\beta=\theta_{c}$, we work in the up basis in which
down-type FCNCs are induced via CKM elements while up-type FCNCs are absent.
## III Observables
### III.1 Charged Semi-Leptonic Current
We use the charged current effective Hamiltonian
$\displaystyle\mathcal{H}_{\text{eff}}^{\ell\nu}=\frac{4G_{F}}{\sqrt{2}}V_{jk}{\hat{C}}_{jk}^{e\nu}\big{[}\bar{u}_{j}\gamma^{\mu}P_{L}d_{k}\big{]}\big{[}{\bar{e}}\gamma_{\mu}P_{L}\nu_{e}\big{]}\,,$
(28)
governing semi-leptonic transitions. The coefficients
$\hat{C}_{jk}^{e\nu}=C_{jk}^{\text{SM}}+C^{e\nu}_{jk}$ are the sum of the SM
and LQ contribution. The normalization is chosen such that we have in the SM
$\displaystyle C_{jk}^{\text{SM}}=\delta_{jk}\,.$ (29)
Integrating out the LQs, we obtain the following tree-level matching results
$\displaystyle\begin{aligned}
C^{e\nu}_{11}=\frac{-1}{\sqrt{2}G_{F}}\frac{c_{\beta}c_{\beta-\theta}}{V_{ud}}C_{\ell
q}^{(3)}\,,\\\
C^{e\nu}_{12}=\frac{{-}1}{\sqrt{2}G_{F}}\frac{s_{\beta}c_{\beta-\theta}}{V_{us}}C_{\ell
q}^{(3)}\,,\end{aligned}$ (30)
where we abbreviated $c_{\beta}\equiv\cos(\beta)$, $s_{\beta}=\sin(\beta)$,
$c_{\beta-\theta}\equiv\cos(\beta-\theta_{c})$ and
$s_{\beta-\theta}\equiv\sin(\beta-\theta_{c})$ and neglected effects related
to third generation quarks and charm quarks, which would result in much weaker
limits than the bounds to be discussed now.
The $d\to ue\bar{\nu}_{e}$ transitions contribute to beta decays where the
measured CKM element $V_{ud}^{\beta}$ (extracted from experiment using the SM
hypothesis) is related to the unitary CKM matrix $V_{ud}^{L}$ of the
Lagrangian (including NP effects)
$\displaystyle V_{ud}^{\beta}=V_{ud}^{L}\big{(}1+C_{11}^{e\nu_{e}}\big{)}\,.$
(31)
The element $V_{ud}^{L}$ can then be converted to $V_{us}^{L}$ applying
unitarity
$\displaystyle\big{|}V_{us}^{L}\big{|}=\sqrt{1-\big{|}V_{ud}^{L}\big{|}^{2}-\big{|}V_{ub}^{L}\big{|}^{2}}\,.$
(32)
We find
$\displaystyle\begin{aligned} V_{us}^{L}\approx
V_{us}^{\beta}+\frac{{|V_{ud}^{\beta}{|^{2}}}}{{|V_{us}^{\beta}{|^{2}}}}{\mkern
1.0mu}C_{11}^{e{\nu_{e}}}\,.\end{aligned}$ (33)
$V_{ub}^{\beta}$ is most precisely determined from super-allowed beta decays.
Following Ref. Crivellin et al. (2020e) we have
$\displaystyle
V_{us}^{\beta}=0.2281(7)\,,\;\;V_{us}^{\beta}|_{\text{NNC}}=0.2280(14)\,,$
(34)
where the latter value contains the “new nuclear corrections” (NNCs) proposed
by Refs. Seng et al. (2019); Gorchtein (2019). Since at the moment the issue
of the NNCs is not settled, we will quote results for both determinations.
This value of $V_{us}^{\beta}$ can now be compared to $V_{us}$ from two and
three body kaon Aoki et al. (2020) and tau decays Amhis et al. (2019)
$\displaystyle\begin{split}V_{us}^{K_{\mu
3}}&=0.22345(67)\,,\;\;\;V_{us}^{K_{e3}}=0.22320(61)\,,\\\ V_{us}^{K_{\mu
2}}&=0.22534(42)\,,\;\;\;\;\;V_{us}^{\tau}=0.2195(19)\,,\end{split}$ (35)
which are significantly lower222During finalization of this article, Ref.
Shiells et al. (2020) obtained a value of
$\left|V_{ud}\right|^{2}=0.94805(26)$ which even slightly increases the
disagreement with $V_{us}$.. This disagreement constitutes the so-called
Cabibbo angle anomaly.
Besides $\beta$-decays, tests of LFU in pion and Kaon decays, defined at the
amplitude level and normalized to unity in the SM, result in
$\displaystyle\begin{aligned} \frac{\pi\to\mu\nu}{\pi\to e\nu}&\approx
1-\dfrac{C_{11}^{e\nu_{e}}}{V_{ud}}\,,\\\
\frac{K\to(\pi)\mu\nu}{K\to(\pi)e\nu}&\approx
1-\frac{C_{12}}{V_{us}}\,.\end{aligned}$ (36)
This has to be compared to
$\displaystyle\begin{aligned} \frac{K\to\pi\mu\nu}{K\to\pi
e\nu}\bigg{|}_{\exp}&=1.0010\pm 0.0025\,,\\\ \frac{K\to\mu\nu}{K\to
e\nu}\bigg{|}_{\exp}&=0.9978(18)\,,\\\ \frac{\pi\to\mu\nu}{\pi\to
e\nu}\bigg{|}_{\exp}&=1.0010(9)\,,\end{aligned}$ (37)
from Ref. V. Cirigliano and Passemar , Refs. Ambrosino et al. (2009);
Lazzeroni et al. (2013); Tanabashi et al. (2018) and Refs. Aguilar-Arevalo et
al. (2015); Czapek et al. (1993); Britton et al. (1992); Tanabashi et al.
(2018), respectively. Numerically, $C_{11}^{e\nu_{e}}\approx-0.001$ would
significantly improve the agreement with data. Note that effects in charged
current $D$ decays are not very constraining Dorsner et al. (2009).
### III.2 Tree-Level Neutral Current
Chiral quark-electron interactions can be constrained from atomic parity
violation experiments like APV Wood et al. (1997); Dzuba et al. (2012) and
from the weak charge of the proton as measured by QWEAK Allison et al. (2015);
Androić et al. (2018). The relevant effective Lagrangian reads
$\displaystyle\mathcal{L}_{\text{eff}}^{ee}=\frac{G_{F}}{\sqrt{2}}\sum_{q=u,d}{\hat{C}_{1q}}\big{[}\bar{q}\gamma^{\mu}q\big{]}\big{[}\bar{e}\gamma_{\mu}\gamma_{5}e\big{]}\,,$
(38)
where $\hat{C}_{1q}=C_{1q}^{\text{SM}}+C_{1q}$ with
$C_{1u}^{\text{SM}}=-0.1887$ and $C_{1d}^{\text{SM}}=0.3419$. Again we can
express the Wilson coefficients $C_{1q}$ in terms of the SMEFT matching
coefficients
$\displaystyle C_{1u}$
$\displaystyle=\frac{-\sqrt{2}}{4G_{F}}\Big{(}c_{\beta-\theta}^{2}\big{(}C_{\ell
q}^{(1)}-C_{\ell q}^{(3)}-C_{qe}\big{)}+C_{\ell u}-C_{eu}\Big{)}\,,$
$\displaystyle C_{1d}$
$\displaystyle=\frac{-\sqrt{2}}{4G_{F}}\Big{(}c_{\beta}^{2}\big{(}C_{\ell
q}^{(1)}+C_{\ell q}^{(3)}-C_{qe}\big{)}+C_{\ell d}-C_{ed}\Big{)}\,.$ (39)
This has to be compared to Zyla et al. (2020)
$\displaystyle Q_{W}(p)$
$\displaystyle=-2\left(2{\hat{C}}_{1u}+{\hat{C}}_{1d}\right)=0.0719\pm
0.0045\,,$ (40) $\displaystyle Q_{W}\left({Cs}^{133}\right)$
$\displaystyle=-2\left(188{\hat{C}}_{1u}+211{\hat{C}}_{1d}\right)=-72.{82}\pm
0.42\,.$
For our numerical analysis we combine these constraints in a $\chi^{2}$ fit
with one degree of freedom since each LQ representation predicts a single
direction in $C_{1u}-C_{1d}$ space.
If we are not exactly aligned to the down basis (i.e. $\beta\neq 0$), some
representations generate $s\to de^{+}e^{-}$ transitions which result in LFUV
in $K\to\pi\mu^{+}\mu^{-}/K\to\pi e^{+}e^{-}$. With the current experimental
constraints Batley et al. (2009); Appel et al. (1999); Batley et al. (2003) we
find according to Ref. Crivellin et al. (2016)
$\displaystyle{{{s_{\beta}}{c_{\beta}}\left({C_{lq}^{(1)}+C_{lq}^{(3)}+{C_{qe}}}\right)=\dfrac{{0.0012\pm
0.0046}}{{{\rm{Te}}{{\rm{V}}^{2}}}}}}$ (41)
from $K^{+}\to\pi^{+}\mu^{+}\mu^{-}/K^{+}\to\pi^{+}e^{+}e^{-}$. Similar test
of LFU in $D$ decays are not constraining Bause et al. (2020a).
Similarly, if the LQ representation couples left-handed down quarks to
neutrinos, effects in $K\to\pi\nu\nu$ are generated for $\beta\neq 0$. Here
the the charged mode Artamonov et al. (2008)
$\displaystyle{\text{Br}}\big{[}K^{+}\to\pi^{+}\nu\bar{\nu}\big{]}=\big{(}1.73\begin{subarray}{c}+1.15\\\
-1.05\end{subarray}\big{)}\times 10^{-10}\,,$ (42)
provides better constraints and using the results of Ref. Buras et al. (2005,
2015) we find
$\displaystyle{\rm{Br}}\left[{{K^{\pm}}\to{\pi^{\pm}}\nu\bar{\nu}}\right]=\frac{1}{3}\left({1+{\Delta_{EM}}}\right){\eta_{\pm}}\times$
$\displaystyle\sum\limits_{f,i=1}^{3}\bigg{[}\frac{{\mathop{\rm
Im}\nolimits}{{\big{[}{{\lambda_{t}}\tilde{X}_{\nu}^{fi}}\big{]}^{2}}}}{\lambda^{{5}}}+\\!\bigg{(}\frac{{\mathop{\rm
Re}\nolimits}\big{[}{{\lambda_{c}}}\big{]}}{\lambda}{P_{c}}{\delta_{fi}}+\frac{{\mathop{\rm
Re}\nolimits}\big{[}{{\lambda_{t}}\tilde{X}_{\nu}^{fi}}\big{]}}{\lambda^{5}}\bigg{)}^{\\!2}\bigg{]}\,,$
(43)
with $\lambda_{q}=V_{qs}^{*}V_{qd}$ and
$\displaystyle\begin{aligned}
&\tilde{X}_{\nu}^{fi}=X_{\nu}^{{\rm{SM}},fi}-{s_{W}^{2}}{C_{\nu}^{fi}}\,,\\\
&X_{L}^{{\rm{SM}},fi}=\big{(}{1.481\pm 0.005\pm
0.008}\big{)}{\delta_{fi}}\,,\\\ &{P_{c}}=0.404\pm
0.024\,,~{}~{}~{}~{}{\Delta_{EM}}=-0.003\,,\\\ &{\eta_{\pm}}=\big{(}{5.173\pm
0.025}\big{)}\times{10^{-11}}{\bigg{[}{\frac{\lambda}{{0.225}}}\bigg{]}^{8}}\,.\end{aligned}$
(44)
The LQ effects can again be expressed in the compact form
$\displaystyle C_{\nu}^{fi}$ $\displaystyle=\frac{-\pi
s_{\beta}c_{\beta}}{\sqrt{2}G_{F}\alpha V_{ts}^{*}V_{td}}\Big{(}C_{\ell
q}^{(1)}-C_{\ell q}^{(3)}\Big{)}\delta_{f1}\delta_{i1}\,,$ (45)
by using Eq. (22). Again, the analogous $D$ decays cannot complete in
precision Bause et al. (2020b) and the loop-induced effects in
$D^{0}-\bar{D}^{0}$ turn out to be more relevant.
Figure 1: Feynman diagrams showing the different search channels for LQs at
the LHC.
### III.3 $D^{0}-\bar{D}^{0}$ and $K^{0}-\bar{K}^{0}$ Mixing
Finally, if a LQ representation couples to left-handed quarks with
$\beta\neq\theta_{c}$ $(\beta\neq 0)$ FCNC in $D^{0}-\bar{D}^{0}$
($K^{0}-\bar{K}^{0}$) mixing is generated. We use
$\displaystyle\begin{aligned}
\mathcal{H}_{\text{eff}}^{D\bar{D}}&=C_{1}^{D}\big{[}\bar{u}_{\alpha}\gamma^{\mu}P_{L}c_{\alpha}\big{]}\big{[}\bar{u}_{\beta}\gamma_{\mu}P_{L}c_{\beta}\big{]}\,,\\\
\mathcal{H}_{\text{eff}}^{K\bar{K}}&=C_{1}^{K}\big{[}\bar{d}_{\alpha}\gamma^{\mu}P_{L}s_{\alpha}\big{]}\big{[}\bar{d}_{\beta}\gamma_{\mu}P_{L}s_{\beta}\big{]}\end{aligned}$
(46)
to parametrize NP contributions and we obtain
$\displaystyle C_{1}^{D}$
$\displaystyle=-s_{\beta-\theta}^{2}c_{\beta-\theta}^{2}\Big{(}C_{qq}^{(1)}+C_{qq}^{(3)}\Big{)}$
(47) $\displaystyle C_{1}^{K}$
$\displaystyle=-s_{\beta}^{2}c_{\beta}^{2}\Big{(}C_{qq}^{(1)}+C_{qq}^{(3)}\Big{)}$
(48)
The limits on the coefficients are Bona and Silvestrini (2017)
$\displaystyle{\begin{aligned} |\text{Re}\big{[}C_{1}^{D}\big{]}|&\lesssim
3\times\frac{10^{-7}}{\text{TeV}^{2}}\,\\\
|\text{Re}\big{[}C_{1}^{K}\big{]}|&\lesssim
1.3\times\frac{10^{-7}}{\text{TeV}^{2}}\,.\end{aligned}}$ (49)
Since the SM contribution cannot be reliably calculated in case of
$D^{0}-\bar{D}^{0}$ mixing, we assumed that the NP contribution should not
generate more than the whole measured mass difference to obtain this bound.
### III.4 LHC Bounds
One can search for signals of LQs at the LHC generated via
* •
Pair production (PP): $qq(gg)\to 2{\rm LQ}\to qq\ell\ell$
* •
Single production (SP): $qg\to{\rm LQ}\to\ell\ell q$
* •
Single resonant production (SRP): $\ell q\to{\rm LQ}\to\ell q$
* •
Drell-Yan (DY): $pp\to{\rm LQ^{*}}\to\ell\ell$
as depicted in Fig. 1.
For first generation LQs, PP sets coupling independent limits on their masses.
Here we use the bounds for the neutrino and charged lepton channels of Ref.
Sirunyan et al. (2018a) and Ref. Sirunyan et al. (2019), respectively. Note
that the interactions of gluons with vector LQs depend on the nature of the
LQ, i.e. whether it is a massive Proca field or a massive gauge boson Blumlein
et al. (1997). We choose the latter case (corresponding to $\kappa_{G}=0$) and
rescaled the experimental bounds on the masses of Refs. Sirunyan et al.
(2018a); Sirunyan et al. (2019) by a constant factor $\approx 1.3$ derived
from Ref. Sirunyan et al. (2018b) by comparing the vector LQ to the scalar LQ
limits333Note that the bounds could be weakened in case the LQ is not purely a
gauge boson by minimizing the cross section with respect to $\kappa_{G}$ and
$\lambda_{G}$ Blumlein et al. (1997, 1998). Furthermore, the limits from PP
differ for the various LQ representations Diaz et al. (2017). In case of a
small mass splitting among the $SU(2)_{L}$ components, as realized for $v\ll
m,M$, their contributions add up to the total signal strength. This can be
incorporated in the analysis by choosing an “effective” value of $\beta$
(originally parametrizing the branching fraction to electrons) which can then
however be bigger than 1 (e.g. $\sqrt{2}$ for $\lambda^{LR}_{2}$). Therefore,
we extrapolated the $\beta$ dependence of the limits given in Refs. Sirunyan
et al. (2018a); Sirunyan et al. (2019) to account for these cases.
While the bounds from SP via $qg\to{\rm LQ}\to\ell\ell q$ are quite weak
Khachatryan et al. (2016); Mandal et al. (2015); Schmaltz and Zhong (2019), in
case of first generation LQs much better bounds can be derived from SRP via
$\ell q\to{\rm LQ}\to\ell q$ Buonocore et al. (2020a); Greljo and Selimovic
(2020) using the electron PDF of the proton Buonocore et al. (2020b). Since
Ref. Buonocore et al. (2020a) considers a simplified setup with $ue$ and $de$
interactions separately we have to adapt the limits for several of our LQ
representations. First of all, as for PP, the small mass splitting between the
$SU(2)_{L}$ components leads to overlapping signals (i.e. the cross sections
of the components have to be added). In addition, we have to take into account
the difference between the up and down quark PDFs, which can be obtained for
the relative strength of the $ue$ and $de$ limits given in Ref. Buonocore et
al. (2020a). Furthermore, if the LQ couples to a lepton doublet, we must
adjust the branching ratio as it can decay to neutrinos whose signal is not
included in the analysis. Finally, for VLQs we have to correct for the fact
that, due to the Dirac algebra, the on-shell production cross section is
$\sigma_{\text{VLQ}}=2\sigma_{\text{SLQ}}+\mathcal{O}(\alpha_{s})$ for equal
LQ couplings to fermions.
Limits from DY-like signatures were derived in Ref. Schmaltz and Zhong (2019)
based on the CMS search for resonant di-lepton pairs Sirunyan et al. (2018c),
but they turn out to be less constraining than the bounds from SRP Buonocore
et al. (2020a). Interestingly, the latest non-resonant di-lepton search of
ATLAS444Note that in v1 and v2 of the ATLAS article a factor 2 in the
definition of the Lagranigian in Eq. (1) was missing. We thank the ATLAS
collaboration for confirming this. Aad et al. (2020) can be used to obtain
more stringent bounds555In principle also LEP bounds on ee-qq interactions
Schael et al. (2013) could be used to constrain first generation LQs. Even
though these limits can be directly applied for TeV scale LQs, they turn out
to be weaker compared to LHC searches and low energy precision constraints..
Here we have to take into account that Ref. Aad et al. (2020) assumed quark
flavour universality which is not respected by most of the representations.
This can be done by correcting for the fact that at $2\,$TeV the
$uu\to\ell^{+}\ell^{-}$ cross section is a factor $\approx 1.7$ bigger than
the $dd\to\ell^{+}\ell^{-}$ one for equal couplings. Furthermore, unlike for
the analysis of Ref. Schmaltz and Zhong (2019) which is valid for low LQ
masses, here care has to be taken if the LQ mass is within the LHC energy
range. Following Ref. Bessaa and Davidson (2015), the four-fermion
approximation can be used if the LQ mass squared is bigger than four times the
center of mass energy. As the highest energy used in the analysis of Ref. Aad
et al. (2020) is $\approx 2\,$TeV, the limits can be applied for a LQ mass
above $\approx 4\,$TeV Bessaa and Davidson (2015). If the LQ is lighter, the
limit is weakened. In particular for a LQ mass of $1\,$TeV the bound on the
coupling is a factor $\approx 1.6$ ($\approx 2.1$) less stringent than
extracted in the 4-fermion approximation Bessaa and Davidson (2015) in case of
constructive (destructive) interference666For our numerical analysis we
interpolated the points given in Fig. 2 of Ref. Bessaa and Davidson (2015) to
estimate the correction factor..
## IV Phenomenological Analysis
In our phenomenological analysis we consider each LQ representation
separately. In addition, we only allow for a single non-zero coupling at a
time so that there are two scenarios for $\Phi_{1,2}$ and $V_{1,2}$ each.
Therefore, we have fourteen scenarios in total with three free parameters in
each case: the LQ mass ($m,M$), the coupling ($\lambda,\kappa$) and the angle
$\beta$. The LHC limits and the bounds from parity violation are to a good
approximation independent of $\beta$ (for $\beta=O(\theta_{c})$). Here we will
consider two cases, $\beta=0$ and $\beta=\theta_{c}$, corresponding to the
down and up basis, respectively. While in the first case no effects in Kaon
physics appear, bounds from $D^{0}-\bar{D}^{0}$ mixing are relevant for all LQ
representations involving couplings to quark doublets. On the other hand, if
$\beta=\theta_{c}$, no limits from $D$ physics can be obtained, but
$K^{+}\to\pi^{+}\nu\nu(e^{+}e^{-})$ puts bounds on the parameter space.
In Fig. IV and Fig. IV we show combined constraints (as well as the 3$ab^{-1}$
projection for SRP) on the parameter space of first generation LQs. All cases
are constrained by LHC searches and parity violation experiments (QWEAK+APV)
but bounds from Kaon and $D$ physics only appear in the case of couplings to
quark doublets. In this case it is not possible to avoid both Kaon and $D$
bounds simultaneously, and the resulting limits are stringent. Furthermore,
the ATLAS bounds on non-resonant di-lepton production are also very stringent
and in fact more constraining than the DY bounds Diaz et al. (2017) (not
displayed here) obtained from recasting resonant di-lepton searches Sirunyan
et al. (2018c). Note that the 95% CL limits from QWEAK+APS give quite
different constraints on the various LQ representations since the central
value is about $1\,\sigma$ off the SM prediction.
Only the representations $\Phi_{1,3}$ and $V_{1,3}$ generate a charged current
whose strength is indicated by the black lines. Here the Cabibbo angle anomaly
prefers negative values $C_{11}^{e\nu}\approx{-}0.001$. This disfavours
$V_{1}$ ($\phi_{1}$) with $\lambda_{1}^{L}\neq 0$ ($\kappa_{1}^{L}\neq 0$)
while it would in principle favour $V_{3}$ and $\Phi_{3}$. However, DY
searches as well as $K^{0}-\bar{K}^{0}$ and/or $D^{0}-\bar{D}^{0}$ mixing
exclude sizeable values of $C_{11}^{e\nu}$. Therefore, despite the fact that
LQs can give tree-level effects in (super-allowed) beta decays, they cannot
account for the deficit in first row CKM unitarity.
Figure 2: Limits on the parameter space of first generation scalar LQs. The
region above the colored lines is excluded. While LHC limits and the bounds
from parity violation are to a good approximation independent of $\beta$ (for
$\beta=O(\theta_{c})$) the bounds from kaon and $D$ decays depend on it. We
consider the two scenarios $\beta=\theta_{c}$ or $\beta=0$. In the first case,
the kaon limits arise for LQ representations with left-handed quark fields
while in the second case these limits are absent but bounds from
$D^{0}-\bar{D}^{0}$ arise.
Figure 3: Limits on the parameter space of first generation vector LQs. The
region above the colored lines is excluded. While LHC limits and the bounds
from parity violation are to a good approximation independent of $\beta$ (for
$\beta=O(\theta_{c})$) the bounds from kaon and $D$ decays depend on it. We
consider the two scenarios $\beta=\theta_{c}$ or $\beta=0$. In the first case,
the kaon limits arise for LQ representations with left-handed quark fields
while in the second case these limits are absent but bounds from
$D^{0}-\bar{D}^{0}$ arise.
## V Conclusion
In this article we performed a combined analysis of constraints on first
generation LQs for all ten possible representations (five scalar and five
vector ones). We included the constraints from parity violating experiments
(QWEAK+APV) and LHC searches, in particular PP, SRP and DY searches. For the
latter case, we find that the latest non-resonant di-lepton analysis of ATLAS
provides stronger bounds than resonant searches recasted so far in the
literature. As for left-handed quarks ”first generation“ can only be defined
in the weak basis before EW symmetry breaking, unavoidable effects in Kaon
and/or $D$ physics occur for the LQ representations coupling to quark
doublets. Our results are depicted in Fig. IV and Fig. IV for scalar and
vector LQs, respectively. One can see that all cases are constrained by parity
violating experiments and LHC searches but only the cases which involve quark
doublets are constrained by Kaon and/or $D$ physics. Furthermore, only 4
representations give rise to charged current effects where the Cabibbo angle
anomaly prefers a destructive effect with respect to the SM. Such an effect
can only be generated by $\Phi_{3}$ and $V_{3}$ and the possible size is too
constrained by DY searches as well as $K^{0}-\bar{K}^{0}$ and/or
$D^{0}-\bar{D}^{0}$ mixing to account for the anomaly.
###### Acknowledgements.
A.C. thanks Marc Montull for useful discussions. The work of A.C. is supported
by a Professorship Grant (PP00P2_176884) of the Swiss National Science
Foundation. L.S. is supported by the “Excellence Scholarship & Opportunity
Programme” of the ETH Zürich.
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|
###### Abstract
We study primordial black holes (PBHs) formation in the excursion set theory
(EST) in a vast range of PBHs masses with and without confirmed constraints on
their abundance. In this work, we introduce a new concept of the first touch
in the context of EST for PBHs formation. This new framework takes into
account the earlier horizon reentry of smaller masses. Our study shows that in
the EST, it is possible to produce PBHs in different mass-range, with enhanced
power spectrum, which could make up all dark matter. We also show that in a
broad blue-tilted power spectrum, the production of PBHs is dominated by
smaller masses. Our analysis put an upper limit $\sim\,$0.1 on the amplitude
of the curvature power spectrum at length scales relevant for PBHs formation.
Primordial Black Holes in the Excursion Set Theory
Encieh Erfani${}^{1,\,*}$, Hamed Kameli2 and Shant Baghram${}^{2,\,{\dagger}}$
1Department of Physics, Institute for Advanced Studies in Basic Sciences
(IASBS), Zanjan 45137-66731, Iran
2Department of Physics, Sharif University of Technology, Tehran 11155-9161,
Iran
000∗erfani<EMAIL_ADDRESS>
## 1 Introduction
####
Primordial black holes (PBHs) can form from the collapse of large density
fluctuations in the early Universe [1, 2, 3]. When the density fluctuations
larger than the threshold reenters the horizon after inflation, the region
will collapse to form a PBH with a mass roughly equal to the mass of the
horizon. According to the Hawking radiation [4], PBHs with mass larger than
$\sim 10^{15}$ g have a lifetime longer than the age of the Universe.
Moreover, since they have formed before matter-radiation equality, they are
non-baryonic. Thus they could be a candidate for dark matter (DM).
After the discovery of gravitational waves (GWs) in 2016 by LIGO/Virgo from
mergers of tens of Solar mass black holes [5], the possibility that these BHs
could be primordial rather than astrophysical [6, 7], has led to a great
interest in PBH version of DM. In addition, the abundance of PBHs in our
observable Universe can give us some clues of small-scale density fluctuations
which are not accessible via observations of the cosmic microwave background
(CMB) and the large scale structure (LSS) (i.e. constraints on the primordial
power spectrum for scales between $k\sim(10^{-3}-1)$ Mpc-1) [8, 9, 10]. (For
recent reviews on PBHs, see also refs. [11, 12].) The power spectrum at small
scales relevant for PBHs formation must become orders of magnitude larger than
$\mathcal{O}(10^{-9})$ detected on large scales via the CMB [13] which is a
matter of active research. Since the amplitude of the power spectrum is only
logarithmically sensitive to the PBHs abundance during the radiation
domination, this variation has little to do with the different PBH masses [14,
15]. However, the PBH mass distribution differs when using different
theoretical techniques for their formation [16]111For instance, primordial
non-Gaussianity can have an important effect on the required power spectrum
amplitude [17]. However, we will not consider this issue in this paper.. PBH’s
formation has been studied with the use of Press-Schechter (PS) formalism
[18], peaks theory [19, 20] and recently by excursion set theory (EST) [21,
22]. In this paper, we make the first detailed numerical study of the mass
distribution of PBH in the EST [23] by counting the number of trajectories
that reenter the horizon corresponding to a specific mass. We study the
formation of PBHs when a broad spectrum characterizes the scalar
perturbations, and the goal of the paper is to answer the question of the mass
distribution of PBHs for broad spectra in the EST. We will show that despite
the broadness of the blue-tilted power spectrum, the production of PBHs
enhances towards small masses.
There are various constraints on PBHs abundance which have recently been
compiled in [12, 14]. In our study, we will consider these observational
constraints in mass ranges if they exist. Otherwise, we will assume that PBHs
can comprise all of the DM. We will find the required blue-tilled spectral
indexes for these mass ranges in the EST formalism. Finally, we will translate
these results into upper limit on the amplitude of the curvature power
spectrum at length scales relevant for PBHs formation.
The paper is laid out as follows: in section 2 we elaborate PBH formation in
PS formalism, and we review current constraints on their density being
associated with a variety of gravitational lensing [24, 25, 26, 27] and GW
[28] effects. In section 3 we briefly explain the EST method and we develop
the PBHs formation in this context. Finally, section 4 is devoted to our
conclusions.
## 2 Primordial Black Holes
####
The most common mechanism for PBHs formation is the collapse of large density
perturbations generated by inflation in the very early Universe. These density
perturbations that reenter the Hubble horizon in the radiation dominated (RD)
era would gravitationally collapse into a BH if their amplitude is larger than
a critical value. Since PBHs are not formed by stellar core collapse, they may
be of any size from the Planck mass $\sim 10^{-5}\,$g to $\sim 10^{50}\,$g
[12]. Mass of PBH, $M_{\rm PBH}$ produced at a given time is limited by the
horizon mass at that time
$M_{\rm PBH}=\gamma\,M_{\rm PH}\sim 10^{15}\left(\dfrac{t}{10^{-23}\,{\rm
s}}\right){\rm g}\,,$ (2.1)
where $M_{\rm PH}$ is the particle horizon mass and $\gamma$ is the fraction
of the total energy that ends up inside the PBH222For simplicity we assume
that the PBH mass is a fixed fraction of the horizon mass corresponding to the
smoothing scale; $\gamma\simeq w^{3/2}\simeq 0.2$ during the RD era ($w=1/3$)
[29].. Since PBHs radiate thermally [4], they evaporate on a time scale
$\tau_{{}_{\rm PBH}}(M)\sim
10^{64}\,\left(\frac{M}{M_{\odot}}\right)^{3}\,\rm{yr}\,,$ (2.2)
therefore, the ones with mass greater than $10^{15}$ g survived until now and
would be plausible DM candidates. In this paper, we will focus on DM PBHs
formed in the RD era.
In order to investigate the abundance of formed PBHs, we define a parameter
which represents the mass fraction (the energy density fraction) of PBHs,
$\beta\equiv\frac{\rho_{\rm PBH}(t_{\rm i})}{\rho(t_{\rm i})}\,,$ (2.3)
where the subscript “i” indicates values at the epoch of PBH formation. It is
straightforward to show that this fraction can be related to present day
abundance of PBHs, $f_{\rm PBH}$ [30]
$\beta\simeq 3.7\times
10^{-9}\,\gamma^{-1/2}\,\left(\frac{g_{*,\,i}}{10.75}\right)^{1/4}\left(\frac{M_{\rm
PBH}}{M_{\odot}}\right)^{1/2}\,f_{\rm PBH}\,,$ (2.4)
where $g_{*,\,i}$ is a number of relativistic degree of freedom at the time of
formation. Note that $f_{\rm PBH}\equiv\Omega_{\rm PBH}/\Omega_{\rm DM}$ is a
fraction of PBHs against the total DM component, $\Omega_{\rm DM}$. Thus, for
each mass of PBHs, the observational constraint on $f_{\rm PBH}$ can be
interpreted as that on $\beta$.
The fraction of the energy density of the Universe contained in regions
overdense enough to form PBHs is usually calculated in PS theory [18] as
$\beta=\gamma\int_{\delta_{c}}^{\infty}P(\delta;\,R)\,d\delta\,.$ (2.5)
Here $P(\delta;\,R)$ is the probability distribution function of the linear
density field $\delta$ smoothed on a scale $R$, and $\delta_{c}$ is the
critical threshold for PBHs formation. The estimation of $\delta_{c}$ is under
discussion by different researches both analytically [31] and numerically [32,
33]. In general, the value of $\delta_{c}$ is not unique and depends on the
shape of the primordial curvature power spectrum [34]. The value of
$\delta_{c}$ is varying between 0.4 and 2/3. In this paper, we consider
$\delta_{c}=0.47$ which is the upper bound of the threshold calculated in [34]
for a Gaussian curvature profile 333Note that in general, the value of
$\delta_{c}$ is a function of the equation of state parameter [32]. For
example, a change in the relativistic degrees of freedom at the electroweak
and QCD phase transitions induces a change in $\delta_{c}$. This will have
unavoidable features in the mass function [35, 36].. For the Gaussian
fluctuations, $\beta$ is given by [37]
$\displaystyle\beta(M_{\rm PBH})$
$\displaystyle=\gamma\int_{\delta_{c}}^{\infty}\frac{d\delta}{\sqrt{2\pi\,\sigma^{2}_{\delta}(R)}}\,\exp\left(-\frac{\delta^{2}}{2\,\sigma^{2}_{\delta}(R)}\right)$
$\displaystyle=\frac{\gamma}{2}\,\operatorname{erfc}\left(\frac{\delta_{c}}{\sqrt{2\sigma^{2}}}\right)\,,$
(2.6)
where $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$ is the complementary
error function and, the variance of $\delta$ is given by
$\sigma^{2}_{\delta}(R)=\int\dfrac{dk}{k}\,\mathcal{P}_{\delta}(k)\,\widetilde{W}^{2}(k,\,R)=\int\dfrac{dk}{k}\,\dfrac{16}{81}\left(\dfrac{k}{aH}\right)^{4}\,\mathcal{P_{R}}(k)\,\widetilde{W}^{2}(k,\,R)\,,$
(2.7)
where $\widetilde{W}(k,\,R)$ is the Fourier transform of the window function
used to smooth the density contrast on a comoving scale $R$. In this paper, we
will consider the $k$-space top-hat window function. Recall that
$\mathcal{P}_{\delta}(k)$ is the power spectrum of $\delta$ which is related
to the power spectrum of curvature perturbations on comoving hypersurfaces in
RD era as follows [38]
$\mathcal{P}_{\delta}(k)=\dfrac{16}{81}\left(\dfrac{k}{aH}\right)^{4}\mathcal{P_{R}}(k)\,,$
(2.8)
and we parameterize the curvature power spectrum as
$\mathcal{P_{R}}(k)=A_{0}\,\left(\frac{k}{k_{0}}\right)^{n_{s}(k)-1}\,,$ (2.9)
where $\ln(10^{10}A_{0})=3.044\pm 0.014$ and the spectral index,
$n_{s}(k_{0})=0.9649\pm 0.0042$ are known by the CMB observation at
$k_{0}=0.05$ Mpc-1 [13].
As already mentioned, the mass of PBH, $M_{\rm PBH}$ is related to scale of
horizon at the time of formation when the perturbation reenters the horizon
($k=aH$). It is straightforward to obtain the relation between the mass of
PBHs and the comoving wavenumbers as [39],
$M_{\rm PBH}(k)\simeq
30\,M_{\odot}\,\left(\dfrac{\gamma}{0.2}\right)\,\left(\dfrac{g_{*,\,i}}{10.75}\right)^{-1/6}\left(\dfrac{k_{\rm
PBH}}{2.9\times 10^{5}\,{\rm Mpc}^{-1}}\right)^{-2}\,.$ (2.10)
If the power spectrum of curvature perturbations were scale invariant, using
Eq. (2), and assuming $\delta_{c}\simeq 0.47$ and $\sigma_{\delta}^{2}\propto
A_{0}$, then the initial mass fraction of PBHs formed in the RD era would be
completely negligible. On the other hand, if we assume that all of the DM is
in PBHs with $M_{\rm PBH}\sim M_{\odot}$ (i.e. $f_{\rm PBH}=1$) then, from Eq.
(2.4), the initial PBH mass fraction must be $\beta\sim 10^{-9}$. This means
that the required power spectrum on this scale (see Eq. (2.10)) must be
$\mathcal{P_{R}}({k_{\rm PBH}})\sim 10^{-2}$ which is 7 orders of magnitude
larger than the measured value on cosmological scales $(A_{0}\sim 10^{-9})$.
Thus for DM PBHs formation, the amplitude of power must be enhanced
exponentially in specific wavenumbers corresponding to PBHs mass. One way to
enhance the power is to have a blue-tilted spectral index in the scale of PBHs
formation, $n_{s(b)}(k_{\rm PBH})$. Hence the abundance of PBHs with different
masses (formed in different scales) could put constraints on the power
spectrum beyond the range probed by cosmological observations.
There are various constraints on PBHs abundance. In this paper, we are
interested in PBHs with a mass range from $\sim(10^{16}-10^{46})$ g. Part of
this mass range is probed by the gravitational lensing and the GWs
observations. However, there remain mass windows where PBHs may constitute the
whole DM. We review the potential constraints as follows:
* •
Gravitational Lensing: Long-duration microlensing is a significant probe to
find the contribution of compact objects as DM. These constraints have been
acquired by a) Subaru HSC (Hyper Suprime-Cam) survey observing the
microlensing of stars in M31 (Andromeda galaxy) [24], b) OGLE (Optical
Gravitational Lensing Experiment) microlensing survey of the Galactic bulge
[25], and c) EROS/MACHO [26, 27] which monitors millions of stars in
Magellanic Cloud, respectively.
\- $10^{-11}\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 10^{-6}\,M_{\odot}$ with
$f_{\rm PBH}\sim 10^{-3}\,,$
\- $10^{-6}\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 10^{-3}\,M_{\odot}$ with
$f_{\rm PBH}\sim 10^{-2}\,,$
\- $10^{-3}\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 10^{-1}\,M_{\odot}$ with
$f_{\rm PBH}<0.04$ .
Note that if $f_{\rm PBH}$ is not negligible and $M_{\rm PBH}\sim$ stellar
mass, the Poissonian noise due to the discrete nature of PBHs implies a
boosted formation of PBH at high redshift, compared to the standard model
[40]. In turn, when most PBHs are in massive halos, they can evade the
microlensing limits due to the additional lensing effect of the cluster, as
shown by [36]. However, in this work, we evade these complications and use the
constraints mentioned by [12].
* •
Gravitational Waves: The merger of sub-Solar PBH binaries emits GWs like those
observed by LIGO/Virgo [28]. The observed merger rates of the second
LIGO/Virgo run impose constraints on PBH fraction in the following mass range
\- $0.2\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 1.0\,M_{\odot}$, corresponding
to at most $0.16$ or $0.02$ of the DM, respectively 444For the Poisson noise,
the limits from GWs and BH merging rates still allow $f_{\rm PBH}>0.1$ [41]..
The main constraints on solar mass PBH are GW and lensing, which both are
model-dependent. For example, one can evade the lensing constraints by
clustering [36]. Therefore, we consider the optimistic fraction
$f_{\text{PBH}}=1$ as well.
\- intermediate mass range, $10\,M_{\odot}\lesssim M_{\rm PBH}\lesssim
10^{3}\,M_{\odot}$: In this mass range there are several limits such as ultra-
faint dwarf galaxies, X-rays and radio binaries, wide binaries, and CMB
distortion limits [12]. Thus we choose the average limit of $f_{\rm PBH}\sim
10^{-4}$.
* •
For the following mass ranges there is no (model independent) confirmed
constraints. Therefore, PBHs could be the whole DM, i.e. $f_{\rm PBH}\sim 1$
[12].
\- asteroid mass range $10^{16}\,{\rm g}\lesssim M_{\rm PBH}\lesssim 10^{17}$
g ,
\- sublunar mass range $10^{20}\,{\rm g}\lesssim M_{\rm PBH}\lesssim 10^{24}$
g ,
\- stupendously large black holes (SLABs) $M_{\rm PBH}\geq 10^{11}\,M_{\odot}$
[42].
Thus, in this paper, we will consider eight mass ranges. It is worth to
mention that we only consider the constraints without any assumption about the
distribution of PBHs at their formation, their clustering and the environment
since these conditions can have a significant effect in their abundances.
## 3 Abundance of PBH in the Context of EST
As already mentioned in section 2, the PS formalism can determine the
abundance of PBHs. In this section, we will consider this issue in EST as a
sophisticated development of PS formalism. In subsection 3.1, we will review
the main idea of EST, and in subsection 3.2, we will have our main results on
studying the abundance of PBHs in this context.
### 3.1 Theoretical Background
According to the EST, the spherical/ellipsoidal collapse of a compact object
of mass $M$ forms where the corresponding smoothed linear density contrast,
$\delta$ is larger than critical value, $\delta_{c}$ [43]. In this framework,
the computation of the mass function of a compact object is formulated in
terms of a stochastic process. In other words, if we smooth the density
contrast in an initial box (higher redshifts where perturbations are almost
linear and Gaussian) and plot the density contrast in terms of variance of
perturbations $S$, it executes a stochastic time series. Since $\delta$ is a
function of variance, its evolution is governed by Gaussian white noise which
implies the random walk with respect to $S$. The trajectories start from an
initial value $\delta=0$ at $S=0$ (see Fig. 1).
Figure 1: The Markov trajectories were performed by the smoothed density
contrast. The trajectories which pass the threshold, $\delta_{c}$ for the
first time are called a) First Up-Crossing (FU): at lower $S$ (larger mass) b)
First Touch (FT): at higher $S$ (smaller mass). The blue inverted coordinates
show the comoving scale versus cosmic time. The dashed inclined lines show the
dependence of the comoving horizon $\left(R_{H}=(aH)^{-1}\right)$ to scale and
time. The dash-dotted and dotted blue vertical lines correspond to the
perturbation modes reentering the horizon in radiation dominated (RD) era,
corresponding to FT and FU concepts, respectively.
In this context, the first up-crossing (FU) trajectories that pass
$\delta_{c}$ will give the mass of the collapsed object. Accordingly, the
statistics of FU trajectories will give the distribution of collapsed objects.
If we use the $k$-space top-hat window for smoothing, the random walks become
Markovian and the distribution of FU trajectories is given by
$f_{\rm
FU}(S,\,\delta_{c})=\dfrac{1}{\sqrt{2\pi}}\,\dfrac{\delta_{c}}{S^{3/2}}\exp\left(-\frac{\delta^{2}_{c}}{2S}\right)\,,$
(3.1)
where the variance of density perturbations, $S$ is given by Eq. (2.7).
We will produce Markov trajectories by the method developed in our previous
works [44, 45, 46] where the distribution of DM halos in the (extended)
standard model are addressed in more detail.
It is worth mentioning that there is a fundamental difference between the
statistics of DM halos and PBHs. PBHs form when the scale corresponding to
their mass reenters the horizon. In Fig. 1, the vertical lines represent the
modes that reenter the horizon on a specific scale. We should consider the
trajectories which touch the barrier in larger variance for the first time,
hereafter first touch (FT). This approach is reasonable since the scale of PBH
with lower mass (larger variance) reenters the horizon sooner than the massive
one. The concept of FT is clear from Fig. 1, where the FT trajectories touch
the threshold in larger variance; i.e. smaller mass PBHs form first. For more
clarification, compare the vertical dashed-dotted lines for small mass PBH
corresponding to FT with vertical dotted lines for larger masses corresponding
to FU. Accordingly, the number of FT trajectories at each variance will give
the abundance of PBHs with corresponding mass. Note that according to
correspondence between PS and EST, the cumulative summation of FU distribution
(Eq. (3.1)) is equal to the fraction of PBHs at their formation, $\beta$ (see
Eq. (2.5)). We show that by knowing the FU distribution for any given power
spectrum, one can find $\beta$ and $f_{\rm PBH}$ (see Eq. (2.4)). The fraction
of PBHs is given by counting the FT trajectories. However, statistically, the
FT distribution is a correction to the FU ones with almost similar behaviour.
To clarify this issue, in the schematic Fig. 2, we plot the FU and FT
distribution (top plot) and their cumulative distribution $\beta$ (bottom
plot) for an arbitrary mass range. FT distribution is larger than FU at
smaller masses and vice versa. One could anticipate this result as the FT is
related to the crossings in the smaller mass ranges in comparison to FU. The
cumulative distribution, $\beta$ of both FT and FU is almost the same for the
lower bound of the mass range.
Figure 2: Top panel: The Schematic illustration of both FU (solid line) and FT
(dotted line) for $f$ (fraction) versus mass is plotted. Bottom panel: The
corresponding cumulative fraction, $\beta$ versus mass for both FU (solid
line) and FT (dotted line) is shown.
### 3.2 PBH in the Context of EST
To calculate the abundance of PBHs at different scales, the distribution is
confined with a window function to smooth out modes smaller than the horizon.
In the EST formalism for halo formation, the variance could be calculated at
redshift $z=0$ and density threshold, $\delta_{c}$ changes with growth
function, $D(z)$. However in the case of PBH, we use a constant
$\delta_{c}=0.47$ in RD era [34] and the power spectrum is identified in
various horizon crossing scales, $R_{H}=(aH)^{-1}$. This means the redshift
dependence is encapsulated in power spectrum instead of $\delta_{c}$. So we
have a unique power spectrum for each horizon scale $R_{H}$ which is related
to a specific mass. Therefore, Eq. (2.8) at horizon crossing scale $R_{H}$ is
given by
$\mathcal{P}_{\delta}(k,\,R_{H})\equiv\mathcal{P}_{\delta,R_{H}}(k)=\dfrac{16}{81}A_{0}\,\left(kR_{H}\right)^{4}\left(\dfrac{k}{k_{0}}\right)^{n_{s}(k)-1}\,.$
(3.2)
We use a variable spectral index to apply a blue-tilted power spectrum in a
broad interval $k\sim[k_{s},\,k_{l}]$ where $k_{s}$ and $k_{l}$ are short and
long wavenumbers. This means
$n_{s}(k)=\begin{cases}n_{s}(k_{0})\,,&k<k_{s}\\\
n_{s(b)}>1\,,&k_{s}<k<k_{l}\\\ n_{s}(k_{0})\,,&k>k_{l}\\\ \end{cases}$ (3.3)
Figure 3: Top panel: The enhanced initial power spectrum is plotted versus
wavenumber. The enhancement is for blue-tilted (dashed and dotted lines) and
red-tilted with $A=10^{11}A_{0}$ (solid line) models in wavenumber range of
$k\sim[k_{s},\,k_{l}]$. Bottom panel: The variance, $S$ and the fraction, $f$
versus PBH mass is plotted in left and right panels, respectively.
where $n_{s}(k_{0})=0.9649\pm 0.0042$ and $n_{s(b)}$ are the observed spectral
index and the blue-tilted one, respectively. In Fig. 3, top panel, we show the
blue-tilted power enhancement in the wavenumber interval
$k\sim[k_{s},\,k_{l}]$. Since we have a whole sequence of $S$ for different
scales, $R_{i}$ at any specific horizon crossing scale $R_{H}$, we rewrite Eq.
(2.7) as follows
$S_{R_{H}}(R_{i})=A\,R_{H}^{4}\int_{0}^{\infty}dk\,k^{n_{s}(k)+2}\,\widetilde{W}^{2}(k,\,R_{i})\,,$
(3.4)
where $A\equiv\dfrac{16}{81}\dfrac{A_{0}}{k_{0}^{n_{s}(k)-1}}$. The above
equation means that using each sequence of $S$, we form a whole realization of
trajectories in $(\delta-S)$ plane for each $R_{H}$. Hence, we could count the
FT numerically, i.e. $f_{\rm FT,\,R_{H}}$. Therefore, for PBHs which form at
horizon crossing, $f_{\rm FT,\,R_{H}}$ is only meaningful where the FT is
counted at $R_{i}=R_{H}$.
Note that since PBHs formation at horizon reentry is independent of other
scales, therefore their formation mechanism is history independent. In the
EST, one can get memory independent trajectories by using the $k$-space top-
hat window function which is known as Markov model555Note that the non-Markov
extension of EST has great importance in halo formation due to more physical
smoothing window functions which lead to correction of statistics of FUs,
especially in small scales [44, 45, 47].. For Markov model Eq. (3.4) is given
by
$\displaystyle\frac{S_{R_{H}}(R_{i})}{A\,R_{H}^{4}}$
$\displaystyle=\frac{1}{R_{i}^{n_{s}+3}}\int_{0}^{min(\frac{R_{i}}{R_{s}},\,1)}{dk\,k^{n_{s}+2}}$
(3.5)
$\displaystyle\,+\,\frac{1}{R_{i}^{n_{s(b)}+3}}\int_{min(\frac{R_{i}}{R_{s}},\,1)}^{min(\frac{R_{i}}{R_{l}},\,1)}{dk\,k^{n_{s(b)}+2}}$
$\displaystyle\,+\,\frac{1}{R_{i}^{n_{s}+3}}\int_{min(\frac{R_{i}}{R_{l}},\,1)}^{\infty}{dk\,k^{n_{s}+2}}\,.$
The $k$-space window function applies sharp conditions in upper/lower limits
of the integral.
Note that due to random nature of these trajectories, we only need to apply
one sequence of $S$ and substitute $S(R_{H})\equiv S_{R_{H}}(R_{i}=R_{H})$. We
can analytically calculate $f_{\rm FU}$ and count the first touch of
trajectories to evaluate $f_{\rm FT}$. In counting process, we will exclude
any trajectory which is already collapsed to a PBH.
Figure 4: Different trajectories are shown for different mass ranges. Higher
masses require larger spectral index for their formation. In any mass range,
the number of trajectories that pass the threshold is higher towards the
smaller mass. The FU distribution (Eq. (3.1)) is shown in the inset figure.
PBH formation needs amplification in primordial power spectrum. We modify the
initial power spectrum introduced by early universe physics in a small
wavenumber, where we do not have the constraints of CMB and LSS. We model this
amplification by phenomenological enhancement. In Fig. 3, we show the red and
blue-tilted enhanced curvature power spectrum in wavenumber range of
$k\sim[k_{s},\,k_{l}]$ for a specific case. Note that for the red-tilted power
enhancement, one needs an amplitude of an order $10^{11}A_{0}$. We also show
that the variance and FU statistics change due to these two types of
modifications are almost in the same order of magnitude (bottom panels for $S$
and $f$ versus mass) because the first crossing statistics depend only on the
variance. The variance is a monotonic decreasing function of mass (scale),
which is obtained by an integral over the power spectrum (see Eqs. (3.3) and
(3.4)). In what follows, we use the blue-tilted power spectrum for PBHs
formation, where the modification in spectral index requires the desired
amplification in the mass range more naturally with the same amplitude of
$A_{0}$. Therefore, we need to enhance the power spectrum on a specific scale
(see Eq. (3.5)). In Fig. 4, we show three schematic realizations of
trajectories corresponding to three different blue-tilted spectral indexes at
different mass ranges (different variances). Although all realizations of
trajectories have the same $f_{\rm FU}$ distribution versus $S$ (see Eq. (3.1)
and inset Fig. 4), however, the difference between realizations is due to the
variation of $S$ at different masses. These realizations also show that a
larger spectral index leads to a higher PBH mass. In each mass range, the
number of trajectories that pass the threshold is higher in the smaller mass
range. Note that although $f_{\rm FU}$ has a maximum in variance ($S_{max}$),
however, the maximum fraction of PBHs are formed in declining part of the
$f_{\rm FU}$ which justified by the $f_{\rm PBH}$ constraints proposed in
table 1.
Hence, the obtained value of $S$ from Eq. (3.5) will be larger than variance,
which makes the fraction maximum. This calculated $S$ corresponds to a
specific blue-tilted spectral index required for PBH formation (see Fig. 4).
We should note that $S$ is highly sensitive to spectral index, therefore a
small increase (decrease) in $n_{s(b)}$ leads to a larger (smaller) mass PBH.
For stupendously large masses [42] the obtained value of $S$ could be close to
$S_{max}$.
We report the results of our studies for eight desired mass ranges in table 1.
In each mass range, we consider intervals with different lower limits because
the statistic of FU/FT is higher for the lower mass of the interval. In this
table, $f_{\rm PBH}$ is based on observational constraint if there exist,
otherwise, we suppose that PBHs comprise the whole of DM. We report the
required value of blue-tilted spectral indexes for both EST and PS formalism
for comparison. One can see that these values are close to each other as
expected.
Figure 5: From left for right: $f_{\rm FU/FT}$, $\beta$, and $f_{\rm PBH}$
with respect to the mass of DM PBHs. From top to bottom: OGLE, intermediate
and SLAB mass ranges. Note that $f_{\rm PBH}$ is the required constraint for
each mass range reported in table 1.
| mass range | $f_{\rm PBH}$ | lower bound | spectral index
---|---|---|---|---
| of mass range | EST | PS
asteroid mass range | $\left(10^{16}-10^{17}\right)\,{\rm g}$ | 1 | $10^{16}$ g | 1.490 | 1.616
sublunar mass range | $\left(10^{20}-10^{24}\right)\,{\rm g}$ | 1 | $10^{20}$ g | 1.540 | 1.703
$10^{22}$ g | 1.600 | 1.756
Subaru HSC | $\left(10^{-11}-10^{-6}\right)\,M_{\odot}$ | $10^{-3}$ | $10^{-10}\,M_{\odot}$ | 1.604 | 1.560
$10^{-8}\,M_{\odot}$ | 1.666 | 1.605
OGLE | $\left(10^{-6}-10^{-3}\right)\,M_{\odot}$ | $10^{-2}$ | $10^{-6}\,M_{\odot}$ | 1.729 | 1.757
$10^{-4}\,M_{\odot}$ | 1.845 | 1.835
EROS/MACHO | $\left(10^{-3}-10^{-1}\right)\,M_{\odot}$ | $0.04$ | $10^{-3}\,M_{\odot}$ | 1.862 | 1.947
$10^{-2}\,M_{\odot}$ | 1.942 | 1.970
sub-Solar mass range* | $\left(0.2-1.0\right)\,M_{\odot}$ | $0.02$ | $0.2\,M_{\odot}$ | 2.018 | 2.046
$0.6\,M_{\odot}$ | 2.115 | 2.078
$1$ | $0.2\,M_{\odot}$ | 2.028 | 2.258
$0.6\,M_{\odot}$ | 2.126 | 2.297
intermediate mass range | $\left(10^{1}-10^{3}\right)\,M_{\odot}$ | $10^{-4}$ | $10\,M_{\odot}$ | 2.103 | 1.848
$10^{2}\,M_{\odot}$ | 2.204 | 1.911
SLABs | $\geq 10^{11}\,M_{\odot}$ | 1 | $10^{11}\,M_{\odot}$ | 5.220 | 5.598
$10^{12}\,M_{\odot}$ | 6.660 | 6.891
Table 1: Reported spectral indexes in EST and PS for eight mass ranges based
on $f_{\rm PBH}$ constraints. * Note that the sub-Solar mass range is studied
with conservative limit of $f_{\rm PBH}=0.02$ and optimistic limit $f_{\rm
PBH}=1$.
In Fig. 5, we show the abundance of PBHs by counting both the FT and FU for
OGLE, intermediate, and SLAB mass ranges from top to bottom, respectively. In
each mass range, we show $f_{\rm FU/FT}$, $\beta$, and $f_{\rm PBH}$ from left
to right, respectively. Our numerical results show that despite considering a
broad spectrum characterized by a range of scales running from $1/k_{l}$ to
$1/k_{s}$, production of PBHs will be dominated by smaller masses (i.e.
smaller scales, $1/k_{l}$). For SLAB mass range we show both FU and FT,
however for smaller mass ranges due to computational limits, we only report
FU. According to Eq. (2.4), $f_{\rm PBH}$ is much larger than $\beta$ for
small mass range, which is clear from the middle and the right plots of Fig.
5. This means that to get the desired $f_{\rm PBH}$ for smaller mass, we need
very low FU/FT. And since we calculate FU analytically and FT numerically,
small FT abundance requires enormous computational effort to produce enough FT
trajectories. Since we have already shown in Fig. 2 that there is no
considerable difference between FT and FU, therefore it is convenient to
report only the results of FU for small mass ranges.
The constraints on the spectral indexes (see table 1) can be translated to the
upper limit on the power spectrum of curvature perturbations,
$\mathcal{P_{R}}$ by using Eq. (2.10) to find wavenumbers corresponding to PBH
masses. We obtain $\mathcal{P_{R}}(k_{\rm PBH})\lesssim 10^{-1}$ with some
scale dependence in the range of $k_{\rm
PBH}\sim(10^{5}-10^{15})\,\text{Mpc}^{-1}$. The power spectrum corresponding
to the scale of SLABs ($k\sim(1-5)\,\text{Mpc}^{-1}$) is tightly constrained
by Lyman-$\alpha$ [48]. Hence, the whole DM can not be made of SLAB PBHs.
## 4 Conclusions
Although the Primordial black holes were first theorized decades ago, the
first gravitational wave detection by LIGO/Virgo, the great interest towards
PBHs as a DM candidate has revived. Consequently, there have been significant
improvements in the theoretical calculations of PBH formation and the
observational constraints on their abundance. Traditionally, the Press-
Schechter (PS) formalism has been used for PBHs formation. In this work, we
use the Excursion Set Theory (EST) as a reasonable and sophisticated extension
of PS. In the EST formalism, the trajectories which first up-cross (FU) the
density threshold, $\delta_{c}$ (which changes with growth function), form a
dark matter halo. However, for PBHs, we implement the EST with a constant
density threshold and encapsulate the redshift dependence in the power
spectrum. We also introduce a new concept of first touch (FT) instead of FU
for the first time (see Fig. 1 and related discussion in section 3.1). Since
PBHs form when the scale corresponding to their mass reenters the horizon,
this leads to the use of the Markov trajectories as a memory-independent
method. We showed in any mass range, the small PBHs form first and dominate
the mass profile of PBHs. For this reason, we count FT at smaller masses
instead of FU (see Fig. 2).
Since PBHs formation is only possible by enhancing the power spectrum at small
scales, we use a blue-tilted power spectrum in our analysis. We also show that
a red-tilted power with a larger value of amplitude leads to almost similar
results in PBH formation. Our numerical results showed that in a broad
spectrum, the production of PBHs is dominated by smaller masses. We report the
results of required blue-tilted spectral indexes in table 1 for eight mass
ranges in the EST formalism. As a by-product, we compared EST with PS, and we
showed that they are in good agreement in the considered masses. The
constraints on the spectral indexes were translated to the upper bound on
curvature power spectrum, $\mathcal{P_{R}}(k_{\rm PBH})\lesssim 10^{-1}$ with
some scale dependence in the range of $k_{\rm
PBH}\sim(10^{5}-10^{15})\,\text{Mpc}^{-1}$.
It is worth mentioning that our approach could be applied for any
redshift/scale dependence of the density threshold. For future work, we will
apply the EST to study the merger history of PBHs and their redshift
evolution. Also, it worths extending our work to non-Markov and non-Gaussian
perturbations.
## Acknowledgments
We are grateful to Sohrab Rahvar for his insightful comments on the
manuscript. EE and SB are partially supported by Abdus Salam International
Center of Theoretical Physics (ICTP) under the junior associateship scheme.
They thanks the hospitality of ICTP where this work is initiated. We thank
Nasim Derakhshanian for her collaboration in the early stage of this work. SB
is supported by Sharif University of Technology Office of Vice President for
Research under Grant No. G960202.
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|
# Merger-ringdown consistency: A new test of strong gravity using deep
learning
Swetha Bhagwat<EMAIL_ADDRESS>Dipartimento di Fisica,
“Sapienza” Università di Roma & Sezione INFN Roma1, Roma 00185, Italy
Costantino Pacilio<EMAIL_ADDRESS>Dipartimento di Fisica,
“Sapienza” Università di Roma & Sezione INFN Roma1, Roma 00185, Italy
###### Abstract
The gravitational waves emitted during the coalescence of binary black holes
are an excellent probe to test the behaviour of strong gravity. In this paper,
we propose a new test called the _merger-ringdown consistency test_ that
focuses on probing the imprints of the dynamics in strong-gravity around the
black-holes during the plunge-merger and ringdown phase. Furthermore, we
present a scheme that allows us to efficiently combine information across
multiple ringdown observations to perform a statistical null test of GR using
the detected BH population. We present a proof-of-concept study for this test
using simulated binary black hole ringdowns embedded in the next-generation
ground-based detector noise. We demonstrate the feasibility of our test using
a deep learning framework, setting a precedence for performing precision tests
of gravity with neural networks.
## I Introduction
The detection of gravitational waves (GWs) emitted during the binary black
hole (BH) mergers presents us with an unparalleled opportunity to test the
behaviour of strong gravity around BHs Abbott _et al._ (2020); Nitz _et al._
(2020). GWs are emitted as the BHs slowly spiral in towards the common center
of mass (a.k.a. the inspiral phase); this is followed by a rapid plunge and
merger (a.k.a. the plunge-merger phase) where the two BHs coalesce forming a
remnant BH which then rings down and settles to a final state (a.k.a. the
ringdown phase) Maggiore (2007). While both plunge-merger and ringdown contain
imprints of the dynamics in the strong field regime at few times the horizon-
length scale, the plunge-merger is dictated by non-linear dynamics and the
ringdown is prescribed by linear perturbation theory Kokkotas and Schmidt
(1999); Konoplya and Zhidenko (2011); Berti _et al._ (2009).
Ringdown corresponds to the evolution of linear perturbations on the space-
time metric of the remnant Pani (2013). Given an underlying theory of gravity,
the dynamics in the strong-field regime that sets up the perturbation
conditions for ringdown and the properties of the remnant BH are not
independent. If GR were to be modified in the strong non-linear regime, one
would expect the relative excitation of modes in ringdown as well as the final
BH’s mass and spin to be altered Kamaretsos _et al._ (2012a); Hughes _et
al._ (2019). We propose a novel test that checks if the excitation conditions
set during the plunge-merger phase are consistent with the properties of the
remnant BH formed after the ringdown phase. The proposed test checks for the
consistency by simultaneously using the frequency content and the amplitudes
and phases of excitation in the ringdown signals. Furthermore, we stack the
information from multiple GW observations efficiently to provide a statistical
‘null’ test across a population of binary BH ringdowns. Henceforth, we call it
_the merger-ringdown consistency test_.
We present a complementary test to the already existing battery of tests of
GR. Although we draw our inspiration from the IMR test, the two tests address
conceptually different questions: while IMR test checks for global consistency
of the binary BH evolution Ghosh _et al._ (2016); Abbott _et al._ (2016);
Ghosh _et al._ (2018); Abbott _et al._ (2020), violation of the merger-
ringdown test indicates GR-modifications that alter the perturbations setup
for ringdown in a way that is inconsistent with the expected radiated angular
momentum and energy in a binary BH coalescence. However, note that such GR
modifications (depending on the details of how GR is modified) might also
leave imprints on the global evolution of the binary BH signals, and could be
picked up by the IMR test. Furthermore, the merger-ringdown test aims at
increasing the sensitivity to the strong field dynamics by zooming in solely
on the ringdown phase. Comparing the performance of the two tests in
distinguishing GR from non-GR signals is non-trivial and depends on the class
of modifications in consideration.
Our test is particularly suited for the third-generation (3g) detectors such
as the Einstein Telescope (ET) Maggiore _et al._ (2020), the Cosmic Explorer
Reitze _et al._ (2019) and LISA Amaro-Seoane _et al._ (2017), where the
ringdowns are expected to be loud and the number of detections can be $\sim
10^{3}-10^{4}/$year Berti _et al._ (2016); Bhagwat _et al._ (2016).
Performing prognostic and realistic benchmarking studies on a large number of
events with full Bayesian parameter estimation demands for a rapid and
computationally efficient inference algorithm. To this aim, we demonstrate the
feasibility of our test entirely using a deep learning framework to speed up
the parameter inference by orders of magnitude Green _et al._ (2020); Gabbard
_et al._ (2019); Yamamoto and Tanaka (2020). We train a neural network
architecture called a conditional variational autoencoder (CVAE) Kingma and
Welling (2013); Doersch (2016); Erdogan to infer posterior distributions of
the parameter set $\\{M,\chi_{f},q\\}$ from a set of simulated ringdown
waveforms. Following the deep learning application to GW science — e.g.,
detection Gabbard _et al._ (2018); George and Huerta (2018a, b); Iess _et
al._ (2020); Wei _et al._ (2021) and parameter estimation (PE) Shen _et al._
(2019); Chua _et al._ (2019); Chua and Vallisneri (2020); Gabbard _et al._
(2019); Green _et al._ (2020); Green and Gair (2021),111See Cuoco _et al._
(2020) for a recent review. our work also sets a precedence for precision
tests of GR using neural networks. Finally, we also demonstrate that deep
learning techniques can be efficiently used for population studies for current
and next-generation GW detectors.
## II Merger-ringdown consistency test
### II.1 Theory
Ringdown is modelled as a linear superposition of damped sinusoids with
characteristic BH frequencies ($f_{lm}$) and damping times ($\tau_{lm}$) known
as the quasi-normal-mode (QNM) spectrum. It is generally decomposed in spin-2
weighted spheroidal harmonic basis $\mathcal{Y}^{lm}(\iota)$, where
$(\iota\in[0,\pi))$ is the inclination angle. Ringdowns take the following
analytical form 222For simplicity, we decompose the ringdown signal on to
spin-2 weighted spherical harmonics basis instead of the more natural spin-2
weighted spheroidal harmonics basis. This assumption is reasonable as long as
the spins are not too high and can be estimated following Berti and Klein
(2014).
$\displaystyle
h_{+}(t)=\frac{M}{D_{L}}\sum_{l,m>0}\mathcal{Y}^{lm}_{+}(\iota)A_{lm}e^{-t/\tau_{lm}}\cos(2\pi
f_{lm}t-\phi_{lm}),$ (1a) $\displaystyle
h_{\times}(t)=\frac{M}{D_{L}}\sum_{l,m>0}\mathcal{Y}^{lm}_{\times}(\iota)A_{lm}e^{-t/\tau_{lm}}\sin(2\pi
f_{lm}t-\phi_{lm}).$ (1b)
Here $\\{+,\times\\}$ are the GW polarizations and $D_{L}$ is the luminosity
distance of the system. The QNMs are indexed by the angular multipole numbers
$(l,m)$ and they are determined by the final mass and final spin
$\\{M,\chi_{f}\\}$, i.e., $f_{lm}=f_{lm}(M,\chi_{f})$ and
$\tau_{lm}=\tau_{lm}(M,\chi_{f})$. $A_{lm}$ and $\phi_{lm}$ are the amplitudes
and the phases of excitations of QNMs. For a non-spinning binary, the initial
system is completely characterized by the total-mass $M_{tot}$ and the binary
mass ratio $q$. While $M_{tot}$ sets the overall amplitude scale, $q$
determines the relative excitations of QNMs, i.e.,
$A_{lm}/A_{22}=A^{R}_{lm}(q)$ and $\phi_{22}-\phi_{lm}=\delta\phi_{lm}(q)$.
Thus, the ringdown waveform can be parameterized by a set of three parameters
$\\{M,\chi_{f},q\\}$ and Eq. (1) can be re-written as
$h_{+}(t)=h_{+}(t;M,\chi_{f},q)\,,\quad
h_{\times}(t)=h_{\times}(t;M,\chi_{f},q)\,.$ (2)
Using the ringdown phase of the GW event one can infer $\\{M,\chi_{f},q\\}$ by
treating them as independent quantities in a Bayesian PE setup.
Next, in GR, a given set of $\\{M_{tot},q\\}$ can be deterministically mapped
to $\\{M,\chi_{f}\\}$ for a non-spinning binary BH system.333The remnant BH
can be expressed in terms of the initial binary BH parameters by fitting the
numerical relativity simulations. The relationship can be expressed explicitly
in approximate analytical forms Barausse and Rezzolla (2009); Pan _et al._
(2011); Barausse _et al._ (2012); Hofmann _et al._ (2016); Husa _et al._
(2016); Jiménez-Forteza _et al._ (2017); Healy and Lousto (2017), or
implicitly using machine learning algorithms Varma _et al._ (2019a); Haegel
and Husa (2020); Varma _et al._ (2019b). The three ringdown parameters
$\\{M,\chi_{f},q\\}$ are not truly independent and, in particular, we map
$\chi_{f}$ to $q$ using the fitting formula presented in Pan _et al._ (2011)
(see also Barausse and Rezzolla (2009); Barausse _et al._ (2012); Hofmann
_et al._ (2016))
$\chi_{f}=2\sqrt{3}\eta-3.871\eta^{2}+4.028\eta^{3}+\mathcal{O}(\eta^{3})$ (3)
where $\eta=q/(1+q)^{2}$.
The test checks if the independent measurements of $\\{M,\chi_{f},q\\}$ from
the ringdowns are consistent with the relation between $\chi_{f}$ and $q$ as
predicted by GR. Specifically, we check if the $\chi_{f}$ directly measured
from the ringdown agrees with the $\chi_{f}$ calculated by plugging the
measured value of $q$ in Eq. (3).
### II.2 Prescription for the merger-ringdown consistency
Let a population of non-spinning binary BHs ringdowns be detected by a GW
observatory. Note that the quantities directly _measured_ in PE have a
superscript _‘meas’_ and those _inferred_ using Eq. (3) have _‘infer’_.
1. 1.
Parametrize the ringdown as in Eq. (2) and estimate $\\{M^{\rm
meas},\chi_{f}^{\rm meas},q^{\rm meas}\\}$ for each event. We used the median
of the marginalized posterior distribution as the ‘measured’ value.
2. 2.
For each event, compute the $\chi_{f}^{\rm infer}$ from the median value of
$q^{\rm meas}$ in step 1 using the relation in Eq. (3).
3. 3.
Make a scatter plot with $\\{\chi^{\rm infer},\chi^{\rm meas}\\}$ using all
ringdown observations. In GR, one expects that all the data should lie along
the $\chi^{\rm infer}=\chi^{\rm meas}$ line in a 2-D scatter plot, with the
noise in the data leading to a spread around this line. To perform the merger-
ringdown consistency test, we express
$\chi_{f}^{\rm meas}=a+b\,\chi_{F}^{\rm infer}$ (4)
and fit for the parameters $\\{a,b\\}$. If the best-fit parameters for Eq. (4)
are compatible with $\\{a=0,b=1\\}$ the observations are consistent with GR,
providing a statistical null test.
### II.3 Details of Implementation
For simplicity, we restrict our study to non-spinning quasi-circular binary
BHs. We compute the QNM spectra
$\\{f_{lm}(M,\chi_{f}),\tau_{lm}(M,\chi_{f})\\}$ using the data in Berti .
Further, we focus our attention on the dominant mode $(l,m)=(2,2)$ and the two
most excited subdominant angular modes for the case of non-spinning systems —
$(l,m)=\\{(2,1),(3,3)\\}$ Bhagwat _et al._ (2018); Jiménez Forteza _et al._
(2020). We concentrate solely on the dominant overtone, i.e.,
$n_{\mathrm{overtone}}=0$ Kamaretsos _et al._ (2012b); Bhagwat _et al._
(2020). We use these simplifying assumptions for this proof-of-concept study.
However, note that including more angular modes and overtones is a tangible
extension to our work.
Key to our study is the expression of the QNM excitation amplitudes and
phases, as functions of $q$. Following the prescription in Jiménez Forteza
_et al._ (2020), we express $A^{R}_{lm}$ and $\delta\phi_{lm}$ as
$\displaystyle
A^{R}_{lm}(q)=a_{0}+\frac{a_{1}}{q}+\frac{a_{2}}{q^{2}}+\frac{a_{3}}{q^{3}}\,,$
(5a) $\displaystyle\delta\phi(q)=b_{0}+\frac{b_{1}}{b_{2}+q^{2}}\,,$ (5b)
where we use the convention in which $q>1$. An updated list of coefficients
$\\{a_{i},b_{i}\\}$ is provided in the Supplemental Material. Further, for the
dominant mode’s amplitude, we use $A_{22}=0.86\eta$ Gossan _et al._ (2012).
Lastly, we assume a uniform support in $\phi_{22}\in[0,2\pi]$ and generate the
waveforms expressed in Eq. (1).
## III Deep Learning Framework
We use a deep learning framework to reconstruct the posteriors for the
parameters $\\{M,\chi_{f},q\\}$ from the waveform. We follow Gabbard _et al._
(2019); Green _et al._ (2020); Yamamoto and Tanaka (2020) and train a CVAE, a
neural network architecture well suited to posterior sampling.
### III.1 Details on the CVAE implementation
The CVAE acts as an inverse nonlinear map from the ringdown strain $h=y(x)$ to
the posteriors of $x=\\{M,\chi_{f},q\\}$.
$\mathrm{CVAE}:y\to p(x|y)\,.$ (6)
It has the structure of a variational autoencoder Kingma and Welling (2013);
Doersch (2016): it is made of two serial neural network units (the ‘encoder’
and the ‘decoder’) separated by a stochastic latent layer. The first neural
network (encoder) maps the input $y$ into the latent layer. The second neural
network (decoder) maps the latent representation of the input into the output
probability distribution $p(x|y)$.
The CVAE is trained by introducing a third neural network unit, called the
auxiliary encoder or the ‘guide’, at training time Tonolini _et al._ (2020).
The training consists in optimising a loss function. For the CVAE, the loss
naturally splits into Tonolini _et al._ (2020); Gabbard _et al._ (2019):
1. 1.
the Kullback-Leibler (KL) divergence $\mathcal{L}_{\rm KL}$, measuring the
similarity between the outputs of the encoder and the guide; it quantifies the
ability of the encoder to produce a meaningful mapping of the input into the
latent space;
2. 2.
the reconstruction loss $\mathcal{L}_{\rm recon}$, measuring the probability
that the true values $x_{\rm true}$ falls within the decoder distribution.
The total loss to be optimised is
$\mathcal{L}_{\rm tot}=\mathcal{L}_{\rm recon}+\beta\mathcal{L}_{\rm KL}\,.$
(7)
When $\beta=1$, $\mathcal{L}_{\rm tot}$ coincides with the standard ELBO loss
Kingma and Welling (2013); Doersch (2016). The additional parameter $\beta$
gives flexibility in implementing effective training strategies.
After the training, the guide is dropped out. At production time, only the
encoder and the decoder are used to sample the posteriors $p(x|y)$. Fig. 1
contains a flow-diagram of the training and production steps. More details on
the neural network architecture and our codes are provided in a dedicated git
repository mrt .
Figure 1: A schematic representation of the CVAE architecture. On the left, a
single training step is represented. First, the signal $y$ is mapped by the
encoder into a latent stochastic distribution, which is a multivariate
diagonal Gaussian with means and standard deviations
$\\{\vec{\mu}_{1},\vec{\sigma}_{1}\\}$; similarly, the couple $(y,x_{\rm
true})$ is mapped by the guide into a second Gaussian with parameters
$\\{\vec{\mu}_{0},\vec{\sigma}_{0}\\}$; the two are then combined into the
$\mathcal{L}_{\rm KL}$ loss. Next, a latent variable $z$ is sampled from the
guide distribution and mapped by the decoder into a third Gaussian with
parameters $\\{\vec{\mu}_{2},\vec{\sigma}_{2}\\}$. This final distribution is
eventually used to compute the $\mathcal{L}_{\rm recon}$ loss. On the right, a
single step at production time is shown. Now, the latent representation is
sampled from the encoder and a predicted output $x_{\rm pred}$ is sampled from
the decoder; this step is repeated $n_{\rm samples}$ times to produce an
informative posterior distribution for $x$; in text, we fix $n_{\rm
samples}=10^{4}$. Note the final distribution of $x_{\rm pred}$ is not a
Gaussian, but is a complex distribution resulting from the convolution of the
two serial sampling steps.
The hyperparameters which determine the CVAE training are listed in Tab. 1.
Batch size | $512$
---|---
Epochs | $500$
Optimizer | Adam
Initial lr | $10^{-4}$
lr decay | $\times 0.5$ every $80$ epochs
$\beta$ annealing | $3\times[10^{-5},\frac{1}{3},\frac{2}{3},1,1,1]$
Validation fraction | $10\%$
Table 1: Hyperparameters used for training the CVAE.
We train the CVAE in batches of $512$ waveforms for $500$ epochs, i.e., $500$
forward-backward passes of the entire training set. The loss is minimized
using the Adam optimizer with an initial learning rate set to $10^{-4}$. The
learning rate is decreased by a factor of $2$ every $80$ epochs. To monitor
the convergence of the loss, we set aside $10\%$ of the training dataset and
we use it for validation. Following Green _et al._ (2020), we use $\beta$ to
implement a cyclic annealing schedule. Annealing improves the efficiency of
the training and allows autoencoders to express more meaningful latent
variables Fu _et al._ (2019). We increase $\beta$ from $0$ to $1$ in steps of
$[10^{-5},\frac{1}{3},\frac{2}{3},1,1,1]$ and these steps are repeated $3$
times. After this, $\beta$ is definitively fixed to 1.
Next, the training performances improve when the inputs $y$ are standardized
to zero mean and unit variance, and when the outputs $x$ are normalized to
have support in $[1,100]$. $x$ is then scaled back to the original
normalization at production time.
### III.2 Network training
To train the network, we simulate a dataset of $10^{5}$ ringdowns by sampling
the waveform parameters uniformly in the ranges indicated in Tab. 2. The
ringdown waveforms are sampled at $4096$ Hz with a total signal duration of
$31.25$ ms, thus corresponding to arrays of length $128$. Signal-to-noise
ratio (SNR) is used to set the waveform scale w.r.t. the noise. The SNR is
computed as in Berti _et al._ (2006); Baibhav and Berti (2019). When
performing PE, we only estimate posteriors for $\\{M,\chi_{f},q\\}$.
Parameter | Symbol | Range
---|---|---
Final BH mass | $M$ | $[25,100]~{}M_{\odot}$
Final BH spin | $\chi_{f}$ | $[0,0.9]$
Binary mass ratio | $q$ | $[1,8]$
Phase of the (2,2) mode | $\phi_{22}$ | $[0,2\pi]~{}{\rm rad}$
Signal-to-noise ratio | SNR | $[40,80]$
Table 2: Ranges for the waveform parameters. All the parameters are sampled
uniformly. Note we marginalize over the last two parameters.
For simplicity, we only consider the $+$ polarization and fix the inclination
angle to $\iota=\pi/3$. The ringdowns are embedded in simulated ET-like noise
segments Hild _et al._ (2011); ET: . At each training iteration, we assign
noise instances randomly to the waveforms to prevent the CVAE from learning
spurious correlations between the waveforms and the noise realizations.
Our training takes $84$ minutes on a single GPU. Fig. 2 shows the evolution of
the reconstruction loss $\mathcal{L}_{\rm recon}$ and the KL divergence
$\mathcal{L}_{\rm KL}$ separately. Further, we show the loss evaluated on the
$90\%$ training dataset and on the $10\%$ validation dataset. Notice that the
training and validation losses are consistent, substantiating that the network
is not overfitting.444The initial oscillations which are visible in
$\mathcal{L}_{\rm KL}$ are due to the cyclic annealing.
Figure 2: Evolution of the reconstruction and KL losses across the training
epochs.
To test the network, we generate a new dataset of $10^{3}$ simulated ringdown
waveforms, whose parameters are sampled again from the ranges indicated in
Tab. 2. For each input waveform, the CVAE samples $10^{4}$ distinct points to
build the posterior. The total time to analyze all the samples is
approximately $40$ s on a single GPU, corresponding to $40$ ms per waveform.
For illustration, Fig. 3 shows the contour plot obtained from the PE of one
signal from the test dataset.
Figure 3: Contour plot for the PE of the ringdown signal shown in the lower
panel, with $(M,\chi_{f},q)=(65,0.8,1.5)$ and SNR$=60$. Blue lines indicate
the true values. Dashed lines mark the $95\%$ confidence interval. The plot
titles indicate $1\sigma$ uncertainties.
For a quantitative diagnostic of the network performances, we present the P-P
plot in Fig. 4: the plot shows marginalized cumulative distribution CDF of the
true values $x_{\rm true}$ as a function of $p\,\%$ confidence interval. A
diagonal P-P plot means that $x_{\rm true}$ is contained $p\,\%$ of the times
within $p\,\%$ confidence interval of the marginalized posteriors for $x_{\rm
pred}$. Note that all the CDFs are consistent with the diagonal, demonstrating
that our CVAE recovers the posterior samples for $\\{M,\chi_{f},q\\}$ from the
ringdowns reliably.
Figure 4: P-P plot of test dataset. For each observable, the plot shows the
cumulative distribution CDF of the true values as a function of the $p\,\%$
confidence interval of the posterior distribution.
## IV Results
We present our proof-of-concept study in two parts. In section IV.1 we
demonstrate that when our test is applied to a set of ringdowns consistent
with GR, these signals satisfy the null test. Next, in section IV.2 we use 4
sets of non-GR ringdowns and show that with $\sim 20-50$ events the non-GR
signals violate our null test - allowing us to distinguish GR from non-GR
ringdown.
### IV.1 Proof-of-concept Merger-Ringdown test: GR signals
We simulate a dataset consisting of $10^{3}$ ringdowns with parameter ranges
as presented in Tab. 2 except for $\chi_{f}$. Here, $\chi_{f}$ is inferred
from $q$ by imposing the relation (3).
First, the posteriors produced by our CVAE for a single event can be used to
check if the event validates the spin relation (3). In Fig. 5, we show the
posterior samples for $\chi_{f}^{\rm meas}$ and $\chi_{f}^{\rm infer}$ — where
$\chi_{f}^{\rm infer}$ is determined by $q^{\rm meas}$ as per Eq. (3), for a
randomly chosen event from our dataset. For this event, we see that
$\chi_{f}^{\rm meas}=\chi_{f}^{\rm infer}$ (blue dashed line) lies within the
$68\%$ credible interval of the posteriors, asserting that this event is
consistent with GR evolution.
Figure 5: Contour plot for the posteriors of $\chi_{f}^{\rm meas}$ and
$\chi_{f}^{\rm infer}$ for a signal with $(M,\chi_{f},q)=(52.6,0.66,1.53)$ and
SNR$=60$. The signal is extracted from the second test dataset, where the
relation (3) is enforced. The blue dashed line represents $\chi_{f}^{\rm
meas}=\chi_{f}^{\rm infer}$ line.
Further, we use the scheme outlined in Sec. II.2 to combine the information
across multiple ringdown observations for a more stringent test of GR, as
illustrated in Fig 6. For a noiseless GR ringdown, we expect $\chi_{f}^{\rm
meas}=\chi_{f}^{\rm infer}$. However, our inferences are probabilistic and
contain noise. This leads to measurement uncertainties that translate as a
scatter around the diagonal line.
Figure 6: Scatter plot of $\chi_{f}^{\rm meas}$ vs $\chi_{f}^{\rm infer}$. The
color bar indicates the value $q^{\rm meas}$ for each observation. The black
dotted line marks the $\chi_{f}^{\rm meas}=\chi_{f}^{\rm infer}$.
In Fig. 6, we confirm that our dataset lies around $\chi_{f}^{\rm
meas}=\chi_{f}^{\rm infer}$. Also, as expected, lower values of $q$ give
higher values of $\chi_{f}$. A weighted least squared (WLS) fit for Eq. (4)
gives $a\in[-0.014,0.014]$ and $b\in[0.963,1.013]$ at $95\%$ confidence level,
showing an agreement with the $\chi_{f}^{\rm meas}=\chi_{f}^{\rm infer}$ line.
We weighed each event by $(\sigma_{\chi_{f}^{\rm meas}}\sigma_{\chi_{f}^{\rm
infer}})^{-1/2}$ to emphasize the more confident recoveries.555We verified
that the results of an ordinary (unweighted) least squared fit do not
significantly change. Fig. 6 thus observationally validates Eq. (4).
Next, to assess the efficiency of this test with the number of observations,
we study the convergence of $a$ and $b$. In Fig. 7, we present the (weighted)
best-fit values for $a$ and $b$ with their $2\sigma$ confidence levels as a
function of the number of observations.
Figure 7: Evolution of the best-fit values (continuous lines) and $2\sigma$
contours (shaded regions) for $b$ and $a$ in Eq. (4), versus the number of
observations. The mean values and the confidence intervals are averaged over
10 random realisations. In GR, the noiseless best-fit corresponds to $b=1$ and
$a=0$.
In Fig. 7, we find that the mean value for $a$ and $b$ are $\sim 0$ and $1$,
respectively, even for a small number of observations. We see that the
uncertainties in the measurements of $a$ and $b$, i.e.,
$\\{\sigma_{b},\sigma_{a}\\}$ decrease with increasing number of observations
as a power-law. For instance, with 20 observations we can constrain
$\\{\sigma_{b}(n=20),\sigma_{a}(n=20)\\}=\\{0.0832,0.00465\\}$ and with 100
observations we have
$\\{\sigma_{b}(n=100),\sigma_{a}(n=100)\\}=\\{0.0398,0.0219\\}$. Concretely,
$\sigma_{a}$ and $\sigma_{b}$ scale with the number of observations as
$\sigma_{a}(n)=\frac{0.21}{\sqrt{n}}\,,\qquad\sigma_{b}(n)=\frac{0.41}{\sqrt{n}}\,,$
(8)
which is consistent with our expectations given our noise model.
Thus, while the merger-ringdown consistency test is powerful when combining a
large number of ringdowns, it is also feasible to perform it with just a few
observations ($\sim 20$).
### IV.2 Proof-of-concept Merger-Ringdown test: non-GR signals
In this section, we present the performance of the test on a set of non-GR
ringdown signals. We consider phenomenological deviations from GR without
assuming physical mechanisms responsible for the GR modifications.
Specifically, we generate 4 sets of non-GR ringdowns by heuristically
modifying amplitudes and phases of mode excitations
$\displaystyle A^{R}_{lm}\to(1+\Delta A)A^{R}_{lm,GR}$ (9a)
$\displaystyle\delta\phi_{lm}\to(1+\Delta\delta\phi)\delta\phi_{lm,GR}$ (9b)
with the 4 distinct cases enlisted as
Case 1: $\displaystyle\Delta A=0\quad$ $\displaystyle\Delta\delta\phi=0.1$
(10a) Case 2: $\displaystyle\Delta A=0\quad$
$\displaystyle\Delta\delta\phi=-0.1$ (10b) Case 3: $\displaystyle\Delta
A=0.1\quad$ $\displaystyle\Delta\delta\phi=0.1$ (10c) Case 4:
$\displaystyle\Delta A=0.1\quad$ $\displaystyle\Delta\delta\phi=-0.1$ (10d)
This class of GR-modifications implies an altered relation between the
ringdown perturbation conditions and the final properties of the remnant;
therefore, we expect our test to be naturally sensitive to these
modifications. Furthermore, note that other than the amplitude ratios and
phase differences between the modes, the details of the non-GR signal are
identical to the GR ones — i.e., the QNM spectrum is unaltered.
The best-fit values of $a$ and $b$ in Eq. 4 for each of the non-GR signal-set
are presented in Fig. 8. We remind that any departure from $a=0$ and $b=1$
indicates that the dataset contains ringdown that do not satisfy the GR null
test. We note that the merger-ringdown test successfully excludes the $a=0$
and $b=1$ for a relatively small number of observations. Specifically, GR
values are excluded at a $2\sigma$ level with $\mathcal{O}(20)$ observations
for Case 1-3 and with $\mathcal{O}(50)$ observations for Case 4.
Figure 8: Similar to Fig. 7 but applied on the set of non-GR test signals
described through Case 1-4 in Eq. 10 (depicted as panels, clockwise from the
top-left). Each set contains $10^{3}$ signals. The mean values and the
confidence intervals are averaged over 10 random realisations. Note that for
all of the cases we get best fit $a\neq 0$ and $b\neq 1$ with $95\%$ credible
intervals with less than 50 observations.
## V Conclusion and Outlook
We demonstrated a proof-of-concept study for a novel test of GR called _the
merger-ringdown consistency test_ that checks for statistical consistency
between the plunge-merger-ringdown phase across a set of ringdown detections
using a deep learning framework. The test aims at increasing the sensitivity
to the merger-ringdown by simultaneously incorporating information on both the
amplitude-phase excitations and the QNM frequency spectra in the ringdown. It
uses the fact that the QNM complex excitation amplitudes and the spectrum in
the ringdown are not independent quantities in GR. The test provides an
efficient way of stacking ringdowns. Furthermore, for possible future
detections of heavy mass binary systems (that is set precedence by the
discovery of Abbott _et al._ (2019)), our test can be used to check the
consistency of initial BH parameters with the ringdown alone if the inspiral
is not well measured.
This work illustrates that Bayesian deep learning methods can be applied to
infer the posteriors of ringdown parameters to conduct precision tests of GR.
Unlike the traditional Monte-Carlo sampling, neural networks perform the PE in
fractions of a second, providing a crucial edge when dealing with the large
number of observations that are expected with future GW detectors.
We also highlight that, in its most generic construct, the TIGER
infrastructure allows to identify parametric deviations from GR in the
waveform by applying a Bayesian hypothesis testing Li _et al._ (2012);
Agathos _et al._ (2014). In testing GR with ringdown signals, the most
popular strategy has been to parametrize the deviations in the QNM frequencies
and damping times, followed by evaluating the Bayes factor for GR versus non-
GR hypothesis Meidam _et al._ (2014); Gossan _et al._ (2012); Brito _et
al._ (2018). Our test is conceptually different, because we are not explicitly
checking whether the QNM spectrum of the remnant is consistent with that of a
general relativistic BH. Rather, we focus on the consistency between the
relative amplitudes and phases of the ringdown modes with the spin of the
remnant BH. The modifications to which the two tests are sensitive do not
necessarily overlap; we expect our test to be efficient in scenarios where the
departure from GR influences the relative amplitudes and phase, but the final
remnant is similar to a general relativistic BH. Comparing the efficiency of
our test to the TIGER implementation is not straightforward and needs further
exploration.
Here we have used non-spinning progenitor BHs where the QNM excitation
amplitudes and phases are fully parametrizable by its mass ratio $q$ and the
$\chi_{f}-q$ relation is approximated by the simple analytical expression in
Eq. (3). However, our method can be extended to encompass spinning progenitor
BHs where the QNM excitations depend on both $q$ and $\chi_{1,2}$. The
dependence of the remnant spin $\chi_{f}$ on the binary BH parameters should
then be replaced by implicit non-analytical relations such as those in Varma
_et al._ (2019a); Haegel and Husa (2020). Our WLS fit strategy does not rely
on analytical relations and new parameters can be estimated by increasing the
output dimensions of the CVAE.
This study used stellar-mass BH ringdowns targeting the ET-like data. Similar
results can be expected to hold for CE. However, LISA will detect ringdowns
from supermassive BHs Berti _et al._ (2016); Baibhav and Berti (2019), with
loud SNRs. We plan to extend our analysis to include LISA-like data in the
future. Finally, in this work, we demonstrate the feasibility of a _null_ test
of GR, by implementing our test on GR as well on a class of phenomenologically
constructed non-GR ringdowns. An interesting extension to our work would be an
extensive study on non-GR signals in a parameterized framework such as that in
ParSpec Maselli _et al._ (2020).
###### Acknowledgements.
We thank Paolo Pani and Andrea Maselli for their productive comments on an
early version of this manuscript and Guido Sanguinetti for his advices on
neural networks. We thank Nathan Johnson-McDaniel for useful clarifications
about the IMR test. CP is indebted to Enrico Barausse and Luca Heltai for
their invaluable support in getting familiar with neural networks and
gravitational wave physics. We acknowledge financial support provided under
the European Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480.
We also acknowledge support under the MIUR PRIN and FARE programmes (GW-NEXT,
CUP: B84I20000100001), and from the Amaldi Research Center funded by the MIUR
program “Dipartimento di Eccellenza” (CUP: B81I18001170001).
#### Software.
The PyCBC Dal Canton _et al._ (2014); Usman _et al._ (2016) library was used
to generate the noise realizations and LalSuite LIGO Scientific Collaboration
(2018) to generate the ringdowns. The neural network model used in this work
was developed using the PyTorch library Paszke _et al._ (2019) for Python.
The WLS test was performed with the statsmodels package sta for Python. The
corner plots have been produced with the corner package Foreman-Mackey (2016)
for Python.
#### Code availability.
The code used for this paper is made available in a dedicated git repository
mrt . *
## Appendix A Excitation amplitudes and phases
We update the fits presented in Jiménez Forteza _et al._ (2020) by
additionally requiring $A_{lm}/A_{22}\to 0$ for $q\to 1$ Kamaretsos _et al._
(2012b) and present the coefficients for Eq. (5) in Tab. 3. The start of
ringdown is chosen at $t_{peak}+12M$. Note that the goodness of the fits do
not change significantly between the version here and in Jiménez Forteza _et
al._ (2020).
| $(3,3)$ | $(2,1)$
---|---|---
$a_{0}$ | 0.433253 | 0.472881
$a_{1}$ | -0.555401 | -1.1035
$a_{2}$ | 0.0845934 | 1.03775
$a_{3}$ | 0.0375546 | -0.407131
$b_{0}$ | 2.63521 | 1.80298
$b_{1}$ | 8.09316 | -9.70704
$b_{2}$ | 8.32479 | 9.77376
Table 3: Values of the fit coefficients in Eq. (5) for $(l,m)=(3,3)$ and
$(2,1)$.
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|
# Simultaneous supply and demand constraints in input-output networks: The
case of Covid-19 in Germany, Italy, and Spain
Anton Pichler1,2,3 and J. Doyne Farmer1,2,4
1 Institute for New Economic Thinking at the Oxford Martin School, University
of Oxford, UK
2 Mathematical Institute, University of Oxford, UK
3 Complexity Science Hub Vienna, Austria
4 Santa Fe Institute, US
###### Abstract
Natural and anthropogenic disasters frequently affect both the supply and
demand side of an economy. A striking recent example is the Covid-19 pandemic
which has created severe disruptions to economic output in most countries.
These direct shocks to supply and demand will propagate downstream and
upstream through production networks. Given the exogenous shocks, we derive a
lower bound on total shock propagation. We find that even in this best case
scenario network effects substantially amplify the initial shocks. To obtain
more realistic model predictions, we study the propagation of shocks bottom-up
by imposing different rationing rules on industries if they are not able to
satisfy incoming demand. Our results show that economic impacts depend
strongly on the emergence of input bottlenecks, making the rationing
assumption a key variable in predicting adverse economic impacts. We further
establish that the magnitude of initial shocks and network density heavily
influence model predictions.
Keywords: Covid-19; production networks; input-output models; rationing;
linear programming; economic shocks; shock propagation; economic impact
JEL codes: C61; C67; D57; E23
_Acknowledgments:_ We thank F. Lafond for helpful feedback. This work was
supported by Baillie Gifford, Partners for a New Economy, the UK’s Economic
and Social Research Council (ESRC) via the Rebuilding Macroeconomics Network
(Grant Ref: ES/R00787X/1), the Oxford Martin Programme on the Post-Carbon
Transition, and the Institute for New Economic Thinking at the Oxford Martin
School. This research is based upon work supported in part by the Office of
the Director of National Intelligence (ODNI), Intelligence Advanced Research
Projects Activity (IARPA), via contract no. 2019-1902010003. The views and
conclusions contained herein are those of the authors and should not be
interpreted as necessarily representing the official policies, either
expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for governmental
purposes notwithstanding any copyright annotation therein.
_Corresponding author:_ Anton Pichler<EMAIL_ADDRESS>
## 1 Introduction
An immanent feature of many natural and anthropogenic disasters is that they
affect both the supply and the demand side of the economy. In this paper we
study the Covid-19 pandemic as an exemplary case of simultaneous supply and
demand shocks. Supply shocks from the pandemic arise from different sources.
While deaths and sickness of employees can limit productive capacity, these
effects are minor compared to nation-wide lockdown measures imposed by
governments to curb the spreading of the virus. During lockdown workers
employed in non-essential industries who cannot work remotely are unable to
perform their jobs (del Rio-Chanona et al., 2020; Dingel and Neiman, 2020;
Koren and Pető, 2020).
The pandemic also affects demand in heterogeneous ways (Congressional Budget
Office, 2006; del Rio-Chanona et al., 2020; Chetty et al., 2020; Carvalho et
al., 2020; Chen et al., 2020). While we expect demand shocks to be
comparatively small for some industries (e.g. manufacturing), other industries
are strongly affected by demand shocks. An illustrative example is the
transportation industry which is considered as essential in many countries and
thus would not experience adverse supply side effects. But the transportation
industry faces large demand shocks, since consumers reduce demand for air
travel and public transport to avoid infectious exposure.
Since economic agents are embedded in production networks, we expect that the
overall economic impact is larger than the initial shocks to supply and demand
suggest. Demand shocks will reduce the sales of firms and, by backward
propagation, also diminish the sales of their suppliers. Supply shocks, on the
other hand, will spread downstream and upstream. Downstream effects
materialize if the limited productive capacity of suppliers creates input
bottlenecks for customers. Due to lower productive capacities, firms will also
require less inputs for production, thus adversely affecting upstream
suppliers of these inputs.
Input-output (IO) models are frequently used to model higher-order economic
impacts arising from such exogenous shocks. Most of these models account for
either supply or demand shocks but do not incorporate them concurrently
(Galbusera and Giannopoulos, 2018). In this paper we revisit existing IO
modeling techniques and introduce a novel dynamic approach, to account for
both types of shocks. Our analysis is based on industry-level data but could
easily be extended to firm-level analysis that has become the focus of recent
studies (Diem et al., 2021; Borsos and Mero, 2020; Mandel and Veetil, 2020b;
Inoue and Todo, 2019). In particular, our results relate directly to the
effects of data aggregation on impact estimates.
Several macroeconomic models that account for the production network structure
have been suggested to study the economic impacts of the Covid-19 pandemic.
Inoue and Todo (2020) use an agent-based model calibrated to more than a
million Japanese firms to model nation-wide economic effects of a lockdown in
Tokyo. Using the World Input-Output Database (WIOD), Mandel and Veetil (2020a)
study the effects of national lockdowns on global GDP in a non-equilibrium
framework. Fadinger and Schymik (2020); Baqaee and Farhi (2020); Bonadio et
al. (2020) and Barrot et al. (2020) use general equilibrium models to quantify
economic impacts of social distancing. All these studies focus on the supply-
side shocks of the pandemic and without further considering changes in final
consumption. Exceptions are Pichler et al. (2020) and Guan et al. (2020) who
incorporate both supply and demand shocks in their production network models.
In contrast to these studies, we consider simpler modeling techniques derived
from traditional IO analysis which we extend to account for simultaneous
supply and demand shocks. We follow this approach to better isolate underlying
mechanisms of shock propagation in production networks that can be hard to
disentangle in more sophisticated macroeconomic models. When applying pandemic
shocks to data from Germany, Italy and Spain, we find that existing IO
modeling techniques yield infeasible solutions of economic impacts. To
understand the nature of this problem we introduce a simple linear programming
method that allows us the determine feasible market allocations, representing
a lower bound of minimal shock propagation. To find more realistic impact
estimates, we further introduce a new bottom-up approach based on rationing
outputs. This setup allows us to dynamically model the propagation of shocks
in production networks and uncover several theoretical implications, such as
the effect of alternative behavioral assumptions on economic impacts.
We intentionally keep our modeling approach simple. As mentioned above, even
more complicated models currently applied to assess the economic effects of
the pandemic do not incorporate supply and demand shocks simultaneously. The
focus of this paper is on the theoretical implications of simultaneous supply
and demand constraints for IO-based impact assessment methods. For more
realistic empirical impact assessment of natural disasters, we stress that
further aspects that we do not consider here could play an important role,
e.g. inventory dynamics (Bak et al., 1993; Pichler et al., 2021), price and
substitution effects, adaptive behavior and governmental intervention (see
Oosterhaven (2017) for a recent discussion).
Our paper contributes to the field of IO models in disaster impact assessment,
for which a recent review can be found in Galbusera and Giannopoulos (2018).
For recent demand-side and supply-side focused contributions see Klimek et al.
(2019) and Avelino and Dall’erba (2019), respectively. Similar to our
derivation of a lower bound of shock propagation, Koks and Thissen (2016) and
Oosterhaven and Bouwmeester (2016) rely on optimization techniques to account
for supply constraints in demand-driven models. Constrained optimization
techniques are frequently used in extended IO models (for example Duchin
(2005) and Duchin and Levine (2012)). The modeling of alternative rationing
algorithms is related to the studies of Steenge and Bočkarjova (2007); Li et
al. (2013) and Koks et al. (2015) which allow for imbalanced IO tables
immediately after the disaster strikes and evaluate different recovery paths.
Our findings show that shock amplification is much larger in the bottom-up
approaches than what best case scenarios would suggest. Micro-level
coordination failures can severely exacerbate adverse economic shocks.
Rationing assumptions play a key role in impact estimates and alternative
behavioral rules can lead to very different aggregate outcomes. These effects
strongly interact with the overall shock magnitude as well as the level of
connectedness in the production network. While we find that economic impacts
increase with higher levels of network density in general, the extent to which
this is true strongly depends on the underlying rationing mechanisms. Our
results also imply that estimates of economic impact can be highly sensitive
with respect to data quality and aggregation.
This paper is organized as follows. We first discuss first-order pandemic
shocks to supply and demand, as well as the datasets used (Section 2). In
Section 3 we introduce the basic IO framework and discuss how existing
approaches have difficulties in incorporating simultaneous supply and demand
constraints. Section 4 discusses the main results of this paper. We propose an
optimization method in Section 4.1 and introduce alternative rationing
algorithms in Section 4.2 to model shock propagation in supply and demand
constrained production networks. We show empirical results and discuss further
theoretical implications in Section 4.3 before concluding in Section 5.
## 2 Pandemic shocks to supply and demand
Our analysis is based on the most recent year (2014) of the World-Input-
Output-Database (WIOD) (Timmer et al., 2015).111 All code and data to
reproduce this paper are made accessible online:
https://www.doi.org/10.5281/zenodo.4326815. We use estimates of supply and
demand shocks from the literature to determine maximum final consumption and
production values for 54 industries in three major European economies, i.e.
Germany, Italy and Spain.222 We removed industries _T_ (Household activities)
and _U_ (Extraterritorial activities) from our analysis since they are not
connected to other industries in the data and thus don’t play any role in the
propagation of shocks in the production network. We focus on these economies
because existing research provides us with lists of essential industries and
thus allows us to make reasonable estimates of supply shocks. For this reason,
we only allow for domestic supply shocks in our analysis, despite
acknowledging the importance of international supply chain disruptions.
We follow the approach of del Rio-Chanona et al. (2020) to compute supply
shocks for every industry during a lockdown. A supply shock is caused by the
removal of labor in non-essential industries due to social distancing
measures. In contrast, an industry which is defined as essential will not be
affected. Even if employed in non-essential industries, workers who can
accomplish their work from home will do so and are assumed to keep pre-
lockdown productivity levels. The Remote Labor Index $\text{RLI}_{i}$
indicates the share of an industry’s labor force that can work from home. The
supply shock to industry $i$ is then computed as
$\epsilon_{i}^{S}=(1-\text{RLI}_{i})(1-\text{e}_{i})$, where $\text{e}_{i}$ is
the share of an industry defined as essential. Thus, the supply shock of an
industry during lockdown is the share of labor in non-essential industries
that cannot work from home. We assume that the supply shock to an industry
determines the maximum production of an industry, i.e.
$x_{i}^{\text{max}}=(1-\epsilon_{i}^{S})x_{i,0},$ (1)
where $x_{i,0}$ denotes the production in industry $i$ before lockdown.
Following this approach, every industry faces binding supply side constraints,
limiting its production to $x_{i}\in[0,x_{i}^{\text{max}}]$. The Remote Labor
Index is taken from del Rio-Chanona et al. (2020) and industry-specific
essential ratings for Germany, Italy and Spain are obtained from Fana et al.
(2020).
To determine first-order demand shocks to every industry, $\epsilon_{i}^{D}$,
we use estimates of a prospective Congressional Budget Office (CBO) study
aiming at quantifying the demand-side impact of a pandemic (Congressional
Budget Office, 2006). Demand and supply shock data have been mapped to WIOD
industry categories by Pichler et al. (2021) and are presented in detail in
Appendix A. As for supply shocks, we assume that demand shocks determine
maximum final consumption values according to
$f_{i}^{\text{max}}=(1-\epsilon_{i}^{D})f_{i,0},$ (2)
where $f_{i,0}$ represents pre-lockdown final consumption. Since consumption
cannot be negative, we have $f_{i}\in[0,f_{i}^{\text{max}}]$.333 In principle,
$f_{i}$ could be negative if there is extremely large inventory depletion.
This is not the case in our data. In national accounts inventory adjustment
merely represents a variable to rebalance row and column sums of IO tables. We
therefore do not consider the possibility of negative final demand. Note that
WIOD distinguishes different final consumption categories.444 Given the multi-
regional nature of WIOD, imports from and exports to other industries are
accounted for in the intermediate consumption matrix. Since we treat
international trade as exogenous, we aggregate all intermediate exports to a
single final consumption category, as is usually the case with national IO
tables. Similarly, we treat intermediate imports as an exogenous input that
does not inhibit production, or equivalently, that is always available if
demanded. We apply the CBO estimates to final consumption of households and
non-profit organizations and assume a 10% shock to investments and exports as
in Pichler et al. (2021). Due to our methodological focus we do not further
distinguish between different final consumption categories but for simplicity
only consider total final consumption values for every industry. It is
important to point out that this is a major simplification and we expect the
various types of final consumption to be highly relevant in empirical
assessments. Table 1 shows country aggregates of essentialness scores, Remote
Labor Index, supply and demand shocks.
| $x$ | $f$ | $\epsilon^{S}$ | $\epsilon^{D}$ | RLI | $e$
---|---|---|---|---|---|---
Germany | 7,066.74 | 4,447.11 | 0.31 | 0.09 | 0.42 | 0.49
Italy | 4,075.40 | 2,343.12 | 0.27 | 0.11 | 0.41 | 0.55
Spain | 2,567.91 | 1,552.88 | 0.33 | 0.13 | 0.40 | 0.44
Table 1: Country aggregates of gross output, final consumption and supply and
demand shock inputs. Columns $x=\sum_{i}x_{i,0}$ and $f=\sum_{i}f_{i,0}$ are
country gross output and final consumption in billion USD based on 2014
values. $\epsilon^{S}=1-\sum_{i}x_{i}^{\text{max}}/x$ and
$\epsilon^{D}=1-\sum_{i}f_{i}^{\text{max}}/f$ are total supply and demand
shocks, respectively. $\text{RLI}=\sum_{i}\text{RLI}_{i}x_{i,0}/x$ and
$e=\sum_{i}e_{i}x_{i,0}/x$ denote the country-wide Remote Labor Index and
essentialness score on which supply shocks are based. In all three countries
total direct supply shocks are larger than demand shock.
## 3 IO framework
We first introduce basic national accounting identities that build the basis
for most IO models. Let us consider an economy consisting of $n$ industries,
each producing a unique, industry-specific good. Inter-industry purchases and
sales are encoded in the intermediate consumption matrix $\mathbf{Z}$ where an
element $z_{ij}$ denotes the total monetary value of goods produced by
industry $i$ that are consumed by industry $j$. For the $n$-dimensional
vectors of gross output, total final consumption and value added, we write
$\bm{x}$, $\bm{f}$ and $\bm{v}$, respectively. In this economy the following
identities hold
$\displaystyle\bm{x}=\mathbf{Z}\mathbf{i}+\bm{f}=\mathbf{Z}^{\top}\mathbf{i}+\bm{v},$
(3)
where $\mathbf{i}$ is a $n$-dimensional column vector of ones.
A core assumption in a Leontief-inspired modeling framework is that industries
produce based on fixed production recipes, allowing us to rewrite the first
identity of Eq. (3) as
$\bm{x}=\mathbf{A}\bm{x}+\bm{f}=\mathbf{L}\bm{f},$ (4)
where $\mathbf{A}$ is the technical coefficient matrix with elements
$a_{ij}\equiv z_{ij}/x_{j}$ (the production recipe) and $\mathbf{L}$ the
Leontief inverse $\mathbf{L}\equiv(\mathbf{I}-\mathbf{A})^{-1}$. A
conventional assumption is that gross output is determined endogenously,
whereas final consumption is taken to be as exogenous. It then follows that
value added is obtained as a residual variable.
Eq. (4) is at the core of many IO models that are frequently used for disaster
impact analysis. This specification defines a demand-driven model. However, as
we have already stressed, many economic shocks actually act on the supply side
of the economy. For example, natural catastrophes such as earthquakes or
floods destroy physical capital, putting upper limits on an industry’s
production in the disaster aftermath. Eq. (4), however, neglects supply
capacity constraints and implies an infinite elasticity of supply with respect
to demand. This is particularly problematic given the frequent short-term
focus of IO studies.
We should point out that there are also supply-driven IO models building upon
Ghosh’s model (Ghosh, 1958) that assumes exogenous primary factors and derives
final consumption values endogenously. Ghosh’s model does not comply with a
Leontief production function and has been heavily criticized amongst other
aspects for the assumption of perfect demand elasticities and perfect
substitution of inputs (Oosterhaven, 1988; Gruver, 1989; De Mesnard, 2009).
The IO inoperability model (Haimes and Jiang, 2001; Santos and Haimes, 2004)
is another notable supply-side focused model. Dietzenbacher and Miller (2015)
show, however, that it closely corresponds to conventional IO modeling
approaches.
Neither the demand- nor the supply-driven specification of IO models are able
to incorporate supply and demand constraints at the same time. A potential
remedy is the mixed endogenous/exogenous model (MEEM), which has been applied
in several studies including Steinback (2004); Kerschner and Hubacek (2009)
and Arto et al. (2015). The MEEM acknowledges that not only final demand is
constrained but that for some industries supply constraints are more severe
and therefore binding. A difficulty in empirical analysis is to define which
industries are supply and which are demand constrained. In some cases there
might be “natural” distinctions. For example, Arto et al. (2015) study global
supply chain disruption effects of the 2011 Tōhoku earthquake by solving the
MEEM where only the Japanese transport equipment industry is supply
constrained, whereas for all other industries gross output is endogenous. In
case of simultaneous (severe) supply and demand shocks the line of distinction
will be more blurred.
Note that the MEEM incorporates both supply and demand shocks, but not
_simultaneously_ for a single industry. Instead, an industry is either supply
or demand constrained. As a consequence, solutions to the MEEM might not
comply with exogenous industry-level constraints to supply and demand, i.e.
they can be economically _infeasible_. As is shown in detail in Appendix B,
this is exactly what happens if we apply the MEEM to the pandemic shocks
discussed in the previous section. For all three countries, the MEEM yields
final consumption values that are either negative or larger than the exogenous
constraints.
## 4 Propagation of simultaneous supply and demand shocks
The industry-specific effects to supply and demand during the Covid-19
pandemic and the difficulty of incorporating them in the existing models
motivated us to explore possible extensions of the IO modeling framework.
Rather than using more complicated models that can deal with simultaneous
supply and demand shocks, such as computable general equilibrium (CGE) models
or those discussed in the introduction, we intentionally keep the modeling
approach simple. We remain close to the Leontief framework, the “work horse”
of many more advanced IO-based models. This allows us to better isolate the
basic mechanisms of shock propagation that are easily conflated with other
effects in more sophisticated models.
To determine lower bounds of shock propagation with respect to output and
consumption, we study the idealized case of a social planner who allocates
goods to maximize either total gross output or total final consumption within
the given economic constraints. This will give us a best case scenario of
negative economic impacts, i.e. the minimal decrease in total output and final
consumption necessary to arrive within the set of feasible solutions given the
exogenous constraints to supply and demand. Of course, alternative objectives
could be optimized as well (e.g. employment).
While deriving a lower bound is valuable, it is unlikely to be realistic. We
therefore consider a second approach by comparing alternative rationing
algorithms. If an industry is supply constrained, it will not be able to
satisfy all of its demand and thus needs to make a decision which customer to
serve and to what extent. We implement several decision rules and investigate
how this choice influences the estimated economic impact. In contrast to the
optimization method or the MEEM, which simply compute an equilibrium, this
approach explicitly computes the transient dynamics that lead to a new
equilibrium.
### 4.1 Feasible market allocations
Given exogenous constraints to supply and demand, what is the feasible market
allocation that maximizes final consumption and/or total output? The solution
needs to lie within exogenous bounds on supply and demand and also needs to
satisfy the assumption of Leontief production, Eq. (4). We seek market
allocations $\\{\bm{x}^{*},\bm{f}^{*}\\}$ that (a) respect given production
recipes $\bm{x}^{*}=\mathbf{L}\bm{f}^{*}$ and (b) satisfy basic output and
demand constraints $\bm{x}^{*}\in[\bm{0},\bm{x}^{\text{max}}]$ and
$\bm{f}^{*}\in[\bm{0},\bm{f}^{\text{max}}]$.
We follow a mathematical optimization procedure to map out the solution space
of feasible market allocations. As a first case, we determine the market
allocation that maximizes gross output under the assumptions specified. Large
levels of gross output indicate high economic activity, which in turn entail
high levels of primary factors such as labor compensation. As a second case we
look at market allocations that maximize final consumption given current
production capacities. Due to the linearity of the Leontief framework, the
problems boil down to linear programming exercises.
##### Maximizing gross output:
$\displaystyle\underset{\bm{f}\in[\bm{0},\bm{f}^{\text{max}}]}{\max}\qquad\mathbf{i}^{\top}(\mathbf{I}-\mathbf{A})^{-1}\bm{f},$
(5) $\displaystyle\text{subject
to}\qquad(\mathbf{I}-\mathbf{A})^{-1}\bm{f}\in[\bm{0},\bm{x}^{\text{max}}].$
##### Maximizing final consumption:
$\displaystyle\underset{\bm{x}\in[\bm{0},\bm{x}^{\text{max}}]}{\max}\qquad\mathbf{i}^{\top}(\mathbf{I}-\mathbf{A})\bm{x},$
(6) $\displaystyle\text{subject
to}\qquad(\mathbf{I}-\mathbf{A})\bm{x}\in[\bm{0},\bm{f}^{\text{max}}].$
To maximize gross output of the economy, $\sum_{i}x_{i}^{*}$, requires us to
the find the vector of final consumption $\bm{f}^{*}$. The constraint
$\mathbf{L}\bm{f}\in[\bm{0},\bm{x}^{\text{max}}]$ ensures that industry output
levels lie within the respective production capacities. The problem is similar
when maximizing final consumption where a vector of output levels $x^{*}$ is
chosen to maximize final consumption, $\sum_{i}f_{i}^{*}$. The auxiliary
constraint enforces that final consumption levels do not exceed given demand.
The optimization problem always admits a solution since the trivial allocation
of a full collapse $\\{\bm{x}^{*},\bm{f}^{*}\\}=\\{\bm{0},\bm{0}\\}$ always
exists, although we expect positive values for realistic input data.
Note that the market allocations are “optimal” in a mathematical sense. That
is, they represent maximum values of output and consumption given the
exogenous constraints to supply and demand. Or stated differently, they
determine the minimum level of shock propagation measured in aggregate final
consumption and gross output. Thus, this optimization method is not intended
to give realistic or necessarily desirable economic outcomes, but rather
probes the system boundaries by mapping out the solution space. Any feasible
solution, under the above-specified assumptions, has to lie within this space.
### 4.2 Input bottlenecks and rationing variations
As our second method we implement different rationing schemes for output
constrained industries. In contrast to the optimization methods, this
represents a bottom-up approach for finding feasible market allocations.
Industries place orders to their suppliers based on incoming demand. Since
suppliers can be output constrained, they might not be able to satisfy demand
fully. A supplier therefore needs to make a decision about how much of each
customer’s demand it serves. Intermediate consumers transform inputs to
outputs based on fixed production recipes. Thus, if a customer receives less
inputs than she asked for, she faces an input bottleneck further constraining
her production. As a consequence, the customer reduces her demand for other
inputs as they are not further needed under limited productive capacities. We
iterate this procedure forward until the algorithm converges.
We run this algorithm with four alternative rationing rules: (a) strict
proportional rationing, (b) proportional rationing to intermediate demand but
priority of intermediate over final demand, (c) priority rationing serving
largest customers first and (d) random rationing, where customers are served
based on a random order. We then compare the results obtained from the four
competing behavioral rules.
These rationing approaches are frequently applied in the literature, although
there are differences in the exact specifications. In general it is hard to
calibrate how firms distribute output in case of supply constraints to
empirical data and so the rationing choice is often ad-hoc. Here, we apply all
four rules within a consistent dynamic framework to better understand how
alternative behavioral assumptions affect impact estimates. Strict
proportional rationing is frequently assumed in the literature (Henriet et
al., 2012; Hallegatte, 2014; Guan et al., 2020; Mandel and Veetil, 2020a;
Pichler et al., 2020) and also mixed proportional/priority rationing has been
considered (Battiston et al., 2007; Hallegatte, 2008; Li et al., 2013; Diem et
al., 2021). The agent-based framework of Inoue and Todo (2019) and Inoue and
Todo (2020) assumes a variation of priority rationing while the random
matching of suppliers and customers in the agent-based model proposed by
Poledna et al. (2018) and Poledna et al. (2019) most closely resembles a
random rationing approach.
##### (a) Strict proportional rationing.
If industries are unable to satisfy total incoming demand completely, they
distribute output proportional to their customers’ demand, where no
distinction is made between intermediate and final customers. More
specifically, if an industry’s output, $x_{i}$, is smaller than incoming
demand, $d_{i}$, it will supply $z_{ij}=o_{ij}\frac{x_{i}}{d_{i}}$ to customer
$j$, where $o_{ij}$ denotes the demand from customer $j$ to industry $i$. We
implement the rationing algorithm in the following way: First, industries
determine their total demand as if there were no supply-side constraints, i.e.
$\bm{d}=\mathbf{L}\bm{f}^{\text{max}}$. Industries then evaluate if they are
able to satisfy demand given their constrained production capacities. If an
industry $i$ can satisfy demand only partially, it will create a bottleneck of
size $r_{i}=\frac{x^{\text{max}}_{i}}{d_{i}}$ to other industries due to
proportional rationing. Since industries produce according to fixed Leontief
input recipes, their largest input bottleneck,
$s_{i}=\underset{j:a_{ji}>0}{\min}\\{r_{j},1\\}$ will be the binding
constraint in production. Thus, in case of input bottlenecks where $s_{i}<1$,
production of $i$ reduces to
$x_{i}=\underset{j:a_{ji}>0}{\min}\left\\{x_{i}^{\text{max}},\frac{s_{i}a_{ji}d_{i}}{a_{ji}}\right\\}=\min\\{x_{i}^{\text{max}},s_{i}d_{i}\\}<d_{i}$.
This in turn reduces the amount of goods delivered to the final consumer
$f_{i}=\min\\{x_{i}-\sum_{j}a_{ij}x_{j},0\\}$. The new final demand vector
$\bm{f}$ now implies a new, lower level of aggregate demand,
$\bm{d}=\mathbf{L}\bm{f}$, and we again let industries evaluate whether they
can satisfy total demand within given production constraints. We iterate this
procedure forward until all demand is met and no input bottleneck further
constrains production. We can write the proportional rationing algorithm more
compactly as follows:
###### Algorithm 1
Proportional rationing; industries are not prioritized over the final
consumer. Take an initial demand vector $\bm{f}[0]=\bm{f}^{\text{max}}$ as
given, implying an initial aggregated demand vector
$\bm{d}[1]=\mathbf{L}\bm{f}[0]$. By looping over the index $t=\\{1,2,...\\}$,
the following system is iterated forward:
$\displaystyle r_{i}[t]$ $\displaystyle=\frac{x_{i}^{\text{max}}}{d_{i}[t]},$
(7) $\displaystyle s_{i}[t]$
$\displaystyle=\underset{j:a_{ji}>0}{\min}\\{r_{j}[t],1\\},$ (8)
$\displaystyle x_{i}[t]$
$\displaystyle=\min\\{x_{i}^{\text{max}},s_{i}[t]d_{i}[t]\\},$ (9)
$\displaystyle f_{i}[t]$
$\displaystyle=\max\left\\{x_{i}[t]-\sum_{j}a_{ij}x_{j}[t],0\right\\},$ (10)
$\displaystyle d_{i}[t+1]$ $\displaystyle=\sum_{j}l_{ij}f_{j}[t].$ (11)
The algorithm converges to a new feasible economic allocation if
$d_{i}[t+1]=d_{i}[t]$ for all $i$. In this case output and final consumption
levels are given as $x_{i}=d_{i}[t+1]=x_{i}[t+1]$ and $f_{i}=f_{i}[t+1]$,
respectively.
Eq. (7) indicates whether an industry is output constrained, where $r_{i}$ is
the share of demand that can be met given existing productive capacities. If
$r_{i}\geq 1$, industry $i$ is able to meet demand completely, whereas demand
can only be partially satisfied if $r_{i}<1$. If any supplier of industry $i$
(which is the set $\\{j:a_{ji}>0\\}$) is sufficiently output constrained,
industry $i$ faces an input bottleneck according to Eq. (8). Due to perfect
complementarity of inputs prescribed by the Leontief production function,
industry $i$ can only produce a fraction of total demand as indicated in Eq.
(9), reducing its delivery to final consumers as specified in Eq. (10). The
new total demand to an industry $i$ is then again derived through the weighted
sum of final demand values where the weights are obtained from the Leontief
inverse, Eq. (11).
##### (b) Mixed proportional/priority rationing.
It has been argued that firm-firm relationships are stronger than firm-
household ties, so that intermediate demand should be prioritized over final
demand (Hallegatte, 2008; Inoue and Todo, 2019). While this assumption might
make sense for households, this is not necessarily the case for other final
demand categories. Final demand categories in national IO tables can include
demand by governments and non-profit organizations, exports and investments
and it is debatable whether intermediate demand should be prioritized over
these categories.
To quantify the effect of this assumption on the amplification of initial
shocks, we implement a mixed proportional/priority rationing algorithm. Here,
industries ration intermediate demand proportional analogously to the
proportional rationing algorithm (a), but prioritize intermediate demand over
final demand. We outline this algorithm below.
###### Algorithm 2
Proportional rationing among industries; industries are prioritized over final
consumers. Take an initial demand vector $\bm{f}[0]=\bm{f}^{\text{max}}$ as
given, implying an initial aggregated demand vector
$\bm{d}[1]=\mathbf{L}\bm{f}[0]$. By looping over the index $t=\\{1,2,...\\}$,
the following system is iterated forward:
$\displaystyle r_{i}[t]$
$\displaystyle=\frac{x_{i}^{\text{max}}}{\sum_{j}a_{ij}d_{j}[t]},$ (12)
$\displaystyle s_{i}[t]$
$\displaystyle=\underset{j:a_{ji}>0}{\min}\\{r_{j}[t],1\\},$ (13)
$\displaystyle x_{i}[t]$
$\displaystyle=\min\\{x_{i}^{\text{max}},s_{i}[t]d_{i}[t]\\},$ (14)
$\displaystyle f_{i}[t]$
$\displaystyle=\max\left\\{x_{i}[t]-\sum_{j}a_{ij}x_{j}[t],0\right\\},$ (15)
$\displaystyle d_{i}[t+1]$ $\displaystyle=\sum_{j}l_{ij}f_{j}[t].$ (16)
The algorithm converges to a new feasible economic allocation if
$d_{i}[t+1]=d_{i}[t]$ for all $i$. In this case output and final consumption
levels are given as $x_{i}=d_{i}[t+1]=x_{i}[t+1]$ and $f_{i}=f_{i}[t+1]$,
respectively.
The algorithm is similar to the proportional rationing algorithm (a) but
differs mainly in one aspect. Only intermediate demand affects the extent of
an industry’s output constraints, Eq. (12). As a consequence, final demand
does not play a role in the creation of input bottlenecks, Eq. (13).
##### (c) Priority rationing (“largest first”).
Since it is not obvious that industries should pass on their output
proportionally in case they are not able to meet demand fully, we next
consider a type of priority rationing. Industries rank their customers based
on demand magnitude and serve larger customers before smaller customers. In
the proportional rationing setting an output constrained supplier affects all
customers in the same way. Under a priority rationing scheme, however, supply
shocks propagate downstream heterogeneously. For example, if a supplier cannot
meet demand by only a small margin, most customers will not be affected by the
priority rationing scheme. Only the smallest customers will face input
bottlenecks, whereas every customer would experience the same small shock in
the proportional rationing setup.
Intuitively, a priority rationing rule could make sense, as firms might have
an interest in serving more important (large) customers fully, or at least as
well as possible, before focusing on less important customers. It also seems
closer to practice that firms process orders one-by-one instead of working
through all orders simultaneously and leaving them incomplete to the same
degree. While a priority rationing scheme could be plausible on the basis of
firms or single transactions, it might be less so for more aggregate industry-
level data. A link between two industries in IO tables corresponds to many
firm/establishment level transactions and so it is not clear if large inter-
industry links are due to a few big orders or many small orders. This becomes
particularly evident when considering final consumers. Several industries face
large demand from private consumers which effectively is the sum of many small
orders (e.g. restaurants, grocery, theaters). Since we only consider an
aggregate of final demand representing several distinct categories such as
private consumers, government and investments, we exclude final demand from
the priority rationing scheme. Thus, in the same manner as in the mixed
prop./prior. rationing algorithm (b), we assume that intermediate demand is
always prioritized over final demand. By adopting this convention, we can
formulate the priority rationing algorithm as follows.
###### Algorithm 3
Largest first rationing; industries are prioritized over the final consumer.
Take an initial demand vector $\bm{f}[0]=\bm{f}^{\text{max}}$ as given,
implying an initial aggregate demand vector $\bm{d}[1]=\mathbf{L}\bm{f}[0]$.
Every industry $i$ ranks each customers $j$ based on initial demand size:
$h_{ij}=\\{k_{(1)},k_{(2)},...,k_{(j)}\,:\,a_{ik_{(1)}}d_{k_{(1)}}[1]\geq
a_{ik_{(2)}}d_{k_{(2)}}[1]\geq...\geq a_{ik_{(j)}}d_{k_{(j)}}[1]\\}$. By
looping over the index $t=\\{1,2,...\\}$, the following system is iterated
forward:
$\displaystyle r_{ij}[t]$ $\displaystyle=\frac{x_{i}^{\text{max}}}{\sum_{n\in
h_{ij}}a_{in_{(j)}}d_{n_{(j)}}[t]},$ (17) $\displaystyle s_{i}[t]$
$\displaystyle=\underset{j:a_{ji}>0}{\min}\\{r_{ji}[t],1\\},$ (18)
$\displaystyle x_{i}[t]$
$\displaystyle=\min\\{x_{i}^{\text{max}},s_{i}[t]d_{i}[t]\\},$ (19)
$\displaystyle f_{i}[t]$
$\displaystyle=\max\left\\{x_{i}[t]-\sum_{j}a_{ij}x_{j}[t],0\right\\},$ (20)
$\displaystyle d_{i}[t+1]$ $\displaystyle=\sum_{j}l_{ij}f_{j}[t].$ (21)
The algorithm converges to a new feasible economic allocation if
$d_{i}[t+1]=d_{i}[t]$ for all $i$. In this case output and final consumption
levels are given as $x_{i}=d_{i}[t+1]=x_{i}[t+1]$ and $f_{i}=f_{i}[t+1]$,
respectively.
##### (d) Random rationing.
As our final case, we again consider priority rationing. Rather than using a
fixed ordering scheme based on demand magnitude, we use random priority.
Industries rank their customers randomly and serve customers based on their
position in the ranking. While largest-first rationing makes intuitive sense
in some cases, it is unlikely to be a good approximation for all real-world
settings. In practice, it is likely that other factors such as timing of
orders matter. In that case industries could adopt a first-come-first-serve
principle to process orders. Since IO data rarely comes with granular time
information, we adopt a random ordering of incoming orders that mimics a
first-come-first-serve principle under a uniform prior of which orders are
coming in first. The algorithm is presented below.
###### Algorithm 4
Random rationing; industries are prioritized over the final consumer. The
algorithm is identical to Algorithm 3, except that the ranking of customer $j$
by industries $i$, $h_{ij}=\\{k_{(1)},k_{(2)},...,k_{(j)}\\}$, is randomly
drawn.
While these algorithms are not guaranteed to converge to a steady-state
equilibrium, we observe convergence in the vast majority of our simulation
experiments. If the algorithm converges, the economic allocations obtained are
automatically feasible. This means that no industry has negative output or
produces more than its productive capacities allow, final consumption is non-
negative and below given exogenous maximum consumption levels, and there are
no input bottlenecks left that further constrain production of downstream
industries. The Leontief equation $\bm{x}=\mathbf{L}\bm{f}$ holds, which
implies that all sales add up to total output.
### 4.3 Results
We initialize these four rationing algorithms with the supply and demand
shocks and IO data of Germany, Italy and Spain. We then compare the steady-
state equilibrium economic outcomes predicted by the rationing algorithms with
the optimization results and the direct shock computations (Appendix A, Tables
2 and 3).
Fig. 1 summarizes the main result visually, where a diamond indicates the
aggregate final consumption and gross output levels predicted by the different
approaches. Note that a data point in the top-right corner indicates high
gross output and final consumption values, i.e. limited negative impacts,
whereas diamonds in the bottom-left corner represent extremely adversely
impacted economies. As already reported in Table 1, for all three countries
supply shocks are substantially larger than final consumption shocks, pushing
the _Direct shock_ diamond substantially below the 45 degree line. Note that
the direct shock market allocations are not feasible, as they ignore higher-
order effects, such that a direct supply shock to an industry reduces the
inputs of downstream industries and thus reduces their output too.
The best case feasible market allocations are given by the two optimization
methods (maximize output vs. consumption), which yield exactly the same
predictions on the aggregate and the industry level. This is true even though
the two optimization methods are not equivalent.555 When perturbing the
economic systems we can find cases where the two optimization methods do not
yield exactly the same results. Practically, we find that aggregate
predictions are always similar for the two methods although industry-level
results can sometimes differ significantly. The two optimization methods have
the highest output, but they still amplify the initial supply shocks to reduce
the output from about $69\%$ to $63\%$ in Germany and from $67\%$ to $53\%$ in
Spain.
All the other feasible market allocations, obtained from the rationing
algorithms, lie substantially below the best case scenarios. The wide range of
predicted outcomes is the most striking aspect of Fig. 1. Both the random
rationing scheme and the priority rationing scheme essentially collapse the
entire economy. Proportional rationing substantially collapses the economy
(with an output below $20\%$ of normal for all countries) and the mixed scheme
reduces output for Germany and Spain by more than 70% and about $60\%$ for
Italy. Interestingly, all feasible market allocations lie close to the
identity line, suggesting similar impacts to output and consumption.
Figure 1: Comparison of different shock propagation mechanisms. The y-axis
denotes aggregate gross output levels normalized by pre-shock levels as
predicted by the rationing and optimization methods. The x-axis shows the same
for aggregate final consumption levels. The _Random rationing_ diamond
represents the average taken from 100 samples and the error bars indicate the
interquartile range.
On a sectoral level, we find the overall orderings of economic impacts are
highly correlated across methods and countries, despite a few pronounced
differences. An interesting example is Forestry (A02) which represents the
only industry that experiences less adverse impacts under proportional
rationing compared to mixed rationing. Due to being non-essential and having a
low Remote Labor Index (Appendix A, Table 3), forestry is the industry with
the largest supply shocks. Thus, it virtually always faces stronger supply
constraints than demand constraints. In case of no prioritization of
industries, Forestry receives less demand and thus creates smaller input
bottlenecks for downstream industries. This, in turn, leads to smaller input
bottlenecks for Forestry as its downstream industries can be found upstream as
well.
There is also substantial variation in how close industries are to their
theoretical maximum value as determined by the optimization method. When
considering the proportional rationing algorithm in Italy, for example, the
output levels of the best faring industries (such as Education (P85) or
Telecommunications (J61)) reach around 30% of their theoretical maximum. On
the other hand, the hardest hit industry, Accommodation-Food (I), is only at a
5% level of what is possible. Thus, Accommodation-Food is not only hit
extremely hard by direct shocks, but also experiences large higher-order
effects through the rationing dynamics.
Fig. 1 makes it clear that the behavioral assumptions imposed on suppliers
matter enormously for economic impact predictions. In fact, results vary more
across alternative assumptions than across countries.
In the absence of further modeling refinements (such as inventories, adaptive
behavior, substitution effects), shock amplification is always pronounced if
basic national accounting identities are required to hold. But the actual
extent of shock amplification depends strongly on how input bottlenecks are
created and passed on downstream in the supply chain.
#### 4.3.1 Shock magnitude effects
We next investigate the sensitivity of economic impact predictions with
respect to shock magnitude and network connectedness. While we have seen in
Section 4.3 that different assumptions on rationing behavior can lead to
entirely different estimates of impact, it is not clear whether these results
are specific to the three datasets considered. We therefore conduct a series
of simulation experiments to gain a better understanding of the generality of
the results.
First, we investigate how the estimates of the various methods depend on the
magnitude of shocks. To do this, we rescale supply and demand shocks and we
apply the optimization methods and the rationing algorithms to the new shock
data. We then redo this analysis for various shock scales. To better
differentiate between the qualitative effects of demand and supply shock
propagation, we allow for different scaling factors for demand and supply
constraints, i.e.
$\displaystyle x_{i}^{\text{max}}$
$\displaystyle=(1-\alpha^{S}\epsilon_{i}^{S})x_{i,0},$ (22) $\displaystyle
f_{i}^{\text{max}}$ $\displaystyle=(1-\alpha^{D}\epsilon_{i}^{D})f_{i,0},$
(23)
where $\alpha^{S},\alpha^{D}\in[0,1]$.
Fig. 2(a) shows aggregate output levels for all cases when only demand shocks
are scaled between zero and one and when there are no further supply
constraints being present ($\alpha^{S}=0,\alpha^{D}\in[0,1]$).666 Since
results for aggregate final consumption and aggregate gross output are very
similar, we only present figures of the latter in this section. We show
results for scaling supply and demand shocks concurrently
($\alpha^{S}=\alpha^{D}\in[0,1]$) in Appendix C. It becomes evident that
predictions made by the rationing algorithms and the optimization methods are
identical and scale linearly with the demand shock magnitude. Thus, the
rationing algorithms do not differ with respect to upstream shock propagation
and always arrive at the optimal solution in absence of further supply side
constrictions.
Figure 2: Economic impact as a function of shock magnitude. (a) Aggregate
gross output levels as a function of scaling only demand shocks between zero
and one ($\alpha^{S}=0,\alpha^{D}\in[0,1]$). All methods yield the same
economic impact estimates which scale linearly with $\alpha^{S}$. The black
horizontal line indicates that there is no adverse economic impact from supply
side constraints. (b) Aggregate gross output levels as a function of scaling
only supply shocks between zero and one ($\alpha^{D}=0,\alpha^{S}\in[0,1]$).
The _Random rationing_ line represents the average taken from 100 samples and
the shades indicate the interquartile range. We do not distinguish further
between the two maximization methods as results are almost identical.
The picture becomes very different when rescaling only supply shocks and
turning off demand shocks ($\alpha^{D}=0,\alpha^{S}\in[0,1]$) as demonstrated
in Fig. 2(b). Here, the direct impact reduces gross output linearly as the
shock magnitude is increased and puts an upper bound to the solutions of the
methods considered here. When there are no shocks, $\alpha^{S}=0$, all methods
recover the empirical IO data as expected. For small shocks we observe that
estimated impacts are very similar across different methods, but the
proportional rationing algorithm consistently returns the smallest output
values. The results of the other algorithms (mixed, priority, random) lie
between those proportional rationing and optimization predictions, and are
identical for small shocks up to about $\alpha^{S}=0.5$. As $\alpha^{S}$ is
increased further they deviate dramatically. Under the priority algorithm even
a small increase in $\alpha^{S}$ causes the economy to collapse. The random
algorithm also causes the economy to collapse, though more slowly, and the
mixed algorithm is substantially better.
Fig. 2 thus makes clear that the ranking of the impact assessment methods as
seen in Fig. 1 is not generic, but strongly depends on the shock magnitude. If
there are only small supply shocks present, a priority rationing rule is
better than proportional rationing. In this case most of the demand of
downstream industries can be met and small shocks only propagate to few
customers. In a proportional rationing setup, on the other hand, shocks are
passed on to every customer. Despite small shock magnitudes, the wider breadth
of shock propagation leads to comparatively larger aggregate impacts.
In contrast, priority rationing exacerbates shock propagation compared to
proportional rationing in case of large supply shocks. If industries are
severely output constrained and ration based on a priority rule, the demand of
several customers’ might not be satisfied at all. The Leontief production
function, in turn, implies a complete shutdown of these downstream industries
due to the input bottlenecks created. More input bottlenecks will be created
in further rounds of shock propagation, potentially causing massive collapses.
If supply shocks are that large, shock amplification is milder if passed on
proportionally to customers. While here every customer experiences some shocks
from a constrained supplier, this effect is outweighed by the fact that
proportional rationing avoids the creation of even larger idiosyncratic input
bottlenecks.
#### 4.3.2 Network density
Intuitively, the extent of shock propagation not only depends on the size of
initial shocks but also on how industries are connected with each other. If
the production network is dense, i.e. most of the potential links are present,
idiosyncratic shocks will spread out very quickly to many other industries in
the network. In contrast, if the network is sparse, shock propagation might be
more local, at least initially, and takes more steps to spread out in the
network.
We therefore conduct an experiment where we again apply the different shock
propagation mechanisms to the data, but control for the IO network density. We
do this in the following way. First, we randomly eliminate a given number of
links in the intermediate consumption matrix $\mathbf{Z}$.777 We also
experimented with eliminating smallest links first instead of randomly
selection. Results are similar to the ones presented here and shown in the
Appendix D. We only consider deleting links instead of adding links since the
aggregate IO networks we are using are highly dense
($\sum_{ij}\mathds{1}_{\\{z_{ij}>0\\}}/n^{2}>99\%$). Note that setting a link
$z_{ij}>0$ equal to zero without changing final consumption values reduces
total output of supplier $i$, since the accounting identity
$x_{i}=\sum_{j}z_{ij}+f_{i}$ has to hold. The output of customer $j$ will not
be affected if we assume that the reduction in intermediate consumption is
absorbed by a respective increase of $j$’s value added (which does not affect
the simulations). After deleting nodes we thus rebalance the economic system
by adjusting gross output of the relevant industries such that the basic
national accounting identity is satisfied. Next, we recompute the technical
coefficient matrix $\mathbf{A}$, the Leontief matrix $\mathbf{L}$ and the
vector of productive constraints $\bm{x}^{\text{max}}$ to initialize the
optimization and rationing simulations. We repeat this procedure 50 times for
a given level of density and explore the whole range of possible density
values.
We visualize aggregate output levels predicted by the various impact
assessment methods as a function of network density in Fig. 3. The plot again
confirms that the observed ranking of methods in Fig. 1 is not generic but is
strongly affected by the network density. It becomes clear that shock
amplification is comparatively small if the network is very sparse. On the
other hand, if the network is very dense, as the empirical data would suggest,
economic impacts are substantial for all cases. In particular, the gap between
the optimal solution and the bottom-up rationing approaches widens with higher
network density. Interestingly, the minimal amplification of direct shocks
also increases with higher density values, although the size of this effect is
rather small.
Figure 3: Economic impact as a function of network density. The network
density is changed by randomly eliminating links in the IO network. Solid
lines are predicted output values obtained from averaging over 50 samples and
shades indicate the interquartile range. Since random rationing is stochastic
by itself, we apply this algorithm 50 times to every network sample and take
averages and quantiles over the full (pooled) density-specific sample. The
actual data corresponds to the very right of the x-axis, as indicated by the
diamonds.
If industries have only few suppliers and customers, proportional rationing
yields by far the most pessimistic predictions of aggregate impact. If there
are many connections among industries, on the contrary, proportional rationing
mitigates shock propagation dynamics compared to priority and random
rationing. Shock amplification is always mildest if industries use a mixed
proportional/priority rationing rule, although even in this case economic
impacts increase substantially with higher density.
These results indicate that estimates of economic impact are highly sensitive
with respect to the network structure which, in turn, often depends on the
level of data aggregation. More aggregated data necessarily implies higher
levels of density. For example, industry-level links are the result of pooling
many firm links. Firm-level production networks, on the other hand, are an
aggregation of many individual contractual relationships and payments. Imagine
applying the rationing mechanisms to a firm-level production network in two
ways: first to the actual firm-level data and second to an industry aggregate
of the same data. Our results suggest that predicted economic impacts could be
substantially larger in the second case, despite using the same underlying
data.
Our findings also point out that economic impact predictions can be sensitive
with respect to data quality. Many disaggregate firm-level production network
datasets are substantially biased, as they frequently include only specific
supplier-customer relationships which are subject to specific reporting rules.
Thus, we would expect real world production networks to be substantially
denser than what the data suggests. Our analysis indicates that such biases
could have important consequences for impact assessment.
## 5 Discussion
We have shown that existing IO models have difficulties in dealing with
simultaneous supply and demand shocks. However, both types of shocks play an
important role in situations such as natural disasters and during a pandemic.
We have introduced a simple optimization method that allows us to find best
case market allocations that are consistent with the exogenous shocks. To
obtain more realistic, bottom-up impact estimates, we studied alternative
rationing dynamics which differ in how suppliers serve customers in case of
output constraints. Using IO data for Germany, Italy and Spain, we found that
these bottom-up approaches lead to substantial amplifications of initial
shocks, which are much higher than optimal solutions would suggest. We further
established that different rationing assumptions can lead to dramatically
different economic impact predictions. Moreover, these predictions are highly
sensitive with respect to the magnitude of first-order shocks and the
production network structure.
It is clear that adequate macroeconomic predictions of pandemic impacts
require more sophisticated modeling techniques than the ones studied here. Yet
the downside of more complicated models is that the underlying mechanisms of
predictions can be difficult to isolate. We therefore studied relatively
simple economic models to better carve out key mechanisms of shock propagation
in twofold constrained production networks. The choice for simplicity in this
study also entails limitations which are important to bear in mind. We did not
depart from the assumption of industry-specific Leontief production functions
throughout our analysis, although the choice of production function has been
shown to be a key variable for impact assessment (Pichler et al., 2020). While
input substitutions might be limited in the short-run, fixed production
recipes are nevertheless a strong assumption, in particular for the aggregate
data considered here.
It is also important to stress that we did not take any adaptive behavior of
economic agents into account and did not allow for the possibility of
inventory buildup and depletion. By imposing feasible market solutions, we
essentially forced the economy to converge into a new equilibrium. It is not
clear, however, that a perturbed complex system such as national economies
would quickly approach a new equilibrium state instead of following transient
paths for an extended period. In general, we refrained from making the time
dimension explicit, although we acknowledge its relevance for shock
propagation dynamics.
Despite the caveats mentioned, our analysis makes clear that level of
aggregation and data quality play an important role in aggregate predictions
of shock amplification. Detailed and high-quality production network data are
rare, necessitating researchers to study biased or aggregate data. Inevitably,
this will affect the connectedness of economic agents and thus influence shock
propagation mechanisms. For example, studies based on large-scale firm level
databases report network density values well below 0.01% (Kumar et al., 2021;
Borsos and Stancsics, 2020; Peydró et al., 2020; Demir et al., 2018; Huneeus,
2018; Spray, 2017). We should also mention that estimates of direct shocks to
supply and demand are subject to large uncertainties. This could have
important consequences for model predictions, due to the sensitivity of shock
propagation mechanisms with respect to first-order shock magnitudes.
We find that the number of links between industries strongly influences how
different behavior rules amplify direct shocks. The level of connectedness is
a very simple aggregate network measure, and we would expect that further
aspects, such as community structure, degree/strength heterogeneity or (dis-)
assortative mixing, play an important role too. Understanding how alternative
rationing assumptions interact with structural properties of complex networks
would require more detailed production network data, e.g. at the firm level,
and could be an interesting avenue for future research.
We conclude by stressing that our analysis is not constrained to pandemic
shocks. While the Covid-19 pandemic is a main motivation of our study,
simultaneous supply and demand constraints are ubiquitous features of any
economy, in particular in the short run. Supply shocks are a prominent
characteristic of many natural hazards (floods, earthquakes, hurricanes) and
tools of mostly demand-driven IO analysis are frequently applied for impact
assessment. Our results indicate that adequately modeling shock propagation in
production networks will require a better integration of both types of
economic constraints.
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## Appendix
## Appendix A Details on first-order shocks to supply and demand
| | $f_{i}$ (%) | $\epsilon_{i}^{D}$ (%)
---|---|---|---
ISIC | Industry | DEU | ESP | ITA | DEU | ESP | ITA
A01 | Agriculture | 0.5 | 1.4 | 1.0 | 10.0 | 9.9 | 10.0
A02 | Forestry | 0.1 | 0.0 | 0.1 | 10.0 | 8.6 | 3.8
A03 | Fishing | 0.0 | 0.1 | 0.0 | 10.0 | 10.0 | 10.0
B | Mining | 0.3 | 0.5 | 0.4 | 10.0 | 10.0 | 10.0
C10-12 | Manuf. Food-Beverages | 4.1 | 5.0 | 4.0 | 10.0 | 10.0 | 10.0
C13-15 | Manuf. Textiles | 0.6 | 1.4 | 2.9 | 10.0 | 10.0 | 10.0
C16 | Manuf. Wood | 0.4 | 0.1 | 0.3 | 10.0 | 10.0 | 9.9
C17 | Manuf. Paper | 0.6 | 0.4 | 0.5 | 10.0 | 10.0 | 10.0
C18 | Media print | 0.1 | 0.1 | 0.1 | 10.0 | 9.9 | 10.0
C19 | Manuf. Coke-Petroleum | 1.7 | 2.6 | 1.4 | 10.0 | 10.0 | 10.0
C20 | Manuf. Chemical | 3.4 | 2.1 | 1.5 | 9.9 | 9.9 | 10.0
C21 | Manuf. Pharmaceutical | 1.1 | 1.0 | 1.2 | 8.1 | 9.3 | 9.8
C22 | Manuf. Rubber-Plastics | 1.4 | 0.7 | 1.0 | 10.0 | 9.9 | 10.0
C23 | Manuf. Minerals | 0.6 | 0.5 | 0.7 | 10.0 | 10.0 | 10.0
C24 | Manuf. Metals-basic | 1.5 | 1.3 | 1.4 | 10.0 | 10.0 | 10.0
C25 | Manuf. Metals-fabricated | 1.8 | 1.0 | 1.6 | 10.0 | 10.0 | 10.0
C26 | Manuf. Electronic | 2.0 | 0.4 | 0.9 | 9.9 | 9.9 | 10.0
C27 | Manuf. Electric | 2.3 | 0.8 | 1.4 | 10.0 | 10.0 | 10.0
C28 | Manuf. Machinery | 5.8 | 1.4 | 4.7 | 10.0 | 10.0 | 10.0
C29 | Manuf. Vehicles | 8.2 | 3.7 | 2.1 | 10.0 | 10.0 | 10.0
C30 | Manuf. Transport-other | 1.0 | 1.1 | 1.0 | 9.9 | 9.9 | 10.0
C31-32 | Manuf. Furniture | 1.4 | 0.6 | 1.4 | 9.9 | 9.9 | 10.0
C33 | Repair-Installation | 0.5 | 0.3 | 0.6 | 10.0 | 10.0 | 10.0
D35 | Electricity-Gas | 1.6 | 1.6 | 1.1 | 2.3 | 0.8 | 1.2
E36 | Water | 0.2 | 0.4 | 0.3 | 1.5 | 0.9 | 0.6
E37-39 | Sewage | 0.7 | 0.7 | 0.6 | 3.6 | 2.5 | 1.7
F | Construction | 5.5 | 7.6 | 7.5 | 10.0 | 9.9 | 9.9
G45 | Vehicle trade | 0.9 | 1.7 | 1.4 | 10.0 | 10.0 | 10.0
G46 | Wholesale | 3.4 | 4.1 | 4.6 | 9.8 | 9.5 | 9.9
G47 | Retail | 3.4 | 5.5 | 5.7 | 9.5 | 9.5 | 9.8
H49 | Land transport | 0.8 | 1.7 | 1.8 | 56.9 | 38.8 | 55.2
H50 | Water transport | 0.7 | 0.2 | 0.5 | 11.4 | 27.5 | 47.3
H51 | Air transport | 0.6 | 0.6 | 0.4 | 50.3 | 24.7 | 47.3
H52 | Warehousing | 0.3 | 0.8 | 1.1 | 22.3 | 19.7 | 33.5
H53 | Postal | 0.1 | 0.0 | 0.1 | 3.4 | 2.2 | 3.4
I | Accommodation-Food | 2.4 | 8.6 | 4.7 | 73.1 | 75.5 | 79.1
J58 | Publishing | 0.4 | 0.3 | 0.3 | 3.8 | 5.4 | 4.0
J59-60 | Video-Sound-Broadcasting | 0.6 | 0.5 | 0.3 | 4.3 | 3.8 | 3.5
J61 | Telecommunications | 0.9 | 1.3 | 1.2 | 1.1 | 1.6 | 3.0
J62-63 | IT | 1.5 | 1.6 | 1.1 | 8.8 | 9.5 | 8.5
K64 | Finance | 1.9 | 0.8 | 0.9 | 2.9 | 3.8 | 1.6
K65 | Insurance | 1.4 | 1.1 | 1.0 | 1.3 | 0.8 | 1.1
K66 | Auxil. Finance-Insurance | 0.0 | 0.2 | 0.1 | 2.0 | 1.9 | 4.9
L68 | Real estate | 7.2 | 7.8 | 9.8 | 0.2 | 0.1 | 0.5
M69-70 | Legal | 0.7 | 0.6 | 0.4 | 9.3 | 7.9 | 5.2
M71 | Architecture-Engineering | 1.0 | 1.0 | 0.2 | 9.4 | 9.3 | 8.4
M72 | R&D | 0.9 | 0.5 | 0.6 | 8.3 | 7.9 | 9.8
M73 | Advertising | 0.1 | 0.1 | 0.1 | 10.0 | 9.1 | 9.2
M74-75 | Other Science | 0.3 | 0.1 | 0.3 | 3.5 | 3.5 | 5.0
N | Private Administration | 1.2 | 1.2 | 1.1 | 4.2 | 4.2 | 4.9
O84 | Public Administration | 6.1 | 6.3 | 7.1 | 0.2 | 0.7 | 0.0
P85 | Education | 4.1 | 5.1 | 3.9 | 0.7 | 1.0 | 0.0
Q | Health | 8.3 | 7.2 | 7.5 | 0.1 | 0.1 | 0.1
R_S | Other Service | 3.2 | 3.3 | 3.2 | 4.3 | 4.3 | 4.7
T | Household activities | 0.2 | 0.8 | 1.1 | 0.0 | -0.0 | -0.0
Table 2: Industry-specific demand shock details. $f_{i}$ denotes final consumption per industry as fraction of aggregate final consumption. $\epsilon_{i}^{D}$ is the total demand shock per industry. | | $x_{i}$ (%) | $\epsilon_{i}^{S}$ (%) | $e_{i}$ (%) | $\text{RLI}_{i}$
---|---|---|---|---|---
ISIC | Industry | DEU | ESP | ITA | DEU | ESP | ITA | DEU | ESP | ITA | (%)
A01 | Agriculture | 0.9 | 2.2 | 1.7 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 13.6
A02 | Forestry | 0.1 | 0.0 | 0.0 | 85.0 | 85.0 | 85.0 | 0.0 | 0.0 | 0.0 | 15.0
A03 | Fishing | 0.0 | 0.1 | 0.1 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 35.7
B | Mining | 0.2 | 0.3 | 0.3 | 48.3 | 48.3 | 34.5 | 30.0 | 30.0 | 50.0 | 31.0
C10-12 | Manuf. Food-Beverages | 3.5 | 6.9 | 4.1 | 0.0 | 26.0 | 26.0 | 100.0 | 66.7 | 66.7 | 22.1
C13-15 | Manuf. Textiles | 0.4 | 1.0 | 2.6 | 68.5 | 68.5 | 59.4 | 0.0 | 0.0 | 13.3 | 31.5
C16 | Manuf. Wood | 0.5 | 0.3 | 0.5 | 73.1 | 73.1 | 60.6 | 0.0 | 0.0 | 17.0 | 26.9
C17 | Manuf. Paper | 0.7 | 0.6 | 0.7 | 34.3 | 0.0 | 48.7 | 50.0 | 100.0 | 29.0 | 31.5
C18 | Media print | 0.4 | 0.4 | 0.4 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 39.0
C19 | Manuf. Coke-Petroleum | 1.5 | 2.4 | 1.7 | 0.0 | 6.4 | 0.0 | 100.0 | 90.0 | 100.0 | 36.0
C20 | Manuf. Chemical | 2.6 | 2.5 | 1.6 | 19.0 | 52.5 | 8.2 | 70.0 | 17.0 | 87.0 | 36.7
C21 | Manuf. Pharmaceutical | 0.9 | 0.7 | 0.8 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 40.3
C22 | Manuf. Rubber-Plastics | 1.4 | 0.9 | 1.3 | 35.3 | 70.5 | 47.2 | 50.0 | 0.0 | 33.0 | 29.5
C23 | Manuf. Minerals | 0.8 | 0.8 | 1.0 | 63.9 | 63.9 | 61.4 | 0.0 | 0.0 | 4.0 | 36.1
C24 | Manuf. Metals-basic | 1.9 | 2.1 | 1.8 | 72.6 | 72.6 | 72.6 | 0.0 | 0.0 | 0.0 | 27.4
C25 | Manuf. Metals-fabricated | 2.4 | 1.4 | 2.6 | 66.3 | 66.3 | 66.3 | 0.0 | 0.0 | 0.0 | 33.7
C26 | Manuf. Electronic | 1.4 | 0.4 | 0.7 | 43.1 | 43.1 | 38.8 | 0.0 | 0.0 | 10.0 | 56.9
C27 | Manuf. Electric | 1.9 | 0.8 | 1.1 | 63.1 | 63.1 | 50.5 | 0.0 | 0.0 | 20.0 | 36.9
C28 | Manuf. Machinery | 4.5 | 1.2 | 3.6 | 61.8 | 61.8 | 47.0 | 0.0 | 0.0 | 24.0 | 38.2
C29 | Manuf. Vehicles | 6.3 | 2.5 | 1.5 | 69.7 | 69.7 | 69.7 | 0.0 | 0.0 | 0.0 | 30.3
C30 | Manuf. Transport-other | 0.8 | 0.8 | 0.7 | 59.7 | 59.7 | 59.7 | 0.0 | 0.0 | 0.0 | 40.3
C31-32 | Manuf. Furniture | 0.9 | 0.6 | 1.2 | 65.2 | 59.7 | 58.0 | 0.0 | 8.5 | 11.0 | 34.8
C33 | Repair-Installation | 0.7 | 0.5 | 0.6 | 60.6 | 60.6 | 34.0 | 0.0 | 0.0 | 44.0 | 39.4
D35 | Electricity-Gas | 2.4 | 4.7 | 2.7 | 0.0 | 5.8 | 0.0 | 100.0 | 90.0 | 100.0 | 41.6
E36 | Water | 0.2 | 0.5 | 0.3 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 33.5
E37-39 | Sewage | 0.9 | 0.8 | 1.3 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 29.8
F | Construction | 5.2 | 6.4 | 6.7 | 71.6 | 71.6 | 49.6 | 0.0 | 0.0 | 30.7 | 28.4
G45 | Vehicle trade | 1.1 | 1.4 | 1.1 | 18.0 | 27.3 | 13.7 | 67.0 | 50.0 | 75.0 | 45.4
G46 | Wholesale | 3.9 | 4.9 | 5.3 | 0.0 | 30.0 | 33.5 | 100.0 | 40.0 | 33.0 | 50.1
G47 | Retail | 3.0 | 3.9 | 3.9 | 24.7 | 25.7 | 25.2 | 51.0 | 49.0 | 50.0 | 49.7
H49 | Land transport | 1.8 | 2.5 | 3.0 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 31.4
H50 | Water transport | 0.5 | 0.2 | 0.4 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 35.3
H51 | Air transport | 0.5 | 0.5 | 0.4 | 0.0 | 24.2 | 0.0 | 100.0 | 66.0 | 100.0 | 28.8
H52 | Warehousing | 2.3 | 2.1 | 2.1 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 29.6
H53 | Postal | 0.6 | 0.2 | 0.2 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 35.6
I | Accommodation-Food | 1.6 | 5.8 | 3.3 | 64.6 | 64.6 | 56.6 | 0.0 | 0.0 | 12.5 | 35.4
J58 | Publishing | 0.6 | 0.4 | 0.3 | 0.0 | 7.6 | 0.0 | 100.0 | 75.0 | 100.0 | 69.8
J59-60 | Video-Sound-Broadcasting | 0.6 | 0.6 | 0.5 | 0.0 | 11.0 | 0.0 | 100.0 | 75.0 | 100.0 | 56.1
J61 | Telecommunications | 1.2 | 1.8 | 1.3 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 55.1
J62-63 | IT | 2.1 | 1.4 | 1.6 | 7.2 | 14.4 | 0.0 | 75.0 | 50.0 | 100.0 | 71.1
K64 | Finance | 2.7 | 2.1 | 2.9 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 71.4
K65 | Insurance | 1.4 | 1.0 | 0.8 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 71.3
K66 | Auxil. Finance-Insurance | 0.6 | 0.4 | 1.0 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 71.7
L68 | Real estate | 7.2 | 6.7 | 7.5 | 51.3 | 51.3 | 51.3 | 0.0 | 0.0 | 0.0 | 48.7
M69-70 | Legal | 2.5 | 1.5 | 2.3 | 36.3 | 18.1 | 0.0 | 0.0 | 50.0 | 100.0 | 63.7
M71 | Architecture-Engineering | 1.2 | 1.1 | 1.0 | 45.9 | 45.9 | 0.0 | 0.0 | 0.0 | 100.0 | 54.1
M72 | R&D | 0.6 | 0.3 | 0.4 | 41.1 | 41.1 | 0.0 | 0.0 | 0.0 | 100.0 | 58.9
M73 | Advertising | 0.4 | 0.5 | 0.5 | 39.7 | 39.7 | 39.7 | 0.0 | 0.0 | 0.0 | 60.3
M74-75 | Other Science | 0.4 | 0.3 | 0.7 | 19.6 | 19.6 | 0.0 | 50.0 | 50.0 | 100.0 | 60.8
N | Private Administration | 3.9 | 2.6 | 3.0 | 42.7 | 54.3 | 41.6 | 33.3 | 15.2 | 35.0 | 36.0
O84 | Public Administration | 4.6 | 4.4 | 4.2 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 44.6
P85 | Education | 2.9 | 3.4 | 2.4 | 0.0 | 46.0 | 0.0 | 100.0 | 0.0 | 100.0 | 54.0
Q | Health | 5.4 | 4.8 | 4.9 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 36.0
R_S | Other Service | 2.8 | 2.7 | 2.7 | 61.2 | 56.0 | 49.2 | 0.0 | 8.6 | 19.6 | 38.8
T | Household activities | 0.1 | 0.5 | 0.6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 100.0
Table 3: Industry-specific supply shock details. $x_{i}$ denotes gross output
per industry as share of aggregate output. $\epsilon_{i}^{S}$ is the total
supply shock per industry. $e_{i}$ is the extent to which an industry is
considered as essential following government policies. $\text{RLI}_{i}$ is the
Remote Labor Index.
## Appendix B Details on mixed endogenous/exogenous modeling
A core motivation to study shock propagation based on alternative, bottom-up
approaches was our observation that existing methods such as the mixed
endogenous/exogenous model (MEEM) yields infeasible solution when applied to
pandemic shocks. In the MEEM the $n$ industries are divided into two groups.
The first group is the set of $n^{s}$ supply constrained industries and the
second group the $n^{d}$ demand-constrained industries (we have
$n^{s}+n^{d}=n$). We can use this setup to partition Eq. (4) into
$\begin{pmatrix}\bm{x}^{s}\\\
\bm{x}^{d}\end{pmatrix}=\begin{pmatrix}\mathbf{A}^{ss}&\mathbf{A}^{sd}\\\
\mathbf{A}^{ds}&\mathbf{A}^{dd}\end{pmatrix}\begin{pmatrix}\bm{x}^{s}\\\
\bm{x}^{d}\end{pmatrix}+\begin{pmatrix}\bm{f}^{s}\\\ \bm{f}^{d}\end{pmatrix},$
(24)
where superscripts $s$ and $d$ denote supply and demand constrained
industries, respectively. The matrix block $\mathbf{A}^{sd}$ indicates the
input recipes of demand constrained customers with respect to output
constrained suppliers and analogously for the other blocks. In this framework
vectors $\bm{f}^{s}$ and $\bm{x}^{d}$ are endogenous and $\bm{f}^{d}$ and
$\bm{x}^{s}$ exogenous. Rearranging Eq. (24) such that all endogenous
variables appear on the left-hand side and all exogenous variables on the
right-hand side, yields
$\begin{pmatrix}\bm{f}^{s}\\\
\bm{x}^{d}\end{pmatrix}=\begin{pmatrix}\mathbf{I}&\mathbf{A}^{sd}\\\
\bm{0}&\mathbf{I}-\mathbf{A}^{dd}\end{pmatrix}^{-1}\begin{pmatrix}\mathbf{I}-\mathbf{A}^{ss}&\bm{0}\\\
\mathbf{A}^{ds}&\mathbf{I}\end{pmatrix}\begin{pmatrix}\bm{x}^{s}\\\
\bm{f}^{d}\end{pmatrix}$ (25)
(for details see Miller and Blair (2009), 621-633).
To apply the MEEM in the pandemic context, we categorize industries into a
demand and a supply constrained group based on which shock is larger. If the
supply shock of an industry exceeds its demand shock in absolute terms, we
treat this industry as supply constrained and vice versa. The shock magnitudes
for supply and demand for each industry are given as
$\displaystyle x_{i}^{\text{SS}}$
$\displaystyle=x_{i,0}-x_{i}^{\text{max}}=-\epsilon_{i}^{S}x_{i,0},$ (26)
$\displaystyle f_{i}^{\text{DS}}$
$\displaystyle=c_{i,0}-f_{i}^{\text{max}}=-\epsilon_{i}^{D}f_{i,0},$ (27)
where $d_{i}^{\text{SS}}$ denotes the total _supply shock_ and
$f_{i}^{\text{DS}}$ the _demand shock_. If
$x_{i}^{\text{SS}}>f_{i}^{\text{DS}}$, we consider this industry as supply
constrained and we will use its gross output values on the right hand side of
Eq. (25). Otherwise we treat it as demand constrained.
Following this approach, we apply the MEEM to the IO data of Germany, Italy
and Spain by calibrating it to the estimated pandemic supply and demand
shocks. As shown in Fig. 4, the MEEM does not yield a feasible solution for
any of the three countries. Violations of feasibility conditions are most
frequent for Spain, which faces the largest shocks to supply and demand. The
model does not compute any negative final consumption values for Germany, but
still allocates final consumption values to industries which are larger than
$f_{i}^{\text{max}}$. Note that the results obtained by the MEEM are out of
the solution space delimited by the exogenous constraints on output and
consumption. Thus, these results are difficult to interpret in this context
and we thus refrain from comparing them more systematically with the results
shown in the main text.
Figure 4: Final consumption values from mixed exogenous/endogenous IO
modeling with simultaneous supply and demand shocks. Black circles indicate
initial pre-shock values, red circles demand-constrained industries and blue
circles supply-constrained industries. Note that demand-constrained
consumption values are exogenously determined by the first-order shocks, while
in other cases the change in output depends on the higher order effects on the
whole economy. Red lines indicate the overall change in production for each
sector. Numbers to the right of both panels indicate the value change in
percentages and are colored red if the result is infeasible.
The MEEM can yield infeasible economic allocations and it depends on the
context whether negative final consumption values are meaningful or not
(Miller and Blair, 2009, p. 628). To see why the MEEM can give negative final
consumption values, let us consider the output-constrained part of Eq. (25),
which can be rewritten as
$\displaystyle\bm{f}^{s}=[\mathbf{I}-\mathbf{A}^{ss}]\bm{x}^{s}-\mathbf{A}^{sd}[\mathbf{I}-\mathbf{A}^{dd}]^{-1}(\mathbf{A}^{ds}\bm{x}^{s}+\bm{f}^{d}).$
(28)
Note that the vector $(\mathbf{A}^{ds}\bm{x}^{s}+\bm{f}^{d})$ is always non-
negative by definition, as is every element of the matrix
$\mathbf{A}^{sd}[\mathbf{I}-\mathbf{A}^{dd}]^{-1}$. (This is clear from
invoking the Hawkins-Simon conditions). Thus, for $\bm{f}^{s}\geq\bm{0}$,
$[\mathbf{I}-\mathbf{A}^{ss}]\bm{x}^{s}$ must be non-negative and larger than
the part to the right of the minus in Eq. (28). However, this term can be
negative for supply shocks that are sufficiently heterogeneous. For example,
consider the case of an economy with only two supply-constrained industries
without self-loops, $i$ and $j$. Any supply shocks which lead to
$x_{i}^{\text{max}}<a_{ij}^{ss}x_{j}^{\text{max}}$ yield negative final
consumption values for industry $i$. This demonstrates that the MEEM framework
is unlikely to yield plausible solutions in the current context.
The MEEM can also lead to final consumption values that lie above any given
pre-specified upper limit of consumption $f_{i}^{\text{max}}$. Let us again
consider an example of two industries, $i$ and $j$, where $i$ is supply
constrained and $j$ is demand constrained. If industry $i$ only supplies
industry $j$ and final consumers and industry $j$ only supplies to final
consumers, it can be verified that $f_{i}^{s}>f_{i}^{\text{max}}$ if
$x_{i}^{\text{SS}}-f_{i}^{\text{DS}}<a_{ij}^{sd}f_{j}^{\text{DS}}$. Thus, in
case industry $i$ is only slightly supply constrained (supply shocks are only
slightly larger than demand shocks) and industry $j$ faces comparatively
serious demand constraints, the MEEM would compute larger final consumption
values for industry $i$ than are possible.
Note that gross output values of demand constrained industries always lie
within the feasible range $x_{i}^{d}\in[0,x_{i}^{\text{max}}]$ if only adverse
supply shocks to the economy are allowed ($\epsilon_{i}^{S}\geq 0$). By noting
that
$\bm{x}^{d}=[\mathbf{I}-\mathbf{A}^{dd}]^{-1}(\mathbf{A}^{ds}\bm{x}^{s}+\bm{f}^{d})$,
it can easily be verified that $x_{i}^{d}\geq 0$. The condition that
$\epsilon_{i}^{S}x_{i}^{d}<\epsilon_{i}^{D}f_{i}^{d}$ ensures that output of
demand constrained industries cannot exceed the maximum output value
$x_{i}^{\text{max}}$.
The supply and demand shocks do not need to be large for the MEEM to generate
infeasible solutions. To show this we scale down the size of the supply and
demand shocks by varying a parameter $\alpha\in[0,1]$ to obtain new maximum
output and consumption values, according to
$\displaystyle x_{i}^{\text{max}}$
$\displaystyle=(1-\alpha\epsilon_{i}^{S})x_{i,0},$ (29) $\displaystyle
f_{i}^{\text{max}}$ $\displaystyle=(1-\alpha\epsilon_{i}^{D})f_{i,0}.$ (30)
Since we scale demand and supply shocks by the same proportion this does not
change which industries are supply or demand constrained.
Fig. 5 shows the MEEM results for varying directs shocks. If $\alpha=0$, there
is no direct shock, resulting in a feasible market allocation since in this
case the MEEM simply recovers the pre-shock economy. This is indicated by the
green colors at the very left of all three panels. But even for very small
$\alpha>0$ we obtain infeasible solutions for all three countries as shown by
the gray colors. If we increase shock sizes further, it becomes more likely
that the model computes negative final consumptions values as can be seen from
the transition of red colors into gray when following the x-axis from left to
right.
Figure 5: Infeasible results for final consumption for the MEEM model as a
function of shock size. The parameter $\alpha$ that scales the supply and
demand shocks varies along the x-axis. The color codes indicate the ratio
$f/f^{\text{max}}$ where $f$ is the MEEM result of final consumption. Note
that this ratio is always equal to one (green color) for industries which are
demand constrained. Infeasible values, $f\notin[0,f^{\text{max}}]$, are
indicated in gray.
## Appendix C Details on shock magnitude effects
In the main text we have shown the effect of scaling either supply or demand
shocks on aggregate output. We also explored shock amplification effects when
scaling both, supply and demand shocks, simultaneously. Fig. 6 shows the
effect of different shock magnitudes on aggregate output and final consumption
values. Results are fairly similar for aggregate values of output and
consumption and are also in qualitative agreement with the results presented
in Fig. 2.
Figure 6: Economic impact as a function of shock magnitude. Aggregate gross
output and final consumption levels as a function of scaling demand and supply
shocks equally between zero and one ($\alpha=\alpha^{S}=\alpha^{D}\in[0,1]$).
## Appendix D Details on network density
In Section 4.3.2 we removed links randomly and repeated this procedure
multiple times for any desired network density value. While random edge
removal is one possible approach, other procedures could be followed too. A
natural alternative to random edge deletion is to first delete small links.
While the high aggregation of the data results in almost complete graphs, link
sizes are highly heterogeneous, implying the existence of many very small
links (McNerney et al., 2013; Cerina et al., 2015). It could be argued that
many of these links are rather an artifact of data aggregation instead of
encoding a fixed production recipe. We therefore repeat the procedure of
Section 4.3.2 but eliminate smaller before larger links to achieve a given
level of network density. Doing this results in Fig. 7 which indicates
qualitatively similar results as Fig. 3 of the main text.
Figure 7: Economic impact as a function of network density. The figure is the
same as Fig. 3 in the main text, except that network density is changed by
eliminating links based on their size instead of random deletion.
The elimination of existing IO links changes key properties of the underlying
economic system. As discussed in Section 4.3.2, removing intermediate
consumption values will reduce aggregate output. Figs. 8(a) and 8(b) visualize
the relationship between output and network density following the random link
and “smallest-first” removal approach, respectively. Similarly, the ratio
intermediate consumption over aggregate output will be reduced as a
consequence. A density value equal to zero means that there is no intermediate
consumption left and firms only use primary factors as inputs in production.
Fig. 8 also shows that the average output multiplier decreases when making the
network sparser, although not necessarily monotonously. Overall, the economic
indicators change fairly linearly with respect to network density if links are
randomly removed, whereas these relationships are highly nonlinear if smaller
edges are eliminated before larger ones.
Figure 8: Key economic measures as a function of network density. (a) The
network density is changed by eliminating links randomly (corresponding to
Fig. 3). (b) The network density is changed by eliminating smaller before
larger links (corresponding to Fig. 7). _Multiplier_ refers to the
(unweighted) average multiplier of the economy, $\sum_{ij}l_{ij}/n$, after
rebalancing as percentage of the initial economy. _Intermediate_ denotes the
share of intermediate consumption in total output,
$\sum_{ij}z_{ij}/\sum_{i}x_{i}$ after rebalancing the economy. _Output_ is
total output after rebalancing divided by initial total output.
|
11institutetext: Institute of Theoretical Astrophysics, University of Oslo,
P.O. Box 1029 Blindern, NO-0315 Oslo, Norway 22institutetext: Rosseland Centre
for Solar Physics, University of Oslo, P.O. Box 1029 Blindern, NO-0315 Oslo,
Norway
22email<EMAIL_ADDRESS>
# Spicules and downflows in the solar chromosphere
Souvik Bose , 1122 Jayant Joshi, 1122 Vasco M.J. Henriques, 1122 Luc Rouppe
van der Voort, 1122
(XXXX; accepted XXXX)
###### Abstract
Context. High speed downflows have been observed in the solar transition
region (TR) and lower corona for many decades. Despite their abundance, it has
been hard to find signatures of such downflows in the solar chromosphere.
Aims. In this work, we target an enhanced network region which shows ample
occurrences of rapid spicular downflows in the H$\mathrm{\alpha}$ spectral
line that could potentially be linked to high-speed TR downflowing
counterparts.
Methods. We used the $k$-means algorithm to classify the spectral profiles of
on-disk spicules in H$\mathrm{\alpha}$ and Ca ii K data observed from the
Swedish 1-m Solar Telescope (SST) and employed an automated detection method
based on advanced morphological image processing operations to detect such
downflowing features, in conjunction with rapid blue shifted and red shifted
excursions (RBEs and RREs).
Results. We report the existence of a new category of RREs (termed as
downflowing RRE) for the first time that, contrary to earlier interpretation,
are associated with chromospheric field aligned downflows moving towards the
strong magnetic field regions. Statistical analysis performed on nearly 20,000
RBEs and 15,000 RREs (including the downflowing counterparts), detected in our
97 min long dataset, shows that the downflowing RREs are very similar to RBEs
and RREs except for their oppositely directed plane-of-sky motion.
Furthermore, we also find that RBEs, RREs and downflowing RREs can be
represented by a wide range of spectral profiles with varying Doppler offsets,
and H$\mathrm{\alpha}$ line core widths, both along and perpendicular to the
spicule axis, that causes them to be associated with multiple substructures
that evolve together.
Conclusions. We speculate that these rapid plasma downflows could well be the
chromospheric counterparts of the commonly observed TR downflows.
###### Key Words.:
Sun: chromosphere Sun: atmosphere line: profiles methods: statistical analysis
techniques: image processing proper motions
## 1 Introduction
Figure 1: Full FOV of the SST dataset observed on 25 May 2017 at 09:16:58 UTC.
Panels (a) and (b) show the H$\mathrm{\alpha}$ red wing and blue wing images
observed with CRISP at a Doppler offset of +40 km s-1and $-$40 km s-1,
respectively. Spicules are seen in the two panels as dark and elongated
thread-like structures. Panel (c) shows a dense chromospheric canopy of
fibrils in the corresponding CHROMIS Ca ii K line core image and (d) shows the
map of the LOS magnetic field (BLOS) saturated between $\pm$100 G derived from
inversions of the Fe I 6301 and 6302 Å spectral profiles. The direction to
solar North is pointing upwards. An animation of this figure is available at
https://www.dropbox.com/s/nxnz5twzczjwaac/Context_movie1.mp4?dl=0
Downflows are known to commonly occur in the solar atmosphere. High speed
downflows have been observed in the transition region (TR) that can sometimes
last from several hours to even several days (e.g. Gebbie et al., 1981; Dere,
1982). Moreover, stronger plasma downflows with speeds ranging from $60$–$200$
km s-1 have been observed over or in the close vicinity of active regions and
quiet Sun alike, with the latter revealing relatively weaker downflows
(Doschek et al., 1976; Kjeldseth-Moe et al., 1988; Brekke et al., 1997; Peter
& Judge, 1999). Decades of observations have revealed the prevalence of
predominant downflows (or red shifts) in the spectral lines of the TR (Doschek
et al., 1976; Scharmer, 1981; Dere, 1982; Peter & Judge, 1999; Dadashi et al.,
2011). These downflows are considered to play an important role in the mass
and energy balance of the solar atmosphere and are crucial to further
understand the physics of the lower solar corona.
These downflows are predominantly seen mainly in the lower coronal and TR
passbands. Brekke et al. (1997); Peter & Judge (1999) reported that most of
the high speed quiet Sun TR downflows usually vanish at chromospheric
temperatures. Chitta et al. (2016) have found supersonic downflows in the TR
of sunspots, which are highly intermittent in time and location (Nelson et
al., 2020). Statistical analysis of such downflows in $48$ sunspots by Samanta
et al. (2018) show that at most half of them show signatures in the upper
chromospheric spectral lines such as Mg ii k, whereas the rest are limited to
the TR. Sunspots (and active regions in general) occupy only a very small
fraction of the solar surface compared to the non active regions. Therefore,
it has remained a mystery as to what happens to the remaining high speed
downflows in the chromosphere that are so predominant in the TR of non active
regions. Why have their signatures been so elusive in the chromosphere? A
possible reason for such lack of evidence could be because the plasma density
in the chromosphere is several orders of magnitude higher compared to the
lower corona or the TR that causes these downflows to simply breakdown as soon
as they reach chromospheric heights.
The chromosphere is a layer of the Sun’s atmosphere that is sandwiched between
the visible photosphere and the million degree corona. It is dominated by a
myriad of different features such as spicules, dynamic fibrils, and filaments
to name a few. All these features contribute towards the vigorous dynamical
changes that the chromosphere experiences on time scales ranging from seconds
to minutes. The chromosphere also plays a crucial role in mass loading and
heating the solar corona since all the non-thermal energy that is responsible
for these mechanisms, propagates through the chromosphere. Only a small
fraction of this energy escapes into the corona or the solar wind while the
majority remains trapped within (Carlsson et al., 2019).
One of the most abundant and ubiquitous features that appear in the
chromosphere are spicules. Spicules are thin, elongated, thread-like, and
highly dynamic structures that are omnipresent in the solar atmosphere both in
active and non active regions. To our knowledge, the first lithographed
drawings resembling spicules were performed by Pietro Tachinni (see, for
historic references and a modern reproduction, Chinnici, 2006) and Angelo
Secchi (Secchi, 1871), both in April 1871, with the drawings in the second
edition of Le Soleil (Secchi, 1877) being better known examples of these early
observations. Spicules are visible all over the solar surface when observed in
chromospheric spectral lines and in some cases are known to reach coronal
temperatures. They have been of great interest to the solar physics community
for a long time. An overview of some of such work can be found at Beckers
(1968); Sterling (2000); Tsiropoula et al. (2012); Hinode Review Team et al.
(2019).
The discovery of a more elusive but energetic category of spicules by De
Pontieu et al. (2007) divided them into two different classes: type-I and
type-II, and stirred a fierce debate in the community regarding their
existence, formation mechanisms and role in coronal heating. The type-Is are
mainly used to refer to the dynamic fibrils found in active regions or mottles
in the quiet Sun, and they usually appear in the close vicinity of strong
magnetic fields (De Pontieu et al., 2004). Further, they display
characteristic parabolic paths in H$\mathrm{\alpha}$ line core space-time
diagrams with typical up and down motions of the order of $10$–$40$ km s-1,
lifetimes between $3$–$5$ min and quasi-periodicities of roughly the same time
periods (Rouppe van der Voort et al., 2007). Advanced numerical simulations by
Hansteen et al. (2006); Heggland et al. (2007) show remarkable similarities
with the observations described and they firmly established that type-I
spicules are formed due to the leakage of photospheric magnetoacoustic
oscillations into the solar chromosphere. The type-II spicules, on the other
hand, are comparatively more dynamic with vigorous sideways motions, high
apparent speeds (80–300 km s-1) and shorter lifetimes (~$1$–$3$ minutes) (De
Pontieu et al., 2007; Pereira et al., 2012, 2016). Moreover, they are known to
be heated as they propagate beyond the chromosphere and become visible in TR
(Pereira et al., 2014; Rouppe van der Voort et al., 2015) and coronal
passbands in both active regions and the quiet Sun (De Pontieu et al., 2011;
Henriques et al., 2016; Kuridze et al., 2016; Samanta et al., 2019). These
qualities render the importance of attributing type-II spicules towards mass
loading and heating the solar corona (see, e.g., Martínez-Sykora et al., 2017;
Kontogiannis et al., 2018; Martínez-Sykora et al., 2018), however, their
detailed physical processes remain far from known.
The spectral signatures of the on-disk counterparts of type-II spicules were
observed for the first time by Langangen et al. (2008). They reported sudden
rapid excursions in the blue wing of Ca II 8542 Å spectral line that led them
to be termed as rapid blue shifted excursions (RBEs). Later, Rouppe van der
Voort et al. (2009); Sekse et al. (2012); Pereira et al. (2016); Bose et al.
(2019a) observed high resolution on-disk images and spectra in the
chromospheric H$\mathrm{\alpha}$, Ca ii 8542 Å, and Ca ii K lines associated
with them and established beyond doubt that RBEs are indeed the on-disk
counterparts of the earlier known type-II spicules.
De Pontieu et al. (2012) reported the presence of torsional motions in type-II
spicules that explained the occurrence of spicular bushes in the red wing
images of chromospheric spectral lines, that were morphologically similar to
their blue counterparts. Later, Sekse et al. (2013b) also described the
existence of red wing counterparts of RBEs in H$\mathrm{\alpha}$ and Ca ii
8542Å with very similar properties as the former and termed them rapid red
shifted excursions (RREs). Next, Kuridze et al. (2015a); Rouppe van der Voort
et al. (2015) and more recently Bose et al. (2019a) also confirmed their
existence, both in the chromosphere and the TR, with high resolution on-disk
observations.
The complex twisting and swaying found in type-II spicules, in addition to the
flows along the chromospheric magnetic field lines, are important
characteristics of spicules that represents outward propagating Alfvénic waves
which can be of the order of several hundred km s-1 and can cause heating in
the hotter TR lines as they propagate (De Pontieu et al., 2014). Like De
Pontieu et al. (2012), Sekse et al. (2013b) and later Kuridze et al. (2015a)
used the transverse motion of spicules to argue for the existence and
appearance of RREs. They interpreted that RREs (though less abundant), like
RBEs, are a manifestation of the same phenomenon with very similar statistical
properties and occur when the latter harbor such complex motions that can
sometimes result in a net red shift when observed on the disk. Therefore, RBEs
can transition to RREs and vice-versa depending on the orientation between the
line-of-sight (LOS) and transverse motion of the structures. Moreover, the
torsional motions can also cause RBEs and RREs to be in close association with
each other (Sekse et al., 2013b; De Pontieu et al., 2014). Both RBEs and RREs
are generally seen to originate in the vicinity of strong network regions with
enhanced magnetic fields, and appear to propagate away as they evolve (Rouppe
van der Voort et al., 2009; McIntosh & De Pontieu, 2009; Sekse et al., 2013b;
De Pontieu et al., 2014; Kuridze et al., 2015a)
Movies of high-resolution H$\mathrm{\alpha}$ blue wing filtergrams at Doppler
offsets $\gtrsim$30 km s-1 are generally dominated by RBEs that appear to move
away from the magnetic network. Corresponding red wing movies at equivalent
Doppler offset, are not nearly as much dominated by outward moving RREs, but
also show elongated absorption features that appear to be moving downward to
the magnetic network. In this paper, with the help of high resolution on-disk
observations from the Swedish 1-m Solar Telescope (SST, Scharmer et al.,
2003), we aim to characterize these returning flows that appear
morphologically similar to RBEs and RREs, but seem to be downflowing in
nature. In the following sections, we describe the methods undertaken to
detect and further investigate their dynamical characteristics and spatio-
temporal evolution. We also discuss their interpretation and the possible
physical processes that could be responsible for their appearance and suggest
that these downflows could be a representative of the chromospheric
counterparts of the lower coronal and TR downflows.
## 2 Observations and data reduction
We observed an enhanced network region close to disk center in a coordinated
SST and IRIS campaign on 25 May 2017 shown in Fig. 1. The heliocentric
coordinates were $(x,y)=(45\arcsec,-93\arcsec)$ with corresponding observing
angle $\mu=\cos\theta=0.99$ ($\theta$ being the heliocentric angle). The
temporal duration was close to 97 min starting from 09:12 UT until 10:49 UT.
We acquired imaging spectroscopic data in H$\mathrm{\alpha}$ and
spectropolarimetric data in Fe I 6302 Å from the CRisp Imaging
SpectroPolarimeter (CRISP, Scharmer et al., 2008). The CHROmospheric Imaging
Spectrometer (CHROMIS) was used to obtain imaging spectroscopic data in Ca ii
K. Both CRISP and CHROMIS are tunable Fabry-Pérot instruments installed at the
SST with the first light of the latter being in 2016. CRISP sampled
H$\mathrm{\alpha}$ at 32 wavelength positions between $\pm$1.85 Å and the Fe I
6301 and 6302 Å line pair at 16 wavelength positions respectively, with a
temporal cadence of 19.6 s and a spatial sampling of
0$\aas@@fstack{\prime\prime}$058\. The Fe I spectral profiles were subjected
to an early version of a robust Milne-Eddington (ME) inversion scheme based on
a parallel `C++`/`Python` implementation111
https://github.com/jaimedelacruz/pyMilne (de la Cruz Rodríguez, 2019). We use
the LOS magnetic field component derived from these inversions in our analysis
as shown in panel (d) of Fig. 1. CHROMIS sampled Ca ii K at 41 wavelength
positions within $\pm$1.28 Å with 63.5 mÅ steps. Furthermore, a continuum
position was sampled at 4000 Å. The CRISP H$\mathrm{\alpha}$ red-wing and
blue-wing images for the full field-of-view (FOV) at +40 km s-1 and $-$40 km
s-1 are respectively shown in panels (a) and (b) of Fig. 1. The corresponding
CHROMIS Ca ii K line core image is shown in panel (c). The temporal cadence
and the spatial scale of this data were 13.6 s and
0$\aas@@fstack{\prime\prime}$038\. High spatial resolution down to the
telescopic diffraction limit (given by $\lambda/\mathrm{D}$ =
0$\aas@@fstack{\prime\prime}$08 at 3934Å) was achieved through excellent
seeing conditions, the SST adaptive optics system (Scharmer et al., 2019), and
the Multi-Object Multi-Frame Blind Deconvolution (MOMFBD, van Noort et al.,
2005) image restoration technique.
We used the CRISPRED data reduction pipeline (de la Cruz Rodríguez et al.,
2015) and an early version of the CHROMIS pipeline (Löfdahl et al., 2018) for
further data reduction while including the spectral consistency method
described in Henriques (2012). The images from both the instruments were de-
rotated to account for diurnal field rotation, aligned and de-stretched to
remove warping due to seeing effects before making it ready for scientific
analyses. Later, both the CRISP and the CHROMIS datasets were co-aligned by
cross-correlating the corresponding photospheric wideband channels and were
rotated to align the direction to solar North along the $y$-axis. The wideband
channel for CRISP has a full-width at half maximum (FWHM)=4.9 Å centered at
H$\mathrm{\alpha}$, whereas for CHROMIS the FWHM equals 13.2 Å centered at
3950 Å between the Ca II H and Ca II K lines.
These observations were first presented by Bose et al. (2019a) where the
cotemporal IRIS observations were included in the analysis. The orientation
and total area covered by the dataset matches the IRIS observations. This data
will be made available in the future as a part of the SST and IRIS database
(Rouppe van der Voort et al., 2020). A brief overview of the dataset and
targetted features of interest are given in the following section.
### 2.1 Overview of the dataset
Figure 1 shows an overview of the FOV and the qualification, enhanced network,
is illustrated by extended patches of strong magnetic field that are
predominately of negative magnetic polarity in the $B_{\mathrm{LOS}}$ map. The
Ca ii K line core image is dominated by a dense canopy of chromospheric
fibrils that covers most of the FOV. The H$\mathrm{\alpha}$ far wing images at
$\pm$40 km s-1 Doppler offset show a large number of elongated absorption
features, and a close look at the animation associated with this figure
clearly reveals that many display the rapid and complex dynamical evolution
that is characteristic for the on-disk counterparts of type-II spicules. More
interestingly, when comparing the evolution in the red and blue wing images,
we see that a majority of the dark threads in the red wing move towards the
strong network field concentrations, opposite to the familiar RBE dynamics
that is predominantly outward in the blue wing images.
These structures, which we term as downflowing RREs, are described here for
the first time and form the major theme of this paper. They are characterized
and compared with the traditional RREs and RBEs which have been widely been
observed in the recent past.
Figure 2: All 50 RPs of H$\mathrm{\alpha}$ spectral lines considered in the
final configuration of the $k$-means algorithm. The RP number is indicated in
the lower right corner. RPs 0–3 are colored in blue signifying blue-ward
excursions whereas 4–7 are colored in red indicating red-ward excursions. The
remaining RPs (8–49) are drawn in grey. The dashed gray spectral line in each
of the panels shows the mean H$\mathrm{\alpha}$ profile averaged over the
entire data set and serves as reference. The magenta profiles show the
positive part of the difference between the mean H$\mathrm{\alpha}$ profile
and the respective RP in each panel. The dashed horizontal magenta line marks
the level of the peak (IP-diff) of the differential profiles, and the vertical
solid magenta line marks the COG Doppler offset of the positive value of the
differential profiles ($\lambda_{\mathrm{COG-diff}}$). The width $W$ of the
H$\mathrm{\alpha}$ line core is given for each RP in Å.
## 3 Method of analysis
### 3.1 Characterizing RBEs and RREs/downflowing RREs from their spectral
profiles
The first and foremost task in our analyses was to identify the rapid blue and
red excursions in our data. We followed the $k$-means clustering method as
described in Bose et al. (2019a) to identify the different H$\mathrm{\alpha}$
and Ca ii K profiles that occur during these rapid excursions.
The method works by partitioning similar observations into $k$ number of
groups or clusters. Each observation is assigned to the cluster with the
nearest mean (also called cluster center). It is an iterative algorithm whose
main objective is to minimize the sum of distances between the observed points
and their respective cluster centers. In our data, each observation (pixel) is
a spectral profile which is compared with the mean spectral profiles of each
of the $k$ clusters before assigning it to one. An advantage of using this
technique is that it relies on the complete spectral signature of the
different features on the FOV rather than their appearance at a particular
wavelength position. This enables efficient detection and characterization of
the features as discussed below.
Thus, we applied this algorithm to cluster intensity profiles of each pixel on
the FOV on the combined H$\mathrm{\alpha}$ and Ca ii K data. The detailed
algorithmic steps, data pre-processing methods, followed by finding the
optimum number of clusters, among which the data can be divided, has been
discussed in detail in Bose et al. (2019a). In this paper, we leverage the
analysis performed in Bose et al. (2019a) and proceed with the 50 clusters
among which the data had been grouped. Each cluster has a representative
profile (RP) that corresponds to the mean over all profiles in that cluster.
Naturally, each and every pixel in the FOV is uniquely assigned to a
particular cluster that can be represented in the form of a 2D map, like the
one shown in panel (c) of Fig. 1 in Bose et al. (2019a) (also see panel (d) of
Fig. 5). We base further analyses, performed in this study, solely on the
H$\mathrm{\alpha}$ RPs and the quantities extracted from them. Figure 2 shows
all 50 H$\mathrm{\alpha}$ RPs which resulted from the $k$-means clustering.
With the exception of RPs-$13$ and $21$, that coincide with the gray borders
outside the common CRISP and CHROMIS FOV in Fig. 1, all the remaining RPs
correspond to various features observed in the FOV. In the following
subsections, we describe the methods undertaken to first identify the RBE and
RRE/downflowing RRE RPs and use them further to detect on-disk spicules in our
dataset.
#### 3.1.1 Identifying the RPs
RBEs and RREs have typically been observed in H$\mathrm{\alpha}$ and their
spectral profiles show significant absorption asymmetries in the blue and red
wing positions compared to an average quiet Sun profile (Rouppe van der Voort
et al., 2009). These asymmetries are a sign of velocity gradients that are
commonly observed in spicules. Preliminary analysis with `CRISPEX` (Vissers &
Rouppe van der Voort, 2012), a widget-based analysis tool written in
Interactive Data Language (IDL), suggests that downflowing RREs have spectral
signatures similar to the traditional RREs with an absorption asymmetry in the
red wing of H$\mathrm{\alpha}$. Therefore, the distinction between RREs and
downflowing RREs was not made while identifying their characteristic RPs.
We begin our identification strategy by computing the differential profiles,
i.e. the difference between average H$\mathrm{\alpha}$ profile and the RP,
similar to Rouppe van der Voort et al. (2009) for each cluster. By considering
only the positive values of these differential profiles, we have a measure to
determine the enhanced absorption part. The positive part of the differential
profiles are shown in magenta in Fig. 2. This is followed by determining the
center-of-gravity (COG) and the peak intensity of all these magenta
differential profiles (their values are indicated by the solid vertical and
horizontal dotted magenta lines, respectively). We used the method described
in Uitenbroek (2003) for computing the COG, and its position was used as a
measure of the Doppler offset of an absorption feature in the RPs. The
combination of large values for COG and positive peak of the difference
profile serves as an effective selection criterion to determine enhanced
absorption in the H$\mathrm{\alpha}$ wings.
Pereira et al. (2016) highlighted the importance of H$\mathrm{\alpha}$ line
width in the detection and analysis of spicules. They reported that the
statistical properties measured solely from the H$\mathrm{\alpha}$ line wing
images at selected wavelength positions would result in an underestimate of
the extracted physical quantities because spicules show a large range of
Doppler offsets during their evolution. For such cases line width maps provide
a more robust tool for spicule detection. Typically, RBEs and RREs show
enhanced line widths in addition to large Doppler offsets, with the fastest
ones having even broader line profiles. This indicates that both Doppler
offset and the line width are important properties that should be considered
while detecting spicules. Therefore, we leverage the detection of RBEs, and
RREs/downflowing RREs by combining the two factors. We choose the method
described in Cauzzi et al. (2009) to calculate the widths of the
H$\mathrm{\alpha}$ line core over the full-width at half maximum (FWHM) method
utilized by Pereira et al. (2016), because FWHM tends to mix the photospheric
and the chromospheric signals, which is effectively avoided in the former.
Nevertheless, spectral profiles of spicules show enhancements in both FWHM and
line core width of H$\mathrm{\alpha}$ rendering the validity of both
approaches.
Figure 3: Scatter plot between the COG Doppler offsets and the peak intensity
(IP-diff) of the differential profiles for all RPs. The points are annotated
with the respective RP index and color coded based on the H$\mathrm{\alpha}$
line-core width. The horizontal dashed red and blue lines indicate a Doppler
offset of $\pm$20 km s-1 as a reference.
Figure 3 shows a scatter plot between the COG Doppler offsets and peak
intensities (IP-diff) of the magenta difference profiles, the color of the
data points follow their respective line core widths. Such a representation
takes into account all possible spectral characteristics important for RBE/RRE
qualification, thereby leading to an efficient and a robust characterization.
For visualization purpose, we choose to restrict the lower limit of the
colormap to a line core width of $1.1$ Å. Thereafter, we imposed the following
three criteria to consider RPs belonging to RBEs and RREs/downflowing RREs.
They should: (1) have a minimum COG Doppler offset of 20 km s-1, (2) have IP-
diff $\geq$ 0.2 and (3) have line core widths $\geq$ 1.2 Å. Evidently, RPs
$2$, $3$, $6$, and $7$ stand out quite distinctly in Fig. 3, as they not only
have the highest Doppler offsets, but also high values of peak intensity and
line core widths making them clear RBE and RRE/downflowing RRE RPs. Moreover,
the above criteria are also satisfied by RPs-0, 1, 4 and 5, though not as
distinctly, whereas the rest of the profiles clearly do not make the cut.
Therefore, we chose RPs 0–7 for the analyses presented in the rest of the
paper and assign RPs 0–3 to RBEs and 4–7 to RREs/downflowing RREs in
increasing order of the strength of their spectral properties and they are
accordingly arranged in the first 7 panels of Fig. 2. The remaining RPs,
numbered from 8–49, are indicated in gray.
According to Fig. 3, the computed Doppler offsets of RP-0 and RP-4 are
comparable to RPs-2 and 6 respectively, but, the latter show stronger
absorption in their line wings and a stronger shift in their spectral lines
that in turn also contributes to an increased line core width making them
stronger than the rest. Therefore, RPs-0 and 4 are assigned lower in the order
of labeling the RPs in Fig. 2. Additionally, we also exploited the difference
between the spectral properties of RPs 6 and 7 (2 and 3) and RPs 4 and 5 (0
and 1) described earlier, and segregated the detected RBEs and
RREs/downflowing RREs into two compartments with the purpose of comparing
them. Basically, we grouped RPs 6 and 7 (2 and 3) for the red shifted (blue
shifted) events under stronger red (blue) excursions, whereas RPs 4 and 5 (0
and 1) were grouped together under weaker red (blue) excursions in a way such
that the events in one group were unique with respect to the other. Such a
segregation formed the basis of the investigations carried out in Sects. 4.1
and 4.2, where we discuss them further. In addition to the above properties,
we also note from Fig. 3 that the line-core widths of downflowing RRE/RRE-like
RPs are slightly enhanced in comparison to the RBE-like RPs. The difference is
more pronounced for the RPs belonging to the strong excursions with respect to
the weaker ones. We would like to note that the estimates based on the COG
technique serves well to identify RBE/RRE like profiles, but it fails for some
of the other RPs (such as RP-12 and 31) for which this technique is rather
meaningless. Nevertheless, the method adopted in this paper provides one of
the most comprehensive approach so far in characterizing spicular spectral
profiles.
Now, once the RPs have been identified, a question arises as to how well are
they able to represent individual profiles in a particular cluster? Figure 4
sheds light in this direction and shows the RBE and RRE/downflowing RRE-like
spectral profiles in H$\mathrm{\alpha}$ (top two rows) and Ca ii K (bottom two
rows) for each RP in the form of density distributions, with darker shades
indicating a higher number density of spectral lines. The colored solid line
in the different panels shows the mean over all profiles for a particular
cluster, which, in our case, is equivalent to an RP. We clearly see that the
distribution of the spectral profiles belonging to each cluster is narrow and
is mostly concentrated near their respective mean profiles. This indicates
that the identified RPs are able to efficiently describe the profiles in the
respective clusters for both H$\mathrm{\alpha}$ and Ca ii K.
Figure 4: Density plots of the H$\mathrm{\alpha}$ (top two rows) and Ca ii K
(bottom two rows) spectra for RBE and RRE/downflowing RRE RPs (0–7). The
density (darker meaning higher concentration of spectra) corresponding to each
RPs shows the distribution of profiles over the whole time series. The solid
lines overplotted correspond to the RPs discussed in the text. The blue (red)
color coding indicates the excursions in the blue ward (red ward) side of
H$\mathrm{\alpha}$ respectively.
From the distribution of the spectral profiles in Ca ii K, we see that for a
vast majority of cases, the strongest Doppler shifted K3 (in both the red and
blue excursions) has a significantly stronger opposite K2 intensity
enhancement relative to its line core. In other words, we see that the
absolute difference between the intensities $I$(K2) - $I$(K3) is correlated
with the shift of K3, as was also shown in Bose et al. (2019a). This is the
case exclusively for the RBE and RRE-like RPs, which also formed an additional
basis for the identification of the RPs in Ca ii K. The rest of the profiles
show no such characteristic behavior (refer to Fig. B.2 in the appendix of
Bose et al., 2019a).
The intensity enhancement in the K2 peaks is inherently linked to lower layer
photons, observed due to the presence of strong velocity gradients in spicules
which remove top-layer opacities at those wavelengths. Such lower layers
probably feature enhanced emission as it is due to increased temperature or
other local source function enhancing effects, but the relation between the K2
and the K3 features, set by velocity and reproducible by a simple model,
reveals the role of the velocity gradients. In Bose et al. (2019a) this was
described as an opacity window effect due to the two-dimensional presence of
background features imaged in K2 wavelengths and to distinguish it from other
effects such as the reflector effect (Scharmer, 1981, 1984). Similar
enhancements are generally observed whenever there is a gradient in the LOS
velocity in strong resonant lines such as Ca ii K or Mg ii k (Carlsson &
Stein, 1997; de la Cruz Rodríguez et al., 2015; Kuridze et al., 2015b). A
further example in a very different solar feature is found in umbral flashes
in the Mg ii k spectral line (Bose et al., 2019b). For a recent discussion on
this topic see Sect. 4.1 of Henriques et al. (2020).
### 3.2 Spicule detection and Morphological operations
#### 3.2.1 Halos around spicules and their substructures
Figure 5: Overview of the detected RBEs and RREs/downflowing RREs in
H$\mathrm{\alpha}$ wing and Ca ii K. Panels (a), (b) and (c) show RBEs and
their associated substructures (halos) in blue and RREs/downflowing RREs and
their halos in red colored contours against a background of H$\mathrm{\alpha}$
wing images at +40 km s-1, $-$40 km s-1 and Ca ii K line core, respectively at
time t=09:17UT. Panel (d) shows the corresponding RP index map with gradients
in the color indicating the contribution from the different RPs as shown in
the colorbar. An animation of this figure is available at
https://www.dropbox.com/s/wav35xo38vo6s9d/shadow_movie1.mp4?dl=0. Figure 6:
Overview of the morphological processing techniques described in the text.
Panel (a) shows a portion of the FOV in H$\mathrm{\alpha}$ wing ($-38$ km s-1)
at time t=09:24 UT (scan:50). Blue contours on this image indicate the RBEs
detected with RPs 0–3 as discussed in the text, panel (b) shows the binary
mask of the detected RBEs, and panel (c) indicates the result after performing
connected component labeling of the binary mask in panel (b). The labels are
color coded as seen in the adjoining colorbar. The total number of labels for
RBEs over the full FOV and full duration corresponds to 19,643. Panel (d)
shows the same FOV as in (a) but at time t=10:43 UT (scan:400). Cobalt blue
colored contours in panel (e) show the masks of detected RBEs obtained after
performing morphological labeling, and (f) shows the fitted ellipses with
their centers (green), semi-major axis and semi-minor axis (red) for each
label shown in (e).
Spicules are known to display complex dynamical behavior owing to their: (a)
flows aligned along the chromospheric magnetic field lines, (b) swaying
motions and (c) torsional motions (De Pontieu et al., 2012). Furthermore,
Sekse et al. (2013b) established that many RBEs and RREs exhibit a variation
in Doppler offsets along their lengths and breadths due to which the widths of
H$\mathrm{\alpha}$ spectral lines are enhanced, as was shown by Pereira et al.
(2016). This, in turn is reflected in their spectra which essentially
translates to having multiple spectral profiles (RPs) for the same spicular
body. In this section, we attempt to fully capture this complex behavior by
showing how the different RPs contribute in detecting the complete morphology
of spicules (including the downflowing RREs) and their implications.
Figure 5 and its associated animation shows an overview of the spicules
detected by including multiple RPs (0–7). Panels (a) and (b) respectively show
the contours of the detected RBEs and RREs/downflowing RREs in blue and red
colors against H$\mathrm{\alpha}$ +40 km s-1 and $-$40 km s-1 images. A close
look at panel (b), for example, clearly reveals that in many cases the
overlaid contours enclose a region that appears to be larger than the visible
dark thread like structures, such as the ones around
($\mathrm{X}$,$\mathrm{Y}$) = (27″,5″) or (15″,17″). Similar examples can also
be found in panel (a) for the RREs/downflowing RREs. These lighter shades
around the central dark regions are here termed as spicule halos. They
represent structures with weaker Doppler offsets surrounding the centrally
stronger offset regions. The halos become prominent in the images observed
closer to the H$\mathrm{\alpha}$ core, implying that though they have weaker
Doppler offsets they form a part of the same morphology and evolve together.
Panel (c) shows the same contours as in (a) and (b) but against a background
of a thick chromospheric canopy imaged in Ca ii K line core. As shown in Bose
et al. (2019a), spicules do not have a sharp intensity contrast in Ca ii K
(due to the opacity window effect) unlike H$\mathrm{\alpha}$, which makes it
nearly impossible to observe them as we do in the H$\mathrm{\alpha}$ wing
images. However, looking at the animation of this panel and closely following
the overlaid contours does provide a better impression of the evolution of
spicules in such band-pass.
Further justification of including multiple RPs in the analysis and
visualization become clearer when we look at the RP index map shown in panel
(d) of Fig. 5. It shows the combined RBE and RRE/downflowing RRE RP indices
with varying strengths as set by their their associated Doppler offsets, line
core widths and peak intensities of the respective enhanced differential
profiles. The darker blue (red) shades are an indicator of higher values of
the above three parameters compared to the lighter ones. Clearly, we see that
in several cases the darker shades are accompanied or surrounded by lighter
shades or the previously described halos, in both blue and red shifted
structures. The animation of this figure distinctly indicates that these halos
evolve in conjunction with their corresponding darker cores making it clear
they are a part of the same, but bigger, morphological structure. As seen
earlier for panels (a) and (b), this again strongly indicates widespread group
behavior among spicules. Such behavior was also pointed out by Skogsrud et al.
(2014) among the off-limb spicules. The animation also shows a large number of
downflowing RREs (in red color), co-evolving with their halos, with an
opposite apparent motion compared to the RBEs/RREs (in blue/red color).
The discussion presented in the preceding paragraphs demonstrates that
spicules have multiple substructures that are possible to be detected only
after the inclusion of multiple RPs. It also further evinces that it is
inaccurate to infer about their nature based on detections from a single
wavelength position since their morphology, length and lifetimes could also be
very different in reality. As an example, we show the multi-structural
features among the red and blue excursions belonging to the stronger category
in Appendix A. Such a representation makes the variation in their spectral
properties abundantly clear and prominent.
#### 3.2.2 Morphological processing techniques
The discussion presented in the preceding section justifies the importance of
including multiple RPs in spicule detection. This section further advances and
describes the details of the image processing algorithms employed in detecting
the RBEs and RREs/downflowing RREs. The remainder of this section describes
the detailed steps followed in the detection of RBEs in our data. The exact
same procedure was also employed in the detection of RREs/downflowing RREs.
We started off by creating a 3D binary mask (in spatial and temporal domains)
containing all the pixels in the FOV belonging to RPs 0–3 and assigning them a
value of 1 (bright) and the rest 0 (dark). A morphological opening, followed
by a closing operation, with a 3$\times$3 diamond shaped structuring element
was applied to each of the binary masks on a per time step basis. The opening
operation is analogous to an erosion followed by a dilation, and is useful to
remove tiny bright dots in the binary mask. This helps to get rid of small
(1-pixel) connectivity that might be present between different morphological
structures in the 2D space. A morphological opening operation however, can
also create small dark holes in between bright structures which can be
effectively closed by using a closing operation that is the reverse of
opening. It is however important to keep the same 3$\times$3 structuring
element as before. These operations were performed on a per time step basis,
i.e. in 2D, because we intended to preserve the connections in the temporal
domain that would enable us to effectively label them.
The next step was to perform a 3D connected component labeling (Fiorio &
Gustedt, 1996) so that we could identify components uniquely based on a given
heuristic. This technique is widely used in computer vision and image
processing technology. Two pixels are said to be connected when they are
neighbors and have the same numerical value. In this case, we aim to label
pixels in 3D space that are connected and are similar in spectra as set by the
$k$-mean procedure. To not bias for direction we selected an 8-neighbourhood
connectivity in 3D. The top row of Fig. 6 (panels (a)–(c)) provides an
overview of the steps undertaken for the detection of RBEs at the indicated
scan.
When applied to the complete dataset, the method described above led us to
identify 19,643 RBEs and 14,650 RREs (including the downflowing RREs). The two
panels of Fig. 13 in Appendix B, shows the location of all the detected red
and blue excursions respectively, over the co-spatial CRISP and CHROMIS FOV
for the entire time series. On average, most of the spicular activities are
seen to exist in the vicinity of the strong field regions or the network
fields (shown in black colored contours). The difference between the number of
RBE and RRE/downflowing RRE detection is consistent with Sekse et al. (2013b),
with the RRE:RBE detection ratio being ¡ 1. However, it is important to remark
that our red excursions also consists of the downflowing RREs along with the
traditional RREs, and we report far more number of on-disk red and blue
excursions than any of the preceding works cited in this paper, mainly because
of the unique detection method followed by advanced morphological operations
discussed above. Therefore, this allowed us to perform much more exhaustive
data analysis than many such earlier works.
### 3.3 Dimensional analyses and lifetime statistics
One of the major goals of this study is to statistically compare the
properties of the newly reported downflowing RREs to the traditionally known
RBEs and RREs. The detection method undertaken in Sect. 3.2 yielded a large
number of on-disk spicules which provides a perfectly fertile ground to
explore further in this direction. In this section, we focus on the technique
employed to compute their dimensions and lifetimes.
We began by fitting an ellipse to each of the labels obtained after the 3D
connected component labeling, and then computing their lengths of the major
axis, eccentricities ($e$, that is given by $e=\sqrt{1-{b^{2}}/{a^{2}}}$, with
$a$ and $b$ being the semi-major and semi-minor axis of a standard ellipse,
respectively), and the occupied area. Panels (e) and (f) of Fig. 6 provide an
illustrative example where we show the identified spicules in cobalt blue
contours for a given scan (shown in panel (d)), along with their fitted
ellipses–together with their centers and semi-major axis, for a small area on
the FOV. To avoid erroneous detections, a lower limit for the length of the
major axis of detected blue shifted and red shifted features was set at ~100
km or 4 CHROMIS pixels, such that any label with length below the threshold is
not included. Strictly speaking, spicules are not perfect ellipses. However,
they share a common morphology where their lengths (generally) far exceed
their widths (Beckers, 1968; Pereira et al., 2012), making them appear as dark
elongated streaks when observed in H$\mathrm{\alpha}$ on-disk wing images. The
elliptical fitting is performed solely for the purpose of determining the
length occupied by RBEs/RREs by measuring the length of their respective major
axes which ensures that the ends are located at the widest points of the
perimeter of the rapid excursions. Furthermore, elliptical shapes allow a
certain degree of freedom even for those spicules that are not highly
elongated but rather have their lengths only slightly greater than their
widths. These shapes can readily be fitted by ellipses with $0\leq e\leq 1$,
which would mean that the length of the major axis of the ellipses provide a
good approximation of the length of RREs/downflowing RREs and RBEs.
Earlier studies, such as Rouppe van der Voort et al. (2009) and Sekse et al.
(2012, 2013a), relied on obtaining the morphological skeleton of the detected
RBEs and RREs in order to compute their lengths. In such cases, the skeleton
is a thin version of that shape that is equidistant to its boundaries. The
major axis of an ellipse is very similar to the skeleton because it passes
through the COG of the ellipse and therefore their lengths will be comparable.
Both these techniques are basically approximations and skeletonizing certainly
has its own benefits as it preserves the shape of any given structure.
However, since we are interested in determining the maximum extent from both
ends of a feature, the major axis of an ellipse can prove to be advantageous
for complex morphological structures that are often associated with RBEs and
RREs.
It is however important to recall that the 3D connected component labeling
produces a chain of events that are attached both in space and time.
Therefore, if an event (or a label) lasts for multiple time frames, we only
consider the length, area and $e$ when it is at its maximum extent. A similar
approach is followed for spicules with multiple structures or halos around
them. This is justified because inclusion of multiple dimensions for the same
spicule would lead to incorrect statistics. The results obtained after
performing the analysis are shown and discussed in Sect. 4.3.
The lifetimes of spicules were determined on a per event basis, where we
considered the difference between the first and the last occurrence of the
same event in the temporal dimension. The lower limit of the measured
lifetimes is set by the CRISP cadence of ~19.6 s. The results are further
discussed in the latter half of Sect. 4.3.
### 3.4 Apparent motion of spicules
Evidently, the major difference between a downflowing and a traditional RRE is
their plane-of-sky (apparent) motion with respect to the strong magnetic
network areas. In this paper, we investigate the apparent motion of 19,643
RBEs and 14,650 RREs/downflowing RREs on a morphological event-by-event basis,
so as to statistically analyze and describe their trajectories in the plane-
of-sky. Such an analysis would help us to get an idea as to what extent the
red shifted excursions show an opposite trajectory in our datatset, thereby
revealing the abundance of such events. We computed the area weighted COG of
each morphological label (spicule) and followed their evolution in both space
and time. The weighting by area allows the algorithm to follow the trajectory
of the larger substructures in a label. It starts tracking the COG from the
first occurrence of a label and continues until the last. The $X$ and $Y$
coordinates of the COG were stored for each time step and and were then
plotted to display their apparent motion. The results are described further in
Sect. 4.1.
## 4 Results
### 4.1 Downflowing rapid red shifted excursions
Figure 7: Examples of RBEs, RREs and downflowing RREs (labelled as DRREs). The
top panels show their spatio-temporal evolution in H$\mathrm{\alpha}$ blue and
red wing images, whereas the bottom row shows the spectral evolution in
$\lambda t$-diagrams corresponding to the locations marked with white crosses.
Animations are available at
https://www.dropbox.com/sh/3x3kvx158u57xho/AAAA5Xm0sH8RSxPyxe8vqA5Wa?dl=0
Figure 8: Plane-of-sky trajectories of RBEs, RREs and downflowing RREs mapped
according to the strength of their excursions towards the blue (top row) and
red (bottom row) wings. The background shows a BLOS map saturated between
$\pm$ 500 G. The rainbow colored trajectories map the travel direction from
the origin in blue to the termination in red. Here, only a zoom in on a
network patch is shown, Fig. 15 shows the full FOV.
The animations linked to the H$\mathrm{\alpha}$ red wing images in Figs. 1 and
5 seem to convey the impression that in many cases the apparent motion of the
events is rather inward moving instead of the characteristic outward movement
(with respect to the background network areas) associated with traditional
RREs. Except for their apparent motion, we find that these events are very
similar in their appearance and morphology of RBEs and RREs. In this section,
we report clear observational and statistical evidences of these new class of
events termed as downflowing RREs.
Figure 7 shows two examples each of RBEs, RREs and downflowing RREs (labelled
DRREs in the figures). The temporal evolution of these events are shown along
the column for each case and they immediately suggest that, morphologically,
the downflowing RREs are very similar to both RBEs and RREs, and like RREs
they appear in the far red wings of the H$\mathrm{\alpha}$ line core at +40 km
s-1. However unlike the former, the downflowing RREs clearly move towards the
bright network structures as they evolve, whereas the traditional RBEs and
RREs move outwards in the opposite direction. The online animation associated
with each of the examples shown in Fig. 7 establishes the scenario quite
convincingly.
The temporal evolution of their corresponding H$\mathrm{\alpha}$ spectra shown
in $\lambda t$-diagrams also display typical type-II spicule-like behavior for
the downflowing RREs, with a sudden development of a highly asymmetric line
profile towards the red side of the line core. This is very similar to the
characteristic RRE $\lambda t$ evolution first reported by Sekse et al.
(2013b). For the sake of completeness, we also show extended $\lambda
t$-diagrams in Appendix 14 for the two downflowing RREs stretching well before
(~550 s) the occurrence of the red excursions. They clearly show no signs of
preceding blue shift that are typical for type-I spicules. Moreover, we also
note that the lifetime of their excursions in the red wing are similar to RREs
and RBEs. Furthermore, the redshifts associated with the downflowing RREs, in
addition to their inward apparent motion, strongly suggests that these are
real plasma flows (moving away from the observer) and not simply apparent
motions that are often associated with the TR network jets (Tian et al.,
2014). The high apparent speeds in the network jets (of the order of 100–300
km s-1) are most likely not caused by real mass flows. Instead, they are most
likely due to heating fronts propagating at Alfvenic speeds (De Pontieu et
al., 2017). Therefore, appertaining to their spectral and morphological
similarities, we suggest that the downflowing RREs are basically like RREs
simply with an opposite plane of sky motion.
To rule out the possibility that these are singular events, we performed a
statistical study based on the COG tracking method described in Sect. 3.4,
where each and every event belonging to the blue and red side of the
H$\mathrm{\alpha}$ line core was tracked individually. Both the blue and red
excursions were first segregated into two compartments, according to the
strength of their spectral features, by grouping them in the manner described
in Sect. 3.1.1. Consequently, after morphological processing operations,
events that have at least one stronger RP (belonging to either 2 and 3 or 6
and 7) over their whole lifetime are considered under the stronger excursion
category, whereas the rest are grouped into weaker excursions. Panels (a)–(d)
of Figs. 8 and 15 show the apparent trajectories of these excursions as
indicated in their title. Figure 8 shows a zoom in to the central network
patch and Fig. 15 shows the full FOV. The direction of motion is shown by
drawing the trajectories of the COG in a rainbow colormap, where the blue
marks the origin and red the final destination. For the sake of clarity, we
display only those events that have sufficient apparent displacement: $\geq$
0$\aas@@fstack{\prime\prime}$5\. ($\geq$ 1″for the full FOV in Fig. 15). The
$B_{\mathrm{LOS}}$ map is shown as background in order to enhance the
visibility of the trajectories and to facilitate better understanding of the
apparent motion of the events with respect to the strong magnetic field
regions.
We immediately notice that almost all the events in panels (a) and (b)
originate close to the network region and move further outwards as they
evolve–a behavior that is typical to RBEs. The bottom row, on the other hand,
shows that the paths traversed by the red excursions are predominantly
opposite to their blue shifted counterparts. Panel (c), for example, shows an
inward apparent motion for all the red excursions. Moreover, the origin of the
majority of such events can be seen to lie outside the enhanced magnetic field
region and they tend to terminate within or in close proximity to the boundary
of the strong network regions. The weaker red excursions in panel (d) mostly
shows a mixed-bag scenario where a large number of events show the inward
apparent motion, but many among them trace the traditional RRE trajectories,
such as the ones around ($X$,$Y$) = (22″,32″) or (35″,45″). A similar trait is
also observed for a vast majority of events detected in the full FOV, shown in
panels (c) and (d) of Fig. 15, which strongly suggests that the newly reported
downflowing RREs are widely prevalent in our dataset in conjunction with RBEs
and RREs.
### 4.2 Spatial distribution of the stronger and weaker excursions
Figure 9: Spatial occurrence of the stronger and weaker blue and red
excursions with respect to the strong field network regions. Panels (a) and
(b) show the stronger and weaker blue shifted excursions (RBEs) whereas panels
(c) and (d) shows the distribution of the stronger and weaker red shifted
excursions (RREs/downflowing RREs), respectively. The colors represent the
number density of the events shown. The events are mutually exclusive, meaning
that the excursions shown in any one panel are unique and are not related to
the events in the other. The black contour indicates the regions with an
absolute LOS magnetic field $\geq$ 100 G.
The trajectories of the strong and weak excursions presented in the preceding
section sparks interest in investigating their detailed spatial occurrences
with respect to the background network areas over the full FOV. Traditionally,
RBEs and RREs are known to appear in the close vicinity of strong magnetic
field network regions which are also thought to be their foot-points (Rouppe
van der Voort et al., 2009). Therefore, it is worthwhile to explore if such a
behavior is also seen among the stronger and weaker excursions detected in
this study. Figure 9 shows the occurrence of these excursions in the form of
2D density maps. Once again, the events were grouped in exactly the same way
as in Sect. 4.1, with the stronger excursions belonging to the group with RPs
2 and 3 (6 and 7), shown in panels (a) and (c), and the weaker ones belonging
to RPs 0 and 1 (4 and 5) panels (b) and (d), in such a way that events
belonging to these categories are mutually exclusive with respect to one
another.
A closer examination of Fig. 9 reveals that stronger excursions are located
closest to the enhanced network regions (indicated by black contours), whereas
their weaker category counterparts are located further outwards. Moreover,
panels (b) and (d) also suggest that the weaker spicular excursions appear to
exist all over the FOV, but their number density is, on average, roughly
10–15% lesser than their stronger counterparts. Panels (a) and (c) also
highlight an important difference between the stronger blue and red
excursions. In panel (a) it appears that the density of blue excursions are
mostly concentrated outside the network regions and appear to spread outwards
and away from them, whereas the red excursions (including the downflowing
RREs) in panel (c) are mostly located on or within the boundaries of the
strong network regions.
### 4.3 Statistical properties
Figure 10: Dimensional analysis and lifetime statistics of RREs/downflowing
RREs in our dataset. 1D histograms of length, area covered, eccentricity, and
lifetime are shown in (a), (b), (c) and (d) respectively. Moreover, panel (d)
also shows the ECDF of the lifetime distribution in solid green and the dashed
black horizontal line indicates the 98% mark. The vertical dashed lines in
panels (a), (b) and (d) indicates the spatial and temporal resolution limits
of our data. Panels (e)–(h) show the multivariate JPDFs between various
quantities as indicated in a rainbow colormap with red (blue) indicating
highest (lowest) density regions. The magenta contour overlaid on each of the
density distribution indicates the region within which 70% of the events lie.
Figure 11: Same format as Fig. 10 above but for RBEs.
Following the method discussed in Sect. 3.3, we computed the maximum lengths,
maximum areas, the corresponding eccentricities of the fitted ellipses and the
lifetimes of 14,650 downflowing RREs/RREs and 19,643 RBEs. This section is
dedicated to the description and comparison of the statistical properties of
the newly reported downflowing RREs with respect to the traditional RBEs and
RREs.
From the 1D histograms of length and area shown in panels (a) and (b)
respectively, of Figs. 10 and 11, we find that the maximum length of the RBEs
varies between 0.102 Mm to 7.8 Mm whereas for their red shifted counterparts,
an upper limit of 7.75 Mm was found. In both cases a lower threshold length of
4 CHROMIS pixels was imposed which roughly translates to 0.1 Mm. The area
occupied by RBEs range from 0.003 Mm2 to 5.53 Mm2, while the RREs/downflowing
RREs occupied an area that have a maximum value of 7.38 Mm2, with roughly the
same minimum. Both the area and length distributions appear to be skewed with
a large number of data points clustered around 1 Mm.
Panel (c) shows the 1D histogram of eccentricities of the fitted ellipses to
the red and blue shifted excursions. A first glance at them shows that the
events detected in both cases are rather elongated with $e$ $\geq$ 0.5. This
is well aligned to the known morphology of on-disk spicules that has been
described in many studies in the past. Moreover, we also find that highly
eccentric events tend to be more abundant. A close look at the eccentricity
histograms, however, reveal that the distribution appears to be flatter for
the RREs/downflowing RREs in Fig. 10, compared to the RBEs in Fig. 11. This is
further supported by the fact that the median value for the eccentricities
corresponds to 0.85 for the former whereas for the latter it equates to 0.92.
This indicates that, on average the red shifted excursions are slightly less
elongated than their blue shifted counterparts.
Panels (e)–(g) shows the combined joint probability distribution functions
(JPDFs) between various quantities as indicated. The magenta contours specify
the regions within which 70 % of the events lie. The motivation behind these
JPDFs was to highlight the bi-variate relationships between the extracted
quantities in the form of a 2D probability density function. The JPDF between
maximum length vs. maximum area of the blue shifted and red shifted events in
panel (e) show a strong correlation indicating that spicules that occupy
larger area are also longer. Further investigation from panels (f) and (g)
reveal that the bulk (in this case 70 % or more) of the events are highly
eccentric and are less than 1 Mm in length and 0.5 Mm2 in area. Also in
general, events that are more eccentric also tend to be larger in size. This
is true for both the RBEs and RREs/downflowing RREs.
Panels (d) and (h) in Figs. 10 and 11 show the 1D distributions and the JPDFs
between lifetime and length of downflowing RREs/RREs and RBEs, respectively.
We find RBEs lasting from 13.6 s to 2140 s, and RREs/downflowing RREs lasting
from 13.6 s to 1618 s with a median lifetime of about 27.2 s in both cases.
Moreover, we also show an empirical cumulative distribution function (ECDF) of
the lifetime that clearly indicates that 98% of the events are shorter than
200 s. The events lasting longer than $\geq$ 300 s in the above distributions
make up less than 2% in our data and can be considered as outliers. Part of
these are related to a few surge-like events that are morphologically quite
different from type-II spicules but can have similar spectral properties. It
can further not be excluded that some of these are actually multiple recurring
spicules that the method cannot separate and classifies as single long
duration events. The JPDFs in panel (h) also show that more than 70% of the
events that last shorter than 100 s, are also shorter than 0.7 Mm. The results
described above complies well with the characteristic properties of type-II
spicules and are well in agreement with the values reported in earlier works
such as Rouppe van der Voort et al. (2009); Sekse et al. (2012) and Pereira et
al. (2012). Further analyses showing the relationship between the eccentricity
of spicules vs. their lifetimes are shown in Fig. 16 in appendix B, which
reinforces that the majority of the detected events are both highly eccentric
and rapid.
The results presented above do not explicitly distinguish between the
downflowing and traditional RREs, but they strongly suggest that the newly
reported downflowing RREs have properties similar to their upflowing
counterparts observed in both the blue ward and red ward side of the
H$\mathrm{\alpha}$ line core. This further reinforces our proposition that the
downflowing RREs belong to the family of type-II spicules with an opposite
apparent motion.
## 5 Discussion
### 5.1 Interpretation of the downflowing RREs
The examples shown in Fig. 7 and the statistical analysis of the proper motion
of stronger and weaker excursions described in Fig. 8, indicate that there
exists a new category of RREs that behave contrary to the traditional
interpretation of RREs. De Pontieu et al. (2012); Sekse et al. (2013b); De
Pontieu et al. (2014) and Kuridze et al. (2015a) explained that RREs, like
RBEs, are a manifestation of the same physical phenomenon and appear either in
the blue or red wing of H$\mathrm{\alpha}$ depending on their transverse
motion along the LOS of chromospheric magnetic field lines. In other words,
RREs are generally found in close association with RBEs. The downflowing RREs,
discussed in this paper, do not satisfy this explanation since they are not
often found associated with their blue wing counterparts. Moreover, their
opposite apparent direction of motion indicates that they do not occur due to
the swaying and torsional motions often associated with type-II spicules (De
Pontieu et al., 2012).
The motion associated with chromospheric spicules is quite complex and often
all the three, i.e., the field aligned flows, the torsional motions and the
transverse swaying motions, are at play. So far, the appearance of isolated
RBEs are interpreted to be the result of upflows along the field lines
parallel to the spicule axis and the RREs the result of the combination of the
three. The appearance, morphology and their statistical properties discussed
in this paper suggests that downflowing RREs are similar to RBEs and they are
the result of the field aligned downflows in the solar chromosphere.
These downflows, however, are quite different from the type-I spicules such as
active region dynamic fibrils or quiet sun mottles that are commonly observed
near the H$\mathrm{\alpha}$ line core images (Rouppe van der Voort et al.,
2007). The $\lambda$t slices shown at the bottom of Fig. 7 show that the
downflowing RREs have no a priori blue shift associated with them, a
characteristic that is often linked with type-I spicules. Moreover, various
studies such as (De Pontieu et al., 2007; Pereira et al., 2012) show that
type-Is have much lesser LOS velocities (Doppler offsets) and seldom appear at
wing positions so far from the line core. Furthermore, the morphological
appearance of the downflowing RREs are very similar to RBEs and RREs, which
also provides a strong evidence that the former category is not associated
with the red shifts of the fibrils but are more like the downflowing
counterparts of RREs.
### 5.2 What might cause downflowing RREs?
One of the most important questions that remains to be addressed is what might
be the origin of these downflowing RREs? In this section we discuss some of
the possible mechanisms that could be responsible for such ubiquitous
downflows.
Type-II spicules sometimes exhibit parabolic up-down motion much like the
dynamic fibrils (Pereira et al., 2014). However, unlike the latter, the type-
II spicules are much faster, shorter lived (in the chromosphere) and show
signatures in the transition region (TR) and even in the solar corona (De
Pontieu et al., 2011; Henriques et al., 2016; Samanta et al., 2019). During
their ascending phase they get rapidly heated to TR with signatures in
passbands sampling coronal (1 MK) temperatures, in both active regions and the
quiet Sun, and eventually fall back in the chromosphere. These downflowing
plasma could, in principle, be responsible for the observed downflowing RREs
in the H$\mathrm{\alpha}$ red wing. The combined upflowing and downflowing
lifetimes in such cases can well be above 600–700 s (Pereira et al., 2014;
Samanta et al., 2019) which makes them typically last longer than their purely
chromospheric type-I counterparts (lifetime ~3–5 min).
A supporting proposition in favor of this returning plasma in the spicular
form can stem from the fact that observations of the spectral lines formed in
the temperature regimes between ~15,000 K and 2.5 $\times$ 105 K, reveal the
prevalence of an average red shift or downflowing motion of the order of
$10$–$15$ km s-1 in the TR (Doschek et al., 1976; Gebbie et al., 1981; Peter &
Judge, 1999; Dadashi et al., 2011) as discussed in Sect. 1. These studies
imply the existence of plasma flows or wave motions in the quiet Sun with
amplitudes that are significant fractions of the sound speed in the TR.
Different mechanisms have been proposed in the past to explain the net
downflowing structure of the TR. Peter et al. (2006) synthesized spectra of
the coronal and TR lines with the help of a 3D numerical simulation spanning
the photosphere to the corona and showed that the persistent red shifts in the
TR could possibly be explained by the effect of the flows caused due to
heating by magnetic braiding. Hansteen et al. (2010) expanded on this model
and injected an emerging flux to the 3D model of Peter et al. (2006) and
concluded that these downflows are a result of the rapid episodical heating
between the upper chromosphere and lower corona. Zacharias et al. (2018)
further complemented these earlier studies by establishing that pressure
driven downflows along the magnetic field lines could be identified as one of
the key mechanisms responsible for these observed red shifts in the TR.
Perhaps the most likely mechanism applicable to the observations of
downflowing RREs is the proposition that TR red shifts are caused due to the
emission from the return flows, that had been formerly heated and injected
into the solar corona by spicules (Pneuman & Kopp, 1977; Athay & Holzer, 1982;
Athay, 1984). Based on their modeling efforts, the studies by the above sets
of authors estimated that spicular material heated to coronal temperatures,
carry a large upward flux (upflows) that is almost 100 times the flux measured
due to solar wind at 1 AU. Consequently, they reasoned that the rest of the
mass must return to solar atmosphere and it should happen in the form of
spicular downflows which are then in turn responsible for the observed net red
shifts in the TR. Older numerical calculations such as the one by Mariska
(1987), however, could not obtain these observed red shifts. Moreover, many
earlier numerical models could not even account for any blue shift (upflows)
in the coronal or TR passbands. Therefore, the community at large remained
skeptical about the contribution of spicules in these downflows. Recent
observations by McIntosh & De Pontieu (2009); Rouppe van der Voort et al.
(2015) and state-of-the-art numerical modeling efforts by Martínez-Sykora et
al. (2017, 2018) addressed these concerns and proved beyond doubt that
spicules harbor a clear blue shift when observed in hotter TR and coronal
spectral channels.
The observations of these downflowing features have mainly been limited to the
TR till date. None of the studies in the past found evidences of the
chromospheric counterparts of these TR downflows. In some studies (such as
Mariska, 1987) the non existence of the chromospheric signatures of the TR
downflows raised serious doubts on their impact in spicular plasma in the
past. However, the examples of the ubiquitous downflows and the detailed
analysis presented in this paper seem to suggest that the downflowing RREs
could well be the signatures of the TR downflows in the chromosphere. Whether
they follow a typical parabolic trajectory or not warrants further
investigation but our results strongly revives the possibility that returning
spicular materials can be one of major driving mechanisms for the observed TR
red shifts and downflowing RREs.
Alternate possible explanation for the origin of the downflowing RREs could
lie in the investigation carried out by Rutten et al. (2019) where they
conjectured that RBEs could display return flows in the form of cool plasma
that traces the trails of the preceding type-II spicules. Their statistical
analysis strongly suggests that there is a tight correlation between the
occurrences of RBEs and subsequent H$\mathrm{\alpha}$ fibrils within a certain
time delay. Consequently, they attribute the dark fibrilar appearance around
the chromospheric network regions (seen predominantly in H$\mathrm{\alpha}$)
mainly to preceding type-II spicules. The downflowing RREs, like the ones
presented in this paper, could well be the immediate following state of these
firbils as they continue to evolve. However, so far there are no studies that
could firmly establish this relation.
Finally, the chromospheric counterparts of the low lying loops found in the TR
(Hansteen et al., 2014; Pereira et al., 2018) could also be attributed to the
appearance of these downflowing RREs in H$\mathrm{\alpha}$. The low lying
loops are not found to show prominent signals in the H$\mathrm{\alpha}$ line
wing images, except in the far wings, including the red wing. If such red-wing
signatures are true flows then a footpoint could appear as a downflowing RRE
whereas most of the loop would be ”hiding” in TR passbands. The footpoints of
nearly all the loops observed by Pereira et al. (2018) lie in close proximity
or share the footpoints of chromospheric spicules predominantly rooted in the
network regions which makes them possible to have a relation with the
downflowing RREs as described above. Furthermore, Martínez-Sykora et al.
(2020) also discussed the possibility of spicules forming along the loops in
numerical simulations which could be further compared with the observations.
However, given the fact that these loops are not as ubiquitous as spicules on
the solar surface it is very unlikely that downflowing RREs are always
associated with the chromospheric signatures of these low lying TR loops.
### 5.3 Comparison with flocculent flows
Flocculent flows in the solar chromosphere were first described by Vissers &
Rouppe van der Voort (2012) as distinct small-scale features that move
intermittently towards and away from a sunspot. They bear morphological
resemblance to coronal rain, both qualitatively and quantitatively (Antolin &
Rouppe van der Voort, 2012), but their sizes are somewhat smaller and they
move at much lower average velocities and over shorter distances. Flocculent
flows have been suggested to be a result of a siphon flow driven by pressure
difference between the footpoints in a loop. Compared to downflowing RREs,
flocculent flows typically travel over larger distances and consist more of
distinct and isolated blob-like features. We observe downflowing RREs
exclusively in close vicinity to magnetic network areas while flocculent flows
were observed to travel along extended parts of the sunspot superpenumbra and
long active region fibrils.
### 5.4 Spatial distribution of spicules and their substructures
The results presented in Fig. 9 suggests that, on average, the stronger
excursions in both the redward and blueward side of H$\mathrm{\alpha}$ line
core are located closer to the strong magnetic field regions, whereas the
weaker counterparts appear to be scattered across the FOV. This seems to be
the case for both the red and blue shifted excursions. Moreover, the weaker
excursions occupy lesser area on the FOV on an individual event-by-event
basis. Since most of the events in the stronger red excursion category have
apparent inward motion (refer to Figs. 8 and 15), we conclude that on average
the downflowing RREs are predominantly located in the regions close to the
network areas, whereas their traditional counterparts appear to exist slightly
farther away.
The spicule halos, described in Sect. 3.2.1, are a consequence of the
substructures seen in RBEs and RREs/downflowing RREs. These substructures are
mainly due to the fact that spicules often have a large distribution of
spectral properties such as line core width, Doppler shift (Pereira et al.,
2016) or absorption in the wings of the H$\mathrm{\alpha}$ spectral line, not
only along the spicule axis but also in the transverse direction, and are a
part of the same morphological structure that evolves in a collective fashion.
Figure 5 and its associated animation strongly suggests that many times
spicules do not evolve independently but in groups that maybe difficult to
discern when observed at one wavelength position. This reinforces the fact
that group behavior is common among type-II spicules (Skogsrud et al., 2014).
Another important feature commonly observed in spicules is the rapid
morphological transformations that they undergo both in space and time,
including multiple splitting and branching (Yurchyshyn et al., 2020). We refer
to the examples shown in Fig. 7 where we clearly see that the DRRE-1 initially
starts as one structure but towards the end of its evolution we see it clearly
split into two as they terminate at the network bright points. RBE-1 also
shows clear signs of morphological branching with a ”Y” pattern just before it
disappears. Often these branches have different LOS velocities but as we see
in the examples they are clearly associated with the original parent structure
evolving together. Therefore it implies that, downflowing RREs, like RBEs and
RREs, can have multiple structures either in situ, due to a range of spectral
properties, or due to the morphological transformations they undergo during
their evolution.
### 5.5 Significance of the detection technique and their limitations
The detection method based on $k$-means clustering and morphological image
processing technique presented in this paper enabled us to study and analyze
spicules in an unprecedented detail. Identifying the RBE and RRE/downflowing
RRE RPs on the basis of the strength of their Doppler offset, line core width
and enhanced absorption measure in the H$\mathrm{\alpha}$ spectral line
provides one of the most compendious approach in their characterization.
Nearly 20,000 RBEs and 15,000 RREs (including downflowing RREs) have been
identified in our 97 minute long dataset that allowed us to perform varied
statistical analysis and compare the properties of the downflowing RREs in the
context of traditional RBEs and RREs. Though no distinction was possible
between the downflowing RREs and RREs in terms of their spectral signatures,
we see that out of the 15,000 red shifted events a substantial number of
strongest excursions are downflowing in nature, with their apparent motion
directed towards the strong network areas. Furthermore, the statistical
analysis presented in this paper reveals that downflowing RREs have similar
dimensions and lifetimes when compared against RBEs and traditional RREs. This
makes them more likely to be a part of the family of type-II spicules.
Previous reports (Rouppe van der Voort et al., 2009; Sekse et al., 2012;
Pereira et al., 2012; Kuridze et al., 2015a) found comparatively longer
lengths for RBEs and RREs. This could possibly be due to the fact that, except
for Pereira et al. (2012), most of these earlier works have focused on
performing the statistical analyses based on images at single wavelength
positions far in the blue or the red wing of the H$\mathrm{\alpha}$ line
profile. Moreover, on most occasions they employed a lower length detection
threshold of ~$0.730$ Mm that strictly limited the lower estimate of their
analysis, thereby facilitating the detections of relatively longer events. On
the other hand, the method employed in this paper is compendious both in terms
of spectra as well as in size thereby enabling smaller detections.
The technique presented in this paper exploits the complete spectral profile
(RPs) instead of relying on the morphology of spicular features at single
wavelength positions. To get rid of erroneous detections we also impose a
lower limit on the length of the detected events but it is far lower (~0.1 Mm)
than the ones chosen in the earlier works. Rouppe van der Voort et al. (2009)
reported the presence of special RBEs in the form of ”black beads” in
H$\mathrm{\alpha}$ that appeared as tiny roundish darkenings and were
interpreted as spicules that were aligned closely along the LOS. The dimension
of such features were reportedly between 0.15 and 0.3 Mm. Interestingly, Anan
et al. (2010) reported the shortest mean lengths of spicules compared to all
other works in the recent past. Pereira et al. (2012) interpreted this
discrepancy owing it to the exponential drop in intensity along the body of
the spicule which can possibly render the tops too faint to be picked up in
their detection when viewed against the disk. This explanation is also valid
in our case because, despite the spatial resolution, the top portion of the
spicules most likely has too little opacity to be considered as a part of
either an RBE or an RRE/downflowing RRE. Therefore, there could be an
intrinsic bias in determination of the lengths of spicules as was also
reported by Sekse et al. (2012). The lifetimes, on the other hand, are well in
agreement with most of the former studies mentioned above, despite the
increased capture of events over the FOV and increased sensitivity to smaller
scales.
Despite undertaking advanced methods to characterize and detect RBEs and RREs,
we do encounter a few limitations. In this paper, we have referred to the
detected events as RBEs or RREs/downflowing RREs which are on-disk
counterparts of type-II spicules; but are we clearly detecting the type-IIs?
One of the limitations of $k$-means clustering is that it only accounts for
the spectral lines in our case. Therefore, both type-I and type-II spicules
could very well be included in our detections as they have similar spectral
profiles in H$\mathrm{\alpha}$. However, the results from the detailed
statistical analysis presented in this paper strongly suggest that we are in
the domain of type-II spicules since, most importantly, 98% of the events last
under ~$3$ minutes and the median lifetime is about $27$ s. Furthermore the
morphology and the evolution of the detected events, like the ones indicated
in Figs. 6 and 5, clearly suggest that we are detecting the rapid type-II
spicules.
## 6 Concluding remarks
We report the first observation and characterization of rapid downflows in the
solar chromosphere in the form of spicules. Their rapidity and apparent motion
in the far red wing images of H$\mathrm{\alpha}$ strongly suggests that they
are the downflowing counterparts of the traditional RREs first reported by
Sekse et al. (2013b), unlike coronal rain (Antolin & Rouppe van der Voort,
2012) and flocculent flows (Vissers & Rouppe van der Voort, 2012). We
therefore term them downflowing RREs. In depth statistical analyses performed
on 14,650 RREs and 19,643 RBEs imply that downflowing RREs are similar to the
already known RREs and RBEs. Moreover, they also undergo rapid morphological
transformations during their evolution in the same way as RBEs and RREs. The
only evident difference of this new class of RREs lies in their apparent
motion where they seem to originate away and terminate close to the strong
field network regions. This suggests that they are a result of the field
aligned downflows in the solar chromosphere. Furthermore, the downflowing RREs
could also undergo the transverse and torsional motions that are often
associated with type-II spicules which hints at the possibility of finding
downflowing RBEs in the solar chromosphere. Future work in this direction
could shed more light in this context.
The downflowing RREs could possibly be linked to the return flows of type-II
spicules which ascend rapidly from the chromosphere to the TR and even coronal
heights, and eventually fall back. Moreover, we present arguments suggesting
that these downflowing RREs could well be responsible for the observed TR
downflows as was first hypothesized by Pneuman & Kopp (1977). It would be
interesting to find direct signatures of these downflowing RREs in the corona
and the TR with coordinated observations from the Interface Region Imaging
Spectrograph and Solar Dynamics Observatory because it will enable better
understanding of the mass and energy cycle in the solar atmosphere. Such a
study is currently underway and will be the subject of a forthcoming paper.
###### Acknowledgements.
We thank Ainar Drews for his help with the observations. The Swedish 1-m Solar
Telescope is operated on the island of La Palma by the Institute for Solar
Physics of Stockholm University in the Spanish Observatorio del Roque de los
Muchachos of the Instituto de Astrofísica de Canarias. The Institute for Solar
Physics is supported by a grant for research infrastructures of national
importance from the Swedish Research Council (registration number 2017-00625).
This research is supported by the Research Council of Norway, project number
250810, and through its Centers of Excellence scheme, project number 262622.
VMJH is also funded by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (SolarALMA, grant
agreement No. 682462).
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## Appendix A Spicule substructures
Figure 12: Discerning the substructures of the RBEs, RREs and, downflowing
RREs but only for the spicules harboring at least one of strong excursion RPs
(2, 3, 6 and 7). Panels (a) and (b) show the strong (RPs 2 and 3) and weak
(RPs 0 and 1) offset counterparts of the stronger excursion RBEs,
respectively; whereas panels (c) and (d) shows the strong (RPs 6 and 7) and
weak (RPs 4 and 5) offset counterparts belonging to the stronger
RREs/downflowing RREs. The color gradient indicates the density of the events
detected over the entire duration of the data. The black contour indicates the
regions that have an absolute value of the LOS magnetic field $\geq$ $100$ G.
We describe the substructures of spicules belonging to the stronger excursions
in blueward and redward side of H$\mathrm{\alpha}$ line core. We reiterate
that both the stronger and weaker excursion spicules can have spatial
substructures due to variation in their spectral properties which causes them
to have different RPs (stronger excursion RPs along with weaker excursion
RPs). As described in Sect 3.1.1 and 4.1, we considered those spicules as
stronger excursion that have at least one of stronger excursion RPs, i.e., RPs
2, 3, 6, and, 7, at least once during their entire lifetime. However, as
discussed earlier in the paper, even the strongest excursions have a spatial
variation in their Doppler offset along and perpendicular to the spicule axis,
hence weaker RPs, i.e., RPs 0, 1, 4 and, 5, could also be present within their
morphological structure (see Sect. 3.2.1). The major motivation of the
analysis presented in this section is to discern the substructures of strong
RBEs, RREs, and downflowing RREs depending on RPs and see if stronger and
weaker excursion RPs have any spatial preference. These variations add to the
structural complexity of spicules as shown in Fig. 12. Panel (b) shows the
”weaker” Doppler offset counterparts of the stronger blue excursion events in
panel (a). The arrangement is identical for the red excursions in panels (c)
and (d). Therefore, unlike Sect. 4.2, only the events common to the stronger
excursion category are identified and represented in panels (b) and (d) of
Fig. 12, while all others are excluded. Consequently, Fig. 9 shows the
distribution of spicules belonging uniquely to the strong and weak excursion
categories whereas Fig. 12 shows the spatial distribution of the spicules
belonging only to the stronger excursion category, but with different
excursion RPs. Naturally, the events in the different velocity compartments
are now no longer unique.
Upon closer investigation, we find that the number density of events increases
roughly by 45% for the strong blue excursions (panels (a)–(b)) and 27% (panels
(c)–(d)) for the strong red excursions. Furthermore, this increase is
predominantly reflected in the regions away from the strong network areas in
the former, whereas for the red excursions this is also linked to an increase
within and inside the strong field contours such as within the network patch
located at ($X$,$Y$) =(45″,33″) or close to the bottom of the FOV at $X$=20″in
panel (d). The increase in the number density can be attributed to the
variation in the Doppler offsets, and line widths of spicules in different
parts of their body.
## Appendix B Supplementary figures
This section provides supplementary resources for the results presented in
Sects. 3.2, 4.1 and 4.3. The details of the figures shown in this section are
discussed in the main text.
Figure 13: Overview of the location and density distribution of the total
detected events over the whole FOV. Panel (a) shows the occurrence of
RREs/downflowing RREs over the whole time series against an H$\mathrm{\alpha}$
red-wing image at 93 km s-1; and panel (b) shows the occurrence of RBEs
against a background of CHROMIS continuum image at 4000 Å. Figure 14: Extended
$\lambda t$ diagrams corresponding to the two downflowing RREs shown in Fig. 7
displaying the lack of any blue shift preceding the downflowing RREs. The
maximum excursion towards the red-wing of H$\mathrm{\alpha}$ is indicated by
blue horizontal markers which also corresponds to the time shown in the
$\lambda t$ diagram of Fig. 7. Figure 15: Apparent motion of spicules in the
same format as Fig. 8 but for the full FOV. The behavior explained in the text
is also seen to be followed all over the FOV. Displacements smaller than
1″have been removed in order to improve the visibility. Figure 16: Bi-variate
joint probability density between lifetime and eccentricity (of the fitted
ellipses) of (a) RREs/downflowing RREs and (b) RBEs in rainbow colormap as in
Fig. 11. The magenta contour indicates the region within which 70 % of the
events lie.
|
# The Devils in the Point Clouds: Studying the Robustness of Point Cloud
Convolutions
Xingyi Li, Wenxuan Wu, Xiaoli Z. Fern, and Li Fuxin
Oregon State University
{lixin, wuwen, xfern<EMAIL_ADDRESS>
###### Abstract
Recently, there has been a significant interest in performing convolution over
irregularly sampled point clouds. Since point clouds are very different from
regular raster images, it is imperative to study the generalization of the
convolution networks more closely, especially their robustness under
variations in scale and rotations of the input data. This paper investigates
different variants of PointConv, a convolution network on point clouds, to
examine their robustness to input scale and rotation changes. Of the variants
we explored, two are novel and generated significant improvements. The first
is replacing the multilayer perceptron based weight function with much simpler
third degree polynomials, together with a Sobolev norm regularization.
Secondly, for 3D datasets, we derive a novel viewpoint-invariant descriptor by
utilizing 3D geometric properties as the input to PointConv, in addition to
the regular 3D coordinates. We have also explored choices of activation
functions, neighborhood, and subsampling methods. Experiments are conducted on
the 2D MNIST & CIFAR-10 datasets as well as the 3D SemanticKITTI & ScanNet
datasets. Results reveal that on 2D, using third degree polynomials greatly
improves PointConv’s robustness to scale changes and rotations, even
surpassing traditional 2D CNNs for the MNIST dataset. On 3D datasets, the
novel viewpoint-invariant descriptor significantly improves the performance as
well as robustness of PointConv. We achieve the state-of-the-art semantic
segmentation performance on the SemanticKITTI dataset, as well as comparable
performance with the current highest framework on the ScanNet dataset among
point-based approaches.
## 1 Introduction
Convolution is one of the most fundamental concepts in deep learning.
Convolutional neural networks (CNNs) have redefined the state-of-the-art for
almost every task in computer vision. In order to transfer such successes from
2D images to the 3D world, there is a significant body of work aiming to
develop the convolution operation on 3D point clouds. This is essential to
many applications such as autonomous driving and virtual/augmented reality.
PointConv [60] seems to be one of the promising efforts. Similar to earlier
work [43, 18, 59, 14, 58, 64], PointConv utilizes a multi-layer perceptron
(MLP) to learn the convolution weights on each point implicitly as a nonlinear
transformation from the point coordinates, basically a Monte Carlo
discretization of the continuous convolution operator on irregular point
clouds, but with the efficient version of PointConv it now can scale to modern
deep networks with dozens of layers. PointConv is permutation-invariant and
translation-invariant, and to our best knowledge is the only approach that
achieved equivalent performance to 2D CNNs on images as well as having one of
the highest performances on 3D benchmarks.
Figure 1: Examples of learned weight functions from PointConv: (a) MLP-based
PointConv trained on MNIST; (b) MLP-based PointConv trained with Faster R-CNN;
(c) Sobolev-regularized cubic polynomial (Best Viewed in Color)
Networks based on point clouds introduce a new complication on the
neighborhoods used in convolution. In 2D images, we are accustomed to having
fixed-size neighborhoods such as $3\times 3$ or $5\times 5$. PointConv and
other point-based networks instead adopt k-nearest neighbors (kNN), which may
potentially make it harder for point cloud networks to generalize from
training locations to testing locations, as sufficiently smooth weight
functions need to be learned. Usually, point cloud networks augment the data
by randomly jittering point locations, but such jittering only provides local
generalization. We attempted to plot one of the typical learnt weight function
on MNIST and on a faster-RCNN detector as shown in Fig. 1 (a-b). As one can
see, due to the nonlinearity in the weight functions, PointConv could
potentially generalize poorly if the testing neighborhoods are substantially
different from those of the training. Indeed, even in 2D images we rely on re-
scaling of all the images to the same scale to avoid this generalization
issue. Such a simple shortcut is, however, unlikely to suffice for point
clouds as each kNN neighborhood may be significantly different from others in
terms of scale.
In this paper we study empirically the generalization of PointConv under scale
changes (resulting in different sampling densities) and rotations, which would
induce very different neighborhoods between training and testing. The basic
methodology is to train the network with a certain set of scales and
rotations, and test it on out-of-sample scales/rotations that are
significantly different from the training ones. Experiments are done both on
the 2D MNIST [25] & CIFAR-10 [22] datasets, and the 3D SemanticKITTI [2] &
ScanNet v2 [8] datasets. Multiple design choices are tested, including
different neighborhood selection methods, activation functions, input feature
transformations and regularization methods to examine their impact on
generalization under scale changes and rotations.
From the experiments, we identify two strategies that have not been applied to
point cloud networks to be the most effective. In 2D images, we propose to
utilize cubic polynomials as the weight functions, with a Sobolev norm
regularization similar to thin-plate splines. This restricts the flexibility
of the weight functions (Fig. 1) and improves generalization. We additionally
find that using an $\epsilon$-ball neighborhood is more robust than kNN in 2D.
With these improvements, we have found PointConv to be more robust than
traditional raster CNNs on scale changes and rotations, which suggests a
potential of applying PointConv on 2D images for the sake of robustness in
future work.
For 3D point clouds, we introduce a novel viewpoint-invariant (VI) feature
transformation to the 3D coordinates. The results show that our novel feature
transformation is not only rotation-invariant, but also robust to neighborhood
size changes and achieves significantly better generalization results than
simply using 3D coordinates as input. VI enables coordinates to generalize to
neighborhood of different sizes, hence we have found that $\epsilon$-ball
neighborhoods become no longer necessary on top of VI. This is welcome news
since $\epsilon$-ball neighborhoods (e.g. utilized commonly in KPConv [49])
are more expensive to compute than kNN neighborhoods.
## 2 Related Work
Volumetric and Projection-based Approaches A direct extension from convolution
in 2D raster images to 3D is to compute convolution on volumetric grids [61,
33, 39, 57]. In densely sampled point clouds, sparse volumetric convolutions
such as MinkowskiNet [5] and Submanifold sparse convolution [10] currently
obtain the best performance. However, they depend heavily on being able to
locate enough points in a local volumetric neighborhood of each point, hence
are difficult to apply to cases where the sampling density of point clouds is
low or especially uneven (e.g. LIDAR). Some other approaches that project
point clouds onto multi-view 2D images [46, 36, 30] or lattice space [45] may
suffer from the same issue.
Point-based Approaches [35] first attempted to directly work on point clouds,
and PointNet++ [37] improved it by adding a hierarchical structure. Following
PointNet++, some other studies also attempted to utilize hierarchical
architectures to aggregate information from neighbor points with MLPs [26,
31]. PointCNN [27] utilized a learned $X$-transformation to weight the local
features and reorder them simultaneously. FeaStNet [52] utilize a soft-
assignment matrix to generalize traditional convolution on point clouds. Flex-
convolution [11] introduced a convolution layer for arbitrary metric spaces,
along with an efficient GPU implementation. A-CNN [21] proposed a annular
convolution that assigned kernel weights for neighbour points based on their
distances to the center point. PointWeb [68] considered every pair of points
within the neighborhood for extracting contextual features. A-SCN [62] adopt
the dot-product self-attention mechanism from [51] to propagate features.
PointASNL [65] proposed an adaptive sampling strategy to avoid outliers before
extracting local and global features from each point cloud.
Generally, convolutional approaches on point clouds performed better than the
approaches listed above. [43, 18, 59, 14, 58, 64] proposed to learn
discretizations of continuous convolutional filters. [18] utilized a side
network to generate weights for 2D convolutional kernels. [43] generalized it
to 3D point clouds, and [58] further extends it to segmentation tasks [58],
along with an efficient version. However, the efficient version in [58] only
achieves depthwise convolution rather than full convolution. EdgeConv [59]
encodes pairwise features between a neighbor point and the center point
through MLPs. [14] takes densities into account. Pointwise CNN [16] located
kernel weights for predefined voxel bins, so it was not flexible. SpiderCNN
[64] proposed a polynomial weight function, which we experiment with in this
paper. However, they did not utilize regularization to control the smoothness.
The formulation of PointConv [60] is mathematically similar to [14], and it
encompasses [43, 59] and [58], since those can be viewed as special cases of
PointConv, removing some of the components (e.g. density, full convolution).
The main contribution in PointConv [60] is an efficient variant that does not
explicitly generate weight functions, but implicitly so by directly computing
the convolution results between the weights and the input features. Such a
variant removed the memory requirement to store the weights and networks,
allowing for scaling up to the “modern” deep network size, e.g. dozens of
layers with hundreds of filters per layer. It is also the only paper showing
results on CIFAR-10 matching those of a 2D CNN of the same structure.
The main competitive point-based convolutional approach to PointConv is KPConv
[49]. In KPConv, convolution weights are generated as kernel functions between
each point and anchor points, points in the 3D space that are pre-specified as
parameters for each layer separately. KPConv enjoys nice performance due to
the smooth and well-regularized kernel formulation, but it introduces
significantly more parameters in the specification of anchor points and their
$\epsilon$-ball neighborhoods are computationally costly. Similar to KPConv,
PCNN [1] also assign anchor points with kernel weights, but it does not take
neighbor points into account for convolution.
Scale and rotation invariance in convolution We did not find significant
amount of related work in studying scale and rotation robustness on point
clouds. [56, 66] build a spatial transformer side network (STN) to learn
global transformations on input point clouds. The main difference comparing
with our work is that we aim to improve the robustness against transformations
through directly working on the network itself, instead of designing
additional structures to accommodate them.
A significant amount of work have been published in 2D CNNs on scale and
rotation invariance. A standard approach has been data augmentation [44, 12,
50, 24, 4, 13], where the training set is augmented by including objects with
random rescaling or rotations. A group of studies attempted to integrate deep
CNNs with side-networks [29, 69, 55, 63, 67, 19] or attention modules [54,
42]. [70] convolved the input with several rotated versions of the same CNN
filter before feeding to the pooling layer. Some techniques proposed to learn
transformations directly [17, 28] on the input or intermediate outputs from
convolutional layers in a deep network [9, 40]. There has also been interest
in combining group concepts with CNNs to encode scale and rotate
transformations [6, 3, 7].
Non-deep Approaches [41] encodes relations between local 3D surface patches as
well as global patches in a viewpoint invariant manner, which is a good hint
for this work.
## 3 Methodology
The main goal of this work is to investigate how to best enable the learned
weight function to generalize from known local locations to unseen ones, to
make PointConv [60] based networks more robust against unseen scales and
rotations of objects. Toward this, we attempted three methods. First, we
replace kNN with $\epsilon$-ball based neighbor search to unify the receptive
field for each PointConv layer. Next, we introduce a much simpler hypothesis
space of third degree polynomials to replace MLP for weight functions to avoid
overfitting. To further enforce the smoothness of this hypothesis space, we
utilize the Sobolev norm for thin-plate splines as a regularizer. Finally, for
3D point clouds, we introduce a viewpoint-invariant (VI) descriptor for the
MLP that utilizes geometric properties of the data to be less sensitive to
local scale and rotation changes.
Below we will first introduce PointConv [60], followed by the description of
$\epsilon$-ball based neighbor search, third degree polynomials, and the VI
descriptor for the weight function, respectively.
Figure 2: (a) We perform robustness experiments of PointConv on 2D images
where the training kNN neighborhood is significantly different from the
testing; (b) For a given local center point $p_{\mu}$ and $p_{\alpha}\in
N_{\epsilon}(p_{\mu})$ for a pair of points, a set of viewpoint-agnostic basis
$(\vec{r},\vec{w},\vec{v})$ can be generated from $\vec{r}_{\mu}^{\alpha}$ and
$n_{\mu}$ with the Gram-Schmidt process, and viewpoint-invariant features such
as the angles between $\vec{n_{\mu}}$ and $\vec{v}$ can be extracted from them
### 3.1 Background: PointConv
A point cloud can be denoted as a set of $3D$ points
$P=\\{p_{1},p_{2},...,p_{n}\\}$, where each point $p_{i}$ contains a position
vector $(x,y,z)\in R^{3}$ as well as a feature vector (RGB color, surface
normal, etc.). A line of work including PointConv generalizes the convolution
operation to point clouds based on discretizations of continuous 3D
convolutions [43, 14, 59, 60]. For a center point $p_{xyz}=(x,y,z)$, its
PointConv is defined by:
$\small\begin{multlined}PointConv(S,W,F)_{xyz}=\\\
\sum_{(\delta_{x},\delta_{y},\delta_{z})\in
G}S(\delta_{x},\delta_{y},\delta_{z})W(\delta_{x},\delta_{y},\delta_{z})F(x+\delta_{x},y+\delta_{y},z+\delta_{z})\end{multlined}PointConv(S,W,F)_{xyz}=\\\
\sum_{(\delta_{x},\delta_{y},\delta_{z})\in
G}S(\delta_{x},\delta_{y},\delta_{z})W(\delta_{x},\delta_{y},\delta_{z})F(x+\delta_{x},y+\delta_{y},z+\delta_{z})$
(1)
where $(\delta_{x},\delta_{y},\delta_{z})$ denote the coordinate offsets for a
point in $p_{xyz}$’s local neighborhood $G$, usually located by kNN.
$F(x+\delta_{x},y+\delta_{y},z+\delta_{z})$ represents the feature of the
point, and $W(\delta_{x},\delta_{y},\delta_{z})$ is the convolution kernel
generating the weights for convolution and is approximated by an MLP, called
WeightNet in [60]. Finally, $S(\delta_{x},\delta_{y},\delta_{z})$ represents
the inverse local density to balance the impact of non-uniform sampling of the
point clouds.
PointConv uses an efficient approach to avoid the computation of the function
values of $W$ at each point, which is extremely memory-intensive. The
computation approach does not change the final output of eq. (1), hence we
omit it. A deep network can be built from PointConv layers similar to 2D
convolutions. For stride-2 convolution/pooling, one can just subsample the
point clouds [37]. [60] also provides a deconvolution/upsampling approach to
increase the resolution of point clouds. Hence, classification and semantic
segmentation tasks can be solved with PointConv networks. It is also
straightforward to incorporate other commonly used 2D convolution operations,
e.g. residual connections. Dilated convolution can be approximated by first
sampling a larger kNN neighborhood, and then subsampling from the
neighborhood.
### 3.2 $\epsilon$-ball neighbor search and activations
The neighborhood $G$ in PointConv is usually defined by kNN. Fig. 2 (a)
illustrates the robustness issue for K-nearest neighbors search. Namely, the
equivalent receptive field for a sparse point cloud is much larger than the
one for a densely distributed point cloud. If trained only on dense (high
resolution) point clouds, the learned weight function may not generalize well
to much larger (unseen) receptive fields when dealing with sparse point clouds
during testing.
An $\epsilon$-ball based neighborhood (e.g. commonly used in KPConv [49]) on
the other hand would be robust to different sampling rates. For a point
$p_{i}$, denote $N_{\epsilon}(p_{i})=\\{p_{j}\in P|d(p_{i},p_{j})<\epsilon\\}$
as its $\epsilon$-ball neighborhood. To ease the computation burden, we
(randomly) select at most $K$ neighbors from $N_{\epsilon}(p_{i})$. The actual
chosen neighbors from $N_{\epsilon}(p_{i})$ are denoted as
$C_{\epsilon}(p_{i},K)$. Compared with kNN, $\epsilon$-ball neighborhood
retains the maximal distance of the neighbors w.r.t. the center point. Since
different $\epsilon$-balls may contain different number of neighbors, we
replace the normalizer $S(\delta_{x},\delta_{y},\delta_{z})$ in equation 1
with $\frac{1}{|C_{\epsilon}(p_{i},K)|}$. Note that the flexibility of the
PointConv framework allows for variable number of neighbors in each
neighborhood.
We also investigate the robustness over several different activations in the
MLP-based WeightNet, such as ReLU [34], SeLU [20], Leaky ReLU [32], and Sine
$(\sin(x))$. Sine is included because its connections to random Fourier
features [38]. In [38], it was proved that a basis of
$\cos(\mathbf{Wx}+\mathbf{b})$ with random $\mathbf{W}$ and $\mathbf{b}$ could
be a universal function approximator. Hence we thought learned $\mathbf{W}$
and $\mathbf{b}$s could improve over the pure random one. Empirically we have
found that using sine worked better than cosine, maybe due to the fact that
$sin(0)=0$ hence it does not introduce additional constants.
### 3.3 Convolutional Kernels as Cubic Polynomials
The set of functions MLPs can represent is very large, which may introduce
overfitting to the training point locations. Hence, we experiment with much
simpler weight functions $W(\delta_{x},\delta_{y},\delta_{z})$ in the form of
cubic polynomials of $(x,y,z)$. This was investigated in [64], however with
some arbitrary additional functions multiplied that actually reduced the
performance on 2D [60]. We utilize a plain version, with proper weights to
regularize for smoothness. In 2D, this results in a feature space:
$\small\phi(x,y)=[x^{3},y^{3},\sqrt{3}x^{2}y,\sqrt{3}xy^{2},\sqrt{3}x^{2},\sqrt{3}y^{2},\sqrt{6}xy,x,y,1]$
(2)
and in 3D:
$\begin{split}\phi(x,y,z)=[x^{3},y^{3},z^{3},\sqrt{3}x^{2}y,\sqrt{3}x^{2}z,\sqrt{3}xy^{2},\sqrt{3}y^{2}z,\\\
\sqrt{3}xz^{2},\sqrt{3}yz^{2},\sqrt{3}x^{2},\sqrt{3}y^{2},\sqrt{3}z^{2},\sqrt{6}xyz,\sqrt{6}xy,\\\
\sqrt{6}xz,\sqrt{6}yz,\sqrt{3}x,\sqrt{3}y,\sqrt{3}z,1]\end{split}$
The coefficients of each term ensures that an $L_{2}$ regularization on the
feature descriptors (excluding linear and constant terms) correspond to
regularizing with the Sobolev $S_{2}$ norm:
$||f||_{s_{2}}^{2}=\lambda\int\int[(\frac{\partial^{2}f}{\partial
x^{2}})^{2}+2(\frac{\partial^{2}f}{\partial x\partial
y})^{2}+(\frac{\partial^{2}f}{\partial y^{2}})^{2}]dxdy$ (3)
in 2D, where $\lambda$ is a parameter. The 3D form can be written similarly.
Sobolev-norm regularizations are commonly used in thin-plate splines [53] but
to our knowledge they have not been used in point cloud networks in the past.
Note that the choice of a third degree polynomial is common in the smoothing
splines community. Quadratic functions have undesirable symmetries and fourth-
order polynomials introduce too many terms, e.g. $35$ terms for a 3D space.
### 3.4 A Viewpoint-Invariant Input Descriptor
PointConv relies on the $(x,y,z)$ coordinates to compute the weights, which is
sensitive to the rotation of the object as well as the sampling rate of the
point clouds. We hypothesize that by using viewpoint-invariant descriptors for
the weight function, we can achieve better generalization.
We develop a viewpoint-invariant (VI) descriptor for each point $p_{\alpha}$
in a local neighborhood as an $8$-dimensional vector utilizing its geometric
relationship with the center point $p_{\mu}$. Denote the surface normal of
$p_{\mu}$ as $\hat{n}_{\mu}$ and its difference with $p_{\alpha}$ as
$\vec{r}_{\mu}^{\alpha}=p_{\alpha}-p_{\mu}$. When the scene is rotated, the
angle between $\hat{n}_{\mu}$ and $\vec{r}_{\mu}^{\alpha}$ remains the same.
To discriminate between different directions, we generate an orthonormal basis
from $\\{\hat{n}_{\mu},\vec{r}_{\mu}^{\alpha}\\}$ and compute the angles
between $\hat{n}_{\mu}$, $\vec{r}_{\mu}^{\alpha}$ and $\hat{n}_{\alpha}$ as
well as the projection lengths of $\hat{n}_{\alpha}$ and $\hat{n}_{\mu}$ onto
the orthonormal basis. With a global rotation of the scene, the basis and
normal vectors are identically rotated. Hence, our descriptor is rotation
invariant and provides a complete characterization of the vectors
$\hat{n}_{\mu}$, $\vec{r}_{\mu}^{\alpha}$ and $\hat{n}_{\alpha}$.
Formally, we utilize the Gram-Schmidt process on
$\\{\vec{r}_{\mu}^{\alpha},\hat{n}_{\mu}\\}$ to generate a 3D basis
$\\{\hat{r},\hat{v},\hat{w}\\}$ where
$\hat{r}=\frac{\vec{r}_{\mu}^{\alpha}}{||\vec{r}_{\mu}^{\alpha}||}$,
$\hat{v}=\frac{\hat{n}_{\mu}-(\hat{r}^{\top}\hat{n}_{\mu})\hat{r}}{\sqrt{1-(\hat{r}^{\top}\hat{n}_{\mu})^{2}}}$
is orthonormal to $\hat{r}$, and $\hat{w}$ is derived by
$\hat{r}\times\hat{v}$, the outer product of $\hat{r}$ and $\hat{v}$, as
illustrated in Fig. 2. Note that this basis is seldom degenerate, because it
is unlikely for $\hat{n}_{\mu}$ and $\vec{r}_{\mu}^{\alpha}$ to be collinear
in 3D surface point clouds.
With the basis defined, for each point $p_{\alpha}$ in the neighborhood of
$p_{\mu}$, we extract the following viewpoint invariant feature vector:
$\begin{split}\beta_{\mu}^{\alpha}=[\hat{n}_{\alpha}\cdot\hat{n}_{\mu},\frac{\vec{r}_{\mu}^{\alpha}\cdot\hat{n}_{\mu}}{\|\vec{r}_{\mu}^{\alpha}\|},\frac{\vec{r}_{\mu}^{\alpha}\cdot\hat{n}_{\alpha}}{\|\vec{r}_{\mu}^{\alpha}\|},\hat{n}_{\alpha}\cdot\hat{v},\hat{n}_{\alpha}\cdot\hat{w},\\\
\vec{r}_{\mu}^{\alpha}\cdot\hat{n}_{\mu},\vec{r}_{\mu}^{\alpha}\cdot(\hat{n}_{\alpha}\times\hat{n}_{\mu}),||\vec{r}_{\mu}^{\alpha}||]\end{split}$
(4)
where $\times$ represents the cross product. The first $5$ features are both
scale and rotation invariant, and the last $3$ features are rotation invariant
only. We believe that a weight function with this input will be invariant to
rotation and more robust against scale changes. We also anticipate this to
alleviate the need for rotation-based data augmentation.
## 4 Experiments
### 4.1 MNIST
MNIST contains $60,000$ handwritten digits in the training set and $10,000$
digits in the test set from 10 distinct classes. The original resolution of
each image is $28\times 28$. We trained a 4-layer network. Table 1 illustrate
the general architecture for our proposed framework and all other baselines.
They only differ in the structures for convolutional layers. The $\epsilon$ is
fixed to $\frac{1}{10}$ for Conv1 in Table 1, $\frac{1}{5}$ for Conv2, and
$\frac{1}{2}$ for Conv3 as well as Conv4. The numbers of output points from
FPS for Subsampling1 Subsampling2 layer in Table 1 are $196$ and $49$
respectively. The coefficient of the Sobolev norm regularization is set to
$10^{-6}$ or $10^{-5}$ for our proposed approach. For other ablation variants,
it was tuned to the optimal one. We utilize cross entropy as the loss
function. The batch size is $60$ and the optimizer is Adam with learning rate
$0.001$. For traditional 2D CNN baselines, we adopt max pooling with $2\times
2$ stride for the sub-sampling layer. Global average pooling is used in the
last layer to account for different input resolutions. For point cloud sub-
sampling, we attempt two different methods. The first one is named as PC-2D
subsampling, which achieves the equivalent number of sampled points as 2D max-
pooling with stride $2\times 2$, but in point cloud formats. The next one is
Farthest Point Sampling (FPS), a commonly used sampling approach in point
cloud networks [37, 60].
Layer name | Layer description
---|---
Conv1 | $3\times 3$ conv. (or PointConv) 64 w/ ReLU
Subsampling1 | max-pooling with stride $2\times 2$ (or farthest point sampling)
Conv2 | $3\times 3$ conv. (or PointConv) 128 w/ ReLU
Subsampling2 | max-pooling with stride $2\times 2$ (or farthest point sampling)
Conv3 | $3\times 3$ conv. (or PointConv) 128 w/ ReLU
Conv4 | $3\times 3$ conv. (or PointConv) 128 w/ ReLU
Pooling | global average pooling
FC-10 | fully connected layer
Softmax layer
Table 1: The general network architecture for MNIST experiments. Every
convolutional layer is followed by a ReLU layer.
To convert a 2D raster image with the resolution of $m\times n$ to a point
cloud, we generate a point for each pixel with normalized coordinates
$p_{i}(x,y)=(\frac{m_{i}}{m},\frac{n_{i}}{n})\in R^{2}$ as its spatial
location. Normalization constrains the input range of each coordinate for the
weight function to $[0,1]$. Thus, images of different scales are converted to
point clouds with different sampling densities. The original RGB feature is
also normalized to $[0,1]$.
Three 2D CNN baselines are considered. The first one is exactly the same
structure as the PointConv network. The other two utilize the Deformable CNN
[9], and CapsNet [40] respectively. Both claimed to be robust to
scale/rotation changes. Every baseline is tuned to the optimal performance
based on the validation accuracy.
Robustness to Scaling The first experiment we conduct is to train with images
of limited known scales and test on larger/smaller objects outside of the
known scales. To construct the training set, all MNIST images are rescaled to
$20\times 20$, $28\times 28$, and $36\times 36$, respectively. The validation
set is built by rescaling the original test images to $24\times 24$ and
$32\times 32$. We trained all baseline models on this new MNIST dataset, and
tuned parameters on the validation set. Test accuracies are measured on images
with $34\times 34$, $38\times 38$, $44\times 44$, $56\times 56$, $72\times
72$, $18\times 18$, $14\times 14$, and $10\times 10$ resolutions, which
creates a discrepancy with training.
Results are shown in Table 2 and 3. PC-2D sampling is generally inferior to
the furthest point subsampling. We note several takeaways:
* •
Naive kNN-based PointConv is less robust to scale changes than conventional
CNN. On most out-of-sample scales it does not generalize as well as the 2D
CNNs.
* •
Cubic polynomials with Sobolev regularization outperforms MLP with any
activation at most scales. Sobolev regularization improves robustness
especially at small scales.
* •
$\epsilon$-ball neighborhood is generally more robust than kNN neighborhood,
especially at scales significantly different from the training.
* •
With appropriate design e.g. cubic polynomial WeightNet and $\epsilon$-ball
neighborhood, PointConv is significantly more robust to scale changes than
traditional 2D CNN.
This builds a case to use PointConv in 2D spaces with an $\epsilon$-ball
neighborhood, and a cubic polynomial WeightNet with Sobolev regularizations.
Especially, the significantly improved robustness w.r.t. 2D CNNs may justify
the use of PointConv even for 2D raster images, if robustness is of concern.
# NBR | Neighborhood | WeightNet architecture | $34\times 34$ | $38\times 38$ | $44\times 44$ | $56\times 56$ | $72\times 72$ | $18\times 18$ | $14\times 14$ | $10\times 10$
---|---|---|---|---|---|---|---|---|---|---
$16$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $95.37\%$ | $94.35\%$ | $\mathbf{94.56}\%$ | $\mathbf{94.01}\%$ | $\mathbf{93.23}\%$ | $\mathbf{95.82}\%$ | $\mathbf{96.15}\%$ | $47.4\%$
$9$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $91.96\%$ | $93.31\%$ | $91.00\%$ | $92.20\%$ | $91.04\%$ | $95.51\%$ | $95.27\%$ | $\mathbf{68.83}\%$
$9$ | kNN | cubic P + Sobolev Reg. | $93.78\%$ | $93.84\%$ | $90.21\%$ | $88.45\%$ | $69.41\%$ | $92.79\%$ | $84.56\%$ | $54.67\%$
$9$ | kNN | cubic P | $90.31\%$ | $94.35\%$ | $89.77\%$ | $90.42\%$ | $68.46\%$ | $91.37\%$ | $81.73\%$ | $24.02\%$
$9$ | kNN | MLP w/ Sine | $91.35\%$ | $92.91\%$ | $84.69\%$ | $79.45\%$ | $66.29\%$ | $86.06\%$ | $74.61\%$ | $31.6\%$
$9$ | kNN | MLP w/ SeLU | $91.77\%$ | $90.52\%$ | $87.5\%$ | $82.66\%$ | $69.5\%$ | $91.74\%$ | $82.9\%$ | $38.18\%$
$9$ | kNN | MLP w/ Leaky ReLU | $95.57\%$ | $88.89\%$ | $80.69\%$ | $49.8\%$ | $37.89\%$ | $94.61\%$ | $81.22\%$ | $26.85\%$
$9$ | kNN | MLP w/ ReLU | $71.38\%$ | $75.04\%$ | $75.41\%$ | $51.33\%$ | $18.85\%$ | $69.23\%$ | $59.54\%$ | $29.88\%$
CNN | $96.30\%$ | $95.68\%$ | $89.19\%$ | $51.23\%$ | $25.77\%$ | $75.6\%$ | $24.25\%$ | $11.2\%$
Deformable CNN | $91.31\%$ | $16.05\%$ | $17.99\%$ | $11.56\%$ | $11.35\%$ | $74.40\%$ | $28.18\%$ | $14.74\%$
CapsNet | $\mathbf{98.53}\%$ | $\mathbf{97.52}\%$ | $93.28\%$ | $43.25\%$ | $16.50\%$ | $80.07\%$ | $21.41\%$ | $8.65\%$
Table 2: MNIST performance comparison across different scales under the setting of farthest subsampling. P is short for polynomials, and NBR is short for neighbor. The first row indicates the resolution of test images. # NBR | Neighborhood | WeightNet architecture | $34\times 34$ | $38\times 38$ | $44\times 44$ | $56\times 56$ | $72\times 72$ | $18\times 18$ | $14\times 14$ | $10\times 10$
---|---|---|---|---|---|---|---|---|---|---
$16$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $95.83\%$ | $96.21\%$ | $\mathbf{97.84}\%$ | $\mathbf{97.92}\%$ | $\mathbf{98.23}\%$ | $90.8\%$ | $64.47\%$ | $21.78\%$
$9$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $94.76\%$ | $95.58\%$ | $96.56\%$ | $96.89\%$ | $97.27\%$ | $\mathbf{93.51}\%$ | $\mathbf{88.20}\%$ | $\mathbf{23.59}\%$
$9$ | kNN | cubic P + Sobolev Reg. | $97.64\%$ | $96.83\%$ | $95.92\%$ | $54.06\%$ | $23.99\%$ | $53.84\%$ | $17.98\%$ | $10.11\%$
$9$ | kNN | cubic P | $97.85\%$ | $97.31\%$ | $95.38\%$ | $42.71\%$ | $13.77\%$ | $14.77\%$ | $12.02\%$ | $7.28\%$
$9$ | kNN | MLP w/ Sine | $95.76\%$ | $94.35\%$ | $94.47\%$ | $47.26\%$ | $19.02\%$ | $76.5\%$ | $30.01\%$ | $10.86\%$
$9$ | kNN | MLP w/ SeLU | $92.03\%$ | $82.49\%$ | $56.90\%$ | $17.09\%$ | $6.01\%$ | $33.78\%$ | $10.51\%$ | $11.59\%$
$9$ | kNN | MLP w/ Leaky ReLU | $66.27\%$ | $24.39\%$ | $13.69\%$ | $10.11\%$ | $10.1\%$ | $65.66\%$ | $16.50\%$ | $8.11\%$
$9$ | kNN | MLP w/ ReLU | $16.54\%$ | $16.66\%$ | $11.35\%$ | $11.35\%$ | $11.35\%$ | $10.63\%$ | $7.83\%$ | $6.43\%$
CNN | $96.30\%$ | $95.68\%$ | $89.19\%$ | $51.23\%$ | $25.77\%$ | $75.6\%$ | $24.25\%$ | $11.2\%$
Deformable CNN | $91.31\%$ | $16.05\%$ | $17.99\%$ | $11.56\%$ | $11.35\%$ | $74.40\%$ | $28.18\%$ | $14.74\%$
CapsNet | $\mathbf{98.53}\%$ | $\mathbf{97.52}\%$ | $93.28\%$ | $43.25\%$ | $16.50\%$ | $80.07\%$ | $21.41\%$ | $8.65\%$
Table 3: MNIST performance comparison across different scales under the setting of PC-2D subsampling. P is short for polynomials, and NBR is short for neighbor. The first row indicates the resolution of test images. # NBR | Neighborhood | WeightNet architecture | $+10\degree$ | $-10\degree$ | $+20\degree$ | $-20\degree$ | $+40\degree$ | $-40\degree$
---|---|---|---|---|---|---|---|---
$12$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $96.93\%$ | $96.63\%$ | $96.2\%$ | $95.86\%$ | $85.27\%$ | $\mathbf{85.51}\%$
$9$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $95.52\%$ | $95.58\%$ | $94.63\%$ | $94.55\%$ | $81.27\%$ | $84.27\%$
$9$ | kNN | cubic P + Sobolev Reg. | $98.11\%$ | $97.68\%$ | $95.96\%$ | $97.2\%$ | $64.79\%$ | $63.71\%$
$9$ | kNN | cubic P | $98.50\%$ | $98.37\%$ | $95.52\%$ | $97.45\%$ | $56.23\%$ | $61.75\%$
$9$ | kNN | MLP w/ Sine | $98.11\%$ | $97.65\%$ | $93.84\%$ | $93.95\%$ | $20.21\%$ | $29.27\%$
$9$ | kNN | MLP w/ SeLU | $97.75\%$ | $97.56\%$ | $43.27\%$ | $94.66\%$ | $10.44\%$ | $11.62\%$
$9$ | kNN | MLP w/ Leaky ReLU | $98.09\%$ | $97.56\%$ | $92.34\%$ | $95.71\%$ | $36.16\%$ | $35.49\%$
$9$ | kNN | MLP w/ ReLU | $98.03\%$ | $97.62\%$ | $92.65\%$ | $95.53\%$ | $37.03\%$ | $34.39\%$
CNN | $96.47\%$ | $96.9\%$ | $95.06\%$ | $95.51\%$ | $75.71\%$ | $74.28\%$
Deformable CNN | $98.21\%$ | $98.60\%$ | $96.59\%$ | $97.60\%$ | $80.82\%$ | $81.01\%$
CapsNet | $\mathbf{99.8}\%$ | $\mathbf{99.9}\%$ | $\mathbf{99.6}\%$ | $\mathbf{99.5}\%$ | $\mathbf{92.5}\%$ | $83.8\%$
Table 4: MNIST performance comparison with different rotations under the
setting of farthest subsampling. P stands for polynomials, and NBR stands for
neighbor. The first row indicates the rotation angles of test images.
Robustness to Rotations The next experiment we conduct is to train with
objects of limited known rotations and test on objects with various rotated
angles. We define counter clockwise as the positive rotation direction. The
training set is constructed through rotating the original $28\times 28$
training images (or point clouds) by $-15^{\circ}$, $0^{\circ}$, and
$+15^{\circ}$ around the center. We rotate the original testing images (or
point clouds) by $-15^{\circ}$ and $+15^{\circ}$ to create the validation set.
All parameters are tuned on the validation dataset. Test accuracies are
measured on the rotation angles of $\pm 10^{\circ}$, $\pm 20^{\circ}$, and
$\pm 40^{\circ}$.
Trends on results shown in Table 4 are similar to those mentioned in the
previous subsection, e.g. $\epsilon$-ball and cubic polynomials outperform kNN
and MLPs. However, here the regular ReLU seems to show the most robustness
among the activation functions, although cubic polynomials still outperform
all the MLP-based WeightNets. With $\epsilon$-ball and cubic polynomials,
PointConv is again significantly more robust than regular CNNs. Interestingly
CapsNet is able to show better rotation robustness than PointConv. Given that
CapsNet is orthogonal to PointConv, it maybe possible to combine them in
future work.
Layer name | Layer description
---|---
Conv1 | $3\times 3$ conv. (or PointConv) 64 w/ BN & ReLU
Subsampling1 | max-pooling with stride $2\times 2$ (or farthest point sampling)
Conv2 | $3\times 3$ conv. (or PointConv) 128 w/ BN & ReLU
Conv3 | $3\times 3$ conv. (or PointConv) 128 w/ BN & ReLU
Subsampling2 | max-pooling with stride $2\times 2$ (or farthest point sampling)
Conv4 | $3\times 3$ conv. (or PointConv) 256 w/ BN & ReLU
Conv5 | $3\times 3$ conv. (or PointConv) 256 w/ BN & ReLU
Subsampling3 | max-pooling with stride $2\times 2$ (or farthest point sampling)
Conv6 | $3\times 3$ conv. (or PointConv) 512 w/ BN & ReLU
Conv7 | $3\times 3$ conv. (or PointConv) 512 w/ BN & ReLU
Pooling | global average pooling
FC-10 | fully connected layer
Softmax layer
Table 5: The general network architecture for CIFAR-10 experiments. BN stands
for batch normalization. Every BN layer is followed by a ReLU layer.
### 4.2 CIFAR-10
In CIFAR-10 [22], there are $60,000$ RGB images in $10$ classes in the
training set, and $10,000$ images in the test set. The original size of each
image is $32\times 32$. The general architecture of all baselines are shown in
Table 5. We adopt the same hyperparameters and preprocessing procedures as
detailed in section 4.1, besides the Sobolev norm regularizer. The $\epsilon$
is fixed to $0.05$ for Conv1 in Table 5, $0.1$ for Conv2 and Conv3, $0.2$ for
Conv4 and Conv5, and $0.5$ for Conv6 as well as Conv7. The numbers of output
points from FPS for Subsampling1, Subsampling2, and Subsampling3 layer in
Table 5 are $256$, $64$, and $16$, respectively. The coefficient of the
Sobolev norm regularization is set to one of $\\{10^{-6},10^{-5},10^{-4}\\}$
for our proposed approach. We does not adopt PC-2D subsampling since it is
experimentally proven not robust in 4.1.
Two 2D CNN baselines are considered. The first one is the traditional CNN as
shown in Table 5, and the other one is Deformable CNN [9] which has the exact
architecture except for that the last four convolutional layer are
deformable.We did not including CapsNet [40] in this experiment, since it has
no pooling layers and performs poorly on this dataset.
Robustness to Scaling The first experiment tests robustness with respect to
scaling. The original training set is augmented to three different
resolutions, which are $24\times 24$, $32\times 32$, and $40\times 40$,
respectively. The validation resolutions are $28\times 28$, and $36\times 36$.
We trained all baseline models on this new CIFAR-10 dataset, and tuned
hyperparameters on the validation set.
Results are shown in Table 6. We note several takeaways:
* •
Naive kNN-based PointConv is less robust to scale changes than conventional
CNN. On most out-of-sample scales it does not generalize as well as the 2D
CNNs. Even for the resolution $32\times 32$, the generalization is less well
than traditional CNNs. This is interesting since we have verified that with
the same architecture PointConv is able to match CNN performance if trained
only under a single resolution, which might show that further work is needed
for PointConv to be robust to mixing resolutions in training.
* •
Cubic polynomials outperform MLP with any activation at most scales. Sobolev
regularization does not universally improves the robustness against scaling on
this dataset.
* •
$\epsilon$-ball neighborhood is generally significantly more robust than kNN
neighborhood when testing scales are significantly different from the
training.
* •
With appropriate design e.g. cubic polynomial WeightNet and an $\epsilon$-ball
neighborhood, PointConv is significantly more robust than traditional 2D CNN
when the testings scales that are significantly larger/smaller than training
scales.
# NBR | Neighborhood | WeightNet architecture | $32\times 32$ | $48\times 48$ | $60\times 60$ | $76\times 76$ | $88\times 88$ | $22\times 22$ | $18\times 18$ | $16\times 16$
---|---|---|---|---|---|---|---|---|---|---
$16$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $73.43\%$ | $70.24\%$ | $\mathbf{68.56}\%$ | $\mathbf{65.7}\%$ | $\mathbf{65.78}\%$ | $66.15\%$ | $\mathbf{67.12}\%$ | $\mathbf{71.89}\%$
$9$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $70.56\%$ | $68.00\%$ | $67.15\%$ | $64.41\%$ | $65.31\%$ | $59.89\%$ | $63.36\%$ | $66.41\%$
$9$ | kNN | cubic P + Sobolev Reg. | $78.59\%$ | $61.67\%$ | $46.32\%$ | $32.48\%$ | $24.91\%$ | $64.67\%$ | $52.50\%$ | $59.65\%$
$9$ | kNN | cubic P | $78.09\%$ | $61.87\%$ | $47.12\%$ | $30.91\%$ | $27.63\%$ | $63.17\%$ | $54.48\%$ | $59.65\%$
$9$ | kNN | MLP w/ Sine | $71.29\%$ | $39.62\%$ | $30.58\%$ | $26.69\%$ | $25.17\%$ | $46.68\%$ | $21.99\%$ | $23.81\%$
$9$ | kNN | MLP w/ SeLU | $72.04\%$ | $47.27\%$ | $37.48\%$ | $34.81\%$ | $34.54\%$ | $35.21\%$ | $18.51\%$ | $23.80\%$
$9$ | kNN | MLP w/ Leaky ReLU | $76.06\%$ | $16.85\%$ | $9.85\%$ | $11.37\%$ | $11.21\%$ | $41.56\%$ | $30.11\%$ | $27.47\%$
$9$ | kNN | MLP w/ ReLU | $69.58\%$ | $20.81\%$ | $15.00\%$ | $12.01\%$ | $12.36\%$ | $38.89\%$ | $27.17\%$ | $25.32\%$
CNN | $\mathbf{88.47}\%$ | $\mathbf{79.21}\%$ | $59.21\%$ | $38.43\%$ | $27.13\%$ | $\mathbf{69.11}\%$ | $44.99\%$ | $39.80\%$
Deformable CNN | $82.82\%$ | $72.03\%$ | $52.07\%$ | $31.91\%$ | $25.68\%$ | $60.20\%$ | $51.20\%$ | $45.67\%$
Table 6: CIFAR-10 performance comparison across different scales under the setting of farthest subsampling. P is short for polynomials, and NBR is short for neighbor. The first row indicates the resolution of test images. # NBR | Neighborhood | WeightNet architecture | $0\degree$ | $+10\degree$ | $-10\degree$ | $+20\degree$ | $-20\degree$ | $+40\degree$ | $-40\degree$
---|---|---|---|---|---|---|---|---|---
$12$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $71.26\%$ | $67.63\%$ | $67.55\%$ | $64.58\%$ | $65.36\%$ | $40.13\%$ | $42.64\%$
$9$ | $\epsilon$-ball | cubic P + Sobolev Reg. | $69.72\%$ | $66.24\%$ | $66.56\%$ | $63.13\%$ | $63.71\%$ | $39.42\%$ | $39.42\%$
$9$ | kNN | cubic P + Sobolev Reg. | $75.57\%$ | $62.95\%$ | $60.12\%$ | $59.96\%$ | $54.35\%$ | $42.94\%$ | $39.13\%$
$9$ | kNN | cubic P | $75.94\%$ | $62.27\%$ | $59.12\%$ | $57.18\%$ | $54.68\%$ | $38.85\%$ | $37.61\%$
$9$ | kNN | MLP w/ Sine | $68.32\%$ | $45.11\%$ | $44.11\%$ | $35.92\%$ | $35.52\%$ | $15.35\%$ | $16.34\%$
$9$ | kNN | MLP w/ SeLU | $69.04\%$ | $43.81\%$ | $41.26\%$ | $40.83\%$ | $37.97\%$ | $21.53\%$ | $18.28\%$
$9$ | kNN | MLP w/ Leaky ReLU | $78.01\%$ | $65.68\%$ | $64.84\%$ | $59.22\%$ | $57.85\%$ | $34.67\%$ | $34.14\%$
$9$ | kNN | MLP w/ ReLU | $76.79\%$ | $64.57\%$ | $62.25\%$ | $57.43\%$ | $55.21\%$ | $31.07\%$ | $33.98\%$
CNN | $\mathbf{86.61}\%$ | $83.72\%$ | $\mathbf{84.48}\%$ | $\mathbf{78.03}\%$ | $78.25\%$ | $45.71\%$ | $46.14\%$
Deformable CNN | $86.53\%$ | $\mathbf{83.74}\%$ | $83.96\%$ | $77.69\%$ | $\mathbf{78.31}\%$ | $\mathbf{47.21}\%$ | $\mathbf{47.72}\%$
Table 7: CIFAR-10 performance comparison with different rotations under the
setting of farthest subsampling. P stands for polynomials, and NBR stands for
neighbor. The first row indicates the rotation angles of test images.
Robustness to Rotations The next experiment we conduct is to train baselines
with images with a few predefined rotations, and test them on images with
various rotated angles. The counter clockwise direction is defined as the
positive direction. To construct the training set, we rotate the original
$32\times 32$ training images (or point clouds) by $-15^{\circ}$, $0^{\circ}$,
and $+15^{\circ}$ around the center. The validation set is created through
rotating the original testing images (or point clouds) by $-15^{\circ}$ and
$+15^{\circ}$. All parameters are tuned on the validation dataset. Test
accuracies are measured on the rotation angles of $\pm 10^{\circ}$, $\pm
20^{\circ}$, and $\pm 40^{\circ}$.
The corresponding results are shown in Table 7. We observe that
$\epsilon$-ball still outperforms kNN in all nonzero rotations but
underperform it at no rotation. However, Leaky ReLU is overall comparable with
cubic polynomial with Sobolev regularization as the best approach, and SeLU
performed significantly worse. Besides, PointConv is generally significantly
less robust as regular or deformable CNNs in this case, showing that more work
needs to be done in terms of robustness to rotations in 2D in more complex
datasets.
# layers | MLP input | activation | NBR setting | rotation augmentation | $100k$ | $60k$ | $40k$ | $20k$ | $10k$
---|---|---|---|---|---|---|---|---|---
16 | VI + $(x,y,z)$ | ReLU | KNN | Y | $\mathbf{68.2}$ | $\mathbf{64.4}$ | $\mathbf{63.0}$ | $\mathbf{57.6}$ | $45.4$
16 | VI | ReLU | KNN | Y | $63.3$ | $60.7$ | $59.7$ | ${55.0}$ | $44.7$
16 | VI | SeLU | KNN | Y | ${63.7}$ | ${61.5}$ | $57.7$ | $53.1$ | $40.2$
16 | VI | Cubic First Block | KNN | Y | $63.5$ | $61.4$ | ${60.2}$ | $54.0$ | $42.0$
| | \+ ReLU in others | | | | | | |
16 | $(x,y,z)$ | ReLU | KNN | Y | $61.7$ | $58.7$ | $53.4$ | $34.6$ | $17.8$
16 | VI | ReLU | KNN | N | $60.4$ | $58.6$ | $57.8$ | $54.2$ | $46.3$
16 | $(x,y,z)$ | ReLU | KNN | N | $56.4$ | $54.0$ | $51.2$ | $40.6$ | $27.4$
16 | VI | ReLU | $\epsilon$-ball | Y | $61.61$ | $58.6$ | $58.0$ | $52.3$ | $40.6$
16 | $(x,y,z)$ | ReLU | $\epsilon$-ball | Y | $48.9$ | $43.3$ | $39.7$ | $30.6$ | $20.7$
16 | surface normal | ReLU | KNN | Y | $60.2$ | $57.5$ | $56.8$ | $54.8$ | $50.9$
| \+ $(x,y,z)$ | | | | | | | |
16 | surface normal | ReLU | KNN | Y | $53.1$ | $50.6$ | $50.2$ | $47.6$ | $43.3$
4 | VI + $(x,y,z)$ | ReLU | KNN | Y | $64.5$ | $61.3$ | $60.6$ | $57.3$ | $\mathbf{51.2}$
4 | VI | ReLU | KNN | Y | $61.0$ | $58.8$ | $57.5$ | $50.8$ | $39.4$
4 | $(x,y,z)$ | ReLU | KNN | Y | $55.3$ | $53.3$ | $47.0$ | $30.7$ | $16.1$
4 | VI | ReLU | $\epsilon$-ball | Y | $59.2$ | $57.5$ | $55.12$ | $44.8$ | $31.1$
Table 8: Performance results (mIoU,%) for the ScanNet dataset. The first
column shows the configurations of the approach, and the top row contains the
number of subsampled points. The default number of neighbors is $8$, and the
default activation for weight functions is ReLU.
### 4.3 ScanNet
We conduct 3D semantic scene segmentation on the ScanNet v2 [8] dataset. We
use the official split with $1,201$ scenes for training and $312$ for
validation.
We implemented 2 PointConv architectures. One is the 4-layer network in [60],
the second one is the 16-layer PointConv network that achieved $66.6\%$ on the
ScanNet testing set. Network architectures are provided by the authors. Our
main results are reported on the ScanNet validation set as the benchmark
organizers do not allow ablation studies on the testing set.
We adopt regular subsampling [48] for feature encoding layers with grid sizes
$0.05$, $0.1$, $0.2$, and $0.4$ (in meter). $\epsilon$ is set to be
$\frac{1}{1.3}$ of the grid size for the corresponding subsampling layer. We
enumerated $\epsilon$ in
$\\{\frac{1}{1.0},\frac{1}{1.1},\frac{1}{1.2},\frac{1}{1.3},\frac{1}{1.4},\frac{1}{1.5}\\}$,
and it turned out the network is not sensitive to those choices (see
supplementary for experiments). The surface normal for a subsampled point is
computed through averaging all surface normals of its corresponding grid.
During training, we randomly subsample $100,000$ points from each point cloud
for both the training and validation sets, and the mini-batch size is set to
be $3$. The learning rate is set to $10^{-3}$, with a decay multiplier of
$\frac{1}{2}$ every $40$ epochs. Moreover, the rotation augmentation is
applied by randomly rotating every mini-batch with an arbitrary angle in
$[0,2\pi)$, as in [60].
To study the robustness to scales for all baselines, we re-subsample each
validation point cloud to less than $100k$ — $\\{60k,40k,20,10k\\}$. This is
equivalent to downsampling the image in the 2D space as it increases the size
of KNN neighborhoods with a fixed $K$. Also, each sub-sampled point cloud is
further rotated with $4$ different predefined angles around $z$-axis —
$\\{0\degree,90\degree,180\degree,270\degree\\}$. Such operations could
significantly change both local scales and rotations. We also evaluate the
performance when the rotation augmentation is not applied during training. We
found performance variation between different rotation angles to be less than
$1\%$ (see supplementary), hence the mIoUs averaged from all angles are
reported. Experiments are performed with the proposed VI descriptor, as well
as configurations that are promising from 2D experiments: $\epsilon$-ball,
cubic polynomial WeightNet, and SeLU activation.
Results are shown in Table 8, which shows that the proposed VI descriptor
significantly improved the performance as well as robustness under every
setting. Especially, it is significantly more robust to input downsampling
than the $(x,y,z)$ coordinates as input. For example, at $10k$ testing points
(reflecting $10x$ downsampling from the training), the VI descriptors still
maintain a $44.7\%$ accuracy while the $(x,y,z)$ coordinates version has its
performance dropped to $17.8\%$, marking an improvement of $251\%$. Besides,
with VI descriptor, the need to use rotation augmentation reduced
significantly. Interestingly, rotation augmentation still improved performance
by $2\%$. We believe the main reason is that the input feature of our
framework consists of both $(x,y,z)$ coordinates and RGB colors, and thus
rotation augmentation creates more $(x,y,z)$ coordinates variations for the
training, and hence improved performance. We further conduct ablation studies
by replacing VI descriptors with surface normals. The mIoU drops by
$8\%-10\%$, which indicates that VI is a better representation of local
geometry than the commonly used surface normal.
Finally, when we combined the VI features with $(x,y,z)$ inputs, it generated
the best performance of all – $68.2\%$ on the original validation set, and
better on almost all subsampled scenarios. This shows that a combination of
scale-invariant, rotation-invariant and non-invariant features is beneficial,
potentially letting the network choose the invariance it requires.
Furthermore, on the test set, we achieved comparable mIoU with KPConv [49],
which is the current state-of-the-art among point-based approaches (Table 9).
Note that PointConv [60] has significantly less parameters than KPConv [49],
and KNN used in PointConv is significantly more efficient than the
$\epsilon$-ball in KPConv.
Otherwise, none of the other tested choices were helpful, including
$\epsilon$-ball, SeLU or cubic polynomial. Our takeaway is that the proposed
VI descriptor is very powerful and improved robustness significantly w.r.t.
neighborhood sizes, hence, none of the other improvements is needed.
From our experience, it is very difficult to find a good set of parameters for
$\epsilon$-ball in $3D$. It also slows down the algorithm because the non-
uniform neighborhood size is not easily amenable for tensor computation. Hence
not needing to use it is a significant bonus. Nevertheless, $\epsilon$-ball
neighborhoods are still very useful in $2D$ settings, where locating them is
much easier and the VI descriptor is not applicable.
Method | mIoU(%)
---|---
PointNet++ [37] | $33.9$
SPLATNet [45] | $39.3$
TangentConv [47] | $40.9$
PointCNN [27] | $45.8$
PointASNL [65] | $63.0$
PointConv [60] | $66.6$
KPConv [49] | $\mathbf{68.4}$
VI-PointConv (ours) | $67.6$
Table 9: Semantic Scene Segmentation results for point-based approaches on the ScanNet test set Method | mIoUs(%) | road | sidewalk | parking | other-ground | building | car | truck | bicycle | motorcycle | other-vehicle | vegetation | trunk | terrain | person | bicyclist | motorcyclist | fence | pole | traffic-sign
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
PointNet[35] | $14.6$ | $61.6$ | $35.7$ | $15.8$ | $1.4$ | $41.4$ | $46.3$ | $0.1$ | $1.3$ | $0.3$ | $0.8$ | $31.0$ | $4.6$ | $17.6$ | $0.2$ | $0.2$ | $0.0$ | $12.9$ | $2.4$ | $3.7$
SPG[23] | $17.4$ | $45.0$ | $28.5$ | $0.6$ | $0.6$ | $64.3$ | $49.3$ | $0.1$ | $0.2$ | $0.2$ | $0.8$ | $48.9$ | $27.2$ | $24.6$ | $0.3$ | $2.7$ | $0.1$ | $20.8$ | $15.9$ | $0.8$
SPLATNet[45] | $18.4$ | $64.6$ | $39.1$ | $0.4$ | $0.0$ | $58.3$ | $58.2$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | $71.1$ | $9.9$ | $19.3$ | $0.0$ | $0.0$ | $0.0$ | $23.1$ | $5.6$ | $0.0$
PointNet++[37] | $20.1$ | $72.0$ | $41.8$ | $18.7$ | $5.6$ | $62.3$ | $53.7$ | $0.9$ | $1.9$ | $0.2$ | $0.2$ | $46.5$ | $13.8$ | $30.0$ | $0.9$ | $1.0$ | $0.0$ | $16.9$ | $6.0$ | $8.9$
TangentConv[47] | $40.9$ | $83.9$ | $63.9$ | $33.4$ | $15.4$ | $83.4$ | $90.8$ | $15.2$ | $2.7$ | $16.5$ | $12.1$ | $79.5$ | $49.3$ | $58.1$ | $23.0$ | $28.4$ | $8.1$ | $49.0$ | $35.8$ | $28.5$
PointConv[60] | $53.0$ | $86.2$ | $68.6$ | $57.7$ | $16.0$ | $89.9$ | $94.2$ | $30.2$ | $29.5$ | $33.9$ | $30.5$ | $78.9$ | $60.8$ | $63.7$ | $48.8$ | $45.7$ | $20.4$ | $59.9$ | $53.4$ | $38.6$
RandLA-Net[15] | $53.9$ | $\mathbf{90.7}$ | $\mathbf{73.7}$ | $60.3$ | $20.4$ | $86.9$ | $94.2$ | $40.1$ | $26.0$ | $25.8$ | $38.9$ | $81.4$ | $61.3$ | $66.8$ | $49.2$ | $48.2$ | $7.2$ | $56.3$ | $49.2$ | $47.7$
KPConv[49] | $58.8$ | $88.8$ | $72.7$ | $61.3$ | $31.6$ | $90.5$ | $\mathbf{96.0}$ | $33.4$ | $30.2$ | $\mathbf{42.5}$ | $44.3$ | $\mathbf{84.8}$ | $\mathbf{69.2}$ | $\mathbf{69.1}$ | $\mathbf{61.5}$ | $\mathbf{61.6}$ | $11.8$ | $64.2$ | $56.5$ | $47.4$
VI-PConv(ours) | $\mathbf{59.6}$ | $88.8$ | $72.5$ | $\mathbf{63.5}$ | $\mathbf{32.7}$ | $\mathbf{91.4}$ | $95.9$ | $\mathbf{41.8}$ | $\mathbf{38.6}$ | $35.0$ | $\mathbf{45.7}$ | $83.9$ | $68.0$ | $66.9$ | $51.2$ | $50.1$ | $\mathbf{27.6}$ | $\mathbf{66.6}$ | $\mathbf{57.4}$ | $\mathbf{54.8}$
Table 10: Semantic Scene Segmentation results for point-based approaches on
the SemanticKITTI test set
### 4.4 SemanticKITTI
We also evaluate the semantic segmentation performance on SemanticKITTI [2]
(single scan), which consists of $43,552$ point clouds sampled from $22$
sequences in driving scenes. Each point cloud contains $10-13k$ points,
collected by a single Velodyne HDL-64E laser scanner, spanning up to
$160\times 160\times 20$ meters in 3D space. The officially training set
includes $19,130$ scans (sequences $00-07$ and $09-10$), and there are $4,071$
scans (sequence $08$) for validation. For each 3D point, only $(x,y,z)$
coordinate is given without any color information. It is a challenging dataset
because faraway points are sparser in LIDAR scans.
We adopt the exact same $16$-layer architecture as showed at the first row of
Table 8, together with the mini-batch size of 16. The initial learning rate is
$10^{-3}$, and it is decayed by half every $6$ epochs. We do not integrate
with any subsampling preprocessing. As reported in Table 10, we achieve the
state-of-the-art semantic segmentation performance among point-based
baselines, improving by $0.8\%$ over KPConv and $6.6\%$ over standard
PointConv, which demonstrate the effectiveness of VI descriptor for PointConv
[60].
## 5 Conclusion
This paper empirically studies several strategies to improve the robustness
for point cloud convolution with PointConv [60]. We have found two
combinations to be effective. In 2D images, using an $\epsilon$-ball
neighborhood with cubic polynomial weight functions achieved the highest
robustness, and outperformed 2D CNNs for MNIST dataset. In 3D images, our
novel viewpoint-invariant descriptor, when used instead of, or in combination
with the 3D coordinates, significantly improved the performance and robustness
of PointConv networks and achieved state-of-the-art. Our results show that kNN
is still a viable neighborhood choice if the input location features are
robust to neighborhood sizes.
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|
An improvement of a saddle point theorem and some of its applications
BIAGIO RICCERI
Dedicated to the memory of Professor Wataru Takahashi
Abstract. In this paper, we establish an improved version of a saddle point
theorem ([4]) removing a weak lower semicontinuity assumption at all. We then
revisit some of the applications of that theorem in the light of such an
improvement. For instance, we obtain the following very general result of
local nature: Let $(H,\langle\cdot,\cdot\rangle)$ be a real Hilbert space and
$\Phi:B_{\rho}\to H$ a $C^{1,1}$ function, with $\Phi(0)\neq 0$. Then, for
each $r>0$ small enough, there exist only two points points $x^{*},u^{*}\in
S_{r}$, such that
$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$
for all $x\in B_{r}\setminus\\{x^{*}\\}$,
$\|\Phi(u^{*})-u^{*}\|=\hbox{\rm dist}(\Phi(u^{*}),B_{r})$
and
$\|\Phi(x)-u^{*}\|<\|\Phi(x)-x\|$
for all $x\in B_{r}\setminus\\{u^{*}\\}$, where
$B_{r}=\\{x\in H:\|x\|\leq r\\}$
and
$S_{r}=\\{x\in H:\|x\|=r\\}\ .$
Keywords. Saddle point; Hilbert space; ball; $C^{1,1}$ function; variational
inequality; best approximation point.
2010 Mathematics Subject Classification. 41A50, 41A52, 47J20, 49J35, 49J40.
In the sequel, $(H,\langle\cdot,\cdot\rangle)$ is a real Hilbert space. For
each $r>0$, set
$B_{r}=\\{x\in H:\|x\|\leq r\\}$
and
$S_{r}=\\{x\in H:\|x\|=r\\}\ .$
The aim of this paper is to establish the following result jointly with two
meaningful applications of it:
THEOREM 1. - Let $Y$ be a non-empty closed convex set in a Hausdorff real
topological vector space, let $\rho>0$ and let $J:B_{\rho}\times Y\to{\bf R}$
be a function satisfying the following conditions:
$(a_{1})$ for each $y\in Y$, the function $J(\cdot,y)$ is $C^{1}$ and
$J^{\prime}_{x}(\cdot,y)$ is Lipschitzian with constant $L$ (independent of
$y$) ;
$(a_{2})$ $J(x,\cdot)$ is upper semicontinuous and concave for all $x\in
B_{\rho}$ and $J(x_{0},\cdot)$ is sup-compact for some $x_{0}\in B_{\rho}$;
$(a_{3})$ $\delta:=\inf_{y\in Y}\|J^{\prime}_{x}(0,y)\|>0\ .$
Then, for each
$r\in\left]0,\min\left\\{\rho,{{\delta}\over{2L}}\right\\}\right]$ and for
each non-empty closed convex $T\subseteq Y$, there exist $x^{*}\in S_{r}$ and
$y^{*}\in T$ such that
$J(x^{*},y)\leq J(x^{*},y^{*})<J(x,y^{*})$
for all $x\in B_{r}\setminus\\{x^{*}\\}$, $y\in T$ .
PROOF. Fix $r\in\left]0,\min\left\\{\rho,{{\delta}\over{2L}}\right\\}\right]$
and a non-empty closed convex $T\subseteq Y$. Consider the function
$\varphi:B_{r}\times T\to{\bf R}$ defined by
$\varphi(x,y)={{L}\over{2}}\|x\|^{2}+J(x,y)$
for all $(x,y)\in B_{r}\times T$. Notice that, for each $y\in T$, the function
$\varphi(\cdot,y)$ is continuous and convex in $B_{r}$ (see the proof of
Corollary 2.3 of [3]). Consequently, if we consider $B_{r}$ endowed with the
relative weak topology, the function $\varphi$ satisfies the assumptions of a
classical minimax theorem ([1], Theorem 2) from which we infer
$\sup_{T}\inf_{B_{r}}\varphi=\inf_{B_{r}}\sup_{T}\varphi\ .$ $None$
The function $x\to\sup_{y\in T}\varphi(x,y)$ (resp. $y\to\inf_{x\in
B_{r}}\varphi(x,y)$) is weakly lower semicontinuous (resp. sup-compact).
Therefore, there exist $x^{*}\in B_{r}$ and $y^{*}\in T$ such that
$\sup_{y\in T}\varphi(x^{*},y)=\inf_{x\in B_{r}}\sup_{y\in T}\varphi(x,y)\ ,$
$\inf_{x\in B_{r}}\varphi(x,y^{*})=\sup_{y\in T}\inf_{x\in B_{r}}\varphi(x,y)\
.$
So, in view of $(1)$, we obtain
$\varphi(x^{*},y)\leq\varphi(x^{*},y^{*})\leq\varphi(x,y^{*})$ $None$
for all $x\in B_{r}$, $y\in T$. Notice that the equation
$J^{\prime}_{x}(x,y^{*})+Lx=0$
has no solution in the interior of $B_{r}$. Indeed, let $\tilde{x}\in
B_{\rho}$ be such that
$J^{\prime}_{x}(\tilde{x},y^{*})+L\tilde{x}=0\ .$
Then, in view of $(a_{1})$, we have
$\|L\tilde{x}+J^{\prime}_{x}(0,y^{*})\|\leq\|L\tilde{x}\|\ .$
In turn, using the Cauchy-Schwarz inequality, this readily implies that
$\|\tilde{x}\|\geq{{\|J^{\prime}_{x}(0,y^{*})\|}\over{2L}}\geq{{\delta}\over{2L}}\geq
r\ .$
From this remark, we infer that the set of all global minima of the function
$\varphi(\cdot,y^{*})$ is contained in $S_{r}$ and so, being convex, it
reduces to $x^{*}$ (recall that $X$ is a Hilbert space), in view of $(2)$.
Therefore, for every $x\in B_{r}\setminus\\{x^{*}\\}$, $y\in T$, from $(2)$ we
obtain
${{1}\over{2}}\|x^{*}\|^{2}+J(x^{*},y)\leq{{1}\over{2}}\|x^{*}\|^{2}+J(x^{*},y^{*})<{{1}\over{2}}\|x\|^{2}+J(x,y^{*})\leq{{1}\over{2}}\|x^{*}\|^{2}+J(x,y^{*})$
and so
$J(x^{*},y)\leq J(x^{*},y^{*})<J(x,y^{*})\ .$
The proof is complete. $\bigtriangleup$
REMARK 1. - Theorem 1 was obtained in [4] ([4], Theorem 2.1) under the
following additional assumption: for each $y\in Y$, the function $J(\cdot,y)$
is weakly lower semicontinuous. This was due to the fact that, instead of
applying the classical minimax theorem in [1] to $\varphi$ (as we did above),
we applied Theorem 1.2 of [2] to $J$.
We can now revisit two applications of Theorem 1. The first one concerns
variational inequalities.
THEOREM 2. - Let $\rho>0$ and let $\Phi:B_{\rho}\to H$ be a $C^{1}$ function
whose derivative is Lipschitzian with constant $\gamma$. Set
$\theta:=\sup_{x\in B_{\rho}}\|\Phi^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$M:=2(\theta+\rho\gamma)$
and assume also that
$\sigma:=\inf_{y\in
B_{\rho}}\sup_{\|u\|=1}|\langle\Phi(0),u\rangle-\langle\Phi^{\prime}(0)(u),y\rangle|>0\
.$
Then, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{2M}}\right\\}\right]$, there
exists a unique $x^{*}\in S_{r}$ such that
$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$
for all $x\in B_{r}\setminus\\{x^{*}\\}$.
PROOF. Consider the function $J:B_{\rho}\times B_{\rho}\to{\bf R}$ defined by
$J(x,y)=\langle\Phi(x),x-y\rangle$
for all $x,y\in B_{\rho}$. Of course, for each $y\in B_{\rho}$, the function
$J(\cdot,y)$ is $C^{1}$ and one has
$\langle
J^{\prime}_{x}(x,y),u\rangle=\langle\Phi^{\prime}(x)(u),x-y\rangle+\langle\Phi(x),u\rangle$
for all $x\in B_{\rho},u\in H$. Fix $x,v\in B_{\rho}$ and $u\in S_{1}$. We
then have
$|\langle J^{\prime}_{x}(x,y),u\rangle-\langle
J^{\prime}_{x}(v,y),u\rangle|=|\langle\Phi(x)-\Phi(v),u\rangle+\langle\Phi^{\prime}(x)(u),x-y\rangle-\langle\Phi^{\prime}(v)(u),v-y\rangle|$
$\leq\|\Phi(x)-\Phi(v)\|+|\Phi^{\prime}(x)(u)-\Phi^{\prime}(v)(u),v-y\rangle+\langle\Phi^{\prime}(x)(u),x-v\rangle|$
$\leq\theta\|x-v\|+2\rho\|\Phi^{\prime}(x)-\Phi^{\prime}(v)\|_{{\cal
L}(H)}+\theta\|x-v\|$ $\leq 2(\theta+\rho\gamma)\|x-v\|\ .$
Hence, the function $J^{\prime}_{x}(\cdot,y)$ is Lipschitzian with constant
$M$. At this point, we can apply Theorem 1 taking $Y=B_{\rho}$ with the weak
topology. Therefore, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{2M}}\right\\}\right]$, there
exist $x^{*}\in S_{r}$ and $y^{*}\in B_{r}$ such that
$\langle\Phi(x^{*}),x^{*}-y\rangle\leq\langle\Phi(x^{*}),x^{*}-y^{*}\rangle<\langle\Phi(x),x-y^{*}\rangle$
$None$
for all $x,y\in B_{r}$, with $x\neq x^{*}$. Notice that $\Phi(x^{*})\neq 0$.
Indeed, if $\Phi(x^{*})=0$, we would have
$\|\Phi(0)\|=\|\Phi(0)-\Phi(x^{*})\|\leq\theta r$
and hence, since $\sigma\leq\|\Phi(0)\|$, it would follow that
$r\leq{{\|\Phi(0)\|}\over{2M}}<{{\|\Phi(0)\|}\over{\theta}}\leq r\ .$
From the first inequality in $(3)$, taking $y=x^{*}$, we get
$0\leq\langle\Phi(x^{*}),x^{*}-y^{*}\rangle$. So, in view of the strict
inequality, we infer that $x^{*}=y^{*}$ (since, otherwise, we could take
$x=y^{*}$, obtaing a contradiction). Thus, $(3)$ actually reads
$\langle\Phi(x^{*}),x^{*}-y\rangle\leq 0<\langle\Phi(x),x-x^{*}\rangle$
for all $x,y\in B_{r}$, with $x\neq x^{*}$. In particular, we infer that
$x^{*}$ is the unique global minimum in $B_{r}$ of the linear functional
$y\to\langle\Phi(x^{*}),y\rangle$. Hence
$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$
for all $x\in B_{r}\setminus\\{x^{*}\\}$. Finally, to show the uniqueness of
$x^{*}$, argue by contradiction, supposing that there is another $\tilde{x}\in
S_{r}$, with $\tilde{x}\neq x^{*}$, such that
$\max\\{\langle\Phi(\tilde{x}),\tilde{x}-x\rangle,\langle\Phi(x)),\tilde{x}-x\rangle\\}<0$
for all $x\in B_{r}\setminus\\{\tilde{x}\\}$. So, we would have at the same
time
$\langle\Phi(\tilde{x}),\tilde{x}-x^{*}\rangle<0$
and
$\langle\Phi(\tilde{x}),x^{*}-\tilde{x}\rangle<0\ ,$
an absurd, and the proof is complete. $\bigtriangleup$
REMARK 2. - Theorem 2 was obtained in [4] ([4], Theorem 2.2) under the
following additional assumption: for each $y\in B_{\rho}$, the function
$x\to\langle\Phi(x),x-y\rangle$ is weakly lower semicontinuous.
A remarkable corollary of Theorem 2 is as follows:
THEOREM 3. - Let $\rho>0$ and let $\Phi:B_{\rho}\to H$ be a $C^{1}$ function
with Lipschitzian derivative.
Then, the following assertions are equivalent:
$(i)$ for each $r>0$ small enough, there exists a unique $x^{*}\in S_{r}$ such
that
$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$
for all $x\in B_{r}\setminus\\{x^{*}\\}$ ;
$(ii)$ $\Phi(0)\neq 0$ .
PROOF. The implication $(ii)\to(i)$ is obvious. So, assume that $(i)$ holds.
Notice that the function
$y\to\sup_{\|u\|=1}|\langle\Phi(0),u\rangle-\langle\Phi^{\prime}(0)(u),y\rangle|$
is continuous in $H$ and takes the value $\|\Phi(0)\|>0$ at $0$. Therefore,
for a suitable $r^{*}\in]0,\rho]$, we have
$\inf_{y\in
B_{r^{*}}}\sup_{\|u\|=1}\langle\Phi(0),u\rangle-\langle\Phi^{\prime}(0)(u),y\rangle|>0\
.$
Now, we can apply Theorem 2 to the restriction of the function $\Phi$ to
$B_{r^{*}}$, and $(i)$ follows. $\bigtriangleup$
Also, it is worth noticing the following further corollary of Theorem 2:
THEOREM 4. - Let $\rho>0$ and let $\Psi:B_{\rho}\to H$ be a $C^{1}$ function
whose derivative vanishes at $0$ and is Lipschitzian with constant
$\gamma_{1}$. Set
$\theta_{1}:=\sup_{x\in B_{\rho}}\|\Psi^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$M_{1}:=2(\theta_{1}+\rho\gamma_{1})$
and let $w\in H$ satisfy
$\|w-\Psi(0)\|\geq 2M_{1}\rho\ .$ $None$
Then, for each $r\in]0,\rho]$, there exists a unique $x^{*}\in S_{r}$ such
that
$\max\\{\langle\Psi(x^{*})-w,x^{*}-y\rangle,\langle\Psi(y)-w,x^{*}-y\rangle\\}<0$
for all $y\in B_{r}\setminus\\{x^{*}\\}$.
PROOF. Set $\Phi:=\Psi-w$. Apply Theorem 2 to $\Phi$. Since
$\Phi^{\prime}=\Psi^{\prime}$, we have $M=M_{1}$. Since $\Phi^{\prime}(0)=0$,
we have $\sigma=\|\Phi(0)\|$ and so, in view of $(4)$,
$\rho\leq{{\sigma}\over{2M}}$
and the conclusion follows. $\bigtriangleup$
The second application of Theorem 1 is as follows:
THEOREM 5. - Let $Y\subseteq H$ be a closed bounded convex set, let $\rho>0$
and let $f:B_{\rho}\to H$ be a $C^{1}$ function whose derivative is
Lipschitzian with constant $\gamma$. Moreover, let $\eta$ be the Lipschitz
constant of the function $x\to x-f(x)$, set
$\theta:=\sup_{x\in B_{\rho}}\|f^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$L:=2\left(\eta+\theta+\gamma\left(\rho+\sup_{y\in Y}\|y\|\right)\right)$
and assume that
$\sigma:=\inf_{y\in Y}\sup_{\|u\|=1}|\langle f^{\prime}(0)(u),y\rangle-\langle
f(0),u\rangle|>0\ .$
Then, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$ and for each
non-empty closed convex set $T\subseteq Y$, there exist $x^{*}\in S_{r}$ and
$y^{*}\in T$ such that
$\|x^{*}-f(x^{*})\|^{2}+\|f(x)-y^{*}\|^{2}-\|x-f(x)\|^{2}<\|f(x^{*})-y^{*}\|^{2}=(\hbox{\rm
dist}(f(x^{*}),T))^{2}$ $None$
for all $x\in B_{r}\setminus\\{x^{*}\\}$ .
PROOF. Consider the function $J:B_{\rho}\times Y\to{\bf R}$ defined by
$J(x,y)=\|f(x)-x\|^{2}-\|f(x)-y\|^{2}$
for all $x\in B_{\rho}$, $y\in Y$. Clearly, for each $y\in Y$, $J(\cdot,y)$ is
of class $C^{1}$. Moreover, one has
$\langle J^{\prime}_{x}(x,y),u\rangle=2\langle x-f(x),u\rangle-2\langle
f^{\prime}(x)(u),x-y\rangle$
for all $x\in B_{\rho}$, $u\in H$. Fix $x,v\in B_{\rho}$ and $u\in H$, with
$\|u\|=1$. We have
${{1}\over{2}}|\langle
J^{\prime}_{x}(x,y)-J^{\prime}_{x}(v,y),u\rangle|=|\langle
x-f(x)-v+f(v),u\rangle-\langle f^{\prime}(x)(u),x-y\rangle+\langle
f^{\prime}(v)(u),v-y\rangle|$ $\leq\eta\|x-v\|+|\langle
f^{\prime}(x)(u),x-v\rangle+\langle
f^{\prime}(x)(u)-f^{\prime}(v)(u),v-y\rangle|$
$\leq\eta\|x-v\|+\|f^{\prime}(x)(u)\|\|x-v\|+\|f^{\prime}(x)(u)-f^{\prime}(v)(u)\|\|v-y\|\leq\left(\eta+\theta+\gamma\left(\rho+\sup_{y\in
Y}\|y\|\right)\right)\|x-v\|\ .$
Therefore, the function $J_{x}^{\prime}(\cdot,y)$ is Lipschitzian with
constant $L$. Moreover, we clearly have
$\inf_{y\in Y}\|J^{\prime}_{x}(0,y)\|=2\sigma\ .$
So, if we fix
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$ and a non-
empty closed convex set $T\subseteq Y$, by Theorem 1, there exist $x^{*}\in
B_{r}$ and $y^{*}\in T$ such that
$\|f(x^{*})-x^{*}\|^{2}-\|f(x^{*})-y\|^{2}\leq\|f(x^{*})-x^{*}\|^{2}-\|f(x^{*})-y^{*}\|^{2}<\|f(x)-x\|^{2}-\|f(x)-y^{*}\|^{2}$
$None$
for all $x\in B_{r}\setminus\\{x^{*}\\}$, $y\in T$. Clearly, $(6)$ is
equivalent to $(5)$, and the proof is complete. $\bigtriangleup$ REMARK 2. -
Theorem 5 was obtained in [3] ([3], Corollary 2.5) assuming, in addition, that
$f$ is sequentially weakly-strongly continuous.
Here is a remarkable consequence of Theorem 5.
THEOREM 6. - Let $\rho>0$ and let $f:B_{\rho}\to H$ be a $C^{1}$ function
whose derivative is Lipschitzian with constant $\gamma$. Moreover, let $\eta$
be the Lipschitz constant of the function $x\to x-f(x)$, set
$\theta:=\sup_{x\in B_{\rho}}\|f^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$L:=2(\eta+\theta+2\gamma\rho)$
and assume that
$\sigma:=\inf_{y\in B_{\rho}}\sup_{\|u\|=1}|\langle
f^{\prime}(0)(u),y\rangle-\langle f(0),u\rangle|>0\ .$
Then, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$, there
exists a unique $x^{*}\in S_{r}$ such that
$\|f(x^{*})-x^{*}\|=\hbox{\rm dist}(f(x^{*}),B_{r})$ $None$
and
$\|f(x)-x^{*}\|<\|f(x)-x\|$ $None$
for all $x\in B_{r}\setminus\\{x^{*}\\}$.
PROOF. Fix $r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$.
Applying Theorem 5 with $Y=B_{\rho}$ and $T=B_{r}$, we obtain $x^{*}\in S_{r}$
and $y^{*}\in B_{r}$ such that
$\|x^{*}-f(x^{*})\|^{2}+\|f(x)-y^{*}\|^{2}-\|x-f(x)\|^{2}<\|f(x^{*})-y^{*}\|^{2}=(\hbox{\rm
dist}(f(x^{*}),B_{r}))^{2}$ $None$
for all $x\in B_{r}\setminus\\{x^{*}\\}$ . From this, we infer that
$y^{*}=x^{*}$. Actually, if $y^{*}\neq x^{*}$, we could take $x=y^{*}$ in
$(9)$, obtaining
$\|x^{*}-f(x^{*})\|<\hbox{\rm dist}(f(x^{*}),B_{r})$
which is absurd. Now, $(7)$ and $(8)$ follow directly from $(9)$. Finally,
concerning the uniqueness of $x^{*}$, assume that $x_{0}\in B_{r}$ is such
that
$\|f(x)-x_{0}\|<\|f(x)-x\|$
for all $x\in B_{r}\setminus\\{x_{0}\\}$. Then, if $x_{0}\neq x^{*}$, we would
have
$\|f(x^{*})-x_{0}\|<\|f(x^{*})-x^{*}\|$
which is absurd in view of $(7)$. $\bigtriangleup$
Finally, reasoning as in the proof of Theorem 3, we get the following
corollary of Theorem 6:
THEOREM 7. - Let $\rho>0$ and let $f:B_{\rho}\to H$ be a $C^{1}$ function with
Lipschitzian derivative.
Then, the following assertions are equivalent:
$(i)$ for each $r>0$ small enough, there exists a unique $x^{*}\in S_{r}$ such
that
$\|f(x^{*})-x^{*}\|=\hbox{\rm dist}(f(x^{*}),B_{r})$
and
$\|f(x)-x^{*}\|<\|f(x)-x\|$
for all $x\in B_{r}\setminus\\{x^{*}\\}$ ;
$(ii)$ $f(0)\neq 0$ .
Acknowledgement. The author has been supported by the Università di Catania,
PIACERI 2020-2022, Linea di intervento 2, Progetto “MAFANE” and by the Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
References
[1] K. FAN, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A., 39 (1953), 42-47.
[2] B. RICCERI, On a minimax theorem: an improvement, a new proof and an
overview of its applications, Minimax Theory Appl., 2 (2017), 99-152.
[3] B. RICCERI, Applying twice a minimax theorem, J. Nonlinear Convex Anal.,
20 (2019), 1987-1993.
[4] B. RICCERI, A remark on variational inequalities in small balls, J.
Nonlinear Var. Anal., 4 (2020), 21-26.
Department of Mathematics and Informatics
University of Catania
Viale A. Doria 6
95125 Catania, Italy
e-mail address<EMAIL_ADDRESS>
|
# The induced permittivity increment of electrorheological fluids in an
applied electric field in association with chain formation: A Brownian
Dynamics simulation study
Dávid Fertig Dezső Boda<EMAIL_ADDRESS>Center for Natural Sciences,
University of Pannonia, Egyetem u. 10, Veszprém, 8200, Hungary István Szalai
Research Centre for Engineering Sciences, University of Pannonia, Egyetem u.
10, Veszprém, 8200, Hungary Institute of Mechatronics Engineering and
Research, University of Pannonia, Gasparich Márk u. 18/A, Zalaegerszeg, 8900,
Hungary
###### Abstract
We report Brownian Dynamics simulation results for the relative permittivity
of electrorheological (ER) fluids in an applied electric field. The relative
permittivity of an ER fluid can be calculated from the Clausius-Mosotti (CM)
equation in the small applied field limit. When a strong field is applied,
however, the ER spheres are organized into chains and assemblies of chains in
which case the ER spheres are polarized not only by the external field but by
each other. This manifests itself in an enhanced dielectric response, e.g., in
an increase in the relative permittivity. The correction to the relative
permittivity and the time dependence of this correction is simulated on the
basis of a model in which the ER particles are represented as polarizable
spheres. In this model, the spheres are also polarized by each other in
addition to the applied field. Our results are qualitatively similar to those
obtained by Horváth and Szalai experimentally (Phys. Rev. E, 86, 061403,
2012). We report characteristic time constants obtained from bi-exponential
fits that can be associated with formation of pairs and short chains as well
as with aggregation of chains. The electric field dependence of the induced
dielectric increment reveals the same qualitative behavior that experiments
did: three regions with different slopes corresponding to different
aggregation processes are identified.
## I Introduction
In electrorheological (ER) fluids Winslow (1949) fine non-conducting solid
particles are suspended in an electrically insulating liquid with the
particles having larger relative permittivity than the solvent. Then, an
applied electric field induces polarization charges at the arising dielectric
boundaries that can be expressed as a multipole expansion with dipoles being
the dominant terms.
The interactions of these dipoles lead to a structural change in the ER fluid
known as the ER response. This structural change is the aggregation of ER
particles first into shorter, then into longer chains due to the fact that the
head-to-tail position of two dipoles along the direction of the applied field
is a minimum-energy configuration. In the case of strong applied fields, the
chains form larger clusters, for example, columnar structures.
This structural change results in changes in major physical properties of the
ER fluid. The externally controllable, fast and reversible change in
viscosity, for example, makes ER fluids a central component of various
devices, such as brakes, clutches, dampers, and valves Duclos _et al._
(1992); Havelka and Filisko (1995).
Another physical quantity whose change can be relatively easily tracked by
measuring the change in the capacitance of a measuring cell when the electric
field is switched on is the relative permittivity, $\epsilon$. Several
experimental studies have been reported for the nonlinear dielectric
properties of ER fluids. Adolf and Garino (1995); Tao and Roy (1994); Wen _et
al._ (1997, 1998); Rzoska and Zhelezny (2006) Horváth and Szalai Horváth and
Szalai (2012) have proposed a new method that made the measurement of
continuous changes in the increments in permittivity possible. They determined
the field dependence of the change in dielectric permittivity
$\Delta\epsilon(E)=\epsilon(E)-\epsilon(E=0),$ (1)
with $t\rightarrow\infty$ and also the time dependence
$\Delta\epsilon(t)=\epsilon(t)-\epsilon(t=0),$ (2)
where the electric field is switched on from 0 to $E$ at $t=0$. The time
dependence and electric field dependence are shown in Fig. 3 and Fig. 4 of
Ref. Horváth and Szalai, 2012, respectively.
They found that the time dependence can be described with a bi-exponential fit
$\Delta\epsilon(t)=A\left(1-e^{-t/\tau_{1}}\right)+B\left(1-e^{-t/\tau_{2}}\right).$
(3)
They hypothesized that the time constant $\tau_{1}$ can be associated with the
process of formation of pairs (and, perhaps, short chains), because $\tau_{1}$
is very similar to the characteristic time of pair formation derived from a
kinetic rate theory. Baxter-Drayton and Brady (1996) The time constant
$\tau_{2}$, on the other hand, was heuristically corresponded to formation of
long chains and their aggregation and proved to be an order of magnitude
larger than $\tau_{1}$, $\tau_{2}\approx 10\tau_{1}$.
In this paper, we investigate how change in the dielectric permittivity is
associated with structural changes (formation of chains of various lengths,
and the aggregation of chains into columnar structures) by computer
simulations. Although several simulation studies have been reported for
cluster formation Klingenberg _et al._ (1989); See and Doi (1991); Toor
(1993); Hass (1993); Climent _et al._ (2004); Domínguez-García _et al._
(2007), order parameters Hass (1993); Tao and Jiang (1994a, b); Baxter-Drayton
and Brady (1996); Enomoto and Oba (2002), diffusion constant Klingenberg _et
al._ (1989); Whittle (1990); Hass (1993), pair distribution functions Whittle
(1990); Hass (1993), relaxation times Heyes and Melrose (1990); Toor (1993);
Hass (1993); Cao _et al._ (2006), aggregation kinetics See and Doi (1991),
and stress under shear Heyes and Melrose (1990); Whittle (1990); Bonnecaze and
Brady (1992); Baxter-Drayton and Brady (1996); Cao _et al._ (2006), we are
not aware of any paper addressing the dielectric properties of ER fluids.
The field dependence revealed three regimes with different slopes of the
$\Delta\epsilon$ vs. $E$ function (Fig. 4 of Ref. Horváth and Szalai, 2012).
Horváth and Szalai hypothesized that at low electric fields (below a threshold
value $E_{1}$) the induced dipoles are not strong enough to generate chain
formation and $\Delta\epsilon$ increases with $E$, because the probability of
the ER spheres to approach each other becomes larger. Above $E_{1}$ chain
formation begins with a large slope, because the parts of a chain can find one
another easier at large electric fields. Above the threshold value
$E_{\mathrm{h}}$ the chains start to form columnar structures. This process is
less accelerated by the large electric field, because chains can also repulse
each other when they are not in the appropriate mutual configuration with
respect to each other. Here, we support this hypothesis with computer
simulation results.
## II Models and methods
The ER fluid is modeled as monodisperse dielectric spheres of relative
permittivity $\epsilon_{\mathrm{in}}$ inside the sphere immersed in a carrier
liquid of relative permittivity $\epsilon_{\mathrm{out}}$. The radius of the
spheres is $R$, while their diameter is $d=2R$. If an electric field,
$\mathbf{E}$ is applied on the sphere, a polarization charge density is
induced on the surface of the sphere that can be approximated with an ideal
point dipole placed in the center of the sphere computed as Jackson (1999)
$\bm{\mu}=4\pi\epsilon_{0}\left(\frac{\epsilon_{\mathrm{in}}-\epsilon_{\mathrm{out}}}{\epsilon_{\mathrm{in}}+2\epsilon_{\mathrm{out}}}\right)R^{3}\mathbf{E}=\alpha\mathbf{E},$
(4)
where
$\alpha=4\pi\epsilon_{0}\left(\frac{\epsilon_{\mathrm{in}}-\epsilon_{\mathrm{out}}}{\epsilon_{\mathrm{in}}+2\epsilon_{\mathrm{out}}}\right)R^{3}$
(5)
is the particle polarizability, $E=|\mathbf{E}|$, and $\epsilon_{0}$ is the
permittivity of vacuum. If it is further assumed that the characteristic time
of the rearrangement of the surface charge during the movement of the
particles is much smaller than the characteristic time of the rotation of the
ER particles, the $\bm{\mu}$ dipole always points into the direction of
$\mathbf{E}$ even if the sphere rotates.
If we take a system of $N$ particles at positions $\\{\mathbf{r}_{j}\\}$, the
electric field exerted on dipole $i$ by dipole $j$ is
$\mathbf{E}_{j}(\mathbf{r}_{i})=\frac{1}{4\pi\epsilon_{0}}\frac{3\mathbf{n}_{ij}(\mathbf{n}_{ij}\cdot\bm{\mu}_{j})-\bm{\mu}_{j}}{r_{ij}^{3}},$
(6)
where $r_{ij}=|\mathbf{r}_{ij}|$ and $\mathbf{n}_{ij}=\mathbf{r}_{ij}/r_{ij}$
with $\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$.
In Eq. 4, the electric field at $\mathbf{r}_{i}$ is a sum of the applied
field, $\mathbf{E}^{\mathrm{appl}}$ (points to the $z$ direction), and the
electric field produced by all the other dipoles,
$\mathbf{E}(\mathbf{r}_{i})=\sum_{j\neq j}\mathbf{E}_{j}(\mathbf{r}_{i})$. The
total dipole moment
$\bm{\mu}^{\mathrm{tot}}_{i}=\alpha\mathbf{E}^{\mathrm{appl}}+\alpha\mathbf{E}(\mathbf{r}_{i})=\bm{\mu}^{\mathrm{appl}}_{i}+\bm{\mu}^{\mathrm{part}}_{i}$
(7)
then is induced by these two components and is split into the terms
$\bm{\mu}^{\mathrm{appl}}_{i}$ and $\bm{\mu}^{\mathrm{part}}_{i}$ accordingly.
The dipole moment $\bm{\mu}_{i}^{\mathrm{appl}}$ induced by the applied field
is constant, while the dipole moment $\bm{\mu}_{i}^{\mathrm{part}}$ induced by
all the other ER particles needs to be calculated by an iterative procedure.
Vesely (1977)
In the rheological literature, it is usual to ignore the polarization of the
particles by each other ($\bm{\mu}^{\mathrm{part}}_{i}=0$). There are,
however, important exceptions. Tao and Sun (1991); Tao (1993); Tao and Jiang
(1994a); Bonnecaze and Brady (1992); w. Wang _et al._ (1996); Wang _et al._
(1997); Martin _et al._ (1998); Hynninen and Dijkstra (2005) In this work, we
present the full self consistent solution of Eqs. 6–7.
If we introduce the force exerted on dipole $\bm{\mu}_{i}$ by dipole
$\bm{\mu}_{j}$ (irrespective whether they are induced by $E^{\mathrm{appl}}$
or by other particles) is
$\displaystyle\mathbf{f}^{\mathrm{dip}}_{ij}(\mathbf{r}_{ij},\bm{\mu}_{i},\bm{\mu}_{j})=-(\bm{\mu}_{i}\cdot\nabla_{i})\mathbf{E}_{j}(\mathbf{r}_{i})=$
$\displaystyle=\frac{1}{4\pi\epsilon_{0}}\frac{1}{r_{ij}^{4}}\left\\{3\left[\bm{\mu}_{i}(\mathbf{n}_{ij}\cdot\bm{\mu}_{j})+\bm{\mu}_{j}(\mathbf{n}_{ij}\cdot\bm{\mu}_{i})+\right.\right.$
$\displaystyle+\left.\left.\mathbf{n}_{ij}(\bm{\mu}_{i}\cdot\bm{\mu}_{j})\right]-15\mathbf{n}_{ij}(\mathbf{n}_{ij}\cdot\bm{\mu}_{i})(\mathbf{n}_{ij}\cdot\bm{\mu}_{j})\right\\},$
(8)
we can express the force exerted on dipole $\bm{\mu}^{\mathrm{appl}}_{i}$ by
dipole $\bm{\mu}^{\mathrm{appl}}_{j}$ as
$\mathbf{f}_{ij}^{\mathrm{appl}}=\mathbf{f}^{\mathrm{dip}}_{ij}(\mathbf{r}_{ij},\bm{\mu}^{\mathrm{appl}}_{i},\bm{\mu}_{j}^{\mathrm{appl}})$
(9)
and the force exerted on dipole $\bm{\mu}^{\mathrm{appl}}_{i}$ by dipole
$\bm{\mu}^{\mathrm{part}}_{j}$ as
$\mathbf{f}_{ij}^{\mathrm{part}}=\mathbf{f}^{\mathrm{dip}}_{ij}(\mathbf{r}_{ij},\bm{\mu}^{\mathrm{appl}}_{i},\bm{\mu}_{j}^{\mathrm{part}}).$
(10)
The finite size of the ER particles is taken into account with a short-range
repulsive core potential for wich the cut & shifted Lennard Jones (LJ)
potential also known as the Weeks-Chandler-Anderson (WCA) potential is used.
The WCA force is defined as
$\mathbf{f}_{ij}^{\mathrm{WCA}}(\mathbf{r}_{ij})=\left\\{\begin{array}[]{ll}\mathbf{f}_{ij}^{\mathrm{LJ}}(\mathbf{r}_{ij})&\mathrm{if}\;\;r_{ij}<r_{\mathrm{c}}\\\
0&\mathrm{if}\;\;r_{ij}>r_{\mathrm{c}}\end{array}\right.,$ (11)
where
$\mathbf{f}_{ij}^{\mathrm{LJ}}(\mathbf{r}_{ij})=24\varepsilon^{\mathrm{LJ}}\left[2\left(\frac{d}{r_{ij}}\right)^{12}-\left(\frac{d}{r_{ij}}\right)^{6}\right]\frac{\mathbf{r}_{ij}}{r_{ij}^{2}}$
(12)
is the LJ force ($r_{\mathrm{c}}=2^{1/6}d$ is the cutoff distance at the
minimum of the LJ potential).
The trajectories of the particles can be computed from Langevin’s equations of
motion Lemons and Gythiel (1997)
$m\frac{d\mathbf{v}_{i}(t)}{dt}=\mathbf{F}_{i}\left(\mathbf{r}_{i}(t)\right)-m\gamma\mathbf{v}_{i}(t)+\mathbf{R}_{i}(t),$
(13)
where
$\mathbf{F}_{i}=\sum_{j}(\mathbf{f}_{ij}^{\mathrm{WCA}}+\mathbf{f}_{ij}^{\mathrm{appl}}+\mathbf{f}_{ij}^{\mathrm{part}})$
(14)
is the systematic force, $-m\gamma\mathbf{v}_{i}(t)$ is the frictional force,
$\mathbf{R}_{i}(t)$ is the random force, $\mathbf{r}_{i}$, $\mathbf{v}_{i}$,
$m$, and $\gamma$ are the position, the velocity, the mass, and the friction
coefficient of particle $i$, respectively. The friction coefficient can be
computed from Stokes’ law as $\gamma=3\pi\eta d/m$.
To solve this stochastic differential equation we use the GJF-2GJ version
Jensen and Grønbech-Jensen (2019) of a collections of algorithms proposed by
Grønbech-Jensen and Farago: Grønbech-Jensen and Farago (2013); Farago (2019);
Jensen and Grønbech-Jensen (2019)
$v^{n+\frac{1}{2}}=av^{n-\frac{1}{2}}+\frac{\sqrt{b}\Delta
t}{m}f^{n}+\frac{\sqrt{b}}{2m}\left(R^{n}-R^{n+1}\right)$ (15)
$r^{n+1}=r^{n}+\sqrt{b}v^{n+\frac{1}{2}}\Delta t,$ (16)
where $r^{n}=r(t^{n})$ is any position coordinate of any particle,
$v^{n}=v(t_{n})$ is any velocity coordinate of any particle, $t^{n}=n\Delta t$
is the time in the $n$th time step, $\Delta t$ is the time step,
$a=(1-\gamma\Delta t/2)/(1+\gamma\Delta t/2)$ , $b=(1)/(1+\gamma\Delta t/2)$,
$t_{n+\frac{1}{2}}=t_{n}+\Delta t/2$, and $t_{n-\frac{1}{2}}=t_{n}-\Delta
t/2$. The discrete time noise, $R^{n}$, is a random Gaussian number with
properties $\langle R^{n}\rangle=0$ and $\langle R^{m}R^{n}\rangle=2kT\gamma
m\Delta t\delta_{mn}$ with $\delta_{mn}$ being the Kronecker-delta.
The Brownian Dynamics simulations have been performed in a cubic simulation
cell using periodic boundary conditions. The ensemble can be considered
canonical because $V$ and $N$ are fixed, while the temperature is also
constant because the system is thermostated by the Langevin integrator via the
fluctuation–dissipation theorem. The dipolar interactions were truncated at
the half of the cell width. System size dependence has been analyzed in our
previous work. Fertig _et al._ (2021)
## III Results and Discussion
Table 1: Reduced quantities defined with $T$, $d$, $m$ or with $T$, $d$, $\rho_{\mathrm{in}}$. Quantity | Reduced quantity
---|---
Time | $t^{*}=t\sqrt{kT/md^{2}}=t\sqrt{6kT/\pi\rho_{\mathrm{in}}d^{5}}$
Distance | $r^{*}=r/d$
Density | $\rho^{*}=\rho d^{3}$
Velocity | $v^{*}=v\sqrt{m/kT}=v\sqrt{\pi\rho_{\mathrm{in}}d^{3}/6kT}$
Energy | $u^{*}=u/kT$
Force | $F^{*}=Fd/kT$
Electric field | $E^{*}=E\sqrt{4\pi\epsilon_{0}d^{3}/kT}$
Dipole moment | $\mu^{*}=\mu/\sqrt{4\pi\epsilon_{0}kTd^{3}}$
Polarizability | $\alpha^{*}=\alpha/4\pi\epsilon_{0}d^{3}$
Friction coefficient | $\gamma^{*}=\gamma\sqrt{md^{2}/kT}=\gamma\sqrt{\rho_{\mathrm{in}}d^{5}/6kT}$
In this work, we use reduced units that are collected in Table 1. They express
physical quantities as dimensionless numbers obtained by dividing a quantity
in a physical unit by a unit quantity, $r^{*}=r/d$, for example. In addition
to $T$ and $d$, we can use either $m$ or $\rho_{\mathrm{in}}$ (mass density of
the ER particles) to define the reduced quantities, because the mass depends
on $m$ and $d$ through $m=\rho_{\mathrm{in}}d^{3}\pi/6$. The diffusion
constant in the high coupling limit can be expressed by Einstein’s relation:
$D=kT/m\gamma$. The square of the reduced dipole moment
$(\mu^{*})^{2}=\frac{\mu^{2}/4\pi\epsilon_{0}d^{3}}{kT}$ (17)
is an important parameter because it relates the ordering effect of the
dipolar energy to the disordering effect of thermal motion. It is proportional
to the $\lambda$ parameter used in the literature. If $(\mu^{*})^{2}$ is
large, the dipolar interactions are strong enough to induce chain formation,
while if it is small, thermal motion prevents chain formation.
In this work, our main goal is to study the dielectric response of the ER
fluid and to analyze the relationship of this response to chain formation in
the system. The relative permittivity of an ER fluid can be computed from the
corrected Clausius-Mosotti (CM) equation that can be derived from a
polarization formula: Neumann (1983)
$\dfrac{\epsilon-1}{\epsilon+2}=\dfrac{1}{3\epsilon_{0}}\dfrac{\left\langle
P\right\rangle}{E^{\mathrm{appl}}},$ (18)
where $P$ is the polarization density obtained from the sum of the dipoles
$\mu^{\mathrm{appl}}=\alpha E^{\mathrm{appl}}$ induced directly by the
external field and the average dipole moment
$\left\langle\mu^{\mathrm{part}}\right\rangle$ induced by other particles:
$\left\langle
P\right\rangle=\frac{N\mu^{\mathrm{appl}}+N\left\langle\mu^{\mathrm{part}}\right\rangle}{V}=\rho\mu^{\mathrm{appl}}\left(1+\frac{\left\langle\mu^{\mathrm{part}}\right\rangle}{\mu^{\mathrm{appl}}}\right),$
(19)
here $V$ is the considered volume, and $N$ the number of particles in it. Eqs.
4, 18 and 19 result in the corrected CM equation
$\dfrac{\epsilon-1}{\epsilon+2}=\dfrac{\alpha\rho}{3\epsilon_{0}}\left(1+\dfrac{\left\langle\mu^{\mathrm{part}}\right\rangle}{\mu^{\mathrm{appl}}}\right),$
(20)
where $\rho=N/V$ is the number density, and the correction factor
$\left\langle\mu^{\mathrm{part}}\right\rangle/\mu^{\mathrm{appl}}$ is the
average induced dipole due to other particles normalized by the dipole due to
the external field. The quantity
$\left\langle\mu^{\mathrm{part}}\right\rangle$ is directly provided by the
simulations. We have investigated the correction to the CM equation in the
case of nonpolar fluids (e.g., carbon dioxide) by Monte Carlo simulations.
Valiskó and Boda (2009)
In the low-field-strength limit, the original CM equation is recovered (the
correction factor is zero), because formally an ensemble of ER particles
corresponds to an ensemble of non-polar, but polarizable molecules. The CM
equation is based on the Lorentz formula Lorentz (1909) for the internal field
and ignores the fact that a particle is also polarized by other particles not
only by the external field. Keyes and Kirkwood (1931)
The number of particles is fixed at $N=256$ in our simulations. This number
proved to be sufficient on the basis of the system size analysis provided in
our previous study. Fertig _et al._ (2021)
The value of the reduced friction coefficient is fixed at $\gamma^{*}=100$.
This value made simulations feasible, because the system developed relatively
fast at the fixed value of the time step $\Delta t^{*}=0.005$. The effect of
larger values of $\gamma^{*}$ that are more characteristic of typical ER
fluids is analyzed in our previous work. Fertig _et al._ (2021) The reduced
density is fixed at $\rho^{*}=0.05$. The parameters that we change are the
reduced electric field $E^{*}$ and the reduced polarizability $\alpha^{*}$.
To characterize time dependence, we show values of block averages (denoted by
$\langle\dots\rangle_{\mathrm{b}}$ for various physical quantities as
functions of $t^{*}$. In this work, we performed $M_{\mathrm{b}}{=}5000$ time
steps in a block. We perform $M_{0}=50,000$ time steps ($20$ blocks) in the
absence of applied electric field ($E^{\mathrm{appl}}{=}0$), after which the
electric field is instantaneously switched on. Then we performed
$M_{\mathrm{E}}=450,000$ time steps ($180$ blocks) in the presence of a
constant applied field.
Such a cycle was started over and done several times and averaged to smooth
out noise. When we start a cycle over, we restart from a freshly generated
initial configuration in a completely disordered state without chains. This
way, the subsequent periods are independent and can be averaged.
Fig. 1 shows the time dependence of $\Delta\epsilon$ for $\alpha^{*}=0.03$
(the curves for other values of $\alpha^{*}$ are similar). The numbers near
the curves indicate the values of the reduced electric field, $E^{*}$. This
figure corresponds to Fig. 3 of Horváth and Szalai Horváth and Szalai (2012)
whose data are reproduced in the inset of Fig. 1 for qualitative comparison.
Figure 1: Time dependence of $\Delta\epsilon$ for $\alpha^{*}=0.03$. The
symbols with error bars are simulated data. The error bars have been computed
from the variance of the data in the consecutive and independent periods. The
lines are bi-exponential fits (Eq. 3). The numbers near the curves indicate
the values of the reduced electric field, $E^{*}$. The inset shows the
experimental data of Horváth and Szalai Horváth and Szalai (2012) for
comparison. The numbers in the inset indicate electric field strengths in MV/m
unit.
Direct quantitative comparison with the experimental data is problematic (see
the end of this section) due mainly to the fact that the ER fluid studied by
Horváth and Szalai is polydisperse. They considered nanosized ($10-20$ nm)
silica (SiO2) particles dispersed in silicone oils (polydimethylsiloxane) with
different dynamic viscosities ($0.34$ and $0.97$ Pa s). Qualitative
comparison, however, is possible. By fitting a bi-exponential to our simulated
data (Eq. 3), we obtain the characteristic times (in reduced units) shown in
Fig. 2. Panel A shows the data as functions of $E^{*}$ for different values of
$\alpha^{*}$.
This figure implies that a larger electric field is needed to achieve smaller
$\tau^{*}$ values (faster processes) in the case of smaller $\alpha^{*}$
values. This is obvious, because the important parameter from the point of
view of the dipolar interactions is the induced dipole that is
$\alpha^{*}E^{*}$. Therefore, if we plot $\tau_{1}^{*}$ and $\tau_{2}^{*}$ as
functions of $\alpha^{*}E^{*}$, we obtain a scaling behavior: the curves for
different $\alpha^{*}$ values collapse onto a single curve (Fig. 2B). This
scaling behavior also applies for the time dependence of the normalized
$\Delta\epsilon$; if we plot
$\Delta\epsilon(t^{*})/\Delta\epsilon(t^{*}\rightarrow\infty)$ as a function
of $t^{*}$ for a fixed value of $\alpha^{*}E^{*}$ but for different
combinations of $\alpha^{*}$ and $E^{*}$, the curves collapse onto a single
one.
Figure 2: The characteristic times $\tau_{1}$ and $\tau_{2}$ (the latter is
larger with an order of magnitude) as functions of (A) $E^{*}$ and (B)
$\alpha^{*}E^{*}$ for various values of $\alpha^{*}$. The inset shows the
experimental data of Horváth and Szalai Horváth and Szalai (2012) (from their
Table I.) for the two different values of the viscosity they considered (black
and red colors refer to $\eta=0.34$ and $0.97$ Pa s, respectively). The error
bars of $\tau_{1}$ and $\tau_{2}$ estimated from the Levenberg-Marquardt
algorithm Marquardt (1963) are within the size of the sysmbols.
This figure also shows that $\tau_{2}^{*}$ is an order of magnitude larger
than $\tau_{1}^{*}$ in agreement with the predictions of Hass et al. Hass
(1993) as well as with the experiments of Ly et al. Ly _et al._ (2001) and
Horváth and Szalai. Horváth and Szalai (2012)
The qualitative behavior of the $\tau_{1}$ vs. $E$ function is also similar to
the experimental behavior as seen from comparison to the data in Table I of
Horváth and Szalai Horváth and Szalai (2012) that are reproduced in the inset
of Fig. 2B.
Figure 3: The equilibrium ($t\rightarrow\infty$) values of the one-particle
dipolar energy, $(u^{\mathrm{dip}})^{*}$ (top-left panel), the diffusion
constant, $D^{*}$ (top-right panel), the induced permittivity increment,
$\Delta\epsilon$ (bottom-left panel), and the average chain length,
$s_{\mathrm{av}}$ (bottom-right panel), for various values of $\alpha^{*}$.
The vertical dashed lines indicate the $E_{\mathrm{h}}^{*}$ field strengths at
which the regions with different slopes meet (see Fig. 4 for more
explanation). The error bars are those computed at $t^{*}=4500$ from the
variance of the values in the periods. The error bars for
$(u^{\mathrm{dip}})^{*}$ and $\Delta\epsilon$ are within the size of the
symbols.
The purpose of this study is to look into the black box and to see how the
dielectric behavior observed in experiment and simulation is related to the
structural changes occurring in the ER fluid as the electric field is
increased. These structural changes can be monitored via various physical
quantities such as the diffusion constant, the dipolar energy, and the average
chain length. Therefore, we plot the equilibrium values of these quantities
(along with $\Delta\epsilon$) as functions of the field strength, $E^{*}$, in
Fig. 3. The equilibrium value of a quantity can be obtained either by running
a long simulation and throwing the equilibration period away or from
substituting $t\rightarrow\infty$ into the bi-exponential. We have chosen the
second option. Fig. 1 shows the fitted functions. The $R^{2}$ coefficient of
the fitting is generally above $0.99$ and the residuals (data not shown)
decrease to zero as $t\rightarrow\infty$. These imply that the equilibrium
values obtained this way are trustable.
The equilibrium values of $\Delta\epsilon$ as functions of $E^{*}$ are shown
in the bottom-left panel of Fig. 3. Also, the curve for $\alpha^{*}=0.03$ is
reproduced in Fig. 4 along with the experimental data in the insets. The
qualitative agreement of the simulations and experimental data is apparent.
The behavior of our curve follows the behavior of the experimental curve in
the respect that there is a initial stage with a small slope. When the
electric field is larger than a threshold value ($E_{1}$), pairs and short
chains start to form and the curve in the interval between $E_{1}$ and the
next threshold value $E_{\mathrm{h}}$ has a larger slope. Above
$E_{\mathrm{h}}$, the chains spanning the simulation cell start to aggregate
with a smaller slope of the $\Delta\epsilon$ vs. $E^{*}$ function.
One notable difference between the simulation and experimental results is that
$E_{1}$ is much smaller relative to $E_{\mathrm{h}}$ in the experiment than in
the model. This is valid for all $\alpha^{*}$ values studied. The reason of
this is not clear, but the results may be system-size and density dependent.
It is also possible that the large-particle fraction of the experimental
polydisperse system can form clusters at lower fields than the average-sized
particles, an effect that is absent in our monodisperse model. The
$E_{\mathrm{h}}$ value, however, appears to be a relatively well defined point
separating two characteristic regions with different slopes, so we indicate
the $E^{*}_{\mathrm{h}}$ vales with vertical dashed lines in all the panels of
Fig. 3.
Note that $\Delta\epsilon(E^{*})$ function exhibits a steep nonlinear increase
at large $E^{*}$ values. This is the result of the strong dipolar attraction
overriding the repulsion of the WCA potential. The overlapping spheres lead to
stronger particle-particle polarization. We consider this behavior an artifact
of the model.
Figure 4: The equilibrium ($t\rightarrow\infty$) value of $\Delta\epsilon$ as
a function of $E^{*}$ for $\alpha^{*}=0.03$. The inset shows the experimental
data from Fig. 4 of Horváth and Szalai Horváth and Szalai (2012). The inset of
the inset shows the results for small $E^{*}$. The solid blue lines indicate
the slopes, while dashed blue lines indicate the crosses of the solid lines
defining the threshold field strengths $E_{1}$ and $E_{\mathrm{h}}$.
Next, let us see how the behavior of the other physical quantities correlates
with the behavior of $\Delta\epsilon$. The one-particle dipolar energy,
$(u^{\mathrm{dip}})^{*}$, does not exhibit the behavior of the three (small-
large-small) slopes (top-left panel of Fig. 3). It decreases with increasing
$E^{*}$ at a continuously increasing rate. The explanation is that the dipolar
energy chiefly depends on the interactions inside the chains. If we increase
$E^{*}$, the dipoles become larger, and also their interactions. Clustering of
chains does not seriously influence this dependence.
The diffusion constant is calculated as the slope of the mean square
displacement (MSD) as a function of time:
$D(t_{\mathrm{b}})=\frac{\langle\mathbf{r}^{2}(t)\rangle_{\mathrm{b}}}{6\Delta
t_{\mathrm{b}}},$ (21)
where $t_{\mathrm{b}}$ is the time at the beginning of a block, and $\Delta
t_{\mathrm{b}}{=}M_{\mathrm{b}}\Delta t$ is the length of the block. This way,
$D$ is characteristic of a block and time dependence can be studied. $D^{*}$
decreases as $E^{*}$ increases as shown in Fig. 3 (top-right panel). The ER
particles lose their mobilities as they are organized into chain-like and
columnar structures. The behavior of $D^{*}$ follows the behavior of
$\Delta\epsilon$. Its value starts with the $E^{*}\rightarrow 0$ limit
($0.01$), breaks down around $E_{1}$, and decreases steeply as longer chains
are formed at higher $E^{*}$ values. Around $E_{\mathrm{h}}$, the chains
aggregate, $D^{*}$ decreases at a lower rate, and saturates into a very small,
but non-zero value. At large $E^{*}$, the spheres move together with their
chains that have much smaller mobility than the single spheres.
These results are closely related to the anomalous diffusion behavior of
dipolar chains that has been studied theoretically and experimentally. Furst
and Gast (2000); Toussaint _et al._ (2004, 2006) Those studies imply that the
diffusion of chains is reduced compared to the case in the absence of chains.
The degree of reduction is related to the average length of the chains.
Figure 5: Equilibrium limits of the chain length distribution, $n_{s}$ (left),
and radial distribution functions, $g(r^{*})$ (right) for $\alpha^{*}=0.03$.
The curves are obtained by averaging over $10$ blocks at the end of $M_{E}$
simulation periods and averaging over periods. Snapshots are shown in the
middle in front and top view. The electric field strength increases from top
to bottom.
The average chain length, $s_{\mathrm{av}}$, is computed by identifying the
number of chains of length $s$, $n_{s}$, for every configuration, taking the
average
$s_{\mathrm{av}}=\frac{\sum_{s}sn_{s}}{\sum_{s}n_{s}},$ (22)
and then averaging over configurations. Chain length, $s$, is measured in the
number of particles in the chain. Two particles are defined to be in the same
chain if their distance is smaller than $1.2d$. The choice of $1.2$ does not
influence our qualitative conclusions for the dynamics of chain formation.
Other definitions of chains were analyzed in our previous study Fertig _et
al._ (2020).
The average chain length starts to increase only above the first threshold
value, $E_{1}^{*}$. Above the second threshold value, $E^{*}_{\mathrm{h}}$,
the average length of chains reaches the value $s\approx 18$ that corresponds
to a chain spanning the simulation cell whose length is $L\approx 17.23d$.
As the electric field is increased further well above $E^{*}_{\mathrm{h}}$,
the average number of chains also increases and eventually reaches a limiting
value of 64. At large electric fields characteristically $4$ clusters of
chains are formed each containing on average 64 spheres, but this is just an
average. There is thermal motion, so values different from 64 may occur, but
the average seems solid. Thermal motion at strong field strengths mainly means
the translational motion of chains in the lateral ($x,y$) plane and rotation
of chains about the axes of the chains. Also, the $4$ clusters of chains do
not aggregate further. At high electric fields, the repulsion between these
clusters seems to prevent further aggregation. This finding, however, is
density dependent too.
It is of interest of looking into the black box even deeper to see how these
average chain length values come about. It is done in Fig. 5 in which we show
chain length distributions, $n_{s}$, radial distribution functions, $g(r)$,
and snapshots. The electric field strength increases from top to bottom. The
values are chosen to show the phases in Figs. 3 and 4. We show results for a
value below $E^{*}_{1}$, around $E^{*}_{1}$, between $E^{*}_{1}$ and
$E^{*}_{\mathrm{h}}$, around $E^{*}_{\mathrm{h}}$, and above
$E^{*}_{\mathrm{h}}$.
As $E^{*}$ increases, the chain length distributions show the increased
probability of longer chains. The snapshots clearly show these chains that are
also indicated by peaks in the $g(r)$ functions.
For $E^{*}=33.33$ (below $E^{*}_{1}$), the system is practically a homogeneous
isotropic gas-like fluid regarding the ER particles. For $E^{*}=52.7$ (around
$E^{*}_{1}$), pair formation and, to some degree, formation of short chains
are present. For $E^{*}=57.74$ (between $E^{*}_{1}$ and $E^{*}_{\mathrm{h}}$),
even longer chains and, accordingly, more peaks in $g(r)$ appear.
When we reach $E^{*}_{\mathrm{h}}$ ($E^{*}=66.67$), chains spanning the
simulation cell are clearly present indicated by the peak at $s=18$ in the
$n_{s}$ function. This chain is more stabilized by the periodic boundary
conditions in our cubic simulation cell of length $L\approx 17.23d$ compared
to other chains. In our previous study, Fertig _et al._ (2021) we analyzed
this behavior in detail.
For even larger electric field strength ($E^{*}=78.17$), we find another peak
at $s=36$ that corresponds to two chains stuck together. Chains are
straighter, and the peaks in $g(r)$ are more pronounced. Also, the $n_{s}$
function is more noisy than at lower electric field strengths that indicates
that the system is “more frozen” or “less fluid”. The evolution of the system
is determined by the movements of the much less mobile chains instead of the
movements of individual particles and short chains.
Figure 6: Snapshots for a very large electric field, $E^{*}=149.07$, for
$\alpha^{*}=0.03$.
If we increase the electric field even further ($E^{*}=149.07$), the chains
aggregate into column-like structures. The resulting $n_{s}$ and $g(r)$
profiles are even more noisy and not very meaningful, so we show only
snapshots in Fig. 6.
Finally, we briefly consider the possibility of a quantitative comparison with
the experimental results. If we accept the hypothesis that the meaning of
$E_{\mathrm{h}}$ is the same in the model and in the experiment (it is a
cornerstone), we can estimate the particle diameter, $d$, that brings the
simulation and the experimental data into correspondence. If we take the value
of $E_{\mathrm{h}}=1.18$ MV/m from Fig. 4 of Ref. Horváth and Szalai, 2012 and
relate it to the $E_{\mathrm{h}}^{*}$ values depicted from Fig. 3
($E_{\mathrm{h}}^{*}=52.5$, $72$, $107$ for $\alpha^{*}=0.02$, $0.03$, and
$0.04$, respectively), we obtain the values $d=673$, $517$, and $418$ nm for
$\alpha^{*}=0.02$, $0.03$, and $0.04$, respectively. These values are much
larger than the $10-20$ nm values specified in the paper of Horváth and
Szalai. Horváth and Szalai (2012)
We can explain this in different ways. First, the ER particles provided by the
manufacturer are polydisperse as opposed to our model that includes particles
of the same size. Also, it was observed in the experiments that the particles
tend to stick together forming larger particles resulting in a larger
effective diameter. Water content that may increase polarizability cannot be
excluded. In any case, the values $10-20$ nm are so small that using them in a
simulation (by transforming them to the corresponding reduced units) does not
result in any kind of chain formation. For these reasons, we regard the
diameters calculated and reported here more realistic than the $10-20$ nm
values.
It is also possible to correspond the $\tau_{1}^{*}$ value depicted from Fig.
2 ($\tau_{1}^{*}\approx 68.5$ for $\alpha^{*}=0.03$) to the experimental
$\tau_{1}$ value depicted from the inset of Fig. 2B ($\tau_{1}\approx 0.38$ s
for viscosity $0.97$ Pa s). The resulting diameter is $d\approx 2470$ nm. The
reason of these large values is that our reduced friction coefficient is small
($\gamma^{*}=100$). Because we can extrapolate to larger values of
$\gamma^{*}$ (see Fig. 6 of our previous work Fertig _et al._ (2021)), we can
provide the estimation that by increasing $\gamma^{*}$ with two orders of
magnitude, the resulting diameter decreases with about one order of magnitude.
Changing $\gamma^{*}$ does not change the equilibrium value of
$s_{\mathrm{av}}$ and $\Delta\epsilon$, it only influences how fast the system
converges to these values.
To relate $t^{*}$ to $t$ and $E^{*}$ to $E$ in Figs. 1, 2 and 4, we collect
the unit values $E_{0}=E/E^{*}$ and $t_{0}=t/t^{*}$ in Table 2 for three
representative values of the particle diameter: $d=517$ and $2467$ nm obtained
from the procedures of relating $E_{\mathrm{h}}$ to $E_{\mathrm{h}}^{*}$ and
$\tau_{1}$ to $\tau_{1}^{*}$, respectively, and a value in between ($1000$
nm).
Table 2: Unit values of the electric field strength and particle diameter, $E_{0}=E/E^{*}$ and $t_{0}=t/t^{*}$, respectively, for different values of $d$ on the basis of Table 1 ($T=298.15$ K and $\rho_{\mathrm{in}}=2650$ kg/m3). $d$ / nm | $E_{0}$ / MVm-1 | $t_{0}$ / s
---|---|---
$516$ | $0.016$ | $0.00011$
$1000$ | $0.0061$ | $0.00058$
$2467$ | $0.0016$ | $0.0055$
## IV Conclusions
We performed Brownian Dynamics simulations for ER fluids by taking the cross-
polarization among particles into account in a self consistent way and
computed the induced dielectric increment, $\Delta\epsilon$, as a function of
time after the applied field is switched on and as a function of field
strength at the equilibrium limit. Particle-particle polarization is essential
for computing $\Delta\epsilon$ that is a very useful quantity because it is
measurable well and also can be obtained from simulations with a small
statistical error.
Our results are in qualitative agreement with the experimental results of
Horváth and Szalai Horváth and Szalai (2012) and relate the computed data to
structural features in terms of energy, diffusion constant, average chain
length, chain length distribution, radial distribution functions, and
snapshots. The hypotheses about the correlations between dielectric properties
and structural features put forward by Horváth and Szalai have been confirmed
by our calculations.
## Author’s contribution
All authors contributed equally to this work.
## Acknowledgments
This research was supported by the European Union, co-financed by the European
Social Fund, via the project “Research of autonomous vehicle systems related
to the autonomous test track in Zalaegerszeg” (EFOP-3.6.2-16-2017-00002). We
also acknowledge the support of the National Research, Development and
Innovation Office (NKFIH), project No. K124353. We acknowledge KIFÜ for
awarding us access to resource based in Hungary at Szeged.
## Data avalability
The data that supports the findings of this study are available within the
article.
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|
# View & Perspective, Frontiers of Physics
Tasting Nuclear Pasta Made with Classical Molecular Dynamics Simulations
Bao-An Li Department of Physics and Astronomy, Texas A&M University-Commerce,
TX 75429-3011, USA
###### Abstract
Nuclear clusters or voids in the inner crust of neutron stars were predicted
to have various shapes collectively nicknamed nuclear pasta. The recent review
in Ref. Lopez1 by López, Dorso and Frank summarized their systematic
investigations into properties especially the morphological and
thermodynamical phase transitions of the nuclear pasta within a Classical
Molecular Dynamics model, providing further stimuli to find more observational
evidences of the predicted nuclear pasta in neutron stars.
Known as the densest visible object in the Universe, neutron stars have many
mysterious properties to be understood. From the external magnetic field
through the surface and crust to the core of neutron stars, there are many
fundamental questions to be addressed NAP2011 ; NAP2012 . Thanks to the recent
advances especially in radio, X-ray and gravitational wave observations of
both isolated neutron stars and their mergers, much progresses have been made
in recent years in understanding the maximum mass, radius and tidal
deformation of neutron stars. These global observables provided some of the
much needed constraints on various theories about the equation of state,
internal structure and composition of neutron stars. However, because of the
scarcity of data available especially those from the un-distorted messengers
directly from the interior of neutron stars and the model dependence of their
interpretations, many mysteries associated with neutron stars remain to be
resolved. In fact, to understand the nature of neutron stars and dense
neutron-rich matter has been a long-standing and shared goal of both
astrophysics and nuclear physics communities LRP2015 ; NuPECC .
Among the main challenges, one important task is to understand the structure,
content and formation mechanism of neutron stars. People have imagined that
neutron stars have several layers with distinct features from the atmosphere,
envelope, crust to the outer and inner core in an analogy to the structure of
earth. The outer core of neutron stars is speculated to be some kind of fluid
consisting of neutron, protons, electrons, muons, hyperons, etc. As the
density of this fluid decreases towards the crust, the system is expected to
enter the so-called spinodal region where the matter (either pure nucleonic
matter of neutrons and protons Siemens83 or neutron star matter Lat00 ) is
dynamically unstable against the growth of small density fluctuations when the
temperature is sufficiently low. Consequently, bubbles/voids or
droplets/clusters are expected to develop depending on the trajectory of the
system before entering the spinodal region, marking a transition between the
uniform core and the inhomogeneous crust. This transition is expected to
happen in the inner crust and the exact transition density has been found to
depend sensitively on the characteristics of nuclear symmetry energy encoding
the energy necessary to make nuclear matter more neutron rich, see, e.g.,
Refs. Li19 ; JXu ; Newton12 . Very interestingly, because of the competition
between the surface tension and long-range Coulomb force, the bubbles and
nuclear clusters may have various shapes: from the spherical void near the
core, through the cylindrical void to the plates, cylinders and spheres of
nucleons towards the surface of neutron stars. Because of their similarity to
meatballs, spaghetti, lasagna, macaroni, and Swiss cheese, respectively, these
structures were collectively nicknamed nuclear pasta. For an earlier review of
the physics of nuclear pasta, see, Ref.Pethick . Since the pioneering works of
Ravenhall _et al._ Rav and Hashimoto _et al._ Has , besides the above five
classical shapes, several other shapes, such as the gyroid, double-diamond Ken
, sponge-like Dorso and parking-garage like Hor1 structures have been
predicted using various approaches, see, e.g., Refs. Wil85 ; Oya93 ; Lor93 ;
Che97 ; Wat00 ; Wat02 ; Wat03 ; Mar98 ; Kid00 ; Hor04 . Interestingly, similar
shapes have been found in nano materials and/or biological systems.
The various shapes of the nuclear voids and clusters correspond to the local
minima of potential energies separated by energy barriers. Often, the energy
minima are very close to each other. The realization of various shapes are
thus somewhat model dependent and the transition from one shape to another
depends sensitively on the nuclear physics inputs, such as the nuclear
equation of state especially its symmetry energy term Newton12 ; Newton09 ;
Bao ; Kaz20 ; Xia . Both static and dynamic approaches using various nuclear
interactions and assumptions about the composition of neutron stars have been
used in studying the formation, properties and phase transition of nuclear
pasta by many groups over the last three decades. The recent review in Ref.
Lopez1 by López, Dorso and Frank summarized their systematic investigations
into properties especially the morphological and thermodynamical phase
transitions of the nuclear pasta within a Classical Molecular Dynamics model.
Using several morphologic and thermodynamic tools, they characterized the
morphology of the emerging structures in neutron star crust by varying the
temperature, density, neutron to proton ratio with and without considering
electrons. They constructed the phase diagrams for both nucleonic matter and
neutron star matter. They also investigated the isospin (neutron to proton
ratio) dependence of the critical points and morphologic phase transition as
well as the symmetry energy of clustered matter. Bearing in mind the possible
model dependence, effects of some poorly known but necessary nuclear physics
inputs, finite size effects in simulating infinite matter using finite-sized
unit cells with periodic boundary condition and the lack of quantum effects,
their results obtained with the Classical Molecular Dynamics simulations are
certainly stimulating and useful for further exploring the interesting physics
of nuclear pasta.
As the nuclear pasta may actually exist inside the core of supernovae and the
crust of neutron stars, the physics of nuclear pasta is important for
understanding some astrophysical observations Cham ; Newton14 ; Chuck . For
example, supernova explosions, protoneutron star cooling mechanism and the
associated neutrino transport Hor04 ; Chuck ; Rog ; All ; Newton13 ; Sc ,
pulsar glitches Wat17 ; Josh , quadrupole deformation or formation of
mountains on neutron stars and the associated gravitational wave emission Cap
; Pet20 ; Bis ; And , frequencies of torsional oscillations of neutron stars
and the associated mechanism for generating quasi-periodic oscillations in the
tails of light curves of giant flares from soft gamma-ray repeaters Gea ; Sot
, all depend strongly on properties of neutron star crust especially whether
the nuclear pasta is considered or not. Moreover, the crustal properties are
also important for determining the r-mode stability window of super-fast
pulsars Wen ; Isaac . They are also critical for determine whether GW190814’s
second component of mass (2.5-2.67)M⊙ discovered by LIGO/VIRGO GW very
recently is the lightest black hole or the most massive and fastest rotating
neutron star observed so far Most ; Zhang as pointed out in Ref. rmode . Of
course, the main challenge is to find unique signatures of the nuclear pasta
as most of the astrophysical phenomena mentioned above have alternative
explanations without considering the nuclear pasta in the inner crust of
neutron stars. Hopefully, the review in Ref. Lopez1 by López, Dorso and Frank
will stimulate more work in this direction.
BALI is supported in part by the U.S. Department of Energy, Office of Science,
under Award Number DE-SC0013702 and the CUSTIPEN (China-U.S. Theory Institute
for Physics with Exotic Nuclei) under the US Department of Energy Grant No.
DE-SC0009971.
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|
# Higher-order Fabry-Pérot Interferometer from Topological Hinge States
Chang-An Li<EMAIL_ADDRESS>Institute for Theoretical Physics and
Astrophysics, University of Würzburg, 97074 Würzburg, Germany Song-Bo Zhang
<EMAIL_ADDRESS>Institute for Theoretical Physics and
Astrophysics, University of Würzburg, 97074 Würzburg, Germany Jian Li School
of Science, Westlake University, 18 Shilongshan Road, Hangzhou 310024,
Zhejiang Province, China Institute of Natural Sciences, Westlake Institute
for Advanced Study, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province,
China Björn Trauzettel Institute for Theoretical Physics and Astrophysics,
University of Würzburg, 97074 Würzburg, Germany
(August 27, 2024)
###### Abstract
We propose an intrinsic 3D Fabry-Pérot type interferometer, coined “higher-
order interferometer”, that utilizes the chiral hinge states of second-order
topological insulators and cannot be equivalently mapped to 2D space because
of higher-order topology. Quantum interference patterns in the two-terminal
conductance of this interferometer are controllable not only by tuning the
strength but also, particularly, by rotating the direction of the magnetic
field applied perpendicularly to the transport direction. Remarkably, the
conductance exhibits a characteristic beating pattern with multiple
frequencies with respect to field strength or direction. Our novel
interferometer provides feasible and robust magneto-transport signatures to
probe the particular hinge states of higher-order topological insulators.
Introduction.—Higher-order topological insulators (HOTIs) feature gapless
excitations, similar to traditional (first-order) topological insulators, that
are protected by bulk electronic topology but localized at open boundaries at
least two dimensions lower than the insulating bulk (Benalcazar _et al._ ,
2017a, b; Slager _et al._ , 2015; Peng _et al._ , 2017; Langbehn _et al._ ,
2017; Song _et al._ , 2017; Schindler _et al._ , 2018a; Geier _et al._ ,
2018; Ezawa, 2018; Khalaf, 2018; Park _et al._ , 2019; Trifunovic and
Brouwer, 2019; You _et al._ , 2018; Hirosawa _et al._ , 2020; Franca _et
al._ , 2018; van Miert and Ortix, 2018). For instance, 3D second-order
topological insulators (SOTIs) host 1D chiral or helical states along specific
hinges of the systems. In recent years, HOTIs have triggered widespread
research interest, owing to their discoveries in a variety of candidate
systems, promotion of our understanding of topological states of matter, and
potential applications (Schindler _et al._ , 2018b; Imhof _et al._ , 2018;
Peterson _et al._ , 2018; Serra-Garcia _et al._ , 2018; Chen _et al._ ,
2019; Peng _et al._ , 2020; Ghosh _et al._ , 2019; El Hassan _et al._ ,
2019; Ni _et al._ , 2019; Xie _et al._ , 2019; Qi _et al._ , 2020; Călugăru
_et al._ , 2019; Szabó _et al._ , 2020; Sheng _et al._ , 2019; Li and Wu,
2020; Li _et al._ , 2020a; Li and Sun, 2020; Zhu, 2018; Luo and Zhang, 2019;
Ezawa, 2019; Zhang _et al._ , 2020a, b; Pahomi _et al._ , 2020). So far,
most efforts have been put into the potential realization and electronic
characterization of HOTIs. However, the transport properties of HOTIs remain
largely unexplored, despite of a few works associated with superconductivity
(Queiroz and Stern, 2019; Li _et al._ , 2020b; Choi _et al._ , 2020).
Indeed, for 3D SOTIs, an intriguing open question is whether the emergent
hinge states can exhibit any particular phenomena in normal-state transport.
Figure 1: (a) Schematic of the higher-order interferometer: a SOTI with four
chiral hinge states (solid red and dashed blue lines) are connected to two
leads (yellow). Adjacent chiral hinge states form interference loops in the
presence of finite reflections at the interfaces. A magnetic field ${\bf B}$
perpendicular to $z$-direction is applied to the SOTI (gray). (b) The hinge
states have linear dispersion and are shifted in $k_{z}$-direction by ${\bf
B}$. (c) Density plot of conductance with respect to the field strength $B$
and direction $\theta$. $B_{0}=\phi_{0}/S_{f}$ with $\phi_{0}$ the flux
quantum and $S_{f}$ the area of the front surface of the SOTI.
One appealing route towards this question involves interferometers built of
SOTIs, which enable us to study quantum-coherent transport of hinge states.
Propagating hinge states that form interference loops enclosing a magnetic
flux applied to the system pick up an Aharonov-Bohm (AB) phase (Aharonov and
Bohm, 1959). In presence of quantum coherence, the AB phase will give rise to
quantum oscillations in transport characteristics such as the charge
conductance. Quantum interference patterns in the two-terminal conductance
have been employed to detect topological phases of matter, for instance,
surface states of topological insulators (Bardarson _et al._ , 2010; Zhang
and Vishwanath, 2010; Peng _et al._ , 2010; Bardarson and Moore, 2013),
chiral Majorana modes (Akhmerov _et al._ , 2009; Fu and Kane, 2009; Li _et
al._ , 2012, 2019), and topological Dirac semimetals (Wang _et al._ , 2016).
In this work, we propose a higher-order Fabry-Pérot interferometer to probe
hinge states of SOTIs. Our basic setup, shown in Fig. 1(a), is composed of a
rectangular chiral SOTI in contact with two leads. The chiral hinge states,
existing in 3D space, form interference loops due to finite reflections (not
shown) at the two interfaces, and their energy dispersions split in a non-
uniform manner under magnetic fields, as shown in Fig. 1(b). Particular
quantum interference patterns in the two-terminal conductance, arising from
the AB effect as exemplified by Fig. 1(c), can be observed either by tuning
the field strength $B$ or direction $\theta$. In addition, owing to the
intrinsic 3D nature of the interferometer, there are generally two frequencies
in the magneto-conductance oscillations, leading to a beating pattern. These
features do not depend on the details of the junction, such as the electronic
spectrum of the leads, and are stable against disorder and dephasing. Hence,
they provide robust transport signatures of hinge states in 3D SOTIs.
General analysis based on scattering matrix theory.—Our proposed
interferometer involves a 3D SOTI with four chiral hinge states attached to
two leads in $z$-direction, as sketched in Fig. 1(a). Adjacent chiral hinge
states form interference loops because of finite reflections at the
interfaces, as will be discussed below. A magnetic field ${\bf
B}=B(\cos\theta,\sin\theta,0)$ in $x$-$y$ plane is applied in the SOTI region,
where $B$ measures the field strength and $\theta$ the field direction with
respect to $x$-direction.
Before presenting concrete results based on specific models, it is instructive
to analyze the main transport features of the interferometer using a
phenomenological scattering matrix approach (Büttiker, 1992; Nazarov and
Blanter, 2006; Maciejko _et al._ , 2010). The transport properties of the
setup are encoded in a scattering matrix that directly connects the conducting
channels in the left and right leads. The scattering processes at the two
interfaces between the leads and the SOTI can be described by two scattering
matrices, respectively. Each matrix consists of four components: transmission
from left to right $t_{L/R}$, transmission from right to left
$t_{L/R}^{\prime}$, reflection from the right $r_{L/R}$ and reflection from
the left $r_{L/R}^{\prime}$, where the subscript ($L$ and $R$) distinguishes
the left and right surfaces. At low energies, the only conducting channels in
the SOTI are the four hinge states which have linear dispersion and are
localized at the four different hinges of the cuboid. In the presence of a
magnetic field, their propagation in the SOTI will pick up AB phases that can
be described by a phase matrix $U\equiv
e^{2i\lambda}e^{i\varphi\sigma_{z}\otimes\sigma_{0}/2}e^{i\phi\sigma_{z}\otimes\sigma_{z}/2}$,
where $\sigma_{z}$ is a Pauli matrix, $\sigma_{0}$ the $2\times 2$ identity
matrix,
$\varphi=BLW_{x}\cos\theta\ \text{and }\phi=BLW_{y}\sin\theta$ (1)
are the magnetic fluxes threading the two surfaces, respectively, with $L$ the
distance between the two leads and $W_{x/y}$ the widths of the sample in
$x/y$-directions. Moreover, $\lambda=k_{F}L$ is the dynamical phase with
$k_{F}$ the Fermi wave number in the absence of magnetic fields. By
eliminating the scattering amplitudes in the SOTI region, we derive
analytically an effective $2\times 2$ scattering matrix that directly connects
the two interfaces (Li2, a)
$\mathcal{S}(B,\theta)=\Phi_{+}(e^{-i\lambda}-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1},$
(2)
where the phase matrices $\Phi_{\pm}\equiv e^{i(\varphi\pm\phi)\sigma_{z}/2}$
account for the AB phase differences between the two right-moving and between
the two left-moving hinge channels, respectively. At zero temperature, the
two-terminal conductance of the setup can be evaluated as
$G(B,\theta)=\frac{e^{2}}{h}\mathrm{Tr}[t_{R}^{\dagger}t_{R}\mathcal{S}(B,\theta)t_{L}t_{L}^{\dagger}\mathcal{S}^{\dagger}(B,\theta)],$
(3)
where $h$ is the Planck constant and $e$ is electron charge. According to Eqs.
(2) and (3), if there is no transmission across any of the two interfaces,
i.e., $t_{L}=0$ or $t_{R}=0$, then $G$ vanishes. In the opposite limit, where
the interfaces are completely transparent for the hinge states,
$r_{L^{\prime}}=0$ and $r_{R}=0$, we find that the matrix $\mathcal{S}$ as
well as $t_{R}^{\dagger}t_{R}$ and $t_{L}t_{L}^{\dagger}$ become diagonal. As
a result, $G$ becomes quantized at $2e^{2}/h$ and is independent of the
magnetic field. These results indicate the necessary condition for a
successful interferometer: non-trivial transmission and reflection at the two
interfaces for the hinge states.
When the interfaces are partially transparent, Eq. (2) indicates the formation
of Fabry-Pérot interference loops. Moreover, the matrix $\mathcal{S}$ contains
explicitly two phases $\varphi\pm\phi$ in general. This indicates the
appearance of beating patterns with two frequencies in the magneto-
conductance. Notably, the two frequencies are intimately connected to the
magnetic fluxes threading the different surfaces of the SOTI. They are solely
determined by the geometry of the sample and insensitive to the details of the
interface barriers. The oscillation pattern of $G$ remains qualitatively the
same even in the presence of a dynamic phase. We verified these results by
properly parametrizing the scattering matrices (Li2, a).
Model simulation and method.—To demonstrate these features of the
interferometer explicitly, we consider an effective model for chiral SOTIs
(Schindler _et al._ , 2018a)
$\displaystyle H({\bf k})=$ $\displaystyle\Big{(}m+b\sum_{i=x,y,z}\cos
k_{i}\Big{)}\tau_{3}+v\sum_{i=x,y,z}\sin k_{i}\sigma_{i}\tau_{1}$
$\displaystyle+\Delta(\cos k_{x}-\cos k_{y})\tau_{2},$ (4)
where ${\bf k}=(k_{x},k_{y},k_{z})$ is the wave vector.
$\bm{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ and
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices acting on
orbital and spin spaces, respectively; $m$, $b$, $v$ and $\Delta$ are model
parameters. Without loss of generality, we set the lattice constant and the
velocity $v$ to unity hereafter. When $1<|m/b|<3$ and $\Delta=0$, the model
describes 3D topological insulators with gapless surface states (Zhang _et
al._ , 2009). The surface states are protected by time-reversal symmetry
$\mathcal{T}=i\sigma_{2}\mathcal{K}$, where $\mathcal{K}$ represents complex
conjugation. A finite $\Delta\neq 0$ breaks time-reversal and
$\mathcal{C}_{4}$ rotation (with the rotation axis pointing in $z$-direction)
symmetries individually. It opens gaps in the surface states. However, the
$\Delta$ term preserves the combined symmetry $\mathcal{C}_{4}\mathcal{T}$, as
indicated by
$(\mathcal{C}_{4}\mathcal{T})H(k_{x},k_{y},k_{z})(\mathcal{C}_{4}\mathcal{T})^{-1}=H(k_{y},-k_{x},-k_{z})$.
As a result, the gaps opened by $\Delta$ depend on the surface orientation,
leading to gapless chiral hinge states localized at the hinges connecting
different surfaces.
We take into account the orbital effect of the magnetic field via the Peierls
replacement in the hopping interaction $T_{ij}\rightarrow T_{ij}\exp(2\pi
i\int_{r_{i}}^{r_{j}}d{\bf{\bf r}}\cdot{\bf A}/\phi_{0})$, where $T_{ij}$ is
the hopping amplitude from sites $r_{i}$ to $r_{j}$, $\phi_{0}=h/e$ is flux
quantum. ${\bf A}$ is the vector potential for the magnetic field and it is
chosen as ${\bf A}=B(0,0,y\cos\theta-x\sin\theta)$ for concreteness (Li2, b).
For simplicity, we model the metallic leads with a conventional quadratic
energy dispersion and assume only a few transport channels in both leads such
that considerable reflections for the hinge channels are generated at the
interfaces. Furthermore, we consider the size of the system to be much larger
than the decay length of the hinge states in order to have a well-defined
multiple-loop interferometer based on hinge states. Under these
considerations, we calculate the two-terminal conductance numerically,
employing the standard Landauer-Büttiker approach (Landauer, 1970; Büttiker,
1986; Datta, 1995) in combination with lattice Green functions (see the
Supplemental Material (Li2, a)). We emphasize that our main results
illustrated below remain qualitatively the same if we choose other models for
SOTIs or leads.
Figure 2: (a) Conductance $G$ as a function of field direction $\theta$ at
small field strengths $B=B_{0}$ and $B_{0}/\sqrt{2}$, respectively. (b) $G$ as
a function of $B$ for $\theta=0$ and $\pi/4$, respectively. In these cases,
the oscillations have a single frequency. (c) Particular beating patterns as
varying $B$ at angle $\theta=0.48\pi$. (d) “Irregular” beating patterns as
varying $\theta$ at a large field strength $B=20B_{0}$. (e) The extracted
frequencies (square and circle dots) as a function of $\theta$. The two
frequencies can be described by $\omega_{1}=S|\cos\theta|$ and
$\omega_{2}=S|\sin\theta|$. (f) The low-energy spectrum of the SOTI in the
presence of a magnetic field $B=2B_{0}$ and $\theta=0.15\pi$. Other parameters
are $L_{z}=60a$, $W_{x}=W_{y}=12a$, $m=2,b=-1,v=1$, $\Delta=1$, and the Fermi
energy $E_{F}=0.002$.
Quantum interference pattern.—Now, we analyze the dependence of the
conductance $G$ on the magnetic field, combining general scattering theory and
concrete numerical simulations. Equation (3) implies an oscillation pattern of
$G$ with respect to the field direction $\theta$. As shown in Fig. 2(a),
$G(\theta)$ is periodic in $\theta,$ in accordance with the scattering theory.
Explicitly, we find that for weak magnetic fields $B\leq B_{0}$, $G(\theta)$
is approximately a sinusoidal function of $\theta$ and takes the maximal value
at $\theta=\pi/4+n\pi/2$, $n\in\\{0,1,2,3\\}$, when $W_{x}=W_{y}$. Here,
$B_{0}$ corresponds to the field strength at which the flux enclosed by the
front surface $S_{f}$ is one flux quantum for $\theta=0$. Thus, $G(\theta)$
has a period of $\pi/2$ in $\theta$. Moreover, $G(\theta)$ is minimal at
$\theta=\theta_{c}$ and symmetric in $\theta-\theta_{c}$, where
$\theta_{c}=n\pi/2$. For strong magnetic fields $B>B_{0}$, the number of
conductance peaks increases with increasing $B$, see Fig. 1(c). When
$W_{x}\neq W_{y},$ the period in $\theta$ becomes $\pi$ but $G(\theta)$ is
still symmetric in $\theta-\theta_{c}$.
Equation (3) also indicates an oscillation pattern of $G$ with respect to the
field strength $B,$ which is again fully confirmed by our numerical
simulations. When the magnetic field is applied in $x$\- or $y$-directions, or
at the specific angle $\theta=\pm\text{arctan}(W_{x}/W_{y})$, $G(B)$ exhibits
simple oscillations with a single frequency, see Fig. 2(b). Generally, the
oscillating conductance takes maximal or minimal values when the interference
loop encloses half a flux quantum. In our cases, $G(B)$ takes maximal values
at odd multiples of $B_{0}/2$ for $\theta=0$. The oscillation amplitude is
relatively smaller since only two of the four loops enclose half a flux
quantum at this field direction. For $\theta=\pi/4$, $G(B)$ takes maximal
values at odd multiples of $B_{0}/\sqrt{2}$, where the interference loop also
encloses half a flux quantum, leading to a resonance peak of $G(B)$. These
features signify the interferometer formed by hinge states being of Fabry-
Pérot type, as we further explain below.
Notably, there exist beating patterns, as signified by Eq. (2), where the
matrix $\mathcal{S}$ explicitly contains the two phases $\varphi\pm\phi$. When
the magnetic field deviates away from the special directions at
$\theta=n\pi/2$ (with $n\in\\{0,1,2,3\\}$) and
$\pm\text{arctan}(W_{x}/W_{y})$, beating oscillations of $G(B)$ are clearly
observed, as shown in Fig. 2(c). By performing discrete Fourier transformation
to the beating patterns, we obtain precisely two frequencies $\omega_{1}$ and
$\omega_{2}$. These frequencies depend strongly on the field direction
$\theta$ [dotted lines in Fig. 2(e)]. Explicitly, we find that the two
frequencies can be well described by $\omega_{1}=|S\cos\theta|=|\phi|/B$ and
$\omega_{2}=|S\sin\theta|=|\varphi|/B$ [solid lines in Fig. 2(e)],
respectively, where $S$ is the area of the surfaces of the system (we consider
the case with $W_{x}=W_{y}$ for simplicity). This corresponds exactly to the
two AB phases in Eq. (1), in excellent agreement with the results obtained
from scattering-matrix analysis. When $\theta=n\pi/2$, only one of the two
frequencies survives. When $\theta=\pi/4+n\pi/2$, the two frequencies become
identical. In both cases, the beating behavior in the oscillations disappear.
Similarly, $G(\theta)$ also shows beating-like patterns with respect to the
field direction $\theta$ with irregular peaks and dips for large magnetic
fields $B\gg B_{0}$, as shown in Fig. 2(d). This direction-induced beating
behavior is another manifestation of the two AB phases.
Higher-order Fabry-Pérot interference.—Next, we clarify, in which sense our
quantum interference pattern is a higher-order Fabry-Pérot type interference.
The two frequencies in the beating patterns correspond physically to two areas
of interference loops. As rotating the magnetic field, the two frequencies
match the effective areas of front surface $|S\cos\theta|$ and top surface
$|S\sin\theta|$ quite well, see Fig. 2(e). This fact indicates: (i) the
adjacent hinge states with opposite chirality form effective interference
loops and the interference is typically of Fabry-Pérot type; and (ii) there
are totally four interference loops but any two opposite surfaces of the
sample (namely, the front and back surfaces, or the top and bottom surfaces)
have the same effective area because of the chosen symmetry of the system
(Li2, c). The interference loops are made of chiral hinge modes located in 3D
space, protected by higher-order topology. When rotating the magnetic field,
one of the frequencies increases, whereas the other one decreases. Moreover,
the ratio between the two frequencies depends on $\theta$ as
$S_{f}/S_{t}=W_{y}|\cot\theta|/W_{x}$. Thus, the two frequencies coincide at
the critical field directions $\theta_{c}=\text{arctan}(W_{y}/W_{x})$ and
$\pi-\text{arctan}(W_{y}/W_{x})$, as shown in Fig. 2(e). These features
indicate the 3D nature of the interferometer.
The mechanism of the interferometer can be better understood by analyzing the
splitting of hinge states under magnetic fields. In the absence of magnetic
fields, the four chiral hinge states have a double degenerate linear spectrum
in $k_{z}$-direction, i.e., $\pm vk_{z}$. The magnetic field gives rise to a
spatially varying vector potential. Note that the hinge states are localized
at different hinges of the system. The local vector potential splits the
linear spectrum of the hinge states. Under the chosen gauge, the spectra of
hinge states are split as $+v(k_{z}\pm\delta k_{1}^{z})$ and
$-v(k_{z}\pm\delta k_{2}^{z})$, where the splittings are determined by $\delta
k_{1}^{z}=BW_{x}|\sin(\theta-\pi/4)|/2$ and $\delta
k_{2}^{z}=BW_{y}|\cos(\theta+\pi/4)|$/2 (Li2, a). Thus, the hinge states
acquire finite momenta even for vanishing Fermi energy [Fig. 2(f)]. When
propagating across the SOTI region, the hinge channels pick up extra phases,
$\pm\delta k_{1/2}^{z}L_{z}$. Such phases turn out to be exactly the AB phases
$\phi$ and $\varphi$, stemming from the magnetic flux enclosed by each loop.
Explicitly, the flux enclosed by front surface $S_{f}$ and top surface $S_{t}$
of the central SOTI region in Fig. 1(a) are given by $(\delta k_{1}^{z}+\delta
k_{2}^{z})L_{z}$ and $2\delta k_{1}^{z}L_{z}$, respectively. Plugging
$\phi,\varphi=\delta k^{z}L_{z}$ into Eq. (2), this indicates that the
dependence of $G$ on ${\bf B}$ can be attributed to the higher-order Fabry-
Pérot interference of the four hinge states. At special values of $\theta$,
say $\theta=\pi/4$ or $5\pi/4$, one kind of splitting vanishes whereas the
other one remains, $\delta k_{1}^{z}=0$ and $\delta k_{2}^{z}\neq 0$ (similar
results occur for $\theta=+3\pi/4,-\pi/4$). In these cases, we have only one
frequency.
Figure 3: (a) Low-energy spectrum of the SOTI in the presence of a magnetic
field. (b) The three frequencies in the case of $\mathcal{C}_{6}$ symmetric
SOTIs as functions of $\theta$. The lattice model and related parameters can
be found in the Supplemental Material (Li2, a).
Generalization to $\mathcal{C}_{6}$ symmetric SOTIs.—So far, we have focused
on the case of chiral SOTIs with four hinge states and a sample with
(effective) $\mathcal{C}_{4}$ symmetry. However, our scattering theory can be
generalized and applied to SOTIs with more pairs of hinge states. As an
example, we consider a $\mathcal{C}_{6}$ symmetric SOTI with three pairs of
chiral hinge states (Zhang _et al._ , 2020c) and show the spectrum in Fig.
3(a). The geometry considered here is a hexagonal prism with $\mathcal{C}_{6}$
symmetry in $x$-$y$ plane. The hinge states split generally with different
amounts of momenta under a magnetic field. If we consider an interferometer
similar to the setup in Fig. 1(a), we can also observe characteristic
oscillations and beating patterns in the conductance which depend sensitively
on the field direction $\theta$. In this case, the conductance $G(\theta)$ is
$\pi/3$ periodic in $\theta$. Since there are three pairs of counter-
propagating hinge states, the oscillations can exhibit three frequencies in
general (Li2, a). Figure 3(b) illustrates the three frequencies as a function
of field direction $\theta$. Particularly, the oscillations are described by a
single and two frequencies for $\theta=0$ and $\theta=\pi/6$, respectively.
Discussion and summary.—In realistic samples, disorder and dephasing (Jiang
_et al._ , 2009) due to environmental noises may be detrimental to the
interference pattern of hinge states. However, we show numerically that the
oscillation patterns of the conductance in our setups persist under weak
disorder and dephasing (Li2, a). This indicates the robustness of our
proposal. Our results based on chiral SOTIs can also be applied to helical
SOTIs, which can be regarded as two copies of chiral SOTIs related by time-
reversal symmetry. Recently, SOTIs have been proposed in many candidate
materials. Among these candidates, bismuth (Schindler _et al._ , 2018b) and
axion insulators including EuIn2As2 and MnBi2Te4 (Xu _et al._ , 2019; Chen
_et al._ , 2020) provide potential platforms to test our predictions.
In summary, we have proposed a higher-order Fabry-Pérot interferometer and
revealed unique Aharonov-Bohm oscillations arising from topological hinge
states by tuning either strength or direction of an applied magnetic field.
Due to higher-order topology, the interferometer is intrinsically three-
dimensional and features particular beating patterns in the magneto-
conductance. Our results are robust and provide unique transport signatures of
hinge states in higher-order topological insulators.
###### Acknowledgements.
This work was supported by the DFG (SPP1666 and SFB1170 “ToCoTronics”), the
Würzburg-Dresden Cluster of Excellence ct.qmat, EXC2147, project-id 390858490,
and the Elitenetzwerk Bayern Graduate School on “Topological Insulators”. JL
acknowledges support by NSFC under Grants No. 11774317.
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Supplemental material for “Higher-order Fabry-Pérot Interferometer from
Topological Hinge States”
## Appendix S1 General scattering matrix analysis
In this section, we present the details for the scattering matrix analysis of
the setup in the main text. Suppose there are $p_{L/R}$ conducting modes at
the Fermi level in the left/right leads. We can define generally
$(p_{L/R}+2)\times(p_{L/R}+2)$ scattering matrices, $S_{L}$ and $S_{R}$, to
describe the scattering at the left and right interfaces, respectively,
$\displaystyle\begin{pmatrix}b_{L}\\\ b_{L^{\prime}}\end{pmatrix}$
$\displaystyle=S_{L}\begin{pmatrix}a_{L}\\\ a_{L^{\prime}}\end{pmatrix},\ \
S_{L}=\begin{pmatrix}r_{L}&t_{L^{\prime}}\\\
t_{L}&r_{L^{\prime}}\end{pmatrix},$ (S1.1)
$\displaystyle\begin{pmatrix}b_{R^{\prime}}\\\ b_{R}\end{pmatrix}$
$\displaystyle=S_{R}\begin{pmatrix}a_{R^{\prime}}\\\ a_{R}\end{pmatrix},\ \
S_{R}=\begin{pmatrix}r_{R}&t_{R^{\prime}}\\\
t_{R}&r_{R^{\prime}}\end{pmatrix}.$ (S1.2)
Here, $a_{L/R}$ and $b_{L/R}$ indicate the incoming and outgoing modes that
propagate in the leads and scatter at the left/right interface, respectively;
$a_{L^{\prime}/R^{\prime}}$ and $b_{L^{\prime}/R^{\prime}}$ indicate the
incoming and outgoing hinge modes that propagate in the SOTI and scatter at
the left/right interface, respectively. The scattering matrix $S_{L/R}$
consists of four components $t_{L/R},$ $t^{\prime}_{L/R},$ $r_{L/R},$ and
$r^{\prime}_{L/R}$, corresponding to the transmission from left to right,
transmission from right to left, reflection from the right, and reflection
from the left, respectively. In the center SOTI region, the conducting chiral
hinge states pick up an AB phases when applying an external magnetic field.
Thus, the incoming and outgoing modes in the SOTI can be connected by a phase
matrix as
$\displaystyle\begin{pmatrix}a_{R^{\prime}1}\\\ a_{R^{\prime}2}\\\
a_{L^{\prime}1}\\\ a_{L^{\prime}2}\end{pmatrix}$
$\displaystyle=e^{i\lambda/2}\begin{pmatrix}e^{i(\varphi+\phi)/2}&0&0&0\\\
0&e^{-i(\varphi+\phi)/2}&0&0\\\ 0&0&e^{i(\varphi-\phi)/2}&0\\\
0&0&0&e^{-i(\varphi-\phi)/2}\end{pmatrix}\begin{pmatrix}b_{L^{\prime}1}\\\
b_{L^{\prime}2}\\\ b_{R^{\prime}1}\\\ b_{R^{\prime}2}\end{pmatrix},$ (S1.3)
where $\lambda=k_{F}L$ is the dynamic phase with $k_{F}$ the Fermi wave number
in $k_{z}$-direction and $L$ the length of the SOTI, and the two phases are
given by
$\phi=BS_{1}\sin\theta,\ \ \varphi=BS_{2}\cos\theta,$ (S1.4)
with $\theta$ the angle between the magnetic field direction and $x$-axis.
Substituting Eq. (S1.3) into Eq. (S1.2), we obtain
$\displaystyle\begin{pmatrix}e^{-i\lambda/2}\Phi_{-}^{\dagger}a_{L^{\prime}}\\\
b_{R}\end{pmatrix}$
$\displaystyle=S_{R}\begin{pmatrix}e^{i\lambda/2}\Phi_{+}b_{L^{\prime}}\\\
a_{R}\end{pmatrix},$ (S1.5)
where $\Phi_{\pm}\equiv e^{i(\varphi\pm\phi)\sigma_{z}/2}$ and the Pauli
matrix $\sigma_{z}$ acts on (pseudo-)spin space for two left- or right-moving
hinge states. Writing Eqs. (S1.5) explicitly, we have
$\displaystyle e^{-i\lambda/2}\Phi_{-}^{\dagger}a_{L^{\prime}}$
$\displaystyle=e^{i\lambda/2}r_{R}\Phi_{+}b_{L^{\prime}}+t_{R^{\prime}}a_{R},$
(S1.6) $\displaystyle b_{R}$
$\displaystyle=e^{i\lambda/2}t_{R}\Phi_{+}b_{L^{\prime}}+r_{R^{\prime}}a_{R}.$
(S1.7)
From Eq. (S1.6), we find
$a_{L^{\prime}}=e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}b_{L^{\prime}}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R}.$
(S1.8)
Writing Eq. (S1.1) explicitly, we have
$\displaystyle b_{L}$ $\displaystyle=r_{L}a_{L}+t_{L^{\prime}}a_{L^{\prime}},$
(S1.9) $\displaystyle b_{L^{\prime}}$
$\displaystyle=t_{L}a_{L}+r_{L^{\prime}}a_{L^{\prime}},$ (S1.10)
Plugging Eq. (S1.8) into Eq. (S1.10), we obtain
$b_{L^{\prime}}=t_{L}a_{L}+r_{L^{\prime}}(e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}b_{L^{\prime}}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R}),$
(S1.11)
and hence,
$b_{L^{\prime}}=(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}(t_{L}a_{L}+e^{i\lambda/2}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}}a_{R}).$
(S1.12)
Plugging this result into Eq. (S1.7), we find $b_{R}$ as
$\displaystyle b_{R}$
$\displaystyle=t_{R}e^{i\lambda/2}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}(t_{L}a_{L}+e^{i\lambda/2}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}}a_{R})+r_{R^{\prime}}a_{R}$
$\displaystyle=e^{i\lambda/2}t_{R}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}t_{L}a_{L}+[e^{i\lambda}t_{R}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}}+r_{R^{\prime}}]a_{R}.$
(S1.13)
Plugging Eq. (S1.10) into Eq. (S1.6), we obtain
$e^{-i\lambda/2}\Phi_{-}^{\dagger}a_{L^{\prime}}=e^{i\lambda/2}r_{R}\Phi_{+}(t_{L}a_{L}+r_{L^{\prime}}a_{L^{\prime}})+t_{R^{\prime}}a_{R},$
(S1.14)
and hence,
$a_{L^{\prime}}=(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}(e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}t_{L}a_{L}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R}).$
(S1.15)
Plugging Eq. (S1.15) into Eq. (S1.9), we find $b_{L}$ as
$\displaystyle b_{L}$
$\displaystyle=r_{L}a_{L}+t_{L^{\prime}}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}(e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}t_{L}a_{L}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R})$
$\displaystyle=[r_{L}+e^{i\lambda}t_{L^{\prime}}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}\Phi_{-}r_{R}\Phi_{+}t_{L}]a_{L}+e^{i\lambda/2}t_{L^{\prime}}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}\Phi_{-}t_{R^{\prime}}a_{R}.$
(S1.16)
Rewriting Eqs. (S1.13) and (S1.16), we obtain the effective scattering matrix
of the junction
$\displaystyle\begin{pmatrix}b_{L}\\\ b_{R}\end{pmatrix}$
$\displaystyle=S\begin{pmatrix}a_{L}\\\ a_{R}\end{pmatrix},\ \
S=\begin{pmatrix}r&t^{\prime}\\\ t&r^{\prime}\end{pmatrix},$ (S1.17)
where
$\displaystyle t$ $\displaystyle=t_{R}\mathcal{S}t_{L},\ \
r^{\prime}=r_{R^{\prime}}+e^{i\lambda/2}t_{R}\mathcal{S}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}},$
$\displaystyle r$
$\displaystyle=r_{L}+e^{i\lambda/2}t_{L^{\prime}}\mathbb{\mathcal{S}}^{\prime}r_{R}\Phi_{+}t_{L},\
\ t^{\prime}=t_{L^{\prime}}\mathbb{\mathcal{S}}^{\prime}t_{R^{\prime}},$
$\displaystyle\mathbb{\mathcal{S}}$
$\displaystyle=e^{i\lambda/2}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1},$
$\displaystyle\mathbb{\mathcal{S}}^{\prime}$
$\displaystyle=e^{i\lambda/2}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}\Phi_{-}.$
(S1.18)
With this general scattering matrix, the two-terminal conductance can be
written as
$G(B,\theta)=\frac{e^{2}}{h}\mathrm{tr}(tt^{\dagger})=\frac{e^{2}}{h}\mathrm{tr}(t_{R}^{\dagger}t_{R}\mathcal{S}t_{L}t_{L}^{\dagger}\mathcal{S}^{\dagger}).$
(S1.19)
## Appendix S2 Chiral hinge states under magnetic fields
In this section, we demonstrate the splitting behavior of the chiral hinge
states when rotating the magnetic field. In the absence of magnetic fields,
the four chiral hinge states have a double degenerate linear spectrum in
$k_{z}$-direction. The left-moving hinge states cross with the right-moving
ones at $k_{z}=0$. The magnetic field gives rise to a spatially varying vector
potential. Remember that the hinge states are localized at different hinges of
the system. The local vector potential splits the linear spectrum of hinge
states. Under the chosen gauge for the vector potential, ${\bf
A}=(0,0,B[y\cos\theta-x\sin\theta])$, the spectrum of hinge states is split as
$+v(k_{z}\pm\delta k_{1}^{z})$ and $-v(k_{z}\pm\delta k_{2}^{z})$, where the
splitting strengths are determined by $\delta
k_{1}^{z}=BW_{x}|\sin(\theta-\pi/4)/2|$ and $\delta
k_{2}^{z}=BW_{y}|\cos(\theta+\pi/4)/2|$. Thus, the splitting of hinge state
spectrum strongly depends on the field direction $\theta$.
We focus on the splitting of the chiral hinge states in the lower-energy
spectrum presented in Fig. S1. At $\theta=0\pi,$ two pairs of the hinge states
split by equal value. At $\theta=\pi/4$, only one pair of the hinge states can
be split, whereas the other one remain unaltered, i.e., $\delta k_{1}^{z1}=0$.
The spectrum at $\theta=\pi/2$ looks the same as for $\theta=0$. Another
special field direction is at $\theta=3\pi/4$ at which we have instead $\delta
k_{2}^{z}=0$. At $\theta=\pi$, the spectrum is the same as that at $\theta=0$.
This evolution with rotating the magnetic field is consistent with the
analytical results $\delta k_{1}^{z}=BW_{x}|\sin(\theta-\pi/4)/2|$ and $\delta
k_{2}^{z}=BW_{y}|\cos(\theta+\pi/4)/2|$.
Figure S1: Evolution of the hinge states spectrum when rotating the field
direction from $\theta=0$ to $\pi$. Here, we choose parameters: $L_{z}=60a$,
$W_{x}=W_{y}=12a$, $m=2,b=-1,v=1$, $\Delta=1$. The field strength is fixed at
$B=2B_{0}$.
## Appendix S3 Numerical simulation details
To calculate the conductance, we employ the Landauer-Büttiker formalism
(Landauer, 1970; Büttiker, 1986; Datta, 1995) in combination with lattice
Green functions. The two-terminal conductance is evaluated as
$G=\frac{e^{2}}{h}\mathrm{Tr}[\Gamma_{L}G^{r}\Gamma_{R}G^{a}],$ (S3.1)
where the line width function
$\Gamma_{\beta}=i[\Sigma_{\beta}-\Sigma_{\beta}^{\dagger}]$ (S3.2)
with the $\Sigma_{\beta}$ being the self-energy due to coupling of the lead
$\beta\in\\{L,R\\}$ to the central region of interest. The retarded and
advanced Green function, $G^{r}$ and $G^{a}$, are obtained as
$G^{r}=(G^{a})^{\dagger}=(E_{F}-H_{c}-\Sigma_{L}^{r}-\Sigma_{R}^{r})^{-1}.$
(S3.3)
Here, both the self-energy $\Sigma_{\beta}$ and the Green function $G^{r/a}$
can be calculated by using the recursive method (MacKinnon, 1985).
In the numerical simulations, we choose the parameters $L_{z}=60a$,
$W_{x}=W_{y}=12a$, $m=2,$$b=-1$, $v=1$ and $\Delta=1$ for the SOHI, and $m=3,$
$v=0,$ $b=-1$, and $\Delta=0$, and chemical potential $\mu=-0.1$ for the two
leads. Without loss of generality, we set the Fermi energy in the SOHI at
$E_{F}=0.002$.
## Appendix S4 Trivial cases of perfect transmission
In this section, we demonstrate the behaviors when the interfaces of the
proposed setup are totally transparent. Under this condition, the chiral hinge
states do not talk to each other and thus no interference loop is forming. As
a result, the two-terminal conductance $G$ is quantized at $2e^{2}/h$ and
independent of the magnetic field. Let us consider two scenarios responsible
for such transparent interfaces:
* •
Case 1: the leads are also made of the same SOTIs. Chiral hinge states exist
in all regions of space and pass from one lead to the other lead directly;
* •
Case 2: the leads are made of conventional semiconductors or topological
insulators but highly doped. In this case, there are too many channels in the
leads such that chiral hinge states loose quantum coherence once entering the
leads.
The results for these two cases are presented in Fig. S2. The two-terminal
conductance is fixed at $2e^{2}/h$ and has no dependence on neither the field
strength $B$ nor the direction $\theta$.
Figure S2: For the two trivial cases 1 (a) and 2 (b) as discussed in this
section, the conductance is quantized at $2e^{2}/h$ and has no dependence on
either the field strength $B$ or field direction $\theta$.
## Appendix S5 Parametrizing the scattering matrix
In this section, we parameterize the scattering matrix with the help of the
numerical method. There are two interfaces in our proposed setup. Each
interface is described by a $4\times 4$ scattering matrix, i.e., $S_{L}$ and
$S_{R}$ as listed above, respectively. Due to time-reversal symmetry breaking,
$S_{L}$ and $S_{R}$ are unitary matrices.Parametrizing these scattering
matrices is cumbersome because of the choice of at least 16 free parameters.
Instead, we obtain the scattering matrices directly from numerical simulations
as explained below.
Figure S3: Transport properties obtained using the analytical formula, Eq. (3)
in the main text, after we parametrize the relevant scattering matrix
according to the corresponding parameter settings. (a) Conductance oscillation
pattern as function of field strength $B$ for different field directions
$\theta=0$ and $\theta=0.25\pi$, respectively. (b) Conductance oscillation
pattern as function of field direction $\theta$ for different field strengths
$B=B_{0}/\sqrt{2}\thickapprox 0.71B_{0}$ and $B=1B_{0}$, respectively. (c)
Beating pattern of conductance as function of field strength $B$. (d) Beating
pattern of conductance as function of field direction $\theta$.
Let us consider a simpler junction with two semi-infinite regions in $z$
direction: one region is made of a trivial insulators in the region $z<0$, and
the other regions made of the SOTI in the region $z>0$. The parameters for
lead and SOTI are taken the same as those in our interference setup. Then, the
interface of this simpler setup mimics the left interface of our interference
setup. The scattering matrix at this interface can be obtained numerically by
calculating the retarded Green functions for the two regions and then
employing the Fisher-Lee relation, or directly using the Kwant algorithm
(Groth _et al._ , 2014). A similar procedure applies for the right interface.
Known from the conductance, described by Eqs. (2) and (3) in the main text,
the relevant four matrices are $t_{L},t_{R},r_{L^{\prime}}$ and $r_{R}$ (or
another group $t_{L},t_{R},r_{L}$ and $r_{R^{\prime}}$ ). Under the same
parameter setting with the original setup, we obtain the four matrices as
$\displaystyle t_{L}$
$\displaystyle=\left(\begin{array}[]{cc}-0.06198685+0.18465009i,&-0.7498761-0.16515456i\\\
0.60757993+0.469507i,&0.10194116-0.16596995i\end{array}\right),$ (S5.3)
$\displaystyle t_{R}$
$\displaystyle=\left(\begin{array}[]{cc}0.55777311+0.34346096i,&-0.0156968+0.44520283i\\\
-0.26134039-0.36076744i,&-0.27139131+0.59617366i\end{array}\right),$ (S5.6)
$\displaystyle r_{L^{\prime}}$
$\displaystyle=\left(\begin{array}[]{cc}0.17964098-0.37617273i,&0.39605429+0.20453843i\\\
0.25086872-0.36845604i,&-0.33709527-0.2452419i\end{array}\right),$ (S5.9)
$\displaystyle r_{R}$
$\displaystyle=\left(\begin{array}[]{cc}-0.40202362-0.44148246i,&-0.06268933-0.1095996i\\\
0.05582996+0.1132477i,&-0.14131563-0.58013761i\end{array}\right).$ (S5.12)
Figure S3 presents the transport properties obtained using the analytical
formula Eq. (3) in the main text after we parameterize the relevant scattering
matrix according to the corresponding parameter settings. We see that the main
features of the conductance are qualitatively the same as the numerical
results in Fig. 2 of the main text. As shown in Fig. S3 (a), each pattern has
single frequency; the oscillation amplitude is larger at $\theta=\pi/4$; and
the period of red line is about $\sqrt{2}$ times that of the blue line. In
Fig. S3 (b), there are two peaks and the oscillation amplitude is more
pronounced when $B=B_{0}/\sqrt{2}\thickapprox 0.71B_{0}$. In Fig. S3 (c), the
conductance shows a beating pattern of $B$ at $\theta=0.48\pi$. Finally, in
Fig. S3 (c), the conductance shows an “irregular” beating pattern as a
function of $\theta$ for large $B$.
## Appendix S6 Model of $\mathcal{C}_{6}$ symmetric SOTIs
The effective model for a $\mathcal{C}_{6}$ symmetric SOTI on a stacked
hexagonal lattice can be written as (Zhang _et al._ , 2020c)
$H_{\text{hex}}=\begin{pmatrix}h+ms_{z}\sigma_{0}&h_{AB}\\\
h_{AB}^{\dagger}&h+ms_{z}\sigma_{0}\end{pmatrix},$ (S6.1)
where
$\displaystyle h$ $\displaystyle=\tilde{C}-\dfrac{4}{3}C_{2}(\cos k_{1}+\cos
k_{2}+\cos k_{3})+\dfrac{v}{3}(2\sin k_{1}+\sin k_{2}+\sin k_{3})\Gamma_{1}$
$\displaystyle\ \ \ +\dfrac{v}{\sqrt{3}}(\sin k_{2}-\sin
k_{3})\Gamma_{2}+w[-\sin k_{1}+\sin k_{2}+\sin
k_{3}]\Gamma_{4}+\Big{[}M-\dfrac{4}{3}M_{2}(\cos k_{1}+\cos k_{2}+\cos
k_{3})\Big{]}\Gamma_{5}$ $\displaystyle h_{AB}$ $\displaystyle=-2C_{1}\cos
k_{z}+2v_{z}\sin k_{z}\Gamma_{3}-2M_{1}\cos k_{z}\Gamma_{5},$ (S6.2)
and $k_{1}=k_{x},$ $k_{2}=(k_{x}+\sqrt{3}k_{y})/2$ and $k_{3}=k_{1}-k_{2}$.
$h_{AB}$ describes the hopping between neighboring layers. The $\Gamma$
matrices are defined as $\Gamma_{i}=s_{i}\sigma_{1}$ with $i\in\\{1,2,3\\}$,
$\Gamma_{4}=s_{0}\sigma_{2}$ and $\Gamma_{5}=s_{0}\sigma_{3}$ with ${\bf s}$
and ${\bf\sigma}$ the Pauli matrices and $s_{0}$ and $\sigma_{0}$ the
corresponding identity matrices. The model parameters are defined as
$\tilde{C}=C_{0}+2C_{1}+4C_{2}$, and $\tilde{M}=M_{0}+2M_{1}+4M_{2}$. In the
numerical calculations, we set the parameters $C_{0}=0,$ $C_{1}=C_{2}=0.5,$
$M_{0}=-2.5$, $M_{1}=M_{2}=1,$ $v=v_{z}=1$, $m=0.5$ and $w=2$. The perimeter
of the hexagonal prisms is $6\times n_{s}$ with side length $n_{s}=25a$ and
$L_{z}=50a$.
Figure S4: Left panel: influence of disorder on the interference pattern. We
average over 100 disorder configurations. Right panel: influence of dephasing
on the interference pattern. We choose parameters: $L_{z}=60a$,
$W_{x}=W_{y}=12a$, $m=2,b=-1,v=1$, $\Delta=1$. The magnetic field strength is
fixed at $B=B_{0}/\sqrt{2}\thickapprox 0.71B_{0}$. Figure S5: (a) Cross
section of the SOTI in a trapezoid geometry. (b) Four frequencies as functions
of field direction $\theta$. Here, S1 (S2, S3, S4) indicates the bottom
(right-side, top, left-side) surface of the trapezoid. Solid lines are
analytical results, and the dotted lines are obtained by Fourier
transformation from the conductance beating pattern. Discrepancy between them
maybe due to the finite-size effects. We choose parameters for the SOTI as:
$L_{z}=200a$, $m=2,b=-1,v=1$, and $\Delta=1$.
## Appendix S7 Disorder and dephasing
In this section, we show that the interference pattern of our interferometer
is robust against disorder and dephasing.
To mimic disorder, we consider the onsite type $V_{\mathrm{dis}}=V({\bf
r})I_{4\times 4}$ with random function $V({\bf r})$ distributed uniformly
within the interval $[-U_{0}/2,U_{0}/2]$ and $U_{0}$ being the disorder
strength. It is shown in Fig. S4 that as increasing the disorder strength
$U_{0}$, the oscillation amplitude decreases gradually. However, the
interference pattern of the conductance remains even when the disorder
strength is quite strong (comparable with the bulk gap).
We also consider dephasing in the SOTI region in our setup by attaching each
site in the discretized lattice model with a virtual lead (Jiang _et al._ ,
2009). These virtual leads are coupled to the system via the self-energy
$-i\Gamma/2$ with $\Gamma$ measuring the dephasing strength ($1/\Gamma$
signifies the quasiparticle life time). It is shown in Fig. S4 that the
interference pattern of the conductance remains under weak dephasing strength.
As increasing dephasing strength $\Gamma$, the electrons loose their phase
memory quickly and thus the oscillation amplitudes decrease accordingly.
The above results shows the oscillation pattern basically remains under weak
disorder and dephasing, which indicates the robustness of our proposal to show
quantum interference of hinge states.
## Appendix S8 Multiple frequencies when the cross section is a trapezoid
In this section, we present the multiple-frequency case when the cross section
of SOTI is a trapezoid, as shown in Fig. S5(a). In this case, there are
generally four frequencies in the conductance oscillation as function of field
strength $B$. Figure S5(b) shows the four frequencies as varying the field
direction $\theta$. The numerical results are basically consistent with the
analytic ones (obtained by the effective surface areas).
One can see from Fig. S5(b) that the four frequencies do not increase or
decrease simultaneously, which stems from the 3D nature of the interferometer,
as discussed in the main text. Beside, we find that there always exist two
field directions (the field directions along the diagonal of the trapezoid),
at which only two of the four frequencies will survive.
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